Principles of Planar Near-Field Antenna Measurements

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Principles of Planar Near-Field Antenna Measurements

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Geometrical theory of diffraction for electromagnetic waves, 3rd edition G.L. James Aperture antennas and diffraction theory E.V. Jull Adaptive array principles J.E. Hudson Microstrip antenna theory and design J.R. James, P.S. Hall and C. Wood The handbook of antenna design, volume 1 A.W. Rudge, K. Milne, A.D. Oliver and P. Knight (Editors) The handbook of antenna design, volume 2 A.W. Rudge, K. Milne, A.D. Oliver and P. Knight (Editors) Corrugated horns for microwave antennas P.J.B. Clarricoats and A.D. Oliver Microwave antenna theory and design S. Silver (Editor) Waveguide handbook N. Marcuvitz Ferrites at microwave frequencies A.J. Baden Fuller Propagation of short radio waves D.E. Kerr (Editor) Principles of microwave circuits C.G. Montgomery, R.H. Dicke and E.M. Purcell (Editors) Spherical near-field antenna measurements J.E. Hansen (Editor) Handbook of microstrip antennas, 2 volumes J.R. James and P.S. Hall (Editors) Ionospheric radio K. Davies Electromagnetic waveguides: theory and applications S.F. Mahmoud Radio direction finding and superresolution, 2nd edition P.J.D. Gething Electrodynamic theory of superconductors S.A. Zhou VHF and UHF antennas R.A. Burberry Propagation, scattering and diffraction of electromagnetic waves A.S. llyinski, G. Ya.Slepyan and A. Ya.Slepyan Geometrical theory of diffraction V.A. Borovikov and B.Ye. Kinber Analysis of metallic antenna and scatterers B.D. Popovic and B.M. Kolundzija Microwave horns and feeds A.D. Olver, P.J.B. Clarricoats, A.A. Kishk and L. Shafai Approximate boundary conditions in electromagnetics T.B.A. Senior and J.L. Volakis Spectral theory and excitation of open structures V.P. Shestopalov and Y. Shestopalov Open electromagnetic waveguides T. Rozzi and M. Mongiardo Theory of nonuniform waveguides: the cross-section method B.Z. Katsenelenbaum, L Mercader Del Rio, M. Pereyaslavets, M. Sorella Ayza and M.K.A. Thumm Parabolic equation methods for electromagnetic wave propagation M. Levy Advanced electromagnetic analysis of passive and active planar structures T. Rozzi and M. Farinai Electromagnetic mixing formulae and applications A. Sihvola Theory and design of microwave filters L.C. Hunter Handbook of ridge waveguides and passive components J. Helszajn Channels, propagation and antennas for mobile communications R. Vaughan and J. Bach-Anderson Asymptotic and hybrid methods in electromagnetics F. Molinet, I. Andronov and D. Bouche Thermal microwave radiation: applications for remote sensing C. Matzler (Editor) Propagation of radiowaves, 2nd edition L.W. Barclay (Editor)

Principles of Planar Near-Field Antenna Measurements Stuart Gregson, John McCormick and Clive Parini

The Institution of Engineering and Technology

Published by The Institution of Engineering and Technology, London, United Kingdom © 2007 The Institution of Engineering and Technology First published 2007 This publication is copyright under the Berne Convention and the Universal Copyright Convention. All rights reserved. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers at the undermentioned address: The Institution of Engineering and Technology Michael Faraday House Six Hills Way, Stevenage Herts, SG1 2AY, United Kingdom www.theiet.org While the authors and the publishers believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them. Neither the authors nor the publishers assume any liability to anyone for any loss or damage caused by any error or omission in the work, whether such error or omission is the result of negligence or any other cause. Any and all such liability is disclaimed. The moral rights of the authors to be identified as authors of this work have been asserted by her in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing in Publication Data Gregson, Stuart Principles of planar near-field antenna measurements. (Electromagnetic waves series ; v. 53) 1. Antennas (Electronics) I. Title II. McCormick, John III. Parini, Clive IV. Institution of Engineering and Technology 621.3’824 ISBN 978-0-86341-736-8

Typeset in India by Newgen Imaging Systems (P) Ltd, Chennai Printed in the UK by MPG Books Ltd, Bodmin, Cornwall

Contents

Preface

xi

1

Introduction 1.1 The phenomena of antenna coupling 1.2 Characterisation via the measurement process 1.2.1 Free space radiation pattern 1.2.2 Polarisation 1.2.3 Bandwidth 1.3 The organisation of the book 1.4 References

1 1 4 6 7 8 11 12

2

Maxwell’s equations and electromagnetic wave propagation 2.1 Electric charge 2.2 The EM field 2.3 Accelerated charges 2.4 Maxwell’s equations 2.5 The electric and magnetic potentials 2.5.1 Static potentials 2.5.2 Retarded potentials 2.6 The inapplicability of source excitation as a measurement methodology 2.7 Field equivalence principle 2.8 Characterising vector EM fields 2.9 Summary 2.10 References

13 13 14 16 18 24 24 24

Introduction to near-field antenna measurements 3.1 Introduction 3.2 Antenna measurements

35 35 35

3

28 28 30 33 33

vi

Principles of planar near-field antenna measurements 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

4

Forms of near-field antenna measurements Plane rectilinear near-field antenna measurements Chambers, screening and absorber RF subsystem Robotics positioner subsystem Near-field probe Generic antenna measurement process Summary References

Plane wave spectrum representation of electromagnetic waves 4.1 Introduction 4.2 Overview of the derivation of the PWS 4.3 Solution of the scalar Helmholtz equation in Cartesian coordinates 4.3.1 Introduction to integral transforms 4.3.2 Fourier transform solution of the scalar Helmholtz equation 4.4 On the choice of boundary conditions 4.5 Operator substitution (derivative of a Fourier transform) 4.6 Solution of the vector Helmholtz equation in Cartesian coordinates 4.7 Solution of the vector magnetic wave equation in Cartesian coordinates 4.8 The relationship between electric and magnetic spectral components 4.9 The free-space propagation vector k 4.10 Plane wave impedance 4.11 Interpretation as an angular spectrum of plane waves 4.12 Far-field antenna radiation patterns: approximated by the angular spectrum 4.13 Stationary phase evaluation of a double integral 4.14 Coordinate free form of the near-field to angular spectrum transform 4.15 Reduction of the coordinate free form of the near-field to far-field transform to Huygens’ principle 4.16 Far-fields from non-planar apertures 4.17 Microwave holographic metrology (plane-to-plane transform) 4.18 Far-field to near-field transform 4.19 Radiated power and the angular spectrum 4.20 Summary of conventional near-field to far-field transform 4.21 References

40 43 44 47 52 56 58 60 60 63 63 64 65 65 65 78 79 81 83 84 87 88 90 92 95 101 104 106 107 108 112 115 117

List of contents 5

Measurements – practicalities of planar near-field antenna measurements 5.1 Introduction 5.2 Sampling (interpolation theory) 5.3 Truncation, spectral leakage and finite area scan errors 5.4 Antenna-to-antenna coupling (transmission) formula 5.4.1 Attenuation of evanescent plane wave mode coefficients 5.4.2 Simple scattering model of a near-field probe during a planar measurement 5.5 Evaluation of the conventional near-field to far-field transform 5.5.1 Standard techniques for the evaluation of a double Fourier integral 5.6 General antenna coupling formula: arbitrarily orientated antennas 5.7 Plane-polar and plane-bipolar near-field to far-field transform 5.7.1 Boundary values known in plane-polar coordinates 5.7.2 Boundary values known in plane-bipolar coordinates 5.8 Regular azimuth over elevation and elevation over azimuth coordinate systems 5.9 Polarisation basis and antenna measurements 5.9.1 Cartesian polarisation basis – Ludwig I 5.9.2 Polar spherical polarisation basis 5.9.3 Azimuth over elevation basis – Ludwig II 5.9.4 Copolar and cross-polar polarisation basis – Ludwig III 5.9.5 Circular polarisation basis – RHCP and LHCP 5.10 Overview of antenna alignment corrections 5.10.1 Scalar rotation of far-field antenna patterns 5.10.2 Vector rotation of far-field antenna patterns 5.10.4 Rotation of copolar polarisation basis – generalized Ludwig III 5.10.5 Generalized compound vector rotation of far-field antenna patterns 5.11 Brief description of near-field coordinate systems 5.11.1 Range fixed system 5.11.2 Antenna mechanical system 5.11.3 Antenna electrical system 5.11.4 Far-field azimuth and elevation coordinates 5.11.5 Ludwig III copolar and cross-polar definition 5.11.6 Probe alignment definition (SPP) 5.11.7 General vector rotation of antenna radiation patterns

vii 119 119 120 121 125 136 137 138 139 143 148 150 151 156 159 159 160 161 163 165 169 169 171 173 174 175 176 177 178 178 178 178 179

viii

Principles of planar near-field antenna measurements 5.12

5.13 5.14 5.15 6

7

Directivity and gain 5.12.1 Directivity 5.12.2 Gain – by substitution method 5.12.3 Gain-transfer (gain-comparison) method Calculating the peak of a pattern 5.13.1 Peak by polynomial fit 5.13.2 Peak by centroid Summary References

Probe pattern characterisation 6.1 Introduction 6.2 Effect of the probe pattern on far-field data 6.3 Desirable characteristics of a near-field probe 6.4 Acquisition of quasi far-field probe pattern 6.4.1 Sampling scheme 6.4.2 Electronic system drift (tie-scan correction) 6.4.3 Channel-balance correction 6.4.4 Assessment of chamber multiple reflections 6.4.5 Correction for rotary errors 6.4.6 Re-tabulation of probe vector pattern function 6.4.7 Alternate interpolation formula 6.4.8 True far-field probe pattern 6.5 Finite element model of open-ended rectangular waveguide probe 6.6 Probe displacement correction 6.7 Channel-balance correction 6.8 References Computational electromagnetic model of a planar near-field measurement process 7.1 Introduction 7.2 Method of sub-apertures 7.3 Aperture set in an infinite perfectly conducting ground plane 7.3.1 Plane wave spectrum antenna–antenna coupling formula 7.4 Vector Huygens’ method 7.5 Kirchhoff–Huygens’ method 7.6 Generalized technique for the simulation of near-field antenna measurements 7.6.1 Mutual coupling and the reaction theorem 7.7 Near-field measurement simulation 7.8 Reaction theorem 7.8.1 Lorentz reciprocity theorem (field reciprocity theorem)

180 180 181 182 183 183 185 186 187 189 189 189 191 193 194 197 198 200 202 205 209 211 213 217 217 218 219 219 220 223 225 227 229 233 234 237 239 240

List of contents

7.9 7.10 8

9

7.8.2 Generalized reaction theorem 7.8.3 Mutual impedance and the reaction theorem Summary References

Antenna measurement analysis and assessment 8.1 Introduction 8.2 The establishment of the measure from the measurement results 8.2.1 Measurement errors 8.2.2 The sources of measurement ambiguity and error 8.2.3 The examination of measurement result data to establish the measure 8.3 Measurement error budgets 8.3.1 Applicability of modelling error sources 8.3.2 The empirical approach to error budgets 8.4 Quantitative measures of correspondence between data sets 8.4.1 The requirement for measures of correspondence 8.5 Comparison techniques 8.5.1 Examples of conventional data set comparison techniques 8.5.2 Novel data comparison techniques 8.6 Summary 8.7 References Advanced planar near-field antenna measurements 9.1 Introduction 9.2 Active alignment correction 9.2.1 Acquisition of alignment data in a planar near-field facility 9.2.2 Acquisition of mechanical alignment data in a planar near-field facility 9.2.3 Example of the application of active alignment correction 9.3 Amplitude only planar near-field measurements 9.3.1 PTP phase retrieval algorithm 9.3.2 PTP phase retrieval algorithm – with aperture constraint 9.4 Efficient position correction algorithms, in-plane and z−plane corrections 9.4.1 Taylor series expansion 9.4.2 K-correction method 9.5 Partial scan techniques 9.5.1 Auxiliary translation 9.5.2 Rotations of the AUT about the z-axis

ix 244 247 247 248 249 249 249 250 253 256 259 259 260 261 261 263 263 267 282 283 285 285 285 287 289 291 296 297 301 303 305 311 315 315 319

x

Principles of planar near-field antenna measurements 9.5.3

9.6 9.7

Auxiliary rotation – bi-planar near-field antenna measurements 9.5.4 Near-field to far-field transformation of probe corrected data 9.5.5 Applicability of the poly-planar technique 9.5.6 Complete poly-planar rotational technique Concluding remarks References

320 329 335 338 342 344

Appendix A: Other theories of interaction A.1 Examples of postulated mechanisms of interaction

347 347

Appendix B: Measurement definitions as used in the text

354

Appendix C: An overview of coordinate systems C.1 Antenna mechanical system (AMS) C.2 Antenna electrical system (AES) C.3 Far-field plotting systems C.4 Direction cosine C.5 Azimuth over elevation C.6 Elevation over azimuth C.7 Polar spherical C.8 Azimuth and elevation (true-view) C.9 Range of spherical angles C.10 Transformation between coordinate systems C.11 Coordinate systems and elemental solid angles C.12 Relationship between coordinate systems C.13 Azimuth, elevation and Roll angles C.14 Euler angles C.15 Quaternion C.16 Elemental solid angle for a true-view coordinate system

357 357 357 358 358 360 361 362 364 365 366 367 368 371 373 374 377

Appendix D: Trapezoidal discrete Fourier transform

380

Appendix E: Calculating the semi-major axis, semi-minor axis and tilt angle of a rotated ellipse

384

Index

389

Preface

So often, it is the very everyday nature of the physical phenomena around us that blind us to their universality and their importance, both in how we understand and use them, in our environment. The list of technological advances over the ages, engineered by exploiting these so-often ignored or unappreciated phenomena would run to a work of thousands of pages crammed with ingenuity, inventiveness and insight. Our entire technological society is riddled with examples of devices, tools and mechanisms that are based on the existence of these physical phenomena, designed and manufactured by engineering techniques based on, and exploiting, the fundamental physical laws that govern these phenomena. As with so many other of the technological wonders of the present day that are taken for granted, countless generations must have dreamt of gazing down on the dark side of the moon. Only a few decades separate us from that day when the crew of Apollo 8 were the first humans to see that sight, so permanently hidden from other humans, by a manifestation of one of the most universal of all observed physical phenomena, the coupling of harmonic systems. Every school boy and girl knows that despite their physical isolation, the harmonic oscillation of the earth rotating on its axis is coupled to the periodic orbit of the moon so that the same side of the moon always points towards the earth. Of course, what the crew of Apollo 8 saw was conveyed to us back here on the earth by making use of the same universal phenomena of coupled harmonic systems, except that in this case they were coupled electronic systems, as opposed to massive gravitational systems. No one who has studied electronic engineering, to any appreciable level, has escaped from hours spent in the pursuit of the solution to problems concerning the arrangement of resistances, capacitances and inductance in circuits, to produce harmonic systems that have in turn their associated resonant frequencies, Bode plots and Q factors. However, much of what is involved in the modern electronic technologies is based on the existence of harmonic circuits and the universally observed phenomena that these circuits couple together. By way of illustration, in essence, the entire field of electromagnetic compatibility (EMC) is an attempt to minimise the extent to which systems couple. Conversely, the fields of communications and radar

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Principles of planar near-field antenna measurements

in turn both involve attempts to maximise this coupling. So, the extent to which electronic systems interact, as a result of this coupling, is fundamental to large swathes of electronic engineering and therefore also to our modern technological society. This interaction can be minimised by using a variety of strategies, for example, physical isolation, screening or the judicious choice of systems components to separate resonant frequencies are all viable, but this coupling can never be completely removed. However, for many systems the existence of this coupling and its exploitation for the transfer of information in the form of a signal is imperative to the successful operation of the technologies. This of course means that for these technologies, techniques and components must be developed that maximise the coupling between harmonic electronic systems. Many strategies have been employed to maximise this coupling and the subsequent transfer of information between the systems. However, if the systems are physically isolated from each other and no fixed or transmission line can be established between them, as with the earth and the moon, the free space between them must be exploited as a medium to facilitate the interaction between these apparently isolated systems. The most commonly used strategy for enhancing the interaction between such isolated electronic systems is the inclusion of circuit elements within the electronic systems that enhance this interaction, these individual circuit components are usually referred to as antennas. It is not the purpose of this book to hypothesise or examine in great detail the mechanisms by which the interaction facilitated by the antennas between electronic systems occurs, although a variety of such mechanisms and the basis of their associated mathematical algorithms is briefly discussed in Appendix A. In this text, the interaction will primarily be described in terms of the propagation of an angular spectrum of transverse (to take account of polarisation) waves propagating in a non-dispersive medium, these waves being consistent with solutions to Maxwell’s equations. In a large variety of circumstances, this is a particularly successful algorithm for the description of the interaction in question, but the treatment, in this volume, will be such that other hypothesised interaction mechanisms and their attendant mathematical algorithms will not be precluded by the explanations introduced. One of the most common techniques adopted to characterise, predict and quantify this coupling between electronic circuits is to attempt to reduce the problem of circuit coupling to that of antenna performance. Thus, by characterising antennas in a known circuit configuration, the extent to which they enhance coupling in other situations can be predicted. This is the fundamental procedure adopted in antenna test ranges, where the inclusion of antennas in a configuration of two coupled circuits, usually referred to as the transmit, (Tx), and the receive, (Rx), circuits allows this measurement process to be performed. This means that the characterisation of the antennas in this circuit configuration can be used to predict the response of other circuit configurations that include the same antennas. The accurate characterisation of how the presence of antennas will affect the coupling of electronic circuits can be accomplished using a number of different range configurations, one of the most accurate being the antenna near-field range. This technique allows the characterisation of antennas where measurements are made in close

Preface

xiii

physical proximity to the antennas and thus these measurements can be performed in small highly controlled environments where extraneous noise and interference, mechanical, environmental and electromagnetic, can be eliminated or effectively suppressed. This means that highly stable, repeatable measurements from which the antenna characteristics can be extracted are possible. All measurement techniques have their limitations and ranges of applicability, not least the near-field antenna measurements. However, the necessary information required to inform and influence the design of systems in which antennas are used to enhance the coupling between electronic circuits can be obtained by the skilful and expert use of such antenna test ranges. Therefore, what is intended, in the following chapters, is an initial examination of the properties of antennas that allow them to enhance the free-space interaction of electronic systems. This will then be followed by the description of the theory of an effective, efficient and accurate methodology for characterising these properties using the antenna measurement technique of planar near-field scanning. This will be followed by a review of the practical implications of making such measurements in terms of techniques, instrumentation, processing and analysis of data. The utility of the planar methodology is then illustrated with example measurement campaigns. These include a discussion of the characterisation of high-gain instruments, electrically large reflector assemblies and planar array antennas along with the ability to transform back to array elements in the aperture plane, to confirm element excitations and to optimise the overall antenna performance. Some of the latest advances in such methodologies will be examined particularly with respect to the introduction of statistical image classification techniques that aim to assess the accuracy, sensitivity and repeatability of given data. These techniques are applicable to all types of antenna pattern, both measured and theoretical and so are of interest to a wide range of readers who undertake, or are required to interpret, antenna radiation pattern data. Finally, the most recent advances in the technique, which deal with the introduction of partial scan techniques based on auxiliary translations and rotations to produce poly-planar near-field data sets, will be described. This will involve an explanation of the measurement techniques, the assessment of the additional terms introduced in the error budget associated with the technique and the theoretical basis of the transforms developed to allow their deployment. A large number of facilities exist worldwide and the poly-planar near-field technique will be of interest to current planar near-field users, as it enables the maximum size of the antenna that can be measured in a given facility to be significantly increased. In summary, the volume will provide a comprehensive introduction and explanation of both the theory and practice of planar near-field measurements, from its basic postulates and assumptions, to the intricacies of its deployment in complex and demanding measurement scenarios. The International System of Units (SI) is used exclusively. Numbers in parenthesis ( ) denote equations while numbers in brackets [ ] denote references. Underlined quantities are vectors. Our thanks goes to the many individuals who generously gave assistance, advice and support. We gratefully acknowledge the invaluable suggestions, corrections and

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Principles of planar near-field antenna measurements

constructive criticisms of the many people who gave freely of their time to review the manuscript at various stages throughout its preparation. However, any errors or lack of clarity must remain the responsibility of the authors, who would welcome any and all such mistakes being brought to their attention. The authors are grateful to their wives (Catherine Gregson, Imelda McCormick and Claire Parini) and children (Elizabeth Gregson and Robert Parini), whose unwavering understanding, support, encouragement and good humour were necessary factors in the completion of this work. A special vote of thanks must be devoted to Catherine for her tireless work on the manuscript. We also thank the companies and individuals who generously provided copyright consent. There are many useful and varied sources of information that have been tapped in the preparation of this text; however, mention must be made of four books that have been of particular help to the authors and will be referred to throughout, but in no particular order. H.P. Hsu, Applied Fourier Analysis, Harcourt Brace College Publishing H. Anton, Calculus with Analytic Geometry, John Wiley & Sons, Inc M.R. Spiegel, Theory and Problems of Vector Analysis and an Introduction to Tensor Analysis, Schaum Publishing Company R.H. Clarke and J. Brown, Diffraction Theory and Antennas, Ellis Horwood Ltd Although the nomenclature and development of the theory of planar near-field measurements as presented within this text has not followed that of the National Institute of Standards and Technology (NIST) format, the technical publications originating from that organisation have been a rich source of valuable information. In particular, D.M. Kerns, Plane wave Scattering-Matrix Theory of Antennas and AntennaAntenna Interactions, Monograph 162, National Bureau of Standards A.C. Newell, Planar Near-Field Antenna Measurements, Electromagnetic Fields Division, National Institute of Standards and Technology, Boulder, Colorado Stuart Gregson, John McCormick and Clive Parini London and Edinburgh

Chapter 1

Introduction

1.1

The phenomena of antenna coupling

This first brief chapter will describe the phenomenological basis of antenna measurements and will attempt to set out the processes and techniques that will be developed in this text against the background of what can actually be observed about the action of antennas as circuit elements. As is abundantly clear from the title, this volume has been penned for a very specific purpose; to explain clearly, concisely and in an understandable form the theory and practice of planar near-field antenna measurements. Again, as stated in the preface, to do this the volume will confine itself to considering the radiative coupling between electronic systems in free space for a number of very sound reasons. First, in almost every practical engineering circumstance this is the mode in which antennas are utilized. If coupling between systems that are not physically separated by large distances is required various forms of transmission lines can be utilized, however, large separation distances almost invariably require the use of antennas. Communication systems contain transmit (Tx) and receive (Rx) subsystems, which use at least two antennas, and broadcast systems typically use considerably more than two antennas. Radars may, or may not, use the same antenna for their Tx and Rx subsystems and the coupling may well be profoundly affected by the scattering from some target, but in essence we are still considering coupling between electronic subsystems. Careful consideration of all electronic systems that utilize antennas as components reveal that it is the extent of this coupling that is fundamental to their operation. Replace Rx antenna with other transducer, for example, bolometer for power detection, rectifier for rectenna (power transmission by microwave signal) and you have summed up almost every possible engineering circumstance except for those systems designed to detect transmissions from naturally occurring radiation sources, for example, radiometers.

2

Principles of planar near-field antenna measurements

Second, the fundamental electromagnetic (EM) properties of antennas are very limited. If you place a passive antenna in an incident field it will scatter energy and dependant on the power flux incident it may get a little hot and that is about all. It is a common fallacy to assign properties to antennas that actually belong to the systems within which they are embedded. By way of example, antennas are often described as being reciprocal devices, in fact by definition reciprocity is a property that can only be applied to objects that are either sources or sinks for energy [1]. Thus, it is correct to say that electronic systems that are reciprocal, for example, those that include generators and loads and satisfy the usual requirements of isotropy, will not have this property affected by the inclusion of passive antennas within them. However, to talk of antennas in the absence of well defined terminals, where load impedances and source voltages are attached, as being reciprocal is to misinterpret the concept of reciprocity. In fact, almost all of the properties usually assigned to antennas, and which are of engineering importance, are in fact properties of systems containing antennas. Antennas do not by themselves transmit or receive energy across free space and they are simply bidirectional transducers that can be included as circuit components in electronic systems. It is those systems that include antennas as components that have the emergent property that they can radiatively couple power across large ranges in free space. Thus the characterisation of antennas via measurements to quantify their performance is really the characterisation of highly specified electronic systems containing Tx and Rx antennas that can be used to predict the response of other systems in which the same or similar antennas will be utilized. Finally, as stated in the preface, this volume will not attempt to hypothesise or examine the variety of postulated physical mechanisms by which the interaction, which is optimised by the antennas, between the electronic systems occurs. However, as it will attempt to develop a thorough theoretical explanation of near-field antenna measurements it must, at least, be consistent with physical law. Very little that is encountered within the discipline of electronic engineering is of a more relativistic nature than the operation of antennas. By definition, the interaction facilitated by the presence of antennas occurs at the speed of light and thus relativity cannot be completely ignored when considering the action of antennas. In this text only Tx and Rx antennas that are in translational equilibrium will be considered and their velocities relative to each other will be specified to be zero. Additionally, the EM interactions will be observed from an inertial reference frame coincident with the fiducial mechanical datum of the Tx antenna. These conditions make it possible to consider the antenna characterisation without reference to any relativistic effects associated with a multiplicity of reference frames or to any known inertial effects. This simplifies the explanations, without invalidating them in more complex situations and makes it convenient to consider the measurement process in terms of classical electromagnetic field theory (EFT) based on the work of the famous Scottish physicist, James Clerk Maxwell. Maxwell’s equations, whose mathematical form as we know them today, have much to do with simplification provided by the renowned German experimentalist Heinrich Hertz, are themselves relativistically invariant. Therefore, the solution of the Helmholtz wave equation that will be developed within the text is automatically compliant with this facet of physical law.

Introduction

3

Thus it is largely possible to develop a consistent theoretical explanation of near-field antenna measurements on the basis of classical EFT. However, classical EFT has limitations built into its basic structure and, as with all other postulated mechanisms for the interaction of antennas across space and time, will produce inconsistent answers when used outside of its range of applicability. One of the most striking limitations in the theory becomes manifest when dealing with any single antenna. Calculations of the power required to accelerate the electrons within antennas and to produce currents that provide values for the energy and momentum transferred to these distributions of electrons can be achieved fairly easily. However, if the amount of energy and momentum transferred to or from the EM field as a result of these accelerations is calculated, it is found that the values calculated from circuit theory and the values calculated from field theory do not agree. Thus the radiation resistance of the antenna will define an equivalent resistance required to dissipate a given amount of power that is not equal to the power transferred to the EM field around the antenna, meaning that, for an individual antenna, energy and momentum are not conserved locally. Globally for the level of energy to fall at one point in a space, it must increase elsewhere; for energy to be conserved there must therefore be a change in the energy distribution and density in the space. This rate of change of energy with respect to time, dE/dt = power will, as a result of the requirement for local conservation, be manifest as a flux between the regions in the space where the energy density is changing. Thus the existence of an energy density at a point in space, in the absence of a flux, does not imply the development of power, an important factor when consideration is given to the near-fields of any antenna. This local aspect of conservation requires that the power dissipated in the antenna via its radiation resistance be equal to that developed in the flux around it and this is not the case for calculations based on simple classical EFT. Therefore, the energy and momentum dissipated from the antenna does not equate to that in the flux propagating away from the antenna. Nothing could be more at odds with all of classical physical science than the concepts that, the law of energy conservation in time and the law of momentum conservation in space, can be violated. Many theoretical mechanisms have been postulated to account for this apparent anomaly, some of which invoke the concept of non-point-like charged particles being involved, for example, Poincare stresses [2]. Others involve the retention of the advanced wave solution to the Helmholtz wave equation, most famously the Wheeler Feynman absorber theory of radiation [3]. The absorber theory is particularly attractive as it is by definition relativistically invariant, as will be expanded upon in Chapter 4 and work extending and eliminating some of its limitations by John Cramer at Washington State University in Seattle [4] has made it also applicable within the realms of quantum as well as classical mechanics. However, making use of the reciprocal nature of systems including antennas allows the consideration of systems where the power launched into the EM field is coupled out by the presence of the same or other antennas. Therefore, the local power imbalance identified in a system containing one antenna will be eliminated when considering any set of systems where energy is coupled into then out of free

4

Principles of planar near-field antenna measurements

space. Thus any physical system containing a Tx antenna and at least one Rx antenna will satisfy the local conservation laws of energy and momentum for the systems containing the antennas. In summary • •

Almost every imaginable engineering situation will involve coupling between Tx and Rx antennas (this coupling is what engineers are actually interested in). Almost every possible measurement scenario will involve both Tx and Rx antennas (it is what can actually be measured).

Finally, classical electromagnetism is applicable in situations that involve both Tx and Rx antennas (it is a situation that can actually be described rigorously and accurately using theory). For all these reasons this volume will confine itself to antenna measurements designed to characterise the radiative coupling between electronic systems in free space, a homogeneous linear isotropic dielectric medium, facilitated by the presence of antennas within the systems, but what exactly do we mean by measurements?

1.2

Characterisation via the measurement process

Measurement is often defined as follows: ‘The quantitative determination of a physical magnitude adopted as a standard or by means of a calibrated instrument. The result of a measurement is thus a numerical value expressing the ratio between the magnitude under examination and a standard magnitude regarded as a unit’ [5]; a definition originally made popular by Lord Kelvin in the ninteenth century. However, the passage of years resulting in an increased level of understanding of the physical process of measurement, along with the introduction of systems centred concepts and the development of information theory, has led to a more generalized view of the measurement process. This gives an information conversion definition of measurement as follows: Consisting of information transfer with accompanying energy transfer. Energy cannot be drawn from a system without altering its behaviour, hence all measurements affect the quantity being measured. Measurements therefore are a carefully balanced combination of physics (energy transfer) and applied mathematics (information transfer) [6].

This information extraction concept is particularly applicable to antenna measurements where the test range holds many similarities to a communications system. Although the procedure involves a Tx signal, which initially contains no information, as it is completely predictable, information can be extracted from the input Tx signal via its comparison with the output Rx signal. Whatever conceptual model of the measurement process is adopted, as will be highlighted in this volume, the raw data from any near-field antenna measurement must be processed to provide predictions of the parameters of the antenna that are required. Thus the measurement process must be consistent with the formal mathematical definition of measurement [7], which is included for completeness as Box 1.1.

Introduction

5

Box 1.1 For a well-defined, non-empty class of extra-mathematical entities Q, let there exist upon that class a set of empirical relationships R = {R1 , R2 , . . . , Rn }. Let us further consider a set of numbers N (in general a subset of the set of real numbers Re) and let there be defined on that set a set of numerical relationships P = {P1 , P2 , . . . , Pn }. Let there exist a mapping M with domain Q and a range in N , M : Q → N which is a homomorphism of the empirical relationship system < Q, R > and the numerical relationship < N , P >. The triplet P =< Q, N , M > constitutes a scale of measurement of Q. It is required that M be a well-defined operational procedure, it is called the fundamental measurement procedure of Q. ni ∈ N the image of qi ∈ Q under M will be denoted by ni = M (qi ), ni will be called the measure of qi on the scale P, qi the measurand and Q the measured class. There will be in general other procedures of mapping Q onto N denoted by M  : Q → N such that M (qi ) = M  (qi ) either for all qi ∈ Q or qi ∈ Q where Q ⊂ Q any such procedure is a measurement procedure Q or Q on the scale P.

The statistical nature of establishing the measure in any practical circumstance from the measurements (measure [Box 1.1] empirically being defined as the limit in the behaviour of the measurement procedure as the number of measurement trials tends to infinity) will be discussed further in Chapter 8. So, whether the measurement process is thought of in purely physical terms or also as the extraction of knowledge from a system containing information, it is clear that there must be a very well-defined and controlled measurement procedure. Additionally the process must have a clearly defined measurand and a recognisable scale or standard against which the response or behaviour of the system during the measurement procedure can be assessed. For near-field antenna measurements the measurand is the power coupled into the Rx system, the source of both the energy and information bearing signal transferred in the process. The scale is relative to a reference signal, which can be calibrated against a standard and the measurement procedure is the stated subject of this volume, near-field antenna measurements. Having defined exactly what the measurand in the measurement system is, how can measurements of the power coupled between such systems be used to characterise the antennas embedded within them and give a measure of the antennas’ performance in another circumstance. In electronic engineering, the interaction between such systems is usually described in terms of a number of useful design parameters that are ascribed to the Tx and Rx antennas. The most important free space parameters are considered next.

6

Principles of planar near-field antenna measurements

1.2.1 Free space radiation pattern Antennas do not radiate equally in all directions, the concept of an isotropic radiator is useful as a standard relative to which any other antenna’s performance can be quantified, but in theory and practice it is impossible to construct. Therefore the variation in the ratio of the radiated power, as a function of angle relative to the fixed mechanical datum of any antenna is an important parameter. Figure 1.1 illustrates a coordinate system against which this variation can be judged with the z or φ axis conforming to the mechanical datum, often referred to as boresight, of the Tx antenna. Figure 1.2 illustrates the relative angular position of the Tx and Rx systems as Rx moves around a circular path at a fixed value of φ with θ varying along the circular path and with Tx at its centre. For any sufficiently large fixed value of |R| the extent to which power is transferred between the two antennas relative to the value at θ = 0 would vary as a function of the angle θ. Clearly this variation in relative power would also be a function of the angle φ so, the so-called radiation pattern of the antenna would actually be a function of θ and φ the angles that define the direction of the displacement R, the path between Tx and Rx antennas. The pattern function is an important parameter of any antenna and, assuming the magnitude of R, the distance between the two antennas is large, for two such antennas the variation along a segment of the circular path shown in Figure 1.2 could be of the form shown in Figure 1.3. Where, the maximum recorded signal is normalized to unity, zero on a dB scale. The measurement of absolute levels of power coupled or the levels relative to calibration standards are also possible but again detailed discussion of this will be delayed until a simple model of radiating structures is developed in Chapter 2. There

Theta axis

yAMS

θ φ Chi axis

kAMS

φ

θ zAMS Phi axis

Figure 1.1

xAMS

Illustrating the antenna coordinate system, where the square represents the antenna aperture

Introduction

7

YAMS ZL R

Zg

θ

~

ZAMS

Figure 1.2

Orientation of Tx and Rx antenna, including all circuit components

10 0

dB

–10 –20 –30 –40 –50 –60 –60

–50

–40

–30

–20

–10

0

10

20

30

40

50

60

Theta (degrees)

Figure 1.3

Recorded power normalized to zero as Rx is moved around the circle

also, those concepts relevant to the characterisation of antennas as circuit elements that cannot by definition be measured by, but are relevant to scanners, will be addressed. These include scattering parameters and their relevance to definitions of gain in terms of accepted as opposed to delivered power in any circuit and their possible use in scattering matrix descriptions of Tx and Rx antennas.

1.2.2 Polarisation Having measured this pattern function you might be forgiven for assuming that you now know everything about the angular variations in coupled power between the Tx and Rx systems. However, for any position of Tx and Rx, where either of the antennas is to be rotated about their mechanical datum (that is, about φ axis for the Tx antenna as per Figure 1.1), a variation in the amount of power coupled as a function of φ would be observed. This variation is ascribed to the polarisation of the antenna and various polarisation bases that can be used to describe this polarisation will be developed over the course of the text. Figure 1.4 being a typical measured response of what would be termed a linearly polarised Rx antenna’s response if Tx had the same so called

Principles of planar near-field antenna measurements

dB normalized

8

3 0 –3 –6 –9 –12 –15 –18 –21 –24 –27 –30 –33 –36 –180 –150 –120

–90 –60

–30

0

30

60

90

120

150

180

phi (degrees)

Figure 1.4

The variation in coupled power as a function of φ

polarisation. This concept of polarisation will be further examined in the light of the development of the Helmholtz wave equation developed in succeeding chapters.

1.2.3 Bandwidth This is the range of frequencies, (f ), in hertz over which the antenna is effective in facilitating the EM interaction. Figure 1.5 illustrates a plot of the measured power in a receiver as a function of frequency for boresight Tx and Rx, where a 3 dB and 10 dB bandwidth for an antenna is marked. For full characterisation such a bandwidth response would of course be required for each combination of pattern and polarisation. By systematically varying the position, orientation and frequency of excitation of the Tx antenna relative to the Rx antenna in terms of the parameters θ, φ, f and R, it is possible to characterise the interaction between the Tx and Rx subsystems in terms of, pattern, polarisation and bandwidth. However, one other additional class of measurement that is directly related to the action of antennas as initiators of harmonic coupling between electronic circuits can be included. The basis of all free space antenna measurement techniques and indeed much of electromagnetism, is the assumption that the antennas under test (AUTs) and the systems used in any test procedures behave in a linear fashion; in fact lack of linearity will be a source of uncertainty in the measurement process and that will be referred to in Chapter 8. Such linear systems can be described by linear differential equations such as Equation (1.1), shown below cn ·

dn f2 (t) dn−1 f2 (t) df2 (t) + c0 · f2 (t) = f1 (t) + c · + · · · + c1 · n−1 n n−1 dt dt dt

(1.1)

Introduction

9

0 –3

dB

–6 –9

–12 –15 –18 –21 8

8.5

9

9.5

10

10.5

11

11.5

12

Frequency (GHz)

Figure 1.5

Showing normalized pattern at θ = φ = zero from 8 to 12 GHz

For any simple system that can support harmonic oscillations this can be truncated to a second-order equation of the form df2 (t) d2 f2 (t) (1.2) + c0 f2 (t) = f1 (t) + c1 · 2 dt dt Where the constants and functions in the equation can be related to the usual circuit parameters of capacitance (C), inductance (L), voltage(V ), charge (q) and resistance (R), to give c2 ·

d2 q(t) 1 dq(t) + · q(t) = V (t) (1.3) +R· 2 dt C dt This is an equation relating the circuit parameters, with a harmonic solution that will be familiar to any student of alternating current (a.c.) theory. However, such an equation is inadequate to describe the harmonic solutions present in a circuit at radio or microwave frequencies. If a voltage is applied to such a circuit this voltage will be propagated through the circuit at approximately the speed of light, 30 billion m/s. Thus at 50 Hz this will produce a spatial harmonic variation in the circuit voltage, which will be cyclic over some 6 million metres. Therefore, for any circuit harmonically oscillating in time at 50 Hz, it is reasonable to assume that the voltage and currents are constant at all points in the circuit at any specified time. However, at the microwave frequency of 10 GHz the associated cyclic spatial variation of the currents and voltages in any circuit will be repetitive over a distance of the order of 3 cm. At this frequency, assuming the circuit itself is at least of the order of a few centimetres in length, the currents and voltages in that circuit will vary harmonically both as a function of when and where they are observed. A harmonic system in which such oscillations are functions of space and time will be described by a partial differential equation. Such an equation that linearly relates L·

10

Principles of planar near-field antenna measurements

the rate of change of the rate of change of the variables with respect to time, to the rate of change of the rate of change of the same variables with respect to space is a wave equation. This means that measurements of the instantaneous power made at different points in the circuit at the same time will give different results and these results can be related to provide a measure of the relative phase of the harmonic oscillation at the different points in the circuit. These measurements, which can be used to assign a phase to the harmonic coupling, are the other additional class of measurement that can be made on antennas. As will be discussed in Chapter 3, these measurements are the source of the in phase and at quadrature data that will be fundamental to the near-field measurement process. All of the above measurements can be made in an effort to characterise the coupling between antennas However, one point that has been briefly mentioned will need further explanation. Figure 1.2 shows an Rx system including antenna placed at a position along the circular path and Figure 1.3 shows the variation in the antenna pattern with angle θ. The text then goes on to state that provided |R| is large enough, the pattern will just be a function of θ and φ, but how large must |R| be for this to be true? When |R| is small, of the order of a few wavelengths in free space, the extent of coupling between the circuits is profoundly affected by the instantaneous distribution of charge on the surfaces of the antennas. The ratio of the power coupled is strongly dependent on |R|. This is the so-called reactive region around an antenna where reactive coupling dominates radiative coupling. As the distance |R| increases, the power coupled between the circuits is no longer dominated by this charge distribution – Figure 1.6 illustrates such a situation. In Figure 1.6 the displacement from Tx to Rx is again labelled R but many displacements, for example, R are also paths between the antennas, meaning that in the situation illustrated, there is no unique path between the two antennas. Only when |R| is infinitely large will the displacements R and R be effectively the same. Therefore, only when |R| is infinitely large can we define a unique path with a definite length, where all parts of the two antennas are effectively at equal distances apart; when this is so, the two antennas are in each other’s far-field. In practice, since in the vast YAMS Z R′ R Θ

Zg

~

Z AMS

Figure 1.6

Two of the possible paths between the Tx and Rx antenna

Introduction

11

majority of engineering situations we are concerned with antennas that are at large but finite distances apart, the far-field is defined as being when all parts of the Tx antenna are effectively at the same distance from the Rx antenna. At this distance the angular field distribution is essentially, but not strictly, independent of |R|. The distances at which |R| can be considered to be large enough to define a far-field region will be examined in the following chapters when a mechanism of interaction based on classical EFT will be expounded. Finally, that region in which the coupling is dominated by radiation but the distance |R| is not sufficiently large to uniquely define a single path is called the radiating near-field; this is the region surrounding the antennas that will be, in many senses, the practical focus of this text. In engineering situations, as a result of how they are employed, almost invariably it is the far-field performance of antennas that is of interest. However, for practical reasons that will be examined in this text, the ability to make near-field measurements that can be used to predict far-field parameters is extremely important in antenna engineering. This short description of the observables and hence measurable parameters associated with antennas concludes the initial phenomenological description of the action and characterisation of antennas. In the rest of the following chapters a particular model explaining the interaction between Tx and Rx systems via the application of classical EFT will be developed and used to explain and examine antenna measurements, procedures and analysis [8].

1.3

The organisation of the book

The mathematical nature of the predictive algorithms associated with theory will be rigorously examined and included in the text. However, if the reader wishes to take as read certain of the key assumptions and results, he will find that full derivations are only included within numbered boxes such as Box 1.1, which mathematically defines measurement. The arguments put forward in the text can be followed without recourse to these numbered boxes on an initial or subsequent examination but they are included to provide a full, complete and rigorous explanation within the body of the text. This separation of much of the fundamental theory behind the plane wave spectrum and its use in near-field scanning also has other advantages. It allows these derivations to be utilized out with the main thrust of the text, and makes them readily available to the reader who is less interested in the measurement process and more concerned with plane wave techniques and its application in other engineering areas. Throughout the text, as the arguments developed move from the theoretical nature of interaction of antennas to the reasons behind the choice of parameters (which are chosen to characterise antennas along with the implications and practicalities of antenna measurement procedures), the text will continue to be rigorously illustrated described and explained. Along with the complete mathematical development of the theory of near-field measurements, which the reader can choose to follow or take as assumed from the contents of the boxes, the text will attempt to inform and advise on the practical implications of the use of near-field antenna measurements. This will

12

Principles of planar near-field antenna measurements

extend to the assessment of near-field measurement data as an input to engineering tools and the development of practical methodologies for the analysis of the results of such measurements. Chapter 2 will mainly concern itself with the fundamental relationship between field and charge; this forms the basis for Chapter 3, which will introduce the nearfield scanning technique. Chapter 4 examines the theory of the plane wave spectrum, the theoretical basis of the near-field to far-field processing concept and Chapter 5 deals with the practicalities of near-field measurements. Chapter 6 explains the nature of and requirement for probe characterisation in near-field scanner measurements, while Chapter 7 develops effective modelling concepts that can be used to assess any near-field scanning procedure. Chapter 8 describes a representative theory of measurements and how the impact of this theory on the accuracy of measured data sets can be assessed. Finally, while throughout attempting to develop consistent logical explanations of all the relevant aspects for near-field scanning, the very latest developments in near-field scanning will be discussed and the bases of a new poly-planar technique will be established and explained in Chapter 9. However, the explanations of these methods and techniques will have to be delayed until a model of radiating structures – and how the near-field scanning measurement procedure relating to it – based on EFT is developed and explained.This is the subject of the following chapters in this volume.

1.4

References

1 Olver, A.D.: The Handbook of Antenna Design, Vol. 1 (Peter Peregrinis Ltd, UK, 1986), pp. 11–12 2 Feynman, R.P.: The Feynman Lectures on Physics, Vol. 2 (Addison Wesley Publishing, Reading, MA, 1964), pp. 28.1–28.10 3 Wheeler, J.A., and Feynman, R.P.: ‘Interaction with the absorber as the mechanism of radiation’, Reviews of Modern Physics, April–July 1945;17(2 and 3):157–81 4 Cramer, J.G.: ‘The transactional interpretation of quantum mechanics’, Reviews of Modern Physics, July 1986;58:647–88 5 Scrivenor, P. (ed.): New Caxton encyclopaedia (Caxton Publishing Company, London, 1964) 6 Stein, P.K.: Measurement Engineering, 1st edn (Stein Engineering Services, Phoenix, AZ, USA, 1964) 7 Sydenham, P.H.: Measuring Instruments Tools of Knowledge and Control. (Peter Peregrinus Ltd in Association with Science Museum London, Steleaus UK and New York, 1979) 8 IEEE Standard 145-1993 (Revision of IEEE Standard 145-1983), IEEE Standard Defintion of Terms for Antennas. Sponsor Antenna Standards Committee IEEE of the Antennas and Propagation Society. Approved 18 March 1993 IEEE Standards Board

Chapter 2

Maxwell’s equations and electromagnetic wave propagation

2.1

Electric charge

It is an empirical fact that under investigation electric charge appears to exist in two forms, usually but not exclusively referred to as positive and negative. This chapter will attempt to explain the action of antennas in terms of an explanation of the nature of the interaction of these different types of charge when they are in motion within the structure of an antenna. This explanation of this interaction will be developed via the concept of the electromagnetic (EM) field. The text in the main attempts to confine itself to the classical representations of field concepts, although as antenna theory concerns the propagation of EM energy between physically remote antennas at the speed of light it is not possible to completely ignore the relativistic aspects of antenna theory. This does not invalidate the approach that will be adopted as the principle equations used, based on the seminal work of James Clerk Maxwell, are, as will be seen in the text, themselves relativistically invariant in form. However, as the concept of relative motion is fundamental to any understanding of the concept of magnetism, it can be helpful to bear in mind during any descriptions the following a priori principles of special relativity, these being: •

The principle of the constancy of the speed of light, which states that the speed of light in vacuum will be measured to be c ≈ 3 × 108 m/s in all inertial frames of reference, irrespective of the state of motion of the frame. • The principle of relativity, which states that the laws of physics can be expressed in the same form in all inertial frames of reference. Therefore, any description or explanation of any physically observable phenomena that is not invariant between inertial reference frames is not consistent with physical law.

14

Principles of planar near-field antenna measurements

Only very occasionally within the arguments constructed within this text will it be necessary to return to these principles but without them being explicitly stated it is unclear that they do in fact underpin the entire theory of classical electromagnetism within the framework of physical law. Another empirically established fact is the conservation of electric charge. This means that electric charge can be neither created nor destroyed. Thus any change in its distribution within space must involve the motion of charged particles. This can be summed up by a continuity equation ∇· j+

∂ρ =0 ∂t

(2.1)

where ρ = the charge density and j = the current density. It can be seen from (2.1) that any change in charge density within a volume constitutes a current density flowing out of or in to that volume.

2.2

The EM field

An EM field can be thought of as constituting that ‘state of excitement’ induced in space by the presence of a, possibility time dependant, distribution of electric charge that has the potential to act on other charges if they are present within the field. The action on any test charges, [1] present will be such that it alters or tends to alter, their state of motion. Although there is only one field, the EM field associated with charge distributions, historically has been split into the concept of an electric and a magnetic field due to the different circumstances under which both are most easily observed. The electric field E can be expressed in terms of the Coulomb force law as q1 · q2 R (2.2) F= 4πR2 where F = force that acts on q1 , by definition equal and opposite to that which acts on q2 , R = distance between charges q1 and q2 , R = unit displacement vector in the direction defining the displacement between q1 and q2 , q1 and q2 = two distinct point like distributions of charge. Thus F = q1 E where E is defined as q2 R E= 4πR2

(2.3)

(2.4)

Thus in this case we are defining E as the field producing the force acting on q1 , that is, the field produced by the presence of q2 . These simple formulas quantify the forces acting on any stationary point charge, a test charge, at a point in space where

Maxwell’s equations and electromagnetic wave propagation

15

the field E is present. Where force is itself defined as that which alters or tends to alter the motion of bodies. Of course, as a direct result of the principle of relativity and the equivalence of inertial reference frames all states of transitional equilibrium are equivalent. Therefore it must be possible to define the force that acts on the test charge even if it is initially in a state of uniform motion, as opposed to being stationary, in a similar fashion. However, although a distribution of separate charges may be individually in translational equilibrium they may be in motion relative to each other and therefore there will be no inertial frame relative to which all of the charges constituting the distribution are stationary. To take account of this relative motion a second vector B is defined. This vector relates the force that acts on the test charge in the presence of fields at the points in space that the charge instantaneously occupies when it is in motion. From experiment, it is found that these fields apply a force F on the charge q moving with velocity v which is given by the Lorentz force law as
 F =q E+v×B

(2.5)

Clearly as v → 0 the Lorentz force law tends to the Coulombic formulation. In this formulation relativistic effects can be taken into account by modifying the mass of the particle so that it becomes a function of the relative velocity and the rest mass. Additionally, the vectors E and B will vary as a function of the inertial state of any observer. However, a range of Lorentz invariant parameters, for example E · H [2], can be defined which allow transformations of the E and B fields between inertial reference frames. This allows the Lorentz force law to be considered fundamental law of physics and to act as the definition of the vector quantities E and B. Another inevitable consequence of special relativity, with respect to the principle of the constancy of the speed of light, has to be taken into account in any theory that attempts to describe a mechanism for the interaction of physically remote antennas. At any point in time the field produced at a point in space remote from the charge distribution, that is, its source, is not equal to the field that would be produced by the charge distribution at that instant in time. Since time elapsed is equal to distance divided by speed it is in fact the field that would be created by the charge distribution; a period of time equal to the magnitude of the displacement of the test charge from the source divided by the speed of light. Thus at any point in time the field at a point in space mirrors the charge distribution that was present at a point in the past equal to the magnitude of the displacement from the source divided by the speed of light. This in turn means that the effect of any change in the charge distribution will take a finite amount of time to act on the test charge. Therefore any change in the field will be retarded by a period of time directly proportional to the magnitude of the displacement of the test charge from the source. The concept that the finite velocity of propagation retards the effects of the variation of any field source across space is crucial to the development of classical EM field theory. This is especially important when electromagnetism is framed in terms of a theory of potentials that will also be retarded by the constant but finite speed of light.

16

Principles of planar near-field antenna measurements

2.3

Accelerated charges

Figure 2.1 illustrates the well-known tool for the representation of electric fields in free space, field lines [3]. It shows a positive charge situated at point A at time t1 . As illustrated in Figure 2.2 the charge is accelerated to point B at a time t1 + τ/2. It is then moved back to point A at a time t1 +τ and Figure 2.3 illustrates the pattern of field lines around the point charge after it has arrived back at A. If a further period of time equal to τ is allowed to elapse with the charge held stationary a situation similar to Figure 2.4 would be observed. Clearly if the point charge was subject to alternating, sinusoidal, time-harmonic displacements between points A and B an arrangement of field lines similar to Figure 2.5 could be expected.

A

Figure 2.1

Fields lines around an isolated positive charge

A

Figure 2.2

B

B

Curvature of field lines associated with accelerating charge

Maxwell’s equations and electromagnetic wave propagation

A

Figure 2.3

B

Curvature of field lines associated with propagation after t1 + 2τ

A

Figure 2.5

B

Curvature of field lines associated with zero total displacement at t1 + τ

A

Figure 2.4

17

B

Field lines associated with propagation of harmonic displacement

18

Principles of planar near-field antenna measurements

From the Figures 2.1 to 2.5 it can be seen that accelerated motion of a charge will result in curvature in the field lines (those areas of the figures that are shaded in grey). As a result of the finite speed of propagation the retardation of the transverse disturbance of the field can be seen to radiate outwards with a speed of c. It will be shown in the next section where we examine the relationships between E and B as described by Maxwell’s equations that this retarded transverse disturbance is in fact the basis of an EM wave, where the changing electric field and its associated changing magnetic field form a propagating harmonic disturbance through space.

2.4

Maxwell’s equations

Classically, the relationships between the components of any EM field are described by Maxwell’s field equations and by the equations representing the properties of the medium in which that field exists (Box 2.1). Maxwell’s equations can be written in differential form as ∇ ·D =ρ

(2.6)

∇ ·B=0

(2.7)

∇ ×E =−

∂B ∂t

∂D ∂t The definitions and units of these quantities are ∇ ×H =J +

E H J D B ρ

(2.8) (2.9)

is the electric field intensity in volts per metre is the magnetic field intensity in amperes per metre is the current density composed from the impressed or source electric current and the conduction electric current density, all of which are in amperes per square metre is the electric flux density in coulombs per square metre is the magnetic flux density in webers per square metre is the charge density in coulombs per cubic metre.

For these differential equations to hold, the field vectors must be well behaved, that is, continuous functions of position and time with continuous derivatives that are both single valued and bounded. By classically we imply that the problems discussed require consideration only of lengths that are large compared to atomic dimensions and charge magnitudes so that recourse to quantum field theories, such as quantum electrodynamics (QED), can be avoided. More exactly, classical concepts can be characterised by assuming that: 1. The world is divisible into distinct continuous elements. 2. The state of each element can be described in terms of dynamic variables that can be specified with infinite precision.

Maxwell’s equations and electromagnetic wave propagation

19

Box 2.1 In this text we are taking Maxwell’s equations to be the fundamental postulates or axioms from which we will be developing classical EM field theory. However, it should be borne in mind that they themselves can be derived as theorems for other sets of postulates, for example, Coulombs law and the already stated postulates of special relativity can be used as a starting point to derive Maxwell’s equations [2].

3. Exact laws that define the change of the system in terms of the dynamic variables can describe the interdependency between parts of a system. Since only alternating, sinusoidal and time-harmonic quantities are to be considered, the time dependency of the complex representations of the EM field vectors jωt can be √ taken to be of the form e where to six decimal places e = 2.718282 and j = −1 is the imaginary unit. Here, ω = 2πf is the angular frequency and f represents the temporal frequency measured in hertz. This complex exponential form of spatial and time variation of the fields is used for convenience where it is understood that the actual field quantities are obtained by taking the real part of the complex quantity; thus   (2.10) E = E0 cos(ωt) = Re E0 ejωt   jωt H = H0 cos (ωt) = Re H0 e (2.11) Although in principle it is equally valid to take the imaginary part of these complex quantities throughout, it offers no obvious advantage and is not commonly adopted. Essentially then, and purely for mathematical convenience, we are simply utilizing a complex representation of a real wave. When using this notation, the time factor is usually suppressed, that is, the complex exponentials are cancelled on either side of the relevant expressions and this convention is adopted throughout. Although it is conventional in electromagnetism and optics to adopt a positive time dependence, in the study of quantum mechanics and solid-state physics, the opposite time dependence is more often adopted. Clearly, as the EM field vectors are of the form ejωt , the following operator substitution can be utilized Dn =

∂n ≡ (jω)n ∂t n

(2.12)

This simply states that differentiating EM field vectors with respect to time is equivalent to multiplying the field vectors by the imaginary unit and the angular velocity of the field which is assumed to be fluctuating in a sinusoidal fashion. Crucially, the simplification afforded by restricting ourselves to considering purely monochromatic waveforms in no way restricts the analysis since any angular

20

Principles of planar near-field antenna measurements

frequency may be considered to be a component of a Fourier series or in the limit a Fourier integral, thus enabling this analysis to be applied to arbitrary waveforms. Some of the field components contained within Maxwell’s equations can be related to one another through the properties of the medium in which the fields exist B = µH

(2.13)

D = εE

(2.14)

J = σE

(2.15)

Here, µ is the magnetic permeability of the medium, ε is the permittivity of the medium, that is, the dielectric constant and σ is the specific conductivity. In general, ε and µ are complex tensors that are functions of field strength, however, for the case of free space antenna problems they can usually be approximated by real constants. A vacuum, in classical EM field theory, can be taken to consist of a source and sink free, simple linear homogeneous and isotropic free space region of space in which harmonic time varying fields are measured. In such an environment no charges are present, the current density will necessarily be zero and the resistance of the medium is infinite, that is, zero conductivity; thus ρ = 0, J = 0 and σ = 0. In this case, Maxwell’s simultaneous differential equations reduce to two homogeneous, that is, equated to zero and two non-homogeneous expressions namely ∇ ·E =0

(2.16)

∇ ·H =0

(2.17)

∇ × E = −jωµH

(2.18)

∇ × H = jωεE

(2.19)

Eliminating the magnetic field intensity from these equations yields
 ∇ × ∇ × E = −jωµ ∇ × H = ω2 µεE

(2.20)

Thus, the most general solution of Maxwell’s equations in terms of the material constants and the angular frequency of the EM radiation is ∇ × ∇ × E − ω2 µεE = 0

(2.21)

Similarly, eliminating the electric field intensity, the magnetic field can be expressed as ∇ × ∇ × H − ω2 µεH = 0

(2.22)

These expressions are often referred to as complex vector wave equations, which constitute the most general forms of the wave equation. These wave equations are usually expressed in a simpler form that is particularly convenient for problems involving Cartesian coordinate systems. Using the vector identity
 (2.23) ∇ 2A = ∇ ∇ · A − ∇ × ∇ × A

Maxwell’s equations and electromagnetic wave propagation and recalling that ∇ · E = 0 then 
∇ ∇ · E = ∇ (0) = 0

21

(2.24)

Thus, the complex vector wave equation can be rewritten as ∇ 2 E + ω2 µεE = 0

(2.25)

This is known as the vector Helmholtz equation. Similarly, as ∇ · H = 0 the magnetic field can be expressed as ∇ 2 H + ω2 µεH = 0

(2.26)

Crucially and a point that is easily (and often) overlooked, is that the vector operator substitution used to obtain the Helmholtz equation
from
 wave  the general equation is only valid in Cartesian coordinates. Thus, ∇ 2 E r, t and ∇ 2 H r, t must be calculated in terms of x, y and z. If instead the problem is cast in another coordinate system, for example, spherical, the general wave equation must be solved instead. In general, the one-dimensional transverse wave equation can be expressed as 1 ∂ 2 u (x, t) ∂ 2 u (x, t) − =0 (2.27) ∂x2 c2 ∂t 2 Where, c is taken to denote the velocity of the wave. Assuming again that the wave is sinusoidal in form this can be expressed as d 2 u (x) ω2 + 2 u (x) = 0 (2.28) dx2 c Through a comparison with the Helmholtz equation we find that the phase velocity c of the EM wave can be expressed in terms of the properties of the medium through which the wave is propagating as 1 =fλ c= √ µε

(2.29)

Where f is the frequency in hertz, λ is the wavelength in metres and c is the velocity in metres per second. Here the radical or root, is assumed positive. For convenience, a positive constant k is defined as √ (2.30) k = ω µε The exact reason for adopting this definition of k is presented below, however, for the time being the adoption of this definition of k can be justified on the basis of notational convenience. Clearly then the constant k, often termed the wave number or propagation constant, is simply related to the wavelength as 2π λ If the medium in which the field exists is a vacuum then, √ k = ω µ 0 ε0 k = 2πf

√

µε =

(2.31)

22

Principles of planar near-field antenna measurements

Here, µ0 and ε0 are the permeability and permittivity of free space respectively and k0 is thence used to denote the free space propagation constant. Thus, the Maxwell equations can be transformed into the following vector Helmholtz or wave equations ∇ 2 E + k02 E = 0 2

∇ H

+ k02 H

=0

(2.32) (2.33)

The velocity of an EM wave is unambiguous when considering simple solutions, that is, plane waves. However, as the wave equation also admits solutions representing standing waves, the concept of a velocity of an EM wave can become a little ambiguous. In 1975 the Fifteenth Conférence Générale des Poids et Mesures (CGPM), Resolution 2 (CR, 103 and Metrologia, 1975, 11, 179–80) entitled recommended value for the speed of light, adopted the speed of propagation of EM waves in vacuum as c = 299,792,458 m/s

(2.34)

Where the estimated uncertainty is ±4 × 10−9 . As this uncertainty principally corresponded to the uncertainty in the characterisation of the metre, in 1975 the Seventeenth CGPM decided that the metre should be defined to be the length of the path travelled by light in a vacuum during the time interval of 1/299,792,458 s. Thus, by redefining the unit of length the velocity of light in a vacuum could be defined to be exactly 299,792,458 m/s. The choice of ε or µ will define a system of units. Again by agreement, the value of the permeability of free space µ in henry per metre has been chosen to be exactly µ = 4π × 10−7 = 8.854187817 . . . × 10−7 henry/m

(2.35)

Thus for the metre–kilogram–second–coulomb (mksc) system of units, it follows that the permittivity of free space which is measured in units of farad per metre is exactly ε=

1 = 8.854187817 . . . × 10−12 farad/m µc2

(2.36)

Assuming a Cartesian coordinate system and noting that if we are to utilize the Helmholtz equation we have no choice in this matter, the electric field may be expressed as E(x, y, z) = eˆ x Ex (x, y, z) + eˆ yEy (x, y, z) + eˆ z Ez (x, y, z)

(2.37)

Maxwell’s equations and electromagnetic wave propagation

23

The Laplacian operator ∇ 2 when expressed in Cartesian coordinates can be obtained from ∇2 = ∇ · ∇     ∂ ∂ ∂ ∂ ∂ ∂ = eˆ x + eˆ y + eˆ z · eˆ x + eˆ y + eˆ z ∂x ∂y ∂z ∂x ∂y ∂z =

(2.38)

∂2 ∂2 ∂2 + + ∂x2 ∂y2 ∂z 2

Hence, we may separate the field components and write the vector Helmholtz equation as three equivalent uncoupled scalar Helmholtz equations 





 ∂ 2 Ex (x, y, z) ∂ 2 Ex (x, y, z) ∂ 2 Ex (x, y, z) 2 y, z) eˆ x = 0 + + k E + (x, 0 x ∂z 2 ∂x2 ∂y2

∂ 2 Ey (x, y, z) ∂ 2 Ey (x, y, z) ∂ 2 Ey (x, y, z) + + + k02 Ey (x, y, z) eˆ y = 0 ∂x2 ∂y2 ∂z 2 ∂ 2E

(x, y, z) + ∂x2

z

∂ 2E

(x, y, z) + ∂y2

z

∂ 2E



(x, y, z) + k02 Ez (x, y, z) eˆ z = 0 ∂z 2

z

(2.39)

(2.40)

(2.41)

Similar expressions also hold for Hx (x, y, z), Hy (x, y, z) and Hz (x, y, z). Therefore, all of the components of the EM field obey the scalar differential wave (Helmholtz) equation. ∂ 2 u (x, y, z) ∂ 2 u (x, y, z) ∂ 2 u (x, y, z) + + + k02 u (x, y, z) = 0 ∂x2 ∂y2 ∂z 2

(2.42)

Using vector notation this can readily be expressed in a more compact form as ∇ 2 u + k02 u = 0

(2.43)

This differential equation can be solved by direct integration using Green’s theorem to yield the Kirchhoff integral theorem. Additionally, however, Chapter 4 will set out an alternative methodology for utilizing this equation that is more convenient when considering most, but not all, problems encountered in the study of planar near-field antenna metrology. However, a methodology that will be instructive and gives insight in to the nature of the EM interaction that is often successfully adopted to evaluate the EM fields produced and subsequently radiated by moving charges is to consider the fields E and B to be the resultants of potentials.

24

Principles of planar near-field antenna measurements

2.5

The electric and magnetic potentials

2.5.1 Static potentials The electric potential, φ is a scalar potential that is a function of position and time defined by  ρ(r  ) 1 dτ (2.44) φ(r, t) = r − r  4πε0 where r is point at which the potential is being evaluated and r  is location of the charge density element and the static electric field is given by E = −∇φ

(2.45)

Alternatively the magnetic potential, A is defined as being a vector given by  j(r  ) µ dτ (2.46) A(r, t) = r − r  4π And the static magnetic field is given by B=∇ ×A

(2.47)

Equations (2.44) and (2.46) for the potentials give the static electric and magnetic fields however, as explained in Section 2.2, at any point that is spatially separated from the source of the potential a finite amount of time must pass before the influence of the source can affect the potential.

2.5.2 Retarded potentials Figure 2.6 illustrates that the potential at point P at time t is determined by the position of the potential source at position A at the earlier time of t − τ where the distance s = cτ = [r − r  (t − τ )] and B is the position of the source at time t. Thus for a distributed charge of density ρ the potential at position r and time t due to the charge in the vicinity of r  depends on the value of ρ at the previous time c A

B

P

s r ′(t −
)

r ′(t )

r O

Figure 2.6

Figure showing potential at time t from point charge moving from A to B

Maxwell’s equations and electromagnetic wave propagation t − [r − r  ]/c, therefore the potential of the entire charge is  ρ(r  , t − r − r  /c)  1 φ(r, t) = dτ r − r  4πε0

25

(2.48)

This formula for the potential that is calculated to take account of the finite speed of light is referred to as the retarded scalar potential and a similar argument can be followed to establish a retarded vector potential.  j(r  , t − r − r  /c)  µ A(r, t) = dτ (2.49) r − r  4π Therefore (2.45) does not apply for time dependant systems and must be modified to take account of the finite but constant speed of light to be ∂A ∂t Since the curl of any grad ≡ 0, this satisfies Maxwell’s third equation as E = −∇φ −

(2.50)

∂ (2.51) (∇ × A) ∂t ∂ ∂B ∇ × E = − (∇ × A) = − (2.52) ∂t ∂t which is in agreement with (2.8). Additionally (2.47) is still correct for time dependent systems as the div of any curl ≡ 0 ∇ × E = −∇ × (∇φ) −

∇ · (∇ × A) = ∇ · B = 0

(2.53)

From (2.6) ε −1 ∇ · D = −∇ 2 φ −

∂ (∇ · A) = ε−1 ρ ∂t

(2.54)

If 1/v2 · ∂t 2 φ/∂t 2 is inserted into (2.54), where v is the propagation velocity 1 ∂ 2φ ∂ 1 ∂ 2φ ρ (∇ · A) + 2 2 = + 2 2 ∂t ε v ∂t v ∂t Rearranged to give −∇ 2 φ −

1 ∂ 2φ ρ ∂ 1 ∂ 2φ = − − 2 2 − (∇ · A) 2 2 ε ∂t v ∂t v ∂t This can be simplified to give ∇ 2φ −

∇ 2φ −

1 ∂ 2φ ρ ∂F =− − ε ∂t v2 ∂t 2

(2.55)

(2.56)

(2.57)

where F =∇ ·A+

1 ∂φ v2 ∂t

(2.58)

26

Principles of planar near-field antenna measurements

By substitution into (2.9) in a similar fashion it can be shown that ∇ 2A −

1 ∂ 2A = −µj + ∇ · F v2 ∂t 2

(2.59)

Both of these equations can be simplified if we make F =∇ ·A+

1 ∂φ =0 v2 ∂t

(2.60)

Equations (2.57) and (2.59) then become 1 ∂ 2φ ρ =− ε v2 ∂t 2 2 1 ∂ A ∇ 2 A − 2 2 = −µj v ∂t

∇ 2φ −

(2.61) (2.62)

Clearly if the condition set out in (2.60) where F = 0 is met then (2.61) and (2.62) are decoupled in that (2.61) now defines φ in terms of charge density without reference to current density and (2.62) does likewise for A and current density. The condition that F = 0, referred to as the Lorentz condition, can in fact always be satisfied due to the nature of the definitions of φ and A. If A is transformed to A A → A = A + ∇χ

(2.63)

∂χ ∂t

(2.64)

and φ → φ = φ +

where χ is a function of position and time. Since ∂ ∂χ − ∇χ = E E = E + ∇ ∂t ∂t

(2.65)

and B = B + ∇ × (∇χ ) = B

(2.66)

The use of either φ or φ  or A or A is arbitrary as E and B will remain unchanged. Transformations such as (2.63) and (2.64) are referred to as gauge transformations and potentials that satisfy the Lorentz condition are said to belong to the Lorentz gauge. The retarded potential (2.48) and (2.49) are solutions of the decoupled (2.61) and (2.62) in combination with the Lorentz condition. Therefore they provide a consistent method for the solution of Maxwell’s equations. They allow the sources ρ and j to be the inputs that can be used to calculate φ and A, which in turn allow the calculation of E and B. This may appear to be a long and convoluted methodology for the calculation of fields from their charge and current density sources but it is usually a much easier process than attempting to evaluate E and B directly from Maxwell’s equations.

Maxwell’s equations and electromagnetic wave propagation

27

Box 2.2 In fact this procedure of moving from sources to potentials to electric and magnetic fields can be avoided if EM theory is expressed in terms of vector quantities that are themselves Lorentz invariant. And by definition the scalar product of four vectors and four vectors with the d’Lambertian vector operator are Lorentz invariant. This requires that the quantities to be used to describe the EM interaction are all four vectors, these being: 1. The interval between any two events in spacetime: R1,2 = [r 1,2 , c(t1 −t2 )]. Its four components being the ordinary three-dimensional distance vector r and the time difference. 2. The four-potential A = [A, φ/c], which contains the vector potential A and the scalar electrostatic potential. 3. The current density J = [J , cρ], which contain the current density and the charge density. In fact this vector describes J as a current in the three spatial dimensions and ρ as a current in the temporal dimension. 4. The propagation vector k = [k, ω/c], composed of the ordinary propagation vector k and the frequency ω of a relativitistic wave. The use of these terms means that the entire EM interaction can be summarised as   1 ∂2 ∇ 2 − 2 2 A = −µ0 J = ∇ 2 A (2.67) c ∂t in terms of a four-vector A and J . This expression allows the calculation of all EM phenomena without recourse to either E or B, thus confirming the conclusions of the Aharonov– Bohm thesis as to the reality of the A field and the Aharonov–Bohm effect that has been subsequently experimentally confirmed [4]. Additionally this formulation of EM theory is particularly effective in Lagrangian formulations of EM theory where the Schwarzschild invariant S = (J · A) = (J · A − ρφ) can be interpreted when integrated with respect to all four coordinates as the action of the EM system. In fact in the seminal work [5], Feynman, Morinigo and Wagner make the comment, ‘The guts of all electromagnetism is contained in the specification of the interaction of current and field as J · A.’ In Chapter 1 the action of antennas was described in terms of the observation that spatially separated physical systems that include antennas will form coupled harmonic systems. These can be used to transfer power in the form of induced voltages and currents and thus develop a signal in the Rx subsystem that contains information as to the nature of the harmonic excitation of the Tx subsystem. Thus although the action of the antennas is described in terms of voltages and currents, the physical manifestations

28

Principles of planar near-field antenna measurements

of φ and j, developing power in spatially remote antennas as shown above, a model of the propagation between them can be constructed in terms of abstract fields and potentials.

2.6

The inapplicability of source excitation as a measurement methodology

The application of this methodology that provides an algorithm for the calculation of the free space EM field propagation away from or to an antenna structure in terms of currents and charge distributions excited on a geometrical structure is widely used in antenna design. As basic circuit theory and methods designed to cope with guided wave paths in systems are based on the concepts of current and voltage the interface of these techniques with antenna design methods is extremely fortuitous. This is particularly so in the areas of antenna feeds where a guided wave structure must be interfaced with an antenna designed to radiate the power delivered by the feed into or out of free space. However, in terms of metrology the concept of excitation currents on a structure has considerable limitations. The direct probing of the surface of any radiating structure to determine charge and current distributions, as a result of the constraints associated with the transfer of signal along a conducting path from any single or array of sensors, is an extremely intrusive measurement methodology. Although to some extent all measurement procedures are intrusive, in general the end result of such a radio frequency (RF) surface probing procedure is a very considerable alteration in the existing state of the system in the course of the measurement procedure. This alteration can in theory be compensated for via the use of theoretical models. However, if the measurement procedures results are essentially the results of a modelling procedure that is no more accurate that the original modelling procedure included as part of the design process then measurements cannot be used to confirm the accuracy or effectiveness of any design. The accuracy of any compensating mathematical processing can also be brought into question owing to its possible inconsistency with theory. In that, as a result of space contraction and therefore the non-invariance of volume under the Lorentz transformation, current and charge density are not separately relativistically invariant quantities. Therefore, any extrapolation from raw measurement data from only one of these to processed measured data may be done without a firm basis in the physical processes involved. However, an alternative strategy can be used to determine the radiated fields from an EM source that can be more amenable to the constraints of metrology.

2.7

Field equivalence principle

The field equivalence principle is the process of replacing the actual sources that create an EM field over some closed surface, S, with equivalent sources located on that same surface. It is in fact a theoretical statement of Huygen’s principle that

Maxwell’s equations and electromagnetic wave propagation

29

Wave front at time t + ∆t

Wave front at time t

Figure 2.7

Huygen’s principle

Sources

E, H •P (E1, H1)

NO sources

E, H

Js

•P (E1, H1) Jm

Surface S

Figure 2.8

The field equivalence principle

any wavefront can be viewed as being made up of secondary sources of spherical waves. Each point on a primary wavefront can be considered to be a new source of a secondary spherical wave and that a secondary wavefront can be constructed as the envelope of these secondary spherical waves, at the same frequency as illustrated in Figure 2.7. An illustrative computer animation of Huygen’s principle can be found at Reference 6. Figure 2.8 defines the field equivalence principle, where a set of electric and magnetic current sources create the radiated electric, E and magnetic, H , fields over an arbitrary closed surface S. The wavefronts that create the radiated field E 1 , H 1 at point P in the left-hand diagram can be alternatively created by equivalent electric and magnetic current sources J s and J m on the surface S of the right-hand diagram, so creating the same radiated field at point P and indeed everywhere outside the enclosing surface S which now contains no sources. In order that the total field throughout the whole of the volume space (both internal and external to S) is a valid solution to Maxwell’s equations the equivalent sources must conform to the proper boundary conditions at S between the internal and external E and H field at the surface S as well as the radiation condition at infinity. By postulating a null field inside S the equivalent surface currents are given by (2.68) and (2.69) J s = an × H (s)

(2.68)

J m = −an × E(s)

(2.69)

where the unit vector an represents the surface normal and E(s) and H (s) represent the tangential electric and magnetic fields at the surface S. A particularly valuable modification to the field equivalence comes about when we note that the zero field with S cannot be disturbed by changing the material

30

Principles of planar near-field antenna measurements Infinite electric conductor Images Js

Js

Js

Js = 0

Jm

Jm

Jm

2J m (b)

(a) S

Figure 2.9

S

(c) S

An application of image theory to the radiating magnetic current in presence of a perfect infinite electric conductor

properties within S, for example, that of a perfect electric conductor. In this case at the moment the electric conductor is introduced the electric current on the surface S, J s , is short-circuited. This leaves just the magnetic current, J m , over the surface S and it radiates in the presence of the perfect electric conductor to give the correct fields E 1 and H 1 at point P in Figure 2.8. Similarly the dual of this process can be enacted such that the material properties are replaced by a perfect magnetic conductor, so short-circuiting the magnetic current, J m , thus leading to a purely electric current J s radiating in the presence of a perfect magnetic conductor. The real utility of the field equivalence approach comes when the surface S becomes an infinity flat plane, since here the problems reduce to determining how an magnetic surface current, J m , radiates in the presence of a flat perfect electric conductor of infinite extent. From image theory this problem is reduced to that of Figure 2.9, where in (a) the presence of the electric conductor short-circuits J s and the removal of the infinite electric conductor by image theory (b) leads to the doubled magnetic surface current in an unbounded medium (c), from which the radiating fields can be determined to the right of the conducting plane. By duality the use of a perfect magnetic conductor reduces the problem to that of a doubled electric surface current radiating in an unbounded medium. The above process of equivalent fields thus leads to a convenient way forward in that we can measure the electric field on an infinite plane close to the antennas physical structure, from which we can derive the generating magnetic current on this scanned plane, which can then be used to determine the far-field characteristics of the antenna. This process of planar near-field measurement will be developed in the next chapter.

2.8

Characterising vector EM fields

We have seen in Chapter 1 (Figure 1.1) that it is convenient to represent the radiating plane aperture of an antenna in rectangular coordinates and the radiated far-field in spherical coordinates. Although the choice of spherical coordinates for the far-field is near universal, the choice of coordinate system to represent the antenna in order

Maxwell’s equations and electromagnetic wave propagation

31

to calculate its radiation pattern very much depends on the structure of the antenna. Considerable mathematical simplifications in calculating the radiation pattern of a given antenna can be achieved by choice of a matching coordinating system. For example the use of cylindrical coordinates to calculate the radiation of a circular open-ended waveguide offers considerable simplification over the use of a rectangular aperture coordinate frame. In this section we will illustrate the process of determining the radiated field by considering the z directed Hertzian dipole and this will also serve to define the concept of the plane wave. The fundamental solution for the wave equation in vector potential A (2.62) is the retarded vector potential of (2.49) at a single source point with current j s is   µ j s (r , t − r − r /c) (2.70) A(r, t) = r − r  4π This is clearly a function of both source point and field point and defines the ‘action at a distance’ property of the EM wave. Equation (2.68) is often called the Green’s function because by definition a Green’s function is the solution to a differential equation for a unit source. We will now consider the radiation from this infinitesimal small (with respect to the radiation wavelength) current element, often termed the Hertzian dipole, as shown in Figure 2.10. For this dipole we have a constant current (2.71) j s (r  , t − r − r  /c) = J0 aˆ z e−jkr and so the vector potential can be written as A = aˆ z

µJ0 l −jkr e 4πr

(2.72)

z

Ar Aφ r θ

φ

Aθ

y

x

Figure 2.10

Coordinate system for infinitesimal dipole over its length l

32

Principles of planar near-field antenna measurements

Since the far-field is to be expressed in spherical coordinates the following vector identity (see Chapter 5) 

  Ar sin θ cos φ  Aθ  =  cos θ cos φ − sin φ Aφ

sin θ sin φ cos θ sin φ cos φ

  cos θ Ax − sin θ   Ay  0 Az

(2.73)

yields  µJ0 l −jkr
A = aˆ r cos θ − aˆ θ sin θ = Ar (θ, r) + Aθ (θ , r) e 4πr

(2.74)

As j s is located in the z direction it is independent of the far-field angle φ and so δ/δφ = 0 thus H =∇ ×A

(2.75)

gives the magnetic field at the observation point r. Expressing the cross product in spherical coordinates (for example, see Reference 7) with δ/δφ = 0 gives   1 ∂(rAθ (θ, r)) ∂Ar (θ, r) H= − aˆ φ r ∂r ∂θ

(2.76)

Yielding only an Hφ component in the far-field. For the electric field we have E=

 1
∇ ×H jωε

(2.77)

with Hr = Hθ = δ/δφ = 0, which yields   ∂(Hφ sin θ ) 1 1 jωε r sin θ ∂θ   1 1 ∂(rHφ ) Eθ = jωε r ∂r Er =

(2.78) (2.79)

Evaluating the three non-zero field components gives   jkJ0 l sin θ 1 1 Hφ = + 2 e−jkr 4π r jkr   Z0 J0 l 1 1 Er = cos θe−jkr + 2π r2 jkr 3   1 jkZ0 J0 l sin θ 1 1 Eθ = + 2 − 2 3 e−jkr 4π r jkr k r

(2.80) (2.81) (2.82)

Maxwell’s equations and electromagnetic wave propagation

33

Near to the dipole 1/r 2 and 1/r 3 terms dominate, whereas in the far-field only 1/r terms are significant. Thus the far-fields are given by jkJ0 l sin θ −jkr e (2.83) 4π r jkZ0 J0 l sin θ −jkr Eθ = e (2.84) 4πr and we note that Eθ /Hφ = Z0 , the free space wave impedance. Thus the infinitesimal dipole radiates a locally plane wave in the radial direction r. It should be noted that computing the average power flow (Poynting vector) in the case of the near-field (1/r 2 and 1/r 3 terms only) results in zero power flow indicating the field is reactive. For the far-field case real power flow is achieved. Hφ =

2.9

Summary

Thus this chapter dealt with a theory that describes propagation through free space as a process of the propagation of EM waves, these waves being directly related to the acceleration of charged particles. A description of these waves is then provided based on Maxwell’s equations and the derivation of the scalar Helmholtz equations. Then a description of the sources of these waves as being retarded potentials that then produce fields was provided. Although the basis of this explanation is charge and current densities on the geometrical structure that constitutes an antenna, direct measurement methods to assess these sources are not viable, therefore an alternative equivalent fields model was developed which in turn suggested other measurement methodologies. The next chapter will go on to investigate and describe this alternative near-field measurement technique.

2.10 References 1 Manners, J., and Ross, S.: Discovering Physics Block B Unit 1 (Open University, Milton Keynes, 1994), pp. 9–12 2 Lorrain Corsor: Electromagnetism (Freeman and Co., San Francisco, CA, 1978), pp. 261–2 3 Schmitt, R.: EMs Explained (Newnes, Amsterdam, 2002), pp. 25–6 4 Tonomura, A.: The Quantum World Revealed by Electron Waves (World Scientific, Singapore, 1998) 5 Feynman, R.P., Morinigo, F.B., and Wagner, W.G.: Feynman Lecturers on Physics, Vol. 3 (Addison–Wesley, Reading, MA, 1964), p. 36 6 Demonstration of Huygen’s principle on http://www.sciencejoywagon.com/ physicszone/lesson/otherpub/wfendt/huygens.htm [Accessed 01 Aug 2007 ] 7 Balanis, C.: Antenna Theory and Design, 2nd edn (John Wiley & Sons, New York, 1997), p. 920

Chapter 3

Introduction to near-field antenna measurements

3.1

Introduction

In the previous chapter we found that if we can determine the near-field on an infinite plane close to a radiating antenna we can subsequently determine the radiated farfield. We shall see in subsequent chapters the mathematical process by which this transformation can be undertaken in the most efficient way and consider the limitations imposed by a finite planar scan and how such limitations can be mitigated. In this chapter we consider the practicalities of measuring the near electric field to provide a data set that can be processed.

3.2

Antenna measurements

By way of introduction we will start by considering conventional far-field antenna measurement. The far-field radiation pattern is characterised by • • •

Spatial amplitude variation Spatial phase variation Spatial polarisation variation.

Ideally these can be determined by placing the antenna under test (AUT) in a perfect plane wave field and mechanically rotating the antenna about the relevant coordinate whilst measuring the received amplitude and/or phase. Using the coordinate frame shown in Figure 3.1 and with the antenna aperture electric field polarised in the y direction, the measurement configuration takes the form of Figure 3.1. To acquire the E-plane all that is required is to rotate the source polarisation by 90◦ and the AUT about the φ-axis by 90◦ ; this is shown in Figure 3.2. These two φ cuts are often termed the principal plane cuts and along with the φ = 45◦ (shown in Figure 3.3) cut are often used to characterise the antennas far-field radiation characteristics.

36

Principles of planar near-field antenna measurements

Perfect incoming plane-wave

AUT

Aperture field of AUT

Incoming plane-wave
=90°

Position controller

Receiver

Received amplitude

Figure 3.1

θ

Azimuth position

H -plane radiation pattern acquisition

AUT Perfect incoming plane-wave Aperture field of AUT

Incoming plane-wave
= 0°

Position controller

Receiver

Received amplitude

θ

Azimuth position

Figure 3.2

E-plane radiation pattern acquisition

Introduction to near-field antenna measurements

AUT

Perfect incoming plane-wave

Aperture field of AUT

Incoming plane-wave
= 45°

Position controller

Receiver

Received amplitude

Figure 3.3

37

θ

Azimuth position

45◦ -plane radiation pattern acquisition

By using two axes of movement, such as that shown in Figure 3.4, the incoming plane wave can be fixed in polarisation in which case: • •

Azimuth pattern is then E-plane pattern Elevation pattern is the H -plane pattern.

Azimuth

Co

AUT

po

lar

Efie

ld

Incoming plane wave

EL Elevation

AZ

Sphere surrounding the AUT

Figure 3.4

Principal plane cut acquisition using azimuth over elevation turntable

38

Principles of planar near-field antenna measurements

Although the E-plane and H -plane terminology is widely used and is helpful in this context, it is, however, ambiguous when describing circularly polarised antennas. By taking azimuth cuts for many different values of elevation one can build up a set of radiation pattern cuts and using interpolation a three-dimensional view of the radiation pattern can be achieved. This can then be viewed as a contour or isometric plot, the latter being shown in Figure 3.5. By way of a comparison, Figure 3.6 contains a threedimensional polar plot of the same antenna radiation pattern illustrating another style of presentation. Although there are a great many ways in which the plane wave illumination of the AUT can be achieved in practice, their mechanisms can be considered to divide into two categories, direct and indirect collimation. Those that rely upon direct collimation include free-space ranges, reflection ranges, that is, compact antenna test ranges (CATRs) and refraction, that is, dielectric lens ranges. Indirect techniques include all forms of near-field ranges, that is, planar, cylindrical and spherical. The most basic direct method is to generate the plane wave from a portion of a spherical wavefront. This can be achieved by having a source antenna at a long distance from the AUT so the AUT aperture sees a nearly plane wave when R is large, see Figure 3.7. To ensure near-plane wave conditions at the AUT aperture, the phase taper across the AUT aperture is controlled to be a maximum variation of 22.5◦ . Referring to Figure 3.8 this can be expressed as    2 2π  2 D φ = R + − R (3.1) 2 λ

−5 0

−10

−5 −10

−15

Power (dB)

−15 −20

−20 −25

−25

−30 −35

−30

−40 −45 −50 1

−35 0.8

Figure 3.5

0.6

0.4

0.2

0 −0.2 −0.4 −0.6 −0.8 −1 −1 v

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

u

Isometric view of three-dimensional radiation pattern

0.8

1

−40 −45

Introduction to near-field antenna measurements

39

Y –5 –10 –15 –20 –25 –30 X –35 Z

–40 –45

Figure 3.6

Three-dimensional polar plot of radiation pattern

Wave-fronts Source antenna

Nearly plane-wave

R = range length

Figure 3.7

Plane wave created from far-field source

   2 12 D 2π R  − 1 1+ φ = 2R λ

(3.2)

Taking the first term of the Taylor series yields π D2 (3.3) 4λR If φ < π/8 or 22.5◦ , then we get the far-field distance. The value of 22.5◦ stems from the optical analogue having been proposed by Lord Rayleigh [1] and is empirically derived. Broadly, this equates to the minimum distance at which a null can be observed φ ≈

40

Principles of planar near-field antenna measurements ∆φ

D

R

Figure 3.8

Far-field phase taper geometry

between the main beam and the first sidelobe. Thus, in practice, this leads to an optimistic assessment of the far-field distance. R≥

2D2 λ

(3.4)

This value of R is approximately equivalent to the displacement where only one unique path exists between the AUT and the test antenna, as per Chapter 1, in the limit if only one path existed then φ = 0. However, this book is concerned with the indirect approach of illuminating the AUT via a near-field probe.

3.3

Forms of near-field antenna measurements

Three commonly employed coordinate systems that are utilized for taking near-field antenna measurement are spherical, cylindrical and planar. These systems are generally considered preferable since not only is the vector Helmholtz equation separable in each of these systems, but also, in practice, the positioner subsystems that employ them can be conveniently constructed. These systems are illustrated schematically in Figure 3.9. In principle, spherical, cylindrical and planar techniques are endeavouring to derive a complex vector field function at a large distance from the antenna, from the sampling of similar complex data over a well-understood surface at a much smaller distance. This facilitates the testing of electrically large antennas in a controlled, indoor, environment. In all cases, the acquisition of the near-vector field is accomplished by placing a probe at a particular position pointing in a particular direction and allowing the electric field that surrounds the probe to generate an observable excitation current. The difference in potential between the probe and a reference is sampled in phase

Introduction to near-field antenna measurements

41

nts

Spherical near-field acquisition geometry

Figure 3.9

Planar near-field acquisition geometry

Near-field acquisition geometries

Plane bipolar

Figure 3.10

Cylindrical near-field acquisition geometry

Plane polar

Plane rectilinear

Planar acquisition geometries

and at quadrature. Provided that two such orthogonal complex voltages are sampled over a well-defined surface at regular intervals, the principle of modal expansion can be utilized to determine the amplitudes and phases of an angular spectrum of plane, cylindrical or spherical waves. This enables the computation of the electric and magnetic fields at any distance from the AUT, and, hence, the computation of the fields when infinitely far removed from the radiator, which results in a true far-field vector pattern. Despite the obvious similarities between the theoretical descriptions at the generic level, the differing geometries result in a significant divergence in the specific implementation of each. In order for the spherical near-field range (SNFR) to characterise the propagating near-field component, a test probe is held at rest, whilst the AUT is nodded in φ and rotated in θ, where θ and φ are conventional spherical coordinates. This results in the path of travel of the probe describing a spherical surface that is attached to the AUT. This experimental set-up is in stark contrast to the planar nearfield scanner (PNFS), where it is the AUT that remains at rest, whilst a small light probe is scanned across the aperture of the AUT. The cylindrical near-field scanning (CNFS) technique utilizes a hybrid measurement configuration in which the AUT is rotated in azimuth (x–y plane), whilst a scanning probe is moved linearly in z. Planar measurement configuration can be subdivided into three mechanically convenient acquisition systems, which are again illustrated schematically in Figure 3.10.

42

Principles of planar near-field antenna measurements

The sampled, usually electric, field components are typically chosen to coincide with the tangent to the measurement surface and are orientated such that samples are taken along paths that are parallel to lines of increasing ordinate. As stated in Chapter 2, Helmholtz equation from the general wave equation is only valid in Cartesian coordinates. So, strictly, the vector Helmholtz equation is not necessarily solved in either plane polar or plane bipolar coordinate systems [2], rather the sampled field components are resolved onto an equivalent Cartesian polarisation basis, whereupon the transformation to the angular spectrum can be performed either directly or by use of approximation. It is possible to solve the wave equation in plane polar coordinates [3] and bipolar coordinates [4]; but this approach offers limited advantage over the ‘cost’ of the more mathematically complicated transformation process. The plane bipolar configuration is similar to plane polar, except that the probe sweeps out an arc in two dimensions rather than just one. Mechanically this is the simplest positioning system to construct, as only two rotary joints are required to connect the radio frequency (RF) output from the probe to the input of the network analyser, with no variation in phase over the scan plane. Thus, this design is costeffective and consequently popular. Probe pattern correction is complicated by the fact that the probe is rotated by a different amount for every measurement point. This can be addressed in three ways; (1) mechanically by counter-rotating the probe; (2) algorithmically by utilizing the generalized transmission formula developed below or (3) by utilizing a rotationally symmetrical probe and deploying the probe pattern correction in the usual way. We will consider this process more in detail in Chapter 5. The plane polar scanning technique offers a similar simplification in mechanical complexity, as only a single linear probe trajectory is required. Probe pattern correction is handled using one of the techniques described above for plane bipolar geometry. Again, an increased computational complexity results from utilizing a curvilinear coordinate system; however, the angular spectrum can be obtained from a fast Fourier transform (FFT)/Bessel or a Jacobi–Bessel series representation of the sampled electric field. Mechanically, the plane rectilinear configuration is the most demanding, as the accuracy of the far-field pattern depends crucially upon the accurate positioning and orientation of the probe whilst sampling near-field data. The measured phase is the relative phase between the signal channel and the reference channel. However, as the electrical path of the test channel must pass through moving parts of the robotics positioner, it must not be allowed to change in electrical length for risk of disturbing the value of the measured phase. Although this is in common with all forms of nearfield measurements, it is particularly challenging for planar measurements as the acquisition planes tend to be larger and the probe is displaced linearly. Rotary joints can be used to form a pantograph arrangement, however, this requires the use of a large number of rotary joints. This can be undesirable as each joint introduces loss, discontinuities, will not yield perfect phase stability and can limit the upper frequency limit of operation. Alternatively, a more popular moving discontinuity arrangement, that is, rolling bend, can be employed. However, in practice, maintaining the form of the discontinuity is extremely difficult, although improvements in both RF cables and rotary joints have eased this problem. On the positive side of the balance sheet, the

Introduction to near-field antenna measurements

43

transformation and probe pattern correction processes are straightforward and highly efficient. In conclusion, although measurements can be made utilizing mechanically convenient plane polar or plane bipolar geometries, these systems attain a considerable degree of mechanical simplicity at the expense of either computational efficiency or the loss of rigour, that is, the introduction of numerical approximations, for example, polynomial interpolation. Alternatively, measurements can be made using the mechanically inconvenient plane rectilinear geometry that yields considerable advantages in terms of computational efficiency and accuracy. Despite the mechanical difficulties inherent within the plane rectilinear geometry its mathematical and computational simplicity has made it by far the most popular of the near-field techniques used in industry today.

3.4

Plane rectilinear near-field antenna measurements

Conventionally, planar near-field measurement systems operate by sampling the amplitude and phase of the propagating near-field at regular intervals, on a plaid monotonic grid over a planar surface, which is tangential to that of the antenna aperture plane and located a few wavelengths in front of it. This arrangement is illustrated in Figure 3.11. The probe must be in the propagating near-field region, not the reactive nearfield, as evanescent coupling is omitted from the antenna–antenna coupling formulae

y

x Ey E

x

r9 0 Z=Zt

AUT

Figure 3.11

Sampling plane

z

Measurement point

Coordinate system for planar scanning

44

Principles of planar near-field antenna measurements

(Chapter 5). An electrically small, that is, a low gain, low scattering cross section antenna, commonly referred to as a near-field probe is used to radiate at each of the preselected points within the planar surface. Typically, the measurements are made on a lattice that corresponds to a regular rectangular Cartesian grid along paths that are parallel with the x- and y-axes. Acting as a transmitter, the field from the probe produces a quasi-spherical wave within the free-space port of the AUT. Amplitude, phase and polarisation of the radiation from the near-field probe is held constant for all positions of the probe and a plane wave is synthesised at the surface by the superposition of these quasi-spherical waves. This process is repeated, but with the probe antenna rotated through 90◦ about the normal to the plane to form an orthogonally polarised plane wave since (Section 4.6, Chapter 4), two orthogonal tangential field components are required to determine the complete polarisation properties of the AUT. Just as the free-space port plane wave is formed through the superposition of the large number of quasi-spherical waves, the response of the AUT to the synthesised plane wave is formed by the superposition of the responses of the AUT to the quasispherical waves. In essence, any change in the spatial distribution of the field in free space on the plane over which the probe is scanned cannot occur at a rate greater than that decided by the free-space wavelength. Therefore, at a specific frequency, spatial rates of change on the surface will all be the same if the propagation is only in one direction through the surface. Given that the only variable that can change the spatial rate of change on the plane, if the frequency is constant, is the angle at which the propagation passes through the plane, and hence its projection onto the plane, each different rate of change represents propagation at a different angle and is the source of the concept of a spectrum of monochromatic plane waves all propagating through the plane at different angles. The synthesised plane wave can then be steered to other directions by linear phase shifting. This process of phase shifting and summing can be recognized as the Fourier transform and is carried out using either the discrete Fourier transform (DFT) algorithm or when appropriate, the efficient FFT algorithm.

3.5

Chambers, screening and absorber

The prospect of testing an antenna indoors and conducting antenna experiments in an environmentally controlled anechoic chamber is attractive, since it eliminates some of the severe problems associated with direct far-field testing. These include the cost of antenna transport, test configuration and the cost of real estate. Additional expenses are introduced through schedule disturbance caused by precipitation, temperature instability, high winds causing transmit (Tx) and receiver (Rx) tower movement and the more subtle climactic effects on antenna performance induced, for example, by variations in direction and level of solar radiation. These difficulties often result in the poor repeatability between measurements and consequently result in an inability to calibrate and correct these uncertainties. However, using conventional far-field measurements means that the 2D2 /λ range length limitation restricts this to only

Introduction to near-field antenna measurements

45

electrically small antenna apertures. The compact antenna test range (CATR) overcomes this length limitation by using a collimating reflector to convert the spherical wave from a feed horn to a pseudo-plane wave (the so-called quiet zone) that illuminates the AUT. However, the need to control reflector edge diffraction and the cost of the reflector surface with root mean square (r.m.s.) surface accuracy of the order of λ/100 make this a costly alternative. Alternatively the near-field approach offers a controlled physical environment (such as temperature, humidity and cleanliness for spacecraft applications) as well as full electromagnetic (EM) screening to avoid outside interference for the experimental arrangement, within a relatively small anechoic chamber. This full EM screening may also have beneficial effects as certain antennas, for example, active electronically scanned arrays, may be required to be tested in transmit. Thus the use of a screened environment may have health and safety and/or security advantages. An anechoic chamber is designed to simulate a reflectionless free-space environment for EM testing. This permits the test environment to be carefully controlled in terms of scattering (RF multipath), EM compatibility (RF noise), temperature, seismic stability, humidity and cleanliness. However, this tends to limit the maximum size of antennas that can be accommodated within the test environment. In practice, the anechoic environment is not perfect, as typical high-quality wide-band radar absorbing material (RAM) has a normal incidence monostatic reflection coefficient of −40 dB. The bistatic reflection coefficient of most RAM will degrade as the angle of incidence becomes larger, whilst the monostatic backscatter is always present. A measure of RAM performance is its reflectivity, which is a ratio of received signal with absorber to that of signal received when absorber replaced by metal plate (Figure 3.12). Typical values of reflectivity range from −20 to −50 dB for normal incidence depending on cone size relative to operational wavelength. Chambers are often constructed inside fully shielded enclosures, such as shown in Figure 3.13.

T

R

θ

Figure 3.12

Measurement of RAM reflectivity

46

Principles of planar near-field antenna measurements (a)

(b)

(c)

Figure 3.13

(d)

(a) Fully screened chamber, (b) details of door, (c) details of corner construction and cable entry and (d) door seal detail

Box 3.1 Pyramidal absorber, as illustrated in Figure 3.14 provides a smooth transition from free-space 377- impedance to that of the metal plate that backs the absorber. The taller the absorber the smoother the transition and better the reflectivity.

Figure 3.14

Pyramid absorbing material

Introduction to near-field antenna measurements

47

They are used where a high level of magnetic, electric and microwave shielding is required. Typical shielding values when tested to IEEE299 are • •

3.6

−100dB at 1–500 MHz in the electric field −100dB at 1–18 GHz in microwave (far-field).

RF subsystem

A typical RF subsystem is shown in Figure 3.15 and is essentially a two-arm microwave interferometer where a probe and AUT are inserted into the test arm. In practice, this is realized with a standard vector network analyser (VNA) measurement system controlled via general-purpose interface bus (GPIB) or Ethernet via a central computer (Figure 3.15). This configuration is based around a VNA operating in remote mixing mode. For small systems it is possible to work without remote mixing if cables are short. Remote mixing is merely a way of redistributing the receiver within the chamber to minimise cable losses and improve the system dynamic range. Anechoic chamber

X,Y,Z scanning frame

Test fixture AUT

Roll axis Mixer

Probe

Mounting plinth

Test

Power control unit

Mixer

LO source

Ref

LO/IF test set

Coupler

RF source

GPIB

Measurement and control computer

Figure 3.15

Receiver

Schematic of near-field measurement system

Plotter

48

Principles of planar near-field antenna measurements

In the far-field case the necessary dynamic range is principally determined by the type of antenna being tested and must have sufficient dynamic range that the entire far-field antenna pattern (main beam peak down to lowest sidelobe of interest) can fit within the system dynamic range. Processing gain of the near-field to far-field transform can suppress noise and increase the usable system dynamic range. This is a consequence of the phenomenon of incoherent random noise signals tending to cancel one another, whilst coherent signals tend to add up together providing an apparent increase in the apparent system signal to noise ratio [5]. This is why near-field ranges can often use poorer absorber than would be the case for an equivalent far-field system. Although a swept signal generator can be used for local oscillator (LO) and RF sources, nowadays it is more common to use a synthesiser as a signal source. The RF output from this source is fed via a (typically) 20 dB directional coupler to the transmit port of the AUT using phase-stable, but relatively high loss, typically 1 dB m−1 of coaxial cable. The test mixer is connected to the single port of the scanning probe by means of a short coaxial cable and is often padded with a 10 dB attenuator to reduce the effect of multiple reflections between the probe and the mixer. This mixer is fed with an LO signal that is 20 MHz above the RF signal, so that a 20 MHz intermediary frequency (IF) can be passed to the VNA receiver. When using a swept LO source this is phase locked, by means of a 10 MHz reference signal, 20 MHz above the RF source by convention. If the mixer is linear then locking the LO source 20 MHz below the RF source will conjugate the phase of the IF signal, which can lead to confusion when processing the measured phase data. Using a synthesiser for the LO is more popular today and the external clock option permits the source synthesizer clock to act as the master for both synthesizers. Occasionally, third, fifth or even seventh harmonic mixing is utilized to reduce the attenuation resulting from an electrically long LO test path. However, this is undesirable as the sensitivity of the RF system is inevitably degraded, for example, utilizing third harmonic mixing can result in a reduction in the sensitivity of the receiver of 20 dB. For high microwave frequencies and millimetre wave operation a more complex configuration is necessary and a typical system is shown in Figure 3.16. In Figure 3.16 fundamental millimetre wave mixers are used to generate a microwave (1.4 GHz) IF, which is then fed to a VNA system that is again mixed down to the VNA’s IF. Great care is now required to handle the three microwave cables from the probe carriage to ensure that phase stability is achieved. It should be noted that harmonic mixing does not help ease the phase stability requirements for the moving guided wave path (cable). Although the cables will change the phase of the signal less when we take the nth harmonic this will also similarly multiply up the phase change. The amplifier in Figure 3.16 is likely to be a solid state low noise amplifier (LNA) to keep weight low, as the probe carriage is generally limited in permitted mass. In Figure 3.17 a less complicated system is employed where the ninth harmonic mixes directly down to the VNA’s IF (typically 20 MHz) is used. Here the synthesizer is set to 11.11333 GHz such that 9×11.11333 = 100.02 GHz so producing a +20 MHz IF when mixed with the 100 GHz transmit signal. Returning to the microwave system of Figure 3.15 the multiplexed LO and IF signals are carried from test mixers on flexible phase-stable RF cables, one required

Introduction to near-field antenna measurements

49

Mounted on probe carriage Synthesizer 90 GHz

Gunn 91.4 GHz

Optional AUT

LO

Probe RF amp

Coupler

Mixer

Attenuator

Coupler −10 dB

IF

−10 dB

Harmonic mixer

Harmonic mixer

Synthesizer 11.335 GHz 0.68 GHz

0.72 GHz

Mixer DC bias

Mixer 1.4 GHz

Mixer 1.4 GHz Reference channel

Test channel VNA

Figure 3.16

Millimetre wave RF subsystem configuration based on eighth harmonic mixing at front end

for each channel, to the RF equipment via flexible conduit. This conduit confines the two RF cables for test channels to constant rolling bends to achieve phase stability. The mechanical dependence arises since the dielectric constant is a function of mechanical stress and strain. When the cables are twisted, the inside of the bend will be compressed and the outside will be stretched, thereby affecting the characteristic impedance that will vary in a complex fashion. Over recent years, the use of a pantograph and rotary joint assembly has become less popular, as the performance of rotary joints degrades with time and imposes limits on the frequency range that the facility can usefully operate. However, such systems can offer remarkably high performance when engineered to a very high standard, such as the system used for the 550 GHz (0.8 × 0.8 m) planar scanner for the NASA submillimetre wave astronomy satellite (SWAS) where a two arm mechanism interconnected with three specially selected rotary joints was used to meet the performance requirement of a 12 µm r.m.s. path length stability for the 5 GHz RF signal (an RF system similar to that of Figure 3.16 being employed) [6]. A potential problem with cables is that polytetrafluoroethylene (PTFE) dielectric can suffer a non-linear volumetric change at certain temperatures,

50

Principles of planar near-field antenna measurements Mounted on probe carriage Optional

100 GHz Multiplier head

AUT

Probe RF amp

Coupler −10 dB

Harmonic mixer IF

LO

100 GHz LO

Synthesizer 20 GHz

Harmonic mixer IF

RF amp

Isolator

−16 dB Synthesizer 11.1089 GHz

Reference channel 20 MHz

Coupler

Test channel 20 MHz VNA

Figure 3.17

A 100 GHz RF system based on ninth harmonic mixing directly down to the VNA’s 20 MHz IF

which will make phase stability at and around these temperatures difficult to control. The need for temperature stability is thus again re-enforced. Generally, such phase errors in the RF-path act in a very similar manner to z-plane errors (Chapter 9) and, provided that they are highly repeatable in nature, they can be measured and corrected during the transformation process. The reference RF signal obtained from the directional coupler is similarly mixed down to 20 MHz by the reference mixer. The lengths of the two LO paths are usually balanced, that is, path I and path II are of equal length to minimise the impact of phase variations resulting from thermal fluctuations. The concept is that if the two path lengths are the same, the relative phase variation between the respective RF paths will be zero, irrespective of how the temperature fluctuates within the facility. As one arm of the interferometer contains a probe, an AUT and free space this can never completely succeed; however, the concept has been found to be of some use. The IF signal is relatively low in frequency, that is, 20 MHz and for convenience it is usually carried from the test mixer to the receiver within the same coaxial cable as the LO signal. The simplified RF subsystem can be found represented in Figure 3.18. Many VNAs use the time convention ej(wt−kz)

(3.5)

Introduction to near-field antenna measurements

51

Test mixer RF RF

RF Source

Probe AUT

LO

20 dB directional coupler RF

Receiver IF

Reference mixer IF LO

LO Source Path I

Figure 3.18

Path II

Simplified schematic representation of RF subsystem

However, published literature, that is, IEEE antennas and propagation society, has adopted the opposite time convention namely ej(kz−wt)

(3.6)

Thus care must be taken to insure that the correct phase convention is adopted within the transformation process. This book, in common with the majority of works that utilize the angular spectrum, has adopted the positive time dependency. Reciprocity (c.f. Section 7.8) can be invoked to show that it makes no difference whether the AUT is characterised whilst in transmit or receive mode. However, for passive antennas, it is convenient to transmit from the AUT as the RF source can be placed directly behind the stationary AUT so that signal losses can be minimised. In practice, the bandwidth over which the facility as a whole can operate is determined by other factors, principally the planarity and precision of the robotics positioner and the absorption characteristics of the RAM placed within the chamber. A number of VNA vendors offer LO/IF distribution units and for clarity these have been omitted from these discussions as their function is merely to amplify, level and distribute the RF and LO signals. Receiver phase drift over the period of time taken by a complete planar scan is an important issue. It is, however, easily overcome by undertaking tie-scans or repeat measurements, where a single horizontal planar cut of phase data is used to correct the phase drift at the same vertical point in the set of vertical scans forming the planar measurement. The process is covered in detail in Chapter 6. Two further corrections that are often required to be compensated for are channel balance, which is usually required when using dual-port probes, and port displacement error, which is sometimes required when using a single-port probe and a mechanical rotation stage. These correction techniques are discussed in detail in Chapter 6 in Sections 6.6 and 6.5 respectively.

52

Principles of planar near-field antenna measurements

Figure 3.19

3.7

Photograph of 1.5 × 1.5 m, 50 GHz, frame scanner (Courtesy of NSI)

Robotics positioner subsystem

Currently there are two main designs of planar near-field scanner (PNFS), the box or frame scanner and the tower and rail inverted T-scanner. Whilst frame scanners can be used in either the horizontal or vertical orientations, T-scanners are usually limited to a vertical configuration. A typical robotics positioning subsystem is shown in Figure 3.19 and comprises of a 2.5 m high by 2.5 m wide vertical frame plane rectilinear near-field scanner. A well-designed frame scanner can potentially offer improved rigidity and positional accuracy over tower and rail designs, however, the metallic frame inevitably introduces additional unwanted scattering sources. Thus, one of the most attractive advantages of the inverted T-design is the reduction in multipath. A tower and rail design is shown in Figure 3.20(a) and in Figure 3.20(b) it is fully absorber covered and is being used to measure a corrugated horn at 10 GHz using a 0.9 × 0.9 m scanner. In most test systems a vertical configuration is chosen, since it affords easy access to the AUT and near-field probe that is crucial when configuring complex active array antennas and acquiring the alignment of the AUT in the range. However, there are special cases where the horizontal plane is employed as the scan plan as this enables the AUT to have uniform gravitational pull about its radiating aperture which can be crucial for ensuring deployable spacecraft antennas do not distort under earth based

Introduction to near-field antenna measurements (a)

(b)

AUT

Figure 3.20

53

Probe antenna

A 0.9 × 0.9 m 100 GHz, tower and rail scanner (courtesy of NSI and QMUL)

gravity. An example of a large horizontal planar scanner is shown in Figure 3.21 (see p. C1). Both types of scanner consist of two orthogonal carriages, X and Y , which are supported on a rigid framework to enable the scanning probe to be moved within the x-y scan plane. The probe carriage is mounted on the Y carriage that is located within the X carriage. The placement of absorber on the front face of the frame is crucial in minimising multiple reflections between the AUT and the probe, however, scattering from the sides of the frame is still an issue and this is an area where the tower and rail scanner offer an advantage. Frequency of operation of near-field systems vary enormously, with millimetre wave and submillimetre wave scanners requiring high planarity inert structures. This often involves granite bases and towers such as the state-of-the-art structure shown in Figure 3.22, which has a 2.8 × 2.8 m acquisition plane and operates at 650 GHz [7]. Sizes of scanner vary enormously, from small 0.6 × 0.6 m scanners used for small antennas through to the massive 33 × 16 m scanner shown in Figure 3.23 (see p. C1). As described in Chapter 6 particular attention needs to be paid to the placement of absorber behind the probe, as this passes through the region of greatest field intensity. See, for example, the planar facility shown in Figure 3.24 where a full anechoic chamber has not been used. Clearly, the mechanical vibrations experienced toward the top of the scan plane will be different to those lower down. However, in this instance, the frame design keeps this variation within acceptable limits, although larger vertical ranges can be found to be prone to thermal stratification. This can therefore be a motivation for adopting a horizontal configuration where the entire robotic positioner can be placed within a single thermal strata. The position of each of the axes is usually determined

54

Principles of planar near-field antenna measurements

Figure 3.22

Large aperture 650 GHz near-field measurement system (courtesy of NSI)

Figure 3.24

3 × 3 m planar scanner with absorber in critical locations only (Courtesy of QinetiQ)

Introduction to near-field antenna measurements

55

using rotary optical encoders, stepper motors or laser interferometers, all of which send a train of pulses as the positioners are moved. For example, a typical encoder system transmits 12,192 pulses per inch in the x- and y-axes and 80,000 pulses per inch in the z-axis. The in-plane resolution of the encoders corresponds to an upper frequency limit of 2.88 THz assuming that a positional tolerance of λ/50 is required. In practice, this is not realized as the planarity orthogonality and linearity of the axes is very much poorer than this. Crucially, these systems measure how far the positioner has travelled, rather than where the positioner is and are thus termed relative position encoders. Hence, in the event that the power supply is disrupted, which could occur whenever the control computer is restarted, the absolute position of the scanner is irrevocably lost. Generally, the positional error is specified as being smaller than λ/50. If this is not achievable, that is, either the scan plane is large or the measurement frequency is too high, then often this requirement is relaxed so that the repeatability is better than λ/50. The positional error is then calibrated, often with the use of laser interferometers and the positional error corrected within the transformation software, for example, k-correction. Unfortunately, unless very carefully implemented [8], this approach is often unsuccessful as many commercially available laser interferometers cannot acquire the position of the probe to the required degree of accuracy whilst the probe is in motion. In these cases, the tabulated positional errors do not correspond with the actual positional errors whilst the probe samples data on the fly as is usual in near-field antenna measurements. Here ‘on the fly’ implies that the probe does not stop at each data point but continuously moves at constant speed measuring the field value as the probe moves past the desired location. A data acquisition software suite is used to control the robotics and RF subsystems. The acquisition software provides four degrees of freedom, as it controls the electric motors that are used to drive the moving parts of the scanner by means of a power control unit (PCU) via a GPIB card or Ethernet connection. Each motor is used to drive one of the axes of motion x-, y-, z- or polarisation. The probe carriage is often moved with a parabolic velocity profile to minimise mechanical strain although for larger scanners a constant velocity profile is often maintained for the majority of the travel which eases timing issues. Generally, in order that acquisition times can be minimised, measurements are made whilst the probe carriage is in motion, that is, the measurements made on the fly. Although a finite time is taken to make a measurement once the probe carriage is in position, this is usually of sufficiently small duration that the positional error introduced is negligible. Many techniques are available for determining the relative phase between two waveforms, however, most require that the waveform be sinusoidal in form. Of these, one of the simplest techniques detects when the waveform passes through zero. This requires at least one full wavelength and in practice many are required because of averaging. So, if the IF is 20 MHz and the maximum velocity of the probe is 0.5 m s−1 the probe will have travelled 2.5 × 10−8 m, that is, 0.025 µ m during the sampling process. Thus, even if several wavelengths are required, the averaging effect introduced by the movement of the probe during this interval is generally considered to be unimportant. Generally, the largest positioning error results from the time delay between the acquisition computer

56

Principles of planar near-field antenna measurements

detecting that the probe has reached a sampling node and signalling the receiver to store the current measurement, as this can take many clock cycles to process. This is also the most commonly quoted reason for not running the acquisition software on a multitasking operating system.

3.8

Near-field probe

The near-field measurement technique places several requirements upon the characteristics of the scanning probe. These are 1. Time invariant gain and mechanically rigid. 2. No pattern nulls in the forward hemisphere, that is, low directivity, that is, electrically and physically small. 3. Wide bandwidth. 4. Low scattering cross section and reflection coefficient – that is, well matched with a small return loss. Unfortunately this requirement cannot usually be satisfied over a wide bandwidth. 5. Good polarisation purity that is, better than that of the AUT. 6. Good front to back ratio to minimise sensitivity to probe placing and multiple reflections. This is at odds with item 2. Typical near-field probes can include cylindrical waveguide, rectangular waveguide, corrugated horns and pyramidal horns. Two common probe antennas are the dual-port choked cylindrical waveguide probe and an open-ended rectangular waveguide probe (OEWG). Figure 3.25 shows a waveguide probe installed in a measurement facility. Here, the aperture of the waveguide probe has been carefully positioned to coincide with the intersection of the θ- and φ-axes of the range. In Figure 3.25, the probe assembly can be seen to consist of a chamfered rectangular waveguide section and a SWAM cone. The SWAM cone is an absorbent

Figure 3.25

Near-field probe installed in spherical range (Courtesy of SELEX)

Introduction to near-field antenna measurements

57

cone designed to minimise the diffraction around the probe area. The cylindrical section is included to displace the flat circular RAM sheet that is used to screen the azimuth positioner from the waveguide probe. Figure 3.26(a) contains a picture of an alternative x-band OEWG and absorber whilst Figure 3.26(b) contains a picture showing a family of OEWG probes each designed to span a different waveguide frequency band. Figure 3.27 contains a picture of a customised millimetre wave 100 GHz OEWG probe assembly. One of the most significant contributions, albeit an extremely systematic one, to the overall error budget of conventional planar near-field measurement technique results from inaccuracies in the characterisation of the near-field probe. (a)

(b)

Figure 3.26

Rectangular waveguide probe with different cartridges for a wide frequency range coverage (courtesy of NSI)

Figure 3.27

A 100 GHz open-ended waveguide probe (courtesy of QMUL)

58

Principles of planar near-field antenna measurements

Conventionally, as the measured main-component pattern is proportional to the maincomponent probe pattern, errors in the corrected main polarisation pattern arising from probe characterisation errors will be a one-to-one mapping. That is, they have the same magnitude and direction as the errors in the probe pattern. However, for some implementations of the plane polar, plane bipolar and auxiliary rotation nearfield measurements this error is not local. Rather, errors in one direction in the probe pattern affect the characterisation of the AUT over an extended angular region. Thus in order that the novel measurement configuration and transformation algorithms (see, for example, Chapter 9) can be successfully verified, errors contained within the knowledge of the probe pattern must be small when compared to those errors within the measurement scheme. The characterisation and modelling portion of this work is discussed in detail in Chapter 6. A generalized three-antenna technique is usually used for precise calibration of the on-axis gain and polarisation of a near-field probe and the resulting gain values can be generally certified to have an uncertainty of approximately 0.10 dB [9]. Once known the probe gain knowledge can then be used to determine the gain of the AUT as part of the near-field measurement process.

3.9

Generic antenna measurement process

The measurement of the antenna radiation pattern relative to the antenna mechanical interface necessitates accurate determination of the alignment of the mechanical interface that is associated with the antenna, relative to the range coordinate system during the acquisition of the near-field data. The known alignment is then compensated for during the transformation process. Once the alignment data has been captured, the near-field scanning process is performed. Following this, the measured near-field data is processed to yield corrected far-field results. These results are typically presented in Ludwigs III (copolar and cross-polar) polarisation basis [10] referenced to a specified electrical boresight system (that system which defines the copolar direction) which may or may not be coincident with the antenna plotting system (the output system in which the field quantities are tabulated). The measurement of absolute gain is a topic in itself and it is not the purpose of this book to describe it in detail. Typically, the substitution method using a calibrated standard is implemented. Figure 3.28 illustrates schematically the general near-field measurement process in the form of an overview. Near-field measurements present other difficulties. The sampling of data over a finite measurement interval results in the failure to sample a portion of the propagating near-field. This introduces truncation errors in the resulting far-field data that restricts antenna pattern coverage to less than 180◦ . The failure to account for the multiple reflections between probe and AUT within the theoretical description of near-field measurements results in the appearance of a series of concentric circles in the calculated far-field pattern that constitute a microwave equivalent of Newton’s rings. Furthermore, they introduce a loss in precision with which the gain can be determined. Additional uncertainties are introduced by incorrect reporting of the

Introduction to near-field antenna measurements

59

AUT

Measure AUT-range alignment

Acquire AUT nearfield pattern

Transform

Correct AUT-range -alignment

Copolar definition

Resolve polarisation

Standard antenna acquired data

Apply absolute gain or directivity offset

Probe pattern data

Plot

Figure 3.28

Generic near-field measurement process

position at which the amplitude and phase measurements have been taken. Such positional errors typically result from timing errors within the control subsystem, as well as from mechanical imperfections within the robotics of the positioner subsystem. In general, if the antenna under consideration is linear, of finite extent, operates at a single fixed frequency, radiates a constant power and, assuming that Maxwell’s equations accurately describe the region of space in which the antenna and measuring equipment is situated then, the only approximations involved within the formulation are 1. The fields outside the finite sampling interval are zero. 2. The AUT is aligned to the scanner with infinite precision. 3. The robotics subsystem positions the probe at the designated points in space with infinite precision.

60

Principles of planar near-field antenna measurements

4. The RF interferometer measures the impinged signal with infinite precision. 5. There are no multiple reflections between the AUT and the probe. 6. There are no truncation or rounding errors introduced by the data processing and plotting machinery. 7. There are no reflections (multipath) from the chamber enclosing the measurement. Subsequent chapters will consider some of these key issues.

3.10 Summary In this chapter, we have considered the practical aspects of measuring the near-field of an antenna in terms of scanning and RF subsystems. In the next chapter we will tackle the mathematical process of taking this measured near-field data and determining the radiated far-field. Subsequent chapters will address key issues associated with the full implementation of a planar near-field measurement system. Chapter 9 (Advanced planar near-field antenna measurements) addresses ways to tackle some of the major limitations associated with the planar near-field technique. Of particular note is the poly-planar approach where a number of separate planar scans taken at different orientations to the AUT can be ‘stitched together’ to offer wide angular far-field coverage with a modest sized planar scanner, enabling such facilities to offer a wider range of antenna measurement services.

3.11

References

1 Rayleigh, L.: ‘On images formed without reflection or refraction’, Philosophical Magazine, 1881;11:214–18 2 Rahmat-Samii, Y., Williams, L.I., and Yaccarino, R.G.: ‘The UCLA bi-polar planar-near-field antenna-measurement and diagnostics range’, IEEE Antennas and Propagation Magazine, December 1995;37:16–35 3 Yaghjian, A.D.: ‘Antenna coupling and near-field sampling in plane polar coordinates’, IEEE Transactions on Antennas and Propagation, March 1992;40 (3):304–12 4 Yaccarino, R.G.: ‘The bipolar planar near-field measurement technique, Part II: near-field to far-field transformation and holographic imaging methods’, IEEE Transactions on Antennas and Propagation, February 1994;42 (2):196–204 5 Newell, A.C., and Stubenrauch, C.F.: ‘Effect of random errors in planar nearfield measurements’, IEEE Transactions on Antennas Propagation, June 1998; AP-36:769–73 6 Slater, D.: ‘A 550 GHz near-field antenna measurement system for the NASA submillimeter wave astronomy satellite’, AMTA Conference, Norwood, MA, 3–7 October 1994

Introduction to near-field antenna measurements

61

7 Slater, D., Stek, P., Cofield, R., Dengler, R., Hardy, J., Jarnot, R., and Swindlehurt, R.: ‘A large aperture 650 GHz near-field measurement system for the earth observing system microwave limb sounder’, Proceedings of the Antenna Measurement Techniques Association 23rd Annual Meeting, Denver, CO, October 2001 8 Slater, D.: ‘Wide range straightness measuring system using a polarized multiplexed interferometer and centered shift measurement of beam polarization components’, US Patent Number 5,408,318, April 1995 9 Newell, A.C., Ward, R.D., and McFarlane, E.J.: ‘Gain and power parameter measurements using planar near-field techniques’, IEEE Transactions on Antennas Propagation, June 1998;AP-36 (6):792–803 10 Ludwig, A.C.: ‘The definition of cross-polarisation’, IEEE Transactions on Antennas Propagation, January 1973;AP-21 (1):116–9

Chapter 4

Plane wave spectrum representation of electromagnetic waves

4.1

Introduction

Somewhere, within almost every treatise on antenna theory and design can be found a passage, or more typically several passages, that describe techniques for determining the far-field properties of a given radiator using Maxwell’s equations, and utilizing a detailed knowledge of the mechanical structure of the antenna and the enforced current distribution. Although effective, such methodologies are overly restrictive for antenna measurements as such detailed information is not typically available to the antenna metrologist. Thus, recourse to more generic treatments becomes unavoidable. The approach usually adopted within the field of near-field antenna measurements is to sample the, potentially very complicated, electric field produced by an antenna over a convenient enclosing surface and then to try and determine the properties of that field elsewhere in space. In general, if the properties of an arbitrary, potentially complex, field distribution are known in one region of space it is difficult to determine the properties of that field elsewhere. However, this is not the case for certain types of elementary field, that is, mode. If for example, the amplitude, phase and direction of propagation of a plane wave are known, then the properties of this field can instantly be determined everywhere in space. Thus, if the complex field distribution associated with some radiator can be decomposed into a summation of plane waves propagating in diverse directions then similarly, the properties of the complex field distribution can be determined throughout space from a summation of the properties of the respective plane waves. Furthermore, solving this ‘modal expansion’ for the fields at an infinite distance results in the far-field pattern. Such modal expansions are not limited to the use of plane waves but rather, any elementary solution of the wave equation and Maxwell’s equations can be utilized, although a degree of mathematical convenience can be obtained from selecting a modal basis that matches the measurement geometry,

64

Principles of planar near-field antenna measurements

that is, by utilizing plane waves, cylindrical waves or spherical waves respectively for the case where the measurements are taken over planar, cylindrical or spherical surfaces. The theoretical development of planar near-field antenna measurements is usually based on this plane wave spectrum (PWS) representation of electromagnetic (EM) fields. This generalized interpretation can be shown to stem from the free-space solution of the scalar wave equation, which itself follows directly from classical EM theory and Maxwell’s equations where the four Maxwell equations are postulated, mathematical generalizations of a great many macroscopic experimental observations of electricity and magnetism. The validity of these phenomenological physical laws is attested to by the extraordinarily good agreement attained between measurement and prediction. This chapter aims to provide a clear, detailed derivation of the coordinate free form of the near-field to far-field transform that is central to the planar near-field antenna measurement methodology. The introduction of various assumptions and limitations within this derivation are highlighted and their impacts on this technique are discussed in some detail. As the development of the PWS representation is lengthy, an overview of the derivation is presented in the following section.

4.2

Overview of the derivation of the PWS

The PWS representation can be viewed as a generic algorithm for obtaining the particular solution of a problem in terms of specific boundary conditions. The interest in the angular spectrum method principally follows from the very close relationship between the asymptotic far-field pattern of the radiating antenna and the PWS representation of the radiated field. For the casual reader who is aware of this fact and whom merely wishes to obtain reliable far-field data, they may turn directly to Section 4.20 which contains the necessary algorithm and expressions required to accomplish this task. However, the intervening sections present a detailed derivation of the PWS representation of EM waves. This derivation takes the scalar wave equation of Chapter 2 as its starting point. The solution of this partial differential equation is sought using standard, but powerful, techniques for solving boundary value problems. Specifically, the particular solution of a given problem is sought to be expressed in terms of a Fourier series or in the limit a Fourier transform. This is accomplished by taking the two-dimensional Fourier transform of the three-dimensional wave equation (the time dependency is assumed to be sinusoidal and is thus suppressed). This with the use of a simple operator substitution, which is derived later on within this chapter, reduces the wave equation to a readily soluble ordinary differential equation. The particular solution to a given problem can be obtained applying the boundary conditions. This is shown to be accomplished by transforming the boundary values that are usually, but not always, specified in the spatial domain to the spectral domain. These equivalent spectral boundary values are then shown to be readily applicable to the general solution to obtain the particular solution. Thus, the full three-dimensional electric field is shown to be obtainable from the two-dimensional boundary conditions. As the scalar wave equation is equally applicable to electric and magnetic fields, similar results

Plane wave spectrum representation of EM waves

65

are quickly obtained for the magnetic spectra. Maxwell’s equations are then used to relate these symmetrical treatments thereby establishing that the entire six-vector EM field can be obtained from either the electric or magnetic, two-dimensional boundary conditions.

4.3

Solution of the scalar Helmholtz equation in Cartesian coordinates

In Chapter 2, it was found that all of the components of the EM field obey the scalar differential wave (Helmholtz) equation. ∇ 2 u + k02 u = 0

(4.1)

This differential equation can be solved by direct integration using Green’s theorem to yield the Kirchhoff integral theorem. Instead, however, the following sections will set out an alternative methodology for utilizing this equation that is more convenient when considering most, but not all, problems encountered in the study of planar near-field antenna metrology.

4.3.1 Introduction to integral transforms One explanation for the appearance of a Fourier integral in the relationship between both propagating and evanescent (disappearing) field components and the spectral components is; if the scalar wave equation, a hyperbolic equation, is solved using the method of separation of variables and the principle of superposition, the resulting summation of orthogonal functions can be seen to be a Fourier series or in the limit, a Fourier transform. However, the method of separation of variables is a powerful but not universally applicable technique for solving differential equations [1]. If the differential equation in question is non-homogeneous or if the domain of the variables concerned is infinite or even semi-infinite, then an alternative, more generally applicable, integral transform technique may prove successful [2, 3]. Although integral transforms are a commonly used tool in the study of differential equations, they are not routinely applied directly to antenna theory. The integral transform is used to separate variables and reduce a partial differential equation to an ordinary differential equation. When transformed, it is often found that such problems reduce to simple algebraic expressions, for example, a convolution in the spatial domain becomes a multiplication in the transform domain. If the boundary conditions are also transformed, then this differential equation may be readily solved in the transform domain before the inverse transform is taken to recover the general solution. This apparently complicated procedure can often be utilized to circumnavigate the requirement to tackle a relatively difficult problem in the original problem space.

4.3.2 Fourier transform solution of the scalar Helmholtz equation Maxwell’s equations are linear if the medium itself is linear. That is, the properties of the medium do not depend upon the magnitude of the electric or magnetic fields

66

Principles of planar near-field antenna measurements

propagating within them. The principle of linear superposition may thus be invoked and an integral transform may be used. Although an infinite number of possible transforms exist, one of the most useful is the Fourier transform, where the kernel is ejαt , as the inverse of this operation, that is, the inverse transform, is easily obtained. The utility of an integral transform approach lies within the possibility that the original problem can be solved only with difficulty in the original coordinate system whilst the transform of the problem is more readily soluble. To solve the vector Helmholtz equation which is a partial differential equation in four coordinates, three spatial and one temporal (although this time dependency is suppressed through an operator substitution, that is, an additional transform) do the following: 1. Express the Laplacian operator in Cartesian coordinates that reduces the Helmholtz equation to three uncoupled scalar wave equations. 2. Take the Fourier transform of the Helmholtz equation so that it is reduced to an ordinary differential equation. 3. Transform the boundary conditions and solve the differential equation in the transform domain. 4. Take the inverse transform to obtain the particular solution in the original domain. If the field can be represented as a region of a planar surface with a radiating field distribution or aperture illumination function across it we can write u(x, y, z = 0) = f (x, y)

(4.2)

Here, u(x, y, z) is the solution of the scalar Helmholtz equation with the boundary condition specified by f (x, y). Let the two-dimensional Fourier transform of the solution u(x, y, z) with respect to x, y be
∞
∞ U (kx , ky , z) = {u(x, y, z)} =

u(x, y, z)ej(kx x+ky y) dx dy

(4.3)

−∞ −∞

Here, the  symbol is used to denote the Fourier transform operation. As this is a Fourier transform its inverse can instantly be written as −1

u(x, y, z) = 

1 {U (kx , ky , z)} = 4π 2


∞
∞ −∞ −∞

U (kx , ky , z)e−j(kx x+ky y) dkx dky (4.4)

where the variables x, y and kx , ky are assumed to be real whilst U (kx , ky , z) and u(x, y, z) may be complex, that is, analytic functions. Occasionally, functions that are analytic are termed holomorphic or regular. Here, the functions U (kx , ky , z) and u(x, y, z) are commonly referred to as a Fourier transform pair and in this case both 1/2π normalization factors have been lumped together within the inverse transform. Here, the forward and inverse transforms are defined so that applying an inverse transform to the transform of a function will return the original function and applying

Plane wave spectrum representation of EM waves

67

the transform to the inverse transform of a function will also return the original function. Clearly then u(x, y, z) = −1 {{u(x, y, z)}}

(4.5)

The condition for the existence of U (kx , ky , z) is usually given by requiring that the function u(x, y, z) should not be ‘pathological’, that is, that it does not oscillate with infinite rapidity and that it should be absolutely integrable. Thus, the following condition is satisfied
∞
∞ |u(x, y, z)|dx dy < ∞

(4.6)

−∞ −∞

Although it is possible for functions to fail to satisfy this condition and still have Fourier transforms, for example, functions of slow growth, such functions shall not be considered hereinafter, as such behaviour is not usually encountered in the study of EM systems. Often, the general conditions that apply to Fourier analysis are referred to as Dirichlet. By applying the Fourier transform to the wave equation, that is, by multiplying by ej(kx x+ky y) and integrating x, y from −∞ to ∞ in each dimension, obtains
∞
∞  −∞ −∞

∂ 2 u(x, y, z) ∂ 2 u(x, y, z) ∂ 2 u(x, y, z) + + ∂x2 ∂y2 ∂z 2 

+ k02 u(x, y, z)

ej(kx x+ky y) dx dy = 0

(4.7)

Here, x, y and kx , ky are conjugate variables. Expanding this yields
∞
∞  −∞ −∞

∂ 2 u(x, y, z) j(kx x+ky y) ∂ 2 u(x, y, z) j(kx x+ky y) e + e ∂x2 ∂y2  ∂ 2 u(x, y, z) j(kx x+ky y) 2 j(kx x+ky y) dx dy = 0 + e + k0 u(x, y, z)e ∂z 2

(4.8)

This complicated looking expression can be considerably simplified with the use of the following operator substitution which is derived in Section 4.5 in Box 4.3 

∂ n u(α, γ )  ∂α n



= (−jβ)n U (β, γ )

(4.9)

Again, the symbol  is taken to represent the Fourier transform operator that acts upon the functions contained within the curly braces immediately to its right. When written

68

Principles of planar near-field antenna measurements

explicitly, the necessary operator substitutions are
∞
∞ −∞ −∞


∞
∞ −∞ −∞

∂ 2 u(x, y, z) j(kx x+ky y) e dx dy = −kx2 U (kx , ky , z) ∂x2

(4.10)

∂ 2 u(x, y, z) j(kx x+ky y) e dx dy = −ky2 U (kx , ky , z) ∂y2

(4.11)

And from definition,
∞
∞ U (kx , ky , z) =

u(x, y, z)ej(kx x+ky y) dx dy

(4.12)

−∞ −∞

Then, − kx2 U (kx , ky , z) − ky2 U (kx , ky , z) +


∞
∞

−∞ −∞

+ k02 U (kx , ky , z)

∂ 2 u(x, y, z) j(kx x+ky y) e dx dy ∂z 2

=0

(4.13)

Provided that the order of integration and differentiation may be interchanged we may write that
∞
∞ −∞ −∞

∂ 2 u(x, y, z) j(kx x+ky y) ∂2 e dx dy = ∂z 2 ∂z 2


∞
∞

u(x, y, z)ej(kx x+ky y) dx dy

−∞ −∞

∂ 2 U (k

x , ky , z) (4.14) ∂z 2 Thus, the Fourier transform of the wave equation can be expressed in the transform domain as       d2 U kx , ky , z   2 2 + k02 U kx , ky , z = 0 −kx U kx , ky , z − ky U kx , ky , z + 2 dz (4.15)

=

Since only derivatives with respect to z are present, it is legitimate to replace the partial derivatives with ordinary derivatives. Factoring yields      d2 U kx , ky , z 2 2 2 + k − k − k (4.16) U kx , ky , z = 0 x y 0 dz 2 As no derivatives with respect to x or y appear, the problem of solving a partial differential equation has been reduced and thus simplified, to that of solving an ordinary differential equation. In order to preserve the validity of this operator substitution we have been compelled to assume that the solution u(x, y, z) and its derivatives ∂u (x, y, z)/∂x, ∂u (x, y, z)/∂y are small for large values of x, y and approach zero

Plane wave spectrum representation of EM waves

69

as x, y → ±∞. This is not found to be too overly restrictive an assumption for most physical situations (that is, consider the medium to be slightly ‘lossy’), however, it renders the general solutions invalid in these limits. The characteristic or auxiliary, equation of this is  ζ 2 + k02 − kx2 − ky2 = 0

(4.17)

whereupon the solutions of this auxiliary equation can be obtained either from the quadratic equation or by factoring. In this case, the factorisation is trivial and yields

  1/2 

1/2  2 2 2 2 2 2 ζ − k0 − k x − k y j ζ + k0 − k x − k y j =0

(4.18)

 1/2 Hence, the roots of this expression are distinct, that is, ζ = k02 − kx2 − ky2 j and 1/2  j which can be seen to form a purely imaginary complex ζ = − k02 − kx2 − ky2

conjugate pair. Two linearly independent solutions are e(a+jb)x and e(a−jb)x . As neither of these functions are constant multiples of the other, the general solution is obtained from a linear superposition thus y = d1 e(a−jb)x + d2 e(a+jb)x

(4.19)

 1/2 where d1 and d2 are arbitrary constants. As a = 0 and b = k02 − kx2 − ky2 we may write that 







U kx , ky , z = d1 kx , ky e

 1/2 −j k02 −kx2 −ky2 z





+ d2 kx , ky e

 1/2 j k02 −kx2 −ky2 z

(4.20)

    where, d1 kx , ky and d2 kx , ky are constants with respect to z. Now −1

u (x, y, z) = 

  1 U kx , ky , z = 4π 2


∞
∞



  U kx , ky , z e−j(kx x+ky y) dkx dky

−∞ −∞

(4.21)

thus 1 u (x, y, z) = 4π 2


∞
∞ 





d1 kx , ky e −∞ −∞

  j + d2 kx , ky e



k02 −kx2 −ky2

 1/2 −j k02 −kx2 −ky2 z

1/2

z



e−j(kx x+ky y) dkx dky

(4.22)

70

Principles of planar near-field antenna measurements 1 u (x, y, z) = 4π 2


∞
∞





d1 kx , ky e

 1/2  −j kx x+ky y+ k02 −kx2 −ky2 z

−∞ −∞

1 + 4π 2


∞
∞





d2 kx , ky e

dkx dky

1/2   −j kx x+ky y− k02 −kx2 −ky2 z

−∞ −∞

dkx dky (4.23)

In the above expression, d1 and d2 are called plane wave spectra because the expressions 



d1 kx , ky e and 



d2 kx , ky e

 1/2  z −j kx x+ky y+ k02 −kx2 −ky2

 1/2  −j kx x+ky y− k02 −kx2 −ky2 z





= d1 kx , ky e





= d2 kx , ky e

 1/2  z −j kt ·r+ k02 −kx2 −ky2

(4.24)

1/2   −j kt ·r− k02 −kx2 −ky2 z

(4.25)

in the integrands represents uniform plane waves propagating in the same  tangential, k t , direction but in opposite longitudinal directions. Thus, d1 kx , ky represents an angular spectrum composed of waves travelling with a component in the positive  z-direction, that is, representing the retarded wave solution, whilst d2 kx , ky represents an angular spectrum composed of waves travelling with a component in the negative z-direction, that is, the advanced wave solution. Importantly, both components are valid as they are solutions of Maxwell’s equations, the vector Helmholtz equation, they are continuous, single valued and finite over all values that x, y and z can take [4]. Such auxiliary postulates are usually applied in mathematical physics to functions representing physical quantities. The extent to which the terms in the general solution enter into the particular solution is dependent on the nature of the source of the fields’ that is, the boundary conditions that constitute the geometry of the system under consideration. For these discussions, the source can be considered to be located in the region z ≤ 0 and to be radiating into the half-space z > 0, so that the waves are travelling in   the +z-direction only implying that the second term is not present thus d2 kx , ky = 0. If the situation were to be reversed so that the waves were propagating in the −z-direction, then d1 kx , ky = 0 [5]. This is in agreement with the convention usually adopted in classical electrodynamics, as opposed to relativistic field theories (see Box 4.1), that only solutions representing the retarded potential are to be considered. Thus, in agreement with convention hereinafter we assume the general solution to be of the form 1 u (x, y, z) = 4π 2


∞
∞





d1 kx , ky e −∞ −∞

 1/2  z −j kx x+ky y+ k02 −kx2 −ky2

dkx dky (4.26)

Plane wave spectrum representation of EM waves

71

Box 4.1 It is a fundamental requirement of all physical laws that they conform to the principle of relativity. This states that ‘the laws of physics must be such that it is possible to state them in the same form for observers in all inertial frames of reference’ and this has implications for any thorough theoretical explanation. The energy transfer between, say, the transmit (Tx) antenna and the receiver (Rx) antennas follows a light like path, which constitutes an event horizon between time like and space like displaced events in spacetime. Thus, along this path time is infinity dilated, meaning that in this frame of reference no time passes between transmitting and receiving a signal. This in turn means that in this frame of reference transmission and reception are simultaneous and therefore cannot be causally related. If they are not causally related in this frame of reference then the principle of relativity requires that they are not causally related in any frame of reference. Thus any interpretation of EM field theory which conventionally abandons the advanced wave solution of the wave equation as not representing physical reality is doing so arbitrarily and not as a result of any constraint implied by physical law.

In order that we may evaluate these constants, we will need to apply some rudimentary boundary conditions. Using the initial condition that u (x, y, z = 0) = f (x, y)

(4.27)

in the transform domain this can be expressed as   F kx , ky =  {f (x, y)} =


∞
∞

f (x, y) ej(kx x+ky y) dx dy

(4.28)

−∞ −∞

     Here, F kx , ky are the Fourier spectra of f (x, y), so that F kx , ky  represent the    magnitude (amplitude) spectra of f (x, y) and the imaginary parts of loge F kx , ky denote the phase spectra. When z = 0 we have       U kx , ky , z = 0 = d1 kx , ky = F kx , ky (4.29) So, the Fourier transform of the solution of the wave equation for any value of z can be obtained from the Fourier transform of the solution of the wave equation at z = 0 as 







U kx , ky , z = F kx , ky e

 1/2  −j kx x+ky y+ k02 −kx2 −ky2 z

(4.30)

By taking the inverse two-dimensional Fourier transform, the general solution can be expressed in terms of the Fourier transform of the boundary conditions so that   u (x, y, z) = −1 U kx , ky , z (4.31)

72

Principles of planar near-field antenna measurements Then clearly

  u (x, y, z) = −1 U kx , ky , z 1 = 4π 2


∞
∞





F kx , ky e

 1/2  z −j kx x+ky y+ k02 −kx2 −ky2

dkx dky

(4.32)

−∞ −∞

  Here u (x, y, z) is a solution of the scalar Helmholtz equation, F kx , ky is an analytic function where the integration can be carried out over any path in the kx , ky plane. Motivated by a requirement to relate the variables k0 , kx , ky let the three-dimensional Fourier transform of the solution u (x, y, z) with respect to x, y, z be
∞
∞
∞

  U kx , ky , kz =  {u (x, y, z)} =

u (x, y, z) ej(kx x+ky y+kz z) dx dy dz

−∞ −∞ −∞

(4.33)

and its inverse be

  u (x, y, z) = −1 U kx , ky , kz 1 = 8π 3


∞
∞
∞

  U kx , ky , kz ej(kx x+ky y+kz z) dkx dky dkz

(4.34)

−∞ −∞ −∞

where again the normalization constant has been lumped together in the inverse transform. As before, it is assumed that the solution u (x, y, z) and its partial derivatives, ∂u (x, y, z)/∂x, ∂u (x, y, z)/∂y, ∂u (x, y, z)/∂z are small for large values of x, y, z and approach zero as x, y, z → ±∞. By taking the three-dimensional Fourier transform to the wave equation ∂ 2 u (x, y, z) ∂ 2 u (x, y, z) ∂ 2 u (x, y, z) + + + k02 u (x, y, z) = 0 ∂x2 ∂y2 ∂z 2

(4.35)

that is, by multiplying by ej(kx x+ky y+kz z) and integrating x, y, z from −∞ to ∞ in each successive dimension we obtain         −kx2 U kx , ky , kz − ky2 U kx , ky , kz − kz2 U kx , ky , kz + k02 U kx , ky , kz = 0 (4.36) where the following operator substitution  n  ∂ u (α, γ )  = (−jβ)n U (β, γ ) ∂α n has been utilized. Factoring yields    k02 − kx2 − ky2 − kz2 U kx , ky , kz = 0

(4.37)

(4.38)

Plane wave spectrum representation of EM waves

73

This equality can only hold if the summation in the brace equates to zero or if the transform of the solution is equal to zero. Ignoring the trivial solution of  U kx , ky , kz = 0 then we obtain the relationship between the various ‘separation’ constants, that is, k02 − kx2 − ky2 − kz2 = 0

(4.39)

k02 = kx2 + ky2 + kz2

(4.40)

or

Thus, for a fixed frequency only two components may vary independently and the third may be obtained as  kz = ± k02 − kx2 − ky2 (4.41) As all real values of kz are admitted, the following radicals are chosen to ensure that the solution is finite over all space and to insure that the radiation condition is satisfied, that is, for each homogeneous plane wave of the spectrum, the solution must ensure that the time averaged power flux is directed away from the plane z = 0. Thus   k 2 − k 2 − k 2, if kx2 + ky2 ≤ k02 x y 0  (4.42) kz = −j k 2 + k 2 − k 2 otherwise x y 0 For kx2 + ky2 ≤ k02 the positive root is chosen to ensure that the waves propagate outward in the positive z-direction, whilst the negative imaginary root is chosen when kx2 + ky2 > k02 so that the waves decay exponentially as they propagate outward in the positive z-direction and remain finite as z → ∞. If instead the opposite sign had been chosen, then this would represent an exponentially increasing wave that would become infinitely large as r became infinitely large. This can be seen to be true because if kz contained a positive imaginary component namely kz = kzr + jkzi where kzr and kzi are both real then e−j(kx x+ky y+{kzr +kzi }z) = e−j(kx x+ky y+kzr z) ekzi z

(4.43)

which clearly contains an exponential increasing amplitude factor. If however, this solution was finite in this limit, then this solution could not be rejected. In the case of z < 0, −j should be changed to +j. Thus, we may write that the solution of the scalar Helmholtz equation is derived from the Fourier transform of the boundary conditions in a more compact form as 1 u (x, y, z) = 4π 2


∞
∞

  F kx , ky e−j(kx x+ky y+kz z) dkx dky

(4.44)

−∞ −∞

This is clearly a physically satisfying result. By way of illustration let us consider the behaviour of the free-space Greens function, as in the limit of the far-field

74

Principles of planar near-field antenna measurements

it behaves in a manner which is similar to that of a general far-field antenna pattern function. The free-space Greens function, which is a radial spherical mode and experiences a soft singularity when r = 0, can be expressed as ψ=

e−jk0 r r

(4.45)

Clearly, as the amplitude of the free-space Greens function reduces by an amount that is inversely proportional to distance, this implies that the rate of change of amplitude with distance will tend to zero in the far-field as the distance tends to infinity. Conversely, the rate of change of the phase of the free-space Greens function is obviously independent of distance as it is a constant. Consequently, the corresponding angular spectra of two coplanar field distributions, that is, two antenna apertures, spaced apart by a few wavelengths in z will differ only by a phase factor. Displacing one plane by a distance z0 in the z-axis will result in a change in path length of z0 cos θ where θ is the polar angle, that is, that angle measured away from the positive z-axis, compare with Figure 4.1. The free-space electrical length, that is, the phase, is related to the physical length l by ϕ = k0 l

(4.46)

where ϕ denotes the electrical length. Thus, the difference in phase between the two angular spectra will be characterised by ϕ = k0 z0 cos θ = kz z0

(4.47)

which is in agreement with our rigorous analysis presented above. By way of a check, we may substitute the integral solution into the left-hand side of the scalar Helmholtz

Y-axis

P (far-field point)

∆z θ

Z-axis Z0 1st measurement

2nd measurement plane

plane

Figure 4.1

Schematic representation of plane-to-plane translation

Plane wave spectrum representation of EM waves

75

equation and check to ensure that the sum of the terms is zero. This yields  
∞
∞  −j(k x+k y+k z)  ∂2  1 x y z dk dk  0= 2 F kx , ky e x y ∂x 4π 2 

−∞ −∞

∂2 1 + 2 2 ∂y 4π 

∂2 1 + 2 2 4π ∂z  + k02 

1 4π 2


∞
∞ −∞ −∞


∞
∞

  −j(k x+k y+k z)  x y z dk dk  F kx , ky e x y   −j(k x+k y+k z)  x y z dk dk  F kx , ky e x y

−∞ −∞


∞
∞

   −j(k x+k y+k z) x y z dk dk  F kx , ky e x y

(4.48)

−∞ −∞

where, if this solution satisfied the wave equation S, which represents the sum of these terms, it will be identically zero. Utilizing the operator substitution for the derivative of a Fourier transform obtains 0

= − kx2

− ky2

− kz2

+ k02

1 4π 2 1 4π 2 1 4π 2 1 4π 2


∞
∞

  F kx , ky e−j(kx x+ky y+kz z) dkx dky

−∞ −∞


∞
∞

  F kx , ky e−j(kx x+ky y+kz z) dkx dky

−∞ −∞


∞
∞

  F kx , ky e−j(kx x+ky y+kz z) dkx dky

−∞ −∞


∞
∞

  F kx , ky e−j(kx x+ky y+kz z) dkx dky

(4.49)

−∞ −∞

Whereupon, dividing by the Fourier transform of the solution of the scalar Helmholtz equation yields 0 = k02 − kx2 − ky2 − kz2

(4.50)

Clearly these solutions satisfy the scalar Helmholtz equation. This can be seen to follow from the fact that each component wave satisfies Maxwell’s equations, consequently any resultant obtained by superposition will likewise satisfy the field equations. Although, as illustrated above, the Helmholtz equation is a direct consequence of Maxwell’s equations, the converse is not true. Consequently, care must be taken to ensure that any solution of the wave equation also constitutes a solution of Maxwell’s equations. An alternative, more conventional, but ultimately

76

Principles of planar near-field antenna measurements

less universally applicable technique for solving the scalar Helmholtz equation is presented in Box 4.2.

Box 4.2 As shown above, the scalar Helmholtz equation in rectangular coordinates is ∂ 2 u (x, y, z) ∂ 2 u (x, y, z) ∂ 2 u (x, y, z) + k 2 u (x, y, z) = 0 + + ∂x2 ∂y2 ∂z 2

(4.51)

The method of separation of variables seeks to find a solution of the form of u (x, y, z) = X (x) Y (y) Z (z)

(4.52)

Substituting this expression into the Helmholtz equation yields 0=

∂ 2 (X (x) Y (y) Z (z)) ∂x2 +

∂ 2 (X (x) Y (y) Z (z)) ∂y2

+

∂ 2 (X (x) Y (y) Z (z)) ∂z 2

+ k 2 X (x) Y (y) Z (z)

(4.53)

or 0 =Y (y) Z (z)

∂ 2 X (x) ∂x2

+ X (x) Z (z)

∂ 2 Y (y) ∂y2

+ X (x) Y (y)

∂ 2 Z (z) ∂z 2

+ k 2 X (x) Y (y) Z (z) Dividing by the solution yields 0=

1 ∂ 2 X (x) X (x) ∂x2 +

1 ∂ 2 Y (y) Y (y) ∂y2

(4.54)

Plane wave spectrum representation of EM waves

Box 4.2 Continued +

1 ∂ 2 Z (z) Z (z) ∂z 2

+ k2

(4.55)

Consider rearranging the above expression to obtain 1 ∂ 2 X (x) 1 ∂ 2 Z (z) 1 ∂ 2 Y (y) − = −k 2 − 2 2 Y (y) ∂y Z (z) ∂z 2 X (x) ∂x

(4.56)

The left-hand side is a function of x alone, whilst the right-hand side depends upon y and z only. So, a function of x is equated to a function of y and z, however, x, y and z are all independent variables. This can only be true if each side is equal to a constant, a constant of separation. Thus let 1 ∂ 2 X (x) = −kx2 X (x) ∂x2

(4.57)

1 ∂ 2 Y (y) = −ky2 Y (y) ∂y2

(4.58)

1 ∂ 2 Z(z) = −kz2 Z(z) ∂z 2

(4.59)

where, kx2 , ky2 , kz2 are referred to as separation constants and for convenience they are chosen to be the square of another constant. By substituting these expressions into the expression above, we obtain the relationship between the separation constants − kx2 − ky2 − kz2 + k 2 = 0

(4.60)

k 2 = kx2 + ky2 + kz2

(4.61)

The separated ordinary differential equation may be written in a more convenient form as ∂ 2 X (x) ∂ 2 Y (y) + kx2 X (x) = + ky2 Y (y) 2 ∂x ∂y2 ∂ 2 Z (z) + kz2 Z (z) ∂z 2 =0

=

(4.62)

As no derivatives with respect to x or y appear, we have succeeded in reducing a partial differential equation to an ordinary differential equation. These

77

78

Principles of planar near-field antenna measurements

Box 4.2 Continued expressions are all of the same form and are referred to as harmonic equations. Any solution of the harmonic equation we shall call a harmonic function. ∂ 2 U (u) + ku2 U (u) = 0 ∂u2

(4.63)

One possible solution of this differential equation is, ejku u . As, any linear combination of harmonic functions is also a harmonic function, more general wave functions may be obtained by summing over all possible choices for the two independent separation parameters, that is, u (x, y, z) =

∞ 

∞ 

  F kx , ky ejkx x ejky y ejkz z

(4.64)

kx =−∞ ky =−∞

Finally, the most general wave functions can be constructed by integrating over all values of the separation parameters, that is,
∞
∞ u (x, y, z) =

  F kx , ky ejkx x ejky y ejkz z dkx dky

(4.65)

−∞ −∞

4.4

On the choice of boundary conditions

Within the above derivation, it has been assumed that the aperture illumination function f (x, y) is known over the entire z = 0 plane and is small when x and y are large. In practice, all real sources, that is, sources that occupy a finite region of space and radiate finite power, must be of finite extent so that their illumination function can be expressed mathematically with no loss of accuracy  f (x, y) When, x1 ≤ x ≤ x2 and y1 ≤ y ≤ y2 (4.66) f (x, y) = 0 Elsewhere as

Clearly then, the limits of integration can be collapsed from infinite to finite values 




y2
x2

F kx , ky =  {f (x, y)} =

f (x, y) ej(kx x+ky y) dxdy

(4.67)

+y1 +x1

Now, depending upon which of the field components f is taken to represent can result in the imposition of various characteristics on the resulting angular spectrum. The most commonly used choice is to assume that f is taken in turn to denote each of the two orthogonal electric field components that are tangential to the antenna aperture

Plane wave spectrum representation of EM waves plane so that the following Fourier transform pairs are formed   Fx kx , ky z = 0 ⇔ fx (x, y, z = 0)   Fy kx , ky z = 0 ⇔ fy (x, y, z = 0)

79

(4.68) (4.69)

In this case, the boundary conditions are akin to specifying that the aperture illumination function is set in an infinite, perfectly conducting ground plane, that is, a perfect electrical conductor (PEC). Thus, outside the aperture, that is, on the infinitely conducting plane the following relationship holds nˆ × E (x, y, z = 0) = 0

(4.70)

Here nˆ is taken to denote the surface unit normal, which in this example has a component purely directed in the z-axis. Thus, it is more convenient to determine the total electric and magnetic field vectors from the tangential components of the electric field, rather than from the magnetic field. However, outside the aperture on the surface of the infinitely conducting material the normal electric field component will not necessarily be identically zero that is nˆ · E (x, y, z = 0)  = 0

(4.71)

Thus, if the normal component of the angular spectrum is obtained directly from the Fourier transform of the normal electric field component the limits of integration will not collapse to the area of the aperture and instead must be obtained from 




∞
∞

Fn kx , ky =

fn (x, y) ej(kx x+ky y) dxdy

(4.72)

−∞ −∞

If the limits of integration are reduced to a finite region of the x–y plane then this is necessarily equivalent to setting the aperture in a perfect magnetic conductor (PMC) where nˆ × H (x, y, z = 0) = 0

(4.73)

nˆ · E (x, y, z = 0) = 0

(4.74)

and The introduction of this additional and often unnecessary, assumption can cause confusion when comparing values for the normal field component obtained by applying the plane wave condition and from direct integration.

4.5

Operator substitution (derivative of a Fourier transform)

Within the development of the PWS representation, a general operator substitution was utilized within the integral transform method of solution of the scalar Helmholtz equation which is a second order differential equation, that is, a hyperbolic equation where n = 2. The derivation of this operator substitution introduces some additional

80

Principles of planar near-field antenna measurements

restrictions that are not generally noted within the literature and that have an importance for noncoplanar analysis. Provided that the reader is prepared to accept the operator substitution  n  ∂ u (x, t)  = (−js)n U (s, t) (4.75) ∂xn and only utilize this relationship in cases where ∂ n−1 u (x, t)/∂xn−1 and all lower derivatives exist and tend to zero as |x| → ∞ then Box 4.3 can be omitted.

Box 4.3 Let the Fourier transform of the solution u (x, t) with respect to x be
∞ U (s, t) =  {u (x, t)} =

u (x, t) ejsx dx

(4.76)

−∞

We shall be required to assume that the solutions u (x, t) and its derivative ∂u (x, t)/∂x are small for large values of |x| and approach zero as x → ±∞. Let ux (x, t) =

∂u (x, t) ∂x

(4.77)

Consider
∞

ux (x, t) ejsx dx

 {ux (x, t)} =

(4.78)

−∞

By using the method of definite integration by parts when u = ejsx then du =  jsx jse dx and dv = ux (x, t) dx then v = ux (x, t) dx = u (x, t) we obtain that
∞

∞ udv = e u (x, t)−∞ − jsx

 {ux (x, t)} =


∞

u (x, t) jsejsx dx

(4.79)

−∞

−∞

As u (x, t) → 0 when x → ±∞ then
∞  {ux (x, t)} = −js

u (x, t) ejsx dx = −jsU (s, t)

(4.80)

−∞

Hence



∂u (x, t)  ∂x

 = −jsU (s, t)

(4.81)

Plane wave spectrum representation of EM waves

81

Box 4.3 Continued Now, a repeated application of this formula will yield  2  ∂ u (x, t)  = −s2 U (s, t) ∂x2

(4.82)

where we have been forced to assume that ux (x, t) → 0 when x → ±∞. Through the repeated application of the formulae above the general operator substitution can be obtained and written as  n  ∂ u (x, t)  = (−js)n U (s, t) . (4.83) ∂xn The assumption that ∂ n−1 u (x, t)/∂xn−1 and all lower derivatives exist and are small for large values of |x| and approach zero as x → ±∞ is strictly adhered. This can readily be extended to two or more dimensions to obtain the general operator substitution quoted in Section 4.3.2 above.

4.6

Solution of the vector Helmholtz equation in Cartesian coordinates

As both Maxwell’s equations and the Helmholtz equation must be satisfied, some additional restrictions are necessarily placed upon any possible solution. The implications of this can be investigated further by substituting the above Fourier integral solution into ∇ · E = 0. Thus, the divergence of the electric field can be expressed in terms of the general solution as  
∞
∞   −j(k x+k y+k z) ∂  1 x y z dk dk  · e 0= ˆx F kx , ky e x y ∂x 4π 2 −∞ −∞

 
∞
∞   −j(k x+k y+k z) ∂  1 x y z dk dk  · e ˆy + F kx , ky e x y ∂y 4π 2 −∞ −∞

 +

∂  1 ∂z 4π 2


∞
∞







F kx , ky e−j(kx x+ky y+kz z) dkx dky  · eˆ z

(4.84)

−∞ −∞

Essentially then we are attempting to solve Maxwell’s equations by trying to obtain a solution of the wave equation (which is the Fourier transform part of the above equation) whose divergence is zero. The divergence of the field is zero because we are attempting to find a solution in free space thus no ‘sources’ or ‘sinks’ are

82

Principles of planar near-field antenna measurements

present. So utilizing the operator substitution for the derivative of a Fourier transform yields 
∞
∞   −j  kx Fx kx , ky e−j(kx x+ky y+kz z) dkx dky 0= 2 4π −∞ −∞


∞
∞ + ky

  Fy kx , ky e−j(kx x+ky y+kz z) dkx dky

−∞ −∞


∞
∞ + kz







Fz kx , ky e−j(kx x+ky y+kz z) dkx dky 

(4.85)

−∞ −∞

Differentiating both sides with respect to kx , ky and dividing by common factors obtains kx Fx (kx , ky ) + ky Fy (kx , ky ) + kz Fz (kx , ky ) = 0

(4.86)

When rewritten using vector notation this can be expressed succinctly as k ·F =0

(4.87)

Alternatively, this can be expressed in an equivalent form in terms of the radial spherical unit vector eˆ r · F = 0

(4.88)

Hence, only two field components may be specified independently, the third being fixed since Fz (kx , ky ) = −

kT · FT (kx , ky ) kx Fx (kx , ky ) + ky Fy (kx , ky ) =− kz kz

(4.89)

Thus, the longitudinal spectral component is derived from the two-dimensional aperture field by enforcing this plane wave condition. This requires special treatment when the propagation vector lies within the xy-plane since the denominator tends to zero (kz = 0). Thus at first sight it would appear that the longitudinal electrical field component would become infinite. By examining the rate of change of the numerator and the denominator in this limit, the value of the longitudinal field component is found to be finite over all space. This can be shown from L’Hôpital’s rule which states that in the limit the ratio of the numerator and the denominator is equal to the ratio of the derivatives of the numerator and the denominator      f (x) f (x) lim (4.90) = lim x→a g (x) x→a g  (x)

Plane wave spectrum representation of EM waves

83

Thus when expressed in terms of spherical coordinates L’Hôpital’s rule, when applied to the longitudinal spectral component, can be seen to be   sin θ cos φEx + sin θ sin φEy lim {Fz } = lim − θ→π/2 θ→π/2 cos θ   

 d   sin θ cos φEx + sin θ sin φEy  = lim − dθ d  θ→π/2    (cos θ ) dθ Hence the angular spectrum remains finite over all space as   cos θ cos φEx + cos θ sin φEy lim {Fz } = lim =0 θ→π/2 θ→π/2 sin θ

(4.91)

(4.92)

By using the plane wave condition, it is possible to write the total, that is, Ex , Ey , Ez , electric field in free space purely in terms of the tangential spectral components as 1 E (x, y, z) = 4π 2

−ˆez


∞
∞



  FT kx , ky

−∞ −∞

  kT · FT kx , ky kz

e−j(kx x+ky y+kz z) dkx dky

(4.93)

where FT



 kx , ky , z = 0 =


∞
∞

ET (x, y, z = 0) ej(kx x+ky y) dxdy

(4.94)

−∞ −∞

Thus it is crucial that all spectral components remain finite for all values that kx and ky can take. If this were not the case, due to the antireductionist Fourier relationship between the spatial and spectral domains, the longitudinal component of the electric field would become infinite everywhere in space which is obviously unsatisfactory.

4.7

Solution of the vector magnetic wave equation in Cartesian coordinates

It is important to note that as the scalar wave equation, ∇ 2 u + k02 u = 0, is equally valid for both electric and magnetic fields, then exactly the same integral transform procedure that was utilized to solve the electric wave equation can be used to solve the magnetic wave equation. Hence by an exchange of the relevant variables it can be shown that it is possible to write the total, that is, Hx , Hy and Hz , magnetic fields

84

Principles of planar near-field antenna measurements

in free space purely in terms of the tangential magnetic spectral components as 1 H (x, y, z) = 4π 2

−ˆez


∞
∞ 

  GT kx , ky

−∞ −∞

  kT · GT kx , ky kz

e−j(kx x+ky y+kz z) dkx dky

(4.95)

Where 




∞
∞

GT kx , ky , z = 0 =

HT (x, y, z = 0) ej(kx x+ky y) dxdy

(4.96)

−∞ −∞

and are the magnetic spectral components which are the direct analogue of the electric spectral components F. Clearly then all observations made regarding the nature of the integral transform solution of the electric field are equally valid for the integral transform solution of the magnetic field. It will be shown below that a simple relationship exists between the electric spectral components and the magnetic field thus the magnetic spectra are not often considered.

4.8

The relationship between electric and magnetic spectral components

The magnetic field anywhere in the forward half-space can be obtained from the angular spectra by using Maxwell’s equation H (x, y, z) = −

1 ∇ × E (x, y, z) jωµ

(4.97)

to determine the magnetic field from the electric field. Hence, the magnetic field can also be determined from the spectral components. Correspondingly, the specification of two orthogonal tangential electric field components over the surface of a plane uniquely determines the magnetic field and thus the entire electromagnetic six-vector is uniquely specified that is 1 E (x, y, z) = 4π 2


∞
∞

  F kx , ky e−jk·r dkx dky

(4.98)

−∞ −∞

1 1 H (x, y, z) = − ∇× jωµ 4π 2


∞
∞ −∞ −∞

  F kx , ky e−jk·r dkx dky

(4.99)

Plane wave spectrum representation of EM waves

85

Expressing the magnetic field in terms of the magnetic spectra yields 1 4π 2


∞
∞ −∞ −∞

  1 ∇ G kx , ky e−jk·r dkx dky = − jωµ 1 × 4π 2


∞
∞

  F kx , ky e−jk·r dkx dky

−∞ −∞

(4.100)

Exchanging the order of integration and differentiation and cancelling like terms yields
∞
∞





G kx , ky e

−jk·r

−∞ −∞

1 dkx dky = − jωµ


∞
∞

  ∇ × F kx , ky e−jk·r dkx dky

−∞ −∞

(4.101)

In order that this can be converted into a more convenient form, let us first consider the divergence of the electric spectra namely   eˆ x eˆ y eˆ z  



∂Ay ∂Ax ∂Az ∂Az  ∂ ∂ ∂ =e ˆ − − e ˆ − ∇ × A =  ∂x  x y ∂y ∂z ∂y ∂z ∂x ∂z A x A y Az

 ∂Ay ∂Ax + eˆ z − (4.102) ∂x ∂y where for convenience the new vector A has been defined to be   A = F kx , ky e−jk·r

(4.103)

  Here, F kx , ky is a constant with respect to the variables x, y and z. Hence     ∇ × A = eˆ x −jky Az + jkz Ay − eˆ y (−jkx Az + jkz Ax ) + eˆ z −jkx Ay + jky Ax &    ' ∇ × A = −jk0 eˆ x βAz − γ Ay − eˆ y (αAz − γ Ax ) + eˆ z αAy − βAx

(4.104) (4.105)



eˆ x ∇ × A = −jk0  α Ax

eˆ y β Ay

 eˆ z γ  Az

(4.106)

86

Principles of planar near-field antenna measurements

Since the unit vector in the radial direction is eˆ r = α eˆ x + β eˆ y + γ eˆ z

(4.107)

the curl of the electric field is related to the electric field through a 90◦ phase change, a linear scaling of amplitude and by taking a cross product with the propagation vector as ∇ × A = −jk0 eˆ r × A = −jk × A

(4.108)

     ∇ × F kx , ky e−jk·r = −je−jk·r k × F kx , ky

(4.109)

Or

Thus
∞
∞





G kx , ky e

−jk·r

−∞ −∞

1 dkx dky = ωµ


∞
∞

  k × F kx , ky e−jk·r dkx dky

−∞ −∞

(4.110)

Differentiating both sides with respect to kx and ky yields     1 k × F kx , ky e−jk·r G kx , ky e−jk·r = ωµ

(4.111)

    1 k × F kx , ky G kx , ky = ωµ

(4.112)

or

Thus the electric and magnetic spectra are related to one another through a cross product with the propagation vector and a simple linear scaling. Taking the inverse Fourier transform of this will yield an expression for the magnetic field in terms of the electric angular spectra hence 1 H (x, y, z) = 2 4π ωµ


∞
∞

  k × F kx , ky e−jk·r dkx dky

(4.113)

−∞ −∞

Or when expressed in terms of the tangential components this becomes 1 H (x, y, z) = 2 4π ωµ 


∞
∞ k −∞ −∞





× FT kx , ky − eˆ z

  kT · FT kx , ky kz

e−j(kx x+ky y+kz z) dkx dky (4.114)

These expressions thus predict the field everywhere in space once the field has been specified over a plane. Within the formulation of this transform method, an assumption is made that the region under consideration is source or sink free, simple

Plane wave spectrum representation of EM waves

87

homogeneous and isotropic free-space region where a source is taken to represent a current in time. Hence, although it is possible to derive the fields for the whole space using spectral techniques, those results obtained for the region z < 0 will be in error, as this is conventionally taken to be behind the antenna aperture plane. That region of space directly behind the antenna aperture plane contains the radiating structure and will necessarily contain both sinks and sources. Thus in practice, these expressions are limited to specifying the field distribution within only the forward half-space, that is, where z ≥ 0. Clearly the electric spectra can be obtained from the magnetic spectra. The transverse nature of the spectral components can be readily established by substituting the respective Fourier solutions into the two remaining Maxwell equations. Thus since we have previously shown that ∇ · E (x, y, z) = 0 implies that   k · F kx , ky = 0 (4.115) and similarly, ∇ · H (x, y, z) = 0 will imply that   k · G kx , ky = 0

(4.116)

Thus both the electric and magnetic spectral components are transverse, that is, no component of F or G lies in the direction of propagation k.

4.9

The free-space propagation vector k

The free-space propagation constant k was introduced above in Chapter 2 as a convenient way of bundling together the angular frequency and the material properties of the medium through which the field is propagating. In this section we derive the length of the propagation vector and show that it has a magnitude equal to the freespace propagation constant. In other words, for a wave to propagate, the electric and magnetic fields must be related in a way that is dependant upon the properties of the material through which the fields exist. Motivated by this, let us first consider the relationship between the electric and magnetic angular spectra namely     1 k × F kx , ky G kx , ky = ωµ

(4.117)

Taking the vector cross product of another vector, k and this term yields   (   k × k × F kx , ky = ωµk × G kx , ky

(4.118)

Using the vector identity       A× B×C =B A·C −C A·B

(4.119)

yields

  (      k × k × F kx , ky = k k · F kx , ky − F kx , ky (k · k)

(4.120)

88

Principles of planar near-field antenna measurements

 2 However, as k · k = k  = k02 and as shown previously, k · F = 0 then     −F kx , ky k02 = ωµk × G kx , ky

(4.121)

Recalling that     1 F kx , ky = − k × G kx , ky ωε Then clearly     F kx , ky k02 = ω2 µεF kx , ky Factorising yields    ω2 εµ − k02 F kx , ky = 0

(4.122)

(4.123)

(4.124)

  Hence for nontrivial solutions, that is, solutions for which F kx , ky  = 0, this is satisfied when k02 = ω2 εµ

(4.125)

Hence the vector k has length   2π √ = k  k0 = ω εµ = λ Where again the positive radical is chosen.

(4.126)

4.10 Plane wave impedance The expressions derived above can be conveniently simplified if it is noted that the ratio of E and H for a transverse electromagnetic (TEM), that is, a plane wave is a constant with units of ohms namely ) µ E Z0 = = = 376.730313461  (4.127) H ε Thus, the magnetic field can be expressed more conveniently in terms of the electric field and the propagation vector as 1 H (x, y, z) = 2 4π k0 Z0


∞
∞

  k × F kx , ky e−jk·r dkx dky

(4.128)

−∞ −∞

or simply 1 H (x, y, z) = 4π 2 Z0


∞
∞

  eˆ r × F kx , ky e−jk·r dkx dky

−∞ −∞

A more detailed treatment of this can be found in Box 4.4.

(4.129)

Plane wave spectrum representation of EM waves

Box 4.4 A useful scaling factor can be obtained by examining the ratio of the plane wave solutions of the electric and magnetic fields. Consider a purely x-polarised plane wave propagating in the direction of the positive z-axis thus   E = E  eˆ x = E0 e−jkz z eˆ x (4.130) Thus, the corresponding magnetic field would be  k0 1  k0  H= k ×E = eˆ z × E0 e−jkz z eˆ x = E0 e−jkz z eˆ y ωµ ωµ ωµ

(4.131)

Now consider the ratio of the electric and magnetic field vectors for a TEM wave namely C=

E k ×E E = = T H k ×H HT

(4.132)

Here, E is the electric field intensity in units of volts per meter and H is the magnetic field intensity in units of amps per meter. E T and H T are the tangential components of the electric and magnetic fields respectively. Thus as C is the ratio of transverse (TEM waves have no electric or magnetic field component in the direction of propagation) orthogonal components of E and H it has units of volts per amp which, by definition, is an ohm, that is, a measure of resistance or more generally, impedance. Explicitly then this ratio, which has units of ohms, is the impedance that a plane wave would experience when propagating in a perfect vacuum when the electric and magnetic field vectors are oscillating in phase, that is, TEM, is usually designated the symbol Z0 where ) µ E E0 e−jkz z ωµ 2π f λµ µ Z0 = = = *  = = = cµ = √ εµ H k0 2π ε k0 ωµ E0 e−jkz z (4.133) Thus, Z0 is the impedance of free space, has units of ohms and in general is a function of the properties of the material in which the field exists. The impedance of free space can be defined exactly in terms of π and the speed of light in a vacuum, c. When quoted to nine decimal places Z0 is Z0 = 376.730313461  However, this is often quoted approximately as )  µ = µ2 c2 ≈ 120π Z0 = ε

(4.134)

(4.135)

Conversely, if the electric and magnetic fields are not orthogonal and are out of phase with one another, the wave impedance will contain a non-zero reactive

89

90

Principles of planar near-field antenna measurements

Box 4.4 Continued component thus this quantity represents an impedance, not a resistance. The inverse of impedance is admittance. Thus, the plane wave admittance of free space Y0 is Y0 =

1 Z0

(4.136)

and is measured in units of ohm−1 (−1 ) that is more often termed siemens (S).

4.11

Interpretation as an angular spectrum of plane waves

Classically, a wave can be thought of as being a complex function of two (or more) coordinates of space and time. It is possible to classify a wave by noting the nature of the surface over which the phase remains constant. Waves are referred to as being plane if their equal-phase surfaces are planar. Each of the infinite number of perpendiculars to these surfaces is called a wave normal and defines the direction in which the wave propagates. If  (x, y, z) = constant describes an equal-phase surface, then ∇ (x, y, z) will be the direction in which the phase varies most rapidly. This quantity is referred to as the vector phase constant τ . By convention, this will be the direction in which the phase decreases most rapidly, that is τ = −∇

(4.137)

Applying this to the term Fe−jk·r within the integrand above where k is real yields   (4.138) τ = −∇ −k · r = k Here the quantity Fe−jk·r can be thought of as representing a uniform homogeneous plane wave of amplitude F, propagating in the direction k, and as shown above, constitutes an exact solution of Maxwell’s fundamental field equations. The function   F kT can therefore be thought of as a superposition of plane waves travelling in different directions with different amplitudes, that is, an ‘angular spectrum’ of plane waves or, a ‘plane wave spectrum’ (PWS). However, when k is complex, that is, k = β − jα, then the phase constant will also become complex as   (4.139) τ = −∇ −k · r = k = β − jα Hence Fe

−jk·r

= Fe

 −j β−jα ·r

= Fe−jβ·r e−α·r

(4.140)

This represents equal-phase surfaces that are perpendicular to β and equal amplitude surfaces that are perpendicular to α that is, inhomogeneous plane waves where the

Plane wave spectrum representation of EM waves

91

z Equiphase surface

(x,y,z) r θ

(kx,ky,kz)

k y

x

Figure 4.2

φ

Coordinate system for plane wave propagation

wave propagates in the direction β and are attenuated in the direction α. Here, the position vector is described by r = xˆx + yˆy + zˆz and the direction of propagation is  k = kx xˆ + ky yˆ + kz zˆ = k0 uˆx + vˆy + wˆz

(4.141)

(4.142)

where in spherical coordinates, u, v, w are the direction cosines sin θ cos φ, sin θ sin φ and cos θ respectively where, θ is the polar or zenith angle and φ is the azimuthal angle. This is represented schematically in Figure 4.2. This, therefore, is the basis for the principle that ‘the electric or magnetic field at any point in space can be constructed from an infinite number of plane waves of different amplitudes propagating in different directions’. It is of interest to note that although the interpretation of the particular solution in transform space as a PWS is of utility, this interpretation was not relied upon within the above derivation. Consequently, the question as to the existence of the PWS and the physical reality of unbounded plane waves in general bears little relevance to this discussion. This is crucial, as the infinite plane wave is physically impossible since the total energy transported across an equiphase surface would be infinite. The attraction of the PWS method stems from the relative simplicity with which plane waves can be expressed and manipulated mathematically as building blocks to synthesise real arbitrary field distributions. The relation between the angular spectrum and the aperture distribution involves no approximations or aperture size restrictions. Consequently, the PWS method is inherently more accurate than other older theories, as the approximations that must inevitably be introduced occur at a later stage of the analysis, that is, during the asymptotic derivation of far-field parameters. The field produced by a bounded current distribution is in any direction outgoing when infinitely far removed from the source. Thus, a single analytic representation as a superposition of plane waves travelling in diverse directions can not be valid over more

92

Principles of planar near-field antenna measurements

than a half-space. A plane wave that propagates outward in a given direction in one half-space will necessarily travel inwards in the opposite direction in the other halfspace. Similarly, unbounded inhomogeneous plane waves that decay exponentially in one direction are matched by an exponential growth in the opposite direction that eventually becomes infinite. It is a violation of physical reality, and therefore unacceptable to allow a solution to increase without limit. As the general solution is formed from the superposition of homogeneous and inhomogeneous plane waves, it is necessary to restrict the PWS representation to the field radiated into a half-space by a surface current density flowing on a bounding infinite plane.

4.12 Far-field antenna radiation patterns: approximated by the angular spectrum The beauty of the angular spectrum approach lies within the simplicity with which far-field parameters can be obtained from plane wave mode coefficients. The method of stationary phase is used to simplify the two-dimensional inverse Fourier transform that is required to obtain spatial quantities from the angular spectra. This results in the reduction of the inverse transform to a multiplication by a trigonometric obliquity factor. For the case of conventional near-field antenna measurements, the obliquity factor is centred about the range boresight which happens to be coincident with the antenna boresight. For the case where the antenna under test (AUT) is not aligned to the axes of the range, care must be taken to insure that the cosine term applied is in the correct coordinate system. In order that this issue can be resolved, the necessary integrals are evaluated whilst paying particular attention to the coordinate systems. As illustrated above and assuming that the integrals involved can be evaluated, the specification of two tangential electric field components or two tangential magnetic fields, can be used to determine the electric and magnetic fields anywhere in the forward half-space. Consequently, no assumption is introduced concerning the relationship between the aperture electric and aperture magnetic fields as either can be used to determine the fields over an entire half-space. This provides a convenient mechanism for the derivation of radiation patterns in the far-field, by recognizing that the far-field can be represented by a spherical surface of large radius, centred about the radiator. Here two assumptions are required; first that the dimensions of the aperture are of finite extent, that is, the field sources lying in the half-space behind the aperture plane are finite and occupy a finite volume. Consequently, the equivalent current distribution over the aperture is of finite area. Second, the radius must be large compared to the wavelength and the greatest dimension of the aperture. In the majority of cases these requirements are satisfied and the double integral 1 E (x = ru, y = rv, z = rw) = 4π 2 −ˆez


∞
∞



  FT kx , ky , z = 0

−∞ −∞

  kT · FT kx , ky , z = 0 kz

× e−jr (kx u+ky v+kz w) dkx dky (4.143)

Plane wave spectrum representation of EM waves

93

can be evaluated asymptotically using a two-dimensional form of the stationary phase algorithm and FT



 kx , ky , z = 0 =


∞
∞

ET (x, y, z = 0) ej(kx x+ky y) dxdy

(4.144)

−∞ −∞

Obtaining asymptotic far-field parameters equates to evaluating the electric and magnetic fields over the surface of a sphere or hemisphere in the case of a half-space, described by x = r sin θ cos φ = ru

(4.145)

y = r sin θ sin φ = rv

(4.146)

z = r cos θ = rw

(4.147)

Here r is fixed and the spherical angles θ and φ are varied such that 0≤r≤∞ π 0≤θ ≤ 2

(4.148) (4.149)

The polar spherical grid can be thought of as being that grid that is most closely related to a positioner that consists of an upper roll rotator, that is, φ, to which the AUT is attached and a lower rotator, that is, θ, upon which the upper φ rotator is attached. As the AUT is attached to the roll positioner, the AUT will rotate about the roll axis that is therefore the polar axis. The field point is obtained by rotating the horizontal θ positioner and vertical roll positioner through the angles θ and φ where the order is unimportant. For antenna measurement, this arrangement has the advantage that it moves the AUT through only a small portion of the test zone, and it places the blockage that results from the AUT mount entirely in the back hemisphere. Moving the AUT by only a small amount minimises errors associated with imperfections in the illumination of the test zone and can render probe pattern correction unnecessary (X , Y ) = (θ , φ)

(4.150)

Here, θ and φ define the direction to the field point through rˆ = sin θ cos φ eˆ x + sin θ sin φ eˆ y + cos θ eˆ z

(4.151)

Hence θ = arccos (w) v φ = arctan u

(4.152) (4.153)

The definition of an equatorial spherical coordinate system is identical to the polar spherical case. The difference results purely from the application of a 90◦ rotation in θ, in order that the main beam of the radiator points along the positive x-axis (through the equator), rather than along the positive z-axis (through the pole). Although this system is commonly used for acquiring antenna radiation patterns in

94

Principles of planar near-field antenna measurements

spherical near-field facilities, it is not used within this text and is only included here for the sake of completion. The far-field is defined such that (4.154) E far = lim E r→∞

and

H far = lim H r→∞

(4.155)

The double integral above, with the spectrum function known, is in general difficult to evaluate. However, the approximate asymptotic method of stationary phase may be used to evaluate this integral, of which a detailed derivation is presented below, so that we may write the solution of the vector wave equation in the far-field as derived in Section 4.3, Box 4.5 E (ru, rv, rw) ≈ j

  e−jkr cos θ F kx , ky λr

(4.156)

Here, the polar angle θ is measured away from the normal vector, (the zenith or positive z-axis). Crucially in the limit of the far-field, the two-dimensional inverse Fourier transform is essentially reduced to a multiplication with a simple trigonometric function. However, this expression is only valid for certain stationary points which can be expressed as kx = k0 sin θ cos φ

(4.157)

ky = k0 sin θ sin φ

(4.158)

kz = k0 cos θ

(4.159)

Alternatively, this can be expressed in terms of the x-, y- and z-axis direction cosines as kx = k0 α

(4.160)

ky = k0 β

(4.161)

kz = k0 γ

(4.162)

Conventionally, the 1/r term and the unimportant phase factor e−jkr are divided out of far-field antenna patterns. From the above expression and recalling that k·F = 0 then clearly k ·E =0

(4.163)

Hence, in the far-field, the electric and magnetic fields are locally planar and tangential to the direction of wave propagation. Furthermore, the magnetic field may be obtained from the electric field as H=

1 eˆ r × E Z0

(4.164)

Plane wave spectrum representation of EM waves

95

This functional relationship between the angular spectrum and the radiation pattern is exact only when the observation point is infinitely far removed from the aperture. Importantly, no assumption is introduced concerning the relationship between the electric and magnetic fields in the aperture, as only the electric field or the magnetic field is required to completely specify the radiation pattern. Here, the principal variation of the far-field pattern is governed by the Fourier transform of the aperture illumination function. The slowly varying geometric terms are conventionally interpreted as ‘obliquity factors’ and are associated with the unit vectors that arise from the aperture polarisation. This formulation implies that there is no one-to-one correspondence between any field point P and the field at any point on the aperture plane; rather, the field at P is an integrated effect of the contributions from every point on the aperture plane. Crucially, the angular spectrum was obtained directly from the sampled fields. The inverse transform was thence obtained from these angular spectra implying that the cosinusoidal term is referred to the normal of the sampling surface, that is, the obliquity factor should be applied in the sampling coordinate system, irrespective of the orientation of the radiator.

4.13 Stationary phase evaluation of a double integral Lord Kelvin initially formulated this principle for use with one-dimensional integrals [6]. The derivation of this asymptotic solution is lengthy and rather involved so merely the result is quoted here and the detailed workings are instead consigned to Box 4.5.   e−jkr (4.165) cos θ F kx , ky λr Essentially then, in the far-field, at the stationary points, the two-dimensional inverse Fourier transform reduces to a simple functional geometric term and a normalization constant. Box 4.5 contains a detailed derivation of the PWS to far-field transform. E (x = ru, y = rv, z = rw) ≈ j

Box 4.5 The method of stationary phase is in effect an algorithm for asymptotically evaluating the following integral

  1 E (x, y, z) = F kx , ky e−jk·r dkx dky (4.166) 2 4π D

   Here, F (x, y) and k ·r are real and continuous on D and k  r  is a large positive number. The remainder of this section is concerned with the evaluation of this

96

Principles of planar near-field antenna measurements

Box 4.5 Continued sole integral. Consequently, placing the final result aside, this section adds little to the understanding of the underlying physics and can thus be ignored on first acquaintance. Assuming that the integrand oscillates rapidly along the integration path except at the points where k · r is stationary at the single point (k1 , k2 ), then the exact value of the integral may only be given by the contributions arising from the stationary points. A point is considered stationary when   ∂k · r  ∂k · r  =0 (4.167) = ∂kx kx =k1 ∂ky kx =k1 ky =k2

ky =k2

Strictly, such a point within the domain of the integration can be referred to as a critical point of the first kind. Thus, the stationary points can be found by applying these conditions to the following relationship k · r = kx x + ky y + kz z = r[kx sin θ cos φ  + ky sin θ sin φ + k02 − kx2 − ky2 cos θ ] Thus

  d k ·r kx cos θ = sin θ cos φ −  =0 dkx 2 k0 − kx2 − ky2

(4.168)

(4.169)

Hence, the stationary points can be expressed as kx = kz

sin θ cos φ = k1 cos θ

(4.170)

ky = kz

sin θ sin φ = k2 cos θ

(4.171)

and

Using sin2 θ cos2 φ sin2 θ sin2 φ + kz2 + kz2 2 cos θ cos2 θ + , 2 2 sin θ = kz +1 cos2 θ

k02 = kx2 + ky2 + kz2 = kz2

(4.172)

Thus by choosing the positive radical kz = k0 cos θ

(4.173)

Plane wave spectrum representation of EM waves

Box 4.5 Continued Expanding, k · r in a Taylor’s series in two variables about these stationary points yields  k · r = k · r kx =k1 ky =k2

 ∂ 2 k · r  1 (kx − k1 )2 2 ∂kx2 kx =k1 ky =k2   2 2 ∂ k · r  1 ky − k2 +  2 ∂ky2 kx =k1 +

ky =k2

  ∂ 2k · r    + (kx − k1 ) ky − k2 + ··· ∂kx ∂ky kx =k1

(4.174)

ky =k2

Hence, the integral becomes

  1 E (x, y, z) = F k , k x y 4π 2 

D

    −j  k·r |kx =k + (1/2)(kx −k1 )2 ∂ 2 k·r/∂kx2 kx =k 1 1 ky =k2 ky =k2 ×e



+ (kx − k1 ) ky − k2

,   ∂ k · r/∂kx ∂ky kx =k1 + · · · dkx dky



2

ky =k2

(4.175)

Let us define the following substitutions A=−

1 ∂ 2k · r 2 ∂kx2

(4.176)

B=−

1 ∂ 2k · r 2 ∂ky2

(4.177)

C=−

∂ 2k · r ∂kx ∂ky

(4.178)

and u = kx − k1

(4.179)

v = ky − k2

(4.180)

97

98

Principles of planar near-field antenna measurements

Box 4.5 Continued Recalling that kz = k0 cos θ then at the stationary points  sin θ cos φ sin θ sin φ y + kz z k · r kx =k1 = kz x + kz cos θ cos θ ky =k2  = k0 r sin2 θ cos2 φ + sin2 θ cos2 φ + cos2 θ = k0 r (4.181) Taking the first three terms in each dimension the integral becomes

 j−kr+u2 A+v2 B+uvC   1 E (x, y, z) ≈ , k dkx dky e F k y x 4π 2

(4.182)

D

Exchanging the variable using, du/dkx = 1, dv/dky = 1 and bringing the independent variables outside the integral yields

   −jkr  1 2 2 E (x, y, z) ≈ e F k , k ej u A+v B+uvC dudv (4.183) x y 2 4π D

Let

√ x=u A √ y=v B

(4.184) (4.185)

Then E (x, y, z) ≈

1 √

4π 2 AB





F kx , ky e

−jkr



e

  *√ j x2 +y2 + Cxy AB

D

dx dy (4.186)

By completing the square for the x variable in the exponential, we may write   1 E (x, y, z) ≈ √ F kx , ky e−jkr 2 4π AB     * √ 2 2   2 *

+y 1− C 4AB j x+ Cy 2 AB × e dx dy (4.187) D

Here we may extend the domain of the integral to infinity, as the contribution outside the domain is negligible   1 E (x, y, z) ≈ √ F kx , ky e−jkr 2 4π AB
∞
∞ × −∞ −∞

e

  * √ 2   *  j x+ Cy 2 AB jy2 1− C 2 4AB

e

dx dy

(4.188)

Plane wave spectrum representation of EM waves

Box 4.5 Continued Performing on the x variable first using the exchange of variable  the integration √ ξ = x + Cy/2 AB where dx = dξ then using the standard integral
∞

2

ejα(x−x0 ) dx =

)

−∞

π j(π / 4) e α

(4.189)

where α = 1 and x0 = 0 we may express the electric field as E (x, y, z) ≈

1 √

4π 2 AB

  √ F kx , ky e−jkr π ej(π / 4)


∞

ejy

2



 *  1− C 2 4AB

−∞





dy

(4.190)

Using the standard integral again where α = 1 − C 2 /4AB and x0 = 0 hence  −jkr √ j(π 4)  π 1 /  ej(π / 4)  * πe E (x, y, z) ≈ √ F kx , ky e 2 1 − C 2 4AB 4π AB (4.191) Thus with no further approximation E (x, y, z) ≈

1 √

4π 2 AB

  F kx , ky e−jkr πej(π / 2) 

1

*  1 − C 2 4AB 

(4.192)

Hence E (x, y, z) ≈

  2π j 1 F kx , ky e−jkr √ 2 4π 4AB − C 2

(4.193)

The constants A, B and C may be evaluated as follows A=− =−

1 ∂ 2k · r 2 ∂kx2

 r ∂2  2 − k 2 − k 2 cos θ k sin θ cos φ + k sin θ sin φ + k x y x y 0 2 ∂kx2 (4.194) 

A=−

=



r ∂  kx cos θ  sin θ cos φ −   2 ∂kx k02 − kx2 − ky2  

r  2 

kx2 cos θ k02 − kx2 − ky2

3/2 + 

cos θ

   2

k02 − kx2 − ky

(4.195)

99

100 Principles of planar near-field antenna measurements

Box 4.5 Continued Hence

+

sin2 θ cos2 φ 1+ cos2 θ

r A= 2k0 Similarly

+

sin2 θ sin2 φ 1+ cos2 θ

r B= 2k0

, (4.196)

, (4.197)

Furthermore ∂ 2k · r ∂kx ∂ky  ∂2  kx sin θ cos φ + ky sin θ sin φ + k02 − kx2 − ky2 cos θ = −r ∂kx ∂ky (4.198)

C=−

 C = −r



∂  kx cos θ  sin θ cos φ −   ∂ky 2 2 2 k0 − k x − k y 

  = rkx cos θ  

ky k02

C=

− kx2

− ky2

  3/2 

(4.199)

r sin2 θ cos φ sin φ k0 cos2 θ

So r 4AB − C = 4 2k0 2

−

+

sin2 θ cos2 φ 1+ cos2 θ

(4.200) ,

r 2k0

+

sin2 θ sin2 φ 1+ cos2 θ

r2 1 r 2 sin4 θ cos2 φ sin2 φ = k0 cos4 θ k02 cos2 θ

,

(4.201)

Hence j√

2π 4AB

− C2

=j

2πk0 cos θ r

(4.202)

Plane wave spectrum representation of EM waves 101

Box 4.5 Continued Finally, the asymptotic evaluation of the electric field, E far = lim E , can r→∞

be expressed as E (x = ru, y = rv, z = rw) ≈ j or

  e−jkr k0 cos θ F kx , ky 2π r

(4.203)

  e−jkr (4.204) cos θ F kx , ky λr Within the above derivation, certain assumptions have been made, the most significant of these being: E (x = ru, y = rv, z = rw) ≈ j

• Truncation of the Taylor series to only the first three terms in each dimension. • Assumption that k · r is a smoothly varying function about the stationary points (without a strict definition of smooth). • An implicit assumption is made that no singularities are present.

4.14 Coordinate free form of the near-field to angular spectrum transform Thus far, the analysis has been restricted to the consideration of problems where the boundary conditions f (x, y) are known in the same coordinate system, as that required by the solution u (x, y, z). Techniques for the removal of this restriction constitute the topic for this section. If the field can be represented as an area of a planar surface located at the origin with a radiating field distribution across it we may express two tangential and mutually orthogonal field components as       (4.205) ET rT = uˆ x Ex rT + uˆ y Ey rT Here, the tangential component of the position vector is expressed as rT = uˆ x x + uˆ y y

(4.206)

Thus, the equivalent tangential spectral components may be evaluated using   F T kT =


∞
∞

    ET rT e j kT ·rT dxdy

(4.207)

−∞ −∞

where the tangential components of the propagation may be expressed as kT = uˆ x kx + uˆ y ky

(4.208)

102 Principles of planar near-field antenna measurements Hence, the Cartesian far-field electric components can be written in terms of the tangential spectral components as           αFx kT + βFy kT e−jkr uˆ z γ Fx kT uˆ x + Fy kT uˆ y − E (ru, rv, rw) ≈ j γ λr (4.209) & −jkr         '  e E (ru, rv, rw) ≈ j Fx kT γ uˆ x + Fy kT γ uˆ y − αFx kT + βFy kT uˆ z λr (4.210) or     ( e−jkr  (4.211) γ FT kT − uˆ z uˆ r · FT kT λr Here, the tangential spectral components have been written in vector form as E (ru, rv, rw) ≈ j

FT = Fx uˆ x + Fy uˆ y + 0ˆuz

(4.212)

The unit vector from the sampling plane to the observation point is uˆ r = α uˆ x + β uˆ y + γ uˆ z

(4.213)

To place this in a coordinate free form this expression must be modified to account for any displacement and rotation of the plane with respect to the origin of the coordinate system. Here primed coordinates are used to denote those coordinates associated with the source, whilst those associated with the observation point are not primed, see Figure 4.3. Angular spectra are obtained by integrating the complex tangential electric field components over the sampling plane. As in general, the source and observation frames of reference will not be coincident and synonymous account must therefore be made for this when determining equivalent angular spectra. As Sampling plane

y

Ey′ r′ nˆ

O

Ey′

θ

x

r z

Ey (P)

r ′′

P

Figure 4.3

Ex (P)

Generalized coordinate system for solution of vector Helmholtz equation

Plane wave spectrum representation of EM waves 103 can be seen from Figure 4.3 the relationship between the position vectors can be expressed as r  + r  = r

(4.214)

A displacement between the origin of the coordinate system and the sampling plane can be accommodated through the introduction of a differential phase change  of the form ejk uˆ r ·r . The introduction of this phase factor can be seen to follow directly from the shifting property of the Fourier transform [7]. The ‘illumination’ factor, γ = cos θ , where θ is measured from the outward-facing unit normal to the sampling plane normal, can be expressed generally as uˆ r  · nˆ . Hence 

     ( e−jkr  uˆ r  · nˆ FT kT − nˆ uˆ r  · FT kT E (ru, rv, rw) ≈ j λr 

(4.215)

Using       A× B×C = A·C B− A·B C

(4.216)

then 

E (ru, rv, rw) ≈ j

    e−jkr  uˆ r  × FT kT × nˆ  λr

(4.217)

Hence

   
  e−jkr   E (ru, rv, rw) ≈ j uˆ r  ×  ET rT ejk0 uˆ r ·r da  × nˆ  (4.218) λr  S

Here, the electric field is correctly resolved onto a Cartesian polarisation basis in the sampling coordinate system. These unit vectors can be readily rotated so that the polarisation basis can be rewritten in terms of any preferred observation frame of reference. This expression is only valid for the positive half-space with respect to the sampling plane or when nˆ · uˆ r  ≥ 0. Near-field data can be recovered directly from the rotated spectral components using 1 E (x, y, z) = 4π 2


∞
∞

  ( F kx , ky e−j(kx x+ky y+kz z) dkx dky

(4.219)

−∞ −∞

Here, there is no requirement for any further isometric transformations and all usual numerical techniques for improving the efficiency of the transformation can be utilized. As illustrated in Figure 4.4, the recovered plane will not be coplanar with sampling surface. Furthermore, even if those issues associated with polarisation are ignored then the differences between the illumination functions are not characterised with a simple linear phase taper.

104 Principles of planar near-field antenna measurements Translated measurement plane

AUT

Reconstructed aperture plane Measurement plane

Figure 4.4

Schematic representation of plane-to-plane transform performed with and without alignment correction

4.15 Reduction of the coordinate free form of the near-field to far-field transform to Huygens’ principle As described in Chapter 3 in some detail, Huygens’ principle [8–10] can be stated as each point on a primary wave front can be considered to be a new source of a secondary spherical wave, of the same frequency and that a secondary wave front can be constructed as the superposition of these secondary spherical waves with due regard to their phase differences.

When expressed mathematically the electric field, at a point P, radiated by a Huygens’ surface S, is [11]
    ( e−jkr  j e uˆ = uˆ r  × E × nˆ da (4.220) λ r  S

r 

Here, is the distance between element and field point, λ is the wavelength, k0 is the wave number, ur  is the unit vector from element to field point, n is the unit normal vector at the element and E is the total or tangential, aperture electric field at the element. This statement of Huygens’ formula is rigorous, provided that the observation point is more than a few wavelengths from the source and that the scalar product of n and ur  is positive. The geometry for this statement of the Huygens’ formula can be found illustrated in its conventional form in Figure 4.5. Importantly, the derivation of this relationship requires the observation point to be placed in the far-field of the infinitesimal Huygens’ element. Thus, this expression can not be used to model the behaviour of the fields in the reactive near-field where evanescent fields become significant. This is unimportant in the simulation of nearfield measurements as these measurements are only taken outside of this region of

Plane wave spectrum representation of EM waves 105 P

r S

r′

O

Figure 4.5

O′

r″

ur ″ n

Coordinate system for Huygen’s principle

space. As such, this form of Huygens’ principle is utilized in Chapter 6 within the near-field measurement simulation software section, where this formulation can be utilized in cases where the field is known over a non-planar surface. Huygens’ principle can be obtained directly from the coordinate free form of the near-field to far-field transformation by reducing the area of the aperture plane until in the limit, it constitutes an elemental, that is, infinitesimal, Huygens’ source. From Section 4.14 above, it can be established that the near-field to far-field transform can be expressed as    
  e−jkr   E (ru, rv, rw) ≈ j uˆ r  ×  ET rT ejk0 uˆ r ·r da  × nˆ  (4.221) λr  S

Thus in the limit when the aperture is reduced to a Huygens’ element, that is, when da = lim {S} and the observer is in the far-field so that r  and r are parallel S→0

dE (ru, rv, rw) ≈ j

  e−jkr   uˆ r × ET rT ejk0 uˆ r ·r da × nˆ λr

(4.222)

Hence integrating over the complete surface yields e−jkr E (ru, rv, rw) ≈ j λr






(   uˆ r × ET × nˆ ejk0 uˆ r ·r da

(4.223)

S

Here, it is important to stress that the coordinate free form of the angular spectrum representation of an electromagnetic field is the most general representation, as it contains information pertaining to both propagating and evanescent fields. Furthermore, as Huygens’ principle essentially constitutes a special case of the angular spectrum approach, the same assumptions and limitations bind it; that is, any deficiency within the Fourier approach will also represent a deficiency with this Huygens’ method.

106 Principles of planar near-field antenna measurements

4.16 Far-fields from non-planar apertures As shown above, the far electric field can be obtained from the near electric fields using the following expression  e−jk0 r  uˆ r · nˆ E (rα, rβ, rγ ) = j λr


∞
∞

f (x, y) ejk0 (αx+βy+γ z) dxdy

(4.224)

−∞ −∞

Often, it is convenient to express the surface profile on which the near-field is defined as a function of two coordinates, x and y, where the coordinates are plaid, monotonic and equally spaced. For a smooth surface defined by the function g (x, y, z) = z − f (x, y) = 0

(4.225)

Thus, z = f (x, y) then the outward-facing, positive, normal is given by  *   *  ∂g ∂x ∂g ∂y 1 eˆ x − eˆ y + eˆ z n = nx eˆ x + ny eˆ y + nz eˆ z = − ∇g ∇g ∇g where



∇g =

∂g ∂x

2

+

∂g ∂y

2

+1

(4.226)

(4.227)

Multiplying by −1 produces the negative orientated, that is, inward facing, normal vector. Generally, the partial derivatives are obtained by taking finite differences. The elemental surface area da is given by    dxdy   da =  (4.228) nz  Hence da = dxdy

-



∂g ∂x

2

+

∂g ∂y

2

+1

(4.229)

Thus the far electric field can be obtained from + 
∞
∞   e−jk0 r ∂g 2 E (rα, rβ, rγ ) = j f (x, y) ejk0 (αx+βy+γ z) uˆ r · nˆ λr ∂x −∞ −∞



∂g + ∂y

2

,1 2

+1

dxdy

(4.230)

  Crucially, the obliquity factor uˆ r · nˆ has been brought inside the integral, as the unit surface normal nˆ will vary over the surface of integration and uˆ r = α eˆ x + β eˆ y + γ eˆ z

(4.231)

Plane wave spectrum representation of EM waves 107 This formula can be used in cases where the field is known over a non-planar surface, provided that the surface is smooth. Specifically, the function describing the surface must be continuous, as must all of the first partial derivatives. Here, the arbitrary but smooth surface is expressed as a function of two coordinates x and y where the coordinates are plaid, monotonic and equally spaced. Finally, within visible space, the angular spectrum can be obtained from + 
∞
∞   ∂g 2 1 jk0 (αx+βy+γ z) F (k0 α, k0 β) = uˆ r · nˆ f (x, y) e γ ∂x −∞ −∞



∂g + ∂y

2

,(1/ 2) +1

dxdy

(4.232)

Although useful, these expressions are presented here merely for the sake of completion as they are not automatically in a form that is amenable for use with probe uncorrected data.

4.17 Microwave holographic metrology (plane-to-plane transform) The theoretical framework for the plane-to-plane transform has already been developed within Chapter 4. It is given special attention here in part because it is of practical utility and in part because it has received such a great deal of attention in the open literature. As shown above, knowledge of the tangential components of the electric field enable the entire electromagnetic six-vector to be determined everywhere within a half-space. This, of course, enables the fields over one plane in space to be used to determine the field over another plane in space. The reconstruction of near-field data over a plane in space, other than the measurement plane, is accomplished by the application of a differential phase change. This can be seen to be analogous to a defocusing of the far-field image. Thus in summary   F kx , ky , z = 0 =


∞
∞

f (x, y, z = 0) ej(kx x+ky y) dxdy

(4.233)

−∞ −∞

and 1 u (x, y, z) = 4π 2


∞
∞

  F kx , ky , z = 0 e−j(kx x+ky y+kz z) dkx dky

(4.234)

−∞ −∞

Hence the field over one plane can be used to calculate the field over the surface of another, parallel plane displaced by an amount z in the z-axis. When expressed compactly using  operator notation this reduces to 0 1 u (x, y, z) = −1  {f (x, y, z = 0)} e−jkz z (4.235)

108 Principles of planar near-field antenna measurements This reconstructed plane can be located at any of an infinite number of planes that are in the region of space at or in front of the instrument’s aperture. It is when the fields are reconstructed at a plane that is coincident with the AUT’s ‘aperture plane’ that this process is of most utility. The antenna aperture can be conveniently thought of as that surface in space, which represents the transition between the majority conduction current and displacement current regions defined by the presence of a charge distribution. A near-field measurement is typically constructed so that the field produced by the antenna is sampled over a region of space in which there is an absence of divergence contained within that field. Therefore, the plane-to-plane transform process results in knowledge of only the radiating components and provides no knowledge as to the stored energy component. Thus, the field is reconstructed with a resolution of at best half a wavelength only. Although the field can be reconstructed with an infinitely fine sample spacing the amount of information contained within the resulting pattern is still limited. The technique of microwave holographic diagnosis, that is, the recovery of the antenna aperture illumination function, is now a well established method of non-intrusive characterisation of antenna assemblies as it can clearly show faulty radiating elements or incorrectly adjusted transmit and receive modules within a phased array antenna or an incorrectly aligned feed or reflector within a reflector antenna assembly. Sometimes the aperture illumination function is referred to as being a hologram. This is perhaps a little confusing as a hologram is usually taken to mean an image that contains both amplitude and phase information. As such, the measurement plane could equally well be referred to as being a hologram.

4.18 Far-field to near-field transform It is customary to present far-field antenna pattern functions in terms of spherical angles. Often, far-field pattern functions are presented tabulated as a function of the spherical angles azimuth and elevation rather than the polar spherical angles θ and φ. The azimuth over elevation grid can be thought of as being that grid that is most closely related to a positioner that consists of an upper azimuth rotator, to which the AUT is attached and a lower elevation positioner upon which the azimuth rotator is attached. As the AUT is attached to the azimuth positioner, the AUT will rotate about the azimuth axis that is therefore the polar axis. The field point is obtained by rotating the horizontal azimuth positioner and vertical elevation positioner through the angle Az and El where the order is unimportant. A detailed overview of coordinate systems, including the azimuth over elevation system can be found in Appendix C. Here, it is sufficient to recall that Az and El define the direction to the field point through rˆ = − sin (Az) cos (El) eˆ x + sin (El) eˆ y + cos (Az) cos (El) eˆ z So that Az = arctan



−u w

El = arcsin (v)

(4.236)

 (4.237) (4.238)

Plane wave spectrum representation of EM waves 109 The inclusion of the minus sign in the x coordinate is useful, as it signifies that the observer is positioned behind the antenna looking out at the far-field pattern. Other choices are possible, however, this is in agreement with that which is usually adopted in space and airborne RADAR applications. It is important to keep in mind the difference between the range coordinate system and the antenna coordinate system. These azimuth and elevation angles are measured with reference to the AUT and are fixed to it. Thus one should try and imagine being bolted to the antenna and seeing the chamber rotate as the rotators move. This is not the usual way we conceptualise these things as we are used to standing stationary on the floor of the chamber and watching the AUT rotate within the fixed chamber and not the converse. This is discussed in greater detail in Chapter 5. Let us assume that the far-field pattern is tabulated on a plaid monotonic equally spaced azimuth over elevation grid. The integration required to obtain near-field parameters from far-field pattern functions must therefore be modified accordingly. As 1 E (x, y, z) = 4π 2


∞
∞

  A kx , ky e−j(kx x+ky y+kz z) dkx dky

(4.239)

−∞ −∞

Since, kx = k0 α and ky = k0 β then by differentiation we obtain the following relationships 2π dα λ 2π dky = dβ λ dkx =

(4.240) (4.241)

Then 1 E (x, y, z) = 2 λ


∞
∞

A (α, β) e−jk0 (αx+βy+γ z) dαdβ

(4.242)

−∞ −∞

The application of the change of variable formula for double integrals can be applied to the Fourier transform of the boundary conditions so that all quantities can be rewritten in terms of the angular spectrum. Using the initial boundary condition specified in a regular azimuth over elevation coordinate system and the transformation between direction cosine and azimuth and elevation angles, the integration required to obtain the near-field can be obtained. The necessary transformation can be expressed as follows, compare with Figure 5.16 in Chapter 5. α = − sin (Az) cos (El)

(4.243)

β = sin (El)

(4.244)

γ = cos (Az) cos (El)

(4.245)

110 Principles of planar near-field antenna measurements Thus E (x, y, z = 0) =

1 λ2


∞
∞ A (− sin (Az) cos (El) , sin (El)) −∞ −∞

×e



 ∂ (α, β)   ∂ (Az, El)  dAz dEl

        −jk − sin Az cos El x+sin El y 

Where

    ∂α   ∂ (α, β)   ∂Az     ∂ (Az,El)  =  ∂β   ∂Az

∂α ∂El ∂β ∂El

     = ∂α · ∂β − ∂α · ∂β  ∂Az ∂El ∂El ∂Az  

(4.246)

(4.247)

Thus ∂ ∂α = (− sin (Az) cos (El)) = − cos (Az) cos (El) ∂Az ∂Az ∂β ∂ = (sin (El)) = cos (El) ∂El ∂El ∂α ∂ = (− sin (Az) cos (El)) = sin (Az) sin (El) ∂El ∂El ∂β ∂ = (sin (El)) = 0 ∂Az ∂Az Hence    ∂ (α, β)     ∂ (Az,El)  = − cos (Az) cos (El) cos (El) − 0

(4.248) (4.249) (4.250) (4.251)

(4.252)

So that 1 E (x, y, z) = 2 λ


∞
∞ A (Az,El) −∞ −∞

× exp (−jk (− sin (Az) cos (El) x+ sin (El) y+ cos (Az) cos (El) z)) × cos (Az) cos2 (El) dAz dEl

(4.253)

This expression will allow the determination of the propagating near-field, on a planar surface from far-field data that has been tabulated on the surface of a sphere, using an azimuth over elevation positioner system, that is, the basis of spherical microwave holographic metrology. When implemented numerically, the integration is separable, if the azimuth integral is evaluated first, as alpha is a function of azimuth

Plane wave spectrum representation of EM waves 111 and elevation whereas beta is a function of elevation only. When the range of azimuth and elevation angles represents a full sphere, this formula is required to be applied separately for the forward and back half-spaces. A similar procedure can be used to derive similar expressions for the case where the far-field pattern has been tabulated on a regular spherical grid. Here the direction cosines are related to the polar spherical angles through α = sin θ cos φ

(4.254)

β = sin θ sin φ

(4.255)

γ = cos θ

(4.256)

So that ∂α ∂θ ∂α ∂φ ∂β ∂θ ∂β ∂φ

∂ (sin θ cos φ) = cos θ cos φ ∂θ ∂ = (sin θ cos φ) = − sin θ sin φ ∂φ ∂ = (sin θ sin φ) = cos θ sin φ ∂θ ∂ = (sin θ sin φ) = sin θ cos φ ∂φ =

(4.257) (4.258) (4.259) (4.260)

Hence    ∂ (α, β)  ∂α ∂β ∂α ∂β    ∂ (θ , φ)  = ∂θ · ∂φ − ∂φ · ∂θ = cos θ cos φ sin θ cos φ + sin θ sin φ cos θ sin φ Or simplifying with trigonometric identities yields    ∂ (α, β)     ∂ (θ , φ)  = sin θ cos θ

(4.261)

(4.262)

Hence 1 E (x, y, z) = 2 λ


π

π /
2+j∞

−π

A (θ, φ) e−jk0 (αx+βy+γ z) sin θ cos θ dθ dφ

(4.263)

0

For the case where we wish to recover propagating fields only, the infinite range of integration in θ reduces to the real finite limit 1 E (x, y, z) = 2 λ


π
π / 2 A (θ, φ) e−jk0 (αx+βy+γ z) sin θ cos θ dθ dφ −π 0

(4.264)

112 Principles of planar near-field antenna measurements This expression is very useful. In addition to enabling microwave holography to be applied to spherical measurement, whether they are taken using a far-field range, a compact antenna test range or a spherical range, they show how to relate the plane wave and spherical wave expansions to one another, they also show how to circumnavigate the soft singularity in the normal field component that is encountered on the kz = 0 (θ = 90◦ ) circle. To illustrate this final point, the plane wave condition can be expressed as k ·A=0

(4.265)

Thus the normal plane wave component can be expressed in terms of the tangential plane wave components as Ax sin θ cos φ + Ay sin θ sin φ cos θ Thus the normal near electric field component can be obtained from Az = −

1 Ez (x, y, z) = − 2 λ


π
π / 2 −π 0

(4.266)

Ax (θ, φ) sin θ cos φ + Ay (θ, φ) sin θ sin φ cos θ

× e−jk0 (αx+βy+γ z) sin θ cos θdθ dφ

(4.267)

Simplifying yields 1 Ez (x, y, z) = − 2 λ ×e


π
π / 2 

Ax (θ, φ) cos φ + Ay (θ, φ) sin φ



−π 0

−jk0 (αx+βy+γ z)

sin2 θdθ dφ

(4.268)

Here, the soft singularity at the unit circle has been removed and the normal near-field component can be recovered without encountering an inconvenient divide by zero error.

4.19 Radiated power and the angular spectrum The time averaged power flux that is transmitted per unit area across an interface, in this case an aperture plane, radiated by an arbitrary current density can be expressed as [12]    
   1 ∗ ET × HT · eˆ z dx dy (4.269) Pr = Re  2    aperture where ET = Ex eˆ x + Ey eˆ y

(4.270)

Plane wave spectrum representation of EM waves 113 and HT = Hx eˆ x + Hy eˆ y

(4.271)

This is equivalent to considering the power radiated by an arbitrary current density flowing in the plane. Here, E and H are assumed to be peak values of the field. If instead E and H had denoted root mean square (r.m.s.) values of the field, then the factor of one half must be omitted. Taking the cross product of the tangential electric field and the complex conjugate of the tangential magnetic field yields    eˆ x eˆ y eˆ z     ET × HT∗ · eˆ z =  Ex E 0  · eˆ z = Ex Hy∗ − Ey Hx∗ eˆ z · eˆ z  H∗ H∗ 0  x y = Ex Hy∗ − Ey Hx∗ Hence Pr =

  




1 Re 2  

Ex Hy∗ − Ey Hx∗ dx dy

aperture

=



  


1 Re 2  

Ex Hy∗ dx dy

aperture

    

(4.272)

    

  


1 − Re 2  

Ey Hx∗ dx dy

aperture

  

(4.273)

 

Using Parseval’s theorem in two dimensions [13]
∞
∞ −∞ −∞

1 f (x, y) g (x, y) dx dy = 4π 2 ∗


∞
∞

    F kx , ky G ∗ kx , ky dkx dky

−∞ −∞

(4.274)

Obtains

   ∞ ∞  ∞ ∞   



1 1 ∗ ∗ Pr = − Re E H dk dk Re E H dk dk x x y y x y y x  8π 2   8π 2  −∞ −∞

Now

−∞ −∞

 1 1 0  H=   k ×E=   eˆ k E − kz Ey + eˆ y (kz Ex − kx Ez ) k  Z0 k  Z0 x k z   +ˆez kx Ey − ky Ex

(4.275)

(4.276)

Now as k ·E=0

(4.277)

114 Principles of planar near-field antenna measurements then Ez = − So

kx Ex + ky Ey kz

(4.278)

2 + ,   k y kx E x + k y E y 1 1 eˆ − − k z Ey H=   k ×E=   k  Z0 k  Z0 x kz + + ,3 ,  k x k x Ex + k y Ey + eˆ y kz Ex + +ˆez kx Ey − ky Ex kz

So Pr =

 ∞ ∞ 



1   Re 8π 2 k  Z0 

−∞ −∞

+

(4.279)

    kx kx Ex∗ + ky Ey∗  dkx dky Ex kz∗ Ex∗ + ∗  kz

 ∞ ∞ 



1   Re 8π 2 k  Z0 



 Ey 

 ky kx Ex∗ + ky Ey∗

−∞ −∞

kz∗

 + kz∗ Ey∗  dkx dky

 ∞ ∞ 

Ex Ey∗ kx ky |Ex |2 kx2 1 2 ∗   Re |E | Pr = k + + x z kz∗ kz∗ 8π 2 k  Z0  −∞ −∞    2 2  Ey  k   Ex∗ Ey kx ky y Ey 2 k ∗  dkx dky + + + z  kz∗ kz∗

  

(4.280)

(4.281)

 ∞ ∞ + ,

 

 2 ∗ ky2 1 kx2 2 ∗   Re |Ex | kz + ∗ + Ey  kz + ∗ Pr = kz kz 8π 2 k  Z0  −∞ −∞

  k x ky  ∗ ∗ + ∗ Ex Ey + Ex Ey dkx dky kz

(4.282)

As k 2 = kx2 + ky2 + kz2

(4.283)

Then kx2 + ky2 = k 2 − kz2

(4.284)

Therefore kz∗ +

k 2 − ky2 kx2 kz2 + kx2 = = kz∗ kz∗ kz∗

(4.285)

Plane wave spectrum representation of EM waves 115 and kz∗ +

ky2 kz∗

=

kz2 + ky2 kz∗

=

k 2 − kx2 kz∗

(4.286)

Hence Pr =

× Re

1   8π 2 k  Z0       ∞ ∞  

|Ex |2 k 2 − ky2 + Ey 2 k 2 − kx2 + kx ky Ex Ey∗ + Ex∗ Ey kz∗



−∞ −∞

dkx dky

  

(4.287)

However, as the contribution made to the integral by evanescent plane waves is purely imaginary as the longitudinal component of the propagation vector is complex, to satisfy the requirement that power is real, that is, the power delivered by real magnetic photons gives Pr =

× Re

1  

8π 2 k  Z0    




|Ex

|2



k2

− ky2



  k 2 +k 2 ≤k 2 x

  2   + Ey  k 2 − k 2 + kx ky Ex E∗ + E∗ Ey x

y

x

kz

y

dkx dky

      

(4.288) The evaluation of this integral outside visible space would recover the power delivered by virtual, columbic photons that can, as a result of the position momentum form of the Heisenberg uncertainty principle, exist only over short distances.

4.20 Summary of conventional near-field to far-field transform What follows is a general overview of the conventional manipulation of data to obtain asymptotic far-field parameters from planar near-field data for the forward half-space. The angular spectrum is obtained directly from the sampled tangential near-field components using 




∞
∞

FT kx , ky , z = 0 = −∞ −∞

ET (x, y, z = 0) ej(kx x+ky y) dx dy

(4.289)

116 Principles of planar near-field antenna measurements The propagating electric field everywhere in the forward half-space can be obtained from the tangential angular spectra as 1 E (x, y, z) = 4π 2


∞
∞ 





FT kx , ky − eˆ z

  kT · FT kx , ky kz

−∞ −∞

× e−j(kx x+ky y+kz z) dkx dky

(4.290)

Here, the longitudinal component of the electric field has been obtained from the tangential components using the plane wave condition Fz (kx , ky ) = −

kT · FT (kx , ky )

(4.291)

kz

For the case where only propagating plane wave mode coefficients are considered, as is the case for near-field antenna measurements, the normal component of the propagation vector is obtained from the tangential components  (4.292) kz = k02 − kx2 − ky2 As only propagating plane wave mode coefficients are considered then kx2 + ky2 ≤ The propagating magnetic field everywhere in space can be obtained from the tangential components of the angular spectra as k02 .

1 H (x, y, z) = 2 4π ωµ




∞
∞





k × FT kx , ky − eˆ z −∞ −∞

× e−j(kx x+ky y+kz z) dkx dky

  kT · FT kx , ky kz (4.293)

At the stationary points kx = k1 and ky = k2 the far-field tangential electric field components can be obtained from the tangential angular spectra using    e−jk0 r kz  F kx , ky E kx , ky ≈ j λr k0

(4.294)

Alternatively, when expressed in terms of the propagation vector the magnetic far zone fields can be obtained from the far zone electric field at the stationary points kx = k1 and ky = k2 as   H kx , ky =

   1  k kx , ky × E kx , ky Z0 k0

(4.295)

This transformation algorithm can be found represented schematically in Figure 4.6. The subject of probe pattern correction, which is crucial to the success of the planar near-field measurement technique, is discussed in detail in Chapter 5. Probe pattern correction is included for the sake of completeness cf. Chapter 5.

Plane wave spectrum representation of EM waves 117 Read in tangential field components for near-field scan.

Convert from polar amplitude and phase to rectangular I and Q.

Transform to the far-field using a two dimentional discrete Fourier transform.

Apply probe pattern correction to the tangential spectral field components.

Obtain the normal field component using the plane wave condition.

Convert from spectral field components to electrical field components.

Calculate magnetic fields if required.

Resolve far-field pattern onto desired polarisation basis, usually Ludwig III.

Inverse transform to aperture plane if required.

Write data to disk and post process.

Figure 4.6

Block diagram of conventional near-field to far-field transform algorithm

4.21 References 1 Zill, D.G., and Cullen, M.R.: Differential Equations with Boundary Value Problems (Brooks/Cole Publishing Company, Pacific Grove, CA, 1997), p. 531 2 Hsu, H.P.: Applied Fourier Analysis (Harcourt Brace College Publishing, San Diego, CA, 1984), p. 193 3 Arfken, G.B., and Weber, H.J.: Mathematical Methods for Physicists (Academic Press, San Diego, CA, 1995), p. 846 4 Pauling, L., and Wilson, E.B. Jr: Introduction to Quantum Mechanics with Applications to Chemistry (Dover Publications, Inc., New York, 1963), p. 58 5 Balanis, C.A.: Advanced Engineering Electromagnetics (John Wiley & Sons, New York, NY, 1989), p. 354

118 Principles of planar near-field antenna measurements 6 Kelvin, Lord: Math. Phys. Papers. 1910, IV, pp. 303–306 7 Hsu, H.P.: Applied Fourier Analysis (Harcourt Brace, San Diego, CA, 1984), p. 76 8 Silver, S.: Microwave Antenna Theory and Design (Mc Graw Hill, New York, 1949), p. 108 9 Hansen, R.C.: Microwave Scanning Antennas (Academic Press, Burlington, MA, 1964) 10 Baker, B.B., and Copson, E.T.: The mathematical theory of Huygens’ principle (Oxford, New York, 1939) 11 Clarke, R.H., and Brown, J.: Diffraction Theory and Antennas (Ellis Horwood Ltd, Chichester, 1980), p. 87 12 Dobs, E.R.: Basic Electromagnetism (Chapman Hall, London, 1993), p. 187 13 Hsu, H.P.: Applied Fourier Analysis (Harcourt Brace, San Diego, CA, 1967), p. 45

Chapter 5

Measurements – practicalities of planar near-field antenna measurements

5.1

Introduction

In Chapter 4 we presented a detailed derivation of the coordinate free form of the nearfield to far-field transform employing the plane wave spectrum representation. We now take this result and consider the various practical issues needed to transform this into a viable planar near-field measurement process. In Chapter 6 we will consider the auxiliary issues of probe pattern characterisation and in this chapter we will address the remaining issues. We commence by considering efficient evaluation techniques for Fourier integrals. Then a mathematical formulation of the truncation problem is presented that will subsequently be utilized in Chapter 9 to motivate the development of the poly-planar approach. An antenna-to-antenna transmission formula for the conventional planar near-field process is then developed. Following this a detailed derivation of a coordinate free form of the antenna-to-antenna transmission formulae is presented, removing the need for probe-to-AUT alignment (AUT – antenna under test). Coordinate systems, conventions and transformations are introduced and are used to illustrate the effects of alignment errors in antenna measurements. Vector isometric rotations are then used as the basis for a novel active alignment correction technique that permits planar near-field measurement without probe-toAUT alignment. The practicalities of determining gain and the related requirement of determining the peak radiated power of the measured three-dimensional radiation pattern are then considered and solutions offered. Finally alternative planar scan approaches are presented, these being the plane-polar and plane-bipolar techniques.

120 Principles of planar near-field antenna measurements

5.2

Sampling (interpolation theory)

If fT (x, y) is band limited in the x- and y-axes to kx0 and ky0 , then a sample spacing of x, y, as per (5.1) and (5.2) will be sufficient to allow
the entire function to be reconstructed from sampling theory [1]. That is, F T kx , ky , z = 0 = 0 when     |kx | ≥ kx0 and ky ≥ ky0 . Hence the limits of the integration become finite and the continuous field can be reconstructed from the samples, 1 E (x, y, z = 0) = 4π 2

ky0 kx0


F kx , ky , z = 0 e−jk0 (ux+vy) dkx dky

−ky0 −kx0

The lattice spacing in each axis may be determined from π x = kx0 and y =

(5.1)

π ky0

(5.2)

(5.3)

where 2π (5.4) = ky0 λ Hence, the sample spacing required to guarantee this, when related to the wavelength is given by kx0 =

x =


π λ   = = y 2 2π λ

(5.5)

Therefore, for a band-limited function, the conventional sampling criterion is sufficient for the case of those measurements that are taken over non-tangential planar surfaces. If the maximum angle of coverage is less than ±90◦ then the expressions are modified as λ x = y = (5.6) 2 sin θmax In one dimension, the ideal band-limited interpolation procedure required to reconstruct the continuous function from the samples taken at a set of grid points can be expressed as f (x) =

 x  n −n f (xn ) sinc π x n=−∞ ∞ 

Here, the series is convergent and the sinc function is defined as sin(x) when x = 0 x sinc (x) = 1 elsewhere

(5.7)

(5.8)

Measurements – practicalities of planar near-field antenna measurements 121 The value of the sinc function in the limit as x tends to zero has been established from an application of L’Hôpital’s rule. Here, x is the sample spacing and xn are the sampling nodes. The process of using such a reconstruction function to compute the values of a field at some given point is referred to as Whittaker interpolation [2]. Thus f (x) =

∞ 

f (xn ) sinc (k0 xn − nπ )

(5.9)

n=−∞

Thus, any band-limited function can be reconstructed from its samples at integer spacing. The summation process can be substantially increased by utilizing trigonometric identities to obtain a recurrence relationship that negates the requirement to evaluate the sinc function, which is a time-consuming operation. Whittaker interpolation can be seen to be exactly equivalent to Fourier interpolation. The extension of this interpolation scheme to the two-dimensional case follows from the two-dimensional form of the inverse Fourier transform (FT). Thus f (x, y) =

∞ 

∞ 

f (xn , ym ) sinc (k0 xn − nπ )sinc (k0 ym − mπ )

(5.10)

n=−∞ m=−∞

This form of interpolation is extensively utilized within this text directly and indirectly within the active alignment correction, pattern function regularisation and within the hybrid pattern recognition processes. From the sampling theorem, the double integral that relates angular spectra to aperture field can therefore be replaced with a double summation without loss of generality. For the majority of this text, the derivation of angular spectra from field components shall be expressed in terms of a double integral. However, this choice is arbitrary and as such it represents a personal preference.

5.3

Truncation, spectral leakage and finite area scan errors

Truncation occurs when the angular spectrum is computed from a partial knowledge of the tangential components of the aperture field distribution or from the tangential components of the field distribution over a finite parallel plane at z = d in front of the radiator. As shown in Chapter 4, the angular spectra can be immediately obtained from the tangential electric field components using




∞ ∞

FT kx , ky , z = d =

ET (x, y, z = d) ej(kx x+ky y) dxdy

(5.11)

−∞ −∞

Let the region over the xy-plane in which the field is sampled be bounded by the curve C . Furthermore, let the area outside C that extends to infinity in all directions be denoted by A , let the aperture be bounded by the curve C and let the area of the aperture be A as illustrated in Figure 5.1.

122 Principles of planar near-field antenna measurements

y A′

AUT A

C

O

z C′

x Figure 5.1

Schematic representation of regions of field integration

If the field is sampled outside the aperture over a plane of finite extent, that is, bounded by the curve C , the error introduced into the angular spectra can be computed from evaluating the following integral  
(5.12) FT kx , ky , z = d = ET (x, y, z = d) ej(kx x+ky y) dxdy A

The far-fields can be obtained by taking the inverse FT of this expression. This is usually accomplished by using the algorithm for the method of stationary phase and calculating the electric field components over a hemisphere of infinite radius centred about the radiator. Thus, the electric field in the far zone of the source can be expressed as
 e−jkr cos θ F kx , ky λr ∞ ∞ e−jkr E (ru, rv, rw) ≈ j cos θ ET (x, y, z = d) ej(kx x+ky y) dxdy λr

E (ru, rv, rw) ≈ j

(5.13) (5.14)

−∞ −∞

Conversely, the error in the electric field in the far zone of the source can be expressed as
 e−jkr cos θF kx , ky λr  e−jkr E (ru, rv, rw) ≈ j ET (x, y, z = d) ej(kx x+ky y) dxdy cos θ λr

E (ru, rv, rw) ≈ j

A

(5.15) (5.16)

Measurements – practicalities of planar near-field antenna measurements 123 Importantly, the truncation error term has an effect over the entire far-field pattern, even on boresight. This is termed the second-order truncation effect and is an artefact of spectral leakage within the transform process. For these expressions to be valid, two important restrictions must be observed: 1. The tangential components of the electric field are zero in the limit, as x and y tend to infinity. 2. The observer must be in the far-field of the source. Importantly, the derivation of the Fourier integral transform as a solution to the wave equation necessitated that the function and its first derivative tend to zero as x, y tend to infinity. Thus, although A constitutes all of the area outside the contour C, which clearly extends to infinity, the tangential field components are required to be finite and bounded. Furthermore, for the method of stationary phase to be valid, the radius of the far-field hemisphere must be large when compared to wavelength and the maximum dimension of the aperture. Strictly in the limit, when the error F in the angular spectrum becomes zero, the sampling plane becomes infinitely large. If the medium in which the radiator is situated is lossless, then the field on the periphery will not be exactly zero and the FT will not exist. Furthermore, as the sampling interval is infinitely large, the method of stationary phase will not be applicable as all observation points can be considered to be in the near zone of the source. Although strictly from a mathematical perspective this is a problem, for all practical situations this is not an issue. The fractional error in the far-field can be expressed as ε (ru, rv, rw) = or

E (ru, rv, rw) E (ru, rv, rw)



ε (ru, rv, rw) =

A

(5.17)

ET (x, y, z = d) ej(kx x+ky y) dxdy

ejkz d



A

ET (x, y) ej(kx x+ky y) dxdy

(5.18)

Clearly, the fractional error, ε, will only become zero when the fields contained within the region of integration A are identically zero. When stated mathematically, the objective of most practical antenna measurement campaigns is to minimise the error function ε (ru, rv, rw) by minimising E (ru, rv, rw), whilst avoiding trivial solutions, that is, making ET (x, y, z = d) = 0 for all x and y. Thus, although the pattern is suppressed by a cosine obliquity factor, the error term is not. From a geometric optics perspective it can be shown asymptotically that in the far-field, the magnitude of the error function ε (ru, rv, rw) is larger than the far-field vector-pattern function for angles larger than the principal angle of truncation θmax [3]. Here, the approximate angle θmax can be related to the near-field measurement geometry through  L−D (5.19) θmax = arctan 2z

124 Principles of planar near-field antenna measurements L Probe

Measurement plane

z

θ

D

AUT

Figure 5.2

Primary angle of truncation for planar measurements

where the lengths, L, D and z and the angle θmax are represented schematically in Figure 5.2. Here, the dimensions of the near-field probe have been ignored as they are assumed to be small when compared to the other dimensions. This would not be possible if a high gain probe is to be used. The primary angle of truncation, as predicted with the use of this geometrical formula, yields indicative results that in practice tend to be optimistic. To illustrate this Figures 5.3 and 5.4 contain measurements of the cardinal cuts of a small circular slotted x-band waveguide array antenna as acquired using a compact antenna test range (CATR) and a planar near-field scanner employing the processing chain developed within Chapter 4 and following sections in this chapter. These patterns have been resolved onto a Ludwig III polarisation basis. For consistency, the phase patterns have been plotted using the phase convention of the receiver, that is, the conjugate of that usually adopted by theory. Although the far-field pattern is encouraging for near in polar angles, at wider angles the pattern is clearly truncated. The separation between the AUT and the probe, that is, range length z, was minimised, typically 3λ ≤ z ≤ 5λ. This separation was chosen to maximise the angular range over which far-field data would be reliable. From inspection, these measurements can be seen to fail for polar angles larger than approximately 65◦ . The range length was sufficiently small that multiple reflections between the AUT and the probe became significant. This can be seen in the discrepancy observed for some near in side lobes. Subsequent measurements in which the range length was increased confirmed this as the level of these side lobes could be seen to change. This was exacerbated by the relatively poor monostatic reflection coefficient that, from reciprocity, is implied by the poor voltage standing wave ratio (VSWR) measured at the coaxial port of the probe at x-band. These results reaffirm that it is not always possible to increase the angle of validity by reducing the separation between the AUT and the probe.

Measurements – practicalities of planar near-field antenna measurements 125 Phi = 0 Freq. = 9 (GHz) CATR PNFS

0 –10

Power (dB)

–20 –30 –40 –50 –60 –70

–80

–60

–40

–20

0

20

40

60

80

Theta (deg) Phi = 0 Freq. = 9 (GHz) CATR PNFS

150

Phase (deg)

100 50 0 –50 –100 –150 –80

Figure 5.3

5.4

–60

–40

–20 0 20 Theta (deg)

40

60

80

Copolar power and phase horizontal cardinal cut

Antenna-to-antenna coupling (transmission) formula

The following derivation of the transmission formula for planar near-field antenna measurement constitutes an extension of that presented by Brown [4]. Here, the theory has been extended to include two additional generalizations, cross-polarisation of

126 Principles of planar near-field antenna measurements Phi = 90 Freq. = 9 (GHz) CATR PNFS

0 –10

Power (dB)

–20 –30 –40 –50 –60 –70

–80

–60

–40

–20

0

20

40

60

80

Theta (deg) Phi = 90 Freq. = 9 (GHz) CATR PNFS

150 100

Phase (deg)

50 0 –50 –100 –150 –80

Figure 5.4

–60

–40

–20 0 20 Theta (deg)

40

60

80

Copolar power and phase vertical cardinal cut

transmit and receive antennas and misalignment between antennas. However, as gain is usually determined by substitution, only relative coupling coefficients are obtained as pattern normalization is avoided. Consider Figures 5.5 and 5.6. As can be seen from the diagrams above both probes are x-polarised in their local coordinate systems. If an angular spectrum of plane waves is incident on an antenna

Measurements – practicalities of planar near-field antenna measurements 127 y AUT Transmit

O z0 x

z

Port1 z9 O9

x9

y9

Figure 5.5

Receive

Port 1 coordinate system measurement of AUT x-polarisation (Probe B) AUT

y

Transmit Port 2 x0

O z

O0

x

z0 Receive

y0

Figure 5.6

Port 2 coordinate system measurement of AUT y-polarisation (Probe C)

then the complex signal received is given by an integral, over all directions, of the scalar product of the incident angular spectrum and the vector-pattern function of the antenna on transmission. Usefully, the reciprocity theorem can be used to establish that the transmitting and receiving patterns of an antenna are identical providing that within the makeup of the coupled antenna system there are no magnetised ferrites, for example, microwave isolators, and so on. This follows from observing that if the source and receiver are interchanged, the transmitted signal remains unchanged. As this must also be true irrespective of the state of the orientation between the two

128 Principles of planar near-field antenna measurements antennas, that is, the angle and separation, this implies that the transmitting and receiving patterns must be identical in both the near- and far-field regions. From Figures 5.5 and 5.6, the relationship between the respective frames of reference can be expressed as follows. For the principally x-polarised probe (probe B) eˆ x = eˆ x

(5.20)

eˆ y = − eˆ y

(5.21)

eˆ z = − eˆ z

(5.22)

For the principally y-polarised probe (probe C) eˆ x

= eˆ y

(5.23)

eˆ y

= eˆ x

(5.24)

eˆ z

= − eˆ z

(5.25)

Alternatively, when expressed in matrix form this can be expressed as 

  eˆ x

1  eˆ y  =  0 0 eˆ z

0 −1 0

   eˆ x 0 0  ·  eˆ y  −1 eˆ z

(5.26)

And 

  eˆ x

0  eˆ y

 =  1 0 eˆ z

1 0 0

   eˆ x 0 0  ·  eˆ y  −1 eˆ z

(5.27)

Assume that a transmitting antenna T and a receiving antenna R face each other in an unaligned way across a semi-infinite slab as shown above. Let the unknown plane wave spectral components of the AUT, at position x0 , y0 , z0, when fed with and radiating unit power, be Ax (α, β) and Ay (α, β), where α, β are the Cartesian components of the unit vector, kˆ = k0 (α, β, γ ) is in the direction of propagation of the elemental plane wave component de (α, β) = Ax (α, β) eˆ x + Ay (α, β) eˆ y A

 β α − Ax (α, β) + Ay (α, β) eˆ z dαdβ γ γ 

(5.28)

Measurements – practicalities of planar near-field antenna measurements 129 Here, the point O has been chosen as the phase reference. Now, consider a plane wave, propagating in the direction k, from the AUT to the probe (probe B), that is k = −k

  = −k0 α eˆ x + β eˆ y + γ eˆ z   = −k0 α eˆ x − β eˆ y − γ eˆ z

  = k0 −α eˆ x + β eˆ y + γ eˆ z

(5.29)

Hence, the relationship between the direction cosines is α = −α

(5.30)



(5.31)



(5.32)

β =β γ =γ Or in matrix form      −1 0 0 α α

 β  =  0 1 0  ·  β  0 0 1 γ γ



(5.33)

Let the far-field
vector-pattern function of the receiving antenna, when receiving  unit power, be eB α , β . Hence, the far-field vector-pattern function (Chapter 4) when expressed in rectangular coordinates and suppressing the unimportant phase factors and 1/r term 
 
 
 j 

eB α , β = γ eˆ x − α eˆ z PxB α , β + γ eˆ y − β eˆ z PyB α , β

λ (5.34)

Here, the phase
reference
is point  O . Here, as this is a predominantly x-polarised B B

probe, Px α , β  Py α , β . Thus when transformed into the coordinate system of the AUT the vector-pattern function becomes

eB (−α, β) =

    j  γ eˆ x − α eˆ z PxB (−α, β) + −γ eˆ y + β eˆ z PyB (−α, β) λ

(5.35)

130 Principles of planar near-field antenna measurements Changing the phase reference to the origin of the receiving antenna, that is, point O yields  j  γ eˆ x − α eˆ z PxB (−α, β) λ    + −γ eˆ y + β eˆ z PyB (−α, β) × e−jk(αx0 +βy0 +γ z0 ) j  eB (−α, β) = γ PxB (−α, β) eˆ x − γ PyB (−α, β) eˆ y λ    + βPyB (−α, β) − αPxB (−α, β) eˆ z

eB (−α, β) =

× e−jk(αx0 +βy0 +γ z0 )

(5.36)

(5.37)

The scalar product of the elemental transmitted wave and this vector-pattern function is j e · de = λ B

A



α2 γ+ γ



PxB (−α, β) Ax (α, β)

βα B P (−α, β) Ax (α, β) γ y αβ B P (−α, β) Ay (α, β) + γ x   β2 B Py (−α, β) Ay (α, β) − γ+ γ −

× e−jk(αx0 +βy0 +γ z0 ) dαdβ

(5.38)

Or since α 2 + β 2 + γ 2 = 1 then

eB · deA =

 j  1 − β 2 PxB (−α, β) Ax (α, β) λ − βαPyB (−α, β) Ax (α, β) + αβPxB (−α, β) Ay (α, β)    − 1 − α 2 PyB (−α, β) Ay (α, β) ×

e −jk(αx0 +βy0 +γ z0 ) dαdβ γ

(5.39)

Measurements – practicalities of planar near-field antenna measurements 131 Hence integrating over the forward hemisphere yields the total amplitude received by probe B j e ·e = λ B



A



 1 − β 2 PxB (−α, β) Ax (α, β)

α 2 +β 2 ≤1

− βαPyB (−α, β) Ax (α, β) + αβPxB (−α, β) Ay (α, β)    − 1 − α 2 PyB (−α, β) Ay (α, β) ×

e −jk(αx0 +βy0 +γ z0 ) dαdβ γ

(5.40)

Now, eB · eA will clearly be a function of x and y, that is, a function of where the near-field probe is within the acquisition plane. Thus eB · eA = s (x, y)

(5.41)

Here, s(x,y) is used to denote the complex coupling coefficient. Thus j s (x = x0 , y = y0 , z = z0 ) = λ



 1 − β 2 PxB (−α, β) Ax (α, β)



α 2 +β 2 ≤1

− βαPyB (−α, β) Ax (α, β) + αβPxB (−α, β) Ay (α, β)    − 1 − α 2 PyB (−α, β) Ay (α, β) ×

e −jk(αx0 +βy0 +γ z0 ) dαdβ γ

(5.42)

Clearly, since this is a two-dimensional FT, that is, of the form 1 s (x, y) = 2 λ

∞ ∞

S (α, β)e−jk0 (αx+βy) dαdβ

(5.43)

−∞ −∞

where in this instance we are integrating over direction cosine space rather than over propagation vector (k) space, this can be immediately inverted to yield ∞ ∞ S (α, β) = −∞ −∞

s (x, y)ejk0 (αx+βy) dxdy

(5.44)

132 Principles of planar near-field antenna measurements Therefore, this is also true for the near-field coupling formula thus ∞ ∞

s (x, y, z = z0 )ej(kx x+kk k) dxdy =

−∞ −∞

 j  1 − β 2 PxB (−α, β) Ax (α, β) λ − βαPyB (−α, β) Ax (α, β) + αβPxB (−α, β) Ay (α, β)    e −jkz z0 − 1 − α 2 PyB (−α, β) Ay (α, β) γ (5.45)

In practice, the left-hand side of this expression is calculated by taking the FT of the measured signal s(x, y, z = z0 ) utilizing a discrete Fourier transform (DFT) or fast Fourier transform (FFT). Alternatively, this can be expressed more compactly in matrix notation as  j  B Sx (α, β, z = z0 ) = Px (−α, β)PyB (−α, β) λ   
  e −jkz z0 1 1 − β2
αβ 2  · Ax (α, β) × Ay (α, β) γ −αβ − 1−α γ (5.46) This is the general formula for the signal coupled from the AUT into the ‘xpolarised’ probe antenna. The measurement of the AUT is repeated for the case where the probe is rotated for ‘y-polarisation’ (probe C). Now, consider a plane wave, propagating in the direction k, from the AUT to the probe, that is k

= −k

  = −k0 α eˆ x + β eˆ y + γ eˆ z   = −k0 α eˆ y

+ β eˆ x

− γ eˆ z

  = k0 α

eˆ x

+ β

eˆ y

+ γ

eˆ z

(5.47)

Hence, the relationship between the direction cosines is α

= −β

(5.48)



β = −α

(5.49)

γ

= γ

(5.50)

Or in matrix form 

  α 0  β

 =  −1 0 γ

   −1 0 α 0 0 · β  0 1 γ

(5.51)

Measurements – practicalities of planar near-field antenna measurements 133 Let the far-field function of the receiving antenna when receiving 
vector-pattern unit power be eC α

, β

. Hence, the far-field vector-pattern function, compare with Section 2.7, Chapter 2, when expressed in rectangular coordinates is 

 j 


   γ eˆ x

− α

eˆ z

PxC α

, β

+ γ

eˆ y

− β

eˆ z

PyC α

, β

eC α

, β

= λ (5.52)

Where the phase is
point O




reference   . Here, as this is a predominantly x-polarised



C C probe, Px α , β  Py α , β . Thus when transformed into the coordinate system of the AUT the vector-pattern function becomes

eC (−β, −α) =

    j  γ eˆ y − β eˆ z PxC (−β, −α) + γ eˆ x − α eˆ z PyC (−β, −α) λ (5.53)

Changing the phase reference to the origin of the receiving antenna, that is, point O yields  j  γ eˆ y − β eˆ z PxC (−β, −α) λ    (5.54) + γ eˆ x − α eˆ z PyC (−β, −α) × e−jk(αx0 +βy0 +γ z0 )  j eC (−β, −α) = γ PyC (−β, −α) eˆ x + γ PxC (−β, −α) eˆ y λ    − βPxC (−β, −α) + αPyC (−β, −α) eˆ z × e−jk(αx0 +βy0 +γ z0 ) eC (−β, −α) =

(5.55) The scalar product of the elemental transmitted wave and this vector-pattern function is j e · de = λ C

A



αβ Ax (α, β) PxC (−β, −α) γ  γ + α2 PyC (−β, −α) Ax (α, β) + γ  γ + β2 PxC (−β, −α) Ay (α, β) + γ  βα C + Py (−β, −α) Ay (α, β) γ

× e−jk(αx0 +βy0 +γ z0 ) dαdβ

(5.56)

134 Principles of planar near-field antenna measurements Or since α 2 + β 2 + γ 2 = 1 then j  αβAx (α, β) PxC (−β, −α) eC · deA = λ   + 1 − β 2 PyC (−β, −α) Ax (α, β)   + 1 − α 2 PxC (−β, −α) Ay (α, β)  +βαPyC (−β, −α) Ay (α, β) ×

e −jk(αx0 +βy0 +γ z0 ) dαdβ γ

(5.57)

Hence integrating over the forward hemisphere yields the total amplitude received by probe C   j αβAx (α, β) PxC (−β, −α) eC · eA = λ α 2 +β 2 ≤1

  + 1 − β 2 PyC (−β, −α) Ax (α, β)   + 1 − α 2 PxC (−β, −α) Ay (α, β)  +βαPyC (−β, −α) Ay (α, β) ×

e −jk(αx0 +βy0 +γ z0 ) dαdβ γ

(5.58)

Again, this can be expressed more compactly in matrix notation as  j  C Py (−β, −α)−PxC (−β, −α) Sy (α, β, z = z0 ) = λ   
  1 1 − β2 αβ  Ax (α, β) e −jkz z0
· × Ay (α, β) γ −αβ − 1 − α2 γ (5.59) This is the general formula for the signal coupled from the AUT into the ‘ypolarised’ probe antenna (probe C). The expressions for the field coupled into the xand y-polarised probes can be combined to yield the general transmission function     j PxB (−α, β) PyB (−α, β) Sx (α, β) = Sy (α, β) λ PyC (−β, −α) −PxC (−β, −α)   
  1 1 − β2 e −jkz z0
αβ 2  · Ax (α, β) × · Ay (α, β) −αβ − 1−α γ γ (5.60)

Measurements – practicalities of planar near-field antenna measurements 135 Or [S] =

j [P] · [M ] · [A] e−jk0 z0 λ

(5.61)

where [S] is the received probe field; [P] is the probe ‘B’ and ‘C’ pattern functions, [M ] is the angular dependant trigonometric factors and [A] is the AUT plane wave spectral components. This can readily be inverted in order that the probe pattern may be de-coupled from the measurement [A] =

λ [M ]−1 · [P]−1 · [S] ejkz z0 j

where −1

[M ]

1 = γ

 
1 − α2 −αβ

(5.62) 


αβ  − 1 − β2

(5.63)

and [P]−1 =

1 PxB (−α, β) PxC (−β, −α) + PyB (−α, β) PyC (−β, −α)  C  Px (−β, −α) PyB (−α, β) × PyC (−β, −α) −PxB (−α, β)

(5.64)

At first sight, it appears that the above expression offers the possibility of sampling the AUT within the reactive near-field region and deriving probe corrected propagating and evanescent field components. Unfortunately, this is not so. In order that the near-field AUT–probe coupling convolution can be expressed as a multiplication in the far-field, it is required that the probe pattern remain constant over the sampling surface. Assuming that multiple reflections between the AUT and the probe are negligible and do not upset the field distribution within the aperture of the scanning probe, then the visible portion of the probe pattern will remain constant. However, the invisible portion of the pattern will not, as evanescent coupling will be greater toward the centre of the AUT than at the perimeter. Hence in practice, near-field measurements cannot be compensated for the evanescent coupling between the probe and AUT. Consequently, near-field measurements are limited to being made in the propagating near-field. This process corresponds to the deconvolution of the response of the input probe from the measured spectra to determine the response of the AUT as though it had been characterised with a Huygens’ source. The factor containing the geometric terms corresponds to the convolution of these spectra with the pattern of a lossless and reciprocal electric Hertzian dipole output probe, which will yield results that are in agreement with theoretical predictions. If however, the output probe had been specified as a magnetic Hertzian dipole then magnetic fields would be produced. Thus, if an ideal electrical dipole antenna is used as a probe then no pattern correction is required.

136 Principles of planar near-field antenna measurements Two probe patterns are required in general and these are termed probe B and probe C (A being the AUT), where each are x copolar in their local coordinate systems. The intention of this is that if a single port probe (SPP) has been used to characterise the AUT, then the same probe pattern can be used for probe B and probe C. This formulation is particularly useful when considering plane-polar or plane-bipolar nearfield measurements as the requirement to employ a rotationally symmetric measuring probe may be lifted. Within this derivation we have essentially considered the coupling between fields associated with a transmit (Tx) and receiver (Rx) antenna. These expressions can be modified to consider the transmission between the input guided wave port and the output guided wave port by incorporating a factor for the mismatch associated with each antenna. More importantly, no account for multiple reflections between the AUT and the probe have been made within the transmission formula derived above. Thus, in order that reliable near-field measurements can be made, multiple reflections between the AUT and the probe must be negligibly small.

5.4.1 Attenuation of evanescent plane wave mode coefficients The region in which near-field antenna measurements can be made can be determined by examining the rate at which an evanescent mode will decay with distance. The attenuation when expressed in decibels can be obtained from the propagation vector as   (5.65) AdB = 20 log10 ejkz z For evanescent modes the propagation vector kz will be complex as  γ = −j α 2 + β 2 − 1

(5.66)

Hence

√  √ 2 2    2 2 AdB = 20 log10 ek0 z α +β −1 = 20 log10 e2π(z/ λ) α +β −1

(5.67)

Since 40π log10 (e) ≈ 56.4

(5.68)

and assuming that the attenuation, when expressed in decibel form, is represented by a negative number and from the law of logarithms  z AdB = −56.4 α2 + β 2 − 1 (5.69) λ Hence, the evanescent portion of the near-field suffers a very large attenuation with distance from the source. Consequently, sampling the near-field of an AUT beyond a few wavelengths from the aperture should ensure that the probe is not reactively coupled to the AUT. To understand the concept of reactive coupling consider the radiation pattern of an isolated dipole, then bring a metal rod of similar length close to it. We have now of course changed the antenna structure (it’s a two element array with passive coupling for the second element formed by the rod – the principle by which the Yagi antenna operates) and so in turn its radiation pattern. If the probe or

Measurements – practicalities of planar near-field antenna measurements 137 any structure associated with the planar scan has the same effect on the AUT we will never recover the desired AUT far-field pattern.

5.4.2 Simple scattering model of a near-field probe during a planar measurement Inevitably, the match between the near-field probe and an incoming plane wave will be imperfect. For example, if the magnitude of the reflection coefficient of a waveguide probe was, at a frequency 10 GHz, approximately −8.5 dB this implies that at every point at which a measurement was taken, −8.5 dB of the incident radiation will be scattered back toward the AUT. Thus, the acquisition plane would appear to the AUT to constitute an imperfect reflector, cf. a semi-silvered mirror, with dimensions exactly equal to those of the acquisition plane. For a perfectly reflecting flat rectangular plate, physical optics yields a monostatic backscatter radar cross section of [5] σPO (u, v) =

 4πa2 b2  2 2 1 − u sinc2 (ak0 u) sinc2 (bk0 v) − v λ2

(5.70)

Where, the result is independent of the polarisation of the incident field and is only valid for angles of incidence approaching the normal to the plane and where the smallest dimension of the plate is large compared with the wavelength. Here, a and b denote the lengths of the sides of the plate, whilst the thickness is taken to be infinitely small. Hence the scattering cross section of the scan plane can be approximated as a two-dimensional sinc function weighted by the, in general complex, reflection coefficient of the near-field probe, . σPO (u, v) =

 4πa2 b2 2  2 2

− v 1 − u sinc2 (ak0 u) sinc2 (bk0 v) λ2

(5.71)

However, the imperfect mirror analogy must be modified to take into account the fact that the near-field measurement is formed from a discrete set of samples spaced at half wavelength intervals in the x- and y-axes. Thus, although half wavelength sample spacing is sufficient to ensure that the far-field pattern function is free from aliases, this is not the case for the scattering cross section where grating lobes will be centred about |u| = 1 or |v| = 1. For this first order analysis the signal coupled from the AUT to the scanning probe can be expressed as A (u, v) · P B (u, v)

(5.72)

Furthermore, the signal reflected at the aperture of the probe and coupled back into the AUT can again be expressed as A (u, v) · P B (u, v)

(5.73)

Thus, the total signal reflected back into the AUT can be expressed as σPO (u, v) =

  2 4π a2 b2 2 

1 − u2 − v2 A2 (u, v) · P B (u, v) sinc2 (ak0 u) sinc2 (bk0 v) 2 λ (5.74)

138 Principles of planar near-field antenna measurements A more exact expression can be obtained if the general antenna-to-antenna coupling formula is utilized. However, the above formulation is sufficient for certain general observations to be made: 1. AUT-to-probe multiple reflections are minimised if the probe is well matched to an incoming plane wave. 2. The scattered signal will contain grating lobes at ±90◦ in azimuth and elevation if the measurements are made with half wavelength sample spacing. 3. The scattered signal is highly oscillatory in nature where the exact function will depend upon the dimensions of the acquisition plane. Multiple reflections between the probe and AUT can be revealed by changing the separation between the AUT and the probe. This works well, provided the repeatability between successive scans is good. If the separation distance is changed by a quarter wavelength, the reflected path length will change by half a wavelength and the interfered signal will change by a maximum amount. Thus taking multiple acquisitions as a function of separation can reveal the sinusoidal variation in the signal level, which can be used to calculate the underlying value and so correct it.

5.5

Evaluation of the conventional near-field to far-field transform

In Chapter 4 (Section 4.19) a general overview of the conventional manipulation of data to obtain asymptotic far-field parameters from planar near-field measurements was undertaken. This led to the result that the far-field tangential electric field components can be obtained from the tangential angular spectra using  
e−jk0 r kz
E kx , ky ≈ j F kx , ky λr k0

(5.75)

It was noted that when expressed in terms of the propagation vector the magnetic far zone fields can be obtained from the far zone electric field at the stationary points kx = k1 and ky = k2 as
 H kx , ky =


 1
k kx , ky × E kx , ky Z0 k0

(5.76)

From Section 5.4, when ignoring unimportant scaling factors the effects of the probe pattern can be removed from the computed field parameters again at the stationary points kx = k1 and ky = k2 as 
 
−1 
−1 
 A kx , ky = M kx , ky · P kx , ky · S kx , ky (5.77)

Here, S kx , ky is the FT of the measured tangential field components and A kx , ky are the probe pattern corrected spectral components of the AUT. As shall be shown subsequently, this is far removed from the generalized transformation algorithm developed within this text.

Measurements – practicalities of planar near-field antenna measurements 139

5.5.1 Standard techniques for the evaluation of a double Fourier integral The necessary FT required within the evaluation of the angular spectrum can be approximated by −1 N −1 M  
 F kx , ky , z = 0 ≈ xy E (nx, my, z = 0) ej(kx nx+ky my) n=0 m=0

(5.78)

where E and F may be x-, y- or z-polarised. When this summation is evaluated in this form, it is often referred to as a DFT (discrete Fourier transform). A DFT can be computed directly or via the efficient FFT. Both of these techniques are utilized within this text and the relative merits of these techniques are compared and contrasted in the following sections. 5.5.1.1 The FFT The FFT obtains its great efficiency at the expense of requiring that the data in both spatial and spectral domains be constrained to lie on a plaid monotonic equally spaced grid. Thus the values that kx and ky can take shall be expressed as kx = kx0 + ikx

(5.79)

Since

 π    |kx0 | =   x Then

(5.80)

π (5.81) x Here, i = 0, 1, 2, 3, . . . , N − 1. Furthermore, the increments can be obtained from kx = ikx −

2kx0 2π = (5.82) N N x Similar expressions can be obtained for the y-dimension. Thus  −1 N −1 M   π π Fx ikx − Ex (nx, my, z = 0) , lky − , z = 0 ≈ xy x y kx =

n=0 m=0

× ej(kx nx+ky my)

(5.83)

which is in a form suitable for evaluation with the FFT. The inverse transform can be expressed as  I −1 L−1 kx ky   π π Ex (nx, my, z) = Fx ikx − , lky − ,z = 0 x y 4π 2 i=0 l=0

×e

−j (kx x+ky y+kz z )

(5.84)

140 Principles of planar near-field antenna measurements Since



kx x + k y y =

 π l2π π i2π − nx + − my N x x M y y

in2π lm2π − nπ + − mπ N M  in lm = 2π + − π (m + n) N M

(5.85)

=

then L−1 I −1    1 π , Fx ikx − ejπ(m+n) x xyMN i=0 l=0 π lky − , z = 0 × e−jkz z e−j2π((in/ N )+(lm/ M )) y

Ex (nx, my, z) =

(5.86)

which again is amenable for evaluation by the FFT. Here, for homogeneous plane wave modes the longitudinal propagation vector can be expressed in terms of the tangential propagating vectors as    π 2 π 2 − lky − (5.87) kz = k02 − ikx − x y Factoring these expressions can be seen to substantially increase the efficiency of the algorithm; thus, the two-dimensional transform is usually evaluated using  N −1  π π Fx ikx − , z = 0 ≈ xy ejkx nx , lky − y x n=0

×

M −1 

Ex (nx, my, z = 0) ejky my

(5.88)

m=0

Alternatively the inverse transform can be evaluated by observing that it can be computed with the transform where N is used to denote the number of elements in the data set F. f =

1 conj (FFT {conj (F)}) N

(5.89)

Most, but not all, FFT algorithms impose a number of peculiarities on the transformed data set that have to be corrected or allowed for. These are: • Place the phase reference at the zero-time, that is, to the bottom left hand corner of the temporal data set, that is, rather than at the centre. • Do not place the zero-frequency component in the centre of the spectrum. • Impose a checkerboard 0, π phase pattern on the spectrum.

Measurements – practicalities of planar near-field antenna measurements 141 • •

The dimensions of the spatial and spectral arrays (cf. time and frequency) must be the same size. If a finer resolution is required on the output data, then the input data can be padded with zeros prior to evaluating the FFT. The number of elements within the array must be a power of two. This is negated if using a more sophisticated mixed-radix transform.

In the latter case we swap the first and third quadrants and the second and fourth quadrants of the pattern. Alternatively we can apply a phase shift prior to taking the FFT. Essentially, this is analogous to the phase change that is applied to the radiating elements of an electronically scanned planar phased array antenna. Thus the spatial function, a is pre-multiplied by the complex factor exp (jk0 (u0 x + v0 y))

(5.90)

The resulting spectral function A will have the zero-frequency component conveniently placed at the centre of the spectrum providing that u0 = v0 = 1 where this is the direction, that is, spatial frequency, in which we want the zero-frequency component steered to. Note, the sign in the complex exponential will depend upon the sign convention adopted within the FFT algorithm employed. In point of fact, we are free to steer the zero-frequency component in other directions. Assuming that the output arrays are of a size equal to a positive integer power of two, the number of rows and columns will logically be an even number. By following this procedure the zero-frequency component will not be placed exactly in the centre of the array of data. Instead, it will be very slightly asymmetrical in kx and ky . This phase factor pre-multiplication procedure enables the boresight direction to be displaced by half an element in each axis so that the main beam direction is symmetrically placed within the array. As stated above, the near-field data must be tabulated on an equally spaced rectangular grid. Without further processing, this renders the FFT inappropriate for applications where the data is tabulated regularly in another coordinate system, that is, polar or bipolar. Furthermore, the angular spectrum resulting from the FFT will be tabulated on a regular direction cosine grid rather than the required angular or other, coordinate system. Although this data can be converted to another coordinate system with the use of interpolation, if this interpolation is performed rigorously the interpolation will take as long as a conventional DFT or if it is accomplished approximately with piecewise polynomial interpolation, it will inevitably contain errors. The conditions of the FFT can be summarised as 1. The spatial function must be periodic. 2. The spatial function must be band limited. 3. The sampling rate must be at least two times greater than the highest frequency component contained within the spatial function otherwise aliasing will result. 5.5.1.2 Discrete Fourier transform The DFT algorithm is flexible, as no constraints have been placed on the values that x, y, kx and ky can take. This extends to not being tied to having the same number

142 Principles of planar near-field antenna measurements of points in the spatial and spectral domains. Thus, the zero padding that is required within the FFT to attain a finer sample spacing in the output array is not required here. The evaluation of the complex exponential in the expressions above is often accomplished with the use of the Euler relation that relates algebra to geometry and is expressed as ejθ = cos θ + j sin θ

(5.91)

The efficiency of the evaluation of the two-dimensional summation can be enhanced greatly by the use of trigonometric recurrence relationships, to avoid the requirement to directly evaluate these trigonometric functions. Both the discrete and FFT algorithms utilize such techniques. The efficiency of the algorithm can be substantially increased by replacing the computation of the cos and sin functions with a standard trigonometric recurrence relation. However, this relation imposes the requirement that the near-field measurements are made on a regular rectangular grid thus x = xstart + nx

(5.92)

Having initialised xj and yj the following recurrence relation can be formed xj+1 = xj + x

(5.93)

Hence




 cos kx xj+1 = cos kx xj + x  


sin kx xj+1 = sin kx xj + x

(5.94) (5.95)

By utilizing standard trigonometric identities this can be expressed in the desired form as cos (kx xstart + kx x) = cos (kx xstart ) cos (kx x) − sin (kx xstart ) sin (kx x) (5.96) sin (kx xstart + kx x) = sin (kx xstart ) cos (kx x) + cos (kx xstart ) sin (kx x) (5.97) Similar expressions can be formed for the y-integration. Here, all the terms can be evaluated before entering the summation loop and only two multiplications and one subtraction or addition are required to compute the required cos or sin. In practice, this greatly increases the efficiency of the operation. Although recurrenence relations that exist are less prone to the cumulative effects of numerical truncation and round off errors, these tend to be more computationally expensive in their evaluation. These forms of recurrence relations are usually utilized within FFT algorithms. 5.5.1.3 FTs with improved accuracy Both the DFT and the FFT described above evaluated the necessary integrals by utilizing zero-order, that is, rectangular integration scheme. Unfortunately, although

Measurements – practicalities of planar near-field antenna measurements 143 the spatial function may be a smoothly varying quantity, the complex exponential factor will inevitably become highly oscillatory as the parameters, u and v tend towards larger values. Utilizing integration schemes of higher order, that is, first order (trapezoidal) or third order (cubic), can improve matters; however, this is obtained at the expense of greater complexity and computational resource. References 6 and 7 describe methods for improving the accuracy of integrals evaluated using the DFT or the FFT and these are verified in Appendix D. The improvement in accuracy offered by ‘integration’ rather than the simple addition process of the basic FFT can be understood when considering the spectral domain of the radiation pattern, where due to sampling the base spectra is repeated periodically. The integration process can then be viewed as a form of filter, removing these spatial spectral ‘harmonics’. Effectively, the errors associated with the conventional rectangular DFT become most apparent at angles away from the direction of the normal vector of the sampling plane. Thus in the case of the poly-planar approach (Chapter 9), as every partial plane is inclined at a different angle to every other partial plane the resulting error term associated with each plane will be different to that error associated with every other plane. Thus the overall error signal will be complex in form and could potentially upset the superposition that is required in order for the entire far-field pattern to be recovered.

5.6

General antenna coupling formula: arbitrarily orientated antennas

The principal assumption made within Section 5.4 was that the probes and AUT are ideally aligned. This is not the case in general. The three main alignment differences that may occur are as follows: 1. The AUT is not aligned with the axes of the range. 2. Probe 1 is not aligned with the axes of the range. 3. Probe 2 is not aligned with the axes of the range. The orientation of the AUT to the range is unimportant in these considerations, as strictly, the convolution can be seen to be between the patterns of the sampled illumination function and probe. The matrix representation of the relationship between the probe B and probe C and the AUT indicate the form of a generalized antenna–antenna transmission function. Let the relationship between the axes of the transmitting antenna A and the two receiving antennas B and C be represented by    eˆ x

eˆ x  eˆ y  = [B] ·  eˆ y  eˆ z eˆ z



(5.98)

144 Principles of planar near-field antenna measurements and



   eˆ x

eˆ x  eˆ y

 = [C] ·  eˆ y  eˆ z eˆ z

(5.99)

where [B] and [C] are orthogonal and correctly normalized, that is, orthonormal, direction cosine matrices. Previously, these matrices have corresponded to a 180◦ rotation in elevation and a 180◦ rotation in elevation followed by a −90◦ roll, respectively. However, in principle, they could describe any orientation in which case   k = −k = −k0 α eˆ x + β eˆ y + γ eˆ z (5.100) Thus

     k = k0 −α B11 eˆ x + B12 eˆ y + B13 eˆ z − β B21 eˆ x + B22 eˆ y + B23 eˆ z

  −γ B31 eˆ x + B32 eˆ y + B33 eˆ z

(5.101)

Clearly   k = k0 α eˆ x + β eˆ y + γ eˆ z

(5.102)

Hence α = − (B11 α + B21 β + B31 γ )

(5.103)



(5.104)



(5.105)

β = − (B12 α + B22 β + B32 γ ) γ = − (B13 α + B23 β + B33 γ ) This can be expressed compactly in matrix form as     α α  β  = − [B]−1 ·  β  γ γ

Similarly 

   α α  β

 = − [C]−1 ·  β  γ γ

(5.106)

(5.107)

Although these substitutions are algebraically unwieldy, they are readily amenable for processing numerically with a digital computer. Here, the process of probe compensation for a rotated probe is in principle identical to that of an ideally aligned probe. For example, probe pattern correction for the case where probe B and probe C are rotated about the normal to the measurement plane can be accomplished within

Measurements – practicalities of planar near-field antenna measurements 145 the existing formulation by 1. Rotating the αRFS βRFS through the angle that the probe was rotated through. 2. Performing the pattern interpolation in the conventional manner to the adjusted positions. 3. Ensuring that the polarisation vector rotates with the probe pattern, that is, a scalar rotation. 4. Deconvolving the response of the probe from the measured antenna pattern in the conventional way. 5. Projecting the measured field components back onto the range vectors. Electromagnetic (EM) fields are vector-valued functions. Thus, the field has both a position and a direction. Therefore, within this text, data is presented tabulated in one of a number of coordinate systems and resolved onto one of a number of polarisation bases. The application of isometric rotations is central to this work so a detailed introduction to the coordinate systems, polarisation basis, rotation conventions and mathematical transformations utilized are presented in Appendix C. The following orthogonal coordinate systems are utilized within this text: 1. 2. 3. 4.

Plane rectilinear Direction cosine Azimuth over elevation Polar spherical.

As an example, consider the case of probe pattern correction used to correct near-field measurements that have been acquired using a rotated probe. In Section 5.4 a formula (5.57) that can be used to correct planar near-field measurements was developed namely    
 λ Ax (α, β) αβ 1 − α2
 = Ay (α, β) −αβ − 1 − β2 jγ D     C Px (−β, −α) PyB (−α, β) Sx (α, β) ejk0 γ z · · Sy (α, β) PyC (−β, −α) −PxB (−α, β) (5.108) Where the matrices are multiplied out according to the usual rules of linear algebra and D=

1 PxB (−α, β) PxC (−β, −α) + PyB (−α, β) PyC (−β, −α)

(5.109)

The x-orientated and y-orientated probe spectral components, that is, PxB , PyB , and PyC can be found plotted in Figures 5.7 and 5.8, tabulated in an azimuth over elevation coordinate system which is the required plotting system for the final antenna pattern in this example. Thus, although the expressions are couched in terms of direction cosines this does not impose a restriction on the plotting or tabulating, coordinate system. PxC

146 Principles of planar near-field antenna measurements Probe port 1, Fx (dB)

40 –0.5

30

EI (deg)

20

–2

0

–2.5 –3

–10

–3.5

–20

40

–15

10

–20

0

–25

–10

–30 –35 –40

–30

–4.5

–45

–40 –40 –30 –20 –10 0 10 20 30 40 Az (deg)

–40 –40 –30 –20 –10 0 10 20 30 40 Az (deg)

Figure 5.7

–10

–20

–4

–30

–5

20

–1.5

10

Probe port 1, Fy (dB)

30

–1

EI (deg)

40

Spectral components of probe B Probe port 2, Fx (dB)

40 –0.5

30

Probe port 2, Fy (dB) –5

30

–1 10

–2

0

–2.5 –3

–10

–3.5

–20

–4 –30

–4.5

–40 –40 –30 –20 –10 0 10 Az (deg)

Figure 5.8

–10 20

–1.5

20 30

40

EI (deg)

EI (deg)

20

–15

10

–20

0

–25

–10

–30 –35

–20

–40

–30 –40 –40 –30 –20 –10 0 10 Az (deg)

–45 20 30

40

Spectral components of probe C

The patterns look very strange here. One might perhaps have expected that the principal patterns swap over between the B and C probes. However, the B and C probes are both principally x-polarised in their local coordinate systems. The difference in the patterns merely results in the scalar rotation that is implemented by the direction cosine mapping. The polarisation rotation that completes the necessary vector rotation is essentially implemented within the pattern correction equations themselves. If one looks back to the derivation of the probe pattern correction equations in Section 5.4, the transformation from local coordinates to antenna coordinate system is evident and the difference between the way the respective polarisations are coupled between the AUT and the probe for each case is clearly different. In order that the probe pattern correction technique can be extended to accommodate the case where the probe has been rotated by an arbitrary fixed angle φ about the z-axis of the range all that is required is to rotate the pattern of the probe and the pattern of the Hertizan dipole and then to resolve the corrected fields back onto the range polarisation basis. Strictly then, the probe pattern correction formula requires

Measurements – practicalities of planar near-field antenna measurements 147 Probe port 1, Fx (dB)

40

40 –0.5

30

Probe port 1, Fy (dB) –5

30

–1

–10 20

–1.5

10

–2 –2.5

0

–3

–10

–3.5

–20

–20

0

–25

–10

–30 –35

–20

–4

–30

–4.5

Figure 5.9

–40 –30 20

30

40

Spectral components of rotated probe B

Probe port 2, Fx (dB)

40

–45

–40 –40 –30 –20 –10 0 10 Az (deg)

–40 –40 –30 –20 –10 0 10 20 30 40 Az (deg)

40 –0.5

30 20

0

–2.5 –3

–10

–3.5

–20

–10 –15

10

–20

0

–25

–10

–30

EI (deg)

–2

–5

20

–1.5

10

Probe port 2, Fy (dB)

30

–1

EI (deg)

–15

10

EI (deg)

EI (deg)

20

–35

–20

–40

–4

–30

–4.5

–40 –40 –30 –20 –10 0 10 Az (deg)

Figure 5.10

20

30

40

–30 –40 –40 –30 –20 –10 0 10 Az (deg)

–45 20

30

40

Spectral components of rotated probe C

no modification, only the pattern data that passes through the formula requires modification. Probe B and probe C spectral components can be found plotted in Figures 5.9 and 5.10 for the case where the probe has been rotated by 53◦ . The rotation of direction cosines is accomplished using the usual expressions namely 

α

β



 =

cos φ − sin φ

sin φ cos φ

   α · β

(5.110)

and the resulting patterns are essentially the same as those shown above only here a simple scalar isometric rotation has been applied to the patterns. Here φ has been used to denote the probe rotation angle. These patterns, together with the rotated direction cosines are used in the probe pattern correction formula to obtain the probe pattern corrected field. Thus, this

148 Principles of planar near-field antenna measurements process can be expressed as    
λ Ax (α, β) 1 − α 2 = Ay (α, β) −α β

jγ




α β  − 1 − β 2



· [P]−1 ·



Sx (α, β) Sy (α, β)



ejk0 γ z

(5.111)

Where [P]−1 =

1 PxB (−α , β ) PxC (−β , −α ) + PyB (−α , β ) PyC (−β , −α )  ×


 PxC −β , −α


 PyC −β , −α


  PyB −α , β


 −PxB −α , β

(5.112)

Finally, the corrected field component can be resolved back onto the range polarisation basis using       Ax (α, β)|RFS cos φ − sin φ Ax (α, β) (5.113) = · Ay (α, β)RFS Ay (α, β) sin φ cos φ Any further correction required as part of the antenna to range alignment correction can now be implemented in the usual way. Clearly and as noted previously, the P matrix essentially contains the vector-pattern function of the two near-field probes, that is, the input probes. Conversely, the M matrix can easily be shown to contain the vector-pattern function of two elementary Hertzian dipole antennas, that is, the vector-pattern functions of an x- and y-axis orientated current element, which can be taken as representing the patterns of the two output probes. Finally, these expressions are in a form that is amenable for processing data acquired using, for example, a plane-polar range. Here, the pattern correction must be implemented within the φ integral so that similarly orientated measurements can be corrected before being added to measurements taken with the next, dissimilar, probe orientation. This is also suitable for processing plane-bipolar data although in this case the probe patterns must be rotated differently for each and every near-field sample point.

5.7

Plane-polar and plane-bipolar near-field to far-field transform

The above concept of taking account of probe and AUT misalignment can be extended to undertaking the planar scan using coordinate planes other than Cartesian. The general benefits and drawbacks of the plane-polar and plane-bipolar near-field measurement systems were described within Chapter 3. Chapter 4 mainly dealt with how asymptotic far-field parameters could be obtained from plane rectilinear nearfield measurements. This section draws these two themes together and describes how far-field data can be obtained from near-field measurements taken using either

Measurements – practicalities of planar near-field antenna measurements 149 plane-polar or plane-bipolar measurement systems. Two methods for accomplishing this task are 1. Interpolate the data from a plane-polar or plane-bipolar measurement grid to a plane rectilinear grid and then utilize the standard techniques of Chapter 4 to obtain the far-field pattern. 2. Recast the near-field to far-field integral transform in terms of the plane-polar or plane-bipolar coordinate systems. The principal advantage of utilizing interpolation is that the FFT can be used to significantly improve the efficiency of the transform, however, this is sought at the price of having to interpolate rapidly varying complex near-field data at a very early stage within the transformation processing chain. Conversely, deploying the near-field to far-field transform in each of these measurement geometries removes the requirement for such approximation but will inevitably increase the amount of computational effort required to obtain far-field data. There is an added benefit of adopting this later approach, which is that it can be used to alleviate the need to counter-rotate the probe or the need to use a probe with a rotationally symmetrical pattern. This stems from the fact that both the planepolar and plane-bipolar techniques inherently rotate the probe about the range zaxis during an acquisition. Thus, the AUT–probe coupling will change differentially across the acquisition surface. This is not the case with plane rectilinear scanning where the angular relationship between the probe and the acquisition window is unchanged whilst characterising each near electric field component. This difficulty can be resolved if the probe is rotated by an equal and opposite amount to counter the rotation and to maintain the angular alignment between the probe and AUT. However, if this additional rotation of the probe does not perfectly counter the inherent rotation then additional inaccuracies will be introduced into the measurement. Worse, these will be different for each rotation angle. Thus, the additional positioning complexity and its associated expense, renders this an inelegant solution. If the probe is not counter-rotated and does not have a rotationally symmetrical pattern then the rotation of the probe about the z-axis must be accounted for within the compensation process. However, as shown below, this additional complication can be resolved if the probe pattern correction formula are modified in a fairly straightforward way to include this φ rotation. When the data is interpolated from the polar to rectangular grids the properties of the probe are included within the measured field points. If the orientation of the probe changes from field point to field point, then the interpolation formula may not be able to properly account for this variation in the underlying function when estimating an interleaving point. Thus, whilst it is quite possible and necessary, to resolve the measured polarisation basis back onto the usual Cartesian basis prior to performing the interpolation, variations in the rotational pattern of the probe are still manifest within the data. Since the interpolated, that is, interleaving, point is formed from an estimate of a number of nearby sampling nodes, each of which could have a different probe rotation angle associated with it, the resulting estimate could clearly be in error. Even in the event that the underlying function has been approximated

150 Principles of planar near-field antenna measurements reliably after processing with the FFT, the associated probe rotation angle is lost and the resulting angular spectra are uncorrectable. However, if each set of rotated nearfield data (that is, each individual cut taken at a fixed φ angle) are transformed to the equivalent angular spectra, then they can be corrected using the generalized probe correction formula before being linearly superimposed on the other corrected data sets. In this way, probe corrected far-field data can be obtained with no loss in rigour. The following sections show first how the near-field to angular spectrum transform can be adapted for use with these measurement systems before going on to show how the probe pattern correction can be modified to work for the case where a probe has been rotated through an arbitrary, but known, angle.

5.7.1 Boundary values known in plane-polar coordinates Plane-polar scanning involves taking measurements on a plane-polar grid. Incremental samples are taken by varying the polar angle, φ and radial displacement, r (Figure 3.10). The principal advantage of the plane-polar scanning technique is that there is a reduction in the mechanical complexity of the positioning system and only a single linear probe trajectory is required. The test antenna and probe are orientated such that they always point in an antiparallel direction where this axis can be either vertically or horizontally orientated. The antenna rotation is avoided by using the antenna in a transmit mode and the probe which is mounted on the carriage of a linear scanner consists of a corrugated feed and an orthomode transducer. The rotation of the probe is avoided by the orthomode transducer and by successively measuring the two orthogonal polarisations at every scan point. The following section shows how the angular spectrum can be obtained from measurements taken in a plane-bipolar measurement system without recourse to approximation. Using the initial condition specified in a plane-polar coordinate system where the condition that r ≥ 0 applies then, u (r, φ, z = 0) = f (r, φ) where x = r cos (φ)

(5.114)

y = r sin (φ)

(5.115)

z=z

(5.116)

Using the exchange of variable formula for double integrals      ∂ (x, y)   dAu,v  f (x, y) dAx,y = f (x (u, v) , y (u, v))  ∂ (u, v)  R

(5.117)

S

With the two-dimensional FT we may then write that




2π ∞

F kx , ky =

f (r, φ)e 0

0

   (x, y)    ∂ (r, φ)  drdφ

jk0 r(α cos(φ)+β sin(φ))  ∂

(5.118)

Measurements – practicalities of planar near-field antenna measurements 151 where  ∂x   ∂ (x, y)  ∂r =  ∂y ∂ (r, φ)   ∂r

∂x ∂φ ∂y ∂φ

       cos φ = sin φ  

   −r sin φ  2 2 φ + sin φ =r = r cos r cos φ  (5.119)

Hence




2π ∞

F kx , ky = 0

f (r, φ)ejk0 r(α cos(φ)+β sin(φ)) rdrdφ

(5.120)

0

Finally, the relationship between the Cartesian and plane-polar unit vectors can be expressed as eˆ r = cos φ eˆ x + sin φ eˆ y

(5.121)

eˆ φ = − sin φ eˆ x + cos φ eˆ y

(5.122)

5.7.2 Boundary values known in plane-bipolar coordinates The plane-bipolar system is very similar to the plane-polar case except here, the probe sweeps out a circular arc in two dimensions (Figure 5.11) and not just one as was the case previously. Mechanically this is the simplest system to construct as the requirement for a linear translation stage is removed. The guided wave path is also simplified as only two rotary joints are required to connect the radio frequency (RF) output from the probe to the input of the network analyser whilst maintaining the phase stability over the scan plane. For these reasons, this design is very cost-effective. Unfortunately, this mechanical simplification is obtained at the expense of increased computational effort. The following section shows how the angular spectrum can be obtained from measurements taken in a plane-bipolar measurement system without recourse to approximation. Using the initial condition in a bipolar coordinate system, Figure 5.11, where the condition that β ≥ 0 applies then u (α, β, z = 0) = f (α, β) where x = L (sin (β − α) + sin α)

(5.123)

y = L (cos (β − α) − cos α)

(5.124)

z=z

(5.125)

where L is taken to represent the arc radius and is fixed for the duration of an acquisition and for the purposes of evaluating this integral can therefore be taken to be a constant.

152 Principles of planar near-field antenna measurements y

 = 3  = 2  =  x

=0


Figure 5.11

L Probe arm

Bipolar coordinate system

Clearly ∂x ∂α ∂x ∂β ∂y ∂α ∂y ∂β

∂ (L (sin (β − α) + sin α)) = L (− cos (α − β) + cos α) ∂α ∂ = (L (sin (β − α) + sin α)) = L cos (α − β) ∂β ∂ = (L (cos (β − α) − cos α)) = L (− sin (α − β) + sin α) ∂α ∂ = (L (cos (β − α) − cos α)) = L sin (α − β) ∂β =

(5.126) (5.127) (5.128) (5.129)

As  ∂x   ∂ (x, y)  =  ∂α ∂ (α, β)  ∂y  ∂α

∂x ∂β ∂y ∂β

   ∂x ∂y ∂x ∂y  · − · =  ∂α ∂β ∂β ∂α 

(5.130)

So ∂ (x, y) = L (− cos (α − β) + cos α) · L sin (α − β) − L cos (α − β) ∂ (α, β) × L (− sin (α − β) + sin α)

(5.131)

Using the trigonometric identities sin (α − β) = sin α cos β − cos α sin β

(5.132)

cos (α − β) = cos α cos β + sin α sin β

(5.133)

Measurements – practicalities of planar near-field antenna measurements 153 Hence

 ∂ (x, y) = L2 − cos α cos β 2 sin α + cos2 α cos β sin β − sin2 α sin β cos β ∂ (α, β)  + sin α cos α sin2 β + cos α sin α cos β − cos2 α sin β  − L2 − cos α sin α cos2 β + cos2 α cos β sin β − sin2 α cos β sin β  + cos α sin α sin2 β + sin α cos α cos β + sin2 α sin β (5.134)

or

  ∂ (x, y) = L2 − cos2 α sin β − sin2 α sin β ∂ (α, β) 2

(5.135)

= − L sin β Hence the multidimensional exchange of variable can be written as    ∂ (x, y)    dαdβ = L2 sin βdαdβ dxdy =  ∂ (α, β) 

(5.136)

Thus by using the formula for a multidimensional exchange of variable we obtain
 A kx , ky = L2

π 2π 0

E (α, β) ejL(kx (sin(β−α)+sin α)+ky (cos(β−α)−cos α)) sin βdαdβ

0

(5.137)

For the sake of clarity within this section, the nomenclature of denoting direction cosines as α, β, γ has been avoided. Next comes the question of the relationship between the unit vectors of the plane-bipolar and Cartesian coordinate systems. This can be established using the usual formula
  ∂r ∂u1 eˆ 1 =    (5.138) ∂r ∂u1  Where r = L (sin (β − α) + sin α) eˆ x + L (cos (β − α) − cos α) eˆ y

(5.139)

Again, L can be considered to be a constant. Thus for the β orientated unit vector ∂r = L cos (α − β) eˆ x + L sin (α − β) eˆ y ∂β So that     ∂r    = L cos (α − β)2 + sin (α − β)2 = L  ∂β 

(5.140)

(5.141)

154 Principles of planar near-field antenna measurements Hence eˆ β = cos (α − β) eˆ x + sin (α − β) eˆ y

(5.142)

Taking the cross product of the z and β directed unit vector will determine the ‘effective’ α orientated unit vector. Thus, eˆ α = − sin (α − β) eˆ x + cos (α − β) eˆ y

(5.143)

The reason the approach of using a cross-product was taken is that the plane bipolar coordinate system is not orthogonal at all points. That is to say, the α and β directed unit vectors are not mutually orthogonal for all values of α and β. However, when taking near-field antenna measurements we generally use a single probe and mechanically rotate it through 90◦ about its axis between scans or alternatively, use an orthogonal mode transducer (OMT) to sample two orthogonal tangential near electric field components. When either of these procedures is adopted, the sampled field components are resolved onto the polarisation basis developed here, rather than onto a true plane bi-polar polarisation basis. The necessary probe rotation angle is obtained from β (5.144) φ=α− 2 Figure 5.12 illustrates the schematics of the plane-polar and the plane-bipolar measurement schemes and their respective polarisation basis. Examples of near-field plane-bipolar measurements can be seen presented in Figure 5.13. The samples are taken across an irregular plane-bipolar coordinate system so that the half wavelength sampling criteria is satisfied on each of the concentric rings thus maximising the efficiency of the measurement process, that is, each concentric ring contains a different number of samples. Here, although the orthogonal field components were sampled on a plane-bipolar grid they have been plotted, tabulated on a regular Cartesian coordinate system to

y

Figure 5.12

Schematics of plane-polar and plane-poly-planar bipolar polarisation basis

Measurements – practicalities of planar near-field antenna measurements 155 Eβ (dB)

0.3

Eα(dB)

0.3 −5

−5

−10

0.2

−10

0.2

−15

−30

0

−35

–0.1 –0.2

–0.2

Figure 5.13 40

–0.1

0 x (m)

0.1

0.2

−20

0.1 y (m)

y (m)

−25

–0.3 –0.3

−15

−20

0.1

−25 −30

0

−35

−40 –0.1

−40

−45

−45

−50 –0.2

−50

−55

−55 –0.3 –0.3

0.3

–0.2

–0.1

0 x (m)

0.1

0.2

0.3

Measured plane-bipolar near-fields Eco (dB)

40 –5

30

Ecr (dB) –5

30

–10

–10 20

–15

10

–20

0

–25

–10

–30 –35

–20

EI (deg)

EI (deg)

20

–15

10

–20

0

–25

–10

–30 –35

–20

–40 –30

–45

–40 –40 –30 –20 –10 0 10 Az (deg)

Figure 5.14

20 30

40

–40 –30 –40 –40 –30 –20 –10 0 10 Az (deg)

–45 20 30

40

Copolar and cross-polar far-field patterns

aid in interpretation. Obtained using the techniques developed within this chapter, the corresponding far-field patterns can be found presented in Figure 5.14 tabulated on a regular azimuth over elevation coordinate system resolved onto a Ludwig III polarisation basis. Figure 5.15 shows a comparison of the cardinal cuts obtained from the planebipolar measurement plotted together with a cardinal cut taken using a quasi far-field range. From inspection, it is evident that the agreement attained is not perfect. However, the plane-bipolar results oscillate around the direct far-field measurement. The plane-bipolar measurement required an extended period of time to perform the measurements and during this time the temperature and therefore the phase of the system was unfortunately able to drift. As the near-field probe continually rotates across the measurement surface it is crucial that the two measured field components are balanced in order that reliable copolar and cross-polar patterns are attained. The errors observed within the copolar and cross-polar patterns are primarily artefacts

156 Principles of planar near-field antenna measurements Eco (dB)

5

θ = 0° θ = 90° Far-field

0 –5

Power (dB)

–10 –15 –20 –25 –30 –35 –40 –45 –50 –40

Figure 5.15

–30

–20

–10

0 θ (deg)

10

20

30

40

Comparison of horizontal and vertical cuts with direct far-field measurement

of this error. However, it is a testimony to the robustness of the technique that the accordance shown was achieved.

5.8

Regular azimuth over elevation and elevation over azimuth coordinate systems

Within this text, far-field vector-pattern functions are tabulated almost exclusively using a regular azimuth over elevation spherical coordinate system. Although a regular direction cosine grid is a more natural partner for use with spectral techniques, this equatorial spherical system is adopted as it is commonly employed within other antenna measurement facilities. The choice is unimportant provided that the definitions and implications are understood. The schematic representation contained in Figure 5.16 contains a comparison of the spherical angles azimuth, elevation and θ , φ. Here, the line OR lies within the xy-plane whilst the line OQ lies within the xz-plane, RP is parallel with the z-axis and QP is parallel with the y-axis. An alternative arrangement is illustrated in Figure 5.17. Here, OS is the projection of P onto the yz-plane, Az (termed α to avoid confusion in the mathematics) is the angle between OS and OP and El (termed ε to avoid confusion in the mathematics) is the angle between the z-axis and OS. In both of these cases, the observer is assumed to be standing behind the antenna looking in the positive z-axis, of the antenna coordinates thus as the y-axis is vertical and is directed up, this means that to have a right-handed set, the x-axis must be horizontal and increase to the left. This is the reason for the introduction of the negative sign in the x-axis of some of the

Measurements – practicalities of planar near-field antenna measurements 157 Theta axis

yAMS R ˆ ˆ

-xAMS

Chi axis

kAMS

P

φ El



O

zAMS Az AUT

Phi axis

xAMS

Q

Figure 5.16

Comparison of polar spherical and azimuth over elevation angles in the antenna coordinate frame Theta axis

yAMS R S

ˆ

ˆ

Chi axis

-xAMS P

Az

kAMS El

O

zAMS xAMS Phi axis

Figure 5.17

AUT

Comparison of polar spherical and elevation over azimuth angles in the antenna coordinate frame

commonly used coordinate system definitions. To help the reader visualize the two coordinate frames based on azimuth over elevation and elevation over azimuth are shown in Figure 5.18. The implication of choosing a curvilinear coordinate system and plotting it on a rectangular grid over the forward hemisphere is illustrated in Figures 5.19 and 5.20.

158 Principles of planar near-field antenna measurements y

y

E ε z

z

Figure 5.18

x

x

A

α

Azimuth over elevation (left) and elevation over azimuth (right) base on the antenna coordinate frame (coordinate system moves with the antenna) 1 0.75

–10

0.5 –20 0.25 V

–30 0 –40 –0.25 –50

–0.5

–60

–0.75 –1 –1

Figure 5.19

–0.75 –0.5 –0.25

0 u

0.25

0.5

0.75

1

Vector-pattern function tabulated on a direction cosine grid

Here, Figure 5.19 contains a far-field pattern function tabulated on a plaid, monotonic, equally spaced direction cosine grid. Conversely, Figure 5.20 contains a plot of the same vector-pattern function tabulated on a plaid, monotonic, equally spaced azimuth over elevation grid. A detailed description of several alternative coordinate systems can be found presented in Appendix C. Although in the region around boresight the pattern appears to be unchanged, towards the north, El = 90◦ and south,

Measurements – practicalities of planar near-field antenna measurements 159 90 –10

60

–20 EI (deg)

30 –30 0 –40

–30 –50 –60 –90 –90

Figure 5.20

–60 –60

–30

30 0 Az (deg)

60

90

Vector-pattern function tabulated on an azimuth over elevation grid

El = −90◦ , poles the pattern is distorted to the extent that round objects will appear square, compare this with, θ = 90◦ contour. Here, only directions corresponding to propagating portion of the vector-pattern function have been plotted, this interval is sometimes referred to as visible space.

5.9

Polarisation basis and antenna measurements

In this section, the polarisation convention used within this text is described. As shown above, in the near zone the electric or magnetic field is completely characterised by specifying three vector components, whilst the far-field is defined unambiguously by specifying two transverse vector components since the vector component in the direction of propagation is identically zero. Although in what follows, the definitions are presented in terms of the electric field, similar definitions can be constructed to specify the magnetic field. It is customary to define cross-polarisation as ‘the polarisation orthogonal to a reference polarisation’. Unfortunately, this leaves the direction of the reference polarisation undefined and thus is ambiguous for all but circularly polarised waves. The definitions presented below can be used to resolve this ambiguity. These transformations are only valid in the true far-field or in the special case where the field has been sampled using an infinitesimal Hertzian dipole.

5.9.1 Cartesian polarisation basis – Ludwig I The electric field is resolved onto three unit vectors, one aligned to each of the three Cartesian axes. When the electric field is decomposed onto these three unit vectors

160 Principles of planar near-field antenna measurements the total field can be expressed mathematically as follows. E (Az,El) = Ex (Az,El) eˆ x + Ey (Az,El) eˆ y + Ez (Az,El) eˆ z

(5.145)

This definition is valid in the near and far zones. Figure 5.21 (see p. C2) shows the copolar (magenta) and cross-polar (red) directions superimposed on a spherical (blue) grid. This clearly shows that this definition is effectively defining the polarisation on a flat screen normal to the boresight direction of the AUT. Clearly around the boresight direction the copolar and cross-polarisation is as one would normally measure.

5.9.2 Polar spherical polarisation basis For the polar spherical basis, the electric field is resolved onto three unit vectors, one aligned to each of the three spherical unit vectors, eˆ θ , eˆ φ , eˆ r . This is illustrated in Figure 5.22 (see p. C2) where the red arrows represent the eˆ θ -orientated unit vectors and the magenta arrows represent the eˆ φ -orientated unit vectors. The eˆ r orientated unit vectors are not shown, as there is no field in this direction since in the far-field all radiation is transverse and propagates in the radial direction. The blue grid lines represent lines of constant azimuth and elevation upon which the far-field vector-pattern function is tabulated. Here, the pole of the azimuth over elevation tabulation grid lies north and south where the ±y-axis pierces the unit sphere. Unfortunately, this polarisation basis displays a soft singularity, that is, a discontinuity, in the z-direction in both the forward and back hemisphere where θ = nπ and n = 0, 1, 2, . . .. Thus to implement this definition, a choice must be made as to the orientation of eˆ θ and eˆ φ at the poles. Typically, eˆ θ is chosen to be aligned to either φ = 0◦ or φ = 90◦ and eˆ φ is chosen to be mutually orthogonal to eˆ θ and eˆ r . Within this text the φ = 0◦ convention has been adopted. When the electric field is decomposed onto these unit vectors and the radial component is assumed zero, the total field can be expressed mathematically as follows E (Az,El) = Eθ (Az,El) eˆ θ + Eφ (Az,El) eˆ φ

(5.146)

If the position vector to a point can be expressed as r = r (u1 , u2 , u3 ), a tangent vector to the curve u1 = u1 (x, y, z) at that point for which u2 = u2 (x, y, z) and u3 = u3 (x, y, z) are constant is, ∂r ∂u1 . Thus, a unit tangent vector in this direction is
  ∂r ∂u1 (5.147) eˆ 1 =    ∂r ∂u1  Similar expressions can be written down for the remaining unit vectors eˆ 2 and eˆ 3 . Thus, if the position of the point in space is expressed in spherical coordinates r = sin θ cos φ eˆ x + sin θ sin φ eˆ y + cos θ eˆ z

(5.148)

Measurements – practicalities of planar near-field antenna measurements 161 then ∂r = cos θ cos φ eˆ x + cos θ sin φ eˆ y − sin θ eˆ z ∂θ

(5.149)

    ∂r    = cos2 θ cos2 φ + cos2 θ sin2 φ − sin2 θ = 1  ∂θ 

(5.150)

Thus

Hence the unit vector in the direction of increasing θ that is tangential to the surface of a sphere eˆ θ = cos θ cos φ eˆ x + cos θ sin φ eˆ y − sin θ eˆ z

(5.151)

Similarly ∂r = − sin θ sin φ eˆ x + sin θ cos φ eˆ y ∂φ     ∂r    = sin2 θ sin2 φ + sin2 θ cos2 φ = sin θ  ∂φ 

(5.152) (5.153)

Hence eˆ φ = − sin φ eˆ x + cos φ eˆ y

(5.154)

The relationship between the Cartesian and spherical field components can be expressed in matrix notation as       Ex cos θ cos φ cos θ sin φ − sin θ Eθ (5.155) = ·  Ey  Eφ − sin θ cos φ 0 Ez As the three column by two row matrix is both orthogonal and correctly normalized to unity, the inverse matrix is equal to the matrix transpose and the converse transformation can be expressed again in matrix notation as follows:       Ex cos θ cos φ − sin φ Eθ  Ey  =  cos θ sin φ  cos φ · (5.156) Eφ − sin θ 0 Ez This clearly does not match the normal copolar and cross-polar definition around the AUT boresight.

5.9.3 Azimuth over elevation basis – Ludwig II It is possible to define Ludwig II in any of the three spherical polarisation basis, however, one that suits the practical measurement, particularly about the boresight (z) direction is the azimuth over elevation basis. This is shown plotted in Figure 5.23 (see p. C2) using the same process as used for Figure 5.22 (see p. C2).

162 Principles of planar near-field antenna measurements Clearly around the boresight direction the copolar and cross-polar components are naturally orthogonal and consistent with Ludwig I. The coordinate transforms for this system are 



EAz EEl

1 = cos El



cos φ cos θ sin φ

− cos θ sin φ cos φ

   Eθ · Eφ

(5.157)

Where   1 − sin2 El = 1 − sin2 θ sin2 φ

cos El =

(5.158)

Where the inverse transformation can be expressed as 

Eθ Eφ

 =

1



cos2 φ + cos2 θ sin2 φ

cos φ cos El − cos θ sin φ cos El

cos θ sin φ cos El cos φ cos El

   EAz · EEl (5.159)

There is another possible system which comes under the Ludwig II definition and that is the elevation over azimuth polarisation basis. This is depicted in Figure 5.24 (see p. C3). Again around the boresight direction the copolar and cross-polar components are naturally orthogonal and consistent with Ludwig I. The coordinate transforms for this system are 

Eα Eε

 =

1 cos α



cos θ cos φ sin φ

− sin φ cos θ cos φ

   Eθ · Eφ

(5.160)

Where α and ε are the azimuth and elevation angles respectively and the inverse transformation can be expressed as 

Eθ Eφ

 =

1 cos2 θ cos2 φ + sin2 φ



cos θ cos φ cos α − sin φ cos α

sin φ cos α cos θ cos φ cos α

   Eα · Eε (5.161)

From inspection of the figures, it can be seen that this definition is broadly equivalent to the polar spherical definition described above only here the antenna boresight is equatorial pointing rather than polar pointing. Essentially, there are three useful choices to be had when adding a triad of axes to a spherical grid. It is possible to align the pole of these coordinate systems with the z-axis, the y-axis or the x-axis and the polar spherical, Ludwig II azimuth over elevation and Ludwig II elevation over azimuth definitions corresponds to each of these choices respectively. Outside boresight both Ludwig I and II deviate from what one would normally measure and indeed use in a practical scenario. Clearly, it is preferable to utilize a definition that applies over all angles and the resolution of this difficulty motivated the adoption of the Ludwig definition III as described below.

Measurements – practicalities of planar near-field antenna measurements 163

5.9.4 Copolar and cross-polar polarisation basis – Ludwig III This definition was formulated mathematically by Ludwig after its usage had become commonplace in the antenna measurement community [8]. This definition corresponds physically to rolling the range antenna or the remote source antenna (RSA) in χ as the AUT is rotated in φ. It has the inherent advantage that it removes the singularity at θ = 0 present within the polar spherical definition. Here, the electric field is resolved onto three unit vectors, one aligned to each of the three unit vectors, eˆ co , eˆ cross , eˆ r . This is illustrated in Figure 5.25 (see p. C3) where the red arrows represent the copolar eˆ co -orientated unit vectors and the magenta arrows represent the eˆ cross -orientated unit vectors. The eˆ r -orientated unit vectors are not shown as in the far-field all radiation is transverse and propagates in the radial direction. Again, the blue grid lines represent lines of constant azimuth and elevation upon which the far-field vector-pattern function is tabulated. For the RSA and the AUT to remain polarisation matched, the RSA must be rotated through an angle -φ as the AUT is rotated through an angle φ. Thus let the polarisation-matched orientation be eˆ co and an orientation that is mutually orthogonal to this and the direction of propagation be eˆ cr . When the electric field is decomposed onto these unit vectors the total field can be expressed mathematically as follows E = Eco eˆ co + Ecr eˆ cr

(5.162)

where eˆ cross = eˆ r × eˆ co

(5.163)

and obviously eˆ co · eˆ cross = 0

(5.164)

The so-called copolar and cross-polar unit vectors are related to the spherical unit vectors through eˆ co = cos φ eˆ θ − sin φ eˆ φ

(5.165)

eˆ cr = sin φ eˆ θ + cos φ eˆ φ

(5.166)

Since, when expressed in matrix notation this is clearly an orthogonal and unit normalized matrix the matrix inverse is equal to matrix transpose; hence eˆ θ = cos φ eˆ co + sin φ eˆ cr

(5.167)

eˆ φ = − sin φ eˆ co + cos φ eˆ cr

(5.168)

E = Eθ eˆ θ + Eφ eˆ φ

(5.169)

As

Thus



 E = Eθ cos φ eˆ co + sin φ eˆ cr + Eφ − sin φ eˆ co + cos φ eˆ cr

(5.170)

164 Principles of planar near-field antenna measurements Hence



E = Eθ cos φ − Eφ sin φ eˆ co + Eθ sin φ + Eφ cos φ eˆ cr

(5.171)

Finally Eco = Eθ cos φ − Eφ sin φ

(5.172)

Ecr = Eθ sin φ + Eφ cos φ

(5.173)

The relationship between the spherical and Ludwig III field components can be expressed in matrix notation as       cos φ − sin φ Eθ Eco (5.174) = · sin φ cos φ Ecross Eφ Again, as the two column by two row matrix is both orthogonal and correctly normalized to unity the inverse matrix is equal to the matrix transpose and the converse transformation can be expressed again in matrix notation as follows       cos φ sin φ Eco Eθ (5.175) = · Ecross − sin φ cos φ Eφ Thus, the soft singularity encountered on boresight (θ = 0) is removed. However, this is achieved at the expense of doubling the severity of the singularity encountered at θ = ±π, ±3π, ±5π , . . .

(5.176)

Often, these equations are modified so that a reference angle is included that represents the angle from the x-axis to the major axis of the polarisation ellipse. This convention has not been adopted within this text. Instead, the Ludwig III definition has been generalized by transforming the copolar and cross-polar field components to Cartesian field components and then applying an isometric rotation to the polarisation basis before resolving the field components onto the now rotated, copolar and crosspolar polarisation basis. Thus, any vector-pattern function can be easily resolved onto a Ludwig III polarisation basis pointing in any direction in space, not just about the direction of the positive normal. This unconventional but extremely useful generalized definition is utilized throughout this text. Often, it is desirable to determine the polarisation purity of an antenna, typically in order that its applicability for use with a frequency reuse scheme can be judged. In such circumstances, it is preferable to know how much smaller or larger, the cross-polar signal is than the copolar signal thus the relative cross-polar power, that is, the cross-polar discrimination, is calculated. Here, the relative cross-polar field is taken to be a scalar quantity that can be related to the Ludwig III copolar and cross-polar fields through the following expression Erelcross (Az,El) =

Ecross (Az,El) Eco (Az,El)

(5.177)

The ratio between two polarisation components is often of interest as it provides a measure of the discrimination or isolation, between the respective polarisations. This

Measurements – practicalities of planar near-field antenna measurements 165 ratio is often called cross-polarisation discrimination (XPD) and is the ratio of the two polarisations, for example, Eco /Ecr , Eθ /Eφ , and so on and, their reciprocals.

5.9.5 Circular polarisation basis – RHCP and LHCP Circularly polarised fields can be produced by combining the two linear polarisations with a 90◦ phase difference. An antenna radiates a right-hand circularly polarised (RHCP) field, when its x-polarised component leads its y-polarised component by 90◦ . If the converse it true, the field is described as being left-hand circularly polarised (LHCP). The preferred method of planar near-field measurement is with linear polarisation, which requires no special probe antennas employing orthomode transducers, with their inherent narrow bandwidth. To derive circular polarisation performance for an AUT one can transform mathematically the linear polarisation results. At a given point in space let Eθ = E1 cos ωt

(5.178)

Eφ = E2 cos (ωt + γ )

(5.179)

Here, γ is the phase difference between the two orthogonal field components. For γ = +90◦ the resultant vector at any point in time rotates in a clockwise direction with period T = 1/fo where fo is the frequency of the EM wave and for γ = −90◦ an anticlockwise direction. This pair of counter rotating fields form an orthogonal pair and when E1 = E2 are usually termed RHCP and LHCP. We define RHCP using the classic right hand rule rotating in the direction of propagation in a clockwise direction. This is the IEEE standard definition of polarisation [9, 10]. Removing the time dependence from these equations will yield the path of the electric or magnetic, field vector. Thus Eθ = cos ωt E1 Eφ = cos (ωt + γ ) = cos ωt cos γ − sin ωt sin γ E2

(5.180) (5.181)

Thus Eφ Eθ = cos γ − sin ωt sin γ E1 E2 Hence 

Eφ Eθ − cos γ E2 E1

2

= sin2 ωt sin2 γ

Expanding yields  2  2 E θ Eφ Eφ Eθ −2 cos γ + cos2 γ = sin2 ωt sin2 γ E2 E1 E2 E1

(5.182)

(5.183)

(5.184)

166 Principles of planar near-field antenna measurements Using trigonometric identities  2  2   Eφ E θ Eφ Eθ −2 cos γ + 1 − sin2 γ = sin2 ωt sin2 γ E2 E1 E2 E1 Thus   Hence 

Eφ E2 Eφ E2 Eθ E1

2 2

2

−2

E θ Eφ cos γ + E1 E2

−2

E θ Eφ cos γ + E1 E2

E θ Eφ −2 cos γ + E1 E2

 

Eθ E1 Eθ E1



2 2

Eφ E2



Eθ E1

2

(5.185)

sin2 γ = sin2 ωt sin2 γ

(5.186)

− cos2 ωt sin2 γ = sin2 ωt sin2 γ

(5.187)

−

2

− sin2 γ = 0

(5.188)

which is in the same form as the equation for an ellipse, namely, Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where the coefficients are 1 A= 2 E1 B=−

2 cos γ E1 E2

(5.189)

(5.190) (5.191)

1 E22

(5.192)

F = − sin2 γ

(5.193)

C=

Here, there is no x or y term which implies that the ellipse is not translated in the xor y-axis as D = E = 0. However, there is an xy term, that is, B = 0, which implies that the ellipse could be rotated. If the semi-major and semi-minor axes are denoted as a and b then from Appendix E it can be shown that      1
2 2 4 4 2 2 p + q + p + q + 2p q cos 2γ (5.194) a= 2      1
2 2 4 4 2 2 b= (5.195) p + q − p + q + 2p q cos 2γ 2 Where p and q are thus semi-radii of the ellipse measured in the θ and ϕ directions (see Appendix E). The tilt angle φ is obtained from  2pq 1 cos γ (5.196) φ = arctan 2 2 p − q2 Where γ is the phase angle between the vectors p and q. The tilt angle is measured from the x-axis and is unrotated when φ = 0. Following IEEE-standard the axial

Measurements – practicalities of planar near-field antenna measurements 167 ratio is the ratio of the major axis to the minor axis of the polarisation ellipse. Thus, when denoted with AR, axial ratio is defined to be the ratio of a and b a AR = (5.197) b Here 1 ≤ |AR| ≤ ∞. This is often presented in a logarithmic form AR|dB = 20 log10 (|AR|)

(5.198)

Figure 5.26 schematically represents the tilted polarisation ellipse. Often the inverse is calculated as this avoids the divide by zero that would be encountered at points in space where the field is linearly polarised. In this case from the law of logarithms the AR in dB can be seen to be the negative of the IEEE definition. A detailed proof of these equations can be found in Appendix E. For plane waves the two cases of particular interest are those of plane and circularly polarised waves. For the case of linear polarisation the phase difference, γ , between the two components is a multiple of 180◦ whereas for the case of circular polarisation p = q and the phase difference between the two components is an odd multiple of 90◦ . The same value of semi-major and semi-minor radii and therefore axial ratio, will be obtained irrespective of the linear polarisation basis that the field is resolved on to when calculating the values of a and b. Therefore any of the linear polarisation basis described above can be used, for example, Ex , Ey ; Eθ , Eφ ; and so on. However, and as ever, it is advantageous to choose a polarisation basis that is free from a singularity in the direction of greatest interest. Figure 5.27 contains an example of a typical axial ratio pattern for a medium gain circularly polarised antenna. In the past, the testing of circularly polarised antennas on far-field ranges was often accomplished by continuously rotating a linearly polarised source antenna while the y E2 a

Ey

b

xis

ra

no

Mi O z

φ

Ex E1

x

is

r ajo

ax

M

Figure 5.26

Polarisation ellipse showing semi-major and semi-minor axes

168 Principles of planar near-field antenna measurements Axial ratio (dB) IEEE definition 20 15 25 10 20 EI (deg)

5 15

0 –5

10 –10 5 –15 –20 –20

Figure 5.27

–15

–10

–5

0 Az (deg)

5

10

15

20

Two-dimensional axial ratio pattern

far-field radiation pattern was measured [11]. This rotation was fast when compared to the motion of the AUT positioner so that essentially the probe will turn through at least one complete rotation for each far-field measurement point. The intent therefore is to vary the signal source in such a way that the full polarisation ellipse can be obtained at each far-field angle. This procedure has two principal advantages. It yields a real time direct measurement of the axial ratio of the AUT and can be accomplished with the use of a scalar network analyser. This form of far-field plot is still popular in some quarters and can be easily obtained from two complex orthogonal linear field components which thereby enables such results to be obtained from near-field measurements. The measured spin linear measured value can be obtained from the semi-major and semi-minor axes using  (5.199) r = (a sin (χ ))2 + (b cos (χ ))2 Here χ has been used to denote the spinning polarisation angle. Thus, the signal that a purely linearly polarised antenna would receive when expressed in dB form is rotating linear = 20 log10 (r)

(5.200)

As AR is defined to be the ratio of the major to the minor axis and as the maximum envelope is 20 log10 (a) and the minimum envelope is 20 log10 (b) then, from the law

Measurements – practicalities of planar near-field antenna measurements 169 5 0

35 30 Power (dB)

AR (dB)

25 20 15 10

–15 –20 –25 –30 –35

5 0 –20 –15 –10

Figure 5.28

–5 –10

–5

0 5 Az (deg)

10

15

20

–40 –45 –50 –20 –15 –10

–5

0 5 Az (deg)

10

15

20

Comparison of axial ratio cut and a traditional spin linear measurement

of logarithms, it is clear that AR will be the difference between these values when expressed in dB form. In the preparation of Figure 5.28, the χ angle was varied over a full 360◦ range for each far-field angle. By way of a comparison, an axial ratio plot of the same pattern cut can also be found presented in Figure 5.28. From inspection, it can be seen that the axial ratio values correspond exactly with the difference in the envelope of the synthesised spin linear measurement.

5.10 Overview of antenna alignment corrections The application of generalized angular alignment corrections to far-field vectorpattern functions is a subject that is of significance to this text and is often avoided both in practice and in the literature. The application of isometric rotations via the use of bespoke non-rectangular coordinate systems is introduced by illustrating the effect of applying scalar, vector and finally polarisation rotations.

5.10.1 Scalar rotation of far-field antenna patterns A scalar rotation constitutes an isometric rotation of the antenna pattern in which the reference polarisation is rotated with the pattern. Figure 5.29 contains the copolar and cross-polar far-field vector-pattern function of an AUT, this is nominally aligned to the equator of the azimuth over elevation grid on which it has been tabulated. Figure 5.30 contains the same far-field vector-pattern function as that shown above only here, a scalar rotation of 90◦ in roll has been applied. Clearly, although the antenna pattern has been rotated the field vectors have not changed. Physically, this is equivalent to applying a rotation to the antenna whilst holding still the coordinate system in which the pattern is tabulated. Furthermore, an identical rotation has been applied to the polarisation basis onto which the pattern has been resolved. This situation is illustrated schematically in Figure 5.31.

170 Principles of planar near-field antenna measurements Copolar power (dB)

Copolar power (dB)

40

40

30 20

–20

10

–30

0 –40

–10 –50

–40 –40 –30 –20 –10

0 10 Az (deg)

20 30

–50 –60

–40 –40 –30 –20 –10 0 10 Az (deg)

20 30

40

Nominally aligned far-field vector-pattern function scalar rotation Az = 0◦ , El = 0◦ , Roll = 0◦ Copolar power (dB)

40

–40

–10 –20

–50

–30

–60

–40 –40 –30 –20 –10 0 10 Az (deg)

20

30

40

–20

10 EI (deg)

–30

–10

20

–20

10 0

Copolar power (dB)

30

–10

20 EI (deg)

–40

–30

40

30

Figure 5.30

–30

0

–20

–60

–30

40

–20

10

–10

–20

Figure 5.29

–10

20 EI (deg)

EI (deg)

30

–10

–30

0 –40

–10

–50

–20

–60

–30 –40 –40 –30 –20 –10 0 10 Az (deg)

20

30

40

Scalar rotation of a far-field vector-pattern function scalar rotation Az = 0◦ , El = 0◦ , Roll = 90◦

Here, the AMS1 axis triad represents the coordinate system attached to a mechanical datum that in turn is attached to the antenna. The AES1 axis set represents the polarisation basis onto which the pattern function has been resolved. The AMS2 axes set represent the coordinate system attached to the rotated antenna whilst the AES2 axes set represents the rotated polarisation basis onto which the field has been resolved. In this example, both the dashed axes have been rotated by 90◦ . The introduction of the white region in the far-field pattern of Figure 5.30 is an artefact of choosing a coordinate system that has a pole for the purposes of plotting radiation pattern functions. If a direction cosine or phase space coordinate system had been utilized, this effect would not have been observed, as these are manifold coordinate systems. This situation is illustrated schematically in Figures 5.32 and 5.33 (see p. C4) where clearly the poles of the coordinate system onto which the pattern is

Measurements – practicalities of planar near-field antenna measurements 171 Y 1AES

X 1AES Y 2AES

Z 1AES Z 2AES

Y 1AMS

X 1AMS Y 2AMS

Z 1AMS Z 2AMS

X 2AES

X 2AMS

Figure 5.31

Schematic representation of a scalar rotation of Az = 0◦ , El = 0◦ , Roll = 90◦

tabulated have remained located to the north and south of the diagrams. The red axis sets shown in the figure represent the mechanical datum associated with the antenna. Thus, from inspection of Figures 5.30, 5.32 and 5.33 (see p. C4) it can be seen that, in general, an isometric rotation is not equivalent to a transposition.

5.10.2 Vector rotation of far-field antenna patterns Figure 5.34 contains the far-field vector-pattern function for that antenna shown in Figure 5.33 (see p. C4) after a vector rotation of 90◦ in roll has been applied. Clearly, although the pattern has been rotated the reference polarisation has remained unchanged thus the old copolar reference has become the new cross-polar reference and vice versa. Again, this situation is represented schematically in Figure 5.35. Here, although as previously the mechanical datum has been rolled by 90◦ the polarisation basis onto which the pattern is resolved has remained unchanged. This corresponds to applying a scalar rotation to the pattern. By means of a by-product of this pattern rotation, the polarisation has also been rotated by the same amount. Thus an inverse rotation must be applied to the polarisation basis in order that it can be returned to its original position.

172 Principles of planar near-field antenna measurements Copolar power (dB)

40

Cross-polar power (dB)

30

–10

30

–10

20

–20

20

–20

10

EI (deg)

EI (deg)

40

–30

0 –40

–30

0 –40

–10

–10 –50

–20

–30 –20 –10 –40 10 Az (deg)

Figure 5.34

20 30

–50

–20

–60

–30 –40 –40

10

–60

–30 –40 –40 –30 –20 –10 0 10 Az (deg)

40

20 30

40

Vector rotation of a far-field vector-pattern function vector rotation Az = 0◦ , El = 0◦ , Roll = 90◦

Y 1AES Y 2AES X1AES X2AES

Z 1AES Z 2AES

Y 1AMS

X 1AMS Y 2AMS

Z 1AMS Z 2AMS

X1AMS

Figure 5.35

Schematic representation of a vector rotation of Az = 0◦ , El = 0◦ , Roll = 90◦

Measurements – practicalities of planar near-field antenna measurements 173 Copolar power (dB)

40

30

EI (deg)

20

–30

0 –40

–10

–20

10

–30

0 –40

–10 –50

–20

–60

–30 –40 –40 –30 –20 –10 0 10 Az (deg)

Figure 5.36

–10

20

–20

10

Cross-polar power (dB)

30

–10

EI (deg)

40

20

30

40

–50

–20

–60

–30 –40 –40 –30 –20 –10 0 10 Az (deg)

20

30

40

Rotation of polarisation basis of far-field vector-pattern function Az = 0◦ , El = 0◦ , Roll = 90◦

5.10.3 Rotation of copolar polarisation basis – generalized Ludwig III This generalized definition of polarisation is of particular utility when handling farfield data obtained from an instrument with two or more polarisations, that is, a corrugated horn fitted with an orthogonal mode transducer or a single spacecraft with many antennas each orientated such that their respective main beams point in different directions. Figure 5.36 contains the far-field vector-pattern function of the pattern shown in Figure 5.31 after a rotation of 90◦ in roll has been applied to the polarisation basis. Clearly, although the pattern has not been rotated the reference polarisation has thus again; the old copolar reference has become the new crosspolar reference and vice versa. Figure 5.37 illustrates the rotation of the polarisation basis. Matters are somewhat complicated by the choice of a Ludwig III polarisation basis for presentation. In general, it is necessary to apply the isometric rotations to the Cartesian, that is, Ludwig I, field components instead. A detailed description of the rotation process is now given: 1. Define the polarisation boresight system as a rotation from the nadir-centred azimuth over elevation plotting system. 2. Convert from a Ludwig III copolar and cross-polar polarisation basis to a Ludwig II polar spherical, that is, Eθ and Eφ polarisation basis. 3. Convert the Eθ and Eφ components to Ludwig I Cartesian components in the same system. 4. Transform these Cartesian field components into the new rotated basis specified by the polarisation boresight. 5. Calculate the Cartesian coordinates of the field point from its azimuth and elevation coordinates in the unrotated system. 6. Transform these coordinates into the new rotated system.

174 Principles of planar near-field antenna measurements Y 1AES

X 1AES X 2AES

Z 1AES Z 2AES

Y 1AMS Y 2AMS

X 1AMS X 2AMS

Figure 5.37

Z 1AMS Z 2AMS

X 2AES

Schematic representation of a polarisation rotation of Az = 0◦ , El = 0◦ , Roll = 90◦

7. Calculate the new spherical angles of the data point in the new system. 8. Calculate spherical field components in the rotated coordinate system. 9. Calculate Ludwig III linear components using standard polar definition in rotated system.

5.10.4 Generalized compound vector rotation of far-field antenna patterns Figure 5.38 contains the far-field vector-pattern function plotted using a nadir-centred polarisation basis using an azimuth elevation gridding system. Figure 5.39 contains the rotated far-field vector-pattern function. Clearly, the pattern has been rotated and the polarisation basis has remained unchanged, as a polarisation mismatch is clearly evident. The rigorous application of an inverse vector rotation constitutes the basis of the active alignment correction technique. This vector rotation is illustrated schematically within Figure 5.40 where it is clear that only the AUT has been rotated. To verify that the alignment information had been applied correctly, the rotated data set shown above was further transformed by an inverse rotation and compared against the original nominally aligned data set. This comparison can be seen presented in Figures 5.41 and 5.42. Here, dotted contours represent the original data set whilst the black contours represent the twice rotated data set. The contours are plotted at 5 dB steps from 0 dB to −70 dB. Clearly, the agreement is encouraging with only minor differences resulting in inaccuracies introduced by approximations inherent within the interpolation algorithm. The clear errors observed at the edges of the recovered data set result from a failure of the interpolation algorithm to correctly handle edge and corner elements. In practice when an antenna is acquired, whilst its axes are misaligned to the axes of the range, we obtain a pattern resembling Figure 5.39. The objective of

Measurements – practicalities of planar near-field antenna measurements 175 Copolar power (dB)

90

–10

60

–40

–30

Figure 5.38

–30 0 30 Az (deg)

60

–90 –90

90

–50 –60 –60

–30 0 30 Az (deg)

60

90

Far-field vector-pattern nominally aligned Az = 0◦ , El = 0◦ , Roll = 90◦ Copolar power (dB)

90

–40

–60

–60 –60

–30 0

–30

–50

–60

–20

30 EI (deg)

EI (deg)

–30 0

–10

60

–20

30

–90 –90

Cross-polar power (dB)

90

90 –10

60

Cross-polar power (dB) –10

60

–20

–20 30

–30 0 –40

–30 0 –40

–30

–30

–50

–60 –90 –90

EI (deg)

EI (deg)

30

–60 –60

Figure 5.39

–30

0 30 Az (deg)

60

90

–50 –60 –90 –90 –60

–60 –30 0 Az (deg)

30

60

90

Generalized vector rotation of a far-field pattern. Vector rotation Az = 0◦ , El = 0◦ , Roll = 90◦

the active alignment correction developed within this text is to acquire the necessary alignment information and then to obtain corrected results resembling those contained in Figure 5.38. Although in principle these approximate methods could be utilized in the application of alignment information, it is preferable to perform the interpolation within the transform as this approach is entirely rigorous.

5.11

Brief description of near-field coordinate systems

The measurement and transformation algorithms are discussed above in terms of a number of coordinate systems. For clarity, these are represented schematically in Figure 5.43 and explained in detail in the subsequent sections. The antenna aperture is represented by the black square centred about the origin and defining the xy-plane of the antenna mechanical system (AMS) as described below.

176 Principles of planar near-field antenna measurements Y 1AES Y 2AES

X 1AES X 2AES

Z 1AES Z 2AES

Y 1AMS

Y 2AMS

Z 1AMS X 1AMS Z 2AMS X 2AMS

Figure 5.40

Schematic representation of a vector rotation of Az = 10◦ , El = 20◦ , Roll = 40◦ 40

Copolar power (dB)

30

EI (deg)

20 10 0 –10 –20 –30 –40 –40

Figure 5.41

–20

0 Az (deg)

20

40

Verification of vector rotation algorithm copolar power

5.11.1 Range fixed system The range fixed system (RFS) coordinate axes form a right-handed set. Looking in the +ZRFS direction, they are orientated as follows: • +XRFS axis is horizontal and increases towards the left • +YRFS axis is vertical and increases upwards.

Measurements – practicalities of planar near-field antenna measurements 177 40

Cross-polar power (dB)

30

EI (deg)

20 10 0

–10 –20 –30 –40 –40

Figure 5.42

–20

0 Az (deg)

20

40

Verification of vector rotation algorithm cross-polar power YAMS

YAES YRFS

ZRFS

ZAES XAMS XAES

ZAMS XRFS

Figure 5.43

Near-field coordinate systems

This forms the basic reference system in the planar range. The RFS system is used for the acquisition, that is, tabulation, of the near-field Cartesian field components.

5.11.2 Antenna mechanical system The AMS coordinate axes form a right-handed set nominally orientated coincident and synonymous with the RFS axes. Thus, looking in the +ZAMS direction, they are

178 Principles of planar near-field antenna measurements orientated as follows: • +XAMS axis is horizontal and increases towards the left • +YAMS axis is vertical and increases upwards. This system is used for plotting the far-field patterns.

5.11.3 Antenna electrical system The antenna electrical system (AES) coordinate axes form a right-handed set nominally orientated coincident and synonymous with the AMS axes as follows: • • •

+XAES parallel to +XAMS +YAES parallel to +YAMS +ZAMS parallel to +ZAPS . Thus, looking in the +ZAES direction, the nominal orientation is:

• +XAES axis: horizontally orientated and increases towards the left • +YAES axis is vertical and increases upwards • +ZAES axis increases towards the far-field. The +ZAES axis defines the electrical boresight of the antenna and the copolar and cross-polar patterns are resolved onto this system. The AES is attached to the AMS and moves with it. Furthermore, the AES may also be rotated about any or all of its axes.

5.11.4 Far-field azimuth and elevation coordinates The antenna plotting system (APS) is defined as follows: •

Azimuth is positive from +ZAPS axis and is measured in XAPS − ZAPS plane, that is, a negative rotation around the +YAPS axis • Elevation is positive towards +YAPS measured from XAPS − ZAPS plane, that is, a negative rotation around the +XAPS axis.

5.11.5 Ludwig III copolar and cross-polar definition The copolar and cross-polar power patterns are calculated according to the Ludwig III definition, using only the electric field components.

5.11.6 Probe alignment definition in single port probe (SPP) In the case of an SPP the probe is rotated clockwise through 90◦ so that two tangential field components can be sampled. The nominal position of this principal polarisation (probe B) looking towards the probe aperture is horizontal and pointing left. When rotated, the principal polarisation points upwards.

Measurements – practicalities of planar near-field antenna measurements 179

5.11.7 General vector rotation of antenna radiation patterns The relationship between kT and kT and E and E can be expressed conveniently in
 matrix form. Let xp , yp , zp represent the AES, the Cartesian polarisation basis onto which the vector antenna radiation pattern is to be resolved. Let (xa , ya , za ) represent the AMS, which is the system with which the antenna patterns are to be plotted (observation system). Finally, let (xr , yr , zr ) represent the RFS, that is, the system in which the measurements are made. The following definitions can be formed from the nine possible scalar products between the respective axes. Let the mechanical alignment direction cosine matrix be represented by   x a · x r xa · y r xa · z r  ya · xr ya · yr ya · zr  = [A] (5.201) za · x r za · y r za · z r Furthermore let the polarisation direction cosine matrix be represented by   xp · xa xp · ya xp · za  yp · xa yp · ya yp · za  = [P] zp · x a zp · y a z p · z a Hence, the antenna to range alignment can be corrected using     kxr , kyr , kzr = kxa , kya , kza · [A] or

   kxa kxr  kyr  = [A]−1 ·  kya  kzr kza

(5.202)

(5.203)



(5.204)

The measured field components, when transformed to the far-field can readily be resolved onto the desired polarisation basis using     Exp Exr  Eyp  = [P] · [A] ·  Eyr  (5.205) Ezr Ezp Finally 

     kxp kxr kxa  kyp  = [P] · [A] ·  kyr  = [P] ·  kya  kzr kza kzp

(5.206)

In general, it is preferable to handle rotations in terms of direction cosine matrices as the requirement to know the sense and order of a sequence of angular rotation is avoided. The determinant of the above direction cosine matrix will be unity as we are only considering right-handed coordinate systems (the determinant will be negative if the transformation changes a right-handed coordinate system to a left-handed one or vice versa). Such vector rotations can be thought of as being constructed from a scalar rotation of the pattern and an inverse scalar rotation of the polarisation vector, where the order of application is unimportant.

180 Principles of planar near-field antenna measurements

5.12 Directivity and gain 5.12.1 Directivity The maximum directivity of an antenna is equal to the ratio of its maximum radiation intensity (often defined as the boresight direction) over that of an isotropic antenna radiating the same power. Thus D=

radiation intensity at maximum radiation pattern direction (W/unit solid angle) radiation intensity of an isotropic antenna (W/unit solid angle) (5.207)

Or D=

radiation intensity at maximum radiation pattern direction (W/unit solid angle)
  total radiated power 4π (W/unit solid angle) (5.208)

The radiation intensity (power per unit area) can be expressed as U = r2W

(5.209)

Here r is the radius and W is the radiation density (power per unit solid angle). The radiation intensity is related to the far electric field through U (θ, φ) =

2 r 2  E (r, θ , φ) 2Z0

(5.210)

In the far-field, it is possible to describe the electric (and magnetic fields) as E (r, θ , φ) =

e−jk0 r E (θ , φ) r

Where E (θ , φ) =

π jk0 λ2





(5.211)

  

rˆ × n × E s + Z rˆ × n × H s × rˆ ejk0 rˆ ·r0 ds0

S

Thus, the radiation intensity can be expressed as 2 1  E (θ , φ) U= 2Z0 Or, in terms of the conventional orthogonal spherical polarisation basis 2  2  1  U (θ, φ) = Eθ (θ , φ) + Eφ (θ, φ) 2Z0

(5.212)

(5.213)

(5.214)

The total power is obtained by integrating the radiation intensity over the far-field sphere thus Prad =

U d 

(5.215)

Measurements – practicalities of planar near-field antenna measurements 181 Here,  is the elemental solid angle. Now, the directivity can be expressed as D=

U U0

(5.216)

Here, U0 is taken to denote the radiation intensity of an isotropic source. Prad =

U0 d = U0 

d = 4π U0

(5.217)



Finally D=

4π U Prad

(5.218)

Within this definition it has been assumed that the field points represent peak values. If the root mean square (r.m.s.) values had been used the expression for directivity would remain unchanged, however, other quantities would differ by a factor of a half.

5.12.2 Gain – by substitution method The near-field theory as described within this text will not automatically enable the gain of an antenna to be determined. In most cases, we deal with relative gain, which is defined as ‘the ratio of the power gain in a given direction to the power gain of a reference antenna in its reference direction’. The power input must be the same for both antennas.

Box 5.1 Power gain is direction (θ , φ) is given by G(θ,φ) = 4π

power radiated per unit solid angle in direction (θ ,φ) Total power accepted from the source (5.219)

This includes dissipative losses in the antenna and does not include transmission line mismatch losses. Directivity in direction (θ , φ) is given by D(θ,φ) = 4π

power radiated per unit solid angle in direction (θ ,φ) Total power radiated by antenna (5.220)

This does not include dissipative losses in the antenna.

In general, gain is measured and directivity is calculated.

182 Principles of planar near-field antenna measurements

5.12.3 Gain-transfer (gain-comparison) method There are three methods for obtaining absolute power gain from planar near-field measurements. In no particular order these are: • • •

Direct gain measurement Gain-comparison measurement Three-antenna measurement technique.

Both the direct gain measurement and the gain-comparison method require that the gain of one of the antennas be known a priori, either the gain of the probe in the direct measurement technique or the gain of the reference antenna in the case of the gain-comparison method. Here, only the most commonly used technique is briefly discussed; further details can be found in the literature [12]. Using the Friss transmission formula [13] for reflection and polarisation-matched directional radiation and reception for two antennas which are in the far-field of one another we can write  λ 2 Pr G0t G0r = (5.221) 4πR Pt
 2 where the factor λ 4π R is the free-space loss factor. Therefore we can write for the gain standard  
 Prprobe 4πR (5.222) + 10 log10 (G0tSGH )dB + Gorprobe dB = 20 log10 λ PtSGH and for the test antenna


(G0tAUT )dB + Gorprobe

 dB

 = 20 log10

4πR λ



+ 10 log10

Prprobe PtAUT

(5.223)

If the separation between the probe and the gain standard is the same as that between the probe and the test antenna and further supposing that the frequency remained constant and that the same amount of power was supplied to the gain standard as to the test antenna then, these expressions reduce to (G0tSGH )dB − 10 log10 (PtSGH ) = (G0tAUT )dB − 10 log10 (PtAUT ) hence (G0tAUT )dB = (G0tSGH )dB + 10 log10



PtAUT PtSGH

(5.224)

(5.225)

where G0tAUT is the gain of the test antenna, G0tSGH is the gain of the gain standard, PtAUT is the transmitted power into the test antenna, PtSGH is the transmitted power into the gain standard.

Measurements – practicalities of planar near-field antenna measurements 183 This technique utilizes a gain standard (antenna with known gain) to determine absolute gain of the test antenna. Therefore by transforming an acquisition of the standard gain horn and noting the power on boresight and by performing the same operation on the test antenna the gain of the test antenna can be computed by normalizing further test antenna far-field patterns by the quantity  10 log10

PtAUT PtSGH

(5.226)

5.13 Calculating the peak of a pattern Very often, one is required to determine the direction and level of the peak of a pattern. This is required for verifying the alignment of an antenna system or for determining gain values. If a function is represented as a table of values at discrete points in space, as is most often the case with antenna measurements, it is unlikely that any of the points of inflection are coincident with a sampling node. Thus, a method for locating the maximum or minimum, value of the underlying function is required. If the maximum value of the underlying function is assumed to be the maximum tabulated value then a pointing error of up to ± (1/2) h, where h is the sample spacing, can be introduced. Reducing the sample spacing h, however, can reduce the magnitude of this error although this is potentially computationally expensive. Thus, an alternative technique for determining the location and maximum value of a function is clearly required. Clearly, a minimum can be located using identical techniques as f (xmin )|min = − max (−f (x))

(5.227)

There are many techniques for determining a maximum of a multidimensional function, best fit ellipsoid, and so on. However, two techniques that have been found particularly useful are a peak find by polynomial fit and a peak find by centroid fit. Owing to the utility of these techniques they are briefly discussed in the following sections.

5.13.1 Peak by polynomial fit It has often been found to be preferable to approximate the underlying function with a quadratic or even a cubic, function and to extrapolate the location of the beam peak. Standard interpolation algorithms can then be used to determine the value of the beam peak. Assuming the underlying function is locally quadratic in form then we can approximate the function using y = ax2 + bx + c

(5.228)

184 Principles of planar near-field antenna measurements Thus dy = 2ax + b dx

(5.229)

At a point of inflection, that is, at the peak of the pattern the slope will be zero 0 = 2ax + b

(5.230)

Thus xy max = −

b 2a

(5.231)

The coefficients a and b can be accurately obtained from a least squares fit with a Vandermonde matrix [14]. Although the coefficients a and b can be obtained from taking numerical differences, in practice the least squares fit approach is often found to yield greater accuracy. For a vector x, a Vandermonde matrix has the form   m x1 x1m−1 · · · x1 1  xm xm−1 · · · x2 1    2 2 (5.232) V = . .. .. ..  ..  .. . . . .  xnm

xnm−1

· · · xn

1

Here, where m is the order of the polynomial (m = 2 for quadratic) and n is the length of the vector of x-positions which must contain at least m + 1 points. For the case of a quadratic fit, the vector of positions, x and the vector of values, y, will each have three elements (n = 3). Let [15]  −1 Vp = V T V VT

(5.233)

Here, the superscript T is used to denote the complex transpose whilst the superscript −1 denotes the matrix inverse then in order to solve this by the least squares method p = Vp y

(5.234)

Here, the vector p contains the coefficients to the least squares fit quadratic polynomial and y contains the values of the tabulated function at the locations specified by the vector x. As an aside for readers familiar with the programming language and visualisation tool Matlab this is equivalent to using p = V \y. Thus     yi−1 a  b  = Vp  yi  (5.235) yi+1 c Where i is used to denote the index of the element of y that has the largest value and a, b and c are the polynomial coefficients. This can be readily extended to two or more,

Measurements – practicalities of planar near-field antenna measurements 185 dimensions. The peak find algorithm first locates the coordinates of the maximum element in the data set. The location of the coordinates of the maximum value of the underlying function can be locally determined using the expression presented above. The peak value of the underlying function can then be readily obtained for example by interpolation. It is possible to use this (least squares) technique to fit higher order polynomial functions too. For example, for a cubic function y = ax3 + bx2 + cx + d

(5.236)

Thus, a point of inflection will be found when dy = 0 = 3ax2 + 2bx + c dx Using the quadratic formula yields the following two solutions √ −2b ± 4b2 − 12ac x= 6a

(5.237)

(5.238)

Again, the coefficients are determined from the least squares fit approach described above. The cubic polynomial has two solutions so we need to evaluate the value of the underlying function for each of these combinations and then choose the correct solution. That is generally assumed to be the solution that lies closest to the elemental peak. It is perhaps worth giving a warning at this juncture. The success of these techniques essentially rely upon our ability to correctly perform numerical differentiation. Clearly, numerical differentiation describes the slope of a function at a point from a small number of neighbouring points and is essentially a localized, microscopic, operation. Thus, any small change in the value of the neighbouring points as a result of noise can significantly change the calculated gradient. Conversely, integration describes the area under a curve from knowledge of all points and is essentially a global, macroscopic, operation. Thus any small change resulting from noise is essentially averaged out over the entire data set rendering integration comparatively insensitive to these effects. As a consequence of this inherent sensitivity, numerical differentiation is a more difficult task to accomplish accurately than numerical integration and this is particularly true of data acquired experimentally. Our only solace is that in this instance we are attempting to evaluate the gradient of a function where the signal is largest and the detrimental effects of noise should be at a minimum.

5.13.2 Peak by centroid Here, the location of the peak of the underlying two-dimensional function is approximated by the weighted mean coordinates of a contour that circumscribes the peak. Initially, the location and level of the largest element in the two-dimensional tabulated function is determined. Next, a single contour, that is, a closed constant level curve, is drawn at a level m dB below that of the elemental peak. The value of m is chosen so that the contour is a single loop with as many line sections as is practical. Often, m is

186 Principles of planar near-field antenna measurements chosen to be −3 dB. The coordinates of the peak are taken to be the ‘weighted’ arithmetic average of the coordinates of the line sections where the weighting is dependent upon the length of the line sections. Thus the coordinates of the beam peak can be obtained from a closed contour data set with N points by N 1  x¯ = xn w n N

(5.239)

N 1  yn w n N

(5.240)

n=1

y¯ =

n=1

where Ncn wn = #N

n=1 cn

and cn =

 (xn − xn−1 )2 + (yn − yn−1 )2

(5.241)

(5.242)

Generally, if the contour contains a significant number of line sections this technique is more reliable, for patterns with small gain slope, than the quadratic fit estimation presented above. However, in order for this technique to be useful it is presumed that an algorithm for drawing contours is available [16].

5.14 Summary In this chapter, the practicalities of undertaking planar near-field measurements were considered. This commenced by considering efficient evaluation techniques for Fourier integrals. A mathematical formulation of the truncation problem was presented and will subsequently be utilized in Chapter 9 to develop the poly-planar approach. A detailed derivation of a coordinate free form of the antenna–antenna transmission formulae is presented, removing the need for probe-to-AUT alignment. Coordinate systems, conventions and transformations are introduced and used to illustrate the effects of alignment errors in antenna measurements. Vector isometric rotations are used as the basis for a novel active alignment correction technique that permits planar near-field measurement without probe-to-AUT alignment and is a crucial component in being able to undertake the poly-planar measurement technique. The practicalities of determining gain and the related requirement of determining the peak radiated power of the measured three-dimensional radiation pattern are considered and solutions offered. Finally, alternative planar scan approaches were considered, these being the plane-polar and plane-bipolar techniques.

Measurements – practicalities of planar near-field antenna measurements 187

5.15 References 1 Dunlop, J., and Smith, D.G.: Telecommunications Engineering (CRC Press, Florida, 1994), p. 77 2 Whittaker, E.T.: ‘On the functions which are represented by the expansions of the interpolation theory’, Proceeding of the Royal Society of Edinburgh, 1915;35:181–94 3 Yaghjian, A.D.: ‘Upper-bound errors in far-field antenna parameters determined from near-field measurements, Part 1: analysis’, National Bureau of Standards, Bolder, Colorado, Technical Note 667, 1975, p. 20 4 Clarke, R.H., and Brown, J.: Diffraction Theory and Antennas (Ellis Horwood Ltd, Chichester, 1980), p. 87 5 Ruck, G.T., Barrick, D.E., Stuart, W.D., and Krichbaum, C.K.: RADAR Cross Section Handbook (Plenum Press, New York, 1970), p. 523 6 Narasimhan, M.S., and Karthikeyan, M.: ‘Evaluation of Fourier transform integrals using FFT with improved accuracy and its applications’, IEEE Transactions on Antennas and Propagation, April 1984;AP-32 (4):404–8 7 Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P.: Numerical Recipies in FORTRAN, the Art of Scientific Computing, 2nd edn (Cambridge University Press, Cambridge, 1992), pp. 577–84 8 Ludwig, A.C.: ‘The definition of cross-polarisation, IEEE Transactions on Antennas and Propagation, January 1973;AP-21 (1):116–19 9 Born, M., and Wolf, E.: Principles of Optics, 7th edn (Cambridge University Press, Cambridge, 1999), p. 35 10 IEEE, Standard Test Procedures for Antennas (John Wiley & Sons Inc., New York, 1979) 11 Balanis, C.A.: Antenna Theory Analysis and Design, 2nd edn (John Wiley & Sons Inc., New York, 1997), p. 879 12 Newell, A.C., Ward, R.D., and McFarlane, E.J.: ‘Gain and power parameter measurement using planar near-field techniques, IEEE Transactions on Antennas and Propagation, June 1988;36 (6):792–803 13 Stuzman, W.L., and Thiele, G.A.: Antenna Theory and Design, 2nd edn (John Wiley & Sons Inc., New York, 1998) p. 79 14 Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P.: Numerical Recipes in FORTRAN, 2nd edn (Cambridge University Press, Cambridge. 1992), p. 114 15 Anton, H.: Elementery Linear Algebra (John Wiley & Sons Inc., New York, 2000), p. 314 16 Monro, D.M.: FORTRAN 77 (Edward Arnold, London, 1981), p. 247

Chapter 6

Probe pattern characterisation

6.1

Introduction

As shown in Chapter 5, for plane rectilinear near-field scanning, probe pattern characterisation errors contribute to the overall facility error budget as a singular mapping. Essentially then, this means that an error in the probe pattern at a particular direction in space will correspond to a similar error at that angle being introduced into any antenna pattern that has been corrected with this data. Furthermore, this can potentially constitute one of the largest and most repeatable, measurement uncertainties. Thus it is clear that in order to obtain reliable measurements, the electromagnetic (EM) properties of the near-field probe must be known very accurately indeed. Throughout this work, it has been assumed that relative measurements are taken, that is, without reference to absolute gain. If an absolute gain is required, then it is assumed that a substitution method is used employing a calibrated gain standard in order that the measurements can be correctly normalized. The importance of this for the characterisation of the near-field probe is that if the same probe and probe pattern used to correct the measurement of the antennas under test (AUTs) and the standard gain horn (SGH) then the gain of the probe will be unimportant. Thus, within this chapter, characterisation of the gain of the probe is not discussed and the gain of the probe will be normalized to unity for convenience. This chapter briefly describes the properties that are desirable in a near-field probe before proceeding to describe methods for obtaining and then using reliable probe pattern data.

6.2

Effect of the probe pattern on far-field data

The probe pattern can be thought of as a device that spatially filters the fields received from different parts of the AUT. In a planar range, the effects include something similar to a direct multiplication of the far-field probe pattern with the far-field AUT

190 Principles of planar near-field antenna measurements Phi = 0 (deg) Freq. = 9 (GHz)

5

Corrected Uncorrected Probe pattern

0 –5 –10

Power (dB)

–15 –20 –25 –30 –35 –40 –45 –50 –55 –60 –90

Figure 6.1

–60

–30

0 Theta (deg)

30

60

90

Comparison between probe pattern and corrected and uncorrected farfield pattern, φ = 0◦

pattern and can be shown to be a direct result of the nature of the convolution theorem and can be visualized directly from the mechanical operation of the scanner. It is not usually possible to neglect these effects in a planar range because of the large angles of validity required and the short measurement distance employed. However, the effects of probe pattern correction can be minimised by utilizing a probe of similar pattern to a Hertzian dipole, that is, a linearly polarised source with an aperture small in comparison to the half wavelength sample spacing. The general effects of the directivity pattern of the near-field probe on the resulting far-field pattern function of the AUT are illustrated in Figures 6.1 and 6.2. From inspection of these figures, the general effect of the probe pattern (black dotted) on the far-field pattern of the AUT (grey dotted) can be broadly determined. The pattern of the probe suppresses the copolar pattern of the AUT by an amount equal to the directive loss of the probe pattern at the angle of observation. As has already been shown, this is not a general statement as effects associated with crosspolarisation and polarisation purity have been ignored. Furthermore, it has been assumed that the AUT and the probe are perfectly aligned. However, it is true to say that in the far-field the effect of the probe on the AUT pattern is a one-to-one mapping that is, that the pattern of the probe at a given angle will only affect the AUT pattern at that same angle. Pattern nulls in the forward hemisphere of the probe antenna are best avoided, as they are difficult to characterise accurately and they correspond to large correction terms in the far-field antenna pattern that introduce additional uncertainties.

Probe pattern characterisation 191 Phi = 90 (deg) Freq. = 9 (GHz)

Corrected Uncorrected Probe pattern

5 0 –5 –10

Power (dB)

–15 –20 –25 –30 –35 –40 –45 –50 –55 –60 –90

Figure 6.2

6.3

–60

–30

0 Theta (deg)

30

60

90

Comparison between probe pattern and corrected and uncorrected far-field pattern, φ = 90◦

Desirable characteristics of a near-field probe

The near-field measurement technique places several requirements upon the characteristics of the scanning probe. These are 1. Time invariance, usually equating to mechanically rigid – resilient against gravitational deformation, and so on. Any variation in the EM performance of the probe between the time when it was characterised and the time when it was used to measure an AUT would essentially correspond to the near-field measurements being corrected with the wrong probe pattern data. 2. No pattern nulls in the forward hemisphere, corresponding to a low directivity, electrically and likely physically small probe. Pattern nulls correspond to angles in which the probe is insensitive, that is, blind, to incoming radiation. This would necessarily correspond to the introduction of large correction terms within the probe compensation process that would render corrected far-field pattern susceptible to spurious signals, that is, noise. This noise could be introduced either within the antenna measurement process itself or within the original probe characterisation, for example, uncertainties associated with the measurement of the null depth, and so on. 3. Low scattering cross section and reflection coefficient, that is, well matched with a small return loss. This is required to minimise the magnitude of the multiple reflections that are set up between the near-field probe and the AUT. Such

192 Principles of planar near-field antenna measurements multiple reflections are omitted from the theoretical treatment of the near-field measurement process and therefore cannot be corrected for; they can only be assessed as part of an empirical error budget calculation. From a practical standpoint, such multiple reflections can result in the introduction of ghost side lobes and can upset the excitation of radiating elements within array antennas. 4. Good polarisation purity is required in order that the various field components can be resolved. Although in principle it is possible to use the probe pattern correction process to effectively improve the polarisation response of the range; it is less demanding and therefore often more reliable, if a probe with good polarisation purity is employed in the first place. 5. Good front-to-back ratio to minimise sensitivity to probe placement and multiple reflections; unfortunately, in practice this is at odds with item 2. 6. Wide bandwidth; this is purely for convenience as it minimises the necessity to use a multitude of probes to cover the operational bandwidth of the facility. Unfortunately, it is not usually possible to satisfy all of the previous requirements over an extended bandwidth, that is, of greater than 20 per cent. Typical near-field probes can include cylindrical waveguides, rectangular waveguides, corrugated horns and pyramidal horns. One of the most significant contributions, albeit an extremely systematic one, to the overall error budget of the conventional planar near-field measurement technique is the inaccuracy in the characterisation of the near-field probe. Conventionally, as the measured main component pattern is proportional to the main component probe pattern, errors in the corrected main polarisation pattern arising from probe characterisation errors will be one-to-one mappings. That is, they have the same magnitude and direction as the errors in the probe pattern. However, for auxiliary rotation near-field measurements (a topic that will be expanded upon in Chapter 9), this error is not local. Rather, errors in one direction in the probe pattern affect the characterisation of the AUT over an extended angular region. Although a number of simple analytical expressions that provide functional forms of the far-field pattern of simple probe antennas are available, if truly accurate farfield probe patterns are required so that highly accurate antenna measurements can be taken there is little recourse other than to use the highly accurate predictions attained from three-dimensional full-wave solvers or to perform very careful auxiliary measurements of the near-field probe. Once this precious data has been acquired a certain amount of post processing is required in order that it can be converted into a form that is amenable for use with the probe pattern correction equations developed above. A treatment of circularly polarised probes is not given here but can be found in Chapter 5. The expressions developed above for implementing probe pattern correction are presented in a form that is amenable for the correction of linearly polarised probe. However, in the event that a circularly polarised probe is used to acquire the near-field data they can still be used as all of the polarisation information associated with the probe is contained within the Cartesian spectral field components. Circularly polarised probes are not commonly used to take near-field data as typically they are expensive to manufacture and are comparatively narrow band

Probe pattern characterisation 193 devices; however, on some occasions they are used as they can offer an improvement in sensitivity and can relax the requirement on the measurement positional accuracy required. As described above, measurement of circularly polarised antennas are made by combining two orthogonal polarisations in quadrature and then scaling the result. For the case of a circularly polarised probe this is merely being implemented in hardware, rather than in software, which is where the limitation in bandwidth is usually introduced. The remaining sections of this chapter are devoted to acquisition and manipulation of far-field probe pattern data.

6.4

Acquisition of quasi far-field probe pattern

The following sections describe one potential technique for characterising a typical near-field probe. The example of a standard proprietary, that is, off the shelf, rectangular waveguide probe is used. However, the techniques discussed are general and are therefore equally applicable to the characterisation of other near-field probes. Though the discussion is not exhaustive, as this would require the preparation of a specialist text devoted to this subject alone, it is intended to provide the reader with sufficient information to enable them to comprehend what perhaps are the most important points. It is highly recommended that all such ‘standard’ measurements are performed in a temperature and humidity controlled screened anechoic chamber. In cases such as these where the field is required to be known over a complete half space a ‘roll-overazimuth’ configuration is often adopted. This useful polar spherical configuration is chosen in order that truncation, resulting from blockage from the robotics subsystem, can be minimised whilst ensuring that the crucial, rapidly varying ‘boresight’ cross-polarisation pattern is comfortably oversampled. In practice, this is conveniently realized with the robotic measurement system illustrated in Figure 6.3. This arrangement of rotators is often referred to in the literature as being a ‘model tower’ positioner. Here, the θ - and φ-axes of the range intersect and are mutually orthogonal. The axis set described above is attached to the range positioner and is the system in which the AUT radiation pattern is tabulated. A ‘polar pointing’ configuration should be chosen when mounting the AUT, that is, one in which the main beam of the AUT is in the direction of the phi-axis of the range thus the boresight is aligned such that it is coincident and synonymous with the positive z-axis of the range system. This configuration has the advantage that it minimises blockage from the AUT mount, although it places considerable demands upon the polarisation purity of the range antenna. In this configuration, the radius, r, is held constant and the angles θ and φ are varied with χ sequentially set to 0◦ and 90◦ to sample two orthogonal polarisations and thus resolving the propagating electric field onto a polar spherical polarisation basis, that is, Eθ and Eφ . Here Eθ and Eφ correspond to field components aligned with the conventional polar spherical unit vectors θˆ and φˆ respectively. When the remote source antenna is assumed to be in the far-field of the radiator, the radial component

194 Principles of planar near-field antenna measurements AUT (Rx)

RSA (Tx)

phi axis


φf

 chi axis

theta axis


θ

Figure 6.3

Schematic of robotic measurement system

is not sampled, as this is taken to be zero. If the probe and the remote source antenna are not sufficiently far removed from one another, a spherical near-field to far-field transform will be necessitated in order that the far-field pattern can be recovered.

6.4.1 Sampling scheme For data to be taken over the entire forward hemisphere the spherical angles are allowed to vary over the range, −90◦ ≤ θ ≤ 90◦ and −180◦ ≤ φ ≤ 180◦ . The inclusion of negative polar angles enables the pattern to be tabulated twice, once over the forward hemisphere and again over the alternative forward hemisphere. As the pattern of the instrument under test is time invariant, differences between the respective patterns can be used to estimate the extent by which chamber multiple reflections are influencing the measured performance. Although undesirable, in practice, this sampling scheme has the added benefit that in the event of a measurement ‘drop out’, data from one hemisphere can be used to complete the other. The spherical angles θ and φ are varied such that the electric field is tabulated on a plaid, monotonic and equally spaced grid, where the acquisition can be formed from taking a series of θ or φ cuts, the choice being unimportant. Such a polar spherical measurement system has the advantage that each angular sampling point is approached from the same side, that is, the sign of the increment is always the same, which not only removes the requirement for a retrace scan, but also removes positional errors associated with backlash in the positioner system. Backlash is the term used to describe the ‘slack’ in a gearing mechanism. When the direction of rotation in a gear chain is reversed, the first cog will turn by a few degrees before

Probe pattern characterisation 195 the last cog in the chain begins to rotate. This lag is a result of the finite gap that exists between teeth on adjacent cogs and directly equates to an angular error in the positioning of the AUT. Thus such a monochromatic measurement scheme should be adopted to remove positional inaccuracies that are inevitable when taking multiple frequency measurements on the fly. The sampling interval is defined according to the general sampling criteria. Usually, the AUT is mounted so that the radius of its minimum sphere rt is made as small as possible. The minimum sphere is a conceptual sphere, that is, not the sampling sphere, which is centred about the origin of the measurement system and completely encloses the AUT. This parameter is necessarily measured prior to the acquisition of the near-field data with, for example, a theodolite. The highest significant spherical wave mode present in the test antenna near and far-field is usually taken to be [1] N = k0 rt + 10

(6.1)

Here, rt is in metres and k0 is the wave number. The addition of 10 extra spherical mode coefficients allows for the possibility of a small, but non-zero, amount of power being contained within a few higher-order spherical modes. The maximum permissible sampling increment in θ and φ is related to the highest significant wave mode through θ = φ ≤

2π (2N + 1)

(6.2)

This sampling theorem is clearly angular in nature and independent of distance, implying that it is equally applicable in both the near- and far-field. Figure 6.4 contains the sampling increment plotted as a function of the radius of minimum sphere for the highest frequency acquisition. At 10000 (MHz)

Theta/phi increment (deg)

6 5 4 3 2 1 0 0.1

Figure 6.4

0.2

0.3 0.4 0.5 0.6 0.7 0.8 Radius of minimum sphere (m)

0.9

1

Sampling increment as a function of radius of minimum sphere

196 Principles of planar near-field antenna measurements At 10000 (MHz)

Maximum angular uncertanty (deg)

0.25

0.2

0.15

0.1

0.05

0 0.1

Figure 6.5

0.2

0.3

0.4 0.5 0.6 0.7 0.8 Radius of minimum sphere (m)

0.9

1

Positional error as a function of radius of minimum sphere

In this case the radius of minimum sphere when measured directly was found to be approximately 30 cm. Thus for the measurement configuration employed here, a sampling increment of θ = φ = 2◦ amply satisfies the sampling theorem and this can be applied to the quasi far-field pattern as measured here. The angular accuracy of the positioning system is typically taken to correspond to one fiftieth of a wavelength over the circumference of the minimum sphere, plotted in Figure 6.5, that is, approximately 25 times smaller than the required angular increment shown above. Such a precision is readily attainable with the rotary encoders available today. Typically, with encoders fitted, digital readouts are assumed accurate to ±0.002◦ and the positioner is assumed to be accurate to ±0.003◦ which is two orders of magnitudes better than required by this measurement configuration. Positional errors in the azimuthal angle on boresight result in the incorrect alignment between the polarisation axis of the AUT and the remote source antenna. This will introduce a polarisation error that is, most obvious on the cross-polar pattern function. This cross-polar error signal is related to the angular positioning error by ErrordB = 20 log10 (|sin φerror |)

(6.3)

Such errors are expected to be negligible as an angular error of ±0.01◦ , that is, three times larger than the expected positioning error, will introduce an error signal of −70 dB on the cross-polar pattern.

Probe pattern characterisation 197 Minimising the radius of the minimum sphere rt has other benefits. Placing the AUT over the centre of rotation results in the antenna occupying only a small portion of the test zone during the measurement procedure. Assuming that the remote source antenna is electrically small with a relatively low directivity, it is possible to approximate the source antenna with an ideal Hertzian dipole antenna and avoid the requirement for probe pattern correction within the spherical measurement system. This is further eased if the range length, that is, the separation between the remote source antenna and the centre of the measurement coordinate system, is maximised. This follows from the fact that the variation in the illuminating amplitude and phase over the test zone attributable to the remote source antenna will be negligibly small. Figure 3.25 shows a waveguide probe installed on a model tower positioner. Here, the aperture of the waveguide probe has been carefully positioned to coincide with the intersection of the θ- and φ-axes of the range. In the photograph the probe assembly can be seen to consist of a chamfered rectangular waveguide section and a SWAM cone. The cylindrical section is included to displace the flat circular radar absorbing material (RAM) sheet that is, used to screen the azimuth positioner, from the waveguide probe and minimise multiple reflections. Unfortunately, the front-to-back ratio of a probe such as this will not be very large and a considerable amount of field could be illuminating this absorber. Indeed, the measured pattern can, in certain circumstances, be seen to vary by as much as a few dB at specific angles by merely varying the distance between the tip of the probe and the front of the absorber ring. Figures 6.6 and 6.7 (see p. C5) illustrate the measured data presented in the form of a polar plot where the radius and colour represents the measured field intensity and the angles correspond to the angles at which the measurement was taken. As the gain of the probe is not required within the transformation software, the pattern functions have been normalized to 0 dB, 1 V, on boresight. The pinch effect observed on boresight is an artefact of the soft singularity present in the spherical polarisation basis at that direction. The adoption of a scan in phi, step in theta test procedure can offer several advantages. In addition to minimising the acquisition time by spinning the smaller and usually faster top axis, this procedure enables additional information to be gathered which can be used to improve the quality of the final measurements. Ways in which this additional information can be used are discussed in the following sections.

6.4.2 Electronic system drift (tie-scan correction) This form of raster scan involves rotating the test antenna about the θ axis to the first θ angle. Next, a continuous rotation is performed about the φ axis, whilst the field is sampled at regular angular intervals. Once a complete revolution was performed the test antenna was rotated to the next θ angle ready for the next φ scan to be taken. When data had been collected over the entire forward hemisphere the remote

198 Principles of planar near-field antenna measurements source antenna was rotated through 90◦ in χ , so that both orthogonal spherical field components could be recorded. In practice, this measurement procedure can often require relatively long measurement durations. Coherent measurements such as these require that the phase remain stable for the duration of the measurement. This translates to a requirement that the temperature within the chamber should also remain constant. A change in temperature will necessarily result in a change in volume for most materials. For air, an increase in temperature will result in a decrease in density that will change the velocity of the EM radiation propagating within it. The corresponding variation in refractive index will introduce a change in the electrical path length that will result in an undesirable change in the recorded phase. Temperature changes will also affect the guided wave section of the radio frequency (RF) subsystem. Most low-power RF cabling currently in use at x-band consists of a coaxial transmission line with a polytetrafluoroethylene (PTFE) dielectric insulator. Clearly, the PTFE dielectric will change in volume as a function of temperature. However, such an expansion will result in a change in the diameter of the outer cylindrical conductor, which will inevitably change the capacitance of the transmission line per unit length. This change in capacitance results in a change in the propagation constant that introduces a variation in the electrical length of the line, which will significantly change the recorded phase. Unfortunately, this volumetric change is a nonlinear effect with the largest variations typically occurring around 20◦ C, that is, room temperature. For this reason, most facilities are air-conditioned either warmer or cooler than this. As even relatively modest changes in temperature, for example a degree or so, will result in a comparatively large variation in the recorded phase, even in facilities with balanced electrical length test and reference paths; techniques to correct for thermally induced phase drift are usually required. Generally the time that is required to complete a φ-scan is short; typically a few minutes or less and it is assumed that the drift measured over this time is negligible during this time. A single tie-scan measurement is made after each hand of polarisation has been acquired. This involves taking repeat measurements on the fly by scanning the axis that had previously been stepped through the region of largest field intensity. Thus for these measurements, a single θ cut can be used to ‘tie’ the φ cuts together and thus compensate for the electronic system drift. It is possible to use more than a single θ cut to tie the φ cuts together. If instead a number of tie scans are taken then a weighted average can be used to determine the tie scan correction factor. If multiple frequency measurements are taken on the fly, then the measurement points may not coincide for more than a single frequency potentially rendering this technique inaccurate. Furthermore, in the event that the cut passes through regions of lower field intensities the correction terms obtained can become inaccurate and susceptible to noise.

6.4.3 Channel-balance correction If the reference signal is obtained by means of an external directional coupler, accurate rotary joints are used on the χ positioner and a singularly polarised remote source

Probe pattern characterisation 199 antenna is used, then channel balance is not usually required if a singularly polarised remote source antenna is used to sample the spherical (ˆeθ and eˆ φ ) field components. However, if for whatever reason, for example thermal drift, there is a difference in the reported signal when the same signal is incident on the remote source antenna when measuring (ˆeθ and eˆ φ ), then the channels are out of balance and a channel-balance correction is required. When a polar spherical measurement system is utilized, an auxiliary channelbalance measurement is not usually required, as the crucial correction parameters contained within the boresight scan are sampled within the pattern acquisition itself. For a polar pointing spherical measurement, the boresight scan involves taking φN measurements of essentially the same parameter with only a rotation of the polarisation reference. This is repeated for each orthogonal polarisation, that is, once when the principal polarisation of the probe is aligned with the x-axis and again with the principal polarisation of the probe aligned with the y-axis. Thus, the channel-balance correction factor can be determined from the difference between equivalent measurements. Figure 6.8 shows a boresight φ cut. The dotted trace corresponds to the measurements taken when the principal response of the remote source antenna was vertically polarised, that is, eˆ φ and the black trace was recorded when the remote source antenna was horizontally polarised, that is, eˆ θ . The channel-balance correction factor was determined from the ‘weighted’ mean difference between the traces. Here, the weighting term is chosen to be proportional to the square of the field present in each cut, that is, proportional to the power which is in accordance with the principle of least squares. Before Power (dB)

–10 –20 –30 –40

Phase (deg)

–50

Figure 6.8

–150

–100

–50

0 Phi (deg) Before

50

100

150

–150

–100

–50

0 50 Phi (deg)

100

150

100 0 –100

Boresight Eθ and Eφ field component at 9 GHz before channel-balance correction

200 Principles of planar near-field antenna measurements Table 6.1

Frequency (GHz)

Amplitude (dB)

Phase (deg)

9.000 10.000

3.887672 −0.587971

−30.052816 −13.716436

N 


CB|dB =

n=1

   eθ |dB (φn ) − eφ dB φn + π2 wn N


N 


CB|deg =

Derived channel-balance coefficients

n=1

n=1

(6.4)

wn

   eθ |deg (φn ) − eφ deg φn + π2 wn N
n=1

(6.5)

wn

Where the weighting coefficient can be expressed as wn = (|eθ (φn )| − max (|eθ (φn )|))

    π  π     − max eφ φn +  eφ φn + 2 2 (6.6)

The weighting coefficient is of particular importance from a practical perspective, as it protects the channel-balance coefficient from being disturbed by the large phase differences encountered within the pattern nulls. The channel-balance figures were determined for each of the measured frequencies and can be found contained in Table 6.1. In general these parameters will be different for each of the measured frequencies. These values are large and arise from the use of a secondary antenna, rather than a directional coupler to sample the reference signal. Figure 6.9 contains an equivalent plot that illustrates the channel-balance corrected boresight eˆ θ and eˆ φ cuts. Figure 6.10 (see p. C5) contains the forward hemisphere Ludwig III copolarisation and cross-polarisation far-field pattern for an open-ended rectangular waveguide probe, after tie-scan and channel-balance corrections have been applied, in the form of a three-dimensional surface plot. From inspection, the effective noise floor of the measurement can be seen to be approximately 45 dB below the peak of the signal, that is, boresight. Although this constitutes a reliable measurement for such a low-gain device, the noise floor suggested by the boresight cross-polarisation level, which should be a constant value, is higher than perhaps expected.

Probe pattern characterisation 201 After Power (dB)

–10 –20 –30 –40 –50

Phase (deg)

–150

–50

0 50 Phi (deg) After

100

150

100

150

100 0 –100 –150

Figure 6.9

–100

–100

–50

0 Phi (deg)

50

Boresight Eθ and Eφ field component at 9 GHz after channel-balance correction

6.4.4 Assessment of chamber multiple reflections The nature of this very low-gain instrument will inevitably result in the facility walls being illuminated by a relatively large amount of power. If not all of this power is absorbed, the scattered signal could become comparable with the direct signal that would raise the facility noise floor. Additionally, the characterisation of the waveguide probe was found to be further complicated by sensitivity to the configuration of absorber placed behind the instrument indicating the existence of large side lobes in the rear hemisphere. This is important, as it suggests that the performance of the probe is dependent upon its local environment that will clearly differ when installed on the probe carriage in the planar near-field scanner. The probe pattern can be characterised by scanning the azimuthal angle φ over the range −180◦ ≤ φ ≤ 180◦ and stepping the polar angle θ over the range −90◦ ≤ θ ≤ 90◦ . Thus, the forward hemisphere of the probe pattern is sampled twice, conventionally and again in the alternate hemisphere configuration. Here, the mapping used can be expressed as and is expounded further in Appendix C.  θ when θ ≥ 0 θ= (6.7) −θ otherwise  φ when θ ≥ 0 φ= (6.8) φ − π otherwise In Figure 6.11, dotted contours represent data taken in the conventional hemisphere, whilst black contours represent equivalent data taken using the alternate

202 Principles of planar near-field antenna measurements Ephi (dB) Freq. 9 (GHz)

150

150

100

100

50

50 Phi (deg)

Phi (deg)

Etheta (dB) Freq. 9 (GHz)

0

0

–50

–50

–100

–100

–150

–150 0

Figure 6.11

20

40 60 Theta

80

0

20

40 60 Theta

80

Comparison of conventional and alternate sphere pattern of waveguide probe

hemisphere that have been mapped onto the forward hemisphere. Contours are plotted in 3 dB increments from −50 dB upwards. The average multipath level for the θ and φ polarisations were found to be encouraging although not overly impressive at −52.36 dB and −51.43 dB respectively. This can be thought of as representing the maximum change in room effects between configurations and is sufficiently small that further pattern correction should not be necessary. However, it implies that the performance of the probe for signal levels smaller than these is unknown. As we have essentially measured the probe pattern twice, once in the alternate sphere and once in the conventional sphere, it is possible to average the two patterns in an effort to suppress these room related effects.

6.4.5 Correction for rotary errors In essence, the θ = 0 cut represents a sequence of repeat measurements of the same part of the probe pattern. The sinusoidal variation with the φ angle that is observed across this cut is merely an artefact of the polarisation basis that the pattern is resolved onto, that is, the probe is rotating in phi, whilst the remote source antenna remains fixed. If instead the pattern was resolved onto a Ludwig III polarisation basis, this corresponds to the case where the probe and remote source antenna are rotating in unison and this sinusoidal variation is removed. Thus, in a perfect system, the

Probe pattern characterisation 203 boresight copolar components, amplitude and phase, should be constant for all φ angles. In practice, small variations will be seen and these can arise from imperfections in the orientation of the antenna, alignment errors, multipath errors or rotary joint errors, as channel-balance errors should have been corrected previously. Provided the rotary errors are repeatable, then this information can be used to correct the entire acquired pattern. Another, more sophisticated correction technique is presented in Box 6.1. The rotary error can be defined to be f (φ) = Eco (θ, φ)|θ=0

(6.25)

Thus, the rotary joint error can be corrected using Eco (θ, φ) f (φ) Ecr (θ, φ) Ecr (θ , φ)|corrected = f (φ)

Eco (θ, φ)|corrected =

(6.26) (6.27)

Strictly this method only works for cases where the measurements are taken in the far-field. Here, we have taken only quasi far-field measurements.

Box 6.1 APC – Circumscribing three nonlinear points on a plane If the errors are not repeatable an advanced pattern correction (APC) scheme can be followed where the two hemispherical measurements are treated as two of a minimum of three data sets that can be used to provide an APC correction. At least one other data set would be required; this could be obtained from making a repeat measurement with the AUT displaced from the centre of rotation, then correcting for the resulting phase change in the far-field pattern and then averaging over three such measurements using the APC method as described below. The equation of a circle can be expressed as a special case of the equation of an ellipse (see Chapter 5): (x − h)2 + (y − k)2 = r 2

(6.9)

Here, r is the radius of the circle and h and k are the x and y coordinates of the origin of the circle respectively. This can be expanded to yield

or

x2 − 2xh + h2 + y2 − 2yk + k 2 − r 2 = 0

(6.10)

  x2 + y2 − 2hx − 2ky + h2 + k 2 − r 2 = 0

(6.11)

204 Principles of planar near-field antenna measurements

Box 6.1 Continued Let c1 = −2h

(6.12)

c2 = −2k

(6.13)

c3 = h2 + k 2 − r 2

(6.14)

Thus x 2 + y 2 + c1 x + c 2 y + c 3 = 0

(6.15)

Consider the problem of determining the origin and radius of a circle that passes through three non-collinear points. Let the points be described by, x1 , y1 , x2 , y2 and x3 , y3 respectively. Thus, three simultaneous equations can be formed, namely, x12 + y12 + c1 x1 + c2 y1 + c3 = 0

(6.16)

x22 + y22 + c1 x2 + c2 y2 + c3 = 0

(6.17)

x32 + y32 + c1 x3 + c2 y3 + c3 = 0

(6.18)

Alternatively, this can be rewritten using matrix notation as  2        x1 + y12 0 x 1 y1 1 c1  2   x2 + y22  +  x2 y2 1  ·  c2  =  0    0 x3 y3 1 c3 x2 + y2 3

3

Thus

  x1 c1  c2  =  x2 x3 c3 

where  x1  x2 x3

y1 y2 y3

(6.19)

y1 y2 y3

 −1  −x12 − y12 1 1  ·  −x22 − y22  1 −x32 − y32

(6.20)

−1 1 1 1  = (y2 − y3 ) x1 + x3 y1 − x3 y2 − x2 y1 + x2 y3 1   − (y1 − y3 ) (y2 − y3 ) (y1 − y2 ) − (x1 − x2 )  (x1 − x3 ) ×  − (x2 − x3 ) (x2 y3 − x3 y2 ) − (x1 y3 − x3 y1 ) (x1 y2 − x2 y1 ) (6.21)

Probe pattern characterisation 205

Box 6.1 Continued Thus the coordinates of the origin of the circle and its radius are 1 h = − c1 2

(6.22)

1 k = − c2 2  r = h2 + k 2 − c3

(6.23) (6.24)

This algorithm was verified by locating the origin and radius of a circle that passes through three points on a plane as illustrated in Figure 6.12, where the black diamonds represent the three points, the black cross represents the origin of the circle and the black plus represents the mean coordinate of the three points. Thus if repeatable measurements cannot be obtained the use of APC as it might be deployed in cases of high multipath, and so on, can be used to assign a nominal value to the measurement data point that would be an average of the three measurements. Of course this implies that three times as many measurement will be needed to assign a value to any given data point.

0.8

0

0.7 –1 0.6 y –2

y 0.5

–3

0.4 0.3

–4

0.2

–5 0.4

0.5 0.6 0.7 x

Figure 6.12

0.8

0.9

1

–1

0

1

2

3

4

x

Two examples of best fit circle

6.4.6 Re-tabulation of probe vector pattern function The probe pattern can be measured and tabulated on a plaid monotonic equally spaced polar spherical grid. Although convenient from a measurement perspective, this grid

206 Principles of planar near-field antenna measurements is not coincident and synonymous with the grid required by the planar transformation software. Invariably, far-field data is required to be plotted in a multitude of coordinate systems, sampling intervals and with varying numbers of sampling points. Thus, a choice is required to be made. Either a single coordinate system, sampling interval and sample spacing can be selected and all measured far-field data corrected using this single definition before the corrected pattern can be converted into the desired form or, the probe pattern data will need to be converted into the form required to present the measured antenna. Selection of the former option is undesirable, as this would inevitably require converting antenna pattern into a form amenable for presentation. This post processing will require that the antenna pattern be re-tabulated which is usually accomplished by means of approximation, specifically by interpolation. Interpolation can often yield unreliable results in the cases where the pattern function is anything other than grossly over sampled. The planar methodology is essentially a very general measurement technique implying that any transformation and data post processing chain will need to function correctly for a wide variety of antenna pattern functions. This clearly places significant demands on any approximation that will likely prove unsuccessful in certain cases. Conversely, if instead the probe pattern is interpolated onto the grid required to present the probe corrected far-field antenna pattern this difficulty is resolved. The underlying probe pattern function is necessarily low gain and thus it is a slowly varying function that is, typically grossly over sampled; all of which make it ideally suited to approximation with a variety of interpolation schemes. Thus, the problem of post processing is essentially the problem of interpolating the probe pattern from the measurement grid onto a grid demanded by the probe pattern correction. When the nature of the underlying function is not too pathological, this is usually accomplished by approximation, that is, piecewise polynomial interpolation. Many techniques are available for this purpose however, bicubic convolution [2] has been found to be both efficient and robust, supplying data that is smooth (possessing continuous first and second derivatives). This algorithm uses bicubic-convolution interpolation to find the values of the underlying two-dimensional function at the desired points from a set of tabulated values of the function at regular grid points, that is, one where the tabulated grid is plaid, monotonic and equally spaced. When xk < x < xk+1 the cubic convolution interpolation function in one dimension can be expressed as f (x) = f (xk−1 ) + f (xk )

−p3 + 2p2 − p 2 3p3 − 5p2 + 2 2

−3p3 + 4p2 + p + f (xk+1 ) 2 + f (xk+2 )

p3 − p2 2

(6.28)

Probe pattern characterisation 207

h

Interpolating node

k

qk

Desired point

ph

Figure 6.13

Schematic representation of the interpolating scheme

where p=

x − xk h

(6.29)

  and, k = 0, 1, 2, . . . , N . Here, the uncertainty is of the order of O h3 or proportional to the cube of the sample spacing. At the boundary of the tabulated array additional points can be extrapolated using f (xk=−1 ) = 3f (xk=0 ) − 3f (xk=1 ) + f (xk=2 )

(6.30)

f (xk=N +1 ) = 3f (xk=N ) − 3f (xk=N −1 ) + f (xk=N −2 )

(6.31)

and

This is readily extended to two (or more) dimensions and is accomplished by one-dimensional interpolation in each successive dimension, that is, g (x, y) =

2 2   l=−1 m=−1



f xj+l , yk+m





x − xj+l u hx

   y − yk+l v hy

(6.32)

where u and v represent the interpolating kernel illustrated above when deployed for each dimension as shown in Figure 6.13. One further choice is available when implementing this interpolation scheme, whether to interpolate on the rectangular (real and imaginary) or polar (amplitude and phase) form of the field. Generally, it is preferable to interpolate using the amplitude and phase patterns as more reliable results are obtained in the presence of a phase taper across the sampled function. This follows from the fact that in polar form, even a large phase taper only equates to a constant amplitude function and a linear

208 Principles of planar near-field antenna measurements Tabulated function Correct value Incorrect value

150

Phase (deg)

100 50 0 –50 –100 –150 0

2

4

6

8

10

x

Figure 6.14

Illustration of phase interpolation in the presence of a phase discontinuity

phase function, whilst in rectangular form this equates to a highly oscillatory real and imaginary function that is, difficult to approximate reliably with piecewise polynomial functions. Figure 6.14 illustrates that when interpolating phase data care must be taken to wrap all local phase points into the same phase range before estimating the value of the intervening point. The black diamond contained within the above plot shows the correct value of the interpolated point, whilst the black circle shows the results obtained when the phase wrapping is not implemented. In practice, the unwrapping of two-dimensional phase data is a demanding task and in part still remains an unsolved problem. Consequently, the task is significantly simplified by unwrapping the phases of only the nearest sixteen points (those points required by the bicubicconvolution interpolation algorithm) to the location of the desired interleaving point. The algorithm can be expressed as 1. 2. 3. 4.

Locate 16 nearest neighbour points. Unwrap all points into the same phase range as the nearest neighbour point. Apply conventional interpolation algorithm. Wrap interpolated value back into the conventional ±180◦ range.

If the rate of change in the phase function is large, that is, greater than ±180◦ between sampling points, this phase interpolation scheme will become unreliable. However, this is not generally a problem as this would also invalidate the sampling theorem. Generally, attempting to perform any form of analysis on interpolated data is dangerous unless, as is the case here, the underlying function is grossly over sampled.

Probe pattern characterisation 209 The only outstanding issue relates to the polarisation basis onto which the probe pattern is resolved prior to interpolating. As polynomial interpolation is approximate, it will introduce errors. As antenna patterns are vector quantities when the pattern is converted from one polarisation basis to another, a small error on a large quantity may equate to a large error on a smaller quantity. Also, selecting the polarisation basis that contains the minimum number of pattern nulls is advantageous. For these reasons, care must be taken when choosing the polarisation basis on which to interpolate and it often turns out be advantageous to use a Ludwig III copolar and cross-polar polarisation basis. Once these choices have been made the task of re-griding the pattern is essentially a simple one. The interpolation scheme involves taking a point in the desired coordinate system and calculating the equivalent coordinates in the coordinate system in which the probe pattern is known. This point will likely not directly correspond to any one tabulating node but will instead lie somewhere in-between. The value of the underlying function is then approximated from knowledge of the function at the sampling nodes and the interpolation scheme. This process is repeated for each of the points in the desired tabulating coordinate system.

6.4.7 Alternate interpolation formula In the preceding section the probe pattern was interpolated using bicubic-convolution interpolation. A great many interpolatory schemes are available to workers each offering a different mix of advantages and disadvantages that may prove beneficial in different circumstances. Several alternate algorithms for accomplishing this task are discussed briefly within this section. As noted above interpolation is the art of ‘making up’ numbers so no one algorithm will be ideal in all circumstances. 6.4.7.1 Nearest neighbour This algorithm uses nearest neighbour interpolation to find the values of the underlying two-dimensional function at the desired points from a set of tabulated values of the function at regular grid points. Here, the underlying function at the desired location is approximated by the value of nearest tabulated point, cf. Figure 6.13. This is defined as f (x0 + ph, y0 + qk) = f0,0

(6.33)

Here, the uncertainty is of the order of O (h). This algorithm is only really useful in the case where we wish to reorder the elements within an array as, for example, in the case where we wish to map from the alternate to the conventional spheres. 6.4.7.2 Three-point formula (linear) The three-point linear formula is f (x0 + ph, y0 + qk) = (1 − p − q) f0,0 + pf1,0 + qf0,1

(6.34)

210 Principles of planar near-field antenna measurements where x − x0 h y − y0 q= k

(6.35)

p=

(6.36)

where 0 ≤ p ≤ 1 and 0 ≤ q ≤ 1, and   xmin − x +1 x0 = fix h   ymin − y y0 = fix +1 k

(6.37) (6.38)

Here, the operator fix will round the argument   towards the nearest integer towards zero. With an uncertainty described by O h2 . This expression approximates the underlying function with a triangular plane that passes through each of the three local points. One potential advantage of this scheme is that the data need not be sampled on an equally spaced grid. In practice this expression offers a less satisfactory approximation with no significant improvement in computational efficiency. Clearly then, unless the data is tabulated over an irregular grid, this algorithm is unsatisfactory for the case when one is considering EM fields and it is suggested that this scheme should only be used as a last resort. 6.4.7.3 Bi-linear This interpolation algorithm is included for the case where the phase function is sufficiently pathological that the degradation in those results obtained from the bicubic-convolution algorithm becomes intolerable. This algorithm uses bi-linear interpolation to find the values of the underlying two-dimensional function at the desired points from a set of tabulated values of the function at regular grid points. Bilinear interpolation is of great utility when working with more pathological functions than those assumed here. This is defined as f (x0 + ph, y0 + qk) = (1 − p) (1 − q) f0,0 + p (1 − q) f1,0 + q (1 − p) f0,1

(6.39)

+ pqf1,1   Here, the uncertainty is of the order of O h2 . 6.4.7.4 Six-point formula (bicubic) This is an alternative bicubic interpolation formula [3] to the cubic-convolution formula described above. As it relies upon fewer points than the cubic-convolution alternative it is perhaps easier to implement sucessfully in the case where the phase

Probe pattern characterisation 211 function is changing rapidly. This is defined as q (q − 1) f0,−1 2 p (p − 1) + f−1,0 2   + 1 + pq − p2 − q2 f0,0 (6.40)   p (p − 2q + 1) f1,0 + 2   q (q − 2p + 1) + f0,1 + pqf1,1 2   with an uncertainty described by O h3 . In practice, this expression is relatively reliable and is computationally efficient to evaluate. Here however, the data must be sampled on an equally spaced and monotonic grid. Unfortunately, for data that is approaching the lower limit of the sampling criteria it can yield results that clearly exhibit discontinuities in their first derivatives, which could result from the asymmetrical nature of the expression. Such difficulties are not encountered when using the bicubic-convolution interpolation scheme. f (x0 + ph, y0 + qk) =

6.4.7.5 Phase interpolation Essentially all of the phase interpolation algorithms above rely upon a ‘path following’ unwrapping algorithm. For the simplest case, this reduces to merely wrapping all of the phase values into the range centred about the value of the nearest neighbour sample. Although unwrapping phase in two, or more, dimensions remains an unsolved problem, provided the patterns are well sampled and not too pathological, a localized wrapping of phase values into a given range is often found to be satisfactory. This section presents a simple generic algorithm for wrapping a phase value into a given range centred about a single phase value as required by the phase interpolation algorithms described above. 1 xmin = xc − p 2 xmax = xmin + p

(6.42)

 x = x − xc

(6.43)

(6.41)

Let

So that



|x | xclosest = x − psign (x ) int p

 (6.44)

212 Principles of planar near-field antenna measurements Here sign is the sign function so that the returned value is equal to 1 if the argument is greater than or equal to zero or −1 otherwise, that is,  1 arg ≥ 0 sign = (6.45) −1 otherwise and the function int returns the whole number closest to the argument. As an example, if the angle −181◦ is to be shifted into a range of 360◦ , centred about the angle 0◦ , then the closest angle will be 179◦ .

6.4.8 True far-field probe pattern As described above, the quasi far-field vector pattern function is assumed to have been characterised using a spherical measurement system where the radius of the imaginary sphere over which the measurements are taken while finite has sufficiently large radius such that the quasi far-field pattern could be recovered, that is, for the case where 2D2 r≥ when r D and r λ (6.46) λ Here, r is the distance from the antenna, D is the largest overall dimension where D > λ and λ is the wavelength. However, the true far-field can be obtained by utilizing a spherical mode expansion to calculate the asymptotic far-field pattern of the probe. It is not the purpose of this text to present a detailed treatment of this technique, which can be found presented in Reference 1. Instead the discussion herein will be limited to an overview of the practicalities of this process. The probe characterisation as described above has been designed in such a way that the measured data is immediately in a form that is amenable for processing using spherical near-field to far-field transform. Several spherical transforms are available commercially and any of these would be suitable for accomplishing this task. The two orthogonal tangential spherical field components have been sampled on a plaid monotonic, equally spaced polar spherical grid with a point on θ = φ = 0◦ using a sample spacing that is sufficiently small to satisfy the Nyquist criteria. The input probe can be specified as being an electric Hertzian dipole, as by design of the measurement no probe pattern correction is required. Probe pattern correction was avoidable when the AUT subtended a small angle as seen from the remote source antenna so that the probe pattern remains essentially unchanged over the minimum sphere. The nearfield data was transformed to the far-field using azimuthal and polar spherical mode coefficients, N and M , respectively. Assuming an electric Hertzian dipole antenna is specified as being the output probe, then, in the far-field, the pattern will be presented as resolved onto a polar spherical polarisation basis. The output polarisation basis was immediately changed to a Ludwig III definition. If the pattern is tabulated on a regular azimuth over elevation coordinate system then a comparison can be made with the existing quasi far-field results. The results of such a comparison can be found presented in Figure 6.15. Here, dotted contours are the true far-field pattern whilst black contours are re-tabulated 12 m quasi far-field data, where the copolar contours are plotted at

Probe pattern characterisation 213 Cross-polar power freq. 10 (GHz)

80

80

60

60

40

40

20

20

El (deg)

El (deg)

Copolar power freq. 10 (GHz)

0 –20

0 –20

–40

–40

–60

–60

–80

–80 –50

0 Az (deg)

Figure 6.15

50

–50

0

50

Az (deg)

Far-field and 12 m quasi far-field copolar and cross-polar power pattern of probe

every dB from −25 dB to −1 dB and the cross-polar contours are plotted at every 5 dB from −70 dB to −5 dB. The differences observed between the two patterns is an artefact, which results from the fact that the measured pattern is not spatially band limited, that is, the function does not tend to zero at the periphery of the sampling interval. Such a discontinuity can only be represented reliably by utilizing a large number of spherical mode coefficients. However, the actual number of modes used within the transformation process is determined from the sampling theorem and equated roughly to two samples per wavelength over the surface of the minimum sphere, that is, 2◦ , in this case. Thus, the finite series representation of spherical wave functions could not accurately represent the function in the vicinity of these discontinuities. This manifested itself with an absence of signal at large angles. A detailed description of the algorithm for converting the measured probe pattern into that shown above will be given in a following section and is essentially identical to the sort of post-processing that is generally required before measured probe pattern data can be used to correct data that has been acquired using a planar system.

6.5

Finite element model of open-ended rectangular waveguide probe

A number of three-dimensional full-wave EM solvers are available from various developers that are well suited to the task of simulating electrically small antennas such as near-field probes. The advantage of these computational electromagnetic (CEM) programs is that they enable the user to customise the model to include small features that are peculiar to the probe in question and that will inevitably be omitted from generic theoretical models.

214 Principles of planar near-field antenna measurements Copolar power freq. 9 (GHz) 0 –10

Power (dB)

–20 –30 –40 –50 –60 –70

Figure 6.18

Copolar modelled Copolar measured EMPL

–80

–60

–40

–20

0 20 Az (deg)

40

60

80

Comparison of azimuth cuts of measured and modelled copolar probe patterns at 9 GHz

Far-field data can be obtained from a range of CEM codes employing either time domain or frequency domain solvers. The advantage of a time domain solver in this instance is that the probe pattern can be obtained at all frequencies from a single run of the solver. Such data can be tabulated on a regular polar spherical grid and resolved onto a polar spherical polarisation basis yielding results that are directly comparable with measured data. Figures 6.16 and 6.17 (see p. C6) illustrate the modelled data presented in the form of a polar plot where the radius and colour represents the measured field intensity and the angles correspond to the angles at which the measurement was taken. A comparison of measured and quasi far-field pattern for the cardinal cuts can be found presented in Figures 6.18 and 6.19 together with the equivalent multipath level (EMPL) for a 9 GHz case. Here, errors within the measurement and the theoretical model have been lumped together into a single error term, plotted dashed black. The slightly higher EMPL present within the elevation cut is probably attributable to the larger field intensities that illuminate the walls of the measurement facility in this plane. The very low EMPL is encouraging as this can be used to confirm both the modelling technique and the measurement process. For the case of conventional planar near-field antenna measurements, errors within the characterisation of the probe pattern can be seen to correspond to errors in the measured pattern of an AUT in the form of a one-toone mapping. Thus, planar measurements that have been taken using this probe and corrected using either the measured or modelled probe patterns can not be

Probe pattern characterisation 215 Copolar power freq. 9 (GHz) 0 –10

Power (dB)

–20 –30 –40 –50 –60 –70

Figure 6.19

Copolar modelled Copolar measured EMPL

–80

–60

–40

–20

0 20 El (deg)

40

60

80

Comparison of elevation cuts of measured and modelled copolar probe patterns at 9 GHz

expected to have a relative amplitude pattern error with an EMPL of better than −50 dB. Figures 6.20 and 6.21 contain overlaid contour plots of the copolar and crosspolar power patterns. Here, dotted contours represent the modelled data whilst black contours represent measured data. Contours have been plotted in 1 dB increments from −30 dB to 0 dB for the copolar plot whilst contours have been plotted in 5 dB increments from −70 dB to 0 dB for the cross-polar patterns. The degree of agreement can be seen to be encouraging over the entire forward hemisphere for both the copolar and cross-polarisations. However, discrepancies are clearly seen in the cardinal planes of the cross-polar patterns where the model shows the response tending to zero. This is expected as, in practice, imperfections in the construction of the probe, that is, the walls of the waveguide, will not be perfectly orthogonal and the end section will not be perfectly smooth; this will disturb the cardinal cross-polar probe pattern. Conventionally, theoretical models such as this are augmented with auxiliary polarisation correction measurements that are used to modify the response of the probe in these regions. This model can be used to predict the pattern over the frequency band; in this particular case of 8–12 GHz. Crucially, a similar degree of agreement can be obtained over the entire band. An interesting philosophical point worth expounding is although science is inherently empirical, which is preferable in this case, the theoretical prediction with its associated assumptions, approximations and numerical errors or the measurement with its uncertainties, noise and truncation? As the theoretical predictions and the empirical measurements can be in such close agreement, for example,

216 Principles of planar near-field antenna measurements Copolar power freq. 9 (GHz) 80 60

El (deg)

40 20 0 –20 –40 –60 –80 –50

Figure 6.20

0 Az (deg)

50

Comparison of measured and modelled copolar probe pattern at 9 GHz

Cross-polar power freq. 9 (GHz) 80 60 40 El (deg)

20 0

–20 –40 –60 –80 –50

Figure 6.21

0 Az (deg)

50

Comparison of measured and modelled cross-polar probe pattern at 9 GHz

Probe pattern characterisation 217 at 9 GHz and 10 GHz, the measured quasi far-field probe pattern can be employed as the probe pattern correction as it automatically contains the polarisation correction data. However, no significance can be placed on relative amplitude pattern for signals that are 50 dB below the peak of the pattern.

6.6

Probe displacement correction

Probe displacement correction is a correction that sometimes has to be applied to correct for errors introduced by poorly designed near-field measurement systems. This is principally a difficulty associated with the use of a singularly polarised nearfield probe and a polarisation positioner. The 90◦ rotation in χ of the near-field probe that is required to sample two tangential near-field components can introduce an undesirable linear phase taper across the equivalent far-field vector pattern function. Such an error arises from the probe being displaced within the acquisition plane when rotated about a point other than the mechanical datum, that is, the geometric centre of the waveguide section. This error can be resolved easily by applying a differential phase change to the far-field pattern function of the measurements taken, using the rotated probe and follows directly from the shifting property of the Fourier transform. This correction process is clearly not required when measurements are made using an orthogonal mode transducer (OMT). This error is often overlooked if predominantly high-gain linearly polarised antennas are being measured and only near-in or cardinal planes are examined. However, its effects will become significant away from the cardinal planes and may become obvious when measuring low-gain or especially circularly polarised antennas.

6.7

Channel-balance correction

Thus far, only the case of singularly polarised probes has been considered. However, two methods exist for characterising two orthogonal field components over a planar surface. These are 1. To measure both field components simultaneously with an OMT, that is, to use a dual polarised probe fed with a two-channel receiver or switched via a positive– intrinsic–negative junction diode (PIN diode) switch. 2. To measure the field components separately. Here, a singularly polarised probe is used to sample the first field component. This probe is then rotated by 90◦ about the range z-axis before the second field component is sampled. Channel-balance correction is used to compensate for any imbalance that may exist between the two ports in the OMT. Generally, this is of particular relevance when characterising circularly polarised instruments where comparable field intensities are sampled at each port. However, it has been found to be of considerable importance when characterising an instrument when the AUT and probe are not perfectly polarisation matched. This process is not required when measurements are taken using a singularly polarised, that is, a single-port probe.

218 Principles of planar near-field antenna measurements If a signal is presented to port1 of an OMT and the resulting response noted, then if the same signal is presented to port2 and again the response is noted the correction of the channel imbalance, in rectangular form, can be expressed as Eport1 = Eport1

(6.47)

 EPort11 measurement Eport2 = Eport2 (6.48) EPort12 measurement    Here, the quantity EPort11 measurement EPort12 measurement is referred to as the channel balance. Generally, this parameter is derived within the auxiliary measurement of the probe and its magnitude can be obtained from the boresight phi cut. Often, the effects of channel imbalance render OMT probes unsuitable for the characterisation of very low cross-polar antennas; however, it is often advantageous to use OMTs as the side port can be loaded so that the cross-polar field is sunk in the port rather than being reflected and adding to the multiple reflections between the AUT and the probe. 

6.8

References

1 Hansen, J.E. (ed.): Spherical Near-field Antenna Measurements (Peter Peregrinus Ltd., London, UK, 1988), p. 189 2 Keys, R.G.: ‘Cubic convolution interpolation for digital image processing’, IEEE Transactions on Acoustics, Speech and Signal Processing, December 1981;29 (6): 1153–60 3 Abramowitz, M., and Stegun, I.: Handbook of Mathematical Functions with formulas, Graphs and Mathematical Tables, 9th Printing (Dover, 1972), p. 882.

Chapter 7

Computational electromagnetic model of a planar near-field measurement process

7.1

Introduction

The development of general-purpose tools for the simulation of near-field antenna measurements is of interest for several reasons. It would enable an engineer to do the following: 1. Plan and optimise a measurement campaign before committing valuable facility time. 2. Assess error terms within the facility error budget. 3. Verify correction algorithms, for example, probe pattern correction and position correction, which can constitute crucial constituents of any transformation algorithm. However, in general, it is difficult to obtain closed form and functional solutions for the electromagnetic (EM) field at an arbitrary point in space from knowledge of the tangential electric or tangential magnetic fields over a closed surface for anything but the simplest of configurations. This is especially true when the closed surface, is not coincident with the aperture of the radiating structure, as is the case for nearfield antenna measurements. Matters are further complicated if the sampling surface is not a canonical surface (for example, sphere, cylinder or plane) as is often the case when simulating data that includes imperfections in the measurement system. As such, recourse to alternative methods, typically numerical techniques, becomes unavoidable. Any near-field measurement can essentially be simulated by evaluating the complex coupling coefficient between the antenna under test (AUT) and the near-field probe. This must be done at each point over the simulated acquisition surface for each sampled polarisation and frequency at which the measurement is to be taken. In principle then, it would be possible to obtain the mutual coupling coefficient (S21 ) between a given mode in the waveguide port on the AUT and a given mode in the

220 Principles of planar near-field antenna measurements waveguide port on the scanning probe from a three-dimensional EM full wave solver for each of the positions at which near-field samples are to be recorded. This approach would have the advantage of potentially introducing the least number of assumptions and approximations and therefore could, in theory, yield the most accurate predications. Unfortunately, although many solvers are available employing say, the finite difference time domain (FDTD) method, the finite element methods (FEMs), the method of moment (MoM), and so on, they are generally inappropriate for simulating problem spaces as electrically large as those needed to enclose a complete near-field measurement system, including chamber, absorber, cranes, and so forth. This is a direct consequence of the extended processing times and the amount of computer resources required. Hence, alternative, less general, but more efficient, techniques are often required. The remainder of this chapter is devoted to the development of a variety of simulation techniques that become successively more sophisticated as their ability to mimic near-field measurement facilities improves. However, this chapter is not intended to be an exhaustive survey of all the possible simulation techniques available, instead the authors have concentrated on a small selection of those methods that extend the near-field method and offer the greatest utility whilst requiring the minimum of computational and intellectual effort.

7.2

Method of sub-apertures

Perhaps the simplest technique for the simulation of near-field measurements is to employ the highly efficient method of sub-apertures. Here, the radiating structure is divided into an assembly of isolated small individual radiators (the sub-apertures). The field at a particular point in space is then formed from the linear superposition of the each of the far-field patterns of these small radiators with due regard to phase. Here, an implicit assumption is made that the field point is in the far-field of each of the individual radiators. The validity of this assumption is clearly dependent upon how large the sub-aperture is and how far removed the field point is. The sub-apertures can be considered to represent distinct discrete antennas, for example, radiating elements within an array or as small sections of a larger single radiating structure, for example, elemental pieces of a waveguide aperture. A simple example of this, and one that is easily imagined, is where each of the individual radiating elements within an array antenna is taken to be a sub-aperture. In principle, it is possible to take a different far-field pattern for each of the individual sub-apertures so that the effects of mutual coupling can be accounted for; however, in practice this is seldom done as the differences can be small and the computational effort and resources required can be considerable. Provided that the field point is in the far-field of the individual radiating elements, the sub-aperture method will provide reliable predictions, even for the case of interest here, where the field point is simultaneously placed in the near-field of the complete array. The field at a point in space can be obtained from the linear superposition of the fields radiated by each of the finitely many sub-apertures

Computational electromagnetic model of a planar near-field 221 thus E (θ , φ, r) =

N
n=1

E n (θ, φ, r)

(7.1)

Here, the subscript, n, is included to account for the case where different patterns are used for each radiating element. Now, provided that the field point is in the far-field of the sub-aperture then this can be expressed as E (θ , φ, r) =

N
n=1

e−jk0 r E n (θ, φ) r



(7.2)

Here, r  denotes the magnitude of the separation between the sub-aperture and the field point. As the field point is in the far-field of the sub-aperture we may assume that all parts of the sub-aperture are equally far removed from the field point. Thus, for convenience, r  can be assumed to be the displacement between the field point and the phase centre, that is, reference of the sub-aperture. If all of the sub-apertures are assumed to be identical, then the subscript on the far-field element pattern can be suppressed. However, to retain generality we shall retain it. The element pattern angle can be easily determined from geometry as r  x = mx − r0x

(7.3)

r  y = my − r0y

(7.4)

r  z = mz − r0z

(7.5)

    2 r  = m − r 0  = (r  x )2 + r  y + (r  z )2

(7.6)

Thus

The spherical angles that represent the direction of the vector linking the field point with the sub-aperture can be obtained from  θ = arccos 

rz r

ry φ = arctan  rx

 (7.7)  (7.8)

The vector pattern function E (θ, φ) can either be determined analytically or from measurements where the electric field is recorded in a tabulated form. Unless an analytical function is used to represent the pattern of the sub-aperture, the choice of spherical angles will likely be determined by the coordinate system that was used to tabulate the far-field pattern. Other spherical angles can be readily obtained from these direction

222 Principles of planar near-field antenna measurements cosines. An approximate value for the magnitude and phase of the field at a specific angle can be obtained by using approximation, for example, piecewise polynomial interpolation. As the sub-aperture is necessarily small the corresponding far-field pattern will be a slowly varying function, as the small effective area corresponds to a low directivity. This significantly eases the demands being placed on the chosen interpolation scheme and minimises the likelihood that wildly unreliable predictions will be obtained. Also, as we have knowledge of the complete electric field, that is, all of its field components, we are free to resolve the field onto any polarisation basis we choose. This enables simulations to be produced that have polarisation bases and imperfections if so desired that are commensurate with the desired measurement system. This method can be readily used to simulate near-field measurements taken on planar, cylindrical or spherical configurations. Furthermore, as we are essentially free to place the field point wherever we like with only very few restrictions; this technique can easily be used to simulate positional inaccuracies within a measurement system. Knowledge of the magnetic field can also be obtained easily as in the far-field the electromagnetic wave is transverse. Thus the electric and magnetic field are related through H (θ , φ) =

1 rˆ × E (θ , φ) Z0

(7.9)

Here, Z0 denotes the impedance of free space. The mutual tangential relationship between the direction of propagation, the electric and magnetic fields is only valid in the far-field. Thus, we are only able to utilize this relationship whilst we are considering the sub-apertures. Clearly then, the total magnetic field at the desired field point can be obtained from knowledge of the electric field by summing over each of the individual sub-apertures H (θ , φ, r) =

 N e−jk0 r 1
rˆ × E n (θ , φ) r Z0

(7.10)

n=1

Here rˆ =

r = sin θ cos φ eˆ x + sin θ sin φ eˆ y + cos θ eˆ z |r  |

(7.11)

As essentially we are summing, that is, in the limit ‘integrating’, over a finite number of finite apertures this method is probably one of the most efficient and flexible methods discussed within this chapter. Unlike some of the methods discussed below which obtain their efficiency from utilizing the fast Fourier transform, this technique can be readily utilized to simulate measurements taken over arbitrarily shaped surfaces. Within this formulation, and in common with several of the simpler implementations detailed below, we have failed to take account of the influence of the measuring probe as we have assumed here that the field is sampled with an infinitesimal Hertzian dipole.

Computational electromagnetic model of a planar near-field 223

7.3

Aperture set in an infinite perfectly conducting ground plane

Another of the simple methods of producing a near-field simulation is effectively to use the method of microwave holographic metrology (MHM) to derive a near-field measurement from an aperture illumination function. This aperture illumination function being initially derived either from measurement or, with the necessary constraints imposed, from an analytical model. The simplest possible aperture illumination function is the uniformly illuminated, singularly polarised, for example, purely y-polarised, rectangular aperture cut from an infinitely thin perfectly conducting sheet that is coincident with the z = 0 plane and centred at the origin of the coordinate system. As an aside, whilst we are free to impose the condition that the x-polarised electric field is identically zero everywhere we are not simultaneously free to set the z-polarised field to zero and have not assumed that this field component is zero. This is because we have made no assumptions about the form of the magnetic field which is of importance as the tangential components of the magnetic field are not shorted out, that is, identically zero, over the surface of a perfect electric conductor (PEC). As the sheet is perfectly conducting, the tangential components of the electric field outside the aperture will be exactly zero and the boundary conditions may be expressed as  b a E0 when, |x| ≤ and |y| ≤ (7.12) fy (x, y) = 2 2 0 elsewhere Here, a and b are used to denote the width and height of the rectangular aperture respectively. The corresponding two-dimensional y-polarised angular spectra can be obtained by evaluating the following finite integral 



b/ 2 a/ 2

Fy kx , ky =

E0 ej(kx x+ky y) dx dy

(7.13)

−b/ 2 −a/ 2

As the aperture illumination function is separable in form it can be evaluated simply as 



a/ 2

Fy kx , ky = E0

e

jkx x

b/ 2 dx

ejky y dy

−a/ 2

−b/ 2 a/ 2

b/ 2 ejkx x ejky y = E0 jkx jky −a/ 2 −b/ 2  

2 λ π bv πau = E0 2 sin sin λ λ π uv   πau π bv = E0 ab sinc sinc λ λ



(7.14)

224 Principles of planar near-field antenna measurements The sinc function has been used to condense the expressions and is understood to have the following properties  1 when θ = 0 sinc (θ ) = (7.15) sin(θ) elsewhere θ The sinc function can also be defined in other ways, see Box 7.1.

Box 7.1 Occasionally, some authors choose to include a factor π within the argument of the sin function and on the denominator. As this offers no advantage here this convention has not been adopted. However, the reader is cautioned that when comparing texts that differences between expressions can arise from the different functional definitions.

Hence, the far-field electric field can be expressed as a function of the direction cosines u and v as   π au π bv E0 ab  eˆ y sinc 1 − u2 − v2 sinc (7.16) Ey (ru, rv, rw) eˆ y ≈ j λ λ λ Here u and v are the direction cosines in the x and y-axes respectively. Adhering to convention, the r −1 term and the unimportant phase factor e−jkr have been divided out of the far-field pattern. With reference to the sub-aperture method discussed above, in the case where a sub-aperture forms a small rectangular section of another larger aperture, for example, a section of a waveguide aperture, then this equation can be used as the far-field pattern of the sub-aperture. Indeed, by including an amplitude and phase factor, to denote the change in amplitude and phase over the complete aperture, a very useful analytical pattern function is obtained thus     πa sin θ cos φ π b sin θ sin φ En ab cos θ sinc sinc eˆ y E (θ , φ, r) eˆ y ≈ j λ λ λ (7.17) Here, En has been used to denote the excitation of the nth sub-aperture. The z-polarised field component can be obtained from the plane wave condition, that is, the scalar product of the field vector and the propagation vector is zero. Thus   Ex (θ , φ, r) sin θ cos φ + Ey (θ, φ, r) sin θ sin φ eˆ z (7.18) Ez (θ , φ, r) eˆ z = − cos θ However, as the x-polarised field is zero everywhere     πa sin θ cos φ πb sin θ sin φ En ab sin θ sin φ sinc sinc eˆ z Ez (θ, φ, r) eˆ z ≈ −j λ λ λ (7.19)

Computational electromagnetic model of a planar near-field 225 Thus the total EM six-vector can be determined over the complete forward half space. As the aperture is set in an infinite perfectly conducting ground plane, the field in the back half space is assumed to be identically zero everywhere. As was described in Chapter 4, the angular spectra of an identical radiator that is merely displaced in space can be determined with the use of a simple differential phase change. Thus, if the radiator is displaced by an amount d in the z-direction then the angular spectra of the displaced aperture can be expressed as     F kx , ky , z = d = F kx , ky , z = 0 e−jkz d (7.20) A similar expression exists for the magnetic fields. The near-field simulation can be obtained by taking the inverse Fourier transform of the angular spectrum to recover the near-fields; hence 1 E (x, y, z = d) = 4π 2

∞ ∞

  F kx , ky , z = d e−j(kx x+ky y) dkx dky

(7.21)

−∞ −∞

Here, E (x, y, z = d) represents the field resulting from the uniform aperture illumination function over the surface of a plane separated from the aperture plane by an amount z = d. In practice, it is often convenient to limit the region of integration to visible space, as only the propagating portion of the near-field is required. Although this results in a failure to recover the reactive portion of the near electric field, actual near-field measurements are very seldom deliberately taken in the presence of evanescent fields. Alternatively, using the  notation this can be simplified as follows     E t (x, y, z = d) = −1  E t (x, y, z = 0) e−jkz d (7.22) Here, E t denotes both orthogonal and tangential components of the electric field. As usual and with the proper precautions, the Fourier transform and its inverse can be evaluated efficiently with use of the fast Fourier transform. Although any aperture illumination function can be used to produce a simulation, by choosing a simple canonical case such as the case of the rectangular uniformly illuminated aperture, subtle errors that may otherwise have gone unnoticed in the processed far-field patterns can be highlighted. Again, as was the case for the sub-aperture method, probe pattern effects have been omitted from this analysis. In general, this need not be the case and the probe pattern correction expressions can be inverted so that the effects of the probe can be included. This significantly increases the complexity of this simulation method and can easily be seen to be rather incestuous. However, for the sake of completeness this method is described below.

7.3.1 Plane wave spectrum antenna–antenna coupling formula The simulation techniques described above enable simple near-field simulations to be constructed from knowledge of the field radiated by a single antenna. However, for truly useful simulations to be produced artefacts in the measured data sets arising from the scanning near-field probe should be included. Actually, all antenna problems

226 Principles of planar near-field antenna measurements consider the coupling between two antennas. This follows directly from the fact that EM fields, that is, photons, cannot exist in isolation. Instead, these ‘fields’ are merely the manifestation of the coupling between distributions of charged particles, for example, electrons. The modelling techniques, set out above, model the coupling between the test antenna and a single electron performing simple harmonic motion, in a given direction, about a single mean position, which is otherwise described as an infinitesimal Hertzian dipole antenna and represents the simplest possible arrangement. More specifically, the force acting on the dipole can be taken to be the definition of an electric field (2.5) in Chapter 2, that is, the coupling between some radiator that is an ensemble of electrons and an electron that is constrained to traverse a given path, that is, a Hertzian dipole orientated in a particular direction. This idea of a vibrating electron that is constrained to move with only a very limited degree of freedom, that is, one dimension, is essentially the basis for the concept of polarisation. This is why the probe pattern correction procedure deconvolves the response of the scanning near-field probe and then convolves in the pattern of an ideal Hertzian dipole antenna. If the Hertzian dipole pattern were not included the measurements would be of indeterminate polarisation and in disagreement with any theoretical prediction with which they would be compared. In summary then all antenna measurement simulations set out to determine the amount of coupling between a current source, that is, the AUT and the scanning probe, for example, a Hertzian dipole antenna or waveguide probe, at a series of locations across a predetermined sampling surface for two or more orientations, that is, polarisations. The simplest approach is to combine the plane wave spectrum (PWS) method for propagating fields from one plane to another with an inversion of the PWS-based probe pattern correction algorithm. As shown in Chapter 4, if the electric field is known over a planar surface it can be propagated to a parallel planar surface by mean of the application of a differential phase change in the spectral domain. As can be surmised from (7.22), the electric field component over the displaced scan plane can be expressed mathematically as     E (x, y, z = d) = −1  E (x, y, z = 0) e−jkz d (7.23) Here,  is taken to denote the Fourier operator and −1 its inverse. For example, with the time dependency suppressed, a two-dimensional Fourier transform of the field function can be defined by the integral 







∞ ∞

F kx , ky =  E (x, y) =

E (x, y) ej(kx x+ky y) dx dy

(7.24)

−∞ −∞

Similarly, the inverse can be expressed as −1

E (x, y) = 

   1 F kx , ky = 4π 2

∞ ∞ −∞ −∞

  F kx , ky e−j(kx x+ky y) dkx dky (7.25)

Computational electromagnetic model of a planar near-field 227 Here, E and F are vector analytic functions and the limits of integration collapse to the region of the partial scan plane for the spatial field function and to the region of visible space, that is, where k02 > kx2 + ky2 , for the spectral field function. If the expressions for the plane-to-plane transform that were developed in the preceding sections are used to propagate the tangential components of the electric field and provided that the angular spectrum of the sampling near-field probe are known, then the antenna-to-antenna coupling formula     j PyB (−α, β) PxB (−α, β) Sx = Sy λ PyC (−β, −α) −PxC (−β, −α)       e −jk(αx0 +βy0 +γ z0 ) 1 1 − β2 Ax (α, β) αβ   × · · Ay (α, β) −αβ − 1 − α2 γ γ (7.26) can be used to incorporate probe pattern effects. Here the nomenclature of Chapter 5 has been maintained. When this is incorporated into the plane-to-plane transform expressions the x- and y-polarised measured field can be obtained from      PyB (−α, β) PxB (−α, β) sx (x, y, z = d) −1 j = sy (x, y, z = d) λ PyC (−β, −α) −PxC (−β, −α)     1 αβ  1 − β2  × −αβ − 1 − α2 γ    e −jkz d  {Ex (x, y, z = 0)} × (7.27) ×  Ey (x, y, z = 0) γ Here, sx and sy are the measured, that is, probe coupled, near-field data. Unfortunately, whilst this method is efficient, as all of the transforms can be accomplished by means of the fast Fourier transform, it is both limited to a planar geometry and is somewhat incestuous; using as it does an inversion of the MHM and probe-correction formula. A general-purpose method for computing simulations of near-field measurements that include the effects of the measuring probe is presented in the final section of this chapter. The utility of this more advanced and significantly more complex simulation methodology stems from noting that it does not rely upon either the angular spectrum method or the probe pattern correction expressions used till this point in the text.

7.4

Vector Huygens’ method

The angular spectrum method that is employed above to produce an initial very useful, but somewhat limited, near-field simulation can be deployed in a very similar fashion to produce a far more effective near-field simulation tool. As was shown above in Chapter 4 the vector Huygens’ method is essentially an alternative deployment of the angular spectrum method that, for the purposes of simulating near-field measurements, is perhaps a more useful one. The far-field form of Huygens’ principle can be obtained directly from the coordinate free form of the near-field to far-field transformation of Chapter 4 by

228 Principles of planar near-field antenna measurements reducing the area of the aperture plane until in the limit, where it constitutes a single elemental, that is, infinitesimal Huygens’ source. From Chapter 4, the general coordinate free near-field to far-field transform can be expressed as      −jkr   e uˆ r  ×  ET rT ejk0 uˆ r ·r  da  × nˆ  (7.28) E (x, y, z) ≈ j λr  S

Now, if the size of the radiating aperture is made infinitesimally small, two convenient things happen. First, the observer will necessarily be situated in the far-field of the infinitesimal radiating aperture provided only that they are a displaced by any finite amount, that is, by a couple of wavelengths or so; and second, as the aperture is an infinitesimally small distance across, it is reasonable to assume that the electric field is constant over the surface of the aperture. Thus, in the limit when the aperture is reduced to a Huygens’ element, that is, when da = lim {S} and the observer is in S→0

the far-field, that is, more than a few wavelengths away from the element 

E (x, y, z) ≈ j

    e−jkr  uˆ r  × ET rT da × nˆ  λr

(7.29)

Now, this logic can be easily extended to consider apertures of finite extent. If the finite aperture is divided up into infinitesimal Huygens’ sources then the total field can be obtained by integrating the contribution over each of these radiators. Specifically, the field at a point in space resulting from an infinitesimal Huygens’ elemental can be expressed as 

    e−jkr  dE (x, y, z) ≈ j uˆ r  × ET rT da × nˆ λr  So that the total field at a point in space can be expressed as      e−jkr j E (x, y, z) ≈ uˆ r  × ET rT × nˆ da λ r 

(7.30)

(7.31)

S

where S denotes the finitely large aperture. The magnetic field can be obtained from       e−jkr j H (x, y, z) ≈ uˆ r  × uˆ r  × ET rT × nˆ da (7.32) λZ0 r  S

Usefully, as the surface unit normal varies over the aperture S with aperture electric and aperture magnetic fields these expressions can be used to calculate the electric and magnetic fields at the observation point (x, y, z) from smooth non-planar apertures. Furthermore, provided that the observation point is displaced by more than a few wavelengths from the radiating aperture, the field points can be distributed arbitrarily throughout space. Thus, this field propagation algorithm can be used to produce near-field simulations over any desired sampling surface, that is, plane rectilinear, plane polar, plane bipolar, cylindrical, polar spherical, equatorial spherical, and so

Computational electromagnetic model of a planar near-field 229 forth. Practically, the elemental far-field requirement manifests itself within these expressions with the inverse r term that will become large as r becomes small and in the limit will tend to infinity as r tends to zero. This aside, provided r is of the order of a couple of wavelengths these expressions can be used to produce reliable pattern predictions. As the derivation of these relationships requires the observation point to be placed in the far-field of the Huygens’ element, neither of these expressions can be used to model the behaviour of the fields in the reactive near-field where evanescent fields become significant. Again, this is unimportant in this area of application as near-field antenna measurements are only made outside this region of space. Here, it is important to stress that the coordinate free form of the angular spectrum representation of an EM field is the most general representation, as it contains information pertaining to both propagating and evanescent fields. Furthermore, as the Huygens’ principle essentially constitutes a special case of the angular spectrum approach, the same assumptions and limitations bind it; in other words, any deficiency within the Fourier approach will also represent a deficiency with this implementation of Huygens’ method. Here, r  is the distance between element and field point, λ is the wavelength, k0 is the wave number, ur  is the unit vector from element to field point, n is the unit normal vector at the element and E is the total, or tangential, aperture electric field at the element. This statement of Huygens’ formula is rigorous, provided that the observation point is more than a few wavelengths from the elemental source (that is, in the far-field of the elemental source) and that the scalar product of n and ur  is positive. The geometry for this statement of the Huygens’ formula can be found illustrated in its conventional form in Section 4.5, Chapter 4.

7.5

Kirchhoff–Huygens’ method

The Kirchhoff–Huygens’ principle is a powerful technique for determining the field in a source free region outside a surface from knowledge of the field distributed over that surface and is in effect a direct integration of Maxwell’s equations. It is applicable to arbitrary shaped apertures over which both the electric and magnetic fields are prescribed. The Kirchhoff–Huygens’ principle has the important benefit that the field point will contain both the propagating and reactive fields. Unfortunately, this added rigour is sought at the expense of an increase in the requisite computational effort and the added requirement that the magnetic field must be known over the radiating surface in addition to the electric field as was the case previously. When expressed mathematically, the electric field, at a point P, radiated by a closed Huygens’ surface S [1] is 1 Ep = 4π

S



      −jωµ n × H ψ + n × E × ∇ψ + n · E ∇ψ da

(7.33)

230 Principles of planar near-field antenna measurements Here, ψ represents the following spherical function ψ= Now

e−jk0 r r



(7.34)

  r  = r − r0 

(7.35)

Thus ∇0 ψ =

∂ψ ∂ψ ∂ψ eˆ + eˆ + eˆ ∂x0 x0 ∂y0 y0 ∂z0 z0

(7.36)

By multiplying the numerator and denominator by ∂r  yields ∇0 ψ =

∂ψ ∂r  ∂ψ ∂r  ∂ψ ∂r  e ˆ + e ˆ + eˆ x y ∂r  ∂x0 0 ∂r  ∂y0 0 ∂r  ∂z0 z0

Now ∂ ∂ψ =   ∂r ∂r Also ∂ ∂r  = ∂x0 ∂x0

!

e−jk0 r r



"





jk0 e−jk0 r e−jk0 r = − − r r2

(7.37)





  1 = −jk0 −  ψ r



1  x − x0 2 2 2 2 =−  (x − x0 ) + (y − y0 ) + (z − z0 ) r

(7.38)

(7.39)

Similarly y − y0 ∂r  =− ∂y0 r

(7.40)

z − z0 ∂r  =− ∂z0 r

(7.41)

Combining these results yields     x − x0 1 y − y0 1 −jk ψ e ˆ −jk ψ eˆ y0 − − − ∇0 ψ = − 0 0 x0 r r r r   z − z0 1 −jk0 −  ψ eˆ z0 − r r

(7.42)

Thus

     x − x0 1 y − y0 z − z0 1 r ψ = jk ∇0 ψ = jk0 +  e ˆ + e ˆ + e ˆ + ψ 0 x0 y0 z0 r r r r r r (7.43)

or



1 ∇0 ψ = jk0 +  r



rˆ  ψ

(7.44)

Computational electromagnetic model of a planar near-field 231 Hence, the general vector Kirchhoff–Huygens’ formula becomes        1 1  −jωµ n × H ψ + n × E × jk0 +  rˆ ψ Ep = 4π r S

     1 + n · E jk0 +  rˆ  ψ da r

(7.45)

Upon factorising we obtain the required result  −jk0 r           e 1 1 da −jωµ n × H + n × E × rˆ  + n · E rˆ  jk0 +  Ep = r r 4π S

Since 1 Hp = 4π Then similarly 1 Hp = 4π

(7.46)





      jωε n × E ψ + n × H × ∇ψ + n · H ∇ψ da

(7.47)

S

 S











1 jωε n × E ψ + n × H × jk0 +  r

     1 + n · H jk0 +  rˆ  ψ da r



rˆ  ψ (7.48)

As before, factorising yields the required result  −jk0 r           e 1 1 jωε n × E + n × H × rˆ  + n · H rˆ  jk0 +  Hp = da 4π r r S

(7.49)

This expression yields the field at a point in space outside the radiating Huygens’ surface from an integral of the electric and magnetic fields over the closed surface S and da is an elemental area of S. The geometry for this statement of the Kirchhoff–Huygens’ formula can be found illustrated in its conventional form in Figure 7.1. The Kirchhoff–Huygens’ theory is exact, provided that the field is known exactly over an entire closed surface. The closed surface can take the form of an infinite plane together with an infinite radius hemisphere. If the source is finite then, from the radiation condition, it can be seen that no contribution to the total field arises from any part of the hemispherical portion of the surface. For these equations to be implemented numerically, we need to have an effective way of calculating the unit normal and elemental area for a general surface; Box 7.2 contains a simple method for evaluating these quantities from knowledge of the surface profile.

232 Principles of planar near-field antenna measurements P

r S

r9

u O

r0 n

Figure 7.1

Coordinate system for Kirchhoff–Huygens’ formula

Box 7.2: Calculation of surface normal and elemental area Let the surface over which the electric and magnetic fields are known be expressed as g (x, y, z) = 0

(7.50)

The outward pointing surface normal can be formed from the cross product of two nonparallel tangent vectors. Thus, provided that the Cartesian coordinates are tabulated such that x = f (u, v)

(7.51)

y = f (u, v)

(7.52)

z = f (u, v)

(7.53)

Then two tangential vectors a and b can be formed from dx dy dz ˆi + ˆj + kˆ du du du dx dy dz ˆ b = ˆi + ˆj + k dv dv dv Hence the surface unit normal can be obtained from a×b  nˆ =  a × b a=

(7.54) (7.55)

(7.56)

The inward-pointing unit surface normal will be antiparallel with this vector. Finally, the elemental area can be found by evaluating   da = a × b (7.57)

Computational electromagnetic model of a planar near-field 233 Again, as with the vector Huygens’ formula, this field propagation algorithm can be used to produce near-field simulations over any desired sampling surface. Usefully, the requirement for the field point to be displaced from the Huygens’ surface by a few wavelengths is removed. In contrast then, these expressions yield reliable predictions within the reactive near-field region. However, the inverse r term is still present and although these expressions yield reliable fields for smaller ranges than is the case for the vector Huygens’ formula, it will still cause these formulas to become unstable for very small values of r. This is an unfortunate consequence of the fact that electromagnetic field theory is essentially a classical theory and as such ignores quantum effects. If r becomes very small then the field point and the source are essentially separated by distances that approach the atomic scale and instead a quantum-mechanical formulation should be adopted. Thus, whilst there exist algorithms that seek to resolve the singularity encountered when r is exactly zero [2] no physical theory is known to yield valid unambiguous results on such subnuclear scale. The utility of these expressions when it comes to the simulation of near-field measurements is that they permit the field everywhere in space to be computed from knowledge of the fields over a finite closed surface. This enables full wave threedimensional computational electromagnetic (CEM) simulation tools to be used to solve for the fields around some, comparatively small tractable and radiating structure whereupon the Kirchhoff–Huygens’ method can be used to calculate the fields resulting from this radiator over a somewhat larger surface. Such simulation techniques can include the FDTD method, the FEM or the MoMs. In this way, measurement simulations of great accuracy can be produced comparatively simply and easily using essentially rigorous, but computationally intensive, solvers.

7.6

Generalized technique for the simulation of near-field antenna measurements

The following simulation technique essentially entails a MoM solution of a two antenna-coupling problem. Essentially, this only differs from a standard MoM solution by the choice of large, complex sources and the necessary use of generalized field propagation formula. The advantage of a MoM-type solution over, say, FDTD or FEMs is that with these techniques, the space between the antennas, even if it is a vacuum, must be considered (that is, meshed.) Clearly, this can become prohibitive for the case where the radiators are separated by any finite distance. In contrast, a MoM approach only considers the radiators so the separation between the antennas is unconnected to the amount of computational effort required in obtaining the solution. Provided that the antennas in conjunction with the circuits in which they are placed, including the source and load are reciprocal, then the mutual coupling between two antennas can be found from knowledge of the fields radiated by these antennas in isolation and from the reaction theorem [3]. For the sake of clarity, a detailed treatment of this subject is reserved for Section 7.8. If the fields radiated by an antenna are known over a convenient, arbitrary, closed surface that surrounds the antenna,

234 Principles of planar near-field antenna measurements Fields known over this surface (1)

Fields to be evaluated over this surface (2)

Tx origin (x0, y0, z0)

Figure 7.2

origin (x1, y1, z1)

Geometry of Kirchhoff–Huygens’ reaction theorem

then the field radiated by this antenna at any point in the region of space outside this surface can be obtained from the Kirchhoff–Huygens’ principle. As illustrated schematically in Figure 7.2, if the Kirchhoff–Huygens’ formula is used to obtain the fields radiated by antenna 1 over a closed surface that surrounds antenna 2, provided that the fields radiated by antenna 2 are also known over this surface; then the surface integral form of the reaction theorem can be used to calculate the mutual impedance between these antennas. The mutual admittance between the antennas can then be found from the mutual impedance. These admittances can be used to populate an admittance matrix from which the equivalent normalized scattering matrix can be easily obtained. The transmission scattering coefficient, when evaluated with the two antennas suitably displaced, can be recognized as constituting a single sampling node within a near-field measurement. By repeating this calculation for every point in the near-field measurement, a full acquisition can be constructed. As the displacement and orientation of the coupled antenna system can be chosen arbitrarily, provided only that the enclosing surfaces do not intersect, any near-field (or quasi far-field) measurement system can be simulated. Hence, simulations produced from this procedure can be used to rigorously verify the corrections made for the modal receiving coefficients (for example, plane wave) of the scanning probe during the near-field to far-field transformation process. The Kirchhoff–Huygens’ method has been described in detail in Chapter 4 and an introduction to the reaction theorem and how it can be used to obtain the mutual coupling coefficient between two antennas is presented in the following section.

7.6.1 Mutual coupling and the reaction theorem Provided that the electric and magnetic field vectors (E 1 , H 1 ) and (E 2 , H 2 ) are of the same, that is, monochromatic, frequency, then the mutual impedance, Z21 , between antennas 1 and 2 in the environment described by ε, µ can be expressed, from the reaction theorem, in terms of a surface integration as given in Reference 3, refer to

Computational electromagnetic model of a planar near-field 235 Section 7.8 for derivation. Z21 =

V21 1 =− I11 I11 I22





 E 2 × H 1 − E 1 × H 2 · nˆ ds

(7.58)

S2

Here, n is taken to denote the outward pointing surface unit normal. The subscript 1 denotes parameters associated with antenna 1 whilst the subscript 2 denotes quantities associated with antenna 2, that is, S2 is a surface that encloses antenna 2, but not antenna 1. Here, I11 is the terminal current of antenna 1 when it transmits and similarly, I22 is the terminal current of antenna 2 when it transmits. From reciprocity, the mutual impedance Z12 = Z21 and is related to the coupling between two antennas. Clearly then, the mutual impedance will also be a function of the displacement between the antennas, their relative orientations and their respective polarisation properties. The terminal currents of these transmitting antennas I11 and I22 can be obtained from the knowledge of the power injected at the port P1 and P2 that is typically taken to be unity and is specified within the modelling tool and the port impedance Z1 or Z2 using # P1 I11 = (7.59) Z1 The power injected into the space that lies outside the Huygens’ surface can be readily obtained by integrating the complex pointing vector over the Huygens’ surface S using       E × H ∗ · nˆ ds (7.60) Prad = Re   S

Here and as in common with many commercially available EM modelling tools E and H are assumed to be root mean square (r.m.s.) values of the field. If instead E and H denote peak values of the field, then a factor of one half must be included. The self-impedance Z11 and Z22 can be obtained in many ways. However, they are perhaps most easily obtained from whichever three-dimensional full wave EM solver was used to obtain the radiated fields from the isolated antennas. As an admittance is merely the reciprocal of an impedance, an admittance matrix [Y ] representing this two port coupled systems can be readily populated so that   Y11 Y12 [Y ] = (7.61) Y21 Y22 It is well known that the re-normalized scattering matrix [S ], can be calculated from this admittance matrix and is used to describe what fraction of the signal is transmitted or reflected at each port   [S ] = [Y ] ([Z] − [Z ]) ([Z] + [Z ])−1 [Z ] (7.62)

236 Principles of planar near-field antenna measurements Here [Y ] = ([Z ])−1 is a diagonal matrix with the desired normalizing admittance as the diagonal entries, that is, the admittance of the attached transmission line, which in this case will be equal to the port impedance Z1 = Z2 = ZTE . This can be expressed mathematically as [Y ] = Y δij with δij , denoting the Kronecker delta where i and j are positive integers  1 (i = j) δij = 0 (i  = j)

(7.63)

(7.64)

The elements S1,2 = S2,1 of [S ] are the complex transmission coefficients for the coupled antenna system that are taken to represent a single point in the nearfield measurement. This then completes what is essentially a MoM analysis of this system. To summarise, the near-field measurement simulation algorithm consists of the following steps: 1. 2. 3. 4. 5.

Use (7.46) and (7.49) to translate the fields of antenna 1 to antenna 2. Use (7.58) to evaluate the mutual impedance and thus the mutual admittance. Populate the admittance matrix for the two-antenna system. Use (7.62) to determine the complex mutual coupling coefficient. Repeat steps 1–4 inclusive for each sample point in the near-field scan.

Rotate the probe to simulate the sampling of a second orthogonal near-field component and repeat steps 1–5 inclusive to generate a second scan and thus complete the near-field acquisition. The magnitude of the mutual coupling coefficient between a pair of polarisation matched lossless dipoles that are perfectly matched to their respective source and load and that are in the far-field of one another can be obtained from the Friis transmission formula [4].   PR λ 2 |) = G T GR (7.65) lim (|S21 = R→∞ PT 4πR Conversely, the mutual coupling coefficient can be obtained by taking the E- and H -fields from a half wavelength dipole from a full wave solver and using the physical optics reaction theorem algorithm set out above. As the antennas that are transmitting and receiving are of exactly the same design, GR = GT . The gain of the dipole at 10 GHz along the x-axis was approximately 2.16 dB. Figure 7.3 contains a comparison of the mutual coupling obtained using these two contrasting methods. From inspection, it is clear that the agreement is encouraging with differences only becoming more pronounced as the separation becomes smaller, that is, in the region where the far-field approximation within the Friis transmission equation is most unreliable. To illustrate the utility of this generalized, but rather involved and computationally intensive, simulation method the following section presents a comparison between a near-field measurement and a near-field simulation.

Computational electromagnetic model of a planar near-field 237 –10 PO–RT algorithm Friis

–15

S21(dB)

–20

–25

–30

–35

–40 0

Figure 7.3

7.7

0.05

0.1

0.15 Distance (m)

0.2

0.25

0.3

Mutual couping between adjacent dipoles

Near-field measurement simulation

Figure 7.4 contains a schematic representation of the near-field probe and a standard gain horn (SGH) that was used as an AUT. An electrically small AUT was chosen as its properties could be obtained directly from a full wave CEM solver. The faint grey ellipsoidal surface that can be seen to enclose these instruments represents the Huygens’ surface that was used with the physical optics reaction theorem CEM simulation. The scanning probe consisted of an WG16 chamfered open-ended rectangular waveguide probe combined with a surface wave absorbing material (SWAM) cone that was designed to minimise reflections from the mechanical interface located towards the rear of the probe. A detailed description of the modelling and verification thereof can be found within the literature [5]. Again similarly good agreement was attained between the CEM model of the SGH and measurements taken using a compact antenna test range (CATR). A near-field measurement of the SGH was taken using a planar near-field antenna test range. The acquisition window was chosen to be ±0.8 m in the x–y plane whilst the distance between the SGH and the near-field probe in the z-direction was set at 10 cm, that is, approximately 3 wavelengths (3λ) at 10 GHz. This separation ensured that the probe was outside the reactive near-field, made the first-order truncation angle as large as possible (≈ ±83◦ in azimuth and elevation) and attempted to minimise the detrimental effects arising from multiple reflections that can be set-up between the AUT and the scanning near-field probe. Although phenomena arising from the first two of these effects will be included within the measurement simulation, multiple reflections between the AUT and the probe are ignored.

238 Principles of planar near-field antenna measurements

Figure 7.4

Near-field probe (left) and AUT (SGH) (right)

The separation between adjacent measurement points was one half wavelength as this satisfied the Nyquist sampling criteria and, in the absence of errors in the position at which measurements are taken, will guarantee alias free far-field patterns over the entire forward half space. The SGH was installed in the range so that it was principally ‘y-polarised’ with respect to the axes of the near-field range. Figures 7.5 and 7.6 contain a comparison of horizontal and vertical cuts through the simulated near-field measurement and the actual near-field measurement. The agreement between the simulation and measurement is encouraging in both planes and it suggests that the simulation technique is capable of creating a simulation of a measurement taken on a near-field range. In particular, the cut in the y-axis shown in Figure 7.6, where there is less energy as you move away from the centre shows very good agreement. It is only when the magnitude of the coupling is at lower levels, that is, <−40 dB, that the characteristics of the two lines begin to diverge. Inevitably, the measurement contains uncertainties arising from imperfections in alignment (both in translation, x, y, z and rotation in azimuth, elevation and roll), multi-path (scattering from the chamber walls and the frame of the robotic positioner), multiple reflections between the antenna and the probe, and imperfections in the manufacture of both the probe and AUT. None of these error terms is included within the near-field simulation. Unfortunately, the evaluation of the transmission coefficient S21 requires the computationally expensive evaluation of a sextuple integration to obtain each nearfield sample point. However, the simulation of an entire near-field measurement constitutes a coarsely granular problem. Specifically, each sampling node can be evaluated independently of every other sampling node and in this way, provided only that a sufficiently large array, that is, cluster, of computers is available, the total processing time is in principle equal to the time taken to evaluate a single measurement point. Finally, it is important to highlight that this technique neither relies on the modal expansion method for the representation of electromagnetic waves nor does it utilize inversions of conventional probe pattern correction algorithms. Thus, it provides an independent approach to the verification of other near-field to far-field transformation algorithms.

Computational electromagnetic model of a planar near-field 239 0 Measured Simulated

Magnitude (dB)

–10

–20

–30

–40

–50

–60 –0.8

Figure 7.5

–0.6

–0.4

–0.2

0 0.2 x-axis (m)

0.4

0.6

0.8

Near-field cut in x-axis 0 Measured Simulated

–10

Magnitude (dB)

–20 –30 –40 –50 –60 –70 –80 –90 –0.8

Figure 7.6

7.8

–0.6

–0.4

–0.2

0.2 0 y-axis (m)

0.4

0.6

0.8

Near-field cut in y-axis

Reaction theorem

The generalized MoM solution outlined in the preceding section made extensive use of the reaction theorem and used several expressions, as is, without explanation or proof. The following sections are unnecessary if all that is required is to implement the techniques developed above; however, they are included here to aid and add detail

240 Principles of planar near-field antenna measurements to the technique whilst not disturbing the development of the near-field modelling technique developed above. First, the reciprocity theorem is developed. Then, this is used as the basis for the reaction theorem that is used to relate circuit parameters to field parameters, which is in the form of the reaction integral that is central to the near-field modelling technique set out above.

7.8.1 Lorentz reciprocity theorem (field reciprocity theorem) Consider two elemental current sources, I1 at r = r1 and I2 at r = r1 where both sources oscillate with the same angular frequency ω. Here, I1 produces E1 , H1 and I2 produces E2 , H2 . For I1 ∇ × E1 = − M1 − jωµH1

(7.66)

∇ × H1 = J1 + jωεE1

(7.67)

For I2 ∇ × E2 = − M2 − jωµH2

(7.68)

∇ × H2 = J2 + jωεE2

(7.69)

Let us first consider the divergence. First, dot H2 with ∇ × E1 = −M1 − jωµH1 to obtain   H2 · ∇ × E1 = −H2 · M1 − jωµH2 · H1 (7.70) Now dot E1 with ∇ × H2 = J2 + jωεE2 to obtain   E1 · ∇ × H2 = E1 · J2 + jωεE1 · E2

(7.71)

Subtracting these expressions yields     E1 · ∇ × H2 − H2 · ∇ × E1 = E1 · J2 + H2 · M1 + jωεE1 · E2 + jωµH2 · H1 (7.72) Now since       ∇ · A×B =B· ∇ ×A −A· ∇ ×B

(7.73)

Hence

        E1 · ∇ × H2 − H2 · ∇ × E1 = ∇ · H2 × E1 = −∇ · E1 × H2

(7.74)

Thus

  −∇ · E1 × H2 = E1 · J2 + H2 · M1 + jωεE1 · E2 + jωµH2 · H1

Now dot E2 with ∇ × H1 = J1 + jωεE1 to obtain   E2 · ∇ × H1 = E2 · J1 + jωεE2 · E1

(7.75)

(7.76)

Computational electromagnetic model of a planar near-field 241 Now dot H1 with ∇ × E2 = −M2 − jωµH2 to obtain   H1 · ∇ × E2 = −H1 · M2 − jωµH1 · H2

(7.77)

Subtracting these expressions yields     E2 · ∇ × H1 − H1 · ∇ × E2 = E2 · J1 + H1 · M2 + jωεE2 · E1 + jωµH1 · H2 (7.78) Now since       ∇ · A×B =B· ∇ ×A −A· ∇ ×B

(7.79)

Hence

        E2 · ∇ × H1 − H1 · ∇ × E2 = ∇ · H1 × E2 = −∇ · E2 × H1

(7.80)

Thus

  −∇ · E2 × H1 = E2 · J1 + H1 · M2 + jωεE2 · E1 + jωµH1 · H2

(7.81)

Subtracting these equations yields

    −∇ · E1 × H2 + ∇ · E2 × H1 = E1 · J2 + H2 · M1 + jωεE1 · E2 + jωµH2 · H1 −E2 · J1 − H1 · M2 − jωεE2 · E1 − jωµH1 · H2 (7.82)

Upon cancellation this becomes     −∇ · E1 × H2 + ∇ · E2 × H1 = E1 · J2 + H2 · M1 − E2 · J1 − H1 · M2 (7.83) or

  −∇ · E1 × H2 − E2 × H1 = E1 · J2 + H2 · M1 − E2 · J1 − H1 · M2

(7.84)

This is the Lorentz reciprocity theorem in differential form. Taking the volume integral of both sides of this expression yields    − E1 · J 2 + H 2 · M 1 ∇ · E1 × H2 − E2 × H1 dv = V

V

 −E2 · J1 − H1 · M2 dv

Now recalling the divergence theorem ∇ · D dv =  D · nˆ ds V

(7.86)

S

So that      E1 × H2 − E2 × H1 · nˆ ds = ∇ · E1 × H2 − E2 × H1 dv S

(7.85)

V

(7.87)

242 Principles of planar near-field antenna measurements Thus

−  E1 × H2 − E2 × H1 · nˆ ds = E1 · J2 + H2 · M1 − E2 · J1 − H1 · M2 dv S

V

(7.88)

This is the Lorentz reciprocity theorem in integral form. So summarising yields

∇ · E1 × H2 − E2 × H1 dv  E1 × H2 − E2 × H1 · nˆ ds = S

V

=



E2 · J1 + H1 · M2 − E1 · J2 − H2 · M1 dv

V



=  E2 × H1 − E1 × H2 · nˆ ds

(7.89)

S

Here, the closed surface S encloses the volume V and the surface normal points out of the volume. Suppose that each of the sources is contained within non-intersecting finite, closed volumes V1 and V2 respectively. Thus, V1 contains only J 1 and M 1 whilst V2 contains only J 2 and M 2. Hence      E1 × H2 − E2 × H1 · nˆ ds = E2 · J1 − H2 · M1 dv S1

(7.90)

V1

Similarly     E2 · J1 − J2 · E1 dv  E1 × H2 − E2 × H1 · nˆ ds = S1

(7.91)

V1

This is a commonly used form of reciprocity and it is derived by assuming that J 1 and J 2 are the only sources of E 1 , H 1 , E 2 and H 2, that is, there are no assumed magnetic sources. Alternatively, by assuming that both sources are outside the volume of integration yields    E1 × H2 − E2 × H1 · nˆ ds = 0

(7.92)

S1

Within Box 7.3 further attention is given to the physical interpretation of this relationship.

Computational electromagnetic model of a planar near-field 243

Box 7.3 If this assumption is not made and the sources are in each others far-field since      E1 × H2 − E2 × H1 · nˆ ds = E2 · J1 − J2 · E1 dv (7.93) S1

V1

Then in the far-field r·E =0=E·r

(7.94)

H = r × E/z

(7.95)

and Here, z is used to denote the impedance of the medium in which the field is propagating. From       A× B×C = A·C B− A·B C (7.96) *   *   *   E1 × H2 = E1 × r × E2 z = E1 · E2 z r − E1 · r E2 z (7.97) Since E1 · r = 0 And

  *  *  E1 × H2 = E1 · E2 z r − 0 = E1 · E2 z r

(7.98)

  *   *   * E2 × H1 = E2 × r × E1 z = E2 · E1 z r − E2 · r E1 z

(7.99)

Since E2 · r = 0

*  *    E2 × H1 = E2 · E1 z r − 0 = E2 · E1 z r

(7.100)

Therefore in this case

*  *    E1 × H2 − E2 × H1 = E1 · E2 z r − E2 · E1 z r = 0

(7.101)

Therefore if J1 and J2 are localized and, as the cross product terms are zero, that is, there are no incoming waves from infinitely far away. (E2 · J1 − E1 · J2 ) dv =  0 · n ds (7.102) V1

Therefore + + (E1 · J2 ) dv (E2 · J1 ) dv =

(7.103)

This is the usually-quoted form of the Rayleigh–Carson Reciprocity theorem. However, if the theorem is considered in the simplest of physical situations,

244 Principles of planar near-field antenna measurements

Box 7.3 Continued i.e. for dipoles as described in Section 7.3.1, if it is averaged to remove any time dependency and thus represents an energy as opposed to a power equation then dimensionally the integrals will give only the products of fields and dipoles, that is E2 · P1 = E1 · P2

(7.104)

Further insight can be gained by performing a dimensional analysis on these products in terms of units where   volt E1 · P2 ≡ · (coulomb · metre) metres = (volt · coulomb)   joule · coulomb = coulomb

(7.105)

= joule Thus the reaction theorem can be viewed as an energy conservation equation where the dot product terms relate to the partition of energy between the physical structures carrying the currents, that is, Newton’s third law relating actions to equal and opposite reactions and the cross product terms represent a flux across the boundary of a surface containing these structures.

7.8.2 Generalized reaction theorem Maxwell’s equations for the fields within S2 but outside the source region can be expressed as ∇ × E 1 = − jωµH 1

(7.106)

∇ × H 1 = jωεE 1

(7.107)

∇ × E 2 = − jωµH 2

(7.108)

∇ × H 2 = jωεE 2

(7.109)

Now, the ‘reaction’ of antenna 1 on antenna 2 is defined to be   2, 1 = E 2 × H 1 − E 1 × H 2 · nˆ ds S2

(7.110)

Computational electromagnetic model of a planar near-field 245 Here, S2 is a surface that encloses antenna 2, but not antenna 1. For the sake of completion, the reaction of antenna 2 on antenna 1 can similarly be defined as   1, 2 = (7.111) E 1 × H 2 − E 2 × H 1 · nˆ ds S1

Similarly, S1 is a surface that purely encloses antenna 1 but not antenna 2. For simplicity, the reaction formula can be expressed as   2, 1 = (7.112) E 2 × H 1 − E 1 × H 2 · ds S2

where ds = nˆ ds

(7.113)

Now, let the surface of integration be changed from S2 to Sm + ST , Thus     2, 1 = E 2 × H 1 − E 1 × H 2 · ds + E 2 × H 1 − E 1 × H 2 · ds Sm

ST

(7.114)

If the metal surface Sm is assumed to be perfectly conducting then E 1 and E 2 will only have normal components thus the term E × H will only have tangential components that are orthogonal to the surface normal thus the integral over the metal surface will vanish. Hence   2, 1 = (7.115) E 2 × H 1 − E 1 × H 2 · ds ST

Now let E 2 = V22 e

(7.116)

H 2 = I22 h

(7.117)

Then



2, 1 =



 V22 e × H 1 − E 1 × I22 h · ds

(7.118)

ST

Now, the voltage and current induced at antenna 2 when antenna 1 transmits are defined in a similar way by setting E 1 = V21 e

(7.119)

H 1 = I21 h

(7.120)

246 Principles of planar near-field antenna measurements Thus



2, 1 =



 V22 e × I21 h − V21 e × I22 h · ds

(7.121)

ST

As the vector mode functions are related by h = nˆ × e

(7.122)

where they are normalized to unity through e · e ds = 1

(7.123)

ST

Thus

2, 1 =



    V22 I21 e × nˆ × e − V21 I22 e × nˆ × e · ds

ST



(7.124)

  e × nˆ × e · ds



2, 1 = (V22 I21 − V21 I22 )

(7.125)

ST

Now as

      A× B×C = A·C B− A·B C

Then



(7.126)

   e · e nˆ − e · nˆ e · ds



2, 1 = (V22 I21 − V21 I22 ) ST





 2, 1 = (V22 I21 − V21 I22 )  Now since,

. ST



 e · e nˆ · ds −

ST



 

  e · nˆ e · ds

(7.128)

ST

e · e ds = 1 and as ds = nˆ ds then

nˆ · ds = nˆ · nˆ ds = ds Thus

(7.127)

(7.129) 

 2, 1 = (V22 I21 − V21 I22 ) 1 −



 

  e · nˆ e · ds

(7.130)

ST

Now, as the fields are tangential then   e · nˆ e · ds = 0 ST

(7.131)

Computational electromagnetic model of a planar near-field 247 Thus 2, 1 = V22 I21 − V21 I22

(7.132)

Now, as the received current I21 is assumed to be zero as the terminals are assumed to be open circuited when receiving then, this becomes 2, 1 = −V21 I22 Hence



V21 I22 = −

(7.133)

 E 2 × H 1 − E 1 × H 2 · nˆ ds



(7.134)

S2

This is of importance as it constitutes a ‘mixed type’ reciprocity theorem. Finally then   1 E 2 × H 1 − E 1 × H 2 · nˆ ds (7.135) V21 = − I22 S2

7.8.3 Mutual impedance and the reaction theorem The mutual impedance Z21 between antenna 1 and antenna 2 in the environment (ε, µ) can be expressed in terms of the electric and magnetic fields as [3] Z21 =

V21 I11

Z21 =

−1 I11 I22

Thus

(7.136)



 E 2 × H 1 − E 1 × H 2 · nˆ ds

(7.137)

S2

Here, I11 is the terminal current of antenna 1 when it transmits and nˆ is the outward pointing unit normal. Similarly, I22 is the terminal current of antenna 2 when it transmits. The mutual impedance Z12 = Z21 represents the coupling between two antennas and will therefore be a function of the distance separating the two antennas, the relative antenna orientations and their respective polarisation properties. Similarly   −1 (7.138) E 1 × H 2 − E 2 × H 1 · nˆ ds Z12 = I11 I22 S1

7.9

Summary

Within this chapter a number of simulation techniques have been developed, each offering a different balance of sophistication and effort. Ultimately, it would perhaps be preferable to utilize a full wave three-dimensional EM solver to tackle these problems and with the passage of time this is becoming an ever more feasible option.

248 Principles of planar near-field antenna measurements However, until these methods can be deployed on a sufficiently large scale and can provide results in a sufficiently compact timescale, other alternative solutions will remain attractive. For many applications, the comparatively simple vector Huygens’ method is sufficient and indeed it was used in the preparation of many of the simulated data sets that are utilized within Chapter 9 to illustrate and verify some of the more advanced transformation and correction techniques. Unfortunately, it is not possible to use these techniques to model every phenomenon that can be observed in a near-field measurement system, for example, multiple reflections between the AUT and the probe, but there is perhaps sufficient choice detailed above that many situations can be accommodated.

7.10 References 1 Silver, S.: Microwave Antenna Theory and Design, 1st edn (McGraw Book Company Inc., New York, 1949), p. 83 2 Mills, R.: Space Time and Quanta an Introduction to Contempory Physics (W.H. Freeman, New York, 1994), pp. 327–36 3 Richmond, J.H.: ‘A reaction theorem and its application to antenna impedance calculations’, IRE Transactions on Antenna and Propagation, November 1961;515–320 4 Balanis, C.A.: Antenna Theory Analysis and Design, 2nd edn (John Wiley and Sons Inc., New York, 1997), p. 86 5 Lyon, R.W., Gregson, S.F., Mitchelson, C., and McCormick, J.: Computational Electromagnetic Modelling of a Probe Employed in Planar Near Filed Antenna Measurements, ICAP 2003, Exeter

Chapter 8

Antenna measurement analysis and assessment

8.1

Introduction

Until this point, this volume has described and explained the techniques and concepts involved in performing near-field antenna measurements in a systematic form. This was done by • • • • •

Addressing the fundamental phenomenological action of antennas. Developing a conceptual model of the interaction of antennas based on classical electromagnetic (EM) field theory. Deriving a methodology for the solution of the differential equations of classical EM field theory based on the plane wave spectrum approximation. Describing the details of a measurement technique (near-field scanning) that can be used to make measurements of the coupling between two antennas, the antenna under test (AUT) and a probe antenna. And finally making predictions of the far-field radiative performance of the AUT based on the plane wave spectrum approximation.

However, the problem of deriving the measure, that is, the true value of the magnitude of a physical quantity, like the power flux density transmitted in a given direction relative to an antenna, from measurements has not as yet been addressed. Note that this is strictly the primary measure of the measurand in question; other secondary measures can be derived from the measurement results but unless specifically stated, in this chapter measure will be taken to mean primary measure.

8.2

The establishment of the measure from the measurement results

The concept of measurement is interpreted differently in different sciences and therefore by definition in different areas of engineering and technology. As described in Chapter 1 the information extraction model is a particularly applicable concept

250 Principles of planar near-field antenna measurements for the cognitive evaluation of antenna measurements. In near-field scanning we are primarily concerned with microwave frequencies and as millions of years of human evolution have left our species without sense organs that respond to such frequencies, sensory cognition is largely irrelevant, leaving only rational cognition as a tool for the evaluation of microwave phenomena. Rational cognition involves analysis, synthesis translation and inference of and from images or representations of reality that are not sensory and is thus performed in the domain of abstraction. By definition, observation, experimentation and evaluation are used in rational cognition and when the observations of physical phenomena are designed to extract quantitative information these observations are of course termed measurements. However, in order to meet the requirement that the results of measurements are quantitative representations in the abstract domain of physical phenomena severe restrictions as to the form of the measurement procedure must be imposed and adhered to, to reduce ambiguity. The minimisation of the effects of error sources in measurements will result in the reduction of ambiguity in the mapping that transforms the magnitude of the real physical quantity to its representational state in the abstract measurement domain. As well as the usual empirical definition of a scale, such a mapping is also often referred to as ‘the scale of the measurement’ (see Appendix B). The possibility/practicality of establishing a non-ambiguous empirical scale as a ratio, relative to a standard in multiples of the resolution of the measurement system, will be addressed in this chapter. Before it is possible to devise strategies that identify, quantify and/or eliminate the error sources, which introduce ambiguity into the measurements, it is necessary to examine the nature and mode of their occurrence.

8.2.1 Measurement errors Figure 8.1 illustrates the relationship between the measured physical quantity and the measurement representation that constitutes the result of the measurement. Clearly Representational symbol set

Physical state set Measurement process M: Q → N q1

n1

q2

n2

Q

Figure 8.1

N

Pictorial representation of the set theoretic model of measurement

Antenna measurement analysis and assessment 251 the integrity of the mapping M , which defines the scale over which the measurements are made, is fundamental to the accuracy of the measurement process and the accuracy with which the measurements can be used to define the measure of the measurand. Where again we would define the measurand (q) the physical quantity to be measured, the measurement result as the image of the measurand in the representational set and the measure (n) as the true value of the physical quantity being measured using the scale (see Appendix B). The operation including the mapping (M ) where, the elements of set Q are mapped to set N , can be viewed as a transfer function where an input non-mathematical physical quantity is mapped to an output signal, which is a representation of the physical state on a scale, that is N = M (Q)

(8.1)

where, Q = (q1 , q2 , q3 , . . . , qi ) is the set of physically realisable values of the measurand, the domain set of the transfer function and N = (n1 , n2 , n3 , . . . , ni ) is the image of Q under the mapping M , the image set of the transfer function.

Box 8.1 Note: This representation of the measurand in a symbol set can take many forms. Examples are as follows: • • • •

A number in a table or array A point on a graph A colour on a graphic or a pseudocolour plot An arrangement of raised indentations on a Braille script.

These are all examples of representative symbols that could be used, where the guiding principle of effective representation in measurement theory is that the relationship between the symbols representing the measured values and the actual true values of the measurand should be of the same logical form [1].

For individual elements of the sets n = m(q)

(8.2)

where an equivalence exists of the form n ≡ q under the mapping m, remembering that ni = m(qi ) will only be true if m is a one-to-one mapping. If, as is usually the case for antenna measurements, the representation is numerical the level of the measurement of any variable describes how much information the numbers associated with the variables contain and whether these are represented on nominal/categorical ordinal/ranked, interval/scaled or ratio/metric scales [2]. The advantages to be gained by considering the different types of information in measurement data will be illustrated

252 Principles of planar near-field antenna measurements in Section 8.3 when data assessment techniques based on the different measurement levels will be discussed.

Box 8.2 Note: For antenna measurements an interval, as opposed to categorical or ordinal scale, calibrated to make it metric, is usually used where these different types of scale can best be described in terms of level of measurement by the following example. Conventionally, two antennas AUT1 and AUT2 may have their boresight gains measured and the results for AUT1 be quoted as 9 dBi and for AUT2 as 21 dBi. These results could have been mapped onto a categorical scale where the representation of AUT1 might have been categorised as low gain and AUT2 as high gain. In this case a nominal level of measurement would have been made defining a categorical variable. These might have been appropriate categories on the nominal scale or not, for example, AUT1 may in fact have higher gain than AUT2 just not on boresight. Measurements of AUT1 and AUT2 could have been mapped into an ordinal scale where under the scale mapping AUT1 and AUT2 would have been ranked. In this case an ordinal level of measurement would have been made defining a ranked variable, with only the ordinal level of information represented. In fact as already stated in near-field antenna measurements the AUT gain is usually measured relative to an arbitrary reference signal, so AUT1 and AUT2 gains are mapped onto an interval scale, where, AUT2’s gain is represented by not only a larger value, but by a value that falls 16 equal intervals above the unitary value assigned to AUT1. In this case an interval level of measurement would have been made defining the scaled variable, with only the interval level of information represented. Note: Although in practice AUT2’s gain might be thought of as 16 times greater than AUT1’s since the zero point on any interval scale is arbitrary, the ratios between numbers on that scale are not meaningful. Thus strictly, the operations of multiplication and division cannot be defined at this level of measurement, which is more accurately described by the complete Presburger as opposed to finitely specified, consistent, but incomplete Peano arithmetic [3]. Finally if the measurements of AUT1 and AUT2 are compared directly with a defined standard, in this case the theoretical gain of an isotropic radiator, these measurements can be considered to consist of a mapping onto a representative metric scale, where, as already stated, AUT1 might have a gain of 9 dBi and AUT2 of 21 dBi relative to the isotropic standard. In this case, a ratio level of measurement would have been made defining a metric variable, with the ratio relative to the isotropic standard level of information represented.

Antenna measurement analysis and assessment 253

Clearly the information required to represent the boresight gain of AUTs 1 and 2 on these categorical, ordinal or interval scales is contained within the metric data. However, if there is a requirement to analyse the data, the different aspects and levels of measurement represented on the different scales are amenable to very different statistical methodologies. As a result considerable insight into the nature of any measurement procedure can be obtained by an approach that does not limit itself to just examining the interval or metric levels of information represented.

Figure 8.2 again shows how a theoretical measurement procedure can be viewed as the operation of a transfer function on an input member of a physical state set, the measurand, to map it onto an output representative abstract symbol set the measurement result [4]. In this theoretical case the measurement result = measure of the measurand. Thus, such a measurement system would provide a measurement result that would unambiguously and accurately define the measure of the measurand.

q

Figure 8.2

n = m(q)

n

Simple functional block diagram of a measurement

However, under no practical circumstances can the result of a single measurement on a scale give the absolute magnitude of a measure as in fact all measurement procedures have inherent measurement errors associated with them irrespective of the extent of control exercised within them. This means that the result of any measurement cannot be taken as representative of the measure of the measurand unless the extent and nature of the ambiguities introduced by error sources into the measurement procedure are analysed, assessed and quantified.

8.2.2 The sources of measurement ambiguity and error The introduction of errors into any measurement process can be considered to be the result of two main factors. First the transfer function can introduce an error into the measurement result where the mapping itself is only approximately equivalent to the true scale mapping, both as a result of systemic errors in the analogue mapping process and as a result of external possibly random error sources, that is, me ∼ =m

(8.3)

254 Principles of planar near-field antenna measurements where me is the actual as opposed to the theoretical scale mapping of the measurement system. For example, the response of the receiver in a near-field antenna measurement system is assumed to be linear; in practice it will only be approximately linear over a given range of values or under a given range of external environmental conditions. This will cause the scale mapping of any practical measurement system to be, at least to some extent, in error. Thus as illustrated in Figure 8.3 possibly as a result of external influences, error sources em1 and em2 , the mapping is the cause of ambiguity.

q

ne = me(q)

em1

Figure 8.3

ne

em2

Measurement ambiguity introduced by mapping errors sources

Second the input member of the physical state set, the measurand, may not be the only input that the system is mapping onto the output as a signal. Again, for example, all antenna ranges attempt to measure the antenna pattern as the direct path relative power flux density propagating in a given direction with the antenna in free space. In practice the antenna will be mounted within a test range that will be the source of scattering; this scattered field will produce multipath within the range, which also acts as an error source inducing an ambiguous input to the measurement system. Thus in the example, shown in Figure 8.4 qe  = q

(8.4)

where, qe is the actual ambiguous input value and q is the value if no other error sources were present.

qe

eq1

Figure 8.4

ne = m (qe)

ne

eq2

Measurement ambiguity introduced by input error sources

Antenna measurement analysis and assessment 255 This means that the model of a measurement system represented by the transfer function (8.1) may actually be represented by ne = me (q)

(8.5)

ne = m(qe )

(8.6)

or

or as illustrated in Figure 8.5 as result of both input and mapping error sources most likely ne = me (qe )

(8.7)

Where the output is the measurement result, ne has an error associated with either or both of the input and scale mapping being inaccurate.

ne = me(qe)

qe

eq 1

Figure 8.5

eq2

eq 1

ne

eq2

Measurement ambiguity introduced by mapping and input errors

Whereas for an unambiguous measurement procedure an equivalence exists of the form n ≡ q under the mapping m

(8.8)

For a practically realisable measurement system ne ∼ = qe under the mapping me

(8.9)

The end result is that, unless the errors associated with the scale mapping of the measurand input can be systemically quantified and removed by post processing of the output signal, in any practical measurement system there will be ambiguity in the measurement results. This means that the measurement result cannot accurately and unambiguously represent the measure of the measurand. How then can a measurement process be used to establish the true value of any physical quantity? This will be considered further in subsequent sections.

256 Principles of planar near-field antenna measurements

8.2.3 The examination of measurement result data to establish the measure In cases where the ambiguity is introduced by the mapping me the result will be that the element in the domain is mapped to a possible set of values, that is q ⇒ Nq ≡ {n}

(8.10)

where Nq ⊂ N . Formally the mapping transforms q into a random variable and Na is the set of values of this variable. This can be illustrated by the functional block diagram shown in Figure 8.6, where the transfer function maps the input member of the physical state set onto a distribution of possible values each separated by the a multiple of the scales resolution, ε. n e = me (qe)

qe

n – 2ε n–ε n

eq 1

eq 2

eq 1

eq 2

n+ε n + 2ε

Figure 8.6

Measurement ambiguity introduced by mapping onto a distribution

The random variable n, which is an image of the element q in the domain Nq , will have a distribution, pq (n), where the assignment of a number of elements in Nq is possible but one state in Nq will be preferred. If there is a preferred value it can be supposed that the probability of the mapping n = me (q)

(8.11)

is either greater or smaller than the mapping n + ε = me (q)

(8.12)

where ε > 0. Therefore, the selection of the elements of the set Na , which maps state q, is described by a measure p ∈ 0, 1

(8.13)

For which
pq (n) = 1

(8.14)

n∈Na

Or

 (Na )

pq (n) dn = 1

(8.15)

Antenna measurement analysis and assessment 257 Dependant on the discrete or continuous nature  of the  domain. Thus, the mapping takes the form of the assignment relation q ⇒ n, pq (n) as opposed to the expression q ⇒ n, where n ∈ N . This illustrates that, for all theoretical and practical measurement systems, measures are in fact probabilities [5], calculated from mappings of random variables onto a distribution which defines a probability density.

Box 8.3 Of particular interest in low resolution measurement systems is the very definition of the Lebesgue measure of a set, where the determination of the measure is based on the creation of closed intervals; and thus can be applied to a non-dense distribution within an image set. Theoretically this methodology of integrating the content or measure of a non-dense set is very important as any practically realisable measurement scale will be composed of rational multiples based on the resolution of the scale and thus will by definition will not be a dense set. Thus strictly any probability calculus based on the Peano–Jordon content derived from a Rieman integral is inaccurate [6]. However, under most practical circumstances the resolution of the scale will be such as to render the scale effectively continuous and not discrete over the rational intervals of the resolution. Therefore in practice and in particular for near-field antenna measurement systems this theoretical limitation of the probability calculus is not significant.

The assignment relation describes a complete set of possible events that will be assigned to q with a probability of 1 as per (8.14). To define a measurement scale only one possible element n∗ , being the best possible mapping must be selected from Nq to define the measure of the measurand. This selection will be based on the decision D where n∗ = D(pq )

(8.16)

Many different decision rules can be followed dependant on the type of measurement and nature of the measurand. However, by following the measurement procedure to produce measurement results that can be analysed statistically in terms of their probability, a maximum likelihood value and an accompanying confidence interval can be assigned to a given value and distribution in the measurement domain. Thus n∗ represents a maximum likelihood measure and can be taken as the true value of the measurand. As already stated, in this text the primary measure defines the maximum likelihood value but other secondary measures that refer to tendencies of spread, kurtosis and skewness are all calculable. For any useful measurement system therefore the spread of possible values defined by a confidence interval at a given level of probability is a vital secondary measure of the data set.

258 Principles of planar near-field antenna measurements For cases where the ambiguity is introduced by input errors, a similar argument can be followed to describe the errors introduced by the measurement process. In turn this distribution relating to the ambiguity in the input, allows the inclusion of this and all other error sources in an overall total distribution pt (n) which can be used in the assignment of a maximum likelihood value and any other measure that is required. This is not the only methodology for deriving the total error or combined uncertainty for a measurement process. In practice for ease of evaluation the errors can be viewed in terms of orthogonal type A (statistically derived) and type B (deterministically derived) error sources as opposed to input and mapping sources where the random and deterministic sources are evaluated to produce orthogonal distributions. Other methodologies can be adopted to calculate the combined uncertainty dependant on how the correlation between the input and output are evaluated and compared to the theoretical assignment relations between them; however, in practice all depend on a statistical analysis of measurement data. From the above discussion it is clear that the measurement process tentatively illustrated in Figure 8.2 consisting primarily of a mapping of the physical state onto a representative symbolic set is over simplistic. The mapping is in fact an assignment relationship that assigns the measure of the measurand to a the maximum likelihood value within the symbolic set according to a distribution pt (n) where the mapping and input are subject to measurement errors as given in Figure 8.6. Therefore in order to use a measurement system accurately, a knowledge of the distribution pt (n) will be necessary to assign the maximum likelihood value and any confidence interval associated with measure to the measurement results. However, in practice a priori knowledge of any distributions is very rare, this means that in any practical measurement system the distribution must be established using the measurement procedure. Since the measurement procedure consists of mapping the element q into a random variable n with a distribution pt (n) the only possible practical methodology for establishing pt (n) is to repeat the measurement process until the measurement results display the nature of the distribution. This, theoretical, necessity of characterising the probability distribution is the basis of the requirement that all measurement procedures need repetition to assign a measure with any degree of confidence. Only measurements repeated infinitely can assign a measure with 100 per cent confidence; hence the requirement to express measurement results with a ± confidence level at a given level of significance (see Appendix B). If there is no assignment of uncertainty, that is, the establishment of secondary measures of central tendency or spread of the results of a measurement, then it is not possible to define any useful level of confidence in the accuracy of the measure, that is, the maximum likelihood value of the measurand. Thus without repetition of the measurement process the ambiguities introduced into the measurement process either by lack of fidelity in the transfer function or as additional unwanted inputs to the system will not allow the derivation of the measure. However, in practice only a limited number of measurements will be possible, perhaps only one, so some method of assigning a measured value to the measurement results must be put in place. The most usual method of doing this is to try and establish the nature and extent of the distributions that will be produced by the measurement

Antenna measurement analysis and assessment 259 process by characterising the measurement system a priori to the measurements being performed.

8.3

Measurement error budgets

The usual pragmatic approach adopted is to establish an error budget for any measurement facility that is indicative of the error levels that can be expected and to quote the uncertainty and thus accuracy of the measurements performed within the facility in terms of this indicative error budget. This error budget consists of a quantitative assessment of the impact of the combination of individual error sources on the measurement result along with an estimation of the likely impact on the accuracy to which the measure of the measurand can be predicted. Thus an accurate and complete examination of the measurement errors associated with a given antenna or set of antennas can be used to assign an expected uncertainty, via extrapolation, to other antennas measured in the same facility.

8.3.1 Applicability of modelling error sources The assessment of measurement and test facilities to establish the diversity and magnitude of possible error sources that can introduce ambiguity into measurement results is a well developed subject. It does not depend entirely on empirical evidence and the use of simulations and other assessment techniques in error analysis can be valuable. However, it should be recalled that all simulation techniques are based on models and their associated algorithms all of which, at least under particular circumstances, are more or less inaccurate. In this text a model of the EM interaction based on Maxwell’s equations has been developed, as it is particularly applicable to the action of antennas and antenna measurements. However, it is important to realize that accurate models and measurements of the interaction of EM radiation with solid objects, for example, antennas must at least to some extent acknowledge the discrete particulate nature of the solid state. For macroscopic structures like antennas and antenna test ranges the errors produced by the uncertainty relations associated with momentum/position and energy/time could be assumed to be negligible and within the noise of the system. For example in his classic text Quantum Theory, David Bohm [7] calculates the inaccuracies associated with the standard macroscopic voltmeter pointing needle to be of the order of 10−26 cm, that is, 10−12 times smaller than a nucleus, a very small error indeed. However, although the quantum-mechanical aspects of the discrete nature of matter may only produce small error terms; the discrete nature of matter does contribute larger random fluctuations that produce errors in the system. These fluctuations referred to as noise have a variety of sources. The parameter that describes the distribution of energy amongst the discrete particles, that is, temperature gives rise to noise via a number of different thermal mechanisms. The most important of these within electronic measurement systems being Johnson noise, which results from the thermal motion of the conduction electrons in the solid. Flicker

260 Principles of planar near-field antenna measurements noise due to quality and stability of components may also be present and noise solely due to the discrete nature of the electrons carrying any current can at small signal levels also produce noise. Although cooling and quality control of components can reduce these effects, in any practical measurement system they cannot be completely removed. Thus the error of the resolution of the scale being the limiting error only applies above a certain noise floor level and for any electronic measurement system this system noise floor is a fundamental limitation in the accuracy of any measurement. These noise error sources can to an extent be modelled statistically and as such can be accommodated with classical EM theory. However, the addition of thermodynamic aspects into the modelling of a measurement system’s response can only be approximate as at root classical EM theory is based on differential equations and the continuous charge distributions associated with extended objects. The actual behaviour of discrete charged matter, in the circumstances we are primarily concerned with, is characterised by statistical concepts based on Fermionic half integer spin particulate distributions and these are concepts that have no classical interpretation. Thus accurate modelling of such effects requires either the abandonment of classical concepts or an ad hoc marriage of incompatible concepts to produce a hybrid model of the system response. If very high degrees of confidence in the results are required this is a process that only has limited applicability. The same limitations implied the application of individually robust but collectively incompatible concepts could also be seen to be present for a range of other environment aspects of any practical measurement system.

8.3.2 The empirical approach to error budgets This suggests that in the absence of some overriding definitive standard or infallible model, although modelling can be used to identify error sources, the only practical methodology for assessing the ability of any test facility to make measurements is by way of repetition of the measurement procedure. This repetition can be accomplished without alteration in the measurement configuration, to simply address repeatability and precision or with the inclusion of parametric variations to assess sensitivity. The parametric variations can also be used to assess the accuracy of the measurement if enough thought is devoted to the nature and extent of the parametric variations to be used, along with the types of analysis that are to be employed in the assessment process. In essence the desirability of such empirically based schemes is based on the same three practicalities highlighted as the basic reasons for attempting to characterise antenna performance in Chapter 1: • •

With careful control and design of the measurement procedures, it will produce results that reflect what it is that we are actually interested in. Since the assessment scheme is based on utilizing the measurement procedure it reflects what we can actually measure.

Antenna measurement analysis and assessment 261 • Since the data sets produced will have the same structure as the measurement data, it is particularly amenable to a range of rigorous mathematical/statistical examination techniques that can be justified rigorously and accurately using theory. The ingenious use of measurement repetition methodologies within a near-field range to examine precision, sensitivity and accuracy can be used to assess the impact of the main possible error sources for a near-field range. These being 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

probe relative pattern probe polarisation ratio probe gain measurement probe alignment error (angular errors) normalization constant (for specific types of gain measurement) impedance mismatch factor AUT alignment data point spacing (aliasing) measurement area truncation probe x, y, z positional errors AUT/probe multiple reflections receiver nonlinearly receiver dynamic range system phase error (flexing cable, rotary joints and so on) room scattering leakage and cross-talk unspecified random errors in amplitude and phase.

Some of these errors being inherently random, for example, error (17) and some are systemic, error (9). In this volume it is not proposed that these individual errors be examined along with methodologies for they are quantitative acquisition and classification as type A or type B. Ample literature exits as to the identification [8] and quantitative acquisition [9] of error sources. However, the implications of these error sources on the integrity of data sets recorded during antenna measurements and therefore their applicability as sources of information as to antenna performance must be addressed. To do this the output data sets generated in repeated trials, either with or without deterministic parameter variation, must be quantitatively analysed, compared and assessed to establish the level of correspondence between them.

8.4

Quantitative measures of correspondence between data sets

8.4.1 The requirement for measures of correspondence In this text, as with other measures, the measure of correspondence is the true value of some quantity that characterises the similarities or differences between data sets. Attempts to produce objective quantitative measures of correspondence

262 Principles of planar near-field antenna measurements between data sets that can be used to assess the accuracy, sensitivity and repeatability associated with the production of such antenna data has been widely reported [10–12]. The utility of such comparisons or, measures of adjacency between data sets, lies not only in their ability to determine the degree of similarity between various data sets but also in their ability to categorise the way in which these sets differ. Without the ability to produce such metrics of similarity any assessment as to the integrity of a data set is necessarily reduced to subjective value judgements. The data sets produced by near-field scanners comprise near-field measurements and post processed far-field predictions. Almost invariably it is the accuracy of the far-field predictions that are of importance so in this section we will confine ourselves to assessment of these data sets, although all the techniques could be deployed for the assessment of the raw near-field data. Classically the antenna pattern is considered as defining the relative power flux density propagating to or from an antenna in a specific direction, usually confined to power associated with a field of a given polarisation. However, the limitations imposed by this scheme, particularly with regard to the antinomy of the electron mass/energy and its implications for radiation resistance [13], mean that this interpretation of the action of antenna-to-antenna coupling can have limited applicability and can also in certain circumstances only provide limited insight. Although this concept has demonstrated its advantages in this text, alternative interpretations consistent with physical law, can provide additional insight as to how an antenna pattern might be interpreted and thus assessed and compared with other data sets representing antenna patterns. Descriptions of antenna patterns based on more fundamental physical interpretations, which concentrate on the irreversible macroscopic process of measurement, can be useful in assessing the process of radiative emission/absorption and are compatible with classical EM theory [14]. The Schrodinger wave equation, the Dirac equation or quantum electrodynamics (QED) can all be useful as conceptual models when attempting to assess data sets produced as a result of measuring antenna-toantenna coupling. This is because they all are empirically based interpretations that concentrate on the process of measurement prediction as opposed to the mechanism of EM interaction. Here, the antenna pattern is described by electron–photon–electron interactions that can only be specified by the probability of interaction, where this resultant probability is formed by the superposition of complex probability amplitudes. Thus, the antenna pattern, which is classically considered as defining the relative power flux density propagating to or from an antenna, is more correctly described as the probability of discrete electron–photon–electron interactions. Here, the probability of interaction is given over known solid angles, relative to the AUT placed at the centre of the inertial frame of reference. Consequently, the AUT pattern can be legitimately interpreted as a frequency distribution for these interactions that, when normalized to unity, can be recognized as an angular probability density function describing the process of EM interaction.

Antenna measurement analysis and assessment 263

8.5

Comparison techniques

Previously, the comparison of such large data sets that can be recognized as probability density distributions has been significantly simplified by the techniques of statistical pattern recognition [11]. The application of statistical techniques is particularly appropriate to antenna patterns as stated above when the nature of the pattern is not constrained to the conventional classical interpretation, that is, it is not restricted to being considered as an angular spectrum of EM waves propagating in diverse directions. Furthermore, the statistical approach has the inherent advantage that it can be used to consider the global, that is, non-local features of the data set and distils the complexity of the pattern into an alternative, dimensionally reduced, set of virtually unique features that can be utilized to describe the data. This extraction of global features is of particular relevance for antenna patterns as it takes account of the inherently anti-reductionist and holistic nature of the integral transforms that relate the aperture excitation to the angular far-field pattern. The holistic nature of the respective domains can be readily expounded as a change in any part of the spatial domain will result in a corresponding change to every part of the spectral domain and vice versa.

8.5.1 Examples of conventional data set comparison techniques An often-adopted technique for the quantitative comparison of antenna pattern data sets is the calculation of an equivalent multipath level (EMPL). This can be thought of as the amplitude necessary to force the different pattern values to be equal. If no account is to be taken of the phase of the patterns, as is often the case when assessing far-field data, then the EMPL can be expressed in terms of the amplitude of the samples as   ||I1 (i, j)| − |I2 (i, j)|| (8.17) EMPL|dB = 20 log10 2 Here, the factor of a half has been included as it is assumed that the ‘correct’ value lies between the two measured samples. Figure 8.7 illustrates the calculated EMPL comparison of cuts through the pattern, for two repeat scans of a high-gain antenna. Scan 1 and 2 in Figure 8.7 agree very closely; however, the calculated EMPL shown on the same plot clearly shows the extent and the nature of the differences between them. The differences can be seen to be a function of the signal strength as the EMPL closely mirrors the variation in the pattern. Also a left to right EMPL trend can be seen implying an asymmetry in the measurements. The EMPL level is an easily evaluated metric that is conceptually simple; broadly, it represents the size of signal required to make the two different signals the same. Thus, this technique is highly sensitive to the presence of constant or varying

264 Principles of planar near-field antenna measurements 10 0 –10 Normalized gain/dB

–20 –30

Scan 1 Scan 2 EMPL

–40 –50 –60 –70 –80 –90 –60

–50

Figure 8.7

–40

–30

–20

–10 0 10 20 Azimuth angle/degrees

30

40

50

60

Cuts through two scans and their calculated EMPL

amplitude displacements between the comparison data sets. The techniques can also be extended to take account of phase information by taking the difference between the respective complex in-phase and quadrature voltages separately. It is often used to represent the uncertainty associated with a given data set, that is, analogous to an error bar. The principal limitation of this technique is that it is a local feature comparison and as such it fails to produce a single or small number of coefficients that can be used to describe the data set. Instead, it produces a value for each element in the comparison data sets. This not only results in the EMPL having to be presented graphically, but also requires that the comparison data sets should contain an equal number of elements; although this difficulty can often be resolved with the use of interpolation. However, the difficulties of displaying the EMPL of a two-dimensional data set like a full antenna pattern means that graphical representation can be problematic. It is for this reason that only a cut through a full data set is shown in this text. Additionally as this is a local interval assessment technique, the results are often sensitive and discontinuous obscuring subtler underlying features. Smoothing, that is, by taking a ‘boxcar average’ can mitigate such effects, although this is undesirable as the fidelity of the response is compromised. Other conventional measures of correspondence, not so closely associated with antenna measurements, can also be used to assess the alignment of the data sets. The peak-signal-to-noise ratio (PSNR) is used to measure the difference between two data sets where the elements have values that lie in the range 0 ≤ |I (i)| ≤ 1. The PSNR is often given in decibel units (dB), which can be used to measure the ratio of the peak signal 1 V and the difference between two data sets I1 (i, j) and I2 (i, j), using

Antenna measurement analysis and assessment 265 the formula





PSNR|dB

      1  = 20 log10 

  N  
  (I1 (i, j) − I2 (i, j))2 (1/N )

(8.18)

i=1

Clearly when I1 (i, j) = I2 (i, j) for all values of (i, j) the two data sets are identical and thus, the PSNR in this case will be infinite. Although there are several different definitions for the signal-to-noise ratio, this choice is commonly employed for the purposes of digital image processing. For the data displayed in Figure 8.7 the PSNR is 54.5 dB a single numerical result that can be used to describe the correspondence of the two displayed data sets. Thus the entire data sets illustrated in Figure 8.7 can be reduced to a single value quantitatively defining the correspondence between the sets. This has the advantage of brevity and accuracy, however, much useful information is lost in reducing the dimension of the data. PSNR is the ratio of the largest signal to the arithmetic root mean square of the differences between the respective data sets and as such the presence of a constant offset between data sets will dominate the value of the PSNR. For the case of nearfield antenna measurements where the system is essentially acting as an interferometer between a reference and test path, an accurate absolute reference can only be obtained by way of a gain calibration, which is difficult and often inaccurate. Although the PSNR approach yields a single coefficient, it has the complication of having an infinite range, that is, 0 ≤ k ≤ ∞, when expressed in dB. In practice, this metric is found to be enormously sensitive with patterns that are essentially very similar yielding very large differences. Although this technique is global, that is, it takes account of differences between every part of each data set; it fails to take account of phase information and as it is a purely interval technique it is sensitive to the influence of outlying points. Finally, the evaluation of the PSNR requires that the respective data sets contain the same number of elements. If two signals, such as antenna patterns, vary similarly point for point then a measure of their similarity may also be obtained by taking the sum of the products of the corresponding pairs of points. If the two sequences of numbers are independent and random the sum of the products will tend to zero as the number of pairs of points is increased to infinity, as all numbers positive and negative are equally likely. If, however, the sum is finite and non-zero this will indicate a degree of correlation. A negative result will occur if one sequence increases as the other decreases. Thus the cross-correlation coefficient r between two data sequences I1 and I2 of equal length can be expressed as N 1
r= I1 (n) I2 (n) N n=1

(8.19)

266 Principles of planar near-field antenna measurements The 1/N term is included in the definition of the cross-correlation to ensure that the result is independent of the number of sampled points. Unfortunately, however, the value of the correlation coefficient will greatly depend upon the absolute values of the respective data sets. This can be overcome by normalizing the coefficient to the range −1 ≤ r ≤ 1. This in turn can be accomplished by normalizing the cross-correlation coefficient by the factor



      N    N N N  1



1 1 I12 (n) · I22 (n) = I12 (n) · I22 (n) N N N n=1

n=1

n=1

n=1

(8.20)

Thus the normalized correlation coefficient can be expressed as N


I1 (n) I2 (n)

n=1

r =    N   N 

2 2 I (n) · I (n) n=1

1

n=1

(8.21)

2

This is usually known as a cross-correlation coefficient and, as shown above, it is normalized so that its value always lies in the range −1 ≤ r ≤ 1 where, +1 implies perfect correlation, 0 signifies no correlation and −1 represents opposite signals, that is, signals out of phase by π. For the data sets shown in Figure 8.7 the cross-correlation coefficient is 0.999941 highlighting the very close correlation between the two data sets. The cross-correlation coefficient is a computationally expensive, general-purpose technique for obtaining a single quantitative correctly normalized measure of adjacency, a technique that is often used to calibrate time delays or offsets between theoretically identical signals. For the case of antennas, this would equate to determining the pointing error of a known antenna pattern function. It is a holistic metric that compares data over the entire extent of the two data sets although, unlike the previous techniques, zero padding the smaller data set can accommodate data sets of differing sizes. The cross-correlation coefficient can take account of amplitude and phase data provided that the data sets are represented in rectilinear form, however, as it is a purely interval technique that essentially relies upon a summation process, it is both potentially numerically unstable and sensitive to the presence of outlying points. In practice, minor differences between otherwise similar patterns are not well discriminated. For the case of the data sets presented in Figure 8.7 where a cross-correlation coefficient of 0.999941 was calculated, as with many observed levels of uncertainty for near-field antenna facilities the differences were mainly reported in the third or fourth decimal place.

Antenna measurement analysis and assessment 267

8.5.2 Novel data comparison techniques The measurement data sets that are produced in near-field antenna test ranges are complex and the integral transforms that are used on them are holistic. This suggests that assessment methods that are based on extracting features from the patterns that are global to the entire pattern, as opposed to specific to localized areas, would be useful in the assessment process. Therefore, the identification of pattern features that are a function of the entire pattern that can then be analysed to calculate a measure of comparison or adjacency in the feature space is desirable. A statistical interval measurement of correspondence based on calculating the moments of the antenna pattern when it is treated as a probability distribution has already been reported [11]. Moments of a probability density function describing area, centroid, variance, kurtosis and skewness, as shown below, yield 15 numerical values that characterise the data. These calculated numerical values effectively and dimensionally, reduce the data set to 15 numbers derived from the first 5 moments of the data which represent 15 global or universal features of the entire two-dimensional (θ , φ) pattern. A similar technique being possible with a smaller number of moments for data sets that only comprise cuts, for example (θ ). For data points i, j with signal value E(i, j): 0th moment (area) A=

J I



E (i, j)

(8.22)

i=0 j=0

1st moment (centroid) Centroid in x direction. I

mx =

J

1

(kx (i, j) · E (i, j)) A

(8.23)

i=0 j=0

Centroid in y direction. I

my =

J

 1

 ky (i, j) · E (i, j) A

(8.24)

i=0 j=0

where kx (i, j), ky (i, j) represent two coordinate points for the data point (i, j) 2nd moment (variance) Variance is a measure of the width or variation of a distribution. The variance of a distribution in the x direction is mx2 =

J  I

i=0 j=0



(kx (i, j) − mx)2 · E (i, j)

(8.25)

268 Principles of planar near-field antenna measurements The covariance is mxy =

J I



   (kx (i, j) − mx) · ky (i, j) − my · E (i, j)

(8.26)

i=0 j=0

The variance in the y dimension is my2 =

J  I



ky (i, j) − my

2



· E (i, j)

(8.27)

i=0 j=0

3rd moment (skewness) Skewness is a measure of asymmetry of a distribution around the sample mean. If the skewness is negative, the data are spread out more to the left of the mean than to the right of the mean. If the distribution is perfectly symmetrical then the skewness is zero. The skewness in the x dimension is mx3 =

J  I





(kx (i, j) − mx)3 · E (i, j)

(8.28)

i=0 j=0

The x dimension co-skewness is mx2 y =

J  I

  (kx (i, j) − mx)2 · ky (i, j) − my · E (i, j)

(8.29)

i=0 j=0

The y dimension co-skewness is 2

mxy =

J  I



  2 (kx (i, j) − mx) · ky (i, j) − my · E (i, j)

(8.30)

i=0 j=0

The skewness in the y dimension is my3 =

J  I



ky (i, j) − my

3



· E (i, j)

(8.31)

i=0 j=0

4th moment (kurtosis) Kurtosis is a measure of the peakness or flatness of a distribution relative typically to a normal distribution, that is, how outlier-prone a distribution is. A distribution with positive kurtosis is termed leptokurtic (clear peak to the distribution) whilst a distribution with negative kurtosis is termed platykurtic (flat top to the distribution)

Antenna measurement analysis and assessment 269 and intermediate values are termed mesokurtic. 4

mx =

J  I





(kx (i, j) − mx)4 · E (i, j)

(8.32)

   (kx (i, j) − mx)3 · ky (i, j) − my · E (i, j)

(8.33)

 2  (kx (i, j) − mx)2 · ky (i, j) − my · E (i, j)

(8.34)

 3  (kx (i, j) − mx) · ky (i, j) − my · E (i, j)

(8.35)

i=0 j=0

mx3 y =

J  I

i=0 j=0

mx2 y2 =

J  I

i=0 j=0

mxy3 =

J  I

i=0 j=0

my4 =

J  I



ky (i, j) − my

4



· E (i, j)

(8.36)

i=0 j=0

This set of 15 separate calculations can therefore be used to condense and reduce the dimensionality of the data in the far-field predicted files from arrays containing thousands of numerical entries to column vectors with 15 elements. Although the column vectors are far more difficult to interpret from the point of view of visual observation, since they are virtually unique to any given pattern they contain sufficient condensed information to define and compare data sets numerically. This 15-dimensional column vector virtually uniquely describes pattern data sets, however, some method of condensed data set comparison must be attempted. Any vector in a space can have another vector defined for which the inner product of the two is zero, that is, the two vectors are orthogonal. Therefore the product of a vector a with another vector b is zero if a and b are orthogonal to each other. This can be illustrated in the three-dimensional case for a and b. If two vectors in three-dimensional space are orthogonal to each other ax bx + ay by + az bz = 0

(8.37)

where the subscripts refer to the three spatial dimensions. If we are required to find b such that it is orthogonal this is simply accomplished since by and bz can be chosen arbitrarily, then bx will be given by bx =

−ay by − az bz ax

(8.38)

Therefore, if a is known, it is a simple task to calculate another vector b that is orthogonal to it. As already stated the product of any vector with another that is orthogonal to it is zero so this operation of taking the inner product can be used to compare vectors. For the vectors above and another vector c, if b·c = 0 then a = c. However, if b·c  = 0 then a  = c. Additionally the extent to which the inner product of

270 Principles of planar near-field antenna measurements b and c varies from zero can form the basis of a metric that can be used to compare the differences between vectors, a vector correlation error can be assigned to the patterns. The angles the vectors present to each other can be used as a measure of similarity, thus the magnitude and the argument are mag = bx cx + by cy + bz cz

(8.39)

From the definition of the dot product b·c cos θ =     b c

(8.40)

Thus cos θ = 

bx cx + by cy + bz cz  b2x + b2y + b2z cx2 + cy2 + cz2

(8.41)

Here, the extension to the 15 or in principle to an arbitrary, dimensional case is obvious. These two numbers can be treated as the abscissa and ordinate for a point on a plane [11]. The rectangular representation of the feature plane possesses an ambiguity, that is, soft singularity, as when the magnitude is zero the argument can take on any value. If, however, the plot is plotted in its polar form this can be avoided as   x = b · c cos θ (8.42)   y = b · c sin θ (8.43) In practice, this technique has been found to yield a reliable, sensitive measure of the degree of similarity between two patterns. Although there are an infinite number of choices for b that are all at a normal to a, an ambiguity is not introduced as the inner product between these b vectors and the c feature vector will be the same. The introduction of the b feature vector insures that a correlation results in 0 whereas the inner product between the two features vectors would guarantee that a match would result in 1. The method is similar to a simple correlation based on normalizing two vectors to unity and taking their product, where a product of unity means that the two vectors are identical. Both methods can be expanded to take account of the angles that the feature vectors present to each other and basically each method defines the extent of similarity between the data sets; in one case, identical data sets being defined by unity and in the other by zero. However, the establishment of an orthogonal test vector to define any differences as being a product greater than zero has the advantage that the numerical range of values for the vector components can be calibrated in terms of the components of a standard antenna. Thus the correlation can be defined relative to a standard antenna and the vector components need not necessarily be normalized. Figure 8.8 illustrates the side lobe levels of four measurements made of the same antenna. The plots represent cuts along the azimuth axis, scaled in wave vector space and dB below maximum, of the predicted far-field patterns from measurements made over 3×3 m and 4×4 m scan planes. The smaller scans (16) and (19), represent

Antenna measurement analysis and assessment 271 –35

Gain (dB)

–40

–45

–50

–55

–60 –0.7

–0.6

–0.5

–0.4

–0.3

–0.2

–0.1

Azimuth angle (radians) Ba_16

Figure 8.8

Ba_17

Ba_18

Ba_19

Cuts through the four data sets

repeat measurements separated by a period of hours and the scans (17) and (18) are each the product of scans made over a larger scan plane, to reduce truncation, but separated in time by a number of days. By inspection it can be seen that the scans appear to separate into two groups, suggesting that truncation effects are involved; but the extent of the variation between scans for such small differences is difficult to extract. Thus, by inspection the variation in the patterns between measurements is difficult to judge; it could, however, be estimated using the statistical image classification technique based on moments. For the entire patterns from which the four cuts shown above are taken, 15 component feature vectors were constructed from their moments. A feature vector for an equivalent isotropic pattern can be calculated and used as a metric to scale the measured pattern vectors. Since the individual moments of any pattern were not linearly related to each other, being summations of higher powers of the measured signal, they therefore vary greatly in size. However, they can be compared in absolute terms if first the individual feature vectors are scaled in terms of an equivalent isotropic radiator. Knowing that the product of two orthogonal vectors is zero, the inner product of the scaled pattern vectors was taken with a test vector which was orthogonal to the isotropic feature vector. The product of the isotropic vector and the test vector would be zero and the product of the pattern vectors with the test vector would define an interval data level of correlation between them. The angle that the vectors present to each other and the modulus of each of the four resulting scalar products were calculated. Their values were plotted against one another as a single point, respectively for each pattern. Figures 8.9 and 8.10 show the points for measurements

272 Principles of planar near-field antenna measurements Angle versus Modulas data sets (16) and (19) 0.28

Angle

0.279 0.278 0.277 0.276 0.275 309

309.1

309.2

309.3

309.4

309.5

Modulas

Figure 8.9

Points for patterns (16) and (19) Angle versus Modulas data sets (17) and (18) 0.275

Angle

0.2748 0.2746 0.2744 0.2742 0.274 410

Figure 8.10

415

420 Modulas

425

430

Points for patterns (17) and (18)

(16) and (19), the 3 × 3 m data sets, and measurements (17) and (18), the 4 × 4 m data sets, respectively. When plotted on the same graph, Figure 8.11, the four points are clearly consistent with the nature of the measured patterns, as, for example, (16) and (19) are the two closest, in fact indistinguishable on this scale, as they were derived from the data of two identical subsequent measurements. Measurements (17) and (18) represent data sets acquired over identical larger scan planes but they also show the possible error effects of drift and reproducibility (see Appendix B) as data sets (17) and (18) do not cluster as closely as (16) and (19). Therefore, very similar measured patterns should be characterised by points that cluster closely together and alternatively, distant points should be features of very different patterns. This means that, by means of the method described above, the complexity of this classification problem can be reduced down to a single point per pattern and the quantitative comparison of data sets can be accomplished using the well developed statistical technique of cluster analysis [15]. Whether conventional or novel, all the data assessment so far examined have addressed the interval nature of the data sets. However, there are two specific aspects

Antenna measurement analysis and assessment 273 Angle versus Modulas data sets (16), (17), (18) and (19) 0.277

Angle

0.2765 0.276 0.2755 0.275 0.2745 0.274 300

Figure 8.11

320

340

360 380 Modulas

400

420

Clustering of the four points

of the near-field antenna measurement methodology that handicap any interval pattern assessment of antenna patterns produced by near-field scanning. They are • •

The very high dynamic range of the measurement system. The interferometric nature of the measurement and the lack of uniformity of the reference source.

Both of these mean that interval assessment of the data sets can lead to misleading results as such an interval methodology depends on absolute signal levels, while the measurement technique is based on relative interferometric test and reference signal level measurements. An ordinal measure of association that overcomes this limitation can be derived if the interval nature of the data is ignored. If ranked, in terms of the amplitude, all antenna data sets sampled over the same intervals and containing the same number of elements are bijections between the set and itself, that is, permutations of the same elements [16]. The only possible variation is where these elements are to be found in the data sets, therefore all data sets containing the same number of elements represent different permutations of the same data. Thus it is the similarity of the permutations that is assessed and by inference, the data from which the permutations are constructed. This provides the opportunity to construct a measure of association based on the inverse permutation of data sets with respect to each other. This will produce a metric of correspondence that is immune to many of the pathological inconsistencies of such large interval data sets that affect interval assessment techniques. Any proposed objective measure of correlation or association, between data sets based on this methodology would be desired to be as follows: • • • •

A single coefficient, independent of scaling or shift due to the differences in reference levels. Insensitive to the large dynamic range of the data. Normalized, that is, give correlation value ranging between –1 and 1. Symmetrical or commutative to the operation of correspondence.

274 Principles of planar near-field antenna measurements If we assume a suitable methodology of defining a single coefficient and normalizing it, a value between −1 and 1 can be found. As the range of values in the permutation is limited to the number of elements in the set, the dynamic range is also limited but not restrictive. Additionally permutations mirror Abelian symmetry under a group operation [16] and therefore are by definition symmetrical and commutative. Thus such a measure of association based on the correlation of the permutations derived from the data sets is possible. Within the image processing community such a measure has already been devised and implemented [17] and it can be applied to the assessment of antenna patterns [12]. Following the development of [17], this measure is expressed in terms of a rank permutation which is obtained by sorting the data in ascending order and then labelling each element with integers accordingly, that is, [1, 2, 3, . . . , n] where n is the number of elements in the set. The correlation between two rankings can be considered to constitute a measure of closeness or distance. For a set of amplitude values I1 and I2 , let π1 be the rank of element I1i among I1 and π2 be the rank of element I2i among I2 . If the ranks are not unique, that is, if two elements have the same value then the elements are ranked so that the relative spatial ordering between elements is preserved. A composition permutation s is defined such that si is the rank of the element in I2 that corresponds to the element with rank π1i in I1 . Hence, for the case of a perfect positive correlation, s = (1, 2, 3, . . . , n), where n is the number of elements in the set. The definition of a distance metric to assess the distance between s and the identity permutation u = (1, 2, 3, . . . , n) will result in a measure of the distance between π1 and π2 . The distance vector dmi at each si is defined as the number of sj where j = 1, 2, 3, . . . , i which are greater than i. This can be expressed as dmi =

i
  J sj > i

(8.44)

j=1

where, J (B) is an indicator function which is defined as  1 when B true J (B) = 0 when B false

(8.45)

Here, dmi can be thought of as a measure of the number and the extent to which the elements are out of order. If I1 and I2 were perfectly correlated, then the distance measure will become a vector of zeros, that is, dm (s, u) = (0, 0, 0, . . . , 0)

(8.46)

The maximum value that any component of this distance vector can take is n/2, which occurs for the case of a perfect negative correlation. Finally, a coefficient of correlation can be obtained from the vector of distance measures as k (I1 , I2 ) = 1 −

2 maxni=1 dmi n/2

(8.47)

Antenna measurement analysis and assessment 275 First antenna pattern

Second antenna pattern

Obtain ranking

Obtain ranking

Compute permutation

Compute distance vector

Evaluate ordinal correlation coefficient

Figure 8.12

Flow chart for ordinal assessment technique

Here, if I1 and I2 are perfectly correlated, then (s = u) and k = 1. When I1 and I2 are perfectly negatively correlated then k = −1. A flow chart that schematically represents the procedure involved in computing either k is presented as Figure 8.12 along with examples of the use of the techniques on a few small simple data sets. Consider the two small data sets I1 and I2 where I1 = [10, 20, 30, 40, 60, 50]

(8.48)

I2 = [10, 20, 30, 40, 50, 60]

(8.49)

and Let the rank of I1 and I2 be π1 and π2 , respectively. Clearly π1 = [1, 2, 3, 4, 6, 5]

(8.50)

π2 = [1, 2, 3, 4, 5, 6]

(8.51)

and Now, s is a composition permutation. To find the first element of s, search through the elements of π1 to find the element containing the value 1 and make a note of its index.

276 Principles of planar near-field antenna measurements We use the value of the element of π2 with the index corresponding to the index of the element already found in π1 as the value of the first element of s. This is then repeated for each element in s. So consider finding the first element of s; here, the first element in π1 is equal to 1. So take the first element in π2 and place it in the first element of s. Now consider trying to find the fifth element of s, for example. So, here the sixth element of π1 is equal to 5. Now the sixth element of π2 contains the value 6 thus the fifth element of s contains the value 6. Repeating this procedure for each of the elements of s in turn yields s = [1, 2, 3, 4, 6, 5]

(8.52)

dmi is a distance vector where the value of the ith element of dmi depends upon the sum of a function that counts the number of out of order elements. This function contains the number of concurrent out of order elements, that is, if one element is out of order, the function takes the value 1, if two elements are next to one another and out of order, the function takes the value 2, and so on. Thus as s contains only one out of order element then dmi = [0, 0, 0, 0, 1, 0]

(8.53)

Since the maximum value of dmi is 1, the coefficient of correlation is k (I1 , I2 ) = 1 −

2×1 1 = 6/2 3

(8.54)

Several more examples are presented below with just the results shown. 8.5.2.1 Example 1 – perfect negative correlation Let, π1 = [6, 5, 4, 3, 2, 1] and π2 = [1, 2, 3, 4, 5, 6]. Then s = [6, 5, 4, 3, 2, 1]

(8.55)

Hence dmi = [1, 2, 3, 2, 1, 0]

(8.56)

Thus k (I1 , I2 ) = 1 −

2×3 = −1 6/2

(8.57)

8.5.2.2 Example 2 – partial negative correlation Let, π1 = [1, 2, 6, 5, 4, 3] and π2 = [1, 2, 3, 4, 5, 6]. Then s = [1, 2, 6, 5, 4, 3]

(8.58)

Hence dmi = [0, 0, 1, 2, 1, 0]

(8.59)

Antenna measurement analysis and assessment 277 Thus k (I1 , I2 ) = 1 −

2×2 1 =− 6/2 3

(8.60)

8.5.2.3 Example 3 – insensitivity to texture Let, π1 = [1, 3, 2, 4, 6, 5] and π2 = [1, 2, 3, 4, 5, 6]. Then s = [1, 3, 2, 4, 6, 5]

(8.61)

Hence dmi = [0, 1, 0, 0, 1, 0]

(8.62)

Thus k (I1 , I2 ) = 1 −

2×1 1 = 6/2 3

(8.63)

Here we obtain the same coefficient of correlation as we did for the case when π1 = [1, 2, 3, 4, 6, 5] and π2 = [1, 2, 3, 4, 5, 6] although clearly there are more out of order elements, that is, two, in this case. Such an occurrence is an example of insensitivity to ‘texture’ within a data set. Although this is clearly a disadvantage, it is not thought to be of primary importance when considering antenna radiation patterns. The ordinal process of ranking the data to produce permutations takes no account of, the absolute amplitude, the interval nature of the data or spatial angles at which the data is found. Thus every region of the pattern is judged to be equally important in the calculation of k irrespective of the amplitude of the measurement result. However, the ordinal measure of association can be readily modified to take account of different regions of interest by re-tabulating the data in such a way as to attribute more samples to regions of greatest interest prior to ranking the data. This approach minimises the impact of numerical instabilities as observed when using a purely interval assessment technique, whilst also minimising the impact of low level spurious signals as discussed above. It will in fact, produce a permutation that is weighted to take more account of the specific property of the patterns that is judged to be important, for example, higher signal levels. Assuming that the patterns are sufficiently well sampled this re-tabulation, which can readily be determined for the case of antenna radiation patterns, can be accomplished rigorously through the application of the sampling theorem, that is, Whittaker interpolation. Alternatively, this can be performed efficiently albeit with approximation, using piecewise polynomial functions, that is, cubic spline or cubic convolution interpolation. This will produce data sets that are biased towards the characteristics of the areas of interest by having more data points within these areas. An example of this technique would be to produce a hybrid interval/ordinal assessment technique based on placing more data points in the set in areas where the signal strength is higher. The idea of choosing to place the most samples where the field intensity is greatest is equivalent to choosing a sampling increment that is, at least to some extent, inversely proportional to the intensity of the field at that point.

278 Principles of planar near-field antenna measurements 0

Normalized gain (dB)

–10 –20 Ant 1

–30

Ant 2

–40

EMPL

–50 –60 –70 –50

–40

–30

–20

–10

0

10

20

30

40

50

Azimuth angle (degrees)

Figure 8.13

Cuts through antenna patterns and EMPL for Ant 1 and Ant 2

Figure 8.13 illustrates cuts through two data sets and the calculated EMPL between them. The measurements results antenna 1 (Ant 1) and antenna 2 (Ant 2) are of the same antenna with the measurement set-up altered so that the noise level in Ant 2 is greater at angles off boresight than for Ant 1. Clearly for the figure the EMPL reflects the presence of the noise in the measurements as the EMPL does not appear to be related to the signal level and in fact is smaller in regions where the recorded signal level is greatest. The use of the ordinal assessment technique illustrated above defines a k value of 0.8800 for the comparison between the two data sets. If, however, we are more interested in the regions of the pattern where the signal level is high the data could be re-tabulated prior to the ordinal assessment in terms of its absolute interval values. Figure 8.14 illustrates, what were initially the same data sets and their EMPL, re-tabulated such that the data sampling interval is a function of the signal level and thus there are far more data points in the regions of high signal level where we wish to concentrate our analysis. Clearly the main beam of the antenna where the signal level is highest now has far more data points than was previously the case. Thus an ordinal assessment will be biased to reflect the relationship between the two data sets in the regions of high signal level. In this case this results in a new k value of 0.9158 implying a closer correspondence between the two data sets in the regions in which we have most interest. Thus a hybrid interval/ordinal technique has been demonstrated that can be used to assess the data if a bias based on signal level is required. Clearly the introduction of bias need not be based on signal amplitude, angle or rate of change of signal level; these are examples of parameters that can be used to bias the data sets prior to assessment. The choice of variation in sample rate as a function of parameter of course depends on the nature of the correspondences that are being sought.

Antenna measurement analysis and assessment 279 0

Normalized gain (dB)

−10 −20 Ant 1 Ant 2 EMPL

−30 −40 −50 −60 −70 −50 −40 −30 −20 −10 0 10 20 30 Azimuth angle (degrees)

Figure 8.14

40

50

Re-tabulated cuts through antenna patterns and EMPL for Ant 1 and Ant 2

The ordinal and hybrid interval ordinal methodologies both overcome many of the disadvantages displayed in traditional and novel interval assessment strategies but they do place constraints on the types of data sets that can be compared. The comparison of permutations requires the two data sets to be, either identical in terms of sampling interval and extent and number of data points or for it to be possible to interpolate the sets to arrive at a situation where these conditions hold. For complex multidimensional data sets containing many different angles and frequencies this is often impossible. Some of these difficulties can be over come and different data set structures can be compared if prior to the ordinal assessment the data sets are categorised and then the relative frequencies associated with the categorisation are the subject of the ordinal measure of correspondence. Although there are a great many ways of categorising a given data set, one of the simplest is to divide the interval data set into a number of amplitude bins and to count how many elements fall within each bin, that is, a categorical interval methodology. Each data set that is compared will provide a single histogram that can be normalized before subsequently being processed to provide the measure of correspondence. Normalization would usually be accomplished by ensuring the total summation of the frequencies of the two sets to be compared was equal while the relative frequencies for the bins in each data set remained constant. Figure 8.15 illustrates two simple histograms constructed from the data sets illustrated in Figure 8.13. Ten bins placed linearly between −50 dB and 0 dB were used to construct the histograms and clearly the two histograms are similar; therefore the deployment of the range of assessment techniques already illustrated in the text would lead to objective measures of correspondence between these data set histograms. However, if the comparison of different data sets can be reduced to a comparison of their amplitude histograms a range of highly developed techniques primarily

280 Principles of planar near-field antenna measurements

Numbers of elements in each bin

Histograms of Ant 1 and Ant 2 30 25 20

Ant 1

15

Ant 2

10 5 0 1

2

3

4

5

6

7

8

9

10

Bins

Figure 8.15

Histograms of data sets for antennas 1 and 2

Levels

Sorted data and required bin levels 0 –5 –10 –15 –20 –25 –30 –35 –40 –45 –50 –55 0

10

20

30

40

50

60

70

80

90

100

Numbers of data points

Figure 8.16

Sorted amplitudes and required bin levels for Ant 1

developed in the areas of image processing can be deployed. Particularly accurate and effective are techniques based on the concept of histogram equalisation [18]. Histogram equalisation is concerned with producing a histogram where there are equal numbers of entries in each bin. Usually, this is done by varying the levels and sizes of the bins until an equal number of points are to be found in each. For example if the data for Ant 1 is sorted in terms of the number of data points at given levels, Figure 8.16 shown will be produced where 10 points are required to be in each bin. From this figure the levels that will be required to equalize a histogram of the data can be calculated and are shown on Figure 8.16. If a histogram of Ant 2 were constructed using the same bins then unless the patterns of Ant 1 and Ant 2 were identical the histogram for Ant 2 would not be equalized. Figure 8.17 shows the equalized histogram for Ant 1 using the values shown on Figure 8.16 to define the bins and the resulting unequalled histogram of Ant 2.

Numbers of elements in each bin

Antenna measurement analysis and assessment 281 Equalized histogram for Ant 1 18 16 14 12 10 8 6 4 2 0

Ant 1 Ant 2

1

Number of elements in total bins

Figure 8.17

2

3

4

5

6

7

8

9

10

Antenna 1 histogram equalized using levels shown in Figure 8.15

Cumlative best fit for Ant 1

100 80 60

Ant 1

40

Ant 2

20 0 0

Figure 8.18

1

2

3

4 5 6 Number of bins

7

8

9

10

Best fit line for Ant 1 cumulative data and Ant 2 scatter around it

Set 1

5

Set 2

5

0

0

–5

–5

–10

–10

–15

–15

–20

–20

–25

–25

–30

–30

–35

–35

–40

–40

–45

–45

–50

–50 50

Figure 8.19

100

150

200

250

300

50

100

Different patterns with identical histograms

150

200

250

300

282 Principles of planar near-field antenna measurements From Figure 8.17 it is clear that the bin levels calculated to equalize Ant 1 have not equalized Ant 2 and thus again there are clear quantifiable differences identified in the two patterns. If the above data is plotted, as a cumulative frequency as shown in Figure 8.18 it is clear that many standard regressive techniques can be deployed on the data to calculate measures of correspondence. The expression of the difference between two antenna patterns as the regression of a linear function allows a great deal of freedom as to which data assessment methodologies should be used; however, care should be taken when deciding on which assessment techniques to use. Figure 8.19 illustrates two very different theoretically possible antenna patterns that will have the same histograms; so care must be exercised at an early stage in the choice of assessment methodology.

8.6

Summary

The theory of measurement that is described but not rigorously illustrated in the early parts of this chapter highlights the requirement for obtaining quantitative, holistic and or local, measures of similarity between data sets that have been acquired within any measurement system. In addition to more conventional interval techniques usually used in antenna pattern assessment, that is, PSNR, EMPL and cross-correlation coefficient and, other newer and more sophisticated techniques have been presented. All of the conventional and novel comparison techniques have particular areas of applicability where their specific characteristics are suited to the abstraction of the large data sets to distil their important or relevant features so that these can be quantitatively assessed. However, of the conventional assessment techniques, the EMPL has been found to be particularly useful for graphically illustrating the differences between two one-dimensional pattern cuts, whilst the ordinal measure of association has been found to be particularly adept at describing differences between two-dimensional pattern functions. In principle, the hybrid ordinal-interval technique should offer significant advantages over the ordinal technique as it takes account of the interval nature of the data set, an aspect that is primary to the statistical moments methodology. However, it is clear that the new and novel antenna measurement techniques being pioneered at present offer an assessment challenge if the large volumes of data these techniques generate are to be quantitatively, effectively and concisely analysed and summarised. All the data assessment techniques at root depend on reducing the dimensionality of the data sets to make them more easily accessible. Antenna patterns acquired in test ranges may contain tens or hundreds of thousands of individual data points and the quantitative assessment of such large data sets is close to impossible without distilling the data down to manageable levels. However, it should be remembered that all data reduction techniques will involve the loss of some information from the data sets. Thus a combination of the techniques is recommended so that inaccurate conclusions are not drawn; for example, Figure 8.19 shows two plots that would yield identical histograms but if the interval methodology based on moments was also used on the data that exact nature of their differences could be extracted. Although these

Antenna measurement analysis and assessment 283 techniques, for the ease of explanation, have been illustrated on small data sets or cuts through much larger data sets as will be illustrated in Chapter 9, they are particularly effective when the nature and size of data sets makes conventional assessment techniques problematic. All of the above arguments confirm the applicability of such assessment methodologies to the measurement process. They also show how they can be used to enhance the understanding and interpretation of measurement data and thus lead to the extraction of information from measurement data. However, the use of such objective quantitative methods of assessment additionally allows the interpretation of the results by a whole new variety of users. Essentially a near-field antenna range, equipped with all of its shielding, absorbing, positioning, radio frequency (RF) and processing equipment, can • Sense its EM environment, the measurement process is essentially a quantified sensing process in which the system records a signal level as a function of the position/orientation of the probe. • Quantitatively assess the data it produces using the types of procedures discussed above. • Deploy a range of decision-making algorithms based on the output of the processing that allow the test range. • Without outside intervention, pursue certain courses of action, for example, a low level of correlation between repeat scans that can be highlighted by objective quantified assessment techniques could initiate a self calibration procedure. Although not at present comprehensively implemented in any range known by the authors, this could provide the antenna test range itself with the ability to instigate actions in response to the decisions it takes as a result of examining the processed data. By definition [19], a machine that senses its environment, processes information from its sensors and also processes other internal information, decides what to do next and then executes that response is a robot. Thus the new variety of users mentioned above could include intelligent, as the decisions may need to be made in the presence of levels of uncertainty, autonomous robotic systems. As well as running the measurement procedure, these systems can, by using objective measures of correspondence, examine data for inconsistencies and decide to run diagnostics on the instrumentation, on the measurements procedure or the AUT in response to decisions arrived at as a result of data processing and examination. Such intelligent autonomous measurement systems, acting on decisions made on the processed data, by removing the requirement for human involvement at every stage of course, offer many prospective advantages with respect to control, reliability and cost.

8.7

References

1 Langer, S.K.: An Introduction to Symbolic Logic, 3rd edn (Dover Publications Inc., New York, NY, 1967), pp. 42–3

284 Principles of planar near-field antenna measurements 2 Stevens, S.S.: Handbook of Experimental Psychology (G. Wiley, New Jersey, USA, 1951) 3 Barrow, J.D.: Impossibility, The Limits of Science and the Science of Limits, comments on the doctorial thesis Godel, K. (Oxford University Press, 1998), pp. 222–3 4 Boros, A.: Measurement Evaluation, English Translation Gabor, 1989 (Elsevier Amsterdam, 1989), pp. 34–42 5 Andrews, P.P., and Grewal, M.S.: Kalman Filters Theory and Practice Using Matlab, 2nd edn (John Wiley and Sons, New York, 2001), pp. 102–3 6 Geilert, W., Gottwald, S., Hellwich, M., Kastner, H., and Kustner, H.: VNR Concise Encyclopaedia of Mathematics (Van Nostrand Reinhold, New York, 1989), pp. 686–7 7 Bohm, D.: Quantum Theory (Prentice Hall, New York, 1951), p. 589 8 Newell, A.C.: ‘Error analysis techniques for planar near-field measurements’, IEEE Transactions on Antennas and Propagation, June 1988; 36(6):754–68 9 McCormick, J.: The Use of Secondary Spatial Transforms in Near-field Antenna Measurements, Ph.D. Thesis, The Open University, Milton Keynes, April 1999 10 McCormick, J., and Da Silva, E.: ‘The use of an auxiliary translation system in near-field antenna measurements’, Proceedings of the IEE International Conference on Antennas and Propagation, April 1997, Edinburgh, Vol. 1, p. 1.90 11 Gregson, S.F., and McCormick, J.: ‘Image classification as applied to the holographic analysis of mis-aligned antennas’, 22nd ESTEC Antenna Workshop on Antenna Measurements, Noordwijk, the Netherlands, May 1999 12 Gregson, S.F., McCormick, J., and Parini, C.G.: Measuring Wide Angle Antenna Performance Using Small Planar Scanners, IEE ICAP 2001, UMIST 13 Feynman, R.P.: The Feynman Lectures on Physics, Vol. II (Addison Wesley Publishing Company, California, 1964), chapter 28, pp. 28.1–28.10 14 Field, J.H.: ‘Relationship of quantum mechanics to classical electromagnetism and classical relativistic mechanics’, European Journal of Physics, March 2004;25:385–97 15 Day, F., Hand, D.J., Jones, M.C., Cunn, A.D., and McConway, K.J.: The elements of statistics (Open University/Prentice Hall, Harlow, 1995), Section 14.2, pp. 528–47 16 Stewart, I.S.: Concepts of Modern Mathematics (Dover Publications Ltd, New York, NY, 1995), pp. 100–9 17 Bhat, D.N., and Nayar, S.K.: ‘Ordinal measures for visual correspondence’, Technical Report, CUCS-009-96, Columbia University Centre for Research in Intelligent Systems 1996 18 Russ, J.C.: The Image Processing Handbook, 3rd edn (IEEE Press, New York, NY, 1999), pp. 233–9 19 Course Team, Robotics and The Meaning of Life, Section 1.2 (The Open University, Milton Keynes, 2004), pp. 3–4

Chapter 9

Advanced planar near-field antenna measurements

9.1

Introduction

This chapter presents a brief introduction to a number of the more advanced, and more recently developed topics associated with planar near-field antenna measurements. These include (1) active alignment correction which seeks to improve the accuracy with which the boresight direction of an antenna is known, (2) position correction techniques for improving the flatness and straightness of the sampling grid, (3) phase recovery which enables measurements to be made, where obtaining a direct phase reference would be impractical, (4) microwave holographic metrology (MHM) which is used for performing aperture diagnostics, (5) auxiliary translation and auxiliary rotation which are used to minimise truncation and (6) the poly-planar technique that is used to mitigate measurement truncation.

9.2

Active alignment correction

The importance of accurately determining the electrical boresight direction of an antenna with respect to a mechanical datum can be illustrated by considering the implications of an error in the pointing of a high gain space telecommunications antenna aboard a geostationary satellite. By way of illustration, Figure 9.1 contains a schematic representation of Europe and North Africa as seen from an orbit radius of 42,164.14 km, that is, an altitude of 35,786 km, a mean Earth radius of 6378.14 km with a subsatellite latitude of 0◦ and a subsatellite longitude of 13◦ . Clearly, an error in the pointing of the antenna of as little as 1◦ will displace the pattern by many kilometres over the surface of the earth. Consider, for example, the distance between London (51.500◦ , −0.083◦ ) and Paris (48.833◦ , 2.333◦ ), shown in Figure 9.1. The distance between London and Paris over the surface of the Earth can be calculated from Napier’s spherical trigonometry cosine rule for sides [1] and is

286 Principles of planar near-field antenna measurements 10 9 8 London + + Paris

El (deg)

7 6 5 4 3 2 –6

Figure 9.1

–5

–4

–3

–2

–1

0 1 Az (deg)

2

3

4

5

6

7

Plot of Europe and North Africa as viewed from geostationary orbit

found to be 343.2 km. This corresponds to an angle of less than 0.4◦ as seen from orbit. As edge-of-cover (EOC) gain slopes of 10 or 20 dB per degree are becoming common place and as coverage regions are typically defined with reference to geographical features, that is, coastlines, rivers, political borders, and so on; fractions of a degree error in antenna pointing between design and in-service performance can have significant implications on mission effectiveness. This is further complicated as the spacecraft mission lifetime is dictated by fuel usage that in turn, depends on the tolerances set on spacecraft attitude control and by the fact that the antenna alignment cannot be adjusted in orbit. Similarly, the polarisation purity of linear antennas, required for polarisation reuse schemes, dictates that antenna alignment in test and installation is known to fractions of a degree. Another example that requires accurate antenna-to-range alignment is the configuration and pattern measurement of large active antennas. A recent example was the test and measurement campaign of the advanced synthetic aperture RADAR (ASAR) antenna that had a dimension of 10 m by 1.3 m and utilized an array of 320 active transmit/receive (TR) modules. The antenna weighed approximately 750 kg and was required to be a deployed structure designed for a zero gravity environment. Figure 9.2 (see p. C7), shows the ASAR antenna deployed in a planar facility together with its crucial zero-g mechanical support. Testing such an array therefore poses challenges for the measurement facilities that are not encountered in typical applications. Specifically, obtaining the electrical boresight relative to a fixed mechanical datum is crucial as the null-to-null beamwidth is just a little more than half a degree. The combined mass of the instrument and its zero gravity support is such that accurately aligning the antenna to the axis of the

Advanced planar near-field antenna measurements 287 range was found to be quite impossible. Thus, existing alignment techniques for planar near-field testing are unable to deliver the required degree of accuracy and are generally inappropriate for use with such large, heavy gravitationally sensitive antenna assemblies. The normal aim of a range measurement process is to characterise the radiation pattern of the antenna under test (AUT) at a very great or infinite, distance with reference to an angular or other coordinate system defined with respect to the mechanical interface. This data can then be utilized to establish the extent to which the instrument fulfils its requirements. The angular accuracy required is usually in the order of ±0.02◦ , particularly where the antenna is to be mounted on a spacecraft intended for low earth or especially geostationary orbit. If an instrument is characterised whilst its axes are not perfectly aligned to those associated with the range, a non-rectilinear correction must be applied to the data set so that range independent predictions can be made. The coordinate free form of the near-field to far-field transform detailed in Chapter 4 together with the probe pattern correction algorithm that was set out in Chapter 5, automatically enable far-field parameters to be determined relative to a fixed mechanical interface that is not necessarily aligned to the axes of the range. Thus, if the orientation of this mechanical interface is known the sampled near-field data can be used to provide alignment corrected far-field patterns. In the sections that follow, three techniques are set out that can each be used to acquire this data. Essentially for the case of the planar near-field to far-field transformation the application of alignment correction data is handled rigorously by expanding the plane wave spectrum (PWS) on an irregular grid in the range system. This irregular space corresponds to a regular angular domain in the antenna mechanical system (AMS). With the transformation of the measured Cartesian field components from the range polarisation basis into the antenna polarisation basis the required isometric rotation is completed. The scanning probe can be thought of as a device that spatially filters the fields received from different parts of the AUT. In a planar range; the effects include something similar to a direct multiplication of the far-field probe pattern with the far-field antenna pattern. This can be seen to be a direct result of the nature of the convolution theorem [2] and can be visualized directly from the mechanical operation of the scanner. It is not usually possible to neglect these effects in the planar range because of the large angles of validity required and the short measurement distance employed.

9.2.1 Acquisition of alignment data in a planar near-field facility The antenna-to-range alignment can be acquired by optical means through the use of optical coordinate measuring devices such as a laser tracker, via theodolites, mirrors and optical cubes or by utilizing a precision mechanical contact probe. A mobile optical three coordinate measuring machine determines points in space relative to the tracker by means of reflecting a laser beam off an optical target usually consisting of a corner reflector, that is, a trihedral, of various sizes depending upon the accuracy required. The larger the size of the trihedral, the greater the accuracy

288 Principles of planar near-field antenna measurements obtained. The device is aligned with the local gravity vector before the polar and azimuth angles are obtained from high precision angular encoders within the tracking head of the device. The distance r, however, is measured with an interferometer by counting interference fringes between the reflected and transmitted waves. In this way, the Cartesian coordinates of a point in space can be determined relative to the coordinate measuring machine directly from the measured spherical coordinates. Such a system is typically capable of acquiring a stationary target over an angular range of ±235◦ in φ and ±45◦ θ with an angular accuracy of 0.14 s, and a distance of between 25 mm and 35 m with an accuracy of 1.26 mm in ideal conditions. As the distance ordinate is acquired by counting wavelengths, the uniformity of optical density of the medium in which the laser is propagating is crucial in order that reliable data can be obtained. Typically, such laser interferometers can suffer from 50 µm of noise per 3 m distance and this can restrict the maximum dimension of the object to be measured. By acquiring a number of points, typically at least three across the measurement plane and a similar number of points across the aperture of the AUT, the alignment of the AUT with respect to the range can be readily determined. Although, by acquiring a large number of points this technique can be seen to be flexible, accurate and repeatable, the cost of such devices at the current time can be prohibitive. Consequently, this technique was precluded from further investigation. Alternatively, an antenna can be aligned optically to the axes of a planar facility with the use of a theodolite. Although a theodolite can be used to acquire the spherical angles to a target, that is, a tooling ball, unlike the laser tracker, the distance to the target is not measured. However, if a precision length bar with targets mounted at each end is used with suitable software, this method can provide a full three-dimensional coordinate measurement of a point with reference to three fixed targets forming a Cartesian coordinate system. A theodolite can also be used to acquire the normal of a mirror, or other optically flat object, that is, optical cube, a practice commonly termed auto collimating. In this case, an antenna can be aligned to the axes of the range if the AUT has attached a mirror whose normal is parallel with the mechanical boresight and the range has a similarly well-aligned reflector. Thus, the object of the alignment process is to make the reference and AUT mirrors parallel. The roll can then be determined relative to the local gravity vector by means of an inclinometer. In practice, however, this can become both impractical and inaccurate, as it often proves impossible to view simultaneously both the reference and antenna reflectors. In this eventuality, an auxiliary mirror is placed on the probe carriage and aligned to the reference mirror that can subsequently be translated in the xy-plane to a position that is convenient. However, without the provision of a mechanism for applying fine mechanical adjustments, such alignments are impossible to perform with the required degree of accuracy. A detailed description of such techniques can be found in Reference 3. Alternatively, if the scanning probe is replaced with a precision mechanical contacting probe an alternative accurate, cost-effective alignment process can be effected. However, as the interface for alignment information is by way of correctly normalized orthogonal direction cosine matrices, the method of derivation

Advanced planar near-field antenna measurements 289 of alignment information is unimportant and any of these techniques can be accommodated.

9.2.2 Acquisition of mechanical alignment data in a planar near-field facility The following acquisition of alignment data in a planar facility is based upon the premise of being able to acquire four points on the antenna mechanical interface plane in the range coordinate system. In practice, this is achieved by replacing the nearfield probe with a precision mechanical contacting probe. Such probes are commonly used for surface profiling and coordinate measuring. From these four points we can construct four normal vectors where the average angle between each can be used to calculate a root mean squared (r.m.s.) angle that can be taken as an indication of the measurement error. The correct projection of each Cartesian component of the antenna system onto each Cartesian component of the range system determines the antennato-range mechanical alignment direction cosine matrix. For the case where there is a suitable datum available on the antenna, a roll angle can be deduced from any of the edge vectors. In Figure 9.3 the antenna mechanical system is denoted by the abbreviation AMS whilst the range fixed system is denoted the abbreviation RFS. Consider a planar-aperture antenna as shown in Figure 9.3. We measure the range coordinates of four points on the antenna aperture plane. From these four points, we construct four normal vectors, one from each of the permutations of the three points. The angles between each normal and the others are calculated and the r.m.s. of these angles taken as an indication of measurement error. We can thus obtain a single mean normal from the average of the four individual normal vectors. This vector is

YRFS

YAMS

XAMS

AUT aperture plane Z AMS XRFS

Z RFS

Figure 9.3

Coordinate system of AUT installed in planar facility

290 Principles of planar near-field antenna measurements za = ZAMS and hence, we enter this as the bottom row of the [B] matrix       xa B11 B12 B13 xRFS  ya  =  B21 B22 B23  ·  yRFS  za B31 B32 B33 zRFS

(9.1)

that is B31 = Za · XRFS

(9.2)

B32 = Za · YRFS

(9.3)

B33 = Za · ZRFS

(9.4)

Now, we assume initially that the antenna is not rolled around the Za -axis. Hence B12 = Xa · YRFS = 0

(9.5)

From this we obtain 2 2 + B13 =1 B11

(9.6)

Xa · Za = 0

(9.7)

B11 B31 + B13 B33 = 0

(9.8)

and so Hence, we may deduce that B11 = 



1

2 /B2 1 + B31 33

B12 = 0 B13 =

−B11 B31 B33



(9.9) (9.10) (9.11)

This gives us the top row of the B matrix. The middle row is then found from the cross product of the top and bottom rows. If as illustrated in Figure 9.4, we now obtain a measure of the AUT roll around Za , we may incorporate this by pre-multiplying [B] by the Zroll matrix   cos φ sin φ 0  − sin φ cos φ 0  (9.12) 0 0 1 Here, φ is the angle of roll around the Za -axis. For the case where a suitable datum is available on the antenna, this angle may be deduced from the information we already have. For example, if the four points acquired are the corners of the rectangular aperture of a horn, then we may deduce the AUT roll from the top edge vector. We may also use the bottom edge vector and take an average to improve accuracy.

Advanced planar near-field antenna measurements 291 YRFS

YAMS

XAMS Vt

AUT aperture plane Z AMS

q nˆ

XRFS

P Z RFS

Figure 9.4

Coordinate system of AUT installed in planar facility including roll

Here, nˆ is a unit normal to the aperture plane, P is orthogonal to nˆ and yRFS , q is orthogonal to P and nˆ and VT is the top edge vector. Clearly, from Figure 9.4, we may write that    xˆ yˆ zˆ   (9.13) p = n × yˆ =  nx ny nz  = −nz xˆ + nx zˆ  0 1 0  Here, nˆ is a unit normal to the aperture, hence    xˆ yˆ zˆ 

 q = p × n =  −nz 0 nx  = −nx ny xˆ + n2z + n2x yˆ − nz ny zˆ  nx ny nz  Hence

 q · vt roll = arcsin          q vt

(9.14)



(9.15)

These expressions that are used to derive the antenna-to-range direction cosine matrices can be used within an active alignment verification programme.

9.2.3 Example of the application of active alignment correction In order that the validity of the alignment techniques discussed above can be thoroughly tested, a low gain instrument, standard gain horn (SGH), can be acquired at a variety of different orientations with respect to the range axes. A low gain instrument

292 Principles of planar near-field antenna measurements is preferable here, as the signal levels are relatively high at large angles so that errors in the isometric rotation are clearly observable. The antenna-to-range alignment is measured, as described above, in each case and then the data transformed. The typically four antenna-to-range alignments used were Set 1. AUT nominally aligned to the range Set 2. AUT nominally aligned to the range but the scanning probe rotated around the range z-axis Set 3. AUT misaligned in azimuth to the range Set 4. AUT grossly misaligned in azimuth, elevation and roll to the range. The coordinates of the four corner points of the aperture of the SGH are acquired whilst the AUT was orientated in each of the positions described above. The acquired coordinates can be found contained within Tables 9.1, 9.2 and 9.3 for each of the antenna orientations. Set 1 and Set 2: Table 9.1

Bottom left Top left Top right Bottom right

Acquired coordinates of corners of SGH for Set 1 and 2 x

y

z

268.985 268.967 280.447 280.453

−0.265 8.2470 8.2510 −0.291

3.2612021 3.2622638 3.2297205 3.2211745

Direction cosine matrix for Set 1 and Set 2.  9.99994536E-01 −9.62894467E-04 [A] =  9.64654645E-04 9.99999381E-01 3.16202475E-03 −5.58136719E-04

 −3.16256120E-03 5.55088973E-04  9.99994845E-01

Set 3: Table 9.2

Bottom left Top left Top right Bottom right

Acquired coordinates of corners of SGH for Set 3 x

y

z

266.840 266.770 278.236 278.294

−3.403 5.104 5.062 −3.420

1.7299882 2.5996785 3.3406798 2.4692953

Advanced planar near-field antenna measurements 293 Direction cosine matrix for Set 3.   9.97777254E-01 1.05650315E-02 6.57946161E-02 [A] = −1.72130108E-02 9.94705988E-01 1.01309969E-01 −6.43759556E-02 −1.02217306E-01 9.92676865E-01 Set 4: Table 9.3

Acquired coordinates of corners of SGH for Set 4

Bottom left Top left Top right Bottom right

x

y

z

271.346 267.075 277.059 281.300

1.175 8.546 14.210 6.868

3.0288871 2.1037310 1.5844619 2.5065814

Direction cosine matrix for Set 4.  8.69528605E-01 4.91830703E-01 [A] =  −4.93672376E-01 8.62891045E-01 −1.44079562E-02 1.16282431E-01

 −4.49729277E-02 −1.08197175E-01  9.93111679E-01

The corrected far-field data can be found in terms of greyscale (checkerboard) plots Figures 9.5–9.8. The far-field plots consists of Ludwig III vertical copolarisation and crosspolarisation data tabulated on a regular 81-element by 81-element grid in an azimuth over elevation coordinate system. The data has only been plotted out to ±40◦ in azimuth and elevation in order that the entire far-field data set should be free from first-order truncation effects. Copolar power (dB)

Cross-polar power (dB)

40

40 –10

–10 –20

20

–20

20

–30

–40

0

–50 –60

–20

El (deg)

El (deg)

–30

–40

0

–50 –60

–20

–70

–70 –40 –40

–80 –20

Figure 9.5

0 Az (deg)

20

40

–40 –40

–80 –20

0 Az (deg)

Far-field copolar and cross-polar pattern of Set 1

20

40

294 Principles of planar near-field antenna measurements Copolar power (dB)

Cross-polar power (dB)

40

40 –10

–10 –20

20

–20

20

–30

–40

0

–50 –60

–20

El (deg)

El (deg)

–30

–40

0

–50 –60

–20

–70

–70 –40 –40

–80 –20

Figure 9.6

20

0 Az (deg)

40

–40 –40

–80 –20

20

0 Az (deg)

40

Far-field copolar and cross-polar pattern of Set 2 Copolar power (dB)

Cross-polar power (dB)

40

40 –10

–30 –40

0

–50 –60

–20

–20

20 El (deg)

20 El (deg)

–10

–20

–30 –40

0

–50 –60

–20

–70 –40 –40

–80 –20

Figure 9.7

0 Az (deg)

20

40

–70 –40 –40

–80 –20

0 Az (deg)

20

40

Far-field copolar and cross-polar pattern of Set 3 Copolar power (dB)

40

40

Cross-polar power (dB) –10

–10 –20

El (deg)

–30 –40

0

–50 –60

–20

–20

20

–30 El (deg)

20

–40

0

–50 –60

–20

–70

–70 –40 –40

–80 –20

Figure 9.8

0 Az (deg)

20

40

–40 –40

–80 –20

0 Az (deg)

Far-field copolar and cross-polar pattern of Set 4

20

40

Advanced planar near-field antenna measurements 295 MHM or aperture diagnostics is a powerful technique that can also be usefully extended with the incorporation of active alignment correction. Figures 9.9 (see p. C7) and 9.10 (see p. C8) contain plots of reconstructed near-field planar illuminations derived from the same near-field data set which was acquired whilst the antenna was grossly misaligned within the range. The small vertical phase taper evident on the corrected aperture phase function results from the small elevation pointing error which was not compensated for. Figure 9.9 illustrates the aperture illumination function of the circular array antenna in the absence of active alignment correction. Conversely, Figure 9.10 show the reconstructed aperture illumination function once the near-field data has been corrected. The differences between the two sets of results are not characterised by a simple linear phase taper. The uncorrected results are focused when x =0, where the aperture plane and the translated measurement plane intersect, and become progressively diffracted as the magnitude of x increases and the two planes diverge in space. However, when corrected, the reconstructed image is focused, that is, free from diffraction effects, as only then is the reconstructed plane coincident and synonymous with the antenna aperture plane and shows a clear resemblance with the physical aperture as shown in Figure 9.11. Obtaining a quantitative verification of an aperture diagnostics algorithm is difficult in the absence of detailed design information. However, as the aperture plane is

Figure 9.11

Photograph of circular slotted waveguide array antenna (Courtesy of SELEX)

296 Principles of planar near-field antenna measurements Freq. 9 (GHz)

Freq. 9 (GHz)

−10

−30

Measured Translated EMPL

−40

Phase (deg)

Power (dB)

−20

50 0

Measured Translated

−50

−50

−100

−60 −70

15 0 10 0

−150 −.2

Figure 9.12

−.1

0 x (m)

0.1

0.2

0.3

−0.2

−0.1

0 x (m)

0.1

0.2

0.3

X-polarised horizontal cut of measured and translated near-field data

merely one of an infinite number of planes in space, the verification of a plane-to-plane (PTP) translation would yield confidence in the procedure. Figure 9.12 contains a comparison of the measured and reconstructed that is planeto-plane translated, amplitude and phase of the x-polarised near electric fields through a horizontal cut in space. The differences between the translated and measured data sets are small with an equivalent multipath level (EMPL) of typically −50 dB in both planes. The phase functions are very similar with a small constant phase offset observable that results from small amounts of drift in the system that is an artefact of a gradual change in temperature within the facility between the two acquisitions. The agreement shown would be less impressive if the measurements were made at the larger AUT-to-probe separation that is a direct consequence of increased measurement truncation. By way of an illustration, as the PWS method essentially enables the entire electromagnetic (EM) six vector to be determined over an entire half-space, it is not only possible to reconstruct the fields over a plane, but also over a sphere or any other surface of interest. This is illustrated in Figure 9.13 (see p. C8) where the near-field has been computed over the xz- and yx-planes and which show the field as it radiates away from the antenna.

9.3

Amplitude only planar near-field measurements

The phase retrieval problem arises in applications of EM theory in which wave phase is apparently lost or is impractical to measure and only intensity data are available. The planar near-field methodology as treated herein requires holographic measurements to be made. In other words, in order that the angular spectrum can be obtained knowledge of both the amplitude and phase of the field must be available. However, direct measurement of phase becomes progressively more difficult as the frequencies concerned become higher. Matters are further compounded since even antennas that are comparatively modest in physical size when operating at high frequencies, will constitute electrically large instruments.

Advanced planar near-field antenna measurements 297 Electrically large scan planes pose significant difficulties for the planar methodology which requires that a probe be moved across a correspondingly electrically large (many hundreds of wavelengths across) scan plane whilst maintaining the same positional tolerances and phase stability of the moving guided wave path, both as a function of position and time. Thus, an alternative method that removes the requirement to measure phase is often thought desirable in circumstances such as these. Also, losses inherent within the guided wave path limit the dynamic range of the receiving system increasing the noise within the measured signal and degrading the accuracy of the measured phase. Many alternatives are available for recovering the phase from amplitude only measurements; however, in the following sections only the most applicable and only those that are most readily implemented are discussed. Specifically, the use of reference beam addition is not considered; instead the use of multiple intensity distributions which permit the use of iterative computational procedures are developed, as this experimental arrangement is perhaps more convenient for the majority of experimentalists.

9.3.1 PTP phase retrieval algorithm The PTP algorithm essentially entails taking two amplitude only measurements over parallel planes in the near-field of the antenna that are separated by a known distance as illustrated in Figure 9.14. Essentially, we are using the PTP transform as used for MHM to calculate the field over one plane from knowledge of that field over another. Now as was shown in Chapter 4 the angular spectrum and the boundary conditions, that is the measurement, are related to one another through the Fourier relationship  ∞ ∞   F kx , ky = {u (x, y, z = 0)} = u (x, y, z = 0) ej(kx x+ky y) dx dy −∞ −∞

and

    u (x, y, z) = −1 F kx , ky e−jkz z  ∞ ∞   1 F kx , ky e−j(kx x+ky y+kz z) dkx dky = 2 4π −∞ −∞

(9.16)

(9.17)

Thus, if the field is known over one plane, which can be defined to be at z = 0, then the field over another parallel plane, that is the PTP transform, can be expressed as   (9.18) u (x, y, z) = −1 {u (x, y, z = 0)} e−jkz z Here, as only knowledge of the propagating field is known, the limits of integration in the spectral domain are truncated to visible space. This equates to imposing a filter function on the spectral field components that removes all evanescent components. As these components decay exponentially away from the aperture then as the field is propagated towards the aperture they will exponentiate and could cause the algorithm

298 Principles of planar near-field antenna measurements 2nd measurement plane 1st measurement plane AUT aperture plane

AUT (Rx) RSA(Tx)

Figure 9.14

Measurement configuration of the PTP algorithm

to become unstable. In practice as the measurements are made outside the reactive near-field region, fields in invisible space will only enter into the algorithm through numerical noise. Using this PTP transform that can be implemented very efficiently using the fast Fourier transform (FFT), the PTP phase retrieval algorithm can be described as follows: 1. 2. 3. 4. 5. 6. 7. 8.

Measure the amplitude of the field over plane 1. Measure the amplitude of the field over plane 2. Use PTP transform to propagate the AUT aperture fields to plane 2 from plane 1. Replace the amplitude estimation at plane 2 with the measured amplitude at plane 2. Use PTP transform to propagate the fields back to plane 1. Replace the amplitude estimation at plane 1 with the measured amplitude at plane 1. Repeat steps 3–6 until amplitude on plane 1 (or plane 2) has converged to within a prescribed tolerance. Transform the fields to the far-field using standard algorithm.

Figure 9.15 contains example plots of the measured amplitude taken across two parallel planar surfaces with a separation between the AUT and the probe of z = 0.105 m and z = 0.235 m respectively taken at mm wave frequencies.

Advanced planar near-field antenna measurements 299 Ex (dB)

Ex (dB) 0.3

0.3

–10

–10 0.2

0.2

–20

–20 0.1

y (m)

–30 0

–40

y (m)

0.1

–30 0

–40

–0.1

–0.1

–50

–50

–0.2

–0.2

–60 –0.3 –0.2 –0.1

Figure 9.15

0 0.1 0.2 0.3 x (m)

–0.3 –0.2 –0.1 0 0.1 0.2 0.3 x (m)

Near-field measurements taken over two different parallel planes

1

1

0.8

0.8

–10

0.6

0.2

–20

0.4

–30

0.2 v

0

–40

–0.2 –0.4

–50

–0.6

–20 –30

0 –0.2 –0.4

–40 –50

–0.6

–60

–0.8 –1 –1 –0.8 –0.6–0.4–0.2 0 0.2 0.4 0.6 0.8 1 u

Figure 9.16

–10

0.6

0.4

v

–60

–0.3

–0.3

–0.8

–60

–1 –1 –0.8–0.6–0.4–0.2 0 0.2 0.4 0.6 0.8 1 u

Comparison of angular spectrum obtained through direct measurement (left) and phase retrieval (right)

These amplitude patterns can be used with the PTP phase retrieval algorithm as described above to reconstruct the associated phase patterns. When the phase retrieval algorithm has converged sufficiently the measured amplitude and reconstructed phase functions can be transformed to the angular spectrum. Figure 9.16 contains a comparison of the angular spectra obtained from a direct holographic measurement and from the PTP phase retrieval algorithm. The figure on the right was derived using retrieved phase function. Clearly, the angular spectra as recovered by conventional amplitude and phase, that is, coherent, measurements agree with those obtained from phase recovery. However, the phase retrieval patterns clearly contain a greater amount of speckle noise. This is perhaps more apparent in Figure 9.17 which contains cardinal cuts of the respective patterns together with the equivalent multipath level. Although the cuts agree, the impact of the noise is clearly illustrated in the EMPL that is perhaps as

300 Principles of planar near-field antenna measurements BT antenna

BT antenna Plane 1 Plane 2 Phase retrieval EMPL

0 –10

–10

Power (dB)

Power (dB)

–20 –30 –40

–20 –30 –40

–50

–50

–60

–60

–70 –1 –0.8 –0.6 –0.4 –0.2

Figure 9.17

0 u

0.2

0.4

0.6

0.8

Plane 1 Plane 2 Phase retrieval EMPL

0

1

–70 –1

–0.8 –0.6 –0.4 –0.2

0 v

0.2

0.4

0.6

0.8

1

Measured and phase recovery result from 76 wavelength diameter cassegrain antenna Ex (dB)

Ey (dB)

0.3

0.3

150

0.2

100

0.1

50

–10 0.2 –20 –30 0 –40 –0.1

y (m)

y (m)

0.1

0

0

–0.1

–50

–0.2

–100

–50 –0.2 –60 –0.3

–150

–0.3 –0.3 –0.2 –0.1

Figure 9.18

0 0.1 0.2 x (m)

0.3

–0.3 –0.2 –0.1

0 0.1 0.2 x (m)

0.3

Aperture illumination function derived from coherent measurement; amplitude (left) and phase (right)

little as 20 dB below the pattern functions, out to approximately ±25◦ and then less beyond this angular region. Figure 9.18 contains the reconstructed aperture illumination function of the antenna as recovered from the coherent, amplitude and phase, measured data. Conversely, Figure 9.19 contains the reconstructed aperture illumination function as recovered from the amplitude only measurement and an application of the iterative PTP algorithm. Although encouraging, to obtain results with this limited degree of agreement took 20,000 iterations of the PTP phase recovery algorithm. In general as the separation between the two measurement planes is increased the convergence rate increases, as the difference between the respective patterns is more significant; however, the additional separation also increases truncation within the measurement which will inevitably degrade the quality of the measurements. Clearly maintaining good alignment between the measurements is crucial to the success of the technique. The

Advanced planar near-field antenna measurements 301 Ex (dB) phase recovery

Ex (deg) phase recovery

0.3 –10 0.2 –20 y (m)

–30 0 –40 –0.1 –50 –0.2 –60 –0.3

y (m)

0.1

0.3

150

0.2

100

0.1

50 0

0

–50

–0.1

–100

–0.2

–150

–0.3 –0.3 –0.2 –0.1

Figure 9.19

0 0.1 0.2 x (m)

0.3

–0.3 –0.2 –0.1

0

0.1

0.2

0.3

x (m)

Aperture illumination function derived from amplitude only measurement

convergence rate of this algorithm is clearly very slow and an alternative algorithm, with a faster rate of convergence would be highly desirable. This, therefore is the motivation for the inclusion of an aperture constraint that is currently the most widely used form of the phase recovery technique, which is discussed in the following section.

9.3.2 PTP phase retrieval algorithm – with aperture constraint The PTP algorithm essentially entails taking two amplitude only measurements over parallel planes in the near-field of the antenna and separated by a known distance. However, if some additional information concerning the construction of the antenna is available then the convergence rate of the algorithm can be significantly improved. One simple modification of the PTP algorithm is to incorporate knowledge of the aperture illumination function of the antenna, providing of course that the antenna has a well-defined aperture. However, many higher gain antennas, such as reflector type antennas or phased array antennas do have well-constrained apertures and as such filtering out the fields outside of the physical aperture is both possible and convenient. The modified PTP phase retrieval algorithm can be described as follows: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Measure the amplitude of the field over plane 1. Measure the amplitude of the field over plane 2. Guess the amplitude and phase of the antenna aperture illumination function. Truncate the fields to the physical extent of the antenna aperture. Use PTP transform to propagate the AUT aperture fields to plane 1. Replace the amplitude estimation at plane 1 with the measured amplitude at plane 1. Use PTP transform to propagate the fields back to the AUT aperture plane. Truncate the fields to the physical extent of the antenna aperture. Use PTP transform to propagate the AUT aperture fields to plane 2. Replace the amplitude estimation at plane 2 with the measured amplitude at plane 2.

302 Principles of planar near-field antenna measurements 11. Use PTP transform to propagate the fields back to the AUT aperture plane. 12. Repeat steps 4–11 until amplitude on plane 1 (or plane 2) has converged to within a prescribed tolerance. 13. Transform the fields to the far-field using standard algorithm. Figure 9.20 contains cardinal cuts of the conventional and phase recovered patterns together with the equivalent multipath level. Here, the degree of agreement attained between the two sets of data can be seen to be significantly better than was achieved previously which is further demonstrated by the reduction in the EMPL. Figure 9.21 contains a comparison of the angular spectra obtained from a direct holographic measurement and from the plane-to-plane phase retrieval algorithm with the aperture constraint imposed. Here, the figure on the right was derived using retrieved phase function. The reconstructed aperture illumination function, shown in Figure 9.22, further illustrates the success of the technique as the phase recovered results are clearly BT antenna

BT antenna Plane 1 Plane 2 Phase retrieval EMPL

0 –10

–10

Power (dB)

Power (dB)

–20 –30 –40

–20 –30 –40

–50

–50

–60

–60

–70 –1 –0.8 –0.6 –0.4 –0.2

Figure 9.20

0 u

0.2

0.4

0.6

0.8

1

–70 –1 –0.8 –0.6 –0.4 –0.2

0 v

0.2

0.4

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0.8

1

Measured and phase recovery result from 76 wavelength diameter cassegrain antenna, using aperture constraints 1

1 0.8

0.8

–10

–10

0.6

0.6 –20

0.4 0.2

0.4 0.2

–30

0

v

v

Plane 1 Plane 2 Phase retrieval EMP

0

–0.2 –0.4

–20 –30

0

–40

–0.2

–40

–50

–0.4

–50

–0.6

–0.6 –60

–0.8 –1 –1 –0.8–0.6–0.4–0.2 0 0.2 0.4 0.6 0.8 1 u

Figure 9.21

–0.8 –1 –1 –0.8–0.6–0.4–0.2 0 0.2 0.4 0.6 0.8 1 u

Comparison of angular spectra for Figure 9.20

–60

Advanced planar near-field antenna measurements 303 Ex (dB) phase recovery

Ex (deg) phase recovery

0.3

0.3

150

0.2

100

0.1

50

–10 0.2

y (m)

–30 0 –40

–0.1

y (m)

–20 0.1

0

0

–0.1

–50

–0.2

–100

–50 –0.2 –60 –150

–0.3

–0.3 –0.3 –0.2 –0.1

Figure 9.22

0 0.1 x (m)

0.2

0.3

–0.3 –0.2 –0.1

0 0.1 x (m)

0.2

0.3

Comparison of aperture illumination function

in good agreement with the phase measured results. However, the effect of the application of the aperture plane spatial filter can clearly be seen with the field outside the physical aperture having been being suppressed which is most clearly seen in the horizontal plane. The importance of this can perhaps be best illustrated by comparing the number of iterations required in order to obtain these results. Previously, we needed 20,000 iterations whilst with the modified algorithm only 200 iterations were required. This algorithm is very similar to the previous one (same building blocks merely put together in a slightly different way); only here we apply an additional aperture constraint. In this case, this merely corresponds to truncating fields that are outside the antenna aperture that is assumed to be a disk of 10 inches in diameter. Essentially these algorithms are the microwave analogue of attempting to reconstruct a hologram from two related, but different photographs. Although impressive, the improvement in efficiency has been gained at the cost of a loss in generality. For both this and the previous phase recovery algorithm the image comparison techniques of Chapter 8 can be implemented to aid in the assessment of the convergence. Phase retrieval algorithms do find application in a number of specialist applications but currently are not found widely deployed in industry. Even at millimetre wave frequencies and above, it is common practice to use specialized radio frequency (RF) subsystems and optimised guided wave paths in order that phase information can be obtained.

9.4

Efficient position correction algorithms, in-plane and z−plane corrections

Ideally, the robotic positioning and control system would be capable of placing the near-field probe with infinite precision at the desired location. This, however, is clearly impossible. Positioning errors either arise from imperfections within the

304 Principles of planar near-field antenna measurements mechanical structure of the scanner or from timing errors within the control subsystem. The experimentalist really has only two options. One, design and construct the measurement system to the precision required, or two, construct the system as well as possible and record the position of the probe to the required accuracy. The ‘calibrate and correct’ option yields two further choices. These are, to correct the position of the probe real time using an active control system [4] or to correct the data within the transformation and data processing chain. As the former option essentially constitutes a real time control problem it is outside of the scope of this text and instead herein only techniques for correcting the measured data will be considered. For the experimentalist to have any chance of compensating for positional inaccuracies within the measurement the position of the probe must be known to the accuracy required and the measurement system must be capable of sampling the near-field whilst satisfying the Nyquist criteria. If this requirement is not satisfied then almost all hope is lost and recourse to statistical techniques is unavoidable. This is undesirable as such techniques inevitably require that additional, repeat measurements, are taken which are costly in terms of facility time and post-processing effort. The discrete near-field to far-field transform itself has no requirement for the data points to be sampled on a planar, plaid, monotonic and equally spaced grid. Instead, this requirement is imposed by the methods that are used to improve the efficiency of the transformation algorithm, namely any recurrence relationships utilized and the FFT algorithm if it is used. Thus, provided the elemental area, surface unit normal and actual measured coordinates are used within the necessary transforms; a direct implementation of the discrete algorithms will be sufficient to accommodate data taken over an imperfect, but known, sampling plane. The salient question here then, is how can we still utilize our efficient transformation algorithms with errorcontaminated data? Furthermore, any ‘more sophisticated’ technique that requires greater computational effort to implement than a direct transformation is unhelpful. Several interpolation-based approaches exist. Some simply fit functions locally so that the data can be interpolated from an irregular grid onto a regular grid. Unfortunately, as the near-field data will be sampled at close to Nyquist and the sampling nodes will be irregularly spaced, this places significant demands on any approximating formula. Others use approximation to regularise the grid and then use rigorous Whittaker interpolation to calculate the field over the irregularly spaced grid. By taking the difference between the original measurement and the recomputed data an error signal can be used to correct the field values over the regularised grid. By iterating around this two-step process increasingly accurate values for the field over the regular grid can be obtained. Unfortunately, rigorous Whittaker interpolation is a computationally intensive procedure that is usually considered to be unsuitable for including within an optimisation loop. Instead, the most widely used correction techniques involve either expanding the measured field in a Taylor series or utilizing the K-correction techniques. These techniques are introduced in the following sections.

Advanced planar near-field antenna measurements 305

9.4.1 Taylor series expansion Following Reference 5 and from Chapter 4, consider the following integral  ∞ ∞   1 A kx , ky e−j(kx x+ky y+kz z) dkx dky E (x, y, z) = 2 4π −∞ −∞ where 



A kx , ky =



∞



∞

−∞ −∞

E (x, y, 0) ej(kx x+ky y) dx dy

(9.19)

(9.20)

Here, the field E is assumed to be defined over a plaid, monotonic and equally spaced grid, so that the Fourier transform (FT) can be evaluated efficiently with the use of the FFT. However, in practice this may not be the case. If we assume that the probe’s position is known accurately we can express the measured fields as E (x + δx, y + δy, 0)

(9.21)

Then, with the use of a series expansion, we can recover the field on a regular grid with arbitrary accuracy. We will need to assume that no angular errors are introduced, as this would change the way that the probe pattern effects the measurements. We can write the Taylor expansion in two dimensions as f (x + δx, y + δy) = f (x, y)   + fx (x, y) δx + fy (x, y) δy  1  + fxx (x, y) (δx)2 + 2fxy (x, y) δxδy + fyy (x, y) (δy)2 2! 1  + fxxx (x, y) (δx)3 + 3fxxy (x, y) (δx)2 δy 3!  +3fxyy (x, y) δx (δy)2 + fyyy (x, y) (δy)3 + · · · (9.22) where ∂ (f (x, y)) ∂x ∂ fy (x, y) = (f (x, y)) ∂y

fx (x, y) =

∂2 (f (x, y)) ∂x∂x ∂2 fxy (x, y) = (f (x, y)) ∂x∂y fxx (x, y) =

fxxx (x, y) =

∂3 (f (x, y)) ∂x∂x∂x

(9.23) (9.24) (9.25) (9.26) (9.27)

306 Principles of planar near-field antenna measurements fxxy (x, y) =

∂3 (f (x, y)) ∂x∂x∂y

(9.28)

fxyy (x, y) =

∂3 (f (x, y)) ∂x∂y∂y

(9.29)

fyyy (x, y) =

∂3 (f (x, y)) ∂y∂y∂y

(9.30)

Thus the corrected field can be expanded using the Taylor series to obtain   ∂E (x, y) ∂E (x, y) δx + δy E (x + δx, y + δy) = E (x, y) + ∂x ∂y   1 ∂ 2 E (x, y) ∂ 2 E (x, y) ∂ 2 E (x, y) 2 2 + δxδy + (δx) + 2 (δy) 2! ∂x∂x ∂x∂y ∂y∂y  1 ∂ 3 E (x, y) ∂ 3 E (x, y) + (δx)3 + 3 (δx)2 δy 3! ∂x∂x∂x ∂x∂x∂y  ∂ 3 E (x, y) ∂ 3 E (x, y) +3 δx (δy)2 + (δy)3 + · · · ∂x∂y∂y ∂y∂y∂y (9.31) Factorising yields

   ∂ ∂ δx + δy E (x + δx, y + δy) = 1 + ∂x ∂y   2 1 ∂2 ∂2 ∂ + δxδy + (δx)2 + 2 (δy)2 2! ∂x∂x ∂x∂y ∂y∂y  3 3 1 ∂ ∂ + (δx)2 δy (δx)3 + 3 ∂x∂x∂y 3! ∂x∂x∂x   ∂3 ∂3 2 3 +3 δx (δy) + (δy) + · · · E (x, y) ∂x∂y∂y ∂y∂y∂y (9.32)

Let T denote the differential operator   ∂ ∂ δx + δy T = ∂x ∂y  2  1 ∂ ∂2 ∂2 2 2 + δxδy + + 2 (δx) (δy) 2! ∂x∂x ∂x∂y ∂y∂y  1 ∂3 ∂3 ∂3 + δx (δy)2 (δx)3 + 3 (δx)2 δy + 3 3! ∂x∂x∂x ∂x∂x∂y ∂x∂y∂y  ∂3 + (δy)3 + · · · ∂y∂y∂y

(9.33)

Advanced planar near-field antenna measurements 307 Alternatively, for ease of use later on, this can be expressed compactly as T = t1 + t2 + t3 + · · ·

(9.34)

Here, t1 contains partial derivatives of first order; t2 contains partial derivatives of second order; and so on. Thus, the electric field at an arbitrary, but local, point can be expressed as E (x + δx, y + δy) = {1 + T } E (x, y)

(9.35)

E (x, y) = {1 + T }−1 E (x + δx, y + δy)

(9.36)

or

Now, the inverse of a summation can be found from the binomial expansion as 1 1 (x + 1)−n = 1 − nx + n (n + 1) x2 − n (n + 1) (n + 2) x3 + · · · 2 6

(9.37)

This is valid for integer n. Hence (T + 1)−1 = 1 − T + TT − TTT + TTTT − · · ·

(9.38)

Thus the necessary correction, when expressed in terms of the differential operator T is E (x, y) = {1 − T + TT − TTT + TTTT − · · ·} E (x + δx, y + δy)

(9.39)

Now, this can be an approximation to any chosen order. So, for a third-order approximation, that is, the case where the highest partial derivative is a third derivative it can be obtained easily. E (x, y) = {1 − T + TT − TTT + TTTT − · · ·} E (x + δx, y + δy)

(9.40)

Now to third order T = t 1 + t2 + t3

(9.41)

TT = (t1 + t2 + t3 ) (t1 + t2 + t3 ) ≈ t1 t1 + t1 t2 + t2 t1

(9.42)

TTT = (t1 + t2 + t3 ) (t1 + t2 + t3 ) (t1 + t2 + t3 ) ≈ t1 t1 t1

(9.43)

Here, we have omitted any terms, that is, product of ti that are of greater than third derivatives. Clearly, TTTT contains terms of fourth order and above only, which is why it is not included here. Hence, to a third-order approximation E (x, y) ≈ {1 − t1 − t2 − t3 + t1 t1 + t1 t2 + t2 t1 − t1 t1 t1 } E (x + δx, y + δy) (9.44)

308 Principles of planar near-field antenna measurements or  ∂ ∂ δx + δy E (x, y) ≈ 1 − ∂x ∂y   2 1 ∂2 ∂ ∂2 − δxδy + (δx)2 + 2 (δy)2 2! ∂x∂x ∂x∂y ∂y∂y  1 ∂3 ∂3 ∂3 − δx (δy)2 (δx)3 + 3 (δx)2 δy + 3 3! ∂x∂x∂x ∂x∂x∂y ∂x∂y∂y     ∂ ∂ ∂ ∂ ∂3 3 δx + δy δx + δy + (δy) + ∂y∂y∂y ∂x ∂y ∂x ∂y   2  2 1 ∂ ∂ ∂ ∂2 ∂ 2 2 + δxδy + δx + δy (δx) + 2 (δy) ∂y ∂x∂x ∂x∂y ∂y∂y 2! ∂x   2  1 ∂ ∂2 ∂2 ∂ ∂ 2 2 + δxδy + δx + δy (δx) + 2 (δy) 2! ∂x∂x ∂x∂y ∂y∂y ∂x ∂y     ∂ ∂ ∂ ∂ ∂ ∂ − δx + δy δx + δy δx + δy E (x + δx, y + δy) ∂x ∂y ∂x ∂y ∂x ∂y (9.45) 



Other workers have implemented solutions of fifth order [6]; however, for the sake of brevity, we shall restrict ourselves to the third-order solution here. In general, the order of solution required will depend upon the magnitude of the deviation from the regular grid with larger error terms requiring higher-order correction formula. However, as the convergence of this series depends upon the successive and asymptotically converging, cancellation of increasingly large terms, numerical instabilities, which are increasingly likely to be encountered when evaluating higher-order derivatives, mean that this series solution can fail to converge under certain circumstances. Also, as can be seen from the third-order expansion, the amount of additional effort is significant and higher-order expansions further add to this burden. Now, we turn to the matter of determining the partial derivatives. Fortunately, as shown elsewhere within this text, where, the necessary derivatives can be found efficiently from spectral techniques that can take advantage of the FFT to aid efficiency. It was shown above that   ∂ n u (x, y, z) = −1 (−jkx )n {u (x, y, z)} ∂xn

(9.46)

 n  ∂ n u (x, y, z) = −1 −jky {u (x, y, z)} ∂yn

(9.47)

and

Advanced planar near-field antenna measurements 309 Also from repeated application of these formulas, a more general expression can be obtained, namely,  n   ∂ (m+n) u (x, y, z) = −1 (−jkx )m −jky {u (x, y, z)} m n ∂x ∂y

(9.48)

However, these derivatives are found from the error-contaminated field function rather than from the error-free field function that is clearly an additional source of error. Although for simplicity these expressions have been derived for the case of inplane errors, that is, imperfections in planarity have not been considered; this analysis is general and can be used to account for these errors also. Figure 9.23 contains a plot of ideal simulated near electric field data, shown in grey, plotted together with the same data recorded on a grid with error-contaminated positions plotted in black. The illuminating aperture was a simple uniformly illuminated square cut in a perfectly conducting infinite ground plane. Here the magnitude of the displacement error varied sinusoidally across the measurement plane, that is, in-plane position measurement errors, where the magnitude of the positional error was 0.4λ. This is significantly larger than would be anticipated in practice; however, Uncorrected

0.3

0.2

y (m)

0.1

0

–0.1

–0.2

–0.3

-–0.3

–0.2

–0.1

0

0.1

x (m)

Figure 9.23

Near electric field shown uncorrected

0.2

0.3

310 Principles of planar near-field antenna measurements 3rd order correction

0.3

0.2

y (m)

0.1

0

–0.1

–0.2

–0.3

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

x (m)

Figure 9.24

Near electric field shown with 3rd order correction

an extreme case was chosen to highlight both the effectiveness and the limitations of the technique. The third-order form of the Taylor series correction formula can be used to calculate the equivalent corrected near electric field from the error-contaminated near electric field data. In general this procedure must be used on each separate measured field component. A comparison of the corrected and ideal data sets can be seen presented in Figure 9.24. Here, grey is used to represent the ideal, whilst black contours are used to represent the third-order corrected pattern. Here, the necessary partial derivatives have been evaluated using the efficient spectral method. This tends to introduce a degree of ripple on the calculated partial derivates. Conversely, numerical differencing, particularly when used to calculate higher-order derivatives can become inaccurate when performed in the presence of local noise. Here, the agreement between the corrected and ideal patterns is evident, particularly in regions of largest field intensity. This comparison can be repeated in the spectral domain which in general is the domain of greatest interest. Figure 9.25 contains the angular spectra of the ideal data set as calculated from the simulated error-free measurement.

Advanced planar near-field antenna measurements 311 Error free

1 0.8

–10

0.6 –20

0.4 0.2

–30

v

0 –40

–0.2 –0.4

–50

–0.6 –60

–0.8 –1 –1

Figure 9.25

–0.5

0 u

0.5

1

Angular spectra of error-free measurement Uncorrected 1

0.8 –10

0.6 –20

0.4 0.2

–30

v

0 –40

–0.2 –0.4

–50

–0.6 –60

–0.8 –1 –1

Figure 9.26

–0.5

0 u

0.5

Angular spectra of uncorrected measurement

1

312 Principles of planar near-field antenna measurements 3rd order correction 1 0.8

–10

0.6 –20

0.4

v

0.2

–30

0 –40

–0.2 –0.4

–50

–0.6 –60

–0.8 –1 –1

Figure 9.27

–0.5

0 u

0.5

1

Angular spectra of third-order corrected measurement

Figure 9.26 contains an equivalent angular spectrum plot; only here the errorcontaminated data was transformed as though it had been sampled without error. A significant amount of distortion is clearly evident within the pattern on both the main beam and the side lobes. Certain side lobes have become displaced and misshapen, pattern nulls have been partially filled in whilst the symmetry of the pattern has been lost. Conversely, Figure 9.27 contains a plot of the corrected angular spectra. Here, the corrected angular spectrum resembles far better the angular spectra obtained from the error-free near-field data. The shape and location of the main beam and side lobes is far better than before and the pattern nulls can now be seen to be both deeper and lie in the correct directions. Although here the Taylor series correction technique has been developed to correct in-plane errors, it is also possible to use a slightly modified form of these equations to correct for errors in the flatness, that is, planarity, of the acquisition plane.

9.4.2 K-correction method An alternative method for correcting small imperfections in the location of the sampling grid has been developed by [7, 8] and is referred to as a K-correction method. This method, in the form that it is most commonly implemented utilizes a phase change to correct for small displacements caused by imperfections in the flatness of

Advanced planar near-field antenna measurements 313 the sampling grid. E (x, y, z = 0) = E (x, y, z + δz) ejk0 δz cos θ0

(9.49)

Here, δz is a function of the x and y coordinates and denotes the imperfection in the flatness, that is, planarity, of the acquisition plane and θ0 is the angle that the main beam of the antenna is steered to. Here, it is assumed that as the displacement is small no observable change in the amplitude of the field will be observed. Clearly this zerothorder compensation procedure is very inexpensive computationally, especially when compared with the more sophisticated Taylor series correction procedure described above. However, it is very effective. The same aperture illumination function and measurement configuration was simulated; only here a cosinusoidal disturbance was introduced into the z coordinate of the measurement plane simulating an imperfectly flat measurement system. The resulting electric field was then used to determine the equivalent angular spectra that would include these errors. The result of this can be found presented in Figure 9.28. Here, the effect of the comparatively modest positional errors can be seen to have seriously altered the angular spectra. Pattern nulls can be seen to have filled in and it is evident that energy has been moved from the direction of the main beam to other directions altering the level and location of the side lobes. Conversely, Figure 9.29 contains the angular spectra derived after the Kcorrection has been implemented. Uncorrected 1 0.8

–10

0.6 –20

0.4

v

0.2

–30

0 –40

–0.2 –0.4

–50

–0.6 –60

–0.8 –1 –1

Figure 9.28

–0.5

0 u

0.5

1

Angular spectra of third-order uncorrected measurement

314 Principles of planar near-field antenna measurements K-correction 1 0.8

–10

0.6 –20

0.4

v

0.2

–30

0 –40

–0.2 –0.4

–50

–0.6 –60

–0.8 –1 –1

Figure 9.29

–0.5

0 u

0.5

1

Angular spectra of k corrected measurement

Again, this can be compared with the angular spectra of Figure 9.25 which were determined from near electric field located perfectly on the plaid, monotonic, equally spaced and perfectly flat sampling grid. Although not perfect, the improvement in the resulting pattern is clearly significant with the main beam and side lobes regaining much of their symmetry with the well-defined pattern nulls becoming apparent once again. Only comparatively minor errors are evident within the wideout side lobes in the inter-cardinal region of lower field intensities. In summary then, the K-correction technique, when implemented in this way, is a very simple, very economic and very effective technique for correcting for small probe displacement errors that have a component in the direction of the main beam. Although as impressive as both these compensation techniques are, it is perhaps worth remembering that they are approximate and when putting cost and mechanical practicalities aside, it is nearly always preferable to take the near-field measurements at the correct location in space rather than to try and correct for numerous positional inaccuracies. Provided the location at which the measurements were taken is known, then the conventional discrete FT can be used to obtain the error-free angular spectra whereupon, for any given practical case, these more efficient techniques can be compared. Assuming that the respective results are found to be in sufficiently good agreement, then these position correction techniques can then be utilized for the reminder of the test campaign.

Advanced planar near-field antenna measurements 315

9.5

Partial scan techniques

The concepts discussed so far in this chapter have been limited to techniques designed to correct or compensate for systematic errors in near-field antenna measurement procedures: • • •

Active alignment allows measurements to be performed when the AUT cannot be corrected aligned in the range. Amplitude only measurements allow far-field patterns to be calculated when accurate phase data cannot be acquired. Position correction algorithms compensate for inaccuracies in the probe-position when the near-field is being sampled.

However, advanced techniques can also be used to enhance the accuracy of measurements and to extend the applicability of the planar near-field methodology to the measurements of a wider range of antennas than would normally be possible.

9.5.1 Auxiliary translation The difficulties of truncation have, in the past, limited the size and types of antennas that can successfully be characterised in planar scanners. However, the requirement that the field that is radiated by the AUT, that is, that the near-field pattern and its gradient is continuous over the extent of the spatial sampling plane, can be met by adopting another strategy. This strategy is based on the construction of a scan plane, big enough to capture all the required data in more than one acquisition. It is the measured near-field that is actually processed to produce the far-field prediction of the AUT that needs to be periodic and this data set need not be the product of a single scan. If the data from more than one scan could be spliced together to form a composite data set, this composite data set can be sufficiently large to have captured all of the incident radiation required to successfully predict the AUTs farfield. This composite data set would then fulfil the requirement of being continuous over the sampling interval, and thus could be used to successfully predict far-field patterns. The partial scans that can be combined to produce the composite data set will require that the position of the AUT, relative to the scan plane, be spatially transformed between scans so that the combined data set represents a larger scan area than that actually available via a single scan. Clearly as the actual scan plane of the scanner is at a fixed location it is the AUT that must undergo the spatial transformation. This can be accomplished using a variety of different AUT spatial transforms. Figure 9.30 illustrates how a large AUT could be sequentially scanned at three positions separated by a spatial translation in the x direction relative to a scan plane. Translations like those illustrated do, however, depend on the availability of a translational mechanism that can be used to mount the antenna and then move it to the predetermined positions, like that shown in Figure 3.23. Figure 9.30, specifically shows the translation of an AUT between three x coordinate positions, in two meter stages, across the 3 × 3 m scan plane thus

316 Principles of planar near-field antenna measurements Probe carriage

AUT Position 1

Shot bolt positions

Position 2

Position 3

3 × 3m Scan plane 9 × 3m Effective scan plane x direction

Figure 9.30

Position of AUT between translations in a partial plane data acquisition

producing a much larger effective scan plane, in this case 7 × 3 m. Clearly the AUT shown above could not have been measured on a 3 × 3 m scan plane but the secondary translation of the AUT between scan planes has allowed the production of a composite scan plane. This scan plane would be large enough to acquire data such that truncation will not adversely affect the far-field prediction. AUTs can also be translated in the y direction relative to the scan plane this can substantially increase the area of the composite scan plane and allows more accurate measurements of larger AUTs. However, the secondary translation would be expected to introduce further errors into the measurement procedure. If the AUT can be manipulated and repositioned with sufficient accuracy then the construction of the combined scans should not involve any

Advanced planar near-field antenna measurements 317 87310D coupler 37203

AUT

Probe

83642A RF source 84906L attenuator

85320B-H50 ref mixer

8341B LO source

Chan. 1 test mixer

Chan. 2 test mixer

LO

37203 IF Trigger 8510C vector network analyser

85309A LO/IF Distribution unit

Figure 9.31

Typical RF subsystem for a near-field scanner

loss of accuracy, and if the AUT is correctly repositioned the spatial transformation should not affect the variation in phase and amplitude of the signal from the scanner. Unfortunately, other factors do affect the phase and amplitude across the scan plane as a result of the spatial transformation of the AUT and these factors cannot be ignored if accurate measurements are to be made [9]. The fundamental mechanism behind the RF system used in the near-field scanner (NFS) to measure antenna patterns is the use of the receiver as an interferometer. The measured signal is the IF mixed down from two mixers which are supplied with RF and local oscillator (LO) from the same sources, but only one of the RF/LO paths includes the AUT and test probe. Figure 9.31 illustrates the signal from the RF source passing via the reference mixer and alternatively to the test mixers via the AUT and probe for a typical near-field scanner range RF subsystem. Variations in the signal that is detected at each sample point by the probe is representative of the spatial variation in the field that is produced by the AUT. Thus any factor which could influence this variation in signal amplitude and phase, which is not a function of the probes position, could be mistaken for a variation in the field produced by the AUT that is not in fact present. The movements of the AUT associated with the spatial transforms involves the disturbance of the reference part of the RF path and this must therefore create some variation in the measured amplitude and phase as a result of disturbing the cables in which these signals are carried. However, the areas over which the probe is scanned, in each of the individual partial scans which constitute the combined scan, can be allowed to overlap to a predetermined extent. This means that certain areas of the near-field composite scan will be measured more than once, and the individual measurements of the overlap

318 Principles of planar near-field antenna measurements areas will have been made before and after the AUT is spatially transformed. Thus direct comparisons of values in the overlap areas can be used to evaluate and correct for any variations in the measured signal that are produced by moving the AUT as opposed to moving the probe. In addition, small extra scans can be made that while they themselves are of insufficient size to be used for transforms can include points that will also be measured in all the partial scans and so they can be used to correct the data in these scans. This means that by quantitative comparison of the points in the partial data sets that correspond spatially to the same point it should be possible to ascertain the variations in the phase and amplitude produced by the movement of the AUT. This makes corrections, to take this variation into account, a simple process to perform. The authors have found by a process of experimentation, that an overlap of two/three columns or rows, is all that is necessary to acquire the information required to perform the correction of the data. Usually both the amplitude and phase corrections are taken to be the mean of the differences between the measured values of I and Q in the data sets at the same actual spatial points, although they are at different points relative to the different partial scans. These calculated correction values are added to the scans to correct the data before it is transformed to produce the far-field pattern of the AUT. The use of a mean weighted in terms of the signal levels has in the past been considered but the agreement of the corrected terms without recourse to such levels of sophistication was such as to render this unnecessary. The change in the phase and amplitude associated with the action of correction does not mean that all the values in the overlap columns/rows are absolutely identical after the correction. However, in experiments performed by the authors a large degree of correlation has been observed. For example, in a 3 GHz scan of 53 data points down to –30 dB of the maximum of the signal level the value of the mean difference between the original and corrected data, with its overall phase corrected by 2.4◦ , was 0.007 dB of amplitude and 0.02◦ of phase. These values are so small as to make negligible difference to the computed far-field pattern and indeed if all the overlap points down to –45 dB of the maximum recorded signal were included these figures only changed to 0.008 dB of amplitude and 0.07◦ of phase. The effectiveness of the technique is demonstrated in Figure 9.32. It illustrates the comparison of part of a primary cut through two data sets, one acquired as a single data set and the other as a result of a composite scan, where both of the single and composite scans were of the same size. The data shown above was acquired for an X band slotted waveguide array antenna that was measured accurately out to 60◦ off boresight. The AUT was measured in a single scan and then in two partial scans which, combined produced a composite scan of the same dimensions as the original scan. The original single scan data is shown as a line and the composite scan data is superimposed as a series of crosses at the predicted far-field points. As can be seen by inspection of this pattern, that is typical for results using the technique, the differences in the two far-field patterns are very small indeed.

Advanced planar near-field antenna measurements 319 –35

–40

Y-axis (dB)

–45

–50

–55

–60

–65 30

35

40

45

50

55

60

Azimuthal axis 30 to 60 degrees off boresight

Figure 9.32

Comparison of data from composite and single scans (composite scans ‘+’)

9.5.2 Rotations of the AUT about the z-axis Mounting an AUT translated to the corner of the scan plane and its transformation by 90◦ rotations about the normal to the AUT between scans, as shown in Figure 9.33, can be used to construct a composite scan. Again an overlap area of data acquired in each of the four partial scans would be required to access the impact on the accuracy of measurements produced by this partial scan processes. An examination of the error terms introduced into the measurement procedure by this process can be found in Reference 10. It should be noted that the movement of the antenna, whether it be translational or rotational, would be expected to degrade the accuracy of the measurements as it must introduce other sources of error associated with the repositioning. This strategy of increasing the size of the affective scan plane by constructing a composite scan plane from a series of partial scans conducted over a smaller region of the same plane can significantly improve the performance of the measurement facility. However, it can never entirely succeed in removing all truncation errors from the data as any finite number of translations or rotations can never synthesise a plane that is infinite in extent. Mathematically, such auxiliary translation is based upon the following theorem of calculus. If f is integratable on a closed interval containing the three points a, b and c, then  b  c  b f (x) dx = f (x) dx + f (x) dx (9.50) a

a

c

320 Principles of planar near-field antenna measurements

Y direction

AUT

Scan plane

Centre of AUT and of rotation X direction

Figure 9.33

Position of AUT and scan plane when antenna is to be rotated

Irrespective of how the points are ordered [11]. When expressed as an integral in two dimensions the derivation of angular spectra from sampled data can be expressed as               F T kT = ET rT e−j kT ·rT dx dy + ET rT e−j kT ·rT dx dy R1

Or

R2

      FT kT = FT kT R + FT kT R 1

2

(9.51) (9.52)

This follows directly from the linearity property of the FT. Here, R1 and R2 are two intersecting regions of a planar surface. Alternatively, when expressed in terms of electric field components and extended over N partial acquisitions this can be expressed as N      E k T R E kT = i=1

i

(9.53)

The transformation process is summarised in Figure 9.34. However, the active alignment explained in Section 9.2 of this chapter illustrates that the usual condition for planar near-field measurements that the boresight of the AUT is coincident with a normal to the scan plane is not a necessary requirement for planar measurements. This opens up the possibility of spatial transformations of the antenna that produce partial scans that are not all in the same plane.

9.5.3 Auxiliary rotation – bi-planar near-field antenna measurements Figure 9.35 illustrates the concept of auxiliary rotation schematically. By considering only the simplest case where rotating the AUT around an axis that is tangential to the measurement plane it becomes clear that it is possible to increase significantly

Advanced planar near-field antenna measurements 321 Acquire partial scan

Apply co-planar translation to AUT

Repeat for each Repeat for each partial scan

Read in tangential field components for partial scan and add to previous data

Convert from polar amplitude and phase to rectangular I and Q

Transform to the far-field using a twodimensional discrete Fourier transform

Apply probe pattern correction to the tangential spectral field components

Obtain the normal field component using the plane wave condition

Convert from spectral field components to electrical field components

Calculate magnetic fields if required

Resolve far-field pattern onto desired polarisation basis, usually LIII

Inverse transform to aperture plane if required

Write data to disk and post process

Figure 9.34

Block diagram of conventional auxiliary translation near-field to farfield transform algorithm

the proportion of the forward hemisphere over which the measurement is made. Figure 9.35 illustrates schematically the difference between auxiliary translation and auxiliary rotation around a tangent to the scan plane. From Figure 9.35 it can be clearly seen that θ2 > θ1 and that θ2 could exceed 90◦ . The FT relates the spatial and angular domains in terms of angles being defined as rates of change of phase with respect to space (PWS). Clearly then, the traditional methodology of combining data sets in the near-field will be unsuccessful for the case where an angular variation between partial scans exists. Here, the fundamental

322 Principles of planar near-field antenna measurements y

A

x x

x

z zr 1

y x B

C

x

x

z

2

2

Figure 9.35

>

1

Comparison of auxiliary translation and auxiliary rotation

requirement of a new transform is that it must allow the combination of the data in the partial scans such that specific vectors in each of the partial data sets represent a single vector in the combined data set. 9.5.3.1 Simple transform The bi-scan auxiliary near-field to far-field transform can be expressed as follows. For Plane 1:  ∞ ∞     F1 kx , ky = f1 X1 , Y1 ejk(αX +βY +γ Z) dX1 dY1 (9.54) −∞ −∞



  e−jkr jk uˆ r ·r1 E1 (ru, rv, rw) ≈ j uˆ r × FT1 kT × nˆ 1 e λr   E = [A1 ] = E1 kt R 1

(9.55) (9.56)

Advanced planar near-field antenna measurements 323 For Plane 2:   F2 kx , ky =



∞



∞

−∞ −∞

  f2 X2 , Y2 ejk(αX +βY +γ Z) dX2 dY2



  e−jkr jk uˆ r ·r2 uˆ r × FT2 kT × nˆ 2 e λr   E = [A2 ] = E2 kt R

E2 (ru, rv, rw) ≈ j

2

(9.57) (9.58) (9.59)

The total field can be reconstructed from the principle of superposition thus       E kT = E1 kT R + E2 kT R 1

2

(9.60)

The transformation process is essentially the same as that depicted in Figure 9.35; however, the near-field to far-field transform is modified in accordance with the expressions contained above. Unfortunately there is a difficulty associated with this approach which will be expounded below in the remaining sections of this chapter. 9.5.3.2 Continuity of field function at intersection Unfortunately, in practice a complication exists, as the two partial scans intersect and as such those points at the interface between the two partial scans are sampled twice, once in each partial scan. Such difficulties can be resolved with the use of the pointwise convergence theorem. This theorem can be applied to the functions f and f  , as they are piecewise continuous functions everywhere except at a finite number of points and have only ordinary, that is, finite, discontinuities at these points. This theorem states that the Fourier series of f over an interval converges to f (x) at a point of continuity whilst at a point of discontinuity the Fourier series converges to the average 1 (f (x+) + f (x−)) 2

(9.61)

where, f (x+) and f (x−) denote the limit of f at x from the right and left respectively, that is, f (x+) = lim {f (x + h)}

(9.62)

f (x−) = lim {f (x − h)}

(9.63)

h→0 h→0

For a proof of this theorem see Reference 12. This modification has been successfully used within the transformation algorithm, where encouraging results have been attained. In three dimensions it is possible for the function to approach a point from three different orthogonal directions, that is, at the point where three orthogonal planes intersect. In this case, the averaging process is extended to include the third point.

324 Principles of planar near-field antenna measurements 9.5.3.3 General near-fields – deficiency of simple transform Initially, the near-field to far-field transform was accomplished simply by first obtaining the tangential angular spectra from the tangential near-field components. The normal component of the angular spectrum was obtained by applying the plane wave condition, whereupon the polarisation basis could be resolved onto that of the AUT. Finally, the angular spectrum of the AUT could be obtained by combining the partial angular spectra using the principle of superposition. Unfortunately, this approach was found to be incorrect for arbitrarily polarised sources and the source of the error was traced to the incorrect determination of the field component that was at a normal to the scan plane. The normal field component is obtained from the tangential field components by applying the plane wave condition to the far-field electric field or to the angular spectra. En = −

kt · Et

(9.64)

kn

where, En is the electric field component at a normal to the measurement plane, kn is the component of the propagation vector that is normal to the measurement plane. Similarly, Et is the component of the electric field vector that is tangential to the measurement plane. This will be successful if the near-field data set is free from truncation. If, however, the sampled data set is truncated, and by definition partial scans are truncated, then the reconstructed normal field component will be in error. These difficulties can be resolved, if the normal field component is sought only over the surface of each partial plane as windowing functions can be utilized, whilst maintaining the integrity of the underlying function. All three orthogonal field components can then be transformed to the far-field whereupon the partial data sets can be combined by the principle of superposition. The normal field component over the partial scan plane can be expressed mathematically as    {E y)} + k y) k E (x, (x, x y y x (9.65) Ez (x, y) = − −1 kz where again   A= a = a=

−1



∞



∞

−∞ −∞

  1 A = 4π 2



a (x, y) ejk·r dx dy ∞



∞

−∞ −∞

  A kx , ky e−jk·r dkx dky

(9.66) (9.67)

Here, a and A are vector analytic functions and the limits of integration will collapse to the region of the partial scan plane. The near-field data set is first windowed before it is transformed to the angular spectrum. The normal field component is obtained by applying the plane wave condition before the inverse transform is taken whereupon the windowing function can be removed from the normal component. A proof of the validity of applying windowing functions to the integral transform of differential operators is presented in the following sections.

Advanced planar near-field antenna measurements 325 9.5.3.4 Reconstruction of normal field component The following derivation aims to establish analytically that the normal field component can be obtained from the tangential field components when the angular spectrum is disturbed by the presence of an arbitrary, but known, windowing function. The plane wave condition can be expressed in terms of the angular spectra as     β  α  Fz kx , ky = − Fx kx , ky − Fy kx , ky γ γ

(9.68)

Consider the effect of applying a windowing function to each electric field component in the near-field on the plane wave condition in the angular spectrum Exr (x, y) = w (x, y) Ex (x, y)

(9.69)

Eyr (x, y) = w (x, y) Ey (x, y)

(9.70)

Ezr (x, y) = w (x, y) Ez (x, y)

(9.71)

Here, the subscript r is used to denote a spatially windowed quantity. Thus by using the convolution theorem [2] the plane wave condition when expressed in terms of windowed electric near-field components becomes        α  W kx , ky ⊗ Fz kx , ky = − W kx , ky ⊗ Fx kx , ky γ    β  − W kx , ky ⊗ Fy kx , ky γ

(9.72)

Hence     β   α Fzr kx , ky = − Fxr kx , ky − Fyr kx , ky γ γ

(9.73)

where       Fxr kx , ky = {Exr (x, y)} = {w (x, y) Ex (x, y)} = W kx , ky ⊗ Fx kx , ky (9.74)           Fyr kx , ky = Eyr (x, y) = w (x, y) Ey (x, y) = W kx , ky ⊗ Fy kx , ky (9.75)       Fzr kx , ky = {Ezr (x, y)} = {w (x, y) Ez (x, y)} = W kx , ky ⊗ Fz kx , ky (9.76) Thus    β  α Fzr kx , ky = − {w (x, y) Ex (x, y)} − w (x, y) Ey (x, y) γ γ

(9.77)

326 Principles of planar near-field antenna measurements Hence Ez (x, y) =

   1 β  α −1 − {w (x, y) Ex (x, y)} − w (x, y) Ey (x, y) γ w (x, y) γ (9.78)

This holds for any windowing function that is both absolutely integrable and non-zero. 9.5.3.5 Modified probe pattern correction The derivation of the expressions that are used to correct the measured near-field data for the directive properties of the measuring probe relied upon the validity of the plane wave condition. The normal field component will be erroneously recovered for the case where the near-field data set is truncated. Thus, truncated measurements that are corrected with these expressions will also be in error. Again, such difficulties can be avoided with the application of a suitable windowing function. It can be established analytically that the normal field component can be obtained from the tangential field components when the probe-corrected angular spectrum is disturbed by the presence of an arbitrary, but known, windowing function. It has been shown in Chapter 5 that the corrected tangential electric near-field components can be obtained from the sampled tangential electric field from 

Ax (α, β) Ay (α, β)

 =



1

γ PxB (−α, β) PxC (−β, −α) + PyB (−α, β) PyC (−β, −α)     C   Px (−β, −α) PyB (−α, β) 1 − α 2 αβ  · × PyC (−β, −α) −PxB (−α, β) −αβ − 1 − β2   Sx (α, β) × (9.79) Sy (α, β)

Where A denotes the probe-corrected fields of the AUT, B and C represent the angular spectra of the measuring probes whilst S are the angular spectra derived from the measured fields. When expanded this can be expressed as Ax (α, β) =



1

γ PxB (−α, β) PxC (−β, −α) + PyB (−α, β) PyC (−β, −α)



× 1 − α 2 PxC (−β, −α) Sx (α, β) + PyB (−α, β) Sy (α, β)

 +αβ PyC (−β, −α) Sx (α, β) − PxB (−α, β) Sy (α, β) (9.80)

Advanced planar near-field antenna measurements 327 and Ay (α, β) =



1

γ PxB (−α, β) PxC (−β, −α) + PyB (−α, β) PyC (−β, −α) 

× −αβ PxC (−β, −α) Sx (α, β) + PyB (−α, β) Sy (α, β)



 − 1 − β 2 PyC (−β, −α) Sx (α, β) − PxB (−α, β) Sy (α, β) (9.81)

Consider the effect of applying a windowing function to each of the measured and corrected tangential field components in the near-field. axr (x, y) = w (x, y) ax (x, y)

(9.82)

ayr (x, y) = w (x, y) ay (x, y)

(9.83)

sxr (x, y) = w (x, y) sx (x, y)

(9.84)

syr (x, y) = w (x, y) sy (x, y)

(9.85)

Again, the subscript r is used to denote a spatially windowed quantity. Thus from the convolution theorem



1  W ⊗ Ax = 1 − α 2 PxC W ⊗ Sx + PyB W ⊗ Sy γD

 + αβ PyC W ⊗ Sx − PxB W ⊗ Sy (9.86) where D = PxB (−α, β) PxC (−β, −α) + PyB (−α, β) PyC (−β, −α)

(9.87)

Here, the angular variables have been suppressed for clarity. Alternatively, this can be expressed as W ⊗ Ax =

 

1  1 − α 2 PxC {wsx } + PyB wsy γD    +αβ PyC {wsx } − PxB wsy

(9.88)

The corrected x-polarised tangential electric field component can be obtained from 

 

1 −1 1  ax = 1 − α 2 PxC {wsx } + PyB wsy w γD    +αβ PyC {wsx } − PxB wsy (9.89)

328 Principles of planar near-field antenna measurements Similarly ay =

  

1 −1 1  −αβ PxC {wsx } + PyB wsy w γD

   − 1 − β 2 PyC {wsx } − PxB wsy

(9.90)

Again, this holds for any windowing function that is both absolutely integrable and non-zero. 9.5.3.6 Spectral leakage and the suitability of windowing functions Generally, windowing functions are chosen to be real functions of real variables so that the introduction of an unnecessary phase change is avoided. Within the analysis presented above the windowing function is assumed to be non-zero and absolutely integrable. Other than this, no assumptions have been made as to the form that the function can take. Spectral leakage results from processing a finite duration record that exhibits a finite discontinuity at the boundary of the observation. This stems from the periodic extension of a signal with a period that is not commensurate with the period of the sampling interval. Figure 9.36 illustrates the case where a sinusoidal signal has been sampled over a period that is different from the period of the signal. This results in the introduction of a discontinuity at the boundary of the observation interval. Such discontinuities are responsible for the introduction of spectral components, that is, leakage, over the entire data set. The windowing process can be thought of as a multiplicative weighting that is applied to reduce the order of the discontinuity at the boundary of the sampling

Sampling interval

Periodic signal

Observed signal Periodic extension Discontinuity

Figure 9.36

Periodic extension of a sinusoidal signal not periodic in the sampling interval

Advanced planar near-field antenna measurements 329 interval. Ideally, as many orders of derivative of the weighted function are matched to zero at the sampling boundary as is possible. A detailed review of data windowing functions can be found in Reference 13. Although the use of windowing functions is commonplace in the field of signal processing it is comparatively rare in near-field antenna measurements. This stems from the fact that whilst a windowing function can be used to suppress the highly oscillatory pattern that often results when truncated data is transformed, ultimately, it often fails to maintain the integrity of the underlying function. Thus, to be sure that the patterns are reliable an un-truncated measurement must also be made. These difficulties are clearly avoided here. 9.5.3.7 Modified near-field to far-field transform Figure 9.37 contains a block diagram of the modified transformation algorithm. The necessary two-dimensional FTs and inverse FTs can be evaluated efficiently with the FFT. In principle, the probe-corrected Cartesian electric and magnetic fields can be recovered over a planar surface from knowledge of the tangential electric fields. This introduces the possibility of constructing a more complex sampling surface from a set of intersecting planes that has not previously been possible. However, one final difficulty remains which is addressed in the following section.

9.5.4 Near-field to far-field transformation of probe corrected data In general, modal expansion techniques are inappropriate for use with bespoke sampling surfaces, as the sampling surface must correspond to a constant coordinate surface in the coordinate system for which the harmonic function series solutions are available. Instead, the derivation of far-field parameters from probe-corrected nearfield data can be performed using, a plane wave modal expansion, described above, as well as by Huygens’ principle or Kirchhoff–Huygens’ method. The Kirchhoff– Huygens’ method requires that the electric and magnetic fields be specified over a closed surface whereas, the Fourier and Huygens’ methods require that only the electric (or magnetic) field be specified over an aperture. However, the aperture is required to be set in a perfect electric (or magnetic) conducting surface. Both of these solutions and the angular spectrum method are exact and identical for exact aperture fields. However, both the angular spectrum method and the vector Huygens’ principle, in the form that was derived above from the PWS is inappropriate for use in obtaining the fields outside of the conceptual measurement box, as the sampling surface is not smooth. By smooth, it is meant that the surface and its first derivative are continuous everywhere. Clearly, at the edges and corners of the box, the first derivative of the surface is discontinuous. However, as the Kirchhoff–Huygens’ formula as introduced in Chapter 7 is essentially a direct integral of Maxwell’s equations, it is free of this difficulty and can be used to obtain reliable far-field data. The following section shows how the Kirchhoff–Huygens’ formula can be modified to yield asymptotic far-field data. As an alternative, the vector and scalar potentials of Chapter 2 could also have been used.

330 Principles of planar near-field antenna measurements Read in tangential field components for partial scan Convert polar amplitude and phase to rectangular I and Q Apply windowing function to tangential field component Transform to the angular spectrum using a two-dimensional fast Fourier transform Apply probe pattern correction to the tangential spectral field components Obtain the normal field component using the plane wave condition Inverse transform all field components to the near-field Remove windowing function and truncate data to the desired sampling interval Change polarisation basis from RFS to the AMS Use pointwise continuity theorem to handle intersection of partial planes Transform to the far-field (referenced to the AMS) Convert from spectral field components to electrical field components Use the principal of superposition to combine partial data sets Calculate magnetic fields if required

Resolve far field pattern onto desired polarisation basis, usually LIII

Inverse transform to aperture plane if required

Write data to disk and post process

Figure 9.37

Block diagram of auxiliary rotation near-field to far-field transform algorithm

Advanced planar near-field antenna measurements 331 9.5.4.1 Kirchhoff–Huygens’ principle The Kirchhoff–Huygens’ principle is a powerful technique for determining the field in a source free region outside a surface from knowledge of the field distributed over that surface. It is applicable to arbitrary shaped apertures over which both the electric and magnetic fields are prescribed. The derivation of the Kirchhoff–Huygens’ method is based upon the Lorentz integral form of the reciprocity theorem, which essentially constitutes a Greens theorem. When expressed mathematically the electric field, at a point P, radiated by a closed Huygens’ surface S is [14]   π e uˆ = 2 jλ



 S

     uˆ × nˆ × E + Z0 uˆ × nˆ × H × uˆ ejk0 uˆ ·r da

(9.91)

This expression yields the far-field vector pattern function from an integral of the electric and magnetic fields over the closed surface S·da is an elemental area of S. The derivation of this equation from the near-field form of the Kirchhoff–Huygens’ formula can be found presented in Box 9.1. This expression is restricted to evaluating the field at a point in space infinitely far removed from the current source. Here, the r −1 term and the unimportant phase factor have been suppressed. The geometry for this statement of the Kirchhoff–Huygens’ formula can be found illustrated in its conventional form in Figure 9.38. Figure 9.39 illustrates the case where the integration surface is that of a regular parallel piped, which in this case is a cube. The integration can be considered to be the superposition of the integration of the field over each of the six planes. The area

P

r r

S

u O

r0 n

Figure 9.38

Coordinate system for Kirchhoff–Huygens’ formula

332 Principles of planar near-field antenna measurements Y Plane 2

Plane 1 Plane 3 Plane 6

Plane 5

Z

X

Plane 4

Figure 9.39

Regular parallel piped integration surface

of the elemental sources and unit normal are detailed below. daplane1 = daplane3 = dy dz

(9.92)

daplane2 = daplane4 = dx dz

(9.93)

daplane5 = daplane6 = dx dy

(9.94)

nˆ plane3 = −ˆnplane1 = eˆ x

(9.95)

nˆ plane2 = −ˆnplane4 = eˆ y

(9.96)

nˆ plane5 = −ˆnplane6 = eˆ z

(9.97)

And

The Kirchhoff–Huygens’ theory is exact, provided that the field is known exactly over a closed surface. The closed surface can take the form of an infinite plane together with an infinite radius hemisphere. If the source is finite then, from the radiation condition, it can be seen that no contribution to the total field arises from any part of the hemispherical portion of the surface. Furthermore, the EM field will

Advanced planar near-field antenna measurements 333 be small over a large portion of the planar surface with the majority of the field being concentrated over a relatively small area in the immediate vicinity of the radiating structure. Thus, it is often assumed that the far-field will be principally determined by the field distribution over this finite area. This approximation is often described as Kirchhoff’s assumption. Importantly, this assumption is not a prerequisite for the success of the poly-planar measurement technique. Subtly, for the case where the sampled field is truncated the field predicted by the Kirchhoff–Huygens’ principle and the angular spectrum method will not agree. As the Kirchhoff–Huygens’ method contains both electric and magnetic tangential fields, truncating the measurement equates to imposing the additional boundary condition that the aperture is set in a perfect magnetic conductor. The PWS method avoids this assumption and the farfields can and will differ as a result.

Box 9.1 These expressions can be further simplified for the far-field case. Since r = r  + r0

(9.98)

    r  = r  = r − r0 

(9.99)

So

Clearly

  2 (9.100) r = (rx − r0x )2 + ry − r0y + (rz − r0z )2  2 + r 2 + r 2 + r 2 + r 2 − 2r r − 2r r − 2r r r  = rx2 + r0x x 0x y 0y z 0z y z 0y 0z 

(9.101)

Thus r =

 r 2 + r02 − 2r · r0

(9.102)

Using the first two terms of the binomial expansion for r  yields r · r0 r = r − = r − rˆ · r0 r Thus 

e−jk0 r = ejk0 Now



rˆ ·r0 −r

 ∇0 ψ = jk0 +

1 r



= e−jk0 r ejk0 rˆ ·r0

(9.103)

(9.104)



e−jk0 r  rˆ r

(9.105)

334 Principles of planar near-field antenna measurements Thus neglecting of small quantities and thus introducing an error of  products  the order of O 1/r 2 then 









jk0 e−jk0 r  jk0 ejk0 rˆ ·r0 −r  r ˆ = rˆ (9.106) ∇0 ψ = r r Since in the far-field r and r  are parallel, then for the purposes of evaluating amplitude the magnitude of r can be considered to be the same as the magnitude of r  with no loss of precision thus 

jk0 e−jk0 r  jk0 ejk0 rˆ ·r0 −r rˆ r ˆ = (9.107) ∇0 ψ = r r These expressions can be substituted into the general formula for the electric field thus Ep =

1 e−jk0 r 4π r          × −jωµ n × H + jk0 n × E × rˆ + n · E rˆ ejk0 rˆ ·r0 da S

(9.108)

Since in the far-field the electric field will be tangential to the direction of propagation        1 e−jk0 r Ep = −jωµ n × H + jk0 n × E × rˆ ejk0 rˆ ·r0 da 4π r S (9.109)     ωµ    1 jk0 e−jk0 r Ep = − n × H ejk0 rˆ ·r0 da rˆ × n × E + 4π r k0 S (9.110) Now −

1 jk0 e−jk0 r k0 e−jk0 r e−jk0 r e−jk0 r k0 e−jk0 r π (9.111) = = = = 4π r jr4π 2jrλ k0 r 2jλ k0 r jλ2

and ωµ 2πf µ 2πf λµ µ = = = cµ = √ = k0 2π εµ k0 Thus e−jk0 r π Ep = k0 r jλ2

 S



!

µ =Z ε

    rˆ × n × E + Z n × H ejk0 rˆ ·r0 da

(9.112)

(9.113)

Advanced planar near-field antenna measurements 335 Now the term n × H will contain a radial component that must be zero in the far-field. Consider the following term    rˆ × n × H × rˆ (9.114) Using

 2         A × B × A = A · A B − A · B A = A − A · B A

(9.115)

Then

         rˆ × n × H × rˆ = rˆ  nˆ × H − rˆ · nˆ × H rˆ      Since rˆ  = 1 and in the far-field rˆ · nˆ × H rˆ = 0 then    rˆ × n × H × rˆ = nˆ × H Thus Ep =

e−jk0 r π k0 r jλ2

 S



(9.116) (9.117)

     rˆ × n × E + Z rˆ × n × H × rˆ ejk0 rˆ ·r0 da (9.118)

Where the magnetic field can be obtained from the electric field using Hp =

1 rˆ × Ep Z

(9.119)

9.5.5 Applicability of the poly-planar technique 9.5.5.1 Bi-scan auxiliary rotation technique To demonstrate the theoretical applicability of such a bi-scan procedure Figure 9.40 illustrates the simulated near-field of a possible bi-scan configuration that could be used to produce far-field patterns. It shows a near-field measurement geometry that, using the field equivalence principle, can be used to construct the surface of a partial plane, initially parallel with the xy-plane. Initially the AUT would be placed centrally to a (2.25 × 4 m) partial scan plane and an acquisition made. Then, the measurement plane would be rotated by ±30◦ in azimuth about the origin of the antenna coordinate system to construct the field distribution plotted in Figure 9.40. Thus, when the AUT is placed centrally 1 m from the initial partial plane two (2.25 × 4 m) scans can be used to produce the equivalent of one 4 × 4 m scan plane. Figure 9.41 is a far-field plot comparing ideal far-field data from a (4 × 4 m) plane and equivalent data derived from the two rotated partial plane near-field data as illustrated above. The agreement between the respective cuts is good with differences only becoming apparent beyond ±80◦ where this is particularly apparent in the phase plot. These

336 Principles of planar near-field antenna measurements 0

–10

Ey (dB)

–20

–30 2

–40

1 1 0.5 Z

–50 0 Y

0 –2

–60

–1

–1

0

1

2

–2

–70

X –80

Figure 9.40

Grey-scale plot of near-field power

0 Theoretical 30 deg –10

Power (dB)

–20 –30 –40 –50 –60 –70

–80

Figure 9.41

–60

–40

–20

0 Theta

20

40

60

80

Comparison of far-field horizontal cuts of polyhedral transform and single planar patterns

Advanced planar near-field antenna measurements 337 differences have been found to result from the discontinuity encountered at the intersection of the two planes. Unfortunately, with any symmetrical bi-scan configuration the intersection between the adjacent partial scans lies in the region of greatest field intensity. 9.5.5.2 Tri-scan auxiliary rotation technique The tri-scan configuration is a practical alternative candidate for a poly-planar system as the intersection between adjacent partial scans can be chosen to be away from regions of high field intensities. To this end the near-field measurement geometry was simulated using the field equivalence principle to construct the surface of a partial plane, initially parallel with the xy-plane. This plane was rotated by ±30◦ in azimuth about the origin of the antenna coordinate system to construct the field distribution shown in Figure 9.42. This constitutes a desirable arrangement where the intersections have been chosen to be across a region of space in which the field intensities are typically more than 30 dB smaller than the largest signal. Figure 9.43 contains a great circle cut of the equivalent far-field vector pattern function compared with the ideal (theoretical) pattern. These results can be seen to be only in error for very large angles that is those angles greater than 87◦ , this stems from the discontinuity in the first derivative of the near-field data across the intersection between the partial scans. However, it can be seen from Figure 9.44 that the equivalent multipath level everywhere within the maximum look angle is low. The largest value being less than −60 dB and the value being typically less than −70 dB over the majority of the forward hemisphere which is approaching the practical noise floor of a typical planar facility. The extent of the differences between the respective far-field patterns can be quantified with the evaluation of the coefficient of ordinal correspondence k, details of which can be found in Chapter 8. Table 9.4 contains values of k, using the ordinal assessment technique, that correspond to a conventional planar configuration, a bi-scan auxiliary rotation scheme (two planes inclined at ±30◦ and where the intersection is through the middle of the antennas peak field) and the modified tri-scan auxiliary rotation scheme. Along with these a pentahedral scan plane is assessed where this procedure yields a poly-planar measurement consisting of five intersecting partial scans that when combined produce a sampling surface that resembles a flat topped pyramid. Note the conventional planar measurement assessed represents a (2.25×4 m) scan plane assessed against the (4 × 4 m) plane. This represents the best measurement that could be made using the maximum scan plan utilized in the poly-planar measurements. The conventional planar measurement can be thought of as constituting the benchmark by which other novel schemes can be compared. The imperfect k value reflects the degradation of the far-field pattern that inevitably results from the introduction of spectral leakage caused by truncation of the near-field data set. Although the biscan configuration successfully increases the ability of a given facility to determine wideout antenna performance, additional errors are introduced that result from the intersection of the partial scans. Clearly the tri-scan scheme can be seen to offer

338 Principles of planar near-field antenna measurements 0 Ey (dB)

–10

–20

–30 2 –40

1 1 0.5

–50

Z

0

0 –2

–60

–1

–1

0 1 2

–2

Y

–70

X –80

Figure 9.42

Grey-scale plot of simulated near-field power for tri-scan configuration

0 Theoretical Tri-scan

Power (dB)

–10 –20 –30 –40 –50 –60 –70

–80

–60

–40

–20

0 20 Theta (deg)

40

Phase (deg)

150

60

80

Theoretical Tri-scan

100 50 0 –50 –100 –150 –80

Figure 9.43

–60

–40

–20

0 20 Theta (deg)

40

60

80

Comparison of far-field horizontal cuts of polyhedral transform and single planar prediction

Advanced planar near-field antenna measurements 339 0 Ideal EMPL

–10 –20

Power (dB)

–30 –40 –50 –60 –70 –80 –90 –100

–80

Figure 9.44

–60

–40

–20

0 Theta (deg)

20

40

60

80

Comparison of far-field horizontal cuts of polyhedral transformed single planar prediction

results that constitute an overall improvement to those supplied by conventional planar techniques. The tri-scan scheme addresses the problem of obtaining wideout azimuthal performance, but does not address the problem in the elevation plane. This can be readily resolved by rolling the AUT through ±90◦ about its mechanical boresight and repeating the measurement. This would not be suitable for all antennas and alternatively the AUT can be ‘nodded’ in elevation, then permitting data to be sampled over the surface of two additional planes. This procedure yields a poly-planar measurement consisting of five intersecting partial scans that when combined produce a sampling surface that resembles a flat topped pyramid. Inspecting the degree of agreement for the pentahedral scan is now found to be encouraging with differences only becoming apparent at the –70 dB level or for large polar angle. Table 9.4 further illustrates the degree of success of this technique with the pentahedral auxiliary rotation scheme offering the best performance. This technique is particularly applicable for instruments with a rectangular aperture plane as little power is delivered to inter-cardinal regions.

9.5.6 Complete poly-planar rotational technique The results illustrated above have been simulations that highlight the potential gained from the polyhedral process, however, in order to provide a full explanation of its applicability the following section describes experimental validations of the technique in a very demanding scenario.

340 Principles of planar near-field antenna measurements Table 9.4

Comparison k-values from various measurement configurations

Measurement

k

Conventional planar measurement Bi-scan auxiliary rotation scheme Tri-scan auxiliary rotation scheme Pentahedral scan auxiliary

0.6372 0.5333 0.8088 0.8154

A poly-planar technique in which the AUT is mounted in one of six discrete orientations: these orientations being the positive and negative directions of three orthogonal positional axes is also possible. This cubic geometry is the most demanding case as the orthogonality between adjacent partial scans would constitute a worst case scenario whilst being relative simple to realize. As already stated such a cubic geometry for the orientation of the partial scans is very demanding as and if the techniques can be shown to be effective in this scenario it can be assumed to be viable in almost all other measurement configurations. 9.5.6.1 Experimental set up To demonstrate the technique a measurement set up was developed that utilized the following components and instrumentation to allow poly-planar measurement to be made and processed to provide predictions of the far-field pattern of an AUT in one of the most demanding scenarios that could be envisaged. 1. A fully absorber lined facility: to minimise multipath. 2. A planar positioner of an inverted ‘T’ design: This design was chosen in order that the scattering cross section of the frame could be minimised; additionally the relatively small physical dimensions of the scan plane, approximately 1 m2 , enable the planarity of the scanner to be maximised. Other advantages include minimising the length of the RF cabling within the facility. As highly phase stable cable is often relatively lossy, that is, 1 dBm−1 , the short length of the cable runs enables the dynamic range of the facility to be maximised, which is crucial, when the AUT is not nominally aligned to the axes of the range. 3. A positioner upon which the AUT could be mounted and aligned precisely in one of six discrete orientations: these orientations being the positive and negative directions of three orthogonal positional axes. 4. An RF subsystem based around a vector network analyser. The scanner used was small, light and very fast with a maximum scan speed of 0.5 ms−1 , thus acquisition times were short, typically of the order of a few minutes so very little thermal drift was present within a given, and between partial scan acquisitions.

Advanced planar near-field antenna measurements 341 The AUT was mounted in one of six discrete orientations: these orientations being the positive and negative directions of three orthogonal positional axes. Finally a relatively low gain corrugated horn was chosen as the AUT as this class of antenna is conventionally thought to be unsuitable for characterisation by planar techniques. 9.5.6.2 Experimental procedure To experimentally validate the technique the x- and y-polarised electric field components were sampled using a square acquisition window of −0.425 m ≤ xRFS , yRFS ≤ 0.425 m with a range length, that is, an AUT-to-probe separation of 0.282 m. Since the intention was that the tangential components of the near electric field were to be sampled over the surface of a cube, this corresponded to an over scan of 12 elements, that is, approximately 6 wavelengths, around the perimeter of the square acquisition window. Once the x− and y−polarised near-field components had been sampled the AUT was rotated by 90◦ in azimuth so the second side of the cube could be measured. All six surfaces of the cube were sampled by performing the following rotations from position: 1. 2. 3. 4. 5.

AUT nominally aligned to axes of range. Positive rotation of 90◦ about y-axis. Negative rotation of 90◦ about y-axis. Positive rotation of 180◦ about y-axis. Positive rotation of 90◦ about z-axis followed by a positive rotation of 90◦ about new x-axis. 6. Positive rotation of 90◦ about z-axis followed by a negative rotation of 90◦ about new x-axis. The requirement for the inclusion of the back plane follows from the requirement to perform the pattern integration over a closed surface. When the partial scans were completed each of the six partial scans were processed using the novel transformation algorithm described. The y-polarised electric nearfield can be found plotted in Figure 9.45 and as expected, the fields at the intersection between adjacent partial scans are continuous. Similarly encouraging results were obtained for other polarisations and for the magnetic fields. These data sets were subsequently transformed to the far-field and resolved onto a Ludwig III polarisation basis. Great circle cardinal cuts are shown in Figures 9.46 and 9.47. Here, the black traces represent patterns obtained from the poly-planar technique whilst the dashed traces denote results obtained from a compact antenna test range (CATR). The high frequency oscillatory behaviour evident within the azimuth cut of the CATR at wide angles is a result of a multiple reflections within the facility and should be ignored. As described previously, the six partial planes will not intersect perfectly and the adverse effects of reflections from scatterers within the near-field chamber will degrade the resulting far-field patterns. Corrugated horns are renowned for their symmetry. Consequently, lack of symmetry can often be used as an indication that a

342 Principles of planar near-field antenna measurements Ey AMS (dB) Freq. 10 (GHz) top view 0 –10 0.3 –20 0.2 –30

z (m)

0.1 –40

0

–0.1

–50

–0.2

–60 –70 0.2 0.1

0.2 0.1

0 y (m)

Figure 9.45

–0.1 –0.2

–0.1 –0.2

0 x (m)

–80 –90

Probe-corrected y-polarised electric field component

measurement is unreliable. Here, although a good degree of symmetry can be observed in the azimuth plane, the elevation cut clearly contains a number of asymmetries. This difference between the cardinal cuts is most probably an artefact of additional rotation required to sample the top and bottom planes of the cube, as the additional 90◦ rotation will inevitably introduce further alignment errors. Clearly, as the AUT is located at the centre of a conceptual measurement cube, classically the angle of validity for the front plane would be ±45◦ in azimuth and elevation. Thus, particular attention should be paid to regions around ±45◦ as if the transformation were in error, this is where it would be expected to be most noticeable. Crucially, no discernible divergence is observable in this region. Finally, the radial component of the far electric and magnetic fields were calculated. As expected these components were found to be reassuringly small with a peak signal of approximately 150 dB below the copolar peak.

9.6

Concluding remarks

Thus, as demonstrated above, such a poly-planar technique is effective in producing accurate far-field patterns even in the most demanding of scenarios. The measurement

Advanced planar near-field antenna measurements 343 Copolar power (dB) 10 Poly-planar QMUL CATR

0

Power (dB)

−10 −20 −30 −40 −50 −60 −70

Figure 9.46

−80

−60

−40

−20

0 Az (deg)

20

40

60

80

Azimuth great circle cut Copolar power (dB)

10 Poly-planar QMUL CATR

0 –10

Power (dB)

–20 –30 –40 –50 –60 –70

Figure 9.47

–80

–60

–40

–20

Elevation great circle cut

0 El (deg)

20

40

60

80

344 Principles of planar near-field antenna measurements processes described in the text of this chapter are new, at the time of writing, in the stage of advanced research and still require a great deal of refinement. However, it is clear that probe-corrected spectral techniques can be combined with the Kirchhoff– Huygens’ method to form a hybrid technique that alleviates the deficiencies that render these techniques useless when used individually. When such measurements are assessed using the advanced measurement methodologies described in Chapter 8 additional confirmation is supplied as to the effectiveness of the technique [15]. Thus, by showing that in measuring the near-field on all six sides of a cubic box an accurate a prediction of the full spherical radiation pattern of the antenna can be obtained, an entirely new range of planar near-field antenna measurements is shown to be possible. This insight into the very newest of scan techniques used in poly-planar near-field antenna measurement represents, as far as the authors are aware, the state of the art in such measurements at the time of writing. However, from the contents of this text, particularly from this chapter it should be clear that planar near-field scanning is a measurement technique that is still under development and that many new concepts and techniques are still to be discovered developed and utilized. Other equally valid and ingenious techniques have unfortunately been omitted due to a lack of space.

9.7

References

1 Spiegel, M.R.: Theory on Problems of Vector Analysis and an Introduction to Tesor Analysis (Schaum’s Outline Series, Schaum Publishing Co., New York, 1959), p. 30 2 Hsu, H.P.: Applied Fourier Analysis (Harcourt Brace, San Diego, CA, 1967), p. 115 3 Newell, A.C.: Planar Near-Field Antenna Measurements (Electromagnetic Fields Divison, Boulder, Colorado, Technical Note OP-US/6632, 1985), p. 55 4 Fooshe, D.A.: ‘Application of error correction technologies to near-field antenna measurement systems’, Proceedings of the 1996 IEEE Aerospace Applications Conference, Aspen, CO, February 1996;1:141–9 5 Muth, L.A., and Lewis, R.L.: ‘A general technique to correct probe position errors in planar near-field measurements to arbitrary accuracy’, IEEE Transactions on Antennas Propagation, December 1990;38:1925–32 6 Francis, M.H.: A comparison of K-correction and Taylor-series correction for probe-position errors in planar near-field scanning, Proceedings of the Antenna Measurement Techniques Association 17th Annual Meeting, Williamsburg, VA, November 1995, pp. 341–7 7 Corey, L.E., and Joy, E.B.: ‘On computation of electromagnetic fields on planar surfaces from fields on nearby surfaces’, IEEE Transactions on Antennas Propagation, March 1981;AP-29:402 8 Agrawal, P.K.: ‘A method to compensate for probe positioning errors in an antenna near-field facility’, Proceedings of IEEE Antennas and Propagation International Symposium, Albuquerque, NM, 1982;20:218

Advanced planar near-field antenna measurements 345 9 McCormick, J., and Da. Silva, E.: ‘The use of an auxiliary translation system in near-field antenna measurements’, IEE 10th International conference on antennas and propagation (ICAP), April 1997, pp. 1.90–1.95 10 McCormick, J.: The Use of Secondary Spatial Transforms in Near-field Antenna measurements, Ph.D. Thesis, Open University, 1999 11 Anton, H.: Calculus With Analytic Geometry (John Wiley & Sons Inc., New York, 1995), p. 281 12 Churchill, R.V.: Fourier Series and Boundary Value Problems (McGraw Hill, New York, 1963), p. 92 13 Harris, F.J.: ‘On the use of windows for harmonic analysis with the discrete Fourier transform’, Proceedings of the IEEE, January 1978;66 (1):51–83 14 Clarke, R.H., and Brown, J.: Diffraction Theory and Antennas (Ellis Horwood Ltd, Chichester, 1980), p. 227 15 McCormick, J., Gregson, S.F., and Parini, C.G.: ‘Quantitative measures of comparison between antenna pattern data sets’, IEE Proc.-Microw. Antennas Prop., December 2005;152 (6):539–55

Appendix A

Other theories of interaction

This list and very short description of postulated alternative mechanisms of interaction is not meant to be exhaustive or comprehensive. It is included to show that a wide range of interpretations of the interaction of charged particles across space and time are available. All of them can appropriately be used over a range of applicable circumstances and they all fail outside their range of applicability. Certain of them lend themselves to more than one formal structure, for example, it is possible to describe classical electromagnetism in terms of the Maxwell tensor in a four-vector relativistic space time and many of them fit more or less neatly into more over arching concepts for example, local gauge invariant field theories or Fisher information exchange concepts. However, the material in this overall text has been written so that while it is specific to classical electromagnetism, as this is anticipated to be the most widely used model recognized by the target readership, none of the above postulated mechanisms are inconsistent with the material in the book.

A.1 Examples of postulated mechanisms of interaction A.1.1 Classical electromagnetism The exchange of the fundamental physical quantities of energy, linear and angular momentum between distributions of point-like charged particles is the subject of classical electromagnetism. The interaction of these charged particles, as they move through three-dimensional space, as a charge distribution in space persisting over a given period of time, is explained via the invocation of the concept of fields. These fields are associated with a charge distribution in three-dimensional space and the rate of change of this distribution with respect to time, that is, currents. These fields themselves have extension in space, persistence in time and extend over threedimensional space and absolute time to expedite the interchange of the fundamental physical quantities between charged particles contained within the space.

348 Principles of planar near-field antenna measurements The fields are defined via the Lorentz’ force law F = ma = q(E + (v × B))

(A.1)

which quantifies the extent of the exchanges of energy, linear momentum and angular momentum between charged particles and electromagnetic (EM) fields via the Newtonian concept of force, that is, that which alters or tends to alter the motion of bodies and inertial mass. Thus this simple physical law defines the extent to which the fields interact with material objects composed of individual massive particles, while the relationship between the fields is most easily illustrated via the application of Maxell’s equations. In classical EM field theory all EM interactions can be thought of as the exchange of these fundamental quantities between point-like charged particles distributed in absolute space and time, which are themselves the sources of the fields, which allow the exchange of these quantities between the particles. All of the concepts, for example, current, charge and so on, used in classical electromagnetism can be derived from and considered as manifestations of the fundamental physical quantities listed above. Indeed, the basic EM SI unit, the ampere, is usually defined and measured in terms of the force between conductors of given lengths separated by specified distances in a three-dimensional absolute space. Thus the fundamental mechanism postulated is that finite velocity retarded waves in the EM field, produced by the acceleration of charged particles, propagate across space to allow the exchange of these physical quantities between charged particles that are displaced from each other in the three-dimensional absolute space.

A.1.2 Classical quantum mechanics As its name suggests in the standard theory of quantum mechanics as expressed mathematically either by wave mechanics or matrix mechanics the fundamental physical quantities that are exchanged across space come in discrete quantised amounts. These quanta are scaled relative to the fundamental quantum of action defined by the Plank constant. The concepts of energy, linear and angular momentum together with their exchange between particles across three-dimensional space and time are still relevant but force as a concept is no longer required. The quanta are characterised by quantum states which refer to the energy and momentum of the quanta and changes in these quantum states are the product of interactions between quanta via a field. However, in the algorithms associated with this theory the quantum states are super positions of the Eigen states of operators which relate to energy and momentum. Thus the theory does not predict the values of these variables; it provides probabilities for their values and only the operation of measurement will collapse the functions to provide an actual value for the variable/observable. Additionally as Eigen states of one variable are not Eigen states of others, certain variables cannot be simultaneously measured to a level of accuracy better than that defined by the Plank constant, for example, the momentum/position and energy/time. This uncertainty in these variables is elevated to the level of a principle upon which the entire mathematical algorithm, which produces all predictions and simulations, can be based.

Appendix A 349 The probability of jumps between quantum states and therefore the exchange of energy and momentum between particles that have an associated electric charge can, within the theoretical framework, also be calculated via the invocation of the concept of an overlap integral. This integral will have an oscillatory solution the frequency of which will align with the frequency that would be assigned to a wave in a classical field that would be have the same values of momentum and energy. Thus no conceptual mechanism of interaction, save that of the quantised interaction that statistically in the limit of large sample sizes is complimentary to a classical situation, is put forward. The theory merely calculates the probability that, given an initial condition of charged particles distributed and moving in absolute space and time therefore characterised by given quantum states, an interaction, internal or with other distributions of charge, will occur. Thus, it provides a prediction of the probable future quantum states and thus energy and momentum of charged particle distributions in absolute space and time if the particles are subjected to observation.

A.1.3 Relativistic quantum mechanics This is usually referred to as quantum electrodynamics (QED), as a result of this being the most successful methodology for the solution of its fundamental equations. This postulated mechanism of interaction is by far the most accurate in terms of the predictions it provides. In fact at the time of writing QED is by far the most accurate, by several orders of magnitude, physical theory and thus predictive tool in the entire realm of science and technology. However, it is a very complex mathematically and conceptually demanding construct based primarily on P.A.M. Dirac’s electron field equation. As such it involves a range of re-normalisation techniques that are required to extract predictions that do not explode into infinities under computation. These techniques being deployed to solve the Dirac equation first postulated by Paul Dirac in 1928 which has the form  
 ∂ (A.2) γ υ i υ − eAυ (x) + m ψ (x) = 0 ∂x Where the Einstein summation conventions are used so   1 0 0 0 0 1 0 0  γ0 =  0 0 −1 0 0 0 0 −1 

0  0 γ1 =  0 1

0 0 0 −1 1 0 0 0

 −1 0  0 0

(A.3)

(A.4)

350 Principles of planar near-field antenna measurements 

 0 0 0 i 0 0 −i 0  γ2 =  0 −i 0 0 i 0 0 0 

0 0 0 γ = 1 0

0 0 0 −1

−1 0 0 0

(A.5)

 0 1  0 0

(A.6)

And the complete equation can be written out as 

i∂0 − eA0 + m 0  i (∂ − ∂ ) − e (A − A ) 1 3 1 3 ∂2 + ieA2   ψe↑ (x) ψe↓ (x)  =0 × ψ (x)

0 i∂0 − eA0 + m −∂3 − ieA2 i (∂1 − ∂2 ) − e (A1 − A3 )

p↑

−i (∂1 − ∂3 ) + e (A1 + A3 ) −∂2 − ieA2 −i∂0 − eA0 + m 0

 ∂2 + ieA2 −i (∂1 − ∂2 ) + e (A1 − A2 )  0 −∂0 + eA0 + m

(A.7)

ψp↓ (x)

where, e is the electron charge, m is the electron mass, A is the vector potential, the subscripts 0 to 3 refer to the three spatial and one temporal dimensions and ψ is the wave function subscripted by e for electron, p for positron, that is, antielectron and ↑↓ relating to either up or down spin states of the matter or antimatter. A number of concepts that do not have classical analogies must usually be invoked to construct the theory including, the idea of the second quantisation (existence of gauge bosons) and therefore EM fields being the classical limit of assemblies of individual coulombic and magnetic real or virtual quanta. This, along with the now experimentally proven concepts of antimatter and energy/matter equivalence, provides the basis for the perturbations imposed on the electrodynamic systems to compute possible outcomes. As with classical quantum mechanics the theory is not deterministic and will only provide the probabilities associated with possible interactions which this time occur in the more physically accurate framework of relativistic four-dimensional space time.

A.1.4 Collective electrodynamics In his original Treatise on Electricity and Magnetism, James Clerk Maxwell asserts the existence of the electro-dynamic momentum A, a vector whose direction and magnitude vary from one part of space to another, and from this he deduces mathematically the magnetic induction B as a derived vector. Maxwell found it necessary to derive such secondary vectors as he was working under the mistaken assumption that electric, E fields and magnetic, B fields were representations of stresses in the ether, the medium that was thought necessary for the propagation of EM disturbances. The development of Maxwell’s equations by Heaviside and Hertz also, as a result of their belief in the existence of the ether concentrated heavily on the concepts of E,

Appendix A 351 D, B and H , A was dismissed as ‘of a metaphysical nature’ and thus ignored or side stepped if possible. However, experimental evidence accumulated over the nineteenth and twentieth centuries, first conclusively invalidated the concept of the ether and second via observations of the Aharonov Bohm effect confirmed the existence of A as the primary field in electrodynamics and that in fact B is the more metaphysical concept. Collective electrodynamics is an effort to derive a theory of electrodynamics based on the physical interpretation of A as being functionally related to the quantum mechanical wave function of charged particles. It takes as its starting place a redefinition of complementarity where classical systems are not defined in terms of size (macro or micro), but in terms of coherence. This is particularly highlighted by the examination of superconducting systems as the archetypal systems that illustrate macroscale coherence that is not statistically obscured by thermodynamic noise. To remove the requirement of postulating the existence of a scalar potential the theory is usually expressed in terms of four vectors in a relativistic space time thus the entire content of Maxwell’s equations, as usually stated, can be encapsulated in one equation, that is,   1 ∂2 2 (A.8) ∇ − 2 · 2 A = −µ0 J c ∂t And the relationship between the vector potential A and ϕ the phase accumulation around a closed path of the charged particles wave function, as originally postulated by Erwin Schrodinger in 1926, can be summarised via the relationships  (A.9) ϕ = (ω1 − ω2 ) dt = k · dl 

= V dt = A · dl (A.10) where

=

h ϕ 2πq0

(A.11)

and, ω1 is the wave function frequency at commencement of path, ω2 is the wave function frequency at completion of path, = flux. It is this relationship between A as the source of the four-vector current density J and characteristics of the electron wave function that forms the basis of collective electrodynamics.

A.1.5 Absorber theory This concept was developed as an attempt to reconcile the local aspects of energy conservation that requires the power dissipated in the antenna via its radiation resistance be equal to that developed via the flux around it. A range of theoretical mechanisms have been postulated to account for the apparent anomaly in this energy conservation process, some of which invoke the concept

352 Principles of planar near-field antenna measurements of non-point-like charged particles being involved, for example, Poincare stresses. Others involve the retention of the advanced wave solution to the Helmholtz wave equation, most famously the Wheeler Feynman absorber theory of radiation. As stated in the original Wheeler Feynman paper on the subject, ‘Reviews of modern physics, April 1945,’ [1] and reprinted with permission from J.A. Wheeler. Copyright 1945 by the American Physical Society. A charged particle on being accelerated sends out EM energy and itself losses energy. This is interpreted as caused by force acting on the given particle given in magnitude and direction by the expression

 2 q2 da

. 3 c3 dt

(A.12)

where q = charge c = speed of light a = acceleration. When the particle is moving slowly, and by a more complicated expression when its speed is appreciable with respect to the velocity of light. The existence of the radiative reaction is well attested: (a) by the electrical potential required to drive a wireless antenna (b) by the loss of energy when a charged particle which has been deflected and therefore accelerated in its passage near an atomic nucleus and (c) the cooling suffered by a glowing object. The origin of the radiative reaction has not been nearly so clear as its existence.

The theory ignores the concept of self-interaction, that is, it specifically states that charged particles do not interact with the total EM in their vicinity; they only interact with the field in their vicinity produced via the interaction with other charged particles. As such the theory removes all the difficulties encountered as a result of the infinities produced by the self-interaction. However, the removal of the selfinteraction apparently leaves the problem of the radiation reaction even less well represented in physical theory. The absorber theory attempts to provide an answer to this problem by considering the interaction between charged particles, although still considered to be mediated by an EM field, to be essentially action at a distance between charged particles, so the existence of both an emitter and an absorber is required before the interaction can occur. Thus, conceptually the theory is separated from classical EM theory in that it does not consider the field that causes the force between particles to exist in the absence of all the particles, that is, a charged particle does not of itself generate a force; the interaction between charged particles generates a force. In classical electromagnetism the advanced wave solution of the wave equations that describe propagation are ignored and only the retarded wave solutions are considered to be physically relevant; in absorber theory this arbitrary removal of these solutions does not occur. In the theory it is suggested that charges interact half via the advanced wave and half via the retarded waves. A detailed description of the theory

Appendix A 353 can be found in the 1945 paper quoted above but the result of treating the radiation in this fashion is that • In the majority of cases no effects that are not consistent with classical EM theory are observed • The interaction does provide a mechanism that explains the radiative reaction force. When a charged particle, p1 the emitter, is accelerated at a time t1 the retarded wave it produces interacts with other charged particles, for ease of understanding we will say only one other particle, p2 the absorber, at a time t2 , where t2 = t1 + r/c and where r = distance between the particles. The acceleration of p2 as a result of the interaction with the retarded wave generated by p1 not only produces a retarded wave generated by p2 , that is, a normal reflected wave, but also an advanced wave which will propagate back to p1 at a time t3 = t2 − r/c. Thus the advanced wave reaction of p2 the absorber will interact with p1 at the instant the retarded wave is emitted from p1 . So the combination of half-advanced and half-retarded waves produces a situation where at the instant of emission p1 feels the effect of the advanced wave produced by the absorber p2 ; this advanced wave being the source of the radiation reaction that acts at the instant of emission. It is a conceptually demanding theory and for some years although it did provide a consistent explanation for many EM formulas it was inconsistent with the precepts of quantum mechanics. However, in recent years John Cramer at Washington State University in Seattle, has made it also applicable within the realms of quantum as well as classical mechanics and the theory although exotic and somewhat esoteric is consistent and does provide accurate predictions of a very wide range of EM phenomena.

A.2 References 1 Wheeler, J.A., and Feynnan, R.P.: ‘Interaction with the absorber as a mechanism of radiation’, Reviews of Modern Physics, April/May 1945; 17 (267):157–81

Appendix B

Measurement definitions as used in the text

Standard: An accepted sample or procedure that is used for establishing a unit for the measurement of physical quantities. Measurement: Highly controlled set of operations having the object of determining a value of a quantity, that is, the execution of a measurement procedure. Measurand: Particular physical quantity subject to measurement. Result of measurement: Value attributed to a measurand, obtained by following the measurement procedure. Uncorrected result: Result of measurement before correction of systematic errors, that is, Type B errors. Corrected result: Result of measurement after correction of Type B errors; only Type A errors remaining. Type A error: Errors that can be evaluated by statistical methods. Type B error: Errors that must be evaluated by other methods. Measurement trial: Repeat of the measurement procedure. Measurement scale: The graduations that, via ranking and comparison with a standard, allow ordinal measurement data to be quantified cardinally, giving the ratio relative to the standard in multiples of the resolution of the measurement. Also refers to the mapping to representational space which maps the values of the measurand to values in the image set, that is, the mapping the assigns the measurand to specific graduations on the physically realisable scale under the measurement procedure. Resolution: Smallest interval on any measurement scale that can be discerned, often referred to as discrimination in disciplines other than electronics in which discrimination is a term usually reserved to describe the action of filters in rejecting input signals with specific spectral characteristics. Sensitivity: Property of a system or part of a system that indicates how the system reacts to stimuli. It is a measure of the variation in the behaviour of the system caused by some change in the original value of one or more of the elements of the system.

Appendix B 355 For a measurement system the variation in the measurement result as the true value of the measurand varies. Precision: Is a statement about the closeness of the agreement of the results of a set of identically performed trials, where it is presumed that the true value of the measurand remains constant over the period of the trials. Repeatability: Closeness of the agreement between the results of successive measurement trials of the same measurand carried out using the same measurement procedure. Reproducibility: Reserved for the same concept as repeatability, where the periods between trials is not small, but are instead spread over a period of days or years that is very much greater than the length of time required to perform the measurement. True value: Value consistent with a definition of a given particular quantity. Measure: True value of a measurand evaluated via a measurement procedure, that is, after correction, the limit in the measurement results as the number of measurement trials tends to infinity. In the absence of infinity many trials, it can only be defined as the maximum likelihood value with a confidence interval at a given level of significance. Confidence interval: An interval in which, one can be confident, with a given level of probability that a parameter lies. In measurement theory the parameter is the measure of the measurand. Level of significance: The quantified probability that defines the level of confidence that a value falls within an interval, for example, 5 per cent significance means that in 95 out of 100 trials the measurement result can be expected to fall within the interval. Specifically the level of significance of the outcome of any trial is the probability, on the null hypothesis of any result being out with a specified interval. Note: A more detailed definition can be found in Reference 1. Error source: Any variable that can affect the highly controlled set of operations, that is, the measurement procedure and thus produce an error in the measurement result. Error (of measurement): Result of measurement minus a true value of the measurand, used to establish a level of confidence in the dispersion of measurement results around the measure. Measurement-true value = error: Error/true value = relative error. Uncertainty in measurement: Parameter associated with the result of any measurement trial that characterises the dispersion of the values that could reasonably be attributed to the measurand, that is, the result of the evaluation aimed at characterising the range within which the true value of a measurand is estimated to fall, with a given level of confidence. Standard uncertainty: Uncertainty in the measurement derived from the error associated with the functional relationship between a single error source and the uncertainty parameter. For Type A errors a statistical correlation will have to be used in the absence of a functional relationship.

356 Principles of planar near-field antenna measurements Combined uncertainty: Uncertainty in the measurement derived from the errors associated with the functional relationship between all the error sources and the uncertainty parameter. For Type A errors a statistical correlation will have to be used in the absence of a functional relationship. Expanded uncertainty the definition of quoted uncertainty in the measure: The total uncertainty in the measure derived from all the errors associated with the functional relationships between all the error sources and the uncertainty parameter taking into account the distributions of the individual uncertainties and a coverage factor to define a confidence interval at a given level of significance. Coverage factor: A factor included in the calculation of the expanded uncertainty in any measurement, assuming normally distributed errors = 2, to approximate a 95 per cent confidence interval in which the measurand could reasonably be attributed to occur. Accuracy of measurement: Closeness of the agreement between a measurement result derived from following the measurement procedure to determine the true value of the measurand. In the absence of an infinity of trials and therefore a definite measure, the probability of proximity to and dispersion around the true value defined via the expanded uncertainty.

B.1

References

1 Kotz, S., Johnson, N.L., and Read, C.B.: Encyclopaedia of statistical sciences, Vol. 8 (John Wiley & Son, New York, 1982), pp. 466–7

Appendix C

An overview of coordinate systems

Within this text reference is made to a great many coordinate systems and the transformations between them. Implicit within this is the assumption that the tabulating grids are plaid, monotonic and equally spaced. Whilst not necessary from a theoretical stand point these conditions greatly simplify the recording process for a robotic positioner as well as simplifying the tasks of numerical integration, differentiation and interpolation. The following section presents a concise description of the most important coordinate systems and then goes on to discuss methods for representing the relationships between them.

C.1 Antenna mechanical system (AMS) The AMS coordinate axes form a right-handed set nominally orientated coincident and synonymous with the range fixed system (RFS) axes. Thus, looking in the +ZAMS direction, they are orientated as follows: +XAMS axis is horizontal and increases towards the left, +YAMS axis is vertical and increases upwards. This system is used for plotting the far-field patterns.

C.2 Antenna electrical system (AES) The AES coordinate axes form a right-handed set nominally orientated coincident and synonymous with the AMS axes as follows: +XAES parallel to +XAMS , +YAES parallel to +YAMS , +ZAMS parallel to +ZAPS . Thus, looking in the +ZAES direction, the nominal orientation is: +XAES axis: horizontally orientated and increases towards the left, +YAES axis is vertical and increases upwards and +ZAES axis increases towards the far-field. The +ZAES axis defines the electrical boresight of the antenna and the copolar and cross-polar patterns are often resolved onto this system. The AES is attached to the AMS and moves with it. The AES may also be rotated about any or all of its axes.

358 Principles of planar near-field antenna measurements

C.3

Far-field plotting systems

The tabulating, that is, recording, grids considered within this text are assumed to be plaid, monotonic and equally spaced. Mathematically, this implies that the output grids can be reconstructed from the expressions X = X0 + X (m − 1)

(C.1)

Y = Y0 + Y (n − 1)

(C.2)

Where m and n are positive integers n = 1, 2, 3, . . . , N

(C.3)

m = 1, 2, 3, . . . , M

(C.4)

Here, X0 , Y0 are the starting values of the grid in the x- and y-plotting axes respectively, X , Y are the incrementing values in the x- and y-plotting axes, respectively. Although a great many, essentially equivalent, far-field plotting systems can be utilized, perhaps the most commonly adopted ones are 1. 2. 3. 4. 5. 6.

Direction cosine (u, v-space) Azimuth over elevation Elevation over azimuth Polar spherical Equatorial spherical True-view (azimuth and elevation).

By way of illustration, Figure C.1 illustrates schematically the relationship between the polar spherical and azimuth over elevation angles. These systems are introduced and discussed in the following sections. Furthermore, relationships between the various coordinate systems can be obtained by equating the respective direction cosine components.

C.4

Direction cosine

The direction cosine grid can be conceptualised as a projection of a sphere on a plane that is orthogonal to the z-axis. (X , Y ) = (u, v) Here, u and v are the first two coordinates to the field point hence   rˆ = uˆex , vˆey , wˆez

(C.5)

(C.6)

Where u and v are related to the spherical coordinates by u = sin θ cos φ

(C.7)

v = sin θ sin φ

(C.8)

Appendix C 359 Theta axis

yAMS R

uˆ fˆ

-xAMS

Chi axis

k AMS

P

u

f

El

O

zAMS xAMS

Az AUT

Phi axis

Q

Figure C.1

Comparison of Az/El and polar spherical angles

For the forward half space w = cos θ =

 1 − u2 − v 2

(C.9)

For the back half-space  w = cos θ = − 1 − u2 − v2

(C.10)

In the true far-field, the electric and magnetic fields will be identically zero for the reactive, that is, non-visible, regions of uv-space. Thus E (u, v) = 0 when u2 + v2 > 1 2

2

H (u, v) = 0 when u + v > 1

(C.11) (C.12)

Figure C.2 illustrates a far-field pattern when tabulated in a regular direction cosine coordinate system. Here, the main beam has been steered to azimuth = 25◦ , elevation = 20◦ to help illustrate the impact that a particular choice of orientation has on the plotted pattern. K-space coordinate system is related to the direction cosine coordinate system through the linear scaling of the free-space propagation constant,

360 Principles of planar near-field antenna measurements 1 0.75

–10

0.5

–20

0.25 –30 v

0 –40 –0.25 –50

–0.5

–60

–0.75 –1 –1

Figure C.2

–0.75 –0.5 –0.25

0 u

0.25

0.5

0.75

1

Pattern plotted in direction cosine coordinate system

thus

C.5

kx = k0 u

(C.13)

ky = k0 v

(C.14)

kz = k0 w

(C.15)

Azimuth over elevation

The azimuth over elevation grid can be thought of as being that grid that is most closely related to a positioner that consists of an upper azimuth rotator, to which the antenna under test (AUT) is attached and a lower elevation positioner upon which the azimuth rotator is attached. As the AUT is attached to the azimuth positioner, the AUT will rotate about the azimuth axis, that is, therefore the polar axis. The field point is obtained by rotating the horizontal azimuth positioner and vertical elevation positioner through the angle Az and El where the order is unimportant. (X , Y ) = (Az, El) where Az and El define the direction to the field point through   rˆ = − sin (Az) cos (El) eˆ x + sin (El) eˆ y + cos (Az) cos (El) eˆ z

(C.16)

(C.17)

Appendix C 361 90 –10

60

–20

EI (deg)

30 –30 0 –40 –30 –50 –60

–60

–90 –90

Figure C.3

–60

–30

0 Az (deg)

30

60

90

Pattern plotted in azimuth over elevation coordinate system

Hence Az = arctan



−u w



El = arcsin (v)

(C.18) (C.19)

The minus sign included with the x-axis coordinate is used to denote that the observer is standing behind the antenna looking in the positive z direction. An alternative and equally valid, choice would be to associate the minus sign with the y-axis. Figure C.3 illustrates a far-field pattern when tabulated in a regular azimuth over elevation coordinate system. Although in the region around boresight the pattern appears to be unchanged, towards the north, El = 90◦ and south, El = 90◦ , poles the pattern is distorted to the extent that round objects will appear square, cf. θ = 90◦ contour.

C.6 Elevation over azimuth The elevation over azimuth grid can be thought of as being that grid that is most closely related to a positioner that consists of an upper elevation rotator, to which the AUT is attached and a lower azimuth positioner upon which the elevation rotator is attached. As the AUT is attached to the elevation positioner, the AUT will rotate about the elevation axis that is therefore the azimuthal axis. The field point is obtained by rotating the horizontal azimuth positioner and vertical elevation positioner through

362 Principles of planar near-field antenna measurements 90 –10

60

–20 EI (deg)

30 –30 0 –40 –30

–50

–60 –90 –90

Figure C.4

–60 –60

–30

0 30 Az (deg)

60

90

Pattern plotted in elevation over azimuth coordinate system

the angle Az and El where the order is unimportant. (X , Y ) = (Az, El) where Az and El define the direction to the field point through   rˆ = − sin (Az) eˆ x + cos (Az) sin (El) eˆ y + cos (Az) cos (El) eˆ z

(C.20)

(C.21)

Hence Az = arcsin (−u) (C.22) v (C.23) El = arctan w The minus sign included with the x-axis coordinate is used to denote that the observer is standing behind the antenna looking in the positive z direction. Figure C.4 illustrates a far-field pattern when tabulated in a regular elevation over azimuth coordinate system.

C.7

Polar spherical

The polar spherical grid can be thought of as being that grid that is most closely related to a positioner that consists of an upper Roll rotator, that is, φ, to which the AUT is attached and a lower rotator, that is, θ, upon which the upper φ rotator is attached. This is sometimes referred to as the ‘model tower’ arrangement. As the AUT is attached to the Roll positioner, the AUT will rotate about the Roll axis that is therefore the polar axis. The field point is obtained by rotating the horizontal theta positioner and vertical Roll positioner through the angles θ and φ where the order is unimportant. For antenna measurement, this arrangement has the advantage that

Appendix C 363 it moves the AUT through only a small portion of the test zone and it places the blockage that results from the AUT mount entirely in the back hemisphere. Moving the AUT by only a small amount minimises errors associated with imperfections in the illumination of the test zone and can render probe pattern correction unnecessary. (X , Y ) = (θ, φ)

(C.24)

Where θ and φ define the direction to the field point through 

rˆ = sin θ cos φ eˆ x + sin θ sin φ eˆ x + cos θ eˆ x

(C.25)

Hence θ = arccos (w) v φ = arctan u

(C.26)

The definition of an equatorial spherical coordinate system is identical to the polar spherical case. The difference purely results from the application of a 90◦ rotation in θ , in order that the main beam of the radiator points along the positive x-axis (through the equator), rather than along the positive z-axis (through the pole). Figure C.5 illustrates a far-field pattern when tabulated in a regular polar spherical coordinate system.

180 150 –10

120 90

–20

φ (deg)

60 30

–30

0 –40

–30 –60

–50

–90 –120

–60

–150 –180 0

Figure C.5

30 60 90 θ (deg)

Pattern plotted in polar spherical coordinate system

364 Principles of planar near-field antenna measurements

C.8

Azimuth and elevation (true-view) (X , Y ) = (Az, El)

(C.27)

where Az and El define the direction to the field point through Az = −θ cos φ

(C.28)

El = θ sin φ

(C.29)

Hence  Az2 + El2   El φ = arctan −Az θ=

(C.30) (C.31)

The polar spherical angles θ , φ are related to the direction cosines through the normal expressions. The minus sign included with the x-axis coordinate is used to denote that the observer is standing behind the antenna looking in the positive z direction. Figure C.6 illustrates a far-field pattern when tabulated in a regular azimuth and elevation (true-view) coordinate system.

90 –10

60

–20 30 EI (deg)

–30 0 –40 –30 –50 –60

–60

–90 –90

Figure C.6

–60

–30

0 Az (deg)

30

60

90

Pattern plotted in azimuth and elevation (true-view) coordinate system

Appendix C 365

C.9 Range of spherical angles For any given direction in space, as referenced to a given frame of reference, it is possible to use an infinite number of different, but equivalent, spherical angles to describe the relationship. However, in most areas of application the spherical angles are limited to modulo 360 or equivalently modulo 2π . However, even if the range of the spherical angles is limited, it is still possible to describe a given direction in more than one way. An implicit assumption has been made within this text concerning the range of the angles (variables) θ and φ. These are 0 ≤ θ ≤ 180

(C.32)

− 180 ≤ φ ≤ 180

(C.33)

An alternative but equally valid choice is − 180 ≤ θ ≤ 180

(C.34)

0 ≤ φ ≤ 180

(C.35)

Typically, this is convenient for displaying cuts, as only one value of φ is required to specify an entire great circle cut. Conversion between the two systems is facilitated through If θ < 0 then θ = −θ , φ = φ + 180

(C.36)

The inverse mapping is If φ < 0 then θ = −θ, φ = φ + 180

(C.37)

These relationships can be justified from Figure C.7 y

−θ

φ +180°

x θ

Figure C.7

φ

Conventional and alternate spherical angles

366 Principles of planar near-field antenna measurements 180 150 120 90

–10 –20

60 φ (deg)

30

–30

0 –30

–40

–60 –90 –120

–50 –60

–150 –180 –90 –60 –30 0 θ (deg)

Figure C.8

Pattern plotted in polar spherical coordinate system – alternate sphere

Finally as an example, the point (θ = 20, φ = −30) is the same point in space as (θ = −20, φ = 150) for a fixed length of r. Figure C.8 illustrates a far-field pattern when tabulated in a regular polar spherical coordinate system.

C.10 Transformation between coordinate systems Table C.1 comprises a summary of the various coordinate systems discussed above and illustrates how the various parameters can be related to one another. For example, the spherical angles can be related to the azimuth over elevation angles as θ = arccos (w) = arccos (cos (Az) cos (El))   v tan (El) = arctan φ = arctan u − sin (Az)

(C.38) (C.39)

Or conversely the azimuth and elevation angles can be related to the spherical angles as   −u = arctan (tan θ cos φ) (C.40) Az = arctan w El = arcsin (v) = arcsin (sin θ sin φ)

(C.41)

Indeed, by transforming via the direction cosines it is possible to convert from any one set of coordinates to any other set of coordinates. As a note of caution, however, when calculating the inverse tangent it is important that the four quadrant inverse

Appendix C 367 Table C.1

Transformation between different parameters

Coordinate system

x-axis

y-axis

z-axis

Direction cosine

u

v

w

kx k0

ky k0

kz k0

− sin (Az) cos (El)

sin (El)

cos (Az) cos (El)

− sin (Az)

cos (Az) sin (El)

cos (Az) cos (El)

K-space Azimuth over elevation Elevation over azimuth Polar spherical True-view (azimuth and elevation)

sin θ cos φ sin θ sin φ cos θ       2 2 2 2 Az + El Az + El Az2 + El2 sin sin cos       El El × cos tan−1 × sin tan−1 −Az −Az

tangent is used. This function will return angles over a full ±180◦ range rather than over the more limited ±90◦ range that returned by the conventional inverse tangent function.

C.11

Coordinate systems and elemental solid angles

The expression for the elemental solid angle for each of the systems described above is presented in Table C.2. A proof of the true-view expression can be found presented in the section below. Table C.2

Expressions for the elemental solid angle

Coordinate system

Coordinates

d


Direction cosine

(u, v)

1 du dv cos θ

Azimuth over elevation

(Az, El)

cos (El) dAz dEl

Elevation over azimuth

(Az, El)

cos (Az) dAz dEl

Polar spherical

(θ , φ)

sin (θ) dθ dφ

Equatorial spherical

(θ , φ)

sin (θ) dθ dφ

True-view∗

(Az, El)

sinc (θ) dAz dEl

∗ Proof of elemental weighting can be found in Section 10.3.1.

368 Principles of planar near-field antenna measurements

C.12 Relationship between coordinate systems Passive transformation matrices are matrices that post-multiply a point vector to produce a new point vector and is merely a change in the coordinate system. The relationship between two coordinate systems can be defined with the use of a fourby-four homogeneous transformation matrix namely 

  x A1,1  y   A2,1   =  z   A3,1 1 0

A1,2 A2,2 A3,2 0

A1,3 A2,3 A3,3 0

  A1,4 x  y A2,4  · A3,4   z 1 1

   

(C.42)

Or 

  x  y      = [A] ·   z   1

 x y   z  1

(C.43)

Here, the elements A1,4 , A2,4 and A3,4 represent a translation between the origins of the respective frames of reference. The three-by-three submatrix 

A1,1

 A  2,1 A3,1

A1,2 A2,2 A3,2

A1,3





eˆ x · eˆ x

 A2,3   =  eˆ y · eˆ x A3,3 eˆ z · eˆ x

eˆ x · eˆ y eˆ y · eˆ y eˆ z · eˆ y

eˆ x · eˆ z



 eˆ y · eˆ z  eˆ z · eˆ z

(C.44)

contains the rotational information relating these frames of reference. This can also be expressed in terms of the cosine of the angles between the various combinations of unit vectors. This is also why these are termed direction cosine matrices. Specifically then 

A1,1  A2,1 A3,1

A1,2 A2,2 A3,2

  A1,3 cos θ1,1 A2,3  =  cos θ2,1 A3,3 cos θ3,1

cos θ1,2 cos θ2,2 cos θ3,2

 cos θ1,3 cos θ2,3  cos θ3,3

(C.45)

Now, in essence we are merely projecting each of the unit vectors of one coordinate system onto each of the unit vectors of the other. Therefore, each row can be considered to represent a vector describing the orientation of the unit vector of the primed coordinate system in terms of the un-primed coordinate system. Similarly, each of the columns can be considered to represent a vector describing the orientation of the unit vector of the un-primed coordinate system in terms of the primed coordinate system. As the rotations that we are considering are isometric, the distance of a point

Appendix C 369 from the origin in one system will be exactly the same in each coordinate system, that is, it is invariant under the transformation. This can be expressed mathematically as l=

  x2 + y2 + z 2 = x 2 + y 2 + z  2

(C.46)

Similarly, the length of a vector will also remain invariant under these transformations. Clearly then, a unit vector will have unit length in every system. Thus, the magnitude of each of the row vectors will be one. Similarly, the magnitude of the column vectors will also be one. This can be expressed conveniently as 1=

 A2i,1 + A2i,2 + A2i,3 where i = 1, 2, 3

(C.47)

1=

 A21,i + A22,i + A23,i where i = 1, 2, 3

(C.48)

and

Also, as the un-primed unit vectors will be mutually orthogonal, which will also be the case for the primed unit vectors then knowledge of any two row vectors will enable the third to be obtained by taking the cross product of the other two. Also, as the unit vectors in each coordinate system are orthogonal, only two of any three vectors can be chosen arbitrarily, the third being recoverable from the cross product of the other two. Here, this implies that knowledge of any two rows will enable the third to be determined and similarly, knowledge of any two columns will enable the third to be deduced. For example, A (3, 1) = A (1, 2) A (2, 3) − A (1, 3) A (2, 2)

(C.49)

A (3, 2) = A (1, 3) A (2, 1) − A (1, 1) A (2, 3)

(C.50)

A (3, 3) = A (1, 1) A (2, 2) − A (1, 2) A (2, 1)

(C.51)

Conversely, the inverse transformation can be accomplished with   x   y     = [A]−1 ·    z  1 

 x y   z  1

(C.52)

The adoption of a four-by-four matrix, with its inherent redundancy, is preferable as the matrix inverse and thus the inverse transformation, only exists for square matrices. An added advantage of this definition is that the four-by-four alignment matrices can be obtained directly from most engineering computer aided design (CAD) packages. By way of illustration, rotations about the x-, y- and z-axes are

370 Principles of planar near-field antenna measurements represented respectively by the three matrices   1 0 0 0  0 cos θx sin θx 0   Rx =   0 − sin θx cos θx 0  0 0 0 1 

0 − sin θy 1 0 0 cos θy 0 0

cos θy  0 Ry =   sin θy 0 

cos θz  − sin θz Rz =   0 0

sin θz cos θz 0 0

0 0 1 0

(C.53)

 0 0   0  1

(C.54)

 0 0   0  1

(C.55)

The derivation of the rotation matrix can either be obtained from the use of trigonometric identities or from geometry. To illustrate this for the case of a positive rotation about the positive z-axis it is clear from Figure C.9 and from trigonometry that x = x cos θz + y sin θz

(C.56)



(C.57)



(C.58)

y = −x sin θz + y cos θz z =z

y y⬘
z

x⬘


z


z

x

z, z⬘

Figure C.9

Illustration of a positive rotation about the positive z-axis

Appendix C 371 A similar construction can be used for rotations about the x- and y-axes respectively. Translation of Tx , Ty , Tz in the x-, y- and z-axes respectively can be implemented using   1 0 0 Tx  0 1 0 Ty   T = (C.59)  0 0 1 Tz  0 0 0 1 A series of transformation matrices may be concatenated into a single matrix by multiplication. If A1 , A2 and A3 are transformation matrices to be applied in order and the matrix A is the product of the three matrices. Thus ((P · A1 ) · A2 ) · A3 ≡ P · ((A1 · A2 ) · A3 ) = P · A

(C.60)

Where the multiplication is noncommutative and A = (A1 · A2 ) · A3

(C.61)

In this way any sequence of rotations can be constructed by sequentially multiplying out the necessary rotations and translations. Often only rotational relationship between two systems is considered. This is often the case when considering far-field patterns. In this case the three-by-three submatrix can be considered alone and is used instead of the homogeneous four-by-four transformation matrix. Utilization of the three-by-three direction cosine matrix has an important benefit. As the direction cosine matrix is orthogonal and normalized the matrix whose elements are all real, the inverse is identically equal to the matrix transpose. This means that obtaining the inverse transformation is essentially reduced to a matter of reordering of the elements within the matrix, which is both trivial and numerically robust. When direction cosine matrices are used, the determinant of the matrix should be calculated and any significant deviation from unity can be treated as being indicative of a bad direction cosine matrix, as the matrix should be normalized to unity. Occasionally a good direction cosine matrix can be reported as faulty, if the number of significant figures used to represent the matrix is insufficient. Typically, all direction cosine matrices should be treated as being of type double precision in order that truncation and rounding errors remain acceptably small. This follows from noting that typically the smallest angular increment that is usually of interest at microwave frequencies from a rotary position encoder is ±0.01◦ or when expressed in terms of a direction cosine this deviates from unity in the eighth decimal place. Furthermore, the act of multiplying out one or more direction cosine matrices can further compromise the data, as the cumulative rounding error can increase appreciably.

C.13 Azimuth, elevation and Roll angles Any number of angular definitions for describing the relationship between the two coordinate systems are available. However, if the angles azimuth, elevation and Roll

372 Principles of planar near-field antenna measurements are used, where the rotations are applied in this order, we may write the equivalent direction cosine matrix as [A] = [A1 ] · [A2 ] · [A3 ]

(C.62)

Specifically 

cos (Roll) [A1 ] = − sin (Roll) 0

 sin (Roll) 0 cos (Roll) 0 0 1

 1 0 0 [A2 ] = 0 cos (El) − sin (El) 0 sin (El) cos (El)

(C.63)





cos (Az) [A3 ] =  0 sin (Az)

 0 − sin (Az)  1 0 0 cos (Az)

(C.64)

(C.65)

These transformation matrices can be easily derived either from geometry or from trigonometric identities. Here, in accordance with the rules of linear algebra, the first rotation matrix is written to the right. When multiplied out [A] can be explicitly expressed as Arow,column A1,1 = cos (Roll) cos (Az) + sin (Roll) sin (El) sin (Az)

(C.66)

A1,2 = sin (Roll) cos (El)

(C.67)

A1,3 = cos (Roll) sin (Az) − sin (Roll) sin (El) cos (Az)

(C.68)

A2,1 = − sin (Roll) cos (Az) + cos (Roll) sin (El) sin (Az)

(C.69)

A2,2 = cos (Roll) cos (El)

(C.70)

A2,3 = − (sin (Roll) sin (Az) + cos (Roll) sin (El) cos (Az))

(C.71)

A3,1 = − cos (El) sin (Az)

(C.72)

A3,2 = sin (El)

(C.73)

A3,3 = cos (El) cos (Az)

(C.74)

where the rotations are understood to have been performed in the following order, 1. Rotate about the negative y-axis through an angle azimuth. 2. Rotate about the negative x-axis through an angle elevation. 3. Rotate about the z-axis through an angle Roll.

Appendix C 373 When Az = 0, El = 0, Roll = 0 the direction cosine matrix will be the identity matrix and specifies that no rotations are to be applied, that is   1 0 0 [A] =  0 1 0  = [I ] (C.75) 0 0 1 Clearly, from this matrix it can be seen that these angles can be obtained from the matrix [A] as   −A31 Az = arctan (C.76) A33 El = arcsin (A32 )   A12 Roll = arctan A22

(C.77) (C.78)

Many other definitions for rotating frames of reference exist. These include the triad of Euler angles (see below) or the yaw pitch and Roll angles. However, the azimuth, elevation and Roll definition is most convenient when presenting data tabulated in a regular azimuth over elevation coordinate system.

C.14 Euler angles As an alternative to the azimuth, elevation and Roll rotations described above the triad of Euler angles are often utilized to represent the relationship between two frames of reference. 1. Rotate about the z-axis through an angle φ. 2. Rotate about the new y-axis through an angle θ. 3. Rotate about the new z-axis through an angle χ. Specifically, if the angles φ, θ and χ are used, where the rotations are applied in this order, we may write the equivalent direction cosine matrix as [A] = [A1 ] · [A2 ] · [A3 ] where



cos (χ) [A1 ] = − sin (χ ) 0 

cos (θ) [A2 ] =  0 sin (θ)

(C.79)

 sin (χ ) 0 cos (χ ) 0 0 1

(C.80)

 0 − sin (θ )  1 0 0 cos (θ )

(C.81)

374 Principles of planar near-field antenna measurements 

 cos (φ) sin (φ) 0 [A3 ] = − sin (φ) cos (φ) 0 0 0 1

(C.82)

When multiplied out [A], can be explicitly expressed as Arow,column A1,1 = cos χ cos φ cos θ − sin χ sin φ

(C.83)

A1,2 = cos φ sin χ + cos χ cos θ sin φ

(C.84)

A1,3 = − cos χ sin θ

(C.85)

A2,1 = − cos φ cos θ sin φ − cos χ sin φ

(C.86)

A2,2 = cos χ cos φ − cos θ sin χ sin φ

(C.87)

A2,3 = sin χ sin θ

(C.88)

A3,1 = cos φ sin θ

(C.89)

A3,2 = sin φ sin θ

(C.90)

A3,3 = cos θ

(C.91)

Here, the three angles are referred to as Euler angles. Conversely, the three Euler angles can be determined from the direction cosine matrix as θ = arccos(A33 ) If θ  = 0 then

(C.92)



A23 −A13   A32 φ = arctan A31

χ = arctan

 (C.93) (C.94)

However, if θ = 0 then a zero divide by zero ambiguity is introduced. In this case we must use χ =0 φ = arctan



A12 A22



(C.95) (C.96)

Clearly, when the θ rotation is zero, the φ and χ rotations are equivalent and thus either rotation may be used. A conversion between the azimuth, elevation and Roll angles and the three Euler angles can be accomplished readily by equating the elements of their respective direction cosine matrices.

C.15 Quaternion As shown above and further illustrated in Section 9.2.2, only four elements of the nine elements within the direction cosine matrix can be chosen arbitrarily. As such, it

Appendix C 375 is sometimes desirable to record the relationship between two coordinate systems in a more compact form. It is not the purpose of this section to give a through presentation of quaternions as this is beyond the scope of this text; instead the discussion will be limited to the use of quaternions in implementing coordinate transforms. Mathematically, quaternions can be considered to be a noncommutative extension of complex numbers. By way of an analogy, complex numbers are represented as a sum of real and imaginary parts. Similarly, a quaternion can also be written as a linear combination of real and hyper-complex parts Q = q0 + q1 i + q2 j + q3 k

(C.97)

Where i2 = j 2 = k 2 = ijk = −1

(C.98)

Thus, as a complex number can be represented as a point on a two-dimensional plane, a quaternion can be considered to be a point in a four-dimensional space. A quaternion can be computed from a direction cosine matrix using q0 =

1 A11 + A22 + A33 + 1 2

q1 =

1 (A23 − A32 ) 4q0

(C.100)

q2 =

1 (A31 − A13 ) 4q0

(C.101)

q3 =

1 (A12 − A21 ) 4q0

(C.102)

(C.99)

Conversely, a direction cosine matrix can be constructed from a quaternion using A11 = 2q02 − 1 + 2q1 q2

(C.103)

A12 = 2q1 q2 + 2q0 q3

(C.104)

A13 = 2q1 q3 − 2q0 q2

(C.105)

A21 = 2q1 q2 − 2q0 q3

(C.106)

A22 = 2q02 − 1 + 2q22

(C.107)

A23 = 2q2 q3 + 2q0 q1

(C.108)

A31 = 2q1 q3 + 2q0 q2

(C.109)

A32 = 2q2 q3 − 2q0 q1

(C.110)

A33 = 2q02 − 1 + 2q32

(C.111)

376 Principles of planar near-field antenna measurements As is the case for vectors, the length or norm, of a quaternion is of utility. This can be calculated from  |Q| = Q∗ Q (C.112) Here, the superscript star is used to denote the complex conjugate of Q so that Q∗ + Q = 2q0 . Thus  |Q| = q02 + q12 + q22 + q32 (C.113) If Pq and Qq are quaternions and are expressed using the vector form of a quaternion then Pq = p0 + P

(C.114)

Qq = q0 + Q

(C.115)

When expressed in this form the quaternions can be multiplied together using Rq = Pq Qq = p0 q0 − P · Q + p0 Q + q0 P + P × Q

(C.116)

Here, a dot denotes the scalar dot product and the cross denotes the vector cross product. When expanded out and expressed in element form this equates to r0 = p0 q0 − p1 q1 − p2 q2 − p3 q3

(C.117)

r1 = p0 q1 + p1 q0 + p2 q3 − p3 q2

(C.118)

r2 = p0 q2 + p2 q0 + p3 q1 − p1 q3

(C.119)

r3 = p0 q3 + p3 q0 + p1 q2 − p2 q1

(C.120)

As was the case for the multiplication of direction cosine matrices, quaternion multiplication is equivalent to the concatenation of several sequential rotations. As was the case for direction cosine matrices, multiplications are non-commutative, that is Pq Qq  = Qq Pq

(C.121)

Inverting a quaternion rotation produces the inverse rotation and the inverse of a quaternion is equal to the complex conjugate of that quaternion thus Qq−1 = (q0 + q1 + q2 + q3 )−1 = q0 − q1 − q2 − q3

(C.122)

Thus, computing an inverse rotation using the quaternion representation requires less effort than accomplishing the same task using direction cosine matrices. All rotations can be represented by a single rotation about an axis in space. The axis and angle of that rotation can be calculated from the quaternion using φ = 2 arccos (q0 ) v = v1 eˆ x + v2 eˆ y + v3 eˆ z =

1 sin (φ/2)

  q1 eˆ x + q2 eˆ y + q3 eˆ z

(C.123) (C.124)

Appendix C 377 Here, φ is used to denote the angle of rotation and v represents the axis of the rotation. Conversely, the quaternion can be computed from the angle and axis of rotation using   φ (C.125) q0 = cos 2   φ q1 = v1 sin (C.126) 2   φ q2 = v2 sin (C.127) 2   φ q3 = v3 sin (C.128) 2 In addition to being a more efficient recording method, quaternions have the advantage that computing inverse rotations is made significantly easier than is the case for direction cosine matrices. Also, whilst the multiplication of two direction cosine matrices can, in the presence of truncation and rounding errors, produce a direction cosine matrix that is not a rotation, that is, the components naturally drift which violates the orthonormality constraints. Quaternions have no such problem, that is, the multiplication of two quaternions will always yield a rotation, all be it, perhaps the wrong rotation. It is also a comparatively simple matter to adjust for numerical drift, one merely needs to compute the norm of the quaternion and then divide each component by it. This takes far fewer operations than matrix renormalization which would be required in order to attempt to correct a direction cosine matrix.

C.16 Elemental solid angle for a true-view coordinate system Consider evaluating the following integral P=

1 Z0



π



−π 0 0

π/2

|ET (θ, φ)|2 sin θ dθ dφ

(C.129)

Now suppose that we have the field tabulated on a regular true-view grid, that is, ET (X , Y ) where X = θ cos φ

(C.130)

Y = θ sin φ

(C.131)

The problem of evaluating the total radiated power of an antenna when the pattern has been tabulated using a true-view coordinate system could easily be encountered. Rather than recalculating the pattern on a more convenient plotting system it would be convenient if it were possible to evaluate this integral directly in this system.

378 Principles of planar near-field antenna measurements Motivated by this, the original integral can be expressed as  ∂ (θ , φ) 1 |ET (X , Y )|2 sin θ P= dX dY Z0 ∂ (X , Y ) X 2 +Y 2 ≤(π 2 /4) Where

  ∂θ  ∂ (θ , φ)  =  ∂X ∂ (X , Y )  ∂φ  ∂X Clearly  θ = X2 + Y2   Y φ = arctan X

∂θ ∂Y ∂φ ∂Y

      

Evaluating the elements of the matrix yields  ∂θ X ∂  2 X + Y2 = √ = 2 ∂X ∂X X + Y2    Y ∂θ ∂ X2 + Y2 = √ = 2 ∂Y ∂Y X + Y2    Y ∂φ ∂ Y =− 2 = arctan X ∂X ∂X X + Y2    ∂φ ∂ X Y = arctan = 2 ∂Y ∂Y X X + Y2

(C.132)

(C.133)

(C.134) (C.135)

(C.136) (C.137) (C.138) (C.139)

Thus ∂ (θ , φ) X Y Y X · 2 +√ · 2 =√ 2 ∂ (X , Y ) X2 + Y2 X + Y X2 + Y2 X + Y2 =

X2 + Y2 3 X2 + Y2 2

(C.140)

1 =√ 2 X + Y2 Recalling the relationship between the polar angle and the plotting axes yields ∂ (θ , φ) 1 = |θ| ∂ (X , Y ) Thus the integral can be expressed as      1 2  sin θ  |ET (X , Y )|  dX dY P= θ  Z0 X 2 +Y 2 ≤(π 2 /4)

(C.141)

(C.142)

Appendix C 379 Or 1 P= Z0



|ET (X , Y )|2 |sincθ | dX dY

(

X 2 +Y 2 ≤

(C.143)

)

π 2 /4

Where in the limit as θ tends to zero sinc θ tends to one. Alternatively, when expressed purely in the plotting coordinates this becomes     1   |ET (X , Y )|2 sinc P= (C.144) X2 + Y2  dX dY Z0 2 2 2 X +Y ≤(π /4)

Appendix D

Trapezoidal discrete Fourier transform

Following Reference 1, consider the following testing integral  2 x2 ejωx dx F (ω) = −2

Integrating this yields
2 jejωx  2 2 F (ω) = − 3 ω x + 2jωx − 2 ω −2

(D.1)

(D.2)

Thus the exact function can be expressed as     j2  F (ω) = − 3 2ω2 + 2jω − 1 e2jω + −2ω2 + 2jω + 1 e−2jω (D.3) ω This was evaluated numerically using the conventional discrete Fourier transform (DFT) and fast Fourier transform (FFT) and then with the trapezoidal transforms, where x = −2 + n x

(D.4)

ω = m ω

(D.5)

n = 0, 1, 2, . . . , N − 1

(D.6)

m = 0, 1, 2, . . . , M − 1

(D.7)

x = 0.05

(D.8)

2π M x Here, N − 1 = 80 and M = 256. The DFT was evaluated as

ω =

F (ω) ≈ FDFT {f (m x)} = e

jω x/2

x

N −1  n=0

f (m x) ejωx

(D.9)

(D.10)

Appendix D 381 The FFT was evaluated as F (ω) ≈ FFFT {f (m x)} = ejω(−2+ x/2) x

N −1 

f (m x) ej2πmn/M

(D.11)

n=0

Some FFTs adopt the opposite sign convention. The trapezoidal DFT was evaluated as F (ω) ≈ FT DFT {f (m x)} = sinc (θ ) FDFT {f (m x)} +

 1 jθ e − sinc (θ) FDFT { f (m x)} j2θ

(D.12)

Either the DFT or FFT algorithm can be employed herein. Furthermore ω x 2

f (n x) = f ((n + 1) x) − f (n x) θ=

(D.13) (D.14)

In practice, the slope of the function is obtained by central differencing, whilst a right difference is taken at the left-hand side and a left difference is taken at the right-hand side, this can be expressed explicitly as follows 1 (f ((n + 1) x) − f ((n − 1) x)) 2

f (n x) = f ((n + 1) x) − f (n x)

f (n x) =

(D.15) (D.16)

f (n x) = f (n x) − f ((n − 1) x)

(D.17)

Results obtained from the DFT and the trapezoidal transforms can be found compared with the exact result in Figures D.1 and D.2. Clearly, the first-order trapezoidal DFT offers a significant improvement over the more commonly employed zero-order rectangular DFT. A further comparison is made below in Table D.1. These values differ from those presented in the reference. 0 –20

Phase (deg)

Mag (dB)

–40 –60 –80 Exact DFT EMPL

50

100

150 m

Figure D.1

50 0

–50

–100 –120

Exact DFT

100

200

250

–100 5

10

15 m

Comparison of exact and discrete rectangular transforms

20

25

382 Principles of planar near-field antenna measurements 0

Exact Trapezoidal EMPL

Mag (dB)

–40 –60 –80 –100 –120

Figure D.2 Table D.1 m

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

Exact Mag dB

Exact Trapezoidal

100

Phase (deg)

–20

50 0 –50 –100

50

100

m

150

200

5

250

10

15

20

25

m

Comparison of exact and discrete trapezoidal transforms Comparison of exact result, trapezoidal, FFT and DFT results

Arg degrees

0.00 −15.28 −18.52 −20.30 −22.34 −25.11 −29.17 −36.17 −60.23 −37.26 −33.13 −31.74 −31.87 −33.31 −36.35 −42.44 −72.27 −42.99 −38.31 −36.51 −36.31 −37.46 −40.23 −46.06 −79.32 −46.42

90.00 −90.00 90.00 −90.00 90.00 −90.00 90.00 −90.00 −90.00 90.00 −90.00 90.00 −90.00 90.00 −90.00 90.00 90.00 −90.00 90.00 −90.00 90.00 −90.00 90.00 −90.00 −90.00 90.00

DFT Mag dB 0.00 −15.28 −18.45 −20.12 −22.00 −24.55 −28.32 −34.90 −57.36 −35.61 −30.93 −28.98 −28.48 −29.20 −31.40 −36.39 −59.19 −35.94 −29.79 −26.50 −24.57 −23.64 −23.71 −25.35 −32.17 −31.27

Arg degrees 90.00 −87.75 91.24 −89.47 89.97 −90.63 88.51 −93.66 −33.75 93.34 −88.59 90.59 −90.00 89.41 −91.42 86.61 −157.50 −86.41 91.47 −89.38 90.02 −90.56 88.67 −92.88 78.75 117.45

FFT Mag dB 0.00 −15.28 −18.45 −20.12 −22.00 −24.55 −28.32 −34.90 −57.36 −35.61 −30.93 −28.98 −28.48 −29.20 −31.40 −36.39 −59.19 −35.94 −29.79 −26.50 −24.57 −23.64 −23.71 −25.35 −32.17 −31.27

Trapezoidal Arg Mag degrees dB 90.00 −87.75 91.24 −89.47 89.97 −90.63 88.51 −93.66 −33.75 93.34 −88.59 90.59 −90.00 89.41 −91.42 86.61 −157.50 −86.41 91.47 −89.38 90.02 −90.56 88.67 −92.88 78.75 117.45

Arg degrees

0.00 −15.28 −18.52 −20.30 −22.35 −25.12 −29.17 −36.17 −60.29 −37.26 −33.13 −31.74 −31.88 −33.32 −36.35 −42.44 −72.61 −43.01 −38.32 −36.52 −36.32 −37.46 −40.24 −46.06 −79.45 −46.42

90.00 −90.00 89.99 −90.01 89.98 −90.03 89.96 −90.08 −89.02 90.04 −89.99 90.00 −90.01 89.97 −90.04 89.91 92.50 −89.94 90.02 −90.00 89.99 −90.02 89.96 −90.06 −89.84 89.76

Appendix D 383 This discrepancy is most likely a result of typing errors within the paper and the use of single precision arithmetic within the original calculation. These results were generated using double precision arithmetic throughout. This is a first-order integration technique so it is exact for functions that are of first or zero order, that is, piecewise linear or piecewise constant. Thus, for a ‘top hat’ function, the results obtained numerically with this formula were found to be in agreement with the exact result to machine precision, that is, approximately –300 dB equivalent multipath level (EMPL). For this reason, this integration method has been verified with a quadratic testing function.

D.1 References 1 Narasimhan, M.S., and Karthikeyan, M.: ‘Evaluation of Fourier transform integrals using FFT with improved accuracy and its applications’, IEEE Transactions on Antennas and Propagation, April 1984; AP-32 (4):404–8.

Appendix E

Calculating the semi-major axis, semi-minor axis and tilt angle of a rotated ellipse

From Chapter 5 Eφ = E2 cos (ωt + γ )

(E.1)

This can be written as Eφ (r, t) = E2 (r) [cos ωt cos γ + sin ωt sin γ ] = pφ (r) cos ωt + qφ (r) sin ωt

(E.2)

where pφ (r) = E2 (r) cos γ

(E.3)

qφ (r) = E2 (r) sin γ

(E.4)

In general it is possible to write E(r, t) = p(r) cos ωt + q(r) sin ωt

(E.5)

pθ (r) = E1 (r);

(E.6)

So qθ (r) = 0; yields Eθ (r, t)

Thus γ = 0 gives linear polarisation and γ = ±90◦ gives circular polarisation (for E1 = E2 ) and elliptical polarisation otherwise. We can write   E(r, t) = Re U (r)e−jwt (E.7) U (r) = p(r) + jq(r)

(E.8)

Now looking at E point r = r0 as time varies, end point of E describes an ellipse defined by p and q. Now we can write    p + jq = a + jb ejφ (E.9)

Appendix E 385 Here, φ is any scalar. Thus we can write a = p cos φ + q sin φ

(E.10)

b = −p sin φ + q cos φ

(E.11)

If we now choose φ so that a and b are perpendicular to each other then by orthogonality     (E.12) a · b = p cos φ + q sin φ · −p sin φ + q cos φ = 0 We can now write

 E(r, t) = a + jb e−j(ωt−φ) = a cos(ωt − φ) + b sin(ωt − φ)

(E.13)

Taking Cartesian axes with origin at r0 and with x and y directions along a and b yields Ex = a cos(ωt − φ)

(E.14)

Ey = a sin(ωt − φ)

(E.15)

which is the parametric equation of an ellipse with semi-major axis = a, semi-minor axis = b and tilt angle = φ. Thus solving for these three parameters will fully specify the ellipse. By simple geometry it can be shown that p and q are thus semi-radii of the ellipse measured in the θ and ϕ directions, see Figure 5.26 in Chapter 5. Simplifying the dot product of a and b yields     (E.16) −p2 + q2 cos φ sin φ + p · q cos2 φ − sin2 φ = 0  1 2 −p + q2 sin 2φ + p · q cos 2φ = 0 2 Hence we obtain the desired result 2p · q tan 2φ = 2 p − q2

(E.17)

(E.18)

If γ is used to denote the angle between the vectors p and q then this can be expressed as tan 2φ =

2pq cos γ p2 − q 2

Now consider 2

2  a = p cos φ + q sin φ

(E.19)

(E.20)

Expanding yields a2 = p2 cos2 φ + q2 sin2 φ + 2p · q cos φ sin φ

(E.21)

386 Principles of planar near-field antenna measurements or a2 = p2 cos2 φ + q2 sin2 φ + p · q sin 2φ

(E.22)

Expanding yields a2 =

p2 q2 (cos 2φ + 1) + (1 − cos 2φ) + p · q sin 2φ 2 2

Simplifying obtains  1  1 2 p2 − q2 cos 2φ + p · q sin 2φ p + q2 + a2 = 2 2

(E.23)

(E.24)

Returning to the expression for the rotation and considering the tangent of an angle yields a useful trigonometric identity, namely, tan2 φ =

sin2 φ

(E.25)

1 − sin2 φ

Thus

2  4 p·q tan2 2φ = 2 p2 − q 2

(E.26)

Hence

 2 4 p · q sin 2φ = 2 1 − sin2 2φ p2 − q 2    2 2 2 p2 − q2 sin2 2φ = 4 p · q − 4 sin2 2φ p · q 2



p2 − q 2

2



+4 p·q

2

=

2  4 p·q sin2 2φ

Thus we obtain the first of our two necessary substitutions.   2 p·q sin 2φ =  2 

2 p2 − q 2 + 4 p · q The second can be obtained from the first using   2 p·q 2p · q 1 =  2  cos 2φ p2 − q 2 2 p2 − q 2 + 4 p · q

(E.27) (E.28) (E.29)

(E.30)

(E.31)

Appendix E 387 Hence we find the second of our two substitutions cos 2φ = 

p2 − q 2 2 

2 p2 − q 2 + 4 p · q

(E.32)

Thus if these substitutions are used to simplify the expression of a2 we obtain

2 2  1 p − q2 1 2 2 2 a = p +q +  2  2 2 2 p2 − q 2 + 4 p · q

+

Or

2  2 p·q

p2

2 − q2

2  +4 p·q

 2 

2  2 2+4 p·q   p − q  1 2  p + q2 +  a2 =     2 2

2 2 2 p −q +4 p·q

(E.33)



(E.34)

Hence

     2

2 1  2 2 p + q + p2 − q 2 + 4 p · q a = 2 2

This can be expressed in terms of the angle between the vectors p and q as
   1  2 p + q2 + p4 + q4 − 2p2 q2 + 4p2 q2 cos2 γ a2 = 2
  

 1  2 2 2 4 4 2 2 2 a = p + q + p + q + 2p q −1 + 2 cos γ 2 Thus the final result is 
   1 2 2 4 4 2 2 p + q + p + q + 2p q cos 2γ a= 2

(E.35)

(E.36) (E.37)

(E.38)

Following a similar procedure we can obtain a similar result for b. Thus b2 = p2 sin2 φ + q2 cos2 φ − p · q sin 2φ So that b2 =

 1  1 2 p + q2 − p2 − q2 cos 2φ − p · q sin 2φ 2 2

(E.39)

(E.40)

388 Principles of planar near-field antenna measurements Hence

    2  

 1 2 b2 = p2 + q 2 − p2 − q 2 + 4 p · q 2

Thus as required 
   1 2 b= p + q2 − p4 + q4 + 2p2 q2 cos 2γ 2

(E.41)

(E.42)

Index

absorber theory 351–3 absorbers, RF field 45–6, 54 accelerated charges 16–18 accuracy of measurement 356 requirements 285–7 see also errors active alignment correction 285–96 acquisition of alignment data 287–91 example application 291–6 advanced pattern correction (APC) 203 AES: see antenna electrical system (AES) coordinate axes Aharonov–Bohm effect 27 alignment data acquisition 287–91 see also coordinate systems alignment corrections 58, 102–4, 143–8, 169–75 active alignment correction 285–96 acquisition of alignment data 287–91 example application 291–6 generalized compound vector rotation of far-field antenna patterns 171–3 rotation of copolar polarisation basis 173–4 scalar rotation of far-field antenna patterns 169–71 vector rotation of far-field antenna patterns 171–3 amplitude only planar near-field measurements 296–303 PTP phase retrieval algorithm 297–303 AMS: see antenna mechanical system (AMS) coordinate axes anechoic chambers 45–7

multiple reflections 58, 124, 136, 138 input error sources 254–5 probe pattern characterisation 201–2 angular definitions 5–6 azimuth, elevation and roll angles 371–3 Euler angles 373–4 angular field distribution 6–7, 10–11 angular spectrum 41, 70, 90–2 approximation of far-field radiation patterns 92–5 near-field to angular spectrum transform 101–4 radiated power 112–15 angular variations, power transfer 6–7 antenna alignment: see alignment corrections; coordinate systems antenna aperture 6, 43, 108 illumination 38–9 see also aperture illumination function antenna coordinate system: see coordinate systems antenna coupling 1–4, 10–11 mutual coupling coefficient 234–7 see also transmission formula antenna electrical system (AES) coordinate axes 178, 357 antenna mechanical system (AMS) coordinate axes 177–8, 357 antenna radiation pattern 6–7 antenna radiation pattern acquisition far-field data 35–9 near-field geometries 40–3 see also plane wave spectrum (PWS) representation APC (advanced pattern correction) 203

390 Principles of planar near-field antenna measurements aperture, antenna 6, 43, 108 illumination 38–9 aperture illumination function 66, 78, 95 computational electromagnetic models 223–5 microwave holographic diagnosis 108 aperture plane 108 approximations: see errors attenuation, evanescent plane wave mode coefficients 136–7 auxiliary rotation 319–44 applicability of poly-planar technique 335–40, 342–4 bi-scan technique 335–7 experimental set up and procedure 340–2 near-field to far-field transform continuity of field function at intersection 323 general near-fields 324 modified transform 329, 330 probe corrected data 329, 331–5 simple transform 322–3 reconstruction of normal field component 325–9 modified probe pattern correction 326–8 spectral leakage 328–9 tri-scan technique 337–9 auxiliary translation 315–19 average power flow (Poynting vector) 33 azimuth and elevation (true-view) coordinates 178, 364 elemental solid angle 377–9 azimuth cuts 38 azimuth over elevation grid 37, 108–9, 145, 155, 156–9, 360–1 azimuth over elevation polarisation basis 161–2 bandwidth 8–10, 51 bi-linear interpolation 210 bi-planar measurements 320–9 applicability 335–40, 342–4 bi-scan technique 335–7 experimental set up and procedure 340–2 near-field to far-field transform continuity of field function at intersection 323 general near-fields 324 modified transform 329, 330 probe corrected data 329, 331–5

simple transform 322–3 reconstruction of normal field component 325–9 modified probe pattern correction 326–8 spectral leakage 328–9 boresight 6, 58 boundary conditions 75 box scanners: see frame scanners cables 48–9, 198 calibration: see correction techniques cartesian polarisation basis 159–60 CEM: see computational electromagnetic models (CEM) chambers, anechoic 45–7 multiple reflections 58, 124, 136, 138 input error sources 254–5 probe pattern characterisation 201–2 temperature stability 198 channel-balance correction 51, 198–201, 217–18 charge: see electric charge charge density, definitions and units 18 circularly polarised probes 165–9, 192–3 classical electromagnetism 347–8 classical quantum mechanics 348–9 CNFS (cylindrical near-field scanning) 40–1 collective electrodynamics 350–1 combined uncertainty, definition 356 compact antenna test ranges (CATRs) 38, 45, 124 complex vector wave equations 20–3 composite scans 315–19 computational electromagnetic models (CEM) 219–48 aperture set in an infinite perfectly conducting ground plane 223–5 generalized simulation technique 233–7 Kirchhoff–Huygens’ method 229–33 method of sub-apertures 220–2 near-field measurement simulation 237–9 plane wave spectrum antenna coupling formula 225–7 reaction theorem 229–33, 244–7 software 213–14 vector Huygens’ method 227–9 confidence interval, definition 355 conservation of electric charge 14 conservation of energy and momentum 3–4

Index 391 coordinate measuring devices 287–8 coordinate systems 6, 30–3, 40–3, 175–9, 357–79 antenna electrical system (AES) 178 antenna mechanical system (AMS) 177–8 azimuth and elevation (true-view) 178, 364 azimuth over elevation grid 37, 108–9, 145, 155, 156–9, 360–1 copolar and cross-polar coordinates 178 direction cosine grid 141, 145–8, 358–60 elemental solid angles 367, 377–9 elevation over azimuth coordinate system 361–2 elevation over azimuth grid 156–9, 361–2 equatorial spherical 93–4 general vector rotation of antenna radiation patterns 179 plane wave propagation 91 plane-bipolar 41, 42 near-field to far-field transform 151–6 plane-polar 41, 42 near-field to far-field transform 148–51, 154 polar spherical grid 93, 156, 157, 362–3 ‘roll-over-azimuth’ configuration 193 range fixed system (RFS) coordinate axes 176–7 range of spherical angles 365–6 for solution of vector Helmholtz equation 102–3 transformation between 366–7 angular definitions 371–4 quaternions 374–7 transformation matrices 368–71 copolar and cross-polar coordinates 178 copolar and cross-polar polarisation basis 163–5 corrected result, definition 354 correction techniques advanced pattern correction (APC) 203 alignment corrections: see alignment corrections channel balance 51, 198–201, 217–18 phase errors 50, 51 position correction: see position correction algorithms probe displacement 217 probe pattern 42–3, 57–8, 144–8

rotary errors 202–5 Coulomb force law 14 coupling, antenna 1–4, 10–11 mutual coupling coefficient 234–7 see also transmission formula coverage factor, definition 356 cross-correlation coefficient 265–6 cross-polarisation 159 current density 14, 18 cylindrical near-field scanning (CNFS) 40–1 data acquisition software 55 data sets comparison techniques 261–82 categorisation of data sets 279–82 choice of technique 282–3 cross-correlation coefficient 265–6 equivalent multipath level (EMPL) 263–4 hybrid interval/ordinal assessment 277–8 novel techniques 267–72 ordinal assessment 273–8 peak-signal-to-noise ratio (PSNR) 264–5 composite 315–19 data windowing functions 326–9 design parameters 5 DFT: see discrete Fourier transform (DFT) dielectric, cable 49–50 dielectric constant: see permittivity dielectric lens ranges 38 direct collimation 38 direction cosine grid 141, 158, 358–60 probe pattern correction 145–8 transformation matrices 371 directivity 180–1 Dirichlet conditions 67 discrete Fourier transform (DFT) 139, 141–2, 380–3 double Fourier integral 121, 139–43 dual-port probes channel-balance correction 198–201, 217–18 choked cylindrical waveguide 56 dynamic range 48 electric charge accelerated 16–18 conservation 14

392 Principles of planar near-field antenna measurements electric charge (cont.) density 14, 18 distribution 10, 14–15 electric field intensity, definitions and units 18 electric flux density, definitions and units 18 electric potential 24–8 electromagnetic field 14–15 accelerated charges 16–18 electromagnetic field theory 13–18, 347–8 electromagnetic field vectors 19, 30–3 electromagnetic screening 45–7 electromagnetic wave propagation 18–33 electronic system: see RF subsystem elemental solid angles 367, 377–9 elevation over azimuth coordinate system 156–9, 361–2 EM solvers 213–14 see also computational electromagnetic models (CEM) encoders 55 energy conservation 3–4 energy transfer 4–5 E-plane radiation pattern acquisition 36 equal-phase surface 90 equatorial spherical coordinate system 93–4 equivalent multipath level (EMPL) 263–4 errors 58–60, 250–5, 258 correction: see correction techniques definition 355 error budget 259–61 finite area scan errors 123–5 multiple reflections 201–2, 254–5 phase errors 50, 51 positional errors 42, 55–6, 58–9 rotary errors 202–5 scale mapping errors 253–4, 255 sources 254–5, 261 definition 355 truncation 58, 121–5 type A and B definitions 354 Euler angles 373–4 evanescent coupling 135, 136–7 excitation currents as a measurement methodology 28 expanded uncertainty, definition 356 far-field, defined 10–11 far-field antenna measurements 35–40 far-field plotting systems: see coordinate systems

far-field radiation pattern approximation 92–5 coordinate free form of the transform 101–4 reduction to Huygens’ principle 104–5 far-field to near-field transform 108–12 for non-planar apertures 106–7 plane-to-plane transform 103–4, 107–8 stationary phase evaluation of the double integral 95–101 far-field to near-field transform 108–12 fast Fourier transform 139–41 field equivalence principle 28–30 field reciprocity theorem 240–4 finite area scan errors 123–5 finite element models, probe pattern characterisation 213–17 Fourier interpolation 121 Fourier transform pair 66 Fourier transforms 44 discrete Fourier transform (DFT) 139, 141–2 fast Fourier transform 139–41 inverse Fourier transform 66–7, 121 scalar Helmholtz equation 65–78 trapezoidal 380–3 frame scanners 52, 53 free space permeability 22 free space permittivity 22 free space propagation constant 22 free space propagation vector 87–8 free space radiation pattern 6–7 frequency range 49 Friis transmission formula 237 gain 181–3 gain-comparison method 181–3 gain-transfer method 181–3 gauge transformations 26 Green’s function 31, 73–4 harmonic coupling 9–10 harmonic mixing 48 harmonic variations 9–11 Helmholtz equation 21–3 Hertzian dipole 31 L’Hôpital’s rule 82–3 horizontal planar scanners 52–3 H-plane radiation pattern acquisition 36 Huygen’s principle 28–30

Index 393 computational electromagnetic models 227–9 Kirchhoff–Huygens’ method 229–33 derivation 104–5 ‘illumination’ factor 103 indirect collimation 38 information extraction model 4–5, 249–50 input error sources 254–5 integral transforms 65 see also Fourier transforms interpolation schemes 206–12 bicubic-convolution 206–8 bi-linear interpolation 210 nearest neighbour interpolation 209 phase interpolation 208, 211–12 six-point formula interpolation 210–11 three-point formula interpolation 209–10 interpolation theory 120–1 inverse Fourier transform 66–7, 121 k constant 21 Kirchhoff–Huygens’ method 229–33, 234 near-field to far-field transform 331–5 kurtosis 268–9 Lagrangian formulations 27 Laplacian operator 23 Lebesgue measure of a set 257 left-hand circularly polarised (LHCP) field 165–9 level of significance, definition 355 local oscillator (LO) 48 LO/IF distribution units 51 Lorentz condition 26, 27 Lorentz force law 15, 348 Lorentz gauge 26 Lorentz reciprocity theorem 240–4 Ludwig’s I polarisation basis 159–60 Ludwig’s II polarisation basis 161–2 Ludwig’s III polarisation basis 163–5, 173–4, 178 magnetic field intensity 18 magnetic flux density 18 magnetic permeability 20 magnetic potential 24–8 mapping errors 253–4, 255 Maxwell’s equations 18–23 general solution 20

in relation to scalar Helmholtz equations 75 measurand 5, 354 measure, definition 355 measurement characterisation via 4 definitions 354–6 errors: see errors field equivalence principle 28–30 source excitation 28 measurement analysis and assessment 249–83 correspondence between data sets 261–2 data set comparison techniques 263–82 categorisation of data sets 279–82 choice of technique 282–3 cross-correlation coefficient 265–6 equivalent multipath level (EMPL) 263–4 hybrid interval/ordinal assessment technique 277–8 novel data comparison techniques 267–72 ordinal assessment technique 273–8 peak-signal-to-noise ratio (PSNR) 264–5 error budget 259–61 establishment of the measure from the measurement results 249–59 examination of measurement result data 256–9 measurement errors 250–3 sources of ambiguity and error 253–5 measurement scale 250, 252–3, 354 measurement trial, definition 354 measurement-true value = error, definition 355 mechanical stability 53, 55 method of moment (MoM) 233–7 method of sub-apertures 220–2 microwave holographic metrology (MHM) 107–8 computational electromagnetic modelling 223 spherical measurement 110–12 millimetre wave scanners 53 modal expansion 63–4 ‘model tower’ positioner 193, 362 MoM (method of moment) 233–7 momentum conservation 3–4 multiple reflections 58, 124, 136, 138 input error sources 254–5 probe pattern characterisation 201–2

394 Principles of planar near-field antenna measurements mutual coupling coefficient 234–7 mutual impedance 234–7, 247 nearest neighbour interpolation 209 near-field 11 near-field antenna measurements acquisition geometries 40–3 generic description 58–60 simulation 237–9 near-field probe 44, 56–8 desirable characteristics 56, 191–3 dual-port channel-balance correction 198–201, 217–18 choked cylindrical waveguide 56 orthogonal mode transducers (OMT) 217–18 scattering model 137–8 types 56–7 see also probe pattern characterisation near-field scanner (NFS) 41, 52 probe antenna: see near-field probe RF subsystem 47–51, 317 cables 48–9, 198 channel-balance correction 51, 198–201, 217–18 phase stability 48, 49–50, 51, 197–8 tie-scan correction 197–8 robotics positioner subsystem 52–6 control 55 correction for rotary errors 202–5 horizontal configuration 52–3 ‘model tower’ positioner 193–4, 362 positional errors 42, 55–6, 58–9 rotary errors 202–5 translational mechanisms 315–16 near-field to angular spectrum transform 101–4, 227–9 near-field to far-field transform 63–4 evaluation 138–9 discrete Fourier transform (DFT) 141–2 fast Fourier transform 139–41 FTs with improved accuracy 142–3 reduction to Huygens’ principle 104–5 summary 115–17 verification 237–9 NFS: see near-field scanner (NFS) non-planar apertures 106–7 normal field component reconstruction 325–6 Nyquist sampling criteria 238

obliquity factors 92, 95 OMT (orthogonal mode transducers) 154, 217–18 open-ended rectangular waveguide probe (OEWG) 56, 57 optical coordinate measuring devices 287–8 ordinal assessment technique 273–8 orthogonal mode transducers (OMT) 154, 217–18 pantograph and rotary joint assembly 49 partial scan techniques 315–42 auxiliary rotation 319–44 applicability of poly-planar technique 335–40, 342–4 bi-scan technique 335–7 continuity of field function at intersection 323 experimental set up and procedure 340–2 near-field to far-field transform 322–4, 329–35 reconstruction of normal field component 325–9 tri-scan technique 337–9 auxiliary translation 315–19 peak of a pattern 183–6 peak-signal-to-noise ratio (PSNR) 264–5 permeability 22 permittivity 20, 22 phase convention 51 phase difference 74 phase drift 48, 49–50, 51, 197–8 phase errors 50, 51 phase interpolation 208, 211–12 phase locking, local oscillator 48 phase recovery from amplitude only measurements 296–303 PTP phase retrieval algorithm 297–301 with aperture constraint 300–3 phase shifting and summing 44 phase stability 48, 49–50, 51, 197–8 phase velocity 21 physical environment 45 planar acquisition geometries 41–3 planar near-field 30 planar near-field scanner (PNFS): see near-field scanner (NFS) plane rectilinear coordinate systems 41, 42–3 plane rectilinear near-field measurements 41–4

Index 395 plane wave formation 44, 45 plane wave illumination methods 38–9 plane wave impedance 88–90 plane wave spectra 70 plane wave spectrum (PWS) representation 63–118 angular spectrum interpretation 70, 90–2 electric and magnetic spectral component relationship 84–7 far-field radiation pattern approximation 92–5 coordinate free form of the transform 101–5 far-field to near-field transform 108–12 for non-planar apertures 106–7 plane-to-plane transform 103–4, 107–8 stationary phase evaluation of the double integral 95–101 free space propagation vector 87–8 method of derivation 64–5 plane wave impedance 88–90 solution in Cartesian coordinates 65–78 boundary conditions 78–9 operator substitution 79–81 scalar differential wave equation 65–78 vector Helmholtz equation 81–4 vector magnetic wave equation 83–4 summary 115–17 plane-bipolar coordinate systems 41, 42 near-field to far-field transform 151–6 plane-polar coordinate systems 41, 42 near-field to far-field transform 148–51, 154 plane-to-plane (PTP) transform 103–4, 107–8 active alignment correction 296 phase retrieval algorithm 297–301 with aperture constraint 300–3 PNFS (planar near-field scanner): see near-field scanner (NFS) polar spherical azimuth over elevation polarisation basis 160–1 polar spherical grid 93, 156, 157, 362–3 ‘roll-over-azimuth’ configuration 193 polarisation 7–8 polarisation conventions 159–69 azimuth over elevation 161–2 cartesian 159–60 circular 165–9

copolar and cross-polar 163–5 polar spherical azimuth over elevation 160–1 poly-planar rotational technique: see auxiliary rotation port displacement correction 217 position correction algorithms in-plane and z-plane corrections 303–14 K-correction method 312–14 Taylor series expansion 305–12 positional errors 42, 55–6, 58–9 positioner subsystem 52–6 control 55 horizontal configuration 52–3 mechanical stability 53, 55 ‘model tower’ positioner 193–4, 362 pantograph and rotary joint assembly 49 positional errors 42, 55–6, 58–9 rotary error correction 202–5 translational mechanisms 315–16 power flow 33 power flux 112–15 power gain 181–3 power transfer bandwidth 8–10 free space radiation pattern 6–7 polarisation 7–8 Poynting vector 33 precision, definition 355 principal plane cuts 35–6 probe alignment: see coordinate systems probe antenna 44, 56–8 desirable characteristics 56, 191–3 dual-port channel-balance correction 198–201, 217–18 choked cylindrical waveguide 56 orthogonal mode transducers (OMT) 217–18 scattering model 137–8 types 56–7 see also probe pattern characterisation probe assembly 56–7 probe displacement correction 217 probe pattern characterisation 57–8, 189–218 acquisition of quasi far-field probe pattern 193–213 alternate interpolation formula 209–12

396 Principles of planar near-field antenna measurements probe pattern characterisation (cont.) assessment of chamber multiple reflections 201–2 channel-balance correction 198–201 correction for rotary errors 202–5 electronic system drift (tie-scan correction) 197–8 re-tabulation of probe vector pattern function 205–9 sampling scheme 194–7 true far-field probe pattern 212–13 desirable characteristics of a near-field probe 56, 191–3 effect of the probe pattern on far-field data 189–90 finite element models 213–17 probe displacement correction 217 probe pattern correction 42–3, 57–8, 144–8 propagation constant 21, 22 propagation vector 27, 87–8 PSNR (peak-signal-to-noise ratio) 264–5 PTP: see plane-to-plane (PTP) transform PWS: see plane wave spectrum (PWS) representation pyramidal absorber 46 quantum electrodynamics (QED) 349–50 quantum theoretical approaches 262, 348–50 quaternions 374–7 radar absorbing material (RAM) 45–6, 54 radiated power 112–15 radiating near-field 11 radiation pattern 6–7 radiation pattern acquisition far-field data 35–9 near-field acquisition geometries 40–3 see also plane wave spectrum (PWS) representation radiative coupling 1–4 radiative paths 10 RAM (radar absorbing material) 45–6, 54 range fixed system (RFS) coordinate axes 176–7 reaction theorem 229–33, 244–7 reactive coupling 10, 136–7 receiver phase drift 51 reciprocity theorem 240–4 reference RF signal 5, 50 relative phase 10 relative position encoders 55

relativistic effects 2, 13, 15, 71 relativistic quantum mechanics 349–50 repeatability 44, 261–2 definition 355 positional errors 55 reproducibility, definition 355 resolution, definition 354 result of measurement, definition 354 retarded potentials 24–8 RF subsystem 47–51, 317 cables 48–9, 198 channel-balance correction 51, 198–201, 217–18 phase stability 48, 49–50, 51, 197–8 tie-scan correction 197–8 right-hand circularly polarised (RHCP) field 165–9 robotics positioner subsystem 52–6 control 55 horizontal configuration 52–3 mechanical stability 53, 55 ‘model tower’ positioner 193–4, 362 pantograph and rotary joint assembly 49 positional errors 42, 55–6, 58–9 rotary error correction 202–5 translational mechanisms 315–16 rolling bend 42 ‘roll-over-azimuth’ configuration 192–3 rotary errors, correction 202–5 rotated ellipse, semi-major axis, semi-minor axis and tilt angle 384–8 rotations of the AUT about the z-axis: see auxiliary rotation sampling (interpolation theory) 120–1 see also interpolation schemes sampling schemes 40–1, 194–7, 206 see also truncation, angular spectrum scalar differential wave (Helmholtz) equation 23 in relation to Maxwell’s equations 75 solution in Cartesian coordinates 65–78 scalar rotation of far-field antenna patterns 169–71 scale mapping errors 253–4, 255 ’scale of the measurement’ 250, 252–3, 354 scanner: see near-field scanner (NFS) scanning probe: see near-field probe scattering model, near-field probe 137–8 Schwarzschild invariant 27 screening, RF field 45–7

Index 397 semi-major axis of a rotated ellipse 384–8 semi-minor axis of a rotated ellipse 384–8 sensitivity, definition 354–5 set theoretic model of measurement 250–1, 257 SGH: see standard gain horn (SGH) shielding, RF field 45–7 signal source 48 simulation: see computational electromagnetic models (CEM) six-point formula (bicubic) interpolation 210–11 skewness 268 SNFR (spherical near-field range) 40–1 software computational electromagnetic models 213–14 data acquisition 55 source excitation as a measurement methodology 28 specific conductivity 20 spectral leakage 123, 328–9 speed of light 22 spherical angles 93, 108, 111 range of 194, 365–6 spherical microwave holographic metrology 110–12 spherical near-field range (SNFR) 40–1 spherical near-field to far-field transform 212–13 standard, definition 354 standard gain horn (SGH) alignment correction 291–6 near-field measurement simulation 237–9 standard uncertainty, definition 355 static electric field 24 static magnetic field 24 statistical techniques 263 sub-aperture method 220–2 submillimetre wave astronomy satellite (SWAS) 49 submillimetre wave scanners 53 SWAM cone 56–7 SWAS (submillimetre wave astronomy satellite) 49 synthesised plane wave 44 synthesizers 48 temperature stability 198 testing environment 44–7 thermal noise 259–60 three-point formula (linear) interpolation 209–10

tie-scan correction 197–8 tilt angle of a rotated ellipse 384–8 time conventions, vector network analysers 50–1 tower and rail inverted T-scanners 52, 53 transformation between coordinate systems 366–7 angular definitions 371–4 quaternions 374–7 transformation matrices 368–71 translational mechanisms 315–16 transmission formula 125–36 arbitrarily orientated antennas 143–8 attenuation of evanescent plane wave mode coefficients 136–7 computational electromagnetic models 225–7 scattering model of a near-field probe 137–8 trapezoidal discrete fourier transform 380–3 tri-scan auxiliary rotation technique 337–9 true value, definition 355 true view coordinate system 364 elemental solid angle 377–9 truncation, angular spectrum 58, 121–5 T-scanners 52, 53 type A error, definition 354 type B error, definition 354 uncertainty in measurement, definition 355 uncorrected result, definition 354 vector Helmholtz equation 21–3, 81–4 vector Huygens’ method 227–9 vector magnetic wave equation 83–4 vector network analysers (VNA) 47, 50–1 vector phase constant 90 vector rotation of far-field antenna patterns 171–5, 179 VNA (vector network analysers) 47, 50–1 voltage standing wave ratio (VSWR) 124 wave equations 10, 20–3 see also electromagnetic wave propagation wave normal 90 wave number: see propagation constant waveguide probes 56–7 Wheeler Feynman absorber theory 351–3 Whittaker interpolation 121 windowing function 326–9