Ultra-Wideband, Short-Pulse Electromagnetics 5
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Ultra-Wideband, Short-Pulse Electromagnetics 5 Paul D. Smith University of Dundee Dundee, Scotland, U.K.
and
Shane R. Cloude Applied Electromagnetics Ltd. St. Andrews, Fife, Scotland, U.K.
KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW
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0-306-47948-6 0-306-47338-0
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Preface The fifth Conference on Ultra-Wideband Short-Pulse Electromagnetics was held in Scotland from 30 May to 2 June 2000 at the Edinburgh International Conference Centre. It formed part of the EUROEM 2000 International Conference under the chairmanship of David Parkes (DERA, Malvern) and Paul Smith (University of Dundee). It continued the series of international conferences that were held first at the Polytechnic University, Brooklyn, New York in 1992 and 1994, then in Albuquerque, New Mexico in 1996 (as part of AMEREM ’96) and more recently in Tel-Aviv, Israel in 1998 (as part of EUROEM ’98). The purpose of these meetings is to focus on advanced technologies for the generation, radiation and detection of ultra-wideband short pulse signals, taking into account their propagation, scattering from and coupling to targets of interest; to report on developments in supporting mathematical and numerical methods; and to describe current and potential future applications of the technology. Since 1996 these meetings have been incorporated into the AMEREM/EUROEM biennial Conference that also includes the High Power Electromagnetics Conference and the Unexploded Ordnance Detection and Range Remediation Conference. This decision taken by the Permanent HPEM Committee in 1996 is a recognition of the interests in technology and methods of these Conferences that are common with those of the UltraWideband Short-Pulse Electromagnetics Conference. It also recognises the benefit in providing an international forum for scientists and engineers in such closely related disciplines. The next meeting will be held as part of AMEREM ’02 in June 2002 at the US Naval Academy in Annapolis, Maryland under the chairmanship of Terence Wieting. The papers in this volume report on newly emerging ideas and develop recurrent themes of earlier meetings. The topics include electromagnetic theory and scattering theory (including papers presented at a special session on fundamental solutions of Maxwell’s equations); ultra-wideband radar systems; ultra-wideband and transient antennas; pulsed power generation and propagation; ultra-wideband polarimetry; ultrawideband and transient metrology; detection and identification studies; RF interactions and chaotic effects; and biological effects. The Chairmen and Editors wish to thank all of those involved in EUROEM 2000 for their assistance and participation, especially members of the National and International Committees and their supporting institutions. We also acknowledge with gratitude the sponsorship of the Summa Foundation and the Permanent HPEM Committee, the European Office of Aerospace Research and Development (Air Force Office of Scientific Research, United States Air Force Research Laboratory), the Defense and Evaluation Research Agency (Malvern), Los Alamos National Laboratory, Lothian and Edinburgh Enterprise Limited and the Edinburgh Convention Bureau, the European Commission, and Dundee University. We acknowledge the technical co-sponsorship of the Institution of Electrical and Electronic Engineers (IEEE), the International Union of Radio Science (URSI), the Institution of Electrical Engineers (IEE), and the Applied Computational Electromagnetics Society (ACES). Paul D. Smith Shane R. Cloude
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Contents Fundamental Solutions of Maxwell's Equations and Electromagnetic Theory From Maxwell to Einstein Van Bladel, J 1 Analytical Methods for Antenna Analysis and Synthesis in the Time Domain Shlivinski, A and Heyman, E 11 Complex-Source-Point Narrow-Waisted Ray-Like Gaussian Beams for Frequency and Time Domain Radiation and Scattering Felsen, L and Galdi, V 21 Diffraction by Arrays of Complex Source Point Beams Jull, E, and Cheong, H 31 Application of Concepts of Advanced Mathematics and Physics to the Maxwell Equations Baum, C 39 Surface Discharge Cellular Automaton Model Hayakawa, H, Korovkin, N, ludin, D, Selina, E and Trakhtengerts, V 53 Green's Functions for Sheet Currents Placed Over Cylindrical Metal Surface Svezhentsev, A and Vandenbosch, G 59 UWB Analysis of EM Fields in Complex Laminates: A Multiresolution Homogenization Approach Lomakin, V Steinberg, B and Heyman, E 67 Time Domain Exact Solution of Problem of UWB Pulse Diffraction on a Conducting Half-Plane Galstjan, E 75 Spherical Wave Expansion of the Time Domain Free-Space Dyadic Green's Function Alp Azizoglu, S, Sencer Koç, S and Merih Büyükdura, O 83 On the Localization of Electromagnetic Energy Schantz, H 89 On Superliminal Photonic Tunnelling Nimtz, G, Haibel, A and Stahlhofen, A 97 Transient Electromagnetic Field of a Vertical Magnetic Dipole on a Two-Layer Conducting Earth Seida, O, Bishay, S and Sami, G 105 Time-Domain Study of Transient Fields for a Thin Circular Loop Antenna Bishay, S and Sami, G 115 Generalized TEM, E and H Modes 127 Stone, A and Baum, C Electromagnetic Wave Scattering by Smooth Imperfectly Conductive Cylindrical Obstacle Tuchkin, Y 137 A Set of Exact Explicit Solutions in Time Domain For UWB Electromagnetic Signals in Waveguide Tretyakov, O 143 Analytical Regularization Method for Wave Diffraction by Bowl-shaped Screen of Revolution 153 Tuchkin, Y Transient Excitation of a Layered Dielectric Medium by a Pulsed Electric Dipole: Spectral Constituents 159 Tijhuis, A and Rubio Bretones, A Transient Excitation of a Layered Dielectric Medium by a Pulsed Electric Dipole: Spectral Representation 167 Tijhuis, A and Rubio Bretones, A Radar Systems A New Ultra Wideband, Short Pulse, Radar System for Mine Detection Gallais, F, Mallepeyre, V, Imbs, Y, Beillard, B, Andrieu, J, Jecko, B, and Le Goff, M
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Ultra-Wideband Ground Penetrating Impulse Radar 183 Yarovoy, A, Van Genderen, P, and Ligthart, L Object Shape Reconstruction at Small Base Ultrawideband Radar 191 Koshelev, V, Shipilov, S, and Yakubov, V UWB Subsurface Radar with Antenna Array for Imaging of Internal Structure of Concrete Structural Elements 199 Boryssenko, A, Boryssenko, O, Ivashchuk, V, Lishchenko, A, and Prokhorenko, V Optimal Short Pulse UWB Radar Signal Detection 207 Immoreev, I and Taylor, J Experimental Results from an Ultra Wideband Precision Geolocation System 215 Fontana, R Recent Applications of Ultra Wideband Radar and Communications Systems 225 Fontana, R A Low Power, Ultra-Wideband Radar Testbed 235 Payment, T Ultra-Wideband Principles for Surface Penetrating Radar 247 Sachs, J, Peyerl, P, Rossberg, M, Rauschenbach, P and Friedrich, J Ray Tracing Assessment of Antenna Arrays and Subsurface Propagation for GPR Systems 259 Pennock, S and Redfern, M Ground Penetrating Radar System for Locating Buried Utilities 267 Pennock, S and Redfern, M Cost Efficient Surface Penetrating Radar Device for Humanitarian Demining 275 Ratcliffe, J, Sachs, J, Cloude, S, Crisp, G, Sahli, H, Peyerl, P and De Pasquale, G Some Problems of GPR soft- and hardware improving in mine detection and classification task Astanin, I, Chernyshov, E, Geppener, V, Jatzyn, A, Kostyleva, V, Nicolaev, V, Sokolov, M 285 and Smirnov, A Antennas Time-Domain Simulation Technique for Antenna Transient Radiation, Reception and Scattering Boryssenko, A, Boryssenko, E and Prokhorenko, V A Collapsible Impulse Radiating Antenna Bowen, L, Farr, E and Prather, W High-Power Ultrawideband Radiation for Radar Application Koshelev, V Broadband Operation of Tapered Inset Dielectric Guide and Bowtie Slot Antennas Hannigan, A, Pennock, S and Shepherd, P A Unified Kinematic Theory of Transient Arrays Shlivinski, A and Heyman, E Powerful Sources of UWB Pulsed Coherent Signals Kardo-Sysoev, A, Brylevsky, V, Lelikov, Y, Smirnova, I, Zazulin, S, Tchashnicov, I, Scherbak, V and Sukhovetsky, B Ultrawideband Solid State Pulsed Antenna Array Kardo-Sysoev, A, Zazulin, S, Smirnova, I, Frantsuzov, A and Flerov, A Admittance of Bent TEM Waveguides in a CID Medium Baum, C Optimization of the Feed Impedance for an Arbitrary Crossed-Feed-Arm Impulse Radiating Antenna Tyo, S Transient Fields of Offset Reflector Skulkin, S and Turchin, V A New Broad Band 2D Antenna for Ultra-Wide-Band Applications Mallepeyre, V, Gallais, F, Imbs, Y, Andrieu, J, Beillard, B, Jecko, B and Le Goff, M Time Domain Array Design Schantz, H
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Recent Developments in Ultra-Wideband Sources and Antennas Prather, W, Baum, C, Lehr, J, Torres, R, Tran, T, Burger, J and Gaudet, J Ultra-Wideband Sparse Array Imaging Radar Crisp, G, Thornhill, C, Rowley, R and Ratcliffe, J Cross-Field Characterization of Dipole Radiation in Fresnel Zone Badic, M and Marinescu, M Parallel Charging of Marx Generators for High Pulse Repetition Rates Lehr, J and Baum, C Pulsed Power Special Lecture: Live Fire Test and Evaluation and the RF Vulnerability Testing Mission O'Bryon, J and Carter, R Compact HPM and UWB Sources using Explosives - The Potential of Future Non-lethal Warhead Systems Ehlen, T, Bohl, J, Kuhnke, R and Sonnemann, F Sub-nanosecond Gas Breakdown Phenomena in the Voltage Regime Below 15 kV Krompholz, H, Hatfield, L, Short, B and Kristiansen, M High-Power, High-PRF Subnanosecond Modulator Based on a Nanosecond All-Solid-State Driver and a Gas Gap Pulse Sharper Yalandin, M, Lyubutin, S, Oulmascoulov, M, Rukin, S, Shpak, V, Shunailov, S, and Slovikovsky, B Upgrading of the Efficiency of Small-Sized Subnanosecond Modulators Yalandin, M, Oulmascoulov, M, Shpak, V and Shunailov, S Characteristics of Trap-Filled GaAs Photoconductive Switches used in High Gain Pulsed Power Applications Islam, N, Schamiloglu, E, Mar, A, Zutavern, F, Loubriel, G and Joshi, R Project of Semiconductor High-Power High-Repitition Rate Compact Current/UWB Pulse Generator Galstjan, E and Kazanskiy, L High Power Subnanosecond Generator for UWB Radar Prokhorenko, V and Boryssenko, A Electromagnetic Noise Emission of Industrial Pulse Power Equipment for Material Treatment Luhn, F, Zange, R, Wollenberg, G, Scheibe, H and Schätzing, W Compact Solid State Pulse Modulators for High Power Microwave Applications Gaudreau, M, Casey, J, Mulvaney, J and Kempkes, M UWB Polarimetry An Introduction to Polarisation Effects in Wave Scattering and their Application in Target Scattering Boerner, W and Cloude, S Unipolarized Currents for Antenna Polarization Control Baum, C Polarimetric Radar Interferometry: A New Sensor for Vehicle Based Mine Detection Cloude, S and Thornhill, C Modelling of the Air-ground Interface for UWB Radar Applications Lostanlen, Y, Uguen, B, Chassay, G and Griffiths, H Ultra-Wideband Polarimetric Borehole Radar Sato, M and Liu, S Buried Mine Detection by Polarimetric Radar Interferometry Sagues, L, Lopez-Sanchez, J, Fortuny, J, Fabregas, X, Broquetas, A and Sieber, A UWB & Transient Metrology An Optical Approach to Determine the Statistical Features of the Field Distribution in Mode Strirred Reverberation Chamber Baranowki, S, Kone, L and Demoulin, B
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423 431 437
445 453 461 467 473 479 485
493 501 519 527 537 545
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Influence of Variations in the Spectral Transfer Function to Time Domain Measurements Garbe, H Influence of the Precursor Fields on Ultrashort Pulse Measurements Oughstun, K and Xiao, H Dihedral Reflector Calibration for UWB Radar Systems Lambert, A and Smith, P Detection, Target Identification and Signal Processing Image Reconstruction of the Subsurface Object Cross-Section from the Angle Spectrum of Scattered Field Vertiy, A, Gavrilov, S, Salman, A and Voynovskiy, I Optimal Acoustic Measurements Cheney, M, Isaacson, D and Lassas, M Parsimony in Signature-Based Target Identification Baum, C Buried Object Identification with an Optimisation of the TLS Prony Algorithm Lostanlen, Y, Corre, Y and Uguen, B Model Problems of Pulse Sensing Velychko, L, Perov, A, Sirenko, Y and Yaldiz, E Simulation of the Transient Response of Objects Buried in Dispersive Media Hernándex–López, M, Gonzalez Garcia, S, Rubio Bretones, A and Gomez-Martin, R Electromagnetic Transient Modelling using Dynamic Adaptive Frequency Sampling Tham, C, McGowen, A, Towers, M and Poljak, D The Time Domain Numerical Calculation of an Integro-Differential Equation for Ultrashort Electromagnetic Pulse Propagation in Layered Media Sherbatko, I, lezekiel, S and Nerukh, A Marching on in Anything: Solving Electromagnetic Field Equations with a Varying Physical Parameter Tijhuis, A and Zwamborn, P Correlation of Antenna Measurements Using the Oversampled Gabor Transform Fourestié, B and Altman, Z On a Rational Model Interpolation Technique of Ultra-Wideband Signals Younan, N, Taylor, C and Gu, J
561 569 577
585 599 605 615 623 631 639 647 655 663 671
Propagation Full-Wave Solution of the Propagation of Generally Shaped Impulses and Wide Band Application in Anisotropic Plasmas 679 Ferencz, O and Ferencz, C Asymptotic Description of Ultrawideband, Ultrashort Pulsed Electromagnetic Beam Field Propagation in Dispersive, Attenuative Medium 687 Oughstun, K Dispersion Reduction in a Coaxial Transmission Line Bend by a Layered Approximation of a Graded Dielectric Lens 697 Bigelow, W, Farr, E, Prather, W and Baum, C RF Interactions and Chaos Optimal Input Signals for Driving Nonlinear Electronic Systems into Chaos Booker, S, Smith, P, Brennan, P and Bullock, R In-band Chaos in Commercial Electronic Systems Booker, S, Smith, P, Brennan, P and Bullock, R An Application of Chaos Theory to the High Frequency RCS Prediction of Engine Ducts MacKay, A Ray Splitting and Chaos in Electromagnetic Resonators Blumel, R
707 715 723
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Biological Effects Ultra-Wideband (UWB) Radio-Frequency (RF) Bioeffects Research at DERA Porton Down Holden, S, Inns, R, Lindsay, C, Tattersall, J, Rice, P and Hambrook, J
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Index of Authors
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FROM MAXWELL TO EINSTEIN
J. Van Bladel Department of Information Technology, Ghent University, and Interuniversity Microelectronics Center (IMEC) Gent, B-9000, Belgium
INTRODUCTION In 1931 Cambridge University Press published a volume commemorating the centennial of Maxwell's birth1. Among the contributors we note J.J. Thomson, Max Planck, Ambrose Fleming, Joseph Larmor, James Jeans, Oliver Lodge, Horace Lamb and Albert Einstein. Some of the authors had known Maxwell personally, and their character sketches of our patron-saint are often delightful. Fleming writes about Maxwell's lectures: "I well remember my surprise at finding a teacher who was everywhere regarded as the greatest living authority on his subject lecturing to a class of two or three students in place of the 100 or more attentive listeners he would have had in any Scottish or German University." As a lecturer, Maxwell was apparently mentally too powerful for his students. Lamb, another former student, writes: "Maxwell’s lectures had a great interest of charm for some of us, not so much for the sake of the subject matter, which was elementary, as in the illuminating glimpses we got of the lecturer’s own way of looking at things, his constant recourse to fundamentals, the humorous and unpremeditated digressions, the occasional satirical remarks, and often a literary or even poetical allusion." Lamb paid occasional visits to Maxwell's house, and remembers that "Maxwell had two toys which he would sometimes bring out to entertain fresh visitors. One was the "dynamical top", intended to illustrate various points in the classical theory of rotation. The other toy was a form of ophthalmoscope which he had independently invented. He was wont to demonstrate the use of this on himself and his friends, including his dog, which he had trained to become a patient and accommodating "subject"." Another author, Oliver Lodge, describes Maxwell’s fight with mechanical models aimed at explaining the propagation of waves in the lightcarrying ether. He writes: "In Maxwell’s model the lines of magnetic force were represented by cylinders rotating round these lines as axes. In a uniform magnetic field the cylinders would all have to be rotating in the same direction. The question was how they were to be geared together to do this. If two adjacent wheels were in contact they would rotate in opposite directions; to make them rotate in the same direction Maxwell introduced between them small spheres like ball-bearings to act as idle wheels". We know that, in an
Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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extraordinary feat of conceptual abstraction, Maxwell dropped these attempts, and decided that reality was best described by his equations.
EINSTEIN'S COMMEMORATIVE ARTICLE Einstein emphasizes the undebtedness of the scientific community to his illustrious predecessor. He writes, in particular, that "The greatest alteration in the axiomatic basis of physics since Newton originated in the researches of Faraday and Maxwell. According to Newton's system the material particle is the sole representative of reality. But the partial differential equation, which came first to theoretical physics as a servant, by degrees became its master. This process began in the nineteenth century, with the wave theory of light. Light in empty space was conceived to be a vibration of the ether, and it seemed uncalled for to regard this ether as itself a conglomeration of material particles. Here for the first time the partial differential equation appeared as the natural expression of the elementary in physics. It is true that Maxwell tried to find a basis of justification for these equations in ideal mechanical constructions, but he took none of these efforts too seriously; it was clear that the equations themselves were all that was essential, and that the field intensities that appeared in them were elementary, not derivable from other simples entities." And Einstein concludes: "Since Maxwell's time, Physical Reality has been thought of as represented by continuous fields, governed by partial differential equations, and not capable of any mechanical interpretation. This change in the conception of Reality is the most profound and the most fruitful that physics has experienced sinde the time of Newton."
TRANSFORMATION EQUATIONS BETWEEN INERTIAL FRAMES At the time of Maxwell's death the problem of including moving media in the theory was left unsolved. It was commonly believed that the ether served as a substratum for the propagation of light, and that it penetrated into bodies like water in a sponge. Some physicists assumed that moving bodies dragged the ether locally. Other believed that the ether was at absolute rest, and that the earth, for example, was swept by an ether "wind" in its motion. In this global picture an "absolute" set of axes existed, in which all true motions should be measured. In a particle left to itself would move in a straight line with constant velocity. Further, this uniform motion would also hold in all other rigid systems K which move with uniform velocity with respect to (the inertial axes). Ether was fundamentally at rest in and electromagnetic waves should therefore have a velocity c with respect to but a value different from c on earth. Experiments such as Michelson and Morley's showed the fallacy of that assumption. Extensive discussions led Lorentz to propose, in 1904, new laws for the transformation of coordinates from an inertial frame (a train for example) to another inertial frame K (a station). Taking v to be the relative velocity of with respect to K, the relevant relationships are
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Space and time are clearly interwoven, and events which are simultaneous in at different are not simultaneous in K. In (1) hence at everyday's low velocities may be neglected with respect to 1. First-order terms in however, must be carefully kept since they play a major role in applications. From (1) it is seen that a velocity c in goes over into c in K, a revolutionary result which does away with the traditional law of addition of velocities. At about the same time, in 1905, Einstein went further2, and erased absolute motion and absolute space by postulating that the laws of electrodynamics and optics have the same form in all inertial frames that light propagates in empty space with a speed c which is independent of the state of motion of the emitting body. As a consequence, Maxwell's equations must have the same form in K and Combined with (1) this leads automatically to the transformation laws for sources, viz.
Here || and fields:
indicate components parallel and perpendicular to v, respectively. For the
At low velocities a relationship used extensively in the determination of motionally-induced currents. The transformation equations (2) (3) are of considerable interest for the electrical engineer, since they allow him to solve the field problem in a frame in which the solution is particularly simple, and to transform the results back to the frame K in which they are actually needed. This process is aptly called frame-hopping.
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Maxwell's equations must be supplemented by constitutive equations. In 1908 Minkowski3, building on Einstein's ideas, concluded that these equations should have the same form in all inertial frames in which the medium is at rest. Thus, if holds in the rest axes it must also hold in every other inertial frame in which the medium is at rest. Transforming from back to K (the "laboratory") by means of (3) gives
At low velocities:
An observer in the laboratory sees an (anisotropic) magneto-electric medium! By a similar argument, the usual boundary conditions must hold in every rest frame. Transformed back to the laboratory these conditions become (Figure 2)
For v = 0 we recuperate the "rest" form, as expected. The same form holds when i.e. at points where the motion takes place in the tangent plane.
A FEW PROBLEMS INVOLVING BODIES IN UNIFORM MOTION When the motion takes place in the tangent plane, as suggested in Figure 3, a solution directly in the laboratory axes K becomes practical4. The boundary conditions are those of a body at rest, and the motion is only felt through the constitutive equations (4).
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In most cases, however, the appropriate strategy is to first determine the fields in the rest axes of the body, and transform them back to the laboratory. This method was already used by Einstein in his fundamental 1905 paper, in his endeavour to determine the fields reflected from a moving mirror (Figure 4). The incident field is:
Frame-hopping gives the reflected field in two easy steps. Thus,
The Doppler shift is apparent. At low velocities, in particular,
MAXWELL'S EQUATIONS IN ACCELERATED AXES Frame-hopping is also applicable to accelerated bodies. The first step is to solve the problem in the (accelerated) rest axes of the body, followed by a transformation back to the laboratory K. This step requires the form of Maxwell's equations in the accelerated axes Fundamental here is the metric tensor obtained by writing the squared elementary distance between neighbouring events - a relativistic invariant - in terms of the new coordinates. Thus, starting from (CT, X, Y, Z) in K:
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For the rotating coordinates of Figure 5
where The general form of Maxwell's equations, given is
where and are tensors4 grouping covariant and contravariant components of resp. (e, b) and (h, d). In rotating coordinates, for example, the curl e equation becomes
It is interesting to note that a tensor also appears when gravitational forces are present. Outside a spherically-symmetric mass distribution, for example,
Here
is a critical radius and m is the total mass. When lies outside the sphere, becomes singular there (the well-known "black hole" phenomenon). The theory further shows that free space acquires locally an equivalent which causes a grazing ray to be deviated (Figure 6).
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This phenomenon, well-observed in astronomy, can be pertinent for radio-communications with or detection of space vehicles. An important problem in accelerated coordinates is the form of the constitutive equations. Firstly, the parameters may be modified by mechanical stresses and deformations. These effects can be accounted for, and corrected. More fundamental is the need for an assumption, namely that at sufficiently low accelerations the rest values of µ, hold at a moving point P when the fields are expressed in the instantaneous rest frame of P. From this working hypothesis it follows that the constitutive equations (4) still hold in the laboratory, but with a v which is now a function of r and t. By the same token the boundary conditions at a boundary point will be the same as in the instantaneous rest frame. In particular, the conditions for a body at rest hold at points where the motion is in the tangent plane. This remark allows solution of quite a few practical problems, as shown in the sequel. The validity of the rest-frame hypothesis has been checked5 on the particular case of a rotating dielectric cylinder. The dielectric is assumed to consist of electrons elastically bound to the nucleus. The analysis shows that the hypothesis holds as long as the rotation frequency is much less than the eigen-oscillating frequency of the electron-nucleus spring, a value which typically lies in the ultraviolet.
A FEW PROBLEMS INVOLVING ACCELERATED BODIES When the motion takes place in the tangent plane a solution in the laboratory axes is the best choice. The motion is now felt through the constitutive equations only. The method is applicable to a variety of configurations, for example to the rotating axisymmetric conductors shown in Figures 7a and 7b. To first order in a volume charge density and an electric field
appear in the conductor, while a voltage
is induced across the Faraday disk.
Another example is afforded by a rotating circular dielectric cylinder of radius a immersed in an E-wave. Figure 7c shows the scattering pattern obtained for and The shift in the pattern is the main effect but, given the smooth contour, it takes nonrealistic peripheral velocities to make it significant.
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When the motion modifies the spatial configuration of the body (e.g. in the case of the rotating blades of a helicopter) it becomes necessary to apply frame-hopping to and from the accelerated axes. For a rotating body this means first solving the problem "on the merry-go-round". Only a few of these problems have been solved, e.g. the rotating circular cylinder carrying a distributed surface reactance, and immersed in an H-wave6. The strong variation of the reactance gives rise to a rich Doppler spectrum (Figure 8).
THE QUASI-STATIONARY METHOD Although the rest axes method is conceptually straightforward, the actual application (for example to a vibrating mirror7) quickly becomes cumbersome. An extensively used substitute is the quasi-stationary approximation, which consists in evaluating the scattered fields in P, at time t, as if the scatterer were frozen in the position it occupies at t. The motion produces amplitude and phase modulation, and therefore an instantaneous frequency shift. The method has its obvious limitations. It would not detect a rotation effect of the type shown in Figure 7c, since all positions of the rotating circular cylinder are equivalent. In addition, the method is theoretically flawed. In the case of a uniformly moving mirror, for example, it would predict a reflected signal (Figure 4)
A comparison with the exact expression (7) shows that (a) the first-order Doppler shift is correctly predicted (b) the amplitude modulation is ignored (c) the reflected field does not satisfy the wave equation in vacuo. In particular, the propagation velocity is instead of c. Although theoretically inconsistent, the approximation yields errors of the order of which are quite acceptable when the scatterer moves little during the characteristic times of the problem. Such times are e.g. the period of a time-harmonic incident wave, of the duration of a short pulse of the type encountered in UWB applications. For for example, an airplane flying at a speed of would move only 0.3 µm during the pulse, a very short distance indeed compared with the dimensions of the target, and one that would hardly change the radar cross-section of the latter!
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CONCLUSION It is clear that Relativity forms an integral, indispensible part of Electromagnetic Theory. The present very simplified survey does not do justice to the beauty of the fourdimensional, tensorial structure of the theory. Its sole ambition was to unfold the mechanisms which lead to a correct formulation of practical problems. From a utilitarian point of view it may be asserted, in all fairness, that most "radio" engineers can live quite happily without Relativity. Approximate methods such as quasi-stationarity suffice for their practical needs. There are exceptions, of course, particularly in the areas of high-velocity electron beams, klystrons, free electron lasers, gyrotrons, relativistic magnetrons or highenergy accelerators. Electromechanical engineers, on the other hand, are much more involved. Let us remember that Einstein worked for the Swiss Patent Office from 1902 to 1909, and that he was routinely confronted with new designs of motors and generators, some of which probably claimed perpetual motion!
He certainly meditated about the relative motion of circuits and magnets when he wrote his fundamental 1905 paper, and we may assume that the circuit law (actually based on the instantaneous rest frame hypothesis) played a role in his travails. Einstein may well have considered the teaser of Figure 9 (current or no current?), where a circuit is connected to a moving magnetized (and conducting) bar through sliding contacts4,8. It is in such an engineering atmosphere that Relativity was born! REFERENCES 1. James Clerk Maxwell, A Commemoration Volume, 1831-1931, Cambridge University 2. 3.
4. 5. 6.
7. 8.
Press (1931). A. Einstein, Zur Elektrodynamik Bewegter Körper, Ann. Phys. (Leipzig), 17: 821 (1905). H. Minkowski, Die Grundgleichungen für die elektromagnetischen Vergängen in bewegten Körpern, Göttingen Nachrichten Math.-Phys. Klasse, 53 (1908), reprinted in Math. Annalen, 68: 472 (1910). J. Van Bladel, Relativity and Engineering, Springer-Verlag, Berlin (1984). T. Shiozawa, Phenomenological and Electron-theoretical Study of the Electrodynamics of Rotating Systems, Proc. IEEE, 61: 1694 (1973). B.M. Petrov, Spectral Characteristics of the Scatterer Field from a Rotating Impedance Cylinder in Uniform Motion, Radio Eng. Electron. Phys. (USSR) (English translation), 17: 1431 (1972). D. De Zutter, Reflection from Linearly Vibrating Objects: Plane Mirror at Oblique Incidence, IEEE Trans. AP, 30: 898 (1982). L.V. Bewley, Flux Linkages and Electromagnetic Induction, Dover Publications Inc., New York (1964).
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ANALYTICAL METHODS FOR ANTENNA ANALYSIS AND SYNTHESIS IN THE TIME DOMAIN
Amir Shlivinski and Ehud Heyman Faculty of Engineering Tel-Aviv University Tel-Aviv 69978, Israel
INTRODUCTION The ever-increasing interest in the radiation and detection of ultra-wideband (UWB) short pulses has made an impact on time domain (TD) analysis and design of shortpulse antennas. One way of analyzing the radiation and detection of these pulses would involve the traditional frequency domain (FD) antenna parameters on a frequency-byfrequency basis. FD parameterization lends itself conveniently to producing a comprehensive transmit - receive system description, yet because of the broad frequency band of the short-pulsed fields, direct treatment in the TD may lead to more efficient and physically transparent representations. The present paper is a brief overview of two formulations which have been introduced recently for TD antenna characterization: The first one is based on plane wave analysis in the TD via the slant stack transform (SST) of the current distribution [1], while the other is based on a TD multipole expansion [2]. The plane wave formulation provides a complete far-zone characterization of transmitreceive antenna systems [1]. The SST proves to be a fundamental tool in TD analysis, and is equivalent to the spatial Fourier transform of the current distribution used to evaluate the radiation field in the FD. The far field is expressed in terms of the effective height operator, which is a characteristic of the antenna, used with the source waveform to produce the far field via a convolution integral. Thus taking the receiving antenna effective height and circuit into account, we arrive at the TD transmit - receive antenna system description, consisting of a succession of convolution integrals. The TD multipole expansion, on the other hand, provides a convenient representation for the near zone properties of the field. Using this formulation we define and explore such concepts as the TD radiative and reactive fields and energies [2]. In the near zone, the TD reactive energy is a relatively strong pulse which, unlike the radiaUltra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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A. SHLIVINSKI AND E. HEYMAN
tive energy, discharges back into the source once the excitation pulse has ended. The radiation effectiveness is thus quantified by the ratio between the total TD reactive energy outside the antenna and the radiative energy. Borrowing from FD terminology, this ration is termed the TD Large implies large reactive energies and thereby a less effective realization. The near and far zone characteristics are, of course, related: As in the FD analysis, highly directive fields consists of high order spherical modes which requires large antennas. Realizing such modes with smaller antennas would generate very large reactive energies and These considerations pose a tradeoff between directivity, size and radiation effectiveness. In [3] we have used the measures above in order to quantify and contrast the effectiveness of several different realizations of highly collimated, short-pulse fields, and in particular, the class of space-time synthesized apertures considered in [4] THE ANTENNA EFFECTIVE HEIGHT IN THE TIME DOMAIN We first introduce the SST which is the fundamental tool in TD plane wave analysis. Given the impressed current distribution on the antenna surface, the radiating vector potential is found via the retarded potential integral where and is the speed of light. In the “TD Fraunhofer zone,” defined by where L is the source dimension and T is the pulselength, we obtain the following expression for the pulsed radiation field
where the unit vector
define the observation direction and (see Fig. 1)
The operation in (2) is the slant stack transform (SST) of be expressed as two cascaded operations [5]
in the
direction. It can
The first operation is a spatial Radon projection of at planes normal to the spectral propagation direction Fig. 1. In the second operation in (3) these planar projections are stacked with a progressive time delay associated with the propagation along the axis at the spectral speed The STT therefore extracts from the source distribution the transient plane-wave information that propagates in the direction. Further details on the SST could be found in [1, 5]. Thus form (2) the radiation pattern depends on the SST in the direction. Next we consider the transmitting antenna circuit in Fig. 2. The effective height of the transmitting antenna defines the relationship between the forward propagating input current-wave at the antenna terminals and the electric field in the far zone via the convolution integral
and we use the notation The effective height is thus the far field impulse response of the transmitting antenna. Note that in contrast to the FD definition, we define the TD effective height with respect to rather than with
ANALYTICAL METHODS FOR ANTENNA ANALYSIS AND SYNTHESIS
13
respect to the total current since the latter also includes the reflected waveform that may include multiple reflections off the antenna structure, etc. In analogy to the far field in the FD, the effective height is independent of the distance and of the radiation delay. Using the relation between E and A we obtain
where is the current distribution due to an impulsive input current and the subscript || denotes the transverse component relative to the observation direction Evaluation of (5) thus requires an electromagnetic solution of which in general needs to be calculated numerically, e.g. via the Finite Difference Time Domain (FDTD) technique, although for electrically small or large antennas some approximate model may apply. The theory above can be applied to a system consisting of a transmitting and a receiving antennas denoted, respectively, by subscripts T and R (Fig. 3). The distance between the antennas is large so that the far zone models apply. The unit vector directions from antenna T to R and from R to T are denoted, respectively, as and Using reciprocity in order to express the effective reception height of the receiving antenna in terms of its effective transmission height we obtain for the receiver load current
where denotes a vector dot product and a temporal convolution. Note that and the convolution operations may commute. Equation (6) describes a complete transmitreceive system. In [1], the theory has been further developed to provide expressions for the energy using TD gain operators. Example: Radiation from a circular disk antenna The formulation above has been applied so far in many configurations. Here we shall briefly demonstrate the use of the SST to derive a closed form expression for the TD radiation pattern from a disk of radius in the plane, carrying a uniform pulsed current distribution
where is a unit vector in the direction. Referring to (1)–(2), the TD radiation pattern in the direction can be expressed as
14
ANALYTICAL METHODS FOR ANTENNA ANALYSIS AND SYNTHESIS
To calculate we note that the component of transverse to is where is the angle of from and is a unit vector along that coordinate. They are given by and Eq. (8) thus becomes
The integral in (9), henceforth termed reduces to the length of the line of intersection between the source disk in the plane and the slanted plane (Fig. 4). For a given the distance of this line from the origin is hence its length is giving
The expression in (10) is the TD analog of the well known FD expression for the a radiation from a circular source disk. However, unlike the FD analysis which involves spatial Fourier transforms and Bessel functions, followed by frequency transform into the TD, the TD analysis above involves only a geometrical projection of the source disk. This expression is used in Fig. 6 to verify the results of the multipole expansion. Full agreement is obtained if a sufficient number of multipoles is taken. TD MULTIPOLE EXPANSION Next we discuss the TD multipole expansion and the near-zone analysis. The analysis may be applied to any spherically stratified configuration bounded by a general conical surface whereon the boundary conditions are imposed (Fig. 5), but here we shall only be interested in radiation in free space. The field in the configuration of Fig. 5 may be decomposed into E- and H- type modes denoted, respectively, by superscript and and expressed as a sum of all these modes, with denoting the mode index and or
with a similar expression for the H field. Here spherical coordinate system, and
denotes the conventional The functions
ANALYTICAL METHODS FOR ANTENNA ANALYSIS AND SYNTHESIS
15
and are the transversal scalar and vector mode functions which depend only on the conical cross section and are independent of The amplitude functions depend only on and account for the source excitation and the medium inhomogeneity via a solution of the radial transmission problem. For The present problem of radiation in free space, and are the well known scalar and vector spherical harmonics (i.e., with being a triple index One finds that outside the source region the field is given by
where and are calculated by projecting
with In (12), are the “TD multipole moments” which onto the spherical basis functions via
An important property of the series in (12) is that only the terms propagate without decay and with pure delay. Expressing the far zone field in the form one finds for the “TD radiation pattern”
Recall that F can also be calculated via the SST as discussed in (8). The large terms, in (12), behave like and thus dominate in the near zone. where they contribute to the “TD reactive field.” These constituents have been used in [2] to define the TD reactive power and energy. It has been demonstrated that the TD reactive power is a pulse with zero mean: At early time it propagates outward and charges the reactive energy around the antenna, while at later time, it propagates inward as the reactive energy discharges back to the source (cf. Fig. 6). The only constituents that carry a net energy are the radiative constituents Using large asymptotics of Eqs. (12)–(13) we have determined in [2] the convergence rate of the multipole expansion and the field structure in the far and in the near (reactive) zone. Example: Radiation from a circular disk antenna The concepts above are demonstrate for the example considered in (7). The excitation pulse is taken to be a twice differentiated Gaussian with pulselength and a unit energy Figs. 6(a,b) shows the radiation-pattern pulse at vs. the normalized time for source disks of radii and The figure compares the SST result (10) (solid line), which is used as an independent closed-form check, with the multipole expansion result obtained by summing up to the and term (dotted, dash-dotted and dashed lines, respectively). Note that the number of modes needed is essentially as has been determined analytically via asymptotic
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A. SHLIVINSKI AND E. HEYMAN
analysis of (12)–(13) in [2]. Finally in Fig. 7 we show typical results for the radiative and reactive power-flows. Note that the radiative power pulse propagates without distortion or decay, while the reactive power pulse decays and distorted as a function of As discussed above the reactive power has a zero mean, implying that the reactive energy discharges back to the source when the excitation pulse turns off. THE TD
As discussed in the Introduction, the radiation effectiveness can be quantified by comparing the total TD radiative and reactive energies. Using the multipole expansion as discussed above, one may identify the following concepts [2]: the radiative power-flow and energy-density and respectively) and the reactive power-flow and energy-density and These quantities may be expressed explicitly in terms of the time-dependent multipole moment functions of (13). The TD factor of the field with respect to the sphere a enclosing the antenna may be defined now as
where the total radiative energy is defined by
The definition of
ANALYTICAL METHODS FOR ANTENNA ANALYSIS AND SYNTHESIS
17
the measure of the total reactive energy is somewhat more complicated. We defined the time-averaged reactive energy outside the source support as where has been discussed above and is the RMS pulse-length of the radiative field. The general definition in (15) applies to any pulse shape with a finite energy. It also reduces to the conventional definitions of for time harmonic sources, given for example in [6]. In order to calculate the general definition in (15) for the total it may be rephrased in terms of modal quantities via [2]
where the summation involves the two mode-types and all mode indexes The modal quantities in (16), namely and are the counterpart of the total quantities defined above. These quantities may readily be calculated from the multipole moment functions which, as mentioned above, are calculated directly from Eq. (16) provides a direct mean for calculating and is the TD analog of the well known FD expression in [7]. Eq. (16) readily clarifies the tradeoff between directivity, antenna size, pulse length and effective realization. For example, large directivity requires large at the higher order modes. This, on the other hand, may cause an increase in the modal which typically increase rapidly for where and T are the antenna size and pulselength (this increase follows from the fact noted after (14) that the modal reactive energies are rapidly growing for The tradeoff noted above will be demonstrated in the simulations below. Example 1: Radiation from a circular disk antenna The concept introduced above are demonstrated here for the example of Eq. (7). We consider the radiation from both “small” and “large” source disks as well as the possible realization of the field due the “large” source by a smaller source.
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A. SHLIVINSKI AND E. HEYMAN
Fig. 8 shows the energy constituents for three different disks with radii and (indicated by and respectively). Fig. 8(a) depicts the radiative modal energies of (16) as a function of the mode index while Fig. 8(b) depicts the total as a function of the highest mode included in the summation (16). Note from the modal energy distribution and from the results for the total quality factor that the relevant modes are those with Beyond this value, the modal energies are small and their contributions to the total are negligible. We explore the possible realization of the field corresponding to the disk, by using a smaller source bounded by (the results are indicated by a +). The moments are taken to be those of the case, leading to the same radiation pattern and mode energies (see the * and the + marks in Fig. 8(a)). The smaller source realization, however, requires larger reactive energies and modal and thereby larger and slowly converging series for (Fig. 8(b)). As a conclusion, super resolution using short pulse fields is not feasible since it involves large TD Q.
Example 2: Collimated, space-time synthesized, source distributions Next consider a class of space-time synthesized pulsed current distributions on a circular disk of radius in the plane, given by (cf. (7))
Where the parameter will be discussed below and by the positive frequency spectrum here a twice differentiated analytic
is any analytic signal, as defined Specifically we shall use
pulse
Where T is a parameter controlling the pulse length. Also the normalization constant A is taken so that the norm of in the space-time domain is normalized to a constant ( in this case). One observes that the current decay away from the center of the aperture is affected by the imaginary part in the argument of in (17): Specifically
ANALYTICAL METHODS FOR ANTENNA ANALYSIS AND SYNTHESIS
19
using (18), one finds that the peak of the pulse at decays like The effective width of the aperture for this distribution, defined as the diameter where the pulse energy decays to one half of its value on the axis, is found to be For large a such that the truncation in (17) may be neglected, giving rise to an iso-diffracting pulsed beam field (PB) of the type considered for example in [4], with being the collimation distance. If however, than the truncation effect is significant and the radiating field is not an iso-diffracting PB. Below we consider three different distributions, all having the same T and but with different values for and In the first and second cases, the effective aperture widths as determined above are and respectively. Since they are smaller than the disk diameter, they approximately generate iso-diffracting PBs. In the third case the aperture distribution is practically uniform over the entire disk as in (7). Fig. 9 shows the far zone electric field pulse on the as computed via the TD multipole expansion (14). Only the first 20 multipole modes have been summed up (dotted lines). In order to validate the TD multipole expansion results and the number of terms needed, the results are compared with the SST formulation (8) for the radiation pattern (full lines), which is exact in the far zone. In the large case the effective aperture is wider and thus more multipoles are needed in order to recover the correct SST result (the number of relevant modes for uniform distributions is approximately
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A. SHLIVINSKI AND E. HEYMAN
The H-plane energy radiation pattern as a function of the angle from the axis (the bore sight) is depicted in Fig. 9(d). One may readily observe that the directivity increases and beam width decreases as and, thereby the effective aperture width, increase. As in Fig. 8, we explore in Fig. 10 the near field effectiveness of the three source realizations by plotting the modal energies as a function of the mode number (Fig. 10(a)) and the total TD as a function of highest mode number in the summation (16) (Fig. 10(b)). From Fig. 10(a), the dominant modes are those with while from Fig. 10(b) the number of relevant modes is (beyond this value, the modal energies are small and their contributions to the total are negligible). One also observe that the PB-type distributions in (a) and (b) are characterized by lower TD as follows from the fact that their effective source diameter is smaller.
Acknowledgements: This work is supported in part by the Israel Science Foundation under Grant No. 404/98, and in part by AFOSR Grant No. F49620-96-1-0039.
References [1] A. Shlivinski, B. Heyman and R. Kastner, “Antenna characterization in the time domain,” IEEE Trans. Antennas Propagat., AP-45, 1140–1149, 1997. [2] A. Shlivinski and E. Heyman, “Time domain near field analysis of short pulse antennas. Part I: Spherical wave (multipole) expansion,” and “— Part II: Reactive energy and the antenna ” IEEE Trans. Antennas Propagat, AP-47, 271–286, 1997. [3] A. Shlivinski and E. Heyman, “Energy considerations in space-time synthesis of collimated pulsed apertures,” Proc. of the URSI Trianum International Symposium on Electromagnetic Theory, Thessloniki, Greece, May 1998, pp. 602–604. [4] E. Heyman and T. Melamed, “Certain consideration in aperture synthesis for ultrawideband/short-pulsed fields,” IEEE Trans. Antennas Propagat., AP-42, 518–525, 1994. [5] E. Heyman, “Transient plane wave spectrum representation for radiation from volume source distribution,” J. Math. Phys., 37, 658–681, 1996. [6] R.E. Collin and S. Rothschild, “Evaluation of antenna ,” IRE Trans. Antennas Propagat, AP-12, 23–27, 1964. [7] R.F. Harrington, “Effect of antenna size on gain, bandwidth and efficiency,” J. Res. NBS, 64D, 1–12, 1960.
COMPLEX-SOURCE-POINT NARROW-WAISTED RAY-LIKE GAUSSIAN BEAMS FOR FREQUENCY AND TIME DOMAIN RADIATION AND SCATTERING
Leopold B. Felsen1,2 and Vincenzo Galdi2,3 1
2
3
Department of Aerospace and Mechanical Engineering Boston University, Boston, MA 02215, USA Also, University Professor Emeritus, Polytechnic University, Brooklyn, NY 11201, USA Department of Electrical and Computer Engineering Boston University, Boston, MA 02215, USA Waves Group, University of Sannio, Benevento, Italy
I. INTRODUCTION Through replacement of the real spatial or spatial-temporal source locations in the frequency domain (FD) or time domain (TD) Maxwell field equations by locations in complex space or space-time, respectively, one may generate a new class of exact field solutions which convert point-source-excited fields in any environment into fields excited by Gaussian-beam-like wave objects in that environment (Deschamps, 1971; Felsen, 1976). While most applications of this elegant and physically appealing complex-source-point (CSP) technique have been concerned with ”high frequency” tracking of wellcollimated ”wide-waisted” beams, we shall be concerned here with utilizing a tight superposition of narrow-waisted ray-like CSP beams (centered on a Gabor lattice) for synthesis of FD and TD distributed aperture radiation, and the interaction of these radiated fields with complex propagation and scattering environments. Previous studies have employed this algorithm for FD distributed phased apertures (Maciel and Felsen, 1989) and for transmission of these fields through focal regions and through plane or cylindrically stratified dielectrics (Maciel and Felsen, 1990a,b). Here, we extend this algorithm to FD scattering by a moderately rough perfectly conducting boundary. We also extend the FD aperture radiation algorithm to the short-pulse TD, utilizing CSP pulsed beam wavepackets. The FD problem is considered first.
II. RADIATION AND SCATTERING IN THE FREQUENCY DOMAIN The problem geometry is shown in Fig. 1a. A two-dimensional electric field with suppressed dependence and spatial profile region in the aperture plane
time-harmonic is assumed to occupy the
This field irradiates a perfectly conducting boundary with sinusoidal height profile
Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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L. B. FELSEN AND V. GALDI
measured from the
plane. We first consider the aperture problem.
II.a Radiation from aperture distributions We summarize here essential results from previous publications (Bastiaans, 1980; Einziger and Shapira, 1986; Maciel and Felsen, 1989; Steinberg et al, 1991). The field radiated into the half-space from the aperture in (1) can be expressed as a line-source superposition (Kirchhoff integration)
where is the free space wavenumber; is the free space wavelength; zeroth order Hankel function of the first kind; is a scalar Debye potential; and
is the
II.b Beam discretization The aperture field is to be parameterized in terms of Gaussian beam basis functions via the rigorous self-consistent Gabor series representation
where
represents the normalized Gaussian window
With
representing the wavenumber, this representation places the beams on a discretized phase-space lattice (Fig. 1b), on which each lattice point gives rise to a Gaussian beam whose spatial and spectral (tilting) shifts are tagged by the indexes and respectively. Spatial and spectral periods are related by the self-consistency relation (configuration-spectrum tradeoff) (Bastiaans, 1980). The expansion coefficients can be computed by introducing an auxiliary function defined through the biorthogonality condition (Bastiaans, 1980),
where * denotes the complex conjugate, while (Bastiaans, 1980),
for
and
for
Accordingly
For Gaussian windows, the biorthogonal function is given in (Bastiaans, 1980). For numerical computation of the Gabor coefficients, see (Einziger and Shapira, 1986). The radiated potential field in the half-space (see (3)) can be represented as (Maciel and Felsen, 1989)
CSP NARROW-WAISTED RAY-LIKE GAUSSIAN BEAMS
where the beam functions
23
are synthesized by Gabor-weighted line-source superposition
R being defined in (4). The integral in (10) (or its spectral counterpart) can be evaluated asymptotically in the beam paraxial far zone, yielding the following complex source point (CSP) approximation (Maciel and Felsen, 1990a)1,
with representing the complex distance between the observer at point (here and henceforth, the tilde denotes a complex quantity),
and the complex source
In accord with the radiation condition, the square root is defined by The displacement parameter (Fresnel length) is related to the beam lattice period and the beam axis angle (Maciel and Felsen, 1990a), whence (11) is valid in the paraxial far-zone of each beam, For large tilt angles with the beam tilt angle is complex and the corresponding beams become evanescent.
II.c Narrow-waisted beams For narrow-waisted beams the Gabor coefficients can be effectively estimated by sampling the aperture field distribution, without performing the integration in (8) (Maciel and Felsen, 1989),
so that from (9) and (11)
where is obtained from (11) with The tilted beams in the Gabor expansion, which generate evanescent ”far fields”, are ignored in this approximation.
II.d Linearly-phased aperture Narrow-waisted beams work very well for nonphased apertures, but usually require finer aperture sampling in the presence of phasing (Maciel and Felsen, 1990a). Here we consider a linearly-phased cosine aperture field,
where is the tilt angle of the main radiation lobe with respect to the axis. In this case, a more effective discretization can be obtained by Gabor-expanding the real function only, and including the linear phasing in the beam integral (10) for the beam propagator. Accordingly, the narrow-waisted beam expansion can be recast as
1 Note that there are some sign changes with respect to (Maciel and Felsen, 1990a), since here we assume propagation into the half-space
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L. B. FELSEN AND V. GALDI
The beam propagator (18) differs from in (15) by the phase shift which produces the propagation-matched tilt in the beam direction. In Fig. 2, the near-zone potential field synthesized using the narrow-waisted tilted beam decomposition in (17) is compared with a calibrated computation-intensive Kirchhoff integration reference solution based on (3), and with the nontilted formulation in (15) when applied to the entire aperture field in (16). The tilted beam synthesis is hardly distinguishible from the reference solution, whereas the nontilted synthesis is less accurate in the magnitude.
II.e Reflection from a periodic perfectly conducting boundary The field radiated by the aperture distribution in (16) is now assumed to impinge on a perfectly conducting moderately rough periodic boundary described by the continuous function which is assumed to vary slowly over a wavelength scale (Fig. 1a). Moderately rough irregular dielectric interfaces separating two dielectric half-spaces are treated elsewhere (Galdi et al, 2000a). The reflected field can be constructed rigorously by complex ray tracing applied to each beam in the aperture decomposition; this requires the analytic continuation, into a complex configuration space, of all geometrical parameters involved (with the exception of the observation point). However, narrowwaisted beams can be tracked accurately and much more efficiently via a beam-tracing paraxial almost real ray-tracing scheme (Ruan and Felsen, 1986; Maciel and Felsen, 1990b)), valid in appropriately calibrated observation ranges. For the new application to a periodic boundary, we first treat the canonical problem of CSP Gaussian beam reflection from a curved segment on a conducting boundary. The problem geometry is illustrated in Fig. 3. An incident Gaussian beam is generated by a CSP at
being the beam axis real departure angle with respect to the axis. For electrically large and smooth scatterers, and when the observation point lies in the paraxial region of a reflected beam in Fig. 3), the field can be approximated in terms of the on-axis field of that beam (at ) and a complex phase correction. Denoting the on-axis parameters by the subscript zero, one finds for the potential field (Ruan and Felsen, 1986)
CSP NARROW-WAISTED RAY-LIKE GAUSSIAN BEAMS
25
Here (see Fig. 3): is the complex phase correction; is the complex (virtual) focus obtained by analytic continuation of the standard ray-optical formulas (Felsen and Marcuvitz, 1973, p. 168); the complex incidence point is approximated by the real beam-axis incidence point is the reflected-beam-axis real departure angle with respect to the axis; is the curvature radius at and is the plane-wave potential field reflection coefficient. As shown in (Ruan and Felsen, 1986), this scheme corresponds to tracing a ray along a complex trajectory from the CSP at to the intersection of the real beam axis with the real surface; from there, the path to the observer proceeds entirely in real configuration space, along the beam axis. Multiple reflections, which may occur in the configuration in Fig. la, can be incorporated by iterating (21), whereby the complex focus (either virtual or real) associated with each iteration becomes the phase reference for the next iteration (Galdi et al, 2000a). Apart from the complex ray connecting the CSP to the first real incidence point, the multi-hop path to the observer proceeds entirely in real configuration space along the beam axes, and the phase correction is applied only on the last path segment leading to the observer.
II.f Application: infinite sinusoidal boundary For a first check on the applicability of the narrow-beam algorithm to surface scattering problems, we have considered the sinusoidal boundary in Fig. 1a illuminated by a nonphased cosine aperture distribution with wavelength Extensive numerical experiments have been performed for various observation heights and aperture heights profiles with various minumum curvature radii and various beam lattice spacings All of these numerical implementations for the scattered potential field have been compared with a numerically integrated, computation intensive Physical Optics-Kirchhoff reference solution based on (3); by previous calibration, Physical Optics has been confirmed to apply to the profile parameters under consideration here. Typical results for the potential are displayed in Fig. 4. In general, we have found that the accuracy of the narrow-waisted beam algorithm improves for greater observation distance (because of the far-zone paraxial approximation), but even at moderate distances the agreement is satisfactory. We also found that the beam algorithm is quite robust with respect to scramblings of the Gabor lattice. As discussed in (Maciel and Felsen, 1990b), this feature can be exploited to obtain a priori accuracy assessments when reference solutions are not available. For the present nonphased aperture example, we found the best tradeoff between accuracy and computational cost to occur for A finer sampling may, however, be required for phased aperture field distributions. On the other hand, the accuracy gets worse as the distance of the aperture from the surface increases. We found that in order to get robust and accurate predictions, we should have where is the maximum of the boundary profile. However, this is not a very restrictive limitation since it is always possible (and computationally cheap because of the simple determination of the Gabor coefficients via (14)) to perform a multi-step Gabor decomposition for greater aperture-to-boundary distances, i.e., project the beam-computed radiated potential field onto a virtual aperture suitably close to the surface and then again apply the narrow-waisted beam algorithm. Concerning degradation of accuracy with increase in wavelength, we found that even for relatively ”low-frequency” geometries, i.e. moderate as in Fig. 4b, the beam algorithm, though no longer highly accurate, still provides reasonably good predictions (for details, see (Galdi et al, 2000a)).
26
CSP NARROW-WAISTED RAY-LIKE GAUSSIAN BEAMS
III. TIME DOMAIN RADIATION FROM DISTRIBUTED APERTURES We shall now explore the extension of the FD results for aperture radiation in Sec. II.a to timedependent excitation, in particular to short pulses. To this end, we consider a space-time aperture field distribution at with separable space-time dependence and linear time delay
where is the speed of light and is a pulse with characteristic width This distribution represents the TD counterpart of the linearly-phased time-harmonic aperture in (16). The present TD formulation is restricted to the radiated field only, and we analyze the propagation into the halfspace, wherein (23) gives rise to a space-time pulse propagating in the direction (Fig. 5). It is assumed that the normalized width of the pulse is much shorter than the aperture dimension i.e.,
III.a Reference solution Using the two-dimensional TD Green’s function
where H(·) is the Heaviside step function, the field radiated into the half-space can be represented as a space-time Kirchhoff integration (Morse and Feshbach, 1973, Sec. 7.3; Kragalott et al, 1997),
Direct numerical integration of (25) is complicated by the square-root (integrable) singularity at the upper limit and requires special care. We have used the Newton-Cotes scheme (Kragalott et al, 1997) for the numerical integration of (25), which represents our reference solution.
III.b Beam discretization The formal extension of the Gabor-based time-harmonic aperture radiation to time-dependent excitation involves a four-index Gabor series set on a discretized lattice in an eight-dimensional phase space (space-wavenumber, time-frequency). For a rigorous treatment and computational issues, see (Steinberg and Heyman, 1991). We shall explore to what extent the narrow-waisted beam approach,
CSP NARROW-WAISTED RAY-LIKE GAUSSIAN BEAMS
27
effective for time-harmonic excitation, can be generalized to TD (short-pulse) excitation. The lineardelay aperture field distribution (23) admits the equivalent spectral representation
A TD beam discretization can be obtained by Fourier-inverting the narrow-waisted tilted beam expansion (17) for the FD linearly-phased aperture presented in Sec. II.d. In order to accommodate the evanescent spectra in the FD beam propagators (18), we use the analytic signal formulation instead of the standard Fourier transform (Heyman and Melamed, 1998). Concerning the beam lattice discretization, one can choose a frequency-independent beam lattice period (resulting in a frequencydependent beam parameter ), or a frequency-independent beam parameter (resulting in a frequencydependent ). We choose frequency-independent because it yields frequency-independent Gabor coefficients (see (17)). The TD counterpart of the narrow-waisted FD beam expansion (17) for the aperture field distribution (26), with reference to the electric field, can be thus written as
The pulsed beam propagator is the TD counterpart of the FD paraxial, far-zone beam propagator in (18), with (19)2 (for simplicity, the subscript is henceforth replaced by )
Via the analytic signal formulation, one has (Re=real part)
While the integral in (30) cannot be evaluated explicitly in general, we have found useful closed-form approximations for the important class of Gaussian pulses. In particular, we consider a Rayleigh (four-times-differentiated Gaussian) pulse
2 Note that (28), (29) are slightly different from (18), (19), since here the electric field is considered instead of the potential, the aperture plane is located at and propagation is into the halfspace.
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L. B. FELSEN AND V. GALDI
but the procedure presented below can be applied to any kind of modulated or differentiated Gaussian. Since the beam lattice period has been chosen frequency independent, the beam parameter and hence the complex distance in (29) are frequency dependent. For we can approximate in the amplitude factor of (28)
rendering the distance
real. In the phase, we retain the first order paraxial correction
valid for
In the TD, the beam parameter
here
are the beam coordinates (see Fig. 5)
must be small over the entire bandwidth
of
in (31),
With these assumptions, the integral in (30) can be reduced to the generic form (the spectrum of in (31) is evaluated readily),
which can be expressed in terms of confluent hypergeometric functions (Abramowitz and Stegun, 1964, Sec. 13). Accordingly, the TD beam propagator can be written explicitly as
where and are defined in (32) and (34); is the gamma function (Abramowitz and Stegun, 1964, Sec. 6); and is the confluent hypergeometric function (Abramowitz and Stegun, 1964, Sec. 13). The above procedure can be applied to any Gaussian pulse; modulation or differentiation only affects the arguments of M. We found simple rapidly converging approximations for the functions and in the form (Galdi et al, 2000b). These functions resemble the functional form of the time pulse excitation in (31). Using these approximations the TD beam propagator in (37) can be computed efficiently. III.c Assessment of accuracy The restriction in (35) is the most serious because, for specified and observation point, it determines the maximum allowable lattice period (i.e., the minimum number of beams). The overall constraint can be expressed as (Galdi et al, 2000b)
where
is the distance of the observation plane scaled by the Fresnel distance of the aperture, is the normalized bandwidth of the pulse and determines the number of beams in the expansion (27). The nondimensional estimator expresses the range of validity of the algorithm in terms of all relevant parameters of the problem. For example, increasing the lattice period (i.e., decreasing the number of beams) can be compensated by a corresponding
CSP NARROW-WAISTED RAY-LIKE GAUSSIAN BEAMS
29
increase of In order to assess the accuracy of the proposed TD beam expansion, we have performed computations for the linear-delay space-time aperture distribution (23) with a sine spatial tapering, excited by the Rayleigh time pulse (31). Figure 6a shows the time evolution of the electric field at a fixed observation point in the near zone of a large aperture without phase delay computed via the space-time Kirchhoff integration (25), and via the TD beam synthesis (27) with various beam lattice periods. As expected, the agreement improves as the beam lattice period decreases, and satisfactory accuracy is achieved for (for this example ). It is observed from the transverse cut in Fig. 6b that, despite the use of the paraxial paraxial far-zone approximation, the TD beam synthesis is quite accurate even in the near zone of the aperture and not only around the main radiation lobes. We found that, as the observation distance increases, a coarser discretization can be used according to (40); even at moderate distances, quite accurate syntheses can be achieved with a relatively small number of beams (~ 30) (Galdi et al, 2000b). The corresponding results for linear-delay are shown in Fig. 7 and the same considerations apply.
IV. CONCLUSIONS In many current forward and inverse scattering scenarios, there is a need for numerically efficient robust
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L. B. FELSEN AND V. GALDI
forward solvers for fields excited by distributed sources in the presence of complex environments. This motivation has led us to re-visit the previously developed FD narrow-waisted Gaussian beam algorithms (Maciel and Felsen, 1989; 1990a,b) and to extend them to new FD applications as well as to the short-pulse TD. The outcomes from the rough sinusoidal scattering example here, as well as the first results in the TD, are encouraging. Further FD extensions to irregularly rough interfaces between dielectrics are already in progress, as are TD interactions with canonical scatterers to learn the new rules.
ACKNOWLEDGEMENTS We acknowledge fruitful discussions with Professor D.A. Castañon (Boston University) on forward and inverse scattering and imaging which motivated this investigation. We also acknowledge partial support by ODDR&E under MURI grants ARO DAAG55-97-1-0013 and AFOSR F49620-96-1-0028. In addition, VG acknowledges a European Union postdoctoral fellowship through the University of Sannio, Benevento, Italy.
REFERENCES Abramowitz, M., and Stegun, I.A., 1964, Handbook of Mathematical Functions, Dover, New York. Bastiaans, M.J., 1980, Gabor’s expansion of a signal into Gaussian elementary signals, Proc. IEEE, 68:538. Deschamps, G.A., 1971, Gaussian beams as a bundle of complex rays, Electron. Lett., 7:684. Einziger, P.D., and Shapira, M., 1986, Gabor representation and aperture theory, J. Opt. Soc. Am. A, 3:508. Felsen, L.B., Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams, Symp. Matemat., Istituto Nazionale, di Alta Matematica,“ Academic, London, XXVIII:40. Felsen, L.B., and Marcuvitz, N., 1973, Radiation and Scattering of Waves, Prentice Hall, Englewood Cliffs, NJ. Classic reissue, IEEE Press, Piscataway, NJ, 1994. Galdi, V., Felsen, L.B., and Castañon, D.A., 2000a, Quasi-ray Gaussian beam algorithm for scattering by, and reconstruction of, moderately rough interfaces - Part I: forward scattering (internal memorandum, in preparation for publication). Galdi, V., Felsen, L.B., and Castañon, D.A., 2000b, Narrow-waisted Gaussian beam discretization for time-dependent radiation from large apertures (internal memorandum, in preparation for publication). Heyman, E., and Melamed, T., 1998, Space-time representation of ultra wideband signals, Advances in Imaging and Electron Physics, 103:1. Kragalott, M., Kluskens, M.S., and Pala, W.P., 1997, Time-domain fields exterior to a two-dimensional FDTD space, IEEE Trans. Antennas Propagat., 45:1655. Maciel, J.J., and Felsen, L.B., 1989, Systematic study of fields due to extended apertures by Gaussian beam discretization, IEEE Trans. Antennas Propagat., 37:884. Maciel, J.J., and Felsen, L.B., 1990a, Gaussian beam analysis of propagation from an extended aperture distribution through dielectric layers, Part I - plane layer, IEEE Trans. Antennas Propagat., 38:1607. Maciel, J.J., and Felsen, L.B., 1990b, Gaussian beam analysis of propagation from an extended aperture distribution through dielectric layers, Part I - circular cyilindrical layer, IEEE Trans. Antennas Propagat., 38:1618. Morse, P.M., and Feshbach, H., 1953, Methods of Theoretical Physics, McGraw-Hill, New York. Ruan Y.Z., and Felsen, L.B., 1986, Reflection and transmission of beams at a curved interface, J. Opt. Soc. Am, A, 3:566. Steinberg, B.Z., Heyman, E., and Felsen, L.B., 1991, Phase-space beam summation for time-harmonic radiation from large apertures, J. Opt. Soc. Am. A, 8:41. Steinberg, B.Z., and Heyman, E., 1991, Phase-space beam summation for time-dependent radiation from large apertures: discretized parameterization, J. Opt. Soc. Am. A, 8:959.
DIFFRACTION BY ARRAYS OF COMPLEX SOURCE POINT BEAMS
Hong D. Cheung and Edward V. Jull Department of Electrical and Computer Engineering University of British Columbia Vancouver, BC. Canada V6T 1Z4
INTRODUCTION Scattering by an object depends not only on the shape of the object but also on the source of the incident field. Most analytical solutions (eg. Bowman et al, 1987), are for plane wave incidence; that is a source so distant that its directivity has no effect. Or, if the source is local, it is omnidirectional. Here a numerical procedure for extending local omnidirectional source solutions to those for local extended sources at any range is described. It can be applied to both low and high frequency scattering solutions with an accuracy dependant only on the number and accuracy of the basis source solutions used. A superposition of solutions for omnidirectional sources closely spaced in the aperture with amplitudes corresponding to the aperture distribution could provide a correct near field scattering solution but more efficient solutions will require larger source spacings. Then it is necessary to use beam rather than omnidirectional sources and arrange the beam sources in Gabor lattice as described by Einziger at el (1986). Both radiative and reactive aperture fields may then be represented to any accuracy at any range. Gabor (1946) proposed a series of time and frequency shifted Gaussian functions as an alternative to Fourier analysis in signal processing. Most of the impediments which delayed the implementation of Gabor analysis appear to have been overcome and its application to aperture radiation has been reviewed by Bastiaans (1998). The translated and phase shifted Gaussian functions of signal analysis become translated and directionally shifted Gaussian beams in aperture analysis. But Gaussian beams are approximate solutions to the wave equation and here it is found preferable to use complex source point (CSP) beams, which rigorously satisfy the wave equation. Then exact scattering solutions for local omnidirectional sources can be converted to exact CSP beam solutions by substituting appropriate complex coordinates for the real source coordinates. These then become the basis functions for extended source scattering solutions. Complex source point beam are paraxially Gaussian and thus fit well into the framework of Gabor analysis. When many are used with the same amplitude coefficients CSP and Gaussian beams provide virtually identical results in aperture analysis. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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H. D. CHEUNG AND E.V. JULL
First a brief review of the theory and the choice of beam arrangements are made. Then some numerical examples of two dimensional radiation from apertures and its scattering by local simple structures are presented to illustrate the power and simplicity of the method.
DISCRETE SOURCES IN APERTURES
Fig. 1 shows an array of linearly shifted and rotated beam sources w(x) which are to represent the two-dimensional E-polarized fields of an aperture in This aperture field can be written as a Gabor series
where L is the beam spacing along the x-axis and radians is the angular spacing of the beam directions. For a Gaussian window function w(x), the beam amplitudes can be determined by convolving the aperture distribution with a biorthogonal function defined by Einziger et al (1986) With these amplitudes, the field in
for an aperture along the
may be written as
are the elementary Gaussian or complex source point
(CSP) beam fields. For a scattering solution they are the solution for a single CSP beam incident on the scatterer. With enough beams included the above can represent the entire field of any aperture at any range, and the scattered fields for any aperture blockage. Usually the reactive fields are significant only within a fraction of a wavelength from the aperture and can be omitted. Then the bound on the summation in n is the first integer Also since the 2M + 1 beam locations spaced L must cover the aperture width the upper bound on is Finally a key parameter is the spacing in wavelengths of
DIFFRACTION BY ARRAYS OF CSP BEAMS
33
bound on m is Finally a key parameter is the spacing in wavelengths of the elemental beam. Einziger et al (1986) proposed that This places the boundary of the visible spectrum midway between the peaks of the last radiating beam and first nonradiating beam and appears to be an efficient arrangement. Einziger et al (1986) and Maciel and Felsen (1989) have investigated numerically various combinations of beam arrangements. An efficient choice of beam spacing reduces the required number beams by selecting only those that are significant. Larger beam spacings require more rotated beams (larger N) but fewer beam locations (small M) and conversely smaller require fewer rotated beams but more beam locations. We have found it most convenient to use a spacing Thus N = 0, so all beam directions are normal to the aperture, and (1) and (3) become simply
PATTERNS OF COMPLEX SOURCE POINT BEAM ARRAYS An omnidirectional electric line source is
where and and the field point at
is the distance between the source at On replacing the source coordinates with complex values
(6) becomes
which is a beam with its maximum in the direction and its half power beamwidth is
its minimum in the direction
While (9) is a far field result, in the near field of the complex source the beam behaves as the near field of an essentially Gaussian distribution in an aperture of width In the above angles are measured counterclockwise off the line of the array. Thus in Fig. 1 and if beam axes are all in the z-direction and spaced along the x-axis with a beam parameter then and each CSP beam has a half power beamwidth of 77.6° from (10). Fig. 2 shows the near and far field radiation patterns of a 5 element beam array with relative amplitudes at x = 0, respectively, representing a uniform aperture distribution. The relative amplitude of the largest coefficients omitted is All the other beam coefficients, including the tilted beams, are substantially smaller and alternate in sign. The parallel five beams produce accurate main beam and first sidelobe level patterns in the near and far field of the
34
H. D. CHEUNG AND E.V. JULL
aperture. For larger apertures more beams are required and then more sidelobes are accurately represented, errors occurring in the height of the furthest sidelobes off the main beam.
SCATTERING BY OBSTACLES NEAR APERTURES
The far field of a line source parallel to a conducting half plane can be written exactly in term of Fresnel integrals. Thus for the beam array of (5) the basis functions
DIFFRACTION BY ARRAYS OF CSP BEAMS
35
become
where
The dashed curve of Fig. 3b is the far field pattern of Fig. 2 and the solid curve is the far field pattern when the aperture is partially blocked by a parallel conducting half plane with its edge at x = 0, that is at a range of The scattered beam is broadened and outward shifted from its original position. With a larger separation between aperture and half plane there is less broadening but slightly more shift. If the array of Fig. 2 is normally incident on a conducting circular cylinder as in Fig. 4a the array elements for the scattered field are
with replaced by in (7) and (8). Fig. 4b shows the total scattered field of a cylinder of radius with its axis at distances of and from a uniform aperture of width Pattern oscillations are due to interference between the direct and scattered fields. These do not occur in plots of the scattered field only. In this situation a single CSP beam of the same far field beamwidth as the aperture pattern provides a good approximation to the scattered field, as has been shown by Cheung and Jull (1999b). This is because only the main beam of the incident pattern in intercepted by the scatterer. The incident main beamwidth of the array broadens in the near field, as shown in Fig. 2. So also, correspondingly, does the CSP beam.
If the array of Fig. 2 is normally incident on the axis of a square conducting cylinder, with the array axis inclined at off the face of the cylinder, as in Fig. 5a,
36
H. D. CHEUNG AND E.V. JULL
If the array of Fig. 2 is normally incident on the axis of a square conducting cylinder, with the array axis inclined at off the face of the cylinder, as in Fig. 5a, the scattered field is symmetrical about The total field in can be found by the methodology of the uniform geometrical theory of diffraction (UTD) if the sides of the cylinder are not small in wavelength. Then the array elements of the total far field can be written as Here
and
are, respectively, the incident field, the field reflected from the upper
illuminated face of the square cylinder and the total diffracted field of the three edges in the angular region Shadow and reflection boundaries of the facets and edges of the square cylinder determine where these incident and reflected fields contribute and for beam sources boundaries between these regions differ slightly from geometrical optics values. These boundaries are defined in Cheung and Jull (2000) along with expressions for the uniform geometrical optics and singly diffracted fields. By comparison with corresponding numerical results obtained by the moment method, it shown that for Epolarization doubly diffracted fields from the edges are insignificant for cylinder facets of two wavelengths or more.
Fig. 5b shows the total E-polarized far field resulting from a single beam and from a beam array representing a uniform aperture distribution, both at a distance from the edge of a conducting square cylinder of side The far field pattern of the beam array is that of the lower curve of Fig. 2 and has a half power beamwidth of about 20°. At the near field range of the scatterer the incident field is similar to the upper curve of Fig. 2 with a wider beamwidth. The single beam source with has a half power beamwidth of about 20° according to (10) and in the near field this broadens corresponding to that of the array, but the pattern remains a single beam which is paraxially Gaussian. The major differences between the patterns of Fig. 5b are mainly due to the differing structures of the incident fields which, together with the scattered field, make up the far field patterns. Both patterns are almost identical in and around the forward direction where the total field is only that diffracted by the edges. The edges are illuminated by very similar incident fields. At the array representing a uniform aperture distribution has a far field pattern null whereas the single beam source does not. Consequently major differences in the total far field pattern occur. Similarly at
DIFFRACTION BY ARRAYS OF CSP BEAMS
37
where the second null of the aperture far field pattern occurs. Short periodic oscillations in the single beam pattern are all due to interference between diffracted fields from edges off the beam axis and the incident field. This is evident from a comparison with corresponding results for a 90° wedge. The 90° wedge comparison for the aperture scattered fields exhibits also the short periodic oscillations over the reflection lobe evident in Fig. 5b and are thus due in part to inherent differences in near field wedge scattering by an extended aperture source and a single beam source.
CONCLUDING REMARKS The procedure described here seems to be the most general and efficient method for accurately dealing with practical near field scattering problems. It can use the repertoire of line or point source low and high frequency diffraction solutions for canonical structures and it can efficiently provide as much accuracy as these solutions if sufficient sources are used. The examples given here for half planes, cylinders and multiple wedges demonstrate its efficiency for these elementary scatterers. Of course a single CSP beam source is simpler and sometimes adequate for the scattered field alone. But unless this incident field closely resembles a single beam, for example if it is that of a cosine-squared aperture distribution, its total field scattering pattern will substantially differ from that of a single beam. REFERENCES Bastiaans, M., 1998, Gabor’s signal expansion in optics, Chap. 14 of Gabor Analysis and Algorithms, Feichtinger, H. G. and Strohmer T. (Eds.) Birkhauser, Boston. Bowman, J. J., Senior, T. B. A. and Uslenghi, P. L. E., (Eds.) 1987, Electromagnetic and Acoustic Scattering by Simple Shapes, revised printing, Hemisphere Publishing Corporation, New York. Cheung, H. D. and Jull, E. V., 1999a, Two-dimensional diffraction by half-planes and wide slits near radiating apertures, IEEE Trans. Antennas Propagat., 47:1669. Cheung, H. D. and Jull, E. V., 1999b, Scattering of antenna beams by local cylinders, J. Electromag. Waves Applic., 13:1315. Cheung, H. D. and Jull, E. V., 2000, Antenna pattern scattering by rectangular cylinders, IEEE Trans. Antennas Propagat., (in press) Deschamps, G. A., 1971, Gaussian beam as a bundle of complex rays, Electron. Lett., 7:684. Einziger, P. D., Raz, S. and Shapira, M., 1986, Gabor representation and aperture theory, J. Opt. Soc. Am. A, 3:508. Gabor, D., 1946, Theory of communication, J. Inst. Elect. Eng., 93III:429. Maciel, J. J. and Felsen, L. B., 1989, Systematic study of fields due to extended sources by Gaussian beam discretization, IEEE Trans. Antennas Propagat., 37:884.
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APPLICATION OF CONCEPTS OF ADVANCED MATHEMATICS AND PHYSICS TO THE MAXWELL EQUATIONS
Carl E. Baum Air Force Research Laboratory AFRL/DEHE 3550 Aberdeen Ave., SE Kirtland AFB, NM 87117-5776
1.
INTRODUCTION
Since the pioneering work of James Clerk Maxwell [17] in establishing what we call the Maxwell equations
including both electric and equivalent-magnetic source terms, these have had a profound effect on the development of science and engineering. (Note that the divergence equations are implied by the curl equations.) In addition, some material-related parameters are needed to relate such as the constitutive parameters, for example in the form
and
to
and
Here we have introduced the common frequency-domain form so that the vector fields are dot multiplied by 3 × 3 dyadic constitutive parameters, which in time domain become convolution operators over time. More general (even nonlinear) forms are sometimes encountered. Various boundary conditions (e.g., perfectly conducting surfaces) are readily derived as limiting cases. People often think of dividing the basic and applied sides of the technological enterprise as between science and engineering, but this can lead to some confusion. I think that there is a better three-part division, which can shed some light on where electromagnetic (EM) theory fits into the structure. First, there is the basic scientific side which has electromagnetics as part of physics, and the fundamental question concerns the replacement of the Maxwell equations by something more accurate, applying to extreme conditions not normally encountered. This is not what we think of as electromagnetic theory in the usual sense. Second, we have what may be called applied science or basic engineering in which we explore the established physical laws (the Maxwell equations in this case) to see what they imply in the sense of discovering what is possible to analyze, synthesize, optimize, etc. This is distinct from the third category which might be termed applied engineering which concerns itself with the routine implementation of what is known from me second category in terms of technological products (“practicing” engineering), for example, by selection of antenna designs from a product catalog. Of course, the reader might prefer some other “diagonalization” but this should suffice for the present. So, concentrating on the second category, the role of the electromagnetic theorist (including sometimes basic experiments, particularly as demonstrations and confirmations) concerns understanding what the Maxwell equations allow one to do in the way of analysis and synthesis of the performance characteristics Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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C. E. BAUM
of various electromagnetic devices as well as understanding the behavior of electromagnetic fields in natural environments. At this point, I would like to emphasize the concept of EM synthesis. One can analyze the interaction of EM waves with arbitrary geometries of various materials. While this is a challenging task, it is not synthesis. Synthesis starts with some desired performance characteristics and asks: “Is this possible within certain general constants (e.g., passivity)?” If it is possible, then one moves on to other questions such as : “What are the best possible values of the appropriate performance parameters?”, and “What are the algorithms for designing (realizing) the device (antenna, scatterer, etc.) with the desired performance parameters?” By analogy one can recall that circuit analysis with passive lumped elements (LRC) was developed into a matrix form based on the Kirchoff laws for voltage and current as written on a network (graph). Circuit synthesis later asked (and answered) questions like [18]: “What kind of input impedances and transfer functions are possible in such networks”, and “How are such things systematically realized?” An important part of EM theory then needs to be concerned with EM synthesis. One might even think of this as a generalization of circuit synthesis In 1976 I published a review paper concerning transient EM theory [3]. In this I outlined some analytic concepts used in mathematics and physics that are not commonly being used, or just beginning to be used, in EM theory for both analysis and synthesis. Since then considerable progress has been made in exploring these concepts and obtaining useful results. In the present paper these analytical concepts and major results are summarized under the following section headings.
2.
INTEGRAL-OPERATOR DIAGONALIZATION Electromagnetic scattering is often formulated as an integral equation of the form
The notation is related to bra/ket notation in quantum mechanics, with here integration over the common coordinates type of multiplication (dot above the comma here), but with no conjugation implied since our operators are not in general Hermitian. For convenience (2.1) uses the symmetric impedance (or E-field) kernel, related to the dyadic Green function (of free space or other linear reciprocal media), but other kernels (e.g., H-field) are also used. The domain of integration can be over a volume or surface (using tangential components) as desired. As with matrices for which one finds eigenvalues and eigenvectors we can form [7,21]
and we can refer to this as the eigenmode expansion method (EEM). For cases of degeneragy (two or more equal eigenvalues) one use the Gram-Schmidt orthogonalization procedure to complete the construction of the orthonormal set. (More on this appears under symmetry.) With (2.2) we can write the kernel in the form
where represents an arbitrary power, including for inverse kernel which in (2.1) solves for the current on the scatterer. This is not the only kind of eigenmodes one can form from the integral equation, but is a natural choice for our purposes. Other kinds with other names (such as characteristic modes) are introduced by others for special purposes. At this point we can recall [3, 7] that having solved for the eigenimpedances and eigenmodes of a perfectly conducting body (for which (2.1) becomes a surface integral equation), one can also solve directly for the body loaded by some uniform, isotropic sheet impedance by the transformation
APPLICATIONS OF CONCEPTS TO THE MAXWELL EQUATIONS while retaining the same eigenmodes. Then
can be synthesized to give desirable characteristics to the
scatterer or antenna described in the form (2.1). Given limitations of circuit synthesis one can make (poles of the response in which
41
for the unloaded body, then within the have desirable characteristics such as roots
appears) at desirable places in the s plane. These roots
can even be made second order in some cases to give critical damping to the response. Here is a clear example of EM synthesis. Here we also note that the can be split into interior and exterior parts (in electrical parallel combination) which separate the internal and exterior resonances (poles) [7]. However, the details are too elaborate to repeat here. Recently [9] a transformation like (2.4) has been found to apply to more general volumetric dielectric bodies, even those consisting of homogeneous isotropic dielectric bodies residing in an inhomogeneous dielectric space. 3.
COMPLEX VARIABLES APPLIED TO FREQUENCY
As discussed in [3] the analytic properties of the solution of the Maxwell equations as a function of the complex frequency, s, lead to several important ways to solve the Maxwell equations. For antennas and scatterers of finite size in three dimensions this leads to three methods based on expansions used in complexvariable theory. 3.1 Low-Frequency Method (LFM) In complex variables functions are often expanded in terms of a power (Taylor) series about some point where the function is analytic. In EM, this is done for scattering by expanding about s = 0. As one expects, the leading terms are related to the induced electric and magnetic dipoles, related to the incident fields by polarizability dyadics. This is extended to antenna input impedance/admittance by inclusion of a pole at s = 0 when appropriate giving leading terms which can be interpreted as inductance, capacitance, and/or resistance [21]. Here, I would like to emphasize an application of importance to antenna design, particularly the lowfrequency characteristics, concerns the matching of the electric- and magnetic-dipole moments ( and ) in transmission [1]. Defining appropriate unit vectors we have
A remarkable property of such combined dipoles is that on the axis from the antenna “center” in the direction (beam center) the electric and magnetic fields are at right angles and related by
even in the near field including and terms, i.e., all the dipole terms. This has important consequences for low-frequency illumination of large areas for EM interaction measurements, such as for simulation of the nuclear electromagnetic pulse (EMP). The pattern of such an antenna (far field) is a cardiod (radiated power proportional to where
is the angle to the observer relative to
remaining
). It has a null in me back direction
but there is a
term there with the same field ratio as in (3.2) (at right angles with Poynting vector still in the
direction, i.e., back to the antenna).
In reception such an antenna also has similarly interesting
directional properties. An important class of low-dispersion antennas (for transient/broad-band radiation/reception) are referenced as impulse radiating antenna (IRAs) [27]. These can be (and are) designed to exhibit this
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C. E. BAUM
combined dipole behavior at low frequencies with pointing in the same direction as the high-frequency beam. This improves the directionality and modestly decreases the low-frequency roll-off frequency. 3.2 Singularity Expansion Method (SEM) Of more recent vintage (1971) there is SEM. There is already an enormous literature on this subject. Here we mention two review papers with lots of references [14, 15]. In this case a related basic complexvariable expansion is the Laurent expansions in which an expansion is found for the neighborhood of a pole. From (2.1) natural frequencies and modes are found via
Immediately we observe that natural frequencies and modes have nothing to do with the incident-wave parameters (direction of incidence, polarization) in a scattering problem. To better appreciate the above, imagine that one is performing a moment method (numerical) computation. The kernel (operator) is replaced by an N × N matrix from which we find the natural frequencies via
with the natural modes subsequently numerically determined. At this point, we can compare (3.3) to (2.2) and observe that the are roots (zeros) of the linking the index as ( root of the eigenvalue). Assuming an incident plane wave as
the current on the body is expanded as
This is the simplest form of coupling coefficient termed class 1, and it contains the information concerning the incident field. The entire-function term is applicable to early times. By judicious choice of the turn-on time it has been shown that, for perfectly conducting bodies, this entire function can be made zero [6]. In time domain the current is
possible entire function (temporal form) convolution with respect to time t so that the pole terms transform to give a simple time-domain form. While the entire function can be made to be zero the sum in (3.7) is not an efficient early-time representation. The scattered far field takes the form
APPLICATIONS OF CONCEPTS TO THE MAXWELL EQUATIONS
43
Using (2.3) one can readily express the scattering dyadic in EEM form. Here we write the SEM form as
In backscattering this takes the symmetric form
In contradistinction to the current in (3.6), except in special cases, the entire-function contribution to the scattering dyadic cannot be made to go to zero by judicious choice of Noting that the entire function is an early-time contribution one can look at the late-time response for target-identification purposes. We can summarize the major areas of SEM development: 1. description of EM response (especially transient) of various structures (currents) modeling electronic systems [19] 2. equivalent circuits for antennas and scatterers [21] 3. target identification (free space) [15] 4. identification of buried targets (mines, unexploded ordnance) [26] There are also various numerical techniques to analyze data for the SEM parameters [15]. Consulting the references one can find a huge list of references. Perhaps other major areas of SEM application will emerge in the future. 3.3 High-Frequency Method (HFM) In complex-variable theory one often deals with asymptotic expansions as the complex variable tends to infinity. In EM we can collectively refer to such techniques as the HFM [3]. This includes geometric, spectral, uniform, etc., theories of diffraction. An enormous literature exists here. While I have had occasion to consider such techniques, these have developed by many others, and I will not dwell on this. 4.
SYMMETRY AND GROUP THEORY
Group theory has long been used in physics to study the quantum mechanical properties of elementary particles, atoms, molecules, and crystal lattices based on the symmetries of the quantum wave functions. One may consider [3] whether something similar would be useful for the analysis and design of antennas and scatterers. Lewis Carroll had the Hatter ask: “Why is a raven like a writing-desk?” One might ask a similar strange question: “Why is an airplane like a hydrogen molecule?” At least the second question has an answer. They are like in two ways. A first way concerns SEM (Section 3.2). The natural frequencies (in general complex) are characteristic of the body (homogeneous problem), and are analogous to the energy levels (typically real (bound states), but also complex (radioactive decay)) of the quantum system. A second way concerns symmetry. Both objects contain a symmetry plane and the EM response (eigenmodes
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C. E. BAUM
and natural modes) and the quantum wave functions naturally divide into two sets (symmetric and antisymmetric) with respect to the symmetry plane. In physics this property is often called parity. While the quantum symmetries are properties found in nature, the EM symmetries are of two kinds: those inherent in the Maxwell equations (duality, reciprocity, relativistic invariance), and geometrical symmetries built into objects by human beings (or aliens). The close connection between the symmetries in antennas and scatterers and the symmetries in the associated EM waves can be used to design antennas and scatterers and to identify radar targets. The reader can consult [23] for a detailed treatment of this subject, concerning which much progress has been made in recent years. Here we take a group in the form of a 3 × 3 dyadic representation as
For the point symmetry groups (rotations and reflections) these are real and orthogonal with
In some cases these can be taken as 2 × 2 dyadics (or even scalars) (e.g., for N-fold rotation axis). By a symmetric body we mean one that is invariant under transformation by each element of the group. Transforming the body by
we require that the body be unchanged after this transformation (applying to every element of the group of interest). For the body constitutive parameters (permeability, permittivity, conductivity) represented as we require
More generally, we can include the symmetries in the Maxwell equations in the transformation. For example, duality (interchange of electric and magnetic fields) can be included with the body symmetry to allow the interchange of permeability and permittivity dyadics (appropriately normalized) upon transformation by the group elements (self-dual body). The EM fields are also transformed as in (4.3) except for a minus sign in the case of the magnetic field when the transformation has a reflection (improper rotation). The eigenmodes (2.2) and natural modes (3.3) are also transformed by the while keeping the eigenvalues and natural frequencies unchanged. This leads to the symmetry-induced condition of eigenvalue (and natural frequency) degeneracy since is also an eigenmode for the same eigenvalue. In general, however, the eigenmodes so generated are not linearly independent. The number of independent eigenmodes for the same eigenvalue is the degree of degeneracy. For example, symmetry for gives a two-fold degeneracy for in the expansion in cylindrical coordinates. Small deviations from such symmetry break the degeneracy by giving small differences to the eigenvalues and natural frequencies, thereby leading to perturbation formulae.
APPLICATIONS OF CONCEPTS TO THE MAXWELL EQUATIONS
45
Some of the recent symmetry results include:
1. placement and orientation of EM sensors on an aircraft to minimize the influence of aircraft scattering on the measurement (reflection symmetry R) ) 2. high-frequency capacitors (dihedral symmetry for e.g., an 3. nondepolarizing axial backscatter (two-dimensional rotation symmetry N-bladed propeller). 4. generalized Babinet principle (for dyadic impedance sheets) and self-complementary structures ( symmetry) 5. vampire signature (zero backscatter cross polarization in h,v radar coordinates) for mine identification (continuous two-dimensional rotation/reflection symmetry ) [10] separation of magnetic-polarizability dyadic into distinct longitudinal and transverse parts, for low-frequency magnetic singularity identification (diffusion dominated natural frequencies) of metallic targets ( symmetry for ) 7. categorization of the scattering dyadic for the various point symmetries, including reciprocity and self-dual case [28].
6.
Other types of symmetry, such as translation, also have important consequences. These include common waveguiding structures and helices, as well as periodic structures (discrete translation). Dilation symmetries (continuous as in conical structures, and discrete as in log-periodic structures and fractal structures) also give special electromagnetic behaviors. 5.
DIFFERENTIAL GEOMETRY FOR TRANSIENT LENS SYNTHESIS
In gravitational theory differential geometry is used as an integral part of general relativity. In that case, one deals with a four-dimensional space/time. One can also use differential geometry in three spatial dimensions. In this case we are looking for coordinate transformations which allow us to take a known solution of the Maxwell equations with desirable properties in a relatively simple medium, and by curving the coordinates have the same solution in a nonuniform and perhaps anisotropic medium. Bending the wave propagation in this manner gives a lens. We think of this as a transient lens because this works equally well for all frequencies (within the limits of the practical realization of such a medium). For the case of a TEM mode (dispersionless) propagating along two or more guiding conductors the conductors are also curved in the coordinate transformation and are thereby positioned as boundaries on or inside the lens medium. The theory with many examples is discussed in [22]. We imagine some as yet unspecified orthogonal curvilinear coordinate system with
The Maxwell equations as in (1.1) are taken in time domain as homogeneous, i.e., without sources. For time domain we then require zero conductivity with frequency-independent and These fields and constitutive parameters are referred to as real (indicating they can be measured), as contrasted to the formal fields and constitutive parameters. These are designated by superscript primes such that
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C. E. BAUM
The formal parameters define a problem in the coordinates. In tensor language the the
coordinates taken as though these were Cartesian
are the covariant components of
are the contravariant components of (applying to The formal and real fields are related by
and
(applying to
and
), while
).
The formal and real constitutive parameters are related by
which for diagonal constitutive-parameter matrices reduce to
For cases considered to date then we have
where the components are referred to the coordinates. The problem is then to take some known formal fields with formal constitutive parameters, and find what coordinates exist in which we have real fields and constitutive parameters subject to constraints (realizability conditions) on the real constitutive parameters. For example, one might be considering TEM waves propagating in the direction, making and irrelevant. But then one might also like the real medium to be isotropic so that and It has been shown in such a case that constant surfaces are planes or spheres, limiting the class of acceptable coordinates. Within this class various acceptable coordinate systems, and hence transient lenses, have been found. As summarized in [22] there are several classes of solutions of these equations:
1. all six components of and nonzero for inhomogeneous but isotropic and (only two possible coordinate systems) 2. TEM waves propagating in the direction for inhomogeneous but isotropic and (coordinate systems constrained by constant surfaces being planes or spheres, examples including converging, diverging, and bending lenses)
3. two-dimensional lenses for TEM waves (only one component each of and nonzero) based on conformal transformations (resulting in only one of and µ being inhomogeneous, but both isotropic) 4. lenses with but anisotropic and inhomogeneous.
APPLICATIONS OF CONCEPTS TO THE MAXWELL EQUATIONS
47
Since the book several new examples have been developed. An important class of these involve but inhomogeneous and isotropic, making them relatively practical for construction. Of these, an important type ofmedium is a cylindrically inhomogeneous dielectric (CID) with the permittivity distributed as
in a cylindrical
coordinate system. This admits as solutions:
5. TEM waves propagating in the direction (bending lens) with very general transmission-line cross sections (e.g., circular coax) [2]. 6.
ELECTROMAGNETIC TOPOLOGY FOR ANALYSIS AND CONTROL OF ELECTROMAGNETIC INTERACTION WITH COMPLEX SYSTEMS
A certain kind of topology, graph theory, is commonly used in electrical engineering to describe electrical networks. For circuit analysis such networks are described by nodes and branches, on which are written the Kirchoff equations which say that the sum of the currents leaving a node are zero and the sum of the voltage drops around a loop are zero. Electromagnetic topology (Fig. 6.1) begins by recognizing mat space can be divided into a set of volumes separated by boundary surfaces. For signals to propagate from one volume to another they must pass through one or more surfaces. Some of these surfaces (closed ones) can take on the role of an EM shield in the usual sense. These can be nested inside one another to form a hierarchical topology. There is a dual topology, the interaction sequence diagram, which is a graph (or network) in which the volumes are replaced by nodes (vertices) and the surfaces separating adjacent volumes by branches (edges), this also being indicated in Fig. 6.1. This is the subject of qualitative (or descriptive) EM topology, which can be used to organize the EM design of complex systems. This is contained (along with quantitative aspects) in [5, 20] which also contain numerous references. Quantitative EM topology is based on the BLT1 equation [4] which was originally stated in a form appropriate to multiconductor-transmission-line (MTL) networks as
48
C. E. BAUM
This is written on an MTL network (Fig. 6.2) consisting of junctions characterized by scattering matrices, and tubes consisting of MTLs with conductors (plus reference) connecting appropriate junctions. Each tube contains two vector waves, one propagating in each direction, indexed by for where is twice the number of tubes. The tubes here are taken as uniform (not varying along the tube) and characterized by
From these we have
APPLICATIONS OF CONCEPTS TO THE MAXWELL EQUATIONS
In turn, the combined voltage waves are defined for each
49
wave by
with positive convention for current in the direction of increasing (For the two waves on a tube (two values of ) the current conventions are opposite.) The distributed combined sources for the uth wave are similarly
giving the source term in (6.1) as
Relating the MTL network to the EM topology, note that by shrinking the tubes to zero length the junctions can represent the volumes, the tubes the connecting surfaces, and the sources lumped equivalent sources at each surface. In this form (BLT2) then
and disappears from (6.1). An alternate way to approach this is to recognize that a tube may be represented by a junction, fitting an MTL network into BLT2 form. A more elaborate form, the NBLT (nonuniform BLT) equation [8], allows for NMTL (nonuniform MTL) tubes for which the per-unit-length parameter matrices are allowed to vary as a function of In this last case the two waves on a tube do not neatly separate, but scatter into each other as they propagate along the tube coordinate. Again this case can also be cast into BLT2 form by defining such a tube as a junction with scattering matrices and equivalent source vectors. A yet further form (BLT3) utilizes the delay property of the tubes to expand the interactionsupermatrix inverse in a geometric series which can be used for early-times in time domain [13]. These BLT networks can become rather elaborate for large electronic systems such as aircraft. Computer codes such as CRIPTE [24] have successfully modeled such systems, and further improvements are anticipated. The computation time has been recently significantly reduced by graph-theoretic techniques in which appropriate portions of the network are reduced to equivalent junctions before inverting the interaction
50
C. E. BAUM
supermatrix [11]. The successful implementation of such calculations has been from DC to several hundred MHz, pushing to a GHz. Further improvements may push this higher by modeling the cavities and cavities with transmission lines in appropriate ways that fit into the topologically-decomposed scattering-matrix formalism. Another potential improvement involves inclusion of the good-shielding approximation to break the full-system problem into smaller problems at shield/subshield boundaries, with simple matrix multiplication to reconnect the subproblems. One can also use SEM concepts to more simply evaluate the late-time behavior of the system in terms of natural frequencies and modes. Closely tied to EM topology (although one could consider this a separate subject) is the subject of the response of NMTLs [25]. For this purpose it is convenient to formulate a single NMTL via a supermatrix differential equation of the form
where currents are referenced to the +z direction. Solving this equation gives relations between voltages and currents at both ends of the tube together with equivalent sources there. This is a chain-matrix-like formulation of the problem which is later (after solution) converted int a scattering-matrix form for insertion into the BLT equation. This equation is related to the supermatrizant differential equation
Provided that we have found the supermatrizant we have the solution of (6.8) as
By choosing as one end of the tube and as the other end the terminal parameters are related and the scattering supermatrix and equivalent sources are obtained. The supermatrizant is expressed as a product integral [12]
This can be thought of as a repeated dot product (increasing
terms multiplying on left of form
by comparison to the usual sum integral. In special cases, this reduces to a sum integral as
APPLICATIONS OF CONCEPTS TO THE MAXWELL EQUATIONS
provided
evaluated at
and
commute with each other for every pair
51
and
in the
interval One example concerns a constant matrix, which gives the result for in (6.1). Another example concerns circulant matrices for the per-unit-length parameter matrices [16]. Special results also apply to the case of uniform modal speeds as occur for nonuniform wires (size, spacing) in a uniform medium [12, 25]. The product integral is suggestive of a numerical way for evaluating the supermarizant, i.e., by dividing the interval into some number of subintervals, approximating the result for each subinterval by assuming a constant matrix there, and multiplying the results for all the subintervals. This is a staircase approximation. One can do better in some cases by allowing a smooth variation (e.g., linear or exponential) of eigenvalues with constant eigenvectors over each subinterval [12, 25]. This allows one to preserve continuity of the line parameters from one subinterval to the nect, thereby reducing reflections at such boundaries. The product integral has various special formulas analogous to those for sum integrals (e.g., integration by parts). What is called the sum rule allows one to separate
into the sum of
two terms. If one term has a readily evaluated product integral (closed form), the problem is changed to a new product integral. If the second term is suitably small, the new product integral can be readily approximated by the first two terms in a series representation of the matrizant (the first term being the identity) [25]. This gives a perturbation formula for approximating the solution of an almost uniform MTL. 7.
CONCLUDING REMARKS
So we now have a collection of modern mathematical techniques to apply to the Maxwell equations (analysis and synthesis). Much has been learned using these and I would expect that much more can be learned. This should lead to new classes of electromagnetic devices. We should continue searching for other mathematical structures which may be of use to electromagnetic theory. Noting the importance of the mathematics used in quantum mechanics, one might consider more esoteric physics such as quantum electrodynamics, string theory, etc. Not included in our discussion here, and still in its infancy is statistical electromagnetics, for which one may expect more important future results. This work was supported in part by the U. S. Air Force Office of Scientific Research, and in part by the U. S. Air Force Research Laboratory, Directed Energy Directorate. REFERENCES 1. 2.
3. 4.
5. 6.
7.
C. E. Baum, Some Characteristics of Electric and Magnetic Dipole Antennas for radiating Transient Pulses, Sensor and Simulation Note 125, January 1971. C. E. Baum, Use of Generalized Inhomogeneous TEM Plane Waves in Differential Geometric Lens Synthesis, Sensor and Simulation Note 405, December 1996; Proc. URSI Int’l Symposium on Electromagnetic Theory, Thessaloniki, Greece, May 1998, pp. 636-638. C. E. Baum, Emerging Technology for Transient and Broad-Band Analysis and Synthesis of Antennas and Scatterers, Interaction Note 300, November 1976; Proc. IEEE, 1976, pp. 1598-1616. C. E. Baum, T. K. Liu, and F. M. Tesche, On the Analysis of General Multiconductor TransmissionLine Networks, Interaction Note 350, November 1978; also contained in C. E. Baum, Electromagnetic Topology for Analysis and Design of Complex Electromagnetic Systems, pp. 467-547, in J. E. Thompson and L. E. Luessen (eds.), Fast Electrical and Optical Measurements, Martinus Nijhoff, Dordrecht, 1986. C. E. Baum, The Theory of Electromagnetic Interference Control, Interaction Note 478, December 1989; pp. 87-101, in J. Bach Anderson (ed.), Modern Radio Science 1990, Oxford U. Press. C. E. baum, Representation of Surface Current Density and Far Scattering in EEM and SEM with Entire Functions, Interaction Note 486, February 1992; Ch. 13, pp. 273-316, in P. P. Delsanto and A. W. Saenz (eds.), New perspectives on Problems in Classical and Quantum Physics, Part II, Acoustic Propagation and Scattering, Electromagnetic Scattering, Gordon and Breach, 1998. C. E. Baum, Properties of Eigenterms of the Impedance Integral Equation, Interaction Note 487, April 1992; Ch. 2, pp. 39-91, in A. Guran, R. Mittra, and P. J. Moser (eds.), Electromagnetic Wave Interactions, World Scientific, 1996.
52 8. 9.
10. 11.
12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.
C. E. BAUM C. E. Baum, Generalization of the BLT Equation, Interaction Note 511, April 1995; Proc. 13th Int’l Zurich Symposium and Technical Exhibition on Electromagnetic Compatibility, Feb. 1999, pp. 131-136. G. W. Hanson and C. E. Baum, A Volumetric Eigenmode Expansion Method for Dielectric Bodies, Interaction Note 517, August 1996. C. E. Baum, Symmetry in Electromagnetic Scattering as a Target Discriminant, Interaction Note 523, October 1996; pp. 295-307, in H. Mott and W. Boerner (eds.), Wideband Interferometric Sensing and Imaging Polarimetry, Proc. SPIE, Vol. 3120, San Diego, Calif., July 1997. J.-P. Parmantier, X. Ferrieres, S. Bertuol, and C. E. Baum, Various Ways to Think of the Resolution of the BLT Equation with an LU Technique, Interaction Note 535, January 1998; Optimization of the BLT Equation Based on a Sparse Gaussian Elimination, Proc. 13th Int’l Zurich Symposium and Technical Exhibition on Electromagnetic Compatibility, February 1999, pp. 137-142. C. E. Baum, Symmetric Renormalization of the Nonuniform Multiconductor-Transmission-Line Equations with a Single Modal Speed for Analytically Solvable Sections, Interaction Note 537, January 1998. C. E. Baum, Extension of the BLT Equation into Time Domain, Interaction Note 553, March 1999. C. E. Baum, The Singularity Expansion Method: Background and Developments, IEEE Antennas and Propagation Mag., Vol. 28, No. 4, August 1986, pp. 15-23. C. E. Baum, E. J. Rothwell, K.-M. Chen, and D. P. Nyquist, The Singularity Expansion Method and Its Application to Target Identification, Proc. IEEE, 1991, pp. 1481-1492. J. Nitsch and C. E. Baum, Analytical Treatment of Circulant Nonuniform Multiconductor Transmission Lines, IEEE Trans. EMC, 1992, pp. 28-38. J. C. Maxwell, A Treatise on Electricity and Magnetism, 3rd Ed., Dover, 1954 (from 3rd ed. 1891). E. A. Guillemin, Synthesis of Passive Networks, Wiley, 1957. C. E. Baum, The Singularity Expansion Method, Ch. 3, pp. 130-179, in L. B. Felsen (ed.), Transient Electromagnetic Fields, Springer-Verlag, 1976. C. E. Baum, The Role of Scattering Theory in Electromagnetic Interference problems, Ch. 13, pp. 471502, in P. L. E. Uslenghi (ed.), Electromagnetic Scattering, Academic Press, 1978. C. E. Baum, Toward an Engineering Theory of Electromagnetic Scattering: The Singularity and Eigenmode Expansion Methods, Ch. 15, pp. 571-651, in P. L. E. Uslenghi (ed.), Electromagnetic Scattering, Academic Press, 1978. C. E. Baum and A. P. Stone, Transient lens Synthesis: Differential Geometry in Electromagnetic Theory, Taylor & Francis, 1991. C. E. Baum and H. N. Kritikos (eds.), Electromagnetic Symmetry, Taylor & Francis, 1995. J.-P. Parmantier and P. Degauque, Topology Based Modeling of Very Large Systems, pp. 151-177, in J. Hamelin (ed.), Modern Radio Science 1996, Oxford U. press, 1996. C. E. Baum, J. B. Nitsch, and R. J. Sturm, Analytical Solution for Uniform and Nonuniform Multiconductor Transmission Lines with Sources, Ch. 18, pp. 433-464, in W. R. Stone (ed.), Review of Radio Science 1993-1996, Oxford U. Press, 1996. C. E. Baum (ed.), Detection and Identification of Visually Obscured Targets, Taylor & Francis, 1998. C. E. Baum, E. G. Farr, and D. V. Giri, Review of Impulse-Radiating Antennas, Ch. 16, pp. 403-439, in W. R. Stone (ed.), Review of Radio Science 1996-1999, Oxford U. press, 1999. C. E. Baum, Target Symmetry and the Scattering Dyadic, Ch. 4, pp. 204-236, in D. H. Werner and R. Mittra (eds.), Frontiers in Electromagnetics, IEEE Press, 1999.
SURFACE DISCHARGE CELLULAR AUTOMATON MODEL
Masashi Hayakawa1, Nikolay V. Korovkin1,4, Dmitry I. Iudin1,2, Ekaterina E. Selina4, Viktor Yu. Trakhtengerts3 1
The University of Electro-Communications 1-5-1 Chofugaoka, Chofu Tokyo 182-8585, Japan 2 Radiophysical Research Institute Bolshaya Pecherskaya st. 25/14, Nizhny Novgorod 603600, Russia 3 Institute of Applied Physics, Russian Academy of Science Ulyanov st. 46, Nizhny Novgorod 603600, Russia 4 Saint Petersburg State Technical University Polytechnicheskaya st., 29, St. Petersburg 125251, Russia
INTRODUCTION The electric charge induced on a dielectric surface of an aircraft or rocket passing through a layer of cloud results in the development of the surface discharge. The distribution of the local non-uniform charge is shown in figure 1. This charge generates short wave electromagnetic radiation (down to X-ray). This radiation penetrates inside the body of an aircraft through the apertures and slits; it causes undesirable action on technical systems and biological objects. When atmospheric pressure decreases then conditions for the discharge initiation are improved, its intensity increases and the apertures are overlapped at lower voltages. The disturbances from the surface discharge in high voltage equipment are also well known. Moreover, the surface discharge impairs the characteristics of insulation and it deteriorates faster. The surface discharge on a dielectric surface is widely used in the various technical applications, for example, in electrophysical devices, pulse light sources, pumping up systems of gas lasers. At the same time the surface discharge is the basic danger for the normal functioning of the numerous dielectric structures.
Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
53
54
HAYAKAWA ET AL.
Though the surface discharge is a frequently registered phenomenon, the methods of its mathematical description and modelling are poorly developed. The reason is that the traditional models in the form of the algebraic or differential equations do not sufficiently represent the charge distribution process. In fact, with the help of the existing models we can make only a crude estimation of mean values of the discharge current, initial voltage and radiation intensity for very simple systems.
So the existing mathematical models of the surface discharge are poorly available for the solution of the practical EMC problems. The purpose of this work is to construct a surface discharge mathematical model that reproduces in details essential features of this phenomenon. Created model should take into account the geometry of the surface, shape of the apertures and slits, characteristics of the dielectric.
PROCEDURE OF CELLULAR AUTOMATON ACTION The sliding discharge in gas along the surface of the solid dielectric appears when the normal component of the electric field intensity exceeds considerably its tangent component. Configuration of the discharge gap where this condition is fulfilled is shown in figure 2. Development of the discharge begins from the appearance of the corona discharge near the upper electrode 1. The magnitude of the initial voltage of the corona discharge initiation depends on the electrical characteristics of dielectric and cleanness of its surface. Then, corona discharge turns into the streamer form when voltage increases. The filaments of the current appear on the surface of the dielectric (see area 1 in figure 2). The surface of the dielectric is charged near the upper electrode through these filaments. Current in the filaments is closed through the dielectric as the displacement current. Transmission of the streamers into the external non-charged area (see area 2 in figure 2) goes with the generation of the branched structures. The results of the experiment show that the distribution of the charge is a treelike structure. This specific structure has the properties of scaling and self-similarity. In the other words the structure of the charge on the surface has the properties of a fractal. Simultaneously with the distribution of charge the process of charge dispersal takes place. The process of discharge of the pre-charged element of the surface occurs significantly slower. In the steady-state condition the processes of charge and discharge are in the
SURFACE DISCHARGE CELLULAR AUTOMATON MODEL
55
dynamic equilibrium. After charging the surface of dielectric by the treelike structure it dies. New treelike structure is growing during the distribution of charge on the uncharged area. The surface discharge modeling is done on a basis of the two-dimensional cellular automaton network where separate automaton cell represents some element of the dielectric surface with square S=d·d (see figure 3). Each element of the network (cell) has four nearest neighbors and may exist in three different states: "A", "B" or "C". Each state can be described by its action procedure. A. Waiting state. Capacitor of the dielectric surface of the cell is not charged. This is the surface of the discharge development. B. Relaxation or tolerance state. Capacitor is pre-charged up to the voltage In the state "B" it is discharged through the linear resistor with the admittance C. Current state. The transposition (distribution) of the charge along the surface takes place. Automaton cell simulates the leader channel of the surface discharge current. Procedure of the automaton action in the state “A” is the simplest among these states. An automaton exists in the state of readiness (waiting) to transition into the state “B” and informs about it by the appropriate announcement. State “A” is the initial state for all automatons except the automaton indicated in figure 3 by the circle. An automaton may turn from the state “A” only into the state “B”.
In the state “B” capacitor of the examined cell S discharges through the admittance of the element that corresponds to the leakage admittance of the dielectric cell. The process of discharge can be described by one of the following difference equations - with the order O(h) of approximation of the initial differential equation
with the order
of approximation
Here - the step of reproduction of the dynamics of an automaton action in the time domain, - initial voltage of the cell, - voltage and current in the surface element S at the point in time, n=0,1, 2, .. .
HAYAKAWA ET AL.
56
Cellular automaton passes from the state "A" to "B" if at least one of its nearest neighbors is in the state "C" and has a superfluous charge The charge satisfies the following equation: Cellular automaton passes from the state "B" to "A" at time when its voltage
becomes less than an arbitrary voltage
If the voltage
in the state "B" satisfies the inequality
is less than
and charge
, then cellular automaton generates the request on receiving a charge
portion This portion of charge can be transmitted only in the case when the nearest automaton in the state "C" has a sufficient superfluous charge. If the request of an automaton in the state “B” at the fixed time step is not satisfied then the request is canceled and is not restored later on. If several automatons in the state “B” are situated near the automatons in the state “C” with the surplus charge then it is distributed randomly between them. If there is enough charge then all neighboring automatons in the state “B” will be charged. When there are both types of automatons in states “B” and “A” among the neighbors of the cellular automaton in the state ”C” then the requests of cellular automaton in the state “B” are fulfilled first and after it the requests of cellular automaton in the state “A” are realized. If there are several cellular automaton of the type “A” among the neighbors of cellular automaton ”C” then the charge distribution passes randomly. However, even though there is an excess of the charge, not surely all requests of cellular automaton in the state “A” will be fulfilled. The number of cellular automaton that will be transferred from the state “A” into the state “B” is determined randomly. The procedure of action of the automaton in the state “C” simulates the dynamics of the surface discharge. Let us consider the continuity of one time step performance. At the beginning of the process one of the automatons (let’s denote it zero, it’s is marked in figure 3) is transferred into the state ”C” and it keeps this state unconverted during all the time of the process simulating. At each time step the null automaton receives the charge equal to m· from the external source. Further, this charge is distributed among the neighboring automatons in the state “A” and they pass into the state “B” in this way. Then, within the framework of the current step, it is determined whether the total charge received from the source is distributed. If it is so, then the time step is finished. Otherwise, the remained charge is distributed repeatedly. Herewith the automatons in the state “B” are transferred in the state “C”. The posterior charge distribution is carried through the newly created automatons in the state “C”.
CHARGE DISTRIBUTION MODELING The process of the charge distribution is made so long as the quantity of the undistributed charge stops changing. In this case the charge flows through the automatons in the state “C” simulating the streamer of the surface discharge. At the end of the process of the charge distribution at each time step the quantity of charge flowed through each automaton in the state ”C” is calculated. If automaton has transferred the more charge than some fixed value then it remains in the state “C”, otherwise it passes into the state “B”. Since the flow of charge is moved from one element S so the automatons of the “C” type generate one associated cluster. The extreme automatons of the “C” type in the cluster that have received charge but haven’t distributed it or distributed the charge less than would transfer into the state “B”. Accordingly the streamers of the surface discharge will change their shape at each
SURFACE DISCHARGE CELLULAR AUTOMATON MODEL
57
time step. These changes will be frequent in the periphery and more occasional in the elements that are near the source. The cellular automaton model parameters: can be simply determined from physical experiments. The model allows us to reproduce not only integrated characteristics of the discharge but also its local properties that are extremely important in applications. For example, the offered model enables us to investigate the influence of the surface irregularities (edges, apertures). The charge transferred into the system during one time step not surely will be distributed completely. In this case the null automaton accumulates it. Let the charge be distributed by the end of the time step then the discharge current and voltage at the n step can be calculated from the equations: , where M is the total number of steps in the time domain,
– the
charge which was not distributed at the step number k.
We obtain cellular automaton in "C" state if it transfers some electric charge. Having transferred a charge greater than fixed, cellular automaton remains in "C" state, otherwise it passes in "B" state. During this process the central cellular automaton is always in state "C".
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HAYAKAWA ET AL.
Temporal evolution of the tolerance cell number is shown in figure 4. Time evolution of the total current and its spectral density are show in figures 5 and 6 correspondingly. The last is power law with a slope of about -1. The cellular automaton cells of the "C" type form a connected cluster. This cluster represents a transport network that distributes electrical charge all over the grid. It is obvious that total discharge current On the other hand where correlates with the loses in the discharge channel and is the network fractal dimension. So, one may obtain that configuration is shown in figure 7.
The surface discharge typical
The leader channel is depicted in figure 7 by red color, dark-blue field correlates with the automatons in the state "A", more light colors up to yellow correspond to the automatons in the state "B". The lighter the field is, the larger the charge of the automaton is. The leader with more branches is shown in red color in figure 8, charged cells in the state "B" are pictured in blue and the cells in the state "A" – in cyan.
CONCLUSIONS 1. Surface discharge can be examined as a distribution of the charge on the surface of dielectric. The mathematical model based on finite automatons describes the behavior of this system. The dynamics of this system is close to the self-organizing chaos. 2. Properties of the surface discharge mathematical model depend greatly from the parameters of the model. These parameters can be determined by simple experiments. 3. Suggested model simulates the real process in the real technical system. This allows changing the technical system in such a way that minimizes the influence of the electromagnetic radiation of the surface discharge on the operation mode of the system.
GREEN'S FUNCTIONS FOR SHEET CURRENTS PLACED OVER CYLINDRICAL METAL SURFACE
Alexander Svezhentsev* and Guy Vandenbosch Katholieke Universiteit Leuven (ESAT-TELEMIC), Kardinaal Mercierlaan 94, B-3001 Heverlee, Belgium
INTRODUCTION In some antenna applications it is preferable to use conformal antennas in which patches are placed on non-planar surfaces, for example, cylindrical ones. In this case a necessary first step is to calculate the Green's Functions (GFs) for sheet currents in such structures. Some aspects of this problem were discussed both in approximate1,2,3 and rigorous4,5 formulations. Mainly, the resonant frequencies and radiation pattern in the far zone were investigated. In this paper the spatial GFs for sheet currents in a cylindrical structure are rigorously calculated on the cylindrical surface where those sheet currents are placed. A special technique is used to obtain the singular part of the spatial GFs in an analytical form. This approach ensures the calculation of the GFs within a reasonable time.
PROBLEM FORMULATION The structure under investigation consists of a metal cylindrical circular conductor (which is infinite in the z-direction) with radius and the current interface The cross section is shown in Fig.l. The current interface at between free space (layer i=0) and layer contains the distribution of the electric and the magnetic sheet currents where 's' stands or The problem is to find the spatial Green's functions, which connect fields and currents at the interface. To solve this problem we will pass two stages. *He has a permanent research position at the Institute of Radio Physics and Electronics of National Academy of Science of the Ukraine and is currently working with the KU Leuven as a postdoctoral researcher.
Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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A. SVEZHENTSEV AND G. VANDENBOSCH
The first one is to find a Green's function in the spectral domain and the second one is to perform the Inverse Fourier Transform (IFT) procedure. The IFT procedure for fields in cylindrical co-ordinates for each layer i (i=0,1) is:
where is the pure imaginary unit. In each layer the z-components of the field are derived as a solution of the 2D Spectral Helmholtz equations.
The solution looks like:
where
GREEN'S FUNCTIONS FOR SHEET CURRENTS
At the current interface
61
the field components obey the boundary conditions:
where is the Fourier Transform (FT) of sheet current In (3) the transverse components of the electric and magnetic field can be expressed in terms of the z-components as:
Note that we can split fields in each layer in two independent systems (TE and TM) with only five components of the field different from zero. Due to the currents the TE and TM waves occur simultaneously. SPATIAL GREEN'S FUNCTIONS Mixed-Potential Integral Expressions for the Electric Field The electric field integral equation (EFIE) in the mixed-potential formulation is one of the forms to which a moment method procedure can be applied to investigate antenna problems. This approach was successfully applied to planar microstrip antennas6,7. Returning to the results of the previous paragraph and applying the IFT procedure to the relations between electric field and current components one can express the spatial electric field components in the form:
where
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A. SVEZHENTSEV AND G. VANDENBOSCH
IFT for the Asymptotes. Spatial Green's Functions. Using handbook9 to solve some IFT integrals we can get analytical formulas for the asymptotic part of the spatial Green's functions. As a result any spatial GF can be represented in the form:
We should use the starting point n=1 for The asymptotic part of the spatial Green's functions
looks like:
where
The asymptotes
can be represented in a form:
where is the modified Bessel function. Signs (+/-) correspond to the respectively. And finally
looks like
GREEN'S FUNCTIONS FOR SHEET CURRENTS
63
the indexes 'J' and stand for current and charge, respectively. Also note that the expressions for the spectral GFs can be easily determined after satisfying the boundary conditions (3). It is seen from (4) that in the common case (dielectric layers) eight spectral GFs need to be Inverse Fourier Transformed instead of four GFs in the planar case6. In our case (without dielectric) it will be shown later that only four GFs need to be calculated. Note also that in the planar case each IFT integral is reduced to a one-dimensional integral of the Sommerfeld type. In the cylindrical case we will have to calculate not only the integral over the h parameter but also we will have to calculate the sum over the angular dependencies. Therefore the spatial GFs in the cylindrical case are always functions of two variables: and This will require more computer recourses than in the planar case. Now the problem is how to calculate the integral over h in the optimal way. Analytical investigation of the spectral GFs showed that there are regions, namely, for large values of h, where the integrand has a so-called 'bad' behaviour. A special technique will be applied. It is described in the next section. Dominant Contribution in the Neighbourhood of the Source. Asymptotes of the Spectral Green's Function. It was shown7 that the asymptotic behaviour of spectral GFs for large h determines the contribution close to the source. In order to speed up the IFT drastically it is very advantageous to subtract the asymptote (in the spectral domain) and to add (in the spatial domain) its spatial equivalent. The spatial equivalent is derived in analytic form. It is very important to stress that the situation with asymptotes (large h) in the cylindrical case is much more complicated than in the planar case because we should take into account also the spectral Green's function dependence over n. Using the approximate expressions for cylindrical functions 8 we get next approximate formulas for the spectral Green's functions which can be divided in three groups. Inside each group all asymptotes are the same. The first group of spectral Green's functions shows the asymptotes:
The asymptotes:
are valid for Signs (-/+) correspond to the The third type of asymptote is:
respectively.
64
A. SVEZHENTSEV AND G. VANDENBOSCH
NUMERICAL RESULTS The dependence of the spatial Green's function on is shown in Fig.2 for fixed value The fact that the function has to be periodical is met. The period is
as it should be. In the plotted case the observation point lies close to the source point and also close enough to the cylinder surface. That is why we can clearly see the region of shadow, which lies opposite to the observation point vicinity with respect to the cylinder. In this region the field is very small. The calculation time mostly depends on the ratio The less this ratio the higher the calculation time. For parameters which correspond to Fig.2, the calculation time for one point is 8 second when For the calculation time of one point is 18 second for the chosen parameters. The calculations were performed on a HP workstation J-5000 (440 MHz); the processor type is PA 8500. The accuracy to calculate the integrals connected with the IFT was Note that we cannot directly sum up the expression (5) in real time without improving its convergence because the common term of this series behaves as when n goes to infinity. This function yields a convergence which is extremely slow. CONCLUSIONS An effective approach was realised for the calculation of the spatial GFs for sheet electric and magnetic currents, which are located at cylindrical interfaces over a cylindrical
GREEN'S FUNCTIONS FOR SHEET CURRENTS
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structure. This approach is based on finding the spectral GFs and calculating the Inverse Fourier Transform equivalent. It is very important to note that the singular part of the Spatial GFs is given in an analytical form. Important is that the spatial GFs depend on two variables in the cylindrical case. This means that the cylindrical case needs considerably more computer resources for this aspect than compared to planar structures. This approach can be generalised to multilayered dielectric structures. REFERENCES 1. K.-M. Luk K.-M., K.-F. Lee and J.S. Dahele. Analysis of the cylindrical - rectangular patch antenna, IEEE Trans. on Antennas and Propagation, vol. 37, N 2, February (1989). 2. M. Hamadallah. Radiation pattern of patch mounted diagonally on cylinder, Electronics letters, vol. 24, N 21, October (1988). 3. C.M. Krowne. Cylindrical-rectangular microstrip antenna, IEEE Trans. on Antennas and Propagation, vol. 31, N 1, January (1983). 4. J. Ashkenazy, S. Shtrikman and D. Treves. Electric surface current model for the analysis of microstrip antennas on cylindrical bodies, IEEE Trans. on Antennas and Propagation, vol. 33, N 3, March (1983). 5. G. Gottwald and W. Wiesbeck. Radiation efficiency of conformal microstrip antennas on cylindrical surfaces, Proceedings of the 1995 IEEE AP-Symposium, pp. 1780-1783 (1995). 6. G.A.E. Vandenbosch and A.R. Van de Capelle. Mixed-potential integral expression formulation of the electric field in a stratified dielectric medium - application to the case of a probe current source, IEEE Trans. on Antennas and Propagation, vol. 40, N7, July (1992). 7. F.J. Demuynck, G.A.E. Vandenbosch and A.R. Van de Capelle. The expansion wave concept - part I: Efficient calculation of Spatial Green's functions in a stratified dielectric medium, IEEE Trans. on Antennas and Propagation, vol. 46, N 3, March (1998). 8. Handbook of Mathematical Functions. Edited by M. Abramowitz and I.A. Stegun. Dover publication, inc., New York, 1965. 1046 p. 9. I. Gradshteyn and I. Ryzhik. Table of Integrals, Series and Products, Academic Press, New York, 1980, (1981 printing).
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UWB ANALYSIS OF EM FIELDS IN COMPLEX LAMINATES: A MULTIRESOLUTION HOMOGENIZATION APPROACH
Vitaliy Lomakin, Ben Zion Steinberg and Ehud Heyman Faculty of Engineering Tel-Aviv Univetsity Tel-Aviv, 69978, Israel
INTRODUCTION Finely layered multi-scale heterogeneity laminates can be found in many man-made structures and natural environments. The study of propagation and/or scattering of EM waves in such laminates is of fundamental importance in diverse areas of applications, ranging from material synthesis, to circuits design, to fault interrogation, and to geophysical exploration, to name a few. As the scales of the laminate heterogeneity can range from the micro- (a fraction of a wavelength) to the macro-scale (wavelength and above), the analysis of the entire set of associated wave phenomena and their detailed structure may constitute a challenge of overwhelming complexity. However, in many applications the detailed (micro-scale) structure of the EM field is of no practical importance. This can be due to the properties of the field itself (e.g. weak microscale field components), or the measurement setup (e.g. far-field or detector size). Thus, a solution strategy that constitutes only the macro-scale behaviour of the field—the field observables—is of great practical importance. The role of homogenization theory is to derive a simpler-to-solve effective formulation for the macro-scale field, which smoothes out the micro-scale heterogeneities while retaining their effect on the macro scale observables. Recently, a new homogenization theory that utilizes multiresolution analysis (MRA) has been developed and applied to time-harmonic propagation in complex laminates [1, 2]. Unlike traditional homogenization schemes that hold only for periodic micro-structures and require a large difference between the micro- and macro-scale (large “scales gap”), the new theory can handle non-periodic structures with scales continuum and enables one to choose the scale on which the fields are homogenized. In the present work we extend the multiresolution homogenization theory by allowing different homogenization scales for the medium and the field observables. We then present a full EM formulation for sources in 3D laminate configurations and explore the properties of the homogenization procedure under UWB conditions.
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FORMULATION OF THE PROBLEM We consider source-excited electromagnetic fields in complex laminates, characterized by multi-scale heterogeneities. The propagation domain (see Fig. (1)) may have any bounded or unbounded cross-section perpendicular to the stratification axis The laminate anisotropic constitutive relations are diagonal tensors whose components and and denote longitudinal and transverse components, respectively) are complex functions of z, i.e., multi-scale functions comprising of both macro and micro scales. The laminate is bounded in and may have penetrable boundaries to the surrounding homogeneous domain. The field is excited by a pulsed current J(r, t). The configuration in Fig. 1 can be analyzed via modal decomposition and transmission line theory [3, Chapter 2]. The field is expressed as a modal synthesis (either discrete or continuous) wherein the transverse and longitudinal field components, denoted by the subscript and respectively, are expressed as [3]
Here the index or denotes the E and H modes, respectively, while and are the corresponding vector and scalar mode functions for the spectral parameter . The spectral (modal) amplitudes and satisfy the spectral transmission line equations
These equations are in general difficult to solve due to the complex (multi-scale) structure of the heterogeneity functions and that are related to the constitutive parameters via
The source terms in the spectral transmission line problem are obtained by projecting the current sources J onto the spectral basis functions via
Note that the component of J, excites only E-type mode via transmission-line source, while the transverse component of J can excite both E and H modes via the source term
MULTIRESOLUTION ANALYSIS (MRA) We would like to express all the field observables on a given resolution (scale) that is typically determined by the physics of the problem (see discussion in connection with (8)). Therefore, referring to the formal theory of multi resolution analysis [4], we decompose in the following form
UWB ANALYSIS OF EM FIELDS IN COMPLEX LAMINATES
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where the resolution space is the linear space of all functions possessing length scales between and infinity, and the wavelet space is the orthogonal complement of in We will be interested in the field components in while smoothing out all small scale field components which are contained in with The scale is a fundamental parameter in our theory, denoted as the homogenization scale (HS). Typically it is chosen such that where is the wave velocity in the medium, but it may also depend on the detector used. The resolution space is spanned by the basis where is the so called scaling function, is the dilation index representing the smallest length scale in and is a running index spanning locations. Similarly, the wavelet space is spanned by the basis with being the wavelet function. Further properties of these functions (regularity, vanishing moments, etc.) can be found in [4]. A function can be decomposed into its macro-scale (smooth) and micro-scale (detailed) components in the form
These components are found by projecting V onto and its orthogonal complement in The corresponding projection operators and are given by where where
EFFECTIVE FORMULATION As denoted above, the electromagnetic field is fully described by the functions that solve the complex-coefficient equation (2). The field observables, however, are described on a scale that is typically determined by the physics of the wave-problem (e.g., frequency, near/far field) and by the measurement setup (e.g., detector size, integration window, dynamic range and polarization). In what follows we choose this scale to be and denote it as the field homogenization scale (F-HS). In many applications we choose
although the theory below is not limited to this choice. Thus, we are only interested in the field observables It is therefore sufficient to consider a simplified (effective) equation whose heterogeneity functions are homogenized (smoother functions),
It is required though that the solutions of (9) will be equal to the true field observables in
It can be shown that this condition is satisfied if the effective (homogenized) heterogeneity functions in (9) consist only of the components of the true heterogeneities in (the index will be discussed after (12) and (13)), i.e.,
Using this and (3) one finds that the effective constitutive parameters are given by
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Following the definition of as the field homogenization scale (F-HS; see (8)) we denote as the medium homogenization scale (M-HS). For a given F-HS (i.e., ) the M-HS (i.e., ) is chosen so that the solution of the effective problem satisfies the requirement (10) with a specified error. The condition for choosing is determined by the error bound in (13). The M-HS is chosen such that i.e., what is left out of the medium after taking away its effective part is finer than the F-HS. Under this condition, the M-HS should be chosen as large as possible in order to obtain an effective medium that is as simple as possible (i.e., expressed by the smallest number of basis functions). To demonstrate these considerations we refer to Fig. 2 describing two typical media. Medium 1 possesses a continuum of scales from the macro to the micro, while Medium 2 has a gap between the macro and micro scales around the F-HS. In Medium 1 as determined by the bound in (13). For Medium 2, the M-HS (i.e. ) can be taken to be the smallest scale above the gap (i.e. M-HS > F-HS). This completely determines the effective problem. The numerical examples below and in [7] demonstrate the numerical efficacy and accuracy of this method.
Outline of the Derivation of the Effective Formulation Step 1: In order to derive the effective formulation we apply the projection operators and to the wave operator in (2). To this end we recast (2) in an integral equation form which is more convenient to handle than the otherwise unbounded differential wave operator. Furthermore, the integral equation formulation is more convenient in handling the transformation of the boundary conditions from those of the complete formulation to those of the effective one [1]. We decompose the heterogeneity functions into background and foreground components defined, respectively, as and (see (11)). One arrives at a Lippmann-Schwinger type integral equation wherein the foreground (detailed) components act as induced sources, in conjunction with the background dyadic Green’s function. The procedure is applied in two steps wherein each step treats only one of the heterogeneity functions (i.e., first and then or vice versa; for the principal idea of this procedure see [2] and references therein). Step 2: We apply and to the integral equations via a Galerkin type procedure and obtain an algebraic matrix equations for the vector of unknown coefficient and The matrix entries are obtained by projecting the Green’s function kernel onto and Via this construction, the matrix is partitioned into submatrices representing field and medium interactions across length scales. Step 3: Since our interest here is the smooth component of the field, we apply the Schur’s complement procedure and substitute the equation for into the equation for obtaining an effective equation for that contains contributions of the micro-scale heterogeneities. Step 4: In order to assess the contributions of the micro-scale heterogeneities we derive bound to the submatrices constructed in Step 2. This is done using the bounds of the inner products of the type in (7) and their dependence on the properties of the expanded kernel and of and One finds an error estimate for the effective formulation, i.e., an estimate in for the difference between the true field observables and those predicted via the effective formulation (9) (see (10)). This result can be expressed as
where (13b) is obtained from (13a) if the the F-HS is chosen according to (8).
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In these equations is the largest scale of the medium left out in the medium homogenization procedure. Referring to the discussion on the M-HS after (12) we have in general (see Medium 1 in Fig. 2) but, if the medium has a scales gap as in Medium 2 in Fig. 2, then is the index of the largest scale of the medium below the gap. Thus the term indicates that the error decays faster than linear with respect to the ratio between the F-HS and the M-HS. Choosing a smaller M-HS (larger ) reduces the error bound at the expanse of having to work with more complicated effective medium.
MRA HOMOGENIZATION UNDER UWB CONDITIONS The effective field solutions constructed via the MRA homogenization procedure discussed in the previous section can be used to construct full 3D solutions for various source configurations. The construction of such solutions involves summation over the plane-wave spectral variable Several alternative spectral formulations for this 3D field constructions have been developed and will be described elsewhere. In this paper we concentrate on the homogenization procedure under ultra wide band (UWB) conditions for a given spectral variable i.e., for a fixed plane wave direction. As discussed above in connection with (8) the F-HS index is typically chosen in connection to the frequency. These leads to two homogenization strategies: (i) A dispersive homogenization approach: Here the F-HS index is chosen for each frequency band according to condition (8). The M-HS index is chosen so that the error bound in (13) is sufficiently small for specific frequency range. The resulting effective medium is frequency dependent. This approach provides the simplest most economical effective medium for each frequency within the source operation band. As a result the field equations have to be solved on a frequency-by-frequency basis and then transformed into the time domain (if time-domain solutions are required). (ii) A non-dispersive homogenization approach: Here the F-HS index and the M-HS index are chosen to comply with condition (8) and (13) at the highest excitation frequency and then used for the entire operation band. Clearly this approach does not provide the most economical description of the medium at each frequency but on the other hand it allows for direct time domain solutions as the effective medium is frequency independent. These two approaches are demonstrated for a complex slab with and
shown in Fig. 3. It comprises a continuum of scales from 1 to The excitation signal is
spanning frequencies from to Assuming that the field observables are determined by (8), the relevant scale range is Recalling the considerations for choosing the M-HS, we show in Figs. 4a and 4b the effective medium for and respectively. Note that the effective medium is anisotropic even though the true medium is isotropic.
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Finally the transient transmitted fields corresponding to the frequency results in Pig. 5(a,b,c) are shown in Fig. 6(a,b,c), respectively. The insets zoom-in on the weak response near ct = 4 corresponding to the first round trip contribution, showing an error in the effective result (dashed line) for case (b), but accurate results for cases (a) and (c).
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EFFECTIVE RESONANCES Another significant observation concerning the behaviour of the solution under UWB excitation has to do with the complex resonances of the laminate. In view of the Singularity Expansion Method (SEM) [5] these resonances may serve as fundamental waveforms in describing the transient response as a series of decaying oscillations and thus may serve for medium classification. For the layered medium case these resonances are obtained as roots of the Wronskian the solutions and of the transmission line equation (2), satisfying, respectively, the boundary (or radiation) conditions on the left and right hand sides of the line. This condition can be stated as
where it can readily be shown that is Equation (16) may be regarded as an equation of the form It roots in the for a fixed define the guided modes supported by the slab laminate. The SBM resonances, however, are the roots in the complex for a given ( i . e . , for a given direction of incidence). Using the effective formulation one can prove the Wronskian equivalence [2]:
where the error is bounded by an expression similar to (13). It thus follows that the physical SEM roots are well described by those obtained via the effective formulation, i.e.,
at least for the frequency range at which the error bound in (13) is small. To demonstrate the equivalence in the complex plane we have used a random number generator to synthesize a laminate with µ = 1 and with a random multi-layer structure for with inner scale of approximately 1/50 (Fig. 7a). Note that due to the term in for the E mode (see (3)), it contains a scales continuum for that case. The laminate total width is 1. Using M-HS=1, we obtain constant effective properties using M-HS=0.1, we obtain the effective medium properties shown in Fig. 7b
Next, the effective poles associated with the two effective media (with M-HS=1 and 0.1) were computed for E-mode and spectral parameter (corresponding to a plane wave incidence of 48.5°). The results are compared to the corresponding true poles of the
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original medium. Fig. 8 shows the relative error for the two M-HS’s. As expected from the bounds, the error increases with frequency and is better for the smaller MHS. The work in [7] explores the dependence of the frequency-plane spectral equivalence on the heterogeneity length scales and its effect on the transient field response.
CONCLUSION In this paper we have presented an MRA homogenization approach for electromagnetic radiation and propagation in the presence of complex laminates. The formulation addresses a general 3D source configuration via plane-wave spectral expansion, yet here we have limited our study to a single plane wave (single spectral parameter ), emphasizing the UWB aspects of the theory. We have demonstrated how the medium homogenization scale (M-HS) should be chosen as to accommodate the wide frequency spectrum of the excitation. In particular we examined two alternative schemes for choosing the M-HS, a dispersive and a non dispersive scheme, and demonstrated how they are used in order to calculate the effective transient response. Finally we have used the effective Wronskian equivalence to calculate the SEM resonances of a randomly laminated complex slab. Acknowledgements The research is supported in part by a grant from the Israeli Science Foundation. References [1] B. Z. Steinberg, J. J. McCoy and M. Mirotznik, “A Multiresolution Approach to Homogenization and Effective Modal Analysis of Complex Boundary Value Problems,” SIAM J. Appl. Math, 60(3) pp. 939-966, March 2000. [2] B. Z. Steinberg, “Homogenization and Effective Properties Formulations for Propagation in Finely Structured Laminates - A Multi-Resolution Approach,” to appear in Wave Motion, special issue on Electrodynamics of Complex Environments.
[3] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, Prentice Hall, 1973. [4] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, SIAM Publ., Philadelphia, 1992. [5] C.E. Baum, “The Singularity Expansion Method,” in Transient Electromagnetic Fields, L.B. Felsen, Ed. New York: Springer Berlag, 1976. [6] B. Z. Steinberg and E. Heyman, “Effective vertical modes and horizontal rays for wave propagation in complex inhomogeneous ducts, ” Proc. of URSI Trianum Int. Symp. on Electromagnetic Theory, Thessloniki, Greece, 1998.
[7] V. Lomakin and B. Z. Steinberg, “Effective Resonance Representation of Propagators in Complex Ducts - A Multiresolution Homogenization Approach,” IEEE Trans. Antennas. Propagat., submitted for publication.
TIME DOMAIN EXACT SOLUTION OF PROBLEM OF UWB PULSE DIFFRACTION ON A CONDUCTING HALF-PLANE
Eugene A. Galstjan Moscow Radiotechnical Institute of RAS Warshawskoe Shosse 132, Moscow 113519, Russia
INTRODUCTION The recent advances in the development of short-pulse (ultra wide band) communication and radar systems have given rise to an increasing interest in the electromagnetic community to formulate time-domain versions of existing frequencydomain techniques for the theory of diffraction. For this reason, exact time-domain solutions of some classical diffraction problems are of grate interest. The purpose of this paper is to obtain an exact time-domain solution of the well-known diffraction task: diffraction on a perfectly conducting half-plane. A plane electromagnetic UWB pulse is under consideration as an incident field. The pulse is arbitrary polarized with respect to the edge of the half-plane and propagates at an arbitrary angle to it. This means that the considered task is a 3D problem. Corresponding 2D problem has been considered (Galstjan, 1999) for the same pulse form and its solution coincided with the known one for the impulsive plane wave (Felsen and Marcuvitz, 1973) in a limit case.
FORMULATION OF THE PROBLEM Consider the perfectly conducting half-plane in Fig. 1. The half-plane is positioned in a rectangular xyz coordinate system with x-axis coinciding with the edge of the half-plane. The half-plane is located in the xz plane (z>0) and is illuminated by a plane pulse described by
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where are real constants, r the position vector, the time, _ the speed of light in vacuum, the polar angle, the azimuthal angle (with respect to the x axis), and the parameter of pulse duration. If the function (1) transforms into the Dirac delta function and the pulse becomes the impulsive plane wave.
First the problem is solved in the frequency-domain representation and then the solution will be transformed into the time-domain one by means of an analytical integration procedure. The frequency-domain representation of the incident pulse (1) is
with ( is the frequency) and the subscript denotes the frequency-domain quantities. Unknown diffraction fields and have to satisfy the wave equation and the boundary conditions for the total fields
Frequency-Domain Solution A frequency-domain solution of the problem under consideration is deriving by using the Wiener-Hopf technique (Noble, 1958) and will be described briefly. In this case, it is convenient to express diffraction fields and in terms of surface current densities induced on the half-plane
as follows
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where are the unknown functions, the complex variable, and the components of the Hertz vector. Applying the boundary condition (3) to the total fields and using the physical condition that the current densities (4) are zero on the half-plane extension (y = 0, z < 0) we may deduce a system of dual integral equations for and . To obtain a single-valued solution of this system we have to apply to the unknown functions, and an additional condition, so called the condition on the edge,
Pass over intermediate calculations and write down the obtained frequency-domain solution for the diffraction fields as follows
TIME-DOMAIN SOLUTION OF THE PROBLEM The time-domain solution can now be derived by applying of the inverse Fourier transform to the frequency-domain solution (6). In other words, we have to take a dual integral with respect to and between the infinite limits. Let us demonstrate this process by the example of the diffraction magnetic field component, First input the substitute, and change the variable of integration from to the following result is obtained
78
where is the complex variable. Now the expression (7) is integrated over limits from 0 to , The result is
E. A. GALSTJAN
between the
It is obvious that the integrand function in this expression possess two branch points, and we have to draw two cuts beginning in the branch points to make this function single-valued. Draw the left cut line from the branch point, to so that the line is parallel to the imaginary axis of the complex variable The right cut line is directed from another branch point, to the upper half-plane of the variable in the same manner. Next step is to modify the integration path into the path along the left cut line. The resulting integral can be taken by the residue method. The final expressions for the timedomain solution are
with the functions defined by
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Pattern of Diffraction Pulse The diffraction fields (8) are the components of a conic pulse with the axis coinciding with the edge of the half-plane and with the apex moving along the edge with the velocity For this reason the expressions (8) and (9) can be simplified and made more illustrative by using a nonorthogonal conic coordinate system. This coordinate system is a combination of the cylindrical coordinate system and a new variable In this "conic" coordinate system the expressions for the diffraction fields are
with The expressions (10) include four terms and these terms have simple physical meaning. First pair of terms describes the incident and reflected pulses (plane pulses) and second one describes properly the diffraction conic pulse. At a constant value an angle distribution of the diffraction fields is defined by a competition of these terms. The value corresponds to a conic surface synchronized with the maximum value of the incident pulse. It is convenient to obtain an approximation of the expressions (10) in a vicinity of this conic surface by expanding (10) in terms of the small parameter This approximation for the diffraction magnetic field, is defined by
with
and for the angle
First term in this
expression is much grater than the second one between the limits from to in other words the diffraction field in this transitional zone has the same nature as the incident pulse. The maximum value of the field in this zone corresponds to and is defined by
and is equal (in absolute value) approximately a half of the maximum magnitude of the incident field.
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In the shadow zone, diffraction magnetic field
we can derive a good approximation for the
The expression (12) demonstrates illustratively the diffraction conic pulse view in the shadow zone. At a given angle, the forward part of the pulse is steep, in comparison with the sloping back part Furthermore, the maximum value of the conic pulse corresponds to
It
means that the conic pulse has a time delay (proportional to the parameter with respect to the incident pulse. Figure 2 illustrates accuracy of the approximation (12) (line 1) in comparison with the exact expression (10) (line 2). The radial magnetic field distributions correspond to the case of (TE - polarization) and the values of the parameters
We have described briefly a qualitative pattern only for reasons of space. It is obvious that all fields can be calculated and illustrated graphically by using the expression (8) or (10). In closing let us describe briefly current distributions on the half-plane. Applying the dual inverse Fourier transforms to the surface current component (4) we can write the timedomain representation of the surface currents. In general case, final expressions are simple but very bulky. For this reason, we centered on expressions for the currents in special case of TE polarization of the incident pulse (1). It is convenient to express the currents
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on the half-plane in terms of currents induced by the incident pulse on a whole plane. The correspondence expressions are
where is an amplitude of the surface current on the whole plane and the superscript denots TE - polarisation.
For the sake of simplicity, the time dependence of the surface current distributions shown in figure 3 have been drown for the values of the angles and In this case, only the z-component of the current (13) is not equal zero. The lines correspond to the time values For the angle values under consideration the current pulse is moving initially to the edge of the half-plane (dashed lines, and later in the reverse direction.
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CONCLUSIONS The exact time-domain solution of the diffraction problem under consideration has been obtained by means of the reverse Fourier transform performed analytically for the special pulse form. In considered case the solution is defined by elementary functions of coordinates and time. Its simplicity makes possible to carry out a complete investigation of all diffraction fields/currents characteristics in the time-domain representation. Besides, this analytical solution can be used as a test for existing numerical or approximate time-domain methods. REFERENCES Galstjan, E.A., 1999, Diffraction of Electromagnetic Impulse on Ideal Conducting HalfPlane, Radiotechnika i Electronika, N10: 1184 (In Russian). Felsen, L.B. and Marcuvitz, N., 1973, Radiation and Scattering of' Waves, Prentice-Hall Inc., New Jersey, USA. Noble, B., 1958, Methods Based on the Wiener-Hopf Technique, Pergamon Press, London.
SPHERICAL WAVE EXPANSION OF THE TIME DOMAIN FREE-SPACE DYADIC GREEN’S FUNCTION
S. Alp
1
, S. Sencer Koç 2 and O. Merih Büyükdura 2
1
Electronic Design Group ASELSAN Inc. Ankara, 06172, Turkey 2 Department of Electrical and Electronics Engineering Middle East Technical University Ankara, 06531, Turkey
INTRODUCTION The importance of expanding Green’s functions, particularly free-space Green's functions, in terms of orthogonal wave functions is practically self-evident when frequency domain scattering problems are of interest. With the relatively recent and widespread interest in time domain scattering problems, similar expansions of Green's functions are expected to be useful in the time domain. In this paper, an expression, expanded in terms of orthogonal spherical vector wave functions, for the time domain free-space dyadic Green's function is presented and scattering by a perfectly conducting sphere is studied as an application to check numerically the validity and to demonstrate the utility of this expression. In the expression derived, in addition to the dependence on and the dependence on and is also 'separated' in the sense that each term in the expansion appears as a function of convolved with a function of Such a dependence in the Green's function is useful in a scattering formulation as it lets one set up an equation (for instance an integral equation) for some unknown quantities which in turn yield the equivalent sources. Throughout the paper the velocity of waves in free-space is taken to be so that in any frequency domain expression which appears, the wavenumber is equal to the angular frequency. are the familiar spherical coordinates of the point of observation and their primed counterparts are those of the source point. An time dependence is assumed and suppressed in the frequency domain expressions. The spherical wave expansion of the time domain scalar free-space Green's function is presented in (Buyukdura et al, 1997); therein a similar expression to our dyadic Green's function is derived for the scalar Green's function. In some previous work on the spherical wave expansion for the radiation from time dependent source distributions using multipole expansion of the sources, (Davidon, 1973), (Heyman et al, 1996) and (Shlivinski et al, Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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1999), the space-time dependence of the source enters via an nth order linear operator (n denoting the order of the associated Legendre functions) while in the present work, the same dependence enters via two (superposition (in space) and convolution (in time)) integrals.
STATEMENT OF THE PROBLEM The problem of interest is to find the dyadic Green's function which satisfies
as well as the radiation condition and causality. In Eq.(l), is the identity dyad which can be represented by a unit diagonal matrix and is the Dirac delta function. This time domain Green's function to a source equation
is useful to find the field
in free-space due
In other words the solution to the inhomogeneous partial differential
is given by
where V is the region occupied by the source.
FORMULATION AND SOLUTION OF THE PROBLEM The solution to Eq.(1) is the inverse Fourier transform of the frequency domain freespace dyadic Green's function which can be written in terms of vector wave functions that are orthogonal over a spherical surface as
In Eq.(4), q is a compact index standing for the indices p , n and
as,
WAVE EXPANSION OF TD FREE-SPACE DYADIC GREEN'S FUNCTION
where the parity index p takes on the "values" either e or
85
(standing for "even" and
"odd", respectively) and when and are the spherical vector wave functions first introduced by (Stratton, 1941). In order to find the inverse Fourier transform of Eq.(4), we write and in terms of the auxiliary spherical vector wave functions and which are independent of . Concerning the case in Eq.(4), we use the following transformation (Buyukdura et al, 1997),
where and denotes the spherical Bessel functions of order n and the spherical Hankel functions of the second kind of order n, respectively (Abramowitz et al, 1972), are the Legendre polynomials, stands for the convolution operation, represents the outgoing wave functions and is a pulse equal to unity when its argument is between -1 and 1, and vanishes elsewhere. However the transformation which is also necessary in finding the time domain expression does not exist. For this reason a new dyadic Green's function is defined in the frequency domain as,
With this definition, the solution to Eq.(2) for the electric field in free-space due to a volume current density is
where is the permeability of the vacuum, Green's function and
In order to find
first Eq.(4) is multiplied by
is the new time domain dyadic
yielding the expression for
and then is inverse Fourier transformed with the help of Eq.(6). After these trivial steps, the spherical wave expansion of the time domain free-space dyadic Green's function for can be expressed as in Eq.(10). One can go through similar steps to get the expression for
as in Eq.(11). As seen in these equations, in
addition to the dependence on and the dependence on and is also 'separated' in the sense that each term in the expansion appears as a function of convolved with a function of and that was the goal in deriving these expressions.
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RESULTS AND DISCUSSION We consider an impulsive plane wave incident from the positive-z axis (from ) which is linearly polarized in the x-direction, i.e., This incident field can be written in terms of the spherical vector wave functions with the aid of the Green's function in (11) and the field scattered by a perfectly conducting sphere of radius a centered at the origin can be expressed inspired by the form of the Green's function in (10). The unknown coefficients in the scattered field expression can be found by imposing the boundary condition on the surface of the sphere and solving the resultant equations either by direct deconvolution or by using system identification techniques. As a numerical example, the plane wave with the waveform of the derivative of a gaussian pulse, i.e.,
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incident to a perfectly conducting sphere of radius a = 1 is considered. The wave function expansion of this incident field can be obtained by convolving the waveform, with the expansion of termby-term and the scattered field can be found by convolving the scattered field due to with the incident waveform. The duration of the incident waveform is 20 ns provided that the wave velocity is that of light in vacuum. In figure 1, two solutions to the scattered field in the back-scatter direction at a distance of R = 3 m are given where the x component of the field is plotted versus time. The solid curve is obtained using the present time domain formulation (direct deconvolution is used in finding the scattered field coefficients), while the cross marks are obtained by inverse Fourier transforming the well known frequency domain solution. Both solutions are obtained by including only 4 terms in the series expansion. An observation made on the time domain solution is that the solution converges rapidly if the dimension of the scatterer is small compared to the wavelength at the highest frequency component of the incident field which is similar to the case in the solution of problems in the frequency domain.
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REFERENCES Abramowitz, M., and Stegun, I.A., 1972, Handbook of Mathematical Functions, Dover, New York. Buyukdura, O.M.,and Koc, S.S., 1997, Two alternative expressions for the spherical wave expansion of the time domain scalar free-space Green's function and an application: Scattering by a soft sphere, J. Acoust. Soc. Am., vol. 101, pp. 87-91. Davidon, W.C., 1973, Time-dependent multipole analysis, J. Phys. A: Math., Nucl. Gen., vol. 6, pp. 1635-1646. Heyman, E., and Devaney, A.J., 1996, Time-dependent multipoles and their application for radiation from volume source distributions, J. Math. Phys., vol. 37, pp. 682-692. Shlivinski, A., and Heyman, E., 1999, Time-Domain Near-Field Analysis of Short-Pulse Antennas – Part I: Spherical Wave (Multipole) Expansion, IEEE Trans. Antennas Propagat., vol. 47, pp. 271-279. Stratton, J.A., 1941, Electromagnetic Theory. McGraw-Hill, New York.
ON THE LOCALIZATION OF ELECTROMAGNETIC ENERGY
Hans Gregory Schantz Time Domain Corporation 6700 Odyssey Drive Huntsville, AL 35806 USA
INTRODUCTION This paper explores the interesting question of whether electromagnetic energy can be localized. Three specific areas will be addressed. First, the historical development of the Poynting – Heaviside theory will be traced. Then, the problems and alleged paradoxes of this theory will be examined. Finally, a method of tracking electromagnetic energy will be presented and applied to some simple examples. The challenge of short pulse electromagnetics is to understand the time evolution of a transient electromagnetic system. The aim of this paper is to demonstrate that understanding electromagnetic energy transfer can be a valuable means to that end. A BRIEF HISTORY OF ELECTROMAGNETIC ENERGY LOCALIZATION In his 1847 treatise on energy conservation, Hermann von Helmholtz introduced the concept that electric energy density depends upon the charge density and the electromagnetic potential (V):1
This approach implies that electromagnetic energy is localized with charges, in accord with the then current “action-at-a-distance” philosophy that underlay physical thought. This action-at-a-distance approach to electromagnetics had already begun to crumble at the hands of Michael Faraday who introduced the concept of a field. The implication of Faraday’s field approach is that electric and magnetic processes are distributed throughout space, not localized with sources. In 1853, William Thompson (later Lord Kelvin) introduced the idea that energy itself might be localized with the fields.2 In the context of the electric field, Thompson’s idea meant that the field carries an energy:
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This idea flowered at the hands of James Clerk Maxwell:3,4 ''The energy in electro-magnetic phenomena is mechanical energy. The only question is, Where does it reside? On the old theories it resides in electrified bodies, conducting circuits, and magnets, in the form of an unknown quantity called potential energy, or the power of producing certain effects at a distance. On our theory it resides in the electromagnetic field, in the space surrounding the electrified and magnetic bodies, as well as in those bodies themselves....'' Not long thereafter, John Henry Poynting and Oliver Heaviside independently arrived at what came to be known as the “Poynting vector” to describe the flow of electromagnetic energy:
The mathematical relationship discovered by Poynting, “Poynting’s Theorem:”
follows unequivocally from Maxwell’s equations. Theorem as a statement of local energy conservation:
The interpretation of Poynting’s
is not so clear cut. With Hertz’s discovery of radio waves (1888-1889) however, the triumph of Faraday’s field point of view was considered complete.5 Having demonstrated that radio waves moved at the speed of light and have optical properties, Hertz proved that they were decoupled from their source. “In the sense of our theory we more correctly represent the phenomena by saying that fundamentally the waves which are being developed do not owe their formation solely to processes at the origin, but arise out of the conditions of the whole surrounding space, which latter, according to our theory, is the true seat of the energy.”6 PUZZLES AND PARADOXES OF ELECTROMAGNETIC ENERGY Later investigators noticed difficulties with the energy flow interpretation advocated by such pioneers as Poynting, Heaviside, and Hertz. Under certain circumstances, the Poynting-Heaviside theory yields seemingly nonsensical results such as closed loops of energy in otherwise static systems. For instance, consider a static point charge “q” with field:
superimposed with a static magnetic dipole with field:
ON THE LOCALIZATION OF ELECTROMAGNETIC THEORY
This “static” system has a non-zero Poynting vector:
corresponding to azimuthal loops of flux.
Further, the Poynting vector is ambiguous to a solenoidal (i.e., divergenceless) term. In other words, if S' = S + G where • G = 0, the new resulting “Poynting vector,” S', ” will still satisfy Poynting’s theorem. Thus, it is often argued that the Poynting vector has no physical significance unless integrated over a completely closed surface.7 Some observers reject the idea that the Poynting vector represents a localized flow of energy while at least accepting that the integral of the Poynting vector over a closed surface has a physical significance – the rate of change of energy in the bounded volume.8 Others, like R.W.P. King, reject the idea that energy has any physical significance whatsoever, beyond being a mathematical quantity that may be useful in calculations.9 More recent investigation has tended to uphold the original vision of Poynting and Heaviside. The seemingly implausible “loops” of Poynting flux are now generally recognized as required by the demands of angular momentum conservation. A simple example (adapted from one offered by Feynman10) serves to illustrate the point. Consider a charged dielectric hoop at rest with a bar magnet along its axis, a physical system whose fields will be similar to the point charge, point magnetic dipole described above. By Lenz’s Law, when the magnet is removed, an e.m.f. will be induced in the hoop, setting it spinning. Since the bar magnet is removed along the axis of the hoop, no angular momentum will be imparted to the system, and yet, it spins. A detailed analysis of an analogous system shows that the electromagnetic angular mometum of the original “static” fields, as predicted by the closed loops of Poynting flux, is the same as the imparted mechanical angular momentum.11,12 In fact, the torque associated with circularly polarized light has been experimentally measured. The results are also in accord with the Poynting – Heaviside interpretation.13
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The seeming ambiguity of the Poynting vector is thus subject to the constraint that any modification must not only satisfy conservation of angular momentum, but also yield the same correct value for the angular momentum. Similarly, there is a linear momentum (source of the so-called “light” pressure) known to be correctly described by the Poynting vector.14 Naturally, any modification must further satisfy conservation of energy. All these constraints severely limit alternatives to the Poynting vector for describing the local electromagnetic energy flow.15 In fact, it has long been realized that any additional solenoidal term added to the Poynting vector cannot be a function of the electromagnetic fields (or their time derivatives) and still satisfy Poynting’s Theorem.16 Since it is reasonable to assume that any measure of local electromagnetic energy flow must be a function of the electromagnetic fields, it is very difficult to imagine how there could be a physically meaningful alternative to the Poynting vector. A final consideration has been largely neglected from the debate on the physical interpretation of the Poynting vector. Virtually every communication link ever designed relies on the Friis transmission formula to predict the received power. The Friis transmission formula in turn relies on our ability to predict the power flux using the Poynting vector, not integrated over a closed surface surrounding the transmitter, but rather, over a very tiny piece of that surface. The undeniable success of the Friis formula suggests that the Poynting vector is the correct local measure of electromagnetic energy flow, or at least places very stringent limitations on alternatives. Open questions may remain, but there is no good reason to unilaterally reject the insights available from localizing and tracking electromagnetic energy.
ON THE LOCALIZATION OF ELECTROMAGNETIC THEORY
CAUSAL SURFACES Some time ago, the author noted the possibility of identifying “causal surfaces,” surfaces on which so that there is no net flow of electromagnetic energy.17 These surfaces partition electromagnetic energy making it easier to track the source of the energy and see how it changes from one form to another. In the exponential decay of a Hertzian dipole, for instance, there is a spherical causal surface of radius (where is the time constant of the decay) on which the Poynting vector is everywhere zero. Thus, this surface bounds the field energy, partitioning it to a region inside the sphere in which energy is absorbed, and a region outside the sphere from which energy radiates away. It has also been shown that the total radiated energy which escapes to the far field is exactly equal to the total energy originally stored outside the causal surface.18
Elsewhere, this technique has been used to describe the flux of energy around a harmonic dipole.19,20 By plotting a space-time diagram of the radial energy density, the time varying causal surfaces, and the local energy velocity, the radiation of energy from a harmonic Hertzian dipole can be understood in detail. Some general observations are possible. First, no energy escapes directly from the dipole to the far field without first becoming temporarily stored or reactive. Second, energy is only emitted by the dipole half the time. The other half of the time the dipole is absorbing energy and the apparent radiation comes from energy already stored in the near fields. Finally, the average distance
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at which reactive energy is converted to radiated energy is around a result consistent with Harold Wheeler’s “radiansphere” concept21 – the observation that the near and far fields around a harmonic dipole are equal in magnitude at around This same space-time portrayal of energy flow was also applied to dipoles subject to three transient excitations: charging, discharging and a Gaussian impulse. One may also apply the concept of causal surfaces to understand how accelerating charges radiate. The process is somewhat different when a charge decelerates, so first consider acceleration. A causal surface appears where the net tangential field goes to zero. In the case of the accelerating charge, this means we need to examine the interplay of the applied external field (that causes the acceleration), and the radiation fields. If we are sufficiently far away from the charge and external field (that together are the radiating system), the dominant field is the radiation field, and the net flow of energy is outward. The radiation fields are opposite in direction to the external fields. A quick comparison of their magnitudes shows that the radiation fields are far weaker than the external fields. Somewhere around the fringe of the region where the external fields exist, radiation and external fields just cancel out yielding a causal surface. The result is that in the region where the external fields exist, the energy flows in toward the accelerating charge, adding to the magnetic field energy of the ever more quickly moving charge (see Figure 5).
Since on a sufficiently local scale the external field must always be uniform and constant, it cannot dominate arbitrarily close to the accelerating charge. A quick calculation in the low velocity limit suffices to learn where that boundary might be. In this limit, the electric field due to an accelerating charge “q” with acceleration (and with mass “m”) is:
while the external field will look something like this:
Summing the tangential is:
fields and noting that the magnitude of the force on the charge
there will be a spherical causal surface about the charge of radius:
ON THE LOCALIZATION OF ELECTROMAGNETIC THEORY
Interestingly, the result is entirely independent of the magnitude of the external field. In fact, this result is just the so-called classical electron radius. Since the net tangential electric field is zero, energy cannot be extracted from within this surface. The surface serves as an electromagnetic analog to the event horizon of a black hole. Whatever energy is absorbed by or radiated from an accelerating charge must reside outside the sphere defined by the classical electron radius. Of course this is strictly a nonrelativistic result, but since the fields in an instantaneously co-moving reference frame should reduce to the low velocity result here, a similar result should hold for the more general case. Further, it should be noted that this is strictly a classical calculation and neglects any quantum mechanical effects. The case of a decelerating charge is slightly different. Although there will still be a causal surface at the classical electron radius, the typical source of the radiated energy is the magnetic field energy around the moving charge. The near and far magnetic fields of a decelerating charge have opposite orientations (see Figure 6). There will be a spherical surface on which the magnetic field goes to zero. The radius of this surface is:
Inside this surface, the magnetic field energy of the decelerating charge is absorbed by the external field that causes the deceleration. Typically, this means the magnetic energy of the decelerating charge is converted to the static field of the applied electric field.
To summarize, electromagnetics is not exempt from the law of energy conservation. Radiant energy does not come into being out of nothing, rather it is converted from other forms of electromagnetic energy. In the case of accelerating charges, the external field loses energy – it is the source of both the radiated energy as well as the magnetic field energy gained by the now more quickly moving charge. In the case of decelerating charges, the more slowly moving charge loses magnetic field energy – it is the source of both the radiated energy as well as the energy gained by the external field.
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CONCLUSION As Oliver Heaviside observed, “However mysterious energy (and its flux) may be in some of its theoretical aspects, there must be something in it, because it is convertible into dollars, the ultimate official measure of value.”22 The ability to track electromagnetic energy is helpful in designing antennas with low reactive field energy. Such antennas would be highly efficient with a low Q and broad bandwidth – ideal for UWB-SP applications. Electromagnetic energy localization is thus a valuable technique for UWB-SP antenna design. REFERENCES 1
Hermann von Helmholtz, The Conservation of Force: A Physical Memoir (1847) collected in Selected Writings of Hermann von Helmholtz, Russel Kahl, ed., Wesleyan University Press, Middletown, Connecticutt, (1971) pp. 3-55. 2 Sir Edmund Whittaker, A History of the Theories of Aether & Electricity,” Vol. 1, Harper and Brothers, New York, (1951) p. 222. 3 James Clerk Maxwell, A Treatise on Electricity and Magnetism Vol. II., Academic Reprints, Stanford, CA, (1953) pp. 270-271. See §631 in particular. 4 James Clerk Maxwell, On Action-At-A-Distance, Proceedings of the Royal Institution of Great Britain, Vol. 7, (1873-5) pp. 48-49. 5 Lord Kelvin, Preface to the English Translation of Hertz’s Electric Waves. 6 Heinrich Hertz, Electric Waves, Macmillan and Co., London, (1893). 7 Ronold W.P. King, Fundamental Electromagnetic Theory (2nd ed.), Dover, New York, 1963, pp. 191-192. 8 Sir James Jeans, The Mathematical Theory of Electricity and Magnetism, Cambridge: University Press, Cambridge, (1933), p. 518. 9 Ronold W. P. King, Op. Cit., p. 180. 10 Richard Feynman, The Feynman Lectures in Physics, Vol. 2, Addison Wesley Pub. Co., Reading, MA, (1964) pp. II-27-8. 11 E.M. Pugh and G.E. Pugh, “Physical Significance of the Poynting Vector in Static Fields,” Am J Phys 35, 153-156 (February 1967). 12 Gabriel G. Lombardi, "Feynman's disk paradox," Am. J. Phys., 51, 213-214 (1983). 13 Richard A. Beth, "Mechanical Detection and Measurement of the Angular Momentum of Light," Phys. Rev. 50, 115-125 (1936). 14 E.E. Nichols and G.F. Hull, “The Pressure Due to Radiation,” Phys Rev17 26-50, 91-104 (1903). 15 Udo Backhaus and Klaus Schäfer, "On the uniqueness of the vector for energy flow density in electromagnetic fields," Am. J. Phys. 54, 279-280 (1986) 16 Kr. Birkeland, “Ueber die Strahlung electromagnetischer Energie im Raume.” Ann Phys 52, 357-380, (1894) 17 Hans Gregory Schantz, “The flow of electromagnetic energy around an electric dipole,” Am J Phys 63 513-520 (1995). 18 Ibid. 19 H. Schantz, “Electromagnetic Energy Around Hertzian Dipoles,” IEEE Antennas and Propagation Society International Symposium (1999), pp. 1320-4 20 Hans Gregory Schantz, “Electromagnetic Energy Around Hertzian Dipoles,” submitted to IEEE Antenna and Propagation Magazine, July 20, 1999. 21 Harold A. Wheeler, “The Radiansphere Around a Small Antenna,” Proc IRE 47 13251331 (1959). 22 Oliver Heaviside, Electromagnetic Theory, Vol. 1, Chelsea Publishing Company, New York, (1971), p. 381.
ON SUPERLUMINAL PHOTONIC TUNNELLING
Günter Nimtz1, Astrid Haibel1 and Alfons A. Stahlhofen2 1
2
II. Physikalisches Institut Universität zu Köln 50937 Köln, Germany Institut für Physik Universität Koblenz 56075 Koblenz, Germany
INTRODUCTION
The analogy of wave mechanical tunnelling are the classical evanescent modes, nowadays often called photonic tunnelling. The Helmholtz equation describing the electromagnetic fields corresponds formally to the Schrödinger equation for the wave function of particles.
Interest in the tunnelling time aroses after the first man-made electronic tunnelling structures were introduced in solid state physics: The tunnelling diode in semiconductor physics discovered by Esaki (1958) and the tunnelling in superconductor-insulatorconductor structures designed by Giaever, Josephson and others (1960). Surprisingly experimental electronic tunnelling time data are not available so far. This serious Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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dilemma has stimulated much the various studies on photonic tunnelling. The mathematical analogy between the Helmholtz and the Schrödinger equations allows to conclude that the photonic results are representative for general tunnelling. Essentially three important kinds of photonic barriers have been studied (Nimtz and Heitmann, 1997): a) Double prism with frustrated total internal reflection (FTIR), b) periodic dielectric heterostructures, i.e. photonic lattices, and c) undersized wave guides. The barriers are sketched in Fig. 1. The three barriers have different transmission dispersion relations, however, all the tunnelling modes are characterized by an imaginary wave number which means that the evanescent field does exponentially decay with distance. According to the phase time approach the field is spread out in no time. This behaviour is illustrated for the case of FTIR at the double prism. Due to the Goos-Hänchen shift D the light beam travels in two paths crossing the forbidden gap in Fig. 2. The first path of the beam is along the surface of the entrance prism, which results in the Goos-Hänchen shift and this wave solution has a real wave number After moving along the surface the beam crosses the gap by an evanescent mode with an imaginary wave number
In the case of FTIR, as sketched in Fig. 2, the electric field E is given by the relation
describing a wave propagating along the prism surface in y-direction and an evanescent mode tunnelling the forbidden gap between the two prisms in x-direction.
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We like to point out that the exponential field decay inside the gap is not caused by absorption since tunnelling is an elastic process. In the following chapters we are going to show that tunnelling of signals and energy can proceed much faster than light in vacuum, i.e. at superluminal velocity. This effect is observed in the .case of opaque barriers and signals containing evanescent frequency components only. (A barrier is called opaque if the transmission is dropped much more than 1/e.) In the next chapter we shall show, that it is a fundamental physical property of signals to be frequency band limited and this implies superluminal propagation. Another important property of the tunnelling process is that the tunnelling time equals approximately the reciprocal frequency of the evanescent modes in question. This property seems to be valid for all opaque barriers as shown below. SUPERLUMINAL SIGNALS The important question is: what is a physical signal? In Fig. 3 an ideal sinusoidal signal is shown. The information, i.e. the signal is presented by both the carrier frequency and the 6 oscillations. This ideal signal is not frequency band limited but it is described by a Fourier spectrum with frequencies between Quite often it has been assumed that a signal has an unlimited frequency band (e.g. Büttiker and Thomas, 1998). Only the poor technical design of transmitter and receiver systems are not able to generate and detect all frequency components.
If signals are considered as being composed of photons with energies h v then the signals finite energy implies that its frequency band has to be limited. Recently Nimtz (1999) has pointed out, that frequency band limitation is a fundamental property of any physical signal in consequence of quantum mechanics. A signal with finite energy has to have a frequency band limitation. According to the Fourier transform such a signal becomes non-causal from the mathematical point of view but not necessarily from the physical one (Nimtz, 1999). A modern signal used in optoelectronics is presented in Fig. 4. The information is given by the carrier frequency which determines the receiver’s address and the pulses halfwidth gives the number of 0 or 1 digits. The presented signal is frequency band limited to only of the carrier frequency. Such a signal can be sent through a photonic barrier without any significant signal reshaping at a superluminal speed of 4.7 times the velocity of light or even much faster (Nimtz et al., 1994; Aichmann et al., 2000). The condition sine qua non for achieving superluminal signal velocities is that all components of the signal have to be evanescent and a signal reshaping is avoided
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by a narrow frequency band width (Nimtz, 1999). The strong attenuation reduces the signals amplitude, however, not the content of information. Of course, as seen in Fig. 4 with decreasing magnitude the noise to signal ratio increases and the signal gets lost eventually. There is an instructive example of a signal in astronomy: A cosmic explosion with an outburst is analyzed by the ”carrier frequency” and by the ”width” of the pulse. For instance the radiation gives the information about the temperature and the halfwidth of the outburst gives the information on the energy involved in the cosmic event. The information (i.e. ”carrier frequency” and ”halfwidth”) is independent of its magnitude and thus independent of the distance between the cosmic event and the observer on earth.
IS THE TUNNELLING TIME UNIVERSAL ? Recently it has been observed that the tunnelling time in the case of opaque barriers has a universal property. The tunnelling time equals approximately the inverse frequency of the evanescent mode or the tunnelling photons (Haibel and Nimtz, 2000). This behaviour has been found experimentally in studies with microwaves as well as with light. Irrespective of the complete different interpretation the behaviour is also in agreement with the theoretical data of evanescent modes or tunnelling solutions of the Helmholtz and the Schrödinger equations (Enders and Nimtz, 1994). In Table 1 data of various studies are summarized. The experiments were carried out at different frequencies as well as with different barriers. Most of the data presented in Table 1 show a good agreement between the measured tunnelling time and the inverse frequency of
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the wave. The deviations are within one order of magnitude. This is not surprising since the experiments are rather sophisticated and from the theoretical point of view (e.g. Hartman, 1962) the tunnelling time depends to some extent on the wave number of the tunnelling wave.
SPECIAL FEATURES OF EVANESCENT MODES The Schrödinger equation yields a negative kinetic energy in the tunnelling case, since the potential U is larger than the particle’s total energy W:
The same happens to evanescent modes. Within the mathematical analogy, their kinetic electromagnetic energy is negative, too. The Helmholtz equation for the electric field E, for instance in a waveguide is given by the relationship
where is the cut-off wave number of the evanescent regime. The quantity plays the role of the energy eigenvalues and is negative in the case of evanescent modes. The dielectric function of evanescent modes is negative and thus the refractive index is imaginary. For the basic mode a rectangular waveguide has the following dispersion of its dielectric function, where holds and is the waveguide width, is the free space wavelength of the electromagnetic wave. According to the last equation the evanescent mode’s electric energy density given by the relationship
is
where is the electric permeability of the vacuum. In the case of tunnelling it is claimed that a particle can only be measured in the barrier with an additional particle, e.g. a photon having an energy (e.g. Gasiorowicz, 1996). This means that the total energy of the system is positive and the
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tunnelling process is stopped. The analogy between the Schrödinger equation and the Helmholtz equation holds again: it is not possible to measure an evanescent mode in analogy to a particle in a tunnel. We may also say that an evanescent mode can not be measured due to an impedance mismatch between the evanescent mode and a probe in the barrier. The impedance Z of the basic mode in a rectangular waveguide is given by the relationship where is the free space impedance. In the evanescent frequency regime the impedance is imaginary. Evanescent modes do not experience a phase shift inside the barrier. Having in mind the phase time relationship
in the evanescent regime there is an absence of time, where and are the phase shift and the angular frequency, respectively. The measured tunnelling time is spent on the entrance boundary due to the interference of the incoming and reflected wave. In the case of frustrated total internal reflection the measured total tunnelling time is spent during the travel along the surface in consequence of the Goos-Hänchen shift, while crossing the gap takes place instantaneously (Stahlhofen, 2000; Haibel and Nimtz, 2000). CONCLUSION Evanescent modes or the tunnelling process are characterized by an imaginary wave number. These special solutions of the Helmholtz and of the Schrödinger equations first noticed in connection with the total reflection were said to have no physical meaning about 200 years ago. Last century the tunnelling problem in quantum mechanics has been described by the phase time approach. The phase time approach yields the group velocity of a wave packet (see e.g. Hartman, 1962). Nowadays the phase time approach is used in network analyzers to determine the group velocity of an electromagnetic wave in devices. We have reported about superluminal signal velocities in photonic tunnelling. It was observed that the superluminal photonic tunnelling time data measured in the time domain are in agreement with the calculated phase time data. This result is also in agreement with the definition of the barrier transition time given in the Federal Standard 1037 C (NTIA, USA 2000) . Thus we have added further credibility to the assumption that the photonic tunnelling results are representative for quantum mechanical tunnelling of particles. Acknowledgements The authors gratefully acknowledge discussions with P. Mittelstaedt and R. Vetter. REFERENCES Aichmann, H., Haibel, A., Lennartz, W., Nimtz, G., and Spanoudaki, A., 2000, Demonstrating superluminal signal velocity, in: Proc. Int. Symposium Quantum Theory and Symmetries, Goslar, 18.-23. July 1999
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Balcou, Ph., and Dutriaux, L., 1997, Dual optical tunnelling times in frustrated total internal reflection, Phys. Rev. Letters, 78:851 Büttiker, M., and Thomas, H., 1998, Front propagation on evanescent media, in: Superlattices and Microstructures, 23:781 Carey, J., Zawadzka, J., Jaroszynski, D., and Wynne, K., 2000, Noncausal time response in frustrated total reflection?, Phys. Rev. Letters, 84:1431 Desurvire, E., 1992, Lightwave Communications: The Fifth Generation, Scientific American, 266:96 Enders, A., and Nimtz, G., 1992, On superluminal barrier traversal, J. Phys. I France, 2:1693 Enders, A., and Nimtz, G., 1994, Evanescent mode propagation and quantum tunnelling Phys. Rev. E, 48:632 Gasiorowicz, S., 1996, Quantum Physics, Second Edition John Wiley & Sons, New York Haibel, A., and Nimtz, G., 2000, On the universality of tunnelling time, to be published. Hartman, Th., 1962, Tunnelling of a Wave Packet, J. Appl. Physics, 33:3427 Mugnai, D., Ranfagni, A., and Ronchi, L., 1998, The question of tunnelling time duration: A new experimental test at microwave scale, Phys. Letters A, 247:281 Nimtz, G., Enders, A., Spieker, H., 1994, Photonic tunnelling times, J. Phys. I France, 4:565 Nimtz, G. and Heitmann, W., 19997, Superluminal photonic tunnelling and quantum electronics Prog. Quantum Electronics, 21:81 Nimtz, G., 1999, Evanescent modes are not necessarily Einstein causal, Eur. Phys. J. B, 7:523. Stahlhofen, A. A., 2000, Photonic tunnelling time in frustrated total internal reflection, Phys. Rev. A in press Steinberg, A., Kwiat, P., and Chiao, R., 1993, Measurement of the single-photon tunnelling time, Phys. Rev. Letters, 71:708 Spielmann, Ch., Szipöcs, R., Stingle, A., and Krausz, F., 1994, Tunnelling of optical pulses through photonic band gaps, Phys. Rev. Letters, 73:2308
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Transient Electromagnetic Field of a Vertical Magnetic Dipole on a Two-Layer Conducting Earth
Samira T. Bishay1, Osama M. Abo Seida2 , and Ghada M. Sami1 1
Mathematics Department, Faculty of Science, Ain Shams University, Abbassia,Cairo, Egypt. 2 Mathematics Department, Faculty of Education, Kafr El-Sheikh Branch, Tanta University, Kafr El-Sheikh, Egypt.
INTRODUCTION The effect of the transient responses of the dipoles on the ground subsurface has a considerable value as means of probing the earth. Sometimes ago, Wait [1] derived the transient fields of a horizontal electric loop on a homogeneous earth in a closed form expressions. An extensive analyses [2], [3] and [4], were carried out for the continuous wave and transient responses of mutual impedance between current-carrying loops or wires. Nevertheless, the transient response of dipoles over horizontally stratified ground and the interpretation of such fields in remote sensing have not been given enough attention. Therefore, a model for the ground which consists of two horizontally stratified layers with contrasting permittivities and conductivities is considered here for the case of a source on the ground with a transient current waveform. The transient electromagnetic fields are found in the frequency-domain. On the other hand, a derivation of the electromagnetic fields of a travelling current line source located over a layered conducting half-space is given [5]. Nabulsi et al. [6], using an exact numerical solution, obtained the transient response of a two-layer earth model for an obliquely incident transverse magnetic plane wave. Using a quasi-static approach, Wait [3] has derived closed-form solutions for the fields of loops above the surface of a two-layered earth which are valid at sufficiently late times. Botros et al [7], using the same method, have obtained simple expressions for the transient responses between loops laying on the surface of a two-layer earth. The late time response is obtained by an approach valid for a sufficiently large time such that all the distances encountered are compared with the significant free space wavelengths. Thus, the displacement currents in the air region may be neglected. Afterwards, Mahmoud et al [8] derived the complete timedomain fields due to a vertical magnetic dipole on the surface of two-layered ground in an analytical form. In their expressions, the displacement currents in the ground have been neglected. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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It is our purpose to obtain the frequency domain full wave solution for the fields of a small horizontal loop, i.e. a vertical magnetic dipole, on a two-layered earth's model, including the waveguide modes propagating within the earth. Concentrating on the solutions of the eignvalue equation of the problem, the displacement currents in the two-layer earth and in the air regions are accounted for.
GENERAL DESCRIPTION Referring to Fig.1, the source is a vertical magnetic dipole situated directly on the earth's surface, where the earth's adopted model consists of a homogeneous overburden slab of thickness d above a half-space. The permittivities and conductivities of the overburden and the lower half-space are ( i = 1, 2 ), respectively. The magnetic permeability is taken equal to that of the free space everywhere. The source represents a small loop of area dA carrying a circulating current I(t) which, in general, is a function of time t. Without loss of generality, we can restrict attention to situations where I(t) = 0 for t < 0. Using cylinderical coordinates for a time dependence exp(st), where in terms of the angular frequency and s is the Laplace transform variable, the frequency domain expression of the vertical magnetic field, on the first boundary where z = 0, i.e. in the air region due to the source, takes the form [9]:
where
TRANSIENT FIELD OF A VERTICAL MAGNETIC DIPOLE
and
107
are the Fresnel reflection coefficients at the interfaces z = 0 and z = d,
respectively, and
is the Hankel function of the second kind and order zero.
The contour integration in (1) can be deformed in the lower half of the complex but it must pass around the two branch points at and and a discrete set of poles, as shown in Fig.2 . Here we may note that there is no branch point at and there is no contribution from the contour at infinity [8], [10] and [11].
FREQUENCY- DOMAIN SOLUTION The frequency domain expression of the vertical magnetic field, for example, can be written as
where
where
is the branch-cuts contribution modes and is given by
and respectively, and
are the branch-cut integrals at the branch points is the poles contribution, written as
and
S. T. BISHAY ET AL.
108
where is the contribution of the pole at Physically, the two branch-line integrals account for a continuous spectrum of radiation in the homogeneous regions above the interface and deep within the earth, respectively. The pole contributions, of course, can be identified with waveguide modes in the stratified region of the earth.
A-Evaluation of the Integral along the Branch -Cut The branch cut integral can be evaluated at the branch point substitution
[8].
Subject
to
the
condition
using the or
the branch cut integral can be expressed as that of a half-space problem multiplied by a sratification factor. Thus, the integral in (1) can be evaluated without approximation and leads to the closed form expression
where the stratification factor
with
defined as
It is worth noting that for a homogeneous ground the stratification factors and expression (9) yields the exact result for this field. Using (10) we can put it in a more convenient form, as
where
Using a similar procedure, the contribution of the branch cut integral at obtained as
is
109
TRANSIENT FIELD OF A VERTICAL MAGNETIC DIPOLE
where
For the other components of the electric and magnetic fields, the contribution of the branch-cut integral at is obtained as
As in the case of a lossy substratum and at a sufficiently large separation between the source and the observation point, i.e., for the overall contribution of the branch-cut integrals can be expressed as that of the integral at
only.
B- Evaluation of the Integral around the Poles The poles of (1) in the complex are the roots of the equation
Looking for the late time response, let us consider the waveguide modes under lowfrequency conditions. Hence, the ratio
and for the first few low-order
modes i.e., those having the least radial attenuation, one may assume that where
n
is
approximations are valid:
an
integer denoting Under
the these
mode order. Hence, conditions the following
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S. T. BISHAY ET AL.
Substituting in (18) gives
where
and
B = 0or
according to whether
or
respectively. A solution of (21) is
where the plus-minus signs apply for the lower and upper halves of the complex s plane, respectively. The corresponding roots in the complex plane are given by
where
and
The residue of the pole
where
for the vertical magnetic field
is obtained as
TRANSIENT FIELD OF A VERTICAL MAGNETIC DIPOLE
111
Taking the summation of the pole residues, the waveguide mode contribution is then given by
where and is the modified Bessel function of the second kind and the n-th order. Using the same procedure to get the other components of the electric and magnetic fields, i.e.,
NUMERICAL RESULTS The magnitude of the frequency rate of change of the vertical magnetic field due to a rectangular pulse is computed for different cases. Plots of the branch-cut contribution, the waveguide mode contribution as well as the overall frequency response are shown in Figs. 3 (a-h) for different frequencies. In these figures, the vertical scale is normalized by the factor while the normalized frequency is different, multiplied by power ten, as shown in the figures. Having determined the location of the poles in the complex the guided modes may be calculated as residues and comapred with the total field. To get the residues of the poles, we need the eigenvalues as defined by equation (18). The poles are located between the branch points and . In Figs 3(a-h) the normalized vertical magnetic field of the dipole is drawn versus the normalized frequencies. From these plots, it is seen that increasing the frequency increases the waveguide mode contribution, Fig. 3 (c,d), up to a certain limit, after which this contribution starts to diminish again Fig. 3(g).
CONCLUSION The complete frequency-domain fields due to a vertical magnetic dipole on the surface of a two-layered ground have been derived in an analytical form. In these expressions, the displacement currents in the air and in the ground are taken into
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S. T. BISHAY ET AL.
TRANSIENT FIELD OF A VERTICAL MAGNETIC DIPOLE
113
consideration. The previously obtained approximate solution [7] is found to be the late time part of the present solution which is of overriding practical significance. It would be interesting to evaluate these fields numerically with different frequency-dependent and which is a diagnostic feature of the subsurface contaminants and hazardous fluid materials.
REFERENCES 1. J.R. Wait, "The magnetic dipole over the horizontally stratified earth", Can. J. Phys., Vol.
29, pp. 577-592, Nov. 1951. 2. J.R. Wait, and K.P. Spies, "Note on electrical ground constants from the mutual impedaance of small coplanar loops", J. Appl. Phys., Vol. 43, no. 3, pp. 810-891, 1972. 3. J.R. Wait, "On the theory of transient electromagnetic sounding over a stratified earth", Can. J. Phys., Vol. 50, no. 11, pp. 1055-1061, 1972. 4. J.R. Wait, L. Thrane, and R. J. King, "The transient electric field response of an array of parallel wires on the earth's surface", IEEE Trans. Antennas Propagat., AP-23, no. 2, pp. 261-264, 1975. 5. J.R. Wait, "EM fields of a phased line current over a conducting half-space", IEEE Trans. on Electromagnetic Comptability, Vol. EM- 38, no. 4, pp. 608-611, 1996. 6. K.A. Nabulsi, and J.R. Wait, " Ray decomposition of the pulse responses of a two-layer half-space", IEEE Trans. on Geosci. Remote Sensing, Vol.GE-35, no. 2, pp. 287-292, 1997. 7. A. Z. Botros, and S. F. Mahmoud, "The transient fields of simple radiators from the point of view of remote sensing of the ground subsurface", Radio Sci., Vol. 13, no. 2, 379-389, 1978. 8. S. F. Mahmoud , A. Z. Botros , and J.R. Wait, "Transient electromagnetic fields of a vertical magnetic dipole on a two-layer earth", Proceedings of the IEEE, Vol. 67, no. 7, pp. 1022-1029,1979. 9. J. R. Wait, "Electromagnetic fields of sources in lossy media", in Antenna Theory, R. E. Collin and F. J. Zucker, part 2, ch. 24, Mc Graw-Hill, New York, 1969. 10. J. A. Fuller and J.R. Wait, "High-Frequency electromagnetic coupling between small coplanar loops over an inhomogeneous ground", Geophysics, Vol 37, no. 6, pp. 997-1004, 1972. 11. A. Ezzeddine, J. A. Kong and L. Tsang," Time response of a vertical electric dipole over a two- layer medium by the double deformation technique", J. Appl. Phys., Vol 53, no. 2, pp. 813-822, 1982.
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Time-Domain Study of Transient Fields For a Thin Circular Loop Antenna
Samira T. Bishay and Ghada M. Sami Mathematics Department, Faculty of Science, Ain Shams University, Abbassia, Cairo, Egypt.
INTRODUCTION In order to probe the earth, the mechanism of the transient responses between dipoles near the ground surface must be understood. Many homogeneous earth models represented by a homogeneous half-space with a specified conductivity and permittivity are usually studied [1], [2]. For transient excitation and with the advent of operational ground penetrating radar [3],it is useful to have benchmark analytical and numerical models of layered half-space. Wait [4] had previously derived the transient fields of a horizontal electric loop on a homogeneous earth in closed form expressions. Also, the study of an elevated loop above the earth’s surface and its use in airborne electromagnetic surveying of the earth has been reported [5]. Due to the complexity of the time-harmonic solutions, closed-form expressions of the fields in the time domain, for a layered earth, are usually lacking. Hence, most frequently numerical or modeling studies have been adopted [6], [7]. Using closed-form solutions, Botros and Mahmoud [8] have obtained simple expressions for the transient responses between loops lying on the surface of a two-layered earth. Afterwards, Mahmoud et al. [9] derived the complete time-domain fields due to a vertical magnetic dipole on the surface of two-layered ground in an analytical form. Recently, Nabulsi et al. [10] have obtained the transient response of a two-layer earth model for an obliquely incident transverse magnetic plane wave using an exact numerical solution. Thus, the decomposition of the general case into ray contributions is an alternative approach that yields consistent results. In the present study, it is our purpose to obtain specific results, without approximation, for the time-domain full wave solution of the field for a small horizontal loop, i.e. a vertical magnetic dipole, on a two-layer earth’s model including the waveguide modes propagating within the earth. Cocentrating on the late time part of the solution, the displacement currents in the two-layer of the earth are accounted for.
Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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S. T. BISHAY AND G. M. SAMI
ANALYTICAL PART OF THE PROBLEM The geometry of the problem is illustrated in Fig.l where the air-earth interface is at z = 0. The earth’s adopted model consists of a homogeneous overburden slab of thickness d above a half-space. The upper layer for has conductivity permittivity and magnetic permeability The corresponding constants of the lower half-space are and The source is a vertical magnetic dipole situated directly on the earth’s surface. Physically, this represents a small loop of area dA carrying a circulating current I(t) which, in general, is a function of time t. Without loss of generality, we can restrict attention to situations where I(t) = 0 for t < 0.
Assuming that the time harmonic variation is the electric and magnetic field components in the time domain may then be dervied from the scalar magnetic Hertz potential at an observation point in the air region, due to the source, and are given by [11]:
and
In the region
where
the scalar magnetic Hertz potential is given as [11]:
TD STUDY OF TRANSIENT FIELDS FOR A CIRCULAR LOOP ANTENNA
where
and
117
are Fresnel reflection coeff- icients at the interfaces z = 0 and z = d,
respectively, and is the Hankel function of the second kind and order zero. The integration contour in (4) now can be deformed to infinity in the lower half of the plane, but it must pass around the two branch points at and and discrete set of poles, as shown in Fig.2. It is worth noting that there is no branch point at and there is no contribution from the contour at infinity. Hence, the time-domain expression of the vertical magnetic field, for example, can be written as
where
where
is the branch-cuts contribution and is given by
and respectively, and
are the branch-cut integrals at the branch points
and
is the poles contribution, written as
where is the contribution of the pole at Physically, the two branch-line integrals account for a continuous spectrum of radiation in the homogeneous regions above the interface and deep within earth, respectively. The pole contributions, of course, can be identified with waveguide modes in the stratified region of the earth.
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S. T. BISHAY AND G. M. SAMI
In the case where both the source and receiver are located above or on the surface of the earth, and when they are separated horizontally by a distance corresponding to several wavelengths in the earth, the contributions to which depend on propagation through the lossy ground, should be negligible [12]. In this case it is sufficient to evaluate
TIME-DOMAIN SOLUTION The time-domain response of the vertical magnetic field can be obtained as the inverse Laplace transform
where in terms of the angular freqency Hence, the overall response is the sum of the time responses of the branch-cut integrals contribution and the waveguide modes contribution.
A-EVALUATION OF THE INTEGRAL ALONG THE BRANCh-CUT Taking the inverse Laplace transform of [13]
where
The impulsive time response of the vertical magnetic field is obtained as
where
TD STUDY OF TRANSIENT FIELDS FOR A CIRCULAR LOOP ANTENNA
The normalized time
the normalized time delay
119
(c is the velocity of light),
and the prime in (15) denotes differentiation with respect to argument. The resulting inverse Laplace transforms of (16) are standard type [14] and neglecting with respect to hence
where is the unit step function which suffers a delay caused by the time delay parameter At sufficiently late time the dominant part of the solution (15) is represented by the time function Using the same previous treatment to the corresponding impulsive time response of the horizontal magnetic field and the circum- ferential electric field leads to
where
B-Evaluation of the Integral around the Poles To evaluate the integral around the poles, we take the inverse Laplace transform of the frequency domain expressions of the waveguide modes which are given by [13]
where and the n-th order.
and
and is the modified Bessel function of the second kind are given as
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S. T. BISHAY AND G. M. SAMI
where
In taking the inverse Laplace transform of the frequency domain expressions of the waveguide modes ( [13] eqs. (29) - (31) ), a difficulty arises from the factor To overcome this difficulty, it is noted that the inverse Laplace transform of these expressions take the general following form:
Expanding this summation by Taylor power separately series about the point collecting the terms of even and odd powers of then:
and
The inverse transform in (28) can be evaluated by resorting to the complex plane, taking into consideration that where is the complex conjugate of F,which take the form
where Re and Im denote the real and Imaginary parts of the integral.
TD STUDY OF TRANSIENT FIELDS FOR A CIRCULAR LOOP ANTENNA
121
Finally, to obtain the time-domain expressions of the electric and magnetic field, for an impulsive current excitation, we use the identity 6.6433 in [15] and the series expansion as given by the identity 9.2371 in [15],as
where
where
and
is the Eulerpsi function [15].
NUMERICAL RESULTS A computed example of pulse response of the time rate of change of the vertical magnetic field due to a rectangular pulse is computed for different cases. The branch cut and residues contributions as well as the total response are shown in Fig.3(a-d) which are denoted by and respectively. These plots for different values of the normalized radial distance
with fixed normalized overburden height
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S. T. BISHAY AND G. M. SAMI
TD STUDY OF TRANSIENT FIELDS FOR A CIRCULAR LOOP ANTENNA
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S. T. BISHAY AND G. M. SAMI
The conductivities ratio is taken as normalized by the factor
The vertical scale is
while the normalized time
is delayed by time
delay factor Fig.3a shows that whenever X/D has small values, the waveguide mode contribution will be only effective at relatively early times. Fig.3b shows that the increase of the separating distance X would lead to the increase of the waveguide mode contribution up till a certain limit after which the reverse would occur (Figs. 3c and 3d). In these latter figures, the oscillations manifested by the branch-cut contribution when the values of X are equal to 10 and 31.6,respectively, deserve an explanation. The observed deviations are due to the second, third and fourth terms in the square bracket of (15). These terms affect the total pulse response considerably, especially at early times, leading to the observed oscillations. It is noted that these oscillations do not accurately represent the response at early times since late time approximations were used in deriving the function However, they could be regarded, on a qualitative basis, as due to the differentiation of the imput pulse. Fig.4 shows the plot of the late time part of the solution and the effect of the earth layer’s conductivities. As the waveguide mode contribution diminshs at the late time, the branch-cut contributins are the only ones plotted for a wide range of these conductivities. The late time decay of the response is also shown to be strongly affected by these conductivities. Fig.5 shows the response on a homogeneous earth
which is also computed
and plotted. It is noteworthy that this half-space response shows two peaks with time locations which largely depends on the seperation parameter X. On the other hand, the time occurrence of the last peak in the layered earth response is dependent largely on the overburden height D. This latter observation is quite important in remote sening applications.
CONCLUSION The time-domain fields originating from a vertical magnetic dipole placed on the surface of two-layered ground have been derived in an analytical form. The displacement currents in both the two ground layers have been taken into consideration. A previously obtained approximate solution [8] thus represents the late time part of the present solution. It is noteworthy to mention that the term "late time" used in the present work is not considered independently. It depends on both the separation and depth parameters X and D. The condition defining the late time term in [eq.(44), 8] thus seems to be overrestrictive in the case of the layered ground. The pulse and step responses of the induced voltage in a horizontal receiving loop would lead to definite conclusions about the properties of the ground. These conclusions could be deduced from Figs. 3-5. Thus, both the overburden height D parameter and the conductivities ratio could be determined. Figs. 3a and 3d, for example, could be used to measure the parameter D as the time occurence of the peak in the pulse response is greatly dependent on it. Also, Fig. 4 could be used to determine the ratio from the late time decay of the pulse response.
TD STUDY OF TRANSIENT FIELDS FOR A CIRCULAR LOOP ANTENNA
125
REFERENCES 1. J.R. Wait and C. Froese, "Reflection of a transient electromagnetic wave at a conducting surface (a half-space)," J. Geophys.Res., Vol. 60, no.l, pp. 97-103,1955. 2. S.H. Dvorak, H.Y. Pao, D.G. Gudley, and M.Sheikh, "Use of the short-space time Fourier transform in the extraction of the electrical properties for a conducting half- space," Porc. Progress in Electromagnetic, Res. Symp, Seattle, WA, July 1995, p. 99. 3. J.A. Pilon,Ed. "Ground penetrating radar," Paper 90 - 4, Geolog. Survey Canada, 1992. 4. J.R Wait, "The magnetic dipole over the horizontally stratified earth," Cand. J.Phys., Vol. 29, pp. 577-592, Nov. 1951. 5. J. R.Wait and R.H. Ott "On calculating transient electromagnetic fields of a small currentcarrying loop over homogenous earth," Rev. Pure Appl. Geophys., Vol. 95, pp. 157162, 1972. 6. J. Ryu, H.F. Morrison, and S. H. Ward, " Electromagnetic field above a loop source of current," Geophysics, Vol. 35, pp. 862-896, 1970. 7. V. K. Gaur, "Electromagnetic model experiments simulating an airborne method of prospecting," But.Nat. Geophys. Res. Inst. (India) Vol. 1, pp. 167-174, 1963. 8. A. Z. Botros, S. F. Mahmoud, "The transient fields of simple radiators from the point of view of remote sensing of the ground subsurface", Radio Sci., Vol. 13, no. 2, pp. 379-389, 1978. 9. S. F. Mahmoud, A. Z. Botros and James R. Wait, "Transient electromagnetic fields of a vertical magnetic dipole on a two-layer earth", Proceedings of the IEEE, Vol. 67, no.7, pp. 1022 - 1029, 1979. 10. K. A.Nabulsi and J. R. Wait," Ray decomposition of the pulse responses of a two-layer half-space," IEEE Trans, on Geosci. Remote Sensing, Vol.GE-35, no. 2, pp. 287292, 1997. 11. J. R. Wait, "Electromagnetic fields of sources in lossy media", in Antenna Theory, R. E. Collin and F. J. Zucker, Eds part 2, ch. 24, McGrawHill, New York, 1969. 12. J. A. Fuller and J. R. Wait," High-frequency electromagnetic coupling between small coplanar loops over an inhomogeneous ground", Geophys., vol. 37, no. 6, pp. 9971004, Dec. 1972. 13. S. T. Bishay, O. M. Abo Seida and G. M. Sami, "Transient electromagnetic fields of a vertical magnetic dipole on a two-layer conducting earth " (Sent for publication). 14. S. R. Murray," Laplace transforms", Schaum’s Outline Series, McGraw - Hill Book Company, New York, 1965. 15. I. S. Gradshteyn and I.M. Ryzhik, "Tables of Integrals, Series and Products", New York: Academic Press, 1965.
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GENERALIZED TEM, E, AND H MODES Carl E. Baum
Alexander P. Stone
Air Force Research Laboratory Directed Energy Directorate
Department of Mathematics and Statistics University of New Mexico Abstract
Previous papers have considered transient lenses for propagating TEM modes without dispersion. This paper considers the properties of E and H modes in such lenses. The presence of longitudinal field components brings in additional constraints on the allowable coordinate systems, limiting the cases of transient lenses supporting E and H modes to a subset of those supporting TEM modes.
1. INTRODUCTION A technique developed by C. E. Baum1 for the design of EM lenses utilizes the expression of the constitutive parameters and and Maxwell’s equations in a general orthogonal curvilinear coordinate system, yielding what we will call the formal quantities. These are customarily denoted by affixing primes as superscripts. The line element is
and the coordinates are The scale factors, relate the formal parameters and to the real world parameters and in the diagonal case via the equations
The scale factors also relate the formal fields via the equations
and
to the real fields
Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
and
127
128
for
C. E. BAUM AND A. P. STONE
Maxwell’s equations for the formal fields are
Here we use the two-sided Laplace-transform variable or complex frequency. This suppresses the time-derivatives for our convenience in notation, and furthermore allows the constitutive parameters to be frequency dependent if desired. This last point is significant only in the case of dispersive media, which need not concern us here. Since we are not going back and forth between the time and frequency domains, we do not need to indicate the fact that all fields are Laplace transforms (functions of complex frequency). Thus if we assume diagonal forms for the tensors and Maxwell’s equations in expanded form become
and
These equations will be the starting point in our search for conditions on the parameters and in the case of E (or TM) and H (or TE) modes. A general discussion of differential geometric lens synthesis appears in Baum and Stone2 .
2. TEM CASE (formal fields) We recapitulate the results obtained,in the formal case, for a TEM wave propagating in the coordinate direction. These results, which are consistent with examples described in earlier work 3,4 , will suggest the approach to be taken in the case of an E-wave or an H-wave. We begin with the assumption that the parameters are in the form
and
GENERALIZED TEM, E, AND H MODES
We will think of our
129
coordinates as though they are cartesian and allow
and to be inhomogeneous and anisotropic. Our TEM plane wave is to propagate in the direction and all fields will be assumed to have propagation factors which account for this. Thus if is the propagation direction, then and are irrelevant. At this point we assume a form for the fields and constitutive parameters that factors the dependence into products of the form a function of and (transverse coordinates) times a function (longitudinal or propagation coordinate). For the constitutive parameters this is (all terms real and positive)
Note that there are not separate functions of for each of the 1 and 2 components, this being an assumption of invariance to transformation (e.g., rotation) of the coordinates. We then seek TEM solutions of the form
with
Maxwell’s equations then take the form
These equations may then be rewritten as, using (9) and (10),
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C. E. BAUM AND A. P. STONE
We note that in the above equations we have “separability” in the sense that these equations may be reexpressed in a form where we have a function of and equal to a function of only. The immediate result is then the fact that both functions are equal to a constant (i.e., independent of the spatial coordinates). This same reasoning, it will be recalled, is used in the “separation of variables” technique in partial differential equations. Thus we define constants and from (13) by
Thus
We also note, from (14), that wave since
and
need not be mutually perpendicular, for a TEM
Instead, orthogonality holds between and and also between and a property associated with allowing the medium to be anisotropic. Whether one considers the uniform formal (prime) medium as anisotropic or isotropic, there is still the question of scaling to the real coordinates and fields. One can then again ask the question of whether the formal permittivity and permeability can be isotropic or anisotropic. In previous work2 , the assumption of isotropic media parameters led to the result that surfaces of constant could only be spheres or planes. Since we now have the possibility of anisotropic and the formal medium need not be isotropic and surfaces of constant may take more general shapes. In particular if then the permeability is both uniform and isotropic, and we have a case of practical significance 2 , that of a bending lens.
3. E-WAVE CASE (TM) We now take parameters and
in Maxwell’s equations and seek conditions on the formal which lead to solutions of the formal Maxwell equations. As usual
GENERALIZED TEM, E, AND H MODES
131
in the case of waveguides we seek solutions for the formal-field components in terms of some operator on which we will later take as some mode function of and (transverse coordinates) times some propagation function of Thus if we obtain
The above equations will lead to restrictions on the formal parameters and as well as solutions for the formal fields and We assume that the formal constitutive parameters have the forms specified in (9) as
for
with
irrelevant since
is the assumed propagation direction. We take
and try solutions of the form
Now for physical realizability and are real, nonzero and frequency independent. The propagation functions and are in general complex functions of the complex frequency (exponential like) and all have derivatives nonzero except possibly at special frequencies like or degenerate cases like propagation perpendicular to (waveguide cutoff). The functions of are taken as independent of the complex frequency Thus we obtain
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C. E. BAUM AND A. P. STONE
and
Moreover, since
we have
and also, from
we have
Constants may be defined since we can separate variables in (21) and (22). Thus we set
Here and are included to give a dimension of inverse length (in the coordinates) to balance the derivatives. Later we will find that these are related to the propagation constant or wave number. As such and may be functions of complex frequency while is not a function of A constant velocity is also included to make the units work out. The equations in (21) can be written in a more compact two-dimensional form (transverse coordinates only for vectors/dyadics) as
where
GENERALIZED TEM, E, AND H MODES
and since there is a function
133
such that
we may find an equation for
which involves only the electric field. It is
where
One is then able to obtain an equation
by a change of variable from
to
given by
with The product is an eigenvalue giving a transverse wave number which can be computed. This applies to waveguide solutions with on some closed contour in the plane. The main result is that we can have E modes in the same media as the TEM modes in Section 2, provided we have an additional constraint on the part of the permittivity. Specifically varies reciprocally with respect to for the special case of a constant. However, the medium can now be both inhomogeneous and anisotropic. The factor is specified by We note that a limiting case of an E mode is a TEM mode. As in a typical waveguide with perfectly conducting walls, the ratio of the longitudinal electric field to the transverse electric field tends to zero for a given mode. This result is derivable under the assumption that two or more independent E modes exist.
4. SCALING TO REAL MEDIUM FOR E-MODES As in the TEM case we may now consider the scaling of the coordinates to something other than cartesian coordinates (for which the results in Section 3 are directly applicable). The scaling relations are given, as in Section 1, by
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C. E. BAUM AND A. P. STONE
where the constitutive-parameter dyadics (matrices) are assumed diagonal in the coordinate system. The scale factors and line element are
We also have
If the real medium is constrained to be isotropic we have
so that we have five relevant constitutive-parameter components to consider, one more than in the TEM case. In particular we have
with surfaces of constant limited to spheres and planes. In the case that (uniform real permeability) we find that and so the surfaces of constant can only be planes. Such a case of a bending lens with constant surfaces is considered in Baum3 . If we have isotropic real and formal media then which gives the case which admits only 2 types of solutions (cartesian coordinates and their inversion). This is a very restrictive case and so we can allow only the formal permittivity to be anisotropic.
5. H-WAVE CASE (TE) In this section we summarize the results obtained in the H wave case. We impose the condition that the formal field component, vanishes and then look for conditions on the formal parameters, and which lead to solutions of the formal Maxwell equations. The analysis is dual to that of section 3, in which the E-wave case was studied, and the results will be dual. Duality is the symmetry on interchange of electric and magnetic parameters. Thus solutions will be sought for the formal field components in terms of an operator on which will eventually be taken as some mode function of the transverse coordinates, and multiplied by a propagation function of The assumptions made in Section 2 on the form of the formal parameters remain in effect. The parameter is irrelevant since we take as the propagation direction, while Just as in the E wave case, and are real, nonzero,
GENERALIZED TEM, E, AND H MODES
135
and frequency independent. Similarly the propagation functions and are in general complex functions of the complex frequency (of exponential type). Thus by separating variables in Maxwell’s equation, a dual collection of constants arise with the interchange of media parameters and electric and magnetic field components. Analogous to (33) there is a differential equation for given by
with appropriate boundary conditions for the magnetic field. Thus we would require that the normal derivative of be zero on some closed contour in the plane. Consequently we have a waveguide problem for with assuming the role of an eigenvalue (a transverse wave number or a propagation constant). Hence we will obtain one result that H modes can propagate in the same media as TEM modes under a constraint, similar to that for E modes, on a parameter The introduction of a scaling in the coordinates then leads to results similar to those obtained in the case of E modes. For example, if the real medium is constrained to be isotropic we have and In this event and surfaces of constant are sphere or planes. In the practical case of uniform real permeability with isotropic the results obtained are
Thus in this case surfaces of constant
can only be planes.
6. CONCLUSIONS We now have a significant set of results for TEM, E, and H modes. The basic form for those is found by separating out the (propagation) coordinate from the (transverse) coordinates, and requiring propagation in one direction without reflection. Various assumptions (constraints) on the constitutive parameters lead to constraints on the allowable coordinate systems. We can note that these results apply only to such modes, and not to all possible solutions of the Maxwell equations which may include additional contributions (e.g., hybrid HE modes). A related problem is treated by Friedman6 . Those results have some similarities to and differences from the present results. There only was considered. His results are based on those of Bromwich7 . The results had the decomposition of the fields into unique E- and H-mode parts. Only cartesian and spherical coordinates were considered. There was found to be a function of only, but our present results allow for more general specifically as a function of and as well. Nevertheless, there are some remarkable similarities in the results. In particular is independent of Constraining the practical case of we all have
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with some freedom for In our case, however can be a function of and This leads to a nontrivial example of a bending lens in which surfaces of constant are nonparallel planes. The present results also allow for more general anisotropic real and/or formal media to be considered, including for the case of TEM modes. This may lead to other interesting cases for transient lens design. Note the fundamental assumption of and each having both and components. This could be relaxed by allowing the fields to have only one transverse component (e.g., and ) as in Baum and Stone5 . So there are various possible other cases that can be considered. An expanded version of this paper will appear in a forthcoming Sensor and Simulation Note. The mathematical details omitted in this paper will appear in the SSN report. This work was sponsored in part by the Air Force Office of Scientific Research, Arlington, VA.
REFERENCES 1. C. E. Baum, A Scaling Technique for the Design of Idealized Electromagnetic Lenses, Sensor and Simulation Note 64, August 1968. 2. C. E. Baum and A. P. Stone, Transient Lens Synthesis: Differential Geometry in Electromagnetic Theory, Taylor and Francis, 1991. 3. C. E. Baum, Use of Generalized Inhomogeneous TEM Plane Waves in Differential Geometric Lens Synthesis, Sensor and Simulation Note 405, December 1996; URSI International Symposium on Electromagnetic Theory, Thessaloniki, Greece, May 1998, pp. 636-638. 4. C. E. Baum and A. P. Stone, Synthesis of Inhomogeneous Dielectric, Dispersionless TEM Lenses for High-Power Application, Electromagnetics, 2000, pp. 17-28. 5. C. E. Baum and A. P. Stone, Unipolarized Generalized TEM Plane Waves in Differential Geometric Lens Synthesis, Sensor and Similation Note, 433, January 1999. 6. B. Friedman, Propagation in a Non-homogeneous Medium, pp. 301-309, in R. E. Langer (ed.), Electromagnetic Waves, U. Wisconsin Press, 1962. 7. T.J.I’a Bromwich, Electromagnetic Waves, Philosophical Magazine, 1919, pp. 143164.
ELECTROMAGNETIC WAVE SCATTERING BY SMOOTH IMPERFECTLY CONDUCTIVE CYLINDRICAL OBSTACLE
Yu. A. Tuchkin 1,2 1
Institute of Radiophysics and Electronics, NAS of Ukraine, Kharkov, Ukraine 2 Gebze Institute of Technology, Gebze, Turkey
INTRODUCTION New strong in mathematical sense and numerically efficient method for investigation of two-dimensional boundary value problem of electromagnetic wave diffraction by infinite and homogeneous in longitudinal direction imperfectly conductive cylinder of arbitrary smooth cross section is suggested. The imperfect conductivity is modelled by the boundary condition of the third kind, for example, by Leontovich condition with impedance, which is supposed to be an arbitrary smooth function of points of the obstacle cross section contour. The method of investigation is based on generalisation of Analytical Regularisation Method, which is developed in our previous papers (Tuchkin, 1985,1987,1997; Tuchkin, and Shestopalov, 1990; Shestopalov at al). A few qualitatively different cases are considered, which can be qualified as regular and a few types of singular perturbation of Neumann boundary value problem (BVP). As a result, the BVPs considered are equivalently reduced to a few different equation of the second kind in of type with the operator H compact in
BOUNDARY VALUE PROBLEM OF THE THIRD KIND Let us consider an infinite cylinder of arbitrary cross-section homogenous in longitudinal direction (see figure 1). This obstacle is illuminated by wave which does not depend on z – coordinate and it is necessary to find the scattering field (evident that it does not depend on z–coordinate either) as a solution of the following two–dimensional BVP of the third kind.
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where D is an open domain bounded by contour and and and are supposed to be (infinitely) smooth functions of point are normalized as
is total field, and they
Boundary condition of type (2) is used widely in diffraction theory for simulation of obstacle’s finite conductive material, small regular or irregular inhomogenities of its surface, its coating by dielectric layers of variable thickness, etc. From mathematical point of view, it is necessary to consider functions and for as uniform on S limits:
where
is unit outward normal to contour S in point
GENERALIZED POTENTIALS OF SINGLE AND DOUBLE LAYER Utilization of the theory of generalized potentials and relevant Green’s formulae technique give us the possibility to obtain following representation of scattering field:
where
and for arbitrary smooth function gle and double layer respectively:
operators
and
are generalized potential of sin-
ELECTROMAGNETIC WAVE SCATTERING BY CYLINDRICAL OBSTACLE
where
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is Green’s function of two dimensional free space:
and
(z) is zeroth-order Hankel function of the first kind. Formulae (4)-(6) enable us to obtain integral representation of functions
and
where
and the dashed operators and are direct value of and on S, and means the limiting value of the same sense as in formula (3). Using notation (5), formulae (2) can be rewritten as follows with evidently defined and known function From other side, substitution of the formulae (8) and (9) into (2) gives the following relation, with known function which incident field defines: Thus, BVP (1) and (2) is reduced to the system (11) and (12) with unknown functions and It has to be pointed out, that all integral transforms in equation (12) have singular kernels of different singularity. Operator has the most singular kernel, and the next, less singular one, has operator Kernels of operators and have the minor singularity. We suppose that some smooth parameterization of contour S is given and where and mean the derivatives of order Such a parameterization can always be constructed numerically with any necessary accuracy. DIRICHLET BOUNDARY VALUE PROBLEM We start with the simplest BVP: with Dirichlet boundary condition, when and for convenience Hence, from (11) and from (12) the unknown function satisfies the equation: where is some known function. By means of parameterization reduced to the integral equation of the following kind
with the unknown function
where
equation can be
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We denote the Fourier transform which maps arbitrary function to infinite vector-column of its Fourier coefficients by F; the inverse transform - by
We will use special infinite diagonal matrix
It can be proved that kernel in (14) is rather smooth function (after its periodical continuation on infinite plane). Thus, operator in (13) describes the most singular part of integral operator in (14). Due to this, equation (14) could be rewritten as follows:
where I is the identical matrix-operator of space and M is matrix-operator formed by Fourier coefficients of function It can be proved, that is a compact (and even Hilbert-Schmidt) operator in space That is why, one may define operators and and infinite vector-column (consequently ). Acting by operator L on left and right-hand sides of (18), following equation of the second kind in can be obtained: with compact operator
in
NEUMANN BOUNDARY VALUE PROBLEM The case of Neumann BVP is (11) immediately follows and function.
and we took for convenience From satisfies the following equation with known
It can be proved that by means of parameterization the integral-differential equation of the type
Kernel has singularity proportional to (21) can be rewritten as follows.
equation (20) can be reduced to
only. That is why, equation
It can be proved that matrix-operator has the same qualitative properties in the operator compact and even Hilbert-Schmidt operator in As above, defining operators reduces (22) to equation of the second kind in
infinite vector-column
BOUNDARY CONDITION OF THE THIRD KIND: REGULAR CASE
like
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ELECTROMAGNETIC WAVE SCATTERING BY CYLINDRICAL OBSTACLE
At first, we consider the simplest, so called regular, case:
where is some (not “very small” - see below) constant. Elimination of function from (3) by means of relation (11) gives after some transformation an equation of the same kind as equation (22), with another kernel which has the same qualitative properties as kernel That is why the considered problem can be reduced to equation just of the same qualitative kind as equation (23).
BOUNDARY CONDITION OF THE THIRD KIND: SINGULAR CASE The alternative condition to (24) is the case
when function
can be very small and even equal to zero on some part of contour S. This situation is arising, for example, in analysis of E-polarized wave diffraction by well conductive cylinder. In this case of small values of boundary condition (2) can be considered as singular perturbed Dirichlet condition: At first, we consider the special case of small values of but when condition (24) is valid for certainly small after that we will consider the general case. Elimination of function from (12) by means of (11) gives after some transformation the following integral-differential equation, with the unknown function
where is a function with the same qualitative properties as kernel in (14). Here is the differential operator of the kind below, with directly connected with function
We introduce small parameter and new normalized function scribes the “shape” of function and parameter describes its values:
where
de-
According to suppositions made above, is infinitely smooth function where is a set of infinitely differentiable functions on with coinciding derivatives of all orders in points Thus, the construction of regularizator of equation (25) can be separated, as it was proved, on two problems. The first one is constructing the resolvent operator of ordinary boundary value problem: The second problem (when the first one is solved) is analytical regularization of the equation of the kind (14) with another but qualitatively the same kernel Consequently, this problem can be solved in the same way as one for equation (14). The resolvent of equation (28) can be constructed numerically (if is not extremely small) by means of standard numerical technique. If is small enough, the resolvent can be constructed analytically on the basis of different well-known asymptotic methods. As a result, equation (25) has been equivalently reduced to the equation of the second kind in of the type with compact in operator which is uni-
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formly bounded and regular one when if The last is natural condition from point of view of possible physical applications. In addition, the corresponding perturbation theory of power series about has been constructed. More general than (24) for small is condition outside some arc (the generalization to a few such arcs is evident):
In this case the only difference is to change the boundary condition in (28) by the following ones: where and corresponds to end points of contour L, parameterized by function and it is evident that equation degenerates into for
CONCLUSION The problem of wave diffraction by impedance cylindrical smooth surface is solved. The initial boundary value problem is reduced to a few different algebraic systems in of the kind This gives relevant basis for efficient numerical algorithm construction for most part of possible physical and engineering applications. The constructed method includes the most complicated case of imperfectly but well conductive cylinder. REFERENCES Yu. A. Tuchkin, 1985, Wave scattering by unclosed cylindrical screen of arbitrary profile with Dirichlet boundary condition, Soviet Physics Doclady, 30. Yu. A. Tuchkin, 1987, Wave scattering by unclosed cylindrical screen of arbitrary profile with Neumann boundary condition, Soviet Physics Doclady, 32. Yu. A. Tuchkin, and V. P. Shestopalov, 1990, A wave diffraction by screens of finite thickness, Soviet Physics Doklady, 35. Yu. A. Tuchkin, 1997, Regularization of one class of system of integral-differential equations of mathematical physics, Doclady of The Ukrainian National Academy of Sciences, ser. A, No. 10, pp.47-51 (in Russian). V. P. Shestopalov, Yu. A. Tuchkin, A. Ye. Poyedinchuk and Yu. K. Sirenko, 1997, Novel methods for solving direct and inverse problems of diffraction theory, vol. 1: Analytical regularization of electromagnetic boundary value problems, Kharkov: Osnova, (in Russian)
A SET OF EXACT EXPLICIT SOLUTIONS IN TIME DOMAIN FOR UWB ELECTROMAGNETIC SIGNALS IN WAVEGUIDE
Oleg A. Tretyakov1,2 1
2
Department of Theoretical Radio Physics Kharkov National University Kharkov-77, 310077, The Ukraine Department of Electronics Engineering Gebze Institute of Technology 41400, P.K. 141, Gebze, Kocaeli, Turkey
INTRODUCTION: TWO APPROACHES TO TIME DOMAIN System of Maxwell’s equations may be shortly written as an abstract operator equation where M is Maxwell’s operator, – electromagnetic field sought for, – given function of impressed forces. Classical approach to study of transient fields corresponds to separation of Maxwell’s operator on two parts as where A is a remainder of M. Unbounded linear differential operator has a set of eigenfunctions that is complete in the sense of Fourier theorem. Just this set serves as a basis for development of Electromagnetics in the Frequency Domain. The solution sought may be presented as a direct Fourier transform
which means projecting of the solution onto the basis elements. Remainder A of M can supply then with a problem for the vector Fourier coefficients (with using of the inverse Fourier transform) provided that A is a linear operator as well, however. The disadvantages of the classical approach were scrutinized by Hillion recently (1993). Evolutionary approach has been developed starting from another separation of Maxwell’s operator (see our cited book). For the waveguide problem under consideration, separation of M has been made as M = W + B where W is liberated as a linear self-adjoint operator acting on waveguide transverse coordinates of the position vector at the argument of the solution sought. Operator B is a remainder of M; it may be linear or nonlinear in the general case as the Constitutive Relations involved in M dictate it. Since operator W is self-adjoint (due to involved Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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in its definition the boundary conditions over the perfectly conducting waveguide surface), it has the eigenvectors specified as the eigensolutions of the operator eigenvalue equation where S – a waveguide cross-section. Spectrum is discrete since operator W is bounded. Eigenvector set is complete and it originates a basis in a Hilbert space chosen as the space of solutions. Projection of the solution sought onto the basis elements looks as follows
where the scalar coefficients are unknowns. A problem for them can be obtained via projecting of Maxwell’s equations themselves onto the same basis elements This procedure supplies with a system of evolutionary (i.e., with time derivative) partial differential equations for Implementation of this scheme with the mathematical details needed the reader can find in our cited book. In this paper some new results will be presented which are obtained within the scope of the evolutionary approach. N.B. Equations (1) and (2) obtained within the scope of different approaches give distinct presentations for the same field sought for.
PROJECTION OF THE FIELD SOUGHT ONTO A MODAL BASIS The vector eigenvalue equation for basis elements in can be scalarized what yields well studied Dirichlet and Neumann boundary eigenvalue problems for Laplacian, supplemented with the proper normalization conditions, namely:
where L is a contour bounding the waveguide cross section S, and subscripts each put the eigenvalues in order of increasing of their values. Then formal equation (2) can be rewritten in terms of potentials and for components of electromagnetic field sought as follows
where are the free space constants. Herein the coefficients with argument are equivalent to from the formal equation (2).
EXACT EXPLICIT SOLUTIONS FOR UWB SIGNALS IN WAVEGUIDE
Terms with a fixed subscript taken from all the series over electromagnetic field of TM waveguide modes as
145
originate jointly
where subscript the same as in equation (3), identifies the mode. The fields of TE– modes reveal themselves in a similar way as
Term from equation (5) generates a specific mode with The basis resulting in the modal decompositions (4), (5) can be named as a modal basis, and the self-adjoint operator W generating as a modal operator, respectively. WAVEGUIDE EVOLUTIONARY EQUATIONS Equations for amplitudes of the modal field components have a simple form when a waveguide is filled with a lossless medium where the following constitutive relations hold
provided that and are arbitrary differentiable functions other than zero at any point in plane Projecting of Maxwell’s equations onto the modal basis yields ultimately a differential equation for the amplitudes of longitudinal mode components and direct formulae for the amplitudes of transversal fields as
where the speed of light in the free space, and the functions of impressed forces we put as zero for simplicity sake. Amplitude from the first equation (5) satisfies a pair of equations resulting in In fact, this system of evolutionary equations is exact form of Maxwell’s equations which they acquire after projecting onto the modal basis. When a waveguide is hollow, i.e., then the differential equations from (9), (10) turn into well studied Klein-Gordon equation (KGE)
where while and while In such a sense, differential equations with variable coefficients exhibited in (9), (10) is a generalization of KGE. Van Bladel (1985) was the first apparently who obtained
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the evolutionary equations for a hollow waveguide but he has made it starting from another position. Evolutionary approach admits to obtain waveguide evolutionary equations as well when the constitutive relations for lossy nonlinear medium hold. However, analysis of this situation needs much more place, therefore it will be considered elsewhere.
SEPARATION OF VARIABLES IN KGE AND ANALYTICAL RESULTS Separation of variables in KGE is evident: it results in monochromatic waveguide waves. A question arise: is it possible to introduce some new variables in such a way which allows to obtain a solution of equation (11) in the form of product as
Direct substitution of
in equation (11) yields
It is clear that substitutions must be very special to give a solution in the form (12). Fortunately, this problem can be solved successfully using study of KGE by means of group theory methods made by Miller (1977). These results will be used below successively. Variables must be real. That is why a pair of substitutions taken separately is unable to cover all the plane completely Therefore an additional dual pair of substitutions is necessary (see Cases 1 and 2 and the rest below). Case 1. These substitutions turn partial differential equation (13) into a pair of standard equations
here and henceforth is a constant (possibly complex-valued) of separation of variables Solution of equation (13) obtained in the form (12) looks as
where are linearly independent Bessel functions, constants; the constant plays role of a free parameter. Example: excitation of a waveguide when of time. Let us first put for and for We may choose then as a set of integers: then with these constants is Bessel function
– arbitrary is a given function
When is complete
EXACT EXPLICIT SOLUTIONS FOR UWB SIGNALS IN WAVEGUIDE
and it may be used for Neumann’s expansion of follows
147
in a series of Bessel functions as
Coefficients can be calculated easily. For example, when is Heavyside step function then For arbitrary function solution of starting KGE (11) can be written as
where coefficients
are the same as in equation (16). In fact, set of functions
plays role of an evolutionary basis here. Some comments may be relevant. Solution (17) is available for amplitudes of longitudinal modal components and both. However, one should take for and for Condition means that either or is given as an arbitrary function of time at the waveguide cross section Since formula for the longitudinal modal amplitudes is obtained as the highly convergent series (17), the amplitudes of transverse modal components can be calculated analytically by differentiations given in (6), (7). Substitutions considered above admit to cover two quadrants only in plane where inequality holds. To cover two other quadrants where inverse inequality holds the following dual substitutions are necessary. Case 2. In terms of these variables, equation (13) has a solution as
where – arbitrary constants and plays role of a free parameter as before; – modified Bessel functions. This solution is convenient for solving of Cauchy problem when function is given at initial moment of time as a function of coordinate This problem has been studied by Kristensson (1995) in another manner. Case 3: In terms of these variables, equation (13) supplies a pair of Bessel equations in the following forms:
which have as their solutions
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with arguments
Case 4: In this case
where
Possibly, a pair of similar substitutions be found useful in some situations. Case 5: when
where
Case 6: when
where
EXACT EXPLICIT SOLUTIONS FOR UWB SIGNALS IN WAVEGUIDE
149
Case 7: This substitutions convert equation (13) to a pair of equations for parabolic cylinder functions as
where
There is an interesting particular case when the parameter where Then
where
take on the values
– Hermite orthogonal polynomials.
Case 8: In this case we have the other forms of equations for parabolic cylinder functions as
where
Case 9: These substitutions give equations for Airy functions
with arguments
Case 10: We have also equations for Airy functions
but their arguments are distinct from previous
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Case 11: These give equation for Mathieu periodic functions
where
and
Next and the last four substitutions yield different versions of modified Mathieu equations.
Case 12:
where
Case 13:
Case 14:
EXACT EXPLICIT SOLUTIONS FOR UWB SIGNALS IN WAVEGUIDE
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Case 15:
Waveguide Modes in a Time-Variant Medium Let the same waveguide be filled with a time variant but spatially homogeneous medium having where are constants and is an arbitrary integrable function. Equations (9) and (10) can be rewritten as
where while and while Its solution is evident: it is a sum of products of the amplitude factors proportional to and ”phase” factors where a constant of separation of variables and When is real, the waveguide modes have instant frequencies depending on time as
In regard to KGE obtained with variables, the same procedure of introducing and successive separation of and variables can be repeated. It supplies a new set of the Cases 1 - 1 5 regarding to the waveguide modes in the time-variant medium.
CONCLUSION Main results may be listed as follows. 1. Evolutionary Approach to Electromagnetics has been proposed for waveguide theory in Time Domain as an alternative to the classical Method of Complex Amplitudes. 2. The Approach opens a prospect for partial evolutionary (i.e. with time derivative) differential equations as a powerful tool for development of Electromagnetics in Time Domain. 3. A set of analytical results valid for hollow waveguides and for waveguides filled with a special time-variant medium as well are obtained via the Approach proposed.
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Acknowledgments The author gratefully acknowledge the support of The Royal Swedish Academy of Sciences by dint of the Contract between Sweden and the former Soviet Union, and the personal support of Swedish Professors S. Ström and G. Kristensson.
REFERENCES Hillion, P., 1993, Some comments on electromagnetic signals; in Essays on the Formal Aspects of Electromagnetic Theory, ed. A. Lakhtakia World Scientific Publ. Co., Singapore. Tretyakov O. A., 1993, Essentials of nonstationary and nonlinear electromagnetic field theory; in Analytical and Numerical Methods in Electromagnetic Wave Theory, ed. by M. Hashimoto, M. Idemen, and O. A. Tretyakov, Science House Co., Tokyo. Van Bladel, J., 1985, Electromagnetic Fields, Hemisphere, Washington. Miller, W., Jr., 1977, Symmetry and Separation of Variables Addison-Wesley Publ. Co., Massachusetts. Kristensson G., 1995, Transient electromagnetic wave propagation in waveguides J. Electromag. Waves Applic., 5/6:645.
ANALYTICAL REGULARIZATION METHOD FOR WAVE DIFFRACTION BY BOWL-SHAPED SCREEN OF REVOLUTION
Yu. A. Tuchkin1,2 1
Institute of Radiophysics and Electronics, NAS of Ukraine, Kharkov, Ukraine 2 Gebze Institute of Technology, Gebze, Turkey
INTRODUCTION New mathematically strong method for solving of the boundary value problem (BVP) of wave diffraction by axially symmetrical bowl-shaped screen of arbitrary profile is suggested. The approach used is based on generalization of Analytical Regularization Method developed in our previous publications (Tuchkin, 1985,1987,1997; Tuchkin, and Shestopalov, 1990; Shestopalov at al, 1997) and on ideas, presented in (Vinogradov at al, 1978,1980,1981), which are devoted to solving the problem of wave diffraction by arbitrary shaped unclosed or closed cylindrical screens. The approach used is based on Green’s formulae technique and reduction of corresponding integral or integral differential equations to the equivalent dual series equations. The regularization procedure for such dual series equations is described in our previous papers. As a result initial Dirichlet and Neumann BVP ‘s are reduced to equivalent infinite algebraic systems of the second kind: with the operator H compact in the space of square summable sequences. DIRICHLET PROBLEM Let us consider bowl-shaped screen of revolution, which is formed by rotation of plane unclosed (or closed) contour L (see Fig. 1). Without loss of generality we suppose that L is a part of smooth non-self-crossing contour S which has ends placed on z-axis. Additionally we suppose, that some smooth parameterization is given, where and are corresponding cylindrical coordinates of point which corresponds to the parameter value and
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Let us consider Dirichlet diffraction problem: it is necessary to find scattering field which satisfies homogenous Helmholtz equation and Dirichlet boundary condition as follows:
where is given incident field which is supposed to be smooth in some vicinity of surface. As well known, the problem considered can be reduced to the integral equation of the first kind
where space:
is unknown function (two side current density) and G is Green function of free
Due to axial symmetry of the screen, the integral equation (5) can be separated into an infinite set of integral equations over contour L, where each equation corresponds to value m=0,+l,±2,±3,... of azimuth index. By means of parameterization each equation can be reduced to the following equation:
with unknown function
where point
is the end-point of contour L and,
ANALYTICAL REGULARIZATION METHOD FOR WAVE DIFFRACTION
In particular, kernel
where,
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for spherical cap is the following:
are orthonormal associative Legendre functions (chosen according to the
rule Let us define new known and unknown functions as follows:
Substituting expressions (11) and (12) into equation (7), one arrives to the following integral equation (m=0,±l,±2,±3,...),
with new unknown function Equation (13) is the subject of Analytical Regularization Method application. The first step of it is reducing equation (13) to dual series equations of special kind. With such a purpose, we define, at first, new function,
that allows us to receive from (13) the relation of the kind,
Let us represent
and
by means of their Fourier-Legendre series:
Substituting the formulae (16)-(18) into equation (13) and using the orthonormal property of functions one obtains the first series equation:
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where
Y. A. TUCHKIN
is matrix-operator formed by matrix elements
are infinite
vector-columns, is n-th component of vector-column The second series equation comes from formula (16) and the second equality in formula (14):
According to standard terminology, equations (19), (20) are dual series equations involving function where are orthonormal Jacoby polynomials. Dual series equations of this type were investigated in our previous papers (Vinogradov at al, 1978,1980,1981; Shestopalov at al, 1997). The relevant regularization procedure reduces equations (19), (20) to an infinite algebraic system of the second kind: where is compact operator in space The compactness of the operator
follows from sufficiently fast decreasing of coefficients
when p,
The last inequality is the result of proper change (11) of unknown function and proper separation of function from the kernel of integral operator in formula (13). NEUMANN BOUNDARY VALUE PROBLEM Let us consider the same diffraction BVP with the exception that condition (4) is changed by Neumann boundary condition:
It can be proved that in the case considered, one obtains integral-differential equation of the following new type:
instead of equation (13). Here is unknown function, function is defined by formula (10) and is some function, which is less singular in comparison with where is the following differential operator (“main” part of Legendre differential operator):
As well known,
ANALYTICAL REGULARIZATION METHOD FOR WAVE DIFFRACTION
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Using the same technique as for equation (13) and after taking relation (25) into account, one obtains dual series equation approximately of the same kind as equations (19), (20). Consequently, these new dual series equations can be regularized by means of the same technique (Vinogradov at al, 1978,1980,1981; Shestopalov at al, 1997). As a result of this regularization, Neumann BVP is reduced to an infinite linear algebraic system of the second kind: where is compact operator in space
CONCLUSION Analytical Regularization Method is generalized on the case of three dimensional diffraction BVP for bowl-shaped screen of revolution. Both Dirichlet and Neumann BVP are solved, i.e. they are reduced to corresponding infinite linear algebraic systems of the second kind in space with compact operator in space H. These systems can be used for construction of numerically efficient algorithms that gives solutions with arbitrary necessary accuracy REFERENCES S. S. Vinogradov, Yu. A. Tuchkin and V. P. Shestopalov, 1978, The effective solving of dual series equation involving associative Legandre functions, Doklady AN SSSR, , v.242, n.1, pp. 80-83. S. S. Vinogradov, Yu. A. Tuchkin, V. P. Shestopalov, 1980, Investigation of dual series equations involving Jacoby polynomials, Doklady AN SSSR, v.253, n.1. S. S. Vinogradov, Yu. A. Tuchkin, and V. P. Shestopalov. 1981, On the theory of wave scattering by unclosed spherical screens, Doklady AN SSSR, v.256, n.6, pp. 13461350. Yu. A. Tuchkin, 1985, Wave scattering by unclosed cylindrical screen of arbitrary profile with Dirichlet boundary condition, Soviet Physics Doklady, 30. Yu. A. Tuchkin, 1987, Wave scattering by unclosed cylindrical screen of arbitrary profile with Neumann boundary condition, Soviet Physics Doklady, 32. Yu. A. Tuchkin, 1997, Regularization of one class of system of integral-differential equations of mathematical physics, Doclady of The Ukrainian National Academy of Sciences, ser. A, No. 10, pp.47-51 (in Russian). V. P. Shestopalov, Yu. A. Tuchkin, A. Ye. Poyedinchuk and Yu. K. Sirenko, 1997, Novel methods for solving direct and inverse problems of diffraction theory, vol. 1: Analytical regularization of electromagnetic boundary value problems, Kharkov: Osnova, (in Russian)
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TRANSIENT EXCITATION OF A LAYERED DIELECTRIC MEDIUM BY A PULSED ELECTRIC DIPOLE: SPECTRAL CONSTITUENTS
Anton G. Tijhuis1 and Amelia Rubio Bretones2 1
Faculty of Electrical Engineering Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, the Netherlands 2 Departamento de Electromagnetismo, Facultad de Ciencias Universidad de Granada 18071 Granada, Spain
INTRODUCTION Spectral methods are the obvious choice for modeling the transient excitation of a continuously layered, plane-stratified dielectric half space. To arrive at an efficient numerical implementation, we consider the evaluation of the spectral constituents, which are governed by two sets of coupled transmission-line equations for the E and H modes. These equations are solved for a fixed space discretization, where the number of subintervals across a slab embedded in between two homogeneous half spaces does not depend on the value of the spectral parameters. The relative error in the constituents increases with increasing frequency and spectral wave number, but this does not affect the accuracy of the obtained space-time results. This general idea was first proposed by Tijhuis et al. (1989), and has since then been applied successfully to a large number of scattering configurations. In this paper, three different schemes are presented to compute the spectral fields. Applying central differences to the transmission-line equations on a staggered grid and eliminating one of the discretized unknowns results in a three-point rule in the interior of the dielectric slab. Boundary conditions at the interfaces are approximated by single-sided three-point rules. Second, we use reciprocity to derive two coupled integral equations and a repeated trapezoidal rule to discretize them. Third, we derive a second-order differential equation for a single unknown. From this equation, we derive a contrast-source integral equation, which is discretized by a combination of central differences and a repeated trapezoidal rule. In all three cases, the discretized equation(s) can be solved by a fully recursive procedure with a computational effort that is proportional to the number of subintervals across the slab. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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FORMULATION OF THE PROBLEM We consider a horizontally stratified medium of thickness embedded in between two homogeneous half spaces. The location of the half spaces and the value of the corresponding constitutive parameters are specified in Figure 1. In all cases, the constitutive parameters may be complex and frequency-dependent.
Transmission Line Equations For this configuration, the spectral electric fields can be decomposed in their transverse and longitudinal parts according to
where the dependence on the spectral parameters and is assumed implicitly. Next, the transverse field components are expressed in components parallel and orthogonal to the direction
and are the spectral amplitudes for E or TM and H or TE modes, respectively. Substituting the definitions (2a) and (2b) into the longitudinal part of Maxwell’s equations directly results in the identification
where we have restricted ourselves to a source-free region. The spectral amplitudes satisfy two coupled transmission-line equations of the form
where when
and where The spectral impedance are defined as and
with with and the corresponding admittance
TRANSIENT EXCITATION: SPECTRAL CONSTITUENTS
161
Exterior Half Spaces and Boundary Conditions In view of the linearity of the problem, we restrict the analysis to the excitation by a properly normalized plane wave incident from In we only have a wave that travels and/or is attenuated in the positive Dropping the superscript and taking as the fundamental unknown, we can express the behavior in the exterior half-spaces as
where depends on the current density in shown in Figure 1. For we have from (4a)
and where
as
From (5) and (6) we can derive the following boundary conditions for which are consistent with the interpretation of Y as a spectral admittance. Since is continuous in (7) can also be used to derive boundary conditions for
The aim of the computation is to solve the system of differential equations (4a,b) subject to the boundary conditions (7) or (8a,b).
METHODS OF SOLUTION For a general stratification, the system of first-order differential equations (4a,b) must be solved numerically. Perhaps the most accurate way to achieve this is to use a Runge-Kutta type scheme with an adaptive step (see e.g. Tijhuis (1987), Section 2.4.2). This leads to a fixed relative error in the obtained result at the cost of an increasing computational effort for increasing values of the spectral parameters and Alternatively, we can use the decrease in the spectrum of the generating current or incident field to compensate an increasing relative error. This means that we can choose a fixed space discretization, independently of and and still arrive at a reliable space-time result. In this category, perhaps the most straightforward choice is to approximate the continuously layered medium by a stack of M homogeneous layers of thickness and use a transfer or scattering matrix formalism (Lekner and Dorf (1987), Ko and Sambles (1988)). This procedure, however, has the disadvantage that the accuracy of the solution is at best of i.e., the results are at best first-order accurate. In this section, we review three methods that are second-order accurate, while the computation time remains of which is the best we can hope to achieve for computing field values.
Finite Difference Spectral Domain In the first approach, we use finite differences to discretize the differential equations
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(4a,b). Considering again as the fundamental unknown, we sample V at with and I at with and we approximate derivatives by central differences. Eliminating the sampled current values then results in the three-term recurrence relation
which is valid for and where a subscript refers to sampling at The difference equations for M follow from the boundary conditions (8a,b). For the evaluation of the derivative, we use a three-term forward difference rule at and a three-term backward difference rule at This results in:
To solve this system of equations, we first remove the element with from the last equation, with the aid of (9) with We then obtain the matrix equation
where for and where A is the right-hand side of (10a). In (11), is an element of the diagonal, and and are elements of the first sub- and superdiagonals. denotes element of row 0 after row operations. These operations use rows 1, 2 , . . . , M to successively remove the leftmost nonvanishing elements of row 0. Since the right-hand sides of (9) and (10b) are zero, the right-hand side of (11) is not affected. The necessary computations can be summarized in terms of a recursive procedure for these elements:
This procedure leads to a matrix equation, where the only nonvanishing elements of the system matrix are located on the diagonal and the first two superdiagionals. From this equation, the sampled values are readily determined by back substitution. That procedure can be written as
The corresponding values for may be obtained from the difference equation corresponding to (4a). At a first glance, it would appear that such an operation enhances the numerical error in the computed values However, (9) was obtained
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by eliminating the and is therefore the exact solution of the discretized firstorder differential equations. As a consequence, the error in the numerical solution of this system of equations will be comparable for both quantities. Finally, it should be observed that the procedures given by (12) and (13) both require M + 1 steps, which confirms the estimate of the computational effort given above. Two Coupled Integral Equations The FDSD approach described in the previous subsection has two disadvantages. First, the differential equations (4a,b) are formally not valid when or has a discontinuity. Second, the boundary conditions (8a,b) are only taken into account in an approximate form. Both problems can be circumvented by deriving two equivalent integral equations. To achieve this, we include source terms and in the righthand sides of (4a) and (4b). For any two solutions of the resulting inhomogeneous equations, referred to as states A and B, we have the following reciprocity relation:
To arrive at the desired integral equations, we make the following choices. State A is the actual state with sources in and material parameters as specified in Figure 1. State B is an auxiliary state with and or and Configuration B is an infinite medium with and for and satisfy the radiation condition as Finally, we integrate (14) over and interchange and For the case with and we then arrive at the integral equation
where the boundary conditions (7) have been used to eliminate I(0) and For and we find a similar equation for In the space discretization, we introduce a spatial grid with with and and we enforce both equations at the grid points. A repeated trapezoidal rule then results in a discretized equation of the form
In (16),
and
are 2 × 2 matrices, is the solution vector, accounts for the excitation in and for the propagation in the reference medium. To solve this system of equations, we perform three row operations in closed form.
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We end up with a matrix equation that closely resembles the one in (11). The difference is that the elements of row 0 and and of rows are now as such 2 × 2 matrices. The tridiagonal elements and can be expressed in terms of the matrices in the discretized integral equation (16):
The method of solution follows the same lines as the one specified in (12) and (13) and can be formulated as
and
In (18) and (19), the “multiplications” and “divisions” are matrix operations that must be carried out exactly in the order as indicated. Each step involves the inversion of a 2 × 2 matrix as well as a number of matrix-vector multiplications. Second-Order Differential Equation and Associated Integral Equation Finally, we present an integral-equation scheme that combines the efficiency of the FDSD with the capability of handling the boundary conditions (8a,b) in closed form. For both types of modes, we can choose either or as the fundamental unknown. However, the computation becomes more efficient when we choose the amplitude that gives rise to a longitudinal field component. We thus obtain a second-order differential equation of the generic form
where either and or and For this choice of the fundamental unknown, the boundary conditions (8a,b) for and the corresponding conditions for can be written as
As in the previous section, Green’s function is introduced as the solution of the secondorder differential equation
that satisfies the radiation condition as In (22), are the constitutive coefficients of an arbitrary homogeneous reference medium. Using Green’s second
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theorem over and substituting the boundary conditions (21a,b) then leads to the integral equation:
Like (15), (23) expresses the unknown amplitude in in terms of plane waves that propagate in the infinite reference medium. The space discretization proceeds along the same lines as in the case of the two coupled integral equations. We introduce a spacial grid with approximate the integrals by a repeated trapezoidal rule, express at in terms of and evaluate at interior points by central differences. This leads to a discretized equation of the form:
where Carrying out the same row operations for (24) as were carried out for (16) results again in a tridiagional system matrix for The equation thus obtained can be solved as described above, but with the difference that all multiplications and divisions are now scalar operations. Special care is needed when the constitutive parameter is discontinuous in and the evaluation of the remaining spectral amplitudes and is no longer straightforward.
A SIMPLE EXAMPLE The algorithms outlined in the previous section have been validated for various choices of and Because of space limitations, we restrict ourselves here to an incident plane wave with To provide a true test for the algorithm based on (23), we must have a configuration where Therefore, we cannot consider the customary example of a homogeneous slab. Instead, we consider a reflectionless configuration with and For the exterior medium we choose free space. The algorithm based on the two coupled integral equations (18) and (19) gives an almost vanishing for any value of M. This can be explained from the property that I, V and are treated in a completely symmetrical manner. Therefore, we consider for which corresponds to nine complete oscillations in and hence to In Table 1, we present for an increasing value of M for each of the three algorithms. It is observed that the error is indeed of for increasing M. For this particular configuration, all three methods deliver a comparable accuracy. However, this is not true in all situations. For the configuration described above, the accuracy test is more severe for the single integral equation, where and are treated in a different manner, while their effects must cancel out. Further, the advantage of having exact boundary conditions becomes more important for increasing Another
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criterion is the flexibility of the algorithms. The coupled integral equations are better in this respect, but their solution requires more computational effort.
CONCLUSIONS In this paper, we have presented three algorithms for the efficient evaluation of spectral fields in continuously inhomogeneous, plane-stratified media. These algorithms have in common that field values are obtained in steps with an accuracy of i.e., doubling the computational effort leads to a four times better result. The effectivity of the approach was demonstrated for a simple test geometry. Possible applications include the analysis of buried objects, antenna arrays, frequency-selective surfaces and feed networks. The algorithms presented in this paper have already been used to analyze the excitation of a plane-stratified medium by a pulsed dipole, one or two straight thin-wire antennas and a circular loop antenna, and the design of an airgap in a rectangular waveguide. In all cases, the spectral representation as such remains to be developed. One example is the excitation of a plane-stratified medium by a pulsed dipole, which is discussed in a companion paper.
REFERENCES Lekner, J. and Dorf, M., 1987, Matrix methods for the calculation of reflection amplitudes, J. Optical Society of America A, 4: 2092. Ko, D. and Sambles, J., 1988, Scattering matrix method for propagation in stratified media: attenuated total reflection studies of liquid crystals, J. Optical Society of America A, 5: 1863. Tijhuis, A.G., 1987, Electromagnetic Inverse Profiling: Theory and Numerical Implementation, VNU Science Press, Utrecht, the Netherlands. Tijhuis, A.G, Wiemans, R., and Kuester, E.F., 1989, A hybrid method for solving time-domain integral equations in transient scattering, J. Electromag. Waves Applic., 3:485.
TRANSIENT EXCITATION OF A LAYERED DIELECTRIC MEDIUM BY A PULSED ELECTRIC DIPOLE: SPECTRAL REPRESENTATION
Anton G. Tijhuis1 and Amelia Rubio Bretones2 1
2
Faculty of Electrical Engineering Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, the Netherlands Departamento de Electromagnetismo, Facultad de Ciencias Universidad de Granada 18071 Granada, Spain
INTRODUCTION Spectral methods are the obvious choice for modeling the transient excitation of a continuously layered, plane-stratified dielectric halfspace. Such methods typically involve an inverse spatial Fourier transformation and the evaluation of the constituents. In this paper, we consider the spectral representation. The idea is to normalize the spatial wavenumber with respect to frequency. Compared with the Cagniard-De Hoop method, our approach is different in the sense that we keep the frequency real, and allow the time variable to become complex. In this respect, our work also resembles the spectral theory of transients. We restrict the temporal Fourier inversion to nonnegative frequencies by expressing the time-domain signal as the real part of a dual analytic signal. Reversing the order of the temporal and spatial Fourier inversions then leads to the so-called time-domain Weyl representation for the reflected field. In this representation, accumulated guidedwave poles give rise to an additional branch cut. The representation thus obtained is used to derive a suitable combination of Gaussian quadrature rules for the evaluation of the spectral integral. FORMULATION OF THE PROBLEM We consider a configuration that consists of an isotropic, linearly and instantaneously reacting, horizontally stratified, lossy dielectric medium embedded between two homogeneous, lossless dielectric half-spaces and (Figure 1). The slab is loUltra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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cated in the domain The configuration is driven by a pulsed electric dipole whose current density is given by
with a unit vector, and a causal time signal. We are interested in the electromagnetic field in the upper half space since that is the field that can be detected by an antenna. Once this field is known, we can apply the superposition principle to determine the effect of a more general current distribution. For this configuration, Maxwell’s equations can be simplified to
Spectral Fields To exploit the fact that the constitutive parameters in (2a,b) depend only on the we solve these equations in the spectral domain. To this end, we introduce the following temporal and spatial Fourier transformations:
where and are vectors in the transverse plane. We decompose the electromagnetic fields into their transverse and longitudinal parts, according to
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and express both parts in terms of spectral amplitudes We arrive at
and
where The spectral amplitudes satisfy the transmission-line equations (Felsen and Marcuvitz (1994))
and a similar system of equations for and where with and when The solutions determined by and and and are indicated as E or TM and H or TE modes, respectively. The systems of first-order differential equations can be reduced to a single secondorder differential equation of the form
where only the dependence on has been indicated explicitly. In (7), is a constitutive parameter, is the unknown function and is the forcing function. In both cases, we choose the quantity that corresponds to a longitudinal flux density as the fundamental unknown. For a vertical dipole, we have For a horizontal dipole pointing in the
we find
where is the derivative of the delta function. For this excitation, the differential equation (7) must be solved twice for each combination of and
LONGITUDINAL BEHAVIOR The first step towards solving the problem formulated above is solving the differential equation (7) for the forcing functions given in (8) and (9a,b) for a set of parameters that allows the evaluation of the integrals in the inverse transformations of the ones given in (3a) and (3b). In this section, we address some aspects of this solution. We restrict ourselves to the longitudinal electric field caused by a vertical dipole, but the procedure runs along the same lines for the remaining field components.
Exterior Half Spaces In the dielectric medium is homogeneous and lossless. For the forcing functions specified above, we can therefore find a closed-form solution of the differential
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equation(7). For the vertical dipole, we have
where is an unknown reflection coefficient. Since (7) is linear, we may restrict the evaluation of the longitudinal behavior to determining the response to a unit-amplitude plane wave. The multiplicative factors in (10) can then be included in the evaluation of the spectral integrals. We therefore consider the normalized solution which, in the homogeneous half-spaces, .behaves as
where and where Obviously, the direct field for must also be adapted. However, for the homogeneous upper half-space considered in this paper, this field is available in closed form. Asymptotic Solution To obtain for we need to solve the system of differential equations (6a,b) or the second-order differential equation (7) numerically. Three possible methods are discussed in a companion paper. For large values of the parameters and/or a first-order asymptotic solution can be derived. By scaling the spatial wave vector according to
where is the speed of light in free space, we can handle both limits simultaneously. We specify the results for i.e. for E-modes. Equation (7) can be written as
where the prime denotes differentiation, where and where is the scaled counterpart of The logarithmic derivative of in (13) is expanded in a geometrical series in powers of and we derive a first-order WKB approximation. Following Erdélyi (1956), we introduce two linearly independent solutions that correspond with
where is the free-space wave impedance. In (14) the superscript ± indicates the direction of propagation, and the wave originates from the point With this result, we can formally write:
The coefficients at
and and
are found by enforcing the continuity of and For the reflection coefficient for example, we
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then obtain the following first-order approximation:
where
and
are asymptotic reflection and transmission coefficients.
Singularities in the Complex As a function of complex the spectral constituent has two types of singularities. In the first place, there are two branch cuts in the upper half of the complex with branch points at and These branch cuts are associated with the choice of the “physical” root in the attenuation coefficients and which occur in the radiation conditions in and In the transform domain, the problem is completely defined by these conditions and the second-order differential equation (7) in the interval This differential equation only contains a term therefore no extra branch cuts are introduced for the interior of In the second place, we need to consider the occurrence of so-called guided-wave poles. For such poles only occur in the interval
where is the local refractive index in provided that this interval exists. Each pole corresponds to a homogeneous solution of Maxwell’s equations that propagates in the transverse direction. The location of the poles, and their number, depends on For the poles occur in the first quadrant of the complex plane, and approach the same interval as Now, the integration contour may be considered as running just below the real Therefore, numerical problems will occur for in that interval, since neither of the equations (6) or (7) has a unique solution for values of that correspond to guided-wave poles. TRANSFORMATION TO THE SPACE-TIME DOMAIN Once has been computed, the longitudinal electric field can be calculated by carrying out the inverse Fourier transformations corresponding to (3a) and (3b). We restrict the discussion to the reflected field in but the ideas presented in this section are also directly applicable to the total fields in and In the transform domain, the longitudinal reflected-field component is given by.
where is the reflection coefficient introduced in (10). To speed up the calculations, we cast the Fourier inversion into a special form that was also used in the half-space configurations considered in Rubio Bretones and Tijhuis (1995, 1997). Weyl Representations First, we restrict the temporal Fourier inversion to nonnegative frequencies by expressing the time-domain signal as
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with Here, represents the dual analytic signal corresponding to This signal is an analytic function in the lower half of the complex and its real part approaches when The restriction to allows us to use the normalized spatial transform vector introduced in (12). Further, since only depends on we also change over to the cylindrical coordinates The spatial Fourier inversion corresponding to (3b) then assumes the form:
where are cylindrical coordinates in actual space. Combining (18), (20) and (21) and reversing the order of the temporal and spatial Fourier inversions then leads to the so-called frequency-domain Weyl representation for the reflected field:
where the complex time delay is given by
and where Because the cosine in (21) is periodical, the variable in the angular integration has been changed to and does not depend on Apart from a constant amplitude and a factor of the term in braces in (22) is of the same form as the right-hand side in (20), and the complex time argument cannot have a positive imaginary part. Therefore, this term can be identified as the time derivative of a dual analytic signal. Contour Deformation and Quadrature This observation is used to derive a suitable combination of Gaussian quadrature rules for the evaluation of the integral over To achieve this, we analyze the situation in the complex We first consider the dual analytic signal
for a fixed with the reflection coefficient
The singularities in the integrand in (24) originate from As remarked above, we have two types.
First, there are two branch cuts in the complex-wavenumber plane. After the normalization of carried out above, these singularities show up as the branch cuts in the normalized attenuation coefficients and These branch cuts are present for all frequencies and, hence, will also be observed in
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Second, there are guided-wave poles. For these poles are located on the interval specified in (17), provided that this interval exists. The integration over in (24) reduces the influence of the poles to an extra jump discontinuity along this interval. For each point in the subinterval is a condensation point for guided-wave poles at different values of Therefore, the behavior along the real will be similar to that in the lossless case. The definition of was chosen such that the integration contour may be regarded as running along the real axis in the fourth quadrant of the complex Therefore, the situation in the complex may be envisaged as indicated in the left half of Figure 2, where it is assumed that To ensure that the transmission-line equations have a unique solution, we deform the contour into the lower half of the complex as shown in the right half of Figure 2. We choose large enough to avoid possible problems in the numerical solution of the transmissionline equations and in the numerical integration along the interval between and To investigate to what extent this is allowed, we write the frequency-domain Weyl representation (22) in terms of the dual analytic signal introduced in (24). This leads to the so-called time-domain Weyl representation for the of the electric field in the case of the vertical dipole:
Since all singularities in the integrand in the right half of the complex are located on or above the real no extra contributions are encountered in this deformation. However, the definition (24) of may only be applied for To satisfy this condition, we take into account the complex time delay in (25). With the aid of the asymptotic expansions it follows that, for a suitable definition of in (1), the imaginary part of the time argument in (25) must be nonpositive for all and This implies that the limiting contour is given by where is a real-valued length parameter. Solution of this equation leads to the Cagniard contour for a point on the interface at with respect to the source point at The representation (25) allows the derivation of a composite Gaussian quadrature rule that is valid for all and and for smaller than a given maximum offset. For ex-
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ample, in the semi-infinite subinterval we substitute and use the asymptotic approximation and the large-argument behavior of analytic signals to derive that the integrand of (25) decays as independently of the values of and This allows the application of a fixed Gauss-Laguerre quadrature rule. Once the quadrature rule is derived, the integral over in (22) is replaced by the discretized form derived for (25). Next, the integral over is expressed as a linear combination of the Bessel functions with Finally, the integral over is truncated, discretized with the aid of a repeated trapezoidal rule, and cast into the form of an FFT operation.
AN EXAMPLE Numerical results were obtained for a variety of configurations. We restrict ourselves to an example for a Gaussian pulsed horizontal dipole located above an inhomogenous slab in free space. In Figure 3, we show the time signature of the of the reflected electric field for two inhomogeneous, nonmagnetic slabs with the same relative permittivity profile with and The dipole is located at a distance above either of the slabs and the observation point is (1, 0, –0.5)m. As expected, the two signals are identical until the arrival of the fields reflected at and beyond.
REFERENCES Erdélyi, A., 1956, Asymptotic Expansions, Dover Publications, New York, Chapter 4. Felsen, L.B. and Marcuvitz, N., 1994, Radiation and Scattering of Waves, Second Edition, IEEE Press, New York, Section 2.2. Rubio Bretones, A. and Tijhuis, A.G., 1995, Transient excitation of a straight thin wire segment over an interface between two dielectric half spaces, Radio Science, 30:1723. Rubio Bretones, A. and Tijhuis, A.G., 1997, Transient excitation of two coupled wires over an interface between two dielectric half spaces, Radio Science, 32:25.
A NEW ULTRA WIDEBAND, SHORT PULSE, RADAR SYSTEM FOR MINE DETECTION
F. Gallais1, V. Mallepeyre1,Y. Imbs1, B. Beillard1, J. Andrieu1, B. Jecko1 M. Le Goff2 1
Institut de Recherche en Communications Optiques et Microondes IRCOM - 7 rue Jules Vallès -19100 Brive la gaillarde (France) E-mail:
[email protected] 2 CELAR (DGA) GEOS/SDM - BP 7419 - 35174 Bruz Cedex (France) E-mail:
[email protected]
INTRODUCTION An experimentation is described for measurement of UWB transient scattered responses from different targets. The measurements are performed using a new UWB Synthetic Aperture Radar (SAR) for the detection of placed atop soil targets at first, and then, buried in soil targets. The aim is to use lower frequencies for penetrating foliage, vegetation, soil and ultra wide band for high resolution SAR image. This study comes after a precedent work which was the realization of a low frequency Radar Cross Section (RCS) measurement facility in the time domain1. This facility is funded by the DGA to meet CELAR requirement2. First results will be presented about lied on soil targets.
MEASUREMENT FACILITY The measurement configuration is an UWB (100 MHz - 1 GHz) transmission and reception system implanted on a mobile boom. This boom which can reach about ten meters high, is installed on a truck and moves sideways along the test area. The system is presented on figure 1 and figure 2. The general synchronisation is carried out by a sequencer. This device is remote by a coder wheel behind the truck which defines measurement step in azimuth along the moving direction. A receiving system between receiving antenna and the digital sampling oscilloscope, is needed to protect the oscilloscope input from high level tension and to optimise the measurement dynamic. The effective signal is insulated by time windows. The receiving Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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system gain is programmable from -20 dB to 40 dB in 1 dB step. This receiving system has got a limited frequency bandwidth. As a matter of fact, its bandwidth is 700 MHz at - 3 dB but it will be upgraded with better components. The oscilloscope must acquire measurements during the moving with the best bandwidth in monopulse mode. It's the Lecroy LC 584 which was chosen for his adequate data transfer rate (GBIP): until 1 Mpoints at 280 Ko/sec. But its bandwidth is DC - 1 GHz (risetime ~ 350 ps) with a 8 Gsample/s rate and a 8 bit dynamic. A tachometer is installed on the area test to provide the position of a near antenna fixed theodolite. This device is a Total Positioning System from Leica Geosystem. It is used in motion compensation to estimate the antenna phase centre position at each acquisition. Then, it is needed to calculate radar distance between pixel and antennas. All the data are sent to a PC recording Unit for storage and data processing is subsequently done with Unix workstation.
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Pulse Generator The generator used is a KENTECH generator based on a PBG3 which has a pulse output voltage of 8 kV, a 10 -90 % risetime better than 120 ps and a 50 % pulse duration (full-width at half-maximum) better than 460 ps. This pulse has a frequency range from DC to about 2 GHz (at -20 dB). See below on figure 3.
Antennas Two identical antennas are used. The dimensions (less than a 60 cm ridge cube) are limited by the mechanical structure of the boom. The choice of the study was to design a 2D antenna to reduce weight and volume. That is why, antennas will be more directive in the antenna arm plan. The dimensions are also 1 m by 0.6 m. Two types of antennas have been selected : "Vivaldi" and "Scissors"3 (see figure 4). Antennas are resistively loaded on the upper half-length of each arm to match impedance in low frequencies, and to limit back scattered field. Only a pair of "Scissors" antennas are currently used in measurements.
The "scissors" antennas have been designed with the space time integral equation method. The computed radiated pulse is shown on figure 5. A differential coaxial balun4 has been designed to feed correctly the antenna. The purpose is to make a transition between a coaxial cable and the two wire feed line and, to feed symmetrically the antenna.
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MEASUREMENT CONFIGURATION A theoretical study enables to estimate the influence of the site angle (from horizontal) on the maximum level scattered field by a buried target for normal and parallel polarisation. The scattered electric field has been calculated with a 2D FDTD method and a plane wave excitation. Dry, fairly wet and wet soil are considered with Debye model5. The normal incidence is the most favourable in free space but with a soil, the specular soil echo is very high and the target response is mixed with this soil echo. The maximum field level globally decreases if the site angle also decreases at a constant radar distance configuration (15 m). This phenomena is less important in case of wet soil for the parallel polarisation. Then, it is better to work with an site angle greater than 30°. Thanks to facilities of this Radar, the first measurements had been done at 8 meter high with targets lied on soil at 6 m site angle ~50°), 10m and 17 m from the truck. Antennas are oriented in VV polarisation. Another study is needed to know the radar displacement L length to collect the most of scattered information from the target for SAR data processing. This distance L depends on the position of the target, the nature of target and the antenna radiation. But, in worse case, the radar and the target may have omnidirectionnal characteristics. The punctual target M is supposed to diffract a pulse as a Dirac pick in order to estimate the received scattered field level v(y) at each position y (see figure 6).
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The radar distance between target and radar is
with d0 the
shortest distance between radar and target If the attenuation distance is only taken into consideration, the electric field level can be expressed by: the maximum level is measured when the radar is in front of the target and it is normalised, so v(ym) = 1 . The SAR data processing used is based on coherently summing pulse responses Sr measured along the area test4. For a pixel I(xi,yi), the algorithm identifies the sample Sr(d) corresponding to the radar distance (radar to pixel) for each measurement Sr(p) and sum them.
For a punctual target M(xm,ym,zm), an analytic approach is presented for calculating the pixel level I(xm,ym) corresponding to the target M : with y1 < y2 y1 start position and y2 end position of radar With this variable change,
the M pixel image of the M point is :
If the radar moves symmetrically along the target on the length L (see figure 6.), then the pixel level is:
In case of the radar displacement is very longer than the shortest radar distance d0, the level pixel should reach : The L distance to get K percent of the maximum pixel level is :
This result shows that if the radar covers twice the shortest distance (d0) between radar and target, the calculated pixel level is half of the maximum level In configuration measurements described before, furthest targets are at the distance (d0) of 19 m. The radar moves on the length L of 90 m, so 75 % of information is collected for the furthest target (see figure 7).
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The radar resolution can be estimated with classical SAR expressions. The distance resolution depends on width impulse and incidence angle.
In the measurement configuration, the distance resolution is about 20 cm ( ns et ). Once proceeded, theoretical signals have been computed to estimate the azimuth resolution The coherently summing treatment allows a good azimuth resolution which is about 20 cm and depends on width impulse.
EXPERIMENTAL RESULTS First measurements have been realised outdoor on January 2000. The test area was grass and the humidity soil rate was high about 30 %. More, roughness and homogeneity were not controlled because of some hole, mole-hill and little water puddle on soil surface. So, measurement conditions were realistic but rather unfavourable for target detection. Several target types have been measured but two types are presented. Nine trihedrons were lied on soil at three distances (x = 6m, 10 m and 17m) with three different length ridges (22 cm, 30 cm, 49 cm). And, three metallic mines which are 11 cm high and a 27 cm diameter large were lied on mown grass. There are also false targets like mole-hill around mines. The scene is presented on figure 7. Measurements show a high low frequency signal (less than 250 MHz). This is repeatable on each position measurements. It means that this signal may be the coupling signal between antennas and soil clutter. This undesirable information will be attenuated by filtering signal with a high pass numerical filter and by subtracting background. After, the coherently summing is proceeded with the correction of the distance attenuation. The trihedron images are accurate and shows the correct position of targets due to motion compensation system (see figure 8). A zone between 5 m and 12 m (along distance x) seems to be more perturbed by soil clutter. The effective signal has been insulated by time window. No signal has been acquired before the distance on soil of x = 6 m. So, noise does not
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appear highly in nearby zone image (x < 5 m). For further zone (x > 12 m), clutter is less important if the angle site is lower than 40°.
The metallic mine image is encouraging. Lied on mines (y = -44m) are detected as well as mole-hills which can be considered like half conductive sphere lied on the soil because of the very high humidity rate (see figure 9). With theses smaller targets, the azimuth and distance can be estimated about 20 cm from mine image on figure 9. Signature on image presents side lobes along the distance axe. It can be explained by the temporal wave form target response and by the high pass filtering operation. But theses side lobes can be a criterion to distinguish small target from noise on image. Theses first results show that contrast environment and permittivity discontinuities are detected.
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CONCLUSION This paper presents the first results obtained in rather unfavourable measurement condition with some metallic targets as trihedron and mine. Lied on soil targets are detected and good localised although roughness soil clearly appears on image. Theses results are encouraging and the next step of works can be decomposed on three points. Data processing is based on coherently summing of measurements and it will be improved with radio frequency interference rejection, specific treatments for rejecting soil clutter and coupling, calibration with a canonical target. Sub banding image will be tested to reduce soil clutter and to discriminate target. Then, some special measurement configurations will allow to analyse and reduce high coupling signal between antennas, truck and soil. The resolution will be estimated. The bandwidth of receiving system will be upgraded. And more exhaustive measurements are planed with more favourable conditions, different type of soil and real mines. Targets will be buried. Results will be analysed and compared to FDTD simulations.
REFERENCES 1. Chevalier Y., Imbs Y., Beillard B., Andrieu J., Jouvet M., Jecko B., Le Goff ML., Legros E., "UWB measurements of canonical targets and RCS determination", Ultra Wide Band Short Pulse Electromagnetics, vol 4. 2. Le Goff M., Pouligen P., Chevalier Y., Imbs Y., Beillard B., Andrieu J., Jecko B., Bouillon G., Juhel B., "UWB short pulse sensor for target electromagnetic backscattering characterization", Ultra Wide Band Short Pulse Electromagnetics, vol 4. 3. V. Mallepeyre, F. Gallais, Y. Imbs, B. Beillard, J. Andrieu, B. Jecko, M. Le Goff, "A new broadband 2D antenna for UWB applications", Ultra Wide Band Short Pulse Electromagnetics, vol 5. In press. 4. The baluns were made by the EUROPULSE company (Cressensac Lot, France). 5. P. leveque, A. Reinex, B. Jecko, "Modeling of dielectric losses in microstrip patch antennas : application of FDTD method", Electronics Letters, vol. 28, n°6, mars 1992, pp 539-541
ULTRA-WIDEBAND GROUND PENETRATING IMPULSE RADAR
Alexander G. Yarovoy, Piet van Genderen and Leo P. Ligthart International Research Centre for Telecommunications-Transmission and Radar, Faculty of Information Technology and Systems, Delft University of Technology, 2628 CD Delft, The Netherlands
INTRODUCTION It is widely believed that ground penetrating radar (GPR) should be a key component of any system designed for humanitarian demining. GPR-part of such sytem should satisfy two crucial demands: high probability of object detection and low false alarm rate. While detectability of the GPR can be improved by means of improving the resolution and the dynamic range, decrease of the false alarm rate can be achieved only via localization, classification and identification of detected targets. Solution of the latter problem requires accurate measurements of the electromagnetic field scattered from the subsurface. This qualitatively new demand makes the principal difference between usual GPR and GPR for landmine detection: the first one should just detect the field scattered from a buried target (i.e. distinguish this field from all other electromagnetic fields) while the second one should measure accurately the scattered field (i.e. determine magnitude of the field as a function of time). Different inverse scattering methods can be used later to determine localization, size, shape and even spatial distribution of dielectric permittivity within the buried target from the measured values of the scattered field. Taking into account this principal demand of high accuracy and using its long experience of near-field antenna measurements the International Research Centre for Telecommunications-transmission and Radar (IRCTR) in the Delft University of Technology has developed two GPR systems dedicated to landmine detection: a video impulse system and a stepped-frequency continuous wave system. In this paper the main guidelines of the video impulse system design are presented. The stepped-frequency system is described elsewhere. The impulse GPR system developed in IRCTR for landmine detection comprises a pulse generator, an antenna system, a receiver (which consists of a signal conditioner and a sampling converter) and a processing software. All these items will be briefly described below. At the beginning of each section we first formulate demands to each item from the Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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system point of view. After that we describe what has been developed in order to satisfy these demands.
GENERATOR For the impulse radar choice of waveform and duration of the pulse is critical. We have chosen a monocycle waveform. The advantage of a monocycle in comparison with a monopulse is that the frequency spectrum of the first one decreases to zero at low frequencies, which cannot be efficiently transmitted via the antenna system, while the frequency spectrum of the second one has a global maximum there. As a result, the magnitude of the field radiated by an antenna system fed by a monocycle is considerably larger than the magnitude of the field radiated by the antenna system fed by a monopulse with the same magnitude. Besides, the energy, which is not radiated from the transmit antenna, reflects back into the feeding line frequently causing additional ringing of the transmit antenna. As far as the pulse duration is concerned we have chosen 0.8ns (a period of approximating ideal monocycle). The frequency spectrum of such pulse covers interval from 210MHz till 2100MHz on 10dB level. At frequencies below 1GHz, attenuation losses in the ground are small (Daniels, 1996) and considerable penetration depth can be achieved. However, landmine detection requires down-range resolution (in the ground) of the order of several centimeters, which can be achieved using frequencies above 1GHz. It was found experimentally that the 0.8ns monocycle satisfies penetration and resolution requirements. The spectrum of this pulse has a maximum at frequencies where the attenuation losses in the ground start to increase. So the spectral content of the monocycle below this maximum penetrates deep into the ground and the spectral content above this maximum provides sufficient down-range resolution.
The pulse generator for the radar has been delivered by SATIS Co. The unique feature of this generator is its small trailing oscillations, which are below 2.4% of the maximum amplitude during the first 2ns and below 0.5% afterwards (Fig. 1a). The generator spectrum covers a wide frequency band from 420MHz till 1.67GHz on 3dB level (Fig. 1b).
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ANTENNA SYSTEM The antenna system is one of the most critical parts of every GPR, because the performance of the whole radar depends strongly on it. The antenna system should satisfy a number of (sometimes contradictory) demands. We list them separately for transmit and receive antennas. In order to achieve sufficient signal-to-clutter ratio and to use same signal processing for detection of surface laid and buried objects the transmit antenna should: 1. radiate short ultra-wideband (UWB) pulse with small ringing; 2. produce an optimal footprint on the ground surface and below it (size of the footprint should be large enough for SAR processing but at the same time it should be small enough to reduce surface clutter and in order to filter out undesirable backscattering from surrounding objects); 3. keep constant the waveform of the radiated field on the surface and in different types of the ground (i.e. with different dielectric permittivity). In order to allow successful SAR processing for the given frequency band the scattered field should be measured with a cross-range step 3cm maximum. The measurement plane should be sufficiently elevated above the ground surface in order to avoid influence of evanescent fields. Together with operational demands for landmine detection it means elevation of the receive antenna at least 10cm above the ground. Thus the receive antenna should: 1. receive the field in a local point (effective aperture should not be sufficiently smaller than 9cm2); 2. provide sufficient sensitivity in order to receive very weak scattered fields; 3. be elevated at least 10cm above the ground surface. Additionally a possibility of isolation of the direct air wave from the ground reflection by the time windowing and a possibility to measure simultaneously backscattered field in two orthogonal polarizations are desirable. To satisfy demands NN.2-3 for the transmit antenna it was decided to implement the far-field approach, meaning that the transmit antenna is elevated sufficiently high above the ground. The demands NN.1-2 can be satisfied if a good transient antenna with reasonably high directivity is used. Such antennas are not commercially available and design of such an antenna is extremely difficult. In close collaboration with SATIS Co. (Russia) a dielectric filled TEM horn (DTEM) has been designed (Yarovoy, Schukin and Ligthart, 2000), which is ultrawideband, has linear phase characteristics over the whole operating frequency band, has constant polarization and possesses short ringing. This antenna is based on a dielectric wedge. Such design reduces the sensibility of the antenna for external EMI and reduces the antenna’s physical dimensions. The shape of the metal flare has been optimized so that the characteristic impedance in each cross-section of the antenna gradually changes from 50Ohm (impedance of the feeding line) near the feed point to (impedance of free space) near the aperture. More specifically we have tried to minimize reflection from all antenna cross-sections, so that only reflection from the aperture can take place. The latter will not cause late time ringing if the antenna is perfectly match to the feeding line and there are no other centers of reflection within the antenna. The waveform of the electric field radiated from this antenna fed by the 0.8ns monocycle generator is presented in Fig. 2a. The footprint of the antenna measured in the plain 54cm from the aperture has an elliptic shape with halfaxes 21cm and 27cm at 3dB level. The waveform of the radiated field remains the same within the whole footprint (on 20dB level). For the receive antenna a small loop antenna has been chosen. This antenna has an aperture of the same order as a linear dipole (so the demand N.1 for receive antenna is satisfied), but unlike the dipole the loop possesses a very small ringing. As the loop is
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transparent for the incident wave, the loop has been placed just below the transmit antenna on its main axis (to satisfy demand N.2). By choosing proper elevation of the loop antenna above the ground we easily satisfy the demand N.3. As a result we have arrived at a new antenna system, which has a number of advantages over usual GPR antenna systems with two (or several) identical antennas elevated to the same height above the ground.
The developed antenna system has an ultrawide frequency band. The spectrum of the signal passed through the antenna system (perfectly conducting flat ground calibration) is presented in Fig. 2b. The developed antenna system has been patented (de Jongh et. al (1999)). The disadvantage of the developed antenna system is a high magnitude signal due to the direct wave from Tx to Rx antenna. The magnitude of this signal is considerably larger than the reflection from the ground, so it determines the upper level of the system dynamic range. RECEIVER According to our simulations and practical experience GPR receiver should satisfy following demands: 1. its bandwidth should be larger than the bandwidth of the received pulse measured at -40dB level; 2. within this bandwidth the receiver should have linear phase characteristic; 3. the linear dynamic range of the receiver should be larger than 60dB to detect both AT and AP mines in typical ground conditions; 4. the sampling time should be considerably smaller than that defined by Nyquist criterion with respect to the highest frequency in the received pulse spectrum (our experience shows that for 0.8ns pulse the optimal sampling time lies between 15ps and 20ps); 5. finally, to perform SAR-like data processing the receiver parameters should be extremely stable (for 0.8ns pulse the time drift should not exceed 5ps within the whole measurement of the C-scan).
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In order to satisfy these demands we developed the receiver, which consists of the signal conditioner and stroboscopic sampling recorder. The signal conditioner should improve signal to noise ratio and should allow to use the whole dynamic range of the ADC. Besides, the received signal (except of its part caused by the direct wave) should be processed linearly otherwise the surface laid target response cannot be distinguished from the surface reflection. These demands can be satisfied if the signal conditioner will clip the high peak due to the direct wave from Tx to Rx antenna and will amplify the ground reflection signal up to the maximal level linearly acquired by the ADC. Thus the developed signal conditioner combines LNAs and a limiter (with very short recovery time) for voltage clipping. Our approach differs from the conventional one in which a variable gain amplifier is used. The main drawback of the conventional approach is that a variable gain amplifier changes the waveform and the spectrum of the received signal. Such changes can be acceptable if the final aim of the radar is target detection, but the task of target identification is not compatible with any nonlinear signal processing. In our approach the signal conditioner behaves linearly from the moment of arrival of the ground reflection. The four channel sampling converter from GeoZondas Ltd. (Lithuania) with a sampling rate 100kHz (by one channel operation) allows to measure transient signals with an accuracy of about 1% in the bandwidth from 100MHz up to 6GHz. Maximal error in time scale linearity is around 1%. The precision of the sampling converter is sufficiently high to do accurate measurements of the scattered transient field. Using averaging the linear dynamic range as high as 80dB can be achieved.
DATA PREPROCESSING Both the direct air wave from the transmit to the receive antenna and the surface clutter create a background which often masks the response of the target. In order to remove this background pre-processing of data has been used. This pre-processing includes subtraction of the system response due to the direct air wave, averaging within the footprint of the transmit antenna and subtraction of the averaged ground reflection from each A-scan. In order to limit the magnitude of artifacts due to subtraction of time domain signals, before the subtraction the time drift is numerically compensated within each A-scan. As a reference signal for the time drift compensation the direct air wave has been used. Despite of fluctuations in the arrival time of the direct air wave due to mechanical vibration of the system and other factors, the quality of compensation was found to be good.
EXPERIMENTAL RESULTS The GPR system has been tested in different environments, e.g. sand, clay, forest ground, etc. (de Jong, Lensen and Janssen, 1999). Examples of the B-scans over flash buried antipersonnel mines and a deeply buried antitank mine are presented in Fig.3 - 4. In Fig. 4 deformation of the ground surface above the mine is clearly visible.
CONCLUSION On the basis of numerical simulations of different GPR scenarios and field experience the technical requirements for the impulse GPR system specialized on detection and identification of small and shallow buried objects have been formulated. These
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requirements have been partly realized in the video impulse ground penetrating radar system developed in IRCTR. Experimental trials confirmed that the system can detect small dielectric and metal targets at a depth up to 50cm. Images build from the acquired data allow to determine position, size, shape and (sometimes) internal structure of the buried objects (Groenenboom, Yarovoy(2000)). In the next step of the program we shall develop polarimetric antenna array system. Using a multi-channel sampling scope we shall simultaneously acquire data in several different positions and for several polarizations. New software for image processing, localization and identification of targets will be also developed and implemented into the system.
Acknowledgements This research is supported by the Technology Foundation STW, applied science division of NWO and the technology program of the Ministry of Economic Affairs of the Netherlands. The authors wish to acknowledge contributions to the system development by G.Hermans, J. van Heijenoort, S.v.d.Laan, I.L.Morrow and B.Sai (all IRCTR), N.Budko
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and J.Groenenboom (TU Delft), I.Kaploun and A.Schukin (SATIS Co., Russia) and B.Levitas and A.Minin (GeoZondas Ltd., Lithuania).
REFERENCES Daniels, D.J., 1996, Surface-Penetrating Radar, IEE, London. Groenenboom J. and Yarovoy A.G., 2000, Data processing for a landmine detection dedicated GPR, in: Proceedings on Ground Penetrating Radar Conference, Gold Coast, Australia, 23-26 May 2000. Jong, W. de, Lensen H.A. and Janssen Y.H., 1999, Sophisticated test facility to detect land mines, Detection and Remediation Technologies for Mines and Minelike Targets IV, SPIE Proc., 3710:1409. Jongh, R.V. de, Yarovoy A.G., Schukin A.D., Ligthart L.P., Morow I.L., 1999, Penetrating air/medium interface microwave radar, Filing number 1013661-NL. Yarovoy, A.G., Schukin A.D. and Ligthart L.P., 2000, Development of dielectric filled TEM-horn, in: Proceedings on Millennium Conference on Antennas & Propagation, Davos, Switzerland, 9-14 April 2000.
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OBJECT SHAPE RECONSTRUCTION AT SMALL BASE ULTRAWIDEBAND RADAR
Vladimir I. Koshelev, Sergey E. Shipilov, and Vladimir P. Yakubov Institute of High Current Electronics RAS, 4, Akademichesky Ave., 634055 Tomsk, Russia INTRODUCTION Recently, owing to the considerable progress in the field of generation and radiation of short ultrawideband (UWB) electromagnetic pulses1,2 the interest to the analysis of the ultrawideband radar potential possibilities including the radar object (RO) recognition has increased. The solution of the object recognition problem is frequently related directly to the possibility to obtain the data concerning its shape. In case of a small-base radar system the application of the well-known methods of RO shape reconstruction becomes problematic. The investigations in this direction previously carried out by the authors3 with the use of a Lewis-Boyarsky transformation allowed to obtain a 10° angular surveillance base being the utmost minimum value for the given approach at a four-fold frequency overlapping of a sounding UWB pulse. The noise level should not exceed 1%. In the paper presented here, the way of the problem solution of the RO shape reconstruction in the time-domain approach at the essential limitation to the angular surveillance base and the utmost minimum number of observation aspect angles is considered taking into account measurement noise. A small angular base doesn’t allow to obtain a spatial RO resolution directly. Application of the short UWB pulses for sounding gives a high temporal resolution of reflected signals. The solution of a shape reconstruction problem is related to a possibility to re-calculate the temporal resolution in the object spatial resolution and makes possible to minimize the angular base. The realization of the given approach for the RO shape reconstruction is based on the use of the so-called genetic functions (GF) as the temporal images of its different fragments. An idea to use the GF for the recognition of the 2D images was suggested previously4. In the given paper, the potential possibilities of a RO shape reconstruction algorithm using the GF are presented. The change of the electromagnetic radiation pulse waveform reflected by the sounded object contains the information concerning the shape of the object and its other parameters. A number of functions approximating the signals reflected by the RO in the given observation aspect angle will be understood as an object set of the GF.
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where S(t) is the signal in a receiver, is the set of the GF for the given aspect angle having the amplitude coefficients and the position at the temporal axis Each GF presents a signal scattered by a definite geometric object. On the basis of the available a priori information concerning the RO a supposition about the composition of GF being a part of the set (1) and about their form is made. An important stage of the suggested reconstruction method is the GF composition determination for description of the signal reflected by the RO. For this operation the distance between the receivers necessary both for the determination of the coordinates of the object fragments with the corresponding GF and for the subsequent RO shape reconstruction is not required. Thus, even application of a monostatic radar scheme allows to evaluate the GF composition and to use this information in the problem of the RO recognition. SHAPE RECONSTRUCTION BY THE SET OF GENETIC FUNCTIONS The problem of the RO shape reconstruction is divided into two parts: 1) the GF composition determination for the approximation of the signals scattered by the RO in the given observation aspect angle; 2) carrying out calculations of coordinates of the object fragments corresponding to the obtained GF set. The first part of the problem comes to the solution of the matrix equation of the type
relative to the unknown values and Here, is the signal reflected by the RO and received by the main receiver of a receiving-transmitting system (Figure 1). A linear part of is found by the solution of the system of linear
algebraic equations
Here, is the pseudoreciprocal matrix from A with the elements obtained with the use of a singular decomposition method. The unknown comes into the equation (2) nonlinearly. In order to find them, the iteration method of successive
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approximation is used where the values of following step of
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obtained a step earlier are used for the
The iteration process is ended when the value becomes less than the given calculation error. At such approach an essential influence on the solution convergence is made by the choice of the initial approximation. The RO parts in which the surface curvature radius essentially differs from the neighboring parts will correspond to the local S(t) maxima. When the RO aspect angle is known and a priori information about the object type is available, the identification of the signal maxima position with the initial approximation of the GF is possible. A set of the GF presents a databank of the previously calculated signals scattered by the RO fragments as well as their variations by the dimensions and shape at a given waveform of the sounding pulse. The GF corresponding to the RO fragments are calculated in the far-field zone. Such presentation makes them independent on the distance to the object. A complete databank is composed of the GF obtained at all the aspect angles of the surveillance. The delays of the signals corresponding to the same GF measured by means of the receivers that are apart from each other can be used to determine the coordinates R of the object corresponding to it. As the initial point of counting it is convenient to use the GF delays in the main receiver which is situated at the beginning of coordinates in Figure 1. The calculation of a relative delay where j is the receiver number is made according to the following equation: Here, corresponds to the main receiver delay. When the receivers have the coordinates (b,0,0), (0,b,0), (0,0,b) relative to the main receiver, the equation for the components of the vector R of the n-th fragment with a corresponding GF is the following
Here, is the distance to the n-th fragment of the sounded object determined by the delay between the pulse radiated by the transmitter and the n-th local maximum of a received signal in the main receiver; c is the velocity of light. The object uniting all the fragments corresponding to the found GF with the coordinates calculated by means of the equation (5) is a RO with the reconstructed shape. COMPUTER SIMULATION To confirm a capacity for work of the suggested algorithm, a computer simulation of the shape reconstruction of the RO in the shape of a stylized 3D model of an airplane (Figure 2) with a linear dimension L at sounding by the UWB pulses of different length has been carried out using a receiving-transmitting system consisting of an ultrawideband pulse transmitter and four space-distributed receivers (Figure 1). To make the calculation more convenient, the transmitter and the main receiver were combined and disposed at the beginning of the coordinates. In the course of the computer simulation the sounding bipolar UWB pulses at the radar output had the length and 4 ns. A situation was considered when a single pulse reflected by the object which had the coordinates X=500 m,
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Y = 1000 m, Z = 1000 m. The distance to the object center was R = 1500 m. The distance between the receivers was given as b = 50 m that corresponded to 2° of the angular surveillance base. A direct problem of calculation of the field reflected by the object with an arbitrary shape was solved by a Kirchhoff method for the nonstationary diffraction problems in the approximation of a single scattering. In the process of calculation the influence of the object self-shadowing was taken into account and the influence of the intershadowing of its separate parts was neglected. Owing to the complex shape of the radar object, the integration by the surface was substituted by the summation by the elementary areas. The dimensions of the areas were chosen much less than the sounding pulse spatial length. Using this approach, all genetic functions for the given object aspect angle composing the databank for each pulse were calculated and used later for the RO shape reconstruction. All the set of the GF was divided into five classes corresponding to the RO main fragments. Each class contained three GF corresponding to the shape variations of the given fragment. One of variations completely corresponded to the fragment by the shape composing the sounded object. The problem of the object shape reconstruction was solved by the reflected signals (Figure 3) calculated for four receivers. To simulate the real
measurements, the uniformly distributed noise with a zero mean value and the given dispersion value was added. The dispersion value was determined relative to the signal maximum in the receiving system. By the signal in the main receiver the positions of its first five (by number of the used classes) local maxima used as the initial approximations of (n = 1, 2, ...5) for solution of the equation (2) were determined. Using the obtained approximations, the amplitude coefficients for the GF were determined from the equation (3). A filtration operation was carried out to minimize the GF set when one GF with a maximum amplitude was chosen from each class. An iteration problem (4) for a more precise definition of the obtained approximation was solved for the resulting set. Then the obtained solution was tested for stability. The delays of calculated in that way were used as the initial approximations to calculate the delays in the rest of the receivers. The same set of the GF obtained as a result of the filtration operation and stability test but with the aspect angle corresponding to the given receiver was used at their calculation. The next stage was the transformation of the obtained delays of the genetic functions into the object fragment coordinates corresponding to the found GF according to the equation (5). The following a priori information was used when reconstructing the shape with the use of the given algorithm: the object type and its aspect angle relative to the receivingtransmitting system were supposed to be known. This allowed to form the GF set for the approximation of the signal reflected by the object and to determine the belonging of the maxima in the reflected signal to one or another GF class. If the information concerning the
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RO aspect angle is absent, its determination is possible through the analysis of the object motion during a definite period of time. In the course of the computer simulation the dependence of the measurement noise influence on the accuracy of the object shape reconstruction was investigated at the different lengths of the sounding pulses. The reconstruction accuracy was determined as
where is the shape projection of the given object to the OXY plane (Figure 2), is the shape projection of the reconstructed object to the same plane. The shape reconstruction accuracy of a model object having the length L = 4.5 m decreased with the increase of the noise level and the pulse length. For the pulse length of 1, 2 and 4 ns and noise level the object reconstruction accuracy was 65%, 55% and 45%, respectively. The accuracy was calculated under condition that the solution of the equation (2) is convergent. The solution convergence denotes that at the given initial approximations the GF composition will be determined at which the summary contribution of the fragments corresponding to it into the reconstructed object shape gives no less than 70% of the whole projection of the given object shape to the plane OXY. In the course of the numerical experiment it was determined by 1000 realizations that the solution comes to the real if the evaluation comes into the interval where when bipolar pulses are used. For such condition is fulfilled approximately in 90% of cases at the noise level not exceeding a 20%. The databank of the GF for the different object fragments should be formed for a whole set of angular directions. The choice of the angular step between the nearest aspect angles with which the databank is formed is directly related to the object dimension. Besides, discretization by the angle results in the error at the determination of a real object aspect angle. Maximum value of such error equals to the half of the chosen angular step. Maximum change in the two neighboring aspect angles of a relative delay between two object points having a dimension L in the far-field zone is determined from a simple geometric consideration by is the angle between the two nearest aspect angles (in radians). At the same time, the error in determination of the coordinates of the object fragments is related to the error in the relative delay determination as The error in the object aspect angle determination doesn’t depend on the distance to the object. The summary error in the determination of the coordinates of the n-th object fragment will be formed from the error in the object aspect angle determination as well as the error of related to the availability of the measurement noise not allowing to determine a relative delay exactly:
Here, is the error of determination of the n-th GF relative delay. The value of is directly proportional to the sounding pulse length and rises with the increase of the noise level and discretization step of the received signal. The equation (6) can be used for the evaluation of the error upper boundary of determination of the object fragment coordinates and hence the accuracy lower boundary of the object shape reconstruction. The numerical experiments at the model presented in Figure 2 have shown that for a complex object the equation (6) is modified as
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Here, is the error averaged by all the fragments; is the coefficient depending on the object shape and aspect angle; is the coefficient dependent on the level of the measurement noise level and discretization step d. Figure 4 presents the dependence of the reconstruction accuracy on the ratio of the object dimension to the mean error of determination of the fragment coordinates for the object used in the calculations. Hence, if
a limitation to the error of the fragment coordinate reconstruction is set, then a maximum step by the angle when forming the GF databank can be determined under condition that the characteristic dimension (as a rule, a priori) of the object is known. E.g., at b=50 m, L=4.5 m, R=1500 m, d=1/20 , the coefficients and were calculated for the given aspect angle by the results of a numerical simulation. In the calculations a step by the angle was At such error of evaluation of the fragment coordinates the accuracy of the object reconstruction was This evaluation of the accuracy is true at where is the dimension of the least RO fragment described by means of the GF. In the course of a numerical experiment, the dependence of the object shape reconstruction accuracy on the ratio of the object dimension to the sounding pulse spatial length at the absence of the aspect angle determination error was obtained (Figure 5). Curves 1 and 2 are calculated at for the noise levels and respectively; curves 3 and 4 are calculated at for the same noise levels. The obtained results show that the reconstruction accuracy has a strongly expressed dependence on that is confirmed by a good match of the curves 1 and 3, 2 and 4. The reconstruction accuracy weakly changes with the rise of ratio of the object dimension to the sounding pulse spatial length at This approach for the object shape reconstruction requires knowing the object aspect angle. By means of the equation (7) and results presented in Figures 4, 5 the dependence of the angular step value on the ratio at a constant accuracy of the object reconstruction can be evaluated qualitatively. This dependence has a form similar to the dependence presented in Figure 4. The investigation of the dependence of on for the fixed values and have shown that at the angular step was
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The decrease of by a factor of two results in the decrease of by an order of magnitude and increase by an order of magnitude gives the increase of by a factor of two. The numerical experiment results confirmed the efficiency of the RO shape reconstruction method based on the GF use. Their application gave a possibility to study the 3D object shape reconstruction. In comparison with the approach previously used by the authors3 and based on the Lewis-Boyarsky transformation, the suggested algorithm allows to minimize the angular surveillance base no less than by a factor of 5 and to decrease it to For the higher noise levels the suggested approach gives rise in the reconstruction accuracy: the accuracy of the given approach for was at and and for the previous method at and At the object dimension of 50 m at using a 1-ns sounding pulse and a 10% noise level, a 1° step by the angle between the two nearest aspect angles in the databank, and a 50-m dimension of the receiving system, the distance to the object at which the reconstruction accuracy exceeds 60% is 30 km. At the distance rise up to 100 km, the reconstruction accuracy decreases to 17% that is insufficient for the object recognition on the basis of its reconstructed shape. In this case, the suggested approach is applicable for the determination of the composition of the GF describing the signal reflected by the object. The information about the GF composition can be used in the tasks of the RO recognition. The decrease of the sounding pulse length, increase of the distance between the receivers in the receiving system, increase of the signal/noise ratio, decrease of the angular step and increase of the object aspect angle determination accuracy allow to increase the accuracy of the RO shape reconstruction and the distance to the object at the given accuracy. CONCLUSION The investigation that had been carried out has shown a possibility to use the genetic functions in a small aspect angle radar for the RO shape reconstruction. The reconstruction accuracy essentially depends on the measurement noise, the ratio of the object dimension to the electromagnetic pulse spatial length, the angular step with which the databank of the genetic functions is formed, the accuracy of the aspect angle determination of the object and the angular base of the receivers in a measuring system. The accuracy of a 4.5-m long object reconstruction at a 1-ns pulse length, a 2° angular base of the receivers and a 10% noise level was 80% for the suggested method. The suggested method is applicable for a monopulse radar when a RO displacement during a pulse can be neglected and it can be generalized for the case of sounding by the series of the UWB pulses with synthesizing the angular base owing to the data summation at the object motion. REFERENCES F.J.Agee, C.E.Baum, W.D.Prather, J.M.Lehr, J.P.O’Loughlin, J.W.Burger, J.S.H.Schoenberg, D.W.Schoflied, R.J.Torres, J.P.Hull, and J.A.Gaudet, Ultra-wideband transmitter research, IEEE Trans. Plasma Sci. 26:860 (1998). 2. Yu.A.Andreev, Yu.I.Buyanov, A.M.Efremov, V.I.Koshelev, B.M.Kovalchuk, V.V.Plisko, K.N.Sukhushin, V.A.Vizir, V.B. Zorin, Gigawatt-power-level ultrawideband radiation generator, in: Proc. of 12 IEEE Inter. Pulsed Power Conf., C. Stallings and H. Kirbie, eds., Monterey, CA, USA. 2:1337 (1999). 3. V.I.Koshelev, S.E.Shipilov, V.P.Yakubov, Reconstruction of the object shape at small aspect angle ultrawideband radiolocation, Radiotechn. Electr. 44:301 (1999). 4. V.M.Ginzburg, Presenting of images by means of genetic geometrical functions, Doklady Akademii Nauk. 244:580 (1979). 1.
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UWB SUBSURFACE RADAR WITH ANTENNA ARRAY FOR IMAGING OF INTERNAL STRUCTURE OF CONCRETE STRUCTURAL ELEMENTS
Anatoliy O. Boryssenko, Elena S. Boryssenko, Vladimir A. Ivashchuk, Alexander N. Lishchenko, Vitaliy P. Prokhorenko Research Company “Diascarb” Kyiv, P.O. Box No. 222, 02222, Ukraine
INTRODUCTION Basic design principals implemented in the advanced time-domain UWB radar including real/synthetic array and full-polarimetric schema with 100 – 900 MHz effective band are considered here. The radar is installing on a moveable platform for remote operation on the territories with radioactive pollution. This probing system should be employed near the Chernobyl nuclear power plant, block No. 4 damaged in 1986, that is situated not far from Kyiv, Ukraine,. The key task is searching and localisation of the nuclear-fuel contained materials (NFCM) inside the Cascade Wall around nuclear reactor. UWB radar described below is under testing now.
PROBLEM BACKGROUND Basically problem of localization if the NFCM inside the Cascade Wall with concrete can be solved by nonstandard ground penetrating radar (GPR) technique rather than by commercially available radars. The challenger features of the treated problem involve: i) sounding medium is layered concrete one up 10 meters and more thickness; ii) this medium has strong signal attenuation determined by internal water/ionic content; iii) it is inhomogeneous medium due to chaotic internal filling with a large variety of different objects including dispersed by explosion elements of the destroyed reactor; iv) the NFCM should be detected can be presented as solidified masses of molted concrete with patches of radioactive substances; v) it is expected that in some cases those patches may have a cylindrical shape as original fuel-assemblies; vi) searching areas on the horizontal plane of the Cascade Wall have rectangular shape where edge regions near vertical walls will cause strong diffraction interference (figure 1); vii)due to radioactive pollution of searching areas only completely automatic radar operation can be applied. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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Actually any GPR maps changes within material under survey due to contrasts in the electromagnetic properties. So the NFCM inside concrete can be recognized by set of its inferential features. For the most of known GPR applications measured data are presented as distorted images or pseudo-cross-sections of the subsurface regions for visual analysis, without any further necessity of processing (Olhoeft, 2000). However, the considered problem requires more detailed and precise information about type of each hidden object, its depth, orientation, composition etc. Such data can be obtained by properly acquired by advanced subsurface radar. One can find some useful approaches to be applied here. There are such fields of GPR applications as land-mine and UXO detection (Daniels, 1999), nondestructive testing of concrete in civil engineering (Boryssenko et al, 2000) and for noninvasive profiling of asphalt and concrete pavement and bridge desks (Wahrus et al, 1994). Key problems that need to solve, when GPR is applied for land mine detection, are connected with small electrical contrast of mines like plastic ones and influence of environment on radar performances. The last is due to irregular stratified structure of sounding media, rough interfaces, the presence of clutters like tree roots, rocks etc. Microwave inspection of concrete structures with is powerful tool with a wide range of applications. This is a rapid technique for nondestructive detection of defects in reinforcement concrete, for multi-layered road structure and so on. Note that all listed above application are connected with concrete probing up to several meters while the considered problem requires more deep down-range operation under specific conditions. We conclude that successful detection and discrimination of the FCM objects deeply inside the concrete construction of the Cascade Wall can be reached by application of the advanced GPR rather than ordinary hand-held GPR. Such advanced GPR should possess a high dynamical range and optimal operating frequency band. Also physical aperture technique should be employed beyond the synthetic aperture method. Polarimetric and other signal processing approaches have to be incorporated in this radar project. High procession of antenna space positioning is also required for exact subsurface mapping. The last can be fulfilled by robotic platform where radar is housed and that is urgently needed for safety operation on the area with radioactive pollution. Some of the formulated above features of the presented GPR project are under consideration in this paper.
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GENERAL DESIGN APPROACHES The key principal employed in this design is that such system must be realised in the short term with available technologies and examined before solutions. The presented radar possesses features originated from previous research and design projects in UWB radar especially for subsurface probing based on the time-domain technology. Optimal Frequency Band The key factor should be care is electrical properties of concrete as medium for electromagnetic wave propagation. Those properties in 1 MHz - 10 GHz band have been studied by many researchers (Robert, 1998). Figure 2 summarizes those data and illustrates the high frequency limit for GPR probing in concrete. Generally there exists a common tradeoff between resolution and depth range for choice of GPR operation frequency band. Increasing of resolution by using of high frequencies, can be achieved by the price of dramatic rise of radar potential and appearance on radar images redundant details and speckle structure (Daniels, 1999). There is also low frequency limit for optimal detection of target with specific shape and size like expected NFCM to minimize signal-to-clutter ratio besides requirement of high resolution (Brock and Patitz, 1994). Due to above reasons 100-900 MHz bandwidth is optimal one for the designed GPR.
Radar Energy Potential Here we express the radar performance factor as usual ratio of the peak radiated power to the smallest detectable receiver signal (Wahrus, et al, 1994; Daniels, 1999). Whereas the first factor is a constant value, additional coherent processing including averaging in time and space can increase the second one. Impulse generator of transmitter forms bell-like pulse on the 10-Ohm impedance antenna as a load with peak power up to 400 kW (56dB). The minimum discernible signal (MDS) is expressed as (Wahrus, et al, 1994): MDS = k·T·BW·NF·SNR, where k – Boltzmann constant, T maximum operation temperature, BW = 800 MHz – bandwidth, NF – Noise Factor of the receiver, SNR – Signal-to-Noise Ratio. Assume that NF = 1.3 dB and SNR = 15 dB. At this case we have MDS = -96 dB. We are considering peak power rather energy because in high-resolution radar system shape of transmitting signal is precisely registered and processed.
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Additional gain of system performance factor is following as stated above from radar response coherent processing like averaging in time with factor up to 2048 (33dB), array processing for physical aperture (9dB) and synthetic aperture (12-20 dB). Therefore the total energetic potential reaches of 180 dB magnitude. Sufficient rise of the last figure by processing is possible due to radar operation with discrete profiling when system has high level of stationary determined by platform positioning at each measurement point. The high system energy potential enables high depth range of radar. In this case for typical 5% internal water content the two-way loss magnitude is about 12 dB/m in accordance to figure 1. The last corresponds to depth range of about 15 m that give opportunity to search through all thickens of the Cascade Wall. Radar signals We paid particular attention to optimising transmitter/receiver array subsystems including optimal choice of signal waveforms and equivalent effective spectra shown in figure 3. Transmitting antenna driving pulse has a rise time of 1 ns and 5-ns fall time. Radiated electromagnetic signal is mainly determined by the current excitation waveform, the antennas and ground-coupling factor. The last has stochastic disturbance effect on both transmitter and receiver radar antennas resulted in registered waveform distortion. Also effect of multiple cooperative /non-cooperative scattering (Astanin et al, 1994) by specific shape targets will further modify received signal making its different from idealised one in figure 3c. Other factors change the registered signal waveform including: roughness of surface interface, antenna shielding from side of upper half-space, non-uniformity of antenna patterns at the angles off vertical directions etc. Finally the common measured two-pass waveform is depicted in figure 4.
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SYSTEM ARCHITECTURE OVERVIEW Generally presented UWB radar system includes two principal components, i.e. electronic and mechanical ones, the multi-channel radar and moveable robotic platform. The multi-channel UWB radar consists of three main subsystems: 1) sensor receiving array; 2) transmitter/receiver electronics; 3) data and command transfer via optical link, 3) operation control and data processing/storing. Transmitter electronics is based on drift step recovery diode diodes and special driving circuits. Lower pulse repetition rate (PRR) are employed in this radar for the reason of increasing of life time of compact high-voltage transmitter units, maintaining its optimal temperature regime for stable operation without external cooler and for power supply minimization. Receivers are built with using low-noise input circuits and stroboscopic time-domain sampling as for conventional pulse GPR. This approach and 16-bit ADC circuit enable high-accuracy registration of time-scaled signal waveforms. Such operation require some time with low PRR and long signal averaging but it is not drawback for the presented system because total data collection time at each point of scanning line is lower that time required for movement and positioning of mechanical part of radar. Radar control and data storing by remote host PC are fulfilled by the data/command transmission subsystem. The last forms a local network between computer on robotic platform and the host computer installed on the safe distance up to 0.5 km from the radaroperating zone. Physical interface of the data exchange between GPR and control PC unit is based on fiber optic communication link with full-duplex data transfer possibility. The radar sensor array is installed on the robotic platform for remote operation. The robotic platform is in fact a distantly controlled vehicle for programmed radar replacement over searching territory. In process of radar survey the sensor array is moving in discrete mode over the searching area. We employed as basic for robotic platform design electrical and pneumatic module components of the FESTO Company (http://www.festo-usa.com/). Table 1 lists the principal characteristics of advanced UWB subsurface radar.
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ANTENNA ARRAY AND PROCESSING TECHNIQUES Inherently the beam pattern of subsurface radar antennas is widely spread (Daniels, 1999) and to improve it the synthetic aperture technique (SAR) technique (Soumekh, 1999) is widely employed. The SAR is based on moving of radar antennas along scanning line laid over searching area (Finkelstein, 1994). In this way processing of radar data forms image of subsurface region. Actually migration processing (Morah, 2000) is basic imaging technique, which rearranges reflected data so that reflection and diffraction events are plotted at the origin locations. Effectiveness of the SAR depends on radar performances and specific features of sites under radar survey. For the considered problem the most unfavorable factor is limited size of the horizontal plane of searching areas due to the edge effects on SAR as shown conditionally in figure 1. In order to overcome limitations of existence GPR technique, such as for hand-held radar, the physical aperture combined with accurate SAR should be employed. In this way a near-field beam-forming with real physical aperture array is implemented by adjusting time delay magnitudes in receiving channels for array focusing on definite space point (spot) inside volume covered by antenna array (Rappaport and Reidy, 1994). Array structure, in figure 5a, implements a high-resolution post-processing array method for radar imaging with improving of signal-to-noise ratio, down-look radar range and suppers interference signal with out of interest arriving. Main functional goal of radar on the survey site is detection and discrimination of the NFCM as stated before. In the case of target with definite geometrical shape the value of scattered response will depend on target position with respect to polarization. Reliable radar system should be insensitive to this factor and designed to separate transmitter and receiver polarization states for arbitrary located buried target. In another words, the polarimetric radar measurement schema is urgently required. Figure 5b shows the dual polarized radar array antenna with separated two transmitting and eight receiving antennas. Besides target characterization radar polarimetric technique improves quality of radar image of subsurface region by reducing its speckle components (Stiles et al, 1999). There are two basic configurations i.e. Vivaldi tapered slot antenna and TEM ridge horn antenna. All antennas are resistively loaded for prevent ringing antenna effect and optimise signal waveform. The transmitter and receiver modules are directly terminated to elements of antenna array. The radar array has upper shielding for improving system interference immunity.
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DATA COLLECTION AND IMAGING Presented radar system implements the following in situ radar measurement that based on described above principal moments. Robotic transport platform after installation, mounting radar on it and connecting to external power, control and data transfer subsystems begins operation. In process of operation platform with radar is moving in step mode for discrete probing. Total trajectory of movement over searching area will cover it by the set of scanning lines. At each step the radar array unit is precisely positioned on surface. Plane coordinates of antenna reference point is measured by platform positioning subsystems and fixed in data file of the measurement records. Then radar measurement is consequently accomplished for both polarizations of transmitters when all eight receivers are working to register the scattered signal. Measurement at each point can be accomplished with different scenario of radar operation including time-varying gain control, data averaging, nonuniform sampling and so on. The host remote PC unit stores results of measurement at each point with coordinate and radar-setting information. Radar team can visualize data received at each point that based high-resolution antenna array technique with time-domain beam forming. In this way searching area covered by antenna physical aperture can be preliminary visualized in the 2-D or 3-D form. Next the transport platform should move along given scanning line and measurement procedure described above would be repeated. Much of the effectiveness of GPR technology is a function of the skill of radar team including operator and data interpreter. GPR equipment must be designed to provide in situ maximum effectiveness of radar data collections. Radar data are analyzed from a graphical presentation in 2-D forms like of B-scan (vertical section) and C-scan (horizontal section) or in 3-D from. Those graphical representations contains focused images of subsurface medium showing its various features originated from the contrast of electrical properties, i.e. dielectric constant, conductivity and magnetic constant rarely. Other factors governing features of scattered signals are attenuation, RCS, polarization etc. Each subsurface scattering phenomenon is characterized by a set of those parameters. The last is not quit unique and actually used for probabilistic discrimination of the scattering events in subsurface medium inside the concrete Cascade Wall. Simulated (left) and real field result (left) for focused radar 2-D image of subsurface scatterer like reinforcement bar in concrete are illustrated in figure 6.
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CONCLUSION Some of the principal components of the presented radar system have been manufactured and are under testing now. Firstly the radar with two-receiver array for copolar and cross-polar operation has been examined. The most resent experiments involve four-receiver array for the signal processing algorithms with TD beam forming. The next will include complete radar installation on platform for remote operation. At the moment of this paper preparation the final decision about using of such GPR system was unknown in the frame of the International Shelter Project (SIP). If this decision will be positive the radar system will be assembled completely on the robotic platform for its final testing, updating and field changing. REFERENCES Astanin, L. Yu., Kostylev A. A., Zinoviev Yu. S., Pasmurov A. Ya., 1994, Radar Target Characteristics: Measurement and Applications, CRC Press, Boca Raton. Boryssenko, A. A., Boryssenko, E. S., Lishchenko, A. N., Prokhorenko, V. P., 2000, Inspection of Internal Structure of Walls by Subsurface Radar, Submitted to 2000 GPR International Conferernce, Queensland, Australia. Brock, B. C., Patitz, W.E., 1994, Factors Governing of Operation Frequency for Subsurface-Imaging Synthetic-Aperture Radar, in: Proceedings of SPIE Conference, 2217: 176. Daniels, D. J., 1999, System Design of Radar for Mine Detection, in: Proceedings of SPIE Conference on Subsurface Sensors and Applications, 3752:390. Finkelstein, M.I, Ed., 1994, Subsurface Radar, Radio i Svjaz, Moscow (In Russian). Morah, M. L., Greenfield, R. J. Arcone, S. A., Delaney, A. J., 2000, Multidimensional GPR array processing using Kirchgoff migration”, J. Applied Geophysics, 43:281. Nathanason, F. E., Ed., 1991, Radar Design Principles, McGraw-Hill, Inc. Olhoeft, G. R., 2000, Maximizing the Information Return from Ground Penetrating Radar, J. of Applied Geophysics, 43:175. Plumb, R.G., Noon D.,A., Longstaff, I. D., Stickley G. F., 1998, A Waveform-Range Performance Diagram for Ground-Penetrating Radar, J. of Applied Geophysics, 40:117. Rappaport, C. M., Reidy, D. M., 199?, Focused Array Radar for Real Time Imaging and Detection, in: Proceedings of SPIE Conference, 2747:202. Robert A., 1998, Dielectric Permittivity of Concrete Between 50 MHz and 1 GHz and GPR Measurements for Building Materials Evaluation, J. of Applied Geophysics, 40: 89. Skolnik, M. I., Ed., 1990, Radar handbook, McGraw Hill, 2nd edition. Soumekh, M., 1999, Synthetic Aperture Radar Signal Processing with Matlab Algorithms, A Willey-Interscience Publications. Stiles, J. M., Parra-Bocaranda, P., Apte, A., 1999, Detection of Object Symmetry Using Bistatic and Polarimetric GPR Observations, in: Proceedings of SPIE Conference on Detection and Remediation Technologies for Mines and Minelike Targets IV, 3710: 992. Wahrus, J. P., Mast J. E., Johnson E. M., Nelson S. D., 1994, Advanced Ground Penetrating Radar, in Proceedings of SPIE Conference, 2275:177.
OPTIMAL SHORT PULSE ULTRA-WIDEBAND RADAR SIGNAL DETECTION
Igor I. Immoreev1 and James D. Taylor2 1
Moscow Aviation Institute Gospitalny val, home 5, block 18, apt 314 Russia 105094, Moscow
[email protected] 2 J.D. Taylor Associates 2620 SW 14th Drive Gainesville FL 32608-2045, USA
[email protected]
INTRODUCTION High-resolution ultra-wideband (UWB) radar systems will produce a multiple time scattered return from large targets. Any individual return will be difficult to detect and meaningless for surveillance radar purposes. Large target detection requires building a filter for integrating multiple time-scattered returns into a single detector output. This filter can provide optimal detection of signals with unknown parameters. We will discuss short duration (1 ns) video pulse type signals, however the concept may apply to other highresolution signals such as pseudorandom noise and FM chirp. References 1,2,3, and 4 describe the advantages of video pulse UWB radar. ULTRA-WIDEBAND RADAR SIGNALS AND LARGE TARGETS For high-resolution video pulse radar, the target geometry and viewing angle determine how each impulse will be reflected toward the receiver. Each target will have a set of N bright points. The target electromagnetic characteristics determine the returned video pulse shape and the effective scattering point radar cross section (RCS) determines the reflected impulse amplitude. If the radar instantaneous bandwidth approaches 100% of the center frequency, and the radiated pulse length is about 1 ns, then the space signal duration is 30 cm, and much
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smaller than targets such as vehicles and aircraft. Some target points may be bright, and others may change the signal waveform according their electrical properties.1 As a result a returned signal is transformed into the sequence of pulses with random parameters such as shown in Figure 1. QUASI-OPTIMAL DETECTORS FOR UWB SIGNALS As shown in Figure 1, a video pulse signal scattered from an extended target has an intricate pattern. Reflected signal parameters, such as the duration and the number, location and amplitude of signal maximums will be unknown. Unknown signal parameter information makes it impossible use correlation processing. G. A. van der Spek proposed a processing algorithm for optimal detection of unknown targets. To understand this approach, suppose the target has a length L and occupies N resolution cells, in space. The signals scattered by bright points are present in K cells, and the other cells are empty. Processing all combinations from N elements on K bright points can provide optimal detection of the unknown signal. This algorithm works because one of these combinations must coincide with the extended target return.. Figure 2 shows a schematic diagram for such an optimal detector. The problem is that a practical realization of 6 this scheme requires many processing channels.
For example, if there are N = 40 resolution cells within the observation interval for a 1 GHz bandwidth signal, and the number of expected bright points is K = 8, then the number of processing channels required will be The structure of such detector is complex and cannot be realized using present day electronic components. G. van der Spek proposed two simpler algorithms for quasi-optimal processing of unknown signals.6 The first algorithm uses the changes of the energy at the detector output when a target scattered signal is received. If N = K, where a scattered signal is present in all resolution cells within the observation interval, the optimal detector of Figure 2 is modified into a quadratic detector with a linear integrator in Figure 3.8 In this case the integration is performed over all N resolution cells, so there is no need to have prior information about the presence and location of K bright points. This detection scheme is called the “energy detector.” If we use this detector when additional losses result from the summation of noise in “empty” resolution cells within an observation interval. By increasing the number of bright points K within this interval, the energy detector approaches the optimal detector performance for fully known signal.
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For case of K<< N, van der Spek suggested a quasi-optimal algorithm to reduce the losses in the energy detector. This algorithm ranks signals scattered by bright points within an observation integral, which approximates the estimated target size. In this interval maximum signal amplitudes are selected using a sliding window. Only K maximum amplitudes from N resolution cells are quadratically processed and linearly summed. Figures 4 a. and b are block diagrams of an example single-channel rank detector.6 Figure 4.c is a procedure for selecting signal maximums for a three bright point target.
If K is unknown, then a multi-channel rank detector is used, which sums the signal maximums scattered by target bright points in each channel. Signals from all detection channels combine into a single output. A more complicated detector scheme can increase the receiver sensitivity. Bakut, Bolshakov et al described the “by-point” detector, which successively compares the signal voltage with a threshold level in each resolution cell, and then combines the results in a logical OR scheme. The detector does not accumulate signals scattered by different bright points, which are partly compensated by the accumulation of logical decisions, or probabilities. Easy construction is this detector’s advantage. 7 To estimate the efficiency of the three quasi-optimal detectors, we can compare them with a detector for a fully known signal. The mathematical modeling of the processing algorithms for these four detectors featured an observation interval of N = 100 resolution
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cells. If the signal duration is 1ns, it corresponds to a target resolution length of 0.15 m. The number of bright points K was varied from 1 to 32. Normally distributed white noises with zero average values and equal dispersions were used for modeling all detectors. 10 Table I shows the energy losses of threshold signal (dB) for three detection algorithms relative to the standard algorithm for a detection probability of D = 0.5 and false alarm rate F = 10-3. Based on these results, the energy detector is less effective than other detectors if the number of bright points is small. If a target configuration becomes more complex so that the number of bright points increases, then the relative losses for this detector are reduced, and for K = 32 it is even more efficient than the “by-point” detector, but still is less efficient than multi-channel rank detector.
This result can be easily explained. If the number of target bright points is small, then the energy detector accumulates many noise samples, while the “by-point” detector selects only one bright point. If the number of bright points is increased, then the number of noise samples accumulated by the energy detector is reduced and its efficiency grows. At the same time the “by-point” detector does not use the full signal energy, because it does not accumulate. In the limit, if the number of bright points approaches N = 100 the energy detector becomes identical to the standard detector for a fully known signal. It should be noted that the “energy detector” and the “by-point” detector can operate effectively in the opposite situations, when the number of bright points is either very small or large. It would help to develop a two-channel system using these two detectors operating simultaneously, and combining the output signals by maximums. Such scheme may be also represented as a simplified multi-channel rank detector comprising only two channels. One of these channels can detect targets with simple configurations, and the other can detect targets with complex configurations. Before combining by maximum, the output signals are normalized to equalize the false alarm rates, as it is performed in a multi-channel rank detector. Figure 5 shows the detection curves for the quasi-optimal detectors discussed.10 INTER-PERIOD CORRELATION PROCESSOR OPTIMAL DETECTORS The pulse repetition period is the only known signal parameter we can use for detecting an over-resolved target. By correlating the returns from each successive interval, we can build an Inter-Period Correlation Processor (IPCP). This scheme has three differences from the conventional correlator: 1. It correlates the target signal with the previous return. 2. Noise signals feed both correlator inputs, but are not correlated because they are received indifferent repetition periods. 3. The integration period T is determined not by the radiated signal duration, but by the observation interval, or the scattered signal duration. For example, if the target length is L, the integration time for the system of Figure 6 is equal to where c is velocity of light and is the radiated signal duration.
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For IPCP detectors, the signal shape determines the modified correlation receiver efficiency. After the correlator combines both the current and delayed signal samples taken from two repetition periods, it performs this process during over the same time interval. CRITERION PROCESSING OF UWB SIGNALS AFTER IPCP Additional criterion processing can improve the IPCP detection characteristics by memorizing the resolution cells having threshold scheme output signals. This operation is performed in several pulse repetition periods and determines the cells in which the signals repeatedly emerge. This scheme only passes signals from resolution cells corresponding to the selected criterion, e.g. “two of two,” etc, and reduces the false alarms at the processor output, but the detection probability deceases at the same time. In practice the detection probability is always higher than the false alarm rate, so the detection probability decreases slower than the false alarm rate. Criterion processing can be effectively used when a low false alarm rate is required, i.e. Consider the case of the “ n of k” criterion. If the scattered signal is absent, and the number of coincidences of threshold voltage samples i in k repetition periods is then the criterion processing scheme output false alarm rate is equal to:
where:
is the number of combinations of k elements taken i at a time, and
is the false
alarm rate in one repetition period. If a signal scattered from a target is present for the same conditions, the detection probability is
Let us apply “two of two,” the simplest criterion processing scheme to threshold signals at the IPCP output, as shown in the criterion processor of Figure 7.
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The signal samples from the same range cells received in two repetition periods are fed to the logical AND scheme. Only coincident samples appear in the output of Figure 8. The IPCP detector approaches the characteristics of a traditional correlator for a fully know signal as the false alarm probability decreases, as shown in Figure 9 a and b.10 The IPCP concept can be expanded to provide optimal detection of targets with unknown characteristics, as shown in Figure 10.
SELECTIVE TARGET DETECTION Given the concept of IPCP processing and multi-channel filters for detecting unknown targets, it is easy to conceive a filter designed to only detect specific targets classes. Given a target characteristic database for aspect angles of interest, the filter could be set reject returns that do not fit those criteria. This could be useful in finding specific targets.
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CONCLUSIONS IPCP provides a new approach to large target detection using high resolution UWB signals, and provides detection nearly equivalent to traditional correlator for high false alarm rates. It is a small step from optimal detection to adjusting the filter for special classes of targets. Selective target detection will be useful for many special radar applications, such as the detection of targets in a high clutter environment.
REFERENCES 1. H. Harmuth, Nonsinusoidal Waves for Radar and Radio Communications, Academic Press, New York, 1981.
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2. L.Yu. Astanin and A.A. Kostylev, Ultra-Wideband Radar Measurements: Analysis and Processing, IEE, London, UK, 1997. 3. J.D. Taylor, (Ed), Introduction to Ultra-Wideband Radar Systems, CRC Press. Boca Raton, Ann Arbor, London, Tokyo, 1995. 4. I.I. Immoreev, “Ultra-Wideband Radars: Main Features and Dissimilarities from Conventional Radars,” Electromagnetic Waves and Electronic Systems, Vol. 2, N1, 1997, pp.81-88. 5. I.I. Immoreev. “Ultra-Wideband Radars: New Opportunities, Uncommon Problems, and System Features,” Proceedings of Bauman Moscow State Technical University. Vol.4, 1998. 6. G.A. van der Spek, “Detection of a distributed target,” IEEE Trans on Aerospace and Electronic Systems, Sept 1971, AES-7, N5, pp. 922-931. 7. P.A. Bakut, I.A. Bolshakov, et al, Problems of Radar Statistical Theory. Vol. 1, Soviet Radio, Moscow, 1963. 8. Y.W. Lee, T.P. Cheatham, and J.B. Wiesner, “Application of correlation analysis to the detection of periodic signals in noise,” Proceedings of the IRE, No. 38, p. 1165, 1950. 9. Levin B.R., Theoretical Foundations of Statistical Radio Engineering, Radio and Communications, Moscow, 1989. 10 . I.I. Immoreev, “Features Detection of Signals in UWB Radars,” J.D. Taylor, ed, Ultra-Wideband Radar Technology, CRC Press, Boca Raton, FL, 2000.
EXPERIMENTAL RESULTS FROM AN ULTRA WIDEBAND PRECISION GEOLOCATION SYSTEM
Robert J. Fontana, Ph.D. President Multispectral Solutions, Inc. Gaithersburg, MD 20855 USA
INTRODUCTION One of the most recent and unique applications of ultra wideband (UWB) technology has been to the determination of precision location in urban terrain and within building structures. In these applications, complex multipath and differential path loss often result in received waveforms in which the direct path is highly attenuated relative to subsequent multipath returns. As a consequence, UWB receiver processing techniques which provide high sensitivity and leading edge detection are critical to successful time-of-flight and, hence, true location measurements. In this paper, we present experimental results taken with an ultra wideband precision geolocation system developed to track soldiers inside buildings and in open terrain. The UWB Precision Geolocation System utilizes a set of untethered, fixed position "Beacons" and an untethered mobile "Ranger". Three-dimensional positioning information is obtained by determining the round-trip time-of-flight from the UWB Ranger to each Beacon transponder. The system utilizes a 2.5 nanosecond (27% fractional bandwidth) burst waveform, and an unique tunnel diode receiver which is sensitive to the received pulse leading edge. The paper begins with a discussion of the design of a UWB transponder which utilizes a quantum, tunneling device as a leading edge detector. Experimental results are then provided for both in-building and open terrain system configurations. RMS ranging errors of less than 1 foot are demonstrated for in-building propagation; while open terrain results produce RMS errors of better than a few inches due to leading edge detection of the received waveform. LEADING-EDGE UWB PULSE DETECTION Among the most significant problems encountered in the measurement of the time-of-flight for a short pulse emission are the deleterious effects of multipath cancellation and reverberation. For example, consider the ultra wideband waveform of Figure 1 which Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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consists of a few cycles of a 1.5 GHz carrier wave. The 3 dB bandwidth of this waveform was designed and measured to be 400 MHz (27% fractional bandwidth).
Figure 2 illustrates this same waveform received at the end of an interior hallway at a distance of approximately 42 feet (12.8 meters). These waveforms were captured on a high-speed sampling oscilloscope (Tektronix 11801B) outfitted with an external low noise preamplifier and wideband horn antenna. For these measurements, the sampling receiver was synchronized to the transmitted burst via coaxial cable.
This response is fairly typical of those encountered with in-building pulse propagation, and serves to illustrate several important points. First of all, the initial line-of-sight response is often not the dominant one. For example, in Figure 2, there are numerous returns significantly stronger in amplitude than the direct path return. Secondly, depending upon the geometry, the first return can itself be corrupted by multipath cancellation with only the leading edge of the first return left uncorrupted. An example of this latter effect is shown in Figure 3 below in which the line-of-sight return is corrupted by at least two additional, close-in responses. Thus, to produce an accurate time-of-flight (and, hence, accurate distance) measurement, one can increase the UWB pulse bandwidth (to further resolve close-in multipath returns) and/or develop techniques for leading edge pulse detection. Propagation anomalies (e.g., frequency selective attenuation) and thermal (kTB) noise ultimately limit the effectiveness of these approaches. In the present system, leading edge detection was performed using a quantum tunneling device as described below.
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Only two devices – the tunnel diode and the avalanche transistor – have been found suitable for triggering on extremely fast, low energy pulses; however, the tunnel exhibits well defined v-i characteristics and an almost order of magnitude improvement in sensitivity. The leading edge detector described below consists of a tunnel diode biased to be near the peak of its transition to a negative resistance region. The point of maximum detector sensitivity is maintained through the use of a constant false alarm rate (CFAR) loop which periodically measures received noise and adjusts the diode bias point until the false alarm rate is at a desired level consistent with a high probability of detection. That is, the loop parameters determine the desired receiver operating characteristic. An unique feature of the tunnel diode operating in this bistable mode is that it is charge sensitive (Ross, 1984), with the diode switching states (i.e., transitioning through its negative resistance region) whenever
where i(t) is the current passing through the diode. If the received signal plus additive white Gaussian noise, s(t) + n(t), is superimposed across the diode's DC bias current, the diode "triggers" whenever
where B(t) is a Brownian motion process (Gikhman and Skorokhod, 1969) whose variance can be shown to increase linearly with time. Thus, for small times t, the variance of the detector noise process is small; and it is this effect which allows for a leading edge detection to occur (Fontana, 1997). The tunnel diode is gated ON, i.e. sensitized, only for an instant of time whose duration is comparable to that of the transmitted pulse width. Tunnel diode gating is synchronized to the received pulse train utilizing a digital phase lock loop in a fashion so as to place the received waveform near the beginning of the integrating "range gate" interval. A photo of the tunnel diode detector and synchronization circuitry is shown below in Figure 4. In the present system, a set of two tunnel diodes is used – one biased for signal detection, and the other utilized for ambient noise detection (CFAR).
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As configured, the CFAR-biased tunnel diode detector is essentially shot noise limited with an overall sensitivity of between -30 and -40 dBm. In order to receive appreciably weaker signals, the detector is preceded by a low noise microwave preamplifier whose gain is chosen to be high enough to overcome the diode shot noise limitation. As currently configured, the detector is capable of directly receiving UWB signals with apparent center frequencies to beyond 6 GHz; however, higher frequency microwave and millimeter wave designs have also been developed. The demonstrated line-of-sight range of the present system is over 2 kilometers with omnidirectional antennas. In-building ranges are, of course, further reduced due to intervening walls and other obstacles. Wall attenuation depends upon building construction techniques and materials; however, typical values are in the range of 5 to 25 dB per intervening wall. PRINCIPLE OF OPERATION The UWB beacon and ranger units (Figure 5) each consist of a UWB transmitter emitting a 2 Watt peak, 2.5 ns burst waveform; low noise amplifier (LNA); and tunnel diode detector and synchronization circuitry (cf. Figure 4 above). The transmitter actually emits a sequence of UWB pulses consisting of a synchronization preamble followed by a short message containing beacon ID.
A block diagram of the UWB transponder is illustrated in Figure 6 below. The receiver portion of the unit consists of a low noise amplifier, CFAR-controlled tunnel diode detector, oven controlled crystal oscillator (OCXO) and time-of-arrival measurement circuitry (precision delay and TOA decoder logic). The transmitter portion of the unit
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consists of a low power, impulse exciter followed by waveform shaping and a time-gated power amplifier to minimize power consumption.
The principal of operation can be described as follows (cf. Figure 7). In this figure, the position to be surveyed is shown at location (x,y,z). In addition, a set of N (fixed) beacon positions will have been previously surveyed using, for example, differential GPS (for outdoor or outdoor-to-indoor applications), topographical chart data, architectural drawings or other measurement techniques. The accuracy to which these fixed positions is measured will, of course, directly affect the resultant accuracy of the position to be measured.
The N UWB beacon positions are selected to provide favorable siting of the person/vehicle/object whose position is to be determined. For example, in an urban terrain experiment performed for the U.S. Defense Advanced Research Projects Agency, a set of four UWB beacons were placed (approximately 1 km apart) on the periphery of an area encompassing several buildings, around and through which the UWB Rover was transported. Once the beacons are sited, measurement proceeds as follows. The UWB Rover initiates an RF burst which is received by a subset (possibly not all due to terrain blockage or excess attenuation) of the available UWB Beacons. Upon receipt of the burst, each Beacon first determines if the message has originated from the Rover (check for ID message); and if so, after a fixed time offset which is assigned to each beacon, transmits a reply message
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containing the Beacon ID. The time offset is selected to avoid simultaneous reception of pulses by the UWB Rover from multiple Beacons. For a fixed beacon geometry, appropriate offsets can be readily determined. Upon receiving the reply message from Beacon i, the Rover determines the round trip timeof-flight to that beacon by subtracting the fixed beacon time offset from the measured elapsed time From the set of time-of-flight measurements and knowledge of the fixed beacon positions, the Rover can now determine its position from the set of N simultaneous equations
where d is Euclidean distance, r = (x,y,z), and c is the speed of light. In the current system design, a Newton-Raphson algorithm was used to find a zero in the gradient of the error functional
The position thus obtained is an unbiased estimate of the true position of the Rover. Advantage is also taken of the relatively large quantity of beacon data (currently set at 100 updates per second) by incorporating a low order, finite impulse response filter on beacon returns prior to optimization. Since the transceiver clocks are effectively resynchronized at each packet burst, only short term oscillator stability is of importance. In the present system, time-of-arrival accuracy is limited to approximately 1 ns due to the manner in which data is clocked into the receiver processor; however, the tunnel diode circuitry has a demonstrated response time of better than 40 ps, ultimately permitting centimeter-type resolutions with appropriate circuitry. In GPS-based rapid-static and kinematic positioning systems, measurement ambiguity is resolved by converting ambiguous carrier-phase measurements into unambiguous ranges. With a UWB-based positioning system, ambiguous arrival time-differences are converted into unambiguous ranges. This may be described as the time-domain "dual" of the carrierphase approach, and permits operation under a wide variety of circumstances for which GPS coverage is either unavailable or seriously degraded. EXPERIMENTAL RESULTS Figures 8 and 9 illustrate results obtained from a single UWB Beacon transponder in an inbuilding environment. For these measurements, the UWB Beacon was moved to 11 different positions and data accumulated for approximately 10 seconds at each point. In these figures, locations through (distances of 22' 8.0", 87' 1.3", 112' 8.4" and 1 1 1 ' 2.6", respectively) are fairly typical of indoor, high signal-to-noise ratio conditions. Measurement standard deviations for these locations were 0.44' (5.3"), 0.52' (6.2"), 0.42' (5.0") and 0.56' (6.7"), respectively. Recall that the UWB pulsewidth for these measurements was approximately 2.5 ns. Location at a range of 152' 6.7", was significantly more noisy – with a standard deviation of 2.55' – primarily due to significant attenuation (as well as reflections) from intervening metal structures (see Figure 9). Location was also of interest, since both direct and multipath returns could be observed depending upon the orientation of the beacon antenna (cf. Figure 10 below).
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Finally, Figures 11 and 12 illustrate the results of indoor and outdoor measurements taken with a set of three UWB Beacon transponders. In Figure 11, the UWB Rover was moved along a set of corridors inside a multi-office suite. As with GPS, it is noted that ranging errors are magnified by the range vector differences between the receiver and the beacons (satellites) – i.e., geometric dilution of precision (GDOP). Poor GDOP results when angles from the receiver to the set of beacons are similar (e.g., circled region in Figure 11). Of course, in-building attenuation effects tend to further degrade performance as the distance from the Rover to Beacons increases. As observed, the geolocation performance within the region bounded by the three beacons was reasonably accurate. Figure 12 illustrates an outdoors experiment in which the UWB Rover was transported on a small cart in the back parking lot at MSSI. The figure "2000." was produced my moving the cart within the lane markings within the lot. As can be observed from the expanded view of the "Period", the range resolution was approximately 0.2 feet for this high signalto-noise ratio case.
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CONCLUSIONS Ultra wideband waveforms can achieve extremely fine, centimeter-level range resolution because of their extremely short signal durations. With the advent of very high speed, high sensitivity, leading edge detectors, the feasibility of such systems has been demonstrated. An ultra wideband (UWB) based system has been described which enables the precision geolocation of a platform over distances of up to several kilometers (line-of-sight) and to more that one hundred meters indoors, depending upon building construction. The system uses a set of N>2, untethered UWB transceivers (Beacons) and an untethered UWB Rover to resolve time-of-flight measurement ambiguities to determine position. The UWB geolocation system is self-synchronizing, and does not require the existence of a clock distribution system and associated cabling. The system operates independently of GPS, thereby providing operation under conditions in which satellite coverage is unavailable or is blocked by obstructions or shielding (e.g., wartime operation, in buildings, urban environments, under heavy canopy, next to large obstructions such as vertical mine walls, etc.). Measurement standard deviations of less than one foot have been demonstrated for inbuilding operation, and less than a few inches for line-of-sight operation, using a 400 MHz bandwidth (2.5 ns duration) ultra wideband waveform. ACKNOWLEDGMENTS The author wishes to thank Dr. Edward Richley for his assistance with the implementation of the optimization algorithms, and Mr. Robert Mulloy for his help with the experimental phase of this effort. REFERENCES Fontana, R.J., 1997, A novel ultra wideband (UWB) communications system, in Proceedings MILCOM 97. Gikhman, I.I. and Skorokhod, A.V., 1969, Introduction to the Theory of Random Processes, Dover Publications, Mineola, NY. Ross, G.F., 1984, Comments on baseband or carrier-free communications, Technical Memorandum, ANRO Engineering, Inc., Lexington, MA.
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RECENT APPLICATIONS OF ULTRA WIDEBAND RADAR AND COMMUNICATIONS SYSTEMS
Robert J. Fontana, Ph.D. President Multispectral Solutions, Inc. Gaithersburg, MD 20855 USA http://www.multispectral.com
INTRODUCTION Ultra wideband (UWB) technology, well-known for its use in ground penetrating radar, has also been of considerable interest in communications and radar applications demanding low probability of intercept and detection (LPI/D), multipath immunity, high data throughput, precision ranging and localization. After a very short introduction to the history and theory of ultra wideband technology, we describe the current state-of-the-art (within the United States) in this emerging field by way of examples of recently fielded UWB hardware and equipment. Multispectral Solutions, Inc. (MSSI) is a pioneer and an established industry leader in the development of ultra wideband systems and has been actively involved in UWB hardware and system development since 1984. AN (ULTRA) SHORT HISTORY OF UWB TECHNOLOGY The origin of ultra wideband technology stems from work in time-domain electromagnetics begun in 1962 to fully describe the transient behavior of a certain class of microwave networks through their characteristic impulse response (Ross (1963, 1966)). The concept was quite simple. Instead of characterizing a linear, time-invariant (LTI) system by the conventional means of a swept frequency response (i.e., amplitude and phase measurements versus frequency), an LTI system could alternatively be fully characterized by its impulse response h(t). In particular, the output y(t) of such a system to any arbitrary input x(t) could be uniquely determined by the well-known convolution integral
However, it was not until the advent of the sampling oscilloscope (Hewlett-Packard c. 1962) and the development of techniques for subnanosecond (baseband) pulse generation Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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(to provide suitable approximations to an impulse excitation) that the impulse response of microwave networks could be directly observed and measured. Once impulse measurement techniques were applied to the design of wideband, radiating antenna elements (Ross (1968)), it quickly became obvious that short pulse radar and communications systems could be developed with the same set of tools. While at the Sperry Research Center, then part of the Sperry Rand Corporation, Ross applied these techniques to various applications in radar and communications (Bennett & Ross (1978)). The invention of a sensitive, short pulse receiver (Robbins (1972)) further accelerated system development. In April 1973, Sperry was awarded the first UWB communications patent (Ross (1973)). Through the late 1980's, this technology was alternately referred to as baseband, carrier-free or impulse – the term "ultra wideband" not being applied until approximately 1989 by the U.S. Department of Defense. By that time, UWB theory, techniques and many hardware approaches had experienced nearly 30 years of extensive development. By 1989, for example, Sperry had been awarded over 50 patents in the field covering UWB pulse generation and reception methods, and applications such as communications, radar, automobile collision avoidance, positioning systems, liquid level sensing and altimetry. Today, literally hundreds of patents, papers, books and bibliographical references exist on all aspects of UWB technology. An excellent, recent compendium of this material was put together by Dr. Robert Fleming of Æther Wire & Location, Inc. and can be found on the Web at http://www.aetherwire.com/CDROM/Welcome.html. As shown by the extensive list of references provided in this compendium, there have been numerous researchers involved with UWB technology over the past 38 years, and it is virtually impossible to even begin to list the other important contributors within the scope of this short paper. Within the United States, much of the early work in the UWB field (prior to 1994), particularly in the area of impulse communications, was performed under classified U.S. Government programs. Since 1994, much of the work has been carried out without classification restrictions, and the development of UWB technology has greatly accelerated. The purpose of this paper is to provide the reader with a short overview of where UWB technology is today, albeit from the perspective of one U.S. company that has been active in this field for the past 15 years. The applications and hardware discussed below are relatively new, all having been developed and demonstrated within the past 5 years. Many of these programs came to exist only because of recent breakthroughs in UWB source, receiver and antenna technologies. UWB APPLICATIONS Figure 1 illustrates an ultra wideband, handheld transceiver that was designed for full duplex voice and data transmission at rates of up to 128 kb/s (CVSD) and 115.2 kb/s (RS232). The radio has an operational center frequency in L-band (1.5 GHz) with an instantaneous bandwidth of 400 MHz (27% fractional BW). Peak power output from the UWB transceiver was measured at 2.0 Watts, with a resultant average power (worst case) of 640 µW. This results in a worst case power density of 1.6 pW/Hz. These units have a range of approximately 1 to 2 km (with small antennas shown and line-of-sight), and an extended range of 10 to 20 miles with small gain antennas.
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Figure 2 illustrates a rather unique UWB radio designed for non line-of-sight communications utilizing surface or ground wave propagation. To excite such propagation modes, the frequency of operation needs to be well below 100 MHz (e.g., Skolnik, 1990). Thus, this system was designed to operate in the frequency band from 30 to 50 MHz (50% fractional BW) and utilized a peak power output of approximately 35 Watts.
As in the above example, this radio was capable of both digital voice and data transmission to 128 kb/s and had an operational range over water of approximately 10 miles using a standard SINCGARS (30-88 MHz) VHF military antenna. Operational range over land depended upon terrain, but was observed to be from 1 to 5 miles with intervening foliage, buildings and hills. [Note that multipath cancellation is a serious problem over water, even with UWB impulse technology, because of the typically low grazing angles (resulting in small differential delays between direct and reflected paths) and strong, outof-phase reflection from the water surface. Thus, both higher frequency UWB systems as well as more conventional narrowband VHF/UHF radios were unable to provide the requisite performance.] Also unique to the receiver design was the addition of anti-jam circuitry to prevent loss of sensitivity due to strong, in-band interferers commonly observed in this frequency range. The units could also be operated in a "digipeater" mode in which packet store-and-forward techniques were used to transmit data from one radio to another via an intermediate repeater link. Figure 3 illustrates a high-speed (up to 25 Mb/s) UWB radio designed for transmission of compressed video and command & control information across an asymmetric, bi-directional link. This system was also designed to operate in the 1.3 to 1.7 GHz region (27% fractional BW) with a 4W peak power output. An earlier design, developed under funding from the U.S. Defense Advanced research Projects Agency (DARPA), operated with a 500 MHz instantaneous bandwidth in the C-band region (5.4 to 5.9 GHz).
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The ultimate goal of this design was to provide for up to 60 nautical mile, line-of-sight transmission to/from an unmanned aerial vehicle (UAV). A small parabolic dish antenna was used at the ground platform. A variant of the above system is illustrated in operation in Figure 4 below. In this figure, a 2 Mb/s asymmetrical UWB link is used to transmit compressed video from a small unmanned ground vehicle (robot) through a UAV (unmanned helicopter) relay to a soldier ground station. The command & control signal (115.2 kb/s) to the robot is relayed through the UAV; while compressed robot video transmissions (1-2 Mb/s) are relayed through the UAV to the soldier. Ranges to UAV and robot were a few kilometers.
Another unique application for UWB communications is illustrated by the tagging device shown below in Figure 5.
This system, dubbed Vehicular Electronic Tagging and Alert System (VETAS), was designed for the U.S. Department of Transportation to provide a means for keeping problem drivers (i.e., drivers who have repeated been convicted of traffic accidents or violations due to driving while under the influence of alcohol) off the road. The concept was to tag the vehicle with a device which relays a picture of the driver, together with
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information on the driver and the vehicle, to a roadside sensor in a police vehicle. The tag would be installed in lieu of impounding the vehicle or placing the convicted driver in jail. Ultra wideband technology was considered for this application because of its ability to transmit large amounts of data at high speed in a mobile, multipath-prone environment. The UWB tag operated in the L-band region (1.4 to 1.65 GHz) and had a peak output power of approximately 250 mW for a demonstrated range of over 300 meters. The image of the driver was stored as a compressed JPEG file, together with additional ASCII data, in EEPROM and periodically transmitted at a 400 kb/s burst rate to a UWB receiver with display. The tag operated off of two AAA batteries (3.0V) and, in an operational scenario, was mounted behind the front grill of the automobile. Figure 6 illustrates a set of prototype UWB transceivers designed for the U.S. Navy to provide a wireless intercom capability on-board Navy aircraft. The prototype UWB transceivers provide multichannel, full duplex, 32 kb/s digital voice over a range of approximately 100 meters. An ultra wideband waveform was selected because of its ability to operate in severe multipath (created by multiple RF reflections inside and around aircraft), and because of its non-interfering, low probability of intercept signature.
Current intercommunications systems (ICS) designs for aircraft utilize lengthy, and often unwieldy, cords to physically attach the crewman’s headset to a distributed audio (intercom) system. Such physical attachment presents a safety hazard to personnel, impedes movement throughout the platform and reduces mission effectiveness. Replacement of these mechanical tethers with wireless RF links is a desirable alternative. Frequency of operation for the WICS transceivers was again in the L-band region (1.2 to 1.8 GHz). One of the unique features of the WICS design was the use of a frequency division multiplex, time division multiple access (FDM/TDMA) strategy for full duplex, multi-user operation. Because of the extremely short duration pulsewidths and resulting low energy densities, UWB systems are much less vulnerable to intercept and ECM attack than conventional RF communications systems. As a consequence, they also minimize interference to other on-board electronics, such as sensitive flight control systems, GPS, etc. With an extremely low duty cycle, a very low power drain can be achieved, thereby providing communications capability for mission life exceeding 12 hours. The WICS program has recently received additional funding to further improve and miniaturize the design. One of the most recent applications of UWB communications technology is to the development of highly mobile, multi-node, ad hoc wireless communications networks. Figure 7 illustrates such a system currently under development for the U.S. Department of Defense. The system is designed to provide a secure, low probability of intercept and
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detection, UWB ad hoc wireless network capability to support encrypted voice/data (to 128 kb/s) and high-speed video (1.544 Mb/s T l ) transmissions.
A parallel effort, currently funded by the Office of Naval Research under a Dual Use Science and Technology (DUS&T) effort is developing a state-of-the-art, mobile ad hoc network (MANET) based upon an Internet Protocol (IP) suite to provide a connectionless, multihop, packet switching solution for survivable communications in a high link failure environment. The thrust of the DUS&T effort is toward commercialization of UWB technology for applications to high-speed (20+Mb/s) wireless applications for the home and business. A UWB application which bridges the gap between communications and radar is that of precision geolocation. Also see accompanying paper (Fontana, 2000). Figure 8, for example, illustrates a system designed to provide 3-dimensional location information utilizing a set of untethered UWB beacons and an untethered, mobile UWB rover. Precision location is derived from round trip, time-of-flight measurements using packet burst transmissions from the UWB rover and beacon transponders.
The system in Figure 8 utilizes a 2.5 ns, 4 Watt peak, UWB pulse, again operating in the 1.3 to 1.7 GHz region. Line-of-sight range for the system is better than 2 kilometers utilizing small, omnidirectional vertically polarized (smaller) or circularly polarized (larger) antennas. Within a building, the range becomes limited by wall and obstacle attenuation; however, ranges exceeding 100 meters inside have been attained. An unique feature of the system is the ability to detect the pulse leading edge through the use of a charge sensitive, tunnel diode detector. Leading edge detection is critical to the resolution of the direct path from the plethora of multipath returns produced from internal reflections. The UWB geolocation system was originally developed to permit a soldier to determine his or her position to within 1 foot resolution in an urban environment. It is currently being used to augment a video capture system for 3-D modeling, and for materiel location onboard a Navy ship.
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Figure 9 illustrates an ultra wideband system designed as a precision altimeter and obstacle/collision avoidance sensor. Originally developed for the U.S. Marine Corps' Hummingbird unmanned aerial vehicle, the sensor has proved capable of detecting small diameter (0.25" or 6.35 mm) suspended wires to ranges beyond 250 feet. With a peak output power of only 0.2 Watts, the system operates in the C-band region from 5.4 to 5.9 GHz (8.9% fractional BW) and has an average output power at 10 kpps of less than Range resolution of the radar was better than one foot utilizing the leading edge detection capability.
For the Hummingbird application, the system incorporated a linear forward-looking phased array (cf. Figure 9 right), and broad beamwidth side-looking antennas, for use in autonomous control. Interestingly, a predecessor of Hummingbird was developed for the U.S. Naval Air Systems Command as a multifunction precision altimeter, collision avoidance sensor and low data rate communications system. A 1 Watt version of the radar operated as a precision (1 foot resolution) radar altimeter to an altitude of better than 5000 feet. Several variants of the Hummingbird radar have also been developed. For example, Figure 10 illustrates an ultra wideband backup sensor for the detection of personnel, vehicles and other objects behind large construction and mining vehicles.
Operating with approximately 250 mW peak in the C-band region from 5.4 to 5.9 GHz, the backup sensor utilizes a dual antenna configuration for the detection of objects as close as 1 foot to beyond 350 feet from the vehicle. Ultra wideband provides a significant advantage for this application because of the ability to provide precision range gating to eliminate clutter which, with conventional Doppler-based sensors, often results in large false alarm rates. This sensor was developed for the National Institute of Occupational Safety and Health. Another variant of the Hummingbird collision avoidance sensor was developed as part of an electronic license plate for the U.S. National Academy of Sciences' Transportation Research Board (Figure 11 below). The UWB Electronic License Plate provides a dual function capability for both automobile collision avoidance and RF tagging for vehicle to
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roadside communications. Collision avoidance functions are achieved with a miniature, 500 MHz bandwidth C-band UWB radar; and RF tagging functions are accomplished with a low power, 250 MHz bandwidth L-band system.
The UWB C-band radar utilized a 0.2W peak power (4 µW average) waveform to achieve a range of better than 100 feet against other vehicles, with an accuracy of better than 1 foot. The L-band tag operated with a 0.3W peak power (500 µW average) packet burst transmission to achieve a data throughput of 128 kb/s over a range exceeding 800 feet. An ultra wideband solution was chosen for the Electronic License Plate because of its precision ranging capability (radar mode) and high multipath immunity (tag mode). Another short range radar, this time operating in the X-band region of the spectrum, is shown below in Figure 12. This prototype sensor was developed for the U.S. Army Missile Command as a low probability of intercept and detection (LPI/D), anti-jam, radar proximity sensor for medium caliber, small caliber and submunition applications. The system exhibited an operational bandwidth of 2.5 GHz with a 10 GHz center frequency. Specifically designed for very short range applications (less than 6 feet), the UWB sensor has a 6 inch range resolution. With an average output power output of less than 85 nanowatts, a -4 dBsm target could be detected at a range of approximately 15 feet using small, microstrip patch antennas.
A variant of the X-band UWB radar fuze is currently being developed for DARPA's Micro Air Vehicle (MAV) program under a Phase II Small Business Innovation Research (SBIR) contract. Figure 13 illustrates a mockup of a 4 inch micro helicopter with an array of four X-band UWB antennas. Weight and size are obviously driving factors for this design, and a UWB chipset is being developed for an onboard collision and obstacle avoidance sensor.
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Figure 14 illustrates a UWB intrusion sensor radar which was designed for throughthe-wall penetration. With an L-band operational frequency and 33% fractional bandwidth, this system utilizes a 1 Watt peak UWB waveform to detect personnel through several intervening walls.
Broad area surveillance coverage was provided for both in-building and outdoor field environments. An extended range system was also developed to detect and track human targets at distances exceeding 1000 feet. Figure 15 illustrates the switched antenna array used with this broad area surveillance system. Target azimuth and distance are determined and used to point a camera in the direction of the target.
CONCLUSIONS Ultra wideband technology has its origins in the development of time-domain (impulse response) techniques for the characterization of linear, time-invariant microwave structures. The advent of the time-domain sampling oscilloscope (Hewlett-Packard c. 1962) and the development of techniques for subnanosecond (baseband) pulse generation provided the requisite tools for further basic research. In the early 1970's, impulse or baseband techniques were applied to a large number of potential applications ranging from low cost, high resolution radar to specialized communications systems having low probability of detection and low interference potential.
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Within the United States, much of the early work in the UWB field (prior to 1994), particularly in the area of impulse communications, was performed under classified U.S. Government programs. Since 1994, much of the work has been carried out without classification restrictions, and the development of UWB technology has greatly accelerated. This paper has illustrated a number of recent UWB developments in the fields of communications, radar and localization. A graphical summary of some of these applications, for both the military and commercial markets, can be seen below in Figure 16.
ACKNOWLEDGMENTS The author wishes to thank Dr. Gerry Ross for his introduction in 1984 to this fascinating technology. He also wishes to thank Mr. Robert Mulloy, MSSI, for his continued support in the development of new applications for this technology over the past 15 years. REFERENCES Bennett, C.L. and Ross, G.F., 1978, Time-domain electromagnetics and its applications, Proceedings of the IEEE, Vol. 66, No. 3, pp. 299-318. Fontana, R.J., 1997, A novel ultra wideband (UWB) communications system, in Proceedings MILCOM 97. Fontana, R.J., 2000, Experimental results from an ultra wideband precision geolocation system, in EuroEM 2000. Robbins, K., 1972, Short Base-band Pulse Receiver, U.S. Patent No. 3,662,316. Ross, G.F., 1963, The transient analysis of multiple beam feed networks for array systems, Ph.D. dissertation, Polytechnic Institute of Brooklyn, Brooklyn, NY. Ross, G.F., 1966, The transient analysis of certain TEM mode four-port networks, IEEE Trans. Microwave Theory and Tech., Vol. MTT-14, No. 11, pp. 528-547. Ross, G.F., 1968, A time domain criterion for the design of wideband radiating elements, IEEE Trans. Antennas Propagat., Vol. 16, No. 3, p. 355. Ross, G.F., 1973, Transmission and reception system for generating and receiving base-band duration pulse signals for short base-band pulse communication system, U.S. Patent 3,728,632.
A LOW POWER, ULTRA-WIDEBAND RADAR TESTBED
Tim Payment Time Domain Corporation Huntsville, Alabama USA
ABSTRACT The availability of custom timer and correlator chips has enabled the design and development of a general purpose UWB research instrument by Time Domain Corporation. One configuration of this tool serves as a basic UWB Radar Testbed suitable for research on antennas and algorithms aimed at specific applications of UWB radar. The system includes the UWB radar unit, an antenna assembly, and a personal computer for user interface and data storage. The radar unit houses the UWB transmitter and receiver under the control of an embedded processor that accepts commands from and provides data to the controlling PC over an Ethernet link. The antenna assembly consists of separate transmit and receive antennas mounted on a back reflector and cabled to the radar unit via connectors to allow experiments with alternate antennas. The system can be viewed as a radar response or received waveform capture device. The waveform is displayed and the user can save the received waveform data to a file for post processing and/or application-specific algorithm development. This radar is particularly useful as a design and development system where a commercial product is envisioned as the ultimate end goal. While the form factor of the Radar Testbed (with its PC, Ethernet, radar unit, and external antenna unit) does not resemble an end product, the fundamental UWB building blocks in this system are applicable to a variety of products. Similar versions of this basic UWB radar research tool are in development to support research into communications and geo-location applications. INTRODUCTION A Radar Testbed system for investigating low power ultra-wideband (UWB) radar applications has been configured as shown in Figure 1. The Radar Testbed consists of a personal computer running a custom application under Microsoft Windows NT, a PulsON™ Application Demonstrator (PAD) in a radar configuration, and a radar antenna assembly. The Radar Testbed operates at a low RF power level. Low power in this context is defined as RF transmissions on the order of 10 µW. Radar Testbeds are operated by TDC Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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under Special Temporary Authority (STA) to support the development of products in compliance with the FCC Waiver of June 1999 [ref. 1]. Waiver compliant devices are limited to 30 nW / MHz (effective radiated power) along with other operational constraints. At these power levels the energy from many transmitted signals must be accumulated to form useful radar returns. This paper provides an overview of the Radar Testbed System, describes its key modules and subsystems, shows sample waveforms, outlines some research directions, then concludes showing how similar subsystems might fit in a candidate product. First, some terms are defined. Definition of Terms and Acronyms
UWB
Ultra-WideBand - RF systems whose signal bandwidth is 25-100% of the center frequency TDC Time Domain Corporation, Huntsville, Alabama, USA PulsON™ Trade name given to TDC’s time modulated, UWB pulse technology PAD PulsON™ Application Demonstrator - Three board set of hardware, software, and firmware in an enclosure designed to interface to a Personal Computer. Provides the basic UWB Radio Frequency (RF) functions needed to investigate communication, geopositioning, and radar applications. SMA Sub-Miniature A connector for coaxial cable Pseudo-random Noise PN STA Special Temporary Authority issued by the US Federal Communications Commission (FCC) to allow otherwise restricted emissions at a specific location for a specific period of time. monocycle short pulse approximated by the first derivative of a gaussian function FPGA Field Programmable Gate Array Low Noise Amplifier LNA Application Program Interface API Application-Specific Integrated Circuit ASIC User Interface UI DESCRIPTION OF RADAR TESTBED MODULES AND SUBSYSTEMS The personal computer used in the Radar Testbed system is a conventional, Pentium™-based computer running Microsoft Windows NT™ 4.0 operating system. A high capacity, removable storage device facilitates data transfer when the testbed is used in a stand-alone mode of operation. A second Ethernet card supports a network connection
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when available. A custom Radar Testbed application, developed in Visual Basic, serves the returned waveform acquisition, review, and archive needs of a radar engineer. The antenna assembly for this system is adapted from a previous, prototype UWB radar unit developed by TDC in 1998 and designated as the RV1000 [ref. 2]. Both the Tx and Rx antenna elements are a magnetic slot design with horizontal polarization. These magnetic antenna elements are mounted 1.3 inches from an aluminum plate, which serves as a back reflector. SMA connectors allow alternate antenna/back reflector assemblies to be substituted for the basic assembly as appropriate for investigation of specific applications. The key module of this testbed is the PulsON Application Demonstrator (PAD). The PAD was first described by Petroff [ref. 3] in Sep 1999. A simplified block diagram of a PAD in a radar configuration is shown in Figure 2. A connection is established between
the embedded processor and the host PC via the Ethernet link. The PAD accepts commands that configure the fixed and programmable logic, executes in accordance with those commands, and returns waveform data to the PC. The gate array provides a register interface between the embedded processor and the hardware. The gate array implements pseudo-random noise (PN) coding and provides 24 digital timing control signals to each of the two Timer subsystems. The PN coding serves to smooth the emitted RF spectrum, mitigate range ambiguity, and enable co-site operation of multiple units with different codes. One Timer subsystem provides the trigger signal to the Pulser which generates a wideband monocycle pulse (add figure??) to the Tx antenna element. The other Timer subsystem provides the trigger signal to the Correlator subsystem for sampling the received waveform at a precise time with respect to the transmitted pulse. Both timers work in conjunction with the 20 MHz master clock. A low noise amplifier (LNA) and a filter provide signal conditioning of the received RF prior to the Sampler subsystem. The Timer and Sampler subsystems under control from the FPGA form the heart of the PAD. Each of these subsystems use first generation, TDC-developed ASICs. The Timer subsystem is depicted in Figure 3. Digital signals from the gate array provide inputs to the Timer ASIC both directly and via high-speed D/A converters such that timing trigger pulses can be placed at a resolution of 3 picoseconds. System jitter limits the timing accuracy to approximately 20 picoseconds. The details of this Timer ASIC and its
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operation are described by Rowe et al. [ref. 4]. For the case of a Tx timer, the trigger is routed to the Pulser circuit which drives the Tx antenna element. A duplicate Timer subsystem provides the trigger to the correlator. The Sampler subsystem is diagrammed in
Figure 4. The trigger from the Rx Timer subsystem is coordinated with the Tx Timer subsystem such that a small (~180 ps) sample of the RF signal is taken by the Correlator ASIC at a precise time difference relatative to the transmitted pulse. More details of the Correlator ASIC are described by Dickson and Jett [ref. 5]. This analog sample is converted to a digital word and read by the gate array as a raw sample. Many (typically 100’s to1000’s) raw samples are summed by the gate array to form an integrated sample for each point in time relative to the transmitted pulse.
The Field Programmable Gate Array (FPGA) provides the real time control signals for the Timer subsystems and reads the raw samples from the Sampler subsystem as shown in Figure 5. Various other control signals from the gate array to the subsystems are omitted for clarity. The gate array is closely coupled to the embedded StrongARM® processor by a number of registers. The embedded StrongARM® processor provides the next higher level of control and gathers a sequence of integrated samples to form the return waveform. These waveform segments are passed to the controlling PC. The embedded software on the StrongARM®
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is linked with an Application Program Interface (API) module on the PC via the two-node Ethernet. From an application viewpoint, the PAD and its API module on the PC can be viewed to a great extent as a black box. Figure 1 is redrawn as Figure 6 to emphasize such a black box view. Other applications are under development at TDC that take a similar view of the PAD and its API, but take advantage of comparable PAD functions for communications and geo-positioning instead of radar waveform functions.
RADAR TESTBED SYSTEM OPERATION The users of this system are Radar Engineers. The Radar Testbed system allows the Radar Engineer to specify how a UWB waveform is to be measured, view the returned waveform (as measured or differenced from a reference), and file the results for use by other analysis tools. The parameters listed in Table 1 are user specified. Pulse Repetition Frequency (PRF) controls the average time between RF transmission of each pulse. A PRF of 5 MHz sets an average of 200 ns between pulse transmissions. In theory, return waveforms could be 200 ns in duration (corresponding to 100 ft or 30 m in range), however the PN coding reduces the useful response time to 175 ns as the instantaneous time between pulses is varied from 175 to 225 ns. The code file values control these instantaneous variations. Code values and the length of the code file are selected to smooth
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the RF spectrum and allow co-site operation of multiple testbeds without interference. The Rx Start and Stop times allow the user to focus on a particular portion of the returned waveform. The units are fundamentally time, however the Radar Testbed application allows the user to alternately specify these values as distances (feet or meters). Similarly, Step Size allows the radar engineer to specify the granularity of the return
waveform sampling in time (or distance). Regardless of how the user specifies Start, Stop, and Step, the values are converted to the nearest integral number of internal clock counts prior to scanning the waveform. Integration specifies the number of pulses and received raw samples to be used in the measurement of each point in the returned waveform. Integration can be viewed as a dwell duration or pulse count for each point. The final user parameter is the Rx Gain. The correlator ASIC includes an RF gain stage that can be varies from 0-30 dB. This gain is adjusted to keep the received signal level at a measurable level. A portion of the measured waveform from a loop-back test with an SMA cable and attnuators from the pulser output to the sampler input is shown in Figure 7. The positive pulse has a width of approximately 700 picoseconds.
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With transmit and receive antennas connected in place of the loop-back cable, Figure 8 shows a typical response with and without a designated target. Annotations on the figure identify the coupling from the transmit to receive antennas, and the response of the
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environment, including the target of interest. With the addition of the Tx and Rx antenna elements and a back reflector, the waveform of Figure 7 gets somewhat more complex as seen in the antenna coupling region of Figure 8. Two waveforms are plotted, a reference waveform taken without the target, and a response waveform with the target. The target in this case is a 14” square aluminum plate perpendicular to boresight at a range of 6 feet. Only minor differences in the two waveforms can be seen in the vicinity of 6 feet. By saving a reference waveform taken without the target present, the Radar Testbed can be set to show only the difference between the current scan and the stored reference. Such a difference waveform is shown in Figure 9.
The Radar Testbed Systems are serve as TDC internal tools for UWB radar research. Data from these systems are being used to complement previously measured values from earlier systems. A number of experiments are planned, including signal propagation through walls, antenna element comparisons, and target polarity response. This information will be compared with alternate UWB propagation results measured using a scanning receiver as reported by Withington, et al. [ref.6] and used to support system design trade studies for specific applications. To support these experiments, Radar Testbed systems are cart-mounted to facilitate movements between labs and positioning of antennas. Figure 10 identifies the major components of a radar testbed system.
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The form factor of the Radar Testbed is appropriate for a research tool, but differs significantly from the form of a commercial product. However, the key subsystems used in the Radar Testbed can be directly applied to candidate products. Algorithms and user interfaces developed on the PC of the Testbed would be embedded in a final product. Figure 10 shows a candidate mapping from the Radar Testbed to a Radar Product. The performance of the key RF subsystems in the product can be directly predicted from their performance in the testbed system. The Testbed provides the means to validate the antenna configuration, algorithms, and user interface prior to the product decision. The areas not proven by the Testbed are the actual performance of algorithms on the embedded processor, the quality of the user interface on the embedded display, the product power system, and the product enclosure. While these items are by no means trivial, their specification and development are not unique to UWB systems. Considerable expertise exists in the implementation of these non-UWB areas.
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CONCLUSION The pieces for UWB products are falling in place. First generation ASICs have given rise to Timer and Correlator Subsystems. These subsystems have been implemented in a general-purpose box under control of software and firmware. This box, under control of a PC and equipped with appropriate antennas, serves as a UWB testbed for (in this case) radar. The path from testbed to product is straightforward. Just as yesterday’s circuits became today’s ASICs, today’s subsystems and applications are earmarked for tomorrow’s ASICs to reduce the size, power, and cost of future products. ACKNOWLEDGEMENTS The author wishes to thank the entire engineering team at Time Domain Corporation. Without the first discrete radios, the timer and correlator circuits would not have existed. Without those circuits, the ASICs could not have been defined. Without those ASICs, the subsystems, firmware, and software being implemented in the PADs and other systems would not be possible. These subsystems and modules allow meaningful research into system applications and the development of products. REFERENCES
1.
FCC Waiver to TDC, Letter to David Hilliard from Dale Hatfield dated 29 June 1999.
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2.
3. 4.
5.
6.
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M. Barnes, “Covert Range Gated Wall Penetrating Motions Sensor Provides Benefits for Surveillance and Forced Entries,” Presentation at 1999 ONDCP International Technology Symposium, Washington, D.C., March 1999. A. Petroff, “Time Modulated UltraWideband Performance on a Chip,” Presentation at UWB Conference, Washington, DC, September 1999. D. Rowe, B. Pollack, J. Pulver, W. Chon, P. Jett, L. Fullerton, and L. Larson, 1999, “A Si/SiGe HBT Timing Generator IC for High-Bandwidth Impulse Radio Applications,” Custom Integrated Circuits Conference, San Diego, CA, May, 1999. D. Dickson and P. Jett, “An Application Specific Integrated Circuit Implementation of a Multiple Correlator for UWB Radio Applications,” S38P6, IEEE MILCOM 1999, Atlantic City, NJ, November 1999. P. Withington, R. Reinhardt, and R. Stanley, “Preliminary Results of an UltraWideband (Impulse) Scanning Receiver,” S38P3, IEEE MILCOM 1999, Atlantic City, NJ, November 1999.
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ULTRA-WIDEBAND PRINCIPLES FOR SURFACE PENETRATING RADAR
J. Sachs*, P. Peyerl**, M. Roßberg*, P. Rauschenbach*, J. Friedrich** *Ilmenau Technical University, Germany
[email protected] **MEODAT GmbH, Germany
[email protected]
INTRODUCTION Surface Penetrating Radar (SPR) uses the properties of RF- and microwave pulses to penetrate into soil and most non-metallic building materials. Obstacles in the way of propagation cause reflections which may be received outside the body of investigation. Thus SPR can be used to detect hidden objects and to investigate the internal composition of manifold structures. The SPR antennas are moved over the surface of the body of interest touching it or retaining a certain distance depending upon the type of employment. The difficulty of the method is in the interpretation of the gathered data as the waves are sensible to all variations of the permittivity and conductivity within the body and not only to the objects searched for. Furthermore the relative long wavelengths (cm and dm range) of the sounding waves provide radar images which are not immediately accessible to the common optical interpretations of the human being. It is to be expected however that these drawbacks will be overcome in the future by using sophisticated software tools for data interpretation and an improved method of data gathering. Thus, SPR will be a powerful tool in many applications such as non-destructive testing in civil engineering, testing and surveillance of transport routes (roads, railways, bridges, tunnels etc.), environmental protection, detection of anti-personal landmines etc. The assumption of successful data processing is based on high quality data which could also include multistatic and polarisation information according to the specific application. In what follows, aspects of wideband measurement technique for SPR data gathering will be emphasised. It should however be noted that high quality data in terms of SPR not only means precision measurements of scattering amplitude as a function of time (or frequency) but also of space. This - the antenna positioning - will not be regarded here. A SPR device will be considered as an LTI-system (linear time invariant). The most important technical parameters will be derived attempting to find a common base in order to compare
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the different working principles. The basic wideband principles will be evaluated and a new principle will be introduced.
LTI-MODEL OF A SPR ARRANGEMENT The aim of a SPR-device is to gather information from an object under test by the use of the scattering properties of electromagnetic waves. In order to do this, the object is touched by a wave and its reaction to this wave is measured. In the simplest case, two antennas are used (see Figure 1) - one for sending and one for receiving. The use of one antenna for both - sending and receiving - is rare because of antenna mismatch. The application of arrays with more than two antennas will however be increasingly seen as an area may be scanned in a faster way and multistatic and polarimetric data may provide more information about the body under test. It will be usual to introduce two interfaces in a SPR arrangement. From the standpoint of an SPR-user and a simple interpretation of the images, the radiators are referred to virtual point sources (see Figure 1) which are considered as sources of spherical waves. In contrast to that, the measurement plane, to which we will restrict further consideration, is more common from the standpoint of the radar electronics. It is defined by the input/output channels and for antenna arrays respectively). The antennas are attributed to the measurement object by embedding the real object of interest. It is referred to [1] for relating between both interface concepts. Regarding the measurement plane, the radar electronics represent a two-port measurement device (N-port-device in case of an array) and the body under test plus its embedding (further called system under test) may be looked upon as a linear two-port (N-port). Assuming the antenna displacement during the measurement time (observation time) is negligible
time independent behaviour can be supposed and classical network theory can be applied. In (1), c means the propagation velocity of the wave, the maximum displacement speed of the antennas and B is the bandwidth of the sounding wave. At fixed antenna positions, the system under test is completely determined by its N by N scattering matrix S: for the frequency domain
or for the time domain Herein a is a column vector of the normalised guided waves incident to the antenna feeds, b is a column vector of the normalised guided waves leaving the antenna feeds and S is the scattering matrix of the system under test. S(f ) represents a set of Frequency Response Function (FRF) and S(t) a set of Impulse Response Functions (IRF). They are mutually referred by the Fourier Transform. Underlined symbols mean complex valued functions and refers to the convolution. The individual functions of the S-matrix will permanently change through the moving of the antennas over the ground. They represent the reflection behaviour (monostatic mode) or of the antenna i at position and the behaviour of the transmission path or between the antennas i and j (bistatic mode) at positions and In practice, these functions form the so called radargrams (B-
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scan) and radar volumes which serve to interpret the inner structure of the body under test. Generally the time representation of the measurement results are preferred because it is more accessible to human imagination. The Fourier transform however permits the change of the domains at will so that software procedures may also act upon frequency domain data if advantageous. Regarding the current state of development, it should be stated that for simplicity only the transfer characteristic or, between one antenna pair is measured. The use of several antenna pairs at the same time is rare in practice. But a further improvement of the SPR-technology supposes antenna arrays and will only arise if sophisticated correction of systematic device errors is applied. Especially the last point demands the knowledge of the full S-matrix of the array. Only the network analyser which is not useful to employ in the field currently meets the stability requirements for error corrections with respect to random fluctuations and drift. Stable and integrated ultra wide band electronics with excellent noise suppression are required for further development of highly sophisticated SPR devices.
PERFORMANCE PARAMETERS The key features describing the performance of the radar electronics in an SPR device refer to its spatial resolution in range and cross range to the observation range R (unambiguity range), its sensibility for detecting weak reflecting objects and to the measurement rate These parameters have to transform to corresponding properties of the IRF or FRF measured by the radar electronics. For details on the performance of the whole radar device, the reader is referenced to [2]. To illustrate the facts, Figure 2 indicates an idealised curve of the IRF of the transmission path resulting from the simple situation in Figure 1. Pulse like sections appear which provide clues as to overall length and attenuation of the individual
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propagation paths. These results are finally used to reconstruct the inner structure of the body. The first two impulses in Figure 2 will merge if the antennas are in contact with the surface.
Spatial resolution. The range resolution depends upon the capability to distinguish between two adjacent pulses of equal amplitude. It is evident from Figure 2 that this may be expressed as:
whereas τ is the half-value width of the pulse envelope and B is the corresponding bandwidth. The effective usable bandwidth B of a SPR device is not only determined by the bandwidth of the stimulus signal and the antennas but also from the time jitter arising from instabilities in the transmitter and receiver circuit. The cross range resolution also strongly depends upon the pulse width if synthetic aperture processing is applied as well as the beamwidth of the antennas.
It should be noted that the carrier frequency does not apparently play any role in equations (4) and (5). It is however very critical since the stimulation band should be at frequencies as low as possible to avoid unnecessary damping of the sounding waves in the body under test. This leads finally to the requirement of a large fractional bandwidth for the electronics and the antennas which is not always easy to achieve in practice. Observation range: The observation range R (unambiguous range) depends upon the length of the time window for which the impulse response is measured. In the case of periodical stimulation signals, it is limited to its period T to avoid time aliasing.
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Measurement rate: The time which is needed to gather all the data which is included in a complete IRF or FRF will be called observation time As such, the repetition rate for the measurement is
In case of an antenna array of L elements the measurements rate factor L [3][4].
may be reduced by the
Detection limit: The detection limit describes the capability to find still small scattering amplitudes in the IRF that are caused either from small scatterers with poor dielectric contrast or by propagation loss. The sensitivity of an SPR device is limited by all deviations from a straight flat base line within the IRF if not any scatterers are present. This means, referring to Figure 2, that only the first impulse may appear. All deviations from that represent errors which limit the sensitivity. Depending upon the error phenomenon, several parameters are usual to characterise these deviations by the notation signal-to-perturbation-ratio. Let us suppose that the maximum detectable amplitude of the IRF is then the following three values can be defined: signal to noise ratio
peak to side lobe ratio
peak to spurious ratio is the effective value of random noise , the maximum side lobe amplitude and the maximum spurious lobe amplitude. Note that the values above are in respect to an IRF which may also result from signal processing, thus can be much higher than the real measured signal amplitudes. Corresponding holds for the perturbation values and which may be reduced in the digital domain by averaging, deconvolution and certain kinds of error correction. It is distinguished between and for the following reason: is caused by linear effects (ripples in the stimulation band, abrupt cutting of frequency band, device internal reflections etc.) and is the result of non-linear effects in the receiver and crosstalk by clock lines or similar. Very hard constraints with respect to may arise for shallow target detection since the scattering peak is located very close to the main lobe where the strongest side lobes can also be found. Note however that the antenna is often the most critical component in this respect. Finally, the maximum depth / minimum size of a scatterer may be estimated from the effective system performance
1 Note, that system performance is seen as a ratio between max. transmitter amplitude and effective noise amplitude, whereas refers to a more usable value respecting receiver overload, non-linearity etc.
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corresponds to the attenuation of the strongest transmission path, e.g. antenna breakthrough or surface reflection. The key to a high side lobe and spurious suppression is sophisticated hardware design. These phenomena are however systematic errors, thus they may be corrected by an appropriate calibration routine. With regards to linear effects, the corresponding methods are well known from network analyser theory (response-, 3-term-, 12-term-calibration etc.). Comparable procedures seems also to be applicable to reduce the influence of spuriousness. It must however be noted that software corrections are only successful if all the required data2 is available and the system is working stable over time – that is equidistant sampling and poor in random noise, jitter and drift.
MEASUREMENT PRINCIPLES Three basic measurement principle are known which are mainly distinguished by the kind of stimulus signal that is applied. In what follows, they will be shortly introduced for the example of a transfer measurement and their advantages and disadvantages will be mentioned. Impulse technique: The impulse technique uses the fact that the convolution (3) may be simplified if the stimulus represents a Dirac-like pulse Referring to the notations in Figure 1 this provides
Since can be measured by an oscilloscope, the image of its screen represents the IRF if the bandwidth of the stimulus pulse is larger than the bandwidth of the antennas. Nearly all commercial available SPR-devices work on this principle because of its simple technical implementation. In order to concentrate the energy of the stimulus in the pass band of the antennas, monocycles are rather used than pulses. The impulse generation is largely based on avalanche transistors or step recovery diodes and sequential sampling circuit serves as receiver front-end. In the following, some problems of pulse systems are summarised. Often, the measurement rate is relatively low because the avalanche transistors need time to recover from the pulse shocks and moreover only one data sample is captured per pulse. This limits the use of the method in large arrays and high-speed applications. It is reported in [6] on a module with improved measurement rates to overcome this drawback. Peak power is however lost. The mean energy of a pulse is very low even for relative high amplitudes and the noise bandwidth of the sampling gates is very large. The method is therefore sensitive to random noise. The noise influence may be suppressed either by averaging (further reducing the measurement rate) or by generating extreme high voltage impulses having a peak power up to 100 MW and more [7]. This however represents no practical solution for an industrial application of the SPR method. The sampling gate control is undertaken by voltage ramps. Thus, any inadequateness of these ramps such as non-linearity, temperature drift or noise translates to errors of the
2
For correction of linear errors this means for example, that the full S-matrix must be known which drastically increases the system complexity. 3
The vectors and of antenna location and the matrix representation for multi-channel systems will be omitted for simplicity in the following
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time axis (drift, jitter and non-equidistant sampling). However, the ability of the sampling control to blank-out the troubling antenna breakthrough and surface reflex is often of advantage in pulse methods for deep sounding. On the other hand error corrections as mentioned above are less promising in cause of the drift susceptibility of sequential sampling. Possibly new system designs may overcome these drawbacks [5]. Sine Wave Technique: The sine wave technique determines the IRF roundabout way over the complex
by the
Herein are IFT{} the inverse Fourier transform, the complex valued cross spectrum between stimulus and receiving signal and is the real valued auto spectrum of the stimulus which should be constant over the bandwidth of the antennas. In the technical implementation, a sine wave is stepped or continuously sweeped over the band of interest and the cross spectra is measured via quadrature modulators in the IFband of an heterodyne receiver (network analyser, vector receiver). The step width of the frequency steps determines the unambiguous range. Sometimes, for deep sounding purposes, gated network analysers are used in order to blank out leakage signals such as ground reflection or antenna coupling [8]. Attention should be paid to the inverse Fourier transformation if the stimulation band is smaller than the antenna bandwidth. The side lobes in the impulse response are no longer determined by the antenna response but rather from the abrupt breakdown of the stimulation spectrum These side lobes may be suppressed by windowing the data before transformation but this results in slightly reducing the range resolution. The potential of the method is its excellent drift stability and random noise suppression because of the narrow band receivers as well as its flexibility within the choice of the stimulation band. It is however also the most expensive and slowest method. The FMCW-radar represents an attractive alternative to the stepped frequency radar because of its simplicity, measurement speed and dynamic range. It is based on a homodyne receiver using a stimulus continuously sweeped over an appropriate band. Particular problems arise by a non-linear VCO characteristic thus further expense is necessary (PLL, reference delay or similar). A FMCW-radar is only able to determine the real part of the FRF due to the lack of a quadrature modulator. Consequently, sophisticated calibration routines such as for network analysers are not available and the IRF can only incompletely calculated. A network analyser is more robust against spuriousness caused by a nonlinearity in the mixers than the FMCW-principle since its narrow IF-filters may partly reject intermodulation products.
Correlation Technique: From the theoretical standpoint, the correlation technique is the most flexible method of system identification since it is not fixed to a certain kind of test signal. Comparable to equation (12) the IRF of a device may also be determined by:
in which the correlation functions
and
are defined according to
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The side condition in equation (14) is much more weaker than that in equation (12), because it does not demand a specific shape of the test signal. It refers only to a flat spectrum: constant. In practice this means that the auto correlation should be short compared to the impulse response and with few side lobes. The opportunity to choose different types of test signals opens the possibility to optimise the measurement method with respect to several aspects. The favourable noise suppression of the correlation is based for example on this feature because high energy signals with low amplitudes (small crest factor) can be applied. The handling of such signals is also often easier than that with high peak power. The problem is however to find an appropriate procedure for solving the correlation integral (15). Several solutions are known from which the matched filtering is the most common but these filters cannot be built for ultra wideband purposes with low centre frequencies. Thus matched filtering by analogue filters is not applicable for SPR purposes. White random noise is a good choice for a stimulus signal if interactions between different radar devices should be prevented or if the device is working in a non-cooperative environment (military application). In that case, the conversion of (15) is based on a mixer, a stepped or sweeped delay line and a low-pass filter [9], [10]. The bottleneck is however the delay line which is expensive to manufacture and its properties limit the overall device behaviour to a large extent. This makes it unattractive for industrial SPR use. The answer for ultra wideband principles can only be to carry out the correlation/matched filtering in the digital domain. The simplest way to do this is the so-called polarity correlator. It captures only the zero crossings of the signals. Multiplication, delay and summation is undertaken by digital circuits (XOR, shift register, counter for bit-wise summing). The method could not be realised up till now despite its simplicity. One reason may certainly be found in its time consuming data capturing. A new ultra wideband principle working on a digital correlation/matched filtering will be shortly introduced in the following (see also [3], [4] for more information). MBC-RADAR As noted above, the correlation technique opens the possibility to freely choose the test signal. Thus a great deal of other constraints may be taken into account in finding an optimum solution: Wideband signal (general requirement), Low crest factor signal in order to generate and handle high (mean) power signals (high signal to noise ratio ) by simple electronics, Periodic signal in order to apply undersampling for signal acquisition and averaging for noise suppression, Simple and stable generation in the RF- and microwave range, Simple and stable generation of acquisition clock, Simple and fast correlation algorithm (digital matched filtering), Ability to simply synchronise and to control multi-channel arrangements, High measurement rate, Integration friendly electronics, and High flexibility with respect to the technical implementation. The Maximum length Binary sequence Correlation Radar (MBC-Radar) meets these requirements. The maximum length binary sequence (MLBS) is a special kind of binary random code. Its time shape, auto correlation function and spectrum is indicated in Figure 3.
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An MLBS of order n may be generated by an n-stage shift register using an appropriate feedback. The MLBS period is where is the period of the system clock. Regarding the spectrum in Figure 3, it is useful to fix the equivalent sampling frequency of the receiver circuit to the clock frequency that means one sample per elementary pulse of the sequence. Thus the usable bandwidth B is limited to the range of the MLBS-spectrum. As long as the maximum clock rate of the shift register is respected, there are no limits in the choice of this rate. As such, the measurement system may be simply adapted to a specific measurement situation by varying the clock rate. Table 1 compares different circuit technologies with respect to the maximum bandwidth within reach.
One of the most important features of the new method is, that the sampling frequency in the receiver may be derived in stable manner from the clock rate by an m-stage binary clock divider so that Since the period length T of an MLBS always differs by one clock period from a power of two periods of the MLBS are needed to acquire the complete data set with the equivalent sampling frequency The practical implementation of the principle is demonstrated in the block diagram of Figure 4. Except for the emphasised part, all components are low cost commercial ICs. The
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whole system is triggered by a stable clock generator that pushes an n-stage shift register and an m-stage binary divider. The shift register generates the MLBS stimulus and the divider delivers the sampling clock which drives the S&H circuit, the ADC and an averager. Finally, a digital signal processor (DSP) calculates the cross correlation which is approximately equal to the IRF of the system under test:
Due to the periodicity of the MLBS, the function represents a cyclic cross correlation function which can be calculated very fast by the Hadamard-Transform.
The p-fold averager matches the signal acquisition rate to the processing speed of the DSP by reducing the data rate to Simultaneously, it increases the dynamic range of the captured signal. The dynamic range with respect to random noise and the overall observation (correlation) time results in:
Here b is the number of effective bits (ENOB) of the acquisition circuitry (S&H and ADC). The last two terms in (17) represent the signal processing gain by which the dynamic of the real signals is improved. is infinity for an ideal MLBS, independent from its order and is mainly determined by the linearity of the receiver circuit (ADC, S&H). The maxi-
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mum measurement rate of the method is mostly fixed by the DSP-hardware. The determination of 16 000 IRFs (consisting of 511 points) per second and more seems to be possible with modern signal processors. The structure of the MBC-radar is divided into three domains each having its own processing speed (see Figure 4). The interconnection between the different sections is formed by programmable digital blocks (divider, averager). This provides a great flexibility to adapt the system parameters to the requirements of the actual measurement and it permits a high flexibility within system design and manufacture. It should be further noted that the simple clock scheme and the ability to enable/disable the MLBS shift register simplifies the creation of multi-channel arrangements like antenna arrays since multiplexing of RF-lines is no longer needed.
CONCLUSION SPRs will achieve wide industrial employment if they succeed in offering adapted solutions for a specific class of tasks. These solutions will be based on sophisticated methods of digital data processing which require stable, high quality data, increasingly gathered by antenna arrays. The future challenge on RF-electronics is to meet these requirements. The main demands on the RF-electronics of a SPR device results from a large fractional bandwidth and a high dynamic range. Several wideband methods were presented. Impulse principles are the most frequently used followed by the FMCW-radar. Sine wave techniques are very flexible. They are particularly suited to laboratory experiments but their low measurement speed may limit field use. A new ultra wide band principle was introduced which is based on a maximum length binary sequence. It is a promising method to capture high quality data and it is suited for application in antenna arrays.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]
[10]
R. Zetik, J. Sachs, B. Schneegast: Evaluation of antenna pattern for radiation in solid media. Proc. of IRS 98, vol. II, p. 629-38 T. Scullion, C. L. Lau, T. Saarenketo: Performance Specification of Ground Penetrating Radar. Proc. of GPR'96, p. 341-6 J. Sachs, P. Peyerl, M. Rossberg: A New UWB-Principle for Sensor-Array Application. Proc. of IMTC/99, vol. 3, p. 1390-5 J. Sachs, P. Peyerl: Ein neues Breitbandmeßverfahren für das Basisband. Workshop of German IEEE/AP Chapter on Short Range Radars, Technical University Ilmenau, July 1999( http://www.meodat.de/veroeff.htm) A. Schukin, I Koploun, A., Yarovoy, L. Ligthart: Evolution of GPR Antennas, Pulse Generators and Sample Recorders. Proc. of AP2000, Davos, Switzerland J. Warhus, J. Mast, S. Nelson: Imaging Radar for Bridge Deck Inspection. http://wwwlasers.llnl.gov/lasers/idp/mir/files/warhus_ spie/spiepaper.html and other publications around the MIR-module. P.R. Bellamy: Ultra Wideband Radar: Current and Future Techniques. Proc. of EUROEM 1995, p. 1620-6 G. F. Stickley, D. A. Noon, M. Cherniakov, I. D. Longstaff: Current Development Status of a Gated Stepped-Frequency GPR. Proc. of GPR'96, p. 311-5 R. M. Narayanan, Y. Xu, P. D. Hoffmeyer, J. O. Curtis: Design and performance of a polarimetric random noise radar for detection of shallow buried targets. Proc. of SPIE, vol. 2496, p. 20-30, Orlando 1995 R. Stephan, H. Loele: Ansätze zur technischen Realisierung einer Geschwindigkeitsmessung mit einem Breitband-Rausch-Radar. Workshop of German IEEE/AP Chapter on Short Range Radars, Technical University Ilmenau, July 1999
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RAY TRACING ASSESSMENT OF ANTENNA ARRAYS AND SUBSURFACE PROPAGATION FOR GPR SYSTEMS
Steve Pennock and Miles Redfern Department of Electronic and Electrical Engineering, University of Bath, Claverton Down, BATH, BA27AY, UK.
ABSTRACT The design of antenna arrays for Ground Penetrating Radar systems has proven to be as much an art as a science. The assessment of the design of the antenna and its housing, together with the mapping of the subsurface propagation has been greatly assisted using the well established technique of ray tracing. In this work its use is extended to include a complex arrangement of electromagnetically different objects, such as is typical of the cluttered subsurface environment that may be encountered by a GPR system.
RAY TRACING ANALYSIS Ray tracing analysis methods are frequently applied in examining radio propagation. The size of the features within the environment, typically buildings or urban areas, are rather smaller than the signal wavelength and the ray tracing analyses are based on geometrical optics assumptions. Dielectric slab waveguides have been studied in the past, with a propagating ray interpretation of the analysis. The Green’s function for the slab can be written in terms of a summation over the finite number of discrete bound modes (N) and an integral over the continuous radiation spectrum [1, 2]:
where
is the transverse wavenumber. As shown in Figure 1 the discrete bound modes
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are guided along the slab by repeated total internal reflection, and the radiation modes can propagate at all possible angles in the region surrounding the slab. Hence by launching rays out from a source at numerous angles, we can then approximate the radiation integral, and explicitly launch the discrete modes.
The environment being studied in the 2-D model used here is divided into a number of equal sized rectangular cells. Each cell is assigned its own permittivity and conductivity. The simulation is based on geometric optics and a large number of rays are launched from the transmitter at equal angle increments. Each ray is traced while its field strength is more than a pre-defined power cut off level. The field of the ray in a cell is:
where K is an amplitude constant, is the cumulative pathlength to that point and is the cumulative attenuation due to reflection and transmission coefficients during transmission of the ray. This is:
where and indicate the reflection coefficient and transmission coefficient respectively whenever the ray is reflected or refracted during its path. and refer to the cumulative attenuation and phase constants as the ray has propagated through the cells. These propagation constants are defined by:
where is the pathlength of the ray in medium is the attenuation constant of medium and is the phase constant of medium i. Both reflection and refraction effects are taken into account. Continuity of propagation constant is used between adjacent cells as in the studies of dielectric slab waveguides and some derivations of the Fresnel equations. This is used to evaluate the wave characteristic impedances in each cell and from this the amplitude and phase of the reflection and refraction coefficients are determined for all combinations of permittivity and conductivity between adjacent cells. The usual spreading loss is applied as the ray propagates and in lossy media the additional conduction loss is also included. The total group delay of each path is also recorded, allowing for simple reconstruction of the delay spread profile at any point in the environment.
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In some instances, particularly close to the transmitter point, many rays will enter a cell that represent the same wavefront. All but one of these are eliminated by examining the effective source positions and cumulative delay of the rays. In addition, refracted rays that originated at adjacent angles from the source, and have the same history of reflection and refraction through the environment, are also reduced to a single ray. This can prevent duplication in later calculations and reduce the filestore space that is used. The information needed to reconstruct the field strength pattern at any point within each cell, such as the angle of entry, distance from source and polarization of each ray are recorded at the boundary of every cell. The effects of all rectangular obstacles, be they lossless dielectric, lossy dielectric or finite conductivity conductors, are included in the model. This model is also able to consider particular transmitter and receiver antenna radiation patterns if needed. The total field at a position is simply the complex sum of the fields of the M individual incident rays:
The model has been seen to compare well with measured radio propagation data in the past [3] and in the study of Inset Dielectric Guide antennas [4]. In addition, as it is not restricted to considering only a finite number of re-reflections, it is very well suited to modeling complex environments.
ANTENNA ANALYSIS The ray tracing and field reconstruction analysis has been applied to the study of an antenna array for a GPR ground mapping system. The system, illustrated in Figure 2, uses one transmitter and 2 receiver antennas mounted within dielectric blocks.
Data available in the literature shown that typical subsurface media, sand, soil, rocks etc have permittivities in the range [5]. A system radiating from air into this will naturally encounter a reflection at the air/ground interface. To reduce this we examine the situation where the antennas are realised in a dielectric block. In particular the standard PCB substrate FR4 has a permittivity of which gives a reasonable match to typical ground media.
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Considering a half wavelength dipole antenna say, its length is reduced by a factor of Hence, a 500MHz half wavelength dipole is about 14 cm long in FR4, rather than 30cm in free space. In addition, if a back reflector is used behind the dipole, the spacing is reduced by the same factor. With this implementation in mind the antenna blocks considered here are typically 24cm × 24cm × 5cm for dipole antennas radiating between 200 M H z and 800 M H z.
A 3 head antenna is considered that consists of 3 blocks with 1.5 cm air gaps between adjacent blocks. These are positioned on the surface of the ground. The ground is modeled using data for relatively wet sand or loam. The calculated field is shown in Figure 3. While a considerable amount of the radiation enters the ground, there is also a significant radiation into the air region over the unit. This gives cause for concern due to its EMC implications and also for the fact that the GPR system would be rather sensitive to objects in the immediate vicinity of the GPR system. In an urban area these objects might well be moving during the measurement cycle and complicate the signal processing. One basic method of alleviating this problem is to place a ground plane on the back of the blocks. The field pattern for this configuration is shown in Figure 4. Clearly the radiation into the air is considerably reduced. The field pattern in the ground is quite complicated and can vary rapidly, indicating a multipath interference situation. The ray path diagram shown in Figure 5 clearly shows that the edges of the blocks are acting as secondary sources. In these diagrams the multipath nature of the signal is apparent, most particularly when there is a target under the unit. The case where there is no target present shows that the multipath interference changes with position under the unit, and so the response to a target must vary with the position of the target. This will complicate the signal processing, as a simple target signature or template cannot
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be used for all target positions.
To alleviate the multipath problem small conductor blocks are placed on either side of the antenna housing dielectric blocks. The field patterns for this configuration are shown in Figure 6. The ray tracing analysis used here does not include diffraction effects at the edges of the conductor blocks and is limited in the prediction that the receiver is perfectly isolated from the transmitter in the absence of a target. The field pattern under the unit is however more uniform as some of the additional effective sources have been reduced or eliminated. Multipath pattern are apparent towards the edges of the illumination area. Introducing the metal target results in a finite signal in the receiver block to the left of the transmitter as desired. Multipath interference is
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more apparent on the left. This is caused by re-reflection of signal from the antenna head back into the ground, and is comparable to the reflection that would occur at the ground/air interface if no antenna head were present. Introducing these conductor blocks into the antenna array has improved the illumination into the ground under the unit, but multipath patterns are apparent towards the edge of the illuminated area. In the system considered there are two receivers, which provides the opportunity to use convolution techniques to enhance the system response to targets directly below the unit center line while reducing the response to off-center targets that lie in the multipath affected areas.
EXAMPLE TARGET SCANS The analysis has been used to simulate the response of the two receiver GPR system over an example environment. In this 3 targets were positioned at horizontal locations of -60cm, 0cm and 40cm, with respective depths of 40cm, 15cm and 40cm. The response of the system was modeled as it was moved in 2.5cm increments over a range of 2 metres. The unit operates as a FMCW radar sweeping from 250MHz to 750MHz. The basic raw data produced by the system is shown in Figure 7. The direct path signal between the transmitter antenna and receiver antenna is apparent at all measurement positions, occurring at a constant frequency bin number of 12. Higher frequency bin numbers correspond to greater depths within the ground. The classic hyperbolic signatures of the targets are apparent with peak signal levels about 10 dB below the direct path. The situation is somewhat cluttered due to the interference between the various signal paths in this environment. Using the data at a horizontal location of 100cm as a reference, which is subtracted from all the other measurements, the influence of the direct path can be virtually removed and concentrates the system on the buried targets. In addition a gaussian symmetry filter is applied where the signal is multiplied by the gaussian function of the difference between the two channels. When the two channels receive the same signal, as is the case when the target lies directly under the center line of the system, the signals are not affected as they are multiplied by one. When there is imbalance, as in the case of off-center targets, the signals are reduced in amplitude by the gaussian function.
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The net response for one of the channels is shown in the right hand plot in Figure 7. Clearly the direct path has been greatly reduced and the hyperbolic signatures of the targets is now clearly visible. The reference was taken at 100cm and the targets were still visible to the system at that point. This produces the constant depth signatures in the plot near frequency bins 100 and 140. Convolving the signals from both of the two measurement channels produces the data plot in Figure 8. The constant depth false targets appear in the two individual channel responses at different depths, and are greatly reduced by the convolution process. The major responses in these plots are then clearly at the target positions, as desired .
CONCLUSIONS The modelling of the GPR radar antenna array, together with the propagation of the microwave signals through the subsurface medium has produced a clear insight into the operation of the system and its response to buried targets. As a result improvements to the antenna housing and signal processing system have been identified and evaluated.
References [1] T. Rozzi & M. Mongiardo, “Open Electromangetic Waveguides”, IEE Electromag-
netic Waves Series 43, London, 1997 [2] S.R. Pennock & P.R. Shepherd, “Microwave Engineering with Wireless Applica-
tions”, MacMillan Press, London, 1998 [3] Ch. Ghobadi, P.R. Shepherd & S.R. Pennock, “A 2D Ray Tracing Model for Indoor
Radio Propagation at MM Frequencies and the Study of Diversity Techniques” IEE Proc.-Microw.Antennas Propag. Vol. 145, No. 4, p349-353, August 1998. [4] S.R. Pennock, Y. Weizhong & T. Rozzi, “Circuit and Antenna Properties of Diodes
Mounted in Inset Dielectric Guide”, Proc. 23rd European Microwave Conference, Madrid, Sept 1993, Paper A10.5 p553-555.
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[5] Daniels DJ et al. Introduction to subsurface radar. IEE Proc. Pt. F, 135(4):278– 320, August 1988.
GROUND PENETRATING RADAR SYSTEM FOR LOCATING BURIED UTILITIES
Steve Pennock and Miles Redfern Department of Electronic and Electrical Engineering, University of Bath, Claverton Down, BATH, BA2 7AY, UK.
ABSTRACT In this paper the design and operation of a ground penetrating radar system based on the Frequency Modulated Continuous Wave (FMCW) technique is described. Simulation and example measurements over a test site containing three targets at separations typical of those occurring for real buried utilities are presented. It is seen that the position of the targets is accurately found, and that quite closely spaced targets can be distinguished. The operational range of the system is approximately up to depths of 2 metres, depending on the ground conditions.
INTRODUCTION With the growth in buried energy and communications utilities in our pavements and roads, the demand for mapping these facilities has increased to ensure both human safety and to minimise disruption to services during trenching operations. More recently, the burying of optical fibre ducting for cable communications has added a further complication that is difficult to detect using established mapping techniques. Unfortunately, although breaking these links presents few safety problems, it can have significant financial implications. This has produced a need for mapping tools that can locate optical fibre ducting as well as pipes and electricity cables in the range of just below the surface to one or two meters in depth. Ground penetrating radar is one of the few technologies capable of locating this variety of buried objects [1, 2]. The use of pulsed radar is well established using the sub-nanosecond pulses that are Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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required for this short range probing. These systems unfortunately tend to be expensive and their operation generally requires skilled surveyors to interpret the results. With the dramatic growth in the availability of microwave electronics for the mobile telecommunications market, inexpensive devices are readily available for low energy radar techniques. The Frequency Modulated Continuous Wave (FMCW) technique offers the promise of inexpensive ground penetrating radar implementations. This coupled with the availability of inexpensive and very powerful portable computing systems makes user-friendly systems a real prospect.
THE SYSTEM A basic FMCW ground penetrating radar system broadcasts a simple linear frequency sweep of an RF signal, and measures the difference between the frequency of the return from the target and the currently transmitted signal. This frequency difference between the transmitted and received signals provides an indication of the presence of buried objects and the distance of the object from the transmitter/receiver antennas. As such it is very amenable to detecting short range targets by suitable choice of the frequency sweep rate.
The addition of a second receiver antenna, as shown in Figure 1, provides a differential facility that is able to highlight an object located directly beneath the transmitter/receiver antenna array, or on an axis at an angle from this centre line. The transmitted frequency range is dictated by the propagation characteristics of the media in which the objects are buried. Although high frequencies are preferred to provide higher resolution of the location of an object, lower frequencies are preferred for their lower signal attenuation through the sub-surface media. A compromise is therefore required in the choice of operational bandwidth. The attenuation characteristics of an infill medium are very variable, however typical data is available [1]. From this, practical trials, and the desire to have a system dynamic range of 70dB between transmitted and received signal, an operating frequency range of from 250 MHz to 750 MHz was chosen. In order to minimise concerns over potential interference with electronic equipment, the transmitter output power was limited to 23dBm.
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The transmitter and receiver antenna use a basic tapered dipole design, with the three antenna embedded in a dielectric of similar characteristics to the infill in which the objects were buried. This ensures a reasonable match between the antenna array and the infill reducing the complications caused by surface reflections.
The electronic system is shown in Figure 2. This and the antenna array were configured to be symmetrical around the transmitter channel. The only compromise to this symmetry was the use of a ’simple’ single end feed to the antenna rather than a balun connection. Experience demonstrated that the consequences of this were minimal. The non-ideal characteristics of the relatively inexpensive VCO required compensation using a non-linear control drive signal. The exact waveform of this control signal was determined using a pre-programmed calibration procedure. This takes measures over a known and matched RF path that is switched in to replace the transmit and receive antennas, and then deduces the required variation in control voltage from a Hilbert transform analysis. The frequency modulation period was chosen to provide a de-modulated signal in the audio frequency range of dc to 2.5kHz for targets buried up to 2 metres in typical infill. The sweep duration was about 20 msec, and this enabled fairly simply signal conditioning, sampling and processing in a commercial personal computer. The use of a PC also provided an easily used man-machine interface. During the sweep the gain of the transmitter and receiver amplifiers are altered to compensate for the fact that the signal propagation through the soil to the target and back suffers from an attenuation that increases with RF frequency. This is a relatively simple process that applies a simple gain control voltage profile with time during the sweep. This can easily be adjusted to suit local soil conditions. In a pulsed system such compensation would require an RF pulse compression filter to be applied to the typically used sub-nanosecond width pulses. Adjustment for soil types would require
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control of the pulse compression filter characteristics.
SIGNAL PROCESSING The central processor unit controls the VCO output frequency, and by using the calibration data derived for the VCO, produces an output signal to the transmit antenna whose frequency varies linearly with time from 250MHz to 750MHz. Amplitude control is realised by controlling the gain of the receiver and/or transmitter amplifiers. The basic aim of this is to produce a linear ramp sweep with constant amplitude. The received and transmitted signals are multiplied in the mixer, and the resulting product is low pass filtered to produce the output If the reflected response from the target is constant with frequency, the output from a single target is a constant frequency sinewave burst whose duration is that of the original ramp. The frequency spectrum of this output signal is the combination of two sinc functions [3, 4]:
where is the delay for the signal to propagate to the target and back over the path length d in the soil medium whose refractive index is The rate of change of output frequency with time is the basic parameter that is set to ensure that the return from the buried targets is within a prescribed bandwidth. Each buried target will produce such a response, with the peak frequency revealing the time delay and hence depth to that target. Combining the outputs of our left and right channel measurements as:
the resulting response concentrates on targets below the center of the unit, while reducing the response from those off the center line. The line of concentration can be made to rotate under the unit, and thereby ‘scan’ the subsurface targets, when the combination of the left and right channels is:
Accurate reproduction of the angular positions of the targets requires a knowledge of the refractive index of the subsurface medium, which is generally unknown. Information is however available as to which side of the unit the target is, allowing the unit to be re-positioned over the target and thereby accurately finding the lateral position of the target. The response from the targets can be further enhanced by suitable matched filtering to suppress random noise, and by clutter filtering to remove particular received signals. Such a signal that can cause difficulty is the direct path signal between the transmitter and receiver antenna. In the experimental system, this direct path signal
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was further suppressed by means of the antenna design [5].
TARGET SCANS The first target scan, shown in Figure 3, shows the results from one receiver channel to three objects buried in a test site at known positions. As such this shows the unit operating as per Equation 2. The three targets were located at horizontal position of 290 cm, 330 cm and 390 cm and at depths of 30 cm, 10 cm and 30cm respectively. To the practiced eye, two of the three objects are identified while the image of the third, although identifiable, is masked by a shallow disturbance in the infill. Such an output can hardly be considered user friendly.
The second scan, shown in Figure 4, shows the result of subtracting a reference signal measured at the site, but displaced to the side. This reduces the contribution of the direct path between the antennas, making the 3 targets more apparent. Figure 5 shows the effect of combining the responses of the two receiver systems and thereby concentrating on the targets directly beneath the antenna array. This shows the unit operating as per Equation 3. In this, the two clear objects are easily highlighted and the third object is more clearly seen than before. The positions of these objects are also more clearly defined. We have also evaluated the improvements offered by matched filter and clutter filters. Figure 6 shows the result of applying clutter filtering to one of the channels. The clutter response used was that of the reference signal taken on one side of the main
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measurements. Clearly the targets can be accentuated, but the process can also throw up extra responses that the user or interpreting software need to take into account. The time involved in the extra signal processing is quite minimal, being less than a second per channel on a DX4-100MHz 486 processor. While the processing time for producing the scans shown here is relatively long, about 1 minute, this is seen as a short term problem. Computer processing speeds are continually improving, and the software can be further optimised. The entire process of data capture and processing is governed more by the time taken to move the system over the targets and capture the data.
CONCLUSIONS An FMCW Ground Penetrating Radar system has been described that is suitable for mapping utilities buried in typical street environments, together with examples of the responses obtained using a real test site. Through appropriate choice of the operating parameters, targets spaced at typical separations for buried utilities are clearly discernible in the display. The images may be improved on by the use of filtering, with some extra processing delay, but this need not be onerous. The system has demonstrated the ability to identify targets, and coupled with the availability of inexpensive RF and microwave frequency electronic components, the prospects for providing a user friendly mapping tool are encouraging.
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References [1] Daniels DJ et al. Introduction to subsurface radar. IEE Proc. Pt. F, 135(4):278– 320, August 1988. [2] Michiguchi Y et al. Advanced subsurface radar system for imaging buried pipes. IEEE Transactions on Geoscience and Remote Sensing, GE-26(6):733–740, November 1988.
[3] Olver AD et al. Portable FMCW radar for locating buried pipes. In Radar 82, volume 1, pages 413–418. IEE Publ. No. 216, 1992. [4] Yamaguchi Y et al. Synthetic aperture fm-cw radar applied to the detection of objects buried in snowpack. In IEEE Transactions on Geoscience and Remote Sensing, volume GE-32(1), pages 11–18, January 1994. [5] D. Park and J. Ra, Decoupled Planar Dipoles for Subsurface CW Interface Radar Microwave and Optical Technology Letters Vol 7, No. 8, pp367-370, 1994
COST EFFECTIVE SURFACE PENETRATING RADAR DEVICE FOR HUMANITARIAN DEMINING
J A Ratcliffe (DERA), J Sachs (TUI), S Cloude (AEL), G N Crisp (DERA), H Sahli (VUB), P Peyerl (Meodat), G De Pasquale (IDS)
INTRODUCTION A consortium of European companies and universities1 are collaborating on an EC supported project to develop a novel hand-held Surface Penetrating Radar (SPR) device which is able to detect Anti-Personnel Landmines (APLs). The sensor utilises a new Ultra-Wideband (UWB) radar principle based on Maximal Length Binary Sequence (MLBS) techniques, already successfully tested in acoustic wide band devices. A 6 element multi-static linear antenna array has been designed to fully exploit the radar technology. The array consists of unique planar bow-tie elements with distributed resistive loading. The data processing techniques investigated include the study of inverse problems, adapted to extract features of the reflecting targets, and to eliminate non-target related influences such as antenna and soil characteristics. Pseudotomographic methods extract measurements of the size and shape of the reflectors. This information is combined in a robust target classification algorithm. Acoustic impulse time-of-flight techniques are being used to register the movement of the antenna array in the X-Y plane. In this paper we present an overview of the key principles and techniques which are being exploited in the development of the device. THE DEMINE PROJECT The DEMINE project is one of many R&D projects partially backed by ESPRIT, an integrated programme of industrial R&D projects managed by the European Commission. Several of these projects are collectively aimed at researching, developing and testing new systems for detecting anti-personnel landmines. This paper specifically relates to one of these projects, entitled DEMINE. The DEMINE consortium consists of 4 companies, 2 universities and 1 demining organisation1. The consortium aims to develop a hand held SPR mine locator, which is
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capable of distinguishing between mine and non-mine targets. The handling of the device is to be as close as possible to that of the commonly used metal detector.
SYSTEM CONFIGURATION The planned operation of the device is shown in figure 1. The common use of a metal detector, with a safe demining lane delineated by a marker stick is adhered to. When the exposed area has been cleared the marker is moved forward. Ultrasonic beacons are located on the marker frame to allow the monitoring of the sensor position. This is useful both in the data processing and for marking the position of a potential mine.
DEMINE SPR PRINCIPLE Wideband Surface Penetrating Radar (SPR) is an attractive method for locating APLs as it is sensitive to metallic and non-metallic materials. Also, with careful selection of the operating frequency, adequate soil penetration can be achieved whilst not compromising the wave scattering from an APL-like object with respect to its size, shape and composition. However, the operating frequencies which may be used are still too low to generate an image which corresponds directly to the known optical appearance of the APL. As a consequence, more sophisticated characteristics of the scattering behaviour must be exploited for classifying buried APLs. The idea underpinning the DEMINE project is to gather a large amount of diverse information about a buried object by use of a sensor array. This then allows for the extraction of features with which to classify the object as ‘No mine’ or ‘Possible mine’. The classical SPR approach is shown in figure 2, using a single bistatic pair of antennas. One radar scan is performed at each position as the sensor is passed over the target. Time records are stacked together to form a Radargram or B-scan. A static signal
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from the ground is usually seen and the target returns form a hyperbola. Only one record is available from each sensor position and thus limited information is available.
The DEMINE approach is illustrated in figure 3. An array of antenna pairs is used which are arranged in a line. At each antenna position, 21 different scans may be made. That means, for each transmitting antenna, radar records may be recorded for all the receiving antennas.
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This yields much more information about the target’s scattering characteristics over different angles, thus allowing a more robust classification to be made. Secondly, the scanning requirements placed upon the operator are greatly eased due to the increased ground coverage of the array. Multi-static array processing may be employed in the lateral direction, while the longitudinal scanning motion allows a Synthetic Aperture process to be employed to build a 3-D image.
DEVELOPMENT TASKS Preliminary Examinations The signature of an R2M2 AP mine target buried in dry sand was simulated. The polarisation response, shape and spectral response have all been considered as potential classifiers of the target. In polarisation analyses it was shown that the signature of the target is dominated by an oblate spheroidal dipole response and hence polarisation is not a strong discriminant between mines and symmetric clutter objects such as stones and boulders. With respect to the shape of the target, it was shown that high resolution 2 or 3-D imaging may be useful for target recognition. The spectral transfer function was found to be similar to a mixed first and second derivative function with no significant resonant behaviour across the bandwidth considered. The full bistatic scattering characteristics of the mine were numerically modelled in detail. The scattering data was analysed in both the time and frequency domains. The conclusion was that an HH antenna polarisation would provide the highest signature variation with respect to bistatic angle. This is a desirable feature, as it will give diverse information about the target, thus aiding classification. The modelling showed that the most useful specular scattering information is only obtained above ~2GHz, and that below this the target behaves like a volume dipole scatterer. The frequency range for the sensor was therefore set at 1-4GHz. System Conception The DEMINE device is designed to be operated in a similar fashion to a metal detector. As such the proposed external hardware is illustrated in figure 4. The headphone and head display are included as a result of discussion with the demining partner who suggested that an audible alarm and a clear visual indication of ‘mine’ or ‘no-mine’ would be the best way to pass information to the user. The final device will utilise a backpack for the power supply and miniaturised computing hardware, but the prototype will instead make use of a separate base station with an umbilical cord to the backpack thus easing the weight considerations.
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UWB Antennas The simplest structure which satisfies the requirements of the DEMINE device is the planar Bow–Tie antenna. These antennas are non-dispersive and the input impedance depends only on the angle of the metal plates in a well defined way. Thus it is possible to match the impedance to the DEMINE radar chip by correct choice of bow-tie angle. However, optimisation of bow-tie antennas for a UWB application requires distributed resistive loading along the antenna plates as shown in figure 5. The resistance has the effect of absorbing the pulse propagating out along the antenna, so that when it arrives at the outer edge there is very little energy left. Thus there is no reflected pulse and the impedance will be flat over a wide frequency range.
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Also, the transmitted pulse, coming from the feed point alone, will be a much more faithful version of the input signal. The optimum antenna blade dimensions and resistive loading profile were found by simulation. As the DEMINE device will utilise an array of antennas, issues of antenna cross-talk and rear lobe screening were also considered. These results were then used as a basis for a practical antenna design, in which the blades were split into discrete radial segments to closely approximate the simulated design. The antennas were manufactured using PCB derived technology and are light, compact, and inexpensive. The antenna elements were then incorporated into a pre-prototype array, for the purposes of testing and gathering training data for algorithm development. The preprototype array was made up from two sub-arrays, each with only one active element. This allowed the simulation of the full linear array by adjusting the relative position of the two active antennas. Parasitic elements were added on either side of the active elements and terminated as though connected to the radar head, to accurately represent the interactions between adjacent antennas in the final prototype array. The sub-arrays were mounted on a computer controlled x-y displacement rig and connected to a network analyser via wide bandwidth baluns. Algorithm training data was gathered for targets in free space, targets on a soil surface and targets buried up to 15cm. Radar Head The DEMINE radar head hardware is fabricated in an integrated circuit using Silicon-Germanium technology. There will be one transmit-receive chip for each antenna pair in the array. Fabricating the radar hardware in this way allows easy integration with the antenna array without numerous lengths of cable and without a complicated R.F. switching network. The radar measurement utilises a Maximal Length Binary Sequence (MLBS) technique, which requires simple transmit and receive electronics. The operation of the radar head is described in detail in another paper2. Position Measurement Measurement of the position of the sensor head during data acquisition is essential for the construction of the 3-D radar image. Accurate instantaneous measurement of the position in the X-Y plane is the most critical aspect, while knowledge of the height change of the sensor head over each scan is also desirable for good quality imaging. The attitude of the sensor head during data acquisition is also an issue. Technologies assessed for feasibility in this application included accelerometer measurements, acoustic methods, RF methods, GPS, mechanical methods and optical methods. An acoustic method was considered the most appropriate for the X-Y measurement in terms of technical suitability, practicality, and cost. Figure 6 shows generically the concept that is employed. Three sets of ultrasonic transceivers measure the distance between 3 points on the ground and the 2 ends of the sensor head. This is achieved by measuring the time of flight of short bursts of ultrasonic energy. By utilising 3 points on the ground separated in height as well as laterally it is possible to remove errors caused by variations in the sensor height.
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However, the height baseline utilised on the ultrasonic system is not large enough to measure height with the accuracy required for the 3-D image construction. An accelerometer/magnetometer system is being investigated for this purpose. The system uses magnetometer data to correct tilt induced accelerometer errors. In order to alert the operator of excessive tilt during the scanning procedure, a simple fixed accelerometer will be used with filtering to remove accelerations caused by height change. Data Processing & Classification The DEMINE system will use a well defined and recognised data processing route as shown in figure 7. Raw data includes all the radar data plus all position data, both with time stamps. Data correction is operated on the radar data and is designed to remove unwanted artefacts before image reconstruction. This includes antenna cross-talk removal, equalisation of the different transfer functions of the different measurement channels and correction of certain known non-linear effects introduced by the radar chips. At the image construction stage the position data is combined with the radar data during multi-static array processing and synthetic aperture processing. Also the ground parameters (permittivity, attenuation, wave speed etc.) are estimated and included in these processes in order to enhance image quality. Certain features in the responses of targets and clutter objects can be exploited in order to tell them apart. These include spatial features which are dependant upon the size and shape of the objects, which are extracted using pseudo-tomographic methods, and time and frequency domain features which can yield other information about the scattering properties.
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A series of feature vectors is then available which may be used to classify the target. These are represented by a set of points in a multidimensional feature space. Each such point represents a given observation that must be classified according to its location in the feature space. The classification process thus reduces to the association of points in feature space with corresponding classes. Several classification processes have been considered, including: tree based classifiers, automatic adaptive classifiers, parametric classifiers, non-parametric classifiers and distribution based classifiers. A combination of parametric and distribution based processes has been considered the most appropriate, although this will be dependant upon well clustered features being found, so that simple classification boundaries can be constructed. However once such features have been found, the use of a parametric form allows relatively sparse training data to be used to train the classifier. Exploitation While the successful operation of an APL detector of this type would provide a great humanitarian benefit, the market for such a specialised device is small, especially given the limited resources and funding available to humanitarian demining organisations. This makes the unit cost of such a system comparatively high. So in order to make a system which is financially viable and affordable to the demining community, other commercial applications for the device or its subsystems must be found. So along with the main product of an SPR mine detector, the following additional products have been identified as having potential for future exploitation:
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Vehicle Based Mine Detector General SPR detector (pipe inspection, NDT, monitoring historic buildings...) UWB antennas and array New software for commercial applications (CAST) Integrated UWB radar chips Positioning system SPR detection / classification algorithms These and other possibilities will be assessed in a future phase of the project. CONCLUSION The current technical status of the project is summarised by the following main points: The concept of the SPR device is developed Preliminary analyses including simulations are complete UWB antennas are designed, manufactured and evaluated RF chips are designed, manufactured and evaluated Positioning system is under development The control system and man-machine interface is designed and under realisation The data processing route has been fully defined and is under development. In the next phases of the project the main focus will be on software development for data processing. Other aspects of work will be the assembly of the full array and connection with the radar chips, completion of the position measurement system and full integration of the sub-systems to complete the SPR device. Laboratory based trials are planned to test the device under known soil conditions and against known targets. Subsequent field tests against real targets in real environments are also planned depending upon the outcome of the laboratory-based trials. REFERENCES
1. The DEMINE Consortium: Applied Electromagnetics (UK) Defence Evaluation & Research Agency (UK) Ingegneria dei Sistemi S.p.A (Italy) Menschen gegen Minen e.V. (Germany) Messtechnik, Ortung und Datenverarbeitung GmbH (Germany) Technische Universitat Ilmenau (Germany) Vrije Universiteit Brussel (Belgium) THATI GmbH Erfurt (Germany) 2. J Sachs et al., Ultra-wideband principles for surface penetrating radar, EUROEM, (2000)
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SOME PROBLEMS OF GPR SOFT- AND HARDWARE IMPROVING IN MINE DETECTION AND CLASSIFICATION TASK
Astanin L.Yu, Chernyshov E.E., Geppener V.V., Jatzyn A.S., Kostyleva V.V., Nicolaev V.A., Sokolov M.A., Smirnov A.B. Radioavionika Corp. P.O.B. 111, St-Petersburg, 198103, Russia (812) 251 3875, (812) 251 2743 E-mail:
[email protected]
Introduction New ultrawideband (UWB) impulse ground penetrating radar (GPR, georadar) has been developed in Radioavionica Corporation. Alongside with broad possibilities of UWB georadars for civil application (pavement evaluation, building control, pipelines and cable detection, soil and water table depth mapping etc.) there are essential premises to use them in mine detection and classification [4]. Georadar is an effective tool for detection of subsurface object which dielectric permeability differs from a permeability of an environment. The main advantage is the possibility of detection not only metal but also small size (from 50 mm) low contrast dielectric objects due to high resolving power of UWB signals. Some results of theoretical and experimental researches carried out in Radioavionica Corporation will be described.
Modeling A processing package "GEO" was developed for synthesis and analysis of sequences processing procedures. It consists of interacted units realise different signal and image processing algorithms [3]. Brief description of program units is indicated below: «Spectral processing» - system of the objects spectral descriptions on basis of FFT spectra, autoregressive methods (Yule-Walker and Berg algorithms) and Prony method; also perform objects selection. «2D filtration» - system of the objects selection based on two-dimensional filtration. «Static subtraction» - system of the objects selection based on static subtraction of a base signal with matching. «Precision subtraction» - system of the objects selection based on precision subtraction of a base signal with matching. «Dynamic subtraction» - system of the objects selection based on dynamic subtraction of a base signal. «Signals modelling» - system of modelling signals reflected from target. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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«Transformation» - system of a grey scale images two-dimensional filtration and histogram transformation. «Segmentation» - system of a grey scale images segmentation and features extraction. «Classification» - system of a classification including discriminant analysis algorithms and a decision making based on fuzzy logic. The carried out researches have shown a preliminary possibility of a mine detection and classification task solution. The worked out optimal technological schemes were practically realise in original UWB radar.
Mine detection problem The problem of mine detection is divided into two stages with rather different requirements. At the first stage the problem of real-time detection is under consideration. The second stage deals with further processing for recognition and classification of detected object. Different methods are used for the solution of this problem. In GPR systems every receiving waveform is a superposition of a number of components: a direct coupling, a ground surface reflection, reflections from target or/and false target and multiple reflections, which are signals that make multiple round trips within layers of soil and antenna aperture. Ground surface reflection amplitude exceeds plastic object reflection amplitude for ten times. Commonly used methods are based on the whole waveform processing and provide the stable successful detection in the experimental setup comprises a sandbox and positioning system but they are not effective in practical demining.
Real-time Signal Processing The new approach has been developed for the solution of mine detection problem – step-by-step decomposition method. The essence of it is the reduction of uninformative components by subtracting model signal of every component from each of receiving waveform. The deducted signal is interpolated in advance for increase processing exactitude. So the term “precise subtracting” is used. The algorithm works with use of correlation technique for minimization of remains. The followed filtration effectively reduce the errors of subtracting. The method under consideration provides level of remains better than - 20 dB relative peak amplitude of raw signal. Step-by-step decomposition procedures for antitank metal and antipersonnel plastic mines are shown on Figure 1,a-c and Figure 2,a-c. Then not a single signal but an ensemble of signals is taken under examination and effected with spatial-time processing procedures. Algorithm of dynamic subtraction based on gradient filtering along the scan line provides the determination of object edges (Fig.1,d; Fig.2,d). Some statistical estimates were made for optimal choice of filter parameters. The segmentation procedure is applied to the energetic envelope of obtained signals. The results of this operation are used for automatic estimating of object size and depth of location. The obtained estimations serve for automatic selection of objects as ‘minelike’ or ‘not minelike’. If ‘minelike’ object is arranged then the system provides an audio alarm signal.
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As target reflection from antipersonnel plastic mines is comparable with noise value and antitank metal mine reflections exceed it to a marked degree the adapted threshold is used. The main advantage of suggested technique is the possibility of real-time processing in developed portable georadar. Hardware improving The signal/clutter ratio increases by hardware improving also. The separation of transmit/receive channels in georadar may be achieved both in space (biposition antenna system) and in feeder line (single antenna). For the second mode wideband microwave bridge is used where one arm is resistive loaded (Fig. 3,4).
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In the single antenna mode there is no direct coupling effect and the level of clutter depends on the matching of antenna aperture to air. Therefore it is rather simple to obtain time window free of clutter (Fig.5).
In spite of waste of power in monoposition antenna mode the levels of received signals prove to be comparable if the square of single antenna aperture is equal to the sum of squares of transmit antenna and receive antenna apertures in bipositional mode. Monoposition antenna mode is notable for more symmetrical antenna pattern and gives the possibility of operating more clearly to ground subsurface as the ground reflection signal prove always to be later in time window than reflection from antenna aperture.
Recognition and classification After ‘minelike’ object is arranged an operator interrupts the real-time processing. The last 100 waveforms stored in RAM are visualized on EL display. The ensemble of waveforms is presented as a gray scale coded intensity plot with depth on the y axis calibrated in time units and linear position on the orthogonal x axis. We implement image enhancement and several filtering techniques to improve the presentation of object. If there is no stable detection of object an operator may repeat scanning in order to get good performance. Multilevel segmentation algorithm is applied to radio image of object to obtain a number of slices corresponding to the different energy levels. The searching objects are man-made ones and they have a sophisticated inner structure (elements of construction). Therefore the obtained slices include the peculiarities which are usual for definite object (Fig..6, 7). It is necessary to note that the low level slices have numerous of peculiarities but such slices are noisy and thus their description is more difficult. The high level slices are similar for all types of objects and look like a spot.
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The problem of classification is rather complicated and includes a number of subtasks [1-2]: spatial-time signal processing, selected object features extraction, generation of standard tables of features, construction of deciding rules. The worked out subsurface object classification methods are based on the analysis of separate segments of image descriptions. The geometric type features are used: square, perimeter, lengths of the circumscribed rectangle sizes; parametrical features of an object contour; curvature features, camber and concavity features of the selected object contour. For the feature description the determined and fuzzy procedures are offered. Figure 8 demonstrates the segmented image samples for four mine types involved in tests. Table 1 gives the brief description of every type.
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The classification procedures are based on discriminant analysis methods, fuzzy logic and neural nets. In Table 2 some results of testing classification are presented. The solutions obtained on the various feature descriptions and various levels of segmentation are integrated.
Conclusion The described methods of step-by-step decomposition and dynamic subtracting provide real-time detection and further classification in portable georadar developed in Radioavionica Corporation. It makes a progress in the solution of demining problem.
References 1. Astanin L.Yu., Kostylev A.A.: ‘Fundamentals of ultrawideband radar measurements’. -Moscow: Radio i Svyaz, 1989; 2. Astanin L.Y. Kostylev A.A. Ultrawideband Radar Measurements: Analysys and Processing. By: The Institution of Electrical Engineering, London, UK, 1997. 3. L.Yu. Astanin, V.V. Geppener, V.A.Nikolaev, V.N. Kaftas'ev and M.A.Sokolov, «Methods for Visualization and Classification of Undersurface Objects and Structures Based on Broad-Band Radar Probing», Pattern Recognition and Image Analysis, Vol.8, No.3, pp.384-386, 1998. 4. Brushini K., Gros B., Guerne F., Piece P-Y., Carmona O. Ground penetrating radar and imaging metal detector for antipersonal mine detection. Journal of Applied Geophysics, v. 40, 1998, p.59-71.
TIME-DOMAIN SIMULATION TECHNIQUE FOR ANTENNA TRANSIENT RADIATION, RECEPTION AND SCATTERING
Anatoliy O. Boryssenko, Elena S. Boryssenko, Vitaliy P. Prokhorenko Research Company “Diascarb” Kyiv, P.O. Box No. 222, 02222, Ukraine
INTRODUCTION This paper gives an insight on some features of transient electromagnetic events related to antenna and scattering problems involving the classical aspects of transient electrodynamics and engineering issues. Such properties like near-field range effects, peculiarities of transient antenna in radiation, reception and scattering modes and others, which are not considered enough in literature, will be treated here. Reaching this goal rigorous and asymptotic analytical bounds for linear and wire-grid modeled antennas will be introduced. There are a variety of intuitively evident definitions here like pulse, ultra-wide band (UWB), transient, non-sinusoidal, non-stationary electrodynamics. Generally those phenomena can be treated from the point of view of energy beams (Zialkowski, 1992) as well as with its time history (Smith, 1997) or time-harmonic presentation. However inherent distortion of signal waveform is principal moment for electromagnetic pulse (BMP) simulators, high-resolution radars, spread-spectrum communications, electromagnetic compatibility (EMC) issue, VLSI and printed board design and so on. Generally each element of such system effects on signal waveform passing through it (Harmuth, 1990). Resulted signal is not rather simple replica of input waveform like in case of narrow-band or sinusoidal signal. Due to these reasons time-domain (TD) modeling of transient electromagnetic events is more preferable than frequency-domain (FD) techniques despite their mathematical equivalence due to the Fourier transform. Traditionally numerical approaches to the transient electromagnetic problems are applied like FD method of moments with the next Fourier transformation or FDTD (Taflove, 1995). Also Baum (1965, 1971) developed analytical approaches with the Laplace transform for some asymptotic cases. Generally numeric studies, mostly applied, have principal drawback followed from sufficient programming and computing efforts. Finally the physical meaning of the most numerical solutions is not initially evident. Therefore we developed simple mathematical models, which enable numerical simulations with universal mathematical software like Maple, Mathcad, Matlab etc. The result of such simulations will illustrate the major points of our study. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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BASIC RESEARCH APPROACHES All known analytical approaches and techniques (Smith, 1996; Martin et al, 1999; Shivinski et al, 1997) are based mostly on: i) far-filed asymptotic solutions; ii) using of first time derivative of exciting signal for radiated field characterization; iii) analysis of electrically short and geometrically simple antennas; iv) consideration of transient antennas in transmitting mode. Sometimes far-field asymptotic solutions may be enough for such applications like wireless and radar systems, while other fields like EMC aspects of VLSI and high-speed printed circuit design, subsurface probing radar etc. demand more deep glance on near-field transient phenomena. Also it valuable involve in exploration, beside canonical structure like monopole and center-fed dipole, other antennas including biconical, V-shape, bow-tie, TEM-horn antennas. Those antennas are schematically shown with their wire-grid models in figure 1a. Analysis of such antenna structures in transmitting, receiving and scattering modes is practically important too. Modes of antenna excitation resulted from its edge loading (Boryssenko and Tarasuk, 1999) should be carefully treated, figure 1b. Please note that number of pulse passing along antenna can be connected with number of time derivatives affected on initial exciting signal discussed by Ziolkowski (1992). This can be done explicitly for electrically short antennas and far-filed range operation while finite length antennas and not far-range operation is characterized by quite different, rather not simple, signal transformations. Our primary goal in the presented study is straightforward expressions for characterization effects in mentioned above antennas and their operations modes. A relatively simple mathematical technique is developed here. Finally minimal programming efforts with Matlab, Maple, Mathcad are required for simulation to receive numerical results with productive physical meaning.
GENERAL VECTOR ANALYTICAL SOLUTIONS We start our exploration from a simple case of radiated monopole by introducing, as usually (Baum, 1968; Martin et al, 1999), the vector magnetic potential. That vector has for linear radiator, figure 2a, only tangential, z-axis, nonzero component (1) with respect to the observation point (2) in the given coordinates. Corresponding magnetic (3) and electric (4) field vectors are followed from Maxwell’s equations (Franceschetti, 1997).
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Let note that we do not consider here the scalar potential function and the result of the Lorentz gauge application that is included in (4). Also we do not study antenna excitation in detail that is specific boundary problem but assume that waveform of current in antenna is same as initial one (Martin et al, 1999). At this point the expressions (1)-(4) can directly transformed in numerically effective Maple code (Boryssenko, 2000) by using rampfunction approximation for antenna current proposed by Thomas et al, 1987. This approach is based on the Maple enhanced opportunities in symbolic computations. Results received in such mode are demonstrated later in this paper later. Also for the problems in figure 2 we developed here other analysis technique based on analytical transformations of (1)-(4). Many researchers (Baum, 1971; Smith, 1997; Martin et al, 1999) have done same before. But we will study numerically the complete range of solutions, not only far-field asymptotic. Following this way one can receive after mathematical manipulations formulas for the magnetic (5) and electric fields (6) in the Cartesian coordinates. Related expressions like (7) and (8) give integro-differential operators applied to the original antenna exciting waveform. Such operators are more general than the slant transform (Shivinski et al, 1997) and define the waveform transformation more exactly than time differentiation (Ziolkowski, 1992).
We introduced also additional definitions (9) including slow-wave factor to modify signal velocity in antenna and a retarded time with respect to observation point given by R quantity and antenna point, q, which is integration variable in (6)-(7):
Introducing now in (5)-(9) a slow-wave factor, and latter in (10) an attenuation factors, are important for studying special class of resistively loaded antennas to control antenna waveform by maintaining its single passing excitation, figure 1b. Next the current (10) in antenna, figure 2a, should be determined for its arbitrary excitation. One can do this by considering equivalent transmission line model with standing wave in straightforward mode or by incorporating the standard analysis technique with the Laplace transform.
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The reflection coefficients characterize antenna edge loading and [...] is a common floor operator, which gives the number of pulse reelections from both antenna ends. Receiving antenna, figure 2b, can be considered with simple boundary condition for conductive monopole illuminated by incident arbitrary, not only plane, electromagnetic wave where tangential component of electric field is taken into consideration. In contrast to dual radiating problem, figure 2a, in this case we have distributed antenna excitation in this case and final integrating along the antenna gives induced current (11) with respect to the point, p on the antenna where this current is evaluated.
The complete mathematical structure of (11a) is similar by main features to that (10) and not shown here due to limitations in paper space. One can use the expressions (5)-(l 1) for numerical simulation with Mathcad, Maple etc. It can be done with numerical integration and differentiation. Some obtained in that way results are shown and discussed later. Approximate, enough frequently for design practice, models for other antenna structures like those in figure 1a can be developed by using linear superposition of vector fields produced by each wire-grid monopole element, which composes entire antenna.
FAR-FIELD RANGE ASYMPTOTIC One can obtain a far-field region asymptotic by limiting transition (Baum, 1968) in (7)-(8). It has simple mathematical presentation in the spherical coordinates, figure 2. In this way elevation electric component of the radiated monopole field with single passing excitation (figure 1b) is expressed with (6).
Similarly, a current (voltage) at a load of monopole receiving antenna, figure 2b, exited in single passing mode by incident plane electromagnetic wave with waveform E(t), is determined with (13). Please note that expressions (12) and (13) are similar by some general features but quite different concerning the transformation of primary waveform of exciting signal due to integration -(13a). Some differences between transmitting and receiving antennas are summarized in Table 1 where two auxiliary functions (14) and (15) are used:
for a pattern factor (14) and a complex pattern function (15) of traveling-wave linear antenna with sinusoidal excitation. and are complex spectra yielded from the Fourier transform of the exciting signals s(t) and E(t). Table 1 illustrates also connection of antenna representation in TD and FD, as well as behavior of electrically short antennas used ordinary as a simplest field probe.
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SOME SIMULATION RESULTS AND THEIR DISCUSSION Near-Field Range Effect Without any loss of generality we consider simple case of monopole antenna with single passing excitation by Gauss-shape pulse of 1-nanosecond (ns) duration. These enables receiving clear physical picture with principal features common for all antennas and not complicated due to more complex geometry or excitation. Results of numerical simulations with Maple and Mathcad are shown in figure 3 and 4. Figure 3 with data computed with Mathcad illustrates near-field range effect when space observation point is chosen at different distances from antenna. One can observe asymmetric waveform of radiated field near antenna due to dc and low-frequency spectral component. Especially the presence or not of dc component can be used to characterize antenna operation range. Traditional Rayleigh criterion valid for sinusoidal signal can not be applied here due to broadband radiation (Zialkowski, 1992). We can introduce some criteria from the physical point of view that far-range field should demonstrate properties of an outward spherical waves. Such properties involve the amplitude change inversely proportional to radial distance R. Also ratio of principal electrical component amplitude to that of magnetic in free space must equal to (Boryssenko, 2000).
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Other illustration of the near-range effect in transient antenna is given in figure 4 for the observation point with elevation angle equals figure 2a. These data are computed with Maple (Boryssenko, 2000) for the case with same distance from antenna to the observation point but antenna length is different. Generally discrimination of antenna near/far field properties depends from ratio of antenna physical length and spatial length of exciting pulse. Notice in case of more complex antenna excitation, different from simple single passing or travelling wave of current (figure 1b) that exciting signal has a long time history. The last can results in more expansion in space of near field radiation as pointed out before by Ziolkowski (1992). Filter Network Presentation For far-field range system formed by pair of center-fed, pulse-driven, linear dipole elements (one terminated to transmitter and other to receiver) Zialkowski (1992) introduced the equivalent network presentation where main feature is a specific number of time derivatives applied to input waveform. So far we concentrated on the near-field range effects in antenna we present transient radio channel model with three same antennas operating in transmitting, scattering and receiving modes without any limitations concerning near or far range, antenna type and its excitation. Such generalized system is shown in figure 5 and can be simulated with presented above models. Each antenna in figure 5 is characterized by its own transformation operator A1,2,3. For example, figure 6 demonstrates results of Mathcad simulation with respect to the notations in figure 5. We have in this case three center-fed dipole antennas with double passing excitation and the effect of near-field range is clear visible in this figure.
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Figure 7 shows cross-link effect for a pair of closely spaced transmitting and receiving bow-tie antennas. Simulated waveform data, figure 7a, has been computed with Matlab wire-grid model while experimentally measured one is given in figure 7b. Both, computed and measured, curves have good agreement in early time behavior but different late time history due to effects in real system do not included in the presented models like non-ideal broadband antenna matching etc.
CONCLUSION Time-domain interpretation of non-stationary electromagnetic events, including nearrange effects, which is not enough shown in literature, is discussed here. Generally the results obtained with time domain simulations demonstrate more physical meaning and are more clearly dependent on the influence of problem parameters than those in frequencydomain. All presented above regularities are important for UWB or transient antenna design. Inherent transformation of signal waveform passing through components of UWB system especially its antennas should be carefully treated. The last is ordinary achieved with complex numerical computing. In this sense the benefits of proposed physically meaningful straightforward technique with easy Matlab, Maple etc. simulation seems valuable for
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research and engineering practice as well for academic goals. Generally the presented above approach allows the next steps in research when time-domain antenna arrays can be considered that are in progress now.
REFERENCES Baum, C., 1968, Some Limiting Low-Frequency Characteristics of a Pulse-Radiating Antenna, Sensor and Simulation Notes, 65. Baum, C., 1971, Some Characteristics of Electric and Magnetic Dipole Antennas for Radiating Transient Pulses, Sensor and Simulation Notes, 125. Boryssenko, A. A., Tarasuk, V. M., 1999, Ultra-Wide Band Antennas for Subsurface Radar Applications, in: Proceedings of Antenna Application Symposium, Monticello, IL, 478. Boryssenko, A. A., 2000, Time-Domain Vector Representation of Monopole Transient Electromagnetic Radiation by Using Maple Software, Submitted to IEEE Antenna and Propagation Magazine. Franceschetti, G., 1997, Electromagnetics: Theory, Techniques, and Engineering Paradigms, Plenum Press, New York. Harmuth, H. F., 1990, Radiation of Nonsinusoidal Electromagnetic Waves, Academic Press, Boston. Martin, G. R., Rubio, A. B., Gonzalez, S. G., 1999, Some Thought about Transient Radiation by Stright Thin Wires, IEEE Antennas and Propagation Magazine, 41: 24. Shivinski, A., Heyman, E., Kastner R., 1997, Antenna Characterization in the Time Domain, IEEE Trans, on Antenna and Propagation, 45:1140. Smith, G., 1997, An Introduction to Classical Electromagnetic Radiation, Cambridge University Press, Cambridge. Taflove, A., 1995, Computation Electrodynamics – The Finite Difference Time-Domain Method, Artech House, Boston. Thomas, D. E., Hutchins, R. L., Wiggins III, Nickei, F., S., 1987, Time-Domain Calculation of Radiated Fields, in: AP-S International Symposium Digest: Antennas and Propagation, Blacksburg, VA, 954. Ziolkowski, R.W., 1992, Properties of Electromagnetic Beams Generated by Ultra-Wide Bandwidth Pulse-Driven Arrays, IEEE Trans. on Antenna and Propagation, 40:888.
A COLLAPSIBLE IMPULSE RADIATING ANTENNA
Leland H. Bowen1, Everett G. Farr1, and William D. Prather2 1
Farr Research, Inc. 614 Paseo Del Mar, NE Albuquerque, NM, USA 87123 2 Air Force Research Laboratory, Directed Energy Directorate 3550 Aberdeen Ave. SE Kirtland AFB, NM 87117-5776
INTRODUCTION A reflector Impulse Radiating Antenna (IRA) consists of a parabolic reflector with a TEM feed. The IRA provides broadband coverage with a narrow beamwidth. This class of antenna has a considerable body of literature associated with both its analysis and measurementsl,2,3. Farr Research, Inc. has developed a Collapsible Impulse Radiating Antenna (CIRA) with outstanding RF characteristics. The approach selected by Fan Research for the FRI-CIRA-1 utilizes an umbrella-like design, with a reflector sewn from a very tough, electrically conductive mesh fabric. The CIRA is lightweight, compact, and easily portable with low wind loading and high mechanical ruggedness. The reflector for the FRI-CIRA-1 is 1.22m (4 feet) in diameter with a focal length of 0.488 m (F/D = 0.4). The antenna, when collapsed, measures 102 mm (4 in) in diameter by 810 mm (32 in) long. The antenna weighs 2 kg (4.5 lb.). We measured the characteristics of the antenna using the time domain outdoor antenna range of Farr Research. The time domain data were processed to obtain the normalized time domain impulse response (TDIR) as described by Farr and Baum4 and summarized in the next section of this paper. We made pattern measurements at 2.5° intervals in both the H and E planes and converted them to effective gain. The conversion from impulse response to effective gain is based on the derivation given by Bowen et al,5 and summarized later in this paper. We present the impulse response characteristics in both the time and frequency domains. We also present the effective gain on boresight as a function of frequency. Finally, we present the effective gain as a function of angle in the principal planes, at multiple constant frequencies.
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Normalized Impulse Response Before we get into the details of the antennas, we first review the parameters used to describe them. We can describe antennas in the time domain with an impulse response, of the form Here we provide a quick review of the derivation by Farr and Baum.4 We use somewhat simplified antenna equations that show only the response to dominant polarization on boresight without the propagation terms. It is straightforward to add these other effects later. In transmission mode, the antenna radiates a field on boresight, which is described by
where is the impedance of free space, is the impedance of the feed cable, r is the distance to the observation point on boresight, is the source voltage measured into a load, c is the speed of light in free space, and the “ ° ” symbol indicates convolution. In reception mode, the antenna is described by
where is the incident electric field on boresight. Note that the normalized impulse response, completely describes the behavior of any antenna in both transmission and reception. If we have both a transmitting and receiving antenna, we can relate the received voltage to the source voltage by combining the above two equations as
where is the normalized impulse response of the receive antenna and is the corresponding response of the transmit antenna. To calibrate our measurement system, we use two identical TEM sensors. In this case, the combined antenna equation becomes
The normalized frequency domain impulse response of the sensors can be extracted from (4) as
Once a calibration has been performed with two identical antennas, then we can measure the response of an antenna under test (AUT) by replacing one of the sensors with the antenna under test. The impulse response of the antenna then becomes
and the time domain normalized impulse response is found with an inverse Fourier transform. When making measurements on a focused aperture antenna, we normally extract an aperture height, which can be related to the physical parameters of the AUT. However, is difficult to measure directly, without making assumptions about the antenna’s feed impedance, so we will first find the effective height, The effective
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height is useful since (at midband) it relates the incident electric field strength (in V/m) to the voltage into a scope ( in volts) by a simple proportionality This expression is valid only when the full-width half-max (FWHM) of is much less than the measured signal. The midband effective height can be determined from (2) and the integral of the impulsive portion of the normalized impulse response, and it is calculated as
where the integral is over the impulsive part of it to using
Once we have
we can convert
where
In these equations, is the antenna feed impedance on the TEM portion of the feed arms, and is the transmission coefficient between the TEM feed arms and the feed cable. Note that the quantity is meaningful only in the context of a wideband antenna with a TEM feed. Note also that the quantity is impossible to measure directly – one can only approximate it after assuming that the feed impedance is a constant across the relevant bandwidth, and is purely resistive. On the other hand, we can measure without making approximations, as long as the time domain impulse response has an impulse-like wave shape. For this reason, provides a more useful description of antenna performance than Note also that when there is a balun in the circuit, a different form of the transmission coefficient must be used. For example, in the case of the IRA, there is a balun that matches the impedance of the antenna to a cable, using two sections of transmission line connected in parallel at one end and in series at the other end. In this case, the voltage is halved, so the transmission coefficient, is 0.50. The use of avoids this difficulty, by avoiding the need to define the transmission coefficient. EFFECTIVE GAIN It is frequently desirable to convert the impulse response developed in the previous section to frequency domain gain as defined by in IEEE Std 145.7 As we will see, for wideband applications the effective gain is a more useful quantity, because it accounts for impedance mismatch between the antenna port and feed cable. We provide here the derivation of effective gain from the normalized impulse response, We begin with the standard expressions in the frequency domain. Thus, the power received into a 50-ohm feed cable is where is the incident power density in is the effective area, and is a power transmission coefficient that accounts for the impedance mismatch between the antenna port and 50-ohm feed cable. Absolute gain is related to effective aperture by
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where is the effective gain, the gain after accounting for the impedance mismatch between the antenna port and the 50-ohm feed cable. The term “effective gain” has not yet been recognized by the IEEE Std. 1457, but it is in common use8. Combining the above two equations, we have
By taking the square root, and recasting into voltages, we find
where space,
is the cable impedance (generally
) and
is the impedance of free
Let us now compare the above equation to the standard equation for reception. Thus, we convert (2) of this paper into the frequency domain, obtaining
where is the normalized antenna impulse response expressed in the frequency domain. The normalized impulse response, is already known. To convert it to effective gain, we combine equations (14) and (15),
This formula allows us to convert our time domain normalized impulse response to effective gain, and this is the formula that is used in the remainder of this paper. Effective gain is simply absolute gain, as defined by IEEE Std. 145, multiplied by a transmission coefficient that accounts for mismatch between the antenna and feed line. For ultra-wideband (UWB) antennas, this is a far more useful version of gain than simple antenna gain (or absolute gain) as defined by IEEE Std. 145. This is due to the fact that impedance mismatch between the antenna and feed line is a large part of the challenge inherent in UWB antenna design.
CIRA DESCRIPTION Let us now provide details of the design of the CIRA. The cross section of the CIRA is shown in Figure 1, and a photograph of the open CIRA is shown in Figure 2. The reflector for the FRI-CIRA-1 is 1.22 m (48 in) in diameter with a focal length of 0.488 m (F/D = 0.4) The parabolic reflector is constructed of a very tough electrically conductive mesh fabric. This fabric is silver and nickel plated and has a resistivity of less than The wind loading on the antenna is low due to the high air permeability of the fabric. The reflector has 12 sections or panels that are supported on an umbrella-like frame with fiberglass stays. The stays are connected to the support at the rear of the antenna by aluminum pivots. The antenna is opened by sliding the yoke on the center rod toward the rear of the antenna. Turning the knob on the yoke locks the antenna open.
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An aluminum enclosure located at the rear center of the antenna houses the RF splitter and acts as the support for the antenna. The splitter was provided by Prodyn Technologies. It consists of a input impedance connector, which then splits into two cables. The cables attach to the feed arms at the feed point in a series/parallel configuration as is standard for IRAs with 4 feed arms. The feed arms are made from a combination of conductive rip-stop nylon and resistive polyester fabric. The feed arms are attached to copper tips that facilitate the necessary solder connections at the feed point. The copper tips are attached to a Teflon support on the end of the center rod. A cover made from ultra-high molecular weight polyethylene (UHMW) protects the electrical connections at the feed point. An SMA connector on the side of the splitter enclosure provides a connection to the antenna. A bracket attached to the splitter enclosure provides a standard 3/8"-16 thread tripod connection. The antenna can be rotated easily to either horizontal or vertical polarization by repositioning the tripod support bracket. The backside of the antenna with the splitter enclosure and tripod mount is shown in Figure 3. The large black knob shown to the right of the picture can be loosened to reposition the tripod bracket. Also shown in this figure is one of the coaxial cables between the splitter and the feed point. The collapsed antenna is shown in Figure 4. When collapsed, the antenna measures 102 mm (4 in.) diameter by 810 mm (32 in.) long. The antenna weighs 2 kg (4.5 lb.) and can be easily transported and set up by one person.
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CIRA DATA We measured the characteristics of the CIRA-1 using the time domain outdoor antenna range of Farr Research. The FRI-TEM-2-100 horn sensor used for these measurements is a standard sensor manufactured by Farr Research. These sensors are ultra-wideband electric field sensors designed for low dispersion and high sensitivity. They are calibrated using two identical sensors and the normalized impulse response procedure described earlier. The impulse response of this sensor has a FWHM of 47 ps. The clear time is 4 ns. The midband effective height of the sensor is 42 mm. These sensors overcome the problem of making fast impulse field measurements with derivative sensors, which have very low sensitivity and small effective areas at high frequencies. The TEM sensors are a half TEM horn mounted on a truncated ground plane. Four versions of TEM horn sensors are available from Farr Research9. We used a Picosecond Pulse Labs 4015C step generator to drive the TEM horn antenna. This step generator has a 4 V output with a 25 ps risetime. The response of the CIRA-1 was recorded using a Tektronix 11801B Digital Oscilloscope with a SD-24 TDR/Sampling Head. The distance between the antennas was 20 m and the height was 3 m. We measured the antenna pattern in the H and E planes at 2.5° increments from 0° to 45° off boresight. Also, the effective gain is computed and plotted on boresight as a function of frequency and at various frequencies as a function of angle in the principal planes. The test data for the CIRA-1 are as follows. The TDR of the CIRA is shown in Figure 5. The TDR at the feed point and along the feed arms is the best (flattest) we have achieved on this type antenna. In Figures 6–10 we show the on-boresight characteristics of the CIRA-1. The data were clipped just before the arrival of the ground bounce signal and then zero-padded out to 20 ns to provide frequency domain information down to 50 MHz. The FWHM of the normalized impulse response (Figure 7) is 70 ns. The CIRA proved to be usable from below 50 MHz to above 8 GHz, as shown in Figures 8 and 9. When deciding the distance at which to place the sensor, one has to realize that the far-field begins at a distance that is dependent upon the smallest FWHM one expects to measure. We expected a FWHM of around 100 ps, so we expected that a distance of 20 meters would be adequate. However, we were pleasantly surprised by the 70ns FWHM measurements of the improved CIRA. This narrower impulse width extends the far field to around 25 m, using the formula antenna radius, c is the speed of light in free space, and
where a is the is the FWHM of the
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radiated impulse response. While there was no opportunity to make new measurements at a greater distance, we believe the measurement error due to antenna spacing is small. Next, we provide the effective gain vs. frequency in Figure 10. These data show that at lower frequencies the response of this antenna is quite flat and that the highfrequency response is approximately smooth to 8 GHz. The peak effective gain of the CIRA is 23 dB at 4 GHz. The midband effective height of the antenna is found from the integral of the normalized impulse response shown in Figure 9 to be approximately 0.28 m This is 71 % of the theoretical value of 0.396 m.
In Figure 11 we show the cross-polarization (crosspol) response of the CIRA. The effective gain on boresight for the crosspol case is shown in Figure 12. The crosspol response is 10–20 dB below the copol response from Figure 10. This data may be of interest due to recent work suggesting improvements in the IRA that would result in improved gain and reduced crosspol.6 This is accomplished by placing the feed arms
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at ±30° from vertical, instead of ±45°, which we currently have in the CIRA. We have recently demonstrated this principle on an 18-inch diameter IRA with an aluminum dish with great success. Since each panel of the CIRA is 30° wide, it will be straightforward to incorporate the new feed arm positions into future versions of the CIRA.
Next, in Figures 13–14 we show the antenna pattern in the H and E planes, based on the peaks of the raw voltage measurements. The half-voltage beamwidth is 5.1° in the H plane and 6° in the E plane. If we choose to use the half-power beamwidth, we have ~3° in both the H and E planes. Samples of the raw data from the H and E plane scans are shown in Figures 15–16.
In Figures 17–18 we show the principal plane pattern cuts of the antenna at various frequencies. At low frequencies, the pattern is quite smooth and flat as expected. At high frequencies the high gain and narrow beam width become evident.
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CONCLUSION The FRI-CIRA-1 provides broadband antenna coverage in a single compact package that is easily portable. The antenna has outstanding RF characteristics in both the frequency and time domains. In the time domain, it has an impulse response with FWHM of 70 ps and mid band effective height of 30 cm. In the frequency domain, the peak gain at 4 GHz is 23 dB, and the antenna is usable from 50 MHz to 8 GHz. The reflector for the FRI-CIRA-1 is 1.22m (4 feet) in diameter with a focal length of 0.488 m (F/D = 0.4). The umbrella-like frame covered with conductive mesh fabric provides a very practical, lightweight, and easy-to-use antenna. The collapsed antenna measures 102 mm (4 in) diameter x 810 mm (32 in) long. The antenna weighs 2 kg (4.5 lb.). The antenna can be easily transported and set up by one person and can be attached to a variety of military and COTS transmitters and receivers. The FRICIRA-1 is now commercially available from Farr Research, Inc.
Acknowledgements We wish to thank Drs. Carl E. Baum and George H. Hagn for helpful comments on this work. This work was sponsored in part by the Air Force Office of Scientific Research, Arlington, VA, and in part by Air Force Research Laboratory, Directed Energy Directorate, under contract F29601-98-C-0004.
Patent Notice A patent is pending on the antenna described in this note.
REFERENCES 1. C. E. Baum and E. G. Farr, Impulse Radiating Antennas, pp. 139-148 in H. L. Bertoni et al (eds.), Ultra-Wideband, Short-Pulse Electromagnetics, New York, Plenum Press, 1993. 2. E. G. Farr, C. E. Baum, and C. J. Buchenauer, Impulse Radiating Antennas, Part II, pp. 159-170 in L. Carin and L. B. Felsen (eds.), Ultra-Wideband, Short-Pulse Electromagnetics 2, New York, Plenum Press, 1995. 3. E. G. Farr, C. E. Baum, and C. J. Buchenauer, Impulse Radiating Antennas, Part III, pp. 43-56 in C. E. Baum et al (eds.), Ultra-Wideband, Short-Pulse Electromagnetics 3, New York, Plenum Press, 1997. 4. E. G. Farr and C. E. Baum, Time Domain Characterization of Antennas with TEM Feeds, Sensor and Simulation Note 426, October 1998. 5. L. H. Bowen, E. G. Farr, and W. D. Prather, Fabrication and Testing of Two Collapsible Impulse Radiating Antennas, Sensor and Simulation Note 440, November 1999. 6. J. S. Tyo, Optimization of the Feed Impedance for an Arbitrary Crossed-Feed-Arm Impulse Radiating Antenna, Sensor and Simulation Note 438, November 1999. 7. IEEE Standard Definition of Terms for Antennas, IEEE Std 145-1993. 8. George Hagn, Personal communication. 9. L. H. Bowen and E. G. Farr, Recent Enhancements to the Multifunction IRA and TEM Sensors, Sensor and Simulation Note 434, February 1999.
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HIGH-POWER ULTRAWIDEBAND RADIATION FOR RADAR APPLICATION Vladimir I. Koshelev Institute of High Current Electronics RAS 4, Akademichesky Ave., 634055 Tomsk, Russia INTRODUCTION In the recent years, investigations directed to the creation of high-power ultrawideband (UWB) radiation sources are made intensively in many countries. In a considerable degree, this is due to the interest to the UWB radar development intended to solve various tasks. Each of the areas of radar application makes its requirements to the parameters of the UWB sources. However, there is a common base of the UWB radars and namely obtaining the information concerning the objects from a comparison of the sounding and reflected pulse waveforms. This information can be used for reconstruction of the object shape. This is a principal feature of the UWB radars which causes the high interest to the development of such radars alongside with a high spatial resolution due to a short pulse length. Our research team used to develop methods and technology of the UWB radars since 1993. The development of methods for reconstruction of the object shape allows to specify the requirements to the parameters of the UWB radiation sources. Understanding the physical and technological limitations of the UWB source parameters allows to specify the direction for searching the methods acceptable for solution of the practical tasks. The main efforts were concentrated on the research and development of new UWB radiators with a constant phase center, high-power UWB sources on the basis of single radiators and antenna arrays, methods of shape reconstruction of single complicated objects with a perfectly conducting surface. This paper presents a short review of the results of these investigations. UWB COMBINED ANTENNA High-power UWB radars require radiation sources on the basis of multielement antenna arrays. An antenna in such arrays should be compact, it should have a constant phase center and radiate high peak power The antenna pattern should be either cardioid for electron steering by a wave beam in a wide angle range or high-directed for the arrays with mechanical steering. Theoretical investigations1 allowed to suggest a new approach to the creation of a combined antenna satisfying the requirements made to the element of the UWB steering antenna arrays. The approach consists in the combination of the near-field zones of two small-dimensional radiators with a constant phase center having a common input but different reactive energies. If the electrical energy prevails in a near-field zone of one Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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radiator, then magnetic energy should prevail in a near-field zone of another radiator. At the fulfilment of definite energetic, frequency and phase conditions, the realization of a combined antenna having a wide band matching with a feeder, constant phase center, cardioid pattern of linearly polarized radiation is possible. Theoretical investigations of a combined antenna consisting of one electrical and two magnetic dipoles have shown a possibility to match the antenna with the feeder at the voltage standing-wave ratio (VSWR) in the frequency band ( is the lower frequency boundary determined by VSWR, is the upper frequency boundary determined by the pattern distortion). It is wellknown that the utmost matching band of the electrical dipole with a feeder by the level at the reactive energy store in the near-field zone equal to zero is A theoretical analysis of the idealized model of a combined antenna has shown that the interval of the parameters at which the matching band of such antenna with a feeder is close to the utmost one is sufficiently narrow. Therefore, the experimental realization of combined antennas with the utmost matching band is a sufficiently difficult and interesting task.
Figure 1 presents the variants of a combined antenna design1. The antenna consists of a plane electrical monopole 1 with a length and magnetic dipole 5. In contrast to the idealized symmetrical combined antenna consisting of an electrical dipole and two magnetic dipoles, the realized antenna is nonsymmetrical. This is due, first of all, to the necessity to use a high-voltage nonsymmetrical feeder and antenna design simplicity. Note that the matching band of the realized antennas with a feeder by the level was that was far from the utmost one and presented a subject of our further investigations. Special experimental investigations2 confirmed the conclusions of the theory concerning the increase of the matching frequency band of a combined antenna. Before starting our investigations in 1993, we have chosen a bipolar waveform pulse (one cycle of a sinusoid) for the antenna excitation. This is due to the possibility to realize a more high energetic efficiency of the bipolar pulse radiation in comparison with antenna excitation by a monopolar pulse. The latter is explained by the difference in the spectra of a monopolar and bipolar pulses. This was confirmed by the experimental investigations1 carried out for different antenna designs. In all the designs of the antennas the length of the electrical monopole was where is the bipolar pulse length at the antenna input, c is the velocity of light. High-voltage antennas were realized for the bipolar pulses with The experimental investigations were carried out with the use of the monopolar and bipolar low-voltage pulses. The monopolar pulse length was two times less than the bipolar pulse one in one series of experiments. The investigations have shown that the
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energetic efficiency of the bipolar pulse radiation was that was more than two times higher than the energetic efficiency of the monopolar pulse radiation An important parameter is the antenna efficiency by the peak power which is determined as the ratio of the peak power of a linearly polarized radiation to the peak power of the electrical pulse exciting the antenna. The value of depends on power of a cross-polarized radiation, radiated pulse waveform and dependence of the radiated pulse waveform on the radiation direction. The evaluations, taking into account the abovementioned, have shown that is less than by a factor of 1.5-2 and for bipolar pulses is approximately equal to 0.4-0.6. Detailed experimental investigations1 have shown that in a wide angle range relative to the pattern main direction (±60° in an azimuthal plane and ±30° in a meridional plane) the waveform of the radiated pulse is preserved and the antenna phase center is constant in a frequency band of the radiated pulse.
ANTENNA ARRAYS Theoretical and experimental investigations2-4 of radiation of the arrays consisting of the equal combined antennas were directed to the influence evaluation of the array geometry, distribution of the amplitudes of the exciting pulses by the array elements, interaction of radiators in the array on the characteristics of the radiation and arrays. Theoretical investigations3 were carried out in the approach of the absence of the interaction between the antennas in the array with the number of elements 8x8. Linear2 (up to 4 elements) and rectangular4 (2x2) arrays at the antenna excitations by bipolar pulses with the length were carried out experimentally. The following main results have been obtained: i) the pattern width decreases and the background radiation increases with the increase of the distance between the radiators; ii) the radiated pulse waveform depends on the angle relative to the pattern main direction, zeroes and diffraction lobes at the wave beam steering are absent; iii) the background radiation level decreases with the increase of the number of elements in the array and it is minimum at the equiamplitude distribution of the exciting pulses in the array; iv) the interaction of the radiators in the array results in the array efficiency decrease by the energy and peak power in comparison with the efficiency of a single antenna; between v) the increase of the number of radiators in the array at the distance them increases the electromagnetic field strength in the pattern main direction proportionally to the number of radiators. Let’s discuss in short the two last results. The array energetic efficiency drops with the increase of the number of radiators not so essentially as the efficiency by the peak power. The latter is due to the more essential dependence of the pulse waveform radiated by the array on the angle in comparison with a single antenna. For a 2x2 array and the energetic efficiency decreased by 13% in comparison with a single antenna and the efficiency by the peak power decreased by 1.5-2 times. The electrical field strength increase in the pattern main direction is proportional to the number of the radiators in the array due to the time delay of the interaction of radiators and allows to increase the field strength of the sounding pulses at the array efficiency decrease by the energy and peak power. New possibilities for solution of the radar tasks will be opened at the development of the methods and devices for synthesizing the electromagnetic pulses of an arbitrary waveform in free space, in particular, for essential increase of the radiation spectrum width in comparison with the radiation spectrum width of single antennas or arrays consisting of equal antennas. One2 of possible ways to expand the spectrum of UWB pulses is application of the arrays consisting of the unequal combined antennas excited by bipolar
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pulses of different length. The calculations have shown that the least changes of the radiation parameters due to the jitter of the switches are realized at the bipolar pulse synchronization by the moment of time corresponding to the change of polarity of all pulses. HIGH-POWER SOURCES OF UWB RADIATION The first source of high-power UWB radiation on the basis of a single combined antenna excited by a bipolar pulse with a 100-kV amplitude, 4-ns length and 50Hz pulse repetition frequency was created in 1994. The subsequent high-power (0.1-1 GW) UWB sources of this series with the length and a 100-Hz pulse repetition frequency are described in detail in the papers4,5. In all the UWB sources the formers of the bipolar pulses developed on the basis of a Vvedenskii scheme6 with one and two gas switches were used. In the UWB sources with a single antenna and two-element array, the formers with one switch5 and the wave impedance and respectively, were used. In the UWB source with a four-element array, the former with two switches4 and was used. In a primary charging circuit the Tesla transformers were used. To decrease the amplitude instability of the output pulses the formers were charged by the short pulses A multichannel commutation regime was realized due to the high velocities of the voltage rise The radiation power increase from 0.1 to 1 GW resulted in the decrease of the time of continuous operation from several hours to 20 minutes with the subsequent cooling during 1.5 hours. The limitation is due to the heating of the insulators installed in the coaxial lines. To provide a stable operation of the UWB sources, the antennas were placed into the dielectric containers at a low gauge pressure (0.1-0.6 atm) of The thickness of the dielectric containers was much less than the spatial length of the exciting pulse and they practically had no influence on the radiation parameters.
METHODS OF OBJECT SHAPE RECONSTRUCTION AT SMALL ANGLE BASE OF ULTRAWIDEBAND RADAR Recognition of radar objects (RO) is related first of all to the possibility to obtain the information concerning their shape. The efforts were concentrated on the developments of methods of the object shape reconstruction at a small angle base of the UWB radars. This is due to our interest to the application of the UWB radars for recognition of the objects disposed at a large distance. In all our works the objects in the shape of a stylized 2D and 3D airplane models with different linear dimensions L were used. The objects were sounded by the bipolar pulses of electromagnetic radiation having a different length It was suggested that at the propagation of the electromagnetic pulses in the air their waveform was not distorted. The problem of calculation of the electromagnetic field reflected by the object was solved by a Kirchhoff method in the single scattering approach. To simulate the real measurements, the uniformly distributed noise with a zero mean and given value of dispersion was added to the signal scattered by the object. The value was determined relative to the signal maximum in a receiving system. Tomographic Method It is well-known that the relation between the joint scattering coefficient object shape is determined by the Lewis-Boyarsky equation
and
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Here, V(p) is the object transmission function, vector corresponds to the direction to the object, is the unit vector, is the circular frequency. Function is equal to 1 and 0 at and respectively. At the large object sounded by the pulses from the ground we fail to record the signals scattered by the upper invisible object part and that’s why the relation (1) can’t be used directly for the object shape reconstruction. In order to eliminate the stated limitation it was suggested to substitute the unknown shadow part of the object for a fragment of the body for which V*(-p) is known. The tests of the suggested approach were carried out at a 2D airplane model with a linear dimension L = 36 m. The object shadow part was substituted for an ellipsoid (dashed line, Figure 2).At the simulation, two radars with the angle base were used. The UWB pulse spectrum was limited by the frequencies and
Figure 2a, b presents the results of the object shape reconstruction at the noise level and angle base and 10°, respectively. The accuracy of the reconstruction was determined by the equation
where is the reconstructed object shape. For the data presented in Figure is equal to 94% and 70%, respectively. The calculations have shown that the reconstruction accuracies of the sounded (unknown) and shadow (known) object parts are close to each other. This allows to evaluate the reconstruction accuracy of the sounded object part beforehand. The accuracy decreases with decrease and increase and equals to for and This is the boundary for the object recognition by its reconstructed shape. The essential increase of the object shape reconstruction accuracy in a tomographic method is related to the spectrum width increase It can be realized only at the pulse waveform synthesizing in free space with the use of multielement arrays2.
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Method of Brilliant Points It is well-known that at the pulse scattering by a complicated object containing the parts with a different curvature, a sharp increase of the reflected signal into the direction of the receiver from the local object parts getting the name of brilliant points is possible. It is obvious that the measurement of coordinates of the brilliant points allows to evaluate the RO shape. An aperture synthesis of the angle base in the process of the moving object sounding by the pulses with the pulse repetition frequency F allows to use only one radiator, to measure the object motion velocity, its position in space and, respectively, its track, and to increase the accuracy of the object shape evaluation by a large number of measurements. To measure the coordinates of the brilliant points, it was suggested8 to use a radar system installed into a Cartesian coordinate system. At the beginning of a coordinate system there are placed the radiator and the main receiver. Three receivers had the coordinates (b,0,0), (0,b,0), (0,0,b). The equation for the vector R components of the n-th brilliant point is the following:
Here, is the distance to the n-th brilliant point of the sounded object obtained by the delay between the radiated pulse and the n-th local maximum in the reflected signal. The relative delay where j is the receiver number, is the signal delay in the main receiver. When this approach was tested, a 3D airplane model with a linear dimension L= 10 m was used. The aperture was synthesized during 10 seconds with the pulse repetition frequency F = 2 Hz. The object was moving at a 1-km constant height with the velocity V=200 m/s. During calculations the values of b = 10-50 m, varied. The best results presented in Figure 3 were obtained at b = 50 m. Note, that at the moving object synthesizing, the coordinates of the brilliant points were re-calculated relative to the beginning of the coordinates corresponding to the first brilliant point. It is obvious from Figure 3 that the object shape approximation accuracy by means of the brilliant points deteriorates as increases.
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Method of Genetic Functions Any complicated object consists of simple fragments. A genetic function (GF) presents a signal scattered by a fragment. A signal S (t) reflected by the complicated object in the given direction can be approximated by the GF set
where is the GF set for the given aspect angle having the amplitude coefficients and position at a time axis The problem of the RO shape reconstruction consists9 of determination of the GF composition and measurement of the fragment coordinates corresponding to these GF. The GF set presents a databank of the previously calculated signals scattered by the RO fragments at the given waveform of the sounding pulses and all the surveillance aspect angles. A radar system described in the previous section and relation (3) is used to measure the coordinates of the n-th GF. This approach was tested at the shape reconstruction of a 3D airplane model having a linear dimension L = 4.5 m and placed at the distance R = 1.5 km. The object was sounded by the single bipolar pulses and was supposed to be motionless during the pulse length. The distance between the receivers b = 50 m and the angle base Figure 4 presents the dependence of the object shape reconstruction accuracy (shape projection to the plane OXY) calculated in accordance with the relation (2), on the noise level averaged by 10 realizations for It is obvious from the Figure 4 that decreases with the increase of and The accuracy for and that is essentially higher than for a tomographic approach. Calculations have shown that essentially depends on the ratio (Figure 5). The shape reconstruction accuracy weakly changes with the rise of the ratio of the linear object dimension to the spatial pulse length at
CONCLUSION The obtained results present a realization of the first stage of the research program on the development of the methods and technology of the ultrawideband radars. Subsequent investigations are related to the development of combined antennas excited by the bipolar
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long pulses, high-power UWB sources on the basis of multielement arrays with the increased pulse repetition frequency, receiving antennas for investigation of a polarization structure of the UWB pulses. An important aspect of our activity is the development of codes for simulation of the UWB pulses and arrays, scattering of pulses by complicated objects. The development of new approaches for object recognition and more detailed study of the previously suggested approaches for the object shape reconstruction is carried out. Acknowledgments The author is thankful to all colleagues for a successful research team work resulting in presenting the given review paper.
REFERENCES 1.
2. 3.
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5.
6. 7. 8. 9.
V.I.Koshelev, Yu.I.Buyanov, B.M.Kovalchuk, Yu.A.Andreev, V.P.Belichenko, A.M.Efremov, V.V.Plisko, K.N.Sukhushin, V.A.Vizir, V.B.Zorin, High-power ultrawideband electromagnetic pulse radiation, in: Proc. of SPIE Inter. Conf. on Intense Microwave Pulses V,H.E.Brandt, ed., San Diego, CA, USA. 3158:209 (1997). Yu.A.Andreev, Yu.I.Buyanov, V.I.Koshelev, V.V.Plisko, and K.N.Sukhushin, Multichannel antenna systems for radiation of high-power ultrawideband pulses, in:Ultra-Wideband Short-Pulse Electromagnetics 4, J.Shiloh and E.Heyman, eds., Plenum Press, New York, in publication. V.P.Belichenko, Yu.I.Buyanov, V.I.Koshelev, V.V.Plisko, Short electromagnetic pulse formation by plane antenna array, in: Proc. of Conf. on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, M.I.Andiychuk, ed., Pidstryhach Institute of Applied Problems of Mechanics and Mathematics of the Ukrain National Academy of Sciences, Lviv. 43 (1997). Yu.A.Andreev, Yu.I.Buyanov, A.M.Efremov, V.I.Koshelev, B.M.Kovalchuk, V.V.Plisko, K.N.Sukhushin, V.A.Vizir, V.B.Zorin, Gigawatt-power-level ultrawideband radiation generator, in: Proc. of 12 IEEE Inter. Pulsed Power Conf. C.Stallings and H.Kirbie, eds., Monterey, CA, USA. 2:1337 (1999). Yu.A.Andreev, Yu.I.Buyanov, A,M,Efremov, V.I.Koshelev, B.M.Kovalchuk, K.N.Sukhushin, V.A.Vizir, V.B.Zorin, High-power ultrawideband electromagnetic radiation generator, in: Proc. of 11 IEEE Inter. Pulsed Power Conf., G.Cooperstein and I.Vitkovitsky, eds., Baltimore, MD, USA. 1:730 (1997). Yu.V.Vvedenskii, Tiratron generator of nanosecond pulses with universal output, Izv. Vuzov. Radiotekhnika. 2:249 (1959). V.I.Koshelev, S.E.Shipilov, V.P.Yakubov, Reconstruction of object shape at small aspect angle ultrawideband radiolocation, Radiotekhn. Electr. 44:301 (1999). V.I.Koshelev, S.E.Shipilov, and V.P.Yakubov, The problems of small base ultrawideband radar, in: Ultra-Wideband Short-Pulse Electromagnetics 4, J.Shiloh and E.Heyman, eds., Plenum Press, New York, in publication. V.I.Koshelev, S.E.Shipilov, V.P.Yakubov, Use of genetic function method for object shape reconstruction at small aspect angle ultrawideband radiolocation, in:Radiotekhn. Electr., in publication.
BROADBAND OPERATION OF TAPERED INSET DIELECTRIC GUIDE AND BOWTIE SLOT ANTENNAS
Andrew B. Hannigan, Steve R. Pennock and Peter R. Shepherd Department of Electronic and Electrical Engineering University of Bath Bath, BA2 7AY, United Kingdom
INTRODUCTION In this paper two promising broadband radiating structures, the inset dielectric guide tapered slot antenna (IDG-TSA) and the bowtie slot antenna, are evaluated. For each type of antenna, the general radiation behaviour is established and the input match characteristics over a broad frequency band are discussed. A recently developed analytical method for prediction of IDG-TSA far field radiation is described, and is shown to provide results which compare well with measured values across a wide range of observation angles. New types of feed for the IDG-TSA are proposed which extend the useable bandwidth of the structure as well as allowing much easier integration with planar circuitry. The operation of both antennas with wideband pulsed excitation is investigated experimentally and, for the IDG-TSA, using a finite-difference time-domain (FDTD) simulation. The IDG-TSA performs well with such excitation, showing little pulse distortion in the far field and low reflection back into the feed. The bowtie slot antenna shows more pulse distortion than the IDG-TSA, but performance is reasonable.
THE INSET DIELECTRIC GUIDE TAPERED SLOT ANTENNA The IDG-TSA is a slow wave travelling wave structure consisting of a tapered dielectric-filled slot cut into a ground plane, as depicted in figure l(a). As such, the antenna can be described as a tapered length of inset dielectric guide (IDG). The simplicity of this structure provides a number of advantages; as well as being inexpensive to manufacture, the IDG-TSA is lightweight, rugged, and presents a flat profile which allows it to be flush mounted in a surface. The width and orientation of the main radiated beam is determined by the slot and ground plane geometry; main beams at
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elevation angles from near-endfire to near-broadside may be achieved. In this paper, only vertically polarised shallow slot antennas are considered.
Analysis An accurate analytical method for predicting the far field radiation characteristics of the IDG-TSA has recently been developed (Hannigan et al, 1999). This method uses the fields of the H-guide, which can be written as simple closed form expressions, to model the shape of the IDG fields in the slot. The longitudinal propagation constant is obtained using a rigorous transverse resonance diffraction technique. A full model of the fields on the aperture and ground plane surfaces is thus built up, and Fourier transform relationships are used to obtain the far field radiation patterns of the antenna. The H-guide fields are in general an acceptable approximation to those of IDG. The approximation breaks down, however, at the slot edges where the H-guide analysis assumes the existence of vertical metal plates. The new method uses static field theory to describe the singular behaviour of the fields near to the IDG edge and thus to provide an analytical link between the ground plane fields and the fields on the slot surface. The static field description is linked to the H-guide fields within the dielectric close to the slot edge thereby fixing the magnitude of the ground plane fields. Figure 2 serves to demonstrate the accuracy of the new analytical method over a wide range of observation angles, and also to establish the near-endfire radiation behaviour of a long IDG-TSA. In the figure, the elevation angle is measured from the antenna surface and the azimuth angle is measured from the longitudinal axis of the structure. This moderately directive pattern is in contrast to that of the much shorter structure depicted in figure 3, which gives a single broad beam in the forward direction. Figure 2 also includes a comparison to an FDTD result, generated by a model similar to that presented by et al (1996). It can be seen that the new analytical method provides results that are at least as accurate as FDTD, but the computer model runs in a fraction of the time required by FDTD. For instance, the FDTD model for a short IDG-TSA takes 20 minutes to run on a Silicon Graphics Origin 2000 machine, whilst the new analytical method takes approximately one minute on the same machine. Feed Structures Consider now the methods by which the IDG-TSA can be fed. In the past, rect-
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angular waveguide (RWG) has been used for this purpose; this arrangement is shown in figure l(a). This has disadvantages in that it can introduce a sharp discontinuity at the feed, the available bandwidth of the structure can be reduced, and the scope for integration with other circuitry is limited. As a result of these shortcomings, other feeding options have now been considered. These are empirical designs based on a knowledge of the fields of IDG and on the work carried out by Ma et al (1990). In order to couple to the strong vertically directed electric field component in shallow slot IDG, a microstrip line can be placed vertically on the open end of the antenna. Backto-back measurements reveal that this transition has a lower than the RWG feed but that transmission is also lower, being -3 to -4dB between 9 and 12GHz compared to -1 to -2dB for the RWG feed. These results indicate that the microstrip feed is unsatisfactorily radiative. Using the microstrip feed as a basis, a less radiative structure has been developed based on the well known co-axial to RWG transition. This consists of a stripline feed placed over the open end of the IDG-TSA with a 5.5mm length of shorted IDG fixed behind it. Figure 3 shows the elevation plane patterns of a short IDG-TSA fitted with each of the three feed types for the complete 360° angular range. In this figure 0°, coinciding with the antenna surface, is defined from the right hand horizontal axis so that the top right quadrant shows the forward radiation above the plane in which the aperture sits. The measurements show that the RWG feed produces the least back radiation, whilst that from the microstrip feed is high. The stripline feed, however, has significantly cut the back radiation, giving performance close to that of RWG. Measurements of stripline fed antenna impedance show reasonable broadband per-
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formance. For a short IDG-TSA, is better than - l0dB between 7.5 and 13GHz, between -5 and -8dB from 13 to 16GHz, and better than -8dB between 16 and 20GHz. Thus, the use of the stripline feed allows the IDG-TSA to be operated over a wider range of frequencies without serious losses to unwanted radiation. Indeed, a short IDG-TSA has been operated in a predictable manner up to 21 GHz using this feed, thus displaying an operating bandwidth of approximately 3:1. Wideband Pulsed Excitation of the IDG-TSA The performance of the IDG-TSA with wideband pulsed excitation has been investigated using PDTD simulation and by experiment. The FDTD model uses a standard algorithm (Yee, 1966) modified using a contour path technique to model the tapered floor of the IDG (Jurgens, 1992). Twofold Mur’s first order absorbing boundary conditions are used (Mao, 1998) and the radiated far field in the time domain is found using the method of Luebbers et al (1991). Only the top surface of the structure is modelled and the antenna is fed using a section of uniform inset dielectric guide. An electric field wall is used for wave launching; an incident transverse electric field is allowed to appear in the dielectric region at the start of the structure which has the same field pattern as the desired mode at the centre of the frequency band. The absorbing boundary conditions at this end of the model are only enabled after the pulse has left the excitation plane. The time dependence of the excitation is shown in figure 4(a). It is a bandpass Gaussian pulse with a temporal width of 80ps (full width at half maximum, FWHM), centred on 20GHz and with a width of approximately l0GHz (FWHM) in the frequency domain. Figure 4(b) depicts the reflected component (directed across the slot) calculated by FDTD monitored 6.35mm prior to the start of a short IDG-TSA with an of 2.08. The antenna aperture is 30mm long with 14.6mm of ground plane extending beyond the slot end. The uniform IDG feed is 69mm long, so that the FDTD test point is 62.65mm away from the excitation plane. For the first 360 time steps negligible field can be seen at the test point, as the pulse is travelling along the uniform feed section. The excitation pulse can then be observed, the trailing edge of which is distorted due to reflection from the discontinuity
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at the taper start. This reflection is approximately 14dB down on the incident pulse. After 1000 time steps, further reflections can be seen, approximately 20dB down on the incident pulse. These reflections come around 0.36ns after the incident pulse, which corresponds to approximately twice the time required for a wave travelling at the speed of light to reach the end of the structure (0.17ns). Thus, these reflections can be seen to be due to a combination of the discontinuities presented by the slot and ground plane ends. Figure 5 shows the calculated elevation plane far field radiation in the time domain at two angles for the antenna considered above. Comparison to the incident pulse, figure 4(a), shows that it is being radiated without serious distortion. At both elevation angles, radiation from the feed point can be seen to be combining with the main radiated pulse, though at the higher angle there is a greater time separation between these two components.
Finally, figure 6 shows the measured transmission between a short IDG-TSA and an X-band horn of a more narrowband pulse. These results support the calculated data given above in that very little distortion of the transmitted pulse can be seen.
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THE BOWTIE SLOT ANTENNA The bowtie slot antenna is a planar version of the wideband biconical antenna. The structure under consideration here is shown in figure 1(b), and consists of a bowtie etched onto a metalised substrate fed by a microstrip line printed on the reverse side of the board, terminated in a radial stub. In this work, antennas were constructed on substrate with an of 2.2 with two structures fabricated on a 50.8 by 25.4mm board so that isolation between two closely spaced antennas could be tested. It has been found that isolation between two antennas with 90° bow angles whose centres are 40mm apart is better than 25dB between 5 and 20GHz. However, it has been observed that radiation from the microstrip lines on the rear of the board is high. The situation has been improved by enclosing the rear of the board in a metal box, although this is at the expense of reduced isolation between the antennas, which is now closer to 15dB across the 5 to I5GHz band. Measurements have shown that the of a boxed bowtie with 30° bow angle is poor compared to the stripline fed IDG-TSA; better than -5dB between 5.5 and 7GHz, between 9 and 12GHz, and from 15.5 to 19GHz. However, power loss measurements (i.e. of ) show significant loss from the structure, greater than 60%, over the aforementioned frequency ranges. With the rear of the structure boxed, this power is being radiated predominantly in the forward direction. Figure 7 shows the measured radiation pattern of two bowtie slot antennas at different frequencies. The structure tends to produce a broad beam, with 3dB bandwidths of approximately 100° and 40° in the elevation and azimuth planes respectively, although there is significant ripple along the beam in the elevation plane. Figure 8 depicts the measured transmission of a wideband pulse between two bowtie slot antennas with 90° bow angles. There is considerably more pulse distortion than was seen in the IDG-TSA measurement in figure 6, albeit for a pulse with a more limited frequency content, but the shape of the incident pulse has been preserved to a large extent. The ‘echoes’ of the main pulse, following 0.4 and 1.2ns behind it, correspond to the pulse propagating for further distances of between 120 and 360mm. These dimensions are consistent with reflections from within the enclosure on the rear of the antenna, which is 50.8mm long, or from the open circuit end of the microstrip radial stub feed line, which is 32mm long.
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CONCLUSIONS The radiation characteristics of the IDG-TSA and the bowtie slot antenna have been established. The efficacy of a new analytical method for the IDG-TSA has been demonstrated. A novel IDG-TSA feed structure has been proposed which gives performance close to that of rectangular waveguide but which allows greater exploitation of the broadband properties of the antenna, and easier integration with planar circuitry. The operation of both antennas with wideband pulsed excitation has been investigated for the first time. Both structures perform well, with particular promise being shown by the IDG-TSA. REFERENCES Hannigan, A.B., Pennock, S.R., and Shepherd, P.R., 1999, Improved modelling of tapered IDG antennas, in: Proc. 29th European Microwave Conf. Jurgens, T.G., Taflove, A., Umashankar, K., and Moore, T.G., 1992, Finite-difference time-domain modeling of curved surfaces, IEEE Trans. Antennas Propagat., 40:357. Luebbers, R.J., Kunz, K.S., Schneider, M., and Hunsberger, F., 1991, A finite-difference time-domain near zone to far zone transformation, IEEE Trans. Antennas Propagat., 39:429.
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Ma, L., Rozzi, T., and Pennock, S., 1990, Linear arrays realised in IDG, in: Proc. of IEE Colloq. on Components for Novel Transmission Lines, 6:1. Mao, J., 1998, Twofold Mur’s first-order ABC in the FDTD method, IEEE Trans. Microwave Theory Tech., 46:299. V., Pennock, S.R., and Shepherd, P.R., 1996, The use of the FDTD method in the design of IDG antennas, in: Proc. 3rd Conf. on Computation in Electromag. Yee, K.S., 1966, Numerical solution of initial boundary problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propagat., 14:302.
A UNIFIED KINEMATIC THEORY OF TRANSIENT ARRAYS
Amir Shlivinski and Ehud Heyman Faculty of Engineering Tel-Aviv University Tel-Aviv 69978, Israel
INTRODUCTION The main objective of this paper is to explore the kinematic properties of transient arrays, in order to define the parameter range for an antenna-array realization of collimated pulsed beam fields. Such arrays are driven by pulsed waveforms and are controlled by the inter-element time shifts. The properties of the radiated field depend on the problem parameters: the center frequency and the fractional bandwidth, the pulse repetition rate, the inter-element spacing and the total number of elements. Following [1] we present a unified parameterization for the kinematic properties that covers the entire parameter range, from the conventional monochromatic dense array to the ultra wideband sparse array. The lobe structure of the radiation pattern is a parametric interplay of several mechanisms: In addition to the side-lobes (SL) and the grating-lobes (GL), which are mainly quasi-monochromatic phase-interference phenomena extended to the wideband regime, we also identify cross-pulse-lobes (CPL) which are an intrinsic wideband phenomenon. It is also shown that under certain conditions one may design a sparse array (with inter-element spacing much larger than the wavelength for all frequencies in the band) which basically does not suffer from sever side lobe problem. THE PHYSICAL MODEL We consider a linear N-element array of identical antennas along the axis with interelement spacing The elements is driven by the signal where the inter-element delay controls the main beam direction. is a pulse-train consisting of M pulses with pulse repetition rate and possible modulation It can
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have one of the following forms:
or
In (1a) all the pulses share a single (coherent) carrier as in digital communication systems. Note that in general, namely unless where is an integer and the modulation within consecutive pulses in (1a) have different phases. The form in (1b), on the other hand, describes systems where the pulses are formed separately, say by triggering an emitting device. Note that in this model all the pulses have the same shape, including the modulation. It will be convenient to parameterize the pulse in (1) as follows:
i.e., the pulses have unit energy, pulse length T and are centered at the origin. Clearly We shall distinguish between three bandwidth regimes: Narrow band (NB) or quasi-monochromatic regime: Intermediate regime: (Note that different phenomena are obtained here if or if which are both possible here). Ultra-Wideband (UWB) regime: As will be shown below, the intermediate regime posses both NB and UWB kinematic phenomena. THE TIME-DEPENDENT RADIATION PATTERN The time dependent radiation pattern in the
direction is given by [1, eq. (8)]
where denotes the element effective radiation height [1, eq. (10)], measures the angle from the axis, is the interelement delay as defined in connection with (1), and the symbol denotes a temporal convolution. To simplify the presentation we shall consider only the “array pattern” i.e. the summation in (3). A contracted characterization of the array is obtained if we eliminate the temporal dependence by projecting (3) on a normed detector that measures the field. The radiation pattern is then expressed as
where denotes the within a consider only energy and peak detectors,
“observation window.” Henceforth we shall or respectively, with
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It will be demonstrated that the type of the detector may affect the radiation lobes magnitudes but not the grid structure (Fig. 5). KINEMATIC PROPERTIES: LOBE STRUCTURE The far field radiation pattern of the antennas array, whether observed directly from the field in (3) or using the normed detectors in (4), is dominated by an interference between signals emanated by the individual radiating elements. The resulting radiation lobe structure is controlled by the bandwidth, the pulse repetition rate and the interelement distance. There are basically two mechanisms for this interference: The first is a “phase interference” causing the side-lobes (SL), nulls and grating-lobes (GL), while the second is “pulse-interference” that causes what will be termed here “cross pulse lobes” (CPL) and is relevant only if M > 1 in (1). The CPL is dominant only in the UWB regime, where the exact conditions will be discussed below. A general formulation A radiation lobe is formed whenever a constructive interference between (two or more) signals emanating from different sources occurs. The condition for can be states as
and it is tagged by the triple index Here represents the distance between the sources that emit the interfering signals, hence represents the index of the interfering pulses hence it is denoted as the CPL index and is bounded by Finally describes the phase difference between the pulses, hence where is a measure for the number of oscillations within and thus is related to the fractional bandwidth. Relation (5) completely characterize the lobe structure in the far field as a function of and the signal parameters. Without loss of generality we shall only consider henceforth the case Examining eq. (5) we note that the parameters and have the same functional role, hence we may start by considering two extremes. In the NB regime where we consider only the case Eq. (5) then defines the conventional phase interference condition for the GL and SL. In the UWB regime, on the other hand, hence there are no oscillation and no phase interference phenomena. We shall therefore set in (5), obtaining the condition for a CPL. In the intermediate regime all indexes in (5) have to be considered, implying the coexistence of both GL and SL with CPL. below we shall consider these cases in more details. Quasi-monochromatic regime: Side lobes (SL), nulls and grating lobes (GL) First we consider the lobe phenomena associated with phase interference. It is simpler to consider first the limit where the interference occurs within a single long pulse (i.e., in (5)). In this case the interfering oscillations have essentially the same magnitude, hence a pure null occurs when the relative phases of all the element contributions sum up to an integer multiple of as described by the condition
which is the same as (5) with The SL are obtained between two nulls. A GL is obtained when the contributions from all elements are in phase, i.e,
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giving in (5) These monochromatic phenomena can also be discerned in the transition regime as long as there are several oscillations within the finite duration pulse In the main-beam direction all the element-contributions arrives simultaneously hence the radiated signal is At other directions these contributions arrive at slightly different times, yet the oscillations may interfere constructively to form a GL via (7) as long as the pulses are partially overlapping. However, due of the relative delay between the pulses, the interfering oscillations have different magnitude and thus the GLs become weaker and wider. This is demonstrated in Fig. l(a) where the full and the dotted lines describe the contributions arriving at the second GL direction from the first and second elements. From the same reason the sharp nulls become wider local minima and eventually the SL-nulls structure disappears completely. Fig. 1(b) depicts the radiation patterns obtained from a sparse 11-element array driven by a single pulse with either NB or UWB spectral content. UWB regime: Cross pulse lobes (CPL) CPL are formed by an interference of two or more pulses emitted at different times from different elements (i.e. pulses with different indexes in (1)). In these directions the pulses in the radiating signal are enhanced while in all other directions the radiating signals consists of a weak pulse-train. The CPL phenomenon is possible only for sparse arrays in the sense that
A CPL is obtained if the temporal delay between the and pulses emerging from the and sources located at equals to the relative propagation delay The resulting condition is (recall that is assumed
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throughout)
This condition is similar to (5) where we identify as the CPL order, and The main beam is tagged as a CPL order giving The temporal signal and the number of the interfering pulses at a given CPL direction depends on the indexes and on the time. In order to demonstrate this process we shall consider, for simplicity, the case (see Fig. 2). One finds that during the first interval, there are equi-distance pulses arriving from the first sources. During the second there are equi-distance doublesize pulses corresponding to the interference of the second pulse of first group with the first pulse of the second group. At the next the magnitude of the pulses is tripled. The number of time that this process continues depends of the length M of the time series and on the array size N (Fig. 2). After intervals the pulse series arrives to a steady state which lasts after which the decaying signal is symmetrical to the initial transient. Referring to (5), the first CPL near the main beam is the one with index This CPL is the weakest since the interfering signals comes only from the first and last elements ( and N). The strongest CPLs are those with the smallest possible and that still yield in (5) (see . The strongest possible CPL is since then the pulses emitted by all the sources merge on the time axis. Such CPLs are obtained (if at all) far from the main beam (this CPL is outside the visible spectrum in Fig. 3(a)). Increasing the inter-pulse index may yield CPL solutions to (5) which, however, are located further away from the main beam. The CPL can also be viewed as the UWB extension of the GL, if one regards the latter as a constructive interference of sinusoidal pulses emerging from adjacent elements with relative time shift, thus replacing by in (9).
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The lobe structure in the transition zone So far we discussed the CPL independently of the pulse-shape. If the pulse contains oscillations (i.e., if ) then there might also be lobes due to the interference of these oscillations as can be noted in Fig. 3. If the array is sparse in the sense of (8) and supports CPLs, then GL and SL can be generated by phase interference of the modulation across pulses. Specifically, if then there are several GL around each CPL as can readily be observed in Fig.3(b). In the same parameter regime it may also be observed that for the signal model in (1b), all the CPL have the same spatio-temporal shape, whereas for the signal model in (1a) the shapes of each CPL is slightly different since the pulses in (1a) have in general different modulation phases. However, the CPL-grid is mainly unaffected by the choice of the signal models in (1). CPL CONSIDERATION FOR DENSITY AND SPARSITY From the discussion in connection with (9) it follows that a CPL of order (with ) is possible only if Thus in order to avoid CPL of that order it is required that Practically, however, we shall try only to avoid the strongest CPL with and demand that
This condition guarantees that there are no strong CPL. Yet there might be phase interference phenomena (i.e., GL and SL) if around each CPL and in particular around the main beam (see Fig. 3(b)). The well known condition for avoiding these
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GLs is
(this condition is in fact for broad-side radiation with For end-fire radiation with the condition is For UWB signals, however, where the phase-interference phenomena disappear whence there are no lobes as long as (10) holds. Noting that can be much larger than T, this implies that in the UWB regime the array may be ultra sparse in the conventional frequency domain sense. Specifically one finds that for most frequencies within the pulse spectrum such that where from (10) Note that the CPL are formed only if there is a sufficient number of repetitive pulses. Fig. 5 (a) explores the CPL of the normed radiation pattern of (3) for the same array used in Fig. (3(b)), but with several values of M: M = 1,... ,6. One readily note that the CPL’s and their associate GL’s occupy the range while in the range we find the main beam and its GL. The number of CPL’s and their magnitude grow as the number of pulses increases. The figure also compares the radiation pattern for two types of normed detectors: square law detector and peak detector A notable similarity between these detectors is observed, the only difference is in the “shape” of the lobes. CONCLUSION We have present a unified framework for the kinematic analysis of transient arrays. In addition to the side-lobes (SL) and the grating-lobes (GL), which are due to phaseinterference, we also identified cross-pulse-lobes (CPL) which are an intrinsic wideband phenomenon. It has been shown that the lobe structure is an interplay of several mechanisms and controlled by the problem parameters: the center frequency, the fractional bandwidth, the pulse repetition rate, the inter-element spacing and the total number of elements. It has been shown that the conventional frequency domain dense-array condition where is the frequency, which is used to avoid the formation of GL, is irrelevant in the UWB case and in fact the UWB signal spectrum does not have
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to satisfy this condition. Thus, under these conditions one may use sparse array (with inter-element spacing much larger than the wavelength for all frequencies in the band) without suffering from sever GL problems.
Acknowledgements This work is supported by in part by the Israel Science Foundation under Grant No. 404/98, and in part by AFOSR Grant No. F49620-96-1-0039.
References [1] A. Shlivinski, E. Heyman and R. Kastner, “Antenna characterization in the time domain,” IEEE Trans. Antennas Propagat., AP-45, 1140–1149, 1997.
POWERFUL SOURCES OF ULTRAWIDE BAND PULSED COHERENT SIGNALS A.F.Kardo-Sysoev1, V.I.Brylevsky1, Y.S.Lelikov1, I.A.Smirnova1, S.V.Zazulin,1 I.G.Tchashnicov2, V.I.Scherbak2, B.I.Sukhovetsky2 1 Ioffe Physico-Technical Institute RAS St. Petersburg, Russia 2 Pulse Systems Group, St. Petersburg, Russia INTRODUCTION Coherent ultrawide band signals are used in UWB radard and communication systems. A short pulse at low PRF is the such simplest signal to be processed at receiving end, but to get enough average power a very high peak power is needed, which strongly complicates pulser, antenna designs and interference problem. New generation of semiconductor fast power switches, developed by us, opens the way to trade peak power for high repetition rate, which eliminates the problems, mentioned above. New semiconductor switches show many advantages compared to other switches used in UWB such, as high peak and average powers, very high time stability, lifetime and so on. The switches can be separated into two classes: 1. Closing switches with faster turn-on process of transition from opened to closed state. 2. Opening switches with faster turn-off process of transition from closed to opened state. Closing switches should be used with capacitive energy storage systems. Opening switches utilise inductive storage systems, which provides far higher stored energy density than capacitors. Generally speaking, any semiconductor (and even not semiconductor) switch may be thought off as a some medium between two electrodes, which may be filled by highly conducting electron – hole plasma or be depleted. There are only two ways to fill medium that is initially depleted of current carriers, (this medium is usually the Space Charge Region (SCR) of a p-n junction under blocking bias): a. To inject carriers into the SCR from its borders via additional pn and np junctions. In the case of field drift, the minimal turn-on time is proportional to SCR thickness and to the maximum blocking voltage: i.e. the higher voltage, the longer the turn-on time. b. To generate carriers inside the SCR by external irradiation (light) or by internal impact ionisation. The turn-on time may be independent of the thickness of the SCR if the rate of carrier generation is enough high. There are also two ways to remove plasma from medium: a) To pull off plasma into the same borders from which carriers were injected; or b) To let electrons and holes recombine within the material In practice it is possible to make injection and generation processes fast enough and controllable by external means (e.g. the well-known case of generation by intense light). It must be noted that the requirement of good external control of the pulse's time position (coherence) is very important for UWB communication and radar systems. But it is very hard to control the withdrawal of high-density plasma due to the "trapping” effect and, practically, it is impossible to control (to "trigger") the recombination process. Therefore, at short pulse repetition periods, it is plasma decay processes, which cannot be well controlled externally, that determine the position of the each successive pulse and, accordingly, the stability of the pulse position in the coding pulse train. Here we discuss new classes of devices that overcome these drawbacks. I. Drift step recovery devices, which include Drift Step Recovery Diodes (DSRD) and Drift Step Recovery Transistors (DSRT). II. Devices based on Delayed Ionisation - Silicon Avalanche Shapers (SAS). Also the new approach to radiation of short pulses by the Folded Horn type antennae will be considered. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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1. DRIFT STEP RECOVERY DEVICES - OPENING SWITCHES. In these devices the plasma injection and plasma removal processes are organised in such a manner that restoration of the SCR and voltage on the device (turn-off process) occurs due to fast withdrawal of plasma into the SCR borders. Two such devices are discussed in this paper, DSRD and DSRT: 1.1. DRIFT STEP RECOVERY DIODES (DSRD)
Practical use of a semiconductor diode as an opening switch includes two steps (Fig. 1): Initially plasma is pumped to the diode by current in forward direction; Then the current's direction is reversed and plasma is removed. Due to the plasma injected at the first "pumping" step, the diode remains at high conductivity during some time and the voltage drop is small. Then at the moment the space charge region appears the voltage rises, and the current decreases. For good switching properties the transition between high (turn-on) and low (turn-off) conducting states should be as sharp as possible, the turnon state time should be long, and the voltage restoration time should be short. The sharpest transition and the shortest (i.e. the highest dV/dt rate) are possible if two conditions are fulfilled: Cond.1. At the moment all injected non-equilibrium carriers are removed from essentially all of the diode, and the equilibrium carriers occupy all p and n layers. Cond.2. The velocity of the SCR’s widening has maximum possible value (Fig.2) which may be reached at current density where - doping level, q - electron charge. Under these conditions the voltage rise rate dV/dt reaches its theoretical maximum
where – maximum breakdown field. For silicon, Equation (2) yields It should be noted that is independent of the maximum voltage of a diode time of restoration may be defined as For example: for
Equation (3) yields
for
The minimal
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The main question is how to fulfil both of these conditions in practice? One approach has been used since the in step recovery diodes (SRD): the llifetime of the carriers is made short, and so is the diffusion length where D - diffusivity, and plasma accumulates only in a thin layer. So the initial space charge region, appearing during plasma removal may only be thinner than and the additional voltage drop on it is small, as may be seen in Fig.2. This moment of switching cannot be controlled just by an external pumping current, both because of the strong temperature dependence of life time and because of high life time dispersion in different diodes due to poor manufacturing control of life time. Due to poor control of it is very hard to increase switching voltage by connecting many diodes in series. We developed two new approaches that overcome these drawbacks: i. To make plasma pumping time short, while carriers lifetime is very long. In this case the enriched diffusion region is determined by the same relation as in Equation (4), where pumping time replaces life [1], Under this condition the moment of switching is strictly controlled by an external circuit which provides current until the moment With such good control it is possible to connect many pn junctions in series, by assembling them in a stack, to increase voltage hundreds times while keeping the turn-off time of the stack equal to that of single p-n junction. Thus the end user sees such a diode stack as a single two-electrode device with larger size. But still, determined by Equation (4), can be very short and for a high voltage p-n-junction is less than 50 ns for a 1 kV device. ii. To fabricate a special distribution of pumped carriers so that the plasma removal stage converges to the distribution, which satisfies both cond.1 and cond.2. In this distribution the injected plasma should be stored in both the p and n layers in appropriate proportions. Such distribution can be realised in a p-n junction with special impurity profiles. For this case the duration of the pumping pulse and the high conductivity state may be increased several times, which simplifies the circuits and increases efficiency. We named these devices Drift Step Recovery Diodes, after SRD. In Fig. 3 we show the most simple and very efficient circuit to realise the current sequences shown in Fig.1. Initially all energy is stored in C1, C2 capacitors. At the moment the primary switch S1 closes. At the second primary switch S2 closes. Any type of closing switch (thyristor, bipolar or field effect transistor and so on) with suitable parameters may be used for S1, S2. At time the energy is stored in inductors L1, L2. When the diode opens (breaks current) this inductively stored energy is switched into the load R1. Voltage at the load may be many times higher than the initial voltage at capacitors C1, C2, i.e. in this circuit the high voltage exists only for a short time, which strongly decreases the danger of breakdown. Load current's rise time is determined by the diode voltage restoration the current decay time is It may be shown that the minimal decay time is which corresponds to a pulse width (FWHM) close to decay time. In this circuit, until time current's shape is controlled by the LC circuits, which easily may be made very stable. As long as the moment is controlled only by external current, the stability of generated pulses is very good. This circuit (Fig.3) may be considered as a pulse compressing circuit, which compresses energy in time from down to with a corresponding increase of peak power. Other circuits for utilising the advantages of the DSRD have been developed. In some cases to get better efficiency, two or more stages of pulse compression in LC circuits may be used. The minimum pulse repetition period for a DSRD is determined by
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and may be made close to But in practice is controlled by the external circuit as well, which generates the desired current shapes. In the case of the circuit in Fig.3, and must be many times more than the turn-on times of the primary switches Additional limiting factors are: Maximum pulse repetition rate of the primary switches; Time needed to charge capacitors C1, C2. So this circuit, while being very efficient, has severe limitations on maximum pulse repetition rate, i.e. far below the DSRD’s capabilities. Using the DSRT as the primary switch, this limitation may be solved by: 1.2. DRIFT STEP RECOVERY TRANSISTORS (DSRT) The Drift Step Recovery Transistor can be fabricated with an n+pnn+ structure. For step restoration of the voltage on the p+n junction (collector) of a DSRT, the same cond.1 and cond.2 as in a DSRD must be satisfied during the process of plasma removal from the n-layer. But there is large difference in the plasma pumping process. The electron current across the collector p-n junction is controlled by balancing two currents: the emitter gating current Ig and the maximum collector current which is limited by the external circuit. If an electron-hole plasma region appears in the n-layer near the p-n junction. But in the DSRT, unlike the DSRD, the electrical field force, moving holes, works against the diffusion force. So the size of plasma enriched region may be controlled only by balancing the emitter and collector currents. (Recall that in the DSRD this size is controlled by or ). Additionally carriers are stored in the pbase. It turns out that for special geometry of the n+pnn+ structure and certain doping impurities profiles, and by adjustment of both currents it is possible to get a pumped plasma distribution, which is quite similar to that for the DSRD. At the point of plasma removal, thereby satisfying cond. and cond.2 In the collector of the DSRT the base current flows only across part of the p-n junction area, due to the presence of the additional emitter electrode. Therefore the maximum dV/dt for the DSRT is less than in DSRD, per Equation (2), usually by a factor of 2 to 3. The simplest circuit for a high repetition rate pulse generation is shown in Fig.4. A triggering pulse pumps both DSRT (Q1) and DSRD (D2) during the period (Fig.5), then plasma dissi-
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pation begins in the DSRT During time the inductors L1, L2 store energy. Due to the separating diode D1, the DSRD (D2) is pumped only via the inductor L3. When the DSRT opens (breaks current), the recharging of the collector p-n junction begins. The current in L2 starts to flow via the separating diode, D1, and the DSRD, D2. The current in L1 changes its direction and combines with that in L2 (Fig. 5). The combined reverse current removes plasma from the DSRD. At the moment the DSRD opens (breaks current) and all the energy from both inductors L1, L2 is switched into the load Rl. As in Fig. 3, the load pulse rise time is determined by the DSRD turn-off time and for decay one has for the case L1 = L2 and L3>>L1. In the DSRT both the externals triggering circuit and the collector circuit rigidly control plasma pumping and removal, and correspondingly control the voltage restoration moment and the pulse position. To improve the efficiency of the pulse forming circuit, other modifications of the circuit (Fig.4) have been ddeveloped as well. The minimum pulse repetition period for a DSRT/DSRD circuit is determined by In practice some period should be added to damp any stray oscillation of energy between circuit's parts. At high power output levels, the need for cooling may limit maximum repetition rate. A typical output pulse is shown in Fig.5. Output is 2.7 kV into 50 Ohms, front 0.7 ns, FWHM ~ 1.5 ns, total efficiency Maximum PRF is 2 MHz. Many other pulsers with different combinations of DSRT and DSRD have been built; some examples are shown in Table 1.
2 DEVICES BASED ON DELAYED IONIZATION (SILICON AVALANCHE SHAPERS SAS) As mentioned in the Introduction, generation of plasma inside the space charge region (SCR) of a device can result in the fastest switching. We generate high density plasma by the process of delayed ionization, which was made feasible in practice by use of high dU/dt sources based on drift step recovery devices [2]. We can utilize this phenomenon to design a very effective closing switch, the Silicon Avalanche Shaper (SAS). When a fast rising voltage is applied to a diode biased in the blocking direction (Fig.6), the high-density displacement current j appears in the SCR. The field in the SCR increases and the border between the SCR and the neutral region (NR) moves with the velocity which is equal to the velocity of electrons in the NR. It may be shown that, when the condition (where is the initial maximum field at p-n junction) is fulfilled, the field in the NR exceeds the value at which the velocity o f electrons reaches the saturated value When U’ is high enough, the maximum field at the p-n junction can reach and exceed the static breakdown threshold If the diode leakage current is small, the number of initial carriers in the SCR that could start ionization is small as well, and no ionization happens in the SCR, even when is much higher than E (overvoltaged region (OR) appears). In the neutral region due to very high electrons concentration the rate of impact ionization generation may be high in spite of relatively low electric field (compared to the SCR) where coefficient). Holes, created by the ionization, drift into the
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SCR (Fig.6), reach the overvoltaged region of the SCR, where and cause a very high ionization rate due to the very high field. Hole-electron plasma, also created by the ionization, sharply decreases the field in the OR which is thereby displaced toward the neutral region. Due to hole current from the NR, the extensive ionization starts again, the field decreases, the OR is displaced again, and so on. A fast running wave appears, the velocity of which may exceed the carriers saturated velocity by orders of magnitude. So the ionization wave quickly fills the volume of the SCR by highly conducting plasma and turns on the diode, i.e. it acts as a closing switch. The total measured turn- on times for high voltage diodes are 50200 ps for diodes in the 3-15 kV voltage range, and turn-on time strongly depends on dV/dt rate (for higher dV/dt one obtains faster turn-on). It may be seen from Equations (2) and (6) that the value of dV/dt needed to start super fast switching of diodes is close to the value of dV/dt for super fast voltage restoration. So these two effects ideally match each other. Thus the fast voltage restoration provided by a DSRD is ideally suited to switch a diode based on delayed ionization. Such diodes we have named Silicon Avalanche Shapers (SAS), and are very suitable for connecting to the pulse output of DSRT+ DSRD circuits for a final stage of pulse compression. Due to plasma " trapping", the pulse repetition period of the SAS is relatively long, but repetition frequencies up to 100 kHz are possible. Some examples of pulsers utilizing this DSRT+ DSRD+ SAS concept are shown in Table 2.
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3. RADIATION OF SHORT PULSES BY THE FOLDED HORN ANTENNA Antenna theory shows that a uniformly excited aperture radiates the shortest pulse. The beam width is inversely proportional to the size of the aperture. The most efficient embodiment of such a transmitter is a horn antenna. It is evident that the beam width angle cannot be less than the apex angle of the horn, even for large aperture size. So ideally the apex angle would be small and total length of the horn would be large (Fig.7), which limits the use of such an antenna. We have proposed to fold the horn, as shown in Fig.7, by a series of steps; each step folds the initial horn to half the length into smaller subhorns. Each level of subhorns has different length, to help the field distribution recover after distortion at "sharp" points A. All subhorns are fed synchronously via a pulse splitter or by separate synchronised pulsers. We have designed and tested two types of folded horns: 1. Flat (or planar horn), which may be made even on a two- sided printed circuit board. 2. Two-dimensional folded horn in which the periodicity of different sub-horn levels shown in Fig.7 exists in both dimensions and the size in H plane is comparable or larger than pulse length. Such a flat folded horn (FH) was tested while radiating the 2 ns pulse which shape is close to shown in Fig.5. Aperture span is 1.5m, beam width for the E plane ~55°, H plane ~ 150°, and the forward/back ratio is 20 dB. The radiated pulse is shown in Fig.8. With a 20 kV (8 MW) feeding pulse amplitude, the field intensity reached 6.345 kV/m at a distance of 12.4 m. With a 2.7 kV (150 kW) HFPG pulser feeding the FH, the field intensity was 1100 V/m at a distance of 4.5 m from the FH. It is well known that for aperture antennas in far field zone maximum field intensity is where pulse output power, r distance, A - quality factor of an antenna (included efficiency and beam width). For our FH, from the experimental data noted above, we have This factor may be compared with the well-known Impulse Radiating Antenna (IRA) [3], which has a
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slightly lower reported value, It should be noted that the disklike IRA is 3.6m in diameter and dozens of times heavier than the FH. Fig. 9 shows radiated field for two dimensional FH. One sees that radiated pulse is very close to derivative of feeding pulse. The field was measured by E-dot sensor. The pulser generates 50 Volts in to 50 Ohm antenna input at PRF up to 10 MHz average.
Conclusions We have developed new and efficient semiconductor devices and circuits for generation of powerful short pulses in nano- and subnanosecond ranges with practically unlimited lifetime (number of shots) at high repetition rate. We have also developed new portable and efficient pulse radiating antennas. References 1. I.V.Grekhov, V.M.Efanov, A.F.Kardo-Sysoev, S.V.Shenderey, "Formation of high nanosecond voltage drop across semiconductor diode". Sov. Tech.Phys.Lett., vol.9 (1983), n4. 2. I.V.Grekhov, A.F.Kardo-Sysoev, "Sub-nanosecond current drops in delayed breakdown of silicon p-n junction". Sov.Tech.Phys.Lett.,vol.5(1979), n8, pp.395-396. 3.Sensor and Simulation Notes, Note 382, 4 July 1995. A Reflector Antenna for Radiating Impulse-Like Waveforms. D.V.Giri and H.Lackner. Pro-Tech, 1308 Mt.Diablo Blvd, Suite 215 Lafayette, CA 94549. I.D.Smith and D.W.Morton, Pulse Sciences, Inc., 600 McCornic Street, San Leandro, CA 94577 and C.E.Baum, .R.Marek, D.Scholfield, and W.D.Prather Philips Laboratory, Kirtland AFB, NM 87117.
ULTRA WIDE BAND SOLID STATE PULSED ANTENNA ARRAY
A.F.Kardo-Sysyoev1, S.V.Zazulin1, I.A.Smirnova1, A.D.Frantsuzov2, A.N.Flerov2, 1 Ioffe Physico-Technical Institute RAS St. Petersburg, Russia 2 Pulse Systems Group, St. Petersburg, Russia INTRODUCTION Phased Antenna Arrays (PAA) are widely used in all fields of modern electronics. Their main advantages are: 1. High total microwave power levels may be achieved by combining of powers of many, small inexpensive sources, 2. Fast electronic deflection of a radiated beam. It is only PAA, which gives means to build powerful all solid state (semiconductor) microwave radiators due to strong limitations on power of semiconductor microwave devices. At the same time for the case of semiconductor devices PAA approach is natural because: 1. Price for a unit sharply drops with production rate increase, 2. Jitter is very low, especially for bipolar devices. But semiconductor devices have inherent slow temperature drift, which, practically, may be orders of magnitude larger than jitter and can disrupt synchronization some time after initial adjustment. The well known practical approach to suppress drift is Closed Loop Techniques CLT [1] which tracks time difference between time positions of a power output and a reference signal and minimizes it. This approach may be used with both narrow band continuous CW and ultrawide band (gaussian monopulse) signals. Whereas CLT is widely used for CW signals, there is little work on UWB. For example in [1], synchronization of four sources had been considered, each of which radiated 3 rf cycles of 2.6 GHz central frequency at 1 kW peak power. CLT used mixing of radiated signal with 2.3 GHz CW reference oscillation. So this system could track the time difference inside only one cycle = 0.4 ns and could be considered as some intermediate step between CW and real monopulse CLTs. Here we consider experimental pulsed PAA system comprising four 50-Ohm input antennas, individually fed by 1,1 kV, 200 ps FWHM pulses at variable up to 12 kHz PRF. 1. PAA SYSTEM 1.1. GENERAL APPROACH The pulsed PAA system is outlined in Fig. 1a. Externally triggered pulser PG feeds antenna module. A pulser's circuit is shown in Fig.2. Synchrogenerator generates four reference signals Time position of each signal may be fine tuned in respect to others inside 00.5 ns range. A pulser PG has main 1.1 kV into 50 Ohm output and 13 V coupled synchronizing output (Fig.3). The and are fed to time discriminator TD (Fig. 1). TD output level depends on the sign of time position difference between and . The TD output via low pass filter (LPF) and amplifier controls Voltage Controlled Delay (VCD). VCD's output triggers Pulse Generator.
Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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Internal delay between triggering pulser of and output is around 200-210 ns. Internal minimal delays of VCDs are around 40 ns. To compensate the delays and their variations, stable delay ns is added to TD's reference signal. So when the loop is closed all four output pulser's positions are reduced to the time position delayed by 280 ns from SG output It is clear that this approach may be applied to a PAA with any number of elements without degradation of time synchronization. To evaluate time stability, the system shown in Fig. 1, was tested on matched dummy loads. RMS of time deviations around is less than 15 ps at PRF = 12 kHz and still is less than 20 ps at PRF ~ 1 kHz. 1.2. PULSE GENERATOR. Pulse Generating circuit is shown in Fig.2a. Generally, its operation is described in [2]. New fast power switches used in the circuit are Drift Step Recovery Transistor (DSRT), Drift Step Recovery Diode (DSRD) and Silicon Avalanche Shaper (SAS) described elsewhere [3]. Initially DSRT is opened and electron hole plasma is pumped into the collector by driving pulse of ~ 100 ns length via transformer Inductor's L1, L3 current rises linearly at the rate After the end of the driving pulse, DSRT still is opened sometime due to pumped plasma. During all DSRT’s conducting time DSRD D3 is
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pumped via L4, R9 circuit. When, pumped into DSRT, plasma is pulled out, collector's conductivity current stops very abruptly. Inductor current starts to flow via separating diode D4 and DSRD, which is still in conducting state due to previous pumping. Inductor L3 current oscillates in a circuit comprising collector capacitance of DSRT, L3, D4, and DSRD D3. At the second half period of oscillation when both currents of L3 and L1 have the same direction, DSRD breaks this total current and switches it to charge peaking capacitor C4 via L2 inductor. When C4 voltage reaches threshold voltage of SAS D5, SAS turns on and discharges the capacitor C4 into the load. Output pulse is shown in Fig.3. Rise time is less than 70 ps, FWHM is ~200 ps. Maximum pulse repetition frequency 12 kHz is limited by overheating. Addition of heat sinks increased PRF up to 50 kHz. Energy efficiency is better than 15 %. 1.3. ANTENNA Antenna array consists of 8 horns, 2 columns (E plane) by 4 rows, shown in Fig. 4. Each horn's impedance is 100 Ohm. Each two horns in a column are fed by a power splitter (PS) with 50 Ohm impedance. A pulse generator feeds each splitter. All edges of adjacent horns are electrically connected. For the case, when all pulses reaches the aperture simultaneously, therefore low cut off frequency of the array is determined by overall aperture size and total efficiency of radiation is high. Energy, reflected from antenna to PG, is less than 20%. Each connected pair of horns, may be considered as two times folded long horn with the same apex angle. This folded horn (FH) approach provides smaller variations of EM wave time arrival at different points of aperture. In our case the difference is less than 50 ps (1.5 cm in distance). So taking
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into an account the tolerance of synchronization, mentioned above (RMS<15 ps) expected uniformity of the field time distribution at the aperture 600x520 mm is determined mainly by geometry and does not exceed 50 ps, which is less than the pulse front ~70 ps. One sees in Fig.4 that at the apex of horn two electromagnetic waves are exited. The main M travels to the aperture and is radiated along the axis. The second stray S partially radiated, partially got trapped between metal parts of antenna and adds to after pulse oscillations. The ratio of stray and main powers may be determined as where - horn impedance 100 Ohm, - impedance of stray waveguide consisting of a horn plate and outer conductor of a coaxial part of splitter. Estimation shows that K <10%. As Fig.4 shows only one of four connections radiates directly into outer space and it total input into side lobes of a pattern would be four times less than (1). It seems, that antenna may be partitioned into as large as possible number of subhorns down to zero sizes. But these are stray waves, which limit sizes of the smallest subhorns. The size of stray wave-guide formed between adjacent subhorns must be larger than the pulse length in space. 2. RADIATED FIELD Radiated field distribution was measured by use of E field strip line probe with 60 ps time resolution. Fig. 5 shows time dependence of EM field intensity on distance. At long distance (far zone) field shape is close to derivative of the pulse fed into the antenna. Some after pulse oscillations are partially due to the stray waves in the array and partially due to scatter from environment and interference at the receiving end. The field shape gradually changes to bell like at the aperture (near zone). Theoretically for the case of uniformly exited aperture far zone signal should be derivative of near zone. We think that some 30% outstretch of the far zone signal is connected with nonuniform field distribution at the aperture. The dependence of the field on distance is shown in Fig. 6. It is well known that in far zone a radiating system may be characterized by effective potential (where - field intensity at distance R). versus R is shown in Fig. 6. One sees that is nearly constant at distances > 4 m, which may be considered as the beginning of far zone. It may be shown for the case of a uniform aperture, that far zone condition satisfies where S aperture area, 1 - wavelength. From Fig. 4 and Fig. 5 we have: with the value estimated from Fig.6.
1 ~ 0.15 m, R > 2 m, which coincides
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We used the next procedure to check the efficiency of summation of field. We put a switch to lock/unlock loops. Initially a loops were unlocked and pulses took random positions inside the range 25-10 ns prior to the synchronized position not interleaving each with other. Then one loop was locked and corresponding pulse took position with field amplitude Then the next loop was locked and so on. In Fig. 7 the field intensity is shown for different numbers of locked loops. One sees that field summation is perfect. In Fig. 8 fields at different angles in H plane are shown. Pulse width increases, and at large angles (>20°), the pulse splits into two pulses. The splitting angle is determined by condition where 1 - pulse length 10 cm, Z - 26 cm distance between the centers of adjacent horns. Fig.9 shows antenna patterns in H and E planes for both positive and negative field amplitudes. H plane beam width is at level 0.7. Background noise due to stray scattering and interference at is under 20 db level. Forward /back ratio is far better than this 20-
dB level. The difference between positive and negative amplitude is less than 20% and may be explained by interference of the main pulse with after pulse oscillations (see Fig.8). E plane beam width is at 0.7 level. To scan beam, the corresponding delays to triggering pulses were added. Array patterns are shown in Fig. 10 for sets of delays corresponding given scanning angles.
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The picture clearly demonstrates the well known for the case of CW rule: array pattern may be represented as a product of a single element pattern and an array multiplier. For
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two columns array, of course, the scanning angle is not large. CONCLUSION Pulsed synchronized antenna array 2x2 has been built and tested. Array radiates monocycle of 0.5 ns length, ~2 GHz center frequency. Total radiated power ~ 80 kW, efficient potential ~11 kV, beam width at 0.7 level. The array showed perfect summation of field intensity of partial radiators. REFERENCES 1. Gerald F.Ross, “The far field synchronization of UWB sources by closed loop techniques”, Proceeding of an International Conference on Ultra-Wideband, Short-Pulse Electromagnetic, October 8-10, 1992, at WRI, Polytechnic University, Brooklyn, New York.. 2. A.F.Kardo-Sysoev, S.V.Zazulin, V.M.Efanov, Y.S.Lelikov, "High Repetition Frequency Power Nanosecond Pulse Generation ", 11th IEEE International Pulsed Power Conference, Baltimore, Maryland, p. 107, 1997. 3. V.M.Efanov, A.F.Kardo-Sysoev, I.G.Tchashnikov, "Fast Power Switches from Picosecond to Nanosecond Time Scale and Their Application to Pulsed Power ", Tenth IEEE International Pulsed Power Conference, Albuquerque, New Mexico, pp.342-347, 1995
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ADMITTANCE OF BENT TEM WAVEGUIDES IN A CID MEDIUM
Carl E. Baum Air Force Research Laboratory AFRL/DEHE 3550 Aberdeen Ave., SE Kirtland AFB, NM 87117-5776
1.
INTRODUCTION
Various solutions have been developed for the propagation of TEM waves in an inhomogeneous dielectric medium with permittivity proportional to in a cylindrical coordinate system with propagation in the direction [1-5, 7]. Experimental work is underway to approximately synthesize such a medium with guiding conductors to form a TEM-transmission-line bend [6]. There are also cases of TEM waves propagating in other directions in such a medium [8]. The general procedures are differentialgeometric lens synthesis as discussed in [11]. This medium then has very special properties. So let us give it a name: cylindrically-inhomogeneous dielectric or CID for short. (El Cid is a Spanish title, roughly translating as lord or sir.) The particular form of inhomogeneity as
is the form to be implied by this name. Here and are convenient reference permittivity and radius respectively. In the present context, these refer to the middle ofthe lens cross section. With as the center of the waveguide with respect to the coordinate, let the guide extend a maximum distance on either side of In the present cases, the fields are all confined to Then we have
This
is an important parameter and as we shall see the guide admittance has a correction proportional to
for appropriately symmetrical guide cross sections. As discussed in [4, 5] the electric potential for the TEM mode satisfies
which is similar to a Laplace equation on a cross section (constant ). Here is a function of and z, but other coordinates such as cylindrical coordinates centered on the cross section are more appropriate for the circular coax case. Note that there is a propagation function for the coordinate not included in the above, Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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as well as a waveform function. One can also write an analogous equation for the magnetic potential, but we will not need it here. Further details are included in the references. In the limit that the waveguide becomes straight and the dielectric is uniform in the guide cross section, changing (1.3) into a true Laplace equation as
with whatever transverse (subscript t) coordinates one wishes. This reference case is denoted by the subscript 0 and we later have occasion to write
as a perturbation. For our cross-section boundary-value problem we take our electric conductors as
or 0
potential (volts) giving boundary conditions for and having zero for boundary conditions. The current I, found from a line integral of the magnetic field is used to define the characteristics admittance as
The electric field for the cross section is given by
with coordinates as appropriate. The magnetic field for the cross section is given by
where propagtion has been taken in the uniformly in the guide.
direction. Note that for
the wave admittance become
The basic problem considered in this paper is the variation of the characteristic admittance as a function of After two simple canonical cases, we concentrate on the important case of a circular coax. 2.
SYMMETRY CONSIDERATIONS
To aid in the analysis let us consider certain kinds of symmetrical guide cross sections. Specifically we consider symmetries under which reflection and or rotation of the bent guide reverses the direction of bend while conserving the characteristic admittance of the guide. Consider first reflection symmetry in the cross section about a symmetry line ( axis) at as in Fig. 2.1 A. Consider first that as looking into the page the guide is bent to the left (positive positive ). Now consider reflection of the entire bent guide through a plane containing the axis and perpendicular to the page. The guide has a rotation axis a distance to the right of the original rotation axis. The guide now bends right as one moves into the page, which we can consider as a negative normalized curvature i.e., in the transformation. Using on the plane measured from the axis the potential transforms from case 1 (left bend) to case 2 (right bend) as
This is a mirroring of the potential as discussed in [9]. The associated fields transform as
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The permittivity transforms like the potential, i.e.,
The potential and the current I (from a line integral around the center conductor) being unchanged in the transformation, the characteristic admittance is unchanged, i.e.,
which is an even function of the bend curvature. A similar thing happens if the guide cross section on the
plane has a two-fold rotation axis
the plane), as illustrated in Fig. 2.1B. Again, case 1 has the guide bending to the left. Case 2 is found by rotating the entire bent buide by (180°) about Now the guide has rotation symetry about the axis which we can interpret as a negative normalized curvature The potential fields on the cross section transform as
The permittivity transforms as in (2.3). The characteristic impedance is unchanged and the conclusion in (2.4) applies again. From (1.6) and (2.4) we have
Let us define a normalized form of the admittance as
When
the lens reaches the rotation axis where we can assume that
series in
The fact that
is an analytic function of is even in
a singularity occurs in the lens medium. For i.e., that it can be expanded in a power (Taylor)
means that only even powers are allowed in the expansion, giving
where the second index above the summation indicates the increment (2 in this case) in the summation index for successive terms. Keeping the first two terms we have
which can be used as an approximation valid for small
3.
CANONICAL H-PLANE BEND
This case with two electrically conducting boundaries of width 2b and spacing 2d is illustrated in Fig. 3.1 A. The lens region is closed at the edges of the conductors by magnetic boundaries to give the case discussed in [2(Section 4.1)]. Here we have the fields
which are integrated to give
ADMITTANCE OF BENT TEM WAVEGUIDES IN A CID MEDIUM
For
we have a uniform dielectric medium giving the simple result
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from which we find the normalized characteristic admittance
Expanding this for small
we find [10]
Consistent with the symmetry results of Section 2. Figure 3.1B shows the case of three conducting boundaries (coax-like) with the outer conductors at zero potential and the “center” conductor at potential With the dimensions as indicated (same overall dimensions as the previous case) the problem is readily solved as the parallel combination of two 2-conductor problems with spacing d and width 2b. This gives as 1/4 of that in (3.2) and the same as in (3.4) and (3.5). 4.
CANONICAL E-PLANE BEND
Figure 4.1 A shows the case of an E-plane bend where now the electric conductors are of width 2d with spacing 2b. Again the lens region is closed at the edges of the conductors by magnetic boundaries to give the case discussed in [2(Section 4.2)]. The fields are now
which are integrated to give
So the characteristic admittance is not a function of a very simple result. The normalized characteristic admittance is simply
This is the case for all
and the expansion for small
gives
with no remaining error terms. This is a special case which is still consistent with the symmetry results of Section 2. Figure 4.1B shows the more coax-like case with three conducting boundaries with outer conductors at zero potential and the “center” conductor at potential With the dimensions as indicated the problem is solved as the parallel combination of two 2-conductor problems with spacing b and width 2d. However, these two problems are not identical owing to the different permittivities in the two regions. Our previous results can be used by replacing by the value of Y (the wave admittance) at the center of each of the two regions giving and conjunction with (4.2) gives
for the left and right regions respectively. Using this in
ADMITTANCE OF BENT TEM WAVEGUIDES IN A CID MEDIUM
357
Again we have the convenient result of a four-fold increase in the characteristic admittance, but now with a second order correction in
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C. E. BAUM
COAXIAL BEND
Now consider a bend in a circular coax as illustrated in Fig. 5.1. The inner conductor (radius a ) has potential and the outer conductor (radius b) has potential 0. Note the coordinates on the plane centered on the coax with
Here the fields are given by
The characteristic admittance is
where this applies to any between a and b. We note for later use
which applies for all
where
We also write
applies to the case of
and
has boundary conditions 0 on both conductors.
5.1 Straignt Coax For we have the well-known case of a straight coaxial waveguide (cable) with a uniform dielectric medium. In this case we have
5.2 General Solution Through Second Order From (5.3) and (5.6) we have
ADMITTANCE OF BENT TEM WAVEGUIDES IN A CID MEDIUM
Divide through by
and integrate with respect to
Now expand as a simple geometric series
which is valid for all
with
giving
In this form we can look at the terms given by each
to give
359
360
the
C. E. BAUM Next apply symmetry for as in Section 2. The circular coax has cross-section symmetry on plane of all rotations and reflections known as This includes both the reflection with
respect to the axis in Fig. 2.1A and the rotation with respect to from left bend (case 1) to right bend (case 2) as
in Fig. 2.1B. From (2.5) we have
Both cases must give the same result for the admittance, so let us take the average giving
Now change the angular variable as
Noting that any
interval for
and
will do due to the periodicity, we then have
Now we note that
Integrating over gives zero for odd functions of only even terms. We can then write
For the
terms first integrate over
Then all terms for odd
in (5.14) give zero, leaving
and invoke (5.4) giving
The leading term 1 is now separated out and terms for (even) are left as corrections. For the term let us first integrate by parts over to give
ADMITTANCE OF BENT TEM WAVEGUIDES IN A CID MEDIUM
Now write
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as in (5.5) where
We then have [10]
Combining these results we have
Noting the form the
must take as in (2.9) then the
(goes to zero faster than
) is combined with
to give
5.3
Special case of small b- a
An interesting special case has
This limit of 1/2 can also be found from a physical problem of integrating the admittance per unit closely spaced coaxial cylinders with the CID medium between them as
of two
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So the general result is consistent with this simple check. The case of small b-a also gives a bound for since [10]
Note for small a/b we have logarithmically. This is associated with the fact that for a small center conductor the electric field is relatively large near the center conductor where is close to The permittivity near
6.
then dominates this case.
CONCLUDING REMARKS
We now have some canonical results for the characteristic admittance of a bent TEM waveguide in a CID medium. The H-plane and coax bends show an increase of the characteristic admittance over that of the straight waveguide with taken as the permittivity in the center of the bend guide. However, the twoconductor version of the E-plane bend shows no change in the characteristic admittance with bending; the three-conductor version shows some increase with bending. More general guide cross sections may also be considered. Here we have the general result that for certain symmetries in this cross section the correction to the characteristic admittance is second order in the bend curvature. This work was supported in part by the U. S. Air Force Office of Scientific Research, and in part by the U. S. Air Force Research Laboratory, Directed Energy Directorate. REFERENCES C. E. Baum, Two-Dimensional Inhomogeneous Dielectric Lenses for E-Plane Bends of TEM Waves Guided Between Perfectly Conducting Sheets, Sensor and Simulation Note 388, October 1995. 2. C. E. Baum, Dielectric Body-of-Revolution Lenses with Azimuthal Propagation, Sensor and Simulation Note 393, March 1996. 3. C. E. Baum, Dielectric Jackets as Lenses and Application to Generalized Coaxes and Bends in Coaxial Cables, Sensor and Simulation Note 394, March 1996. 4. C. E. Baum, Azimuthal TEM Waveguides in Dielectric Media, Sensor and Simulation Note 397, march 1996. 5. C. E. Baum, Use of Generalized Inhomogeneous TEM Plane Waves in Differential Geometric Lens Synthesis, Sensor and Simulation note 405, December 1996; Proc. URSI Int’l. Symposium on Electromagnetic Theory, Thessaloniki, Greece, May 1998, pp. 636-638. 6. W. . Bigelow and E. G. Farr, Minimizing Dispersion in a TEM Waveguide Bend by a Layered Approximation of a Graded Dielectric Material, Sensor and Simulation Note 416, January 1998. 7. W. S. Bigelow and E. G. Farr, Impedance of an Azimuthal TEM Waveguide Bend in a Graded Dielectric Medium, Sensor and Simulation Note 428, November 1998. 8. C. E. Baum and A. P. Stone, Unipolarized Generalized Inhomogeneous TEM Plane Waves in Differential Geometric Lens Synthesis, Sensor and Simulation Note 433, January 1999. 9. C. E. Baum, Interaction of Electromagnetic Fields with an Object Which Has an Electromagnetic Symmetry Plane, Interaction Note 63, March 1971. 10. H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th Ed., macmillan, 1961. 11. C. E. Baum and A. P. Stone, Transient Lens Synthesis: Differential Geometry in Electromagnetic Theory, Taylor & Francis, 1991.
1.
OPTIMIZATION OF THE FEED IMPEDANCE FOR AN ARBITRARY CROSSED-FEED-ARM IMPULSE RADIATING ANTENNA
J. Scott Tyo Department of Electrical and Computer Engineering US Naval Postgraduate School Monterey, CA 93943
INTRODUCTION Impulse radiating antennas (IRAs) are members of a class of antennas that are designed for the radiation of ultra-wideband (UWB) electromagnetic impulses. Through a combination of a non-dispersive transverse electromagnetic (TEM) feed structure and a focused aperture, IRAs act like differentiators for the early-time portion of the waveform. When excited by a fast-rising step, the radiated field closely resembles a narrow impulse. While the nature of the focusing optic and the feed structure do affect the features of the radiated waveform before and after the prompt impulse, the fast part of the radiated signal for a general IRA is [1]
V(t) is the applied voltage waveform, is the peak of the applied voltage waveform, and the surface integral is over the transverse components of the TEM mode in the aperture defined by The radiated field can also be described in terms of the geometric impedance factor of the TEM transmission and the aperture height [2]
A number of metrics have been proposed to compare the performance of antennas operating in the time domain [3,4,5]. The difficulty in comparing performance arises from the non-unique choice of a norm for time domain comparisons. In [4] and [5], performance metrics are defined in terms of the or the peak radiated field. Farr and Baum define a power normalized gain and a voltage normalized gain
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that are used to compare the performance of antennas under constant input power and voltage conditions, respectively. Optimization of is typically is accomplished by allowing to go to zero, an impractical scenario that requires infinite input power and results in infinite fields in the aperture [4]. For that reason, the power normalize gain is often a better metric; however, as defined in (3) has units of length, and care must be taken when applying it to an optimization problem in order to make a fair comparison between antennas of different sizes, as can be increased simply by increasing the physical size of the antenna. For example, when optimizing the feed impedance of a lens IRA constrained to fit within a circular aperture of fixed radius, Farr and Baum [6] used to conclude that low-impedance horns were undesirable. This result is true given the imposed constraint, but the result is dominated by the fact that the area of the aperture of low impedance horns that fit inside a circle of fixed radius goes to zero as Buchenauer, et al., [5] introduced the dimensionless quantity of prompt aperture efficiency defined as
where A is the area of the aperture defined by and is the principle component of the electric field in the aperture, taken without loss of generality as being parallel to the y-axis. Because of the area normalization in (4), aperture efficiency is the preferred metric for comparing the inherent performance of classes antennas regardless of physical size. In contrast to the result presented in [6], it was demonstrated in [5] that low impedance horns are actually more aperture efficient than high impedance horns, and they can be used to efficiently fill a given aperture by arraying. Aperture efficiency and power normalized gain are related by Regardless of the metric used to compute the optimum, it is clear from (4) that the optimum antenna for a fixed input impedance is the one that maximizes the aperture height. An important class of IRAs is the set of antennas that are fed by self-reciprocal feed structures. Self reciprocal antennas are discussed in [7], and have feed geometries that are unaltered by the reciprocation operation where is the position vector in the aperture plane (after stereographic projection) and a is the radius of the circle of symmetry. The coplanar feed IRAs discussed in this paper are examples of self-reciprocal apertures, as shown in fig. 1. Self reciprocal apertures have a number of interesting properties, but the most important ones for this study are 1) exactly half of the power on the transmission line propagates outside the circle of symmetry, 2) the total charge on the feed arms inside the circle is equal to the total charge on the feed arms outside the circle of symmetry, and 3) all contiguous points on the circle of symmetry that are not occupied by conductor lie on a single field line. For the important class of self reciprocal apertures, which are typically confined to focusing the circle of symmetry, the aperture area A is constant for all configurations, and and are equivalent metrics. Aperture efficiency will be the parameter used in this study to optimize the feed configuration in crossed coplanar fed IRAs, primarily because of its dimensionless property and ready interpretation [5].
OPTIMIZATION OF THE FEED IMPEDANCE
365
COMPUTATION OF THE TEM MODE DISTRIBUTION It is well known that the field distribution of the TEM mode on a multi-conductor transmission line can be computed as the gradient of a scalar potential that satisfies the Laplace equation. For many classes of feeds, the potential can be calculated using a combination of the stereographic projection and conformal transformations [1]. However, for arbitrary geometries, the conformal map may not exist in closed form. In this study, it is assumed that the stereographic projection has already been carried out, and the conically symmetric feed structure has been transformed to a longitudinally symmetric structure as discussed in [8]. The important properties of the stereographic projection for this class of antennas are discussed in greater detail in [8]. The asymmetrically crossed coplanar feed structure depicted in fig. 1 can be described in terms of successive conformal mappings, but the Schwartz-Christoffel transformation integrals for the asymmetric cases have not been performed analytically. When the analytic form of the conformal transformation is not known, a numerical approximation can be obtained by employing a Laplace equation solver such as the method of moments or finite element method (FEM). The properties of self-reciprocal symmetry discussed in section I.B. above make the geometry depicted in fig. 1 ideally suited to analysis by the FEM. After numerical calculation of the fields as described in [9], the integrals in (1) – (4) can be evaluated directly or by casting the aperture integral into one of the alternate contour integral forms presented in [2]. The modeling method was
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validated by comparing the computed solutions for
solution is known, namely
and
for the case where the analytic
[9].
CALCULATION OF ANTENNA DESIGN PARAMETERS Using the FEM method described in the previous section, the feed impedance, aperture height, and aperture efficiency were calculated at values of ranging from 3° to 87° and at values of ranging from 0.02 to 0.97. Figure 2 presents curves of for several values of and fig. 3 presents the value of as a function of to achieve popular values of the feed impedance. As expected, as (limit as the spacing between the electrodes goes to zero) and as (limit as the electrodes approach infinitesimal wires). The family of curves in fig. 2 has been fitted to the functional form
where m is given in [4]. The values of the coefficients A, B, C, and D are tabulated for the values of investigated in this study in table 1. The sum of (5) was obtained by analyzing the solution for The first term is the low-impedance limiting form, the second term is the high impedance limit. The third term represents the error due to the addition of the two asymptotic solutions. Because the actual form of is expected to be given in terms of elliptic integrals and elliptic functions the coefficients in (5) do not have a convenient representation in terms of elementary function of Figure 4 presents curves of for several values of Figure 5 presents the aperture height as a function of for popular values of the feed impedance. As which is one-half of the mean charge separation for a four-wire transmission line [2]. As The family of curves in fig. 4 is fit by the functional form
The tabulated values of valid only over the range DISCUSSION
and
are given in appendix 1. The form given in (6) is and strictly does not fit the solution as
The data presented in figs. 2 - 5 provide all of the information needed to design an antenna with a particular aperture efficiency, feed impedance, or feed arm angle Curves of can be plotted for distinct values of as is done in fig. 6. Analysis of fig. 6 provides two interesting results. First, for any particular value of feed impedance, there is a unique geometry that provides the optimum aperture efficiency. The curves in fig. 6 can be used as a design tool to select a particular feed geometry to match the impedance of an individual source. Second, it is clear from the figure that as increases, the optimum aperture efficiency occurs at higher and higher impedances. The peak value of the aperture efficiency for each is plotted in fig. 7, and the feed impedance corresponding to this peak is plotted as a function of in fig. 8. The relationship between optimum feed impedance and appears to be linear so the data points were fit using a
OPTIMIZATION OF THE FEED IMPEDANCE
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constrained least squares linear regression (setting the intercept to 0). The equation for the line in fig. 8 is
assuming that the medium is free space. This linear relationship between optimum feed impedance and was unexpected, and if closed form expressions for the conformal mapping can be obtained, they might provide some physical understanding of the interaction between feed arm angle, extent of the electrodes, and aperture efficiency. It is worth noting that (7) predicts an optimum impedance of for the case of when the two pairs of crossed coplanar feeds are at the same location, corresponding to the case of a single pair of coplanar feeds. This geometry was optimized by Farr and Baum in [4], and the optimum impedance was found analytically to be a difference of less than 2%. In this paper, the optimization was considered for the aperture efficiency (or equivalently, the power normalized gain of [4]). However, for many UWB systems, the quantity that should be maximized is the prompt radiated field, which scales like or the voltage normalized gain of [4]. As mentioned in the introduction and found in [4], the voltage normalized gain is optimized by allowing Not only is this impractical for current-flow reasons (since current on the antenna goes like ), but the wide feed arms needed to obtain low impedances may be expected to significantly enhance feed blockage [10]. However, the importance of for maximizing the radiated field should not be overlooked in designing a system, and hence the value of (normalized to the aperture radius a) is plotted as a function of in fig. 9 for several important values of the feed impedance. The optimization reported here was for the 4-arm IRA, but there is nothing preventing a similar analysis of N-arm IRAs. The optimum aperture efficiency for the 2arm IRA can be computed using the results from [4] and is 27%. The optimum aperture efficiency for the 4-arm case considered here is 35%. Intuitively, the addition of nonblocking feed arms will continue to optimize the aperture efficiency. This is true because additional feeds cause the field distribution in the aperture to be more uniform, hence increasing aperture efficiency. However, the late-time field will be pulled down more rapidly as more feed arms are added, affecting the nature of the radiated pulse, even in the absence of feed blockage. CONCLUSIONS The study described in this paper has provided three principle results. First, the entire design space for reflector IRAs fed by crossed coplanar feeds with reflection symmetry has been sampled. Curves are presented in figs. 2 and 4 with corresponding empirical fits in (5) and (6), that allow ready prediction of feed impedance, aperture height, and aperture efficiency as a function of the geometric parameters of the antenna. These relationships provide more flexibility in IRA design beyond what was possible using configurations with known analytic solutions [4]. Second, the data presented in this paper allows the optimization of the aperture efficiency for any value of the geomterical properties. It has been shown that a distinct optimum exists for any feed arm angle and that an absolute optimum configuration exists at and ( in free space). Finally, the results presented in figs. 6 and 8 show that for any specific value of feed impedance, there is a unique optimum configuration that will maximize aperture efficiency. The feed arm angle is linearly related to the desired
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impedance by (7). This is important in that once the input impedance of the antenna is specified, the antenna can be optimized without impacting upstream components of the system by selecting the appropriate values of and presented in this paper. The method used in this report are general in that they can be applied to any focused aperture system to calculate feed impedance and aperture height (and hence any of the performance metrics described above). While best suited to the analysis of self reciprocal apertures, iterative boundary condition methods have been developed that allow computation of open TEM modes [5]. The method can be easily modified to include the effects of aperture blockage in the evaluation of (2), allowing analysis of geometries for the feed arms that are not coplanar plates, including circular-cross-sectioned feed arms, curved plates, or other arbitrary configurations. REFERENCES
1.
C. E. Baum and E. G. Farr, “Impulse Radiating Antennas” in Ultra Wideband/Short Pulse Electromagnetics H. L. Bertoni, C. E. Baum, and L. B. Felson, Eds., pp. 139147, Plenum Press, New York, 1993 2. C. E. Baum, “Aperture efficiencies of IRAs” Sensor and Simulation Notes #328 (USAF Phillips Lab, Albuquerque, NM, 1991) 3. E. G. Farr and C. E. Baum, “Extending the Definitions of Antenna Gain and Radiation Pattern Into the Time Domain” Sensor and Simulation Notes #350, (USAF Phillips Lab, Albuquerque, NM, 1992) 4. E. G. Farr and C. E. Baum, “Optimizing the Feed Impedance of Impulse Radiating Antennas Part I: Reflector IRAs” Sensor and Simulation Notes #354 (USAF Phillips Lab, Albuquerque, NM, 1993) 5. C. J. Buchenauer, J. S. Tyo, and J. S. H. Schoenberg, “Aperture Efficiencies of Impulse Radiating Antennas” Sensor and Simulation Notes #421 (USAF Research Lab, Albuquerque, NM, 1997) 6. E. G. Farr, “Optimization of the Feed Impedance of Impulse Radiating Antennas, Part II: TEM Horns and Lens IRAs” Sensor and Simulation Notes SSN#384, (USAF Phillips Lab, Albuquerque, NM, 1995) 7. E. G. Farr and C. E. Baum, “Radiation from Self-Reciprocal Apertures” Chapt. 5 in Electromagnetic Symmetry, C. E. Baum and H. N. Kritikos, Eds., Taylor and Francis, Bristol, PA, 1995 8. E. G. Farr and C. E. Baum, “Prepulse Associated with the Feed of an Impulse Radiating Antenna” Sensor and Simulation Notes #337, (USAF Phillips Lab, Albuquerque, NM, 1992) 9. J. S. Tyo, “Optimization of the feed impedance for an Arbitrary Crossed-Feed-Arm Impulse Radiating Antenna” Sensor and Simulation Notes #438 (USAF Research Lab, Albuquerque, NM, 1999) 10. E. G. Farr, Personal Communication, October 1999
TRANSIENT FIELDS OF OFFSET REFLECTOR
Sergey P. Skulkin 1 , Victor I. Turchin2 1
Radiophysical Research Institute Nizhny Novgorod, Russia 2 Institute of Applied Physics Russian Academy of Science Nizhny Novgorod, Russia
INTRODUCTION Reflectors have figured prominently in the history of electromagnetic radiation, antennas and communications. The latter part of the 20th century has seen significant changes and improvements in the design and practice of reflector antennas. Offset configurations of parabolic reflector antennas are widely used for many applications. Transient fields radiated from such antennas1 (which is fed by transient step or deltapulse) can be useful not only for transient ultra wideband antennas. In many cases time domain calculation techniques are simple and can be used for frequency domain fields calculations 2. In this paper a closed form analytical solution is developed for predicting the transient electromagnetic fields which radiated by a perfectly conducting offset parabolic reflector antenna when it is fed by an elementary dipole. We suppose here that the projection of offset reflector surface to the plane has a circular shape. The calculation technique described below is used for any reflector surface with circular projection and the special case of such geometry is also prime-focus parabolic antenna.
ANALYTICAL DEVELOPMENT IN THE TIME DOMAIN Consider a reflector which is a cut of the symmetrical parabolic surface where F is the focus, is Cartesian coordinate system. Let us assume that the antenna aperture (projection of the cut to plane) is the circle described as where are coordinates of its center and is the radius. It is seen when the origin of coordinate system is placed outside the circle, and the cut of the parabolic surface represents an offset reflector. When the system represents a symmetrical reflector. It is assumed that the reflector is fed by an elementary dipole placed at the focus point (0,0, F) and oriented along the unit vector If the reflector antenna is fed by input signal the electric field at any observation point which is defined by the radius-vector can be written as
where denotes the convolution over the time, and represents the transient field at the observation point which is the response on the input signal at the Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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point The exact calculation and analysis of from any reflector antenna require complex solution of nonstationary diffraction problem. However, we suppose that the most part of energy of the input signal spectrum is concentrated in the interval and In this case the usually approximations may be used. In particularly, the electrical current on the reflector surface can be represented as
where is the magnetic field on the reflector surface, is the normal to the surface. Note that If the input signal represents the current on the reflector surface can be found as where the vectors and determine the coordinates of the reflector surface and focus respectively, For instance, when the dipole is directed along the Thereafter the transient field
can be written in the next form
where
space. Each polarization component of vector
and is the impedance of the free will be represented as:
The dependencies of on time for each polarization can be obtained from (7) immediately, using the integration technique for a of complex argument2. Using this technique the integral over the surface can be represented as the integral over the curve
where is a part of plane contour L, belonging to the plane area is determined by the equation
the contour L
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and
It was found that for parabolic surface (13) represents the circle of the radius depending on time: with the center at the point
where The integral over
where
in (12) is written after substituting of (15) as
where in (19) lie on the circle and if i.e. the current point of L lies inside and in the opposite case. As follows from (19), the behavior of considering as time function depends on mutual location of L and There are four cases: (i) if i.e. the observation time is less then minimal travel time from the focus to the reflector and from the reflector to the observation point; it can be shown that this case is realized if or (ii) the circle L lies inside the aperture (iii) only a part of L lies inside the aperture (iv) the circle L lies outside this case corresponds to as (i). The above simple rules determines the difference between the transient fields of the off-set reflector and symmetrical reflector as it will be demonstrated below. NUMERICAL RESULTS AND DISCUSSION As the first example, consider the pulses when the observation point lies in front of the aperture center, i.e. In the case of symmetrical reflector Taking into account that we obtain that the contour L and the aperture boundary are the concentric circles, and when Because when the time boundaries of the transient field are written as
Note that only the cases (i),(ii), and (iv) are realized for this symmetrical scenario which results in the appearance of two in the transient field of the symmetrical reflector: see Figure 1. Compare this result with the transient function of the offset
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reflector: When observation time increases the radius increases starting with zero and the center of L moves simultaneously from the point to the origin which results in the appearance of the case (iii) and corresponding decreasing of the magnitude of the second pulse: see Figure 1. The above properties are illustrated in Figure 1 showing the normalized transient field for symmetrical and offset reflectors
Figure 2 shows the time domain dependencies of for prime-focus reflector antenna. Here and projection of observation point is placed out of
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aperture.
Figure 3 shows the time domain dependencies of for prime-focus reflector antenna. Here and projection of observation point is placed out of aperture. From these figures we notice that fields of prime-focus and offset reflector antennas in near-field region are different. CONCLUSIONS The transient fields for offset reflector antenna have been obtained for all points of the half-space in front of the aperture. We illustrated that the structure of the spatial-temporal field distribution is quit complex, especially in the near-field region. We show that in this region there is a difference between transient fields of prime-focus and offset parabolic antennas. REFERENCES 1. C. E. Baum, E. G. Farr, and D.V. Giri ”Review of Impulse-Radiating Antennas”, in book Review of Radio Science 1996-1999, ed. by W. R. Stone, Oxford University Press, 1999. 2. S. P. Skulkin, V. I. Turchin, ”Radiation of nonsinusoidal waves by aperture antennas,” Proc. EUROEM ’94 Symposium, Bordeaux, France, part2, pp.1498-1504, May 1994.
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A NEW BROAD BAND 2D ANTENNA FOR ULTRA-WIDE-BAND APPLICATIONS
V. Mallepeyre1, Y. Imbs1, F. Gallais 1, J. Andrieu 1 , B. Beillard1, B. Jecko1 M. LeGoff 2 1
2
Institut de Recherche en Communications Optiques et Microondes IRCOM - 7 rue Jules Vallès - 19100 Brive la Gaillarde (FRANCE) E-mail :
[email protected] CELAR (DGA) GEOS/SDM - BP 7419 - 35174 Bruz Cedex (FRANCE)
INTRODUCTION In this article, the theoretical analysis and the experimental results of a new broad band 2D antenna are presented. This study was supervised by the CELAR (French Technical Centre for Armament Electronics) and the IRCOM (Research Institute of Microwave and Optical Communications) in order to develop a radar demonstrator to localise buried or surfaced targets. The purpose of this study is to design an antenna to radiate and receive ultra short pulses covering the frequency range 100 MHz-1 GHz. Even more this antenna must be contained in a minimum volume. The antenna must be well matched in the frequency range, have maximum gain and also preserve the rise time and duration of the generated pulse signal. PURPOSE AND STUDY The aim of the study is to realise an antenna whose requirements are to radiate and receive ultra short pulses (about 1 ns) with high voltage level (up to 20 kV). One important point is that the antenna does not distort the generator pulse, which will degrade the resolution of the radar if the impulse response of the antenna is significantly extended. We determine a distortion coefficient as the extension of the transient radiated pulse divided by the input pulse duration at the feed point of the antenna. The calculation of the distortion coefficient requires the knowledge of the expressions below: = time when 95% of the entire signal energy of the generator is reached
Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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= time when 95% of the entire signal energy of the radiated signal is reached
Suitable distortion coefficient of the antenna should be less than 1.5 . Broad band antennas like TEM horns, ridged horns or log-periodic, don’t have that quality. For instance the distortion coefficient is greater than 15 for ridged horns, 30 for TEM horns and 120 for log-periodic antennas. Also the new concept(figure 1) is an original conducting wire aerial which covers required bandwidth and could radiate ultra short pulse with a distortion coefficient equal to 1.4 .
THEORETICAL STUDY – INTEGRAL FORMULATION TEM antennas consist in two perfectly conducting wire structures with symmetrical fed. The electromagnetic field is formed in the feeding alimentation wire plane, and propagates along conducting arms. Vertical plane contains electric field (E-plane) and horizontal plane magnetic field (H-plane). The theoretical study of perfectly conducting wire structure behaviour can be resolved in the same way as a problem of transient electromagnetic wave diffraction by an obstacle1. This generates a space-time integral equation 2 verified by the induced currents. For the wire structure, currents are obtained by solving the equation (1).
with
and
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379
s and refer respectively to the observation points and the source points. represents the image source points, a and are the radius and the wire outline. are the unit vectors. is the applied field corresponding to the source voltage. The electric field is deduced from currents with the equation (2).
NUMERICAL PROCESSING The integral equation (1) is discretised with the method-of-moments. It uses a second order Lagrange interpolation polynomial and is numerically resolved in the time domain2. It can be written in a matrix system as :
with : the wire structure is divided into segments of length the time is divided into equal intervals The unknown is the column vector at the time The current at the structure segment at the time depends on diffracted field at the time The latter is calculated from the induce currents on the structure at the anterior times and from the incident electric field which is known at this time. Then, the matricial system is solved numerically by successive time steps.
APPLICATION TO THE NEW ANTENNA The antenna is directly fed by the bifilar line. Electric field is also guided in the line, before being radiated in space. Electric field polarisation is principally vertical rectilinear and a single 90° degrees rotation of the antenna gives the horizontal rectilinear polarisation. The entire device is contained in a single plane (figure 2) , so that there is no crosspolarisation. Electromagnetic properties of the antenna (feed impedance, gain, radiation patters, frequency range, distortion) depend essentially on geometrical dimensions such as length or aperture angle.
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SPECIFICATIONS Aperture angle calculation Adaptive formulas exist in literature for similar antenna design like V-dipole which consists in only two wires. Empirical formulations allow to determine inner optimum angle with maximum axial gain, function of s dimension and wavelength
If
then it’s possible to make an extrapolation from above formulas.
It appears useful to add to V-dipole many additional wires connected at their ends (figure 2) whose geometric forms have been optimised step by step to increase electromagnetic qualities of the device, especially to stabilise feed point impedance on the bandwidth, to improve directivity, to intensify field level in axial direction and to completely eliminate cross-polarisation. Electromagnetic fields stay between the two planar lines. Feed point impedance Feed point impedance depends on antenna geometry, on resistive loads and on wire radius. A small r radius strengthens inductance effects of the wires, whose are increasing the inductance imaginary part with the frequency. At the opposite, a more important radius allows to keep a small imaginary part on the bandwidth. So for the simplicity of the input matching of the system, it’s essential to choose a suitable radius (1 centimeter for instance). End of wires matching Classical antennas present reflections of currents from the ends of wires whose are deteriorating their performances. This resonance is responsible for an important lengthening of transient radiated pulse and a weakening of standing wave ratio at the antenna feed-point. The problem is solved with resistive loads distributed on the upper part of each arm of the antenna. The currents propagation on each conductive wire is progressively reduced until nullifying and decreasing parasite reflections. For instance, the evolution law of distributed resistance uses the Wu and King non-reflecting principle3 which is written below :
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with wire length resistively loaded, position of the resistive element on the wire, first resistance at The value is choosen between and Each resistance takes place every 5 centimetres from each other. Resistive values are not critical, so it’s possible to use a hyperbolic law approximation. Then simple and easy to use devices have been built with the association of many standard resistances in parallel along each extremity of wires. The main disadvantage of this technique is the global efficiency of the antenna which is fairly reduced. So to avoid gain degradation, only the upper part of each wire is resistively loaded. The length of this part is between s/3 and s/2.
EXPERIMENTAL RESULTS Geometric dimensions of the antenna given below, are calculated from design rules presented before: L =
s= s’ = s’’ =
1 m 1,044 m 0,3m 0,744 m
l= l’ = r=
0,65 m 0,35 m 0,01 m
To match this antenna (input impedance ) to the feed line (characteristic impedance ), a differential balun has been constructed by Europulse4. This device has been designed to feed the antenna with a symmetrical pulse too.
Reflection coefficient determination Figure 4 shows reflection coefficient of the antenna with the The delivered signal has a maximum of -13dB for in the frequency range 200 MHz - 1.6 GHz.
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Radiated pulse determination One of the antennas is transmitting, fed by a HMP/F Kentech generator [distributed in France by ARMEXEL society] (output voltage of 4 kV, risetime of 220 ps, 50% pulse duration (full-width at half-maximum) of 700 ps, output impedance). This gaussian pulse has a frequency range from DC to about 1.5 GHz (-20 dB/maximum) (figures 5 and 6).
The other is receiving, connected to a digital sampling Tektronix sequential acquisition oscilloscope TDS820 (6 GHz bandwidth). On figure 7, theoretical and measured signals are presented when two “scissors” antenna are facing each other at a 5.80-meter distance. The graph is normalised to allow comparison. The measured voltage peak at feed point is about 50 Volts. The distortion rate is less than 1.4. The measured signal spectrum (figure 8) exhibits a bandwidth from 80 MHz to 1.2 GHz of -20 dB below the maximum.
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Frequency gain and radiation patterns determination As the two antennas are perfectly identical, the axial gains will be identical too. Thus, it is possible to determine the gain for different azimuthal and elevation angles by the following expression:
with
excitation signal spectrum at the drive point of the emitting antenna, measured pulse spectrum at the drive point of the receiving antenna, R : distance between antennas.
So, the determined gain depends on the measured pulses at the drive point of the emitting and receiving antennas, on the electrical length and on the distance between antennas.
The comparison between computed and measured axial gain is given on the figure 9 into the V-V and H-H polarisation with a good agreement. “Scissors” antenna pattern results from combination of each own radiated field wire. However, the main lobe is maximum in axial direction, but there are, in site, side lobes whose levels are usually smaller. The radiation patterns in E-plane and H-plane are shown in figure 10 and 11. In Hplane, the principal lobe has a half aperture angle of 45° at 500 MHz. In E-plane, the lobe has for the same frequency a half aperture angle of 13°. Side lobes, in this plane, are about 8dB smaller (for 500 MHz) from the maximum level. The backscattered field is -15 dB below field measured in axial direction.
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Antenna performances are in good agreement with requirements for distortion coefficient, for bandwidth matching and for high frequency gain.
CONCLUSION A study was led by IRCOM to design antennas capable to radiate and receive ultra short transient pulses with a minimal distortion in the frequency range from 100 MHz to 1GHz. It has resulted an original conducting wire aerial with suitable geometric dimensions for outdoor applications. Resistive loading used in upper-part of antenna wires improves matching input of the antenna and transmitted transient pulse. This plane-antenna has been successfully and easily integrated in the experimental CELAR radar demonstrator PULSAR5.
REFERENCES 1. Miller E.K., Poggio A.J. and Burke G.J.; «An integrodifferential equation technique for time domain analysis of thin wire structures» . Journ. Of Comp. Physics, pp. 12-24, n°48(1973) 2. «Transient EM fields» Topics in Applied Physics. Ed. : L.B. Felsen-Vol. 10-Springer Verlag, New-York(1976). 3. Maloncy J.G., Smith G.S., «A study of transient radiation from the Wu-King resistive monopole-FDTD analysis and experimental measurements». IEEE Transactions on Antennas and Propagation Vol.41-n°5-May 1993. 4. The baluns were made by the EUROPULSE Company (Cressensac, Lot, France) 5. F. Gallais, V. Mallepeyre, Y. Imbs, B. Beillard, J.Andrieu, B. Jecko, M. Le Goff, «A new Ultra Wideband short pulse, radar system for mine detection» Ultra Wide Band Short Pulse Electromagnetics, vol.5, in press.
TIME DOMAIN ARRAY DESIGN
Hans Schantz Time Domain Corporation 6700 Odyssey Drive Huntsville, AL 35806 USA
INTRODUCTION A short pulse ultra-wideband system imposes different constraints on array design than the usual continuous wave narrow band system. The goals of this paper are threefold: to explore these differences, to show how to model a short pulse array, and to discuss how best to graphically describe the performance of an ultra-wideband antenna system. First, this paper will discuss the differences between narrow band and wide band array design. The implications of wide band/short pulse systems to such array properties as grating lobes will be considered. Second, this paper will discuss how to model short pulse arrays under the assumption that there is no mutual coupling between elements. As an example, the behavior of a particular end fire array will be calculated and compared to experimental measurements. Finally, traditional narrow band depictions of antenna performance are ill-adapted for ultra-wideband radiators. Several alternate methods will be presented, including portrayal of the angular dependence of peak instantaneous power, the angular dependence of average power, and the angular dependence of the time domain pulse waveform. CW LINEAR ISOTROPIC ARRAY Linear arrays are a standard topic in any antenna or electromagnetics text.1 Assume a linear arrany has n isotropic sources as shown in Figure 1. Then, the radiation intensity in the direction of the principal maximum is times the maximum intensity of a single oscillator. Since power is divided among n sources, array gain goes as The total phase difference of the fields from adjacent sources is:
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where
if the sousrces are excited simultaneously.
If the field due to an individual source is given by given by:
then the total field of the array is
In a sparse array (one whose inter element spacing is grating lobes will occur as waves are “aliased” integer and half integer periods apart, thus causing constructive and destructive interference, respectively.
UWB LINEAR ISOTROPIC ARRAY A short pulse, sparse UWB array is inherently grating lobe resistant. In the limit as a single pulse is transmitted, there are no earlier or later cycles to be aliased. Grating lobes will show up as pattern sidelobes. The number and intensity of these sidelobes depend upon the complexity of the waveform being transmitted. Consider the ultra wideband “diamond” dipole invented by Larry Fullerton for example.3 When excited by a broadband source, this antenna emits a waveform similar to a Gaussian third derivative:
where is the center frequency and where the waveform has been normalized to have unitary peak amplitude. A comparison of this model to a waveform measured from a diamond dipole is shown in Figure 2. Because of the short duration of this waveform, grating lobes are strongly suppressed away from the main beam.
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As a specific example, consider a two element broadside array with a 26” separation (about at the pulse center frequency of 1.85 Ghz). The roughly 7° halfpower beamwidth is well modeled by conventional continuous wave (“CW”) array theory, but the predicted grating lobes are absent except for a single minor sidelobe on either side. A surface plot of the pulses propagating out in time is shown in Figure 3.
An experiment was performed to verify this grating lobe suppression in simple two element UWB arrays. A Picosecond Pulse Lab Model 4050 Pulser was used to excite a variety of different two element UWB arrays. A 1-18 GHz ARL Horn antenna was used to receive the resulting waveforms and feed them to an HP 54750 Sampling Digitizing Oscilloscope. The array was placed on a rotator and spun around by a computer which gathered waveform and peak power information for each angular
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position. A custom built data acquisition (DAQ) program running in HP Vee was used. Details of the experiment are shown in Figure 4.
The peak power patterns for three arrays are compared to the CW array predictions in Figures 5a-c. Notice the suppression of all but the closest grating lobes. As in a CW array, the intensity in the direction of the principle maximum is times the maximum intensity of a single element. In the present experiment, the short range (“R”) raised the off main beam background level above the –6 dB that would otherwise be expected for a two element array with d << R.
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MODELING SHORT PULSE ARRAYS These results suggest that while CW array theory does an adequate job of modeling the main beam of a UWB array, a different method is necessary to predict sidelobe levels. The most conceptually simple method is to employ the technique of Figure 3 – to calculate the time domain waveform transmitted in each direction as a result of the superposition of the array elements. Assuming mutual coupling between the n elements is negligible, this is accomplished by adding up multiple copies of the transmitted waveforms with the appropriate delays in the appropriate directions. The method of waveform superposition is particularly appropriate for very short or “distorted” pulses where quasi-harmonic methods will tend to breakdown. Waveform superposition was used to model an eight element 1.3 GHz end fire array created by Larry Fullerton over ten years ago (see Figure 6a, below). Array performance was measured by an experiment similar to the one already described. The endfire array was found to have a half power beam width of roughly 70° - slightly more than the 60° predicted by waveform summation. Timing limitations prevented all eight elements from being perfectly synchronized, and the resulting pattern more closely matched what would be expected from a six element array (see Figure 6b, below).
The directivity of an ultra-wideband array depends upon the aperture size and the center frequency.4, 5 In the limit of a narrow beam width, the directivity of a linear array is proportional to its length. A linear array twice as long will have half the beam width.
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DEPICTION OF UWB ANTENNA PERFORMANCE Traditional narrow band depictions of antenna performance are ill-adapted for ultra-wideband radiators. A variety of schemes have been adopted to describe UWB antenna performance including total energy pattern plots, and peak energy pattern plots. One particularly interesting and innovative method is to create time and frequency domain waterfall plots. This method (pioneered by Jon Young and John Gwynne6) is shown applied (courtesy of Jon Young) to a 2 GHz diamond dipole in Figure 7, below.
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The peak power patterns and waveform plots employed throughout this report offer a simpler portrayal of UWB antenna and array performance. These simpler methods tend to obscure the many details evident in the waterfall plots, however.
CONCLUSION This paper has explored the difference between conventional CW array design and short pulse UWB array design. In particular, the grating lobe suppression of a short pulse array was emphasized. Although conventional CW array theory can adequately predict the main lobe pattern of a broadside linear UWB-SP array, it does not adequately model performance away from the main lobe. The method of superimposing time domain waveforms to model array performance was described and applied to a two element UWB broadside array and to an eight element UWB endfire array. This method is particularly well suited for very short pulses where quasi-harmonic methods may be more difficult to apply. Finally, alternate means of portraying UWB antenna performance were discussed. In conclusion, although traditional CW array analysis techniques can be profitably applied to UWB-SP arrays, a fuller understanding requires specialized UWB techniques. Acknowledgements The author gratefully acknowledges the assistance of Troy Fuqua, Mark Faust, Jon Young, Mark Barnes, Larry Fullerton, Rachel Reinhardt and Ivan Cowie. REFERENCES 1
John D. Kraus, Antennas, 2nd ed., McGraw Hill, New York (1988), pp. 118-200. J. A. Stratton, Electromagnetic Theory, McGraw Hill, New York (1941), pp. 451-3. 3 Larry Fullerton, “Time Domain Radio Transmission System,” U.S. Patent 5,363,108, Nov. 8, 1994. 4 F. Anderson et al. “Ultra-wideband beamforming in sparse arrays,” IEE Proceedings-H, Vol. 138, No. 4, August 1991. 5 Kraus, Op. Cit., pp. 150-151, 6 J.D. Young and J. Gwynne, Report on Antenna Transient Patterns, The Ohio State University, ElectroScience Lab Technical Report 732169-3. 2
Recent Developments In Ultra-Wideband Sources and Antennas William D. Prather, Carl E. Baum, Jane M. Lehr, Robert J. Torres, Tyrone C. Tran, Jeffrey W. Burger, John A. Gaudet Air Force Research Laboratory Kirtland AFB, New Mexico
INTRODUCTION Ultra-wideband (UWB) microwave sources and antennas are of interest for applications such as the location and identification of hidden objects [1,2]. These may include land mines, unexploded ordnance (UXO), or objects hidden by foliage. We have even been asked to assess the feasibility of finding termite damage in structures! Interest in is growing rapidly, and we are now seeing the technology emerging in several countries in Europe, Russia, and the Ukraine. Much of the current research is being performed at the Directed Energy Directorate of AFRL at Kirtland AFB, NM. Significant progress has been made in UWB antenna technology since its infancy in the late 1980s, and the discipline is becoming rather mature. But, new opportunities continue to arise. Recent requirements include antennas that are lighter weight, have extremely wide bandwidths, and operating frequencies reaching down into the UHF, VHF, and even the HF bands for use with low frequency Synthetic Aperture Radar (SAR) systems. Significant progress is evident also in gas and oil switched source development. In a previous series of articles and papers, we discussed the development of UWB technology over the last decade [3-7]. In this paper, we will describe recent progress in sources and antennas and some exciting new developments in both gas and solid state switching and in antenna design. UWB SOURCE TECHNOLOGY The Impulse Radiating Antenna (IRA): Despite the name, the Original IRA, shown in Figure 1, was not just an antenna, but actually a high voltage UWB source developed and fielded in 1994. It used a high pressure hydrogen switch, a focusing lens, and a fourarm TEM horn to produce an extremely powerful UWB pulse from a 4m reflector [8]. With a charge of only ± 60 kV, this system generated a transient signal of 4.6 kV/m at 305m. This gives an field-range (E x R) product of 1.4 Mv. At the time it was built, it was not the most powerful UWB source in existence, but it was without a doubt the most efficient, and it produced an extremely clean waveform with a very flat spectrum that covered the entire range from 35 MHz to 3.5 GHz. Originally the 4m parabolic was chosen in order to get the widest range of performance that we could afford at the time, and to not overly confine the design and construction of the lens, the arms, and the terminating resistors. The result was a radiator that performed extremely well in all respects. However, one must admit that it posed some challenges. The 4m parabolic reflector is difficult to handle, and the far field waveform does not emerge until one is beyond 100m! This made the source
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quite good at long range propagation, but it was difficult to use in a more confined laboratory setting. With even the largest of our anechoic chambers, one cannot get a test object or sensor into the far field. As a result, the IRA has not been used much since it was initially developed and demonstrated. However, within the last year, it was decided to go ahead with plans to upgrade the system by reworking the high voltage power supply and refitting it with a 2m dish. Also, during the refurbishment of the ± 60 kV pulsed power source, the gas thyratron was replaced with a trigatron which allowed the system to be run at ± 75 kV. So, with a reflector only half the size of the original, and the output voltage at 150 kV, the near electric field is more than doubled. It is now near 60 kV/m. The far field, of course, is approximately half of the original, but for applications which are within the near field zone, (now 25m) such as laboratory tests, this design has some advantages. Since it is using basically the same power supply and exactly the same lens and switch, the voltage pulse from the 2m IRA has the same risetime of around 130 ps and 1/e fall time of about 20ns. The near field, of course, has exactly the same waveform. The radiated far field, being a derivative of the voltage waveform, has a risetime of 85ps and a pulse width (FWHM) of about 130ps. The radiated spectrum of the 2m IRA has been measured to be flat within 3 dB from 70 MHz to 4 GHz. The Solid State Shockline and Lensed Horn: For some applications, low-jitter triggered switches which are small, inexpensive, and have very long lifetime are needed. Sub 100-picosecond switching occurs in certain silicon PIN-diode-like structures called Delayed Breakdown Devices (DBDs) that are rapidly overvolted beyond static reverse breakdown. There is a delay of several nanoseconds, the period of which is bias dependent, before a fast breakdown occurs to close the contact between
cathode and anode. Sub-l00-picosecond breakdown occurs as an ionization wave sweeps across an intrinsic material faster than the carrier drift velocity. As with the avalanche transistor and photoconductive switch, the duration of the pulse through the diode must be limited to avoid filamentation and destruction of the device. This two-terminal avalanche technology uses a Drift Step Recovery Diode (DSRD) driving a Silicon Avalanche Shaper (SAS) in a TRAPATT mode. Ivor Grekhov and his colleagues at loffe Physical-Technical Institute, St. Petersburg Russia, pioneered this technology and have been developing it since the late 1970’s [9,10]. Recently, the Air Force Research Laboratory in conjunction with The University of New Mexico, has studied the physical processes of the silicon avalanche sharpeners and its operating characteristics [11,12], This study confirmed the robustness of the technology and validated all the operating parameters. The DSRD opening switch achieved a rise time of 740 ps and 1.9 kV pulse amplitude, providing the transient required to rapidly drive the SAS into reverse bias breakdown. The SAS sharpens the transient to a 100 ps rise time at the DSRD pulse amplitude level into a 50 load. Life times greater than shots have been achieved for this system. The DSRDs have been stacked in such a way as to produce high voltage pulsers with fast transients using a SAS shaping head. One such system which was tested was an loffe Institute PPG-10-5 pulse generator which achieved an output amplitude of 10 kV, a rise time of less than 100 ps (with shaping head), a pulse width of 2 ns, and a pulse repetition rate of 5 kHz into a 50 load. Since the stacking limits of the technology have not been reached, the Air Force Research Laboratory is pursuing a design with the Russian group that will produce 100 kV pulses with a rise time of less than 100 ps.
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Cultivation and improvement of this technology by this team has resulted in a significant advance in high voltage triggering capability which is now being developed into solid state switched shocklines. Recently, these demonstrated the ability to multiply the charge from a 12 volt battery into a 3.2 kV transient signal with a risetime of 100 ps. The shockline device was precisely triggered at 2 kHz with only 16 ps of jitter, a remarkable accomplishment [13]. This very low jitter performance will enable such devices to be combined into high voltage arrays or single, high voltage pulsers capable of competing with today’s gas and oil technology for total power while offering the additional feature of allowing the source to be electronically steered. Further development of this technology using higher energy silicon carbide (SiC) diodes is expected to enable the construction of a 350 kV module within the next year. It is expected that both of these breakthroughs will result in new products and capabilities for both the military and the private sectors. Gas Switching Technology: A.
Triggered Gas Switches Precision switching is a critical technology that will allow the full exploitation of UWB radiation for a wide range of applications. Thus, AFRL is pursing three approaches to reducing the jitter of UWB radiating systems: 1) the development of a low-jitter switching element, the Ferratron, 2) optimization of overall system design parameters and 3) introducing field enhanced triggers or laser triggers into high pressure hydrogen switches that already have high rep rate and fast risetime capability [14,15]. The Ferratron, as shown in Figure 8. uses a ferroelectric trigger as an electron source which, when combined with high gas flow, provides reliable, low jitter triggering at high rep rates. Although the design is not yet optimized, the measured jitter of the prototype Ferratron was only 63 ps using a charge voltage of 2.5 kV with a 500 ps risetime. Triggering was done in low pressure nitrogen gas. For comparison, laser triggering has only produced jitter as low as 130ps. Hyperboloidal Lens One of the difficulties of measuring the output waveform emitted by an in-line coaxial switch is that the wave launched spherically from the switch and suffers some reflections, and hence dispersion, before becoming established in the coaxial waveguide. To solve this problem, Baum and Farr [17] derived a hyperboloidal lens to transition the wave into the coax. Thus, for measurement purposes (development) or transition to an antenna, the process is much improved, especially for very fast switches. B.
Oil Switches and New Materials A liquid dielectric is sometimes a desirable choice as a switching medium because it avoids the mechanical constraints imposed by high pressure gasses. Also, whereas the electrical breakdown field in gases scales with the pulse width of the charging waveform as in liquids, the breakdown field scales as Thus, a fast charging waveform, coupled with the intrinsically higher electrical breakdown fields, allows for very high inter-electrode electric fields. Of course, since the resistive phase of the transmitted pulse risetime is also larger, there is a tradeoff: but risetimes on the order of 100 ps have been achieved at 200 kV and 1 kHz and at lower rep rates up to 700 kV in further tests of the system described in. The typical liquid used is transformer oil, but its drawback is that switch firing leaves behind particles of carbon residue which limit rep rate and will eventually short out the switch. As a result, we are investigating the switching properties of several other types of synthetic dielectric oil.
DIELECTRICS AND ARTIFICIAL DIELECTRICS. Artificial Dielectrics Many HPM applications require the use of dielectric materials for lenses required either for focusing or for guiding wideband signals around bends. Typically, the materials used have been polyethylene, polypropylene, or polycarbonate because they are inexpensive, easy to use and have about the same dielectric constant as transformer oil or other insulators. However, they are fairly dense such that for a lens of more than a foot in diameter, they become prohibitively heavy. One lens we designed for a 3m TEM horn weighed over 300 lbs! Therefore, we have been searching for alternatives that would save weight. Recently, C.A. Frost found a new material made by Emerson-Cummings that was deemed worthy of further investigation. It is an artificial dielectric material that was designed for narrow band microwave systems, but we are investigating it for use in UWB applications. It consists of hollow glass beads coated with aluminum suspended in a dielectric foam or other substance. The beads can be made in various sizes from a few mm in diameter to less than a mm. Thus, different concentrations of beads can be used to make an artificial dielectric in the classical sense or, alternatively, the glass beads without any coating can be used to
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lower rather than raise the dielectric constant of a given material. No test results are available yet, but by this time next year, we should be able to share the results of this investigation. Constantly Varying Dielectrics A significant study has also been published by Baum and Stone [19] on the propagation of UWB waves through lensing material with constant and varying dielectric properties. It is a first-principles analysis of wave propagation in canonical geometries with varying dielectric constants and provides new insight into EM wave behavior under these conditions, Dielectric Bends Another investigation into the application of constantly varying dielectrics to UWB systems involves their use for guiding UWB signals around bends without dispersion. Over the past year, Farr Research has been investigating various configurations of varying dielectrics with a great deal of success [20], The best results to date have been achieved using 5 or 6 parallel channels of different dielectric materials consisting of powder in an epoxy base. Using this technique, Farr Research has been able to propagate a pulse around a 90° bend while preserving a 56 ps risetime.
UWB ANTENNA TECHNOLOGY Impulse Radiating Antennas (IRAs): Ultra-wideband antennas which have extremely wide bandwidth and which minimize both frequency and spatial dispersion are available in numerous sizes and shapes. The most recognizable of these is the IRA shown in Figure 1 which has been built in diameters ranging from 9" to 12' diameter and with various focal lengths depending on the application. These antennas have an extremely wide bandwidth, usually two decades, and a beamwidth of only a few degrees. Collapsible IRA (CIRA): The ultra-wide bandwidth and focused beam characteristics of the IRAs have attracted the attention of many users, one of which was the U.S. Marine Corps. They needed a wideband antenna that could be used by ground troops and therefore needed to be lightweight and portable. The result was the Collapsible IRA (CIRA) as shown in Figure 9 that is made of conducting fabric. This remarkable antenna folds up like an umbrella, but has surprising performance. It has been demonstrated to perform from the high end of the HF band all the way to x-band. As discussed in previous reports, because of the p x m feature of this design, the pattern at lower frequencies is that of a cardioid with a deep null in the back. This has been found to be very useful, because the antenna serves as an unambiguous direction finder, even at HF frequencies [21-24]. SAR ANTENNAS The successful demonstration by the Army Research Laboratory of land mine detection by an UWB synthetic aperture radar (SAR) system high above ground level [2] has raised a considerable amount of interest in UWB technology and particularly in UWB antennas. The SAR community in general has shown interest in increasing the bandwidth of their systems, and in response, AFRL has built and tested two different types of wideband antenna in order to evaluate their suitability for use with SAR systems [25-27], First a narrow beam antenna was built using a shallow 18" parabolic reflector with a focal length to diameter ratio (F/D) of 0.5. This antenna is shown in Figure 10. Then, as an alternative, a reflector antenna called a pulse radiating antenna element (PRAE) was designed, fabricated, and tested. In this antenna, shown in Figure 11, the paraboloidal reflector has been replaced by a flat plate in order to broaden the beam from that normally produced by an IRA. These antennas are being evaluated in both time- and frequency-domain facilities in Albuquerque and in Dayton, Ohio.
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CONCLUSIONS There has been a tremendous amount of progress in the field of Ultra-Wideband technology in the last decade. Innovations in sources, antennas, sensors, and wideband electromagnetic theory have greatly enhanced the capability of UWB electromagnetic systems. In particular, there are many varieties of UWB antennas that have been developed and deployed in sizes ranging from 9" to 12' in diameter with bandwidths ranging from 35 MHz to over 20 GHz. As a result, many new applications are emerging that make use of the unique properties of UWB signals. With continued research and development we should see many of these come to fruition in the next few years.
ACKNOWLEDGEMENT This work was sponsored in large part by the Air Force Office of Scientific Research, Washington, D.C. In particular, the authors would like to thank Dr. Jack Agee, Dr. Clifford Rhoades, Dr. Arje Nachmann, and Dr. Robert Barker for their encouragement and support.
REFERENCES [1] C.E. Baum, ed, Detection and Identification of Visually Obscured Targets, Taylor & Francis, Philadelphia, 1999. [2] L. Carin, et al, "Ultra-Wideband Synthetic Aperture Radar for Mine Field Detection," Antennas & Propagation Magazine, 41 (1), February 1999. [3] F.J. Agee, W.D. Prather, et al, “Ultra-Wideband Transmitter Research,” IEEE Trans on Plasma Science, Special Issue on HPM, 26 (3), June 1998. [4] C.E. Baum, L. Carin, and A.P. Stone, eds., Ultra-Wideband, Short-Pulse Electromagnetics 3, Plenum Press, NY, 1997. [5] E. Heyman, B. Mandelbaum, and J. Shiloh, eds, Ultra-Wideband, Short-Pulse Electromagnetics 4, Plenum Press, NY, 1999. [6] W.D. Prather, et al, "Ultra-Wideband Source & Antenna Research," IEEE Trans. Plasma Science, 8th Special Issue on High Power Microwave Generation, June 2000, to be published. [7] C.E. Baum, "Review of Impulse Radiating Antennas," with E.G. Farr and D.V. Giri, Chapter in Review of Radio Science 1997-1999, Oxford University Press, UK, 1999. [8] D.V. Giri, H. Lackner, I.D. Smith, D.W. Morton, C.E. Baum, J.R. Marek, W.D. Prather, and D.W. Scholfield, "Design, Fabrication, and Testing of a Paraboloidal Reflector Antenna and Pulser System for Impulse-Like Waveforem," IEEE Trans. on Plasma Science, 25, 1997. [9] I.V. Grekhov, and A.F. Kardo-Sysoev, “Subnanosecond Current Drops in Delayed Breakdown of Silicon p-n Junctions,” Sov. Tech. Phys. Lett. 5 (8), 1979. [10] I.V. Grekhov, "New Principles of High Power Switching with Semiconductor Devices," Solid State Electronics, 32 (11), 1989. [11] R.J. Focia, E. Schamiloglu, C.B. Fleddermann, F.J. Agee, and J. Gaudet, “Silicon Diodes in Avalanche Pulse Sharpening Applications,” IEEE Trans. Plasma Sci., 25 (2), April 1997. [12] R.J. Focia, E. Shamiloglu, and C.B. Fleddermann, “Simple Techniques for the Generation of High Peak Power Pulses with Nanosecond and Sub-nanosecond Rise Times,” Rev. Sci. Instrum., 67 (7), July 1996. [13] C.A. Frost, T.H. Martin, R.J. Focia, and J.S.H. Schoenberg, "Ultra-Low-Jitter Repetitive Solid State Picosecond Switching," 12th IEEE Pulsed Power Conference, Monterey, June 1999. [14] L.H. Bowen, E.G. Farr, J.M. Elizondo, and J.M. Lehr, "High-Voltage, High Rep-Rate, Low Jitter, UWB Source with Ferroelectric Trigger," 12th IEEE Pulsed Power Conference, Monterey, June 1999. [15] J.M. Lehr, C.E. Baum, L.H. Bowen, J.M. Elizondo, D.E. Ellibee, E.G. Farr, and W.D. Prather, "Progress in the Development of the Ferratron: A Novel, Repetitively Rated, Triggered Spark Gap Switch with Ultra Low Jitter," Proceedings of the International Society for Optical Engineering, SPIE 2000, Orlando, April 2000. [16] J.M. Lehr, M.D. Abdalla, B. Cockreham, F. Gruner, M.C. Skipper, W.D. Prather, and K.R. Prestwich, "Development of a Hermetically Sealed, High Energy Trigatron Switch for High Repetition Rate Applications," 12 th IEEE Pulsed Power Conference, Monterey, June 1999. [17] E.G. Farr, D.E. Ellibee, J.M. Elizondo, C.E. Baum, and J.M. Lehr, "A Test Chamber for a Gas Switch Using a Hyperboloidal Lens," Switching Note 30, Air Force Research Lab, Kirtland AFB, NM, March 2000. [18] J.S.H. Schoenberg and C.J. Buchenauer, “Artificial Dielectrics for Ultra-Wideband Application,” Ultra Wide-band/ShortPulse Electromagnetics 3, C.E. Baum, L. Carin, and A.P. Stone (eds.), Plenum Press, New York, 1997.
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[19] C.E. Baum and A.P. Stone, "Synthesis of Inhomogeneous Dielectric, Dispersionless TEM Lenses for High-Power Application," Electromagnetics, 20:17-28, 2000. [20] W.S. Bigelow and E.G. Farr, "Minimizing Dispersion in a TEM Waveguide Bend by a Layered Approximation of a Graded Dielectric Lens," Ultra-Wideband/Short-Pulse Electromagnetics 4, J. Shiloh and E. Heyman, eds, Plenum Press, NY, 1999. [21] L.H. Bowen and E.G. Farr, "Recent Enhancements to the Multifunction IRA and TEM Sensors," SSN 434, February 1999. [22] E.G. Farr, C.E. Baum, W.D. Prather and L.H. Bowen, "Multifunction Impulse Radiating Antennas: Theory and Experiment," Ultra-Wideband/Short-Pulse Electromagnetics 4, J. Shiloh and E. Heyman, eds, Plenum Press, New York, 1999. [23] L.H. Bowen, E.G. Farr, and W.D. Prather, "Fabrication and Testing of Two Collapsible Impulse Radiating Antennas," SSN 440, December 1999. [24] L.H. Bowen, E.G. Farr, and W.D. Prather, "An Improved Collapsible Impulse Radiating Antenna," SSN 444 April 2000. [25] E.G. Farr, C.E. Baum, W.D. Prather, and T. Tran, "A Two-Channel Balanced-Dipole Antenna (BDA) with Reversible Antenna Pattern Operating at 50 Ohms," SSN 441, Air Force Research Laboratory, December 1999. [26] E.G. Farr, L.H. Bowen, G.R. Salo, J.S. Gwynne, C.E. Baum, W.D. Prather, and T.C. Tran, "Studies of an Impulse Radiating Antenna and a Pulse Radiating Antenna Element for Airborne SAR and Target Identification Applications," Sensor and Simulation Note 442, Air Force Research Laboratory, Kirtland AFB, NM, March 2000. [27] E.G. Farr, L.H. Bowen, G.R. Salo, J.S. Gwynne, W.D. Prather, C.E. Baum, and T. Tran, "Lightweight Ultra-Wideband Antenna Development," SPIE 2000, Orlando FL, April 2000.
ULTRA-WIDEBAND SPARSE ARRAY IMAGING RADAR
Graeme N Crisp, Christopher R Thornhill, Ray Rowley, Justin Ratcliffe Defence Evaluation and Research Agency St Andrews Road Malvern WORCS UK INTRODUCTION Ultra-Wideband radar has been proposed for a number of applications primarily where the penetration of obscurants, and the detection of physically small targets are of importance. One good example of such an application is the stand off detection of land mines for a route clearance asset. Another is the detection of navigation hazards for the negotiation of hostile terrain by autonomous robotic vehicles. In both cases operational considerations preclude the use of a side looking geometry and a forward looking sensor is essential 1,2. DERA has been engaged in research for both of the applications described above and in this paper we describe a forward looking ultra-wideband radar system developed for this work. In its simplest form a forward looking bistatic impulse radar system comprises a simple transmitter and receiver as shown in figure 1.
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In order to detect and identify small targets such as landmines, the antennas employed must have wide bandwidth and high pulse fidelity. For this reason the system described here uses TEM horns for transmission and reception, and the resulting radiated pulse has a rise time ~ 130 ps. The maximum antenna size is limited by two factors. Firstly for operational reasons compact antennas are desirable. Secondly TEM horns of too wide a plate angle show a degraded high frequency response as a result of differential time delays across the antenna aperture. Thus in a practical system relatively small antennas are employed resulting in a large antenna foot print.
Figure 2 shows the measured azimuth pattern of one of our TEM horn antennas measured in the laboratory. As can be seen from this figure the peak field strength measured in the time domain falls off only slowly with azimuth angle and the two way full beamwidth is approximately 30 degrees. Figure 3 shows a typical range profile measured by such a simple impulse radar in the test area shown in figure 4.
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The data shown in figure 3 have been scaled to compensate for dependence of received signal strength with range, and stationary background signals arising from platform reflections, direct antenna coupling etc. have been removed. The resultant signal clearly illustrates the difficulty in identifying targets with such a system in the presence of natural clutter. The signatures of targets 1,2 and 3 shown in the inserts of fig 3 are hardly discernible amongst the complex of scattered signals arising from clutter. Clearly such a system is only useful at very short range, and for longer range operation measures must be adopted to reduce the sensitivity to clutter. The measurement area shown in figure 4 is representative of a realistic situation where the primary interest is the detection of targets on the track ahead of the measurement platform. Clearly the majority of the extraneous scatterers are located to the side of the track, while targets of interest lie on the road. Additional angular resolution would enable the clutter components of the signal to be largely isolated from the target signatures. In this paper we show how such angular resolution may be obtained and show some results obtained using adapted impulse radar systems.
ANGULAR RESOLUTION ENHACEMENT One way of improving the angular resolution would be the use of larger aperture antennas. However for the reasons given above this is not a practical solution and other techniques are needed to provide clutter rejection using the antennas described above. Two approaches have been considered. Firstly by exploiting the forward motion of the radar platform, an effective aperture may be synthesised thus improving the cross range resolution. Although the geometry of a forward looking radar is not favourable for aperture synthesis, some resolution is obtainable in an analogous fashion to the Doppler beam sharpening methods employed by some narrowband radars. Alternatively the radar aperture may be increased in effective width by deploying a number of antenna elements across the vehicle width. The first approach has the attraction that only two radar antenna elements are needed, but simple symmetry arguments show that for the forward looking geometry of
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interest here, such synthetic aperture processing will result in a left/right ambiguity so that precise target localisation is not possible. The second approach involves greater radar complexity, but in principle allows unique localisation of targets in cross range. Thus both methods are worthy of consideration and it is therefore of interest to estimate their relative benefits. In order to do so we consider a simplified signal model for the impulse radar. We model the radar radiated signal as an idealised Gaussian pulse of prescribed width so that the radiated pulse may be described by:
where
is the half-height full width of the pulse.
While in practice such a signal is not radiateable, on account of its finite DC component, it provides a convenient approximation which is adequate for the purpose of characterising the resolution of an impulse radar system.
In the spirit of the Rayleigh criterion for resolvability, it is readily shown that two such pulses with a relative displacement of give rise to a bimodal response as shown in figure 5. Although the depth of the minimum in the net response differs slightly from that used in the Rayleigh criterion, figure 5 suggests a convenient definition for resolution in the time domain, namely that two pulses are distinguishable if their separation exceeds This definition is used throughout the analysis presented below. It should be noted that some impulse systems radiate pulses that differ markedly from the assumed form given by equation 1, but in general these also can be used to define resolution criteria. In most cases when compact pulses are considered, two pulses become resolvable when their separation is of the order of the pulse width. The analysis presented below thus gives the correct order of magnitude even for systems that employ non-Gaussian waveforms. SYNTHETC APERTURE PROCESSING
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Figure 6 depicts a forward looking radar approaching two targets Tl and T2. The Radar starts at the position O which is equidistant from the two targets. When the radar reaches the position A, the two targets are no longer at equal ranges from the radar and Tl is a distance further from the radar than T2. When exceeds half the pulse length the two targets become time resolved as the two way propagation effectively doubles the path length difference Thus when an aperture of length d is synthesised through forward motion of the radar platform, the cross range resolution may be determined by expressing in terms of d and y, and then solving for y under the condition:
The problem may be solved numerically without approximation and a convenient approximation can also been derived for the case In this case:
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Figure 7 shows the geometry of a sparse array real aperture radar system. One transmitter and receiver is located at A and another transmitter and receiver is located at B. Their separation in cross range being d. T1 and T2 denote two targets which are at the same range from the point of A. However the range of T1 from B exceeds the range of T2 from B by
as indicated. Consequently T1 and T2 can be resolved when
Thus the
cross range resolution may be determined for targets close to the direction of travel by numerically solving
This problem may also be solved numerically without
approximation. For the case of long ranges
is approximated by:
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Figure 8 shows the "exact" numerical solutions as a function of aperture size for the case where is 10m and the radar resolution is 0.05m. The plots show that for model parameters relevant to the design of a practical vehicle mounted system, the approximate analytical formulae derived above agree well with the exact numerical solutions. The plots may also be used to compare the cross range resolution capabilities of synthetic and real aperture methods. As in the case of a real aperture system the distance d represents the antenna separation, this is limited to a distance of the order of the vehicle width. A typical value would be ~1.5m. In the case of the synthetic aperture system d is limited only by the maximum radar range, and by drift in any platform position measuring devices used to correct the data for the vehicles forward motion. Even so it is clear from figure 8 that the integration distance needs to be very large in order to achieve similar resolution by synthetic aperture to that obtainable by the use of real aperture processing. CONCLUSIONS The experimental data presented above indicate that forward looking vehicle mounted impulse radar systems using broad beamed antennas are susceptible to clutter which in some situations arises predominantly from scatterers outside the area of interest for target detection. When physical constraints limit the size of the antennas which can be used in such systems we have shown that cross range resolution can be obtained either by the use of aperture synthesis, or by the use of a sparse radar array. In the limiting case considered here this comprises two UWB radar systems separated on a base line in the cross range direction. A simplified radar model may be used to predict the resolution obtainable with both approaches. For the realistic sensor parameters presented the model shows that aperture synthesis is less effective in providing good cross range resolution than a real radar array.
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Additionally synthetic aperture processing in the geometry considered cannot be used to locate targets uniquely as the symmetry of a single forward looking radar system results in a left/right ambiguity. It is therefore concluded that the use of an array radar for forward looking vehicle mounted applications is advantageous. REFERENCES 1. Amazeen, C, US army Cecom NVESD & Kositsky, J, SRI International. June 2000 Vehicle mounted forward looking mine detection. 2. Boling, R, Institute for Defence Analysis. June 2000 General decision models for electromagnetic detection of objects.
CROSS-FIELD CHARACTERIZATION OF DIPOLE RADIATION IN FRESNEL ZONE
Mihai Badic and Mihai-Jo Marinescu Research Institute for Electrical Engineering - ICPE Splaiul Unirii 313 Bucharest 74299, ROMANIA
INTRODUCTION The paper deals with vectorial and phasorial characterization of electric and magnetic fields in Fresnel zone for a short dipole antenna. As known, in Fresnel zone, these fields have a special behaviour, so that they result in a reactive energy component which flows coming back to the source. This cross-field phenomenon is characteristic to waveguides, reflection and Fresnel zone of radiating systems 1. The accurate description of the vectors/phasors evolution in the near field zone is very important for calculating both the Poynting vector and the characteristic impedances in this area; also, it is of great importance for studying the interaction between electromagnetic field and substance 2,3. The electromagnetic field of an alternating dipole is involved in macroscopic applications (impulse radiation, electromagnetic shielding in Fresnel zone) as well as in molecular theory (polarization and magnetization). On the other hand, the paper may be help for the development of the theory concerning fields of impulse radiating antennas. If spectral analysis used, all the conclusions of the paper may be extended to pulse propagation because each spectral component will propagate according to the same rules 4. So, the paper should be considered as the first stage of complex theoretical and experimental research, in order to obtain an accurate characterization of electromagnetic radiation in Fresnel zone. CLASSICAL APPROACH OF DIPOLE RADIATION Elementary dipole antenna is the simplest, ideal source of polarized radiation, which can be visualized as a tiny oscillating doublet of length much lower than λ. Thus, an important category of antennas may be described starting from this model and applying the principle of effect superposition. The characteristic equations of such current element with harmonic variation are deduced from Maxwell's equations, by applying the magnetic vector potential. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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where:
l = length of the short wire.
Equations (1-3) are written for the spherical system of coordinates according to Fig. 1.
One can observe these fields - electric and magnetic - present a time dependence and a space double dependence: versus angle and distance from source r. Function of r it is formally settled the limit of the radiation zone: at a distance comparable with the wavelength, the radiation component becomes equal to the stationary field component. On the other hand, the dependence on determines near field and far field sidelobes pattern. VECTORIAL AND PHASORIAL CHARACTERIZATION
Usually, the space-time evolution of vectors and is studied only by means of their components in and respectively and impedances and Further on we propose a method for accurate characterization for the phasor vectors described by equations (1-3). Vectorial Approach Because we deal with a double dependence - space and time - we have to analyze them separately in order to reconstitute at the end the complete vectorial-phasorial image.
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For the vectorial analysis versus r, we will apply a variable change in equations (1-3) as follows: which implies meaning and respectively The consideration of an initial moment for the phasor attached to the respective vector doesn't affect the generality of the demonstration, because the phasorial analysis will be carried on. The same thing is valid also for the implicit condition MHz corresponding to Results:
where Using identity: where
and
we obtain:
The above form of equations allows us to print the variation of the phasors' amplitude which represent the electric and magnetic fields. These equations can be normalized as follows:
In Fig.2 it is presented the amplitude variation of these magnitudes versus r :
for two different values of angle
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It is obvious that amplitude doesn't depend on the considered time phase matter to what curve from Fig.2 we refer to.
no
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In Fig. 3 we have presented the variation of and for to equations (12). The resultant E field is calculated as follows:
according
The curves are plotted in order to easily highlight the phase difference between the components, versus r. It may observed that: between and there is, everywhere in space, a phase difference that tends to establish itself at quadrature; between and there is a phase difference in the near field zone, approaching to be in time phase when r increases; between and there is also a phase difference at any distance r, but it tends to establish itself at quadrature when evaluated far from the source; as a consequence of the above, the resultant module is in time phase with excepting an area placed very close to the elementary antenna, and it has also a phase difference of 90° when compared to As a conclusion, it is obvious that the phase difference between
and
is
significant only in the near zone and tends to be in phase time when r increases. This is very important because there can be not reactive power, meaning a power flow to the source, than if there is a phase difference between electric and magnetic fields. So, the necessary condition for the existence of a negative component of the Poynting vector on the direction of is the existence of this phase difference, independent of the angle (Fig. 1). Obviously, in the Fraunhoffer zone,
is indistinct of
It is also important to plot angle, made up by the directions of
It may be observed that, in the near field area (Fig.4) indicating the presence of the component.
and
angle shows a behaviour
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In Fraunhoffer zone (Fig.5), varies rectangular between exclusively on the direction of so that the fronts are vertical lines.
and
but
Phasorial Approach For performing phasorial analysis, we start again from equations (1-3), but we still consider condition These equations become:
Retaining the normalization expressions described by (12) different values may be 0 0 assigned to r together with considering the variation of within 0 and 360 . The expressions of and represent the parametric equations of an ellipse. The result is a curve family corresponding to an oscillation cycle of the phasor zone.
versus r, in Fresnel
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Figure 6.a shows a double-parametric plotting: versus r considering and phase plane for a certain value of r placed in near zone. Based on this theory a complete description in parametric plotting of (Fig.7) may be made. This offers the possibility to describe the electric phasors and respectively their resultant in any point in space and in any moment.
For the diagram in Fig.7, the normalized variables are plotted for distance r within the range 0.5 m - 7 m. Therefore, the evolution in space and time of the electric field, in Fresnel zone, is described by a spiral, which wraps up the ellipse family from Fig.7. In the transition zone the ellipses are stretching significantly, having the long axis on the direction of and, for large values of r (Fraunhoffer zone), tend to collapse to a straight line along this direction. CROSS-FIELD CHARACTERIZATION As, in the near field of antenna, there are electric field components in both the direction of propagation and normal to this direction, cross-field is present. As a function of time the locus of describes cross-field ellipses as shown before. This characteristic is transmitted to the Poynting vector because it's direction and the one of are perpendicular but the corresponding phasors are not in time phase in the near field zone. If we consider a reference system as in Fig.8, the Poynting vector may be expressed as:
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accept a vectorial representation versus r and a
For the parametric representation of the Poynting vector's evolution versus r in equation (21) there were used expressions (12) of the electric and respectively magnetic fields. For the locus of the corresponding phasor equations (18-20) were used. CONCLUDING REMARKS The phenomenon presented in the paper appears always when vector rotates in a plane parallel to the propagation direction: near field of an antenna, radiation systems (travelling waves in perpendicular directions with the same frequency), reflection and waveguides. Cross-field appears in all these situations but it different from the one that characterizes wave polarization, a case when vector E rotates in a plane perpendicular to the propagation direction. Using the proposed algorithm there may be described and quantified the phenomena implied by the 4 applications mentioned above: stationary waves (Fig.3) space-time evolution of the fields (Fig.7) reactive power corresponding to the Poynting vector (Fig.8) It is important to stress that there are two characteristic phenomena in the near zone: angle <900 and the phase difference between and The first one determines the two components of the Poynting vector: and (a flux that slides on the surface of the sphere). is an angular energy, which flows in direction. The second phenomenon determines the reactive energy of radiation that flows out and backs twice per cycle without being radiated. Obviously, Poynting vector travels around the ellipse twice per cycle too. REFERENCES 1. J.D. Kraus, Electromagnetics, McGraw-Hill, Inc., (1953). 2. A. von Hippel, Dielectrics and waves, Second Edition, Artech House, Boston, (1995). 3. D.R. White, Electromagnetic Shielding, Don White Consultants, Inc., Gainesville, Virginia, (1980). 4. C.E. Baum, Intermediate Field of an Impulse-Radiating Antenna, Ultra-Wideband Short-Pulse Electromagnetics 4, Kluwer academic / Plenum Publishers, (1999). 5. C.C. Johnson, Field & Wave Electrodynamics, McGraw-Hill Comp., (1965).
PARALLEL CHARGING OF MARX GENERATORS FOR HIGH PULSE REPETITION RATES Carl E. Baum and Jane M. Lehr Air Force Research Laboratory Directed Energy Directorate Kirtland AFB, NM 87117-5776 INTRODUCTION In designing Marx generators for high pulse repetition rates, many things need to be considered. In addition to providing power with a consistent voltage and current profile, careful attention should be given to the charging network. Specifically, the Marx charging network should be designed with a short interpulse charging time, yet maintaining sufficient reactance so as not to load the Marx during discharge. Additionally, losses in the Marx, both in the charging and discharging circuits, must also be minimized. In this paper, we concentrate on the first point, by calculating how short a Marx charging time is allowed by various charging network approaches. In general, a Marx generator is charged with a series network of resistors, or inductors, charging the energy storage capacitors. In this paper, we have evaluated the conventional charging schemes, as well as considering parallel charging with inductors and resistors. A differential charging scheme, with is assumed, and the number of capacitors, N, is taken as even for convenience. The erected Marx capacitance, neglecting strays, is When erected, the open-circuit voltage presented to the switch to the load is nominally Note that due to stray capacitance and loading from the charging resistors during the Marx erection the actual output voltage is a little less than in magnitude. In designing a Marx for high PRF one can consider the time for a Marx cycle as composed of three parts where is the time to charge the Marx capacitors, is the high voltage time and the switch recovery time. The high voltage time is the time the Marx has some appreciable portion of (the erection time or gap running time) plus ring up time (as into a transfer capacitor) plus discharge time. The switch recovery time, is the time after which the Marx can begin recharging and is not addressed here. Other time variables of interest are the Marx erection time, or gap running time, and the discharge time, the time from switching to load until voltage on load decays to negligible value, compared to Generally, we would like to impose the inequality so that
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for a discharge in the 10 ns regime, the high-voltage time in the 100 ns regime and does not intrude on the discharge time. The left hand inequality is effectively imposed on so that the charging network does not load down the erected Marx discharge. We will calculate the limitation this places on more exactly for various designs. The switch recovery time is not discussed here. For purposes of the present discussion let us use as an example: and For simplicity, let such that and Furthermore, let us choose These will be used later to estimate values of resistance and inductance in the charging networks. The analysis of charging the Marx capacitance with a series charging network can be found (Baum and Lehr, 1999). It can be shown that resistive series charging takes on the order of 100s of µs. Analysis of series charging through inductors, shows that the charge time can be reduced down to the 10s of µs regime. However, a major disadvantage of implementing an inductive charging system is the resonance that can occur (Willis, 1989). The inductive and capacitive elements can resonate if the current through the inductors and charge on the capacitors are not zero at the end of the discharge into the load. PARALLEL RESISTIVE CHARGING Another circuit topology for the charging network is a parallel one as in Fig. 1. In this case, there is a direct connection from each capacitor through charging elements to the or power supply. Consider first the case of parallel pure resistors, by setting to zero. With zero initial charge and a charge voltage, the voltage on each capacitor is given by
This well-known waveform rises to 90% in about so the charging time, or a larger number. During the Marx erection process these resistors load the capacitors, removing some charge. With the Marx switches closed the symmetry of the network allows both negative and positive power supply voltages to be set to zero. We have a resistance of to ground just after each odd numbered capacitor. For the present let us assume that so that we can neglect the presence of the resistors. Let us then consider how much the loading reduces the erected Marx voltage The initial currents in the resistors after full voltage is reached are given by the voltages for odd n as indicated in Fig. 4.2, where now and Here is assumed short enough that the voltages have not been significantly reduced by These currents remove charge from the capacitors and hence decrease the voltages. Considering just
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the leading terms (first time derivatives) we have
for odd n, with
where the summation of odd integers is found in (Dwight, 1961). The second upper index in the summation indicates that m increments in steps of 2. Summing these derivatives we obtain the derivative of the Marx voltage (open circuit) as
which can be shown(Baum and Lehr, 1999) to simplify to
Normalizing this by
we obtain
to define a time constant
Constraining
which can be used
for the loading of the charging resistors,
we have for our example,
and
considering the charging for our example, which is a very modest improvement over series resistive charging. PARALLEL INDUCTIVE CHARGING One can, of course, consider parallel inductive loading where the resistances are replaced by inductances In this case the capacitors are resonantly charged and reach their
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in one half cycle given by
compared with
which we require to be large
This gives
when erected the open-circuit
voltage at the load is
The analysis of the previous section can be modified for this case. Replace where
by
is the complex frequency. The leading term becomes
Inserting this, we obtain
The second time
derivative is significant and Our loading then decreases the voltage proportional to time squared. We then define the time constant as
Constraining
for our example
This more severe constraint replaces the previous one,
which, in turn, implies a
charging time, This parallel inductive charging time can then be in the tens of µs regime, similar to the series-inductive-charging result. As with series inductive charging, parallel inductive charging can have resonance that may be troublesome.
SERIES RESISTORS AND INDUCTORS FOR PARALLEL CHARGING Now let the parallel charging elements comprise the series combination or resistors and inductors as in Fig.4.1. During the charging cycle the voltage on each capacitor is where ~ and
is the complex frequency. If
we have with
Laplace transform over time and
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For the special case of critical damping we have second order pole as [7]
which gives a
with the corresponding time domain
waveform, For comparison, both time domain waveforms are plotted in Fig. 3. As we can see the critically dumped waveform starts more slowly but crosses over the waveform at to give a similar charging time. In more detail we can see that the early-time behavior is better for the resistive/inductive case due to the initial quadratic behavior. This slows the initial charging to keep voltage off the Marx switches longer than the ramp behavior of the resistive case. The late-time behavior is also better for the resistive/inductive charging because, as we can see, the voltage more rapidly approaches the full-charge value. To give a few numbers, for the resistive charging is 0.92 versus 0.96 for resistive/inductive charging, a change from 8% lack to 2% lack from final value. For the resistive charging is 0.963 versus 0.99 for resistive/inductive charging, a change from about 4% lack to 1% lack from final value. During the Marx erection, the current passing through is initially limited by which takes most of the voltage drop. Using the previous results for determining the required inductance to make
large compared to
we have
such that for our
example, Thus,
and This is a significant improvement over purely resistive parallel charging, allowing charging in tens of µs. Moreover, during erection of the Marx, most of the voltage is across instead of offering some protection for the resistors. By increasing (and proportionally) then becomes a smaller portion of and less voltage appears across during this time. Also, the introduction of inductors into the charging circuit give the switches slightly longer to recover, since the current through each inductor is approximately zero at the beginning of the charge cycle.
CONCLUDING REMARKS Of the several techniques for Marx charging, inductive charging, for both series and parallel is faster than resistive charging, and has the additional benefit of a voltage-doubling property. The parallel resistive charging is only slightly faster than the series resistive charging, but parallel charging has the advantage that the Marx capacitors are charged in
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approximately the same time. Parallel charging with series is approximately as fast as inductive charging with the critical damping avoiding resonance. The inclusion of in the parallel charging also has the advantage that after the Marx discharges, the inductors limit the initial charging current, thereby giving more time for the Marx switches to recover. The inductors also protect the resistors by taking most of the voltage during the Marx high-voltage time. The inclusion of inductors in the charging network has a potential problem in that the large magnetic fields produced by a coil can couple to other parts of the Marx including other coils. This coupling can be avoided by designing a coil that suppresses the external magnetic field while retaining a large internal magnetic field (Chen, et al, 1985). A simpler bisolenoidal form that suppresses the magnetic-dipole term has been reduced to practice (Giri, et al 1994). As the foregoing calculations indicate, the charging time needs to be much greater than some multiple of the Marx high-voltage time This may be overly conservative in that the Marx loading during the high-voltage time does not begin (approximately) until the Marx switches leading up to the particular parallel (and ground) loading elements have all closed. For many of the elements, especially the last few, the loading occurs for a time somewhat less than but because the voltage on these last elements is near the full Marx voltage here, this loading is the most significant here. Thus, some average, or effective value of may be more useful in the loading formulae. Acknowledgement: We would like to thank G. J. Rohwein, W. J. Sarjeant, T. H. Martin, and expecially I.D. Smith for discussions and providing some of the references. REFERENCES C.E. Baum and J.M. Lehr, “Charging of Marx Banks”, Circuit and Electromagnetic System Design Note 43, September, 1999. Y. G. Chen, R. Crumley, C. E. Baum, and D. V. Giri, “Field-Containing Inductors”, Sensor and Simulation Note 287, July 1985; IEEE Trans. EMC, 1988, pp. 345–350. Y. G. Chen, R. Crumley, C. E. Baum, and D. V. Giri, “Field-Containing Inductors”, Sensor and Simulation Note 287, July 1985; IEEE Trans. EMC, 1988, pp. 345–350. H.B. Dwight, Table of Integrals and Other Mathematical Data, Fourth Ed., Macmillan, 1961. D. V. Giri, C. E. Baum, and D. Morton, Field-Containing Solenoidal Inductors, Sensor and Simulation Note 368, July 1994 W.L. Willis, “Pulse Voltage Circuits”, Ch. 3, pp87-116, in High Power Electronics, W.J. Sarjeant and R.E. Dollinger, (Eds), TAB Books, 1989.
LIVE FIRE TEST AND EVALUATION AND THE RADIO FREQUENCY VULNERABILITY TESTING MISSION
James F. O’Bryon and Richard J. Carter Deputy Director Operational Test and Evaluation Live Fire Test and Evaluation Office of the Secretary of Defense The Pentagon, Room 1C730 Washington, DC, 20301-1700
[email protected]
INTRODUCTION The Office of the Secretary of Defense (OSD), Director, Operational Test and Evaluation (DOT&E) was created by the United States (U.S.) Congress in 1983. The DOT&E reports directly to the Secretary of Defense and is the principal adviser to the Secretary on operational and live fire test and evaluation (LFT&E) in the Department of Defense (DoD). He is also the senior operational test and evaluation official within the management of DoD. The overall mission of the office is to ensure that weapon systems are realistically and adequately tested and to provide complete and accurate evaluations of operational effectiveness, suitability, and survivability/lethality to the Secretary of Defense, other decision makers in DoD, and Congress. The DOT&E Office is subdivided into four distinct parts: Live Fire Test and Evaluation; Resources and Ranges; Operational Test of Conventional Systems; and Operational Test of Strategic and Command, Control, Communication, and Integration Systems. The Deputy Director, Operational Testing for Live Fire Test and Evaluation is responsible to the DOT&E for LFT&E policy and live fire testing of systems qualifying for oversight. The LFT&E program is oriented towards providing a timely and realistic assessment of the survivability (of systems designed to provide user protection) and lethality (of missiles, rockets, and munitions) of a military system as it progresses through its development cycle and prior to full-rate production. The LFT&E requirement was placed in law by the U.S. Congress in 1986 and now covers air, land, and sea platforms, as well as a variety of weapons from small arms to national missile defense systems. The program is particularly aimed at providing information to decision makers on potential user (i.e., individual soldiers and armor, aircraft, ship, and tactical vehicle crews) casualties, survivability, vulnerability, and lethality, taking into equal Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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consideration susceptibility to attack and combat performance of the system. It is also directed towards ensuring that knowledge of user casualties and system vulnerabilities or lethalities are based upon testing of the system against expected threats under realistic combat conditions. Survivability is defined as the capability of a system and its crew to avoid or withstand a man-made hostile environment without suffering an abortive impairment of its ability to accomplish its designated mission. It is partitioned into two subsets: susceptibility (i.e., the degree to which a device, equipment, or weapon system is open to effective attack due to one or more inherent weaknesses) and vulnerability (i.e., the characteristics of a system that cause it to suffer degradation [loss or reduction of capability to perform the designated mission] as a result of having been subjected to a certain [defined] level of effects in an unnatural [man-made] hostile environment). Lethality addresses the ability of missiles and munitions to defeat their targets.
LIVE FIRE TEST AND EVALUATION There are a number of reasons to conduct live fire tests. They include: to reduce program risk, meet statutory requirements, add discipline to the development process, reduce cost, save equipment, push the frontiers of science and technology, and, foremost, save lives. LFT&E begins early in the development stage with component testing, followed by subsystem tests, system-level tests, and culminating in “full-up system-level tests” using complete systems configured for combat. The term full-up system level test means testing that fully satisfies the statutory requirement for either realistic survivability or lethality testing. As an example of the different types of tests, a helicopter during early component testing might have only its main rotor blades available and tested; while during subsystem testing both its blades and rotor would be investigated. During systemlevel testing the complete operating helicopter with fuel and munitions on board would be tested. Although the LFT&E process is similar for all systems, it is never identical between systems due to the unique nature of each system and its intended use. The Live Fire Test and Evaluation Office is currently responsible for the oversight of more than 80 systems spread across the four services (Army, Air Force, Navy, and Marine Corps). The systems consist of land, air, and sea-based platforms, missiles, rockets, and munitions and have a procurement value in excess of $600 billion. Joint Live Fire The Live Fire Test and Evaluation Office has an oversight role for two main testing programs: LFT&E and joint live fire (JLF) testing. LFT&E deals with systems in development and undergoing product improvement. In contrast, JLF addresses fielded systems. The purposes of JLF testing are to: gather empirical data on the vulnerability of fielded U.S. weapon systems to threat systems and/or the lethality of fielded U.S. weapon systems against threat systems; develop insights into potential design changes; enhance the data base available for battle damage assessment and repair; and validate current vulnerability and lethality methodologies. LFT&E was legislated in fiscal year 1986; JLF was chartered in fiscal year 1984. LFT&E is funded by the four services, while OSD is the funding source for JLF testing.
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Roles of Modeling and Simulation in LFT&E Modeling and simulation are used throughout the entire LFT&E process to support several tasks. These tools are integral and essential to: perform pre-test predictions, guide test planning and instrumentation, achieve economies in testing, assist in shot sequencing, provide a benchmark against which to compare test outcomes, and correct model deficiencies. Modeling and simulation methods are predictive in nature. Models are exercised and predictions provided to the Live Fire Test and Evaluation Office prior to every test. Live Fire Test and Training Program In fiscal year 1997 Congress directed the Live Fire Test and Evaluation Office to develop a program that “will explore, select, and implement alternative uses of simulation and synthetic environment technologies that are being used for education and training for implementation by the live fire community”. The objective of the live fire test and training (LFT&T) program is to exploit the exchange of technology development initiatives and uses between the LFT&T communities that will improve our ability to better serve our ultimate customer – the warfighter. New LFT&T projects must meet the following criteria: applicability to both military training and LFT&E; training technology that has direct application to LFT&E; LFT&E technology that has direct application to military training; cross-service/joint applicability; timely transition and implementation; completion in less than two years; cost savings to the LFT&E/training communities; and anticipated transition of the project to operational use. THE RADIO FREQUENCY VULNERABILITY TESTING MISSION LFT&E requires that testing be done not just against current threats, but also against “expected threats” that would exist when the system under development is fielded and beyond. The term expected threat, in this context, consists of the current and projected threat. LFT&E also addresses the reactive threat to the degree to which a potential adversary might readily implement an expected response. Live fire testing initially was primarily focused on conventional, ballistic threats (e.g., small arms, fragmenting projectiles, shaped charges, kinetic energy rods, selfforging fragments, and high explosives). As the projected threats have changed, so have the test plans to incorporate them. These new, non-ballistic threats include: radio frequency (RF) devices, low, medium, and high-energy lasers, incendiaries, blast/fuel air explosives, charged particle beams, and high-power microwaves (HPM). Policy on Directed Energy Weapons The DOT&E policy on directed energy weapons (DEW) is quite simple. DEWs are treated no differently than other conventional systems. If an U.S. platform qualifies for survivability LFT&E, then the LFT&E program must address survivability to DEW threat weapons, provided there is reasonable expectation that the U.S. system will encounter such threats. Conversely, a DEW being developed or modified by the U.S. must be subjected to lethality LFT&E if the DEW system meets the appropriate LFT&E criteria for cost, or if modification significantly affects lethality. Clearly the ability to test and evaluate DEW threats or threat surrogates requires support from the intelligence community. The intelligence community must provide validated threat information that is both timely and adequate to guide the selection and
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acquisition of threat weapons or their surrogates. For DEW threats, as for other threats, LFT&E programs exercise judgment as to the level of test and evaluation resources to be devoted to each threat class. This judgment takes into consideration the expected likelihood of encounter, the potential consequences to the system given an encounter, and the expected ability to modify the design and/or employment of the system based on results of the test and evaluation. LFT&E addresses the effects of DEWs with regard to system vulnerability or lethality. Typically this excludes what would be termed electronic warfare. LFT&E addresses any effects to systems that persist after the directed energy is removed (i.e., the damage or degradation remains after the target ceases to be engaged by the device) and which will either cause the mission to be degraded or aborted, or cause loss of combat capability if the mission is continued. LFT&E includes effects that might even result in loss of the target system. For example, if continued exposure to directed energy were to affect electronic flight controls and cause an airplane to crash, this would be within the scope of LFT&E. RF Pre-Test Prediction Models A study is underway to identify, describe, and discuss prospective models that could be used in preparing pre-test predictions for RF tests. Preliminary survey results are presented in Table 1. The table exhibits 20 models that have been identified so far by name/proponent, type, and comments. The Live Fire Test and Evaluation Office recently awarded a contract to perform pre-test analysis. The effort consists of two sequential tasks. The objective of the first task is to predict the electric and magnetic field strengths throughout the volume that contains the targeted equipment. Predictions will be made of the transmission of the incident pulse through the surface of the enclosure that contains the equipment, and subsequently computations will be made of the internal reflections of the fields within the object. The second task will be aimed at assessing the malfunctions of electronic information systems caused by the time-dependent electric and magnetic fields calculated in Task 1. A key element of this task will be to convert the electromagnetic signals at selected locations into bit error rates. For each task the contractor will employ the finite difference time domain (FDTD) approach for modeling the interaction of RF pulses with electronic equipment. The contractor has obtained a beta version of an FDTD software application. The code, in addition to being a three-dimensional full wave electromagnetic simulator, has the unique feature of being coupled to the electronic design and simulation tool known as SPICE. This feature will facilitate the modeling of relevant electronic hardware and the incorporation of these models into the FDTD propagation code. Prior RF Vulnerability Testing The LFT&E program has been testing and evaluating the on-target effects of potential RF devices over the past three years. This endeavor has been a small, but pioneering, effort. An open-air RF test was conducted in late 1997 on the U.S. West Coast. It was funded as part of the JLF program discussed above. Sources included both conventional HPM and high power transient electromagnetic devices (HPTED). The testing was performed on a fielded military weapon system, various computer systems, computer networks, and security systems. The test, sponsored by the Live Fire Test and Evaluation Office, OSD, was a cooperative effort between DoD (the Naval Air Warfare Center, Weapons Division, China Lake; the Air Force Research Laboratory; and the Army Research Laboratory) and the Department of Energy (Lawrence Livermore
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National Laboratory). The purpose of the test was to develop and demonstrate the methodology required to perform RF device live fire test survivability testing. A RF source and target used in the test are pictured in Figure 1. A follow-on test was performed during 1998 with an HPTED provided to DoD by a private company. The purpose of the effort was to test and evaluate a number of RF ultra-wideband devices constructed using a “terrorist mind-set”. The contractor was asked to design and build three inexpensive devices (using a natural progression of device maturity) characteristic of what a rogue nation or terrorist could fabricate using only “open source” information and commonly available hardware components. The three devices were characterized at a range on the West Coast of the U.S. Figure 2 shows the three devices. Upon conclusion of the two tests, it was determined that a permanent ultra-wideband RF source should be developed to facilitate future RF vulnerability testing. Specifications included the capability of having a reconfigurable source in regards to waveform characteristics, rise time, pulse rate, and power levels. The source was developed and subsequently delivered in October 1999. Figure 3 exhibits the RF source with a deployed antenna.
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Current RF Vulnerability Testing The Live Fire Test and Evaluation Office plans on conducting two RF vulnerability tests during calendar year 2000. One will take place on the East Coast; the other on the West Coast of the U.S. Both will be outdoor, live fire, open-air tests. The tests will be oriented towards assessing the vulnerability of electronic systems representative of the U.S. commercial and military infrastructure to high power ultra-wideband illumination under “operationally relevant” conditions. The devices to be tested include both military weapon systems and commercial offthe-shelf (COTS) technology. The COTS equipment is subdivided into two distinct sets: industrial control and monitoring technology and medical equipment. The industrial control and monitoring technology will include: portable generators, uninterrupted power supplies, a physical security system, a fire monitoring and alarm system, a telephone switching device, a baggage screening operator-assisted x-ray machine, a stand-alone emergency response system, and a ground-based navigation system. The medical equipment will include: ventilators/respirators, blood pressure monitors, heart monitors, fetal monitors, telemetry systems, multi-parameter monitors, intravenous pumps, heart pacemakers, hospital beds, and call buttons. Any test system effects (upsets, anomalies, and/or failures) observed and recorded will be assigned a specific effect level. The following ranking will be used: 0 – no observable effects; 1 – effects are present only during illumination; 2 – residual effects requiring user intervention (system function reset required); 3 – residual effects requiring user invention (system power recycle required); 4 – effects requiring system maintenance action; and 5 – physical damage. The instrumentation to be employed during the testing will allow for multi-channel, simultaneous data acquisition and near real time data retrieval, storage, and reduction. It will characterize incident electric field and pattern and (for housed assets) characterize actual field incident on the assets. The instrumentation will contain multiple video channels to record visual evidence of upset and audio monitoring of the asset facility and/or device under test (DUT).
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Figure 4 pictures the RF vulnerability test site layout. Table 2 presents the test object exposure protocol that will be employed during the two tests in calendar year 2000.
Future RF Vulnerability Testing The Live Fire Test and Evaluation Office will continue to test the vulnerability of U.S. commercial and military infrastructure technology to RF devices for some period of time. The office will also continue to develop methodologies and tools to do these types of tests. In addition, procedures and techniques to mitigate the identified vulnerabilities and harden the systems will be explored.
COMPACT HPM AND UWB SOURCES USING EXPLOSIVES - THE POTENTIAL OF FUTURE NON-LETHAL WARHEAD SYSTEMS
T. Ehlen 1, J. Bohl1, R. Kuhnke2, F.Sonnemann 1 1 2
Diehl Munitionssysteme & Co.KG, Fischbachstr. 16, (D) 90552 Roethenbach/Pegn. BWB-Koblenz, Germany.
INTRODUCTION Modern electronic systems are susceptible to intense electromagnetic fields. After penetration into the internal system the electromagnetic waves are converted to current and voltage waves on cable systems connected to the electronic components. Dependent on the level either distortion due to demodulation effects occur or the component is destroyed. In either case the system loses the capability it was designed for. So delivering an intense electromagnetic field to an target provides a non lethal defense technique to interrupt missions. In the far field each electromagnetic field decreases inversely proportional to the distance from its source. To overcome this 1/r problem for a large distance between source and sink it is possible to transport the source to the vicinity of the sink. Due to the fact that now less field strength at the source is necessary, either the source power may be reduced with a directional radiation pattern or the antenna gain may be reduced at the same source power to obtain a quasi omnidirectional radiation pattern. For a small non stationary source system the primary energy has to be stored in a small volume. The conversion of the huge stored chemical energy in explosives to electrical energy even with low conversion efficiency is applicable. Either UWB and HPM Subsystems can be realized with the use of explosives. Mainly this concerns the primary energy section. The switching systems in the pulse forming section may be realized by explosives if the switching velocity and the absolute switching time slot need not be too precise. Even the directivity of the antenna might be manipulated by ionized regions in air produced by the detonation front. Explosive devices operate in the microsecond region. So if not using a long time energy storage unit (which is bulky and heavy) the radiation time is also in the microsecond and nanosecond regime. Therefore explosive devices operate primarily as one-shot or short burst devices. This implies that the electronics of the target is either destroyed or upset with a long recovery time by this single field pulse.
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CLASSIFICATION OF EXPLOSIVE SOURCES The energy conversion process usually is from chemical energy to mechanical energy to electrical energy. The last step can be classified as a change of geometry or change of material properties. In the latter case the detonation destroys certain material properties as magnetic or electric prepolarisations with higher intrinsic energy levels in the regular ordered state before the destruction.
In the first case preenergized inductors, capacitors, or cables are reconfigured in such a manner, that mechanical work has to be done against some electrical forces i.e. the reduction in the capacitance of a loaded capacitor by enlarging the distance between the two capacitor plates or – the reduction of the current floated inductance by successively short-circuiting the
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windings. Due to parasitic resistive losses that determine the electrical time constant, as a rough estimate, the electrical time constant has to be bigger than the mechanical operation time for an energy amplification to take place.
Because the explosive inductive energy amplification (flux compression) doesn’t suffer from electrical breakdown problems it is widely used for generating huge currents and magnetic fields. For very fast signals the windings don’t act as an inductor but as a cable. Applying simple cable theory to a current floated cable with a time-varying inductive load one can show that a fast change of inductance generates a fast nanosecond voltage pulse as a backward travelling wave with high frequency components in the gigaherz region when the forward travelling wave is constant. Approximately the time derivative of the inductance acts as a negative resistor enabling the reflection coefficient be less than –1 and generating amplification.
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Irregular pitching of the winding in the production process allows high values of dL/dt. The unrolled windings that produce these high dL/dt values are shown in figure 6. PULSE RADIATION In flux compression configurations fast pulses are partly radiated by the winding itself. The FDTD-analysis of the conical expanded liner together with the windings around the liner show a similar behavior as in non homogeneous cables. When a fast gaussian voltage pulse is applied between liner and windings this pulse travels along the windings. A mode conversion takes place when the distance between the winding and liner is wider than the winding-winding distance. In a long distance from the starting point the field at the winding is similar to the field of two coupled cables.
The travelling pulse on the winding is reflected back at the end points on the winding and the connection point to the liner. According to the simulation the frequency components corresponding to the total length of the winding and to the winding perimeter (circular resonance) are radiated with a quasi cylindrical symmetric radiation pattern.
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SOURCE - SINK - INTERACTION To compare non stationary UWB and HPM systems both source and sink parameters have to be included. For a specific test system the destruction parameter for HPM is assumed to be energy dependent (compare the Wunsch-Bell-Relation) with a destruction-energy density of For UWB a destruction power density of l00kV/m is assumed.
The UWB parameter pulse-length can’t be modified without changing the coupling efficiency to the target. As shown in figure 9, the highest induced current values on a very thin wire model are achieved when the UWB-pulse-length in the source T is about where corresponds to the frequency of the wire length.
The HPM coupling efficiency however is only determined by the carrier frequency (for a given AoI and polarization) which is independent of the HPM-pulse-length. Assuming an HPM source energy of antenna gain of G=20dB and a critical sink energy density of an effective source-sink-distance (ssd) of about ssd=8m can be achieved. Assuming an UWB source power of an pulse specific antenna gain G=8dB and a critical sink power density of an effective ssd of about ssd=10m is achieved. For these specific sources, antennas and sink parameters the ssd for HPM and UWB sources is quite similar. To enlarge the ssd the source power and antenna gain have to be increased in both sources - for HPM sources additionally the pulse length can be enlarged to increase the energy.
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CONCLUSION Using explosive devices to convert chemical / mechanical energy to electrical energy enables non stationary HPM and UWB sources to generate electromagnetic fields for destructive electronic interactions. Because these sources are intended to be near the target, the effective source-sink distance need not be as high as in stationary systems. Whether using the intrinsic antenna of these devices or external ones, the antenna indicates to be a key element to enlarge the ssd. The Comparison of destruction parameters of a test system in combination with typical source- and antenna parameters show similar ssd for HPM and UWB systems. However due to the wide range of destruction parameters of different target systems this statement cannot be generalized. The drawbacks and advantages of either source types according to the coupling efficiency for highly resonant targets or to the variation in the angle of incidence are the same as for stationary systems. REFERENCES [1] J. Bohl, F. Sonnemann, T. Ehlen, ,,A Methology to Determine and to Describe the Destroying HPM-Effects of Electronic Components Using a Generic Electronic Unit (GENEC)“, Euroem 1998, Israel, 1998. [2] C. E. Baum, L. C. Carin, A. P. Stone, “Ultra-Wideband, Short-Pulse Electromagnetics 3”, Plenum Press, N.Y., 1997. [3] L. M. Marin, T. K. Liu, "A simple way of solving transient thin-wire problems", Radio Science, Vol 11 , February 1976. [4] G. A. Mesyats, S. N. Rukin, V. G. Shpak, M. I. Yalandin, “Generation of High Power Subnanosecond Pulses”, Institute of Electrophysics Russian Academy of Sciences, Yekaterinburg, Russia [5] E. V. Chernych, V. E. Fortov, K. V. Gorbachev, E. V. Nesterov, V. A. Stroganov, V. P. Shumilin, “Accelerator for High Power Microwave Generation”, IVTAN, RAS, Beams 96, 1996
SUB-NANOSECOND GAS BREAKDOWN PHENOMENA IN THE VOLTAGE REGIME BELOW 15 KV
H. Krompholz, L. Hatfield, B. Short, M. Kristiansen Pulsed Power Laboratory Departments of Electrical Engineering and Physics Texas Tech University Lubbock,TX 79409-3102
ABSTRACT Fast gaseous breakdown is of interest for both UWB/short pulse electromagnetics, and for plasma limiters to protect devices from high power microwave radiation. A quantitative investigation of fast breakdown phenomena, especially for relatively low voltages and for special geometries, does not exist to the authors’ knowledge. Breakdown in gases is studied in a point–plane geometry with fast high voltage pulsers, covering the parameter range of voltage amplitude 1.7 to 7.5 kV, risetime 400 ps to 1 ns, and pulse duration 1 to 20 ns. The setup consists of a pulser, transmission line, axial needle-plane gap with outer conductor, and load line. The needle consists of tungsten and has a radius of curvature below 0.5 µm. The constant system impedance of (except in the vicinity of the gap) and special transmission-line-type current sensors enables sub-nanosecond current and voltage measurements with a dynamic range covering several orders of magnitude. Digitizing oscilloscopes with sampling rates of 5 ps and 50 ps are used, with analog risetimes of 80 and 240 ps. In addition, the luminosity is measured with a sensitivity of about and a risetime of 800 ps. For pulse amplitudes of 1.7 kV (which are doubled at the open gap before breakdown), long pulse duration, and a gap distance of 1 mm, delay times between start of the pulse and start of a measurable current flow (amplitude > several milliamperes) have a minimum of about 8 ns. The pressure dependence of this delay time was measured, in 10 to 600 torr argon, and a minimum is observed at 50 torr Voltages of 7.5 kV produce breakdowns with a delay of about 1 ns. Statistical delays could not be found for either pulse amplitude, with the tip positively pulsed. With negative pulses applied to
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the tip, at an amplitude of 7.5 kV, breakdown is always observed during the rising part of the pulse, with breakdown delay times below 800 ps. This delay time does not depend on the pressure, and points to a vacuum type discharge with field-emitted electrons as the dominant discharge mechanism for this case. INTRODUCTION Gas breakdown in the sub-nanosecond regime is of vital interest for ultra-wideband radiation sources, for plasma limiters1 (i.e. passive methods for the protection of electronic components using gas breakdown caused by the first half-cycle of incoming high power radiation in the GHz regime), and for fast gas switches in general pulsed power applications. Published data on dielectric breakdown for the sub-nanosecond regime are relatively scarce 2,3,4,5. These empirical data show, that for atmospheric conditions breakdown fields (for quasi homogeneous field conditions) on the order of 100’s of kV/cm are needed to achieve breakdown delay times in the sub-nanosecond regime. For compressed gases at several 100 atmospheres, or for liquids, electric field strengths which are even several orders of magnitude higher than the ones for gases at atmospheric conditions, are needed. Theoretical modeling of fast breakdown is mainly based on the streamer mechanism6, expanded by several other physical mechanisms, such as the existence of fast electrons 4,7 taking part in the build-up of space charges, and the emission and reabsorption of photons leading to fast ionization processes8. For practical applications, high pressure gases as the switching medium, and homogeneous electric fields requiring relatively high voltage amplitudes for subnanosecond breakdown, are technologically demanding. In this paper, we investigate breakdown in tipplane geometries, where the field enhancement at either the cathode or the anode is sufficient to provide fast breakdown, and the applied voltages are in the range of 3 to 15 kV, for gas pressures between zero and one atmosphere. For plasma limiters, this voltage regime would correspond to a microwave power, e.g. for S-band frequencies in a standard waveguide, of several MW, and a much smaller power for typical stripline geometries. EXPERIMENTAL SETUP Several sub-nanosecond risetime high voltage pulsers (1-1.7 kV with risetime of 0.5 ns using standard mercury relay switched pulsers, a Russian PNG-10 solid state pulser with an amplitude of 7.5 kV, and a high pressure switched pulser (Bournlea Instruments Ltd., Type 3148X, with an amplitude of 7 kV) are connected via a high voltage transmission line (two-way transit time 240 ns) and a high voltage vacuum feedthrough to a test gap in a vacuum, see Fig. 1. The other side of the test gap is terminated by another transmission line (load line) with a two-way transit time of 240 ns. A mesh outer conductor surrounds the test gap and the overall system has a constant impedance of which enables fast electrical diagnostics. Some measurements were done with a radial gap arrangement, in which the discharge connects the inner with the outer conductor, and in which the pulser voltage is applied to the gap. The majority of the measurements were performed with an axial gap, where the
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discharge connects the input transmission line via the inner conductors with the output transmission line. Here, the pulser voltage doubles at the open gap before breakdown.
The test gap consists of a tungsten needle (GGB Industries, Inc., Naples, Fl) with a radius of curvature at the tip of down to 0.2 µm (Fig. 2), which has been operated at both polarities, and a brass electrode with a radius of curvature of 5 mm at the other side. The maximum electric field is estimated9 using to about 500 MV/cm. At a pulser voltage of 7.5 kV, the needle tip is destroyed after one discharge (Fig. 2). Main
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measurements were done in argon at pressures between several mTorr 600 Torr. The gap distance was 1 mm in all cases. Diagnostics include several transmission-line type current sensors 10 with a risetime below 400 ps and a sensitivity on the order of 1 V/A, standard capacitive/resistive voltage dividers with a comparable risetime, and a photomultiplier tube with a risetime of 800 ps which measures the spark luminosity. It is planned to expand the current measurement setup to cover a dynamic range of six orders of magnitude using different current sensors in combination with fast amplifiers and attenuators11. All signals were recorded with Tektronix SCD 5000 transient digitizers, with a sampling interval of 5 ps, and with an analog bandwidth of 4.5 GHz which corresponds to a risetime of 80 ps. All diagnostics cables used had an analog bandwidth of at least 8 GHz. RESULTS The majority of the measurements is based on the output of a current sensor located on the pulser-side transmission line, at a two-way transit time of 6 ns away from the test gap. A schematic picture of the expected waveforms for an easier interpretation of the actually measured data is depicted in Fig. 3. For an input pulse shorter than the two-way transit time, without breakdown, the current wave would be reflected with inversion at the open gap, and is recorded after the two-way transit time at the sensor. With breakdown, the gap represents a shorted connection to the terminating transmission line, and the reflection coefficient for the current is zero, i.e. the reflected current signal goes to zero. For an input pulse longer than the two-way transit time, a negative reflection would add to the input pulse after the two-way transit time to a total of zero without breakdown, and breakdown would be indicated by zero reflection, i.e. a rising pulse again.
Some examples for measured pulses are shown in Fig. 4, for an input pulse amplitudes of 1.7 kV and pulse durations longer than the two-way transit time, i.e. with a mercury relay pulsers. Delay times are in the range of 12 to 40 ns, dependent on the pressure (Fig. 4). Fig. 5 (left) shows an example using the PNG-10 pulser, with the breakdown delay measured according to Fig. 3. Oscillations on the top of the pulser output waveform introduce some uncertainty in the determination of the breakdown delay.
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Fig. 5 show some examples of the dependence of the breakdown delay time on pressure, for 15 kV applied voltage (voltage doubled at the open gap). The pulser voltage was 7.5 kV, with positive voltage applied to the tip. Fig. 6 shows an example for the reflected waveform using the 3148X pulser, with the tip at negative polariy. For this case, the breakdown delay time was 800±50 ps, independent of pressure in the range 1 to 600 torr.
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DISCUSSION AND CONCLUSIONS Short breakdown times in the sub-nanosecond regime can be achieved with moderate applied voltages with amplitudes on the order of several kV and doubling at the open end of a waveguide in a tip-plane geometry. With radii of curvature below 0.5 µm, breakdown delays are below one nanosecond, with only slight variations as a function of pressure in argon for a positive tip, and virtually no pressure dependence for a negative tip.
The breakdown mechanism is far from being clear, and an attempt to model this type of discharge would have to treat an extremely inhomogeneous field configuration (where the mean free path of electrons is much larger than the scale length for spatial variation of the field) and a time-varying electric field (where the discharge development time is on the order of the time during which the field is applied). The pressure dependence for positive corona (or better the fact that the breakdown delay time is almost independent of the pressure) points to a vacuum discharge mechanism as the main contribution to breakdown. For negative corona, where virtually no dependence of the delay time on the pressure has been observed, a pure vacuum discharge started by field emission of electrons seems to be the dominating mechanism. Estimates of the breakdown delay times12 are based on the heating time of the emitter tip as a function of the space charge limited electron current density. Previous observations12 cover the electric field range of up to 150 MV/cm, where the delay time is about 1 ns. The estimated electric field amplitude for our case is larger by a factor of 5, for which a delay time far into the sub-nanosecond regime would be expected. For practical applications as plasma limiters, the tip-plane geometry with radii of curvature below 1 µm seems to be a promising scheme. It is expected, that a low impedance discharge will be ignited by the first half cycle of incoming radiation in the GHz frequency regime, at power levels of MW for a standard waveguide, and at much smaller power levels for a stripline geometry. Further advantages of this scheme are the low vol-
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ume needed at gap distances at or below 1 mm, and there is no need for pressurizing the discharge volume. A disadvantage can be seen in the fact that this concept works only for a single discharge, since the needle tip usually is destroyed, providing a much higher breakdown voltage or much longer delay times for successive discharges. An array of needles could possibly provide successive discharges with short delay times. ACKNOWLEDGEMENT This work is supported by the US Army Space and Missile Defense Command. REFERENCES 1. A. Kikel, L. Altgilbers, I. Merritt, M. Brown, L. Ray, T.X. Zhang, Plasma limiters, American Institute of Aeronautics and Astronautics, paper AIAA-98-2564 (1998)]
2. T.H. Martin, An empirical formula for gas switch breakdown delay, Rs. Sci. Pulsed Power, Sandia Nat. Labs., pp. 73-79, (1991)
3. P. Felsenthal, J.M. Proud, Nanosecond pulse breakdown in gases, Phys. Rev. 139, pp. A1796-1804 (1965)
4. E. Kuhnhardt, Nanosecond pulsed breakdown of gases, in NATO ASI Series, Series B: Physics, vol. 89a, 5. 6. 7. 8. 9.
10. 11. 12.
Electrical Breakdown and Discharges in Gases, Plenum Press, New York and London, 1983, pp. 241263 J. Mankowski, J. Dickens, M. Kristiansen, High voltage subnanosecond breakdown, IEEE Transactions on Plasma Science 26, 874 (1998) H. Raether, Electron Avalanches and Breakdown in Gases, Butterworth, London 1964 E.E. Kuhnhardt, W.W Byszewski, Development of overvoltage breakdown at high gas pressure, Phys. Rev. A 21, 2069 (1980) E.D. Lozanskii, Sov. Phys. USP 18, 893 (1976) G.A. Mesyats, Explosive Electron Emission, URO Press, Ekaterinburg, 1998, pg. 24 H. Krompholz, K. Schoenbach, G. Schaefer, Transmission Line Current Sensor, IEEE Instrumentation and Measurement Technology Conference, Tampa, Florida, USA 1985, p. 224 A. Neuber, M. Butcher, L.L. Hatfleld, H. Krompholz, Electric current in dc surface flashover in vacuum, J. Appl. Phys. 85 (1999) 3084 G.A. Mesyats, Pulsed Electrical Discharge in Vacuum, Springer Verlag, Berlin, Heidelberg, New York, 1989
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HIGH-POWER, HIGH-PRF SUBNANOSECOND MODULATOR BASED ON A NANOSECOND ALL-SOLID-STATE DRIVER AND A GAS GAP PULSE SHARPER
M.I.Yalandin, S.K.Lyubutin, M.R.Oulmascoulov, S.N.Rukin, V.G.Shpak, S.A.Shunailov, and B.G.Slovikovsky Institute of Electrophysics, Russian Academy of Sciences, 34 Komsomolskaya Str., Ekaterinburg 620049, Russia
INTRODUCTION Further improvement and practical implementation of generators of high-power ultrawideband electromagnetic pulses1-3 and relativistic generators of coherent subnanosecond microwave pulses4 call for modulators providing high pulsed and average power. While at a fixed output resistance of a device and a fixed duration of the pulses it produces the peak power is increased by increasing the output voltage, the average power is increased, first of all, by increasing pulse repetition rate. Any pulse modulator, including a subnanosecond one, is in fact a series of sections for timeamplitude transformation of pulses, i.e., for energy compression. Each section consists of a transforming device, an energy store, and a switch. At the same time, the subnanosecond highvoltage pulse modulators have some features. First, it is difficult to realize the formation of subnanosecond pulses without a preliminary nanosecond-range compression stage. This is related to controversial requirements of high electric strength of the energy store insulators and a high level of overvoltage of the energy store switch. Second, as distinct from the microsecond and nanosecond compression stages, the subnanosecond pulse former is a system with distributed rather than lumped parameters. The geometrical dimensions of the subnanosecond stage are rather small because of which the engineering approaches conventional for the preceding stages of the modulator are unacceptable. The latter circumstance is most critical for modulators operating at high repetition rates. This paper presents first results of testing of a new high-repetition-rate modulator (Fig. 1) in which a solid-state nanosecond driver with an inductive energy store and a semiconductor opening switch (SM-3NS)5 is combined for the first time with a subnanosecond pulse former based on superhigh-pressure gas gaps6. An experimental mockup of this modulator became feasible after a study of the operating modes of the semiconductor opening switch that provide a duration of the driver output pulses of shorter than The modulator was tested at pulse repetition rates of 100-2000 Hz; nitrogen and hydrogen were used as the fill gases for the spark gaps. Forced circulation of the gas in spark gaps was not used.
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EXPERIMENTAL SETUP Nanosecond driver The SM-3NS nanosecond driver5 is a system comprising several energy compression stages being intermediate energy stores, a pulse transformer, and solid-state thyristor and saturable magnetic switches. The output stage is made as a series circuit (see Fig.l) which contains, besides a capacitive energy store and a magnetic switch, an inductive energy store and a high-current, highvoltage semiconductor opening switch (high-voltage assembly of SOS diodes7). On operation of the switch a voltage of up to 400 kV is achieved with a pulse FWHM of ~6 ns (Fig.2). The driver provides a packet operation of the system at a pulse repetition rate of up to 2 kHz.
In the experiments the output voltage of the nanosecond driver was measured by a resistive voltage divider, and the pumping current of the SOS diodes was monitored by a low-resistance shunt (see Fig.l). The recorder was a Tektronix TDS684 oscilloscope. Measuring the above parameters in accumulation modes ("ENVELOP") and averaging the measurements over a great number of pulses (100-1000) have shown that the output parameters of the SM3-NS nanosecond driver were stable within the rated recording accuracy characteristic of this type of oscilloscope. This was also confirmed by reference measurements of the parameters of test nanosecond pulses produced by a G5-84 low-voltage (10 V) generator. Subnanosecond converter To produce pulses of duration ~1 ns, an additional energy compression section was connected at the output of the SM-3NS driver. This section was a short coaxial pulse-forming line (PFL) switched into a load by an untriggered two-electrode gas gap (Fig.3). Thus, the SN3-NS driver
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was used as a charging device for the PFL whose wave resistance was and capacitance was ~12 pF. This capacitance was calculated numerically with the help of the electrostatic block of the SAM computer code8 and included the capacitance of the PFL itself and the additional capacitance of the coaxial line short section connecting the PFL and the driver output. The additional capacitance increased the duration of the output pulse to 2 ns and lengthened its falltime, although the electric length of the PFL itself was only ~1 ns. The recorded pulse waveform (Fig.4a) was in rather good agreement with the waveform predicted with the use of the axisymmetric version of the KARAT code (Fig.4b)9. The second spike in the oscillogram (see Fig.4a), as shown below, is a reflection from the modulator load.
In the mode of charging the PFL of capacitance 12 pF to a maximum voltage at a pulse repetition rate of up to 100 Hz a voltage of ~300 kV was achieved. The breakdown voltage of the peaking spark gap was adjusted to be close to this maximum value. This became possible for two reasons. First, the output stage of the driver had an electric strength high enough to hold off the voltage of no-load operation which resulted from the spark gap idling. Second, even on occasional idling of the spark gap the probability of breakdown of the output insulator of the subnanosecond converter was rather low due to the reduced time of action of a high voltage. Insulators of this type were tested prior to experiments in the no-load mode with pulsed charging of the PFL at a risetime of It should be noted that in our experiments we did not observe idle operation of the spark gap. Constructionally, the pulse forming line was integrated with the assembly of high-pressure gas gaps. The mechanical strength of the case (see Fig.3) was high enough to withstand operating pressures of up to 100 atm. In the cases where it was necessary to have pulses shorter than the electric length of the PFL together with the lead electrodes, the pulse formation occurred through peaking and chopping on successive operation of two spark gaps. The peaking spark gap operated first which is followed by the operation, with a controllable delay, of the chopping spark gap. The spark gaps were immersed in a common gas medium. The separation of both spark gaps could be smoothly adjusted accurate to 0.05 mm with the help of an eccentric mechanical gear directly during the operation of the modulator, i.e., without release of the pressure in the spark gaps and disassembling of the case. The output pulse parameters (peak voltage, risetime, and duration) were varied by varying the breakdown voltage of the spark gaps. The matched load of the subnanosecond pulse former was a coaxial oil-filled transmission line of electric length ~2 ns which imitated the input feeder of a horn TEM antenna or a transforming line connecting the vacuum diode of an electron accelerator. The high-voltage resistor connected at the output of the line was responsible for the appearance in oscillograms (see Fig.4a) of a reflected pulse with the peak voltage making up to 75% of the main pulse amplitude. A calibrated capacitive voltage divider with a division factor of ~500 and a transient response of no worse than 150 ps was used for measuring voltage.
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EXPERIMENTAL RESULTS The SM3-NS driver was powered from a three-phase rectifier. The capacitance of the filter was 15 mF. In the given case, the driver operated in the mode of the power limited by the resistance of the supply line (3x380V, 50Hz). As the pulse repetition rate was increased to 2 kHz, the voltage across the filter as well as across the driver stages was observed to decrease gradually. As a result, the direct and reverse pumping currents of the SOS diodes decreased and, as a consequence, the voltage across the inductive energy store at the cutoff of the current in the SOS diodes decreased as well. The dependence of the output voltage of the driver on the pulse repetition rate was obtained for two different resistive loads (Fig.5). The corresponding curves 1 and 2 were normalized for the respective maximum amplitude achieved at a pulse repetition rate lying in the range from zero to 100 Hz. These measurements have shown that the dynamics of the decrease in the amplitude of the driver output pulse with increasing pulse repetition rate was independent of the load resistance. This fact was taken into account in determining the actual decrease in the amplitude of the output subnanosecond pulse at elevated pulse repetition rates of the modulator. There are some difficulties in increasing the rate of operation of spark gaps associated with growing instability and decreasing breakdown voltage. This is due to the limited ability of the working gas to recover its electric strength within a time shorter than the interpulse period. The dominant factors are the change in the thermodynamic state of the gas and the drop in the electric strength of the discharge gap resulting from the presence of products formed in the spark. When the system is operated for a long time, changes in the electrode surface and heating of the electrodes should be taken into account. For nanosecond high-voltage generators with gas gap switches, if the erosion of the spark gap electrodes is ignored, there are two ways of increasing substantially the pulse repetition rate: forcing the gas in the discharge gap to flow at a high velocity10 and reducing the prebreakdown overvoltage. The latter is achieved by triggering the spark gap switch11, 12. These both methods are difficult to use in the miniature stage of energy compression. Therefore, hydrogen showing an order of magnitude higher rate of recovery of the electric strength of a spark gap12 compared to nitrogen is preferable for use in high-repetition-rate subnanosecond spark gaps. Figure 6 presents a series of oscillograms of output pulses of the subnanosecond modulator, recorded at different pulse repetition rates. Each waveform has been obtained with a Tektronix TDS820 (6 GHz bandwidth) digital stroboscopic oscilloscope in the accumulation mode for 500 successive pulses. The working gases for the peaking and chopping spark gaps (waveforms in Figs. 6a and 6b) were hydrogen (100 atm) and nitrogen (60 atm), respectively. With hydrogen fill gas (see Fig.6a), the spacings of the spark gaps were adjusted to obtain for the modulator operating at a pulse repetition rate of 100 Hz stable pulses of maximum amplitude and minimum (~400 ps) duration (see Fig.6a, 1). Thereafter (see Fig.6a,2-5), the spark gap spacings were not changed. In view of the mentioned feature of the operation of the nanosecond driver, as the pulse repetition rate was increased from 100 to 2000 Hz, the charge voltage of the PFL of the subnanosecond converter decreased by 20-21% (see Fig.5, curves 1 and 2). An equivalent reduction of the breakdown electric field at the electrodes of the peaking spark gap at a nanosecond (~5 ns) time of charging of the PFL practically had no effect on the output pulse risetime. The latter remained at a level of 300 ps.
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Analysis of the waveforms given in Fig.6a has shown that as the pulse repetition rate was increased to 2000 Hz, the decrease in the amplitude of the output subnanosecond pulse made up ~23%, and, with normalization for a maximum amplitude (at a pulse repetition rate of 100 Hz), the respective curve (curve 3 in Fig.5) practically coincided with curves 1 and 2 in the same figure. For the operation at a pulse repetition rate of 100 Hz, the maximum amplitude of the output subnanosecond pulse reached ~145 kV, which was nearly half the charge voltage of the PFL. With this voltage, the energy stored in the PFL prior to breakdown of the spark gap was 0.5 J. As the pulse repetition rate was increased to 2000 Hz, the stored energy decreased to about 0.3 J. In the limit of the highest pulse repetition rates the average power at the output of the modulator was no less than 0.3 kW. Thus, it has been demonstrated (curve 3 in Fig.5) that for the switched energies ranging up to 1 J the rate of recovery of the hold-off capabilities of the hydrogen peaking spark gap is acceptable for the generation of pulses with a pulse repetition rate of up to 2000 Hz with almost no reduction in amplitude. Attention is drawn to the changes in the waveform of the trailing edge of the generated pulse with increasing pulse repetition rate. In the range 100-1000 Hz (see Fig.6a,l-3) the trailing edge lengthened from 100 to 200 ps. This took place when the voltage pulse amplitude and, correspondingly, the electric field in the chopping spark gap decreased by 13%. At the same time, the pulse duration shortened by the chopping spark gap remained stable: the spread in FWHM was no more than 100 ps. As the prebreakdown voltage was further decreased (by 23%), the delay to the operation of the chopping spark gap became unstable in time. In the oscillogram recorded with the use of the stroboscopic oscilloscope the instability of the pulse duration showed up as a chaotic sequence of spikes on the lengthened trailing edge (see Fig.6a,5). It is clear that the above facts by no means characterize the rate of recovery of the electric strength of the gas medium of the chopping spark gap. For any pulse repetition rate from the range 100-2000 Hz the spark gap could be preadjusted to produce pulses of waveform similar to that shown in Fig.6a,l, but of reduced amplitude. This, for instance, is illustrated by the oscillogram in Fig.7 obtained by accumulation of 200 pulses followed with a repetition rate of 1 kHz in the ENVELOP mode of the Tek-TDS684 oscilloscope. The limited bandwidth (1 GHz) and reading frequency (5 GS/s) reduced the amplitude of the pulse and distorted to some extent its waveform.
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Comparing the operating modes of the subnanosecond converter with hydrogen (see Fig.6a) and nitrogen spark gaps (see Fig.6b), we can make some notes and conclusions. Like in the experiments with hydrogen fill gas, the peaking nitrogen spark gap was adjusted to achieve a maximum amplitude of the output pulse at a high repetition rate of 100 Hz. The delay of the breakdown of the chopping gas gap provided a pulse duration of ~600 ps. In this mode the pulse FWHM was stable within 200-300 ps (see Fig.6b,l), which corresponded to the results obtained earlier in studying the operation of the subnanosecond converter with nitrogen spark gaps6. At a pulse repetition rate of 1000 Hz (see Fig.6b,3) the jitter in the operation of the chopping nitrogen spark gap was equal to that obtained for hydrogen fill gas at 2000 Hz (see Fig.6a,5). The decrease in the amplitude of the output pulse (breakdown voltage of the peaker) with increasing the pulse repetition rate of the modulator turned out to be significant. The absolute decrease in amplitude was 32-33% (see Fig.5, curve 4), while in comparison with the pulse amplitude that was actually provided by the nanosecond driver (see Fig.5, curves 1 and 2) the decrease was 15%. Nevertheless, this value turned out even somewhat lower compared to the decrease in breakdown voltage obtained by the authors of Ref.12 for a hydrogen spark gap which, when operated from an external starting device, switched an energy of ~9 J. We cannot exclude the effect of the difference in the gas pressures used in the present work and in the experiments12. Nevertheless, the factors deserving much attention are the great difference in the switched energies (0.5 and 9 J) and the difference in the times of charging of the capacitive energy stores prior to breakdown of the gap.
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CONCLUSION Preliminary testing of a new subnanosecond modulator has shown that the integration of a charging device based on a nanosecond high-voltage driver with solid-state switches and a pulse peaker with high-pressure hydrogen spark gaps has considerable promise. These two devices, when combined in one system, allow one to produce stable pulses of amplitude ~150 kV and higher and duration shorter than 500 ps at pulse repetition rates over 1000 Hz. Even in the first experiments the number of pulses in a packet at a pulse repetition rate of 2 kHz reached 104. As for now, we see no conceptual physical limitations on the duration of the pulse packet. After additional investigations and modernization of the subnanosecond modulator, research and development in some practically important fields will be quite realistic. It is planned to study the "accumulation" effects responsible for the changes in the electric strength of the air insulation of the feeder devices of shock-excited ultrawideband antennas1 with increasing pulse repetition rate. This is an important stage in the creation of compact sources of ultrawideband electromagnetic pulses of subnanosecond duration with an average power of 0.5 kW and higher. It is expected that the proposed modulator may increase the pulse repetition rate of wideband subnanosecond pulses of superradiation with a carrier frequency of 38 GHz and a peak power of more than 50 MW4.
REFERENCES 1. V.G.Shpak, S.A.Shunailov, M.R.Ulmaskulov, and M.I.Yalandin, Generation of high-power broadband electromagnetic pulses with PRF of 100 pps, in: Proc.of the 10th IEEE International Pulsed Power Conference Vol.1, W.Baker and G.Cooperstein, ed., IEEE, Albuquerque, (1995). 2. F.J.Agee, D.W.Scholfield, W.Prather, and J. W.Burger, Powerful ultra-wide band RF emitters: status and challenges, in: Proc. of SPIE International Symposium Intense Microwave Pulses III Vol. 2557, H.Brandt, ed., SPIE, San Diego, (1995). 3. L.Yu.Astanin, A. A.Kostylev, The Foundations of the Ultra-Wide Band Measurements in Radaring, Radio i Svyaz, Moscow, (1989). 4. N.S.Ginzburg, N.Yu.Novozhilova, I.V.Zotova, A.S.Sergeev, N.Yu.Peskov, A.D.R.Phelps, S.M.Wiggins, A.W.Cross, K.Ronald, W.He, V.G.Shpak, M.I.Yalandin, S.A.Shunailov, M.R.Ulmaskulov, and V.P.Tarakanov, Generation of powerful subnanosecond microwave pulses by intense electron bunches moving in a periodic backward wave structure in the superradiative regime. Phys. Rev. E 60: 3297 (1999). 5. S.K.Lybutin, G.A.Mesyats, S.N.Rukin, and B.G.Slovikovsky, Repetitive short pulse SOS-generators, in: Digest of Technical Papers of the 12th IEEE Int. Pulsed Power Conference Vol. 2, C.Stallings and H.Kirbie, ed., IEEE, Monterey, (1999). 6. G.A.Mesyats, V.G.Shpak, S.A.Shunailov, and M.I.Yalandin, Desk-top subnanosecond pulser research, development and applications, in: Proc.of SPIE International Symposium Intense Microwave Pulses Vol.2154, H.Brandt, ed., SPIE, Los Angeles, (1994).
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7 S.N. Rukin, High-power nanosecond pulse generators based on semiconductor opening switches, Instr. Exper.Tekh.. 42(4): 439 (1999). 8. M.A.Tiunov, B.M.Fomel, and V.P.Yakovlev, SAM -an Interactive Code for Electron Gun Evaluation, INP-87-35, Novosibirsk, (1985). 9. V.P.Tarakanov, User's Manual for Code KARAT, BRA, Springfield, (1992). 10. A.S.Elchaninov, F.Ya.Zagulov, S.D.Korovin, and G.A.Mesyats, Study of high-voltage spark gap stability for its operation with a gas flow between electrodes, Prib. Tekh. Eksper. 4: 162 (1979). 11. N.M. Bykov, O.A.Vashaev, V.P.Gubanov, A.V.Gunin, S.D.Korovin, and A.F. Yakushev, The highcurrent triggerable spark-gap with repetition rate of 100 Hz, Prib. Tekh. Eksper. 6: 96 (1988) 12. M.G.Grothaus, S.L.Moran, and L.W.Hardesty, Recovery characteristics of hydrogen spark gap switches, in: Proc.of the 9th IEEE International Pulsed Power Conference Vol. 1, K.Prestwich and W.Baker, ed., IEEE, Albuquerque, (1993).
UPGRADING OF THE EFFICIENCY OF SMALL-SIZED SUBNANOSECOND MODULATORS
M.I.Yalandin, M.R.Oulmascoulov, V.G.Shpak, and S.A.Shunailov Institute of Electrophysics, Russian Academy of Sciences, 34 Komsomolskaya Str., Ekaterinburg 620049, Russia
INTRODUCTION Compact modulators of the RADAN type1 based on a high voltage pulse forming lines (PFL) and gas spark gaps have found application in high-power ultra-wide-band (UWB) electromagnetic pulse generators equipped with a shock-excited TEM antennae2. These devices producing unipolar pulses of amplitude up to 150 kV and duration shorter than 1 ns include a circuit where the starting pulse generated by a nanosecond driver is peaked and then chopped with high-pressure gas spark gaps3. With this method of formation of subnanosecond pulses, the output pulse amplitude is always less than that of the starting pulse, i.e., resulting efficiency of UWB generator is low. This paper considers two methods for upgrading the efficiency of small-sized subnanosecond UWB generators: by increasing the output power of the modulator and by transforming the shape of its output pulse thereby improving the match with a shock-excited antenna.
ENERGY COMPPRESSION OF NANOSECOND MODULATOR The most efficient way of producing high-voltage pulses of duration ~1 ns and shorter is to provide an amplitude-time transformation (i.e., compression) of the energy of original nanosecond-width pulse. To do that an additional unit is placed at the generator (driver) output, which, in combination with the driver, forms an C-L-C circuit (Fig. la) with two capacitive energy stores (pulse-forming lines, and Fig. lb), a charging inductance coil, L, an intermediate spark gap, and an output spark gap, This circuit is used in one of the two versions as follows: (i) The starting pulse is compressed it time with a high efficiency of energy compression, but with no increase in the voltage of the capacitive energy store (see, e.g., Ref 4). For this case we have: (ii) The voltage of the energy store is increased (up to doubling, in the ideal case). In this mode we have and the energy transformation efficiency (Fig. 1c). Both versions are realized most completely if the characteristic time of charging of the energy store satisfies the condition The latter requirement means that in the process of charging both pulseforming lines can be considered as lumped-constant capacitors. If the charging time satisfied the relation then even for the condition of the increase in the voltage across the capacitor is low. For this regime realized by Jheltov5 the energy of the nanosecond generator was lost in reflections at joints (Fig. lb): and at the open spark gap At the same time, the energy of reflected pulses can be utilized to increase the efficiency of compression, in the case included. The analysis of this possibility given below takes into account that the condition means in fact the necessity to consider the processes of charging of the output energy store in the traveling wave mode. For this case we solved the problem of finding optimum parameters of the circuit given in Fig. lb. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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Optimum parameters of the energy compression unit The simplest way to analyze the process of charging of the output energy store is to consider an example of the charging of such a unit with a short risetime pulse satisfying the condition It was supposed that a pulse of duration and amplitude is produced by a nanosecond driver, passes via a rather long transmission line with a wave resistance through a section of a high-resistance line (Z), and arrives at an energy storage line with an open end. The wave resistance of the later is The representation of the inductance coil L (see Fig.lb) as a section of a homogeneous high-resistance line 2 (Fig.2a) is justifiable since in the general case we have For definiteness, we believed that the lines Z and are identical in electric length and where n=1 ;2;.... Note that the circuit given in Fig.2a represents a typical transmission line with a discrete inhomogeneity. For the case of an infinitely long output line this circuit has been analyzed in detail6 when considering the distortion of a transmitted pulse with a perfect leading edge with the help of the method for calculation of ring circuits. In our case, the circuit (see Fig.2a) consists of two series-connected ring circuits (Z and ). Obviously, as long as the time delay of the line Z equal to makes the summation in the line (and, hence, an increase in the amplitude of the energy store voltage) impossible (see Fig.2b). The situation is different if the pulse produced by the nanosecond driver is twice as long: (Fig.2c). In such a mode, the summation of the amplitudes of three voltage waves is realized in the line beginning from the time until the operation of the output switch (in the ideal case, at ):
In this expression we denote which is always greater than unity for our problem. Figure 2e (curve I) presents the result of the calculation by formula (1) which shows the presence of a maximum in the normalized amplitude at An estimation of the energy balance for the pulses shows that when we attain maximum charging, the line stores 87% of the energy of the starting pulse produced by the nanosecond driver. This is somewhat lower than the value (89%) obtained for an equivalent case in the approximation of a lumped-constant circuit (Fig. la). Putting the pulse duration equal to (see Fig.2d) and performing one more summation of the amplitudes of the pulses in the lines Z and we obtain for the voltage across the output line the multiplying factor close to 1.5 (see Fig.2e, curve II). This mode is realized at The energy efficiency of the charging in this mode decreases to 75%. It can be seen that the optimum value of a increases in correlation with the increase in the required duration of the starting pulse and with the decrease in the efficiency of charging. In terms of a lumpedconstant circuit (see Fig. la), this means nothing but an increase in inductance L until the condition for prolonged charging allowing one, in the limit, to double the voltage across the output capacitive energy store, however with low efficiency, is satisfied.
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Experimental results Preliminary experimental verification of the modes of charging of the energy store with a shortrisetime (~100 ps) pulse was performed with a low-voltage mockup of the circuit presented in Fig.2a, assembled from sections of coaxial lines. Results7 were in a complete agreement with the dependence shown in Fig.2e. In practice, when compressing the energy of a nanosecond high-voltage generator, a mode is generally realized where the risetime of the starting pulse is comparable to the electric length of the energy store The above analysis does not become invalid if the summation of amplitudes is provided for the quasi-flat portion of the top of the pulse having passed from the line Z to the energy store by properly choosing the delays for the lines Z and It is this option that was realized in our experiment on the transformation of the pulse produced by the RADAN-303B compact high-voltage generator1 whose input impedance is 45 ohm and the FWHM pulse duration is The energy stored by the generator double pulse-forming line was 1.9 J for the switching voltage of the spark gap equal to 150 kV. The only distinction of the experimental circuit from the model one (see Fig.2a) was the use of a spiral line instead of the highresistance coaxial section (Z). However, the line parameters (Z ~ 120 Ohm, ) were chosen such that the predicted optimum mode (see Fig.2e) be realized. Figure 3a presents the waveform of the starting pulse of the nanosecond driver taken for the case that the spiral line Z was replaced by a matched coaxial line section with a wave resistance of 45 Ohm and the spark gap was initially closed. When the spiral line with Z=120 Ohm was used, the starting pulse was distorted, as shown in Fig.3b. With precise tuning of the breakdown voltage of the spark gap a compression mode has been realized (Fig.3c) which in general corresponded to the case illustrated by Fig.2c. The time of charging of the energy store can be judged by the duration of the prepulse (2 ns) appearing due to the existence of the interelectrode capacitance of the spark gap The maximum amplitude of the transformed pulse was a factor 1.3 higher than that of the starting pulse (150 kV, see Fig.3a) being ~200 kV. That is, the summation of the voltage across the energy store in this case was indeed accomplished at the very "top" of the starting pulse. In the case that the delay to the spark gap operation was increased on purpose (the prepulse duration therewith increased to 3 ns, see Fig.3d), the summation time corresponded to the mode illustrated by Fig.2d; however, because of the too short duration of the starting pulse, the summation occurred during the falling portion of the pulse. The latter circumstance has made it possible to form a pulse with a nearly rectangular waveform. With that, the amplitude has increased from 150 to 180 kV. Estimates show that for the compression modes presented in Fig.3c;d the efficiency of the energy transformation for the nanosecond driver was no less than 80%.
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Advantages of the energy compression in a traveling wave mode The voltage multiplication factors of 1.3-1.5 achieved in the above modes are comparatively low; therefore, such an increase in pulse amplitude may be considered in the design of a concrete pulser. At the same time, an optional device providing even small compression factors may be highly useful to improve the performance of now available UWB systems. This is all the more obvious if we take into account the opportunity to connect in series several compression stages. The undoubted benefit from the energy compression with a small transformation factor has been demonstrated when using the developed compression unit for preparatory processing of a nanosecond pulse which was further subjected to additional peaking and cutting-off. This made us possible to produce a subnanosecond pulse whose amplitude exceeded the amplitude of original pulse of the nanosecond driver. An increase in the output power attained a factor of 1.7 (Fig.4).
GENERATION OF BIPOLAR SUBNANOSECOND PULSES In producing radiation with shock-excited TEM antennas, a bipolar pulses are preferable owing to the feature of the spectral function whose maximum is shifted toward the high frequencies. The reduced lowfrequency component provides a better match of the high-voltage generator to the antenna. In pulsed power, a conventional technique for the production of subnanosecond bipolar pulses is based on using a shortcircuited line (loop) which is connected in a T-circuit into the output channel of the unipolar pulse generator2, 8. With a starting unipolar traveling pulse of amplitude a bipolar pulse of the same peak-topeak amplitude can be produced. A fundamental advantage of such a transducer is that the output pulse has a well reproducible waveform. The second advantage is the possibility to smoothly control the pulse duration. At the same time, the short-circuited line is a passive transducer of the waveform of the starting pulse whose energy is partly lost in reflections. That is, an obvious disadvantage is associated with the low
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efficiency of energy conversion being not over 0.5 for the case that the pulse portions of unlike polarity are symmetric in amplitude. Another technique for the formation of a bipolar pulse is illustrated by Fig.5. The technique is based on simultaneous operation of two spark gaps placed on the opposite ends of a pulse-forming line8. The line charged to a voltage of is switched at the output into a matched load. As a result, the uninverted portion of the pulse with the amplitude equal to and the duration, equal to the time the wave takes to execute one pass along the line is formed across the load. The opposite end of the line is short-circuited on operation of the second spark gap, providing the formation of the inverted portion of the pulse. This pulse portion arrives at the load with a delay of has the same duration, and its amplitude is As a result, the device produces a bipolar pulse of doubled peak-to-peak amplitude as compare with conventional converter based on a short-circuited line. An obvious requirement for this device is that the switching time of the spark gap should be short and show a small spread around The switching jitter the spark gaps is also responsible for the symmetry of the portions of the bipolar pulse in time.
In forming nanosecond and subnanosecond pulses for the case of dc or slow pulsed charging of the pulse-forming line, an acceptable mutual delay in the operations of the spark gaps can be ensured only by their synchronous controllable triggering9. It is also obvious that the switching jitter of an untriggered switch decreases with increasing charging rate. In our experiments10 two split pulses of peak voltage ~150 kV and risetime 1.5 ns were applied in the traveling wave mode to the identical peaking gas gaps. This made it possible, owing to precision adjustment of the gap spacings, to provide a 100-ps rms spread of the attained 300-ps risetimes. The goal of the study under description was to obtain a similar information about the operation of spark gaps in the circuit presented in Fig.5a on varying the rate of charging of the pulse-forming line. Features of the operating modes of the spark gaps The two-electrode spark gaps of the device under consideration (see Fig.5a) cannot be made identical in principle because of the unlike operating modes and the specificity of design. First, the switching current of the grounding spark gap is twice the current of the output switch operating into a matched load. Second, the charging of the pulse-forming line by a short-risetime pulse in the traveling wave mode calls for a proper match between the pulse-forming line and the input channel. The corresponding configuration of the pulse former (Fig.5c) gives no way of making the electrode systems of the spark gaps and identical in design. The third impediment follows from the features of the layout shown in Fig.5a: in the ideal case ( Fig.5b), the output switch should withstand the double voltage for a time
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of The time during which the traveling wave of voltage acts on the electrodes of the spark gap is twice as much: however, this spark gap should operate near the peak of the voltage immediately on arrival of the reflected pulse.
Design of the pulse former In accordance with the circuit charging the pulse-forming line in the mode of a short-risetime traveling wave (see Fig.5), the duration of the charging pulse should be equal to or longer than the total duration of the bipolar pulse to be formed Since of interest to us was to form bipolar pulses of duration we used for the charging device the RADAN-303B compact nanosecond driver. To shorten the risetime of the charging pulse at the output of the nanosecond driver, an additional peaking spark gap was connected upstream of the pulse-forming line. The pulse former was made as a 50-Ohm coaxial line insulated with nitrogen at 40atm. For precise tuning of the breakdown voltage level for the spark gaps used were eccentric gears allowing a variation of the gap spacings within a range of 0-4 mm. The operating edge of the grounded ring electrode of the spark gap is rounded. The potential electrode of this spark gap was formed by a section of a conical surface on the central conductor of the pulse-forming line. The waveform recording of the pulses produced was performed at the transducer output where a capacitive voltage divider was connected in a 50-Ohm oil-filled transmission line. Calibration has demonstrated that the transient response of the voltage divider was 150 ps. The line ended in a high-power 50-Ohm carbon resistor. The resistor failed to provide a proper match for subnanosecond pulses. Therefore, reflections from the load showed up in the pulse waveforms in about 4 ns after the front of recorded pulse. To measure subnanosecond high-voltage pulses, a type Tek TDS-820 digital stroboscopic oscilloscope with a recording band of 6 GHz, which requires a minimum of 500 readings to take a waveform, was used. With that obtained waveforms allow one to evaluate the pulse-to-pulse stability of the device operation.
Testing results Figure 6a presents the waveform of an output pulse produced by RADAN-303B. Judging by the spread of the points by which the stroboscopic oscilloscope has constructed the waveform, the amplitude rms spread was not over 3-5%. The waveform in Fig.6a was obtained in the conditions where the spark gaps and where short-circuited, while the gap was opened. The pulse risetime was ~1.5 ns. On opening and tuning the spark gap the 0.1-0.9 amplitude risetime was shortened to ~300ps (Fig.6b). The mode of charging of the pulse-forming line even with this short pulse risetime, is far from the ideal case illustrated by Fig.5b. For the length of the pulse-forming line of 17 or 10 cm, the condition was fulfilled.
The features of the waveform of the bipolar pulse were conditioned by the tuning of the spark gaps and and by the length of the line Given in Fig.7a is a waveform with the highest double amplitude between the peaks of unlike polarity, being 270 kV. This value is very close to the double voltage of the traveling wave with a shortened risetime ( see Fig.6b). The long time of the peak-to-peak voltage drop (~500 ps) testified to the fact that the spark gaps and had operated somewhat earlier than the wave reflected from the spark gap arrived at the spark gap With that, the drop time is the sum of the risetimes of the pulses formed by the peaker and by the chopper
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Figure 7b presents a bipolar pulse produced under the conditions where the voltage wave reflected from the spark gap being open reached the spark gap In that case the double voltage amplitude between the peaks was 250 kV. Some decrease in this amplitude as compared to the voltage pulse presented in Fig.7a seems to be due to the fact that the spark gap operated at a voltage somewhat lower than the peak charge voltage At the same time, the increased prebreakdown voltage of the spark gap changed substantially the rate of its closure, and the drop time between the peaks shortened to ~250 ps. Judging by the waveform given in Fig.7b, the total range of switching jitters for the spark gaps and was not over 200 ps. Given in Fig.7c is the waveform of a bipolar pulse taken in the operating mode of the spark gaps similar to that illustrated by Fig.7b. The reduced duration of both half-periods of the pulse in this case is related to the use of a shortened pulse-forming line of length The negative and the positive portions of the pulse are more symmetric than those of the pulses shown above. This is related to the fact that, owing to the shorter pulse duration, the superpositions of reflections on the positive pulse tail could be avoided.
Of fundamental importance in the results above was the rise rate of the starting voltage pulse. For comparison, given in Fig.8a;b are the waveforms of bipolar pulses produced by a generator where the risetime of the charge pulse was not shortened. Acceptable stability of the generator operation and series-toseries reproducibility of the voltage waveforms could be attained only for the case that the spark gaps operated within the risetime of the charging voltage, early in its second half, i.e., when the voltage action time was limited (see Fig.8a). Therefore, the double voltage amplitude between the peaks of unlike polarity was not over 165 kV with the starting pulse voltage being 150 kV. In attempting to form a pulse with a longer time delay to the breakdown of the spark gaps (i.e., near the peak charge voltage), the chopper operated unstably. This showed up in the unevenness of each waveform on the screen of the stroboscopic oscilloscope (see Fig.8b). Thus, the experiments performed have demonstrated the possibility to form bipolar pulses of total duration ~1 ns with the use of two independently operating untriggered high-pressure gas gaps. It has been revealed that the critical parameters responsible for the achievement of an acceptable mutual delay in the operations of the switches is the prebreakdown rise rate of the voltage across the chopping spark gap. Its total switching jitter is of the order of magnitude or shorter than the duration of the front of the traveling wave of the pulsed charge voltage.
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REFERENCES 1. G.A.Mesyats, V.G.Shpak, S.A.Shunailov, and M.I.Yalandin, Compact high-current accelerators based on the RADAN SEF-303 pulsed power source, in: Proc.of the 9th IEEE International Pulsed Power Conference Vol.2, K.Prestwich and W.Baker, ed., IEEE, Albuquerque, (1993). 2. M.I.Yalandin, V.G.Shpak, S.A.Shunailov, and M.R.Ulmaskulov, Compact repetitive generator of highpower broadband electromagnetic pulses, in: Proc. of SPIE International Symposium Intense Microwave Pulses III Vol. 2557, H.Brandt, ed., SPIE, San Diego, (1995). 3. G.A.Mesyats, V.G.Shpak, S.A.Shunailov, and M.I.Yalandin, Desk-top subnanosecond pulser research, development and applications, in: Proc.of SPIE International Symposium Intense Microwave Pulses Vol.2154, H.Brandt, ed., SPIE, Los Angeles, (1994). 4. O.A.Vashaev, V.P.Gubanov, and S.D.Korovin, Compact, high-current nanosecond accelerator with electrons energy of 1.5 MeV, Prib.Techn. Exper. 2: 41 (1991). 5. K.A.Jheltov. Picosecond, High-Current Electron Accelerators, Energoatomizdat, Moscow, (1991). 6. L.A.Morugin, G.V.Glebovich. Nanosecond Pulsed Technique, Sov.Radio, Moscow, (1964). 7. V.G.Shpak, M.R.Oulmascoulov, S.A.Shunailov, M.I.Yalandin, Amplitude compression of high-voltage pulses in subnanosecond formers on gas spark gaps, in: Digest of Technical Papers of the 12th IEEE Int. Pulsed Power Conference Vol.2, C.Stallings and H.Kirbie, ed., IEEE, Monterey, (1999). 8. V.G.Shpak, S.A.Shunailov, M.R.Ulmaskulov, and M.I.Yalandin, Generation of high-power broadband electromagnetic pulses with PRF of 100 pps, in: Proc.of the 10th IEEE International Pulsed Power Conference Vol.1, W.Baker and G.Cooperstein, ed., IEEE, Albuquerque, (1995). 9. V.G.Shpak, S.A.Shunailov, and M.I.Yalandin, Investigations of compact high-current accelerators RAD AN 303 synchronization with nanosecond accuracy, in: Proc. of the 10th IEEE International Pulsed Power Conference Vol.1, W.Baker and G.Cooperstein, ed., IEEE, Albuquerque, (1995). 10. V.G.Shpak, S.A.Shunailov, M.R. Oulmascoulov, and M.I.Yalandin, Synchronously operated nano- and subnanosecond pulsed power modulators, in: Digest of Technical Papers of the 12th IEEE Int. Pulsed Power Conference Vol.2, C.Stallings and H.Kirbie, ed., IEEE, Monterey, (1999).
CHARACTERISTICS OF TRAP-FILLED GaAs PHOTOCONDUCTIVE SWITCHES USED IN HIGH GAIN PULSED POWER APPLICATIONS
N. E. Islam 1, E. Schamiloglu 1, A. Mar 2, F. Zutavern2, G. M. Loubriel2, and R. P Joshi 3 1
Department of Electrical and Computer Engineering University of New Mexico Albuquerque, NM 87131-1356, USA 2 Sandia National Laboratories MS-1153, Post Office Box 5800 Albuquerque, NM 87185-1153 3 Department of Electrical and Computer Engineering Old Dominion University Norfolk, VA 23529-0245
INTRODUCTION The electrical properties of semi-insulating (SI) Gallium Arsenide (GaAs) have been investigated for some time, specifically for its application as a substrate material in microelectronics [1]. Of late this material has found a variety of applications other than as an isolation region between devices or the substrate of an active device. High resistivity SI GaAs is increasingly being used as charged particle detectors and as a photoconductive semiconductor switch (PCSS). PCSSs made from this material and operating in both the linear and non-linear mode have applications as firing sets, as drivers for lasers, and as a high impedance, low current Q-switch or Pockels cell [2-4]. In the non-linear mode, it has been used in a system to generate Ultra-Wideband (UWB) High Power Microwaves (HPM) [5]. The choice of GaAs over Silicon is due to the advantage that its material properties allow for fast, repetitive switching action [6]. Furthermore, photoconductive switches have advantages over conventional switches such as improved jitter, better impedence matching, compact size, and in some cases, lower laser energy requirement for switching action. In PCSS applications the rise time is an important parameter that affects the maximum energy transferred to the load and it depends, in addition to other parameters, on the bias or the average field across the switch. Increased bias operation without changing the distance between the contacts, and high field operation have been an important goal in the PCSS research. Due to “surface flashover” or the premature breakdown mechanism at higher voltages, most PCSSs, specifically those used in high power operation, need to operate at much below the breakdown voltage of the material. The “lifetime,” or the total number of switching operations before breakdown, is another Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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important switch parameter that needs to be considered for operation at high bias conditions. A lifetime of shots has been reported for PCSSs used in UWB-HPM generation [5], while it has exceeded shots as electro-optic drivers [2]. The contact material used and electrode profiles are important areas of study. Although these problems are being looked into through the incorporation of different contact materials and introducing doping near contacts for increased longevity [2,5], it is important that the switch properties and the conduction mechanism in these switches be well understood so that the basic nature of the problems can be properly addressed. In this paper we report on these two basic issues related to the device operation, i.e., mechanisms for increasing the hold-off characteristics through neutron irradiation, and the analysis of transport processes at varying field conditions in trapdominated SI GaAs in order to identify the failure mechanism during device operation. It is expected that this study will result in a better understanding of photoconductive switches, specifically those used in high power operation. PCSS MATERIALS Since the semiconductor resistivity varies inversely as the free carrier concentration, both intrinsic and compensated semiconductors [7,8] (with high density of trap levels near the mid-gap) having low free carrier concentrations are high resistivity materials. Charge transport in these two material types, however, differs. As discussed in [9], compensated materials have better charge transport characteristics because their properties are very similar to “lifetime” semiconductors where the trap concentration defines the “screening length” (a parameter related to the Debye length) which is small compared to the diffusion length. Intrinsic materials are of the “relaxation” type, where the dielectric relaxation time is large and comparable to the minority carrier lifetime, thus affecting the performance of the PCSS as a whole [9]. The compensated materials are made through various growth techniques such as liquid encapsulated Czochralski (LEG), horizontal Bridgeman (HB), the vertical gradient freeze (VGF) as well as other methods. It is possible that any two compensation techniques may give the same value for the materials resistivity; however the various trap levels and impurity types in the materials may affect the conduction properties and the characteristics of the photoconductive switch [8]. Hence, in understanding device parameters such as rise time and switch lifetime, it is important to distinguish the material type, compensation processes, and the resistivity of the material. In this study we have targeted two device types of similar material, both LEC-grown SI GaAs. The switch geometry however is different. One of the devices has lateral (same side) forward-biased p and n-type contacts and operates in a comparatively lower DC field when operated in forward bias for increased device lifetime. The other has two Rogowski-profiled opposed contacts made from refractory materials and is used in high power microwave generation. When reverse-biased, the lateral switch operates at similarly high DC fields as the opposed contact device, but with decreased contact life. NEUTRON IRRADIATED PCSS The effects of neutron irradiation on the properties of conductive GaAs have been extensively studied and reported effects include such phenomena as the degradation of free charge carrier mobility, removal of free charge carriers, and changes in the carrier lifetimes [10]. However, there is not much available data on SI GaAs since, until recently, these materials were used for device isolation and carrier transport was not a major concern. Capacitance voltage analyses of undoped SI material used as detector material have also shown an increase in the effective space charge concentration with fluence when they are subjected to neutron irradiation, thus implying the introduction of deep level defects [11]. The effect of neutron irradiation also results in an increase in the resistivity of the materials of the detectors [12].
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Since increased resistivity would facilitate higher bias operation and is expected to improve the rise time, experiments were conducted with the lateral devices at the Sandia Laboratories facilities. Two switch types (Figure 1A) were studied, one with p and n contacts to the SI and the other with diffused layers below the contacts. Figures 2A and 2B show the forward I-V characteristic for the diffused contact devices before and after neutron irradiation. Hysteresis in the curves is evident, with the circles showing the ascending part of the voltage sweep and the squares showing the descending part.
The device was exposed to a wide energy spectrum neutron flux with a maximum fluence of The SI contact devices were not irradiated and its pre-irradiation IV characteristic showed a steep increase at ~ 400 V, which is well beyond that of the diffused contact type (~ 260 V, Figure 2A). The effects of neutron irradiation result in an improved hold-off characteristic of the diffused contact device (2B). The simulated response of the devices, obtained using the SILVACO [13] device simulation code, is shown in Figure 3. This commercial code provides for a 2- and 3-D semiconductor device simulation and includes a large number of models and parameters to simulate physical conditions in devices. In addition, it also allows for user-defined model parameters as input through a C-interpreter. Figure 1B shows the 2-D device
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profile used in mesh generation. There are approximately 4000 grid points, and simulation parameters include charge trapping and de-trapping at defect sites, carriercarrier scattering, concentration and field-dependent mobilities, etc. The vendor supplied EL2 concentration of the wafer was and its energy level was 0.730 eV. We have assumed that neutron irradiation-induced EL2/EL6 traps have been generated [14], and hence the effective capture cross section has increased. This increase was shown by increasing the electrons capture cross section of the defect levels. Other simulation and model parameters were the same as in reference [14].
Since the diffusion layer below the metals is conductive, the effective distance between the contacts for this PCSS is 0.8 mm. For the SI contact PCSS the distance is 1.0 mm and the steep rise measured voltage is 400 V. Thus the voltage for steep rise for the diffused contact PCSS can be predicted by the relationship, where the subscripts D and S refer to the diffused and SI contacts [15]. Thus which is very near the measured value as shown in Figure 2. Unstable filamentary characteristics [16], high field pockets and trap filled regions, however, are likely to be aggravated by damage [17], such as introduced through neutron irradiation. Thus a post-irradiated analysis involving high power conduction in the on state of the switch may be required for final comparison. CARRIER TRANSPORT AT DIFFERENT BIAS Simulations for the opposed contact PCSS show [Figure 4A] current controlled negative resistivity (CCNR) characteristics (S-shaped I-V). This characteristic is open circuit stable, i.e., if the device of area A is operating at voltage X and the current density is it achieves stability at point [see Figure 4B]. If the current at is and is carried by an area a, one can easily derive Equation (2) [inset, 4B]. Since then a Thus a very small area carries a large amount of current, which is a case for filamentation [17]. For switches that operate at high fields, the CCNR characteristic may be due to impact ionization. For lower fields where negative differential mobility is due to the transfer of electrons from a high-mobility energy valley to a low-mobility higher energy satellite valley the I-V characteristics is of the voltage controlled negative resistivity type (N-shaped). For such cases the device response is a traveling spacecharge accumulation rather than filamentation [15, 18]. However, filamentation has been observed in most of the lateral PCSSs [2] operating at lower bias. The reason for this delay is shown in Figure 5, where an increase in current due to double injection is shown,
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and the I-V characteristic is similar to the current controlled negative resistivity effect [Figure 4B].
The switch material for which the I-V is plotted has a resistivity of
with
EL2 traps near mid-gap. Measurements were performed using a HP-4142 Parametric Analyzer and the unit used for the measurement was a HP41423A, High Voltage Source/Monitor Unit (HVSMU) with a range from 2 mV to 1 kV and a supply current ranging from 2 pA to 10 mA. CONCLUSION We have presented experimental and simulation results for a SI GaAs and discussed the results in terms of the dominance of the trap levels in determining the I-V characteristics of the devices. Experimental results and simulation analyses indicate that, when subjected to neutron irradiation, there is an improvement in the hold-off characteristics of the devices, which may lead to higher voltage operation and thus an improvement in the rise time of the device. The maximum energy transferred to the load would increase, thereby improving the overall device characteristics. The filamentary conduction processes of the PCSS are more susceptible to failure if a large number of defect sites are formed in the device during a radiation transient. This is mainly because defects are expected to bring about inhomogeneous conduction.
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Acknowledgements The authors at UNM and ODU acknowledge the support of the support of the AFRL/AFOSR New World Vistas program for this study. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed-Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000.
REFERENCES [1] C.Y. Chang and F. Kai, GaAs High-Speed Devices, Physics, Technology and Circuit Applications, John Wiley & Sons, Chapters 1 and 2, 1994. [2] A. Mar, G.M. Loubriel, F.J. Zutavern, M.W. O’Malley, W.D. Helgeson, D.J. Brown, H.P. Hjalmarson, and A.G. Baca, “Doped Contacts for High-Longevity Optically Activated, High Gain GaAs Photoconductive Semiconductor Switches,” IEEE Trans. Plasma Sci., Special Issue on Pulsed Power Science and Technology, to be published 2000. [3] K. Zdansky, J. Santana, B. K. Jones and T. Sloan, “Modeling of Particle Detectors Based on
Semi-insulating GaAs,” Proceedings of the Third International Workshop on Gallium Arsenide and Related Compounds, Tuscany, Italy, Editors P. G. Pelfer, J. Ludwig, K. Runge and H. S. Rupprecht, World Scientific, New York, NY, 1995, p. 78. [4] G.M. Loubriel, F. Zutavern, A.G. Baca, T.A. Plut, W.D. Helgeson, M.W. O’Malley, M.H. Ruebush and D.J. Brown, “Photoconductive Semiconductor Switches,” IEEE Trans. Plasma Sci., vol. 25,p.l24, 1997. [5] J.S.H. Schoenberg, J.W. Burger, J.S. Tyo, M.D. Abdalla, M.C. Skipper, and W.R. Buchwald, “Ultra-Wideband Source Using Gallium Arsenide Photoconductive Semiconductor Switches,” IEEE Trans. Plasma Sci., vol. 25, p. 327, 1997. [6] M.D. Pocha and W.W. Hofer, “High-Speed Switching in Photoconductors,” High-Power Optically Activated Solid State Switches, Editors A. Rosen and F.J. Zutavern, Chapter 3, Artech House, Norwood, MA, 1994. [7] R.B. Darling, “Electrostatic and Current Transport Properties of GaAs Junctions,” J. Appl. Phys., vol. 74, p. 4571, 1993. [8] N.E. Islam, E. Schamiloglu, J.S.H. Schoenberg, and R.P. Joshi, “Compensation Mechanisms and The Response Of High Resistivity GaAs Photoconductive Switches During High Power Applications,” submitted to IEEE Trans. Plasma Sci. [9] E. Schamiloglu, N.E. Islam, C.B. Fledderman, B. Shipley, R.P. Joshi, and Z. Zheng, “Simulation, Modeling and Experimental Studies of High Gain Gallium Arsenide Photoconductive Switches for Ultra Wideband Applications”, Ultra Wideband Short Pulse Electromagnetics 4, Edited by Hevman et. al., Kluwer/Academic Publishers, New York, NY, 1999, p. 221. [10] D. Pons and J.C. Bourgoin, “Irradiation Induced Defects in GaAs,” J. Phys. C, vol. 18, p. 3839, 1985. [11] F. Dubecky, J. Betko, M. Morvic, J. Darmo, I. Besse, 1. Hrubein, S. Halvac, M. Benovic, P. G. Pelfer, E. Gombia, and R. Mosca, “Neutron Irradiated Undoped LEC SI GaAs: I. Galvanometic, I-V, PC and Alpha Detection Study,” Gallium Arsenide and Related Compound, Editors P. G. Pelfer et. al., World Scientific, New York, NY, 1996, p. 152. [12] F. Dubecky, F. Dubecky, T. Lalinsky, P. G. Pelfer, S. Halvac, E. Gombia, and R. Mosca, “Neutron Irradiated Undoped LEC SI GaAs: II. C-V and Deep State Analysis,” Gallium Arsenide and Related Compound, Editors P. G. Pelfer et. al., World Scientific, New York, NY, 1996, p. 158, [13] SILVACO International, ATLAS User’s Manual, www.silvaco.com [14] G.M. Martin, E. Esteve, P. Langlade and S. Markam-Ebeid, “Kinetics of Formation of the Midgap Donor EL2 in Neutron Irradiated GaAs Materials,” J. Appl. Phys. vol. 56, p. 2655, 1984. [15] J.J. Mares, J. Kristofik, V. Smid, and K. Jurek, “Impact Ionization and Space Charge Effects in SI-GaAs,”Semi-Insulating III-V Materials, p171, Edited by M. Godlewski, World Scientific, (1994). [16] N.E. Islam, E. Schamiloglu, C.B. Fleddermann, J.S.H. Schoenberg, and R.P. Joshi, “Analysis of High Voltage Operation of Gallium Arsenide Photoconductive Switches Used in High Power Applications,” J. Appl. Phys., vol. 86, p. 1754,1999. [17] S.K. Gandhi, Semiconductor Power Devices, Chapter 1 and 2, John Wiley & Sons, New York, NY, 1977. [18] S.M. Sze, Physics of Semiconductor Devices, 2nd Edition, Chapter 11, John Wiley and Sons, New York, NY, 1981.
PROJECT OF SEMICONDUCTOR HIGH-POWER HIGH-REPETITION RATE COMPACT CURRENT/UWB PULSE GENERATOR
Eugene A. Galstjan, Lev N. Kazanskiy Moscow Radiotechnical Institute of RAS Warshawskoe Shosse 132, Moscow 113519, Russia
INTRODUCTION The tendency of the last years in the field of development of high-power pulse and accelerating facility is the maximum extension of possible areas of its application. It demands development simple in operation, reliable in maintenance, and, principal, whenever possible of compact devices. On the other hand, for the last two decades of research in the field of solid state physics have resulted in creation of semiconductor devices which parameters allowed to create generators of high-voltage high-power impulses working in a repetitive mode (Lyubutin, 1996; Kardo-Sysoev, 1997; Efanov, 1997). However, till now these semiconductor devices have not found the place directly in development of high-power pulse facility. The presented paper is devoted to exposition of idea of possible usage of modern semiconductor devices for creation of a compact device, which, on our opinion, can find rather broad application. In the basis of this device the principle stating lies that the inductive storage of energy combined with possibilities of modern semiconductors, is most perspective for creation of high-power pulse devices working with a high repetition rate. Instead of gas discharges, which limit a pulse-repetition rate and have enough wide jitter in a response time, it is offered to use new semiconducting switches. These switches are capable to reconnect for extremely short time (0,1 - 1 ns) currents by magnitude 1 - 10 kA at an operation voltage 10 - 100 kV. In addition, the process of switching is controlled with split-hair accuracy, as the jitter in operation of keys does not exceed 20 ps. Thus the repetition rate of switching is limited only by conditions of heat rejection from the device and can reach tens of megahertz.
PRINCIPLE OF OPERATION Let's consider more in detail principle of operation of the device. It is expected that the device consists of a few sections. In figure 1 the schematic of one section is shown. A toroidal inductor is inserted in a transmission line TL which generally can be a coaxial Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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line or can be loaded with a beam of charged particles. Originally capacitor C is charged by an external source up to primary voltage reaching several kilovolt. Then it starts to be discharged through walls of the inductor connected by a switch S. There is a transformation of energy accumulated in the capacitor in energy of a magnetic field in the inductor and at
reaching a maximum value of a current in the inductor (when voltage across the capacitor
C becomes equal to zero) the switch S is turned off switching the inductor to a load. Thus there is a pulse high voltage across the gap AB and the generated electric pulse transmits in a coaxial line or accelerates charged particles of a beam.
Simple estimates display that under condition of a small switching time maximum voltage across the gap AB is defined by the relation
the
where are the reference times of charging and discharging of the inductor, and Z the impedance of the load.
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The front of the high-voltage impulse is determined by the rupture time and the mean pulse duration equals to the discharge time of the inductor (see figure 2). Though the schematic of the section reminds to a section using in linear inductance accelerators, actually it works completely in another way. In usual linear accelerator energy transforming in an accelerating impulse is stored in capacitors. Accumulation of energy directly in the inductor of the section allows generating short (10 - 15 ns) high-power electric pulses effectively. Besides, it is possible to shape very short impulses (3 - 5 ns) of rectangular form by using matched radial lines as the inductors. THE PROJECT However incarnation of all these ideas in actual devices requires a solution of a lot of engineering problems. For this purpose now in Institute of Experimental Physics (Sarov, Russia) in cooperation with the authors and Pulse Technology Group (St. Petersburg, Russia) the construction of an accelerator grounded on the above-described principles is developed. This accelerator is designed to operate with the cold explosive-emission cathode, for which it is necessary to create the operation voltage not less than 100 kV. Therefore, a series connection of several described above sections is supposed (see figure 3). As the first variant it is supposed to connect sequentially such three sections. At primary voltage and the relation the output voltage on the cathode should be 120 kV at pulse duration about 15 - 25 ns, working current up to 2 kA and repetition rate of working impulses 1 kHz. On this accelerator it is supposed to decide engineering problems, bound up with a construction, an electrical circuit, matched operation all sections etc.
In figure 4 one of possible variants of the principle electrical circuit of the section of the above-described accelerator is shown. The semiconductor switch, first of all, defines the view of this circuit, as it demands the initial pump by electrical carriers in the forward direction, and the current rupture happens at the opposite direction of a current, when it reaches the maximum value. In addition, the charge, which is flowing past through the
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switch in the forward direction, should coincide with the charge that has flowed past in the opposite direction. This condition also determines choice of the circuit. The circuit operates as follows. Originally the capacitors and (connected in series) are charged through inductances and up to the primary voltage Further the capacitor is used for the direct pump of the semiconductor switch For this purpose the thyristor is turned on at a closed thyristor and the capacitor is completely recharged through the inductor and the switch After the recharge process is over, the thyristor is turned on and, as a result, the capacitors and are connected in parallel. It results in that the above-stated condition of equality of an amount of the flowed past charges is fulfilled for a quarter of phase of oscillation of a reverse current. At a maximum value of a current in the inductor the switch produces cutoff of this current and the voltage pulse is shaped across the load resistance
It is necessary to mark that the selected operational mode of the semiconductor switches in the given circuit is far from limiting on output voltage. For this reason, expected rate of acceleration can reach only moderate magnitude ~ 0.2 MV/m, but this value may be gained. Besides, it is known that an electron beam generated by a cold cathode with the explosive emission has no enough high quality. It limits its applications, for example, in Free Electron Lasers. In our variant of the accelerator the generated beam could be additionally accelerated up to a necessary energy in a set of the same sections, loaded with the beam (see figure 5). In this case, quality of the beam could be essentially improved.
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The above-described device can operate not only as the accelerator. The same structure without the cathode can be used as a UWB pulse generator. CONCLUSIONS Thus, usage of modern semiconductor switches allows creating the compact device both a generator of power electric pulses, and an accelerator of charged particles. The generator of electric pulses can be used in medicine and biology, where a possibility of selective action of short electric pulse on a cell now is researched. The accelerating devices, assembled from above described sections, are good sources of charged particle bunches (electrons, protons, and ions), which can be used for generation of electromagnetic radiation, for surface treatment, in an ecology etc. Besides, it is possible to gather a linear electron accelerator with a possibility of acceleration of bunches up to large energies. REFERENCES Lyubutin, S. K., Mesyats, G. A., Rukin, S. N., 1996, New solid-state opening switches for repetitive pulse power technology. Proc. XI Int. Conf. on High Power Particle Beams. Prague, Czech Republic, V.1: 1211. Kardo-Sysoev, A.F., Zazulin, S.V., Efanov, V.M., Lelikov, Y.S., Kriklenko, A.V., 1997, High repetition frequency power nanosecond pulse generation. Proc. XI IEEE Int. Pulsed Power Conf. Baltimore, USA, P. 420. Efanov, V.M., Karavaev, V.V., Kardo-Sysoev, A.F., Tchashnikov, I..G., 1997, Fast ionization dynistor (FID) - a new semiconductor superpower closing switch. Proc. XI IEEE Int. Pulsed Power Conf. Baltimore, USA, P. 988.
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HIGH POWER SUBNANOSECOND GENERATOR FOR UWB RADAR
Vitaliy P. Prokhorenko, Anatoliy A. Boryssenko Research Company “Diascarb” Kyiv, P.O. Box No. 222, 02222, Ukraine
INTRODUCTION Subnanosecond pulse generator is one of the most important elements of ultrawideband (UWB) radar. Parameters of impulse generator influence to radar performance since it depends on radiation power and receiver sensitivity. There are some ways to get high performance factor of the UWB radar. It can be achieved as increasing of radiation power and multiply sampling data accumulation. In this paper we shall describe solid state, high power subnanosecond generator for portable UWB radar design. PROBLEM BACKGROUND There are two methods to reach high value of UWB radar performance factor (PF) that we understand as ratio of peak radiation power to receiver sensitivity. Usually PF increasing is achieved by additional signal processing and post-processing of acquiring data. Radiation power increasing is more attractive method since lead to linear rise of PF value. However it is difficult way because range of commercial available nanosecond and subnanosecond pulse generators are hardly limited especially for application in portable UWB radar. It should be simultaneously satisfied some conditions: high peak power and pulse repetition rate (PRR), compact size, time stability and long lifetime, high efficiency and reliability.
THEORY OF THE GENERATOR OPERATION It is known a lot of components that is successfully used for nanosecond pulse generation (Meixier, 1991; Litton et al, 1995; Agee, et al, 1998). However everyone has some disadvantages. For example step recovery diodes (SRD) are stable and reliable, form impulse with as small duration as 100 picoseconds but only some dozens volts in magnitude. Krytron or hydrogen thyratrons conversely generate extremely powerful
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impulses with low repetition frequency. There were no pulsers that could generate powerful impulses with high repetition frequency simultaneously. However at the beginning of 1980’s new type of semiconductor opening switches has been discovered by Grekhov et at (1983, 1984). This commutator so called Drift Step Recovery Diodes (DSRD) gave a rise to a new generation of all solid state nanosecond pulsers with peak power up to hundred megawatts (Brylevsky et al, 1996). The main advantages of these switchers are long lifetime, excellence time stability (low jitter) and small size. Besides they have no need restoration time and after pulse generation are ready for the next cycle. Generally speaking it is possible to generate power pulses with megahertz PRR (Kardo-Sysoev et al, 1997). Principle of the DSRD operation is similar to SRD one. However there is essential difference. Since drift diodes function on slow carrier pumping current should to be pulse but not continuous. The main idea of the DSRD operation can be explained as following. Short impulse of current applied in forward direction “pumping” p-n junction or, another words, “charges” p-n junction capacity. Then the current changes direction into a reverse and accumulated charges remove from base region. As soon as accumulated charge is equal zero the diode closes rapidly. Thanks to self-induction effect a high voltage appears impulse on the diode terminal. The bigger commutation current and shorter forward to reverse switching time the higher impulse magnitude and generator efficiency (KardoSysoev et al, 1997). In order to design nanosecond pulse generator based on DSRD structure charge model of the p-n junction has been developed and analyzed (Prokhorenko et al, 2000). A result of the diode modeling is shown Figure 1. It is good seen (Figure 1b) a time correlation between excitation voltage and voltage drop on the diode terminal. Taking into account delay effect of diode switching off in a frame of this model allows analyzing current driving circuit that is used for the DSRD pumping. It has been arranged nonlinear transient, differential equation based on charge behavior in the circuit and compute by using finite-difference time domain approach. As a result of the calculation note that special attention to coil inductance and diode parameters. The main conclusion was possibility to generate powerful nanosecond pulses by help the DSRD diode on both high and low impedance loading.
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SCHEME DESCRIPTION The nanosecond pulse generator was based on two principles: using a ferrite transformer to provide bipolar current for DSRD pumping as it was proposed by Belkin et al (1992) and monostable blocking-oscillator to increase scheme efficiency. Simplified scheme is shown in Figure 2. As a switcher we applied power MOSFET transistor BUK456-60H by Philips Semiconductor that is characterized comparatively fast turn-on time (90 ns) and such high peak current as 240A. Since impulse transformer provides reverse current there is no hard requirements to switch-off time. Transformer was made with soft ferrite and consisted of three windings on doubled cores K7x2x2. Turn number of first and second windings and feedback winding were 4, 12 and 1 ones, accordingly. Power supply voltage was changed from 15 to 50 volts however MOSFET driving voltage was limited in 25 volts level.
The generator was triggered by positive pulse from external oscillator. Minimum pulse duration was 50 ns whereas maximum time may exceed 500 ns. Minimum value is determined by internal delay into the scheme elements and maximum one does not exceed time interval between trigger pulse and output impulse. The nanosecond generator timing is shown in Figure 3. Note that time delay between trigger pulse and output impulse is depended on power supply voltage and varied from 500 ns to 800 ns. Besides the smaller time delay the higher time stability. During the testing was got impulse jitter less than 0.5 ns.
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Special attention was devoted to output circuit construction. It consists of four elements: impulse capacitor, loading, transformer output winding and naturally DSRD structure. Note that capacitor has to be suitable for impulse operation condition that is essentially limited types of components. We used metallized polypropylene film capacitor that has rated voltage pulse slope up to 1300 V/µs and 2200 Volts operation on DC current. As a DRSD was used 1N5408 high voltage impulse rectifier diode (reverse voltage forward current reverse recovery time ). Output circuit operation without DSRD (a) and when DSRD connect to loading terminal (b) are shown in Figure 4. Current reverse time was 80-150 ns when power supply voltage changed from 50 to 15 volts. Peak voltage of output impulse varied from 150 to 500 Volts with insignificant increasing of a rise time.
DISCUSSION During the series of the experiments have been got the following results. Minimum rise time was 1.6 nanosecond. As high maximum peak voltage as 550 Volts on the 50-Ohm loading has been achieved under 30 Volts power supply voltage. Power consumption was less than 6 Watts with 20 kHz PRR. Connection of two identical diodes in parallel improved impulse shape without visible rise time degradation. Maximum PRR volume was 25 kHz and it is determined by output pulse transformer overheating. Note that peak pulse power decreasing did not allow increasing of the PRR value. Evidently higher PRR using proposed schematic can be achieved by utilization of other transformer with improved electrical parameters. CONCLUSION We have described a high power subnanosecond pulse generator based on monostable blocking-oscillator and drift step-recovery diode sharper. Have been discussed principles of the DSRD operation and the scheme functioning. It is shown that the generator can be used to form nanosecond impulse with peak power up to 6kW and more. Portable design and low power consumption make it attractive for hand-held UWB radar application.
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REFERENCES Agee, F. J., Baum, C. E., Prather, W. D., Lehr, M. J., O’Loughlin, J. P., Burger, J. W., Schoenberg, J. S. H., Scholfield, D. W., Torres, R. J., Hull, J. P. and Gaudet, J. A., 1998, Ultra-wideband transmitter research, IEEE Trans. Plasma Science, 3:860. Belkin, V. S., Marin, ,O. Y. and Shulzchenko, G. I., 1992, Forming of high-voltage nanosecond pulses by serial diodes, Pribory i Tekhnika Experimenta, 6: 120 (in Russian). Brylevsky, V. I., Efanov, V. M., Kardo-Sysoev, A. F. and Tchashnikov, I. G., 1996, Power nanosecond semiconductor opening plasma switches, 22th International Power Modulator Symposium, Boca Raton, Florida, June 25-27: 51. Grekhov, I. V., Efanov, V. M., Kardo-Sysoev, A. F. and Shenderey, S.V., 1983, Formation of high nanosecond voltage drop across semiconductor diode, Sov. Tech. Phys. Lett., 4. Grekhov, I. V., Efanov, V. M., Kardo-Sysoev, A. F. and Shenderey, S. V., 1984, Power drift step recovery diodes (DSRD), Solid State Electronics, 6:597. Kardo-Sysoev, A. F., Zazulin, S. V., Efanov, E. M., Lelikov, Y. S. and Kriklenko, A. V., 1997, High repetition frequency power nanosecond pulse generation, Proc. of the 11th IEEE Int. Pulse Power Conf., 420. Litton, A. B., Erickson, A., Bond, P., Kardo-Susoyevn, A. and O’Meara, B., 1995, Low impedance nanosecond and sub-nanosecond risetime pulse generators for electrooptical switch applications, Proc. of the 10th IEEE Int. Conf. On Pulsed Power, 733. Meixier, L., 1991, Fast pulse techniques related to the X-ray laser project at PPPL, Proc. of the 14th IEEE/NPSS Symp. on Fusion Engineering, 1188. Prokhorenko, V. and Boryssenko, A., 2000, Drift step recovery diode transmitter for high power GPR design, Submitted to 2000 GPR International Conferemce, Queensland, Australia.
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ELECTROMAGNETIC NOISE EMISSION OF INDUSTRIAL PULSE POWER EQUIPMENT FOR MATERIAL TREATMENT
F. Luhn, R. Zange, G. Wollenberg, H.-P. Scheibe, W. Schätzing Institute for Fundamentals of Electrical Engineering and EMC Otto von Guericke University Magdeburg Universitätsplatz 2, 39106 Magdeburg, Germany
INTRODUCTION In the industrial field pulse power equipment is used for a wide range of applications, as there are electromagnetic workpiece deformation ('magnetic forming'), material disintegration by impulsive sound, or test purposes. In view of EMC those processes are characterized by a strong emission of transient electromagnetic noise, which may arise as a single event or as events with a very low repetition rate. Here the electromagnetic emission of industrial pulse power equipment will be discussed by means of magnetic forming.
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Conducting workpieces can be formed by pulsed magnetic fields of high energy. It is a special advantage compared to mechanical stamping that magnetic forming leaves the surface of the workpiece unharmed. Even complicated forms, as for instance undercuts, can be realized. In figure 1 two examples are sketched: the compression of a tube by an outer coil and the forming of a sheet by a flat coil. Using an inner coil, the expansion of tubes is also possible. Magnetic forming can be applied for the positive joining of two workpieces or, with the help of a die, for forming a workpiece. Figure 2 shows the equivalent circuit diagram of the equipment. For processing the workpiece is inserted into the coil arrangement, and the high voltage capacitor is discharged via the coil. The eddy currents in the workpiece cause Lorentz forces. The workpiece is deformed by the short-time but very strong repulsion. The resonant circuit formed by the capacitor and the coil leads to a damped middlefrequent oscillation of the process current (figure 3). Because of the high voltage and current levels the switch is carried out as a spark gap.
For workpiece deformation a pressure of some is required. Energy levels of 1kJ up to 100kJ are used. The charging voltage is in the order of 10kV. The process current reaches peak values of some 10kA. Inside the coil a magnetic flux density up to about 50T is achieved. The paper focusses on the unwanted noise emission to the environment. The deforming process will not be discussed in detail.
EMC ANALYSIS EMC problems may be caused by the high level of the middle frequent process current, by the spark gap and by the relay control. The examinations are carried out at the industrial equipment of the Institute for Machine Tools and Factory Management (Technical University of Berlin) and at the experimental equipment of the Institute for Fundamentals of Electrical Engineering and EMC (Otto-von-Guericke-University of Magdeburg). The results are qualitativly the same. Here the measurements gained at the Technical University of Berlin are presented.
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The components of the equipment are spacially separated (figure 4). The energy source includes the high voltage capacitors, the charging set, the spark gaps and the relay control. The discharging process is initiated by ignitrons, which are low pressure spark gaps with a mercury pool cathode. The energy source is connected to the coil by a transmission line with a length of 6m. To achieve a low inductance the line is realized by 30 paralleled coaxial cables. The presented measurement results are gained at an energy level of l,8kJ. The charging voltage is 4,5kV, the peak current reaches 40kA. A compressing coil is used. An aluminum tube is inserted as a workpiece. Because the equipment can operate only in single shot mode, the use of an EMI receiver is not possible. So the measurements are carried out in time domain with the help of a real time sampling oscilloscope. The frequency spectra are calculated using FFT.
MAGNETIC FIELD The magnetic field in the vicinity of the equipment is dominated by the stray field of the coil. The field strength is directly proportional to the processing current. Figure 5 shows a measuring result in time and frequency domain gained at a distance of 3m. Concerning the field orientation and the dependency on the measuring position the short cylinder coil behaves like an infinitesimal magnetic dipole (Balanis, 1982). In the near field area the magnetic field strength decreases almost with the third power of the distance.
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The magnetic field is weakened by the eddy currents in the workpiece. For comparison measurements without workpiece are carried out. Dependent on the design of the coil the field strength reaches peak values which are higher by factor 2...3. Further the losses in the workpiece deliver the main contribution to the damping of the oscillation.
ELECTRIC FIELD In the ignition moment of the ignitrons a transient electric field arises. The electric field is not exactly reproducible because of stochastic effects within the ignitrons. Here typical results are selected. The high noise floor is caused by a strong incidence of radio broadcasting. To demonstrate the time correlation figure 6 shows the antenna voltage during the process phase. A RF signal contribution is resolvable only in the ignition phase. A continuous RF generation, as it is typical for spark erosion machines (Wollenberg, 1999), cannot be observed. During the zero crossing of the current the discharge may become unstable. Here some of the measurement results show a second, much weaker disturbance. Figure 7 shows the spectrum of the electric field. There are strong levels up to the frequency limit of the measurement equipment. A significant radiation can be expected even at much higher frequencies.
Measurements with shorted coil connections show only very low field levels. So the coil proves to be the main source also for the electric field. Indeed, the field contribution of the cable bundle (figure 4) can be neglected. The measurement of the common mode current shows no high frequent components. The low-inductive line arrangement ensures a minimal voltage drop and therefore only very small sheath currents. When activating the equipment and when starting with charging, there is also a transient field caused by the relay control. The relay disturbances are by far not as broadband as the disturbances generated by the ignitrons.
DISTURBANCE VOLTAGE AT THE MAINS At the switching times of the relay control, as there are the activating of the equipment and the begin of charging, a mains disturbance voltage arises (figure 8). There is a strong
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effect of contact bounce. The duration and the level of the disturbances are not reproducible. The interference of the ignitrons is not transmitted to the mains connection.
PHENOMENA, PROTECTION MEASURES AND EMC-STANDARDS The disturbances caused by magnetic forming are of a very short duration. There is a very low or even none repetition rate. So the radio interference is scarcely relevant, but the equipment may interfere with its vicinity. In a distance of some cm from the coil the magnetic field proves to be strong enough to erase magnetic store media, as there are diskettes or hard disks. As experience shows, field levels above 1 A/m may cause flicker of video monitors. Near the coil this level is exceeded, but the duration is extremely short. In experiments no flicker is observed. Before taking any measures against the magnetic field, it should be tested, whether a simple increasing of the distance or a change of the coil orientation leads to a sufficient result. If necessary, the magnetic field can be shielded by a tube or a case around the coil arrangement. As the signal frequency is in the order of at least some kHz, an aluminum or copper shield basing on a field compensation by eddy currents shows a good efficiency (Paul, 1992). An expensive material with a high permeability (e.g. Mu-metal), as it is used at low frequencies, is not required. The shield must be large enough not to influence the process. The electrical field may disturb sensitive electronic equipment in the environment, for example fast logic circuits. As the example of the examined equipment shows, the radiation of the line between the energy source and the coil can be minimized by realizing a low inductive arrangement of paralleled coaxial cables, which are mounted on a metallic rail. The coil can be shielded by a metallic case. The cases of the energy source and of the coil should be connected low-inductively with the outer conductor of the cable. If the equipment is arranged compactly, it can be shielded as a whole. The disturbance voltage at the mains is in no way characteristic for magnetic forming. Such disturbances can be observed for all equipment with mechanical or electromechanical switches. A usual commercial mains filter can be used. The emission limits in the EMC-standards of the EU are mainly based on the protection of broadcasting. So in (EN550081-2, 1993) transient disturbances are not considered, if
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their repetition rate is lower than 5 per minute. The fact that there may be noise sinks, which can be disturbed even by a single event (e.g. fast logic), must be rated by the user. In (EN55014-1, 1997) limits for click disturbances are given, but only for conducted emissions. Additionally to the technical EMC the electromagnetic exposition of the operator must be taken into account. In Germany the concerning industrial safety regulations are still in a draft stadium (BGV B11, 1999). For broadband fields all spectral lines are squared and summed up rated with a frequency-dependent factor. So a weighted measure for the energy is gained. Three frequency ranges are separately evaluated. In the low frequency range there is a limit also for the linear sum. The linear sum proves to be independent of the repetition rate. For magnetic field this limit may be critical to meet. For the examined equipment and the choosen parameters the distance, where an infinite duration of exposition is allowed, is limited to at least 2.5m.
CONCLUSION During the discharge phase the process current yields a strong middle-frequent magnetic field in the vicinity of the coil. The ignition of the spark gap causes a transient broadband field. The disturbance voltage at the mains connection is predominantly generated by the relay control. Concerning the EMC-limits the exposition of the operator will be most critical to meet. The noise emission is controllable with a low expenditure.
REFERENCES Balanis, A., 1982, Antenna Theory. John Wiley & Sons, New York. Paul, C. R, 1992, Introduction to Electromagnetic Compatibility, Wiley, New York. Wollenberg, G., Luhn, F., 1999, RF-Noise Generation and Emission by Electrical Discharge Machines. International Symp. on EMC. Magdeburg 05.10. - 07.10.99, conference report pp. 95... 100. EN50081-2, 1993, EMC; general emission standard; part 2: industrial evironment. EN55014-1, 1997, EMC; requirements for household appliances, electric tools and similar apparatus, part 1: emission. BGV B11, 1999, Berufsgenossenschaftliche Vorschrift Elektromagnetische Felder, Fachausschußentwurf 12/1999.
COMPACT, SOLID-STATE PULSE MODULATORS FOR HIGH POWER MICROWAVE APPLICATIONS Dr. Marcel P.J. Gaudreau, Dr. Jeffrey Casey, J. Michael Mulvaney, Michael A. Kempkes Diversified Technologies, Inc., Bedford, MA 01730 USA
ABSTRACT Diversified Technologies, Inc. (DTI) has successfully developed and demonstrated a highly efficient and reliable new approach to solid state switching that can be used in a wide range of high power microwave systems. Switches using this approach are shown in Figure 1 and Figure 2.
Multiple switch modules can be combined in series and parallel to meet a wide range of power switching requirements. This paper describes how these switches have been used as replacements for vacuum switch tubes in a Navy radar modulator design, and incorporated into fully solid state modulators and power supplies for high power RF tube testing.
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BACKGROUND – HIGH POWER, SOLID-STATE SWITCHING The essential device in a pulsed power application is a pulse modulator, an electronic device used to provide high voltage, high current power bursts with great precision and accuracy. Ideally, a modulator acts as a simple switch between a high voltage power supply and its load, such as a klystron. The desired properties of such an ideal switch are infinite voltage holdoff, infinite off-resistance, zero on-resistance, and full immunity to transients and voltage reversals. One conventional approach to high voltage switching is to use a gridded vacuum switch tube, such as a triode or tetrode, as the series switch for a pulse modulator. This approach has three major drawbacks. First, the switch tube is current limiting which is detrimental to pulse rise-time. Such a switch tube forces a linear risetime slope due to the limited current available for charging of the cable and load capacitance. Second, there is a very large voltage drop across the tube, which may be 10-20% or more of the total switched voltage This means that the power supply must operate at a higher voltage than required. Additionally, this voltage drop, at high current, means a significant amount of power is being dissipated in the tube. Third, tubes, in general, arc, which reduces their lifetime and mandates complex conditioning, arc detection and crowbar protection systems. A second conventional option for switching uses a Pulse Forming Network (PFN) and also has several drawbacks. For example, the requirement for variable pulsewidth at high PRFs is problematic for a thyratron/PFN pulser. In addition, the thyratron typically used to drive a PFN has a finite lifetime, and must be replaced at regular intervals. At high levels of operation, this can be a noticeable cost factor. Also, a thyratron can only serve as a closing switch, and cannot open during a pulse in the event of an arc. The damage thresholds of the target often require additional hardware (opening switch tube or crowbar) to limit arc damage. Finally, the DC power supply required to drive a PFN must typically operate at about twice the voltage desired at the output unless a step up pulse transformer is used. Again, this increases the cost and complexity of the overall system. Nonetheless, vacuum tubes have provided a nearly exclusive solution to the problem of high-voltage switching because no cost-effective alternatives were available. As future systems require higher voltage and power, the use of switch tubes becomes increasingly impractical due to their inherent voltage and current limitations. Recently developed high voltage, high power, solid-state systems have demonstrated benefits such as the following: Efficiency >> 90% Low component cost Very high average and pulse power densities ( peak power) Voltage levels from 1 - 150kV Peak current levels from 0A to 5000A Pulse Repetition Frequencies (PRFs) >40 kHz and above (up to 400 kHz demonstrated) Rise and fall times <100 ns Variable pulse lengths (1µs to DC) High Reliability Achieving, as closely as possible, an ideal pulse is critical to the performance of a number of pulsed power applications. An "ideal pulse" has instantaneous rise and fall time, and a flat top, independent of load current and repetition rate. Very fast rise and fall times minimize the energy provided at voltages other than In a radar transmitter, the rise and fall times must be within the amplifiers' operating parameters, but the flat top is very critical to parameters such as phase noise. Generating pulses that most closely approach the ideal pulse waveform is, therefore, often a critical objective of high pulsed power system design.
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Solid State Modulator Principles In its simplest configuration, solid-state technology provides a fast, high current series switch, or circuit breaker. State of the art switches typically open and close in less than 0.5 µS.
When used as a Pulse Modulator, the opening and closing of the switch is controlled by a command signal at low voltage. The result is a stream of high power pulses into the load, each with rapid rise and fall times, and extremely consistent pulse-to-pulse characteristics.
Because solid state modulators do not use resonant circuits, each pulse can be arbitrarily sized. This allows complete pulse width and separation flexibility - from 1 µ to DC - on a pulse to pulse basis. Historically, solid state devices have been low voltage devices. Recent advances in Insulated Gate Bipolar Transistors (IGBTs) have improved the voltage and current handling characteristics considerably. Today’s typical devices have voltage ratings from 1200V-3300V and current ratings from 50A-1200A continuous. They also feature the very low drive current requirements of Field Effect Transistors (thus the Insulated Gate). This eliminates the need for cascaded stages of bipolar drives required by the low betas of early high current bipolar circuit designs.
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Systems having high voltage and/or high current requirements can be constructed by stacking these switch modules in various series and parallel configurations. Figure 3 shows the basic design of a solid state series modulator. This concept provides the flexibility of a modular design, with no inherent limit to voltage handling. However, it also necessitates the formidable task of ensuring that the load is shared equally between devices so that no single device sees harmful or destructive voltages. The gate drives must be highly synchronized to accomplish this. DTI has developed and patented the technology to achieve this synchronization, and has shipped commercial units with up to 160 IGBTs in series, and up to six IGBTs in parallel. Figure 4 shows the nearly ideal square voltage and current traces from a DTI solid state commercial modulator. The rise time of this pulse is less then 1 µS into a resistive load. The flat top of the pulse results from appropriate sizing of the storage capacitor, since the closed switch acts as a very low impedance connection directly between the capacitor and the load. The turn-off time of the switch is essentially equivalent to the turn-on time - the fall time shown is dominated by the discharge of load and cable capacitance through a pulldown resistor. Similarly, the slight overshoot is a function of inductance in the high voltage circuit rather than the switch performance. Finally, the switch has a voltage drop of less than 3V/kV (< 0.3%) when closed, and a leakage current <1mA when open. The switch is very nearly "ideal" for most high power applications. SOLID STATE AN/SPG-60 MODULATOR UPGRADE The AN/SPG-60 fire control radar, a key component in the GFCS MK-86 shipboard weapons control system, is in service on more than 30 US Navy ships. The Naval Surface Warfare Center, at Port Hueneme, CA contracted with DTI for the development of a replacement solid-state modulator for the radar. The existing vacuum switch tubes, monitoring and housekeeping circuits were a major contributor to the overall AN/SPG-60 failure rate. The goal was to decrease the failure rate of the transmitter by improving its
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overall reliability, reducing maintenance requirements, and improving power handling capabilities. DTI combined two of the solid state switches shown in Figure 1 into a modulator /transmitter assembly rated at 18 kV, 30 A, and 35 kHz. The upgrade included a new 15 kV switching cathode power supply, and improved filament supply, to reduce stress on the klystron. The components of the upgrade replaced a switch tube, pulse transformer, and linear power supply in the existing AN/SPG-60 transmitter. The upgrade also eliminated the need for traditional switch-tube housekeeping circuits, such as filament supplies. The entire modulator upgrade was designed to be field-retrofitted quickly into the existing enclosures (Figure 5). Results Preliminary test results are very promising. The solid state modulator has increased the calculated MTBF of the cabinet by a factor of more than twenty. The limiting component is now the designed service life of the klystron of about 10,000 hours, which means the overall MTBF of the system has improved by a factor of more than six. Further, the increased reliability is estimated to reduce maintenance labor overhead by 200 hours and spare parts costs by $280K annually. Significant improvements in system uptime are also expected. The less complex solid state design has resulted in a reduction of nearly three amps in the current requirement, the elimination of a 440/400Hz cable, and an 80% heat reduction in the cabinet. DTI’s design also allows most maintenance to be conducted on the inner door (see Figure 5), away from the high voltage components. HIGH POWER TUBE TESTING A second new application of high power, solid state switching technology is the testing of high power microwave tubes such as those used in accelerators.
Under a recently completed DOE Phase II contract, DTI developed and installed the world’s highest power, solid state, high voltage modulator at Communications and Power Industries (CPI) in Palo Alto, California (Figure 6). CPI is using the modulator for testing and conditioning high power klystrons, such as the prototype for the Spallation Neutron
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Source (SNS) program. The modulator is rated at 140 kV and 500 A peak, and designed to pulse at up to 10 kHz. This product uses commercial high voltage IGBTs with DTI’s patented gate drive protection and circuits. The 140 kV modulator shown in Figure 6 and in Figure 7 as the fast series switch, has been installed in the MSR test set at CPI. An improved power delivery system that utilizes a modulator similar to a DC-DC converter (buck regulator) has also been installed (Figure 7). In this configuration, the full power of the existing DC supply can be provided at lower
voltages and proportionally higher currents at high efficiency. The modulator at the heart of the buck regulator is identical to the 140 kV switch, with the further addition of more switch units to provide extra voltage handling capability. Pulse Width Modulation (PWM) at 4kHz provides excellent control regulation of the operating voltage. The inductor and capacitor at the output of the buck regulator provide filtering of the DC voltage. In operation, loads are protected from arcs by sensing of modulator output overcurrent and termination of the pulse. After overcurrent is sensed, the switches open in approximately 700 nsec. Prior to klystron testing, DTI successfully showed the modulator was capable of passing the “wire test” where a full voltage pulse into a 40 gauge copper wire is interrupted without damage to the wire/ Figure 8 shows this current protection in action. The tube is pulsed at 110 kV, and a tube arc occurs at 10 µs into the pulse. The overcurrent condition is sensed and voltage is removed, extinguishing the arc. Tests at CPI show that the tube is protected with less arc-
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deposited energy than CPI’s crowbar protection, at significantly less stress on key power supply components. Since the modulator simply opens in the event of an arc, pulsing can resume immediately after the arc clears. Results The buck regulator efficiency is dominated by the resistive losses, primarily in the inductor. At DTI, moderate current tests (~ 10A CW) have yielded efficiencies over 99%. The 160kV buck regulator has a more lossy (solid copper) inductor, and recently operated at 92-93% efficiency in CW 1MW operation. CONCLUSIONS Solid state pulsed power is both feasible and practical at high voltage (over 150kV) and high power (tens of MW pulsed). Solid state modulators provide nearly ideal high power pulses, with sub-microsecond rise and fall times, high efficiency and very low droop Solid state modulators provide a range of flexibility in voltage, pulsewidth and PRF for radar, and high power tube testing and conditioning. As a fast opening switch, solid state modulators eliminate the need for crowbar systems, and provide much higher levels of protection than crowbar systems. A solid state modulator, with minimal modifications, can provide a very high efficiency DC-DC transformer, providing full power over a wide range of output voltages.
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AN INTRODUCTION TO POLARISATION EFFECTS IN WAVE SCATTERING AND THEIR APPLICATIONS IN TARGET CLASSIFICATION
Wolfgang M Boerner1, Shane R Cloude2 1
University of Illinois at Chicago EECS Department,900 W Taylor St., Chicago, IL 60607-7018,USA 2 Applied Electromagnetics 83 Market St., St Andrews Fife, Scotland, KY16 9NX, UK
INTRODUCTION In this paper we consider the subject of scattering polarimetry. While the classical Jones calculus is now widely employed for the analysis of polarization transformation due to propagation through optical devices [Collet 1992], there is a relatively poor analytical appreciation of how to deal with the structure of depolarization effects caused by wave scattering. This despite a long history of depolarization measurements in astronomy [Dollfus 1992], remote sensing [Kouzoubov 1998], atmospheric optics [Mishchenko 2000] and surface roughness studies [Egan 1985]. This paper seeks to review progress in this field, to summarize the main aspects of our current analytical understanding of depolarization and to review important applications of these techniques in radar and optical remote sensing of targets. A key distinguishing feature of scattering polarimetry is the presence of depolarization, to be distinguished from cross-polarization. While the latter is a deterministic effect, which has been used for example in radar topographic mapping [Schuler 1996] and through its absence for buried mine detection [Carin 1998], the former is fundamentally a stochastic process whereby incident radiation is made noise-like due to the randomness of a scattering volume [Brosseaau 1998]. The classical methods for describing such phenomena are based on the degree of polarization, defined from the Stokes vector [Collet 1993]. However, there has been a recent trend towards using active sensors (particularly polarized Radar [Boerner 1992] and Lidar [Mishchenko 2000]) for remote sensing purposes. These systems can fully populate the scattering matrices of polarimetry and this has lead to a more sophisticated understanding of the generalised relationship between depolarization and scattering. This in turn has lead to improved methods for remote sensing using polarized waves [Boerner 1998]
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DEFINITION OF SCATTERING MATRICES IN POLARIMETRY The general vector scattering problem is defined in figure 1. Here we show plane co-ordinates for incident and scattered fields. These define the planes of polarization and the scattering plane ASB with the bisectrix defined by the scattering angle where is backscatter (radar/lidar) and is forward scatter.
Note that for backscatter, when A and B coincide, there is a mismatch between incident and scattered wave co-ordinates. It is normal practice in Radar to compensate for this mismatch (and also for the change in sense of elliptical polarisations due to time reversal) in the backscattering alignment (BSA) convention. The relation between the wave co-ordinates and BSA or sensor co-ordinates is summarised in figure 1. Note that for backscatter, the reciprocity theorem leads under the BSA convention to a symmetric scattering matrix. The next important step in scattering polarimetry is to transform the S matrix into a complex vector [Cloude 1996, Krogager 1992]. Figure 2 summarises one way in which this may be achieved. An expansion of [S] in terms of the Pauli spin matrices facilitates the interpretation of rotation effects in the scattering matrix. This vector formulation then leads naturally to a definition of orthogonal scattering mechanisms, as shown in figure 2. This is an important extension of the concept of wave state orthogonality in conventional polarimetry. We note that for backscatter the scattering vector can be decomposed into four parameters the scattering mechanism which is 0 for a sphere and for nonsymmetric objects such as a helix. In this way can be considered a symmetry index the orientation of any object symmetry axis about the line of sight. Linear Phase (differential propagation or scattering phase between linear polarisations) Helicity Phase (differential propagation or scattering phase between circular polarisations) Figure 3 summarises the transition from coherent problems, described by a matrix [S], to depolarising problems, based on second moments of the scattered field. When collected, these moments form a 4 x 4 Hermitian covariance matrix [C]. Figure 4 summarises the relationship between this matrix [C] and the coherency matrix [T] and Mueller matrix [M]. The matrices [T] and [M] are the two most important in polarimetry. The latter is used in optical scattering because of its relation to the Stokes vector of the scattered light [Lu 1996] whereas the former is widely used in radar polarimetry where it is employed for target classification and parameter estimation [Cloude 1996, Lee 1999].
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A key advantage of the [T] matrix formulation is its eigenvalue decomposition which can be used to quantify the depolarization caused by an object. Equation 1 summarises this decomposition of [T] into 4 eigenvalues and corresponding eigenvectors.
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Note that the normalised eigenvalues can be interpreted as probabilities in a Bernoulli statistical model of the scattering object [Cloude 1997]. There are several important secondary parameters derived from these eigenvalues and eigenvectors. Figure 5 summarises four, the entropy H, the scattering anisotropy A , the mean scattering mechanism and the mean orientation of the objects symmetry axis The parameters are especially important and can be mapped into a plane as shown in figure 6. Here spherical symmetry is mapped into the origin and all depolarizing polarimetric systems can be mapped as a point inside the feasible region of this plane.
RADAR POLARIMETRY AND MINE DETECTION We now present some results illustrating the application of the theory to radar detection of mines. Figure 7 shows example backscatter data collected at the EMSL facility at JRC Ispra. The microwave SAR image on the left was collected for a set of objects laid on the surface as shown on the right.
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The lower portion shows on the left an alpha parameter image for the scene. We can see that the mine targets are well discriminated from non-mines on the basis of their mean scattering mechanisms. OPTICAL AND IR POLARIMETRY FOR SURFACE SCATTERING These techniques have also been applied to active IR and optical systems [Harris 1999, Lewis 1998]. Figure 8 shows an example of a laser based surface backscattering experiment carried out by DERA Malvern and designed to measure the full Mueller matrix [Lewis 1998]. In the lower part of the figure we show some results of how the scattering anisotropy A varies with angle of incidence onto the surface, showing that it depends only on the surface roughness. Such ideas have been developed for the remote sensing of surface roughness and dielectric constant using backscatter measurements and could be used for the classification of different surface textures in target discrimination. UWB RADAR POLARIMETRY An important sensor for mine detection is ultra wide band radar (i.e. a radar with a very large fractional transmitted bandwidth). Figure 9 shows the wide band FDTD prediction of broad band alpha parameter variations for a low metal AT mine. The top figure shows the response at 10 degrees angle of incidence, showing only a weak departure from spherical symmetry. However, at larger angles, the response departs considerably from spherical symmetry and this can be exploited for stand-off detection of surface and buried mine targets. On the left of figure 9 we show spectra of the scattered field in the two Pauli matrix channels and of figure 1. The first is the spherical symmetry channel and dominates the response at low frequencies. However, as the frequency increases we see that the nonsymmetrical-channel grows and can even exceed the spherical channel at certain
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frequencies. These ‘symmetry resonances’ can be seen in the corresponding alpha parameter. This gives a clear physical application of the alpha parameter in the interpretation of target scattering signatures.
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REFERENCES Baum C, H.N. Kritikos (eds.), 1995 Electromagnetic Symmetry, Taylor and Francis, Washington, USA, Boerner W M et al (eds.), 1998,“Polarimetry in Remote Sensing: Basic and Applied Concepts”, Chapter 5 in Manual of Remote Sensing, Vol. 8, 3rd edition, F M Henderson, A J Lewis eds. New York, Wiley Boerner W M et al (eds.), 1992 “Direct and Inverse Methods in Radar Polarimetry”, Volumes I and II, Proceedings of NATO-ARW, Kluwer Academic Publishers, Netherlands Brosseau C, 1998,“Fundamentals of Polarized Light: a statistical approach”, J Wiley, ISBN 0471143022 Carin L, et al, 1998, “Polarimetric SAR Imaging of Buried Landmines”,IEEE Trans. On Geoscience and Remote Sensing”, Vol. GE-36, pp. 1985-1988 Cloude S R, E. Pottier, 1995, "The Concept of Polarisation Entropy in Optical Scattering", Optical Engineering, Volume 34, No. 6, ISSN 0091-3286, pp. 1599 – 1610 Cloude S R, E. Pottier, 1996, "A Review of Target Decomposition Theorems in Radar Polarimetry", IEEE Trans.on Geoscience and Remote Sensing, Vol.34 No. 2, pp. 498-518 Cloude S R, E. Pottier, 1997, "An Entropy Based Classification Scheme for Land Applications of Polarimetric SAR", IEEE Transactions on Geoscience and Remote Sensing, Vol. 35, No. 1, pp 68-78 , January Collet E, 1993, "Polarized Light: Fundamentals and Applications", Marcel Dekker, NY Egan W G, 1985, “Photometry and Polarization in Remote Sensing”, Elsevier, NY Harris M, D J Wilson, J Mallott, M Handley, “Polarimetry of Scattered Light Using Heterodyne Detection”, Journal of Modern Optics, Vol. 46, pp 721-728, 1999 Dollfus A, 1992, “Planetary Investigation by Polarimeter”, Adv. Space Res., Vol. 12, No. 11, pp 167-175 Hovenier J W, C.V.M. van der Mee, 1996, “Testing Scattering Matrices: a compendium of Recipes”, Journal of Quant. Spec. and Radiative Transfer, Vol. 55, pp 649-661 Kouzoubov M J, Brennan, J.C. Thomas, 1998, “Treatment of Polarization in Laser Remote Sensing of Ocean Water”, Applied Optics, Vol.37, No 18, June, pp. 3873-3885 Krogager E., 1992, “Decomposition of the Sinclair Matrix into Fundamental Components with Applications to High Resolution Radar Imaging”, in W.M Boerner et al (eds.),Direct and Inverse Methods in radar Polarimetry, Vol.2 , pp. 1459-1478, Kluwer Academic Publishers, Dordrecht, NL Lee J S, M R Grunes, T L Ainsworth, L J Du, D L Schuler, S R Cloude, 1999, “Unsupervised Classification using Polarimetric Decomposition and the Complex Wishart Distribution”, IEEE Transactions Geoscience and Remote Sensing, Vol. 37/1, No. 5, p 2249-2259, September Lewis G D, D L Jordan, E Jakeman “Backscatter Linear and Circular Polarisation Analysis of Roughened Aluminium”, Applied Optics, 37(25), pp 5985-5992, 1998 Lu S Y, R.A. Chipman, 1996, “Interpretation of Mueller matrices based on polar decomposition”, JOSA A Vol 13, No 5, May, pp. 1106-1113 Mishchenko M I, J.W. Hovenier, L.D. Travis (eds.), 2000, “Light Scattering by Nonspherical Particles”, Academic Press, NY Nghiem S V, S.H. Yueh, R. Kwok, F.K. Li, 1992, "Symmetry Properties in Polarimetric Remote Sensing", Radio Science, Vol. 27. No. 5, pp. 693-711 October Schuler D L et al, 1996, "Measurement of Topography using Polarimetric SAR Images", IEEE Trans on Geoscience and Remote Sensing, Vol.GE-34(5), pp. 1266-1277
UNIPOLARIZED CURRENTS FOR ANTENNA POLARIZATION CONTROL Carl E. Baum Air Force Research Laboratory AFRL/DEHE 3550 Aberdeen Ave., SE Kirtland AFB, NM 87117-5776 1.
INTRODUCTION
For certain types of radars, specifically polarimetric SAR (synthetic aperture radar), control of the polarization of the incident wave on the target and the polarization of the receive antennas is important. This relates to certain signatures in the target scattering which can be used for target identification [13-16]. Symmetry is an important concept in controlling antenna patterns, including polarization [6, 18]. This can be thought of as complementary to the symmetry in the target which strongly influences the scattering pattern, including polarization [8, 9, 11-15, 17]. The antenna symmetry can be used for polarization control, but with limitations depending on the relative orientations of the antenna and target (assumed in the antenna’s far field). In the present paper we consider another technique for antenna polarization control: the restriction of the antenna currents to flow only parallel/antiparallel to a given frequency/time-independent direction designated This gives a particular real frequency/time-independent polarization at each point in the far field. However, this polarization in general differs from point to point. Taking two choices for (conveniently orthogonal) for two such antennas gives two independent polarizations at each point in the far field which can be used to mathematically construct the usual h,v radar polarizations scattering dyadic to the transmit/receive properties of the two antennas. Furthermore, by use of two orthogonal symmetry planes for the two antennas (with currents on thin wires) the two antennas can be made mutually noninteracting so as not to disturb each other’s pattern/polarization. Making the antennas electrically small further simplifies the analysis by making the pattern simply that of an electric dipole. 2. as the
ANTENNA WITH UNIPOLARIZED CURRENTS Consider, as in Fig. 2.1, a coordinate system for an antenna with a preferred axis axis (subscript for these coordinates). These are Cartesian and spherical coordinates related as
which is taken cylindrical
The associated unit (direction) vectors are
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The far field radiated by a current distribution limited to a region of space with finite linear dimensions (say within a sphere of radius b with and centered on is [10]
Here the integration is over the primed coordinates over the domain function
of the antenna. Note the weighting
giving phase from the various positions on the antenna (a complicating factor). The basic idea here is to constrain all the antenna currents to be parallel (including antiparallel) to specifically,
Then noting that
the far electric field takes the form
UNIPOLARIZED CURRENTS FOR ANTENNA POLARIZATION CONTROL
and, voila, the far-field polarization is given by direction in space, perpendicular to and the observer. However, as and
503
which is independent of time and frequency. It is a real
(direction from the antenna), and in a plane containing the axis are varied there is still the complicated variation associated with
As illustrated in Fig. 2.2A one way to realize this condition is to constrain all (net) currents to thin wires, all of which are parallel to the designated
axis, and hence to
and which is one of the planes perpendicular to
Note also that we have a reference plane (say
). Now there can be many thin
wires (say N), all perpendicular to with various currents which can vary along the length of the wires. With various lengths and various coordinates on (i.e., ) this configuration is quite general, constrained only by (2.4) and the limited size of the source (antenna) region. Going a step further, now let be a symmetry plane as illustrated in Fig. 2.2B. In particular let the currents be antisymmetric [7, 18], i.e.,
where is taken as the plane, and positive current is taken in the (or + ) direction. The wires, of course, have equal extent in the and directions but the various wires need not have the same lengths. This is a kind of array which can be driven (with antisymmetric sources, such as illustrated), or parasitic (undriven). These wires can be impedance loaded as well as long as the symmetry with respect to is maintained. The pattern of this antenna is then also antisymmetric with respect to Note that the symmetry plane can now be in part a conducting sheet and no net surface current density (sum from (above) and (below)) will flow on it Furthermore, letting the conducting sheet have some small thickness this can be used to route (hide) various transmission lines connected to the antenna elements and source(s) at the antena terminal(s).
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For the special (but important) case that the antenna is electrically small the antenna is characterized in transmission by its electric dipole moment
which in (2.3) gives
This, of course, assumes that so that we need not go to other moments of the current distribution. Specializing to the case of unipolarized currents gives
This is a very simple pattern function in addition to the simple polarization. (Note that a unipolarized electric dipole moment does not in general require unipolarized currents.) Assuming that the antenna has a single port for transmission/reception we can characterize its performance in various ways, including voltage, current, and wave variables, including reciprocity between transmission and reception [3, 5]. The various forms of these transmission and reception parameters are relatable to each other. For convenience, let us use the wave variables for which we have in transmission
In reception we have
Applying reciprocity we have
which for the simple but useful case of
gives the result
The signal into the antenna results in the currents in (2.3) from which we can identify
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Specializing to unipolarized currents as in (2.4) gives
showing the
constant polarization. Writing this in the form
the assumption of antisymetric currents as in (2.7) gives an antisymmetric radiation pattern as
Note similarly in reception that (2.18) implies
and the antisymmetric currents further imply
For an electrically small antenna in transmission (2.9) gives
For currents polarized in the
direction (2.10) applies and gives
with the same result as for antisymmetric currents. For convenience we can define
as a characteristic of the antenna (an electric-dipole transfer function). Then we have
so that the angular dependence and frequency dependence are completely separated as distinct factors. Note that in general (and hence ) can be different for different antennas. In such a case a superscript a (later taken as 1, 2,...) can be used. For convenience these functions are assumed to be the same for the various antennas, as expressed in appropriate antenna-based coordinates. 3.
TWO COLOCATED ORTHOGONAL ANTENNAS, EACH WITH UNIPOLARIZED CURRENTS
For measuring the target scattering dyadic two polarizations are required. This in turn implies at least two antennas for transmission and two for reception (which may be the transmitter antennas as well). Retaining the unipolarized currents in each antenna it is important that the two antennas do not significantly interact. Note that the two antennas need to have different polarizations for the scattering-dyadic measurement. One approach to this design is as illustrated in Fig. 3.1. Here we take two antennas such as discussed in Section 2. Denoting these as 1 and 2, they are assumed to have symmetry planes and (mutually perpendicular). The polarization direction with
for the currents now takes on two values
and
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Furthermore, let the 1 antenna elements lie on so that the 2 currents do not excite currents on the 1 elements (by symmetry), and conversely. This does not imply that the two antennas are identical except for a rotation by
about the axis designated by
as the line of intersection of
and
with
This axis can be used to define the location of a (thin) conducting tube, inside of which transmission lines can be placed to feed both sets of antenna elements from two ports also located on or near this axis. Various designs are still possible for the two antennas, including a single element pair (thin-wire antenna) or an array of elements such as a log-periodic antenna. As mentioned previously, symmetrically positioned pairs of identical impedances can also be included in the elements. Of course, a convenient choice for the two antennas is to have them identical so as to have the same radiation and reception characteristics except for a coordinate rotation ( now becoming two sets of coordinates, and ). If desired, one antenna can be shifted (translated) along the axis with respect to the other, but this introduces a phase shift (dependent on angles to the target) between the two antennas. 4.
ANTENNA POSITIONS AND ORIENTATIONS WITH RESPECT TO TARGET SITE
The antennas are now positioned and oriented as illustrated in Fig. 4.1. First, establish site coordinates as in Fig. 4.1A. These are the standard Cartesian, cylindrical, and spherical coordinates as in (2.1) and (2.2), except with no subscript The coordinate origin,
is taken at a height h above the
ground surface which is assumed flat. The z axis is taken as perpendicular to The x axis is assumed extended over the target site, an orientation appropriate for a side-looking SAR with antenna motion in the y direction. Note that horizontal polarization on the target site is given by
and vertical polarization is given by
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These vary over the target site, but they are appropriate to characterizing the target scattering, including the lack of a crosspol component for targets with symmetry, including the ground presence (the vampire signature) [13-16]. In general, these polarizations are not the same as those radiated by the antennas to the target location. For later use we have the transverse dyadic with respect to (direction to the target) as
Now take antenna 1 with in the y direction so that the symmetry plane perpendicular to the antenna elements, is the y = 0 plane and is perpendicular to This symmetry then includes the ground as well. Antenna 1 has horizontally oriented elements, but this is not the same as horizontal polarization on the target site, except on As indicated in Fig. 4. 1B the antenna may be canted (rotated) downward toward the target site by an angle with respect to the x axis. This gives the orientation of the other symmetry plane of the antenna. Antenna 2 now has
in the y = 0 plane and oriented at an angle of
from the vertical axis.
Regarding as the nominal forward direction from both antennas to the target site, this is also in the y plane and depressed by the angle from the x axis. This antenna has the same symmetry planes and as antenna 1. While we might think of antenna 2 as approximately vertical, this does not in general give vertical polarization
on the target site, except on
C. E. BAUM
508 For the special case that polarization since
we have the interesting result that antenna 2 produces pure vertical
all over the target site. However, antenna 2 still does not give pure horizontal
polarization, except in the limit of small h so that the incident fields are nearly in the z= 0 plane and Antennas 1 and 2 can be designed for both transmit and receive. If one desires greater isolation between transmission and reception, one can supplement this antenna pair by a second pair: antennas 3 and 4 as indicated in Fig. 4.2. In this case, if we locate the centers of both pairs on the y = 0 plane then we can have as a common symmetry plane for both pairs. is then different from but can be made parallel to it if desired. Furthermore, neither nor is a symmetry plane for the entire array, but a plane centered between and can be. In order to maintain the lack of coupling between the various antennas it is now necessary to make the separation D>>b. Not only does this reduce the coupling from one antenna pair (say 1 and 2 in transmission) to the second pair (say 3 and 4 in reception). It also reduces the scattering from one pair off the second pair back to the first. This reduces the effect of symmetry breaking ( distinct from ) in allowing say antenna 2 to couple to the conductors (thin wires) in antennas 1 and 3, and thereby distort the polarization purity implied by (2.4). Note that the common symmetry plane implies that antennas 1 and 3 can mutually couple, as can 2 and 4. However, this symmetry also means that in terms of signals at the antenna ports neither can antenna 1 mutually couple with antenna 4, nor 2 with 3. One may also wish to constrain D<< r so that the various angles of both antenna pairs with respect to a target are nearly the same, simplifying the analysis, and thereby the data reduction. 5.
FIELDS OF ANTENNA 1 IN SITE COORDINATES
Having the fields of antenna 1 described in coordinates in Section 2, these coordinates can now be related to the site coordinates discussed in Section 4. In Cartesian form, this relation is indicated in Fig. 5.1 and given by
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The electric field incident on the target is polarized in the direction which we can think of as quasi-horizontal (with a minus sign). With horizontal polarization as in (4.1) then we can form
showing the rotation of by This gives the portion of the field with horizontal polarization. Simiarly with vertical polarization as in (4.2) we can form
This gives the portion of the field with vertical polarization. For computing
and
for a given
location, on the target site, one can use (5.1) to compute These in turn via (2.1) can be used to compute the various angles and/or the appropriate trigonometric functions of these angles which are in turn substituted into (5.2) and (5.3). Note that on the symmetry plane we have over the target site
The fields radiated to the target are
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For present purposes we assume that the two antennas are identical except for the rotation previously discussed. The above incident field can now be decomposed into h and v components as
For an electrically small antenna this further reduces as
In this last form we see one advantage of electrically small antennas in the factorization of the dependences on frequency and angles, simplifying the analysis of experimental data. 6.
FIELDS OF ANTENNA 2 IN SITE COORDINATES
Having the fields of antenna 2 described in coordinates, these also need to be related to site coordinates. In Cartesian form this relation is indicated in Fig. 6.1 and given by
The electric field incident on the target is polarized in the direction which we can think of as quasi-vertical (with a minus sign). With vertical polarization as in (4.2) then we can form
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This gives the partial ofthe field with horizontal polarization. With the vertical polarization as in (4.2) we can form
This gives the portion of the field with vertical polarization. As in the previous section one can now compute the various angles from the two sets of Cartesian coordinates for the target. On the symmetry plane we have over the target site
Also we have the special case for
which gives everywhere over the target site
which is perfect vertical polarization. The fields radiated to the targets are just like in the previous section ((5.5) – (5.7)) except with the index 1 replaced by 2. 7.
FIELDS SCATTERED FROM THE TARGET TO THE ANTENNAS AND INFERENCE OF THE BACKSCATTERING DYADIC The target in turn scatters the incident fields back to the two antennas. These fields are naturally
represented in the usual h,v radar coordinates. The far scattered electric field back at the antennas is just [11, 13]
(reciprocity)
backscattering dyadic
For present purposes we can regard this dyadic as 2 × 2 relating to the transverse fields in the h,v coordinates. Note the superscript “a” since the above relation applies to fields incident from both the 1 and 2 antennas. This also applies to the combination of fields from both antennas. At the antennas the reception is described by [3]
With the case of a resistive antenna and load impedance matched as in (2.14) this reduces to
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Again the 1 and 2 antennas are assumed identical except for a rotation. Here the far scattered field is a linear combination of the fields scattered from the incident fields from both antennas 1 and 2. The previous two sections have considered the properties of for the two antennas, as well as its projection onto h,v components. What we need is a 2 x 2 matrix equation relating transmission and reception of the scattering in the 1,2 channels to obtain some effective scattering matrix which can be related to the scattering dyadic in h,v coordinates. Collecting the various terms we have
As one can readily see this matrix is symmetric as is required by reciprocity. The problem is now to calculate from which we can obtain by measurement. We have four matrix components, presumably from measurements from which we can infer the four components of the backscattering dyadic in h,v coordinates. Of course, by reciprocity only one of the off-diagonal components needs to be computed. We can expand the backscattering dyadic as
This can be used to write the matrix elements in the form
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In a shorthand form we can define
giving four factors based on the choices of
All four of these factors are available from measurement (or calculation) of the antenna transmission function as in (5.5), together with its decomposition into h,v coordinates as discussed in Sections 5 and 6. With the four choices of the n, m combination we can form the matrix equation
Inverting this 4 × 4 matrix with the
known from measurement we have the four components of
in h,v coordinates. After solving (7.9) we should have equality of the off-diagonal components of the backscattering dyadic. Of course, this is only approximate in real measurements due to noise. At this stage one might average the off-diagonal components for a hopefully more accurate estimate. An alternate procedure is to impose this (7.5) from the start as well as
which we can impose by averaging the measurements of these two latter parameters, or by measuring only one of them and using this measurement for both. We can then set up a matrix equation in the form
which reduces the problem to the inversion of a 3 x 3 matrix. With the antenna polarizations controlled to be in the Sections 5 and 6
directions we have as discussed in
Here the frequency dependence is separated in a scalar factor giving frequency-inependent polarizations. Writing out the four terms we have
where the angular functions
are tabulated in Sections 5 and 6. Substituting these in (7.9) and (7.11) we
can see that the scalar pattern functions still appear in the matrix in a mixed way such that they cannot be factored out of the matrix which is then still frequency dependent in a complicated way.
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However, we do have the advantage of effectively having scalar antenna patterns times an easily calculable frequency-independent polarization. Going further, let us further simplify the problem by assuming that the antennas are electrically small. Recalling (2.25) we then have
Noting that the two antennas are identical except for the angular rotation, we can now rewrite (7.4) in the form
In this case we know these matrix elements from measurement and the and are purely geometrical and need not be recomputed for each frequency, but only for different target locations in the data processing. The matrix equation corresponding to (7.9) can now be written as
with the sine functions on the left side for simplicity. The matrix to be inverted now is a function of only the angles, and so need not be inverted for each frequency and can apply to a complete temporal waveform. As previously in (7.11) this can be expressed using a 3 × 3 matrix using reciprocity. Adapting the form in (7.16) we have
If desired, a similarly reduced form of (7.16) is just as easily constructed. An important point to note about the electrically-small case is that the electric-dipole transfer function is removed from the matrix as a common factor. Errors in do not appear in the reconstruction of the h,v polarizations via the matrix inversion. The matrix elements are only functions of geometrical parameters (angles and range r) which one can determine accurately. 8.
SPECIAL CASE OF If we choose the special case of
Then from (6.5) we have the important result
this makes antenna 2 be oriented vertically, i.e.,
UNIPOLARIZED CURRENTS FOR ANTENNA POLARIZATION CONTROL What this says is that antenna 2 gives pure vertical polarization and that
515 can be directly found by
transmission and reception from antenna 2 alone. As can be seen in (5.2) and (5.3), such a simple result does not similarly apply to antenna 1. With unipolarized currents so that the antenna 2 polarization is in the
This can be substituted in (7.11) to give
This is readily solved as
which can be expressed in matrix form as
thereby giving the explicit matrix inverse. For electrically small antennas (7.18) becomes
This is also solved as
which can be expressed in matrix form as
direction we have
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again giving the explicit matrix inverse. So this special case of simplifies the algebra. Antenna 2 has the properties of an antenna (including the ground) giving a pure vertical polarization over the target site. Antenna 1 does not have this special property. So a partial separation of the h,v polarizations is achieved. 9.
ELECTRICALLY SMALL ANTENNAS WITH UNIPOLARIZED CURRENTS
A common technique used to make wire antennas resonate at frequencies for which the antenna is electrically small is the addition of inductive loading. However, inductive coils do not have currents all running in one direction. A typical coil is a helix and has a magnetic-dipole moment given by
As illustrated in Fig. 9.1, such a coil can have
parallel or antiparallel to
depending on the sense of
winding pitch. (Other orientations are also possible.) The presence of such a moment is undesirable because of how it distorts the polarization of the far field, a polarization which we would like to be governed by the electric-dipole moment One technique for canceling the magnetic-dipole moment is to make the two oppose each other as illustrated in Fig. 9.1 by making their winding pitches have opposite sense. This restores the plane as a symmetry plane, about which the currents and fields are antisymmetric. However, these coils can present other problems due to their mutual interaction. In arrays of such thin-wire elements as in Fig. 2.2B and Fig. 3.1 there may be many such coils. Consider, for example, a log-periodic antenna in which now, potentially, one element can couple to another via these coils, a transformer effect. For various possible reasons, one may wish to avoid this additional coupling. (Conceivably, one may wish in some instances to utilize this coupling.) Another approach to removing the effects of such magnetic dipole moments is to design the coil(s) to have negligible such moments. This is accomplished by designing a coil such that the magnetic field produced remains internal to the coil structure. The general theory is developed in [2]. A simpler approximate form is given in [4]. In this simpler bisolenoidal (or multisolenoidal) form, for each loop turn its magnetic-dipole moment is cancelled by another loop turn with opposite sense, but displaced as part of a second parallel solenoid so that the magnetic fluxes do not cancel. This is an approximate solution in the sense that higher order magnetic moments (quadrupole, etc.) remain. As illustrated in Fig. 9.2, one can think of this as a figure–8 winding on two parallel cylindrical (circular or otherwise) dielectric forms for supporting the windings. (The forms may be removed for an aircore coil if desired.) With N turns in each solenoid of length with cross-section area A the inductance is
(valid for sufficiently large ) Note that the magnetic flux density reverses direction between the parallel solenoids, and the magnetic flux leaving the end of one solenoid enters the end of the adjacent solenoid (approximately). Besides viewing such inductors as lumped elements, one can make a distributed bisolenoidal inductor by letting be large (even the entire length of the antenna element). In this case we have an inductance per unit length
In this case the antenna element becomes a slow-wave structure. The inductance per unit length can even be variable as based on a variable turns density and/or a variable cross-section area
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One can add as an additional series impedance per unit length in a transmission-line model of a wire antenna [1], and look for useful forms of such loading. This can also be combined with resistanceper-unit length for damping the antenna response for desirable transient response [1]. This opens various possibilities. One need not be limited to a single loaded-wire element, but can have arrays of same, say in logperiodic form. By this technique one can in principle have electrically small arrays with multiple-wavelength waves on the slow-wave structures, if desired. The present considerations are but an introduction to the various design possibilities. 10.
CONCLUDING REMARKS
This basic idea of unipolarized antenna currents has led to various implications for antenna design for frequency-independent far-field polarization. Combination of two such antennas with orthogonal antenna currents has led to the requirement of two orthogonal symmetry planes applying to both of the antennas. Within these constraints there is still much flexibility in designing arrays of parallel thin wires. Furthermore, such wires can be symmetrically impedance loaded for various reasons, including a desire to make them electrically small so as to simplify the analysis in terms of a unipolarized electric-dipole moment. It will be interesting to see where these ideas may lead. While the present analysis is in terms of unipolarized electric currents, what about the electromagnetic dual, i.e., unipolarized magnetic currents? The present analysis still applies with the interchange of electric and magnetic fields. The question is then how to synthesize such magnetic currents from loops to give magnetic moments without significant electric moments. This should be achievable using pairs of coils whose magnetic dipole moments add, but electricdipole moments subtract This work was supported in part by the U.S. Air Force Office of Scientific Research and in part by the U. S. Air Force Research Laboratory, Directed Energy Directorate.
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REFERENCES
1. 2. 3. 4. 5.
6. 7. 8. 9. 10.
11. 12. 13. 14. 15. 16. 17. 18.
C. E. Baum, Resistively Loaded Radiating Dipole Based on a Transmission-Line Model for the Antenna, Sensor and Simulation Note 81, April 1969. Y. G. Chen, R. Crumley, C. E. Baum, and D. V. Giri, Field-Containing Inductors, Sensor and Simulation Note 287, July 1985; IEEE Trans. EMC, 1998, pp. 345-350. C. E. Baum, General Properties of Antennas, Sensor and Simulation Note 330, July 1991. D. V. Giri, C. E. Baum, and D. Morton, Field-Containing Solenoidal Inductors, Sensor and Simulation Note 368, July 1994. C. E. Baum, E. G. Farr, and C. A. Frost, Transient Gain of Antennas Related to the Traditional Continuous-Wave (CW) Definition of Gain, Sensor and Simulation Note 412, July 1997; pp. 109-118, in E. Heyman et al (eds.) Ultra-Wideband, Short-Pulse Electromagnetics 4, Kluwer Academic/Plenum Press, 1999. C. E. Baum, Symmetry and SAR Antennas, Sensor and Simulation Note 431, November 1998. C. E. Baum, Interaction of Electromagnetic Fields With an Object Which Has an Electromagnetic Symmetry Plane, Interaction Note 63, March 1971. C. E. Baum, Scattering, Reciprocity, Symmetry, EEM, and SEM, Interaction Note 475, May 1989. C. E. Baum, SEM Backscattering, Interaction Note 476, July 1988. C. E. Baum, Representation of Surface Current Density and Far Scattering in EEM and SEM With Entire Functions, Interaction Note 486, February 1992; ch. 13, pp. 273-316, in P. P. Delsanto and A. W. Saenz (eds.), New perspectives on Problems in Classical and Quantum Physics, Part II, Acoustic Propagation and Scattering, Electromagnetic Scattering, Gordon and Breach, 1998. C. E. Baum, Target Symmetry and the Scattering Dyadic, Interaction Note 507, September 1994; Ch. 4, pp. 204-236, in D. H. Werner and R. Mittra (eds.), Frontiers in Electromagnetics, IEEE Press,1999. C. E. Baum, Scattering Dyadic for Self-Dual Target, Interaction Note 514, December 1995. C. E. baum, Symmetry in Electromagnetic Scattering as a Target Discriminant, Interaction Note 523, October 1996; pp. 295-307, in H. Mott and W. M. Boerner (eds.), Wideband Interferometric Sensing and Imaging Polarimetry, Proc. SPIE, Vol. 3120, July 1997. C. E. Baum, Splitting of Degenerate Modes for Buried Targets With Symmetry, Interaction Note 545, July 1998. C. E. Baum, Symmetry Analysis of Targets Near an Earth/Air Interface, Interaction Note 554, April 1999. L. Carin, R. Kapoor, and C. E. Baum, Polarimetric SAR Imaging of Buried Landmines, IEEE Trans. Geoscience and Remote Sensing, 1998, pp. 1985-1988. C. E. Baum, SEM and EEM Scattering Matrices and Time-Domain Scatterer Polarization in the Scattering Residue Matrix, pp. 427-486, in W.-M. Boerner et al (eds.), Direct and Inverse methods in Radar Polarimetry, Kluwer Academic Publishers, 1992. C. E. Baum and H. N. Kritikos, Symmetry in Electromagnetics, Ch. 1, pp. 1-90, in C. E. Baum and H. N. Kritikos (eds.), Electromagnetic Symmetry, Taylor & Francis, 1995.
POLARIMETRIC RADAR INTERFEROMETRY: A NEW SENSOR FOR VEHICLE BASED MINE DETECTION
Shane R Cloude 1, Chris Thornhill 2 1
Applied Electromagnetics 83 Market St., St Andrews Fife, Scotland, KY16 9NX, UK 2 DERA Malvern St. Andrews Rd. Great Malvern, WR14 3PS,UK
INTRODUCTION A key challenge in remote mine detection1 is the ability to detect shallow buried targets in a forward looking geometry. In this paper we investigate the application of a new technique called polarimetric radar interferometry 2,3 . The problem of forward mine detection using radar poses several technical challenges. The first is detection of the small backscatter signal level, which results from the large signal loss due the air/ground interface and to propagation losses in the dielectric soil medium. A second problem is that the surface itself is seldom smooth on the radar wavelength scale and so scattering by surface clutter can dominate the return. Polarimetric Interferometry overcomes these problems by combining two important physical identifying characteristics of a buried target: Both the coupling of energy into the ground and the scattering by the target are polarisation dependent 1,4. The first results from the Fresnel equations which show better coupling for polarisation in the plane of incidence than perpendicular. The second results from the fact that many mines of interest have plastic casings with minimal metal content and so the backscatter signal is dominated by internal waves inside the mine. Since these involve reflection and scattering from dielectric materials, they too depend on the polarisation of the incident wave 1. Interferometry is a two antenna array technique whereby the phase difference between elements of the array depends on the position of the phase centre of the target5. For buried mine detection the phase centre of the target is separated from the surface (by the optical depth of the target). Hence interferometric phase can be used to separate targets and hence reduce clutter. Polarimetric Interferometry works by combining these two techniques such that an interferometer measures the phase between all possible polarisation combinations and
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selects those combinations which maximise the separation of target from clutter. An algorithm for adaptively performing this separation has recently been developed 2, 3. FORMULATION OF THE PROBLEM Figure 1 shows the basic geometry of a Radar Polarimetric Interferometer. Two measurement positions, 1 and 2 are separated by a physical baseline B. Radar measurements of the full coherent scattering matrix [S] are made at each position in the range/cross range co-ordinate system m/n. By coherently combining signals from 1 and 2, the range difference may be sensed as a phase shift between the signals. By transforming the data into the surface x/y co-ordinate (through a process called range filtering5) the sensor can be used to locate the z co-ordinate of a scattering point.
In order to analyse such problems we employ a coherency matrix formalism so as to maintain phase shifts between all polarisation channels. This requires generation of a 2N x 2N coherency matrix from the original N x N polarimetric problem. Figure 1 shows how this matrix can be defined from the polarimetric scattering vectors from each end of the
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baseline. The hermitian matrix [P] may be factored into three N x N block matrices as shown. The matrices and are the conventional polarimetric coherency matrices. The matrix is a complex N x N matrix which contains the relative phase information about the structure of the scattering volume. In order to extract information from this matrix, use is made of a scalar parameter, the interferometric coherence defined in figure 1. Here we show from left to right how a vector interferogram can be formed for arbitrary scattering mechanisms2 and The scattering mechanism is defined as a normalised scattering vector as shown. The coherence is a function of the choice of filter Hence, if we can find a for which 1, then we have successfully isolated a scattering centre in the volume which has a well defined phase centre. Hence maximising is a clear objective in determining structure within the scattering volume. It has been shown [3] that the vectors which maximise the coherence are given as solutions of an eigenvalue problem. There are N such maxima, given by vectors and which are solutions of the following equations
Sub-Clutter Visibility Using Polarimetric Interferometry Consider the problem of finding a target embedded in a clutter background. We assume that the target has rotation symmetry (like a plastic mine) so that it generates zero cross-polarisation. In this case the polarimetric coherency matrix is 2 x 2 (only HH and VV are significant). Hence N = 2 and the matrix [P] is 4 x 4 hermitian. The filter then has only two parameters as shown in equation 2
Another way to view the role of the coherence optimiser in polarimetric interferometry is to consider it as a contrast optimisation procedure. In this way the target is defined as the scattering mechanism which has the best phase centre in the data (a mine target for example) and the clutter becomes everything else (surface clutter, vegetation etc). The contrast optimisation can then be represented by the following related matrix eigenvalue equation
where the eigenvectors are the scattering mechanisms which maximise the signal to clutter ratio between the two processes with polarimetric coherency matrices and The eigenvalues are the values of the optimum contrast and hence are important in estimating the sub-clutter visibility. The important point to note is that the vectors which satisfy equation 3 also satisfy the optimiser equation 1. In general terms we can then write the two hermitian coherency matrices, one for the clutter and one for the target in terms of their eigenvalue decompositions so that
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Where is the target scattering cross section is the clutter cross section is the system noise level, which will either be set by the thermal noise or by noise generated by depolarisation of the signal by the clutter. are orthogonal independent scattering mechanisms for the target are orthogonal independent scattering mechanisms for the clutter Note that are independent of The contrast optimisation can then be written from equation 3 in the form shown in figure 2.
Note that there are four terms in the expansion. The whole expression is multiplied by a common factor, which is the raw signal-to-clutter ratio. In practice this may be around 0 dB or worse (some indications for buried plastic mines in surface clutter are around –16 dB S/C ratio). Of the four terms inside the expansion, three have order unity or zero while the highlighted third term will generally be dominant and can provide processing gain. For clarity we have isolated this component and expressed it as the product of three significant terms. It is the net product of these terms (called the contrast) that can provide the gain in sub-clutter visibility using polarimetric interferometry. The three terms are: S/C ratio of the raw data C/N which is the effective clutter to noise ratio of the system the polarimetric gain. The factor is the projection of the target scattering mechanism onto the orthogonal clutter space. This projection term can be 0 dB maximum, in which case we can recover a maximum factor equal to the C/N ratio (which can be very substantial). In the worst case however, (if for example) and we lose any discrimination capability between the clutter and target, We now try and estimate these factors for the problem of wide band detection of buried mine targets. Buried Mine Detection Using Polarimetric Interferometry The problem to be considered is shown in figure 3. Here we see a buried mine target at depth d below a surface interface. The radar system illuminates the surface at an angle and the soil has a dielectric constant which may be complex. The problem has three important components as shown in figure 3.
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The clutter matrix is assumed to be given by a backscatter polarimetric surface model6. Hence the coefficients A, B and C are assumed to be Bragg scattering coefficients while the parameter indicates surface roughness. We shall use rough surface backscatter data from the JRC EMSL facility to estimate the parameters
The coupling of energy through the air/ground interface can be represented as a diattenuation i.e. as a polarisation dependent attenuation of the incident signal. The coupling is given by the Fresnel equations, which depend on the angle of incidence and on the dielectric constant The diattenuation parameter is a function of the ratio of these parameters as shown in figure 3. The effect of the surface on the detection performance is modelled by a matrix product as shown in equation 5. The backscattering from the target embedded in a dielectric background must be obtained through numerical modelling. For plastic mine targets, energy is coupled inside the mine and is reflected and scattered from internal dielectric surfaces. These processes are frequency and polarisation dependent and hence a UWB interrogation pulse is used to obtain the spectral target signature. We used a finite difference time domain (FD-TD) model of a typical plastic anti tank mine target,. At shallow angles the Bragg backscatter yields small α angles for the surface6. However, the target also has small values and so the projection of the target onto the orthogonal clutter space is small. Hence the gain is small, around –10 dB or less. However, as long as the C/N ratio is better than this figure, then some sub-clutter visibility will be obtained. At steeper angles the projection of the target onto the orthogonal clutter space improves and the polarimetric gain increases. Here the gain can be around –2 dB at certain frequencies. Hence we might conclude that steeper angles of incidence would provide improved subclutter visibility and hence better detection of buried mines. However there is one counter argument to this. The problem is that the C/N ratio has two components. The first is the system noise floor which is independent of angle of incidence. The second is the depolarisation by the surface. To illustrate the problem we consider the 2 x 2 surface clutter matrix. Its dominant eigenvector will be a Bragg surface component. However, its second eigenvalue will not generally be zero, due for example to surface roughness causing
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depolarisation (the parameter in figure 3). In this case the effective C/N ratio must be defined as the ratio of eigenvalues of the clutter matrix as shown in equation 7
The problem is that this C/N ratio is a function of and generally the surface depolarisation increases with so acting to counter the increasing with We have used the JRC-EMSL surface scattering data to estimate the C/N ratio. We found that for shallow angles the C/N ratio is good (around 25 dB which, in combination with figure 11, gives a sub-clutter visibility around 15 dB). However, at steeper angles this C/N ratio falls to around 12 dB (which corresponds to a sub-clutter visibility of around 10 dB). Hence, although the factor improves with the effective sub-clutter visibility does not vary very much with It does seem however that around 10-15 dB of sub-clutter visibility could be obtained from such a sensor. SIMULATION RESULTS FOR UWB VEHICLE BASED SYSTEM To illustrate the above concepts, consider the problem of finding two targets located 15m in front of the vehicle. We assume they are separated by 30cm depth and the S/C ratio is –14dB in HH and –8dB in VV. Figure 4 shows the raw data obtained from sensor positions 1 and 2. The radiated pulse is the derivative of a Gaussian with FWHH = 100ps and we have a good S/N ratio of 50 dB. Figure 4 also shows the interferometric coherence and phase, averaged over a decade of bandwidth. The x-axis is obtained by first time shifting one channel relative to the other to scan in depth for the targets. We see that in both the HH and VV channels, the main target has been found ( a target is located if its coherence is 1 and its phase is zero).
However, there is no indication of the second target since the coherence estimation is dominated by the larger signal from the ‘clutter’. It is here that polarimetry plays an
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important role. Figure 5 shows the results of using the coherence optimiser (equation 1) on the same data to obtain the maximum coherence and corresponding phase of the combined signals. Here we see a kind of super-resolution. In this case the optimiser again finds the main surface component at zero shift. However to the left and right it also finds local maxima in the coherence. These correspond to possible secondary targets. The one to the right however has a phase larger than zero and corresponds to a coupling of the surface target from position 1 with the buried target from position 2. However, the phase of this cross term will never be zero (unless the two targets are identical). The maximum to the left corresponds to our desired second target. It has zero phase (i.e. it is the same target from baseline 1 and 2) and it has high coherence. Figure 5 also shows the value selected by the optimiser. The surface clutter has a scattering mechanism with Hence its orthogonal sub-space has We see that the optimiser moves into this sub-space as we shift away from zero time shift. This confirms its ability to adapt to the local polarimetric properties of the target.
EXPERIMENTAL RESULTS FOR VEHICLE BASED UWB RADAR SYSTEM In order to investigate the feasibility of employing polarimetric interferometry from a vehicle, DERA Malvern have developed an array system mounted on a landrover platform. In this way coherent wide band data can be collected at HH and VV polarisations from
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either end of a baseline. Figure 6 shows sample experimental results for a calibration sphere located on the surface in front of the vehicle (the surface is located at zero time shift in figure 6). The coherences were calculated over a decade of bandwidth from 0.3 – 3GHz.
Also shown are phase plots indicating the precise location of the target at the zero phase crossings. We conclude that high coherence can be obtained and precise vertical localisation achieved in all polarisation channels. The results demonstrate the basic feasibility of employing such methods for the detection and location of targets in forward looking geometries. Future studies will address the detection of buried targets. Acknowledgements The authors gratefully acknowledge the support of the Defence Evaluation and Research Agency, DERA Malvern, UK. REFERENCES [1] "UWB SAR Detection of Dielectric Targets" S. R. Cloude, A. Milne, G Crisp, C Thornhill, Eurel International Conference on the Detection of Abandoned Land Mines, BEE publication No. 431, October 1996, pp 114-118 [2] “Polarimetric SAR Interferometry”, S R Cloude, K P Papathanassiou, IEEE Transactions on Geoscience and Remote Sensing, Vol 36. No. 5, pp 15511565, September 1998 [3] "Polarimetric Optimisation in Radar Interferometry", S R Cloude, K P Papathanassiou, Electronics Letters, Vol. 33, N0. 13, June 1997, pp 1176-1178 [4] "Polarization Processing for UWB Signals" S.R. Cloude In : Ultra-Wideband Short Pulse Electromagnetics 3, edited by C E Baum, L Carin & A P Stone, Plenum Press 1997, pp 461-468 [5] R. Bamler, P. Hartl, 1998, “Synthetic Aperture Radar Interferometry”, Inverse Problems, 14,R1-R54 [6] S R Cloude, I Hajnsek, K P Papathanssiou “An Eigenvector Method for the Extraction of Surface Parameters in Polarimetric SAR”, Proceedings of ESA CEOS SAR Workshop, Toulouse, France, October 1999. Paper available at http://www.estec.esa.nl/papers/po26.pdf
MODELLING OF THE AIR-GROUND INTERFACE FOR ULTRAWIDEBAND RADAR APPLICATIONS
Lostanlen Y., Uguen B., Chassay G.1 and Griffiths H.D.2 1
2
Groupe Détection Image Diffraction Laboratoire des Composants et Systèmes pour les Télécommunications Institut National des Sciences Appliquées, Rennes 35043 Cedex, Prance Electronics and Electrical Engineering Department University College London London WC1E 7JE, England
INTRODUCTION The ability of ultrawideband (UWB) radar systems to detect buried objects or objects embedded in materials or merely covered by a shallow layer of vegetation is an undeniable advantage. Indeed, they combine very low frequencies, allowing penetration in the subsoil, and a very wide bandwidth, providing a better resolution. However, the first experiments involving such systems have shown that a high level of clutter is added to the relevant radar signal, and this greatly blurs and disrupts the exploitation of the measurements. The mechanisms leading to this clutter are not known so far and are a lot more complex than in the narrow band case. Understanding the phenomenology of electromagnetic interactions between very short pulses and the complex dielectric ground surface should provide an important input to the design of UWB radar systems leading to improved clutter cancellation and improved detection performance. The analysis of such a physical phenomenon will be achieved only through reliable electromagnetic simulation tools. Yet UWB clutter modelling is a genuine challenge launched for the electromagnetism and radar specialists. Among the existing electromagnetic analysis methods, the techniques directly formulated in the time domain seemed to be the best suited for UWB signals and were naturally first set up. Many contributions have been brought, especially using FDTD (Finite Difference in Time Domain) schemes (Dogaru, 1999; Hastings et al, 1995; Jaureguy et al, 1996) and their improvements (parallelization, multi-resolution). One can-
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not avoid their use for those UWB applications because of their accuracy in describing the dispersive media. However when one wishes to model a full large scale UWB experiment, one has to face a considerable need for computer ressources. This is of course even less appropriate to repetitive simulations required by the statistical analysis of the phenomenon. The method we propose here is different in that it is formulated in the frequency domain and it is bi-dimensional. The frequency domain has many avantages for the considered problem. First, there is no need for a discretization of the entire volume which was leading to an increase of the computer ressources when considering an extended area with the time domain methods. Only areas involving excitation sources are discretized which is a considerable advantage when increasing the scale of the experiment. Secondly, reflection coefficients are very well defined and have a simple form in the frequency domain. Starting from these observations, our idea was to set up a 2D method in the frequency domain, the time domain response being obtained with a kind of Discrete Fourier Transform. This method should provide useful information concerning the generation of the surface clutter. FORMULATION OF THE PROBLEM In this section we explain how our formulation of the problem is derived. Let and be the distances which separate respectively the emission and reception antennas to the origin O of the coordinate system (see Fig. 1). and are the angles formed by the segments joining both antennas to the origin. This origin O plays an important role in our modelling. It will always be associated to the intersection of boresight with the profile; then it will be the temporal origin when considering the time domain formulation.
Using these conventions the distances between the emission and reception antennas
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to the current point P (x,y) of the profile are expressed by:
The incident field at P may be written from the field at the emission antenna:
where the quantity is complex and takes into account the distorsion diagram of the emission antenna. The field scattered by the interface is expressed at the receiver antenna (R) by the following curvilinear integral:
where is a complex quantity including the distorsion of the receiver antenna. This expression of the scattered field is valid in 3D, but the radar case is well suited to a 2D configuration at a first approach. In order to spare computer ressources for a first analysis of our model, we implement a 2D formulation. To this purpose, we start from the expression given in Eq. (4). The scattered field at the receiver antenna may take the simplified form:
where For 2-D distributions the field given by Eq. (5) is valid provided that the correct Green’s function is used, that is the zero-order Hankel function of the second kind We can rewrite the expression of the scattered field as a curvilinear integral:
The scattered field at the receiver antenna has now a 2D form. In the following subsection, we will introduce the expressions of including the Fresnel coefficients. Introduction of the Fresnel coefficients The electric and magnetic source currents and which are present in the vectorial part of the integrand in Eq.(4) may be approached by a local decomposition of the incident field (assumed to be TEM) in parallel and perpendicular polarizations:
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where is a vector tangent to the interface. This is a Physical Optics formulation which includes the local Fresnel reflection coefficients (on the dielectric interface) usually expressed by:
The angle value in Eq. (11) follows the variation of the local normal to the surface. Using the usual relationships:
We can now rewrite the vectorial expressions of the integral equation as:
These equations are the contributions of the induced current sources on the surface by the incident field.
Splitting the polarizations of At this stage it is convenient to rewrite the integrand of the curvilinear integral as a function of the two splitted polarizations. The scattered field alternatively becomes:
with:
We find more readable (and easier to implement) to have vectorial expressions rather than angular expressions. Substituting the angular values:
we get a vectorial formulation:
which alternatively may be written as:
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The scattered field in parallel polarization cannot be expressed directly in a local basis associated to the scattered ray for the vectors are not parallel. A vectorial expression is therefore needed. This is not the case for the perpendicular polarization because the vectors are parallel to Taking this difference into account, we express the scattered field as:
where the split parallel and orthogonal components are:
These components are function of the wavenumber the ad hoc components of the incident field and integrals. We have introduced these curvilinear integrals which take into account the local geometry on the profile, the antennas properties and the 2D radiation function separately.
At this point, we need to give a more explicit form to the Hankel function. In solving the problem for the far zone we can take the asymptotic expansion of the Hankel function:
The integrals are approximated by:
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The previous expressions can be applied to any ground profiles provided that the height of the profile varies. Parametrisation of the interface The roughness of the profile is an important parameter in our simulations because it is the source of the clutter. To be able to take into account the roughness of the interface, here we introduce the profil height as a function of the abscissa. A point P on the interface is defined by its coordinates which are expressed as a function of a parameter depending on the roughness of the soil:
The unit vectors associated to the incidence (respectively observation) direction on the interface at a point P(u) with parameter u is given by:
Returning to Eq. (31) we can introduce the parameter u (coming from our parametrisation of the interface) in the curvilinear integral. This leads to a new integral:
We have modelled gaussian, exponential and power-law rough surface following the work of Dierking, 1999. RESULTS In this section, we present a few results obtained by our electromagnetic tool. In each case, at least a rough profile is generated. The antenna height is 6.7m, its features was given in a previous figure. The incident pulse is shown in Fig.2. We only consider monostatic cases, the emitter antenna is the same as the receiver antenna. The sampling frequency is 10 GHz. Time domain responses First we represent a time-frequency response of a gaussian profile with a height standard deviation of 0.024 cm (Fig. 3). In the time domain response, one can notice that the ground straight below the antenna exhibits a is high response. The response of the boresight is quite interesting as well. The echo is rather dense and concentrates many frequencies as can be seen on the time frequency viewgraph. This last plot shows how non stationary the signal is. As previously explained, we have decomposed the scattered field into its orthogonal and parallel components, a plot of each is presented in Fig. 4.
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The responses are similar except for the point at the vertical of the antenna for the parallel y-component. This is easily interpreted by the fact that the parallel component of the incident field is entirely a x-component. The parallel y-component of the scattered field is therefore null just below the antenna but not in the boresight regions, where the roughness of the ground for these angles returns energy. When 50 time-domain responses are put next to each other, a kind of B-Scan (analogy with GPR images) can be represented in Fig. 5. One clearly sees the straight lines corresponding once again to the points straight below the antenna. A more diffuse area lets appear the clutter produced by the rough interface in the boresight. Later, the echos are weaker until the end of the footprint of the antenna. Monte Carlo Simulations The first plots have shown the time domain response for a single rough ground. Our work makes sense if we perform Monte-Carlo simulations in order to give a statistical description of the resulting signals. Thus we have added a loop over the statistically generated rough profile. Practically, 50 profiles are generated. For each of them the en-
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tire frequency loop is run. Fig. 6 shows the variance of the components of the scattered field for 50 profiles. One notices that the amplitude of the orthogonal response corresponds to the amplitude of the sum vector of the parallel components. The boresight area clearly appears once again in this viewgraph. Finally, we have represented for a progressively increasing height standard deviation, the variance of the scattered field amplitude for two models of roughness in Fig. 7.
It demonstrates the important contribution of the roughness to the surface clutter.The highest amplitude is for the exponential profile, which was the roughest. These results present a very small part of the capabilities of the electromagnetic method explained in this paper. Indeed the effects of the moisture of the ground may be analysed (through the permittivity and conductivity) as well as the type of soil (’normal’ ground, sandy grounds, ...). The influence of the system elements such as receiver filters, antennas (types, pointing angles, location), sampling frequencies, incident pulses are interesting parts
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for which attention may as well be given. CONCLUSION An electromagnetic simulation tool has been described to characterise the clutter generated by a rough surface in UWB radar applications. The numerical approach is based on a Physical Optics approximation of Maxwell’s equations. This is a frequency domain formulation. The time domain scattered fields are evaluated by means of an inverse Fourier Transform. We have demonstrated the ability of this technique to take into account the system elements. Moreover a great flexibility is given by the possibility to tune the simulation parameters. We have also shown the advantage of our method to consider large scale configurations. The non stationarity of the responses was obvious in the time-frequency representation. We have also presented how statistical descriptors could be used to describe the echos. We believe the method will provide a useful input for the design of a receiver chain. We are currently refining our clutter model through comparisons of our simulated data to real measurements. Acknowledgements The authors gratefully acknowledge the support of the Région Bretagne. REFERENCES Dierking, W., 1999, Quantitative Roughness Characterization of Geological Surfaces and Implications for Radar Signature Analysis, Trans. GRS, 35:5. Dogaru, T., 1999, Modeling and Signal Processing for Electromagnetics Subsurface Sensing, PhD. Thesis, Duke University. Hastings F. D. and Schneider J. B.and Broschat S. L., 1995, A Monte-Carlo FDTD Technique for Rough Surface Scattering, IEEE Transactions Antennas Propagation,43:11. Jaureguy M. and Borderies P., 1996, Modelling and processing of ultra wide band scattering of buried targets, Eurel international conference on detection of abandoned landmines, Edinburgh. Young J.D. and Gwynne J., 1997, Report on Antenna Transient Patterns, Technical Report 732169-3 Ohio State University.
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ULTRA-WIDEBAND POLARIMETRIC BOREHOLE RADAR
Motoyuki Sato 1 and Sixin Liu 1 1
Center for Northeast Asian Studies Tohoku University Sendai, 980-8576, Japan
INTRODUCTION Borehole radar is a new technology for subsurface sensing. It is used in various fields such as water permeable subsurface fracture detection, geological survey and mineral exploration. Due to the high attenuation of electromagnetic wave in geological material, borehole radar is normally operated under 100MHz and its radar resolution is not enough in many applications. The diameter of borehole is often less than 10cm, and the restriction to antenna configuration is strict. The authors have introduced a polarimetric approach to borehole radar in order to improve its capability and showed its ability of characterizing fractures. Since most geological structure are complicated and many radar targets are existing at the same location. Therefore, ground penetrating radar (GPR), is operated as ultra wideband radar. Although most GPR systems are operated as an impulse radar system, which uses a short unmodulated pulse as a transmitting signal and acquires a time-domain received signal. When we apply radar polarimetry in a borehole radar system, we need a flexibility of the radar system, because we have to test various kinds of antennas, which have different frequency characteristics, to control the polarization status. Therefore, we selected stepped frequency radar system for this purpose. In this paper we describe a polarimetric borehole radar system, which we have developed, and show the acquired results with polarimetric signal processing for subsurface fracture classifications. RADAR POLARIMETRY FOR SUBSURFACE SENSING Due to high attenuation in subsurface material, only relatively low frequency can be practically used for GPR systems. Borehole radars are operated in deep-drilled boreholes, so the penetration depth is determined as a radial distance from a borehole. In order to acquire information of larger distance from a borehole, longer penetration depth is usually required for borehole radar. Most conventional borehole used frequencies lower than Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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100MHz, although typical GPR systems use 500MHz-1GHz. And the penetration depth of borehole radar can be reached a few ten meters in very hard rock such as granite, but it is less than 10m in soft rock including sedimentary rocks and soils. Subsurface fracture is one of the most important radar targets for borehole radar. It dominates the water flow in hard rock, which is very important factor for many engineering applications including nuclear waste final disposal and environmental problems. Larger scale of fractures and fractured zones are important in civil engineering and prevention of natural disasters. The subsurface fractures to be determined have normally very small aperture. Typical fractures in granite have aperture less than 1mm, and they extend more than 10m. The location and dip of fractures can be determined even by conventional borehole radar. These applications have been very successful for detection of water flow in crystalline rock (Nickel et al., 1983, Olsson et al., 1992). However, at the same time, we found that not all of the fractures detected by borehole radar are water permeable ones, but only some of them function as water path. Since the radar resolution is much less than the fracture surface geometry and aperture size, we cannot obtain detailed information of the fracture. We think fracture surface roughness has relations to the fracture properties, although it does not directly related to water permeability. If we can classify fractures by using the information of surface roughness, it will provide us further information about subsurface fracture. Radar polarimetry is known as an advanced technique of radar. It is also known that the surface structure of scattering objects can be sometimes classified and evaluated by radar polarimetry. We simulated scattering from a 3-D fracture model having a surface roughness by FDTD and found that the cross-polarization component is related to the surface roughness, when the roughness height is small compared to the wavelength (Takeshita and Sato, 1999). We think that radar polarimetry can be a powerful technique for borehole radar. POLARIMETRIC BOREHOLE RADAR SYSTEM Antenna
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In order to obtain full-polarimetric radar data, we have to have sets of orthogonal polarization transmitter and receiver antennas. This can be easily achieved by rotating polarization plane of one antenna, if we use liner polarization. Antennas for airborne and space borne remote sensing radar systems normally have a set of antennas set orthogonal to each other. Most conventional borehole radar uses dipole antennas for transmitter and receiver. They are operated as a half-wave length dipole antenna, so the length of antenna is normally longer than 1m. Therefore, we cannot use an orthogonal set of dipole antennas for radar polarimetry, because the diameter of borehole is normally less than 10cm. We proposed a use of combination of a dipole antenna and an axial slot antenna on a conducting cylinder as shown in figure1, to achieve a set of orthogonal polarization antennas in a thin borehole (Sato et al, 1995). In a vertical borehole, a dipole antenna corresponds to vertical (V) and a slot antenna corresponds to horizontal (H) component, respectively. Full polarimetric borehole radar measurement can be achieved by replacing the dipole and the slot antennas as combinations of VV, VH, HV and HH. Four sets of measurements have to be repeated to acquire the full-polarimetric radar data at one borehole. However, frequency characteristics of a dipole antenna and a slot antennas are quite different. The dipole length dominates a resonant frequency of a dipole antenna. On the other hand, an axial slot antenna on a conducting cylinder has two parameters for the resonance, namely the radius of the conducting cylinder and the length of the slot. When we use a slot antenna on a conducting cylinder, the diameter of the conducting cylinder is limited by the borehole diameter and it is much smaller than the wavelength of the operating frequency. In this case, we found that the resonant frequency is dominated by the diameter of the conducting cylinder and the length of the slot is not a dominant parameter for the resonant frequency. Therefore, even if we use the largest diameter of the conducting cylinder, available in a borehole, the resonant frequency of the slot antenna is much higher than the dipole antenna. Figue2 shows an example of the spectrum of the received signal acquired by the stepped-frequency polarimetric borehole radar system to be described in the next section. Signals are measured by Dipole-Dipole and Slot-Slot combination for a transmitter and a receiver. We can find that the dominant power spectrum of dipole-dipole combination is at around 100MHz and that of slot-slot combination is at around 150MHz - 200MHz. The difference of the dominant frequencies is principally caused by the difference of the frequency characteristics of the two different antennas.
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Radar System We have developed a polarimetric borehole radar system using a stepped frequency radar system due to its advantages of flexibility over an impulse radar system (Miwa et al., 1999, Sato and Miwa, 2000). The antennas, which will be used in the polarimetric borehole radar system, have very different frequency characteristics. Also the antennas have to be replaced during one set of measurement. Therefore the radar system has to have a broadband characteristic. If we use an impulse radar system, we have to use a very short pulse to generate broadband frequency spectrum. This reduces the mean transmitting power. At the same time the dominant power spectrum should be matched to the antenna resonant frequency, but it can be changed depending on the material around the antennas. Therefore designing a tuned radar system is difficult in ground penetrating radar system. The radar system is composed from a vector network analyzer, an analog optical signal link with a 100m optical fiber cable and a downhole radar sonde. A block diagram of the radar system is shown in figure 3. All the downhole equipments are powered by batteries, so they are electrically completely isolated from a cable suspending the downhole sonde. This configuration enables an ideal radiation pattern of antennas. The frequency bandwidth of the developed analog optical link is 0.1-500MHz with a dynamic range over 70dB. The downhole O/R and R/O (RF signal to Optical signal) converters are installed in a cylindrical casing with a diameter of 20mm, which is easy to assemble inside antennas.
FIELD EXAMPLE Mirror Lake Test Site By using this radar system, we have conducted full-polarimetric borehole radar measurements in several fields. In most of the measurements, detection and characterization of subsurface fractures have been important purposes. In October 1998, we have conducted polarimetric borehole radar measurement at Mirror lake test site, NH, USA as a collaborative research work with U.S.Geological Survey. The host rock is granite and this test site has been used for studying water flow in fractured rock. U.S.Geological survey has corrected many radar data and hydraulic data in
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this test site (Lane et al., 1996). We selected this site because we can compare our polarimetric borehole radar data with other information about the subsurface fractures. The radar data was acquired in the frequency range of 2MHz to 400MHz at 2MHz step. Amplitude and phase of the received signal is stored at 10cm depth interval. Time domain radar signal can be obtained by inverse Fourier transformation of the stored spectrum. Although broadband spectrum is stored, the signal to noise ratio changes in frequency. At the same time, the signal to noise ratio changes depending on the antenna combination. Therefore the selection of frequency range for inverse Fourier transformation is important for obtaining clear radar images. Figure 2 shows the example of the acquired raw spectrum. Selection of Frequency Range The radar profiles having the best signal to noise ratio can be obtained by taking the
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Fourier transformation using the frequencies of high power spectrum. However, the frequency range where the signal is maximized is different in the dipole-dipole and the slotslot combinations, as shown in figure2. Figures 4 and 5 show the time domain radar profiles of V-V (Dipole-Dipole) and H-H(Slot-Slot), obtained from the same frequency domain signal as shown in figure 2, but using different band pass filters to select effective frequency range. In both figures, (a) is obtained by using 70-120MHz, (b) is obtained by using 100-150MHz and (c) is obtained by using 130-150MHz. We can find clear difference between the radar profiles depending on the frequency range. For the dipole-dipole combination, (a) 70-120MHz seems most clear, but for slot-slot combination, (c) 130180MHz seems to be the best. We have to use the same frequency range for polarimetric analysis, we selected (b) 100-150MHz as a common frequency range. Polarimetric Analysis Figure 6 shows the polarimetric borehole radar profiles obtained by using the frequency range of 100-150MHz, which we think most suitable for analysis in this specified data sets. The difference of antenna characteristics has been compensated by a technique, which we use a direct coupling of a single-hole reflection data (Sato and Miwa, 2000). Therefore direct comparison of the signal amplitude is possible in figure 6. We can see common reflection waves in the four radar profiles, but we can also find the intensity of the reflection differs in different polarization status. Polarimetric signature is a common technique to represent the characteristics of scattering objects and it can be determined by using the full polarimetric borehole radar data. Reflected signal from a single fracture can be obtained by time-depth gating windowing of the time domain radar profile. The spectrum of the separated signal is estimated by Fourier transformation. For example, the scattering matrixes determined for the fracture crossing the FSE-1 borehole at 30m were obtained as:
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The corresponding polarimetric signatures are shown in figure 7. We observed many polarimetric signature of subsurface fractures, but found that it is not so easy to classify fractures directly from the polarimetric signature. We think it is mainly due to phase error caused by the positioning of radar system in measurement. Even we carefully control the radar depth in a borehole, it cannot be fixed at exactly the same position, when we replace antennas. In order to avoid the error caused by the phase, we proposed a technique to utilize only the amplitude information of the scattering matrix. The ratio of energy in co- and cross-polarization reflection corresponded to other geological information of the fractures (Sato, 2000), and showed that it can be used for fracture classifications.
CONCLUSION Stepped-frequency ultra-wide-band polarimetric borehole radar system was developed and tested for subsurface fracture characterization. The radar was tested for subsurface fracture characterization. Scattering matrix of subsurface fractures can be estimated by using the radar system and examples of polarimetric signature were shown. However, further work is required for understanding of the scattering matrix of subsurface fractures. The radar system we have developed can be assembled to different bistatic radar configurations. We have tested surface GPR and vertical radar profiling (VRP), where antennas are set on the ground surface and in a borehole. The flexibility of the system is a great advantage in most practical field measurement.
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Acknowledgements The authors acknowledge Mr. Moriyasu Takeshita for their contribution to radar signal processing. This work was supported by Grant-in-Aid for Scientific Research (A) l1355041 and 12305067.
REFERENCES Lane, J.W., Haeni, F. and Placzek, G. ,1996, Use of borehole-radar methods to detect a saline tracer in fractured crystalline bedrock at Mirror Lake, Grafton county, New Hampshire, USA, Proc. 6th Int. Conf. on Ground Penetrating Radar, Sendai, Japan, :185. Miwa, T., Sato, M. and Niitsuma, H., 1999, Subsurface fracture measurement with polarimetric Borehole Radar, IEEE Trans. Geoscience & Remote Sensing, 37: 828. Nickel, H. Sender, F, Thierbach,R. and Weichart.H., 1983, Exploring the interior of salt domes from boreholes, Geophysical Prospecting, 31: 131. Olsson,O, Falk,L., Forslund,O., Lundmark, L., and Sandberg,F., 1992, Borehole radar applied to the characterization of hydraulically conductive fracture zones in crystalline rock, Geophysical Prospecting, 40: 109. Sato, M., Ohkubo T., and Niitsuma, H., 1995, Cross-polarization borehole radar measurements with a slot antenna, Applied Geophysics, 33: 53. Sato, M. and Miwa, T., 2000, Polarimetric Borehole Radar System, Subsurface Sensing Technologies and Applications, 1: 161. Taksehita, M. and Sato,M., 1999, Numerical Analysis of Scattering from Rough Surface for Polarimetric Borehole Radar, Technical Report IEE Japan.
BURIED MINE DETECTION BY POLARIMETRIC RADAR INTERFEROMETRY
L. Sagués1 , Juan M. López-Sánchez2, J. Fortuny 3 X. Fàbregas1, A. Broquetas4 , A. J. Sieber3 1
Dpt. Teoria del Senyal i Comunicacions Universitat Politècnica de Catalunya (UPC) Campus Nord UPC, c/ Jordi Girona, 1-3, 08034 Barcelona, Spain 2 Depto. Física, Ing. Sistemas y Teoría de la Señal Universidad de Alicante P.O. Box 99, E-03080 Alicante, Spain 3 Space Applications Institute, Joint Research Centre SAI, TP.272, 21020 Ispra, Italy 4 Institute of Geomatics Parc de Montjuïc, E-08038, Barcelona, Spain
INTRODUCTION Nowadays, subsurface detection of objects is a challenge for the scientific community since it would give a solution to actual humanitarian and civil problems, such as de-mining. A large number of remote sensing techniques has been considered as candidates for this task at a safe stand-off distance. To date, several promising results have shown that ultrawide band SAR sensors can detect most metallic mines. However, buried plastic mines are nearly invisible to the radar and can not be easily detected due to the fact that the dielectric contrast between the target and the soil is insignificant1. Moreover, the weak signal returned from the mine is normally hidden in strong surface clutter, making its detection more complicated. In this context, Polarimetric SAR Interferometry is a new technique that can be applied to enhance the discrimination between targets and clutter. It is well known that conventional SAR interferometry uses relative phase information from two radar images acquired from different viewing angles in order to estimate the height of ground scatterers. On the other hand, radar polarimetry allows us to extract target information from the measurements of the state of polarisation of the scattered wave, making possible to decompose the target into different scattering behaviours. Therefore, by combining both polarimetry and interferometry it would be possible to get the height of different scattering mechanisms that can be present in the same resolution cell, even if one scattering process is much weaker than the other2. The objective of the polarimetric and interferometric Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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approach presented in this paper is to separate the phase centre corresponding to the mine from that related to the clutter and estimate their relative height. Several multifrequency experiments have been carried out in the anechoic chamber of the European Microwave Signature Laboratory (EMSL) at JRC Ispra4 to test the application of this new technique for buried mine detection. Fully coherent polarimetric data were collected at L, S and C band. The target was formed by four plastic mines and a metallic cylinder, that were buried below a thick layer of gravel. The obtained results will be shown and discussed in this paper. MINE DETECTION USING POLARIMETRIC SAR INTERFEROMETRY SAR Interferometry The geometry of an interferometric imaging system is illustrated in Fig. 1. The interferogram is formed by cross-multiplying two SAR images obtained from slightly different viewing angles. The antennas are located at two different positions and illuminating the same patch on the surface at two incidence angles and respectively. The distance between the two antenna positions is known as the baseline B, whereas the distances from each antenna to the centre of the resolution cell are and
The interferogram is generated by multiplying an image by the complex conjugate of the other. Thus, the phase of the interferogram corresponds to the phase difference between both SAR images:
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Since the phase of each SAR image is related to the distance between the antennas and the ground, we can derive a relationship between this travel path difference and the local height of each ground scatterer:
whereby we can generate an elevation map of the scene under observation. Thus, an additional height dimension is introduced in the reflectivity images. The quality of the generated elevation map depends on the quality of the interferometric phase. The parameter used to evaluate the phase quality is the so-called interferometric coherence, which can be interpreted as a useful tool for measuring the resemblance (correlation) between the two SAR images:
If the coherence equals to zero, it would mean the scene is completely uncorrelated and then the interferogram is noisy and not related to the topography. At the other extreme, a coherence close to one corresponds to a noise-free interferogram from which a high quality elevation map can be generated. In a real buried mine scenario, as shown in Figure 1, the mine is confounded into the clutter and there exists an uncertainty in determining the interferometric phase due to the existence of different scattering centres inside the resolution cell. As a consequence of this, the coherence degrades and it is not possible to extract the height of the mine accurately. Therefore, it would be useful to separate the scattering mechanism related to the mine from that corresponding to the clutter surface in order to get their height independently. If the choice of these scattering mechanisms is based on selecting those polarisation states that maximise the interferometric coherence, then the interferometric phase associated with them presents a better quality and the accuracy of their height estimation is higher. Two different polarimetric coherence optimisation algorithms can be applied to extract this height information. The original optimisation algorithm, presented by Cloude2, formulates the exact formal solution of the coherence maximisation problem, leading to two complex 3x3 eigenvalue problems that share the same eigenvalues. The three derived eigenvalues are related to pairs of eigenvectors that can be interpreted as the optimum scattering mechanisms. The second method, called Polarisation Subspace Method (PSM) is described in next section. This alternative approach is based on finding the local maxima of those copolar or crosspolar coherence function that have been calculated from the measurements of all polarisation states of the scattered wave3. Polarisation Subspace Method If fully coherent polarimetric data are collected, the information associated with each pixel of the SAR image is then defined by the following scattering vector:
where is the complex scattering coefficient for n transmitted and m received polarisation expressed in the orthogonal (H,V) basis. In case of interferometric
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measurements, the polarimetric information will be contained in two different scattering vectors, and for image 1 and image 2 respectively, leading to the following hermitian matrices:
where i=1,2 and j=1,2 refers to each SAR image used to form the interferogram. Thus, and are the two covariance matrices associated with the separate images, whereas is a 3 x 3 complex matrix which contains the interferometric information between polarimetric channels. Otherwise, instead of using a linear polarisation basis to describe the vector fields, one can use any other orthogonal elliptic basis. The coordinates transformation of a scattering vector from the linear basis (h,v) to another orthonormal elliptic basis (x,y) can be accomplished by applying a unitary matrix transformation to the polarisation vector3:
where and are the ellipticity and inclination angles that define any polarisation state. Physically, this transformation can be interpreted as a change of the selected scattering mechanisms in both images. After applying this polarisation basis change, we can express the covariance matrices defined in (5) in any different polarisation state
The application of these polarimetric basis transformations enables the formation of interferograms between all possible elliptical polarisation states and polarimetric combinations between both SAR images. In this way, the interferometric coherence between all polarimetric combinations will be given by:
being the elements of the covariance matrices defined in (7). The optimum interferogram will be generated by selecting that combination of polarisation states which maximise the interferometric coherence . Once the interferometric phase is calculated, it can be converted to absolute height by using the conventional interferometric phase-to-height equations described in (2):
We can further simplify this approach by using some a priori knowledge about the values in The first assumption is that, in cases where there is no temporal
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decorrelation and the baseline is short, the highest coherence will be given by the same polarisation state in both images, and therefore the elements of the diagonal of the matrix will be much higher than the rest. Moreover, the definition of the geometrical angles implies the following transformation symmetry:
and, consequently, if the assumption is valid, we must calculate only the first and the second elements of the diagonal of and select the maximum one:
This optimisation method is called Polarisation Subspace Method (PSM) because we are only considering the copolar and crosspolar elements, and the same basis polarisation transformation is applied in both images. This method can not only be used for getting the optimum scattering mechanism that maximise the coherence but also for deriving the height structure of the target. By graphically representing the copolar and crosspolar coherence functions, it is possible to derive the existence of independent scattering mechanisms inside the same resolution cell. In case of a single dominant scattering behaviour, both coherence functions generally present only one absolute maximum. However, in those cases where the target presents a clear multilayer vertical structure, as shown in Figure 1, it is possible to identify various local coherence maxima. These optimum polarisation states will be related to different independent scattering mechanisms that lead to the locally best height estimation. In case of buried mine detection, it is expected to find two coherence maxima, each of them related to a different scattering mechanism located at a different height: the mine and the surface clutter (or air-ground interface). After generating the two interferograms corresponding to these optimum polarisation states, we can extract the height difference between them. The resulting height difference map will exhibit some peaks at those positions where the mines are buried whereas it will remain close to zero at those zones of the image where only a single surface scattering process can be found. EXPERIMENTAL RESULTS Several multifrequency experiments were carried out in the anechoic chamber of the European Microwave Signature Laboratory (EMSL) at JRC Ispra, Italy. The experimental setup is shown at Fig. 2. Four plastic mines and a metallic cylinder were buried below a thick layer of gravel. Mine 1 and mine 4 were buried at 20 cm depth under the surface, whereas the other two mines were buried at 10 cm depth. The separation between mines was 80 cm and their size was different: the diameter of mines 1 and 2 was 20 cm, whereas the diameter of mines 3 and 4 was 25 cm. The cylinder was in the middle of the four mines and it was buried at 10 cm depth. Fully coherent polarimetric data were collected for different baselines, covering the frequency range from 2 GHz to 6 GHz.
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Fig. 3 shows the reconstructed SAR images using HH and VV polarisation for a frequency range between 2 GHz and 3 GHz. The resolution cell was about 20 cm in both dimensions. As can be seen, the metallic cylinder is detected and located 15 cm far from the centre of the scene because the image was focused at the gravel reference level. However, the four plastic mines can not be distinguished using the amplitude reflectivity image because the strong surface clutter masks the mine backscattering response.
After applying the Polarisation Subspace Method, we can notice that the coherence copolar coherence function exhibits two local maxima when the method is applied at those zones of the scene where a mine was buried. As an example, Fig. 4 depicts the obtained coherence function and the resulting absolute height for all polarisation states when calculating them inside a 60 x 60 cm window centred at mine 3 location. The two local maxima appear at and The absolute height associated with the first state of polarisation is -0.7 cm, which is very close to the surface reference height, whereas the height derived from the second one is 8.54 cm. Therefore, we can state that the first one corresponds to the surface scattering mechanism while the second one is related to the buried mine. The height difference between both polarisation states results 9.24 cm, which is very close to the actual depth of mine 3 (10
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cm). It is important to note that this result has been obtained by applying a multi-look procedure (spatially averaging of N = 9 resolution cells) in order to reduce the height standard deviation.
This procedure can be applied to the entire surface by sliding the averaging window on the resulting complex interferograms. However, it is not possible to consider a high number of spatial looks during the averaging process because it implies a loss of spatial resolution. Fig. 5 shows the elevation map obtained by calculating the height difference between those polarisation states that maximise the copolar coherence function, for a frequency range from 2 up to 3 GHz. As can be seen, the resulting height difference increases at those zones where the mines were buried, but it remains close to zero for the rest. Therefore, the four buried mines can be clearly detected. Nevertheless, we can note that the metallic cylinder is not detected at this low frequency band. The reason is that the ground attenuation is low and the resulting cylinder scattering amplitude is too high compared to the surface one. Consequently, the copolar function, calculated at the cylinder zone, exhibits only a single maximum that corresponds to the cylinder height.
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However, if we apply the same method at higher frequencies (4 – 5 GHz), an additional second maximum, which corresponds to the surface response, appears in the copolar coherence function. Hence, after calculating the height difference between both optimum states of polarisation, we obtain the result shown in Fig. 6. Note that in this case the cylinder is well detected whereas the mines can not be distinguished.
Thus, we can conclude that the polarimetric method here presented has a strong dependency on the operating frequency. This can be clearly noticed by representing a cut of the obtained height difference along the ground-range axis where mine 3 and mine 4 were buried, for different frequency ranges (see Fig. 7). The obtained height values are higher than expected due to that no multi-look procedure has been applied in order to preserve the spatial resolution and extract the correct position of the mines. Note that both mines are only detected with the lowest frequency band (2 – 3 GHz). At higher frequencies, the ground attenuation weaken the mine signal, and only one optimum polarisation state can be derived from the copolar coherence function.
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CONCLUSIONS In this paper we have presented a new polarimetric and interferometric approach that can be applied to improve the sub-clutter visibility in de-mining applications. The obtained experimental results have shown that buried plastic mines can not be detected by using only SAR images. However, after applying the method presented here, it is possible to separate the scattering mechanism that is associated with the clutter surface from that corresponding to the mine and estimate their height location. Nevertheless, the success of the proposed method will depend on the geometry of the problem. Indeed, it is necessary the surface clutter to be physically separated from the mine in order to distinguish both scattering mechanisms by optimising the interferometric coherence. It has also been proved that the probability of detection shows a strong dependency on the frequency range. In case of detecting plastic mines, we have seen that it is convenient to operate at low frequencies in order to reduce the attenuation caused by ground propagation. On the other hand, multi-look averaging techniques must be applied in order to reduce the false alarm rate and improve the height estimation accuracy, at the expense of loosing spatial resolution. Acknowledgments The authors wish to thank the CICYT (Spanish Commission for Science and Technology) ref TIC99-1050-C03-01 for their financial support. This work has been carried out in the frame of the European Commission TMR Network on radar polarimetry.
REFERENCES 1. L. Carin, N. Geng, M. McClure, J. Sichina, L. Nguyen, Ultra-wide band synthetic aperture radar for minefield detection, IEEE Antennas and Propagation Magazine, 18:33(1999). 2. S.R. Cloude, K.P. Papathanassiou, “Polarimetric SAR Interferometry”, IEEE Trans, on Geoscience and Remote Sensing, 1551:1565(1998). 3. L. Sagués, J.M. López-Sánchez, J. Fortuny, X. Fàbregas, A. Broquetas, A.J. Sieber, “Indoor Experiments on Polarimetric SAR Interferometry”, IEEE Trans. on Geoscience and Remote Sensing, 671:684 (2000). 4. A.J. Sieber, “The European Microwave Signature Laboratory”, EARSel Advances in Remote Sensing, 195:201 (1993).
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AN OPTICAL APPROACH TO DETERMINE THE STATISTICAL FEATURES OF THE FIELD DISTRIBUTION IN MODE STIRRED REVERBERATION CHAMBER
Sylvie Baranowski, Lamine Kone and Bernard Demoulin Universite des Sciences et Technologies de Lille Laboratoire de Radio propagation et Electronique, Bat P3 F59655 Villeneuve d'ascq cedex, France
INTRODUCTION Mode stirred reverberation chambers (MSRC) are suitable tools for electromagnetic compatibility measurements, especially to test complex but small sized systems. This technique is based on the properties of the field inside the chamber. These are strongly dependant on the geometry (dimensions, shape,...) and on the material constituting the walls (conductivity) of the reverberation cavity but also of the mechanical modes stirrer. Theoretical simulations are today necessary in order to perform an optimal mode stirring, because measurements can only be done after an expensive chamber realization. Classical methods, such as FDTD or Finite Element methods which provide good results at low frequency (Bunting, 1999; Hoeppe, 2000), cannot easily be used for simulation at higher frequency range. Indeed, due to the grid size and the computer performance, the frequency range is limited. Thus, at high frequency, when the wavelength is small compared to the chamber size, an optical approach is preferred to predict the field distribution inside this oversized cavity. In this paper, we present some simulation results obtained with an optical approach at high frequency; as a first stage, an empty cavity is modelized; the electric field is then calculated and the results are statistically analyzed in order to define the simulation parameters.
USE OF RAY THEORY The electric field received at any point R in the cavity, can be considered as the sum of the field of the direct ray issued from the emitter and the multi-reflected rays (the diffracted and scattered fields are neglected). Figure 1.
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By using an image theory, a reflected wave can be easily calculated as the field, radiated by a virtual source, image of the real emitter, weighted by reflection coefficients:
Where: L, M, N are the maximum number of reflections on the walls (parallel to the x0z plane for L, parallel to y0z for M, and parallel to x0y for N), and the associated reflection coefficient (calculated for the convenient angle of incidence). Due to the high conductivity of the walls, these reflection coefficients are closed to 1 then, a too large number of images is required to reach a numerical convergence (Kwon et al, 1998). The aim of this approach being to compare efficiency of various chambers and not to compute exactly the field amplitude, a statistical analysis is used to define the simulation parameters such as the maximum number of reflections per ray. STATISTICAL BEHAVIOR OF THE FIELD INSIDE THE CHAMBER It as been shown (Hill, 1998) that the magnitude of each component of the electric field in a reverberation chamber is distributed with two degrees of freedom (Rayleigh distribution); the received power being distributed (exponential distribution). This property is often used to evaluate experimental results, for example: the stirring efficiency; the obtained cumulative distribution function (CDF) can then be compared with the CDF of a theoretical (or ) distribution. The same approach (comparison with theoretical CDF) is used to evaluate the simulations. In a first step, an empty cavity is studied; its dimensions are equal to 1.8m x 2.6m x 2.9m. The emitter is an omnidirectional antenna; the field is computed at 90 randomly distributed receiving points and then, the cumulative distribution function of the electric field is determined. Figure 2 shows the CDF of the Ez component, calculated at 1GHz for different numbers of reflections, compared to the CDF of a theoretical distribution. The parameter N in all figures corresponds to the maximum number of reflections on each wall of all rays taken into account in the computation. The following table presents the number of images corresponding to N.
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We can remark, in figure 2, that the distribution is better approximated when the number of reflections becomes higher and higher. The difference between the curves (theoretical and simulated) is evaluated by the difference between their integrals. Figure 3 shows this difference versus N for the three components of the electric field. We can remark that from N equal to 20, this number of reflections (531441 rays) is sufficient to evaluate the statistical behavior of the field, in this case, although the convergence is not reached.
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LOW FREQUENCY ANALYSIS The frequency range of the optical approach is limited in the low frequency band since the wavelength must remain greater than the geometrical dimensions of the studied problem; indeed, the wave must be considered as locally plane at the reflection point. In spite of this frequency limitation, low frequencies have been considered because only the statistical aspects of the field are studied and not its exact value at any receiving point. It is verified that low resonant frequencies, in accordance with modal theory, can be found with the rays method as illustrated in figure 4. In this example, the frequency response, computed (with N = 40) for a given emitter-receiver configuration gives a resonant frequency at f = 110.8 MHz, the same as in modal theory.
As an example, figure 5 shows the total received field and each component versus receiver position along the x axis, for this resonant frequency equal to 110.8 MHz. Although the amplitude is not precise, the shape of the curves is correct.
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A statistical analysis of the field computed at 90 randomly distributed receiving points has also been made at 100 MHz, in the same manner as previously. The results are given in figure 6, compared with others computed at l0GHz.
Analysis of these curves shows that all the curves tend to a constant but the "convergence" is slower at low frequency, this remark is in accordance with the fact that at high frequency, the mode density is important and thus the field amplitude is more fluctuant. Note: For the time being, this study at low frequencies has been made in an empty cavity. In that band, the number of modes is small and thus, it is possible that, for a resonant frequency, the comparison with distribution is not an efficient condition to reduce the number of reflections, this problem is now under experimentation.
CONLUSION Frequency limitation of classical method requires the use of asymptotic approach, like ray tracing and image theory, to compute the field at high frequency. Due to the high conductivity of the metallic walls of the cavity (and then, to high reflection coefficients), the main problem of these methods is the difficulty to reach the convergence of the results during a reasonable computation time. As an example, simulation of an empty cavity with our computer can use few minutes for 90 points with N=10 but 6 hours for N=50! Comparison between simulation and theoretical distribution has been chosen in order to reduce the number of reflections per ray taken into account in the model. Nevertheless, this reduction allows the exact determination of the impulse response and then, the quality factor of the chamber (Tesche, 1997). The amplitude of the field computed in these conditions is not exact but its statistical behavior is correctly predicted, thus this model can be used as a tool to compare different characteristics of reverberation chambers: shape, material, ... The first results, in an empty cavity, are interesting but have still to be improved in order to introduce metallic objects inside the chamber and at last the stirrer in the model.
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REFERENCES
Bunting C. F., Moeller K. J., Reddy C. J., Scearce S. A., "A two -dimensional finite element analysis of reverberation chambers"; IEEE trans on EMC vol. 41, n°4, November 1999. Hoeppe F., Gineste P.N., Kone L., Demoulin B. "Simulation numerique du brassage de modes et confrontation avec des mesures de fonction cumulative de distribution d'amplitude du champ electromagnetique"; Colloque international sur la compatibilite electromagnetique, 14-16 mars 2000; Clermont-Ferrand (France). Kwon D.H., Burkholder R.J., Pathak P.H., "Ray analysis of electromagnetic field build-up and quality factor of electrically large shielded enclosures" ; IEEE trans on EMC vol. 40, n°1, February 1998. Hill D. A., "Plane Wave Integral Representation for Fields in Reverberation Chambers" IEEE trans on EMC, vol.40 n°3, August 1998. Tesche F.M., Ianoz M.V. , Karlsson T.K. "EMC Analysis methods and computational models", John Wiley & son inc, 1997.
INFLUENCE OF VARIATIONS IN THE SPECTRAL TRANSFER FUNCTION TO TIME DOMAIN MEASUREMENTS
Heyno Garbe University of Hannover Appelstr. 9A D-30167 Hannover, Germany e-mail:
[email protected]
INTRODUCTION It is often seen, that generating electromagnetic fields as very fast rising transients or ultra wide band signals leads to the fact, that the measured field pulse is different to the output signal of the generator. Within this presentation reasons for this effects are discussed. Therefore the behaviour of the test facility will be described in frequency as well as in time domain. The main goal is to show the links between them and to present relations error margins for time and frequency domain.
GENERAL DESCRIPTION OF FIELD GENERATION Generally the influence of a test facility on a signal can be described as shown in fig. 1.
Using these model, it is well known that the signal y(t) is given by the convolution of the generator output (u(t)) and the transfer function (g(t)) of the test facility
or in the frequency domain as multiplication of the spectrums
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The transfer functions g(t) or describes the behaviour of an amplifier, a test site or a sensor system. Ideally the test facility should not change the shape of the signal u(t) that y(t) should be only the time delayed generator signal u(t). Therefore the ideal transfer function of the test facility has to be a dead time term. The ideal transfer function
should be
for the whole frequency range Unfortunately this requirements cannot be guaranteed by real test facilities. Most transient measurements suffer from one or more of the following limitations: 1. low pass effect This limitation is given by the upper frequency of the measurement equipment. Often it is called the frequency bandwidth. 2. high pass effect Some test facilities require a balun (balanced-unbalanced transformer). A balun is unable to transfer DC. 3. stochastic amplitude variation of frequency response Specially in field generators we can observe amplitude variation throughout the imperfections of the test site. 4. phase shifting of frequency response This effect is well know a dispersion. Higher order field modes have a propagation velocity not equal to speed of light. Therefor we have dispersion effects in field generators. It is very interesting to see that most of the effects are well defined in frequency domain. We are able to measure the transfer function of a system with very high precision. The following chapters shall link the frequency description to time domain.
LOW PASS EFFECT Most of real measurement system are acting as a sume a -System as given by
Let's assume a step-function with a rise-time two ramp-function with one delayed by
-system [1]. For simplicity we as-
as u(t). Eq. 6 shows the step-function
The spectrum of the real step is given by eq. (7) and displayed in fig. 2. The frequency in fig. 2 is displayed in radians.
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Our intention is to measure the real rise-time. Therefor we have to consider the frequency range up to the first zero-point of the si-function. This frequency is given by eq. (8). This gives the upper frequency limit of the test site.
HIGH PASS EFFECTS Similar consideration are done for high-pass systems like baluns. The transfer function of a balun is given by eq. (9). It represents a High-Pass-Filter with the lower frequency limit ofd
Feeding such a balun with a ideal step-function we get eq. 10 as output signal of the system.
To ensure a constant signal like the top of a rectangular pulse for a certain time we assume a system as given in eq. 9. Now we can calculate the lower frequency limit by taking the normalised step-response and taking the amplitude at time has to be smaller than the pre-defined time
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AMPLITUDE VARIATION OF FREQUENCY RESPONSE Field test site standards [2] allow a field variation in the specified test volume of +6dB/-0dB for a certain frequency step. Extending this requirements to transient testing this leads to a similar ± 3dB -criterion for the frequency range. Keeping in mind that 6dB means a factor of 2 this frequency-criterion seems to be not applicable for transient measurement. To model this effect we assumed a system with an amplitude response given by eq. 12. A constant factor of "1" is superposed by a random signal with a given variation. The phase of was always set to zero. An double-exponential pulse as displayed in fig. 3 was used as input signal.
The results for 10 calculations of the pulse response are displayed in fig. 4. The curves and are given too.
One can conclude that for a variation of ± 2dB the pulse response doesn't exceed the 10%-limits. The influence of ± 6dB -variation is shown in fig. 5.
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It looks a little bit worse but we only can see an error in the order of 20 % maximum. It might have been expected that this error is near by 100%. This is not correct. It can be stated that a variation of ± 2dB in frequency domain fulfils the 10%-criterion in time domain. On the other hand, the +6dB/-0dB-criterion from [2] can be seen as ± 3dB -criterion. So the frequency domain definition fits very well to the time domain. Having a more general look to this problem a dependence between the amplitude variation in dB and the variation in time domain in percentage is given in fig. 6. It can be seen that the ± 3dB -variation gives only 8%-variation in time domain. From this figure one is able to estimate the expected error in the other domain.
PHASE SHIFTING OF FREQUENCY RESPONSE As said before (see eq. 4) only time delay is accepted for an ideal transfer function. In this chapter the effects of a non perfect phase response is discussed. For TEM-waveguides such effects occur dispersion of a pulse is observed. Assuming that the TEM-waveguide is filled with a homogenous material the dispersion effects are generated by higher order mode propagation. The propagation velocity of the TEM-mode is equal to the speed of light Higher order modes are propagating with a specific but lower velocity. This leads to the fact that the spectrum of a pulse field is propagated very frequency dependently. Again we tested the system with an double-exponential pulse as displayed in fig. 3. Now the amplitude of the frequency response was set to 1. The phase of was randomly varied within the limits of 5° to 40°.
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The results for 10 calculations of the pulse response are displayed in fig. 7 and 8. The curves and are displayed, too.
Again the dependence between the phase variation in degree and the variation in time domain in percentage is given in fig. 9. It can be seen that the 20°-variation gives only 8%variation in time domain. From this figure one is able to estimate the expected error in the other domain.
Talking about phase shifting in frequency domain means time delay in the other domain. A constant phase shifting leads to a stronger influence to low frequencies than to high frequencies. On the other hand some test systems introduce a frequency dependent time delay. To study this effect the time delay of was randomly varied within the limits of 5ns to 30ns as written in eq. 14.
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The results for 30 calculations and a maximum time delay of 5ns of the pulse response are displayed in fig. 10. The curves and are displayed, too.
As depicted in fig. 11 a dependence between the maximum time delay and the pulse variation in percentage is given. A time delay of 25ns leads to a pulse variation of 8%. It has to be taken into account that a double-exponential pulse with a rise time of and a fall time of is used. Under this aspect fig. 11 can be normalised with the rise time of 5ns. The ratio leads to a pulse variation of
CONCLUSION Values have been shown to predict the time domain behaviour from given frequency data. To ensure a shape inherent pules transition the cut-off frequency has to be chosen greater than
variation of less than ± 3dB in amplitude or 20° in phase results
in a variation of less than 10% for the pulse. A variation in frequency dependent time delay for a pulse with the rise time of less than leads to a pulse variation of
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REFERENCES [1]
[2] [3]
[4] [5] [6]
[7]
Mittermayer C., Steininger A., 1999, On the Determination of Dynamic Errors for Rise Time Measurements with an Oscilloscope, IEEE Transactions on Instrumentation and Measurements, Vol. 48, No. 6, pp1103-1107 Radasky: Immunity Testing in TEM Waveguides, proposed Amendment to IEC 61000-4-20, Nov. 1999 P. Richman: Computer modelling the effects of oscilloscope bandwith on ESD waveform including arc oscillations, IEEE Int. Symposium on EMC, 1986, pp238245 P. Stenlus, B. York: On the propagation of transients in waveguides, IEEE Antennas amd Propagation Magazine, Vol. 37, No. 2, April 1995, pp39-44 R. S. Elliott: Pulse waveform degradation due to dispersion in waveguides, IRE Transactions on Microwave Theory and Techniques, Oct. 1957, pp254-257 D. Kralj, Lin Mei, Teng-Tai Hsu, L. Carin: Short-pulse propagation in a hollw waveguide: analysis, optoelectronic measurement, and signal processing, IEEE Trans. on Microwave Theory and Techniques, Vol. 43, No. 9, Sept. 1995, pp2144-2150 Shi Lihua, Zhou Bihua, Chen Bin: Time domain characterization of a system by using CW-measurement results without phase information, EMC'98 Roma, Sept. 1998, pp752-754
INFLUENCE OF THE PRECURSOR FIELDS ON ULTRASHORT PULSE MEASUREMENTS
Kurt E. Oughstun1 and Hong Xiao2 1
College of Engineering & Mathematics University of Vermont Burlington, Vermont 05405-0156
[email protected] edu 2 JDS Uniphase 1289 Blue Hills Avenue Bloomfield, Connecticut 06002
[email protected]
INTRODUCTION A useful, well-established experimental measure of the fundamental characteristics of ultrashort pulse evolution is provided by interferometric autocorrelation techniques1-3. 4 Optical pulse widths as short as 5 femtoseconds have been measured with this technique. The interpretation of these measurements has been almost exclusively based upon the popular group velocity description5,6 of dispersive pulse propagation phenomena. However, it has recently been shown7-9 that this approximate description breaks down in either the ultrashort pulse or ultrawideband signal limits when the propagation distance exceeds a single absorption depth in the dispersive, attenuative medium and so cannot be used to accurately interpret the experimental results obtained in either of these two extreme domains. In particular, the group velocity description is incapable8,9 of providing an accurate description of the precursor fields that are a characteristic of the dispersive medium and are primarily responsible for pulse distortion 10-15 in both the ultrawideband signal and ultrashort pulse domains. In this paper, we present the manner in which the precursor fields that are characteristic of a double resonance Lorentz model dielectric influence the interferometric autocorrelation function of an ultrashort optical pulse as it propagates through that linear dispersive medium. We hope that these results will stimulate careful experimental studies of this important physical phenomenon and that it may then lead to a proper accounting of the precursor fields throughout the experimental community.
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MATHEMATICAL FORMULATION
Consider a pulse envelope modulated sine wave that is initiated at the plane with constant applied carrier frequency and is propagating in the positive -direction through a double resonance Lorentz model dielectric10 with complex index of refraction (with relative magnetic permeability
Here is the undamped resonance frequency, the plasma frequency, and the phenomenological damping constant of the j th resonance line (j=0,2). This causal model provides an accurate description of both normal and anomalous dispersion in homogeneous, isotropic, locally linear optical materials when the input pulse carrier frequency is situated within the medium passband between the two absorption bands and where and The medium parameters considered here are representative of a fluoride glass and possess an infrared and a visible resonance line, with associated relaxation times and respectively. The propagated optical field is given by the exact FourierLaplace integral representation10
for all where is the temporal frequency spectrum of the initial pulse envelope function u(t), and where a is a constant that is greater than the abscissa of absolute convergence10 for u(t). The spectral amplitude of the optical field satisfies the Helmholtz equation
where
is the complex wavenumber of the plane wave field with propagation factor and attenuation factor The normalized interferometric autocorrelation function of the pulse at any fixed propagation distance is given by2
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which contains both intensity autocorrelation and third-order correlation information. The analytical description of the dynamical pulse evolution is usually determined under the slowly-varying envelope approximation in which the Taylor series approximation of the complex wavenumber about the carrier frequency
where is truncated after only a few terms with some undefined error. Typically, the cubic and higher-order terms in Eq. (6) are neglected5.6, in which case the quadratic dispersion approximation results, as is given by
Within this approximation, the pulse is found to propagate at the classical group velocity where
while the quantity dispersion5.6.
results in the so-called group velocity
NUMERICAL RESULTS It is now known7-9 that the group velocity description of both ultrawideband signal and ultrashort pulse dispersion breaks down as the propagation distance exceeds a few absorption depths into the dispersive, attenuative medium, this failure being due to its inability to properly describe the precursor field structures that are a characteristic of the dispersive medium. The question remains regarding the influence of these precursor fields on interferometric autocorrelation measurements and whether or not these effects are measurable. To that end, consider an infinitely smooth, unit amplitude Van Bladel16 envelope function
and is zero elsewhere, which has compact temporal support with an input full pulse width For an input five cycle pulse at the applied carrier frequency which occurs at the minimum dispersion point in the medium passband the initial pulse width is with equal rise and fall times so that In this case, and 99.98% of the input pulse spectral energy is contained in the medium passband8,9. The
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dynamical field evolution and associated interferometric autocorrelation functions due to this input pulse are depicted in Figures 1-4. The upper diagram in each figure depicts the temporal field evolution at a fixed propagation distance expressed in terms of the absorption depth in the dispersive medium at the input carrier frequency, and the lower diagram depicts the associated interferometric autocorrelation, where the solid curves denote the numerically determined behavior using the full Lorentz model given in Eq. (1) of the frequency dispersion, while the dotted curves denote the approximate behavior using the cubic dispersion approximation
obtained from the four term Taylor series approximation of about This approximation is often used since it includes the effects of pulse asymmetry into the approximation.
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At the only observable differences in the propagated field structures for the full and approximate dispersion relations occur at the leading and trailing edges of the pulse, and there are hardly any discernible differences in the corresponding interferometric autocorrelation function, as seen in Fig. 1. At the group velocity description of the propagated field structure has begun to break down, as clearly seen in Fig. 2, and discernible differences are now evident in the interferometric autocorrelation. At the middle precursor fields have begun to dominate the propagated field structure as the group velocity description rapidly decreases in its accuracy, as seen in Fig. 3. The structure of these middle precursors is reflected in the outer peaks in the interferometric autocorrelation function. This behavior continues at as seen in Fig. 4. At this point, the group velocity description has completely failed in its description of both the field and the resultant interferometric autocorrelation function. At the propagated field is found to be completely dominated by a pair of interfering middle precursors and a pair of interfering Brillouin precursors due to the leading and trailing edges of the pulse. The low frequency Brillouin precursors provide the observed slowly varying envelope structure in the
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interferometric autocorrelation while the high frequency middle precursors lie on top of this structure. This behavior continues for all larger propagation distances as the leading and trailing edge Brillouin precursors dominate the propagated field evolution.
These results clearly show that interferometric autocorrelation measurements can reveal the presence of the precursor field structures that are a characteristic of the dispersive medium for ultrashort pulse propagation when the propagation distance is sufficiently large so that the group velocity description has broken down8,9. For a ten-cycle Van Bladel envelope pulse, this will typically occur when while for a single cycle pulse of this type, this will typically occur when Similar results will also be obtained for other smooth envelope pulses, such as the gaussian envelope pulse. However, for rectangular envelope pulses of arbitrary length, the accuracy of the group velocity description will typically begin to break down when
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ACKNOWLEDGMENT
This research has been supported by the United States Air Force Office of Scientific Research Grant #F49620-94-1-0430. REFERENCES 1. D. J. Bradley and G. H. C. New, Ultrashort pulse measurements, Proc. IEEE 62:313 (1974). 2. J. C. M. Diels, J. L. Fontaine, I. C. McMichael, and F. Simoni, Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy, Appl. Opt. 24:1270 (1985). 3. F. Hache, T. J. Driscoll, M. Cavallari, and G. M. Gale, Measurement of ultrashort pulse durations by interferometric autocorrelation: influence of various parameters, Appl. Opt. 35:3230 (1996). 4. A. Baltuska, Z. Wei, M. S. Pshenichnikov, and D. A. Wiersma, Optical pulse compression to 5 fs at a 1-MHz repetition rate, Opt. Lett. 22:102 (1997). 5. Y. R. Shen. The Principles of Nonlinear Optics, Wiley-Intersclence, New York (1984). 6. G. P. Agrawal. Nonlinear Fiber Optics, Academic, Boston (1989).
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7. K. E. Oughstun and C. M. Balictsis, Gaussian pulse propagation in a dispersive, absorbing dielectric, Phys. Rev. Lett. 77:2210 (1996). 8. K. E. Oughstun and H. Xiao, Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium, Phys. Rev. Lett. 78:642 (1997). 9. H. Xiao and K. E. Oughstun, Failure of the group-velocity description for ultrawideband pulse propagation in a causally dispersive, absorptive dielectric, J. Opt. Soc. Am. B. 16:1773 (1999). 10. K. E. Oughstun and G. C. Sherman. Electromagnetic Pulse Propagation in Causal Dielectrics, Springer-Verlag, Berlin (1994). 11. K. E. Oughstun and G. C. Sherman, Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium), J. Opt. Soc. Am. B 5:817 (1988). 12. K. E. Oughstun and G. C. Sherman, Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium), J. Opt. Soc. Am. A 6:1394 (1989). 13. S. Shen and K. E. Oughstun, Dispersive pulse propagation in a double resonance Lorentz medium, J. Opt. Soc. Am. B 6:948 (1989). 14. K. E. Oughstun and G. C. Sherman, Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium, Phys. Rev. A 41:6090 (1990). 15. C. M. Balictsis and K. E. Oughstun, Uniform asymptotic description of ultrashort gaussian pulse propagation in a causal, dispersive dielectric, Phys. Rev. E 47:3645 (1993). 16. J. Van Bladel. Singular Electromagnetic Fields and Sources, Oxford U. Press, Oxford (1991).
DIHEDRAL REFLECTOR CALIBRATION FOR UWB RADAR SYSTEMS
Adeline P. Lambert and Paul D. Smith Department of Mathematics and Computer Science, University of Dundee, Dundee DD1 4HN, Scotland
INTRODUCTION Dihedral reflectors are widely used to calibrate radar systems, particularly for polarimetric measurements. In transient measurement systems their response must be calibrated across the bandwidth of operation, which generally includes wavelengths both short and long compared to reflector dimensions. As a result, accurate wide bandwidth methods for the prediction of their ultrawideband (UWB) signature are highly desirable. This paper considers the responses of such reflectors, comparing theoretical/numerical predictions with experimental measurements, and demonstrates a successful, fully polarimetric calibration which is important for ultrawideband systems. Both the calculations and the measurements are performed directly in the time domain. When considering the response of canonical scatterers, a variety of methods are available - each with its own advantages and disadvantages. The choice of any particular method is dependent upon the problem specifications, the required accuracy and the resources available. For large scatterers, high frequency (ray) techniques provide good approximations in the frequency domain (see, for example Polycarpou et al, (1995)). However, for targets with dimensions of the order of a wavelength, or when the information is required over a large frequency range, alternative methods must be considered. The time domain electric field integral equation (TD-EFIE) is ideally suited to the direct calculation of the wideband response of small to intermediate size scatterers (see, for example Rynne (1991)). This approach has already been successfully demonstrated for the optimization of the radiated field of TEM horns (see Lambert et al, (1995)) and in the calculation of the characteristic impedance of such antennas (see Booker et al, (1994)). Because of its suitablility and ease of application to problems of this type, this method has therefore been selected for the prediction of the UWB signatures of the reflectors.
"© British Crown Copyright 2000. Published with the permission of the Defence Evaluation and Research Agency on behalf of the Controller of HMSO".
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Description of the Dihedral Reflectors Before going on to detail the experimental and computational methods employed in the analysis, we shall now describe briefly the types of dihedral reflector considered. The reflectors used in the simulations and measurements were square 90° corner reflectors of side dcm (and their ground plane equivalents). The dimensions were chosen in the range 1-4 pulsewidths, the pulsewidth being of the order of l00ps. In addition to considering different sizes of reflector, both horizontal and vertical electric field orientations were examined, as indicated in Figure 1.
EXPERIMENTAL METHOD The measurements were obtained at the D.R.A. facility at Pershore and are detailed below. The facility consists of a large elevated ground plane and the experimental set-up is shown schematically in Figure 2.
The antenna used was a 30cm on side, 30° half-angle triangular plate TEM horn. A 3.46kV Kentech pulser was used to produce a step input to the TEM horn. Prior to measuring the back-scattered field from the reflectors, measurements of the background field were taken. These were later subtracted from the measured waveforms, thereby increasing the sensitivity of the measurement scheme. The electric field reflected from the dihedral was detected using a 5mm D-dot sensor. This size of sensor was selected in order to achieve a balance between resolution of the frequencies of interest in the signal and dynamic detection range. The resulting voltage was then measured on
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the Tektronix 11801A oscilloscope and stored for future analysis. Note that, all of the experimental results presented in this paper have first been filtered to remove responses outside the 20GHz bandwidth of the oscilloscope. In order to compare the experimental results with the numerical predictions, it was necessary to establish the magnitude of the incident field on the dihedral. This was achieved by measuring the voltage on the sensor in the absence of the dihedral, and determining the incident field using the equation
where is the sensor calibration constant. This constant is related to the cone radius and half-angle as follows:
Thus, the maximum electric field at the sensor was taken to be The measured 10 – 90% risetime of the step pulse from the generator was 86ps. In the initial time window, the TEM horn effectively radiates the first derivative of the applied voltage and so, for an 86ps risetime, the corresponding (1/e) pulsewidth is With a sensor/dihedral distance of 2.86m, the electric field incident on the dihedral was therefore taken to be a Gaussian pulse of the form
where
denotes the specified time offset.
COMPUTATIONAL METHOD As explained previously, the computational method selected to predict the signatures of the dihedral reflectors was a time-marching solution of the electric-field integral equation (EFIE). The details of the solution scheme are discussed fully in Rynne, (1991), and only a brief description will be given here. Essentially, the four governing equations are the EFIE, supplemented with the continuity equation and the retarded time definitions of the scalar and vector potentials, as listed in (3)-(6)
In order to produce a discrete system of equations, the surface of the scatterer is first decomposed into a mesh of triangles. The surface current density is then expanded in the form
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where denotes the unknown current density coefficient across the internal edge and denotes the vector basis functions originally introduced by Rao, Wilton and Glisson, (1982), for frequency domain EFIE analysis. The surface charge density is approximated using piecewise constant basis functions on each triangle, such that
A Galerkin-type Method of Moments approach is then used to test equations (3) and (4) with the basis functions and respectively. Finally, after discretizing in time, the transformed equations result in a system in which and J are calculated successively at each time step, from previously calculated values. Time-marching systems of this type are prone to instability, where small errors are amplified at each time step producing a characteristic exponentially increasing instability at later time. As explained in Rynne, (1991), two methods are introduced to circumvent these problems. The first is to use a predictor-corrector stage in the calculation of the surface current density and the second is to apply temporal averaging when necessary. Thus, in order to calculate the predicted backscattered electric field from the dihedral reflectors, the incoming pulse was taken to be a Gaussian of the form (2). The time-marching EFIE was then used to determine the surface current and charge density histories. The far magnetic field at a distance R was then calculated using the far-field approximation
where
denotes the unit vector in the direction of R.
COMPARISON OF RESULTS Having defined the electric field incident on the dihedral by (2), the EFIE code was used to predict the scattered far-field from a series of corner reflectors in both horizontal and vertical configurations. The dimensions chosen were and 12cm, corresponding to effectively 1,2 and 4 pulsewidths respectively. Measurements were also made with 24cm reflectors but the computational effort required to predict the signature of such large reflectors made comparisons unfeasible. In order to compare the results with the experimental data, two options were considered: 1. integrate the measured sensor voltage using (1) and compare the resultant electric field with the far-field computed using the EFIE code 2. differentiate the computed far-field data, to obtain the corresponding predicted voltage waveform at the sensor, and compare the results directly with the experimental data.
Each option has both advantages and disadvantages. Obtaining the electric field by integration of the sensor voltage is preferable from a physical point of view but can cause problems when noise in the measured data results in offsets in the calculated fields. In contrast, differentiating the smooth computed far-field waveform is preferable
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computationally. However, because the corresponding sensor voltage waveforms are the derivative of the incident field, non-ideal features in the radiated pulse can appear more pronounced when this method of comparison is used. In order to demonstrate the problems which can arise when using the first approach, the results for the 3cm dihedral reflector illuminated by a horizontal polarization pulse are presented in both voltage and electric field formats. The remaining results are given only in equivalent sensor voltage format. First consider the 3cm dihedral reflector. Figure 3 compares the measured voltage waveform on the sensor with that calculated by differentiating the computed electric field. In Figure 4, the voltage at the sensor has been integrated and compared directly with the computed electric field, illustrating the offset introduced by integration procedure.
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The results for the 6cm dihedral reflectors, with both horizontal and vertical polarizations are shown in Figures 5 and 6
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Finally, for the 12cm dihedral, Figures 7 and 8 demonstrate the corresponding horizontal and vertical polarization results.
CONCLUSIONS The results presented in this paper highlight several important points about the calibration and measurement of the ultrawideband signatures of dihedral corner reflectors The main specular response of the dihedral targets calculated by the EFIE code is in excellent agreement with the experimental results, for all targets and polarizations considered. The fact that the code reproduces the experimental pulsewidths confirms that the early time portion of the radiated pulse is indeed a Gaussian, whose (1/e) pulsewidth can be calculated directly from the 10 – 90% risetime of the generator pulse.
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The experimental data confirm that, providing background subtraction is used in the measurements, the facility at D.R.A. Pershore can accurately detect the response of small to intermediate size targets. Both the experimental and computed responses resolve the reflection from the edges of the dihedrals as well as the main specular response. The excellent agreement between the experimental and predicted responses means that we can have confidence both in the measurement system and in the use of the EFIE code for calibration exercises on canonical structures such as the dihedral corner reflector.
REFERENCES Booker, S.M., Lambert, A.P. and Smith, P.D., 1994, A determination of transient antenna impedance via a numerical solution of the electric field integral equation, J. of Electromagnetic Waves and Applications, Vol 8, Nr 12, pp 1669-1693 Lambert, A.P., Booker, S.M. and Smith, P.D., 1995, Transient design parameters for optimising radiated pulse, in Agard Conference Proceedings 564 - High Power Microwaves (HPM), 2-5 May 1994, Ottawa, Canada Polycarpou, A.C., Balanis, C.A. and Tirkas, P.A., 1995, Radar cross section of trihedral corner reflectors : theory and experiment, Electromagnetics, Vol 15, Nr 5, pp 457-484 Rao, S.M., Wilton, D.R. and Glisson, A.W., 1982, Electromagnetic scattering by sur faces of arbitrary shape, I.E.E.E. Trans. Ant. Prop., Vol AP30, nr 3, pp 409-418 Rynne, B.P., 1991, Time domain scattering from arbitrary surfaces using the electricfield integral equation, J. of Electromagnetic Waves and Applications, Vol 5, nr 1, pp 93-112
IMAGE RECONSTRUCTION OF THE SUBSURFACE OBJECT CROSSSECTION FROM THE ANGLE SPECTRUM OF SCATTERED FIELD
A. Vertiy * l, 2 , S. Gavrilov 1, 2, A.O. Salman1, I. Voynovskiy1 1
TUBITAK – MRC, Turkish-Ukrainian Joint Research Laboratory, TURKEY Marmara Research Center, P.O. Box 21, 41470 Gebze – Kocaeli, Turkey 2 IRE, National Academy of Sciences of Ukraine, UKRAINE
INTRODUCTION It is well known that the scattered electromagnetic field may be expanded in terms of plane waves. A part of the nonevanescent spectrum of the plane waves may be obtained by measuring of the scattered field amplitude and phase at an arbitrary distance from the boundary “air-medium” (Chommeloux, Pichot, Bolomey, 1986; Vertiy, Gavrilov, 1998). The evanescent spectrum of the plane waves corresponds to nonhomogeneous plane waves in the spectrum of the scattered field with amplitudes exponentially damping with increase of distance from the interface. Use of these various parts of the scattered field spectrum allows obtaining different images of the object investigated. Our experiments were carried out at frequencies from 3 to 4 GHz. The following steps were taken to obtain the images: irradiation of the ground with underground objects by electromagnetic wave at normal incidence( or an excitation of a dipole source, placed in medium); rotation (scanning along circuit) of the receiving antenna in respect to a boundary “airmedium” normal; measurement of real and imaginary parts the total super high frequency scattered signal (spectrum of the plane waves) in dependence on the inclination angle of receiving antenna at all frequency points; storage of experimental data in PC data files; calculation of the object image from the spectrum data. In the experiment, data on back-scattered field (or data on the transmitted field) at 32 frequencies in the operating band with a constant frequency step were used. A box with objects have been filled by the sand and had the sizes of The images of the irradiating dipoles, considered as models of the scattering centers of an object, were reconstructed from the measurement data.
Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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THEORETICAL CONSIDERATION In this part we will briefly consider the image processing. For image reconstruction of objects, a plane wave spectrum of scattered field is used. It means that the scattered field at line (1 -D case, Figure. 1) is represented in the form of Fourier integral
where
is Fourier image of
where
and it is defined as
is wave number of plane wave in free space;
is cyclic frequency; is velocity of light. Function is the angle spectrum of the scattered field. Variable is the space frequency in equation (1) and it also defines a direction of propagation of plane wave in expansion of the scattered field in terms of the plane waves. It is seen from Fig. 1. The following relation connects value and angle
Function
may be written as
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where
T is Fresnel transmittance of the boundary between medium1 and medium 2 with dielectric permittivities (air) and respectively; is dielectric permittivity of vacuum;
is relative dielectric permittivity of medium 2;
respectively;
is magnetic permeability of the vacuum.
are the wave number and conductivity of the medium 2, Function
where
may be written in an integral form
are functions of the variables is an angle of incidence; symbol S denotes that integration is over the
cross section S of object under investigation; function represents normalized the polarization current which is sought for. Note that at obtaining formulae (5), (6) the results of the work (Chommeloux et. al, 1986) were used. From formulae (3), (4) and (6) one can see that function is associated with the image function on a direct line will look like
and this function depends on the angle. If the viewpoints are not and they are on a circle C with a radius R then the function
Our aim is to obtain the function from the measurement of complex amplitude of plane wave (component of the spectrum of the plane waves of scattered field) on a circle C with a radius R (See Figure.2). Figure 2 shows the scheme for such measurement. Let us consider this scheme. We suppose that the x - axes divides free space on two infinitive half-spaces 1, 2 filled by homogeneous dielectrics having dielectric constans (air, y < 0) and (sand, y > 0). The plane y = 0 is the surface of the sand. There is a scattering center D under the surface. Generally, this center can be exited by the field of electromagnetic wave, source of which is placed in region 1. In our case the generator G (from Network analyzer) exits the center D by using coaxial line with the transmission coefficient The dipole D radiates electromagnetic wave into the party of the surface of the half-space 2. We can characterize the propagation of this wave by the transmission coefficient of dielectric layer between the dipole D and the surface of the half-space 2. The wave radiated by the dipole D creates on the surface of the half-space 2 the distribution of electromagnetic field. If we will take Fourier transform of this function we will obtain the angle spectrum of electromagnetic field in the half-space y < 0. Using Network analyzer we can measure the complex transmission coefficient T between the reference planes (generator) and
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(generator) and (detector) for each angle is integer numbers; (2m +1) is the number of angles) at the scanning by the antenna A along the circle C of the radius R. This transmittance for can be written in the form
where
is complex amplitude of electromagnetic field in reference plane
antenna A is placed at
when
is complex amplitude of electromagnetic field in
reference plane is the coefficient of coupling of the field with the plane wave propagating in the direction of the angle is the transmission coefficient of the free-space layer between the plane
and reference plane
of
antenna
is the transmission coefficient of
antenna A . Similar equations can be written for any position C
Here the transmission coefficient
of antenna A on the circle
is defined for the free-space layer between the plane
IMAGE RECONSTRUCTION OF THE SUBSURFACE CROSS-SECTION
and reference plane
of antenna A at the position
is the coefficient of coupling of
the field with the plane wave propagating in the direction of the angle transmission coefficient can be written in the form (Goodman, 1968)
The coefficients of coupling
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The
are defined by equations
where are complex amplitudes of the plane waves propagating in directions respectively. They are the angle spectrum of electromagnetic field in the space y < 0. Let us suppose that in the calibration process we can determine the complex coefficient then
and
as for reference planes tangential to the line C. With using definitions (13), (14), the equation (16) can be rewritten as
From (17) one can see that if we will measure the transmission coefficient in each n - th position of the antenna A, we will determine the angle spectrum of electromagnetic field in half-space y < 0 up to the complex constant
After that it is possible, using tomographic algorithm (Chommeloux, Pichot, Bolomey, 1986; Vertiy, Gavrilov, 1998), obtain the image of the center D.
EXPERIMENTAL SETUP The scheme of experimental setup one can see in Figure 3.
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The experimental setup has next parts: 1) 2) 3) 4) 5) 6) 7) 8) 9)
Network analyzer; Microwave amplifier; Receiver antenna; Radiating source (dipole); Semicircle scanner; Stepping motor control block; Box filled by sand; Table for the fixing of the scanner; Absorber.
The setup operates as follows. Signal from Network analyzer, operating in “transmission “ regime, is increased by the microwave amplifier and after that exited the source The transmitted signal is received by the antenna which is fixed on the scanner and moved along semicircle of the radius with constant the angle step The plane wave component from the plane wave spectrum of the radiated field propagating in direction of antenna is received of antenna. The signal from an output of the antenna acts on an input of the analyzer. The dipole was placed in different points on the surface and under surface of the sand. Scheme of the construction, the VSWRfrequency characteristic and the radiation pattern of the dipole are shown in Figures 4, 5a), b), respectively. Before measurements the antenna is placed in the center of the circle the dipole is placed under antenna on the surface of the sand and the system is calibrated such way, that on all 32 frequencies from the work band taken with constant the frequency step complex transmission coefficient On the
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following step the antenna is reseted (n = –m) and the data acquisition begins. The data are real and imaginary parts of complex transmission coefficient for each angle of the antenna at all 32 frequencies.
RESULTS OF EXPERIMENT Several schemes of the dipole position were studied in experiment. The images of the radiating centers were reconstructed using described above the measurement method and tomography algorithm. The image function is the modulus of the normalized polarization current |K| distributed in the sand around the radiating center. In Figures 6a), b), c) one can see the images of the source placed on the surface of sand. Figure 6a) shows the source shifted on distance from the center of the circle C (See scheme on the right); in Figure 6b) and for Figure 6c). The schemes on the right show the position of source and values of the shift
*
Victor Spetanyuk has done design and realization of this dipole antenna.
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Figures 7a), b), c) illustrate the images of the source in the case when the source is placed under surface of the sand on the depth Schemes of positions and values of and d are shown on the right. It is possible to see that the described method allows reconstructing the image of radiation–producing center and to find the positions of the center. The shape of the investigated source close to tomographic image of the point scattering center (Chommeloux, Pichot, Bolomey, 1986). The images have the sizes in depth large, than in transversal direction. The positions of images of the source are pursuant to actual positions of the source. There is an interest to investigate the resolution of the described method. Results of such experiment are presented in Figures 8a), b), c). This Figures show the images of two identical dipoles placed on the surface (Figure 8a)), under surface (Figure 8b), c)). In Figures 8a), b) dipoles are shifted in opposite directions along the x -axis on the distance (Fig.8a), b)) and in the depth (Figure 8c)). In last case the dipole 1 is placed at
In this figure one can see good resolution “in
cross-range” (along x -axis). The deference in the depth is not enough for the reconstruction of the images of two sources 1, 2. The real positions of the sources are shown on the right.
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Except for described experiments, the experiments with a metal scattering rod were conducted. The rod was placed inside a marble plate near to its surface or on depth Schemes of these experiments are shown in Figures 9a), b), c) on the right. Figures 9a), b), c) show the reconstructed images of scatterers 1, 2. These scatters are 1) 1-empty hole, 2 -hole with the inserted metal rod; 2) 1 - hole with the inserted metal rod, 2-empty hole; 3) 1-empty hole, 2-empty hole. The hole 1 was made close to surface and the hole 2 was made at The holes have diameter Figure 9a) illustrates the case when the metal rod is placed in hole 1; Figure 9b) - the metal rod is placed in the hole 2 and Figure 9c) – the metal rod is absent. In the last case two empty hole can be considered as scatterers and Figure 9c) illustrates this case. It was used rectangular adapter AD for the excitation of electromagnetic wave propagating inside of the marble plate. Into the adapter was inserted the end of the marble plate with thickness The adapter AD associates the coaxial line with the dielectric waveguide and excites inside of the dielectric plate the waveguide mode which propagates along the plate. This wave is scattered by the scatterers 1 and 2. The results of the reconstruction presented in Figures 9a), b), c) show that in the considered case one can see size (shape) of scatterers and they positions. In the case Figure 9c) the two scatterers are not resolved. For observing two scatterers distributed In depth it is need the distance between scatterers larger, then CONCLUSION Thus the conducted experimental investigation in the frequency range of 3÷4GHz showed that described setup and signal processing methods allow to figure the shape and to estimate the cross-section size of objects buried in sand. We can also evaluate depth on which the researched object is loaded using the object cross-section image in the plane perpendicular to the medium surface. The results obtained may be applied in practical microwave imaging systems for detection and observation of different undersurface objects.
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REFERENCES Chommeloux L. Pichot, Ch. Bolomey, J. – Ch., 1986, Electromagnetic modelling for microwave imaging of cylindrical buried inhomogeneties, IEEE Trans. Microwave Theory Tech., Vol. MTT – 34, No. 10, pp. 1064 – 1076. Vertiy A. A., Gavrilov S. P., 1998., Modelling of microwave images of buried cylindrical objects, InternationalJournal of Infrared and Millimetre Waves, Vol. 19, No. 9, p p. 12011220. Vertiy A. A., Gavrilov S. P., Tansel B, Voynovskyy I.V., 1999, Experimental investigation of buried objects by microwave tomography methods, Part of the SPIE Conference on Subsurface Sensors and Applications, Denver, Colorado, SPIE Vol. 3752, pp. 195-205. Goodman J.W., 1968. The book. Introduction to Fourier Optics.-McGraw-Hill Book Company, San Francisco-New York-St.Louis-Toronto-London-Sydney.
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OPTIMAL ACOUSTIC MEASUREMENTS
Margaret Cheney1,2, David Isaacson1, and Matti Lassas3 1
Department of Mathematical Sciences Rensselaer Polytechnic Institute, Troy, NY 12180 USA 2 Department of Electromagnetic Theory Lund Institute of Technology, Lund, S122 00 Sweden 3 Department of Mathematics P.O. Box 4, 00014 University of Helsinki, Finland
INTRODUCTION This paper is motivated by the question “How can we design the best possible system to do acoustic imaging?” If we want to make the best possible images, we must begin with data that contain the most possible information. In particular, since all practical measurements are of limited precision, some scatterers may be undetectable because their scattered fields are below the precision of the measuring instrument: our data will contain no information about them. What incident fields should we apply that will result in the biggest measurements? There are many ways to formulate this question, depending on the measuring instruments. In this paper we consider a formulation involving wave-splitting in the accessible half-space: what downgoing wave will result in an upgoing wave of greatest energy? A closely related question arises in the case when we have a guess about the configuration of the inaccessible half-space. What measurements should we make to determine whether our guess is accurate? In this case we compare the scattered field to the field computed from the guessed configuration. Again we look for the incident field that results in the greatest energy difference. This optimal measurement problem has been studied for fixed-frequency problems in electrical impedance tomography (Isaacson, 1986) and acoustic scattering (Mast, Nachman, and Waag, 1997). For time-domain problems, the issue of optimal time-dependent waveforms in a special 1 + 1 – dimensional case was studied Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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in (Cherkaeva and Tripp, 1997), where a time-harmonic waveform was found to be optimal. In this paper we study the question of optimal time-dependent waveforms in the 3 + 1 – dimensional case. In particular, we consider the half-space geometry: we imagine that a plane divides space into accessible and inaccessible regions, and we assume that we can make measurements everywhere on the plane. We show that the optimal incident field can be found by an iterative process involving time reversal “mirrors”. The connection between optimal measurements and an iterative time-reversal process was first pointed out in (Mast, Nachman, and Waag, 1997), (Prada and Fink, 1994), and (Prada, Thomas, and Fink, 1995); in all these papers, the analysis was carried out a a single frequency. In this work we consider time-dependent fields. We show that for band-limited incident fields and compactly supported scatterers, the iterative time-reversal algorithm converges to a sum of time-harmonic fields. This provides a theoretical foundation for the pulse-broadening observed in the computations of Cherkaeva and Tripp (1997) and in the time-reversal experiments of Prada and Fink (1994), and Prada, Thomas, and Fink (1995). Section 2 contains a careful formulation of the idealized problem: the wave equation model, the measurements, the notion of “biggest”. Section 3 gives the adaptive experimental algorithm that can be used to find the optimal field even if the scatterer is unknown. In general the iterates converge to a time-harmonic field. The paper concludes with a discussion. Full details can be found in (Cheney, Isaacson, and Lassas, 2000).
PRECISE FORMULATION OF THE PROBLEM We consider the constant-density acoustic wave equation
in the case in which everywhere in the upper half-space Here This model includes neither dispersion nor dissipation. We consider the half-space geometry, in which the lower half-space is inaccessible and the upper half-space is accessible. The measurements we consider are those in which we send a downgoing wave into the lower half-space and measure, on the plane the corresponding upgoing wave. “Upgoing” and “downgoing” can be defined by means of Fourier transforms (in time and in the lateral space variables see (Cheney, Isaacson, and Lassas, 2000) for details. We denote by S the scattering operator that maps the downgoing wave to the upgoing wave on the plane The scattering operator for a reference configuration is denoted
by In order to find the downgoing field that maximizes the difference between the scattered waves and we want to maximize the quotient
Here it is necessary to divide by energy we can apply is always finite.
to account for the fact that in practice the
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In (2), we need to define the measured of “size” || · ||. We do this with the help of the energy identity
where the right side corresponds to the increase in energy in the volume V and the left side is the energy flux through the surface We thus use as our measure of “size” the total (time-integrated) energy flux into the region
(In the electromagnetic case, the corresponding quantity would be the integral of the Poynting vector, dotted with the unit normal vector integrated over the plane and integrated over all time.) Thus the best downgoing field is the one that maximizes
AN ADAPTIVE METHOD FOR PRODUCING THE BEST FIELD To maximize (4) in the case when the medium is unknown, we can use the following adaptive method. 0) Begin with any downgoing wave 1) Send into the lower half-space; measure the resulting upgoing field
2) Calculate the corresponding scattering from the reference configuration Calculate the difference field
3) The next downgoing wave is the (normalized) time-reversed difference
add one to j; go to step 1). To understand why this iterative process converges to the right answer, imagine the case of a single point scatterer, and take the reference medium to be empty space. Any incident wave will scatter from the point scatterer into an outgoing spherical wave. If this outgoing spherical wave is time-reversed, it becomes a spherical wave that focuses on the point scatterer. Focusing all the energy of the incident wave on the scatterer gives rise to the largest possible scattered wave.
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For fixed frequencies, the iterative time-reversal algorithm has been analyzed by Prada, Thomas, and Fink (1995). They considered the case of a finite array of transducers and the corresponding “transfer matrix” in which the element of the matrix is the scattered field measured at the transducer resulting from the activation of only the transducer. Prada et al. (1995) showed that the iterative time-reversal algorithm converges to the spatial eigenfunction corresponding to the largest eigenvalue of the transfer matrix. Their paper contains diagrams that show what happens when the scatterering region contains a strong scatterer and a weak scatterer. The strong scatterer returns a stronger wave, which, when time-reversed, becomes a stronger wave focusing on the stronger scatterer. After a few iterations, the signals from the weak scatterer disappear because all energy becomes focused on the stronger scatterer. Here we consider the question of the time-domain waveform of the optimal field. In (Cheney, Isaacson, and Lassas, 2000) we analyze the problem by Fourier transforming to the frequency domain. At each fixed frequency, the scattering operator has eigenvalues, and these eigenvalues depend analytically on the frequency. Suppose the largest eigenvalue is biggest at the frequency Then in general, the iterative time-reversal algorithm converges to a time-harmonic (fixed-frequency) wave with the frequency The spatial shape of the waveform is given by the eigenfunction corresponding to this largest eigenvalue. This prediction is consistent with the experiments of Prada, Thomas, and Fink (1995), in which pulse broadening and frequency shifts are observed as the iterative algorithm proceeds. This prediction is also consistent with the one-dimensional work of (Cherkaeva and Tripp, 1997), in which the optimal time-domain waveform was found to be a time-harmonic wave.
CONCLUSIONS AND OPEN QUESTIONS This approach to the optimal measurement problem decouples the measurement issue from the problem of forming an image. However, this work shows only how to find the single best measurement for determining whether an unknown scatterer is identical to a particular guess. It is not clear how to extend this work to find a full set of measurements that would be necessary for forming an image. This analysis shows that the iterative time-reversal work of Prada and Fink (1994) and Prada, Thomas, and Fink (1995) provides an experimental method to obtain optimal fields. Moreover, this analysis explains the pulse-broadening seen in (Prada, Thomas, and Fink, 1995) and (Cherkaeva and Tripp, 1997): the optimal time-domain waveform is a time-harmonic one. This analysis suggests that the commonly-used pings and chirps are not optimal from the point of view of providing the biggest signal to distinguish an unknown scatterer from a particular guess. There are many open questions related to this work, some of which are probably easy and others hard. In particular, problems that seem to be relatively straightforward are extending this work to Maxwell’s equations and to materials in which the medium parameters depend on frequency. Extending the theory to measurements made in a limited aperture is probably not difficult, but will involve detailed modeling of transducers or antennas. More difficult problems include extending the work to dissipative media, and
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finding a full set of optimal measurements for forming an image. In the fixed-frequency case, it is clear that a full set of optimal measurements are the eigenfunctions of the operator or But the eigenvalues of these operators depend continuously on frequency, which means that there is no “next best” incident field. Thus it is not clear what finite set of measurements contain the most possible “information” in a broader sense.
ACKNOWLEDGMENTS This work was partially supported by the Office of Naval Research. M.C. would like to thank a number of people for helpful discussions: Gerhard Kristensson and his group in Lund, Jim Rose, Claire Prada, and Isom Herron.
REFERENCES 1 Cheney, M., and Isaacson, D., “Inverse Problems for a Perturbed Dissipative
Half-Space”, Inverse Problems 11 (1995) 865–888. 2 Cheney, M., Isaacson, D., and Lassas, M., “Optimal Acoustic Measurements”,
preprint (2000). 3 Cherkaeva, E., and Tripp, A.C., “On optimal design of transient electromagnetic
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waveforms”, SEG97 Expanded Abstracts, 67th Annual Meeting of Soc. Exploration Geophys. (1997) 438–441. Isaacson, D., “Distinguishability of conductivities by electric current computed tomography”, IEEE Trans, on Medical Imaging MI-5(2):92-95, 1986. Lassas, M., Cheney, M., and Uhlmann, G., ”Uniqueness for a wave propagation inverse problem in a half space”, Inverse Problems 14, 679-684 (1998) . Mast, T.C., Nachman, A.I., and Waag, R.C., “Focusing and imaging using eigenfunctions of the scattering operator”, J. Acoust. Soc. Am. 102, Pt. 1 (1997) 715–725. Prada, C. and Fink, M., “Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media”, Wave Motion 20 (1994), 151–163. Prada, C., Thomas, J.-L. and Fink, M., “The iterative time reversal process: Analysis of the convergence”, J. Acoust. Soc. Am. 97 (1995) 62–71.
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PARSIMONY IN SIGNATURE-BASED TARGET IDENTIFICATION
Carl E. Baum Air Force Research Laboratory AFRL/DEHE 3550 Aberdeen Ave., SE Kirtland AFB, NM 87117-5776
1.
INTRODUCTION
In signature-based target identification [20] the scattering dyadic of each type of target is characterized by a set of functions based on a particular scattering model (e.g., complex exponentials for the late-time response, or delta, step, ramp, etc., for the early-time response). Each of the functions is characterized by a small number of parameters (e.g., complex natural frequencies ) including a scaling coefficient (scalar, dyadic) to adjust the amplitude (perhaps including vector orientation). Each of the targets is represented in a target library by an appropriate set of such functions with parameter values particular to the individual target types (e.g., a particular type of aircraft such as a 707). The approach is to associate these parameter values with the characteristics of the electromagnetic waves scattered (usually backscattered, but not necessarily so) from the target by some appropriate radar. The problem then is to distinguish one target from another by the differences in the parameters in the scattering model(s) inferred from the measured scattered fields. Given the presence of noise in any measurement there is some ambiguity in the declaration of a particular target type because of errors in the parameter estimation. Particularly as the parameter values for one target type approach those of another type the discrimination becomes increasingly difficult. Let us distinguish between two types of parameters. One type (fixed parameters) assumes particular values (scalar, vector, dyadic) for each target type. A second type, which we might call variable parameters is adjusted as part of the process of fitting the scattering model to the data. Such variable parameters are typically the coefficients of the fitting functions, which adjust the amplitudes (not necessarily scalars) of the fitting functions to best fit the data. Since we wish to discriminate between various targets we would like it to be difficult to fit the wrong target parameters to the data. So we would like to reduce the number of variable parameters (and the range of their variation) as much as we can. This is aided by constraining these scaling coefficients (e.g., pole residues) to values appropriate to the target type to the degree practical. This leads to a principle of parsimony as [25]: “Use as few feature variables as possible to provide consistent classification.” In the present context feature variables are interpreted as variable parameters, or parameter values that aapply to multiple target types (at least approximately). 2.
FITTING WITH GENERAL FUNCTION SETS To illustrate the problem of fitting data with too many parameters, consider some target response (before or after decomvolution for impulse response) along with some noise
some parametric scattering model for which we have a set of functions infinitely many) appropriate to the mth target type. Then we try to approximate
Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
for
We have (perhaps by these functions as
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where L is available for us to choose. The
are chosen to minimize an appropriate norm of the
difference of the two sides in (2.1). Then define
where the norm || || can be defined in various ways [3, 17], including the use of various weighting functions [18] if desired. While the above functions are written as functions of time t, they can be transformed to complex frequency domain or to a wavelet/window-Fourier-transform domain [9, 31 ] and the norm can be defined in such terms.
Having chosen our norm we define the set of
difference as in (2.2) as the set
that minimizes the
There may be more than one such set in which case
we choose one of these sets at our convenience. Then we have
What now if we try again with L + 1 functions? We obtain some set This gives
Now one choice of the (before minimization) is just the with This gives as in (2.3). this choice might give the minimum in (2.4), but generally gives something larger than the minimum. Hence we can conclude
This does not necessarily imply that as but it does show that adding more functios in our target-signature set with adjustable weights makes the mth target-type signature more closely match the data from the nth target type (with or without noise). Suppose however, that the
for
form a complete set on the support of interest
(time interval, frequency interval, or even some function can be approximated by this set and
phase space). Then any reasonably well behaved
However, the are for the mth target type and we are approximating the waveform for the nth target (plus noise). In this case minimization of the norm for large L cannot distinguish between the nth and mth targets. So it is important that the
not be complete for the domain of time, freq., etc., for successful
target discrimination. We want the to apply to only the mth target (for all m in our library). Note that a complete set of functions need not be orthogonal (zero inner products on the support). Of course a nonorthogonal set can be converted to an orthogonal one (Gram-Schmidt orthogonalization [26]). Alternately, if the function set
is complete, it is important that L be limited (parsimony) so
that only terms that are dominant for target m are included. Similarly the set
needs to include only
terms that are dominant for target n. This is also a question of how to best order the i.e., which is labelled by etc. Presumably they should be placed in the order of decreasing dominance. In (2.2) and (2.3) the norms do not take account of varying signal strength as the same target is measured at various distances from the radar. Using the far-field approximation (incident and scattered fields
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varying as 1/r with the same waveforms) one can normalize these expressions as the dimensionless expressions
However, this does not account for variation in the strengths of the signals between different targets (large vs. small scatterers). So one may wish to instead normalize as (one factor of r for the incident wave, one for the scattered)
assuming one has a measure of the range r to the target. In this form the combinations
also give
information concerning the strength of the various scattering modes and can perhaps be constrained (parsimony) to give better target discrimination. Combining these ideas we can define
so that the are now range invariant. The are, however, in general still aspect (polarization, angle of incidence) dependent. While the discussion in this section has been in terms of scalar functions with scalar coefficients, vector and dyadic forms are readily considered in the same expressions, with appropriate attention to the norm used. Such forms are appropriately used with multiple radar measurements to give the scattering dyadic of the target, which is then fit by the above procedure. 3.
EXPONENTIAL FUNCTIONS
A common set of functions used for representing a time-domain signal f(t) is exponential functions (in general complex). This is readily seen through the two-sided Laplace (or Fourier) transform as two-sided Laplace transform
Bromwich contour parallel to
axis
Here the Bromwich contour is taken to the right of any singularities of f(s) in the s plane. So already we have a restriction that f (t) must be passive, i.e. with Re[s] > 0 is not allowed in representing f(t). However, as discussed in the previous section, our signal from the nth target includes noise which does not necessarily include this constraint. This implies some constraint (parsimony) on the allowable functions to represent a target and discriminate against noise. Note that the Bromwich contour extends over implying an infinite set of functions to represent any target in the library (not parsimonious). The integral over the Bromwich contour can be represented by a sum as
where now the all lie on or to the left of the axis. In this form we can see the effect of a finite sum of exponentials. As the number of suchfunctions any well-behaved passive f (t) can be represented. Instead of an infinite interval in time where the signal is eventually lost in the noise, one might consider a finite interval oftime The transform f(s) of f(t) can then be replaced by a Fourier series with The are now discrete, but generally infinite in number. Furthermore, as is well known any reasonably behaved f(t) (not necessarily passive, and including noise) can be accurately represented by such a Fourier series. So this is also not a good choice.
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Following [27] we can deform the Bromwich contour into the left half plane. In Fig. 3.1 A we see the singularities of a passive system lying in the left half plane (LHP). Since our f(t) is real valued the singularities not on the negative axis must occur in complex conjugate pairs, both for their locations and amplitudes (i.e., pole residues). As we deform our contour to the left these singularities are isolated to give separate functions which for poles (shown as first order, but not necessarily so) gives a representation as
where the entire functions corresponds to the singularity as In time domain it is an early-time contribution to the response [4]. Hence the pole series is used to represent the response for late times after the incident and scattered waves have had time to transit over the target. Already we see some parsimony at work. The poles corresponding to the mth target do not well approximate those belonging to the nth target for So discrete pole locations in the LHP are more parsimoneous than a continuous distribution (or large number) of such locations on the axis. Even more parsimoneous would be some restrictions on the residues Carrying the contour deformation yet further, suppose that our targets of interest have no singularities away from the negative axis. Then the contour collapses as in Fig. 3.1B to include only singularities there. While this can include in principle a branch cut there [4], there is an important class of targets which can be well-approximated by first order poles there. These are the diffusion poles in highly (but not perfectly) conducting metal targets of finite linear dimensions [6]. In this case the response takes the form
where the entire function is now a constant in frequency domain or a delta function in time domain. Comparing (3.4) to (3.3) we can see that (3.4) is more parsimonious in the sense that it cannot represent poles in the third and fourth quadrants of the s plane far from the negative axis. For this special class of targets appropriate to magnetic singularity identification (MSI) the form in (3.4) is more general corresponding to a decomposition of the magnetic-polarizability dyadic [6]. The dyadic residues rotate together with the target allowing constraints on them [13, 15, 16], thereby giving more aspect independent parameters (besides the to aid in the target discrimination. This is then even more parsimonious.
PARSIMONY IN SIGNATURE-BASED TARGET IDENTIFICATION 4.
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RELATION TO PRONY-LIKE FITTING OF DATA
Much work has been done extracting damped sinusoids out of data [19]. Recently, some improvement has been made with what is called matrix pencil [21 ]. Restricting the number of terms and only considering those with large coefficients (residues) helps. A more robust procedure may consist of fitting the data with only the specific exponential sets corresponding to targets in the library to see which fits best. Of course, if the real target is not in the library, consideration of how good is the fit (size of residuals) may be needed to establish this fact. Filters such as the E/K. pulse [9, 19] are one way of doing this. One has for the mth target type which annihilates the late-time response (except for noise) when it is convolved with the response of the nth target only when n = m. This is achieved by setting (for all significant natural frequencies
of the mth target type).
Prony-like fitting may still be needed to generate the natural-frequency sets for the target library from experimental data. Of course, this can be achieved from many careful measurements under more ideal conditions. Such measurements may be eventually supported by accurate numerical computations of natural frequencies from integral and/or differential scattering equations. Preliminary estimates of the can perhaps be refined by adjusting them to optimally annihilate the late time response for many target aspects with one 5.
USE OF POLARIZATION WITH TARGET SYMMETRY
With two choices of polarization one can reconstruct the backscattering dyadic. If one has the target impulse response dyadic (2 × 2) for a particular angle of incidence, one can rotate the target about this incidence direction (or equivalently rotate the radar) to match the stored data and thereby also orient the target by an angle about the incidence direction. Of course the rotation can be accomplished by a rotation of the scattering dyadic in a computer. In the process one can identify this rotation angle modulo since the scattering dyadic is invariant to a rotation by (sign reversal of incident and scattered fields). From a parsimony point of view this is a single real parameter varying over a restricted interval. The constraint of a known angle of incidence has greatly reduced the number of fitting parameters. The amplitude is also assumed to be constrained by independent knowledge ofthe distance to the target. 5.1 Target Symmetry Plane Passing Through Observer:
Symmetry
A special case of interest is that of a target with a symmetry plane (such as a typical fixed-wing aircraft) [1, 2, 29]. Referring to Fig. 5.1, let this symmetry plane lie along the direction of incidence at the target. With the usual h, v radar coordinates we have the directions (unit vectors)
with
forming a right handed system. Note that
The target symmetry plane ized by a reflection dyadic
is rotated from the vertical by an angle
giving
identity (three dimensional)
symmetrycharacter-
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If one measures the backscattering dyadic (delta-function response) by removing the antenna characteristics and range dependence, this will have three separate terms in the 2 × 2 scattering dyadic removed) since the fields of concern only have h and v components and we have
Defining a rotation matrix [30] as
coordinate
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gives a positive (counterclockwise) rotation of the coordinates as in Fig. 5.1. Suppose we have measured in the h,v coordinate system. Then we can compute
where we have rotated the scattering into the as, sy system corresponding to the antisymmetric part and symmetric part
These two parts only (parsimony) are needed to characterize this scattering,
being reduced to vectors parallel to
and
on the symmetry plane. Note that this decomposition is
frequency/time independent. Natural frequencies decompose into two sets and etc. If is not known a priori from some other measurement, then rotating the data by varying discovers to make (5.6) hold, thereby learning the roll angle. Note, however, then adding integer multiples of to also produces this diagonalization. generally has more symmetry than the target, in this case two symmetry planes giving the symmetry group
There is generally some error in target alignment such that the symmetry plane does not pass exactly through the observer (radar). This means that will generally not be perfectly diagonal but the off-diagonal components will have some minimum (norm sense over time/frequency) for particular Noting this alignment error one may wish to restrict the range of frequencies. If the maximum transverse dimension target is d one may wish to restrict radian wavelength such that
where is the angular error. If say D is 10 m and is radians then frequencies to less than about 400 MHz, with even lower uppermost frequencies as related to glint or angle noise in traditional radar systems [28]. 5.2
Body-of-Revolution Target, Including the Nearby Media:
restricting is increased. This is
Symmetry
A yet higher degree of symmetry is that of a body of revolution with axial symmetry planes giving symmetry [11]. Here the axis of revolution
is taken as perpendicular to the earth surface
which we take as the z = 0 plane, as in Fig. 5.2. The earth constitutive parameters are allowed to vary with z (layering) provided the
symmetry is preserved, including the earth.
In this case the h,v coordinates are established with the traditional convention that and
lies in the plane of incidence containing
is parallel to
and the rotation axis. In this case we have [ 11, 24]
for all making it appropriate for synthetic aperture radar (SAR) as the radar is moved with respect to the target. This is called the vampire signature due to the lack of reflection in the h, v “mirror”. Here we use the nulling of a parameter (the cross polarization) as an identifier of a class of targets. However, it does not discriminate among the various target types with this symmetry. One can go on to consider other details of the and (only two remaining now) to achieve yet further discrimination
[11].
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6.
PARSIMONY IN MAGNETIC-SINGULARITY IDENTIFICATION
For magnetic singularity identification (MSI) we have the general form of the magnetic polarizability dyadic as [6, 16]
= real unit vector for mode , (all negative real natural frequencies)
= real scalar
This applies to highly, but not perfectly, conducting targets. The correspond to exponential decays in time domain. The frequencies of interest are rather low, corresponding to diffusion into the metal targets. The incident fields are now not in the form of a plane wave, but the near fields of loops. Similarly the scattered near fields are sensed by loops. As such three-dimensional information concerning the target is available. The and are all aspect independent while the and rotate with the target, i.e. are fixed in a target-based coordinate system. As discussed in [16] by rotating the MSI signature in the target library and moving the target location (say under the earth surface) with respect to the observer location one can attempt to match the library entries to the data. In so doing there are six real parameters to adjust: three Euler angles for target orientation, two angles for direction to target, and one for distance to the target. Thereby the aspect-dependent parameters are constrained by their mutual orientation relationships. The situation is further simplified in the case of target symmetry [7, 13, 15]. For a target with symmetry with (N-fold rotation axis with no assumption of symmetry planes) the magnetic polarizability dyadic becomes
PARSIMONY IN SIGNATURE-BASED TARGET IDENTIFICATION
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where the rotation axis is taken as Here the unit vectors have all conveniently lined up to give two distinct sets, each with common orientations (common aspect dependence) of the unit vectors characterizing the pole residues. In this case there are only needed two real scaling parameters (more parsimonious) to multiply the axial and transverse functions characterized by constrained poles and residues. 7.
EFFECTS OF VARIATION IN EMBEDDING MEDIA
For targets in a uniform well-characterized medium one can use the a priori knowledge of this medium to constrain the target signatures. In particular, aspect independent parameters can be considered constants, i.e., not variables depending on the medium parameters. Such is the case for targets effectively in free space, e.g., flying aircraft and missiles. For targets in a variable medium such as earth the situation is more complicated since the signature in the scattering (e.g., natural frequencies) can be significantly affected by the constitutive parameters of the nearby earth [14]. In this case we are concerned primarily with frequencies such that wavelengths in the external medium are of the order of the target dimensions (used in electromagnetic singularity identification (EMSI)). If one has independent knowledge of the earth parameters (particularly the permittivity ), say by a nearby measurement, then one can attempt to compute the effect of such parameters on the library signatures (e.g., natural frequencies) before fitting these to the radar data. Alternatively, one can use one or more of these parameters as variable fitting parameters with the library signatures to best fit the data (less parsimonious). For targets that can be approximated as perfectly conducting in a uniform isotropic earth, there are exact scaling relationships for natural frequencies and associated modes and residues [5], making the shifting of library parameters fairly simple. For dielectric targets in such a medium the situation is more complicated. However, if the target permittivity is less than that of the surrounding earth there are applicable perturbation formulas simplifying the situation somewhat [8,10]. Realistically, earth is not uniform. In particular the earth surface can be near the target of interest, significantly changing its signature [22,23]. If the target is not too close to the interface (earth surface, either above or below) perturbation formulas can also be used [12]. In this case, distance from the interface is the parameter to be adjusted. 8.
CONCLUDING REMARKS
Parsimony in target identification then seeks to constrain the representations of the target scattering (the target signatures) so as to make it difficult for the target signature of the mth target type represent that of the nth for This implies that there be as few variable fitting parameters as possible. It is generally helpful to have aspect-independent parameters to the extent feasible because these can be constrained as a priori constants (e.g., natural frequencies), not having to assume a large number of different values for the various possible (a priori unknown) directions of incidence and polarizations. To the extent that direction of incidence and polarization with respect to the target orientation are known from other measurements, one can constrain the various scaling constants for the target-signature functions, making it harder to fit the data with the wrong target type. Target symmetry also plays a useful role in parsimony. Symmetry planes allow the 2 x 2 backscattering dyadic to be diagonalized, reducing the number of elements to be considered from three to two and giving orientation information in the process. For low-frequency MSI characterized by the magneticpolarizability dyadic we have found that the number of scaling constants can be reduced from six in the general nonsymmetrical case to two for targets characterized by symmetry for [15]. Part of the problem in target identification is the corruption of the scattering data by noise. In fitting the data with target signatures one is also fitting the noise with such signature functions. Of course we would like the fit to the noise to be poor. Furthermore, if we could distinguish the functional form of the noise (random?) and model it or remove it we might reduce this signature-fitting problem. This work was supported in part by the U. S. Air Force Office of Scientific Research, and in part by the U.S. Air Force Research Laboratory, Directed Energy Directorate. REFERENCES 1. C. E. Baum, Scattering, Reciprocity, Symmetry, SEM, and EEM, Interaction Note 475, May 1989. 2. C. E. Baum, SEM Backscattering, Interaction Note 476, July 1989. 3. C. E. Baum, The Theory of Electromagnetic Interference Control, Interaction Note 478, December 1989; pp. 87-101, in J. Bach Anderson (ed.), Modern Radio Science 1990, Oxford U. press, 1990. 4. C. E. Baum, Representation of Surface Current Density and Far Scattering in EEM and SEM with Entire Functions, Interaction Note 486, February 1992; Ch. 13, pp. 273-316, in P. P. Delsanto and A. W. Saenz (eds.), New Perspectives on Problems in Classical and Quantum Physics, Part II, Acoustic Propagation and Scattering, Electromagnetic Scattering, Gordon and Breach, 1998.
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5. C. E. Baum, The SEM Representation of Scattering from Perfectly Conducting Targets in Simple Lossy 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.
Media, Interaction Note 492, April 1993; Ch. 3, pp. 47-79, in C. E. Baum (ed.), Detection and Identification of Visually Obscured Targets, Taylor & Francis, 1998. C. E. Baum, Low-Frequency Near-Field Magnetic Scattering from Highly, but not Perfectly, Conducting Bodies, Interaction Note 499, November 1993; Ch. 6, pp. 163-218, in C. E. Baum (ed.), Detection and Identification of Visually Obscured Targets, Taylor & Francis, 1998. C. E. Baum, The Magnetic Polarizability Dyadic and Point Symmetry, Interaction Note 502, May 1994; Ch. 7, pp. 219-242, in C. E. Baum (ed.), Detection and Identification of Visually Obscured Targets, Taylor & Francis, 1998. C. E. Baum, Concerning the Identification of Buried Dielectric Targets, Interaction Note 504, July 1994; Ch. 4, pp. 81-102, in C. E. Baum (ed.), Detection and Identification of Visually Obscured Targets, Taylor & Francis, 1998. C. E. Baum, Direct Construction of a from Natural Frequencies and Evaluation of the Late-Time Residuals, Interaction Note 519, May 1996; pp. 349-360, in G. Heyman et al (eds.), Ultra-Wideband, Short-Pulse Electromagnetics 4, Kluwer Academic/Plenum Publishers, 1999. G. W. Hanson and C. E. Baum, Perturbation Formula for the Internal Resonances of a Dielectric Object Embedded in a Low-Impedance Medium, Interaction Note 520, August 1996. C. E. Baum, Symmetry in Electromagnetic Scattering as a Target Discriminant, Interaction Note 523, October 1996; pp. 295-307, in H. Mott and W. Boerner (eds.), Wideband Interferometric Sensing and Imaging Polarimetry, Proc. SPIE, Vol. 3120. G. W. Hanson and C. E. Baum, Perturbation Formula for the Natural Frequencies of an Object in the Presence of a Layered Medium, Interaction Note 532, October 1997; Electromagnetics, 1998, pp. 333351. C. E. Baum, Application of Symmetry to Magnetic-Singularity Identification of Buried Targets, Interaction Note 543, June 1998. C. E. Baum, Target-Vicinity Scattering Parameters, Interaction Note 546, August 1998. C. E. Baum, Use of Residue and Constant-Dyadic Information in Magnetic-Singularity Identification, Interaction Note 547, August 1998. C. E. Baum, Magnetic Singularity Identification of Nonsymmetrical Targets, Interaction Note 549, December 1998. C. E. Baum, Norms of Time-Domain Functions and Convolution Operators, Mathematics Note 86, December 1985; Ch. 2, pp. 31-55, in H. N. Kritikos and D. L. Jaggard (eds.), Recent Advances in Electromagnetic Theory, Springer-Verlag, 1990. C. E. Baum, Energy Norms and 2-Norms, Mathematics Note 89, April 1988; Ch. 11.4, pp. 498-508, in H. Kikuchi (ed.), Environmental and Space Electromagnetics, Springer-Verlag,, 1991. C. E. Baum, E. J. Rothwell, K.-M. Chen, and D. P. Nyquist, The Singularity Expansion and Its Application to Target Identification, Proc. IEEE, 1991, pp. 1481-1492. C. E. Baum, Signature-Based Target Identification and Pattern Recognition, IEEE Antennas and Propagation Mag., Vol. 36, No. 3, June 1994, pp. 44-51. T. K. Sarkar and O. Pereira, Using the Matrix Pencil Method to Estimate the Parameters of a Sum of Complex Exponentials, IEEE Antennas and Propagation Mag., Vol. 37, No. 1, February 1995, pp. 48-55. E. J. Rothwell and M. J. Cloud, On the Natural Frequencies of an Annular Ring Above a Conducting Half Space, J. Electromagnetic Waves and Applications, Vol. 10, 1996, pp. 155-179. S. Vitebskiy and L. Carin, Resonances of Perfectly Conducting Wires and Bodies of Revolution Buried in a Lossy Dispersive Half-Space, IEEE Trans. Antennas and Propagation, 1996, pp. 1575-1583. L. Carin, R. Kapoor, and C. E. Baum, Polarimetric SAR Imaging of Buried Landmines, IEEE Trans. Geoscience and Remote Sensing, 1998, pp. 1985-1988. H. Rothe, Approaches to Pattern Recognition, pp. 1-1 through 1-29, RTO Lecture Series 214, Advanced pattern Recognition Techniques, RTO-EN-2, 1998. J. N. Franklin, Matrix Theory, Prentice-Hall, 1968. C. E. Baum, Toward an Engineering Theory of Electromagnetic Scattering: The Singularity and Eigenmode Expansion Methods, Ch. 15, pp. 571-651, in P. L. E. Uslenghi (ed.), Electromagnetic Scattering, Academic press, 1978. D. D. Howard, Tracking Radar, Ch. 18, pp. 18.1-18.60, in M. I. Skolnik, Radar Handbook, 2nd Ed., McGraw-Hill, 1990. C. E. Baum, SEM and EEM Scattering Matrices and Time-Domain Scatterer Polarization in the Scattering Matrix, Ch. 1-9, pp. 427-486, in W.-M. Boerner et al (eds.), Direct and Inverse Methods in Radar Polarimetry, Kluwer Academic Publishers, 1992. C. E. Baum and H. N. Kritikos, Symmetry in Electromagnetics, Ch. 1, pp. 1-90, in C. E. Baum and H. N. Kritikos (eds.), Electromagnetic Symmetry, Taylor & Francis, 1995. C. E. Baum, Symmetry and Transforms of Waveforms and Waveform Spectra in Target Identification, Ch. 7, pp. 309-343, in C. E. Baum and H. N. Kritikos (eds.), Electromagnetic Symmetry, Taylor & Francis, 1995.
BURIED OBJECT IDENTIFICATION WITH AN OPTIMISATION OF THE TLS PRONY ALGORITHM
Lostanlen Y., Corre Y., Uguen B. Groupe Détection Image Diffraction Laboratoire des Composants et Systèmes pour les Télécommunications Institut National des Sciences Appliquées , Rennes 35043 Cedex, France
INTRODUCTION Radar systems emitting an ultra-short pulse, corresponding to an ultra-wide frequency bandwidth, (UWB-SP) are used in many applications including detection of buried objects (landmines, unexploded ordnances UXO) or stealth targets. (Bertoni, 1993; Carin, 1995; Baum, 1997; Shiloh, 1999). Such systems are designed to operate at low frequencies allowing a deeper penetration in a dispersive ground, while combining a very large bandwidth, hence providing a better resolution. Although these systems have been found to detect buried objects quite well, the resolution is rather poor preventing a good radar imaging of the scene. Many false alarms occur when interrogating the subsoil. Therefore there is a great need for a reliable identification tool. Many targets have a simple shape and only a few complex natural resonances (CNR) dictate the late time behaviour of the scattering response. Many authors have studied this late-time backscattered response in an attempt to identify and discriminate targets by studying the CNR. The first efforts were based on the use of Prony’s method, but this failed to give any satisfying result in real environments. However, improvements of the method were conducted by Rahman (1987). Therefore the purpose of the paper is to present the TLS-Prony method we have implemented (based on Rahman’s work) and to propose some criteria to optimise the input parameters of the method. After having validated our method on a sphere, we will illustrate an application
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of this method to real GPR data. THE PRONY AND TLS-PRONY METHODS For many radar targets, the late-time impulse response may be expressed in the time-domain as the sum of a finite number of resonant modes as follows:
is the early-time duration, are the complex poles on the s-plane, and the residues (or mode amplitudes). Note that where are the damping factors and the resonant pulsations. The complex poles are invariant target characteristics (provided that the target is perfectly conducting). Only the amplitudes depend on the target orientation or the excitation source. The discrete time signal, sampled at interval is written as a function of the complex poles lying in the z-plane:
The object of the Prony method is the resolution of this non-linear system, and the extraction of poles and residues, at least in the absence of any perturbation. By means of the Vandermonde matrix Z, the system is written in its matrix form:
The Prony method leads to a unique and exact solution when the number of sampled data N is twice the number of complex modes P. The basic idea is to consider the characteristic polynomial
whose
roots are the poles of the system. We can easily demonstrate that the signal solution of the following autoregressive difference equation:
Finally, the different steps of the Prony method are: The polynomial coefficients
are determined by solving Eq. 5.
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The poles
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are deduced from the roots of
System 4 is solved to obtain the residues However this method is not very well suited when dealing with data perturbed by noise. In that case the system 4 is ill-conditioned, which in turn introduces substantial errors in the pole estimations. The first way to improve the method performance is to choose N > 2P and solve the polynomial coefficients in the difference Eq. 5 in a least square error sense. That is known as the Least Square (LS) - Prony method. In many cases, especially for data with low Signal to Noise Ratio (SNR), the Total Least Square (TLS) approach is more appropriate (Dowling, 1994; Chen, 1996). This method takes into account the fact that noise introduces perturbation in both sides of Eq. 5 which tends to minimize its effect on the pole estimations. If we note L the order of the characteristic polynomial B(z), chosen greater or equal to the number of poles P, system 7 may be written as follows:
For simplicity, the matrix is noted The TLS solution is found by means of the Singular Value Decomposition (SVD) of the matrix as follows:
where U and V are unitary matrices, and the elements are the singular values of the matrix; they are supposed to be arranged in descending order. If the data under test is free of noise, is not full rank and only part of the singular values are greater than zero. In practical cases, all the values are nonzero, and the effect of noise is reduced by considering the values that are inferior to a threshold as negligible.
The choice of the threshold value is not obvious and will be the object of the following section. In Rahman (1987) it is shown that the polynomial coefficients may be deduced from the matrix where is the column of V.
where
The determination of L pole locations results from the TLS algorithm: P true poles and L – M extraneous poles. A common way to minimise the number of extraneous poles is to choose only the poles lying inside the unit circle (on the z-plane). A second method consists in the selection of the poles corresponding to the most energetic modes. The energy of the resonant modes is given by:
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The described method gives very good results in ’normal’ conditions. However the low SNR, of real signals disturb considerably the extraction process. Thus the TLS-Prony method needs to be modified.
THE MODIFIED TLS-PRONY Many difficulties arise when the TLS-Prony algorithm is applied to real radar signals with low SNR. The choice of the threshold which determines the number of singular values that must be retained after the SVD decomposition is delicate. Indeed, a non-appropriate choice may affect the integrity of the algorithm, or simply gives rise to large errors in the location of the poles. On one hand, the calculation based on a too small number of singular values does not permit the extraction of a sufficient number of poles (only the main resonant modes are recognised). On the other hand, the intervention of the smallest singular values in the calculation increases the effect of noise and degrades the algorithm accuracy. Moreover, as illustrated later in this section, a small variation of the threshold may produce results with extremely variable precision. That gives a random aspect to the conventional TLS-Prony method, and represents a serious drawback. A second difficulty is the presence of extraneous poles. To prevent false alarm, or to permit effective target detection and identification, a perfect distinction between real poles and poles introduced by the algorithm is necessary. The rejection method based on the energy of the poles removes a part of the extraneous poles, but is not sufficient. The impulse response of the perfectly conducting sphere is a good basis for the validation of the TLS-Prony method efficiency. The sphere impulse response illustrated in Fig. 1 presents a highly resonant part comprised between 1.5 ns and 2 ns, resulting from the creeping wave. The Singularity Expansion Method (SEM) shows that the response may be considered as an infinite sum of resonant modes. In the s-plane the complex poles associated to those resonant modes lie into infinite branches centred on the real axis Lavenant (1994), Chen (1981). Whereas an infinite number of poles are needed for an accurate reconstitution of the impulse part, only a few poles located in the first and second branches are necessary to obtain a good approximation of the creeping wave contribution.
To estimate the efficiency of the TLS-Prony method, we compute the sphere re-
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sponse by means of the Mie series. We then compare the poles extracted by the algorithm to the theoretical poles given by the SEM method. Lets consider the impulse response of the sphere of radius computed with the sampling period A gaussian noise is added, so that the SNR is equal to 55 dB (the SNR is given by the ratio of the magnitude of the creeping wave contribution over the noise standard deviation). A 5th order lowpass Butterworth filter is used in order to simulate the effects of anti-aliasing filters. Then the algorithm is applied to the late-time part of the signal, that is the 350 samples placed after instant As we have no priori knowledge about the number of singular values that must be retained after SVD, we calculate the poles and residues for different singular value thresholds. The poles located in the vicinity of a theoretical pole are counted; the errors about the frequency, the damping factor, and the residue are determined. As shown in Fig. 2, up to 9 poles which correspond to the theoretical resonant modes are identified. At first, when the singular value threshold increases, the number of identified poles increases as well, continuously. But, afterwards, that number is fluctuating, and some of the threshold values lead to a very poor identification. Fig. 2 represents also the error on the frequency for the three first poles of the sphere. Note that the frequency accuracy is extremely variable. Nevertheless, intervals such as [25 – 30] or [40 – 45] offer acceptable solutions. Finally, if we combine the information provided by both figures, the threshold indices that lead to the most precise and complete solutions are 27 and 28.
The plot of the singular values with a logarithmic scale (in Fig. 3) presents, in the vicinity of indices 25 and 26, a breaking point separating two linear areas. It accurately indicates the optimal singular value threshold, and the singular values placed after it may be considered as negligible. However, as shown in the next section with experimental data, that point is not always so distinct and so relevant. Nevertheless it remains a useful indicator. We decide to carry out a tracking process on the poles that have been extracted with successive threshold indices, in order to study their evolution. Some of the extracted poles appear only for one threshold value; others are very unstable: they are rejected. This permits to put aside many extraneous poles. The error relative to the frequency, and the residue magnitude of the remaining poles compared to the theoretical ones are calculated. The square of these errors is averaged for each threshold value to obtain the global frequency and residue behaviour. Fig. reffigGlobVar presents
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both frequency and residue global variations. The figures let appear small variations for the threshold indices inferior to 30, and large variations for higher values. From this observation, we have developed a modified and iterative TLS-Prony method, which leads to the extraction of a great number of poles with reasonable accuracy, and to the rejection of most extraneous poles: SVD decomposition of the matrix pole extraction for a range of singular value thresholds, pole tracking, variation measurement, and pole rejection, the optimal threshold is chosen such that the variations are small and the number of poles is maximal
The result for our noisy sphere impulse response appears in Fig. 5. The extracted poles are represented by points, the theoretical ones by circles. (Note that the singular value threshold chosen by the algorithm is 25).
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REAL DATA Detecting targets buried in the ground with a GPR device requires the ability to identify or remove the clutter contribution. Besides, the discrimination of inoffensive stones from hazardous objects such as antipersonnel landmines may be performed by the extraction of features which positively characterise the target. Therefore the modified TLS-Prony method may be found to be a useful processing tool. The method has
been applied to a set of data resulting from measurements. An exhaustive study on the processing techniques to characterise and remove the clutter has been carried out on the same kind of data set by Brooks (1999). The data set consists in a C-scan of GPR returns regularly measured from an antenna moving above a sandbox, in which different objects are buried (5 cm below the surface). Fig. 5 and 6 show the frequencies, residues and poles extracted with the TLSProny algorithm at different antenna positions. We observe that some of the frequencies are constantly present; they are probably due to the correlated clutter and the radar
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system. On the other hand the poles which occasionally appear result from soil local particularities or buried objects. CONCLUSION This paper has presented an application which uses CNR signatures to identify buried objects. In particular, we have explained the improvements we have made in the automatic determination of the number of singular values. Finally we applied our optimised method to real data. In many practical cases, the clutter generated in the vicinity of the target causes major errors in the extraction process of CNR. We are currently working on the generation of this clutter by the interface and in the close surrounding of the targets in order to provide a useful input for the target classification. Acknowledgements The authors gratefully acknowledge the support of the Région Bretagne. REFERENCES Baum, C.E., Carin, L., Stone, A.P., 1997, Ultra-Wideband, Short-Pulse Electromagnetics 3, Kluwer Academic/Plenum Press. Bertoni, H.L., Carin, L., Felsen, L.B., 1993, Ultra-Wideband, Short-Pulse Electromagnetics, Kluwer Academic/Plenum Press. Brooks J. W. , Van Kempen L., Sahli H., 1999, Ground penetrating radar data processing: clutter characterization and removal, IRIS TR 0059. Carin, L., Felsen, L.B., 1995, Ultra-Wideband, Short-Pulse Electromagnetics 2, Kluwer Academic/Plenum Press. Chen H., Van Huffel S., Dowling E. & DeGroat R. D., 1996, TLS based methods for exponantial parameter estimation, 2nd international workshop on TLS and errors-in-variables modeling in Leuven Chen K.-M. & Westmoreland D., 1981, Impulse Response of a conducting sphere based on singularity expansion method, Proc. IEEE 69 Dowling E. M., DeGroat R. D. & Linebarger D. A., 1994, Exponential parameter estimation in the presence of known components and noise, IEE Transactions on antennas and propagation 42:5 Kergall J., Extraction des pôles naturels de résonance de cibles radar, dans le domaine temporel, Mémoire pour DEA (1998) Lavenan T., 1994, Contribution à al discrimination d’obstacle radar en zone de résonance, Thèse de Doctorat Rahman MD. A.,Yu K.-B., Total Least Squares Approach for frequency estimation using linear prediction, IEEE Trans. on acoustics, speech, and signal processing., 35:10 (1987). Shiloh J. , Mandelbaum B., Heyman Ehud, 1999, Ultra-Wideband Short-Pulse Electromagnetics 4, Kluwer Academic Press.
MODEL PROBLEMS OF PULSE SENSING
Lyudmyla G. Velychko,1 Andrey O. Perov,1 Yuriy K. Sirenko,1,2 and Ercan Yaldiz2 1
Department of Mathematical Physics Institute of Radiophysics and Electronics 12 Acad. Proskura st., Kharkov, 61085, Ukraine 2 Department of Electronic Engineering Gebze Institute of Technology P.K. 141, 41400, Gebze/Kocaeli, Turkey
INTRODUCTION The radio-locating means for determination of layered media parameters and visualization of objects or voids hidden in them have gained wide-spread acceptance in various fields, namely, civil engineering, geodesy, use of hydraulic-engineering structures and of oil-pipe-lines, etc. The solution of the engineering problems arising therewith requires the development of essentially new approaches. Traditional theoretical and experimental methods have made possible only the first generation of radars with both small signal penetration depth and incomplete retrieval of useful information from remote sensing data. Quite reasonable hopes for improvement of the situation are widely associated with the construction of radars based on nonsinusoidal waves. A central theoretical problem here consists in correct interpretation of the results of measurements, demanding for its solution adequate mathematical models (and algorithms) describing the transient processes in the part of space under research. Model problems of pulse sensing are initial boundary value ones in unbounded regions with inhomogeneous compact objects. The key to the effective solution of them lies with a proper truncation of the computational domain in finite-difference methods, i.e. in a limitation of such kind that, on the one hand, reduces the essentially open problem to the closed one, and, on the other hand, does not effect on accuracy and reliability of the data obtained. Our paper is just devoted to the analysis of this problem. In the first section, the problem of scattering of nonsinusoidal waves by a perfectly conducting compact object imbedded in an inhomogeneous half-space with an irregular boundary is considered. The second section is devoted to the extension of the results to other situations peculiar for pulse sensing: an inhomogeneous compact object and a stratified dielectric structure in a field of nonsinusoidal waves and a pattern-forming structure. All model problems are two-
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dimensional and scalar ones (E-polarization), and yet the results can be generalized and modified as applied to solution of three-dimensional scalar (acoustic) or vector (electromagnetic) problems. In the third section of the paper, an example of correct incorporation of exact conditions on virtual boundaries into a standard computational scheme of the finite-difference method is given, the results of numerical experiments focused on testing and approbation of the constructed algorithms are presented. The theoretical and methodological foundations for the approach being developed here are laid in the papers by Y. K. Sirenko, A. O. Perov, and N. P. Yashina,1,2 which are devoted to origination of rigorous and efficient techniques for the analysis of electromagnetic transient processes in open periodic resonators (gratings) and waveguide resonators. We follow these papers, extending the results obtained there to another class of topical problems of electromagnetic theory that are of practical significance.
MODEL PROBLEM OF UNDERSURFACE PULSE SENSING Let us consider a problem with the geometry presented in Figure 1 ,a. We have to find a function such that
Here Q is a part of the plane bounded by contours S, and are relative permittivity and specific conductivity of the medium sensing respectively, and
It is supposed that the functions F,
which are finite in the region Q, satisfy the conditions of the theorem on
single-valued solvability of problem (1) in the Sobolev space
.3
The supports of all these functions at all instants considered are concentrated in the region Above the boundary separating the homogeneous and inhomogeneous media, the condition is valid. The tangential components of the field intensity vectors are continuous on For efficient use of the finite-difference method one has to close the area of the analysis by introduction of virtual boundaries. The most-used approach is the construction of approximate absorbing boundary conditions (ABC).4,5 However, these conditions do not “pass” completely the field incident on the boundary. The wave is partially reflected by the imaginary boundary. The free propagation of the field is distorted, so the calculation error caused by it cannot be estimated analytically. Too many factors affect its value. The alternative approach is based on the use of exact radiation conditions for secondary fields.1,2,6 It does not lean upon any heuristic assumption on the field structure near the virtual boundaries and reflects the essence of the simulated process. In our paper we construct the boundary condition using the representation of the field as expansion in elements of an evolutionary basis of nonstationary signals.1,2 In the region the solution of initial boundary value problem (1) represents a nonstationary wave “outgoing” from the area containing both the sources and the effective
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scatterers. Separation of variable
and makes
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in (2) results in the following representation for
elements of the evolutionary basis of wave U subjected the conditions
Multiplying (2) by semiaxis we obtain
and performing Fourier-Bessel transform in
for the images
of functions
Here
is the Bessel function of order
Heaviside function, and
on the
is the
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of the operator
Going back to the originals we obtain
Performing differentiation in (4) and taking into account properties of Legendre functions, we come to the following exact representation of in the region for
This relationship gives one of the possible forms of representation of explicit radiation conditions for the solution of problem (1).
EXACT CONDITIONS ON VIRTUAL BOUNDARIES IN OTHER MODEL PROBLEMS OF PULSE SENSING The investigation of the scattering of E-polarized nonstationary electromagnetic field by a dielectric object in a free space (see Figure 1,b) is reduced to solution of the initial boundary value problem similar to (1) but with the following condition The area of the analysis here is different from that considered above. It covers the whole plane of variables and and condition in this connection, is replaced by the condition of periodicity. Obviously the technique for deriving the exact radiation conditions is not subjected to serious changes. The following transformations are similar to those for problem (1). The resulting condition on the virtual boundary takes the form
Here
and
MODEL PROBLEMS OF PULSE SENSING
are
the
eigen
values
appropriate
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to
the
eigen
functions
Let us consider one more problem in Cartesian coordinates (Figure 1 ,c). The layer ( and are functions of only one variable z) is excited by the sources F, and of nonsinusoidal E-polarized waves with compact in the region supports. It is assumed that in all instants t these supports belong to We have to find the function U(y,z ,t) such that
Here
). By carrying out Fourier-transform in y for (7) we obtain a new problem
whose solution determines U uniquely in taking inverse Fourier-transform. For represents the “wave” outgoing from the area containing the sources and the scatterers and satisfies homogeneous initial boundary value problem (8) with and . With the use of cosine transform in z on the semiaxis z>0, we obtain the Cauchy problem for images and we have for its solution
(the technique is identical with that used for deriving (4)) or
On the virtual boundary z = a, after simple transformations in (17), we obtain
(the region as well as does not contain either sources or scatterers). A few words about an antenna problem whose geometry is exemplified in Figure 1,d. The virtual boundary (dashed lines) closing the area of wave propagation consists of two parts. On the first part of it, which is located in the region z > 0, one can use, without changes, condition (5) obtained for problem (1). On the second part coinciding with the cross section of the waveguide, the standard technique makes it possible obtaining the explicit conditions of the same type, as well as in the case with a periodic structure.1
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FINITE-DIFFERENCE SCHEMES AND SOME NUMERICAL RESULTS Further steps in algorithmization of model problems differ from standard ones only by the set of expedients in the technique.1 The computational experiments, within the framework of which the exact conditions following from (9), (13), (18), and (19) were tested, are practically identical to ones realized by Perov and Sirenko1 for gratings. Omitting technical details we present only one result describing the situation in general. Figure 2 allows us to compare errors caused by the use of both exact conditions (5) (dashed lines) and classical heuristic ABC of the first approximation order4 (full lines). Two problems of type (1) with simple geometry (see Figure 2,a) and are considered. For the data from the left column
is valid, for the data from the right column
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is valid. The local errors D(n,m,l) are calculated at grid points on the imaginary boundary The value D(l) determines the averaged global error on the l th time step within the limits of the whole grid. The discretization step in and (indices n and m) is equal to in time it is equal to The second approximation order is considered. The results of the experiment (Figure 2) clearly demonstrate that the truncation of computational domain by introducing virtual boundaries with exact radiation conditions does not increase calculation errors caused by the difference algorithm proper. Some more examples of implementation of the algorithms considered in this paper we present in Figures 3 and 4. The curves in Figure 3 are the amplitudes of pulses
arising in the reflection zone of the stratified structure (Figure 1,c) excited by the signal
The function
is the solution of the initial boundary value problem (8) with and
The response of the structure from Figure 4,a illuminated by the pulse
is shown in Figure 4,b. The pulse is depicted by a dashed line. The perfectly conducting boundary is described by the function
REFERENCES 1. 2. 3. 4. 5. 6.
7.
A.O. Perov and Y.K. Sirenko, Nonstationary model problems of the electrodynamic theory of gratings, in: Radiofizika i Electronika, Institute of Radiophysics and Electronics, Kharkov. 2, No.2: 66 (1998). Y.K. Sirenko and N.P. Yashina, Nonstationary model problems for waveguide open resonator theory, Electromagnetics, 19,No.5: 419(1999). O.A. Ladyzhenskaya. The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York (1985). B.B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Mathematics of Computation. 31, No. 139: 629 (1977). G. Mur, Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetics field equations, IEEE Tr. on EMC. 23, No.4: 377 (1981). A.R. Maykov, A.D. Poyezd, A.G. Sveshnikov, and S.A. Yakunin, Difference schemes of initial boundary value problems for Maxwell equations in unbounded region, Comp. Math. and Math. Physics, 29, No.2: 239 (1989). V.S. Vladimirov. Equations of Mathematical Physics, Dekker, New York (1971).
SIMULATION OF THE TRANSIENT RESPONSE OF OBJECTS BURIED IN DISPERSIVE MEDIA
M. A. Hernández López 1, S.González García2, A. Rubio Bretones2 and R. Gómez Martín2 1
Departamento de Física Aplicada Universidad de Salamanca Plaza de la Merced, s/n 37008 Salamanca (Spain) 2 Departamento de Electromagnetismo y F. de la Materia Universidad de Granada Fuentenueva s/n 18071 Granada (Spain)
INTRODUCTION The study of the response of dispersive media excited by a transient electromagnetic signal is of interest in areas such as the simulation of Ground Penetrating Radar, the stimulation of biological tissues and broadband communications. This paper is focused on the simulation of a three-dimensional short-pulse ground penetrating radar (GPR) to study the transient response of objects buried in dispersive media. Since these systems often include thin wire antennas (Montoya and Smith, 1999), the hybrid technique combining the Finite Difference Time Domain method (FDTD) and the Method of Moments in Time Domain (MoMTD), described in (Rubio Bretones et al, 1998), has been extended to deal with the frequency dependence characteristics of the media, modeled with a non-uniform discretization and truncated with Berenger’s Perfectly Matched Layer (PML) conditions. An in-house tool based on the commercial program AutoCAD™ is used to specify the geometry and the general parameters of the full problem, thus providing a versatile way to define complex geometries involving several kinds of media.
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DESCRIPTION OF THE HYBRID METHOD In the hybrid MoMTD-FDTD method, the inhomogeneous region of the problem is modeled using the Finite Difference Time Domain Method (FDTD), which is based on the direct solution of Maxwell's curl equations and is capable of dealing with complex geometries with arbitrary electrical properties. However, the application of FDTD to an arbitrarily oriented thin-wire antenna presents some difficulties in modelling the thin-wire antenna features precisely, and so the MoMTD is used. The interaction between the antenna and the object is carried out by means of the surface equivalence theorem (Huygens’ principle). Since the method is implemented entirely in the time domain, it can efficiently generate information over a wide frequency band. The hybridization begins by dividing the original problem into two separate ones. The first of these, the thin-wire antenna, is solved by using the MoMTD, while the second one, the dielectric body, is handled via the FDTD scheme (Fig. 1). The basic steps in the hybridization are: 1) An imaginary closed Huygens’ surface S is located around the thin-wire antenna. Equivalent sources on S are deduced from the fields radiated by the antenna in free space, which are obtained solving the electric field integral equation (EFIE) by the MoMTD for the currents on the antenna. 2) The incident fields on the dielectric body are computed from the equivalent currents on S. The FDTD algorithm is applied to compute the fields at any cell inside the FDTD computational domain, in both the total field zone (outside S) and the scattered field zone (inside S). 3) The FDTD solution inside S when the antenna is not present is, by definition, the incident field on the antenna needed to solve the EFIE by the MoMTD and hence compute the currents induced on the wire. Spatial and temporal linear interpolation is applied to calculate the incident field at specific locations on the antenna and specific times. NON-UNIFORM FDTD-PML FOR DISPERSIVE MEDIA The classical FDTD method is based on the approximation of Maxwell’s curl equations replacing the time and space derivatives by centred differences on a uniformly discretized space using the field distribution given by Yee's cube (Yee, 1966). A drawback of this algorithm appears when high permittivity and permeability media are present, since an overall fine discretization must be used, with the consequent increase in memory and CPU time requirements. The non-uniform FDTD algorithm (Kim and Hoefer, 1998) overcomes this problem by using first-order non-centred differences to approximate the spatial derivatives in a graded spatial mesh with the same relative field distribution given by Yee's cube. This allows us to use a fine discretization where needed, maintaining a coarser one in the rest of the computational space.
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In order to incorporate a realistic model of the soil in FDTD, it is necessary to take into account its frequency-dependent dielectric behaviour, as well as its conductivity, which depend mainly on its water concentration. Let us assume a Debye model in frequency domain
where is the relative infinite frequency permittivity, the relative static permittivity, is the relaxation time, is the free space permittivity, and its constant conductivity. The time domain relationship between equation as
and
can be written in the form of a differential
Then, the source-free Maxwell’s curl equations in time domain for non-dispersive magnetic linear media of permeability µ, are written as
Eqs. (2) and (3) are discretized with second order accurate centred differences for the time derivatives and first order non-centred differences for the spatial ones, obtaining an explicit-in-time advancing scheme. Finally, Berenger’s PML absorbing boundaries (Berenger, 1994) have been chosen to truncate the computational space. Following the formulation described in (González, Villó et al, 1998), a material-independent form (Zhao et al, 1998) of the PML valid for dispersive and conductive media has been obtained. In time domain the governing equations are
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where the subscript s denotes any split field in the PML related
to
the
corresponding
maxwellian
field
at
the
interface
through
It can be proved (González, Villó et al, 1998) that a plane wave with wavevector propagating in a maxwellian medium, and inciding on a PML medium, for instance at is not reflected if the transmitted wave inside the PML is enforced to propagate with the same wavenumbers in the x and y directions as the incident one Since there is no constraint on the wavenumber in the z direction this can be properly chosen to attenuate the transmitted wave at a desired rate by setting
If the PML is backed by a perfect conductor at a depth at
the normal reflection coefficient
when the incidence medium is not dispersive is
with
being the
frequency independent phase speed in the z direction. Since for dispersive media the wavevector is complex and frequency dependent, the normal reflection coefficient now depends on frequency and takes the form So must be chosen to provide the desired reflection coefficient in the less favourable case (maximum frequency). and related by
in Eq. (5) are the electric and magnetic conductivity matrices, which must be
to achieve perfect matching. The matrix elements must fulfill freely chosen positive number in order to match waves impinging at
with
being a
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MoMTD
The method of moments is based on the solution of the EFIE derived by enforcing the boundary condition on the tangential electric field over the surface of the wires. It has been widely used for analyzing thin-wire structures, both in the frequency and directly in the time domain (MoMTD). For a detailed explanation of the method see, for instance, (Rubio et al, 1989). MODELING
The geometry of the problem, as well as the general FDTD parameters, is fully specified in a drawing created with the aid of AutoCAD™, which provides a very flexible way of defining a complex geometry. A set of AutoLISP™ routines translates AutoCAD database information into suitable FDTD input parameters. The tool, which is inspired by the modeling technique used by the EMPIRE™ FDTD simulator1 (González, Baggen et al, 1998), allows among other capabilities: The specification of the materials involved in the problem through rectangular boxes, cylinders and spheres which can be interpenetrated by means of a system of increasing priorities. The definition of voltage and current sources with 3D boxes wherein the fields are enforced to the desired values. The specification of Huygens’ surfaces on which the equivalent currents are placed to separate total from scattered field zones. The definition of the wire antenna geometry inside the Huygens’ boxes. The placement of observation boxes. RESULTS
In order to test the tool, we simulated several typical GPR problems. In the first one a metallic pipe, with a 4 cm square cross section and running indefinitely in the x-direction, is buried 2.5 cm beneath the surface of a wet soil A pair of crosswise wire antennas separated vertically 2.8 cm, are located 3 cm above the soil. A full FDTD model (Fig. 2), incorporating an inverse-with-the-distance field variation at the vicinity of the wires in the FDTD equations, is used. Firstly, the antenna aligned with the pipe was excited with a pulse of 5 GHz of bandwidth, keeping the perpendicular one passive; and secondly their roles were interchanged. Fig. 3 represents the subtraction between the currents at the centre of both antennas when the pipe is buried in the soil and when the soil is empty, showing, as expected, that for the transversal polarization, the current is smaller than for the longitudinal one, thus providing a technique to detect the orientation of the pipe. In none of the cases were the cross-polarized reflections significant. In order to test the hybrid technique, the analytical expression of the fields created by a simple Hertzian dipole was used to obtain the equivalent currents on a Huygens’ surface, and used to illuminate a 24x24x12 cm plastic mine with low permittivity The mine was buried at a depth of 12 cm in the same soil of the previous example, whilst the dipole was placed 24 cm above the soil and excited with a 3 GHz bandwidth gaussian pulse. An observation point was placed inside the Huygens’ box 1
Developed by the Institut für Mobil-und Sattellitenfunktechnik (IMST) in Germany
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(scattered field zone) above the dipole. Fig. 4 shows the reflected copolarized electric field component dispersed when the mine is or not buried. The results obtained for the crosspolarized components were also non-significant with respect to the copolarized ones and are not shown in the figures.
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Finally, the hybrid FDTD-MoMTD method was used to simulate a GPR problem involving a V-antenna. A metallic box of dimensions 20x20x6 cm was buried at a depth of 50 cm. in a purely dielectric soil The antenna was excited at its centre by a 3 ns. gaussian source voltage. An observation point was located 8 cm from the vertex of the antenna, and on a plane parallel to the one containing the antenna. The solid line in Fig. 5(left) represents the copolarized scattered field without the object, and the dashed line shows it when the object is present. Fig. 5(right) shows the same results but when the Vantenna is resistively loaded with a Wu-King profile (Sánchez García et al, 1998). It can be seen that the loading enables the reflections from the mine to be distinguished in time domain quite clearly.
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CONCLUSIONS A time-domain tool to simulate GPR problems has been developed and tested. It uses an automatic geometry preprocessor based on AutoCAD to define the full problem in a versatile way. The kernel of the tool is a non-uniform FDTD simulator which can handle inhomogeneous dispersive media terminated by PML conditions, as well as wire structures, through the incorporation of the MoM solution of the wire problem, coupled to the FDTD zone via the equivalence principle. We have used this tool to study the transient excitation of inhomogeneous bodies by thin-wire broadband antennas and specifically to simulate a short-pulse GPR. Acknowledgements This work is partially supported by the CICYT (Spain) through project TIC99-0624.
REFERENCES Montoya, T. P., Smith, G. S., Land Mine Detection Using a Ground-Penetrating Radar Based on Resistively Loaded Vee Dipoles, IEEE Trans. on Antennas and Prop., vol. 47, no. 12, 1999 Rubio Bretones, A., Mittra, R., and Gómez Martín R., A new Hybrid Method Combining the Method of Moments in the Time Domain and FDTD., IEEE Microwave and Guided Wave Letters, vol. 8, n.8, 281-283, 1998. Yee, K.S., Numerical solution of initial boundary problems in isotropic media, IEEE Trans. on Antennas and Propagation, vol. 14, pp. 302-307, 1966. Kim, I. S. and Hoefer, W. J. R., A local mesh refinement algorithm for the time domain finite difference method using Maxwell's curl equations, IEEE Trans. on Microwave Theory and Techniques, vol. 38, no. 6, pp. 812-815, 1990. González García, S., Villó Pérez, I., Gómez Martín, R., García Olmedo, B., Extension of Berenger's PML to adapt bi-isotropic media, IEEE Microwave Guided Wave Lett. , vol. 8, no. 9, 1998. Zhao, A. P., Renko, A., and Rinne, M. A., Material-independent PML absorbers for arbitrary lossy anisotropic dielectric media, Proc. of 1998 Int. Conf. on Microwave and Millimeter Wave Technology, pp. 978-981, Beijing, CHINA, August 1998. González García, S., Baggen, L., Manteuffel D., and Heberling D., Study of Coplanar Waveguide-Fed Antennas Using the FDTD Method, Microwave and Optical Technology Letters, vol. 19, no. 3, 1998 Sánchez García, I., Rubio Bretones, A., and Gómez Martín, R., Pulse Receiving Characteristics of Resistively Loaded V antennas, IEEE Electromagnetic Compatibility, vol. 40, n. 2, pp. 174176,1998
ELECTROMAGNETIC TRANSIENT MODELLING USING DYNAMIC ADAPTIVE FREQUENCY SAMPLING
Choy Yoong Tham 1, Andrew McCowen 1, Malcolm S Towers 1 and Dragan Poljak 2 1
Department of Electrical and Electronic Engineering University of Wales Swansea Singleton Park, Swansea, SA2 8PP, UK 2 Department of Electronics FESB, University of Split bb, 21000 Split, Croatia
INTRODUCTION In frequency domain transient modelling, the inverse Fourier transform (FT) process is usually implemented by the fast Fourier transform (FFT) algorithm. In the usual approach, the frequency sampling interval is established by the relation where is the time window of interest [1]. In this relation the sampling interval, depends on an arbitrary parameter which requires empirical insight in its selection to yield accurate result. For transients with a relatively smooth spectrum such as the current on a thin wire scatterer [1] or a dipole in free space [2, 3], the empirical approach does not pose serious difficulty. The sampling interval may vary over a wide margin while the result remain reasonably accurate. However with a resonant structure, the spectrum has sharp and narrowly defined peaks. Coarse sampling intervals are not sufficient to resolve the peaks and will result in error. When used to model the transient of a highly resonant structure the conventional FFT technique may fail to give the correct result. This is demonstrated [4] in the case of a long single conductor transmission line lying very close to a ground plane with one end short circuited. The short circuit current induced by an obliquely incident EMP plane wave was modelled with the conventional FFT technique which yielded totally erroneous result. These observations highlight the weakness of the frequency domain FFT method of transient analysis. When modelling a structure without a priori knowledge of its frequency response, guess work is often being relied on to select the values for and which may not yield successful result. In this paper a systematic and objective procedure is proposed. A new technique using dynamic adaptive sampling to take samples of the spectrum at nonuniformly spaced frequency intervals is formulated. A modified inverse discrete FT (DFT) formula is used to process the data into the transient waveform. This proposed technique Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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improves the computational efficiency of frequency domain analysis by reducing the number of samples substantially. ILLUSTRATING THE PROBLEM OF USING UNIFORM SAMPLING Consider the current induced in a parasitic wire by a voltage pulse in an adjacent parallel wire. Fig. 1 shows the geometry of the two parallel wires over a perfect ground plane. The wires are of length L = 1.0 m., radius a = 2.0 mm., separated by a distance s = 0.5 m. and lying at height h = 0.25 m. over the ground plane. One wire is excited at its centre with a Dirac delta function voltage pulse. The coupling effect and the presence of the ground plane induces a current with a resonant spectrum in the parasitic wire.
The frequency response spectrum for the current at the centre of the parasitic wire is shown in Fig. 2. The spectrum exhibits the high Q characteristic of a resonant structure. The spectrum for an identical single wire in free space is also shown to contrast the resonant point of the parasitic wire. To study the transient, a Gaussian voltage pulse of the form where and is used as the excitation source. For a transient waveform duration of 30 ns., the parameters for the inverse DFT, rounded to convenient figures are N = 128 and Figs. 3 and 4 show the frequency response spectra for the imaginary part of the
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current near the resonant point. For the case of the single wire in free space, virtually the same spectrum is obtained with the frequency interval 13.020833 and 6.510417 MHz. corresponding to N = 64, 128 and 256 respectively. However, for the parasitic wire, the frequency response spectrum obtained with where N = 512, shows an error against a more closely sampled spectrum. This error is significant and will lead to an erroneous transient response which will be shown in Section IV. In Section III a method is developed to accurately represent such a frequency response without incurring the penalty of needing a large number of frequency samples.
FORMULATION OF THE METHOD A convenient way to derive a transient waveform from frequency domain data is by taking the inverse FT of the frequency response spectrum. The impulse response so obtained is then used to carry out a convolution with the excitation waveform in the time domain. This procedure avoids the end effect error of using the alternative convolution theorem in the frequency domain. The inverse DFT is given by the expression below [1] –
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The inverse DFT is usually implemented with FFT routines which compute results more rapidly than the DFT. However, standard FFT routines process only uniformly spaced samples where the number of samples N in equation (1), may be large and is usually, though not always, be a power of 2. In the new procedure proposed here, the time step is determined from sampling theory as (2 × bandwidth of the excitation waveform) and the frequency sampling interval by –
The inverse DFT formula in (1) may be written in terms of the frequency samples and expanded by using Euler’s identity as,
Equation (3) implements the Fourier integral over a period from - to by the trapezoidal rule. In computing the inverse FT, the product of each frequency sample with its corresponding sin or cos function contributes to the summation of equation (3). For a resonant spectrum the error from sampling is very much accentuated by the high Q characteristic. Sampling error at the resonant point carries disproportionate weight into the computed inverse DFT. This is due to the fact that samples away from the resonant point have negligible amplitudes and contribute little to the overall sum. A test is necessary to ensure the frequency samples characterise a resonant spectrum accurately at their respective positions. This is done by the method for the adaptive integration of an unknown function using the trapezoidal rule [5]. In trapezoidal rule integration the local error is given by –
where h is the panel width and
are the integration limits. This is illustrated in Fig. 5.
The error expressed in equation (4) is a function of the second derivative which implies large errors at the sharp resonant peaks. The error diminishes rapidly when the sample spacing is reduced. Using this principle a sampling interval is subdivided
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progressively with intermediate samples to achieve the desired accuracy. As the sample spacing is reduced, successive integral values will converge towards the true value. The difference of the integration values obtained by two successive set of samples is used for convergence test. Unlike in standard adaptive integration, there is no a priori knowledge of the amplitude for an unknown electromagnetic spectrum nor even its order, to fix an absolute tolerance limit. Instead a relative criterion as shown in (5) is used for the test.
is the most recently evaluated integral, is the previous value and Tol is the tolerance limit. When the pre-selected tolerance limit is reached further subdivision is stopped. The algorithm is implemented by first computing three initial samples using the interval determined from the Nyquist rate and the time window. Taking three samples at a time, is the area of the single trapezium formed by the first and the third samples and that obtained by adding the two individual trapeziums together as shown in Fig. 5. If convergence is not reached a fresh sample midway between the first and second is computed. The test is then repeated on the two subintervals. When the procedure is used the first time, it is necessary to conduct a numerical experiment to find the optimum convergence criterion. It has been found that 10% - 20% are highly precise limits with which very accurate results can be expected in all cases tested. In this parasitic wire example, it is found that a limit of 50% will produced transient results of good accuracy. With more relaxed convergence limits, the accuracy of the result deteriorates rapidly. It is also found that the saving in the number of samples by specifying a larger tolerance limit is relatively marginal. Table 1 shows the result of the numerical experiment. In the table the accuracy indicated is that of the transient waveform comparing against a benchmarked result validated by using an independent time domain method described later in Section IV.
There are three points in the inverse DFT formula of equation (3) that need modification to process the non-uniformly spaced samples. Firstly each of the samples is assigned a weight inversely proportional to its sampling interval. Secondly the sin and cos phase terms associated with a sample are calculated using the integer index “n” which is the sample number. This is no longer correct with non-uniform intervals and is
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instead calculated by using the ratio of the sample’s frequency to twice the bandwidth; i.e. instead of n/N to obtain the phase displacement. Thirdly, to use the trapezoidal rule on differently weighted samples the integration is carried out piece-wise. In practice the samples although taken at non-uniform intervals often occur in bands in which the interval is constant. This is due to the fact that away from the resonant points most electromagnetic spectra remain relatively smooth. Taking advantage of this nature, the piece-wise integration is implemented by summing the samples in bands with constant weighting. The modified inverse DFT formula is given in equation (6).
NUMERICAL RESULTS The frequency domain electromagnetics code, MoM3D, used in this paper is based on the method of moments applied to the electric field integral equation (EFIE) [6]. A time domain solver for wires which uses the finite element integral equation method (FEIEM) [7], is used to compared the results.
For the single wire radiating in free space, the conventional FFT is able to produce the transient waveform without noticeable error with only 64 uniformly spaced samples of the spectrum. This result together with another waveform obtained with 128 samples is shown in Fig. 6 as compared to that obtained with the FEIEM. It is to be noted that the 64 samples yield only a 19.2 ns. transient, not enough to
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cover the whole specified time window of 30 ns. In the case of the parasitic wire problem, as discussed in Section II, the conventional FFT procedure produces erroneous result with the 128 samples. Increasing the number of samples to 256 or even 512 the transient waveform is still in error as shown in Fig. 7. Examining the set of samples taken with dynamic adaptive sampling it is found that the frequency interval at the resonant point is reduced to to achieve convergence. Using this value of for the FFT over the whole bandwidth would result in 2048 samples. The transient obtained by using the FFT on 2048 uniformly spaced samples is shown in Fig. 8. This result is in good agreement with that obtain by the time domain FEIEM.
Using adaptive sampling, just 176 samples or less than 10% of 2048 are sufficient to produce an accurate transient current waveform. Fig. 9 shows this latter result compared to the waveform derived from 2048 samples using the conventional FFT. With the new technique the sampling intervals ranging from to 13.0208 MHz. There is hardly noticeable difference between the two results. The effectiveness of this technique in saving computer time can be gauged by looking at the time taken to generate each frequency sample. In this problem the solution uses a total of 106 nodal current unknowns on the two wires. The effect of the perfect ground plane is implemented by image theory. The single precision MoM3D code runs on a SUN Spare Ultra 1 workstation and takes about 9 sec. to compute one sample. The total time projected for the 2048 uniformly spaced samples for the conventional FFT is over 5 hours
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as compared to only about 26 minutes using this new technique.
CONCLUSION The main difficulty in modelling transients via the frequency domain in highly resonant structures is to extract the complete information from its frequency response spectrum. The selection of suitable values for the number of samples and the frequency resolution determines the accuracy of the result. However, neither of these parameters are known prior to simulation. A high Q spectrum requires high resolution samples which if taken at uniform intervals for the FFT will result in a large number of samples. The proposed dynamic adaptive sampling technique takes the samples at intervals according to the requirement at different points of the spectrum. It will, without a priori knowledge of the spectrum, concentrate samples around resonant frequencies. The method has successfully overcome the difficulty without incurring the penalty of increasing computer time. REFERENCES 1 2 3 4 5 6 7
Tijhuis, A G, Peng, Zq and Bretones, A R, “Transient excitation of a straight thin-wire segment: a new look at an old problem”, IEEE Trans. on Antennas Propagat., vol. 40, no. 10, Oct. 1992, pp. 1132 -1146. Miller, E K, Poggio, A J and Burke, G J, “An integro-differential equation technique for the time-domain analysis of thin wire structure. II Numerical results”, J. Comput. Phys., vol. 12, no. 1, June 1973, pp. 210 - 233. Miller, E K and Van Blaricum, M L, “The short-pulse response of a straight wire”, IEEE Trans. on Antennas Propagat, May 1973, pp. 396 - 398. Tesche, F M, Ianoz, M V and Karlsson, T, EMC Analysis Methods and Computational Models, John Wiley & Sons, Inc., 1997, pp. 342 - 345. Gerald, C F and Wheatley, P O, Applied Numerical Analysis, 4th ed. Addison-Wesley Publishing Co., 1989. The MoM3D Code, Dept. of Electrical and Electronic Engineering, University of
Wales Swansea, UK, 1997. D Poljak and V Roje, “Time domain calculation of the parameters of thin wire antennas and scatterers in a half-space configuration”, IEE Proc. Pt. H, Microw. Antennas Propag., vol. 145, no. 1, Feb. 1998, pp. 57 – 63.
THE TIME DOMAIN NUMERICAL CALCULATION OF AN INTEGRO-DIFFERENTIAL EQUATION FOR ULTRASHORT ELECTROMAGNETIC PULSE PROPAGATION IN LAYERED MEDIA
Igor V. Scherbatko 1, Stavros Iezekiel 1 and Alexander G. Nerukh 2 1
Institute of Microwave and Photonics, School of E&E Engineering, University of Leeds, Leeds, UK 2 Department of Mathematics , Kharkov Technical University of Radio&Electronics, Prosp. Lenina 14 Kharkov, 310726, Ukraine
INTRODUCTION Continuous progress in computing power and numerical methods has led to considerable growth in the simulation of short and ultrashort pulse propagation in media that exhibit dispersion, loss, gain and nonlinearity 1,2,3 . Finite-difference time-domain (FDTD) methods are popular due to their clarity and elegant realization in numerical algorithms. Indeed, it is difficult to identify areas in electromagnetic theory where FDTDbased methods have not been applied. Unfortunately, an undesirable feature of FDTD schemes is the time instability during long integration times. For electrically large domains and late-time analysis, the classical Yee scheme is limited by accumulation of phase errors, as shown by Young 4 . One possible solution is to provide a specific boundary treatment, by introducing a simultaneous approximation term, or to use a low-pass filter that eliminates the destabilizing Fourier components (numerical noise) from the results 5 . The high-order and leap-frog integrator methods also have been used to overcome these shortcomings 2,4,5. However, the leap-frog scheme is inapplicable to the time-dependent Maxwell equations and filtering leads to loss of high-frequency information in the signal, which is unacceptable for some applications. One reason for insufficient stability of FDTD-based methods, in some cases, lies in the high sensitivity of numerical differentiation to computational errors, especially for higherorder derivatives. Moreover, finite digit representation in computer memory limits the improvement of such calculations. As a result, most calculations need to be supported by double-precision or even higher levels of accuracy, which demands more computer memory and increases computation time. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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The integral equations approach seems to have a more promising future in terms of both reflection-free realization and also stability due to elimination of errors related to differentiation. An advantage of the direct time integration methods is that they can be generalized to nonlinear dispersive materials. Although the integral-equation approach avoids the stability problems, they require the solution of a sparse matrix equation at each time step and need special treatment of the singular and hypersingular integrals that arise 6 . The integro-differential Volterra equation approach takes an intermediate place between the differential and integral methods, because it consists of both differential and integral operations. The most attractive feature of this method is that the Volterra integral equation is of the second kind, which means that unlike integrals of the first kind, it neither possesses singularities nor requires special regularization treatment. In addition, the derivatives presented here have only the lowest order that can lead to higher stability. Using a resolvent method for solving Volterra equations that describe electromagnetic transients allows the formulation of the evolutionary recursion method 7 . The evolutionary algorithm developed by Scherbatko8 for the direct numerical calculation of nonstationary electromagnetic field in active media is based on a spatial-time discretization scheme for the 1-D case. The algorithm imposes restrictions neither on the signal shape and duration nor on the temporal behaviour of the medium and is therefore generic. This scheme was applied recently to simulation of wavelength conversion of infrared optical pulses in a semiconductor9.
VOLTERRA INTEGRO-DIFFERENTIAL EQUATION The propagation of an electromagnetic signal into a dispersive dielectric with conduction is described by the integro-differential Volterra equation of the second kind 7
where
and E(t,r) is the time and spatial dependent electrical intensity,
is the
velocity of light in the initial medium, are the electric and magnetic permittivity of vacuum, is the nabla operator, is the Dirac delta function, is the Heaviside unit function, and x = (t,r). Beginning from an initial time (assumed to be t = 0), the permittivity and conductivity are determined by functions and respectively. Before this initial time, the permittivity and conductivity of the medium were and zero respectively. For the one-dimensional case, where the medium’s parameters depend only on time and the spatial coordinate x, the vectors have components perpendicular to the x-axis. Then after integrating over transverse coordinates in the equation (1) we have the following scalar integro-differential equation:
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In (4) the following notation is used: We can switch to normalized variables in (4): scaling factor with the wave-number dimension. Then we get:
where
is the
where
In terms of complex refractive index n = n'+in" , these parameters have a form 8 :
where
is normalized frequency if the initial electromagnetic wave is taken as Since the electric field exhibits jumps when the medium parameter changes abruptly, it is appropriate to consider the electric flux density:
which remains continuous. Rewriting the integro-differential Volterra equation in terms of electric flux density allows the stability of the numerical algorithm to be improved. Finally we will have
where
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This equation describes the evolution of the process and determines explicitly the field magnitude at a point through its magnitudes at points preceding the given moment on the lines THE MARCHING SCHEME To overcome the weak stability of the previous method8 an enhanced numerical scheme was developed. The essence of the proposed methodology is the sequential resolution of equations (7)-(11) step by step in time. The main difference between the previous technique and the one proposed here is the introduction of an smaller internal subgrid into square cells to satisfy the Courant condition ( or for the normalized variables) and the use of a prediction-correction technique. As in previous developed scheme8, we assume a uniform external grid on the coordinate plane with equal time and spatial steps: The integration paths in equations (8-11) are then straight diagonal lines, which pass through the nodes in the grid. If we assume that electric flux density and functions and are known for the previous time step the electric flux density at node we obtain
where the expressions (8)-(11) were transformed according to the recursive scheme described in 8 and have the form:
The integrals in equations (13) - (16) can be calculated numerically using the internal time subgrid introduced here. The subgrid nodes are chosen according to the Chebyshev quadrature scheme at the time interval
where are tabulated abscissas, and N is the number of points in the subgrid (usually no more that 9). Chebyshev’s quadrature scheme has equal weights and therefore minimizes the error for the case of integrated data that undergo uniformly distributed stochastic
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deviations. The only problem now is that the electric flux density is unknown for An intuitive way to solve this problem is through extrapolation of the function D at the interval to
about the point
If we apply a second-order Taylor series expansion with respect we will have:
where the derivatives at the prediction stage can be estimated through simple two-point scheme.
VALIDATION OF THE PROPOSED SCHEME The test problem relates to plane wave propagation and reflection from a dielectric slab with thickness d. Simulation of such a problem allows simple estimation of amplitude and phase errors in the calculated reflected and transmitted signals, because the phase error is transformed into amplitude error by interference inside the dielectric layer. The analytical solution for a multilayered medium and plane harmonic wave is well known and described, for example, in 10 . In our case, the initial signal is incident on a dielectric layer with complex refractive index n = n'+in" and has the form of a semi-infinite travelling wave with unitary magnitude and wavelength
It is assumed that the medium outside the dielectric layer has refractive index Therefore, the electric flux density of the reflected and transmitted signals is the same as the electric strength that is traditionally used in electromagnetic notation. The reflection and transmission coefficients for the dielectric layer with complex (in the general case) refractive index n and width d are given by the following expressions:
Figure 1 shows the spatial distribution of the electric flux density at for two special cases of electromagnetic wave propagation through the dielectric layer with real refractive index The solid line represents the case of total internal transmission. When the interference inside the layer totally suppresses the back reflection. The value of the layer’s width in this special case corresponds perfectly to the theoretical prediction (with accuracy ). The power of the reflected signal was about 40 dB below the initial one, which also corresponds to perfect coincidence with theory. In contrast with the previous case, the open squares represent the distribution of D for the maximum reflectivity case, when For this situation the calculated absolute values of reflection and transmission coefficients (for magnitude) are and respectively. Note that the expressions (20) give slightly different values, namely 0.303 and 0.956 respectively. This means that the electromagnetic transients still continue inside the dielectric layer. Electromagnetic pulse propagation through a dielectric layer presents a more interesting situation. Comparing the reflection of the continuous wave from a dielectric slab
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and the pulse signal, we see a noticeable difference. For example, the totally suppressed back reflection, which is observed for the continuous wave, does not occur for the pulse signal. If we select a Gaussian pulse
with subcarrier frequency for the incident signal, the minimum in reflection will be the same at However, the reflected power is strong enough. This situation is shown in Figure 2. The maximum magnitude of the reflected signal is 0.15, which corresponds to –16 dB power suppression only.
CONCLUSIONS An improved direct numerical algorithm for solving an integro-differential Volterra equation in the time domain for time- and spatially-varying dispersive dielectric media was developed. The technique is based on a marching scheme and introduces an internal subgrid that results in an enhanced Courant stability condition. The algorithm performs refinement of the results by a prediction-correction scheme. Results show good correlation with existing theoretical solutions and considerable improvement in precision over previous methods. The influence of electromagnetic transients in the lossless dielectric layer on reflection and transmission coefficients is investigated.
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Acknowledgments This investigation has been carried out under the Royal Society/NATO postdoctoral fellowship awarded to Dr. Igor Scherbatko for joint research with the Institute of Microwaves and Photonics at the University of Leeds. REFERENCES 1. A. S. Nagra, and R.A. York, “FDTD Analysis of wave propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. AP, vol. 46, pp.334-340, Mar. 1998.
2. J.L Young, “A higher order FDTD method for EM propagation in a collisionless cold plasma,” IEEE Trans. AP, vol. 44, No.9, pp.1283-1289, Sept. 1996.
3. M. Morgan, “Ultrawideband impulse scattering measurements,” IEEE Trans. AP, vol. 42, No.6, pp.840-846, June, 1994.
4. J.L Young, D. Gaitoude, and J.S. Shang, “Toward the construction of a fourth-order difference scheme for transient EM wave simulation: Staggered grid approach, ” IEEE Trans. AP, vol. 45, No. 11, pp. 1573-1580, Nov. 1997. 5. J.S. Shang, “High-order compact-difference schemes for time-dependent Maxwell equations,” J. of Computat. Physics., No.2, pp.312-333, Aug. 1999. 6. M.D. Pocock, M. J. Bluck and S.P. Walker, “Electromagnetic scattering from 3-D curved dielectric bodies using time-domain integral equations,” IEEE Trans. AP, vol. 46, No.8, pp. 1212 -1219, Aug. 1998. 7. A.G. Nerukh, I.V. Scherbatko and D.A. Nerukh, “Using the evolutionary recursion for solving electromagnetic problem with time-varying parameter”, Microwave and Optical Technology Letters, Vol. 14, No.1, pp. 31-36,1997. 8. A.G. Nerukh, I.V. Scherbatko, and O.N. Rybin, “The direct numerical calculation of an integral Volterra equation for an electromagnetic signal in a time-varying dissipative medium,” J. of Electromag. Waves and Applications, Vol.12, pp.163-176, 1998. 9. I. Scherbatko, “Double-Doppler wavelength conversion of infrared optical pulses by moving grating of refractive index in semiconductors,” Optical and Quantum Electronics, Vol.31, pp.965-979, 1999. 10. S. L. Chuang, Physics of Optoelectronic Devices, -(Wiley series in pure and applied optics), A Wiley-Interscience Publication , 1995.
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MARCHING ON IN ANYTHING: SOLVING ELECTROMAGNETIC FIELD EQUATIONS WITH A VARYING PHYSICAL PARAMETER
Anton G. Tijhuis1 and A. Peter M. Zwamborn2 1
2
Faculty of Electrical Engineering Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, the Netherlands TNO Physics and Electronics Laboratory, P.O. Box 96864, 2509 JG ’s-Gravenhage, The Netherlands.
INTRODUCTION In this paper, we consider the determination of electromagnetic fields for a (large) number of values of a physical parameter. We restrict ourselves to the case where the linear system originates from one or more integral equations. We apply an iterative procedure based on the minimization of an integrated squared error, and start this procedure from an initial estimate that is a linear combination of the last few “final” results. When the coefficients in this extrapolation are determined by minimizing the integrated squared error for the actual value of the parameter, the built-in orthogonality in this type of scheme ensures that only a few iteration steps are required to obtain the solution. The paper is organized as follows. We first describe the general approach. Second, we give an overview of various practical applications. Third, the iterative procedure is illustrated for scattering by a two-dimensional dielectric cylinder in free space. For that example, finally, we outline the use of the algorithm in transient scattering, in linearized and nonlinear inverse-scattering algorithms, and in scattering by an object in a more general environment. Results for all four applications are available, but cannot be included because of space limitations. METHOD OF SOLUTION In the computational modeling of electromagnetic fields for practical applications, typically a large system of linear equations must be solved. This system originates from Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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spatially discretizing Maxwell’s differential equations (in “finite” or “local” techniques) or equivalent integral equations (in “global” techniques). In formal notation, such a system can be written as where = = = =
a linear operator, the unknown field, the forcing function, a physical parameter.
The operator originates from discretizing the integral operator in the continuous equation, is a discretized field and corresponds to an impressed source or an incident field. We are interested in the situation where this problem must be solved for a large number of sampled values of the parameter e.g., with Iterative procedure In this subsection, we consider the iterative procedure that is used to solve the system of equations (1). We summarize the classical description of Van den Berg (1985). The basic idea is to construct a sequence of functions such that the norm of the residual in the operator equation (1), i.e.,
decreases with increasing
At each step of the iterative procedure, we write
where is a suitably constructed correction function. We start the procedure with an initial guess with corresponding residual and a suitably chosen variational function Let We now determine the scalar
such that
is minimized. This leads to
In subsequent steps, we let
for follows that
and
In (6),
is again a suitably chosen variational function. It now is minimized when
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With equations (3)–(8) the iterative scheme based on error minimization has been defined. The interpretation of this procedure is that the forcing function is projected on the set of functions This can also be seen from the fact that
for
and
respectively.
Variational functions
In principle, the iterative procedure outlined above works for any choice of the expansion functions Improvement is obtained as long as the coefficient differs from zero, i.e., when Our choice of expansion functions is inspired by the fact that depends in a wellbehaved manner on the parameter Therefore, it should be possible to extrapolate, by choosing for where N = 2 (linear extrapolation) or N = 3 (quadratic extrapolation). For we follow the conjugate-gradient method, and generate the expansion functions from the residual:
Because of the built-in orthogonality of the iterative procedure, we are certain that this procedure does not search for components of in the space spanned by the “previous” functions The iterative procedure is formulated such that the variational functions can be determined from the residual in each iteration step. However, with (11), the first N variational functions are available at the start. Therefore, the relevant coefficients can be determined directly from the second orthogonality relation in (9). If we substitute
we find directly that the coefficients
with initial estimate
where the
can be found from the system of linear equations
This is equivalent to a conventional conjugate-gradient scheme with
are found by minimizing the squared error
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This modification leads to a more efficient implementation. However, the original version of the procedure provides a better explanation of its effectiveness.
POSSIBLE APPLICATIONS Examples of physical parameters for which this idea can be applied are: Frequency: as in conventional time-marching solutions, a fixed space discretization is introduced for the integral equation. However, the subsequent time discretization, which introduces the well-known stability problem, is replaced by a temporal Laplace or Fourier transformation. This leads to a system of linear equations of a fixed dimension, which must be solved for increasing with integer Afterwards, the desired time signals are obtained by a straightforward FFT operation (Tijhuis and Peng (1991)). Angle of incidence: in EMC applications it may be necessary to obtain the electromagnetic plane-wave response of a – usually conducting – object for a large number of directions of incidence. When such a computation is needed at a single frequency, we can again choose a fixed space discretization. In fact, the operator products required in the minimization procedure for the initial guess are now already available from carrying out the iterative procedure at “previous” angles. Source position: in conventional iterative techniques for solving multi-dimensional inverse-scattering problems, it is often necessary in each step to compute the result of a point- or line-source excitation for a varying position of the source. This can be achieved in the same manner as the variation in angle of incidence mentioned above. Obstacle position: when an object moves with a non-relativistic velocity with respect to a fixed source, the electromagnetic-field computations can be carried out by considering the problem in the rest frame of the obstacle. We then end up with a configuration with a varying source position. Contrast: for penetrable objects, the iterative solution of contrast-source integral equations may be a quite efficient way to determine the electromagnetic field, especially when the convolution structure of the original integral equation is preserved in the space discretization (Zwamborn and Van den Berg (1994)). In that case, the operator products can be evaluated with the aid of FFT operations, which has led to the designation “CGFFT method”. However, the convergence may deteriorate when the contrast in material parameters between the scatterer and the surrounding medium increases. This can be remedied by gradually increasing this contrast, and using the field values for lower contrasts to generate the successive initial estimates until the complete object has been “beamed up”. Shape: in the design of microstrip circuits, it may be necessary to “tune” the dimensions of one or more of the elements of the structure. A similar situation arises when the shape of a scattering object of known constitution needs to be reconstructed or changes gradually in time. The approach was originally conceived as an alternative for marching-on-in-time computations, and has therefore been called “marching on in frequency”. Compared with time-marching computations, the main advantage is that using results for “previous”
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frequencies accelerates the computations, but does not influence the final result. Therefore, error accumulation is inherently avoided. Compared with conventional frequencydomain techniques, the main advantage is that the space discretization is fixed for all frequencies. Therefore, the computational effort does not increase for increasing frequencies. Since its publication, the scheme has been applied as well to march on in angle, source position, contrast and object dimension. Therefore, the name “marching on in anything” now seems more appropriate. 2D DIELECTRIC CYLINDER: FORMULATION The applications mentioned in the previous section have by now all been demonstrated for one or more configurations. In this section, we outline the implementation of the iterative procedure for an infinitely long, axially uniform lossy dielectric cylinder embedded in free space (Figure 1). We follow Peng and Tijhuis (1993). In the next section, various applications will be described. An electrically polarized two-dimensional wave with electric-field strength is normally incident on the cylinder. The aim of the computation is to determine the corresponding total field component
The cornerstone of our method of solution is the integral representation
which holds for all In (16), denotes the modified Bessel function of the second kind of order zero, and When (16) reduces to an integral equation of the second kind for inside the cylinder. The space discretization of (16) is obtained as follows. The logarithmically singular behavior of as is substracted by breaking the integral over on the right-hand side of (16) up into
The first integral in (17) contains an almost regular part of the integrand of (16). The logarithmic singularity is confined to the factor of in the integrand of the second integral.
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The cross section is enclosed within a rectangular region consisting of square subregions with mesh size The grid points of the simple cubic mesh are located a t with for and for Solving (16) now amounts to determining an approximation of at the grid points The discretization of the integrals in (17) is based on approximating suitable parts of the integrands by piecewise-bilinear approximations, and integrating analytically over polygons determined by the boundary of and the grid. For the first integral, approximating the entire integrand results in
as In (18), is a discretized kernel, a weighting coefficient that contains shape information, and a sampled, extrapolated susceptibility. For the second, integral in (17), we use the piecewise-bilinear “filtered” approximation
and we integrate over
The resulting approximation assumes the form
where the discretized kernel is independent of frequency. Combining the approximations (18) and (20) then results in the discretized integral equation
where the convolution-type structure of the continuous equation (16) has been preserved. This makes this equation suitable for the application of the conjugate-gradientFFT method.
2D DIELECTRIC CYLINDER: APPLICATIONS As mentioned above, the solution of the two-dimensional scattering problem described above has been used in various applications. In the present section, we give an overview. Transient fields: the most straightforward application of the procedure is the solution of a transient scattering problem. In the marching-on-in-frequency method, (21) is solved repeatedly for with and
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From the frequency constituents thus obtained, the time-domain field is obtained by numerically evaluating a Bromwich inversion integral of the type
with the aid of an FFT operation. In this procedure, is determined by the duration of the time interval of interest, and N by the bandwidth of the incident pulse. Linearized inverse profiling: in inverse profiling, the cylinder is excited by an electric line source on a contour outside the cylinder (Figure 2). The aim of the computation is to determine from scattered-field values at a single frequency Let be the field caused at by a line source at Linearized inversion schemes like the distorted-wave Born and NewtonKantorovitch iterative procedures essentially proceed as follows. In step we first determine by taking the permittivity equal to i.e., the estimate available from the previous step. Subsequently, we obtain the next estimate by minimizing a cost function involving the difference between the simulated and known scattered fields on In the field computation, we march in angle in the first few iteration steps and in contrast once the inversion begins to converge. Nonlinear optimization: the availability of the full scattering matrix also offers the possibility to overcome some of the limitations of linearized schemes. The idea is to replace the determination of by a line search in a nonlinear optimization scheme. We search in the so-called gradient direction, which can be determined from the scattered field. There is no need of inverting matrices or solving adjoint problems; only straightforward integrations are involved. Until now, the expression for the gradient direction has been rarely used in practice, because it was too time consuming to compute the full scattering matrix with
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conventional solution methods. With the fast forward scheme described in this paper, that is no longer a problem. We march on in angle in the first two steps of each line search, and in search direction in subsequent steps. Embedding: the most recent idea is to consider objects in a more general environment. The CGFFT method can only be applied to objects in a homogeneous environment. Practical measurements, e.g. for biological tissue, must be carried out in a more complicated set-up like a water-filled metal cylinder. Such a set-up is presently being developed at CNRS/Supélec. An impression is given in Figure 3. In that case, a direct computation of the field is relatively complicated from a computational point of view. Hence, the idea is to use the procedure outlined above to determine the complete scattering response of the estimated object in a homogeneous environment, and subsequently use a spectral formulation to “embed” this object in the actual environment. That environment then needs to be characterized only once. This will make it feasible to solve the corresponding inverse problem by one of the approaches outlined above.
REFERENCES Van den Berg, P.M., 1985, Iterative schemes based on the minimization of the error in field problems, Electromagnetics, 5:237. Tijhuis, A.G. and Peng, Z.Q., 1991, Marching-on-in-frequency method for solving integral equations in transient electromagnetic scattering, IEE Proc. H, 138:347. Peng, Z.Q. and Tijhuis, A.G., 1993, Transient scattering by a lossy dielectric cylinder: marching-on-in-frequency approach, JEWA, 7:739. Zwamborn, A.P.M. and Van den Berg, P.M., 1994, Computation of electromagnetic fields inside strongly inhomogeneous objects by the weak conjugate gradient FFT method, JOSA A, 11:1414.
CORRELATION OF ANTENNA MEASUREMENTS USING THE OVERSAMPLED GABOR TRANSFORM
Benoît Fourestié and Zwi Altman France Telecom R&D 92794 Issy les Moulineaux FRANCE
INTRODUCTION Antenna measurements in amplitude and phase inside anechoic and semi-anechoic chambers have important applications in antenna characterization and electromagnetic compatibility. The analysis of antenna radiation characteristics, the characterization of electromagnetic properties of materials such as absorbers, and the analysis of the properties of the test site itself, in terms of spurious reflections and resonances, are only few examples. Typically, a measurement setup of two antennas facing each other is used and a network analyzer performs frequency measurements in amplitude and phase. The receiving antenna measures both the direct propagating wave component as well as the reflected components. By identifying the reflections from the measured signal, one can assess the influence of the test site. In addition, if one can suppress the reflected components, it is possible to retrieve the signal one would have measured in a perfect anechoic chamber. In most cases Fourier analysis fails to provide enough resolution to separate a signal from its propagating components due to the lack of frequency bandwidth, and other powerful signal processing techniques are required, such as super-resolution or multi-resolution techniques. In this work we propose to apply the Oversampled Gabor Transform (OGT) to perform a high resolution analysis of a measured discrete and complex signal. The Gabor Transorm (GT) (Gabor, 1946) is a gaussian windowed short time Fourier transform, and in certain cases it may suffer from instabilities. Recently, the GT has been reformulated using frame analysis for continuous and discrete signals (Zibulski and Zeevi, 1993, 1994, 1997), rendering the GT stable to the condition that it is oversampled. An important gain in resolution has been obtained by allowing the time-frequency windows to be located at small time and frequency intervals from each other, in an overlapping tile-like structure in the time-frequency coefficient domain. Although less resolution is obtained using the OGT analysis compared to other super-resolution techniques such as the Matrix Pencil (Fourestié et al 1999, 2000), the OGT provides a more robust decomposition of the signal into its propagating wave components.
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FRAMES AND OVERSAMPLED GABOR TRANSFORM The OGT is based on frame analysis which is an extension of the concept of bases (Daubechies, 1990). The definition of a frame is given as follows: A sequence in a Hilbert space H constitutes a frame if there exist positive numbers A and B called frame bounds, such that for all we have
were denotes the scalar product corresponding to the Hilbert space, For an orthonormal basis we have A = B = 1. Given a frame in a Hilbert space H, we define the frame operator S as follows
is the dual frame of with frame bounds represented by the frame as follows:
and
Unless the frame is a basis, the representation coefficients
Every
can be
are not unique. The
choice of the dual frame for computing the representation coefficients yields the minimal energy solution for the coefficients. In the context of antenna measurements, discrete representation of frames is of particular interest since measurement results are obtained in a discrete form. Consider a signal f of L samples which belongs to an L-dimensional space of complex vectors. The finite set of vectors is represented as the columns of a matrix X, and the inner product of the elements of and f by a vector d, If is a frame, then is nonsingular, and we can write the frame operator as
Next, we write (3) in matrix form. The dual frame vectors and the vector made of the coefficients
are the columns of can be written as
is the Moore-Penrose or the pseudo-inverse of X (Golub, and Van Loan, 1989) which is a simple and elegant way for finding the dual frame. The continuous GT of a function f(t) is given by
where
The decomposition (5) is stable for ab < 1, and we choose ab = p/q < 1, where p and q are prime integers. This choice is referred to as the oversampled scheme, or the OGT. Assume that constitutes a frame; then the coefficients can be found using the dual frame of which are also referred to as the bi-orthogonal functions
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Having found the
we derive the coefficients
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using the following scalar product:
For the discrete OGT we consider a discrete L-periodic signal f. Let M and N be two divisors of L satisfying L = N ’M = M’N, with M, N, M’ and N’ positive integers. We define the discrete Gabor representation of f(i)
with the discrete Gaussian window function
As in the continuous case, we write and consider the oversampled case with for which the Gabor representation is stable. A small value for N’ is translated into oversampling in the time domain, with big overlapping of adjacent Gaussian windows. Similarly, a large value for N corresponds to a big overlapping of the windows in the frequency domain. The expansion coefficients can be derived as follows:
where (11) can be calculated using a DFT-based algorithm. Equations (9-11) are formulated in the signal domain. The continuous and discrete OGT have first been formulated in the Zak Transform (ZT) domain (Zak, 1967) which is equivalent to the formulation in the signal domain. The ZT of a function f(t) is defined as follows:
with a fixed parameter Oversampling in the GT can considerably increase the resolution of the analyzed signal as explained below. To each coefficient corresponds a time-frequency window of size with constant area of The sides and are the root-mean-square radii of g in the time and frequency domain respectively. By oversampling one can locate the windows close to each other in an overlapping tile-like structure. It is then possible to set the window approximately at the center of any event in the time-frequency plane, such as a maximum of a function. Without oversampling, the maximum of a function may occur at the center of four adjacent windows; its energy will be split amongst the four windows, resulting in a very poor resolution. The application of the OGT to antenna measurements is described in the next section.
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ANTENNA MEASUREMENT ANALYSIS In this section we propose to perform a high resolution analysis by applying the OGT to antenna measurements in amplitude and phase inside a semi-anechoic chamber. The objective is to identify and suppress the reflected wave components in order to remove the influence of the test site from the measured results and to obtain the signal one would have measured in a folly anechoic chamber. The measurement setup consists of two log-periodic antennas with a nominal frequency bandwidth of 100-1,000 MHz in a semi-anechoic chamber. The two antennas are in horizontal polarization, 2 m above the ground and 3 m apart, and the corresponding parameter is measured using a network analyzer. The transmission coefficient is measured in the semi-anechoic chamber in the frequency range of 100-1,000 MHz and with a sampling step of Denote by the transmission coefficient in the same folly anechoic chamber which can be measured by adding absorbing materials on the ground. The propagation delay of the direct wave component decreases logarithmically with frequency. This behavior is due to the migration of the active zones of the log-periodic antennas towards their apexes with the increase of frequency. The propagation path for the direct wave component is illustrated in Figure 1, with the distances and defined in the Figure. The wave travels the distance inside the coaxial line or the waveguide portion of each antenna and the distance in free space. In a first order of approximation, and are logarithmically decreasing functions of frequency (Mittra, 1969).
The magnitude and phase of
are plotted in Figure 2. We can observe that the
phase period of increases with frequency. This behavior is due to the decrease of the propagation path and the propagation time between the two antennas with frequency.
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The complex signal is decomposed using the OGT and the results are presented in Figure 3. The decomposition is performed with M = 20 and N = 700, i.e. an oversampling factor of 20, to obtain the highest possible resolution in time. The width of the gaussian window is chosen with d = 100 samples. Two distinct strips in the timefrequency plane corresponding to the direct and reflected wave components are clearly identified above 300 MHz.
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At low frequencies the OGT coefficients are smeared on the time-frequency plane with insufficient resolution to separate out the propagating components. We now proceed and force to zero the coefficients corresponding to the reflected wave on the ground. Denote by the corresponding reconstructed signal.
The results obtained for the
reconstructed signal will be validated by comparing the reconstructed signal with measured in the fully anechoic chamber. The suppression of the reflected component is performed using the following automatic procedure: be the vector formed by the magnitudes of coefficients for a (i) Let given m, i.e., a given frequency location and of the two peaks of corresponding to the (ii) Calculate the indices direct and reflected components (iii) Find the index of the minimum of between and
(iv) Force coefficients
to zero for It is assumed in (iv) that the width of the reflected component does not exceed 20 samples over the entire frequency range. In Figure 4 the results for the decomposition of in the time-frequency plane is presented. For frequencies above 300 MHz, the contribution of the reflected wave component has been identified and removed using the automatic procedure.
The decomposition of the reference signal, this decomposition is very close to that of the signal
is shown in Figure 5. We can see that in Figure 4.
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Finally, the signal
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is reconstructed in the signal domain from the time-frequency
coefficient data and compared to in the frequency range of 300-1,000 MHz (Figure 6). We can see that the strong oscillations of more than 7 dB in magnitude (Fig. 2) which correspond to the interference pattern of the direct and reflected wave have practically disappeared. The difference between the two signals is less than 1.3 dB, with an average of 0.44 dB, and a standard deviation of 0.33 dB. Since the correction process is fully automatic, it is possible to know the frequency range of validity of this approach, which makes it a very robust technique for antenna measurement analysis.
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CONCLUSION The OGT has been successfully applied to the analysis of antenna measurements in amplitude and phase in the frequency domain. The measured signal can be easily processed in the transform, time-frequency coefficient plane, and its reconstruction from the timefrequency data is straightforward. The OGT allows to perform a robust and high resolution analysis of the measured signal leading to the identification and separation of its propagating wave constituents. The component reflected on the ground has been systematically identified and removed to retrieve measurements performed in a fully anechoic chamber. It is interesting to note that the difference between the reconstructed signal and the signal measured in the anechoic chamber remains less than 1 dB above 350 MHz, i.e. in the major part of the frequency bandwidth under consideration. Since a difference of 1 dB corresponds to the standard criterion for Measurement Test Site equivalence, the proposed method can be used to correlate measurements at different test sites. The OGT seems promising in the context of antenna testing, site characterization and, more generally, signal component discrimination.
REFERENCES Daubechies, I., 1990, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 36:961. Fourestié, B., Altman, Z., and Kanda, M., 2000, Efficient detection of resonances in anechoic chambers using the Matrix Pencil Method, IEEE Trans. Electromagn. Compat., 42:1. Fourestié, B., Altman, Z., Wiart,J., and Azoulay A., 1999, On the use of the Matrix Pencil method to correlate measurements at different test sites, IEEE Trans. Antennas Propagat., 47:1569. Gabor, D., 1946, Theory of communication, J. Inst. Elec. Eng., 93:429. Golub, G. H., and Van Loan, C. F., 1989, Matrix Computation, Hopkins University Press, Baltimore. Zak, J., 1967, Finite translations in solid state physics, Phys. Rev. Lett., 19:1385. Zibulski, M. and Zeevi, Y.Y., 1994, Frame analysis of the discrete Gabor-scheme, IEEE Trans. Signal Processing, 42:942. Zibulski, M. and Zeevi, Y.Y., 1997, Discrete multiwindow Gabor-Type transforms, IEEE Trans. Signal Processing, 45:1428. Zibulski, M., and Zeevi, Y.Y., 1993, Oversampling in the Gabor scheme, IEEE Trans. Signal Processing, 41:2679. Mittra, R., Log-Periodic Antennas, 1969, chap. 22, in: Collin et al, ed., Antenna Theory, Part 2, 22., McGraw-Hill.
ON A RATIONAL MODEL INTERPOLATION TECHNIQUE OF ULTRA-WIDEBAND SIGNALS
N. H. Younan, C.D. Taylor, and J. Gu Department of Electrical and Computer Engineering Mississippi State University Mississippi State, MS 39762
INTRODUCTION Applications of ultra-wideband pulsed RF energy is increasing. These applications range from the treatment of cancer and other maladies to non-destructive testing, remote sensing, and ultra-wideband weapons. Monitoring the time history and the energy spectrum of ultrawideband pulses pushes the state of the art in electronic instrumentation. This paper examines how limited data can be used to infer data outside the range of the instrumentation. Generally, discrete-time or discrete-frequency data are obtained via uniform sampling. In some cases, this may not be practical. For instance, logarithmic spacing is often used to limit the samples to a practical number in frequency domain data, collected for frequencies over several decades. Moreover, for high frequency measurements, gaps in the frequency domain data occur as a result of skipping certain frequency bands. Accordingly, interpolation is used to characterize the frequency domain response. Interpolation techniques require a data model to restore the unknown data samples. Often, the model is quite simple. Consequently, interpolation is related to function approximation. In general, interpolation schemes presume some degree of smoothness for the function to be interpolated. However, this may not be valid for noisy data. Moreover, if the interpolating function is fitted to the known data points with additive noise, significant interpolation error may occur. Various interpolation schemes have been used to restore unknown data samples. However, most of them have been performed on time-series data using models like AR, MA and ARMA1-3. Techniques for interpolating complex frequency data have received little attention. Traditional techniques, such as linear, cubic spline, and Lagrange interpolation, have been shown to be data dependent and are generally not satisfactory. They become highly unstable for data corrupted with noise. Linear interpolation is perhaps the most widely applied and the simplest technique. However, when applied to noisy data, inaccurate results are obtained when a spectral peak or null occurs between samples. The Lagrange technique uses a polynomial function to
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interpolate between data samples, it causes instability at high order degree even for data without noise; therefore, it is not suitable for noisy data4. The Cubic Spline interpolation uses a polynomial fit between each pair of known data points. The coefficients are determined nonlocally. For noise-free data, splines tend to be more stable than polynomials, with less possibility of wild oscillation between the known data points. However, this technique becomes highly unstable for data corrupted with noise5. A regressive rational function interpolation for noisy data is presented. This technique incorporates the use of the singular value decomposition method and a statistical measure of goodness-of-fit to obtain the best estimate of the model coefficients. Additional features include a self-tuning ability to obtain an optimum model order and noise reduction. Compared to traditional interpolation techniques, this method is shown to be more robust. Results are obtained for noisy data with low signal-to-noise ratios to ascertain the validity and robustness of the proposed technique.
REGRESSIVE FIT MODEL INTERPOLATION In general, rational functions interpolation methods have shown superiority over polynomials because of their ability to model functions with poles. Accordingly, rational functions can be viewed as ARMA type models for frequency domain applications. An ARMA model is described as6,
where,
equation (1), represents the actual received frequency data samples and the right-hand side represents the ARMA model used to fit the corresponding frequency data. In traditional interpolation techniques, the coefficients are usually determined by forcing the interpolating function to match the known samples7. For the technique presented here, a rational function interpolation is incorporated with a least-square method that uses a regressive fit to obtain the rational model parameters8. Once the coefficients are known, then the complete frequency response can be reconstructed at any frequency value. In general, the square error is defined as,
Where N is the number of data samples. Although this error criterion can be used to calculate the coefficients, it has some deficiencies. First, if the frequency response has to be determined for frequencies extending several decades, the lower frequencies can not be fitted well since they have very little influence. Second, if has poles in the complex S-plane such that could vary widely throughout the experimental points, large errors would be
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introduced. The stated deficiencies can be overcome by an iterative procedure where, at the stage of iteration, the square error is modified to9,
where the model coefficients, are obtained by setting the derivative of with respect to the coefficients to zero. This yields a set of simultaneous equations of the form of Ax = B. Here A and B are a 2n×2n matrix and a n×1 vector formed from the known frequency data and x consists of the model coefficients. In solving the linear algebraic equations, the singular value decomposition (SVD) technique is incorporated to eliminate the noise components. At each step, the SVD is used to solve for the coefficients via the leastsquare technique and the singular values are truncated according to the Akaike Information Criterion (AIC)10, i.e.,
Here K is the rank of the matrix A and corresponds to the i th eigenvalue. The optimum order is chosen as the value of m that minimizes the AIC. SIMULATION RESULTS To determine the limitations and accuracy of the presented technique, a damped sinusoid with additive white-Gaussian noise is used to function as the received signal, i.e.,
Here, L is the number of sinusoids, T is the sampling period, and w(n) is the noise sequence. The noise level is quantified by the signal noise ratio (SNR), which is defined as
Fast Fourier Transform (FFT) is then used to obtain the frequency samples. To illustrate the validity of the technique, a signal with a spectral peak at 20 MHz is generated with and The sampling rate is chosen to be A worst case data gap, namely points centered around the resonant peak, is considered. Figures 1 and 2 ilustrates the magnitude and phase plots of the noisy simulated data with SNR = 20 dB. The resulting magnitude and phase plots obtained by applying the regressive fit interpolation to the one peak noisy signal with a 3-point gap together with the
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magnitude and phase plots of the noise free signal are shown in Figures 3 and 4. These figures indicate how well the presented technique performed, i.e., the reconstructed signal matches well with the original signal and the noise component is successfully suppressed. Similar results are obtained for the one peak noisy signal with SNR=10dB. The corresponding magnitude and phase plots comparisons are illustrated in Figures 5 and 6 respectively. Note that the relative error for the 20 dB case is approximately 9% compared to 57% and 33% relative errors obtained from Lagrange and Spline interpolation schemes. In addition, a slight degradation in performance is obtained for the 10 dB case, but it is still reasonably acceptable.
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To further demonstrate the applicability of the interpolation technique, simulated data of length of 1024 samples with two spectral peak at 10MHz and 30MHz are. In this case, and and the
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sampling rate is chosen to be 600 MHz. Similar to the single peak signal, a 3-point gap around each spectral peak is considered. The restored magnitude and phase plots for the 20 dB case are shown in Figures 7 and 8 respectively. It is clearly seen that the proposed technique performs well.
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CONCLUSIONS A regressive rational function interpolation technique for noisy frequency domain data is presented. This technique is based on using a modified ARMA model to fit the transfer function of the underlying noisy spectral data. The coefficients of the model are then calculated based on minimizing a weighted least square criterion iteratively. This procedure incorporates the singular value decomposition technique to accurately estimate the order of the model and at the same time to suppress the noise components, which corrupt the actual data samples. Simulation results are presented to demonstrate the applicability of the interpolation technique.
REFERENCES l.
R. Steele and F. Benjamin, “Sample Reduction and Subsequent Adaptive Interpolation of Speech Signals,” The BELL System Technical Journal, Vol. 62, No. 6, July-August 1983. 2. Ronald E. Crochiere and Lawrence R. Rabiner, “Interpolation and Decimation of Digital Signals: A Tutorial Review”, Proceeding of The IEEE, Vol. 69, No. 3, March 1981. 3. A.J.E.M. Janssen, R.N.J. Veldhuis, and L.B. Vries, “ Adaptive interpolation of Discrete-Time Signals That Can Be Modeled as Auto-regressive Processes,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. ASSP-34, No. 2, April 1986. 4. Clayborne D. Taylor, Nicholas H. Younan, and Shinfeng D. Lin, “EMP Data Analysis and Software Development”, Final Report for TRW Inc., Space & Defense Sector, January 1990. 5. Richard L. Burden, J. Douglas Faires, Numerical Analysis, six edition, 1996. 6. S.L. Marple, Jr., Digital Spectral Analysis with Applications, Prentice-Hall, Englewood Cliffs, New jersey, 1987. 7. C. Daniel and F.S. Wood, Fitting Equations to Data, Wiley-Interscience, 1971. 8. C.K. Sanathanan and J. Koerner, “Transfer Function Synthesis as a Ratio of two Complex Polynomials,” IEEE Transcations on Automatic Controls, pp 56-58, January 1963. 9. A.W.M.v.d. Euden and G.A.L. Leenknegt,” Design of Optimal IIR Filters with Arbitrary Amplitude and Phase Requirements,” Proceedings EUSIPCO, Signal Processing III: theory and Applications, 1986. 10. Steven M. Kay, Modern Spectral Estimation: Theory & Application, Prentice Hall, 1988.
FULL-WAVE SOLUTION OF THE PROPAGATION OF GENERALLY SHAPED IMPULSES AND WIDE BAND APPLICATION IN ANISOTROPIC PLASMAS
Orsolya E. Ferencz1 and Csaba Ferencz1 1 Eötvös University, Dept. Geophys., Space Research Group H-1117 Budapest, Pázmány P.s. 1/A., Hungary E-mail:
[email protected]
INTRODUCTION One of the most interesting wave propagational problems is the investigation of arbitrarily shaped signals, i.e. impulses in different media. This problem is actual, because in space research and applications the signals of natural origin detected by space vehicles or ground stations are generated by general shaped sources (e.g. lightening strokes, atmospheric currents, seismic events), or the artificial signals used in man made applications must have a general form with extra-wide frequency band or these signals are very short impulses. The most common media in space applications are the anisotropic plasmas. Therefore the paper deals with propagation of impulses in inhomogeneous anisotropic plasma-models using a plane-wave, applying of which the solution of Maxwell's equations can be obtained in accurate and closed formed time-space function of the impulse-type signal, without any monochromatic or quasi-monochromatic assumption or restriction. The example, in the case of which the model is presented, is the phenomena of magnetospheric (upper-atmospheric) signals excited by lightening stroke impulses and detected in ELF-VLF and HF-UHF bands ("whistlers" and related phenomena). The propagation of impulses and other generally shaped signals in plasmas are commonly described by approximate theories based on the monochromatic solution of Maxwell's equations in the given medium (dielectric tensor, refractive index, AppletonHartree formula, wave packet, group velocity calculations etc.) – e.g. by Budden (1961) or Walker (1993). These descriptions of the wave-propagational situation do not always result in a satisfactory interpretation of the electromagnetic phenomena caused by transient effects, e.g. the physical, theoretical explanation of some observations. The real character of the phenomena contains definite starting point according to time and/or space. The problem is similar to the well known one in the network analysis as transient effects. In these cases the introduction of functionals becomes necessary.
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The solving process is fully analytical, starts directly from the Maxwell's equations and applies the multi-dimensional Laplace-transformation managing the transient character of the signals. The main part of the solution uses the Method of Inhomogeneous Basic Modes (MIBM) deriving the propagating field strength components of the electromagnetic signal. A great advantage of this method is hidden in the fact, that the "transfer function" of the medium – the system-answer given for a Dirac-delta functional – can be available, which makes possible to develope a very flexible model for linear propagational problems, which type of models are often used in network-engineering problems. THEORY The task is to derive a general-shaped, real solution of the Maxwell's equtations, using the MIBM. The main steps of the solution process are presented below. Let the medium be considered in which the medium-parameters suddenly change at some open or closed surfaces having no intersection (Figure 1). The medium-parameters within the volumes are described by continuous functions joining each other with jumps at the surfaces where the medium is characterised by functionals (distributions). The signal in each volume will be disintegrated into n modes. During the derivation process all of the possible modes must be determined in every volume. Let the form of the solution looked for in be the following
where means the components of the electromagnetic field. Furthermore, let the 1(x) Heaviside (or unit step) and the Dirac-delta distributions (functionals) be introduced. denotes the distribution, the value of that varies from 0 to 1 at the surface i.e. where the vector represents the the parametric equation of the surface Using the functionals it is possible to create the following distributions (or "gate-functions") from them
which has 1 value in the region between surfaces everywhere out of that (Figure 2).
and
while it has 0 value
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According to MIBM the complete solution can be obtained using equations (1) and (2) as
where M is the number of the continuous regions. Using this form of solution in the Maxwell's equations the derivative of the "gate function" will appear, which is
where is definitely the normal vector of surface, points outward (to in the present case). Substituting (3) into the Maxwell's equations, they can be automatically decomposed into two groups. One group is valid within the regions and the other has to be fulfilled at the surfaces In the first step the equations valid in the volumes will be solved. Applying the common notations of electrodynamics this equation-system is
where and are the source current and charge densities in the given region, if they exist there. The solutions of eqations (5) can be obtained for each layer in different ways, e.g. using the Laplace-transformation. If the medium is inhomogeneous in these regions known methods of solutions can be used, see Ferencz (1978). If the application of the (generalised) W.K.B. philosophy is necessary, one can apply it at the end of the solving process just before the inverse-Laplace transformation. Using the derivated form of the Laplace-transformed amplitudes a W.K.B.-type relation can be defined between the electric and magnetic amplitudes. The other group of the equations is valid at the boundary surfaces delivering the connection among the regions. This is called coupling equation system
In equation system (6) it is necessary to use every modes existing in the volumes Solving (6), the complete solution can be obtained.
SOLUTION OF THE PROBLEM IN MAGNETOIONIC MEDIA As an example of application the propagation of an electromagnetic signal in an anisotropic plasma and generated by a source current existing beyond the plasma is presented here. This is the case of whistlers and related phenomena. The whistlers are remarkable bursts of electromagnetic energy in ELF~VLF bands produced by ordinary lightning. Related phenomena are energy bursts appearing in higher frequency bands with special character, e.g. the Transionospheric Pulse Pairs (TiPP's) in the 20~100 MHz bands in the Earth's atmosphere. Using the traditional magneto-ionic theory based on the examination of harmonic signals (e.g. Budden, 1961; Walker 1993), numerous hypotheses were born regarding the generation and propagation mechanism of the whistler signals, but an exact time-space waveform of the e.m. field was missed up to now. Because of the
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contradictions between these monochromatic-type model-calculations and the physical reality, many ideas and explanations of the measured effects were fundamentally false. Besides this no acceptable idea was born up to now about the generation mechanism of the TiPP's. This problem can be solved deriving a real full wave solution using the theory summerised above.
For the investigation of this situation one can use a relatively simple model structure (Figure 3). The main atmospheric payers are modelled in Cartesian coordinate system by two volumes, i.e. by two half-spaces and or medium and medium – and the plane-surface between these regions at the place of Medium is a usually vacuum-type, infinite half-space modelling the Earth's surface - ionosphere waveguide, where the arbitrarily shaped source current density generated by the lightning discharge impulse arises in the region It is important to emphasise that there are not any 'traditional' wave-packet models or other monochromatic-type approximations in the model. The shape of in time and in space is general and has a switch-in character. The starting time of the excitation is Medium is an inhomogeneous or homogeneous, anisotropic plasma in the infinite half-space, as a usual model of the magnetosphere. In this half-space no source current density exists. The plasma can be considered to be lossless or lossy (characterised by collision frequency), cool, tempered, anisotropic, magnetised, time-invariant and linear electron-plasma or a plasma-model containing arbitrary ion-distributions too. The superimposed (geomagnetic) field can be parallel to the direction of propagation (i.e. strictly longitudinal propagation; ), or not (oblique propagation; ). The different plasma-effects are included in the motion equtions of the electrons and ions. The supposed wave-pattern of the solution is a plane-wave (i.e. plane-signal) having definite temporal and spatial starting points. At the begining of the deriving process let the Laplace-transformation be applied for the equations (5) written in ( t) domain in medium and respectively. The solution in medium is not too complicated and the result contains the source current density The next step is the inverse Laplace-transformation, which produces the space-time functions of exsisting modes in medium – The solution process in the
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medium is more complicated. If the propagation is oblique, i.e. three independent variables appear (x, y, t) which turn into transformed variables (p, l, s) by the Laplacetrasformation. If the medium is inhomogeneous weakly, the conditions of the W.K.B. approximation are given and the amplitudes of the exsisting modes can be correct by a generalised W.K.B. philosophy: for example, if the pole of an e.m. mode is after an transformation, than the W.K.B. amplitude of the mode is
The inverse Laplace-transformation can be executed in the way known from the engineering applications. In the last step the equation-system (6) will be solved at the boundary plane for all existing e.m. modes. At the end of these investigations it is possible to apply the same signal analysing methods (FFT, curve fitting, matched filtering etc.) for the computed signals, for the computed field strengths, which are commonly used for the signals measured on board of satellites or at terrestrial stations. Further, as an example of the derived solutions, the strictly longitudinal modes will be presented here, propagating in an inhomogeneous, multicomponent plasma along the magnetic field. In every cases two modes propagate into one direction, in this case into the +x direction. Here only one electric component of these two modes is shown:
where
is the wave impedance for vacuum, n = 1, 2 and
and are the plasma frequency of the electron and the different ions in the plasma respectively, and are the giro-frequency of the electron and the different ions respectively; if n=1, the upper signs are valid, if n=2, the lower signs are valid in (9); and the excitation defines the quantity as
The excitation is a Dirac-delta in order to get the transfer function of the medium (of the whole propagational situation), and in this case
Another useful excitation is a simple rectangular impulse at the place x = x *, i.e.
The relation (10) determines the excitational effect independently of the shape of source signals.
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APPLICATION OF THE RESULTS It is a simple fact that using these computed signals it is possible to calibrate and verify the analytical methods, i.e. softwares and systems, used in wave analyses and research. Processing the transfer functions of various propagational models and calculated signals generated by different excitations the real answers and artifacts can be seen, produced by the wave-processing method itself. This is an important new possibility in method-calibration. Another new possibility in wave-investigations that it is possible to separate the propagational and excitational effects. Comparing the measured signals to the transfer functions and signals computed by different excitations, it can be determine the effects of the propagatinal path and the possible (probable) excitation (Figure 4).
Another example: Seeing the relation (12) it is clear that in the case of a rectangularimpulse excitation the signal contains amplitude minima which are strictly periodic on the frequency-axis and the place of the first minimum is If the excitation is a double Dirac-delta, the wave-pattern will be similar, but the places of this strictly periodic amplitude minima will be different. However, if the signal is a resultant of two transfer functions originating from the same source but propagating on slightly different paths, the resultant signal contains amplitude minima which are non-periodic on the frequency-axis. Using the oblique propagational solution one can compute a branch of signals (transfer functions) originating from the same source but propagating with different angles to the superimposed magnetic field. In some cases on board of satellites it was registrated whistler branches with nearly the same character (Figure 5). The satellite registrated these signals on a small sequence of the orbit, nearly at one given place. Therefore these signals were at a given time at a given region and could not propagate longitudinally along different earth magnetic field lines, which was supposed in some former theories. This investigations open the way for new ideas in ELF~VLF magnetospheric propagation.
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The application of these solutions in the wave-investigations is a new an effective way in the research (in space research) as well, as in practical applications.
RESULTS IN EXTREME WIDE-BAND IMPULSE PROPAGATION Using the results presented by equation (8) it was demonstrated (Ferencz, 1999) that the first mode represents the electron-whistler and the second mode belongs to the ionwhistlers in the magnetospheric propagation in the ELF~VLF bands. (This fact gives an essentially new explanation of ion-whistler generation.) However, expanding the frequency bands of investigation up to the high frenquencies (e.g. up to 100 MHz) it can be found a consistent and common explanation of the electron- and ion-whistlers, the Faraday rotation and the TiPP-events. The TiPP events have been registrated by satellites in high frequency bands (Rodger, 1999). In every cases a pair of dispersed signals appear on the FFT dynamic spectrum and both has a doubled fine structure. No acceptable interpretation of this phenomenon was presented up to now. In order to give explanation of these events, the solutions (8) in the extreme wide band from ELF to 100 MHz can be computed, using a rectangular type impulse excitation, where the length of this impulse in time is equal to the time difference of the main signal pairs in the TiPP event. The two exsisting modes will propagate with different time delay. This phenomenon produces the electron- and ionwhistlers in the lower frequencies, and this will appear in the higher frequency-bands above the plasma and giro-frequencies of the medium as a doubleness inside of the first and second main part of the TiPP event. This small time-delay between this two modes produces the Faraday-rotation if the propagating signal is monochromatic (Figure 6). However, seeing the evolution of a signal excited by a rectangular impulse it can be demonstrated that an essential doubleness appear in the signal during the propagation (Figure 7a). This inherent structure of the signal will appear in the high frequency bands in the dynamic spectrum as twin pair of signals with a time difference equal to the temporal length of the impulse (Figure 7b). The time delay of the two modes in the high frequency-band is much smaller than the length of the source impulse. Therefore the upward directed edge of the impulse causes the
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first main part of a TiPP with an inner doubleness and the downward directed edge of this impulse-origined signal forms the second part of these twins.
A final conclusion is that this new method of the solution of Maxwell's equations and the solutions itselfs form very effective and productive way in wave-propagational research and applications. REFERENCES Budden, K.G., 1961, Radio Waves in the Ionosphere, Cambridge Univ. Press, London, E. Ferencz, O., 1999, Electromagnetic wave propagation in different terrestrial atmospheric models, Ph.D. Thesis, Budapest University of Technology and Economics, Budapest, Hungary. E. Ferencz, O., 1999, Origin of ion-cyclotron whistlers; Proc. of XXVIth Gen. Assembly of URSI, pp.534, H4.P9, 13-21 Aug. 1999, Toronto, Canada. Ferencz, Cs., 1978, Electromagnetic wave propagation in inhomogeneous media: Method of inhomogeneous basic modes, Acta Techn. Ac. Sci. H., 86(1-2), 79-92. Rodger, C.J., 1999, Red sprites, upward lightning, and VLF perturbations, Reviews Geophys., 37, 3, 317. Walker, A.D.M., 1993, Plasma Waves in the Magnetosphere, Springer-Verlag, Berlin.
ASYMPTOTIC DESCRIPTION OF ULTRAWIDEBAND, ULTRASHORT PULSED ELECTROMAGNETIC BEAM FIELD PROPAGATION IN A DISPERSIVE, ATTENUATIVE MEDIUM
Kurt Edmund Oughstun College of Engineering & Mathematics University of Vermont Burlington, Vermont 05405-0156
[email protected]
INTRODUCTION
The asymptotic description of the coupled spatio-temporal dynamics of an ultrawideband, ultrashort pulsed electromagnetic beam field that is propagating in a dispersive, attenuative material occupying the halfspace is obtained from the angular spectrum of plane waves the paraxial representation1. The analysis leads to a comparison between approximation and the generalized Sherman expansion2,3 of that exact integral representation. Both approaches rely upon an expansion of some aspect of the propagation kernel that appears in the angular spectrum representation, each approach differing from the other in its particular method of expansion. The generalized Sherman expansion provides a spatial series representation of a pulsed, source-free electromagnetic beam field. This spatial series representation explicitly displays the temporal evolution of the pulsed beam wave packet throughout the halfspace through a single contour integral that is of the same form as that obtained in the Fourier-Laplace integral description of a pulsed plane wave field that is propagating in the positive z-direction in the dispersive medium. In particular, the spatio-temporal pulsed beam-field evolution is shown to depend upon the transverse spatial position in the propagated field through the spatial derivatives of the initial field boundary values at the plane As a consequence, it is found that the Sommerfeld and Brillouin precursor fields, which are a characteristic of the dispersive material, cause the ultrashort pulsed beam field to break up into several localized subpulses which travel at their own characteristic velocity through the dispersive material.
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ANGULAR SPECTRUM REPRESENTATION
Consider the evolution of a freely-propagating electromagnetic field in the half-space that is occupied by a homogeneous, isotropic, locally linear, temporally dispersive material. The term ‘freelypropagating’ is used here to indicate that there are no external sources for the field throughout this half-space, the source then residing somewhere in the half-space It is unnecessary to know what the source is provided that either the electric field vector or the magnetic field vector is a known function of time and the transverse position vector in the plane The freely-propagating electromagnetic field throughout the half-space is then given by the angular spectrum of plane waves representation1
for both the electric and magnetic field vectors. It is assumed here that the two-dimensional transverse spatial Fourier transform and temporal Fourier-Laplace transform of each field vector at the plane exist, as given by
where is the transverse wavevector. If the initial time dependence of the field vectors at the plane is such that for all for some finite value of then the time-frequency transform pair relation appearing in Eq. (2) is a Laplace transformation and the contour of integration is the straight-line path with the real constant a being greater than the abscissa of absolute convergence1,4 for the initial time behavior of the field and with varying from 0 to if not, then it is a Fourier transform. The spatio-temporal spectra of the electric and magnetic field vectors at the plane satisfy the transversality conditions1
so that Here is the complex wavevector for propagation into the positive half-space with the associated complex wavenumber given by
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where and are the plane wave propagation and attenuation factors, respectively, and where is defined as the principal branch of the expression
with
Here µ is the constant magnetic permeability and
is the complex permittivity of the dispersive medium with frequencydependent dielectric permittivity and electric conductivity Notice that both cgs and MKS units are employed here through the use of a conversion factor that appears in the double brackets in each equation affected. If this factor is included in the equation it is then in cgs units provided that while if this factor is omitted the equation is then in MKS units. If no such factor appears, then the equation is correct in both systems of units. The integrand appearing in the angular spectrum representation (1), namely
where with corresponds to a time-harmonic plane wave field that is propagating away from the plane at each angular frequency and transverse wavevector that is present in the initial spectra of the electric and magnetic field vectors at that plane with just one significant difference: the wavevector components and are always real-valued and independent of while is, in general, complex-valued. Hence, each spectral plane wave component appearing in the angular spectrum representation (1) is attenuated in the z-direction alone. The angular spectrum representation (1) explicitly displays the manner in which the temporal and spatial dynamics of the field are coupled. Even for the special case of an initial field whose temporal and spatial properties are separable as with spatiotemporal spectrum the temporal and spatial dynamics remain coupled through the complex factor as
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For a monochromatic field of angular frequency angular spectrum representation of the propagated field becomes
the
for all
Real Direction Cosine Form of the Angular Spectrum Representation The plane wave propagation factor appearing in the angular spectrum representation (1) may be cast into a more geometric form under the change of variable defined by the set of relations3
where
with magnitude
and phase so that the direction cosines p and q are real-valued. With these substitutions, Eq. (5) yields
and the angular spectrum of plane waves representation (1) becomes3
Here
and
for all
Paraxial Approximation of the Angular Spectrum Representation In the paraxial approximation one expands in a binomial series for small with3 and retains only the first two terms, with the result
With this substitution, the angular spectrum representation given in Eq. (12) becomes
for all
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GENERALIZED SHERMAN EXPANSION
Wave fields in attenuative media that contain only inhomogeneous plane wave components with for all in their angular spectrum representation are called source-free wave fields3. In lossless media, these inhomogeneous plane wave components become homogeneous plane wave components with it is for this case that source-free fields were first defined by Sherman2. In this approach, the exponential propagation kernel
is replaced by its Taylor series expansion
where the complex variables and been introduced for notational convenience, and where
have
With this substitution in Eq. (12), one obtains the spatial 3series representation of a pulsed, source-free electromagnetic beam field as
where
Here
and
for integer values of m > 0. The generalized Sherman expansion given in Eq. (18) explicitly displays the temporal evolution of an electromagnetic beam field through a single contour integral that is of the same general form as that obtained in the description of a pulsed plane wave field that is propagating in the positive z-direction in the dispersive medium1. In particular, the temporal pulse evolution is seen to be dependent upon the transverse position in the field through the even-ordered spatial derivatives of the transverse beam profile at the plane The
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remaining contour integral may then be directly evaluated as an asymptotic approximation using well-defined saddle point methods1.
NUMERICAL RESULTS
As a specific example of a dispersive, attenuative material, consider triply-distilled water. The Rocard-Powles model5 of the frequency dispersion of the dielectric permittivity of this material results in the frequency dispersion of the complex wavenumber that is presented in Fig. 1. At the paraxial approximation of the complex quantity as given in Eq. (13), is found to be quite accurate for as seen in Fig. 2. In contrast with the lossless case where there is a discontinuous change from the lossless, homogeneous wave region to the evanescent wave region notice the smooth transition in the exact behavior of from the “low-loss” region to the “high-loss” region where at A comparison of the exact behavior of the real and imaginary parts of the propagation kernel G(p,q, ) with both the paraxial approximation (13) for and the quadratic approximation
of the Taylor series expansion (16) is presented in Fig. 3. The paraxial approximation of the propagation kernel is seen to remain reasonably accurate for all while the quadratic approximation (21) of the Taylor series expansion is accurate only for Unfortunately, this series expansion converges very slowly so that the
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inclusion of each higher-order term only provides a very slight improvement in the accuracy of the resultant approximation over a slightly expanded (p,q) domain.
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Because of the increased material absorption as the frequency is increased further into the absorption band of triply-distilled water (the peak of which occurs at approximately 10THz], improved results are obtained for the paraxial approximation at The paraxial approximation (13) of is now accurate for all as seen in Fig. 4. Finally, the paraxial approximation of the propagation kernel G(p,q, ) is now reasonably accurate for all while the quadratic approximation (21) of the Taylor series expansion (16) is still accurate only over the very limited domain as is clearly evident in Fig. 5.
SUMMARY AND CONCLUSIONS
The results presented here have shown that the Sherman expansion of the angular spectrum representation of a pulsed 3,electromagnetic beam field, while being exact for a source-free wave field 4, is unfortunately of minimal practical use for computational purposes because of its slowly convergent character that only decreases as the propagation distance increases above zero. Nevertheless, it may have some practical application for near-field calculations. The paraxial approximation, on the other hand, has been shown to improve in its accuracy as the material absorption increases. As a consequence, it provides a convenient method with which to evaluate the general features of ultrashort, ultrawideband pulsed beam field propagation in any causally dispersive material. For example, the spatial part of the paraxial approximation (14) of the angular spectrum representation may be directly evaluated for a Hermite-gaussian beam, the remaining contour integral over the angular frequency may then be evaluated using well-defined asymptotic techniques. Because of the
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precursor fields that are a characteristic of the particular dispersive medium1, the propagated ultrawideband, ultrashort pulsed beam field will break up into several spatio-temporally localized packets. For a Rocard-Powles (or Debye) model dielectric, the asymptotic description of the propagated beam field may be expressed as
as where Here describes the low-frequency Brillouin precursor portion of the propagated beam field, and describes the carrier frequency portion of the propagated wave field. Because of the lower frequency content of the Brillouin precursor, the space-time packet will have a larger diffractive spread than does the carrier signal packet For a single resonance Lorentz model dielectric an additional term due to the high-frequency Sommerfeld precursor appears in the asymptotic representation
as Because of its very high frequency content, this additional space-time packet would have near minimal diffractive spreading.
ACKNOWLEDGMENT
This research has been supported by the United States Air Force Office of Scientific Research Grant #F49620-94-1-0430.
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REFERENCES 1. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics, Springer-Verlag, Berlin (1994).
2. G C Sherman, Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves, J Opt Soc Am, 59:697 (1969).
3. K E Oughstun, The angular spectrum representation and the Sherman expansion of pulsed electromagnetic beam fields in dispersive, attenuattve media, J Eur Opt Soc A, 7:1059(1998). 4. J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York (1941). 5. J. E. K. Laurens and K. E. Oughstun, Electromagnetic impulse response of triplydistilled water, in: Ultra-Wideband, Short-Pulse Electromagnetics 4, E. Heyman, B. Mandelbaum and J. Shiloh, eds., Plenum, New York (1999).
DISPERSION REDUCTION IN A COAXIAL TRANSMISSION LINE BEND BY A LAYERED APPROXIMATION OF A GRADED DIELECTRIC LENS
W. Scott Bigelow,1 Everett G. Farr,l William D. Prather,2 and Carl E. Baum2 1
Farr Research, Inc. 614 Paseo Del Mar NE Albuquerque, NM 87123
2
Air Force Research Laboratory / DEHP Directed Energy Directorate Kirtland AFB, NM 87117
INTRODUCTION A bend in an electrically large geometry poses a problem for high-voltage UWB pulse transmission. Dispersion is introduced by the dependence of propagation path length on the local radius of curvature within the bend. This dispersion increases the risetime and broadens the pulse width of the transmitted signal, thus degrading system bandwidth. This source of dispersion can be reduced by use of a gradient index lens to compensate for the electrical path length differences through the bend. This approach has been suggested in a number of papers which develop solutions for a TEM wave propagating in the in a cylindrically inhomogeneous dielectric (CID), with permittivity, proportional to in a cylindrical coordinate system [1, 2, 3, 4, 5, 6, and 7]. We take the particular form of the CID inhomogeneity as
where n is the refractive index, is the relative permittivity, and is the radius at which the index becomes unity. In such a perfect CID medium, the electrical path length and pulse transit time, through a bend, are independent of the radius. In [8 and 9], we described our approximate synthesis of a CID medium to compensate a strip transmission line bend. That experimental effort employed five coarsely graded layers of uniform dielectric materials to approximate a CID lens. The compensated bend achieved a 70 ps reduction in pulse risetime, a 30% improvement, when compared to an identical air-filled line. The performance of the strip line was limited by propagation of fringe fields ahead of and behind the main pulse. Here, we summarize our latest implementation of a layered approximation of a graded dielectric lens, this time, employing square coaxial hardware [10]. Use of a coaxial geometry eliminates the fringe fields that plagued the strip line results. Although our design approach for a layered coaxial bend employed a heuristic technique, we subsequently perUltra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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formed a global optimization, which demonstrated that our design was very nearly optimal. To guide the design effort, we obtained the line impedance from a finite element model of the cross section [6]. In contrast to the modest dispersion reduction achieved with the strip line bend, compensation by a layered approximation of a CID medium dramatically improved impulse transmission through the coaxial transmission line bend.
LAYERED DIELECTRIC LENS DESIGN We began the design of a transmission line bend, approximately compensated by graded layers of dielectric material, by assuming a cross section and maximum radius of curvature for a 90° bend. To simplify manufacture of the layered dielectric material, we chose a square cross section, which we assumed to be 5.08 cm on each side. For consistency with an earlier design approach, we chose 14.68 cm as the outer radius of curvature. With these choices, the centerline path length of the bend was about 19 cm. Next, we specified a performance goal to use in assessing the need for bend compensation. Our criterion was that the signal transit time through the bend should not vary by more than about 100 ps. The transit time at any radius is
where is the bend angle in radians and c is the speed of light. For an uncompensated bend (constant index of refraction), the difference in transit time between the inner and outer radii of curvature is completely determined by the width of the bend and by the index of refraction. For an air-filled, 5.08 cm wide, 90° bend, that time difference is 266 ps. It increases linearly with the index of refraction. Since the 100 ps limit is exceeded for the proposed bend, even in air, compensation is required. To begin the design of a compensating dielectric laminate, we needed to identify the materials to be used. We selected a series of low-loss, cross-linked polybutyldiene resins, commercially produced at half-integer increments of the dielectric constant, from 3.0 to 10.0. Although the materials can be readily machined with sharp carbide tools, they are somewhat brittle, especially at the higher dielectric constants. The manufacturer recommended extreme care if layers thinner than about 0.64 cm were to be machined. Therefore, we chose to avoid layers thinner than this limit. If all layers were of this minimal thickness, eight would be required to fill the radial extent of the bend. Since manufacturing cost increases with layer count, we sought to use the minimum number of layers capable of meeting the assumed 100 ps transit time variation limit. In a layered bend, there are two sources of transit time variation. Across the width of each layer, the transit time changes linearly because the dielectric constant is uniform, while the radius of curvature changes. Across adjacent layers, the transit time changes abruptly because the dielectric constant is discontinuous at the layer interface. In designing the bend, we applied the same 100 ps limit to both sources of transit time variation. Finally, we sought a design in which the laminate of uniform dielectric layers would closely approximate the inverse square relationship between the dielectric constant and radius of curvature that characterizes a CID medium. To this end, we attempted to place the midpoint of each layer as close as possible to the correct radius for the dielectric constant of the material comprising that layer. A heuristic algorithm used to design the dielectric laminate, subject to the constraints outlined above, has been described in detail in [10]. Application of that algorithm led to a six-layer design employing materials with dielectric constants of 3.0, 3.5, 4.0, 4.5, 5.0, and 6.0. The layer-thickness-weighted average dielectric constant for the laminate is 4.1. As the plots in Figure 1 demonstrate, the design represents a good approximation to a CID profile
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with (9.636”). In the plot of transit time variation, note that is defined as the radius-dependent difference between the predicted transit time and the constant transit time through an ideal, 90° CID bend. Thus, in the plot, (2) has been evaluated at and to obtain
where for the 90° bend. Note that, between inner and outer radii of curvature, an uncompensated bend, filled with a material having a dielectric constant of 4.1, would exhibit more than five times the postulated 100 ps limit on transit time variation.
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Although the heuristic design algorithm works well, it provides no indication as to whether its result is anywhere near optimal. It is also tedious to apply. In [10], the rootmean-square of (3), over all dielectric layers, was chosen as an objective function for use with a nonlinear global optimization routine. Specifically, for a bend consisting of
The optimal set of dielectric constants and layer-bounding radii is that which minimizes this objective function, subject to all applicable constraints. The minimizing set was found to be nearly identical to the set generated by the heuristic algorithm. The associated minimum value of is 27 ps.
INTEGRATION OF LENS AND COAXIAL TRANSMISSION LINE BEND We integrated the dielectric lens laminate with a square coaxial TEM transmission line bend test fixture. To minimize reflections and enhance high-voltage performance, we required the bend and its interfaces to have identical conductor sizes in addition to having the same nominal impedance. Since the layer-thickness-weighted average dielectric constant within the laminate is 4.1, material with a dielectric constant of 4.0 was selected to fill straight (tapered) sections interfacing to the bend. There is a parametric expression available for the impedance of a uniform straight square coaxial line with cylindrical center conductor [11]. For a dielectric constant of 4.0, it predicts when the length of the side of the outer conductor is five times the diameter of the inner conductor. Since an outer conductor matching the laminate is 5.08 cm on each side, 1.016 cm was selected for the diameter of the inner conductor. To predict the impedance of the layered transmission line bend, we modeled its cross section using finite elements and calculated the impedance as described in [6]. There we proved that the characteristic impedance could be found (approximately) from
Here, is the impedance of free space, T is the vector transpose operator, and S and U are, respectively, the finite element matrix and solution vector for the two-dimensional potential equation for a rotationally symmetric geometry,
The electrical potential, u, is a function only of
and
and
where and are unit vectors. For a layered bend, (5) is valid to the extent that the layers represent a good approximation of (1). The model result was The cross section of the layered bend model, showing the position of the center conductor, is shown in Figure 2. A top view of the bend layers is shown in Figure 3.
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In addition to the bend test section, we also designed and built a comparable 19 cm long straight coaxial section, as well as a pair of square pyramidal taper sections. The latter provided a constant impedance transition between the 5.08 cm square coaxial test sections and a standard N-type female connector. Details of the taper design are to be found in [10]. A photograph of the complete transmission line test fixture with 90° bend is shown in Figure 4. The diagram in the lower portion of the figure is a z-plane cross section at the fixture centerline, with layered dielectric filling the bend.
EXPERIMENTAL PROGRAM Equipment Setup and Signal Processing The equipment setup for the coaxial transmission line measurements is shown in Figure 5. The device under test (DUT) was either a 90° bend or straight section loaded with dielectric material. After propagation through the test fixture, the raw impulse response was detected by the SD-24 sampling head, and stored on the Tektronics 11801B oscilloscope. To obtain a system response, the DUT was removed and the two tapered sections were directly connected. Signal processing was used to isolate the response of the DUT by deconvolving the system response from the raw transmission data. Impedance measurements employed the 0.25 voltage step of SD-24 operating in TDR mode, while a terminator replaced the PSPL 5210.
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Measurements The TDR measurements of the test fixture are shown in Figure 6. With the straight section of transmission line loaded with material having a dielectric constant of 4.0, the impedance is nominally throughout the fixture. It is extremely flat within the straight section, which is delimited by the small bumps at 10.3 and 12.8 ns. The larger features on either side of the straight section arise primarily from SMA-to-N adapters at the ends of the tapered sections. A minor machining error is responsible for the broad hump at the input taper; the output taper does not exhibit this feature. The TDR with the layered bend installed in the test fixture has similar features, and also exhibits nominal impedance.
Impulse transmission through the straight section of transmission line provides a reference, against which transmission through the 90° bend can be compared. In Figure 7, the top pair of plots shows the raw impulse response of the straight section in both time and frequency domains. The FWHM of the time domain signal is 55 ps. The middle pair of plots shows the impulse response for the 90° bend, filled with the same uniform dielectric.
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Instead of a narrow impulse, the uncompensated bend transmits a broad bipolar signal spanning approximately 500 ps. The bottom pair of plots shows the impulse response for the same 90° bend when compensated by the layered dielectric lens. The FWHM of the time domain signal is 70 ps, only 15 ps wider than observed for the comparable straight section of transmission line filled with a uniform dielectric.
The raw impulse response data for the straight transmission line section and for the 90° layered bend were processed to correct for the system response. The FWHM of the system response, shown in Figure 8, is 50 ps. A 5 th order modified Butterworth filter with a 15 GHz cutoff was used in deconvolving the system response from the raw data. The
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corrected impulse responses are shown in Figure 9. The FWHM of the time domain response of the straight section is 40 ps. For the layered bend, the peak is only slightly wider, 44 ps. Thus, the layered dielectric lens has been extremely successful in compensating the transmission line bend.
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CONCLUSIONS In an electrically large square coaxial transmission line, we demonstrated preservation of a narrow impulse during propagation through a relatively tight 90° bend compensated by a layered approximation of a CID lens. By use of a non-linear global optimization algorithm, we established that the layered dielectric design was essentially optimal. In terms of the observed, raw impulse response, the compensated bend increased the pulse width by only 15 ps, to 70 ps, from the 55 ps observed for an equivalent length of straight transmission line. In contrast, without compensation, the transmission line bend distorted the impulse into a bipolar signal spread over a 500 ps time interval. With a compensating dielectric lens like the one described here, even electrically large transmission line bends can be implemented with minimal dispersion. Use of such compensated bends may permit more compact, efficient designs of high-voltage, UWB systems. ACKNOWLEDGMENTS This work was funded in part by the Air Force Office of Scientific Research, Alexandria, VA, and in part by the Air Force Research Laboratory, Directed Energy Directorate, under contract F29601–98–C–0047.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
10. 11.
C. E. Baum, “Two-Dimensional Inhomogeneous Dielectric Lenses for E-Plane Bends of TEM Waves Guided Between Perfectly Conducting Sheets,” Sensor and Simulation Note 388, 14 October 1995. C. E. Baum, “Dielectric Body-of-Revolution Lenses with Azimuthal Propagation,” Sensor and Simulation Note 393, 9 March 1996. C. E. Baum, “Dielectric Jackets as Lenses and Application to Generalized Coaxes and Bends in Coaxial Cables,” Sensor and Simulation Note 394, 23 March 1996. C. E. Baum, “Azimuthal TEM Waveguides in Dielectric Media,” Sensor and Simulation Note 397, 31 March 1996. C. E. Baum, “Use of Generalized Inhomogeneous TEM Plane Waves in Differential Geometric Lens Synthesis,” Sensor and Simulation Note 405, 5 December 1996. W. S. Bigelow and E. G. Farr, “Impedance of an Azimuthal TEM Waveguide Bend in a Graded Dielectric Medium,” Sensor and Simulation Note 428, 21 November 1998. C. E. Baum, “Admittance of Bent TEM Waveguides in a CID Medium,” Sensor and Simulation Note 436, 2 May 1999. W. S. Bigelow and E. G. Farr, “Minimizing Dispersion in a TEM Waveguide Bend by a Layered Approximation of a Graded Dielectric Material,” Sensor and Simulation Note 416, 5 January 1998. W. S. Bigelow, and E. G. Farr, “Minimizing Dispersion in a TEM Waveguide Bend by a Layered Approximation of a Graded Dielectric Lens,” p. 213-219 in Ultra-Wideband, Short-Pulse Electromagnetics 4, E. Heyman, B. Mandelbaum, and J. Shiloh (eds.), Kluwer Academic / Plenum Publishers, New York, 1999. W. S. Bigelow, E. G. Farr, and W. D. Prather, “Compensation of an Electrically Large Coaxial Transmission Line Bend by a Layered Dielectric Lens,” Sensor and Simulation Note 445, 17 July 2000. Reference Data for Engineers: Radio, Electronics, Computer, and Communications, 8th Ed., p. 20-22, SAMS, Prentice Hall Computer Publishing, Carmel, Indiana 46032, 1993.
OPTIMAL INPUT SIGNALS FOR DRIVING NONLINEAR ELECTRONIC SYSTEMS INTO CHAOS
Stuart M. Booker1, Paul D. Smith 1 , Paul V. Brennan2 and Richard J. Bullock2 1
2
Department of Mathematics University of Dundee Dundee DD1 4HN, UK Department of Electronic and Electrical Engineering University College London Torrington Place, London, WC1E 7JE, UK
INTRODUCTION Nonlinear feedback loops have found a wide variety of applications in modern electronic systems. Many such circuits are vulnerable to chaos if driven by an appropriate input signal. Furthermore, since nonlinear circuits cannot be characterised simply by their spectral response, the onset of chaos is extremely waveform dependent. The question arises: which input signal is the most effective at driving the circuit into chaos? It has been shown recently (Levey and Smith, 1997; Booker, 2000) that an answer can be found to this fundamental question for one particular class of electronic systems. This class comprises those nonlinear electronic systems whose dynamical equation resembles a perturbed Hamiltonian system. In fact, this is a very broad class of electronic systems and includes many practical circuits. For this class of electronic systems a pair of optimal input signals may be determined. These induce the onset of homoclinic chaos in the dynamics of the circuit with the smallest possible amplitude, or power, for which homoclinic chaos can be found. These optimal input signals allow in-band disruption of the nonlinear electronic systems concerned. In this paper we review the approach of Booker (2000) with particular reference to a realistic test circuit, a second order phase-locked loop (PLL).
OPTIMAL FORCING FUNCTIONS Many of the nonlinear electronic circuits used in modern communication systems Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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can be described by a dynamical equation of a particular form. This can be written
where the forcing term is a T–periodic function. It is important to know how small this forcing term can be and still cause disruption of the circuit. Such information would allow the best chance of disrupting the circuit in-band, and would also suggest a design strategy for minimising any unwanted disruption. The other important property of the circuit is that: if and are set identically equal to zero then the resulting dynamical system is Hamiltonian and possesses a homoclinic orbit connecting a saddlepoint to itself. The dynamics of the circuit, as defined by equation. (1), can then be viewed as a perturbed Hamiltonian system. This allows us to make a quantitative analysis of the circuit. Such a homoclinic connexion is depicted in figure 1(a) and results from a coincidence of the unstable and stable manifolds, and of the saddle-point; a useful introduction to these ideas is given by Drazin (1992). If damping and forcing are applied to this system, as in equation (1), the manifolds separate and one of two situations arises. Either, the manifolds intersect nowhere (as depicted by the dash curve of figure 1(a)), or, the manifolds intersect an infinite number of times (as depicted in figure 1(b)). The latter situation is termed a homoclinic tangle and gives rise to complex dynamics, including subharmonic, quasiperiodic and chaotic dynamics. Such dynamics will completely disrupt the behaviour of any circuit designed to exploit a simple periodic signal. If we can determine an input signal which is optimal at inducing a homoclinic tangle, then we will know the best signal for disrupting the operation of the circuit. This fact allows us to determine two important input signals: the signal of smallest amplitude which will disrupt the circuit, and that of smallest power. In order to determine these we assume that has or norm 1, respectively, and that is therefore a direct measure of the amplitude, or power, of the input signal. Our aim is thus: to determine the unit norm waveform, which will induce a homoclinic tangle in the dynamics of equation (1) with the smallest possible value of This can be achieved using an approach due to Mel’nikov (1963) who proved that the distance between the unstable and stable manifolds is proportional to a T–periodic function defined by
The functions and define the homoclinic orbit of the Hamiltonian system which results by setting and to be identically zero; these functions are usually well known for electronic circuits of this form. Obviously, if this function is zero for any value of then the unstable and stable manifolds of the saddle-point intersect and a homoclinic tangle occurs in the dynamics of the circuit. This in turn gives rise to circuit disruption. Since we know that is constrained to have unit norm, we may use some elementary functional analysis to determine the smallest possible value of for which a zero occurs in equation (2). We may also determine the waveform which is optimal at inducing such disruptive dynamics. In general, the optimal forcing waveform, of smallest power is defined by special functions relating to the function however, the optimal forcing waveform of smallest amplitude is usually a square-wave.
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THE PHASE-LOCKED LOOP (PLL) In order to demonstrate the validity and usefulness of the approach described above, we consider a simple test problem: how to best disrupt a second order phaselocked loop (PLL) FM demodulator. The PLL is a simple nonlinear feedback loop and is depicted in figure 2. The input signal is subject to a baseband modulation which it is the purpose of the loop to recover; for a more detailed account of the PLL the reader is referred to the book by Brennan (1996). Whether this can be done or not depends upon the phase error the phase difference between the input signal and that fed back by the loop. If the phase error is a simple periodic oscillation about zero then the modulation can be recovered. More complex dynamics, such as a subharmonic, quasiperiodic or chaotic responses, will effectively disrupt the operation of the PLL. Chaos has been demonstrated in the PLL by Endo and Chua (1988) and other authors (Endo et al, 1988; Chu et al, 1991) but in each instance a sinusoidal modulation was assumed to allow an analysis of the circuit. However, the sinusoid is only one of many possible modulations, many of which may be better at disrupting the PLL. Using the approach described above, the PLL may be analysed subject to the optimal modulation for inducing homoclinic chaos. We consider a typical second order PLL design, for which the phase error satisfies a non-dimensionalised dynamical equation of the form
The baseband modulation which excites the system has form where is a unit amplitude waveform and is the depth of modulation. We assume a phase detector characteristic of the form
It is evident that the PLL satisfies a dynamical equation with the form of (1). Furthermore, if we assume that and are both identically zero, then the dynamics of the undamped, unforced PLL form a Hamiltonian system. In this case we find that a saddle-point exists at the point which is connected to itself by two homoclinic orbits. The underlying dynamics of the undamped, unforced PLL are depicted in Figure 3; and denote the two homoclinic connexions. Note that is and that, hence, the points and are identical. Disruptive dynamics can result from the break-up of either homoclinic connexion. Clearly, the approach of Booker (2000) may be applied to this circuit, in order to determine a waveform of least modulation depth with which to disrupt the PLL. If this approach is taken then it is found that this optimal waveform takes the form of a squarewave. Other effective disrupting waveforms can also be found by this method, such as the sawtooth, which proves to be a particularly effective disruptor at low frequencies of modulation (Booker et al, 2000). In order to demonstrate the effectiveness of our approach for designing disruptive signals, we next contrast the effectiveness of three different modulations. These are: the sinusoid (the signal commonly assumed in the study of nonlinear circuits), the sawtooth (expected to be particularly effective at low frequencies of modulation), and the square-wave (expected to be the optimal disruptor of the PLL).
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In figure 4 we contrast the effectiveness of these three different modulation waveforms at inducing a homoclinic tangle in the dynamics of the PLL. These results are due to an analysis of the Mel’nikov function for each waveform, as given in equation (2). The figure depicts the threshold value of the modulation depth, at which a homoclinic tangle can first be observed in the PLL’s dynamics. The parameters assumed in this analysis correspond to those of the test circuit design described below. The threshold for the sinusoid (solid curve) is very similar to the linear limit for the PLL. This is the limit of operation usually assumed for the circuit; see Brennan’s (1996) account for detail. It may be clearly seen, however, that the sawtooth waveform is much more effective at inducing a homoclinic tangle at low frequencies of modulation than is the sinusoid. It is also clear that the square-wave is a significantly better disruptor than either the sinusoid or sawtooth. This is just what we expected.
Numerical Results In order to demonstrate that the onset of a homoclinic tangle in the dynamics of the PLL really does disrupt the operation of the circuit, numerical simulations were performed. The dynamical equation of the PLL was solved numerically for a wide range of initial conditions across a range modulation frequencies. The parameters assumed here were again taken from the experimental test circuit design discussed below. In figure 5 we depict the threshold for disruption observed in the PLL subject to each modulation. The threshold marks the depth of modulation, which is required to induce complex dynamics in the circuit; below the threshold only period 1 oscillatory responses are observed, but above each curve the relevant modulation induces chaos and other complex dynamics.
Experimental Results A test circuit for the second order PLL was designed, with a damping parameter and a natural frequency of 5 kHz. The response of the PLL was then measured when subject to a range of modulation frequencies from 1 to 10 kHz, with a modulation of sinusoidal, sawtooth and square-wave form. The experimental set-up for measuring the PLL is depicted in figure 6. The measured response of the PLL was found to be in remarkable agreement with the numerical simulations described above; even the finer details of a rich array of dynamics were found to be in agreement. A comparison of the numerical and experimental results obtained for sinusoidal and sawtooth modulation have been reported (Booker et al, 2000) and demonstrate the sort of agreement which was found, in general, between the numerical and experimental data. This agreement clearly demonstrates that our original model for the PLL (3) is a very accurate representation of the PLL’s dynamics. The resulting thresholds for disruption of the PLL, subject to sinusoidal, sawtooth and square-wave modulation, are depicted in figure 7. Below each threshold curve only a period 1, oscillatory response can be observed, whilst above each threshold a wide variety of complex dynamics are found. The results depicted in figures 4, 5 and 7 clearly corroborate the approach to circuit disruption which has been described in this paper.
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CONCLUSION In this paper we have described a novel approach to the problem of disrupting nonlinear electronic circuits; this approach is applicable to a wide range of nonlinear electronic systems. It allows us to calculate input signals of smallest amplitude and power which will induce homoclinic chaos and, hence, disruption in the circuit. Furthermore, as is clearly demonstrated by the results described above, this approach allows us to develop in-band signals with which to disrupt such circuits. It may also be noted that the strategy which we have exploited in this paper, for a certain class of analogue circuit, can be extended to the problem of disrupting digital nonlinear systems. Using the method described above it is possible to: develop optimal waveforms for disrupting certain nonlinear electronic systems; and, develop a design strategy to minimise the risk of unwanted disruption of such circuits.
Acknowledgements The authors gratefully acknowledge the support of the Defence Evaluation and Research Agency, DERA Fort Halstead, UK.
REFERENCES Booker, S.M., 2000, A family of optimal excitations for inducing complex dynamics in planar dynamical systems, Nonlinearity, 13:145. Booker, S.M., Smith, P.D., Brennan, P. and Bullock, R., 2000, The effect of the forcing function on disruption of a phase-locked loop, to appear in Int. J. Bifurcation and Chaos, September, 2000. Brennan, P.V., 1996, Phase-locked Loops: Principles and Practice, McMillan, London. Chu, Y.H., Chou, J.H. and Chang, S., 1991, Chaos from third order phase-locked loops with a slowly varying parameter, IEEE Trans. Circuits and Systems, 37:1104. Drazin, P.G., 1992, Nonlinear Systems, Cambridge University Press, Cambridge. Endo, T. and Chua, L.O., 1988, Chaos from phase-locked loops, IEEE Trans. Circuits and Systems, 35:987. Endo, T., Chua, L.O. and Narita, T., 1989, Chaos from phase-locked loops – Part II: high dissipation case, IEEE Trans. Circuits and Systems, 36:255. Levey, D.B. and Smith P.D., 1997, The core of chaos in the dynamics of phase-locked loops, in: Proc. 5th Int. Workshop on Nonlinear Dynamics in Electronic Systems (Moscow, June 26-27, 1997), Moscow Technical University of Communication and Informatics, Moscow. Mel’nikov, V.K., 1963, On the stability of the center for time periodic perturbations, Trans. Moscow Maths. Soc., 12:1.
IN-BAND CHAOS IN COMMERCIAL ELECTRONIC SYSTEMS
Stuart M. Booker1, Paul D. Smith1, Paul V. Brennan 2 and Richard J. Bullock2 1
2
Department of Mathematics University of Dundee Dundee DD1 4HN, UK Department of Electronic and Electrical Engineering University College London Torrington Place, London, WC1E 7JE, UK
INTRODUCTION In this paper we describe a method for inducing chaos in a standard, commercially available electronic sub-system using an in-band signal. In particular, we show how, with relatively little knowledge of the commerical circuit design, an in-band signal can be determined which will disrupt the operation of the circuit and drive it into chaos. The problem which we consider is the disruption of an INMARSAT standard-A terminal. This device, which employs a phase-locked loop (PLL) FM discriminator, is designed to accommodate a 3 kHz modulation bandwidth with 12 kHz of peak FM deviation. In order to model this commercial system we consider a PLL test circuit with an identical specification. This PLL circuit provides a test problem with which analysis and numerical modelling can be compared, the INMARSAT terminal design being, in essence, a ‘black box’. Having developed a successful strategy for disrupting the test PLL circuit we then employ our knowledge on the INMARSAT terminal. Experimental measurements are reported on an actual INMARSAT standard-A terminal which validate our approach.
A SCHEME FOR CIRCUIT DISRUPTION The problem which we shall investigate is: how best to disrupt the operation of a standard, commercially available electronic system, using only a limited knowledge of the system itself. In particular, we consider how to disrupt the operation of an INMARSAT mobile communications terminal, for use with standard-A analogue voice
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transmissions. The knowledge which we shall exploit is limited: that the terminal comprises an RF/IF module and phase-locked loop (PLL) FM discriminator, and that the device is designed to accommodate 3 kHz modulation bandwidth with 12 kHz of peak FM deviation. The INMARSAT terminal contains two sub-systems for which chaos has been reported in the literature: an automatic gain control (AGC) loop in the IF board, and the phase-locked loop (PLL). Although chaos has been reported for one particular AGC loop design (Chang et al, 1993) the design investigated was highly unusual; more common AGC designs are known to be stable, and robust (Green, 1983). Using a suitably amplitude modulated waveform to induce chaos in the INMARSAT terminal is unlikely to prove successful as a strategy for disruption. By contrast, the PLL is known to be much more prone to chaos. Standard PLL FM demodulator designs have been shown to exhibit chaotic dynamics, if driven by suitable frequency-modulated signals (Endo and Chua, 1988; Endo et al, 1989; Chu et al, 1991). For this reason we choose the following strategy to disrupt the operation of the INMARSAT terminal: (i) design a PLL test circuit with identical specifications to the INMARSAT terminal, (ii) determine a frequency modulation suitable for disrupting the PLL test circuit given our knowledge of its design, (iii) validate the effectiveness of the disrupting signal on the PLL test circuit, (iv) validate the effectiveness of the disrupting signal on a test circuit comprising RF/IF module and PLL, (v) validate the effectiveness of the disrupting signal on the INMARSAT terminal itself.
PLL TEST CIRCUIT The PLL FM discriminator test circuit can be modelled as a nonlinear feedback loop whose dynamical equation resembles a perturbed Hamiltonian system, when modulated by a periodic input signal. For such an electronic system it is possible to determine an input signal of smallest amplitude (or modulation bandwidth, in this context) which will allow the onset of homoclinic chaos in the circuit (Booker, 2000). A review of this approach has been given in an accompanying paper in this volume (Booker et al, 2000a) with particular reference to an example problem: the PLL FM discriminator circuit whose specification matches that of the INMARSAT terminal. For details of the PLL test circuit, and an outline of how the optimal modulation waveform is determined, the reader is referred to the accompanying paper. Analysis of the PLL test circuit design reveals that a square-wave modulation is the optimal signal for inducing a homoclinic tangle in the dynamics of the PLL; a useful introduction to this kind of dynamics is given by Drazin (1992). This feature of the circuit’s dynamics is responsible for inducing a chaotic response in the circuit, together with other complex dynamics such as subharmonic and quasiperiodic responses. All of these dynamics will effectively disrupt the operation of the PLL test circuit and, hence, the onset of a homoclinic tangle in the PLL test circuit corresponds to the threshold at which disruption can be expected. In figure 1 we depict the threshold at which a homoclinic tangle occurs in the test PLL circuit when modulated by: the sinusoid (solid curve), the sawtooth (dot-dash curve), and the square-wave (dash curve). The squarewave is chosen as the optimal disruptor of the PLL circuit, whilst the sawtooth and sinusoid are given for comparison. The sinusoid is the signal conventionally assumed for the investigation of a nonlinear circuit. Below each curve a simple period 1 oscillatory response can be expected, given the appropriate modulation, whilst above the curve complex and disruptive dynamics may be expected. Numerical and experimental tests
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of the PLL test circuit design revealed these predictions to be extremely accurate (Booker et al, 2000a).
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EFFECT OF RF/IF MODULE In an application such as the INMARSAT terminal the PLL FM demodulator is preceded by an IF board. It is necessary, therefore, to ensure that this additional sub-system has no significant effect on the waveforms used to disrupt the PLL. To this end a simple RF/IF test module was designed in order to mimic, as closely as possible, a physical implementation of the PLL circuit. The design of this module is depicted below, in figure 3; it consists of a front-end RF amplifier followed by a frequency convertor, IF filter and IF amplifier with automatic gain control (AGC).
The test/IF input allows a suitable IF signal to be directly introduced to the module before the IF filtering stage; this enables us to directly assess the effect of IF filtering on our disruptive waveforms. The test IF filter was designed for a bandwidth of around 27kHz, to comply with Carson’s rule, and with a centre frequency of 1 MHz. The AGC loop bandwidth must be significantly less than the IF frequency and a value of 5 kHz, equal to the PLL FM demodulator loop natural frequency, was chosen. The series of experimental measurements which had been performed on the PLL test circuit alone were repeated with the RF/IF module preceding the PLL FM demodulator. The limits of disruption found with the RF/IF module included were very close to those for the PLL test circuit alone. The dynamics observed above the limit of disruption were also similar to those observed for the PLL alone. The only significant difference due to the introduction of the RF/IF module was an increase in the amount of chaos observed above the limit of disruption (as opposed to subharmonic, quasiperiodic responses etc) when a sawtooth modulation was applied to the circuit. This may be accounted for by the fact that the sawtooth resembles the zero modulation frequency limit of an important waveform: the optimal disrupting modulation of least rms modulation depth (Booker et al, 2000b). At non-zero frequencies of modulation, however, this waveform resembles a sawtooth waveform with smoothed peaks. The effect of IF filtering is thus likely to render the sawtooth even more effective at inducing chaos then it would otherwise be.
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THE INMARSAT TERMINAL Having determined an optimally disruptive modulation waveform for the test PLL circuit, and having proven that the effectiveness of this disruptor is undiminished by the presence of an RF/IF module, we are now in a position to consider the actual circuit: the INMARSAT standard-A terminal. A series of experimental measurements were performed on the INMARSAT terminal which allowed its limit of disruption to be measured. The test signal was introduced before the IF board of the INMARSAT terminal (rather than immediately before the PLL FM discriminator) ensuring receiver behaviour identical to that encountered in normal operation. Three different modulation waveforms were used: the sinusoidal, sawtooth and square-wave signals which had been used in assessing the PLL test circuit. The range of dynamics observed in the disrupted response of the INMARSAT terminal was identical to that observed with the PLL test circuit, including chaos, subharmonic responses and cycle-slipping (or rotaional-type periodic responses). A summary of these experiments is given in figure 2 which depicts the limit of disruption for the INMARSAT terminal modulated by: a sinusoid (solid curve), a sawtooth (dot-dash curve), and a square-wave (dash curve); triangles are used to denote the data points corresponding to actual measured values for the threshold of disruption. The results of figure 2 can be compared directly with the threshold for which a homoclinic tangle becomes apparent in the dynamics of the test PLL circuit, figure 1. Bearing in mind that neither figure is scaled to fit the agreement between the two is remarkable. The sinusoid is the waveform commonly used to investigate the dynamics of nonlinear circuits; also, the usual calculation for the linear limit of a PLL (Brennan, 1996) gives a limit almost identical to the threshold for disruption of a sinusoidally modulated PLL. The limits of operation for a PLL, or PLL-based circuit, are thus effectively defined by our results for sinusoidal modulation. As may be seen, however, the sinusoid proves to be remarkably poor at inducing disruption in the INMARSAT terminal at low frequencies of modulation. The sawtooth is a much better disruptor in this regime. However; the square-wave proves to be significantly better than either of the two test waveforms across all frequencies of modulation. Indeed, at low frequencies of modulation the square-wave is several times better then the sinusoid at inducing disruption in the INMARSAT terminal. Obviously, the square-wave provides an excellent waveform for inducing in-band disruption of a circuit like the INMARSAT terminal, at relatively low powers. It achieves this highly desirable effect by exploiting to the full the potential of the PLL circuit to be driven into homoclinic chaos. In figures 4, 5 and 6 we compare the limits of disruption of the INMARSAT terminal with those measured experimentally for the test PLL circuit, and with the predictions of numerical simulations of the test circuit. Figure 4 depicts the limit of disruption measured for the relevant circuit modulated by a sinusoidal waveform, whilst figures 5 and 6 depict results for sawtooth and square-wave modulation, respectively. The slight differences between the experimentally measured results for the test circuit and the INMARSAT terminal are easily accounted for by two facts. Firstly, the IF filtering stage will cause some slight distortion of the input signal. Secondly, the INMARSAT PLL FM discriminator may have design parameters which differ slightly from those assumed for our test circuit; in particular, if the damping parameter of the INMARSAT PLL was slightly different from that of the test PLL circuit, then we would expect their respective thresholds for disruption to differ slightly in magnitude, although possessing the same shape.
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CONCLUSION In this paper we have presented an approach to the problem of disrupting effectively a commercial nonlinear electronic system. Our approach is based on a minimal understanding of the circuit concerned, the basic design and operational specifications, rather than a detailed knowledge of the design implementation itself. By designing a test circuit with identical specifications to the commercial circuit and concentrating on one method of in-band disruption, we have been able to determine an input signal which proves to be extremely effective at disrupting the system in practice. Analytical, numerical and experimental study of the test circuit confirms the effectiveness of our approach at inducing disruption in a realistic PLL design. Experimental investigation of the commercial circuit itself demonstrates: that our disruptive signal is effective against commercial equipment as well as a test circuit, that it is the PLL of the INMARSAT terminal which gives rise to disruptive dynamics, and that our approach to circuit disruption is of practical value. From the results presented in this paper it is clear that in-band disruption of commercial systems can be achieved using relatively low power levels. In consequence, the ideas outlined in this paper offer a methodology for two important tasks: the design of signals with which to disrupt commercial systems in-band, and the design of signals with which to ensure that any commercial circuit is robust against such disruption.
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Acknowledgements The authors gratefully acknowledge the support of the Defence Evaluation and Research Agency, DERA Fort Halstead, UK.
REFERENCES Booker, S.M., 2000, A family of optimal excitations for inducing complex dynamics in planar dynamical systems, Nonlinearity, 13:145. Booker, S.M., Smith, P.D., Brennan, P.V. and Bullock, R.J., 2000, Optimal input signals for driving nonlinear electronic systems into chaos, Ultra-Wide band Short-Pulse Electromagnetics 5. Smith, P.D. and Cloude, S.R., ed., Plenum Press, New York. Booker, S.M.; Smith, P.D., Brennan, P. and Bullock, R., 2000, The effect of the forcing function on disruption of a phase-locked loop, to appear in Int. J. Bifurcation and Chaos, September, 2000. Brennan, P.V., 1996, Phase-locked Loops: Principles and Practice, McMillan, London. Chang, F.J., Twu, S.H. and Chang S., 1993, Global bifurcation and chaos from automatic gain control loops, IEEE Trans. Circuits and Systems, 40:403. Chu, Y.H., Chou, J.H. and Chang, S., 1991, Chaos from third order phase-locked loops with a slowly varying parameter, IEEE Trans. Circuits and Systems, 37:1104. Drazin, P.G., 1992, Nonlinear Systems, Cambridge University Press, Cambridge. Endo, T. and Chua, L.O., 1988, Chaos from phase-locked loops, IEEE Trans. Circuits and Systems, 35:987. Endo, T., Chua, L.O. and Narita, T., 1989, Chaos from phase-locked loops – Part II: high dissipation case, IEEE Trans. Circuits and Systems, 36:255. Green, D.N., 1983, Global stability of automatic gain control circuits, IEEE Trans. Circuits and Systems, 30:78.
AN APPLICATION OF CHAOS THEORY TO THE HIGH FREQUENCY RCS PREDICTION OF ENGINE DUCTS
Andrew J. Mackay1 1
Defense Evaluation and Research Agency (DERA), Malvern. St Andrews Road, Great Malvern, WR14 3PS, UK.
INTRODUCTION The accurate prediction of the radar cross section (RCS) of electrically large ducts and cavities is a particularly difficult problem but one of significant interest for a number of applications. Most general purpose methods employ variants of shooting-andbouncing (SB) ray tracing (e.g. Burkholder et al, 1991) since more accurate modal, finite element or finite difference methods are computationally too intensive. I have previously shown (Mackay, 1998; Mackay, 1999) that SB ray tracing in straight ducts of moderate length is inherently non-convergent for most duct geometries. This involves a direct application of well established results in chaos theory used in the study of classical and quantum dynamical systems. In the previous work I established a convergence bound in terms of the characteristic duct dimensions, the angle of incidence and the Lyapunov exponent for a straight duct terminated by a flat conductor. Here, we summarise these findings and extend the investigation to describe the RCS properties of a long stadium cross section duct in a regime where SB convergence is not achievable. I investigate the use of random fields to construct an ensemble of statistically equivalent RCS predictions and compare with an accurate modal solution.
DETERMINISM IN RCS PREDICTIONS Radar engineers seldom require a precise lobe-for-lobe convergence of predicted RCS at high frequencies. Firstly, measurements seldom agree with each other for complex targets at high frequencies since tolerances are rarely maintained to a small fraction of a (short) wavelength. Secondly, high frequency predictions are almost always approximate. This is primarily because such algorithms only provide an approximate solution to Maxwell’s equations, but also because of approximations in the mathematical representation of the geometry for complex structures (commonly using standard
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computer aided design meshes). Quite often it is sufficient only to require the mean RCS over an angular window, small compared with 360° but large compared to the average lobe width. For other applications, e.g. for imaging purposes, it is also necessary to obtain the correct autocorrelation of the RCS taken as a function of incidence or scatter angles. It is our contention that, at high frequencies, deterministic “exact” RCS predictions which provide lobe for lobe correspondence are not required for nearly all applications. In any measurement system there is always a finite sensor resolution and experimental uncertainty associated with structure dimensions and orientations. For service applications, e.g. for vehicles in transit, these uncertainties can be very large and considerably limit any requirement for deterministic predictions. Thus, although we have shown that SB ray methods can not generally achieve convergent RCS predictions, there are grounds for believing that SB methods can probably be employed in a non-convergent setting to obtain statistically correct RCS predictions. In this paper I report some work to support this belief and show how the fields at the end of a long straight stadium duct can be modelled by a sum of randomly directed plane waves. For chaotic structures of this kind it then becomes possible to decompose the exit field by a “deterministic” component predicted using convergent ray tracing and a “random” component to represent the non-convergent behaviour. A more detailed account of these findings is given elsewhere (Mackay, 1998-1999).
DYNAMICAL CHAOS AND RAY TRACING Chaos theory is concerned with the properties of formally deterministic systems (i.e. without an intrinsic random element) which evolve unpredictably with the gradual increase of some parameter such as time. Here, a small perturbation in the initial conditions of the system grows exponentially with time. There is considerable literature proving that ray tracing within a closed or ‘nearly closed’ structure is in general (but not always) a chaotic process (e.g. Berry, 1989; Heller, 1989; Gutzwiller, 1990). Here, by ‘nearly closed’, we mean a system where at least some rays eventually escape after a finite time. This is the case for many scattering systems including engine ducts, open cavities and certain other classes of concave perfectly conducting structures. Not all structures resulting in multiple ray bounces exhibit chaos. In particular, geometries which are fully separable under the wave equation do not (e.g. Lichtenberg and Lieberman, 1983). For example, a straight duct with circular or elliptical cross section is fully separable and hence does not give rise to chaotic ray tracing. Because fully separable geometries allow exact solutions in terms of standard functions, these are just those geometries which are often used for software validation. It is thus rather too easy to arrive at incorrect general conclusions on the use and accuracy of ray tracing algorithms. There are two further points from chaos theory which are important for our application. Firstly, in order to exhibit chaos, the scatterer must be concave or composed of disconnected elements. An example of the latter is the three cylinder problem (Svitanovick and Eckhardt, 1989; Smilansky, 1989) which also gives rise to fractal scattering in the high frequency limit. Straight ducts, as we indicate below, can give rise to chaotic but not fractal scattering. Secondly, the scatterer must have non-zero curvature over at least some of its domain. For example, a duct with polygonal cross section with a finite number of sides may or may not be separable under the wave equation but can not give rise to chaos in the high frequency limit.
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Suppose a wave enters a straight perfectly conducting duct of arbitrary cross section and length L, terminated by a flat perfectly conducting plate normal to the duct axis. This problem can be unfolded, using mirror symmetry, and is equivalent to an open duct of length 2L with a separate exit and entry plane. We consider an incident plane wave entering the duct at the entry plane at an angle to the normal and an azimuth angle as illustrated in figure 1, below.
If we assume an SB first order ray method, then the plane wave can be represented as a set of zero divergence ray pencils each with the same direction as the incident wave contiguously spanning the entry aperture. A given ray pencil will make N bounces before escaping from the exit plane. N will generally be different for each ray pencil but for a straight duct, the total transit time for any ray between the exit and entry plane is given by, where C is the speed of light in free space. This is independent of the duct cross section and it follows that a straight duct can not trap rays; i.e. it is not possible to choose a ray trajectory where the time approaches infinity for a finite length duct away from grazing incidence. A consequence of this is that fractal scattering is impossible. However, it is important to note that for bent ducts (1) is not valid, can approach infinity and fractal scattering is possible for non-separable geometries in the high frequency limit. Within the duct let us define the “billiard ray” as the projection of a ray within the duct on to the cross section, where the cross section will sometimes be referred to as the “billiard plane”. Let be the angle made between the billiard ray and the inward pointing normal and let be the chord distance in the billiard plane between the and ray bounce for a given ray pencil. If we define as the expected (mean) value of then for large N,
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We will now consider a special class of ergodic chaotic systems, where every ray state is equally likely for large N. For an ergodic system the average geometrical path length of a ray between successive bounce points is equal to the average chord length. A result, probably going back to Euler, shows that the average chord length is given by for a general billiard shape where A is the area of the billiard and S is its perimeter. Hence, for large N, While this result is not strictly valid for non-ergodic ducts, it is likely to hold for most straight ducts to a reasonable approximation. In the tangent mapping formulation one can define a deviation matrix (Berry, 1981), which relates a small change in the ray characteristics between one bounce and the next. One deviation matrix is of special use and relates the deviations and where is a deviation in the width of a billiard ray normal to the direction of the ray in the billiard plane at the ray bounce and is the divergence angle of the billiard ray. has a specially simple form and describes the change of the first order ray characteristics between successive bounces, analogous to the Deschamps formulation (Deschamps, 1972). The deviation matrix is defined by,
where
where
is the Abbe invariant at skew incidence at the
reflection defined by
and is the surface curvature in the billiard plane at the bounce point, defined positive if the local boundary and the billiard ray lie on the same side of the tangent plane and negative if they lie on opposite sides of the tangent plane. The Lyapunov exponent (strictly, the maximum Lyapunov exponent) is an important measure of the level of chaos in a dynamical system which can be connected with ray tracing and hence with high frequency RCS prediction. This is defined as the limit of where
where
is the maximum eigenvalue magnitude of the compound deviation matrix defined with respect to the mapping by,
This result is independent of the coordinate system to an accuracy of order O(1/N), so we may employ Suppose that the initial values of a ray, specifying its origin and direction on the incidence plane, are known to a relative accuracy of for some positive This can be interpreted in several ways. In terms of the SB ray algorithm, we can define
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is the width of a ray pencil and D is the maximum distance between any two points on the duct aperture in which case defines the convergence error. Alternatively, we can define where is a small change in azimuth angle associated with the incident plane wave. In this case is a measure of angle of incidence error. We can also relate to a small relative change in the shape of the boundary. Defined in terms of an azimuth angle error or uncertainty in the geometry, and given the general uncertainties in RCS prediction and geometrical design specification, it is unlikely that can exceed 3 or 4 in simulation work and may be rather smaller. Defined as a mesh convergence error the same is true, since on a square grid the number of ray pencils is of the order of resulting in launched rays for In all cases can be regarded as an error estimate of the initial or first reflected rays. The growth of this error from bounce to bounce will allow precise knowledge of the state of the system only as long as where is our estimate of the Lyapunov coefficient for a finite number of ray bounces, N. We can thus set the limit of deterministic predictability when,
This implies that the maximum number of ray bounces permitted, before we can no longer know their position and direction, is given by
This provides an important inequality for straight ducts. In particular, deterministic predictions can only be made if,
The ratio is a measure of the aspect ratio of the duct where can be estimated from (3), although strictly valid only for ergodic ray tracing for large N. The Lyapunov exponent can be determined using the eigenvalue definition, but it may also be estimated from the intensity of a ray. In particular we can show that if is the base-10 logarithm of the ray intensity of the ray at the bounce (defined either per unit ray cross section or per unit divergence angle) then,
with increasing certainty for large N. More accurate estimates are also possible (Mackay, 1998,1999) and we have a simple means to estimate A using the standard Deschamps ray tracing formulation. RANDOM WAVE REPRESENTATIONS In this section we show how a field distribution down a duct can be represented by a sum of randomly directed plane waves, valid in a statistical sense when convergent ray tracing is not possible. We assume a fixed incidence angle, away from any planes of symmetry, and consider the nature of the bistatic RCS resulting from such a distribution. The extension of the model to allow variation with incidence angle, and hence monostatic RCS, is the subject of other work which will not be reported here. Our hypothesis, based on those in the field of quantum chaos (e.g Heller, 1989), is to suggest that in the high frequency limit the fields within a chaos-inducing duct can
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be represented as a coherent sum of randomly directed plane waves with random phase. Such a field can, we suggest, be used to replace the contribution of non-convergent rays assuming incident ray pencil widths which are small but not very small compared to a wavelength. Such fields form members of a statistical ensemble, each member of which is a valid representation of the ‘true’ field in the high frequency limit. The fact that such fields do not satisfy the boundary conditions at the surface of the duct is not important in this limit. We then assume that the random field model can be applied at finite frequency over the frequency regime for which ray tracing is traditionally employed. The random wave component of a transverse electric field in a straight duct is written by,
for independent random variables and to The polarisation is defined by,
uniformly distributed over the interval 0
for the two different states of incident polarisation. The normalisation constant defined by,
is
where is the number of ‘bad’ rays which have diverged past some preset threshold and is the total number of launched ray bundles at the duct entry plane. M must be sufficiently large to satisfy the Nyquist sampling criterion on the boundary, for cross section perimeter and wavelength (not to be confused with the Lyapunov exponent ). The total transverse field is then taken as the sum of the random field and the ray field from the convergent rays. For a sufficiently long ergodic duct, e.g. the stadium duct, the convergent ray contribution will approach zero except at special angles (i.e. when ). As an example we consider a stadium duct which is an example of a structure exhibiting ergodic behaviour and which may be representative of more realistic engine intakes. A stadium duct is defined by a duct whose cross section is composed of two semi-circles of radius a, smoothly joined by two straight sections of length I consider a stadium duct where unfolded duct length 2L = 30.0 a and incident angles for a plane incident wave polarised in the direction. In this special case all rays have diverged beyond computational limits and the field is composed only of the random wave components. Predictions are given assuming a wavenumber Each figure shows a co-polar (top) graph, labled as ‘phi-directed’ and a cross-polar (bottom) graph, labled as ‘theta-directed’. RCS is shown in dB relative to a square metre assuming metre. This is shown as a function of azimuthal scattering angle with elevation scattering angle Figures 2, 3 and 4 show three members of the ensemble of possible random wave predictions whilst figure 5 shows a modal solution (assuming a Kirchhoff approximation) which, neglecting non-Kirchhoff edge diffraction, can be regarded as accurate. These results are typical of those for a long stadium duct and demonstrate the feasibility of using random waves for RCS predictions of chaotic ducts.
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CONCLUSIONS We establish a connection between the Lyapunov exponent and SB ray methods which places a limit on the ability to make numerically deterministic or convergent predictions. When this limit is reached the ray fields are essentially random. One method, valid for ergodic straight ducts, is to represent the random ray field component by a field composed of randomly directed plane waves. We demonstrate that predictions made under this assumption have very similar bistatic RCS characteristics to those of an accurate modal solution. Estimates of the Lyapunov exponent are directly related to average ray divergence and hence average ray intensity. For ergodic ray tracing this can be determined by the tracing of a single ray in order to place limits on the ability to achieve convergence. The computational cost of estimating the Lyapunov exponent is usually negligible compared to the cost of a full RCS prediction where all the rays on the entry plane must be traced. REFERENCES G.A.Deschamps, September 1972, ‘Ray techniques in electromagnetics’, Invited paper, Proceedings of the IEEE, Vol. 60, No. 9. A.J.Mackay, July 1988, ‘Chaos theory and first-order ray tracing in ducts’, IEE Electronics Letters, Vol. 34 No. 14, pp1388-1389 A.J.Mackay, April 1998, ‘Chaos theory applied to first order ray tracing in ducts’, DERA Malvern, UK, report DERA/SN/R/TR980002/1.0, (available through DRIC) A.J.Mackay, March 1999, ‘New representations of ray tracing, chaos theory and random waves in straight ducts’,DERA Malvern, UK, report DERA/S&P/RAD/CR990108/1.0, (available through DRIC) A.J.Mackay, December 1999, ‘Application of chaos theory to ray tracing in ducts’, IEE Proceedings Radar, Sonar and Navigation, Vol 146, No 6, pp298-304 M.C.Gutzwiller, 1990, ‘Chaos in classical and quantum mechanics’,Springer Verlag, Interdisciplinary Applied Mathematics Series. M.Berry, 1989, ‘Some quantum to classical asymptotics’,section 4 pp251, Les Houches 1989 Session LII, Chaos and quantum physics, North Holland publishing 1991. E.J.Heller, 1989, ‘Wavepacket dynamics and quantum chaology’,section 9, pp548, Les Houches 1989 Session LII, Chaos and quantum physics, North Holland publishing 1991. A.J.Lichtenberg and M.A.Lieberman, 1983, ‘Regular and stochastic motion’,Springer Verlag, Applied Mathematical Sciences 38. P.Cvitanovic and B. Eckhardt, August 1989, ‘Periodic quantization of chaotic systems’, Physical Review Letters, Vol 63, No. 8, pp823-826 U.Smilansky, 1989, ‘Theory of chaotic scattering’, section 7 pp371, Les Houches 1989 Session LII, Chaos and quantum physics, North Holland publishing 1991 M.Berry, 1981, ‘Regularity and chaos in classical mechanics, illustrated by three deformations of a circular billiard’, Eur.J.Phys.2 , pp91-102. Doc. ref. DERA/S&E/RAD/CP000180 ©Crown Copyright 2000 /DERA.
RAY SPLITTING AND CHAOS IN ELECTROMAGNETIC RESONATORS
Reinhold Blümel Department of Physics Wesleyan University Middletown, CT 06459-0155, USA
INTRODUCTION Rays are a fundamental concept in any wave theory. They describe the particlelike propagation of radiation in the small-wavelength limit of geometrical optics. There is no conservation law for rays. Quite the contrary. Whenever the properties of the propagation medium change on a scale small compared with the wavelength of radiation, rays proliferate as they split into transmitted and reflected components. This phenomenon is called ray splitting. Ray splitting occurs in many fields of science and technology. The best-known example is the splitting of a light ray as it encounters the interface between two media with different indices of refraction. The implications of ray splitting in acoustics and quantum mechanics were first investigated by Couchman et al (1992). The presence of ray splitting in a wave system has important consequences for the mean and fluctuating parts of the level density of the system. It was shown by Prange et al (1996) that the average number of states in a resonator – Weyl’s famous mode-counting function – has to be modified in the presence of ray splitting. The essence of ray splitting is best conveyed with the help of a simple onedimensional quantum system, for instance the one shown in figure 1 (see also Bauch et al, 1998). A quantum particle is confined between two impenetrable walls a distance apart. A potential step of height and width produces ray splitting of the rays representing the quantum particle bouncing between the walls. If the energy E of the particle is smaller than the particle (neglecting tunneling) bounces between the left-hand wall and the potential step periodically traversing the orbit D. This is precisely what a “Newtonian” particle would do. Therefore the periodic orbit D is called a “Newtonian” orbit (characterized by N in figure 1). At the quantum particle has many more options for its motion. Starting on the left, it might decide to reflect Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002
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off the potential step resulting in the orbit C. This kind of motion does not occur in Newtonian mechanics since a Newtonian particle at passes straight over the potential hump heading for the right-hand wall. Thus orbit C is a non-Newtonian orbit (characterized by NN in figure 1). The orbit B is non-Newtonian as well, while the orbit A corresponds to what we would expect according to Newtonian mechanics.
Does quantum mechanics know about the existence of the non-Newtonian orbits B and C in figure 1? It certainly does. In order to demonstrate, let us consider the scaled quantum problem defined by where is a scaling constant. We will see later that this is not an artificial, academic problem. It is tightly connected with the spectral problem of electromagnetic resonators partially filled with dielectric substances. Defining the Schrödinger equation of the scaled quantum
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problem leads to the transcendental equation
where is the wave number of the particle. The roots of (1) determine the particle’s energy spectrum Choosing and units such that where is the mass of the quantum particle and is Planck’s constant divided by the real part of the discrete Fourier transform
of the first M = 162 roots is shown in figure 2 as a function of We see a regular sequence of peaks at 0.6 and 0.9 which can be associated with multiples of the optical path length of the non-Newtonian periodic orbit B. The peak at is the (optical) path length of the non-Newtonian orbit C. The largest peak in figure 2, at corresponds to the Newtonian orbit A. The remaining two peaks are explained as composites of the Newtonian orbit A plus one or two additional (non-Newtonian) above-barrier bounces of the particle. As a result we obtain that quantum mechanics is able to identify every single one of the classical orbits of the ray-splitting system of figure 1, whether Newtonian or not. Since Schrödinger’s wave mechanics is essentially a classical field theory, and the system shown in figure 1 is essentially a quantum resonator, we expect similar phenomena to be present in other confined-wave systems, for instance in electromagnetic resonators.
ELECTROMAGNETIC RAY-SPLITTING RESONATORS Electromagnetic resonators partially filled with dielectric substances are excellent examples of ray-splitting systems. We focus here on flat cylindrical resonators operated below the cut-off frequency for the onset of the first non-trivial axial mode. The cross section of a typical ray-splitting resonator is shown in figure 3.
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The resonator has a generic shape. One part of it (perimeter L, area A) is filled with a dielectric with dielectric constant the other part (perimeter area ) is filled with a dielectric with dielectric constant The interface between the two dielectric substances is assumed to be sharp. It is called the ray-splitting boundary (“RS boundary” in figure 3). Electromagnetic ray-splitting cavities of the type shown in figure 3 were first considered by Prange et al (1996) who show that the spectral problem of a ray-splitting resonator is formally identical with a scaled quantum problem. This is demonstrated in the following way. Since the resonator is assumed to be thin in axial direction (assumed to be the direction here) and operated below the axial cutoff frequency, the electrical field in direction is homogeneous, vanishes at the boundary of the resonator (Dirichlet boundary condition) and is a function of and only. If we denote the phase velocities in sections A and of the resonator by and respectively, the wave equation for takes the form
where is the Laplacian in two dimensions and is a function that takes the value 0 if is in A, and the value 1 if is in Identifying the term with the scaling constant shows that the dielectric resonator problem is indeed formally equivalent with a scaled quantum problem. Thus electromagnetic raysplitting resonators can be used as analog computers to solve scaled quantum problems. They can also be used to predict new electromagnetic and quantum phenomena.
CHAOS INDUCED BY RAY SPLITTING Suppose the perimeter of the resonator shown in figure 3 is circular and i.e. the resonator is empty. Then there is no problem solving for its resonance frequencies. They are essentially given by the zeros of the Bessel functions of the first kind. We may also imagine a classical point particle bouncing in this resonator imposing perfectly elastic (specular) reflection at the walls of the resonator. The particle will trace out a regular pattern; its motion is simple and perfectly predictable. But now imagine the circular resonator split down the middle and one of its parts filled with dielectric. A narrow laser beam bouncing in this cavity traces out a completely erratic, chaotic path. Thus ray splitting can induce chaos in an otherwise regular resonator. The quantum analog of the split-circle resonator was studied in detail by Blümel et al (1996). Apart from the split-circle resonator, other resonator geometries were studied that are analytically solvable (regular) in the absence of ray splitting, but become chaotic if dielectrica inducing ray splitting are present. Examples are the triangular ray-splitting resonator studied by Kohler et al (1997) and the annular ray-splitting billiard studied by Kohler and Blümel (1998).
LATERAL RAYS There are special types of orbits that occur only in ray-splitting systems. One of them are lateral rays. Lateral rays lead to periodic lateral-ray orbits whose signatures are found in the Fourier transform of ray-splitting resonators. Kohler and Blümel (1998) identified the signature of a family of periodic lateral-ray orbits in the spectrum of a rectangular quantum ray-splitting resonator. Indications of a lateral periodic orbit were
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also found in the split-circle ray-splitting resonator (Blümel et al, 1996). A periodic lateral ray hits the ray-splitting boundary at the critical angle of total internal reflection, runs along the ray-splitting boundary, emerges from the ray-splitting boundary under the critical angle, and completes its path by joining up with its starting point by specular reflections off the resonator’s boundary, if necessary. Lateral periodic orbits are non-Newtonian orbits.
STADIUM-SHAPED RAY-SPLITTING RESONATOR: EXPERIMENTAL RESULTS Are the signatures of Newtonian and non-Newtonian periodic orbits actually observable in the Fourier transform of the frequencies of dielectric-loaded ray-splitting resonators? This question was answered recently by the experiments of Sirko et al (1997) and Bauch et al (1998).
The ray-splitting cavity used in these experiments is shown in figure 4. It consists of a metallic enclosure in the form of a stadium and a movable (and removable) teflon bar insert a distance d away from the tip of the stadium. In a first set of experiments the teflon bar was removed and 50 resonances were measured. The Fourier transform of the measured resonances is shown in figure 5. The signatures of the simplest Newtonian periodic orbits are clearly visible as peaks in figure 5. No peaks associated with nonNewtonian orbits appeared (control experiment). The signatures of non-Newtonian orbits appeared as soon as the teflon bar was inserted. The simplest non-Newtonian periodic orbits are the “internal bounce orbit” that bounces inside of the teflon bar, and the orbit that bounces between the upper edge of the teflon bar and the tip of the stadium. Figure 6 shows that both non-Newtonian orbits can actually be identified in the Fourier transform of the experimental data. The mobile teflon bar makes sure that the peaks in move in the correct direction as
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SUMMARY AND CONCLUSIONS The above discussion shows that flat electromagnetic resonators partially filled with dielectric substances are excellent systems for studying ray-splitting phenomena. Due to a formal equivalence of the stationary Maxwell equations and the quantum Schrödinger equation, flat ray-splitting cavities are also suitable for the study of ray splitting and quantum chaos in mesoscopic devices, important components of the next generation of super computers. The experiments of Sirko et al (1997) and Bauch et al (1998) demonstrate that the Fourier transform of the frequency spectrum of a raysplitting cavity shows peaks that correspond to periodic orbits of a novel non-Newtonian mechanics that forms the semiclassical backbone of ray-splitting systems in the shortwavelength limit. In addition the work of Prange et al (1996) and Kohler and Blümel (1998) showed that ray splitting contributes novel universal terms to the Weyl formula describing the number of modes in electromagnetic and quantum resonators. Raysplitting induced chaos and the emergence of new ray-splitting phenomena, such as periodic lateral rays are the most promising directions for future theoretical and experimental research. Acknowledgement The author gratefully acknowledges financial support of the National Science Foundation, grant number 9900730. REFERENCES Bauch, Sz., A., Sirko, L., Koch, P.M., and Blümel, R., 1998, Signature of non-Newtonian orbits in ray-splitting cavities, Phys. Rev. E, 57:304. Blümel, R., Antonsen, Jr., T.M., Georgeot, B., Ott, E., and Prange, R.E., 1996, Ray splitting and quantum chaos, Phys. Rev, Lett., 76:2476; Phys. Rev. E, 53:3284. Couchman, L., Ott, E., Antonsen, Jr., T.M., 1992, Quantum chaos in systems with ray splitting, Phys. Rev. A, 46:6193. Kohler, A., Killesreiter, G.H.M., and Blümel, R., 1997, Ray splitting in a class of chaotic triangular step. billiards, Phys. Rev. E, 56:2691. Kohler, A., and Blümel, R., 1998, Annular ray-splitting billiard, Phys. Lett. A, 238:271. Kohler, A., and Blümel, R., 1998, Signature of periodic lateral-ray orbits in a rectangular ray-splitting billiard, Phys. Lett. A, 247:87. Kohler, A., and Blümel, R., 1998, Weyl formulas for quantum ray-splitting billiards, Ann. Phys. (N. Y.), 267:249. Prange, R.E., Ott, E., Antonsen, Jr., T.M., Georgeot, B., and Blümel R., 1996, Smoothed density of states for problems with ray splitting, Phys. Rev. E, 53:207. Sirko, L., Koch, P.M., and Blümel, R., 1997, Experimental identification of non-Newtonian orbits produced by ray splitting in a dielectric-loaded microwave cavity, Phys. Rev. Lett., 78:2940.
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Ultra-Wideband (UWB) Radio-Frequency (RF) Bioeffects Research at DERA Porton Down S Holden, RH Inns, CD Lindsay, JH Tattersall, P Rice and JL Hambrook Chemical and Biological Defence, Defence Evaluation and Research Agency (DERA), Porton Down, Salisbury, Wiltshire SP4 0JQ, UK Abstract Current guidelines controlling exposure of personnel to radiofrequency fields are not specifically relevant to situations where subjects may be exposed to ultra-wide band (UWB) pulses. In addition, there is an increasing awareness that pulses of low mean power may induce cellular effects below the thermal threshold, with effects being found in windows of a particular frequency. The Biomedical Sciences group of the Chemical and Biological Defence (CBD) sector undertakes fundamental and applied research on the biological effects of RF radiation. An ongoing programme is addressing whether personnel may be at risk from exposure to radio frequency radiation of low mean power (both pulsed and continuous wave). The main areas of research are: Physical and computer based dosimetry modelling; Physical models using dosimetry phantoms filled with tissue simulants; Computer based models using computational electromagnetic methods such as Finite Difference Time Domain (FDTD), Transmission Line Matrix (TLM) and Quasi-optical methods.
In vitro human and animal tissue studies; Human and animal cell cultures exposed to UWB; Rat hippocampal brain slice exposed to CW (microwave). This paper will describe details of the work programme and provide examples of findings to date. Introduction Currently, RF health risks are attributed to thermal and “sub-thermal” effects. “Subthermal effects are further divided into the general categories of microthermal (no measurable heating, any transient heating is compensated by homeostatic mechanisms) and non-thermal (no measurable temperature rise). Thermal effects due to RF are well known to produce deleterious effects and are used as the basis of UK/US safety guidelines at MHz/GHz frequencies. “Sub-thermal” effects are much more contentious and are not sufficiently proven for guidelines1 to have been issued. There are some established physiological effects at sub-thermal levels (e.g. microwave hearing) but it is not known if they are hazardous. Presently, sub-thermal effects and in particular non-thermal effects are a major source of investigation worldwide. 1
See recent Independent Expert Group an Mobile Phones (IEGMP) report Mobile Phones and Health - “Stewart Inquiry”
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Ultra-wideband (UWB) sources: The MoD2 problem Novel UWB RF sources are under development in the DERA. Current national guidelines in the UK, produced by the National Radiological Protection Board (NRPB) and the IEEE Standard in the USA do not address human exposure to novel sources such as UWB. This presents a problem, as the MoD is required to develop safety guidelines for MoD/DERA staff to demonstrate “duty of care” as an employer. UWB sources are characterised by the production of very short RF pulses (<1nS) with high amplitudes (several kV/m). This class of source does not cause a significant temperature rise in tissue but there could be a health risk due to long-term exposure. In addition, standard RF dosimetry methods, which are based on the thermal effects of RF sources, are not appropriate for assessment of the health risk from this new class of source. Therefore, a programme has begun to address the issues highlighted above and investigate new methods for assessing possible bioeffects of UWB sources in order to develop new safety guidelines. Potential approaches for guideline development. An initial approach taken by many researchers in the RF bioeffects area is to undertake animal experiments and screen for effects of exposure (if any). However, this approach is considered to have several drawbacks. It is expensive, time consuming and difficult to extrapolate results across from one complex system (e.g. animal) to another (human). There are also important ethical issues to address. A better approach is to predict in which sites, organs, tissues or cell types are effects likely to be observed. Using this information, model systems could then be used to determine whether bioeffects occur as a result of exposure This has been the approach taken by DERA Porton Down. Modelling studies are being used to determine power absorption in the human body and predict fields/currents at various anatomical locations. The results of the modelling are then being used to support and inform in vitro studies.
In vitro studies in a number of model systems have already been used to screen for the effects of exposure and, when observed, the underlying mechanism has been investigated. When a sufficient body of information has been accumulated, targeted in vivo studies (e.g. behavioural studies) could then be used to determine the manifestation of in vitro effects. A computer based model of a human exposed to UWB – a hybrid approach Modelling a human being as a series of lossy dielectric materials in an UWB RF environment (spectra > 10GHz) is problematical. In particular, the memory requirements of differential methods are high. The number of discrete cells required in differential methods is normally 10 cells per minimum wavelength modelled. If the frequency to be modelled is doubled this can mean the memory requirement can go up 8 fold! Using expertise within DERA it was decided that a hybrid computer model was the best way forward using differential methods for most of the band of interest and quasi optical methods for the high frequency end of the spectra. Quasi-optical methods can be applied in this regime as they have much lower memory requirements and the electromagnetic properties of human tissue do not change much above at the upper end of the UWB spectra.
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MoD, the Ministry of Defence of the United Kingdom
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The DERA Porton Down hybrid model will consist of a geometric representation of the human being which will have the appropriate material properties assigned to it. Then a series of analysis codes will be applied, consisting of: Differential methods covering most of the band of interest (~ 8GHz): Transmission Line Matrix (TLM) algorithm; Finite Difference Time Domain (FDTD) algorithm. Quasi - Optical methods, based on differential ray tracing, to extend to higher frequencies (>8GHz). To develop codes and assemble the model straight away has an element of risk associated with it as the model could provide incorrect predictions. The approach taken is to undertake model development in conjunction with a validation programme. Validation targets referred to as common target models (CTM) have been designed to allow comparison during model development of: Different computer-based solution methods; Analytical solutions; Direct physical measurements. The CTM will address the two main problem areas encountered when modelling humans over a wide range of frequencies: Modelling frequency dependent lossy dielectrics accurately and; Modelling large dielectric objects at high frequencies. Code development -differential methods At present we expect to have two differential codes developed that will have frequency dependant material models implemented and will be able to perform calculations on parallel machines. The parallel approach has been taken due to the size of the problem and associated memory usage and processing time requirements. To date, the TLM code has an implementation of a Debye, frequency dependent, dielectric model of human tissue (see Figure 1) with 43 tissue types at frequencies of up to 20 GHz, the upper frequency limit being dependent on computer resources. A Computer Aided Design (CAD) interface is available for the software to input human geometry and parallelisation of the code is underway. However, more work is required after parallisation to reduce the memory requirement. This is being investigated by relaxation of the cell requirement to
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Note: the TLM method has a degree of redundancy in the method of calculation. This approach is acceptable for preliminary investigations.
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The FDTD code also has a Debye tissue model under development and, in addition, parallelisation of the code has almost been completed. Preliminary experiments using parallel differential codes on a DERA SGI Origin 2000 are underway. The model being used is a humanoid with homogenous tissue properties (known as CTM-4). A 4th order (spatial) algorithm is being used which should reduce the cell requirement to cell whilst giving the equivalent accuracy. This should reduce the number of cells to be calculated from cells to cells. Given the memory on the Origin 2000 (32 GBytes) calculations up to ~10 GHz are expected.
In-vitro neurophysiology - brain slice tem cell model Computer-based modelling is being applied to in vitro studies. The in vitro studies consist of investigations into RF effects on organised tissue (brain tissue, heart) and individual cells in culture. An important element of the overall work programme is to relate computer modelling results with in vitro test systems. This is to ensure that field strengths derived from computer models are the same as the in vitro systems are exposed to. Studies have been targeted at understanding the processes involved in exposing in vitro systems to RF and associated dosimetry. Initial studies on in vitro neurophysiology have been made using continuous wave sources. Slices of rat hippocampus have been exposed to RF within a TEM cell (see Figure 2).
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Dosimetry of the brain slice has been undertaken using non-invasive thermometers (fibre optic), analytical calculations and now computational methods4 (FDTD, see figure 3).
Figure 4 shows a slice of brain tissue of the type used in the experiments at DERA Porton Down. This part of the brain, the hippocampus, is involved in learning and memory as well as epilepsy. An electric stimulus is delivered via an electrode: this elicits an electrical response in the tissue (marked PS). The top graph shows the amplitude of the PS response plotted against time. The lower graph shows the RF field intensity to which the tissue is exposed. Low exposures, at 700MHz, caused a slight increase in the response, the highest intensity caused a dramatic, reproducible and reversible reduction of the electrical response. This effect is at very low field intensities (SAR5 1.5 mW.kg-1) and no temperature rise associated with gross temperature changes was observed. However, since a metal stimulating electrode was used to evoke the response, there is the possibility of an artefact. This issue was addressed in further experiments using a chemical stimulant rather than electrical by introducing a chemical that causes an epileptic like seizure. It was found the RF at 700 MHz and the same field strength, caused an inhibition of spontaneous seizure activity. This reinforces results from the previous study on electrically evoked potentials. It is unlikely the measurement electrode (used in both experiments) consisting of an ionic solution in a glass tube is causing an electrode artefact. However, we are currently investigating optical methods such as calcium ion imaging to confirm this. Other investigations have taken place on isolated heart and ion channels. No effects were observed on isolated heart by UWB or CW fields and no effects were observed on nicotinic ion channels6 when exposed to CW fields at 10GHz.
4
This study is also being used to look at TEM cells with differing apertures (for measurement) and to relate dose to that observed in published head models. Confusingly for radar engineers the unit of RF energy absorption is the Specific Absorption Rate or SAR. 6 Nicotinic ion channels facilitate the propagation of electrical impulses. 5
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Rf-lnduced stress response in the nematode Caenorhabditis elegans
C. Elegans is a nematode worm widely used in biological research. It has recently had its entire genome sequenced. Genetically modified variants of C.elegans have been used for environmental monitoring. One form of modification causes the nematodes to change colour when their cells produce Heat Shock Proteins (HSP). HSP are produced by cells as a protective response to one (or more) of a range of stressors, e.g. heavy metal ions and not just heat as the name implies.
Figure 5 shows the sensitisation of HSP16 expression in C.EIegans with increasing temperature. The upright triangles show sham exposures and indicate the response in the absence of exposure to radio frequency radiation. The downward pointing triangles indicate results after exposure to CW RF at 750MHz. There was a marked difference between exposed compared to unexposed nematodes over the temperature range of 24.5 – 27°C. The results indicated a reproducible biological effect (in this case a stress response) in a well defined model system. Note that there was no measurable increase in temperature during or after exposure to RFR i.e. the effect was not due to gross heating caused by the radiation. Work is now proceeding to expose the worms to UWB.
In Vitro Cell Culture Studies In addition to studies involving organised tissue experiments have also been performed on cell cultures. This is an important element to understanding the biological effects of RF as it permits use of cells of human origin including growing cell cultures from organs “at risk”. Four cell culture models of relevance to current epidemiological concern were used: 2 human bone marrow leukaemia stem cell lines (KG1 and MEG01; non-adherent); Human foetal fibroblast line (HFFF2; adherent); Rabbit corneal cell line (SIRC; adherent - no human equivalent available). The cultures were placed 70cm from impulse radiating antenna (IRA) and exposed to UWB pulses with peak field 5 kV/m, rise times <1 nsec and repetition rates between 1350 and 2170 Hz. Exposure durations were between 0.5 - 3 hours. A wide range of cell analysis methods were used including:
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Cellular integrity and growth rate indices; Heat shock protein (HSP) analysis; Cellular ultra-structure studies (transmission electron microscopy TEM); 2-D Electrophoresis (a form of protein fingerprinting). The mechanism of action of UWB effects was also investigated by: Cytoskeletal function (cytokeratins), the production of transcription factors and ornithine decarboxylase levels (ODC); Effects on cell growth and DNA repair processes using biochemical approaches and flow cytometry respectively. Results using SIRC or HFFF2 cells exposed to UW8 radiation indicated no significant, reproducible effects except for those on the cell cycle (see Figure 6).
Cell cycle analysis was performed using flow cytometry. Figure 6 shows the output from the flow cytometer (a) and the phases of cell cycle (b). The flow cytometer can interrogate a cell at any phase of its cycle. These phases are: - Resting Phase (cells not cycling) - Chromosomes are unreplicated. High level of biosynthesis underway. S - DNA undergoes replication - Chromosomes are completely replicated M - Mitosis; nuclear membrane reforms, two daughter cells form, cells reenter Cell cycle times can vary from less than 8 hours to greater than 1 year. The flow cytometer works by directing laser beams at single cells in a flow tube. Before
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testing, cells are treated with a dye that attaches to the genetic material of the celll (DNA). By analysing scattered light the amount of DNA in the cell can be determined and the phase in the cell cycle identified. The different heights of the peaks shown in Figure 6(a) summarises the proportions of cells in the various phases. SIRC cells exposed to UWB radiation showed a consistent decrease in cell growth of up to 14% (p<0.01). This was largely a result of an increase in the proportion of cells in phase compared to the phase of the cell cycle. This may be due to changes in factors that control cycle progression. The effects were cell line specific and reversible. Summary DERA Porton Down is developing a computer based model of a human being in an UWB RF environment. The model will use Debye frequency dependent dielectric models to produce an inhomogeneous human model capable of producing simulations up to 10 GHz, the upper frequency of UWB spectra. Results are then to be directed towards providing information for targeted in-vitro studies.
In vitro studies have indicated that CW and UWB RF can induce effects at below thermal levels. Future work will address the: Relationship between computer-based of humans with in-vitro studies; Mechanism of the interaction of RF with biological systems; Assessment of the biological effects of RF exposure; Development of safety guidelines; Investigation of novel detectors of UWB hazards. References The effects of radiofrequency radiation on long-term potentiation in rat hippocampal slices. lain R. Scott and John E.H. Tattersall, 21st annual meeting. The Bioelectromagnetics Society, Long Beach 1999. An In Vitro Investigation Of The Bioeffects Of Pulsed Radiofrequency Radiation In Human And Animal Cells, C.D. Lindsay, R.H. Inns and S. Holden, 21st annual meeting The Bioelectromagnetics Society Long Beach 1999. Transgenic nematodes as biomonitors of microwave-induced stress. Daniells, C., Duce, I., Thomas, D., Sewell, P., Tattersall, J.E.H. and De Pomerai, D. Mutation Research, 399, 1998, 55-64. Effects of stress-inducing microwave radiation on life-cycle parameters in the nematode caenorhabditis elegans. De Pomerai, D., Barker, S.L., Duce, I.R., Thomas, D.W., Sewell, P.D. and Tattersall, J.E.H. Bioelectromagnetics Society Abstracts, 20, 1998, 174-5.
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Suppression of evoked and spontaneous field potentials by radio frequency radiation in rat hippocampal slices. Tattersall, J.E.H., Nettell, J.J. and Wood, S.J. Bioelectromagnetics Society Abstracts, 20, 1998,173. Effects of ultra-wideband microwave pulses on rat hearts in vitro. A.L. Bottomley, S.J.Neely, S.J. Wood, and J.E.H Tattersall, Second World Congress for Electricity and Magnetism in Biology and Medicine, Bologna, Italy, June 8-13 1997. The application of cell culture technology in the assessment of ultra-wideband electromagnetic field hazards. C.D. Lindsay, R.H. Inns and S.J. Holden DERA Port Down, UK EMC York 99, University of York, UK: 12 -13 July 1999.
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INDEX
Alp Azizoglu, S., 83 Altman, Z., 663 Andrieu, J, 175, 377 Astanin, I., 285 Badic, M., 407 Baranowki, S., 555 Baum, C., 39, 127, 351, 393, 415, 501, 605, 697 Beillard, B., 175, 377 Bigelow, W., 697 Bishay, S., 105, 115 Blumel, R., 731 Boerner, W., 493 Bohl, J., 431 Booker, S., 707, 715 Boryssenko A., 199, 291, 493 Boryssenko E., 291 Boryssenko, O., 199 Bowen, L., 299 Brennan, P., 707, 715 Broquetas, A., 545 Brylevsky, V., 335 Bullock, R., 707, 715 Burger, J., 393
Ehlen, T., 431 Fabregas, X., 545 Farr, E., 299, 697 Felsen, L., 21 Ferencz, C., 679 Ferencz, O., 679 Flerov, A., 343 Fontana, R., 215, 225 Fortuny, J., 545 Fourestié, B., 663 Frantsuzov, A., 343 Friedrich, J., 247 Galdi, V., 21 Gallais, F., 175, 377 Galstjan, E., 75, 467 Garbe, H., 561 Gaudet, J., 393 Gaudreau, M., 485 Gavrilov, S., 585 Geppener, V., 285 Gomez-Martin, R., 631 Gonzalez Garcia, S., 631 Griffiths, H., 527 Gu, J., 671
Carter, R., 423 Casey, J., 485 Chassay, G., 527 Cheney, M., 599 Cheong, H., 31 Chernyshov, E., 285 Cloude, S., 275, 493, 519 Corre, Y, 615 Crisp, G., 275 Crisp, G., 399
Haibel, A., 97 Hambrook, J., 739 Hannigan, A., 319 Hatfield, L., 437 Hayakawa, H., 53 Hernándex-López, M., 631 Heyman, E., 11, 67, 327 Holden, D., 739
De Pasquale, G., 275 Demoulin, B., 555
Iezekiel, S,, 647 Imbs, Y., 175, 377 749
750
INDEX
Immoreev, I., 207 Inns, R., 739 Isaacson, D., 599 Islam, N., 461 Iudin, D., 53 Ivashchuk, V., 199
Perov, A., 623 Peyerl, P., 247 Peyerl, P., 275 Poljak, D., 639 Prather, W., 299, 393, 697 Prokhorenko, V., 199, 291, 473
Jatzyn, A., 285 Jecko, B., 175, 377 Joshi, R., 461 Jull, E., 31
Ratcliffe, J., 275, 399 Rauschenbach, P., 247 Redfern, M., 259, 267 Rice, P., 739 Rossberg, M., 247 Rowley, R., 399 Rubio Bretones, A., 159, 167, 631 Rukin, S., 445
Kardo-Sysoev, A., 335, 343 Kazanskiy, L., 467 Kempkes, M., 485 Kone, L., 555 Korovkin, N., 53 Koshelev, V., 191, 311 Kostyleva, V., 285 Kristiansen, M., 437 Krompholz, H., 437 Kuhnke, R., 431 Lambert, A., 577 Lassas, M., 599 Le Goff, M., 175, 377 Lehr, J., 393, 415 Lelikov, Y., 335 Ligthart, L., 183 Lindsay, C., 739 Lishchenko, A., 199 Liu, S., 537 Lomakin, V., 67 Lopez-Sanchez, J., 545 Lostanlen, Y, 527, 615 Loubriel, G., 461 Luhn, F., 479 Lyubutin, S., 445 MacKay, A., 723 Mallepeyre, V., 175, 377 Mar, A., 461 Marinescu, M., 407 McGowen, A., 639 Merih Büyükdura, O., 83 Mulvaney, J., 485 Nerukh, A., 647 Nicolaev, V, 285 Nimtz, G., 97 O’Bryon, J., 423 Oughstun, K., 569, 687 Oulmascoulov, M., 445, 453 Payment, T., 235 Pennock, S., 259, 267, 319
Sachs, J., 247, 275 Sagues, L., 545 Sahli, H., 275 Salman, A., 585 Sami, G., 105, 115 Sato, M., 537 Schamlloglu, E., 461 Schantz, H., 89, 385 Schätzing, W., 479 Scheibe, H., 479 Scherbak, V., 335 Seida, O., 105 Selina, E., 53 Sencer Koç, S., 83 Shepherd, P., 319 Sherbatko, I., 647 Shipilov, S., 191 Shlivinski, A., 11, 327 Short, G., 437 Shpak, V., 445, 453 Shunailov, S., 445, 453 Sieber, A., 545 Sirenko, Y, 623 Skulkin, S., 371 Slovikovsky, B., 445 Smirnov, A., 285 Smirnova, L, 335, 343 Smith, P., 577, 707, 715 Sokolov, M., 285 Sonnermann, F, 431 Stahlhofen, A., 97 Steinberg, B., 67 Stone, A., 127 Sukhovetsky, A., 59 Svezhentsev, A., 59 Tattersall, J., 739 Taylor, C., 671 Taylor, J., 207 Tchashnicov, I., 335
INDEX Tham, C., 639 Thornhill, C., 399, 519 Tijhuis, A., 159, 167, 655 Torres, R., 393 Towers, M., 639 Trakhtengerts, V., 53 Tran, T, 393 Tretyakov, O., 143 Tuchkin, Y., 137, 153 Turchin, V., 371 Tyo, S., 363 Uguen, B., 527, 615 Uguen, B., 615 Van Bladel, J., 1 Van Genderen, P., 183 Vandenbosch, G., 59
751 Velychko, L., 623 Vertiy, A., 585 Voynovskiy, I., 585 Wollenberg, G., 479 Xiao, H., 569 Yakubov., V., 191 Yalandin, M., 445, 453 Yaldiz, E., 623 Yarovoy, A., 183 Younan, N., 671 Zange, R., 479 Zazulin, D., 335, 343 Zutavern, F., 461 Zwamborn, P., 655
