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=c/[l-(Kc/a>o)2]l/2. The phase front > c, and whose envelope propagates at v
(48)
t=z/c[l-(Kc/(Oo)2]l/2
(49)
is plane and propagates at the constant subluminal group velocity v(g> = c[l - (Kc/ca0)2]in .
(50)
96 As affl(r) does not depend on frequency, there are no amplitude spectral changes, and in particular, red or blue shift of the carrier oscillations during propagation. Diffractioninduced spectrum chirp, given by Eq. (46), is always negative with absolute value growing linearly with propagation distance z, as in the propagation of a plane pulse in a medium with anomalous group velocity dispersion (negative second derivative of the propagation constant) k"oa=
1=0
30 ' -10
t-z/c
(fs)
0
t-z/c
1«
10
(fs)
D
10
t-z/c
20
(fs)
30'-ID
D
t-z/c
10
20
(fs)
Figure 7. Field amplitude and envelope for the pulsed Bessel disturbance E(r,0,t)=J
(51) All previous features are contained in the expression
E(r,t) = AGVD[(p'' adz), t - (p'ox, (z)] x exp [-1(0* t - /"(We)2- K2]1/2z - W JrfKr)
(52)
for the propagated field of pulsed Bessel beams, obtained from Eqs. (22) and (8). Eq. (52) represents a pulse whose carrier oscillations propagate at the phase velocity v
b(z)
(53)
97 b2 (z) = b2 - 2i q>"
(54)
In Fig. 7 we see the diffraction changes in temporal form of a pulsed Bessel beam with Gaussian envelope of parameters (Oo = 3.2 fs"1, K = 0.2 coo I c = 2.13 X 103 mm"1 (transversal size of the Bessel function 2/K = 2^aa), b = 3.34 fs (two-cycle) and (j>= 0. The solid curves represent the exact pulse form Re E(r,z,t) and \A(r,z,t)\ at several distances along the beam axis, numerically calculated from Eqs. (42) and (6) [out off axis the pulse form only changes by a factor JrfKr)]. The dashed curves represent the approximate field of Eq. (54), and the dotted curves the initial pulse propagating without change at speed c, for reference. The exact and approximate fields are almost indistinguishable; in particular, they peak at the same times [corresponding to a pulse traveling at the group velocity of Eq. (50) v(s> = 2.94 X 10"4 mm/fs smaller than c], and they also have similar durations and peak amplitudes. The discrepancy in the trailing and leading parts of the pulse at distances of the order or larger than the dispersion length Ldisp = 0.126 mm are due to diffraction-induced third-order dispersion, which turns the envelope into an asymmetric function. As predicted, no significant red or blue shift is found.
4
Conclusions
We have applied basic concepts of the propagation of three-dimensional wave packets to the description of diffraction (finite transversal size) effects on the propagation of ultrashort, fewoptical-cycle pulses. Diffraction-induced pulse transformations in vacuum can be described in the same way as one-dimensional pulse transformations in a dispersive system. Previously described effects as pulse time-delay (pulse front curvature), frequency blue and red shift of the optical oscillations, and pulse broadening, appear here classified as first-order and secondorder diffraction-induced dispersion effects, which can be accurately quantified by simple analytical expressions. By analogy with dispersive pulse propagation, higher order effects due to third-oder dispersion and second-order gain dispersion can be easily introduced. Approximate analytical expressions for the diffracted pulse can be written in terms of the well-known problem of pulse propagation with group velocity dispersion. Although we have restricted our attention to transformations of the pulse temporal form due to finite transversal size of the source, the same method can be used to characterize the effects of ultrashort pulse duration on transversal diffraction pattern. References [1] I. P. Christov, "Propagation of femtosecond light pulses," Opt. Commun. 53, 364-366 (1985).
98
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
C.J.R. Sheppard and X. Gan, "Free-space propagation of femto-second light pulses," Opt. Commun. 133, 1-6 (1997) A. E. Kaplan, "Diffraction-induced transformation of near-cycle and sub-cycle pulses," J. Opt. Soc. Am. B 15,951-956 (1998). G. P. Agrawal, "Spectrum-induced changes in diffraction of pulsed optical beams," Opt. Commun. 157, 52-56 (1998) M. A. Porras, "Ultrashort pulsed Gaussian beams," Phys. Rev. E 58,1086-1093 (1998). C.F.R. Caron and R.M. Potvliege, "Free-space propagation of ultrashort pulses: spacetime couplings in Gaussian pulse beams," J. Mod. Opt. 46, 1881(1999). M. A. Porras, "Pulse correction to monochromatic light-beam propagation," Opt. Lett. 26,44-46(2001). T. Brabec and F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000). M. Mehendale, S.A. Mitchell, J.P. Likforman, D.M. Villeneuve and P.B. Corkum, "Method for single-shot measurement of carrier envelope phase of a few-cycle laser pulse," Opt. Lett. 25,1672-1674 (2000). E. Born and E. Wolf, Principles of Optics, (Pergamon, Oxford, 1975). See for example G. P. Agrawal, Nonlinear Fiber Optics, (Academic, New York, 1995). O. Svelto, Principles of Lasers, (Plenum, New York, 1982). K. E. Oughstun and G. C. Sherman, Electromagnetic pulse propagation in causal dielectrics, (Springer, Berlin, 1997) J. Jones, Am. J. Phys 42, 43-46 (1974). T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997). M. A. Porras, F. Salazar-Bloise, L. Vazquez, "Creation of localized optical waves that do not obey the radiation condition at infinity," Phys. Rev. Lett. 85, 2104-2107 (2000). M. A. Porras, "Diffraction-free and dispersion-free pulsed beam propagation in dispersive media," Opt. Lett. 26, 1364-1366(2001). L. Ronchi and M. A. Porras, "The relationship between the second order moment width and the caustic surface radius of laser beams," Opt. Commun. 103, 201-204 (1993). J. Lu, J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39,441 (1992). D. Mugnai, A. Ranfagni and R. Ruggeri, "Observation of superluminal behaviors in wave propagation," Phys. Rev. Lett. 84,4830-4833 (2000). M. A. Porras, M. Santarsiero, R. Borghi, "Few-optical-cycle Bessel-Gauss pulsed beams in free space," Phys. Rev. Lett. 62, 5729-5737 (2000). Z. Liu and D. Fan, "Propagation of pulsed zeroth-order Bessel beams," J. Mod. Opt. 45, 17-21 (1998).
GAUSSIAN AND BESSEL BEAMS AND PULSES BEYOND THE PARAXIAL APPROXIMATION
COLIN SHEPPARD
Physical Optics Dept., School of Physics A28, University of Sydney NSW 2006 (Australia) E-mail: [email protected]
Abstract The Gaussian beam is a fundamental concept in optical beam propagation, but is based upon assumptions of paraxial optics. There are several ways of generalizing the concept of che Gaussian beam to the non-paraxial case, but some of these have limitations if a solution is required that is valid for a complete three-dimensional space. A solution that satisfies che requirements is based on the complex source-sink representation. This can be applied in a full vectorial treatment, in which a lineary polarized (LPO1) mode is generated from transverse and orthogonal electric and magnetic dipoles. The individual TM and TE modes are represented by electric or magnetic dipoles alone. All three of these beam modes reduce to the ordinary TEMOO mode in the paraxial approximation. Axial dipoles result in radial or azimuthal TM or TE modes. Higher order multipoles generate higher order beams that can be expressed in terms of sums over che LaguerreGauss beams of paraxial theory. Another important beam is the Bessel beam, recognized as the fundamental solution of the wave equation in cylindrical coordinates. This too can be generalized to a non-paraxial vectorial form. Ultra-short pulses can be modeled based on these types of beam. Again there are several ways of constructing solutions corresponding to different assumptions for the spatial distribution of the spectral components. A particulary useful type is the isodiffracting pulse, corresponding to the field, of a mode-locked laser. Beams and pulses can be treated by methods based on Fourier or phase space representations, generalized to the non-paraxial case. It can also be shown that there are relationships between these different representations.
99
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PARTIALLY COHERENT BEAMS IN FREE SPACE AND IN LENSLIKE MEDIA
RAJIAH SIMON
The Institut of Mathematical Sciences C IT Campus Tharamani, Chennai 600113 (India) E-mail: [email protected]
Abstract The behaviour of partially coherent Gaussian beams under free propagation and under passage through lenslike media or first order systems (FOS) is governed by the smplectic group of real linear canonical transformationsrepesented by the ray-transfer matrix of the system under consideration. And this, coupled with the Wigner representation of the cross-spectral density of the beam, reduces the propagation problem to one of 2x2 or 4x4 matrices depending on whether the beam under consideration is rotationally symmetric or asymmetric about its axis, thus often resulting in complete answers to interesting questions, without too much effort. These considerations hold good not only for Gaussian beams, but also for non-Gaussian partially coherent beams as long as one's interest is restricted (as often happens to be the case) to the first and second moments of the beam. The combined power offered by the Wigner representation on the one hand and the waveoptic representation of canonical transformations on the other will be briefly explored in this sequence of three talks. No prior knowledge or experience with group structures is reqired to follow the arguments and to appreciate the results. Lecture 1: The first obvious question in respect of partially coherent Gaussian beams is this: Given a hermitian Gaussian two-point function, how to test if it is ^nonnegative' so that it qualifies to be a physical cross-spectral density in some transvese plane? This nontrivial issue gets resolved relatively easily by the 'combined power' referred to above. The evolution of the beam parameters as the beam propagates is obtained as a byeproduct. In particular, the invariants or quality parameters associated with a beam present themselves with no additional effort. And so are generalizations of the ABCD-law as well as the presence of the twist phase. Lecture 2: Gori [Opt. Commun. vol. 34, 301 (1980)] was probably the first to exploit symmetry consideratons to achieve coherent-mode decomposition of a class of partially coherent beams. Reinterpretation of this approach in the light of the 'combined power' noted above, shows that this approach leads to coherent-mode decomposition of the much broader ten-parameter family of general anisotropic Gaussian Schell-model beams and, as a paticular case, to that of the twisted Gaussian beams. What may be more interesting, perhaps, is the fact that this approach leads quite simply to the first nontrivial twodimensional generalization of Factional Fourier Transform.
101
102
Lecture 3: This 'combined power' becomes also the natural setting to generalize the notion of shape-invariant propagation studied by Gori and collaborators [Opt. Commun. vol.48, 7 (1893); Opt. Lett, vol.21, 1205 (1996)]. There are two types of shape-invariant propagatons: one in which the intensity ellipse representing the beam cross-section simply undergoes a scaling as a function of the propagation distance, retaining invariant its eccentricity as well as its orientation; and another in which only the eccentricity, and not its orientation, is maitained invariant. The 'combined power' leads to omplete characterization in respect of both types of shape-invariant propagations.
BEAM POLARIZATION MODULATION IN WAVE-OPTICAL ENGINEERING
JANI TERVO AND JARI TURUNEN University of Joensuu, Department of Physics, P.O. Box 111, FIN-80101 Joensuu, Finland E-mail: [email protected] The concept of spatially varying polarization state is examined from the viewpoint of the wave-optical engineering. The possibility for increasing the attainable diffraction efficiencies for several types of signals by using a polarization-controlling element is discussed. This type of elements are based on the simultaneous control of the phase and amplitude of the incoming field such that no energy is absorbed. The discussion is then extended to the polarization multiplexing and partial polarization. Possible research topics to be explored in near future are briefly discussed.
1. Introduction The state of polarization of the electromagnetic field is one of the key quantities of concern in numerous applications of wave-optical engineering, and its controlled modulation is therefore an important task. We begin this article with a brief introduction to the mathematical description of fully and partially polarized plane waves. We then proceed to discuss the control of the state of polarization of fully polarized paraxial fields by spatially variable polarization-modulating elements, concentrating on periodic structures that generate a discrete set of plane waves. We show, e.g., that proper modulation of the state of polarization facilitates the design of 100% efficient multiple beam splitters based on diffractive microstructures, which is not possible within scalar diffractive optics. Some comments on polarization multiplexing and partially polarized light are also provided. 2. Fundamentals
In polarization optics it is usual to describe a monochromatic vectorial plane wave with time dependence exp (— iwi) by a column vector, known as a Jones vector
where the elements Ex and Ey denote the x and the y components of the complex amplitude of the electric field vector. The transmission of the plane wave through an arbitrary polarization-modulating interface may be described by multiplying the
103
104
Jones vector J\ associated with the incident field by a matrix T, resulting in the transmitted Jones vector Jt: Jt = TJS.
(2)
The matrix T appearing in Eq. (2) is known as the Jones matrix or the transmission matrix, and its form depends on the optical function of the interface. The Jones matrix for a cascaded system consisting of several polarization-modulating elements is obtained by multiplying the Jones matrices of the individual elements. Let us consider an absorbing wave plate with its optical axis parallel to the re-axis. The Jones matrix of such an element takes the simple form T=M
[0 B\ '
(3) (6)
where A and B describe the complex-amplitude transmittances in the x and y directions, respectively. If the element is rotated by an angle 6 with respect to the x axis, the Jones matrix may be expressed in the form1 L4 cos2 9 + B sin2 0 (A-B) cos 9 sin 01 " [ ( A - B) sin 9 cos B B cos2 9 +A sin2 0]
(
'
It should be noticed that A and B in Eq. (4) are the complex-amplitude transmittances, not in the x and y directions, but in the directions parallel and perpendicular to the optical axis, respectively. For example, by choosing A = I and B = 0 in Eq. (4) we have a rotated linear polarizer. If, on the other hand, A = 1 and B = — 1, the matrix in Eq. (4) describes a rotated half-wave plate. Considering partially polarized (quasimonochromatic) plane waves instead of monochromatic plane waves, the electric-field components Ex and Ey become stochastic quantities. Hence, in the definition (2) of the Jones vector, we can no longer use the complex amplitudes but must consider the time-dependent field. The polarization properties of the field are described by the coherency matrix J defined as
\(Ex(t)Ex(t)) [(E*y(t)Ex(t)}
(Ex(t)Ey(t)) (E*y(t)Ey(t)}
where the angular brackets denote time averages and the asterisk denotes the complex conjugate. It is immediately clear from the definitions (1) and (5) of the Jones vector and the coherency matrix that
J = (J*JT) ,
(6)
where J denotes the transpose of J. The degree of polarization P of a partially polarized plane wave is defined as 4detJ
105
where det and tr denote the the determinant and the trace of the coherency matrix, respectively. The extreme values P = I and P = 0 of the degree of polarization indicate completely polarized and completely unpolarized plane waves, respectively. The effect of a polarization-sensitive interface described by a Jones matrix T on the coherency matrix is, according to Eqs. (6) and (2),
(8) since the Jones matrix T is deterministic. This expression governs the transmission of an arbitrarily polarized axial plane wave through a planar polarization-sensitive element or a cascade of such elements. It is of substantial interest to consider the effect of some basic polarizationsensitive devices on unpolarized light with a diagonal coherency matrix (9)
Considering the polarization-sensitive absorber described by Eq. (4) we see, e.g., that a non-polarized incident plane wave can be transformed into a x-linearlypolarized plane wave with the choices 0 — 0, A — 1, and B — 0. On the other hand, retarders described by Eq. (4) with \A\ = \B\ = I have no effect in the state of polarization of an incident non-polarized plane wave. Thus far we have considered only plane waves, but the formalism presented above can be extended straightforwardly to fully or partially polarized paraxial beams of light with rather arbitrary spatial profiles.2 In this case the quantities 0, A and B are functions of position. 3. Periodic elements with coherent illumination Let us next assume that A(x,y), B(x,y), and 0(x,y) are periodic in the x and y directions with periods dx and dy. Since this kind of an element modulates the state of polarization of the incident field it is called a polarization grating3'4'5 It follows directly from the periodicity of the element that also the electric field E in the half-space after the element must be similarly periodic and hence we may express it as a Fourier series with the Fourier coefficients given by Jm
I dx""y Jo
I
E(x,y,zo)ex'p[—i2'!r(mx/dx
+ ny/dy)]dxdy,
(10)
Jo
where z = ZQ is the output plane of the polarization-modulating element or system. The coefficients rim
106
In the following considerations, we normalize the incoming field with || Ji|| = 1. Diffractive elements are often used to transform an incoming field into a signal, which can be of a very complicated form. To find an element capable of performing the desired transformation numerical methods must usually be used. In this article, we restrict ourselves to consideration of discrete signals, which can be produced by means of periodic diffractive elements. Hence the desired signal consists of a certain set of diffraction orders, denoted by S, and the conversion efficiency is defined by
r>=
The design goal is to maximize the efficiency rj while simultaneously keeping the signal to noise ratio at an acceptable level. Unfortunately, it is not always clear whether or not a result obtained by numerical design methods is close to the best possible one, since the used algorithm may not converge. Thus the quality of the results must be characterized somehow. If the field is uniformly polarized, then the diffraction efficiency may be determined by using only one scalar component, for example by Ex(x,y,z). In that case one may use Wyrowski's upper bound6'7 for the determination the quality of the design result. The best possible conversion efficiency cannot exceed the upper bound and thus one may determine the quality of the numerically obtained results by comparing the efficiency to the upper bound. The best possible conversion efficiency is, apart from rather trivial cases, always lower than the upper bound. For example, in the case of scalar triplicator, i.e., a 1 —> 3 beam splitter, the upper bound is rju K 93.81%,8 whereas the best possible solution gives the efficiency rj « 92.56%.9 However, if the polarization freedom may be used in the sense that the state of polarization of each diffraction order is free, 100% efficiency can be obtained for a wide class of signals,10'11'12 including the triplicator.
4. Polarization multiplexing Since a paraxial field is a combination of two orthogonal scalar fields, one may design an element which produces different signals depending on the state of polarization of the incoming field. This is called polarization multiplexing.13'14'15 With this kind of an approach, one makes separate designs for both polarization states and then combines them into a single element. Hence the maximum attainable diffraction efficiency is equal to the scalar efficiency and is obtained by using a polarization-sensitive element, such as a polarization grating. In some cases one may obtain relatively high efficiencies by using a combination of birefringent substrates and isotropic materials.15 If only anisotropic scalar-domain diffractive elements are used, high diffraction efficiencies cannot usually be obtained.
107
5. Partially polarized light Spatially varying polarization-sensitive diffractive elements have useful applications also in the domain of partially coherent optics. One good example is the possibility to measure the Stokes parameters of a partially polarized beam by using an absorbing polarization grating and a linear polarizer.5 It has been shown some time ago16 that the state of polarization of a light beam can experience dramatic changes upon propagation in free space. It remains to be studied how diffractive elements can be applied to perform useful polarization modulation tasks for partially polarized and spatially or temporally partially coherent light. Prom the perspective of the wave-optical engineering, the most natural starting point in the polarization modulation of partially coherent fields would be fixing the distribution of the state of polarization of the desired signal field. Such a signal could be, for example, a discrete set of diffraction orders, each having fixed degree of polarization and intensity but free phase differences and/or the correlations between the orders. In this case the main design' problem is to find the optimum phase differences and/or correlations such that the diffraction efficiency is as high as possible with minimal signal errors. One could expect that with this kind of approach, the traditional expression of the coherency matrix as the sum of fully polarized and unpolarized parts17 is not necessarily the best possible choice. Instead, one could express the coherency matrix with the help of its eigenvalues \j and eigenvectors V
J = Ai0 1 01+A 2 0 2 01,
(13)
where the dagger denotes the Hermitian adjoint. Equation (13) expresses the decomposition of the partially polarized field into two fully polarized but mutually uncorrelated parts whose intensities are given by the eigenvalues AI and X? in parallel to the coherent-mode decomposition of partially coherent fields.19'20 6. Conclusions In this short account we were able to cover only a minor fraction of possible methods to modulate the state of polarization of a light beam by means of sophisticated surface-relief microstructures. Only some first steps have been taken to fabricate diffractive elements capable of polarization modulation (see, e.g., Ref. 21) and a lot remains to be done to explore the potential of this technique. Acknowledgments J. Tervo gratefully acknowledges the financial support from the Academy of Finland. References 1. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, Boston, 1990).
108
2. F. Gori, M. Santarsiero, V. Vicalvi, R. Borghi, and G. Guattari, Pure Appl. Opt. 7, 941-951 (1998). 3. O. Bryngdahl, J. Opt. Soc. Am. 62, 839-848 (1972). 4. S. D. Kakichashvili, Opt. Spectrosc. 33, 171-174 (1972). 5. F. Gori, Opt. Lett. 24, 584-586 (1999). 6. F. Wyrowski, Opt. Lett. 16, 1915-1917 (1991). 7. F. Wyrowski, J. Opt. Soc. Am. A 10, 1553-1561 (1993). 8. U. Krackhardt, J. N. Mait, and N. Streibl, Appl. Opt. 31, 27-37 (1992). 9. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, Opt. Commun. 157, 13-16 (1998). 10. J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, J. Opt. Soc. Am. A 20, 282-289 (2003). 11. J. Tervo and J. Turunen, Opt. Lett. 25, 785-786 (2000). 12. M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, J. Mod. Opt. 47, 2351-2359 (2000). 13. W. Yu, T. Konishi, T. Hamamoto, H. Toyota, T. Yotsuya, Y. Ichioka, Appl. Opt. 41, 96-100 (2002). 14. I. Richter, P. C. Sun, F. Xu, F. Fainman, Appl. Opt. 34, 2421-2429 (1995). 15. N. Nieuborg, A. Kirk, B. Morlion, H. Thienpont, and I. VeretennicofT, Appl. Opt. 36, 4681-4685 (1997). 16. D. F. V. James, J. Opt. Soc. Am. A 11, 1641-1643 (1994). 17. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995). 18. L. Mandel, Proc. Phys. Soc. 81, 1104 (1963). 19. E. Wolf, J. Opt. Soc. Am. 72, 343 (1982). 20. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, J. Opt. Soc. Am. A 20, 78 (2003). 21. L. H. Cescato, E. Gluch, and N. Streibl, Appl. Opt. 29, 3286 (1990).
SPATIAL COHERENCE: DEFINITIONS AND MEASUREMENTS B. EPPICH, G. MANN, H. WEBER
Optisches Institut Technische Universitdt Berlin
10623 Berlin, Germany 1.
Introduction
Spatial or transverse coherence is of increasing interest for several laser applications in industry, as surface structuring or lithography. Depending on the special application high or low transverse coherence is demanded. Anyway, coherence has to be defined correctly, and reliably set-ups for quantitative measurements have to be developed. Most of the fundamentals on coherence can be find in the book of Born-Wolf,1 but have to be adapted to the special requirements of laser technology. Up to now a beam is mainly characterized by its width, divergence and beam propagation factor M2 according the ISO proposals. A beam with M2 = 1 is always the fundamental mode TEMOO, which is completely coherent. For M2 > 1 the beam can be a completely coherent higher order mode or a mixture of non-correlated modes with different frequencies, which means the field is partially coherent. An example is a laser, oscillating in TEMOO and TEM01 with different eigenfrequencies Cflbm. Each of the two modes is coherent, but the superposition of both is partially coherent. A coherent beam with M2 > 1 can be transformed into a fundamental mode by suitable phase plates. It means that M2 can be reduced to its minimum value without losses (in principle). This is not possible for partially coherent beams, where reduction of M2 always means loss of power. Hence it is necessary to characterize a laser beam by at least one additional number, which refers to coherence. It is called the global degree of coherence K.
2.
Principal procedure to determine coherence
The classical instrument to measure spatial coherence is Young's interferometer, shown in fig. 1. The main element is a double mask with two crossed slits, by which two pinholes with variable positions r,, r2 are generated (see left fig. 2). It is often convenient to use instead of rj, ?2 the distance S and the position f : f12=f±S/2
(1)
The field under investigation is illuminating the masks. The field behind the pin-holes is collimated, and in the focal plane an interference pattern is produced. The resulting field (neglecting polarisation) reads:
E(r,,f 2 ,t)=E 1 (f 1 ,t)+E 2 (f 2 ,t + At) 109
(2)
110 beam cleaner
He/Ne-
CCD
Figure 1 Schematic set up of Young's interferometer3
Figure 2 Left: The masks with variable pin-holes Fj, T2 to scan the transverse field. Right: An example of the resulting cross-correlation function3 Fi2 for a special position T .
The detector (CCD-camera, film) in the observation plane squares and delivers a time averaged signal, the intensity:
(3) The bracket means time average, with an integration time large compared with the time constants of the field fluctuations. From eqs. (2,3) the intensity of the interference pattern is obtained: 1 ,r 2 ) = j 1 (r 1 )+j 2 (f 2 )+r i 2 (r 1 ,f 2 )+r; 2
(r 15 r 2 )
(4)
F12 is the cross-correlation function:
(5)
111 and Ji, J2 the intensities behind the tow pin-holes. It is assumed that the wave is quasimonochromatic, which means the band-width Aco is small compared with the carrier frequency ro0. Under this assumption the variation of the amplitude across the observation plane due to temporal fluctuations and different optical paths At can be neglected. The cross correlation function is normalised and delivers the complex degree of coherence, which reads in the f, S representation:
r,,(f,s '^V
'
/
p.,
I -f
r, H
nn.n<4i.
ff\\
(j)12 (r, s) is the phase difference between the two points f ± S / 2 or the mutual phase. Eq. 4 then can be rewritten:
j(f,s)= J, + J 2 + 2 y , 2 > / J 1 J 2 cos(|>12
(7)
It can be easily show that for a coherent field with E,,2 = A, 2 (f, 2 )• exp[i co 0 t + icp(r, 2 )J the modulus of j\2 is equal one. It is not sufficient to measure J, J b J2, because if the field is not coherent, two unknown parameters exist |yi2|
J = J, + J 2 + 2|y12 • Tjp^' sin (|)12
(8)
Now the cross-correlation function Fi2 can be reconstructed as shown in fig. 2, and amplitude and phase can be evaluated. This does not mean that in any case an unique, time average phase front exists (e.g. vortex fields).
3.
Definition of some coherence parameters
The disadvantage of yi2 is its dependence on the positions f t , ? 2 . The goal is to define some global coherence parameter. Therefore an integral parameter seems more useful, first proposed by Bastiaans4. It is called the global degree of coherence:
112
with P the total power in the scanned field. For a coherent beam the global degree of coherence is K2 = 1 . An interesting relation exists between K and the beam propagation factor M2
K-M2>-
,
(10)
which can be proved by the modal expansion4, discussed in section 4. Other parameters of interest are local and global transverse coherence lengths5.
dr2 The local coherence length still depends on the positions r,, except for special fields as the Gaussian-Schell beam. The overall or global coherence length reads in r, S coordinates.
= 8•
Js 2 r i 2 (r,sj 2 drds J|ri2(r,sf drds
or using eq. 9
r12(f,s)2 dr ds
(12)
The factor 8 is arbitrary, but guarantees that for the Gauss-Schell beam the coherence length is equal or smaller than the beam diameter, defined by the second intensity moment. The far- field divergence of coherence is defined by:
coh
If the field can be factorised E(x,y) = E0 f(x) • g(y) the cross-correlation function can be written , yy ) i2 (r,s) = r x (x,s x )-r yy(y,s
(14)
and eq. 9 delivers 2 |r y ( yi ,y 2 | 2 d yi dy2
P The global coherence length results in
= K x Ky
113
2
_ p2
• glob ~~ *• glob,x
4.
v
xv
+
/2
y T c glob,y
The modal decomposition
The cross-correlation function can be represented by its modal expansion6'7'8'9
(16) m,n
with P the total power and P,™ the eigenfunctions of F12 Jr i 2 (r 1 ) r 2 )F m n (f 2 )df 2 =P.a m n F m i (f 1 ) The eigenfunctions are orthogonal and normalised (FmnFm'n'H JFnJri )' FmV Ol K =5^^,
The coefficients are positive and normalised
For sake of simplicity the following discussion is restricted to one-dimensional fields, assuming Ky = 1, K2 = Kx. Then the x-part of the cross-correlation reduces to
;(x 2 )
(17)
One example is the field of a stable, spherical, non-degenerated and homogeneous laserresonator (gi,g2), which is the super-position of a limited number N of Gauss-Hermite polynomials with different eigenfrequencies co0m
A
N
~' (18)
114
KM
2
(i \ IZ/Z R ;
q m ^x,z,t; = exp
\
J-k/7
-7
(19)
om
^
, 1+ m C00m = - p + H -- arccos.Jg1g2
W
R,x
, . ,. , p = longitudinal mode order
Eq. 18 with eq.5 immediately deliver eq. 14, because the cross terms qn (xi, t) qm (x2, t) vanish in time average, and the coefficients are a m = C m C m . The intensity is given by:
(21) R,x
and the power
P= Jldx=
A
(22)
Now the coherence parameters defined in sect. 3 can be expressed by the modal coefficients am. Global degree of coherence:
(23)
Global coherence length: N-l
' glob.x
(24) m=0
The beam parameters are evaluated from the second intensity moments. Beam radius, beam divergence are and beam propagation factor are: W R,x W 0,x
N-l 0
e R,x
0,x =
7CW 0,x
m=0
In the special case of Gauss-Hermitean functions the parameters read:
115
4w2R,x
I
,2
K,
N-l
w,0,x
e?> ,x R
N-l
5.
Examples of Gauss-Hermite fields
To demonstrate that these definitions make sense, some examples will be demonstrated. The results are compiled in table 1. Beam radius wR>x and coherence length ^giob,x are normalized to the fundamental mode radius w0>x, which propagates accordingly eq. 20. Pure All modal coefficients are zero, except aN = 1. The well known results are obtained. The coherence length in this case is equal the mode diameter. Two Mode Oscillation Two modes are oscillating, TEM0o and TEM0i, with the power amplitudes a0, ai a0 + a] = 1
am = 0 for m > 1
Now it is useful to apply the modal representation and the above derived formulas deliver
beam radius / divergence W
0,x
global degree of coherence K x = a2, + a 2 beam propagation factor
M 2 = 1 + 2a^
product
K x M 2 = 1 - 2af + 4af
116
It is interesting to note that for ai=l/3 this product has its
minimum-value
(K X M X K = /97 < 1 • The dependence of these parameters on the power content ai of the TEMOl-mode is plotted in fig. 3. The global coherence length (see table 1) now depends on the power content ratio of the two modes and varies between the beam diameter 2w of the TEMOO-mode and V 3 • 2w of the TEMOl-mode Gauss-Schell beam10'11 This is the most general partially coherent beam with respect to the second order moments and is characterized by 10 independent parameters. We will restrict to the onedimensional beam with its two parameters WR*,CT,which describe beam dimension and degree of coherence. The Gauss-Schell beam is defined by the cross-correlation function, which can not be factorised. It reads in the one-dimensional case:
x /w ox
*• TEM01 power a1 Figure 3 Coherence parameters of the two-mode case vs the normalised power a, of the TEMOl-mode.
2ixs z
i_
v,X
2w^ xa K.,X
W^Z KX F r
(25)
0
xM x ,z)= jrx(x,sx,z)exp[iksx0Jdsx 1
A •w T x aexp -2-
(26)
(ka6 x w T]X ) 2 w T,x
(27)
117
WT>X is the waist radius of the beam at z = 0. The definitions of the second moments deliver
beamradius
w 2 R x = w 2 j X (l + (z/z R x ) 2 )
divergence
0x = A, /(rcwT x • a)
Rayleighlength z R x =7iw 2 rx o/X, The eigenfunctions of this special cross-correlation function are the Gauss-Hermite polynomials4 with coefficients depending on the coherence parameter a. For the calculation of the coherence parameters it is easier to use the compact presentation of the Gauss-Schell beam, eq. 25. The results are compiled in table 1. The local coherence length in this special caae is independent on the position and equal the global coherence length
The global coherence length scales with WR>X, which means by using eq. 20:
• w TiX Then for the far field coherence angle, one obtains with eq. 13:
,
~ cr Z
R,X
A, ,
The coherence angle in the far field is given by the divergence of the corresponding coherent Gaussian beam, a well known result, which holds for any light source. It is worth to note that the product of the global degree of coherence and the beam propagation factor is equal one KX-M2=1
(28)
N-Modes of Equal Power Finally the case of N-modes with equal power will be discussed, which presents the maximum of entropy2.
fl/N
0
0
n>N
The modal expansion delivers
I I
beam propagation
radius beam
J2N+1
2N+ 1 ~
Two - mod e - laser TEM,,
+ TEM,,
Jm
N modes TEM,, of equal power a m=1/N
1
1+2a,
1
la0 +2a,rl,rlnl
dz-z?dzGz
fi
coherence length
42N + 1
2N-t 1
a: +a:
l/o
1/N
N
Table 1 Some characteristic parameters of Gauss-Hemitean modes. q =
radius.
global coherence
~
a, + a , = 1 Gauss-Schell
degree of coherence
AX /
(3
1
1
1
W R , x , wllX= z-dependent beam radius in x-direction, wo,= z-dependent fundamental mode
119
beam radius, beam divergence
9R U
0,x
^ g iobx :
coherence length global degree of coherence
Kx = 1 / N
beam propagation factor
M^=N
product 6.
W w 0,x
=1
Interferometric measurements
Young's double slit experiment The experiment was already discussed in section 2, and one example of the crosscorrelation function is shown in fig. 2. From all these functions r(?j,r 2 ) the modal content and the intensity profile can be reconstructed. An example is given in fig. 4, which confirms the suitability of Young's set-up to measure beam structures. Instead of two pin-holes, two fibers can be used to build an Young Interferometer12.
m
1
1
1
1
- *-
<> — (>
1 1
-t" ^<
P
1
>
t
1
41 |
1 I
t1 1
_ _ 4
U _ _ 4- - - •k1 1 1 I.-- 0.--
1
1
{-
1
1
I
Figure 4 Left: Modal coefficients amn of a multimode He/Ne-laser3. In this case a two-dimensional field has to be determined. The open circles indicate negative values of amn. Right: Intensity structure.
Shear interferometer The disadvantage of the Young-interferometer is the time-consuming point-to-point measurement. More efficient is the shear interferometer, which at each fixed distance s = ij — ?2 delivers by one shot the information for all values r = (r, + r2 )/ 2 , if a 2dim. CCD-camera is used. The shear-interferometer was first used to measure the phase distortions produced by optical elements14'15. A schematic set-up is shown in fig. 5. The incoming beam is split into two beams, which are reflected by two
120
laser
mirror S
Figure 5 Schematic set-up of a shear-interferometer".
mirrors and recombined in the observation plane. By shifting mirror Si, a transverse shift s between the two beam is generated. An interference pattern appears, whose structure depends on the shape of the phase-front. An example is shown in fig. 6. The phase / intensity structure of a He-Ne-laser was investigated 16. Using the relations of section 2 the degree of coherence y12 can be evaluated. The experimental result is plotted in fig. 7. Another set-up to measure the coherence of a pulsed copper-laser is presented in fig. 8. By inserting a rotating Dove-prism, the complete transverse plane could be scanned. A result is given in fig. 9. The transverse coherence of this Cu-vapour laser strongly depends on the magnification of the unstable laser and the number of round-trips. 7.
Non-interferometric Measurements
The shear-interferometer requires very homogeneous and isotropic optical elements. Especially locally varying can falsify the experimental results. Therefore noninterferometric measurements can be more suitable. For sake of simplicity, fields with only one transverse coordinate E (x,z) are discussed. The cross-correlation function is then a two-dimensional function rx(xi,x2) or Fx (x,sx) using the transformation eq. 1. The field is completely described by Fx in one special z-plane e.g. z = 0. Phase and amplitude can be reconstructed, but also the global degree of coherence eq. (9) and the transverse coherence length Eq. 12. Fx will change due to diffraction if the field propagates. This is described by twice the Fresnel-integral, a double-integral because Fx is the product of two fields. The propagation is easier to describe by a suitable Fourier-transform of Fx, the ambiguity function. Ambiguity-function The intensity J(x,z) will also vary with z, and its special structure depends on the phase and amplitude in the initial plane. This means that the two-dimensional intensity distribution also contains the full information. But how to reconstruct Fx(x,sx) from J(x,z)? One possibility is to use the ambiguity-function18 ZA, which is defined as the Fourier transform of Fx(x,sx) with repect to x:
Z A (s x ,a x )=JT x (x,s x ).e' k a *Mx
(29)
121
s =0
(sm = 0 |jm) 1,00,90,80,7-
30,40,30,20,10,0 -Vm 0
s = 0,08
25
50
75
100 125 ISO 175 200 225 250 275 Ottx/Pixe!
106
(sm = 525 |jm)
0,0 JIT 0 25 SO 75 100 125 150 175 200 223 250 275 Ort >; / Pixel
s = 0,16
(s m = 1025pm) 0,80,70,6-
-0,5-
io,«"0,30,20,10,00
25 SO
75 100 126 150 175 200 Z& 250 27S 306 Oft X / Pixel
Figure 6 Shear interferogram16 of a He-Ne-laser with M2 = 1.2 .
0-
0
—I
1
1
sm/wR
1
1
Figure? Degree of coherence16 |yi2| vs normalized shift S/WR,X, for M 2 = 1 . 2 .
122
Interferogram
R1
M3
Camera eg. Gated Diode Array
Figure 8 A shear-interferometer with a rotating Dove-prism to scan the transverse plane'7.
2 Round Trips
4 Round Trips
25 mm
Figure 9 Transverse coherence of a copper-vapour laser'7. Parameters are the magnification M of the unstable resonator and the number of round trips.
123
It is a representation of the field in the sx, ax-phase space. The second moments of ccx and sx are the far-field coherence angle 0coh x in x-direction and the transverse coherence length ^ g l o b x :
(30)
2 ^ Q Js x Z A (s x ,aJ ds x da, _o
(31)
* glob.x
The global degree of coherence reads:
(32)
dsx da x
Eqs. 30, 31,32 can be proved by inserting eq. 29 into eqs. 9, 12. The propagation of ZA in free space is very simple. It turns out that with increasing z the ambiguity function is sheared as shown in fig. 10. The following transformation18 of the coordinates holds:
s x -»s x -zcc x ax->ax Using this property a simple relation between the intensity J(x,z) and ZA can be deduced
= Jj(x,z)-exp[ikaxx]dx
z=-sv /a.
(33)
Finally the procedure is as follows measure J (x,z) at 10-20 different positions around the focal region calculate ZA by the Fourier transform and replace z by -s/a. from ZA, global coherence, coherence length and far-field divergence can be evaluated by another Fourier transform the cross-correlation function is obtained use the second moments of J(x,z) to calculate beam width and beam divergence Wigner function13'19
Another complete representation of the field in the x, 6X phase-space (position, divergence) is the Wigner-function hx(x,9x), which has a certain similarity with the radiance, the power per angle 9X at a position x. It is related to Fx also by a Fourier transform with respect to sx
h*Mx) = jr x (x,s x )exp[ik0 x sjds x
(34)
124
J(x,z)=rx(x,s=0)
(s x ,cc x )= Jr x (x,s x )exp[ikxa x ]dx - JJ(x, z)exp[ikxax]dx z=-s, /a,
transverse coherence ^ g i 0 b, x Figure 10
Propagation of the ambiguity function in free space-
and is related to the intensity distribution by a three-fold Fourier transformation (equivalent to the Radon-transformation13, well know from tomography). The Wignerfunction propagates in the same way as the ambiguity function and delivers beam width and beam divergence directly
_A Jx 2 h x (x,6 x )dxd9, W
R K,Xx =
Ql =•
4
'
Jh x (x,e x )dxd0 x Jh x (x,e x )dxde x
(35)
(36)
and of course all other parameters as phase, phase gradient. But numerically the construction of the Wigner-function is more difficult and more time-consuming than the ambiguity function.
125
Experimental results Experiments were performed with a mode generator. This is a diode off-axis pumped YAG-laser, producing by mechanical adjustment of the gain profile the Gauss-Hermite polynomials20. The resulting global coherence vs adjustment is shown in fig. 11. The global degree of coherence has always a maximum when a pure TEMom mode oscillates, but never reaches Kx = 1, which means that always a mode mixture oscillates. In table 2 the results obtained with a Young-interferometer are compared with the nonmterferometric measurements. The theoretical values are based on the assumption that only two modes are oscillating. Two-dimensional fields If the field cannot be separated E(x,y) ^ E0-f(x)-g(y) or if the field is not of circular symmetry, the field has two transverse dimensions. In this case a four-dimensional crosscorrelation function r i2 (r 1 ,r 2 ) results, which again includes the full information. The corresponding ambiguity- and Wigner-functions are also four-dimensional functions. But the intensity J(f, z) is only three-dimensional. The missing information is hidden in the azimuthal phase structure. Inserting a cylinder lens and rotating by p, delivers the lacking information. Now the intensity becomes four-dimensional J(r,(3,z), and can be used to evaluate the other functions and all beam parameters.
I
10 -
TEM04
TEM03.
M ., TEM02 TEM01
1,0 0.8
—
0.2
I 0.5
0.6
I 0.7
I 0.3
diode laser adjustment (mm)
Figure 11 Beam propagation factor M^ and global degree of coherence Kx vs diode laser adjustment.
126
TEMOO: TEM01 a 0 :a,
beam propagation theory exp.
transverse coherence
global degree of coherence theory
Young
Wigner
0.96-1.0
1.0
1.0:0
1.0
1.01
1.0
0.75:0.25
1.75
1.71
0.79
0.5:0.5
2.0
2.04
0.25:0.75
2.5
0:1.0
3.0
theory Young
Wigner
1.0
0.99
1.01
0.70-0.75 0.74
0.63
0.63
0.60
0.71
0.66-0.73 0.71
0.71
0.74
0.71
2.39
0.79
0.72-0.77
0.94
0.89
0.87
3.04
1.0
1.0
0.97
0.96
0.75
0.89-0.99 0.95
Table 2 Comparison of beam propagation factor M2X, global degree of coherence Kx, and global transverse coherence length 4oh,x. obtained with Young's interferometer and by non-interferometric measurements.
Notation w0, QO :
WR (z), 9R: WT: ZR = 9R/wT:
TEMOO beam radius / half divergence beam-radius, half divergence waist radius Rayleigh length beam propagation factor
A./71
This research project was supported by the Federal Ministry of Education and Research; EUREKA-CHOCLAB II (E12359; 13EU0153).
References 1. 2. 3. 4. 5. 6. 7. 8. 9.
M. Born, E. Wolf, Principles of Optics, Cambridge University Press (1999) Th. Graf, I.E. Balmer, "Laser beam quality, Entropy and the Limits of Beam Shaping", Optics Comm. 131, 77, (1996) E. Tervonen, J. Turunen, A.T. Friberg, "Transverse laser-mode structure determination from spatial coherence measurements", Appl. Phys. B. 49, 409-414 (1989) M.J. Bastiaans, "Application of the Wigner distribution function to partially coherent light", JOSA , A3, 1227, (1986) B. Eppich, "Definition, meaning and measurement of coherence parameters", Proc. SPIE Vol. 4270, 71, (2001) F. Gori, Opt. Comm. 34, 301 (1980) M.J. Bastiaans, Optica Acta 28, 1215 (1981) H. Laabs, B. Eppich, H. Weber, "Modal decomposition of partially coherence beams using the ambiguity function", JOSA 19A, 497 (2002) E. Wolf, "New theory of partial coherence in the space-frequency domain",
127
10. 11. 12. 13. 14. 15.
16. 17. 18. 19. 20. 21. 22.
JOSA 72, 343 (1982) R.Simon, K. Sundar, N.Mukunda, "Twisted Gaussian Schell-model beams" JOSA, A10,2008, (1993) R.Simon, E.C.G. Sudarshan, N. Mukunda, Phys. Rev. A29, 3273, (1984) R.Simon, N.Mukunda, E.C.G. Sudarshan, Opt.Comm.65, 322, (1988) A. Siegman, Proc. SPIE 1224, 2, (1990) C.M. Warnsky, B.L: Anderson, C.A. Klein, "Determining spatial modes of lasers by coherence measurements", Appl. Optics 39, 6109 (2000) B. Eppich, Dissertation Technical University Berlin, D83, (1998), "Die Charakterisierung von Strahlungsfeldern mil der Wigner-Verteilung und deren Messung" E. Waetzmann "Interferenzmethode zur Untersuchung der Abbildungsfehler optischer Systeme", Ann. Phys. 39, 1043 (1912) M.V.R.K. Mruthy, "A compact lateral shearing interferometer based on the Michelson interferometer", Appl. Optics 9, 1146 (1970) S.M. Jackisch, "Shearing Interferometrie zur Laserstrahl-charakterisierung", Diploma Thesis, Optical Institute, Technical University Berlin, (1999) D.W. Coutts, "A versatile angular shearing interferometer for measurement of spatial coherence", Techn. Digest, Conf. Lasers Electro-Optics-Europe 1998, paper CWI4, pp.217 A. Papoulis, "Ambiguity function in Fourier optics", JOSA 64, 779 (1974) MJ. Bastiaans, "Wigner distribution function and its application to first order optics", JOSA A69, 1710, (1979) H. Laabs, B. Ozygus, "Excitation of Hermite Gaussian modes in end-pumped solid-state laser via off axis pumping", Optics & Laser Technology, 28, 213 (1996)
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INVITED SEMINARS
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OPTICAL BEAMS IN UNIAXIAL CRYSTALS GABRIELLA CINCOTTI Electronics Department University Roma Tre via della Vasca Navale 84,1-00146 Rome, Italy, and National Institute for the Physics of the Matter (INFM) Unity Roma 3, Rome, Italy email: s.cincotti(a),uniroma3.it Url: www.ele.uniromaS.it/fotonica ALESSANDRO CIATTONI Physics Department University Roma Tre, 1-00146 Rome, Italy, and National Institute for the Physics of the Matter (INFM) Unity Roma 3, Rome, Italy HORST WEBER Institute for Optics Technical University of Berlin, D-10623 Berlin (Charlottenburg)
1 Introduction Many different methods have been proposed in literature to evaluate the electromagnetic field inside anisotropic media; some of them are very general and comprehensive, but the mathematics involved are rather cumbersome. On the other hand, the simplest approach of a single plane wave or ray propagating in an anisotropic medium, which can be found in many textbooks, does not take the diffraction effects into account, so it cannot accurately describe the propagation of beams with transverse widths comparable with the wavelength. Moreover, the study of light propagation in anisotropic media requires a vectorial approach, as a complete description of different polarization states and the coupling between the field components is mandatory. The propagation of a paraxial beam along the optical axis of an unbounded uniaxially anisotropic crystal is described by means of a vectorial angular spectrum representation of the electromagnetic field, to evidence the combined effects of anisotropy and diffraction. Within this approach, it is possible to solve the boundary problem in the most general case, i.e. to evaluate the analytical expression of the propagated beams, when the boundary field distributions are assigned. In addition, these results allow us to identify some relevant feature of propagation in uniaxial crystals. The optical field inside the crystal is the superposition of an ordinary and extraordinary part, which satisfy two parabolic decoupled equations. Therefore, the ordinary and the extraordinary beams propagate independently and superimposed along the optical axis. Due to the intrinsic polarization states of the ordinary and extraordinary plane waves, the ordinary component is a solenoidal field, whereas the extraordinary one is irrotational. If the Cartesian components of the boundary field are cylindrically symmetric, the analytical expression of the propagated field presents a very simple dependence on the azimuthal angle. This result clearly shows the way in which the anisotropy changes the state of polarization and how the boundary cylindrical symmetry is lost. In addition, in this case a very simple correspondence between the paraxial propagation formulas inside a uniaxial crystal and in an isotropic medium can be found, so that it is possible to evaluate closed-form expressions for Hermite-Gauss beams, Laguerre-Gauss beams and Bessel-Gauss beams, which are well-known solutions of the paraxial equation in vacuum.
131
132
The effect of the anisotropy on the polarization state of an optical beam can be also described by means of the propagation equations for the Cartesian components of the electric field; in fact, the x- and ^-components of the electric field satisfy a system of coupled 'paraxial-like' equations, that highlights the energy coupling between them. The energy exchange between the x- and y-components undergoes to a saturation mechanism that is due to both diffraction and coupling between the components. For Hermite-Gauss beams of any order, linearly polarized along the x-axis at the entrance facet of the crystal, it is possible to furnish a full characterization of their diffraction properties, evaluating the spot-size evolution formula and the corresponding angular divergence: due to anisotropy, the beam parameters evaluated along the x- and y-axes have different expressions. The analytical results are confirmed by a set of experimental measurements at the output of a calcite crystal, for a fundamental Hermite-Gauss and for the field diffracted by circular apertures. 2
Fundamental Hermite-Gauss mode
Let us consider a uniaxially anisotropic crystal whose optical axis is the z-axis of a suitable reference frame, and with n0 and ne being the ordinary and extraordinary refractive indexes, respectively. Without any loss of generality, we assume that entrance facet of the crystal coincides with the z=0 plane, and that the input fundamental HG mode is linearly polarized along the x-axis, i.e.
— I E x (x,y,0)=Eexp
x (1)
E y (x,y,0)=0, where E is a constant, and w0 the spot size at the waist plane. The field propagated through the crystal has the following expression [6,7] 2 exp(ik 0 n 0 z) = —-
(2) \ =—
2 exp(ik 0 n 0 z)
l^ttr
;
1
-+
(Q e (z)
' i
x2+y
where k0=2n/K is the wavenumber in vacuum, and (3)
are the ordinary and extraordinary propagation parameters, which are related to the complex propagation parameters of Gaussian beams propagating through isotropic media with index of refractive indices n0 and ne2/n0, respectively.
133
Figure 1 shows the intensities \EX\2 and \Ey\~ evaluated at the planes 80=11, 26, 45, and 78, where 80=z/zRo and zRo =7iw02n0/h is the Rayleigh distance in an isotropic medium, with refractive index «„. It is evident that the ^-component of the electric field loses its boundary cylindrical symmetry, due to the anisotropy; besides, the anisotropy gives origin to ay-component of the electric field. Ex and Ey experience both diffraction and mutual coupling, the latter being responsible for the change of polari/ation state; to investigate their energy exchange, we consider the optical power carried by Ex and Ey IE
/
IE
/
IE/
IE/
-20
80 =z/zRo=ll
0 „/*„
20
S0 =z/zRo=26
80 =z/zRo=78
S0 =z/zRo=45
Figure 1: Intensities of the x- and y-components of the electric field originated in a calcite crystal by a fundamental HG mode linearly polarized at the crystal entrance facet: analytical (upper rows) and experimental results (lower rows) for four propagation distances.
P (j) (z)=jdxdy(|E j (x,y,z)|
= x,y.
Substituting Eqs. (2) into Eq. (4), after a simple algebra, we get
(4)
134
3 +-
(5)
where P0=¥t*)(Q) is the input power, and 8e=z/zRe with z/ee =mv02ne2/foi0 being the Rayleigh distance in an isotropic medium, with refractive index ne2/n0
Figure 2: Evolution of the optical power and spot-sizes associated to the x- and y-components of the electric field inside a calcite crystal: numerical (lines) and experimental results (marks).
Figure 2 shows the evolution of P6^ as a function of S0: we observe that the energy associated to the y-component of the electric field monotonically increases and undergoes a saturation: the asymptotic power exchange between Ex and Ey amounts to a quarter of the total power for every linearly polarized circularly symmetric input beam [8]. To investigate the diffraction properties of Ex and Ey, we evaluate the evolution of their spot-sizes, along the x- andy-axes
w
=4
Jdxdyi7 2 E j (x.y > z)| :
(6)
135
getting as a result
wn
-=—+9
- + 2-
I o —O
-In
1 1 1
.
-
1
°
1
1
!
1 2 JJ
! /I 1 7
.
e
1
o 2
JJ
8.80-\
9
6 (7)
1+ 4 +3
w,. wn
- + 21n 1 +
wn
1+ -
Figure 2 reports the spot-sizes of the Cartesian components of the electric field, along with the spot size of a Gaussian beam propagating in an isotropic medium with refractive index n0, that is plotted for reference. From the expressions of Eqs. (7), we can evaluate the angular divergences of the beams and the waist spot-size of ^distribution (x) _
w
x
(X) z
( )_
A
!
(x) ,
(8) 0(y)=0(y)=lim y
~»
w
(y)
z
(z)_ ;ra 0 w 0 > /2\n*
w x (y) (0) = w y (y) (0) = lim w x (y) (z) = V3w0. Of course, all the beam parameters coincide with those of a Gaussian beam propagating in an isotropic medium if we put n0—ne\ let us also observe that the waist plane of the y-
136
component coincides with the z=0 plane and that the corresponding spot size is equal to V 3w0 , even though it is Ey-0 at the entrance facet of the crystal. 2.1 Higher-order Hermite-Gauss modes For a HG10 mode, linearly polarized at the entrance facet of the crystal, the electric field distribution is [6]
IE/ HS,0
IE/ HG0,
("I
ff
HG,,
IE/ HO,,
= z/zRo =26 Figure 3: Intensities of the x- and y-components of the electric field of higher-order HG modes inside a calcite crystal, numerically evaluated for two propagation distances.
137
Qe2l (9)
for a HG0i mode we have
X +y
Q»(z)
x +y"
~ow
(10)
and for a HG. i mode it is
"Q.W
138
E y (x,y,z) = Ew4elkon°:
" x'+y*
-3Q* (z)(x 6 -5x 4 y 2 -5x 2 y 4 + y6)-3Q^ (z)(x 4 -6x 22y2
-3Q200(>
6
y 4 )]
;(z)(x 4 -6x 2 y 2 + y4)]}.
-5x 4 y 2 5x 2 /
30
40
(11)
50
sn=z/zDn
Figure 4: Evolution of the optical power associated to the y-component of the electric field of higher-order laser modes inside a calcite crystal.
Figure 3 shows the intensity distributions of the Cartesian components of the electric field corresponding to the HG10, HG0i and HGu laser modes, propagated for two distances z inside a calcite crystal; from an inspection of this figure, it is evident how the anisotropy modifies the beam profiles. Substituting Eqs. (9)-(ll) into Eq. (4), we obtain the expressions of the energies pertaining to the Cartesian components of the laser modes, that are plotted in Fig. 4: it is worth noting that laser modes of different order present different saturation values; moreover, the energy exchange does not ever exceed the half of the input power [8]. Tables I and II summarize the main results of the present investigation: the angular divergences, evaluated along the .x-and >>-axes, of Ex and Ey are reported, along with the waist spot-size of the y-component and the saturation values of P*y).
139
Table I: x-component of the electric field
mode
3 00
wx(0)
Wy(0)
wn
wn
ex
wn
Px(oo)/P(0)
*- (3)
HG oi
+
« wn
(3)
• 129n
3/4
f
5/8
(5)
w
13/16
11/16
Beam parameters of the x component of the electric field: wx(0)Wy(0): waist spot size along the x and y axes, respectively, 0x, 0 y : angular divergence along the x and y axes, respectively, Px(oo)/P(0): normalized energy.
140 Table II: y-component of the electric field
mode
wx(0)
Wy(0)
HGoo
V3w
V3w
Mx
M
Py(«>)/P(0)
1/4
1/4
HGoi
V3w0
1/4
3/8
HG
V3w
3/16
3/16
5/16
Beam parameters of the y component of the electric field: Wx(0)wy(0): waist spot size along the x and y axes, respectively, MX, My: M factor along the x and y axes, respectively, Py(°o)/P(0): normalized energy.
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3 Experiments To confirm the analytical results for the fundamental HG beam, we used the laboratory setup illustrated in Fig. 5. From the right-hand side to the left-hand side there are: an Argon-Kripton laser tuned at the A=0.514 jum line on the fundamental Gaussian mode, with 1 mm spot size, / mrad divergence and linear polarization; a neutral density filter to attenuate the beam; an additional polarizer, as the linear polarization is strongly requested; a lens that focuses the beam on the entrance facet of the calcite crystal: the input spot size wg depends on its focal length; a calcite crystal with refractive indices n0=L658 and ne=1.486, dimensions 10x10x20 cm, and the optical axis coincident with the longest axis, along which the beam is made to propagate; a rotating polarizer that can be oriented to transmit the components of the electric field along the x-axis, or y-axis; a lens that, together with the objective mounted on the camera, forms a magnified image of the exit facet of the crystal on the CCD detector (3.3x4.4 mm, 576x768 pixels) of a Sony TV camera; a the polarizer that attenuates the intensity on the CCD detector. A computer is connected to the camera to record and process the images, that are reported in Fig. 1. Starting from the intensity distributions of Ex and Ey acquired with the camera, we measured the power dynamics and the beam variances, that are shown in Fig. 2. From an inspection of these figures, it is evident that the experimental results confirm the theoretical approach.
Figure 5: Experimental setup
4 Conclusions We have investigated the propagation of laser modes along the optical axis of a uniaxially anisotropic crystal: starting from the analytical expressions of the Cartesian components, we have characterized their diffraction properties. All the beam parameters, as the spot sizes and the angular divergences have been computed. The combined effect of anisotropy and diffraction gives rise to a change of the input polarization state and an interaction between Ex and Ey: we have determined the energy exchange dynamics and their saturation values. The analytical results have been contrasted with experimental measurements, showing a good agreement.
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References 1 2
3 4 5 6 7 8
9 10 11
12 13 14
15
16 17
A. Ciattoni, B. Crosignani, and P. Di Porto, "Paraxial vector theory of propagation in uniaxially anisotropic media, " J. Opt. Soc. Am. A, 18, 1656-1661 (2001). A. Ciattoni, G. Cincotti, and C. Palma, "Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals, " Opt. Commun., 195, 55-61 (2001). G. Cincotti, A. Ciattoni, and C. Palma, "Hermite-Gauss beams in uniaxially anisotropic crystals," IEEE J. of Quantum Electron., 37, 12, 1517-1524. (2001). A. Ciattoni, G. Cincotti, C. Palma, "Optical beams in uniaxially anisotropic crystals," Sixth International Workshop on Laser Beam and Optics Characterization, Munich (Germany), (2001). A. Ciattoni, G. Cincotti, and C. Palma, "Propagation ofcylindrically symmetric fields in uniaxial crystals " J. Opt. Soc. Am. A, 19, 792-796 (2002). A. Ciattoni, G. Cincotti, and C. Palma, "Nonparaxial description of reflection and transmission at the interface between an isotropic medium and a uniaxial crystal, " J. Opt. Soc. Am., 19, 7, 1422-1431 (2002). G. Cincotti, A. Ciattoni, and C. Palma, "Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals, "' J. Opt. Soc. Am. A, 19, 9, 1680-1688 (2002) A. Ciattoni, G. Cincotti, C. Palma, and H. Weber, "Energy exchange between the Cartesian components ofaparaxial beam in a uniaxial crystal," J. Opt. Soc. Am., 19,9,1894-1900(2002). G. Cincotti, A. Ciattoni, and C. Palma, "Propagation-invariant beams in anisotropic crystals," J. Mod. Opt., 49, 13, 2267-2272 (2002). D. Provenziani, A. Ciattoni, G. Cincotti, C. Palma, "Diffraction by elliptic and circular apertures in uniaxially anisotropic crystals: theory and experiments, " J. Opt. A Pure Appl. Opt., 4,424-432 (2002). D. Provenziani, A. Ciattoni, G. Cincotti, C. Palma, F. Ravaccia, C. Sapia, "Stokes parameters of a Gaussian beam in a calcite crystal, " Opt. Expr., 10, 15, 699-7062 (2002). G. Cincotti, A. Ciattoni, D. Provenziani, C. Palma, and H. Weber, "Characterization of laser beams in uniaxial crystals," 7th International Workshop on Laser Beam and Optics Characterization (LBOC7), Boulder Co. (USA), (2002). A. Ciattoni, G. Cincotti, D. Provenziani, C. Palma, "Paraxialpropagation along the optical axis of a uniaxial medium," Phys. Rev. E, 66, 036614-1-11, (2002). A. Ciattoni, G. Cincotti, C. Palma, "Paraxial beams in uniaxially anisotropic crystals, " 19th Congress of the International Commission for Optics, Firenze (Italy), (2002). A. Ciattoni, G. Cincotti, C. Palma, "Circularly polarized beams and vortex generation in uniaxial media," Journal of the Optical Society of America A, 20, 1, 163-171 (2003). A. Ciattoni, G. Cincotti, C. Palma, "Angular momentum dynamics of a paraxial beam in a uniaxial crystal," Physical Review E, 67, 036618-1-109 (2003). G. Cincotti, A. Ciattoni, C. Sapia, "Radially and azimuthally polarized vortices in uniaxial crystals, " Optics Communications 220, 1-3, 33 - 40 (2003)
LIGHT BEAM SHAPING: THE INTEGRATION METHOD PAOLO DI LAZZARO, SARAH BOLLANTI, DANIELE MURRA ENEA, C.P. 65, 00044 Frascati (Italy) E-mail: dilazzaro@frascati. enea. it http://www.frascati. enea. it/fis/lac/excimer/index-exc. html
1 Introduction The general light beam shaping problem consists of an optical system which operates upon an input light beam to modify its spatial or temporal shape, producing an output beam having the desired space distribution or time evolution. Let us limit our attention to low-loss shaping techniques acting on the beam space distribution. They may be divided in two categories: field mapping and beam integration. Field mapping techniques include optical systems operating on both phase and amplitude of the input beam to produce an appropriate intensity distribution on the output plane1. Beam integration methods mix fractions of the original beam to smooth out intensity spikes. Usually, field-mapping techniques are successfully applied to honest beams, having well defined modes. On the contrary, beam integration is convenient when the beam is partially coherent and its intensity distribution is irregular. As the authors have a fatal attraction to bad laser beams, only beam integration methods will be discussed in the following.
2 Motivation A large number of laser applications need a strict control of the beam profile. Many of them require beam coverage uniformly extended across target areas, and it is necessary using high-quality homogenized profiles, i.e. top-hat spatial energy distributions with steep edges and high uniformity in the plateau region. Typical examples include laserbased material processing (e.g., surface cleaning, ablation, marking, drilling, metal hardening), thin film transistors for flat panel displays, chip-making, medical applications (e.g., corneal reshaping, cosmetic surgery, UV-curing dermatology diseases). Unfortunately, none of the high-power laser systems available in the market emit beams with flat-top, uniform intensity profiles. As a consequence, different beam integration techniques for homogenizing laser radiation are employed. 3 Design elements Figure 1 illustrates the principle of a typical homogenizer along one transverse axis. An array of cylindrical lenses (the "divider") breaks the incident beam into secondary beamlets. A "condenser" lens provides the overlap of the beamlets on the focal plane of the homogenizer. The addition of different portions of the initial (non-uniform) beam smoothes the intensity profile fluctuations, finally creating a flat-top beam. The averaging
143
144
effect is proportional to the number of the secondary beamlets. That is, the larger the number of the divider lenses, the more uniform the beam in the focal plane of Fig. 1. Focal plane
Figure 1. 2-D scheme of a conventional homogeniser. The beam travels from left to right.
A simple exercise shows that the size D of the homogenized beam in the focal plane of the optical system of Fig. 1 is given by:
(D
J D
where S = size of a single divider lens; fD = focal length of each divider lens; fc = focal length of the condenser. Note that D is independent of the distance between divider and condenser. To optimize the homogeniser performance, we have to take into account interference and diffraction effects that can modulate the homogenized intensity shape, thus frustrating the averaging effect of the homogenizer. Let us discuss these issues. Interference: if beamlets exiting by adjacent lenses of the divider are mutually coherent, the output intensity will be dominated by speckles (high-contrast interference patterns). The period P of this interference pattern, in the simplest case of rectangular lenses (1-D case) is P ~ Kxfc/S and its impact is minimized decreasing the value of the ratio fc/S. However, this cannot be directly achieved by increasing 5, as this would obviously lead to a worse averaging effect of the homogenizer. Diffraction: diffraction effects arise from the boundaries of the divider lenses, and create an additional pattern having the same period P above and a modulation depth in the homogenized intensity shape. The modulation depth is minimized by a sufficiently large (> 100) equivalent Fresnel number1 = S2 /(4Xx/D) = SxD/(4Kxfc),
(2)
where A. is the laser wavelength. Again, it is not convenient increasing P by augmenting S, to preserve the essential averaging effect of the homogenizer. When reducing to practice the above considerations, the size S of the divider lenses should be small enough to guarantee an effective averaging of the input beam intensity profile fluctuations, but larger than the transverse coherence length of the input beam to avoid speckles caused by interference. Moreover, the choice of S affects also the equivalent Fresnel number, see Eq. (2). As a consequence, the designer makes a tradeoff between S and P and this usually yields a homogenizer able to integrate only the average fluctuations of the input intensity. Then, relatively poor results are expected in the case of light beams having local intensity fluctuations stronger that the average fluctuations (e.g., beams with asymmetric non-uniformity profiles emitted by discharge-pumped lasers, like
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excimer lasers, copper vapor, TEA CO2 lasers, and beams emitted by diode laser arrays). This is one of the two main inconveniences of conventional homogenizers. The second one is evident from Eq. (1), which shows that, once fixed the geometry and lens power, the energy density on the focal plane can be changed only by changing the energy of the input beam. Sometimes it is not convenient changing the output energy of a laser, either because this may reduce the efficiency and stability of the laser emission, either because this may be hardly done continuously. This means those irradiation processes requiring different energy density values and/or different spot sizes need distinct homogenizer systems. A possibility to overcome the above limitations is given by a recently proposed homogenizer with zoom and asymmetric divider arrays2. This patent pending ENEA homogenizer on the one hand can continuously modulate the spot size (and then the energy density) of the homogenized beam along one or both axes in a fixed target plane; on the other hand, it is able to make homogeneous any beam, including beams having strong local intensity spikes.
4 Advanced design Figure 2 illustrates the principle of the ENEA homogenizer. The two main differences with respect to the scheme of Fig. 1 are, respectively, 1) the asymmetric divider (the more homogeneous part of the beam impinges on the larger size lenses of the divider array, and vice versa) and 2) the zoom lens added after the condenser lens. In spite of their different size, all the secondary beamlets have the same dimension on the focal plane, provided that a constant ratio Si/fa is maintained, where S{ is the size of the i-th lens of the divider and /Di its focal length. The zoom element can be a spherical lens or a couple of cylindrical lenses (according to the condenser lens). In the last case, it is possible to independently change the spot size along the two orthogonal axes.
Intensity profile
Focal plane
Figure 2. 2-D scheme of the ENEA homogeniser. When the beam has an asymmetric profile, the divider array is made by different lenses, such that the smaller lenses intercept the less homogeneous part of the beam, and that each lens size is much larger than the local coherence length of the corresponding portion of the beam.
The size D of the homogenized beam in the focal plane of the optical system of Fig. 2 is given by:
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fc*fz (3) fz+fc~dcz where fz = focal length of the zoom lens; dcz = distance between condenser and zoom. The following equation gives the total length of the homogenizer, from the divider to the focal plane: _,_, , z = d, DC+d cz+-r- ----—,,
(4)
where d^c is the distance between the divider and the condenser. From Eq. (3) it is clear that changing the distance dcz we change the size of the homogenized beam in the focal plane and, from Eq. (4), that a corresponding proper adjustment ofdDC allows keeping fixed the position of the focal plane of the homogenizer (i.e., of the target plane) with respect to the divider and to the laser system. In short, the ENEA beam integrator allows to make homogeneous beams having asymmetric intensity spikes, and to vary the spot size of the homogenized beams (which can be either a square or a rectangle with a fixed or a variable aspect ratio) without changing any optical element and keeping constant the optical path length from the laser to the target plane.
5
Software
In order to fully exploit this homogenizer technology, ENEA developed a proprietary software, useful to design the best optical system to achieve the wished output beam performance (once known the input beam characteristics), and to know how the output beam shape and size change when changing position of each optical element. This software is written in Visual C++ and runs on MS Windows 95/98. It is user friendly, thanks to a properly designed user interface. In particular, this software can: *J* give the optimum choice of the optics parameters once the boundary conditions are known (input laser beam characteristics, mechanical constraints, desired beam shape and desired output beam size); •J» simulate the homogenizer performance by inserting the input beam profile data and by carrying out the ray-tracing of the light beam passing through the optics. The optical elements can be simulated according to the paraxial optics (focal length) or by physical parameters (curvature radius, thickness, refraction index); * show the profile of the beam on any transverse plane along the homogenizer path, including the focal plane; *!* calculate meaningful beam parameters, like the baseline size, the plateau size, the average and peak intensity, the steepness, the plateau fluctuations root mean square (RMS), the threshold uniformity; * give the position of the divider, condenser and zoom elements when a particular final size is wanted, leaving the distance laser - target unaltered.
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6
Experiment
Up to date (November 2002) the ENEA homogenize! has been successfully applied to a number of lasers, namely excimers, HeNe, diode. Let us present the results achieved with the most challenging attempt, made by homogenizing the large-size, high-energy beam (10 cm x 5 cm, 8 J) emitted by the XeCl (A, = 308 nm) excimer laser-facility Hercules4'5. Hercules Near Field - Horizontal
Hercules Near Field - Vertical 7000 6000 H 5000 o 4000 Q 3000 y 2CKH) 1000
0
Figure 3. Typical cross-sections of the near-field profile of the XeCl laser Hercules, 0.5 m behind the laser housing.
Figure 3 shows the typical near field profiles of the laser beam emitted by Hercules. The horizontal profile (along the discharge electric field) is asymmetric because of typical discharge non-uniformity (streamers6) near the grounded electrode. Here and in the following, the beam profiles are measured by using the UV-sensitive CCD camera Andor model DV438-BV. This back-illuminated camera7 has an active sensor area of (1.73 x 2.59) cm2, and the image is digitized in 770 x 1152 pixels. To avoid saturation of pixels, we attenuated the laser beam down to an energy density of 5 x 10"8 J/cm2. The attenuation factor was maintained constant during all the measurement runs. In the case of laser cross sections larger than the CCD sensor size (like in Fig. 3), the overall image was obtained adding images captured in consecutive shots, by rastering the CCD with the help of a step-motor-driven slide (Physik Instrumente, model M521-DD) moved along the cross section profile. We have designed and tested a homogenizer made by two crossed arrays of 4+4 cylindrical lenses as divider, by a condenser along the horizontal direction and by a condenser and a zoom along the vertical one. The transmission factor @ K = 308 nm of this homogenizer is 80%. The subaperture sizes and the focal lengths are: 5H= 2.5 cm, /HD= 22cm;/HC= 120 cm, Sv = 1.25 cm, fVD= 50cm, fvc= 44 cm,fvz.= 21 cm. Here the subscripts H and V mean horizontal and vertical, respectively. By substituting these values into Equation (3) we obtain a fixed 14-cm spot along the horizontal direction, while, along the vertical one, the homogenized beam size is variable from 0.6 cm up to about 7 cm, with a maximum zoom factor larger than 11. Equation (4) gives an homogenizer length dDC + dcz « 1.7 m, while the software provides the suitable distances between dividers, condensers and zoom to keep fixed L (see Eq. (4)) as well as the distance between the laser housing and the focal plane of the homogeniser. Equation (2) gives the equivalent Fresnel numbers along the horizontal (0H = 2200) and vertical (Pv « 4200) axes. Thanks to the short A., both p values are large enough to make negligible the intensity modulation depth due to diffraction effects. Interference effects can be neglected as well, because the period of the pattern (P » 15 ^m) is very small and the associated
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modulation depth is close to zero, being SH and Sy respectively 25 times and 10 times larger than the corresponding transverse coherence lengths of Hercules.
7
Results
Figures 4 and 5 show the vertical cross-sections of the homogenized beams with the smallest and the largest measured size, respectively. The horizontal cross-section of the homogenized beams is shown in Fig. 6: the horizontal length of the beam spots is the same for all the vertical zoom factors. 6 mm-wide Homogenised Spot
66 nun-wide Homogenised Spot
30000 25000 a 20000
s
U 15000
y 10000 5000 0-
)
5 1 Bottom I
10
15
pftp]
nun
Figure 4 Typical cross-section of the Hercules laser beam in the focal plane of the homogeniser along the vertical axis, zoomed from 50mm down to 6mm.
Figure 5 Typical cross-section of the Hercules laser beam in the focal plane of the homogeniser along the vertical axis, zoomed from 50mm up to 66mm.
Homogenised Profile-Horizontal Fig. 6 Typical cross-section of the Hercules laser beam in the focal plane of the homogeniser along the horizontal axis, resized from 100mm to 140mm.
H.V side
A comparison of Figs. 4, 5 and 6 with Fig. 3 shows that the homogenizer made the beam shape very smooth and with steeper edges. A substantial reduction of the high-spatialfrequency intensity fluctuations on the plateau is evident comparing Fig.3 and Fig. 6. The apparent noise increment in the plateau of Fig. 5 with respect to Fig. 4 is a device-related effect, due to the different laser energy density impinging on the CCD in these two cases (note the different vertical scales), which yields a worse signal-to-noise ratio in Fig. 5.
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8
Characterization
There are two important questions one can rise: how "good" are the homogenized beam profiles shown in Figures 4 to 6? How it can be made quantitative the improvement in terms of beam flatness and steepness between the input raw beam (Fig. 3) and the homogenized beam (Figs. 4, 5 and 6)? In order to characterize the quality of homogenized laser beams, the ISO standard 13694 proposes two parameters, respectively named "edge steepness" and "plateau uniformity"8' 9 . The edge steepness is defined as '10%, 90%
(5)
= (A10% - A90%) / A 10%
where AK>% (A90%) is the effective irradiation area at 10% (90%) of the maximum energy density Hmax. The plateau uniformity is defined as
(6)
= AH/Hn
where AH is the full width at half maximum of the peak in the energy density histogram curve N(Hi). The energy density histogram curve is the number of data points plotted as a function of the corresponding energy density H;. As an example, Figure 7 shows the energy density histogram curve of the cross section profile of Fig. 4. The width AH of the high-energy-density peak approaches zero for ideally flat distributions.
Figure 7. Histogram of the homogenised laser beam cross section shown in Fig. 4. Vertical: number of pixels. Horizontal: energy density (% of
40
Hjnax)-
AH is the full width at half maximum of the peak just below the maximum energy density Hmax.
I
0.4 0.6 Energy density (% of Hmax)
0.8
Specializing Equations (5) and (6) to our experimental data we can see, for example, that along the horizontal axis the steepness decreased from 47% (raw beam) to 11.3% (homogenized beam), and the plateau uniformity decreased from 13.5%, to 5.5% (see Fig. 6). Concerning the vertical axis, the results are summarized in the Figures 8 and 9.
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Figure 8. Comparison of the steepness (see Eq. (5)) along the vertical axis between the raw beam (without) and the homogenized beam zoomed in to 6 mm, 7 mm and 13 mm, zoomed out to 66 mm and reshaped to 50 mm.
Vertical axis: steepness
without
6mm
7mm
13mm
50mm
66mm
V e r t i c a l axis: p l a t e a u u n i f o r m i t y
without
6mm
7mm
13mm
50mm
Figure 9. Comparison of the plateau uniformity (see Eq. (6)) along the vertical axis between the raw beam (without) and the homogenized beam zoomed in to 6 mm, 7 mm and 13 mm, zoomed out to 66 mm and reshaped to 50 mm. 66mm
Some trends can be identified from these data. For example, the edge steepness improves when the beam is zoomed to larger spots (see Fig. 8). This effect is due to a not perfect quality of the zoom lens, which is aberrated when working with large numerical apertures. The plateau uniformity of the homogenized beams clearly improves with respect to the one of the input beam, decreasing by a factor between 1.5 and 3 (see Fig. 9). This oscillation of the improvement factor does not exhibit a clear dependence on the zoom factor. It is probably related to a not perfect alignment, which was made difficult by the very strong attenuation factor of the Hercules laser beam to avoid saturation of the CCD camera (see Sect. Experimental). 9 Applications The laser sources equipped to date with the ENEA homogenizer were successfully used by different customers to a number of laser material processing applications. They include the recrystallization of amorphous silicon films, cleaning semifinished aluminum plates, the selective removal of paints and graffiti without damaging different substrates (including glass, weather-strip rubber, marble, and varnished metal). Some illustrative images of irradiated samples are reported in Fig. 10.
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Figure 10: Examples of material processing done by laser beams reshaped by the ENEA homogenizer. a) SEM image of recrystallized silicon film; b) photos of graffiti stripped without damaging the glass (top) and white marble (bottom) substrates; c) SEM images of a semifinished aluminum plate before (top) and after (bottom) laser irradiation. The homogeneous irradiating beam removed the dirty layers, without affecting the original surface structure: note the sub-micrometer scratches and digs on the cleaned sample do not exhibit any material ablation or melting effects.
10 Ut breviter dicam Beam shaping is the process of redistributing the irradiance and phase of a beam of optical radiation. A correct beam shaping is an essential requisite to optimize a large number of laser-based materials processing. We have discussed in some detail the beam integration method, especially suitable to reshape partially coherent beams having an irregular intensity distribution. In particular, we have presented the design elements, the experimental performance and the main characterization parameters of a new beam integrator homogenizer technology (patent pending3), able to homogenize and reshape bad light beams, including those having asymmetric-spike profiles. In particular, the new homogenizer is useful in the following cases: 1) Uniform irradiation processes where the optimum energy density and/or spot size of the irradiation (illumination) process are not known in advance. Operators can vary the output spot size and/or the energy density of the homogenized beam until the optimal working point is reached - all without making the expensive and time-consuming equipment adjustments (e.g., change of optical elements, change of the homogenizer length, change of distance laser-target) necessary with existing beam-handling systems. 2) Homogenization of light beams having local intensity spikes stronger than the average intensity fluctuations, like, e.g., beams with asymmetric non-homogeneity profiles (discharge-pumped lasers, diode laser arrays). The homogenizer prototype presented in this note has been designed using a proprietary software and experimented on the laser beam emitted by the XeCl (k = 308 nm) excimer laser facility Hercules: it was tested up to a zoom factor 11 along the vertical direction. The cross-sections of the homogenized beams were characterized in terms of edge steepness and plateau uniformity.
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Thanks to the licensing agreement between ENEA and Info & Tech S.p.A.10, the ENEA's homogenize! technology is commercially available. At the moment, our transfocal homogenizers have been successfully used in several laser applications, including recrystallization of amorphous silicon films, selective paints and graffiti stripping from several substrates (glass, rubber weather-strip, marble and painted metal), and cleaning of semifinished aluminum plates.
References 1
F. M. Dickey and S. C. Holswade, "Laser beam shaping: theory and techniques" (M. Dekker, Inc. New York Basel, 2000). 2 P. Di Lazzaro, S. Bollanti, D. Murra, G. Felici: "A novel light beam homogeniser" Proc. Int. Conf. on Gas Flow and Chemical Lasers, High Power Lasers (Wrocklaw, August 2002), SPIE, to be published. 3 ENEA patent pending n° IT RM 000229 (UIBM, April 28, 2000); n° 09/727.268 (US PTO, November 30, 2000); n° 00830807.4-2210 (EPO, 6 December 2000). 4 T. Letardi, S. Bollanti, P. Di Lazzaro, F. Flora, N. Lisi, C.E. Zheng: "Some design limitations for large-aperture high-energy per pulse excimer lasers", II Nuovo Cimento D 14, 495 (1992). 5 P. Di Lazzaro: "Hercules, an XeCl laser facility for high-intensity irradiation experiments" 2nd GR-I Int. Conf. on New Lasers, Technologies and Applications, Proc. SPIE vol. 3423, pp. 35 - 43, A. Carabelas, P. Di Lazzaro, A. Torre, G. Baldacchini Eds. (1998). 6 S. Bollanti, P. Di Lazzaro, F. Flora, T. Letardi, N. Lisi, C.E. Zheng: "Space- and time-resolved discharge evolution of a large volume X-ray triggered XeCl laser system", Appl. Phys. B 55, 84 (1992). 7 Andor Technology Limited, 9 Millennium Way, BT12 7AL Belfast Northern Ireland. Web site: http://www.andor-tech.com 8 K. Mann, J. Ohlenbusch, V. Westphal: "Characterization of excimer laser beam parameters" Third Int. Workshop on LBOC, Proc SPIE vol. 2870 pp. 367 - 377, M. Morin, A. Giesen Eds. (1996). 9 ISO/DIS 13694: "Test method for laser beam parameters: Power (energy) density distribution", May 1999. 10 Info&Tech S.p.A. Via Teognide 24, 00124 Roma (Italy). E-mail: [email protected]
BEAM PROPAGATION IN QUADRATIC MEDIA GIUSEPPE LEO NOOEL, INFM & Dept. of Electronic Engineering, University "Roma Tre" Via delta Vasca Navale 84, 00146 Roma, Italy E-mail: [email protected] This lecture is intended to provide a general introduction to quadratic nonlinear optics, along with a view on a few selected research topics of present interest. In Section 1 I will recall the quadratic effects arising from non-resonant light-matter interactions, in the absence of absorption and carriers. Specifically, 1 will illustrate the concept of phase matching and overview the related techniques, with emphasis on guided-wave interactions. In the last decade, novel parametric sources have been conceived and fabricated in both bulk and integrated geometries, and new phenomena and applications have been demonstrated in the domain of soliton propagation. In Section 2, on the side of novel integrated optical sources, I will focus on the recent progress towards a monolithic semiconductor OPO. In Section 3, in the framework of the phase effects associated to the cascading of %(2), I will deal with quadratic spatial optical solitons, i.e. free-propagating optical-field configurations where diffraction is balanced by nonlinear self-focusing.
1.
Introduction
As compared to other nonlinear sciences, nonlinear optics (NLO) is relatively young, since it only started after the demonstration of the laser in 1960. The first observed NLO effect, second harmonic generation (SHG) [1], was a quadratic one. Quadratic nonlinearities stem from the material-dependent valence-electron anharmonic oscillator and occur in non-centrosymmetric molecules or crystals [2]. Far from resonances, the related interactions rely upon virtual transitions, and are intrinsically ultra-fast since they do not depend on material lifetimes (like electron recombination). In the case of a spatially and temporally local medium response, the electric polarization associated to an incident field E takes on the general form
P(r.f) = £ 0 x (l) (r.f )• E(r,Q +£ oX (z) (r.f ) ; E(iy)2 + £0x'3) (r.f ).:E(r,r)3 JIG
p(3)
where EQ is the vacuum permittivity, and j£(1) the linear susceptibility tensor; %(l) is a rank(i+1) tensor defining the ith-order susceptibility, with its structure and components determined by the type of nonlinearity and crystal symmetry. The linear term P(L) is adequate to describe the response to weak natural light, while further terms, involving increasing powers of E, come into play for higher (typically laser) powers. In general the electric field consists in the superposition of n (quasi)monochromatic waves E(r,f) = £E,.(r,0 1=1 where c.c. stands for complex conjugate, and the complex amplitude E.(r,o>.) is the Fourier component with frequency (Oj and wavevector £,=
154
amplitude is given by Ei(x,y,z)=Ej(z). In general, however, it is expressed as Ej(x,y,z)= fi(x,y)ai(z), i.e. as the product of the slowly-varying field amplitude a,fzj [V] and the twodimensional (2D) field profile fi(x,y) [ml] in the transverse plane (the latter is normalized such that
For a non-magnetic medium, in the absence of charges and currents, the electric-field propagation is described by the wave equation
2
_ ec 2
2
c 3r
=J_-L^E-+J-J^l
(2)
where c is the light velocity in vacuum, e is the dielectric permittivity, and the nonlinear polarization acts as driving force. If P(NL) is a small perturbation relative to P
Let us focus now on x<2> effects, neglecting higher-order terms. We can take advantage of the linearity of the wave equation, by introducing in Eq. 2 the above Fourier expansions of E(r,t) and P(NL)(r,t)=P(2)(r,t). If E contains two fields oscillating at GJ, and 6)2, P(2) will consist of waves oscillating at co$=2coh 20% (SHG); co3=(D1±a)2, (sum- and difference-frequency generation: SFG, DFG); 0)3=0 (optical rectification). For SFG and DFG (see Figure 1), the rth amplitude component of the nonlinear polarization is complex valued and can be expressed as: SFG:
Pm(a)3 =0)l+mt) = (E0D/2j£Jk X$(
-co,;co,,w2 )EJ(atl )E,(co2 )
DFG: pW(o,3 = a,, - «2 j = M/2)£ *<2)( ~ a>3 ; «, -«2 )E
(3)
)E'k(co2 )
(4)
with the degeneracy factor D=l(2) for indistinguishable (distinguishable) input fields. While the quadratic susceptibility is often expressed in terms of d-tensor elements dik =Xrk/^' Eq. 3 and Eq. 4 are usually simplified thanks to crystal symmetries and nonresonant operation [3]. In order to review the fundamental concepts of quadratic interactions, let us address Type-I SHG: i.e. SFG with frequency- and polarization-degenerate pump inputs at the fundamental frequency (FF) ®/=c%=ft>, and second harmonic (SH) output at
exp[i(2fl» - k2(a z)]}+ c.c. .
The relevant EjE^ terms in Eqs. 3, 4 induce the polarization terms P®(z,co)=2e0^jkdijk(
-to;2aa,-€o)E'J(z,to)Et(z,2to)exp[-i(kIa
Pt(2) (z,2co)= e0 ^jt dljk< - 2w; to, co)EJ (z,co)Ek(z, co) exp[- 2»Bz]
-ka)z]
155 CO,=03
SFG
SHG
DFG
Figure 1 Schemes for SHG, SFG, DFG.
- 1 0 -
5 0 AkL/2
5 z/L
(a)
(b)
Figure 2 a) Normalized phase-matchinging curve, b) SHG conversion efficiency along propagation.
to be introduced into Eq. 2. Within Kleinman symmetry and slowly-varying envelope approximation, this leads to the coupled mode equations for FF and SH fields:
\dEa(z)/dz = \dE2m(z}/dz = -i
(6)
where Em(z)=E(z,co), Ak=k2a,-2km and deff is the nonlinear effective coefficient, given by the projection of the field eigenvectors onto the the d tensor. In the nondepleted-pump approximation (dE^/dz^O), only the second equation need be integrated. After a distance z=L the SH power, given by P2w = (A/2)cE0n2 E2J\ with A the cross-section area, is generated with the efficiency 1 = P2. (L)/p» (°) = % sine2 (MI/2) where sincC^=sin(3c,)A, % = C2 Z,2 POT(o)/ A , and c2 =8n2deff2 /n^ce
(7)
(with
the
pump wavelength and £Pm the pump power). For an efficient SHG, two conditions are required: 1) Ak&O, i.e. wave-vector conservation, or phase matching (PM); 2) high phasematched efficiency r)0. For the latter condition to occur, we need: a material with high nonlinearity (r]0 <x de/); an intense pump beam (r;0 x ^(OJ/A ); a sufficiently long propagation distance (TJO <xL2). PM is necessary because the ratio of quadratic to linear polarization, being on the order of the ratio of the applied electric field to the interatomic field [3], is extremely low even for an intense pump. Therefore, since nonlinear dipoles only radiate very weakly, any appreciable harmonic generation must result from the construcive interference of
156
photons emitted by a large number of such dipoles. The phased-array character of the nonlinearly-radiating dipoles along z is expressed by the sine2 feature of Eq. 7 (see Figure 2a). Due to normal dispersion, the pump-induced nonlinear polarization at 2a> generally experiences a larger phase velocity than the field at 2a> (Ak^O), and this periodically brings out of phase the coherent contributions to the SHG at different stages of propagation. When the dephasing between such contributions reaches the value of n, power starts to flow back from SH to FF field, giving rise to the oscillatory evolution of Figure 2b. It is common to express this behaviour in terms of the coherence length Lc = 7T/AA: — -^(B/4(n2(U — M m ), corresponding to the distance after which FF and SH field get to phase opposition (half a cycle in the oscillatory evolution of Figure 2b). In Type-I SHG, wave-vector conservation implies that nlo=n2a>. The most common way to fulfill this condition is to use two different FF and SH polarizations in a birefringent nonlinear medium, by exploiting the angular dependence of the extraordinary index. An alternative SHG phase-matching scheme requires two orthogonally polarized pump fields, with Ak=k3(2a>)-kl(a>)-k2((o), and is known as Type-II SHG. In this scheme the condition to fulfill is nj=(ni+n2)/2. Besides angle tuning, either type of birefringent phase matching (BPM) can be achieved through the extra degree of freedom provided by the temperature dependence of material dispersion curves or, if the SHG wavelength is not a tight constraint, by the use of a tunable pump source. Among the most relevant rf2> devices there are optical parametric amplifiers (OPA), optical parametric generators (OPG), and optical parametric oscillators (OPO) [4]. This class of devices were invented in the early days of nonlinear optics [5]. Until the mid-80's, however, they were only considered as research setups, as dye lasers were largely preferred as coherent tunable sources. In the last fifteen years, mostly due to progress in nonlinear materials, research on OPA's and OPO's was greatly boosted. Today large segments of the related technology are at a maturing stage, for the most diverse industrial, militar, environmental and research applications, with new commercial products being frequently launched on the market. Furthermore, parametric devices are used in fundamental research, as valuable sources of squeezed light and twin photons, and to investigate quantum systems. OPA is associated to a DFG process (see Figure 1) where: the intense, higher-frequency input at a>i acts as the "pump" (p), the lower-frequency input at a>2 is the "signal" (s), and the DFG output at a>3 is named the "idler" (i). Besides the DFG at (o^Wp-cOs, the input signal is amplified through the parametric interaction. At PM, for weak parametric gain g, neglecting losses, the signal output power is readily found as Ps(z,)Kps(o)(l + g 2 L 2 ) [2]. In OPG an incident pump photon splits into two lower-energy, idler and signal photons, fulfilling energy conservation. The output frequencies are selected by momentum conservation (PM condition), with possible angle-, temperature-, or pump-wavelength tuning. This spontaneous "decay" of the pump photon, formerly studied in 1961 [6] and also known as Parametric Fluorescence (PF), is a quantum phenomenon associated to zero-point fluctuations in signal-idler inputs. These fluctuations, interacting with the pump field via the %(2), locally induce a nonlinear polarization which radiates in all the possible directions, and at all the possible frequencies between 0 and cq, . However, output radiation only grows for those frequencies and directions which, being close to the PM condition, experience constructive interference of the nonlinearly-radiating dipoles. The OPO is an OPA within a cavity, and it can operate in either cw or pulsed regime. In the most common configurations, the cavity mirrors are transparent at (q,, and highly reflecting at en, and/or
157
or (Of . In the presence of a pump beam, quantum noise triggers PF which, upon repeated reflections on the cavity mirrors, is parametrically amplified at subsequent passes through the nonlinear crystal. An analogy is apparent between the OPO and the laser, with the roles of spontaneous (stimulated) emission played by PF (OPA). Relative to the laser, however, the OPO exhibits a few major differences: two coherent beams are emitted at different wavelengths, which can be tuned by varying crystal angle, temperature, or pump frequency; parametric amplification only occurs as pump, signal and idler co-propagate, implying that the oscillation threshold occurs when single-pass gain g equals the roundtrip losses (as in a ring-cavity laser). Efficient frequency generation requires high intensities and long interaction lengths. For a pump laser beam with a given power, the former requirement implies tight focusing (at the price of large diffraction and small effective interaction length), whereas the latter implies loose focusing (long Rayleigh range). These conflicting requirements are simultaneously satisfied in optical waveguides, where light is confined in a highrefractive-index channel surrounded by a low-refractive-index medium [7]. In such waveguides Maxwell equations result in an eigenvalue problem, with two discrete sets of solutions (fim „} representing the propagation constants along z of two orthogonal sets of guided modes. In each set of modes there is a prevailing polarization component of the electric field, along either the horizontal transverse axis (y : TEmn) or the vertical transverse axis (x : TMmn). The transverse distributions of the guided modes are stationary waves described byfmn(x,y), with m (n) zeroes along x (y). Through the same procedure leading to Eq. 7, guided-wave SHG efficiency is found to be 77 =J]_^(0)L2 sine2 (Aj8L/2), with ^ =7]0/Pffl(o)L2 =C2/Aeff [W'cm2], and the effective area
* = \L K (*> For Type-I SHG, the mismatch Afts^mn(2a))-^m'n{o}) can be expressed in terms of effective indices N=fi/k0 as A/3=ko((a)[Nmn(2(o)-Nm'n'(a])], and the corresponding PM condition is Nmn(2co)=Nm'n'((o). Since N is higher for stronger confinement, and usually waveguides are much more confining in the vertical direction than in the horizontal direction, in general its approximate calculation can be carried out referring to the corresponding planar waveguide. The corresponding one-dimensional (ID) problem is easier to handle: eigenvalues and eigenvectors are identified by one index m, the guided modes have pure transverse-electric (TE) or transverse-magnetic (TM) polarization, and their profiles are as in Figure 3, with m zero crossings along x axis. As the thickness-towavelength ratio h/A. increases, guided modes get more confined, and higher-order modes become guided. This is an effect of modal dispersion, which is also exemplified in Figure 3, for a GaAs waveguide. In principle, PM in waveguides can be pursued with the same techniques that prove useful for bulk media. However, once a waveguide is fabricated, BPM is constrained by the substrate cut and the channel orientation. This often results in waveguides cut for noncritical PM (0=90°), with the consequent need of a temperature control. With respect to bulk, moreover, waveguides offer an additional PM option, which stems from their multimode dispersion. Since e.g. the PM condition for Type-I SHG in (planar) waveguides
158
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
0.1
3.0
0.3
0.5
0.7
0.9
1.1
1.3
1.5
GaAs thickness h (urn)
(a)
(b)
Figure 3 TE modal profiles (a) and dispersion (b) in an air / (1 .5 \un) GaAs / Alo.3Gao.7As planar waveguide. 3.5
-n,(2co)
3.3
£ «L> (D
S
3.2 3 1
-
3.0 n,(2w)
2.9 ns(
2.8 2hkQ(co),
hk 0 (2co)
Figure 4 Two cases of MPM in an air/GaAs/AlAs planar waveguide: a) TEo((o)—*TEi(2a)) for GaAs thickness h=0.5 urn, and b) TEo((0)^>TE2(2
is Nm(2co)-Nm'(ca), matching of the two effective indices can also be fulfilled for FF and SH fields having the same polarization. Two cases of modal phase matching (MPM) are shown in Figure 4, for SHG from 2.128 |j,m to 1.064 urn in a GaAs slab on AlAs substrate. In general, the price to pay for the ease of phase-matchability with MPM is the low conversion efficiency associated to the small nonlinear overlap
between modes of different order. For modes of different parity, the overlap is exactly zero in a symmetric slab waveguide with uniform deff, unless some symmetry-breaking strategy is adopted. Solutions reported to date entail some schemes of de/x) modulation, with an odd parity relative to the symmetry plane of the slab [8]. Such schemes are compatible with both polymer and semiconductor technologies, which allow to deposit thin homogeneous layers and to synthesize a "digital" odd profile of de^x). Another PM scheme for guided-wave SHG is Cherenkov PM, which consists in harmonic generation into a radiation mode: the nonlinear polarization induced by a guided mode with effective index NO, radiates an SH field into the substrate. The latter, in order to cumulatively grow along the waveguide, should keep in phase with subsequent contributions from the propagating guided pump, which only occurs under the condition Naf=nSi2e> cosft Since the substrate modes form a continuous spectrum, this condition is
159
"automatically" fulfilled, e.g. in the shaded regions of Figure 4a. Firstly demonstrated in the early 70's, [9] Cherenkov PM lost much of its interest with the advent of quasi phase matching, due to its limited efficiency and difficult collection of its far field. Quasi phase matching (QPM) is a powerful technique, which allows roomtemperature phase matching between guided modes, irrespective of their modal order and polarization. QPM therefore allows to select not only the waveguide cut that maximizes deff, but also the pair of interacting modes with largest overlap integral: TEoo((o) and TEgo(2o)). QPM was suggested in 1962 and relies on a grating along the propagation direction z [10]. Such grating consists in a longitudinal modulation with period A on either the effective index or the effective nonlinearity, the latter option being largely preferred due to lower scattering losses. In essence, the mth harmonic of the grating provides the quadratic interaction with an additional momentum Km=2mn/k (m = l,2,...), thus allowing QPM as Ap-Km=0 if the grating period fulfills the condition A=2mLc. QPM (0/1)
x(2>
0
x(2)
0
x<2> 0
x(2)
0
A=2LC
QPM (-
A=2LC
x(2) -X' ' 2
X'2'
-x'2>
X'2'
-x<2'
X'2'
-x(2)
Figure 5 QPM rectangular x<2) gratings, with (-1/+1) and (0/1) longitudinal modulation.
Although first-order (m=l) QPM is more efficient, higher-order QPM becomes necessary if the grating fabrication technique cannot provide sufficiently short grating periods. In the most common case, i.e. rf2> rectangular grating with 50% duty cycle and +deg values, dm — 2deff /inn (with odd m). This case is described in Figure 5, where it is compared to the case of a rectangular grating with (0/1) modulation. Historically, QPM has become viable only three decades after its prediction, as it was firstly implemented in ferroelectric LiNbC>3 and KTP waveguides [11,12], through shallow periodic reversal of their ferroelectric domains, based on selective dopant indiffusion. Today, after the advent of electric-field-induced ferroelectric poling one decade ago [13], deep domain inversion can be achieved, thus creating truly new bulk materials like PPLN (periodically-poled lithium niobate). Ferroelectric poling has allowed to extend QPM to bulk media, with impressive performances in both high-power frequency generation and OPO's [14,15]. Finally, QPM has also been demonstrated in polymer [16] and semiconductor [17,18] waveguides. In conclusion, high performances can only be obtained for highly-nonlinear and phase-matchable materials, although several other issues are of practical importance, like a noncritical phase-matchability, a sufficient birefringence for generating over a wide range of frequencies, small spatial and temporal walkoff, a small group-velocity dispersion. In sum, no inorganic, organic, or semiconductor material has been found yet, which is capable to simultaneously satisfy the major desiderata. To date, the most common PM approach is still BPM, with the related wavelengths and temperatures set by fortuitous coincidences in material parameters. A microelectronics-like scenario, where Silicon growth/processing know-how allows to tailor a large amount of functional devices, is clearly lacking. Therefore today, on the way towards a full optoelectronic integration, NLO in direct-bandgap semiconductors is of great interest.
160
2.
Integrated Semiconductor Parametric Sources
Up to two decades ago, nonlinear optics in semiconductors was mostly studied with reference to nonlinear absorption, encompassing phenomena such as two-photon and freecarrier absorption, saturable absorption and gain [3]. Although Kramers-Kronig relations imply refractive counterparts for all these effects, nonlinear phase control through real transitions is inherently slower than virtual-transition-based effects. This also applies to all-optical signal processing through inter-band and inter-subband transitions in quantum confined structures. However, virtual transitions and the related bound-electron nonlinear effects also occur in bulk semiconductors, and cubic compounds such as GaAs have attracted considerable attention due to their strong polarity and the correspondingly high ^2). In spite of its high quadratic susceptibility with respect to LiNbOs, until one decade ago GaAs was not used for parametric generation, for two reasons: it is not phase-matchable through BPM, and its lack of ferro-electricity does not allow periodic poling. Recently, benefitting from the mature technologies of molecular-beam-epitaxy and lithography, design and fabrication of semiconductor waveguides and photonic structures have reached a great flexibility. In the framework of semiconductor engineering for NLO, several approaches have been proposed, which can be grouped in three classes, according to the adopted strategy of tailoring: 1) the periodic domain reversal in bulk structures, 2) the nonlinearity in quantum-confined structures, 3) the material birefringence. The first two classes are QPM-oriented, and the third one is BPM-oriented. 1) Historically, the first QPM in GaAs was implemented through a discrete stack of properly oriented bulk GaAs plates, with a CC>2 laser pump beam propagating normally to the plates [19]. In the mid-infrared, the related scattering losses at each GaAs-air interface could be reduced only several years later, with the diffusion-bonding technique [20,21]. Alternative QPM implementations consist in GaAs epitaxial regrowth over a template substrate with periodically reversed orientations [22]. To date, the templates have been prepared with two main techniques: 1) a "flip-chip" procedure introduced by Yoo et al. [17,18] and 2) a selective GaAs/Ge/GaAs heteroepitaxy [23]. 2) The first method of this class consists in creating an artificial %(2) through asymmetric quantum wells [24,25]. In AlGaAs this asymmetry makes d is =10 pm/V [26], and allows (-1/+1) QPM schemes, through sequences of x(2> domains with alternating sign [27]. (0/1) QPM was also implemented in AlGaAs waveguides through asymmetric-quantum-well disordering, i.e. periodically destroying the asymmetry-induced nonlinearity along propagation [28]. The second method consists in (0/l)-modulating the d14 bulk coefficient along a GaAs/AlAs superlattice waveguide, through selective superlattice disordering along propagation [29]. 3) Reminiscent of band-gap engineering [30], today semiconductor technology grants a true refractive-index-engineering approach to integrated-optics design. Refractive-index engineering is possible in a multilayer stack, according to the textbook notion of form birefringence (FB) [31]. FB was formerly proposed for BPM SHG in a GaAs/AlAs stack, with a CO2 pump [32]. Due to small index contrast nQaAs-nAiAs and high GaAs dispersion, shorter pump wavelengths could not be used until AlAs oxidation technology emerged in the early 90's, providing
161
the low-index AlOx [33]. Besides triggering significant breakthroughs in optoelectronic technology [34,35], AlOx has recently allowed form-birefringent phase matching (FBPM) in AlGaAs multilayer waveguides [36]. The first two approaches share a high fabrication complexity and the related generation efficiencies are disappointing. This is due either to high scattering losses in the near infrared (1), or to poor %(2) modulation (2). We therefore detail briefly the AlOx-based method: the related structure is fabricated through the double etching of a planar waveguide (with AlGaAs claddings and GaAs/ALAs multilayer core), which is epitaxially grown on a (100) GaAs wafer. As can be appreciated in Figure 6, the first etching defines a ridge for mode confinement. The second etching defines a mesa for the lateral oxidation of AlAs layers (see inset). After the oxidation, carried out at 400°-500°C in a water vapour atmosphere, the waveguide exhibits a strongly birefringent GaAs/AlOx core [37]. With AlAs "" ..
guided mode oxidation —+
ridge
AlGaAs / ••*— oxidation
Figure 6 Double-etched AlGaAs multi-layer structure for optical guidance and oxidation.
this type of waveguide, both DFG and SHG have been reported [38,39]. Recently, mainly due to the growing importance of environmental issues, an increasing interest has arisen towards infrared tunable compact sources operating at roomtemperature. Whereas traditional lead-salt diodes are being rapidly superseded by quantum cascade lasers [30] and antimonide lasers [40], waveguide OPO's are an additional competitor, owing to their wide tunability, integrability, and cw operation. To date, due to the unique combination of mature waveguide technology, very low losses and good nonlinearity, all the operating integrated OPO's have been fabricated in LiNbO3 [41,42], but of course their semiconductor counterparts are greatly appealing. Within this framework, parametric fluorescence in GaAs waveguides has recently attracted a great deal of interest, since it constitutes the basic element towards an integrated OPO. Specifically, PF has been demonstrated in the above AlOx-AlGaAs waveguides, with a cw Ti:Sa source and a »3 mm long waveguide [43]. As the pump wavelength was varied around 1.06 urn, the total PF power in the 1.9 to 2.5 um spectral window varied as in Figure 7a, with the output power depending linearly on the pump power (TJPF =
162
major relevance towards an OPO, whose oscillation threshold can be readily calculated, inserting the measured values of efficiency and losses (tjnorm=IOOO% cm"2W"' and a&0.8 cm"1) in standard formulas [44]. In principle the cavity mirrors could simply consist in the cleaved waveguide facets, but their reflectivity would be too low and the OPO threshold would correspondingly be far too high. Integrated mirrors are therefore required, with much higher reflectivity. In the PF experiment, the maximum power that was safely coupled into the pump mode was
-Z. 8 CD 03 6
1056
1058
1060
1062
1064
1900
"1059
1059.5
1060
1060.5
1061
1061.5
*.„. nm (a)
(b)
Figure 7 Towards a GaAs OPO: a) overall PF power vs. pump wavelength, and b) parametric tuning curve.
(a)
(b)
Figure 8 a) Phase-effect of quadratic cascading, b) SHG along propagation in the depleted-pump regime for high (solid line) and low (dashed line) FF input, at zero and non-zero phase mismatch.
163
3.
Quadratic Spatial Optical Solitons
In the last decade, besides the fabrication of novel integrated parametric sources, research in rf2) NLO has considerably developed in the domain of all-optical processing and spatial solitons. A quick insight into the related concepts is facilitated by the analogy with the optical Kerr effect [45], which manifests itself as a linear dependence of the refractive index on the optical intensity (n=n0+n2I), and allows two interesting applications: 1) With plane waves, this index change induces a dephasing which, with the aid of an interferometer, can give rise to an optically-induced intensity modulation. 2) With laser beams, an intensity distribution across the beam spot induces a similar index distribution, which provides a lens effect and self focusing (defocusing) for n2>0 (n2<0). Under proper conditions, self focusing can exactly balance the beam diffraction, with the formation of a spatial soliton, i.e. a spotsize-invariant beam along propagation. In this case the beam writes dynamically its own waveguide, which is also suitable to guide another probe beam. Most of the material nonlinearities that give rise to spatial solitons (e.g. photorefractive and re-orientational ones) can be thought of as Kerr-like saturating nonlinearities [46]. Quadratic solitons, however, are quite different and rely on the cascading of rf2) nonlinearities, which always occurs for Ak^O [47]. At the crystal input, in the first coherence length Lc, a part of the pump is converted to 2co. Due to dispersion, SH propagates more slowly than FF: therefore after 2LC, as photons at 2a> are back-converted to ft>, the total field at a> is given by the sum of these photons and those which were not converted during the first Lc. Therefore after a a>->2a)->co cycle, as suggested in Figure 8a, the total phase at a> is modified according to the strength of these frequency conversions. Such SHG-induced self dephasing can become very relevant after several conversion cycles. Let us focus now on SHG in a depleted-pump regime, which is illustrated in Figure 8b [45]. At phase matching, two obvious effects occur: efficiency saturation, and an increasing conversion rate for a higher input pump intensity Im(0). Such a rate dependence holds for Ak^O, whereby, defining a characteristic nonlinear length LNL x l/^e(f VA»(o)' amphtude and frequency of efficiency oscillations along z increase as LNL decreases. Each oscillation cycle induces a nonlinear dephasing which, after a certain propagation distance, will prove larger for a higher input intensity. Analogously to Kerr effect, this intensity-dependent dephasing can be exploited for all-optical switching, although in the %(2) case there is no refractive-index change [47]. Let us specifically consider Type-I SHG pumped by a laser beam, which in the slowly-varying-envelope approximation is described by
SP
dz -
oz
=0
+ diffraction
=0
(8)
nonlinearity
Now the parametric gain is locally higher where the pump beam is more intense. If the latter has a gaussian profile (see Figure 9), since SHG is proportional to the square of the FF field, the SH beam profile will be narrower than the input FF one. Moreover, at the
164
first 2co-HQ back-conversion, the FF field will grow proportionally to the product EJE2m*, which corresponds to a still narrower beam. As this process goes on, a balance can be achieved between LNL and the Rayleigh distance. The corresponding soliton is actually a two-color simulton of two mutually phase-locked fields, and markedly differs from the afore-mentioned Kerr soliton [48]. Quadratic spatial solitons (QSS) have started to attract a great deal of interest in the mid-90's, a quarter of century after their first theoretical prediction [49], due the availability ,-3/A FF
7
SH
V
co+2co
Figure 9 Basic mechanism of beam narrowing associated to SHG, for an input gaussian beam.
of high-quality materials, and to a mature understanding of tf2) cascading [50]. Since their first experimental demonstration, [51] a considerable amount of both theoretical and experimental work has been performed. While detailed reviews on QSS can be found in Refs. [46,52], a brief chronology of their main experimental demonstrations can be summarized as follows: • (2+l)Dimensional QSS in bulk KTP, obtained above 5 GW/cm2 FF input, associated to Type-II SHG, in the presence of birefringent walkoff at both a> and 2w [51]. • (1+1)D QSS in LiNbOs planar waveguides, obtained with a non-uniform temperature profile along propagation, with threshold in excess of 1 kW [53]. • steering of Type-II bulk QSS [54,55] • QSS associated to OPA, exhibiting peculiar robustness and phase-matching dependence, as compared to SHG-related QSS [56-58]. • QSS collisions in both a LiNbO3 planar waveguide (which behave similarly to phasecontrolled collisions between saturable Kerr-like solitons) and bulk KTP [59,60] • QSS-based shaping of elliptical beams associated to Type-II SHG in KTP, through modulational instability (where noise breaks high intensity beams into filaments which may form new solitons) [61]. • Type-I (2+l)D QSS in bulk PPLN, with lowest soliton threshold («1 GW/cm2) [62]. • Optical bullets, in which both 2D-spatial and temporal self-trapping occurs [63], and spatial trapping of short temporal pulses in planar waveguides [64]. Virtually all of these effects can be exploited for all-optical switching, disclosing novel perspectives for dynamically reconfigurable interconects, spatial multiplexers, and solitonbased all-optical logic [65]. Currently a few experimental groups are active on QSS, with focus on different materials and geometries. In the waveguide (1+1)D case, a novel kind of QSS has been recently reported in uniform Reverse-Proton-Exchanged (RPE) LiNbO3 waveguides, associated to d31-based
165
SHG interaction between two orthogonally polarized modes of two superimposed planar waveguides (see Figure lOa). Transverse light trapping was obtained at room temperature and excitations as low as 340 nJ in 20 ps pulses, as suggested in Figure lOb [66],
TE
TM
2.24
2.19 U
D
Depth (urn)
(a) Figure 10 (1+1)D QSS in RPE LiNbOs waveguide: a) Ordinary (o) and extra-ordinary (e) refractive-index profiles, and b) Typical SH output vs FF input power, with QSS formation as observed by end-fire output spot.
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GENERATION OF POLARIZATION ENTANGLED PHOTONS BY A UNIVERSAL SOURCE OF ENTANGLEMENT M. BARBIERI, F. DE MARTINI, G. DI NEPI AND P. MATALONI Dipartimento di Fisica and Istituto Nazionale per la Fisica della Materia, Universitd di Roma "La Sapienza", Roma, 00185 - Italy A novel high-brilliance universal source of polarization entangled photon states is presented. It consists of a type-I non-linear crystal which operates in a ultrastable interferometric scheme where all the photon couples emitted at a certain wavelength take part in the entanglement and can be measured. In these peculiar conditions we have obtained violation of 213 standard deviations of a Bell's inequalities test. The brightness and the robustness of the source have been also characterized by different experiments. The possible generation of a native entangled Bessel Gauss beam by this source is also discussed. 1
Introduction
The main issue of modern technology is represented by the manipulation of information, its transmission, processing, storing, and computing, with an increasingly high demand of speed, reliability and security. Quantum Physics has recently opened the way to the realization of radically new information-processing devices, with the possibility of guaranteed secure cryptographic communications, and of huge speedups of some computational tasks. The modern science of quantum information squarely lies on the concept and on the applications of entanglement which is the irrevocable signature of quantum nonlocality and represents the basis of the exponential parallelism of future quantum computers 1, of quantum teleportation 2,3,4,5 an(j of some kinds of cryptographic communications6'7. It is of great significance that the most promising applications of entanglement have been obtained so far in the field of quantum optics. At the moment, the most accessible and controllable source of entanglement arises from the process of Spontaneous Parametric Down Conversion (SPDC) in a nonlinear optical crystal 8. In this process, two twin photons are generated at frequencies vi and v-i and momenta kj and ka by annihilation of a pump photon with frequency vp and momentum k p . Conservation of energy and momentum, 1/1 + 1/2 = vp and ki + k2 = k p , leads to two photon entanglement in these two continuous degrees of freedom. As far as polarization is concerned, the possibility of entanglement leads to the four "Bell -states" 9 , |*±) = •$.(\Hl,V3)±\V1,Hi)), and |$±> = ^ (\Hl:H2) ± |Vi,V 2 », where H and V correspond to the horizontal and vertical.direction of polarization. A reliable source of entangled particles is based on SPDC in a Type II noncollinear phase matched crystal in which two orthogonally polarized photons are generated over two sets of correlated directions corresponding to two different emission cones 10 . These intersect each other in two particular directions where the correlated photons are produced in one of the two Bell states |\l/-t} = -4= (\Hi, V2) ± |Vi, H2)), depending on the crystal orientation. By a careful spatial selection of the two directions of intersection of the orthogonal polarization cones, a high purity entangled state may be generated by this source. The typical achievable coincidence count
171
172
rate is of the order of few hundreds sec l in the case of a 1mm thick NL crystal excited by a WOmW UV pump laser. More recently, a different source of polarization entangled photon pairs, an order of magnitude brighter than the standard Type II, based on two thin, orthogonally oriented Type I NL crystal slabs in mutual contact, has been introduced by Kwiat n . In the present work we report on a novel SPDC source of polarization-entangled photon pairs, recently developed in our laboratory 12, that we believe represents the ultimate solution in the framework of quantum optics in terms of state generation and detection flexibility. Moreover, the new solution, besides realizing the maximum attainable brightness, allows the detection of all the SPDC photon pairs participating to the entanglement over the entire set of the wavevectors excited by any scattering process, i.e. the entangled state accessible to the detection apparatus may be considered as a virtually a pure state 13. Bell's inequalities violation tests, together with measurements of brightness and robustness of the source have been performed and are described in the present paper. Far more generally, all possible bi-partite states in 2 x 2 dimensions can be created by this device 14, how it is shown in this paper. This indeed expresses the "universality" of our source. In the last section the possible generation of a native entangled Bessel-Gauss beam by the high brilliance source is also discussed. 2
The high brightness source of polarization entangled photons
The source of polarization-entangled photon pairs is given by a single arm interferometer operating with a Type-I nonlinear crystal which is excited in two opposite directions (kp, — kp) by a back-reflected UV pump beam and generates two degenerate (AI = A2 = A = 727.6nm) SPDC photons. The generation of the 2-photon polarization entanglement is given by the quantum superposition of the product states created by SPDC in the opposite directions k and —k, the last one after back-reflection and suitable phase and polarization transformations. By this source all SPDC photon couples are emitted at a certain wavelength and take part in the entanglement over the entire phase-matching cone, with angle of aperture a between kp and k. Let us give here a detailed description of the apparatus (Fig. 1). A Type I, .5mm thick, /3-barium-borate (BBO) crystal is excited by a ^-polarized cw Ar+ laser beam (Ap = 363.8nro) with wavevector -kp, i.e. directed towards the left in Fig.l. The two photons have common H polarization and are emitted with equal probability over a corresponding pair of wavevectors belonging to the surface of a cone with axis kpand aperture a ~ 2.9°. The emitted radiation and the laser beam are then back-reflected by a spherical mirror M with curvature radius R — 15cm, highly reflecting both A and Ap, placed at a distance d = R from the crystal. A zero-order A/4 waveplate placed between M and the BBO intercepts twice both back-reflected A and Ap beams and then rotates by Tr/2 the polarization of the back-reflected photons with wavelength A while leaving in its original polarization state the back-reflected pump beam Ap K 2A. The back-reflected laser beam excites an identical albeit distinct downconversion process with emission of a new radiation cone directed towards the right in Fig. 1 with axis kp. In this way each pair
173
i4fl
Figure 1. Layout of the universal, high-brilliance source of polarization entangled photon states and of general mixed states. Inset: partition of the half Entanglement-ring into the spatial contributions of the emitted pair distribution to an output Werner-state.
originally generated towards the left in Fig. 1 is made, by optical back-reflection and a unitary polarization - flipping transformation, "in principle indistinguishable" with another pair originally generated towards the right and carrying the state \HH}. The state of the overall radiation, resulting from the two overlapping indistinguishable cones, is then expressed by the pure entangled state: \3>) = -^(\Hl,H2} + ei*\Vl,V2))
(1)
with phase (0 <
174
inequalities
16
. They are expressed as 2},
(2) Q
2
where the complex numbers a and /3 obey to the condition |Q| + |/3| = 1 and Q ^ /3. The way we have adopted to produce non maximally entangled states consists of the insertion of a small zero-order A p /4 waveplate in the optical path M - BBO of the UV beam, in order not to intercept the SPDC photon pairs traveling along the cone surface (Fig. 1). By orienting the waveplate axis at an angle 6P with respect the vertical direction, the polarization pump vector experiences a rotation of 29P before exciting again the NL crystal in the second passage. In this way the generation probability of horizontally polarized photon pairs, is oc cos2 19P and then a < f3. By adjusting Op in the range 0 - Tr/4, the degree of entanglement 7 = f is continuously tuned between 1 and 0. 3
Phase control of the entangled state
Because of its peculiar configuration the high brilliance source overcomes many of the instability problems due to the typical phase fluctuations of a standard two arm interferometer. The spherical mirror M determines a large value of the displacement, |Ad| ~ 60//m in our case, which allows to perform the phase transition $ = 0 —> 4> = TT from the Bell state |
Xp
A
A
function of the distances OAl, OB/ and BlC. Since OA = OB = Of At = OiBl — R, we have
OAl = R + Ad.
(4)
By approximating all = at ~ a and applying the Carnot theorem to the triangle OO/B/, we find the following expression for OB/: OB/ = y(Ad) 2 + R2 + 2R&dcos(a).
(5)
Finally, we have BlC — OllC + OltBl, where O//C and OnBl are obtained from a careful analysis of Fig. 2: 0,,C
where
=
cos (all)
=
cos (a)
'
^'
175
«0
«8
Figure 2. Scheme representing the optical path difference within the single arm interferometer.
Figure 3. Maxima and minima interference fringe distribution vs. the BBO displacement.
The expression of OllBi is again obtained by applying the Carnot theorem to the triangle OB/O/I. By approximating a —> 0, we have (OllBi) = -(OOn) cos(a) + (OBi)
(8)
By the above equations it's easy to find that the transition |$+) —> |^>-} corresponds to a a value of the displacement |Ad| = 60^m, in very good agreement with the experimental results. Note that a value A.d ^ 0 implies a lateral displacement OC of the reflected SPDC beams (Fig. 2). Because of the intrinsic cylindrical symmetry, OC may be viewed as the radius of an annular-shape region which grows with Ad on the BBO plane. This geometrical effect makes distinguishable the two emission cones, it introduces a spatial decoherence which becomes relevant as far as OC becomes comparable with the diameter of the active pumped region of the crystal (~ ISO^m). In our experimental conditions, OC ~ ^>Ad, we have observed that any coherent superposition on the state vanishes for |Ad| ^ 600/^m. This is confirmed by the results of Fig. 3, where the maxima and minima of coincidences are reported as a function of the BBO displacement Ad.
176
4 4-1
Experimental results Violation of Bell inequalities
The Bell state |$_) over the whole Entanglement- ring has been adopted to test the violation of a Bell inequality by the standard coincidence technique 17, by the following angle orientations of the A (1) and B (2) sites l?-analyzers:{^i = Q,0{ — 45°} and {6-2 = 22.5°, 0'2 — 67.5°}, together with the respective orthogonal angles: WiS^i"1 f and 102 i^-i" (• By these values, the standard Bell-inequality parameter could be evaluated 15:
92) + p(0'1,92)\
(9)
where
and C(6i,02) is the coincidence rate measured at sites A and B. The measured value S — 2.5564 ± .0026, obtained by integrating the data over 180s, corresponds to a violation as large as 213 standard deviations respect to the limit value S = 2 implied by local realistic theories 12>18. Fig. 4 shows the "^-correlation obtained by varying the angle 6\ in the range (45° — 135°), having kept fixed the angle 0? — 45°. The interference pattern demonstrates the high degree of polarization entanglement of the source. The measured visibility of the coincidence rate, V > 94%, gives a further strong indication of the entangled nature of the state over the entire cone of emission, while the single count rates don't show any periodical fringe behaviour as expected. The experimental data are compared in Fig. 4 with the dotted line which correspondes to the limit boundary between the quantum and the "classical" regimes and with the theoretical (continuous) curve expressing the ideal interferometric pattern with maximum visibility: V = 1. We have characterized the robustness and the brightness of the source by measuring coincidences for different values of the of the radius r of the iris diaphragms (I.D.) (Fig. 1). This corresponds to select different portions of the entanglement ring, with area A = 2DS arcsin(^). The experimental results of Fig. 5 demonstrate that a coincidence rate of more than 4000 sec"1 are measured over the entire Entanglement ring with a still relevant value of visibility. By taking into account the UV pump power (P ~ lOOmW) and the overall efficiency of the apparatus (optic transmissions + detector efficiencies), we can evaluate that more than 2 • 105s~1 entangled photon pairs are generated. 4-2
Generation and characterization of mixed states
Because of the peculiar spatial superposition property of the output state, the present apparatus appears to be an ideal source of any bi-partite, two-qubit entangled state, either pure or mixed. In particular of the Werner state: pw = £>|*-) (*-| + ^j2! consisting of a mixture of a pure singlet state |\&_) = 2~5 {\HV) - \VH)} with probability p (0 < p < 1) and of a fully mixed-state
177
A-©|t
(Dcg)
Figure 4. Measurement of the polarization entanglement.
The selected state is
•
cat-
•w>-
*
0.6-
20DO
•
tu-
2EXJO 'g
„
'5 U
B
1
»
a-i.
&
\'i^tbf)f(y
" t'<'i»«*^»««"««
100Q
»
GO*
c
4
e
s
1C
Iris radius (mm)
Figure 5. Plot of the fringe visibility (white cirles, left axis) and coincidence rate (black squares, right axis) as a function of the iris diafragm radius r.
expressed by the unit operator I. Because of their structure these states are particularly useful to simulate the presence of noise in a communication channel and the consequent depletion of correlation within the photon pairs. The corresponding
178
density matrix, expressed in the basis \HH), \HV), \VH), \VV) is:
'A 0 0 0 \ 0 BC 0
(11)
0 0 0 DJ with: A=D=\(\ — p), S=j(l + p), C=—p/2. The Werner states possess a highly conceptual and historical value because, in the probability range [1/3 < p < l/v/2], they do not violate any Bell's inequality in spite of being in this range nonseparable entangled states 19. In order to synthesize these states, we have selected a convenient patchwork technique implying three steps which consist of the insertion of three different optical elements, shown in the grey regions of Fig. 1: [1] Making reference to the original source-state expressed by Eq.l, a singlet state |*_) is easily obtained by inserting a lr -flipping, zero-order A/2 waveplate in front of detector B. [2] A anti-reflection coated glass-plate G, 200/^m thick, inserted in the d — section with a variable trasverse position Az, introduces a decohering fixed time-delay Ai > TCOh that spoils the indistinguishability of the intercepted portions of the overlapping quantum-interfering radiation cones: Fig.l, inset. As a consequence, all nondiagonal elements of pw contributed by the surface sectors B + C of the E-ring, the ones optically intercepted by G, are set to zero while the non intercepted sector A expressed the pure-state singlet contribution to pw- [3] A A/2 wp is inserted in the semi-cylindrical photon distribution reflected by the beamsplitting prism towards the detector A. Its position is carefully adjusted in order to intercepts half of the B + C sector, i.e. by making B = C. Note that only half of the E-ring needs to be intercepted by the optical plates, in virtue of the EPR nonlocality. The other, optically non-intercepted, half of the E-ring is not represented in Fig.6, inset. In summary, the sector A of the E-ring contributes to pw with a pure state p\^>-) (\E r _ , the sector B + C = 2B with the statistical mixture: 1=& {[\HV) (HV\ + \VH) (VH\] + [\HH) (HH\ + \VV) (VV\}} and the probability p, a monotonic function of Az, can be easily varied over its full range of values: p oc Az for small p. The two extreme cases are: [a] The G and the A/2 wp intercepting the beam towards A are absent: B = C = 0. The A section covers half of the E-ring and: pw = |*-} (*-|; [b] The G Plate intercepts half of the E-ring and the position of the A/2 wp intercepting the beam towards A is set to make B = C. In this case A = 0 and: pw = |I. Any Werner state has been realized by this technique, by setting B = C and by adjusting the value of p(Az). Far more generally, all possible bi-partite states in 2 x 2 dimensions could be created by this technique. This indeed expresses the "universality" of our source. A set of Werner states has beens indeed synthesized and several relevant properties investigated by our method. A relevant property of any mixed-state, the "tangle"T = [C(p)}2, i.e. the square of the concurrence C(p], is directly related to the entanglement of formation Ep(p] and expresses the degree of entanglement of p 20. Another important property of the mixed-states is the "linear entropy" SL= d(l - Trp2)/(d - 1), Sz,=(l - p2) for Werner states, which quantifies the degree of
179
disorder, viz. the mixedeness of a system with dimensions d 21. In virtue of the very definition of C(p), these two quantities are found to be related, for Werner states, as follows: Tw (SL) = \(l - 3-y/l - SL)2 for 0 < SL < 8/9 20. The Bell inequalities are not violated in the range [| < SL < 8/9], while pw is a separable state for SL > 8/9 then T (SL) = 0 19. The experimental result shown in Fig.6, left inset, of a standard tomographic analysis of the Werner state corresponding to p ~ 0.42 reproduces graphically, and quite accurately the structure of the matrix pw- The properties TW (SL) of a full set of Werner states (full circles) are also reported in Fig. 6. The state represented by Fig.6, left inset does not violate any Bell inequality, in spite of being a non separable one. Indeed, the corresponding Bell-inequality parameter has been experimentally measured: S = 1.048 ± 0.011. The transition to S > 2 has been experimentally determined, and found consistent with theory, by increasing the value of p in order to set SL = (1-P 2 ) < \-
Linear entropy Figure 6. Experimental plots " Tangle-Linear entropy" for Werner states (full circles) and MEMS (open circles). Left inset: Experimental tomographic reconstructions of a Werner-state with p = 0.42; Right inset: A MEMS with p = 0.63.
As a final demonstration of the universality of our method, a full set of "maximally entangled mixed states" (MEMS) has been synthesized by our source and tested again by quantum tomography 22 > 23 . The MEMS are to be considered, for practical reasons, as important resources of modern QI because they achieve the greatest possible entanglement for a given mixedeness, i.e. the one which is the unavoidable manifestation of decoherence. Of course, in order to synthesize the MEMS a convenient partition of the E-ring, different from the one shown for Werner states, has been adopted. The class of MEMS generated and tested by our method are expressed by p — PMEMS, with the following parameters: A=(l-2g(p)), B=g(p), C=-p/2, D=0 and: g(p) = p/2 for p > 2/3 and g(p) = 1/3 for p < 2/3. The tomographic result shown in Fig.6, right inset reproduces graphically, and with fair accuracy the PMEMS structure with the parameter: p — 0.63
180
and then: g(p) — 1/3. The entire set of generated MEMS is given in Fig. 6 (open circles) . In this case the agreement between experimental and theoretical results is less satisfactory than it was for Werner states. This is due to the strict experimental requirement of working just on the boundary with the forbidden region, on the right of fig. 6, where no bi-partite quantum state is allowed 21. 5
Generation of a Bessel-Gauss beam by SPDC
The fact that photon pairs are emitted by parametric fluorescence over a coherent cone and the particular geometry of the source described in the present paper, where any degenerate SPDC event is selected by the annular mask, suggest that a native Bessel-Gauss beam could be generated by this device 24'25. This property might be of particular interest for quantum information pourposes in the case of an entangled photons beam. For istance, it could be necessary to focus entangled photon pairs in optical fibers and transmit them to mirrors, beam splitters or other linear optical devices which have been proposed as the basic elements of logic gates1 . Assuming that the single photon field reproduces the Gaussian distribution of the pump, the SPDC ring may be viewed as the annular distribution of infinite Gaussian functions (Fig. 1). Because of the coherence properties of the two photon emission, this condion corresponds to the generation of a Bessel-Gauss beam (BGB). It represents the best physical solution of the Helmoltz Equation with interesting non dispersive properties 25. A BGB maintains in fact the same field distribution across any plane orthogonal to the direction of propagation z within a proper range \zc . In cylindrical coordinates, in the plane z = 0, the BGB field is given by: u*=o(r) = EQ J0(pr) exPHfcr2/2
(12)
41
where Jo(/3r) is the zero ' order Bessel function and qo represents the complex radius of curvature of the field, -!- = !_ i-^s 26. It can be demonstrated that (12) represents the envelope of the generic gaussian beam whose axis belongs to the surface of a cone of angle a 25 . This field behaves like a pseudo-non-diffracting beam over a distance u>o/a. Regarding our experimental conditions, the field emerging from the anular mask corresponds to the envelope of gaussian beams with wavevectors located on the surface of a cylinder of radius D/2: Uz=0(r]
= E 0 exp[-zfe(r 2 +
-)/2
4
Zq0
(13)
This field belongs to the class of the generalized Bessel-Gauss beams 27. We want to enphasize that the beam expressed by (13) is given by the coherent superposition of the entire set of photon pairs belonging to the ring. The propagation of this beam through a positive lens / can be studied by applying the standard ABCD matrix formalism 28. It is easy to demonstrate in this case that the field distribution in the focal plane is an ordinary BGB. In order to characterize the modal structure of the beam generated by the source we have measured the intensity distribution on the focal plane of a lens / = 30cm. A 100/xrn pin-hole aperture could be micrometrically traslated in the
181
Ay (mm)
Figure 7. Intensity distribution in the lens focal plane of the single photon gaussian beam (dashed line) and of the Bessel Gauss mode (continuous line). The experimental results of the gaussian beam with the relative fit (dotted line) are also shown.
horizontal direction of the lens focal plane. The transmitted radiation was measured by a photomultiplier Burle C31034A connected to a single photon counter Stanford SR400. In this experiment, because of the low quantum efficiency of the detection apparatus, the experiment has been performed by detecting the single photons transmitted through the pin-hole, without looking at coincidences. In this conditions any photon correlation effect is washed out and we expect that the measurement reproduces exactly the intensity distribution oc |exp[—ikr 2 /2q}\ of the generic gaussian beam in the focal plane, where the complex radius q can be calculated by the ABCD matrix formalism 28. This is demonstrated by Fig. 7, where the experimental results and the relative best fit are compared with the theoretical intensity distribution of the gaussian beam. The theoretical distribution corresponding to a focused BGB is also shown for comparison. This result is expected in case that the experiment is performed by measuring the coincidences of photon pairs transmitted through the pin-hole, as said. Acknowledgments
Thanks are due to G. Giorgi for early involvement in the experiment. This work has been supported by the FET European Network on Quantum Information and Communication (Contract IST-2000-29681: ATESIT) and by PRAINFM 2002 (CLON). References
1. Ekert, A., Jozsa, R., Rev. Mod. Phys. 68, 733 (1996); Knill, E., Laflamme, R., Milburn, G.J., Nature 409, 46 (2001).
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2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
14.
15. 16. 17.
18. 19. 20.
21.
22.
Bennett, C.H. et al. Phys. Rev. Lett. 70, 1895 (1993). Bouwmeester, D. et al. Nature 390, 575 (1997). Boschi, D. et al., Phys. Rev. Lett. 80, 1121 (1998). Marcikic, I. et al. Nature 421, 509 (2003). Ekert, A., Nature 358, 14 (1992). Gisin, N. et al. Rev. Mod. Phys. 74, 145 (2002). D. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, New York, 1988). J. S. Bell, Physics (Long Island City, NY) 1, 195 (1964). P. G. Kwiat, K. Mattle, H. Weinfurter and A. Zeilinger , Phys. Rev. Lett. 75, 4337 (1995). P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum and P. H. Eberhard, Phys. Rev. A, 60, R773 (1999). G. Giorgi, G. Di Nepi, P. Mataloni and F. De Martini, Laser Physics, 13, 350 (2003). The concept of pure state must be intended as an asymptotic one in the context of the present experiment. Obviously a real pure state can not exist because of the unavoidable limitations due to the finite bandwidth of the IF filters and the spatial selection due to the annular mask which give the boundaries within the electromagnetic field mode can be defined. In general we can say that any selection process performed within the collection of the total number of photons emitted by SPDC, limits the purity of the state. The particular configuration of the high brilliance source largely increases the collection efficiency of the nonlinear scattered photons emitted over the cone. This effect allows a better approximation of the pure state. M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, "Violation of Bell Inequalities and Quantum Tomography with Pure-states, Werner-states and Maximally Entangled Mixed States by a Universal Quantum Entangler", e-print quant-ph/0303018. J. F. Clauser, M. A. Home, A. Shimony and R. A. Holt, Phys. Rev. Lett., 23, 880 (1969). L. Hardy, Phys. Rev. Lett. 71, 1665 (1993); D. Boschi, S. Branca, F. De Martini and L. Hardy, Phys. Rev. Lett. 79, 2755 (1997). A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 49, 91 (1982); Y. H. Shih and C. 0. Alley, Phys. Rev. Lett. 61, 2921 (1988); Z. Y. Ou and L. Mandel, ibid. 61, 50 (1988). A. Garuccio, Phys.Rev.A, 52, 2535 (1995). A. Peres, Phys. Rev. Lett. 77, 1413 (1996) W. K. Wootters, Phys. Rev. Lett. 80, 2245, (1998); S. Bose and V. Vedral, Phys. Rev. A. 61, 040101(2000); V. Coffman, J. Kundu, W. K. Wootters, ibid. 61, 052306 (2000). A.G. White, D. F. V. James, W. J. Munro and P. G. Kwiat, Phys. Rev. A. 65, 012301(2001);Yong-Sheng Zhang, Yun-Feng Huang, Chuan-Feng Li and Guang-Can Guo, ibid. 66, 062315 (2002). S. Ishizaka, T. Hiroshima, Phys. Rev. A, 62, 022310 (2000), W. J. Munro, D. F. V. James, A.G. White and P. G. Kwiat, ibid. 64, 030302 (2001), F.
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Verstraete, K. Audenaert and B. De Moor, ibid. 64, 012316 (2001). 23. D. F. V. James, P. G. Kwiat, W. J. Munro and A.G. White, Phys. Rev. A. 64, 052312 (2001). 24. J. Durnin, J.J. Miceli Jr, J.H. Eberly, Phys. Rev. Lett, 58, 1499 (1987). 25. F. Gori, G. Guattari, C. Padovani, Optics Commun 64, 491 (1987). 26. B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, Wiley-Interscience (New York), pag. 82, (1991). 27. V. Bagini et al, Journal of Modern Optics 43, 1155 (1996). 28. M. Santarsiero, Optics Commun. 132, 1 (1996).
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POLARIZATION INSTABILITIES IN A QUASI-ISOTROPIC CO2 LASER R. MEUCCI1,1. LEYVA1'2 and E. ALLARIA1 ' Istituto Nazionale di Ottica Applicata. Largo Enrico Fermi 6, 50125 Firenze, Italy. 2
Universidad Key Juan Carlos, Tulipan s/n, 28933 Mostoles, Madrid, Spain.
In this work we present experimental and theoretical results for the polarization competition dynamics in the transient state of a quasi-isotropic low-pressure CC>2 laser. We show that the polarization dynamics during the switch-on is well described by means of a model including optical coherences (intrinsic anisotropy) and extrinsic linear anisotropies. Furthermore, the experiment provides a numerical assignment for the decay rate of the coherence term for a CO2 laser. These results extend a previous work in which competition is interpreted in a phenomenological way as a parametric cross coupling between the matter polarization and electric fields.
1
Introduction
Laser dynamics is commonly studied considering the electric field as a scalar variable, since in most systems the polarization state is imposed by anisotropies of the cavity. For instance, Brewster windows or gratings, generally used in gas lasers to close the laser tube or to select a vibro- rotational transition, impose a linearly polarized state of the laser emission. However, in perfectly cylindrical laser cavities without any elements to select a preferred polarization, the study of the dynamics includes the necessity of considering the vector nature of the electric field. The experimental realizations of the isotropic system show anyway a tendency to prefer certain polarization orientations, which can not be explained sufficiently by considering geometrical anisotropies as the only way of breaking the cylindrical symmetry. Several theoretical works have been devoted to the study of the polarization dynamics of the quasi-isotropic laser, showing the important role played by the material variables in the selection of the polarization state. In particular, the degeneracy of the angular momentum states of the laser transition sublevels has been considered as the coupling source between different polarization states. Initial studies [1,2] considered only stationary solutions, but more recently dynamical models have been developed to explore the role of anisotropy due to the laser medium, that from now on we will call intrinsic anisotropy [3-6]. These models show how the selection of circularly polarized or linearly polarized emission depends on the value of the total angular momentum of the lasing levels, and the relative magnitudes of the magnetic dipole and electric quadrupole relaxation rates of the sublevels. These studies predict a rich dynamics even for the simplest transition, J=l-^ J=0. Later versions of these models include also linear and circular cavity anisotropies that we call extrinsic [7,8], or different level structure [9]. However, few experiments have been performed on this subject, most of them regarding only steady states. Some of the phenomenology predicted by these models has been observed experimentally, as flips between linear [10] or circular [11] polarization states. Experiments carried out on gas lasers reveal that in some systems it is necessary to
185
186
consider the dynamics of the matter variables to fully understand the polarization features of an isotropic system [2,12]. In other cases, the observations could be explained by a nonlinear coupling of the modes and residual cavity anisotropies [13,14]. Up to now, most of the experiments have been performed in class A laser systems with simple level structure as He-Ne or HeXe [6,7], showing a good agreement for the steady state. In this work we present experimental and theoretical results for the polarization competition dynamics in the transient state of a quasi-isotropic low-pressure CO2 laser. We show that the polarization dynamics during the switch-on is well described by means of a model including optical coherences (intrinsic anisotropy) and extrinsic linear anisotropies. These results provide a quantitative comparison between the amount of anisotropy due to the coupling of molecular angular momenta and that induced by cavity anisotropies. The use of a CO2 laser enables us to experimentally observe the case in which the polarization decay rate is much faster than the other variables (class B laser). We can also observe how the level structure modifies the polarization behavior, since our system has a higher order transition (J=19 -> J=20) than those previously studied. In fact, the effective decay rate of the coherences, defined in Ref. [3], is shown explicitly to be closer to the decay rate of the population inversion rather than to the decay rate of the induced polarization. 2
Experimental results
The experiment has been performed using an unpolarized Fabry-Perot cavity as shown in Fig.l. A total reflective flat mirror (M|, reflectivity RI=!) and an outcoupler mirror (M2) with a reflectivity R2=0.914 set the cavity length at L=75 cm. A piezo translator (Pzt) is used to select the P(20) laser emission line and to adjust the laser detuning.
V
ji
1jLL—-
w10
j
1/
M1
Figure 1: Experimental setup: Ml: total reflecting flat mirror, M2: outcoupler mirror, V: Vertical axis of the cavity, H: Horizontal axis of the cavity, Pzt: piezo electro translator, Pol: wire grid polarizer, D1(V): fast detector for the horizontal component, D2(H): fast detector for the vertical component.
The active medium, a mixture of He (82 %), N2 (13.5%) and CO2 (4.5%) at a pressure of 25 mbar, is pumped by a DC discharge fixed at 6.1 mA when the threshold current is 3 mA. As we are interested in the transient dynamics, an intracavity chopper is used to induce a switch-on event at a repetition rate of 200 Hz. The polarization state of the laser emission has been analyzed by means of a wire grid polarizer (Pol), which has the property of reflecting one linear polarization of the incident radiation and transmitting the orthogonal one, with an extinction ratio of 1:180. The reflected and transmitted parts of the beam are directed to two HgCdTe fast detectors (Di, D2, 100 MHz bandwidth), whose sensitive areas (104 urn 2 ) are much smaller than the
187
beam size. Both signals are recorded on a digital oscilloscope (Lecroy LT423L) with 500 MHz bandwidth. In the stationary regime, we observe that the laser has two possible linear polarization directions which are orthogonal as far as we can measure. In Fig. 1 these cavity eigendirections are called H and V respectively.
-8
-4 0 Cavity detuning (% free spectral range)
Figure 2: Experimental intensity of both polarization components along a free spectral range (black triangles, vertical polarization direction; gray squares, horizontal polarization direction) in the quasi isotropic condition.
The polarization direction is determined by the detuning of the cavity, as usual [2,7,13]. This is illustrated in Fig. 2, where we plot the laser intensity along both eigendirections when the detuning is varied around the line center. A residual hysteresis around the transition is masked by the limited resolution of Pzt. At the transition point near the line center there is not a preferred polarization direction and the laser flips spontaneously from one to other due to noise, as shown in Fig.3.
0.0
0.1
0.2
0.3
0.4
Time (sec) Figure 3: Intensity of vertical polarization component when the cavity detuning is set at resonance.
188
Our goal is to study the polarization behavior in this bistable region. In particular, we analyze the switch-on transient state, in which the total intensity presents relaxation oscillations. In our system the pump strength imposes a relaxation frequency of 55 KHz. When the two polarized components are separated, they show oscillations in relative antiphase, which do not appear on the total intensity (Fig. 4). The amplitude of these oscillations depends on the angle between the polarizer axes with respect to H-V, reaching a maximum when the analysis is performed at 45°. In Fig. 4(a) we report an example of these polarization oscillations, when the cavity is tuned at the center of the line. The total laser output intensity (thick solid line) is displayed together with its two orthogonally polarized components analyzed at 45° (thin solid and dashed lines). These oscillations are always damped until they disappear, with a damping rate depending on the cavity detuning. Precisely the oscillations are more persistent the closer is the detuning to the bistable region. This fact points to the competitive origin of the oscillations. In Fig.4(c) we show an example of the transient dynamics when the cavity detuning is slightly moved from resonance.
o.oa
a.as
0.10
a as
aaa
o.is
a.ia
(c):
o.is
(d)
•
rV-
_J",j V " v
oxio
O.QS
0.10
Tims(ms}
o.is
la Q-oa
axis
a.ia
O.IS
Tir»(ms}
Figure 4: Experimental time intensity profiles of the total intensity (solid line) and both polarization components (thin solid and dotted lines) for and slightly different detuning condition : (a) resonance, (c) out of resonance. Numerical generated intensity profiles for a=p=0.01 and : (b) 8=0 and (d) 8=0.05. The curves have been vertically shifted for a better observation.
3
Physical model
Our theoretical approach is based on the theory of the isotropic laser developed in Ref.[3] where the optical coherences between upper levels are considered. This theory was
189
developed for the simplest case (J=1->J=0), while the transition involved in our system is much more complicated (J=19->J=20). However, this theory has been showed to predict also the behavior of lasers with a different level structure, as shown in Ref.[12,7], where an effective value of the coherence decay rate was deduced. Only first order coherences (Am=±l) will be considered. Therefore, independently of the number of sublevels, there are only two kinds of possible transitions, which generate a split of the population in two ensembles, in such a way that an anisotropy is induced in the active medium [6]. Furthermore we introduce an extrinsic linear anisotropy as done in Re f. [7], The field will be decomposed in a circularly polarized basis. Just losses and linear detuning anisotropies along the principal axes of the system will be included, but not circular asymmetries since our system does not show signs of dichroism. It reads as [15]:
ER = k(PR -ER} + i8-ER-(a + ij3)EL , EL = k(PL -EL} + iS-EL-(a + ij3)EK ,
c-rc-c-\y.\El. p^Et.p'\ \-(F • P* -t-
+£
r
+
+£
r
where ER(r, t), EL(r, t) are the slowly varying electric fields in the circular basis. PR(r, t), Pi(r, t) stand for the matter polarization fields, DR(T, t), DL(r, t) are the respective population inversions. The parameter r, fixed to a value of 2.0, stands for the pump strength normalized to its threshold value. The field C(r, t) represents the coherence between the upper sublevels. We recall that, in a density matrix treatment, the polarization corresponds to off-diagonal matrix elements between upper and lower level of the radiative transition, whereas C is proportional to the off-diagonal matrix elements coupling different angular momentum states of the upper level [12]. The parameter S represents the detuning between the cavity and the atomic an ~ d np = ^H ~sv> represent 2 2 respectively the linear anisotropies in the losses and detuning with respect to the cavity HV axes, where AV ,% are the losses in the horizontal and vertical axis, and SH, Sy are the corresponding detunings.
transition frequencies. The parameters
a
=
v
190
In our low pressure CO2 laser, the polarization decay is y_c= 4.4 108 s"1 and the inversion decay rate as y\\ =1.95 105 s"1. The yc parameter represents the coherence decay rate, whose value should be chosen between yj_ and y\\ [5]. However, this parameter cannot be directly measured, and it will be used as control parameter in order to fit the theory to the experimental results [7]. We find that the optimal value is yc KY\\ in all cases, which is also consistent with the observation that just linearly polarized states are found in the experiment. Indeed, a higher value of yc would give rise to a periodic modulation of the total intensity [5] which has never been observed in our experiments. Both the polarization and coherences decay rates are related to molecular collisions, but how these affect the induced polarization and the inner sublevel coherence can be different. A physical reason for the small effective value for yc in the CO2 laser can be found in the complexity of the upper and lower sublevel structures. The elastic collisional processes which interrupt the phase of the emission (contributing to yj) produce a minor effect on optical coherences as compared with that induced in the simplest case (J=1->J=0). In this last situation only a lower sublevel exists and collisions easily induce changes on the two population ensembles. Once we have fixed the value yc » Yy to quantify the effect of the intrinsic anisotropy, we can reproduce all the features observed in the experiment. In Fig. 4 (b) and (d) the numerically generated intensity profiles are compared with their experimental counterparts reported in Fig 4 (a) and (c), respectively. It can be observed that the intensity profiles show antiphase oscillations for both polarization components, while the total intensity remains unmodulated. When 8=0, the polarization oscillations remain undamped for any degree of anisotropy. For S ?6 the oscillations are still undamped only in perfect cavity symmetry conditions a=(3=0. In Fig. 4 (d) it can be seen that for 8=0.05, a detuning or losses anisotropy of 0.5 % is sufficient to damp the oscillation in a few hundred microseconds as observed in the experiment. In the experimental system unavoidable residual anisotropies break the cylindrical symmetry, and therefore the polarization oscillations are always damped.
4
Phenomenological model
The phenomenon of polarization competition in a quasi isotropic CO2 laser was previously interpreted in the framework of the Maxwell-Bloch equations by adding a phenomenological coupling term to the polarization equations that allows the fields to compete while interacting with the same population inversion [16]. Adopting an orthogonal linear basis (Ex, Ey~) and considering the isotropic condition (a, (3=0) the phenomenological model is described by the following equations :
Ey=k(py-Ey}+iS-Ey,
191
where g is the coupling term between orthogonal fields. From a comparison with the physical model (Eq. 1) we infer that the parameter E is related to the coherence term C. 5 Conclusions We study experimentally and theoretically the polarization alternation during the switchon transient of a quasi-isotropic CO2 laser emitting on the fundamental mode. The observed transient dynamics is well reproduced by means of a model which provides a quantitative discrimination between the intrinsic asymmetry due to the kinetic coupling of molecules with different angular momenta and the extrinsic one due to cavity anisotropies. Acknowledgements The authors are grateful to F.T.Arecchi for fruitful discussions. I.Leyva and E.Allaria wish to acknowledge the support the European Project HPRN-CT-2000-00158.
References 1. M. Sargent and W.E. Lamb, Phys.Rev. 164, 436 (1967). 2. D.S. Bakaev, V.M. Ermachenko, V.YuKurochin, V.N. Petrovshil, E.D. Protsenko, A.N. Rurukin, R.A. Shananin, Sov.J.Quantum Electron. 18, 1 (1988). 3. G.P. Puccioni, M.V. Trantnik, J.E. Sipe, G.L. Oppo, Opt Lett.12 (1987). 4. N.B. Abraham, E. Arimondo, M. San Miguel, Opt. Commun.117, 344 (1995). 5. N.B. Abraham, M.D. Matlin, R.S. Gioggia, Phys.Rev. A S3, 3514 (1996). 6. For a review see: G.M. Stephan, A.D. May, Quantum Semiclass.Opt.10, 19 (1998). 7. M.D. Matlin, R.S. Gioggia, N.B. Abraham, P. Glorieux, T. Crawford, Opt.Commun.120, 204 (1995). 8. J. Redondo, G.J.de Varcarcel, E. Roldan, Phys.Rev. A 56, 648 (1997). 9. A. Kulminskii, R. Vilaseca, R. Corbalan, N.B. Abraham, Phys.Rev. A 62, 648 (2000). 10. D. Culshaw, J. Kannelaud, Phys.Rev. 141, 237 (1966). 11. R.L. Fork, W.J. Tomlinson, L.J. Helios, Appl.Phys.Lett. 8, 162 (1966). 12. G.C. Puccioni, G.L. Lippi, N.B. Abraham, Opt.Commun. 72, 361 (1989). 13. C. Taggiasco, R. Meucci, M.Ciofni, N.B. Abraham, Opt. Commun. 133, 507 (1997). 14. A. Labate, R. Meucci, M. Ciofni, N.B. Abraham, C. Taggiasco, Quantum Semiclass. Opt. 10, 105 (1998). 15. I.Leyva, E. Allaria and R. Meucci, Opt. Commun. 217, 335-342 (2003). 16. I. Leyva, E. Allaria, R. Meucci, Opt. Lett. 26, 605 (2001).
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SUPERLUMINAL LOCALIZED SOLUTIONS TO THE MAXWELL EQUATIONS FOR VACUUM AND FOR DISPERSIVE MEDIA (WITH ARBITRARY FREQUENCIES AND BANDWIDTHS) MICHEL ZAMBONI RACKED, K. Z. NOBREGA DMO-FEEC, State University at Campinas, Campinas, S.P., Brazil.
ERASMO RECAMI Facolta di Ingegneria, Universita statale di Bergamo, Dalmine (BG), Italy; and INFN-Sezione di Milano, Milan, Italy. E-mail: [email protected] In this paper we set forth new exact analytical Superluminal localized solutions to the wave equation for arbitrary frequencies and adjustable bandwidth. The formulation presented here is rather simple, and its results can be expressed in terms of the ordinary, so-called "X-shaped waves". Moeover, by the present formalism we obtain the first analytical localized Superluminal approximate solutions which represent beams propagating in dispersive media. Our solutions may find application in different fields, like optics, microwaves, radio waves, and so on.
1
Introduction
For many years it has been known that localized (non-dispersive) solutions exist to the (homogeneous) wave equation 1 ' 2 ' 3 , endowed with subluminal or Superluminal 4'5'6'7'8'9 velocities. These solutions propagate without distortion for long distances in vacuum. Particular attention has been paid to the localized Superluminal solutions like the so-called X-waves5'6'8 and their finite energy generalizations7'8. Such Superluminal Localized Solutions (SLS) have been experimentally produced in acoustics10, optics11 and more recently microwave physics12. As is well known, the standard X-wave has a broad band frequency spectrum, starting from zero8'9 (it being therefore appropriate for low frequency applications). This fact can be viewed as a problem, because it is difficult or even impossible to define a carrier frequency for that solution, as well as to use it in high frequency applications. Therefore, it would be very interesting to obtain exact SLSs to the wave equations with spectra localized at higher (arbitrary) frequencies and with an adjustable bandwidth (in other words, with a well defined carrier frequency). To the best of our knowledge, only two attempts were made in this direction: one by Zamboni-Rached et al.8, and the other by Saari13. The former showed how to shift the spectrum to higher frequencies without dealing with its bandwidth, while the latter worked out an analytical approximation to optical pulses only. In this work we are presenting analytical and exact Superluminal localized solutions in vacuum, whose spectra can be localized inside any range of frequency with adjustable bandwidths, and therefore with the possibility of choosing a well defined carrier frequency. In this way, we can get (without any approximation) radio, microwave, optical, etc., localized Superluminal waves. Taking advantage of our methodology, we obtain also the first analytical
193
194
approximations to the SLS's in dispersive media (i.e., in media with a frequency dependent refractive index). One of the interesting points of this work, let us stress, is that all results are obtained from simple mathematical operations on the standard "X-wave". 2
Superluminal localized waves in dispersionless media
Let us start by dealing with SLSs in dispersionless media. From the axially symmetric solution to the wave equation in vacuum (n — 1), in cylindrical coordinates, one can easily find that i/>(p,z,t) = Jo(kpp)e+lk^ e~lwt
(1)
with the conditions
fcp2-- —2 k
C
!;
k*>o,
(2)
where JQ is the zeroth-order ordinary Bessel function; kz and kp are the axial and the transverse wavenumber respectively, LJ is the angular frequency and c is the light velocity. It is essential to call attention right now to the dispersion relation (2). Positive (but not constant, a priori) values of k^, with real kz, do allow both subluminal and Superluminal solutions, while implying truly propagating waves only (with exclusion of the evanescent ones). We shall pay attention in this paper to the Superluminal solutions. Conditions (2) correspond in the (u>,kz) plane to confining ourselves to the sector shown in Fig.l; that is, to the region delimited by the straight lines w = ±ckz.
x
Figure 1. Geometrical representation, in the plane (ui, fcz), of the condition (2): see the text.
Important consequences can be inferred from (2), when performing the coordinate transformation kz — (w/c) cos 6,
(3)
195
which yields kp = (uj/c) sin 9 .
(4)
With this transformation, solution (1) can be rewritten, in the new coordinates (&,()), as 1>(p,Q = JQ(^psm&)e+i^cose
,
(5)
where C = z — Vt, and where the propagation speed (group velocity) is obviously V = cf cos 6. Equation (5) states that the beam is transversally localized in energy, and propagates without suffering any dispersion. It should be noticed also the relationship between V and B: namely, each value of 0 yields a different wave velocity. This fact will be used in the next Section. Equation (5) represents the well known "Bessel beam". As can be seen, such an equation has two free parameters, u> and d. Considering 9 constant, and making a superposition of waves for different frequencies, one can obtain localized (non-dispersive), Superluminal solutions; namely />oo
= /
Jo
+! cose S(w)Jo(-psin0)e '^ dw. c
(6)
In eq.(6), if an exponential spectrum* like S(u) — e~ ow is considered, one obtains, by use of identity (6.611.1) of ref.14, the ordinary X-shaped wave: (7)
where a is a positive constant. This solution is a wave that propagates in free space without distortion and with the Superluminal velocity V = I/ cos 9. Because of its non-dispersive properties, and its low frequency spectrum, the X-wave is being particularly applied in fields like acoustics5. The illustration of an X-wave, with parameters a = 10~7 s and V = 5 c, is shown in Fig. 2. 3
Superluminal localized waves for arbitrary frequencies and adjustable bandwidths
In the last Section, it has been shown that a superposition of Bessel beams can be used to obtain a localized and Superluminal solution to the wave equation in a dispersionless medium. It is known that it may be a difficult task, it being possible, or not, finding analytical expressions for eq.(6). Its numerical solutions usually brings in some inconveniences for further analysis, uncertainties concerning the fast oscillating field components, etc.; besides implying a loss in the physical interpretation of the results. Thus, it is always worth looking for analytical expressions. *It is easy to see that this spectrum starts from zero, it being suitable for low frequency applications, and has the bandwidth Aw = I/a
196
Figure 2. Illustration of the real part of an X-wave with bandwidth, Au, of 10 MHz starting from
Actually, the kind of solution found by us for eq.(6) is strictly related to the chosen spectrum 5(w). Following previous work of ours8, we are going to present our spectrum together with its main characteristics. 3.1
The S(w) Spectrum
One of our main objectives is finding out a spectrum which can preserve the integrability of eq.(6) for any frequency range. In order to be able to shift our spectrum towards the desired frequency, let us locate it around a central frequency, w c , with an arbitrary bandwidth Aw. Then, let us choose the spectrum (8)
where V is the wave velocity, while m and a are free parameters. For m = 0, it is 5(w) = exp[—aw], and one gets the (standard) X-wave spectrum. After some mathematical manipulations, one can easily find the following relations, valid for m ^ 0: 771 =
(Aw±/u; c ) - In (1 + (Aw±/w c ))
*= =.
(8J)
M
Here, because of the non-symmetric character of spectrum (8), let us call Aw + (> 0) the bandwidth to the right, and Aw_ (< 0) the bandwidth to the left of w c ; so that Aw = Aw + — Aw_. It should be noted however that, already for small values of m (typically, for m > 10), one has Aw+ « — A w _ . Once defined w c and AOJ, one can determine m from the first equation. Then, using the second one, a is found.
197
Figure 3 illustrates the behavior of relation (8.1). From this figure, one can observe that the smaller Au/w c is, the higher m must be. Thus, one can notice that m plays the fundamental role of controlling the spectrum bandwidth.
Figure 3. Behavior of the derivative number, m, as a function of the normalized bandwidth frequency, AU^/WC- Given a central frequency, wc, and a bandwidth, Au>±, one finds the exact value of m by substituting these values into eq.(S.l).
From the X-wave spectrum, it is known that a is related to the (negative) slope of the spectrum. Contrarily to a, quantity m has the effect of rising the spectrum. In this way, one parameter compensates for the other, producing the localization of the spectrum inside a certain frequency range. At the same time, this fact also explains (because of relation (8.2)) why an increase of both m and a is necessary to keep the same w c . This can be seen from Fig.4. In Fig.4, both spectra have the same u>c. Taking the narrow spectrum as a reference, one can observe that, to get such a result, both quantities m and a have to increase. Moreover, this figure shows the important role of m for generating a wider, or narrower, spectrum.
3.2
X-type Waves in a Dispersionless Medium
To illustrate the use of the proposed solutions, let us define the ordinary X-wave, by rewriting eq.(6) with S(u) = exp[— aw]:
(9)
= VJ
198
si (10 1 *Hz)
Figure 4. Normalized spectra for wc = 23.56 X 1014 Hz and different bandwidths. The first with TV =: 27 (solid line), and the second spectrum with TV = 41 (dotted line). See the text.
which is the same as
V
X =
(10)
where n0 is the refractive index of the medium underlying these considerations. Applying our spectrum expressed by eq.(8), equation (6) can be rewritten as [°° U>
JoL
Vv
( UJ 0
I
V1
(11)
P
\\v~ V vV ° ~ cc ^ ~
We have therefore seen that the use of a spectrum like (8) allows shifting it towards any frequency and confining it within the desired frequency range. In fact, this is one of its most important characteristics. It can be seen that eqs.(9) and (10) are equivalent. Deriving eq.(9) with respect to (aV — iQ, a multiplicative factor (w/V) is each time produced (an obvious property of Laplace transforms). In this way, it is possible to write eq.(ll) as :
.\
, - \m
o A. d(aV-i()m
'
^
A different expression for eq.(ll), without any need of calculating the m-th derivative of the X-wave, can be found by using identity (6.621) of ref.14:
F
(12')
where X is the ordinary X-wave given in eq.(10), and F is a Gauss' hypergeometric function. Equation (12') can be useful in the cases of large values of m.
199
Let us call attention to equations (12) and (12'): to our knowledge,* no analytical expression had been previously met for X-type waves, which can be localized in the neighbourhood of any chosen frequency with an adjustable bandwidth. Equations (12) allow getting one or more of them in a simple way: All that has to be done is calculating the m-th derivative of X with respect to (aV — i£). Alternatively, one can have recourse to eq.(12'). Fig.5 shows an example of an X-shaped wave for microwave frequencies. To that aim, it was chosen wc — 6 x 109 GHz and Aw = 0.9w c , and the values of n and a were calculated by using eqs.(8.1) and (8.2): thus obtaining m = 10 and a — 1.6667 x 10~9. As one can see, the resulting wave has really the same shape and the same properties as the classical X-waves: namely, both a longitudinal and a transverse localization.
Figure 5. The real part of an X-shaped beam for microwave frequencies in a dispersionless medium.
4
Superluminal localized waves in dispersive media
We shall now pass to dealing with dispersive media. In Section (2), equations (3) and (4) were written for a dispersionless medium (no = constant, independent of the frequency). However, for a typical medium, when the refractive index depends on the wave frequency, n(w), those equations become13 fit can be noticed that dX/d(aV— t'f) = (iV) 1dX/dt. Time derivatives of the X-wave have been actually considered by J.Fagerholm et al. 15 : however the properties of the spectrum generating those solutions (like its shifting in frequency and its bandwidth) did not find room in that previous work.
200
(13) z(oj)
- |n(w) cos(0) .
The above equations describe one of the basic points of this work. In Section (2) it was mentioned that 0 determines the wave velocity: a fact that can be exploited when one looks for a localized wave that does not suffer dispersion. In other words, one can choose a particular frequency dependence of 0 to compensate for the (geometrical) dispersion due to the variation with the frequency of the refractive index13. If the frequency dependence of the refractive index in a medium is known, within a certain frequency range, let us see how the consequent dispersion can be compensated for. When a dispersionless pulse is desired, the constraint kz — a -\-uib must be satisfied. And, by using the last term in eq.(13), one infers that such a constraint is forwarded by the following relationship between 9 and w:
where a and b are arbitrary constants (and b is related to the wave velocity: b = 1/V). For convenience, we shall consider a = 0. Then, eq.(6) can be rewritten as
(15)
Let us stress that this equation is a priori suited for many kinds of applications. In fact, whatever its frequency be (in the optical, acoustic, microwave,... range), it constitutes the integral formula representing a wave which propagates without dispersion in a dispersive medium. Now, let us mention how it is possible to realize relation (14) for optical frequencies. Although limited to the case of the air, or of low-dispersion media, the axicon5'6'11'16 is one of the simplest means to realize it. Another possibility is using "spectral hole burning filters", or holograms17. More in general, one can follow a procedure similar to the one illustrated in Figure 6. The process illustrated in Fig. 6 is actually simple. In fact, there is a different deviation of the wave vector for each spectral component in passing through the chosen device (axicon, hologram, and so on): and such a deviation, associated with the dispersion due to the medium, makes the phase velocity equal for each frequency. This corresponds to no dispersion for the group-velocity. More details about the physics under consideration can be found in Ref.13. Now, let us consider a nearly gaussian spectrum as that given by eq.(8), and assume the presence of a dispersive medium whose refractive index (for the frequency range of interest) can be written in the form n(ui) = n0 + u8 ,
(16)
201
Figure 6. Sketch of a generic device (axicon, hologram, etc.) suited to properly deviating the wave vector of each spectral component.
where no is a constant, while 6 is a free parameter that makes it possible a linear behavior of n(o>): something that is actually realizable for frequencies far from the resonances associated with the used material. Notice that the linear relationship between the refractive index and the wave frequency assumed in eq.(16) is not necessary: but its existence gets our calculations simplified. In this way, substituting eq.(16) into eq.(15), and considering the spectrum, shifted towards optical frequencies, given by eq.(8), a relation similar to eq.(17) is found:
'.<*) = v (17) To the purpose of evaluating eq.(17), let us make a Taylor expansion and rewrite it as ^(p,<^,S) = \P(/9,£,0) + J —\ s = 0 -f —
I^-Q + —
|5=0 + ... (18)
For the above equation it is known that, if S is small enough, it is possible to truncate the series at its first derivative. For the time being, let us assume this is the case and that there is no problem on truncating eq.(18). One can check Fig.7, which shows typical values of S for SiO2, a typical raw-material in fiber optics. Looking at eq.(18), one can notice that its first term \IJ(/9,C,0) is already known to us, because it coincides with the solution given by our eq.(ll). To complete the expansion (18), one must find -^\S=Q. After some simple mathematical manipulations, one gets that
202
Ijjlnner Window —
Refractive index
^Outside Window.
O . I n n e r Window, n =1.43914, 6 = 5.99792 * 10"18 -
a
, Outside Window, n f l =1,43882. 6=6.12239 * 10
to (10 1 4 Hz)
Figure 7. Variation of the refractive index n(u>) with frequency for fused Silica. The solid line is its behaviour, according to Sellmeir's formulae. The open circles and squares are the linear approximations for TV = 41 and N = 27, respectively.
fe f <7 ' ^o ,,2
-1
(19) This integral can be easily evaluated by using identity 6.621-4 of ref. 14 , so
to obtain _
Us
=0
=
(-1)
m+4
Qm+2
V3n0
(20)
As in the case of eqs.(12), (12'), another form for expressing eq.(19) can be found by having recourse once more to the identity (6.621) of ref. 14 : £* I<5 — 36 l = ° ~
» o P 2 r ( m2+ 4 ) X" 2c V"»
m+4 2
—m— 1 . '
2
(20')
'
where, as before, X is the ordinary X-wave given by eq.(10) and F is again a Gauss' hypergeometric function. Once more, equation (20') can be useful in the cases of large values of m. Finally, from our basic solution (12) and its first derivative (20), one can write the desired solution of eq.(15) as
dnX d(aV -
\n+4
V3n0
Qn+2
[(aV - i()X] S .
(21)
203
However, if one wants to use equations (12') and (20'), instead of eqs.(12) and (20), the solution (21) can be written in the form
L±l _ ™ .
2^\
2^
(21') 2
ro
cn0p r(m+4)X + 2c 2 V™
u
4 p
/m+4 I 2 '
2
-m-1 . o . / 2 V 2 ' ^ ' v*0 c 2
2
-n .2 X \ / " V2 I >
It is also interesting to notice that, e.g., the approximate Superluminal localized solution (21) for a dispersive medium has been obtained from simple mathematical operations (derivatives) applied to the standard "X-wave" . 5
Optical applications
To illustrate what was said before, two practical examples will be considered, both in optical frequencies. When mentioning optics, it is natural to refer ourselves to optical fibers. Then, let us suppose the bulk of the dispersive medium under consideration to be fused Silica (SiOs). Far from the medium resonances (which is our case) , the refractive index can be approximated by the well-known Sellmeier equation18
where Wj is the resonance frequency, Bj is the strength of the jth resonance, and N is the total number of the material resonances that appear in the frequency range of interest. For typical frequencies of "long-haul transmission" in optics, it is necessary to choose N = 3, which leads us18 to the values BI = 0.6961663, B2 = 0.4079426, Bz = 0.8974794, AI = 0.0684043 /urn, A2 = 0.1162414 p,m and A3 = 9.896161 /urn. Figure 7 illustrates the relation between N and u, and specifies the range that will be adopted here. In the two examples, the spectra are localized around the angular frequency wc = 23.56 X 1014 Hz (which corresponds to the wavelength AC = 0.8/um), with two different bandwidth Awi = 0.55wc and Aw2 = 0.4w c . The values of a and TV corresponding to these two situations are a = 1.14592 x 10~14, N = 27, and a = 1.90986 x 10"14, N = 41, respectively. Looking at these "windows" , one can notice that Silica does not suffer strong variations of its refractive index. As a matter of fact, a linear approximation to n = n(uj) is quite satisfactory in these cases. Moreover, for both situations, and for their respective no values, the value of parameter 8 results to be very small, verifying condition (15): which means that it is quite acceptable our truncation of the Taylor expansion. The beam intensity profiles for both bandwidths are shown in Figs. 8 and 9. In the first figure, one can see a pattern similar to that of Fig. 2; but here, of course, the pulse is much more localized spatially and temporally (typically, it is a fentosecond pulse).
204
Figure 8. The real part of an X-shaped beam for optical frequencies in a dispersive medium, with N = 27. It refers to the larger window in Fig.7.
Figure 9. The real part of an X-shaped beam for optical frequencies in a dispersive medium, with TV = 41. It refers to the inner window in Fig.7,
In the second figure, one can observe some little differences with respect to the first one, mainly in the spatial oscillations inside the wave envelope19. This may be explained by taking into account that, for certain values of the bandwidth, the carrier wavelength become shorter than the width of the spatial envelope; so that one meets a well defined carrier frequency. Let us point out that both these waves are transversally and longitudinally localized, and that, since the dependence of \P on z and t is given by £ = z — Vt, they are free from dispersion, just like a classical X-shaped wave.
205
6
Conclusions
In this paper we have first worked out analytical Superluminal localized solutions to the wave equation for arbitrary frequencies and with adjustable bandwidth in vacum. The same methodology has been then used to obtain new, analytical expressions representing X-shaped waves (with arbitrary frequencies and adjustable bandwidth) which propagate in dispersive media. Such expressions have been obtained, on one hand, by adopting the appropriate spectrum (which made possible to us both choosing the carrier frequency rather freely, and controlling the spectral bandwidth), and, on the other hand, by having recourse to simple mathematics. Finally, we have illustrated some examples of our approach with applications in optics, considering fused Silica as the dispersive medium. A cknowledgement s
One of the authors (ER) is grateful to Sergio Martellucci and Massimo Santarsiero for very kind invitation and hospitality. The authors acknowledge the continuous scientific collaboration of Cesar Dartora, Hugo E. Hernandez F., and Amr Shaarawi. For useful discussions they thank also C.Becchi, R.Borghi, M.Brambilla, C.Cocca, R.Collina, G.C.Costa, G.Degli Antoni, F.Fontana, F.Gori, J.M.Madureira, M.A.Porras, M.Villa and M.T.Vasconselos. References 1. J.Durnin, J.J.Miceli and J.H.Eberly: Phys. Rev. Lett. 58 (1987) 1499; Opt. Lett. 13 (1988) 79; J.N.Brittingham: J. Appl. Phys. 54 (1983) 1179. 2. R.W.Ziolkowski: J. Math. Phys. 26 (1985) 861; Phys. Rev. A39 (1989)2005. 3. A.Shaarawi, I.M.Besieris and R.W.Ziolkowski: J. Math. Phys. 30 (1989) 1254; A.Shaarawi, R.W.Ziolkowski and I.M.Besieris: J. Math. Phys. 36 (1995) 5565. 4. R.Donnelly and R.W.Ziolkowski: Proc. Roy. Soc. London A440 (1993) 541. 5. J.-y.Lu and J.F.Greenleaf: IEEE Trans. Ultrason. Ferroelectr. Freq.Control 39 (1992) 441. In this case the beam speed is larger than the sound speed in the considered medium. 6. E.Recami: Physica A252 (1998) 586, and refs. therein. 7. I.M.Besieris, M.Abdel-Rahman, A.Shaarawi and A.Chatzipetros: Progress in Electromagnetic Research (PIER) 19 (1998) 1. 8. M.Zamboni-Rached, E.Recami and H.E.Hernandez-Figueroa: "New localized Superluminal solutions to the wave equations with finite total energies and arbitrary frequencies" [e-print # physics/0109062], Europ. Phys. Journal D21
206
(2002) 217.
9. M.Zamboni-Rached: "Localized solutions: Structure and Applications", M.Sc. thesis; Phys. Dept., Campinas State University, 1999). 10. J.-y.Lu and J.F.Greenleaf: IEEE Trans. Ultrason. Ferroelectr. Freq.Control 39 (1992) 19. 11. P.Saari and K.Reivelt: "Evidence of X-shaped propagation-invariant localized light waves", Phys. Rev. Lett. 79 (1997) 4135. 12. D.Mugnai, A.Ranfagni and R.Ruggeri: Phys. Rev. Lett. 84 (2000) 4830. For a panoramic view of the whole experimental situation, cf. E.Recami: Found. Phys. 31 (2001) 1119 [e-print # physics/0101108]. 13. P.Saari and H.Sonajalg: Laser Phys. 7 (1997) 32. 14. I.S.Gradshteyn and I.M.Ryzhik: Integrals, Series and Products, 4th edition (Ac.Press; New York, 1965). 15. J.Fagerholm, A.T.Friberg, J.Huttunen, D.P.Morgan and M.M.Salomaa: Phys. Rev. E54 (1996) 4347. 16. R.M.Herman and T.A.Wiggins: J. Opt. Soc. Am. A8 (1991) 932. 17. Cf., e.g., H.Sonajalg, A.Gorokhovskii, R.Kaarli, et al.: Opt. Commun. 71 (1989) 377. 18. G.P.Agrawal: Nonlinear Fiber Optics, 2nd edition (Ac.Press; New York, 1995). 19. Cf., e.g., also M.Zamboni-Rached, K.Z.Nobrega, E.Recami & H.E.Hernandez F.: "Superluminal X-shaped beams propagating without distortion along a coaxial guide", Phys. Rev. E66 (2002) 046617.
ANGULAR MOMENTUM IN OPTICAL BEAMS JULIO SERNA Departamento de Optica, Universidad Complutense de Madrid, Facultad de Ciencias Fisicas, 28040 Madrid, Spain E-mail: [email protected] Light beams carry energy and linear momentum, but they also transport angular momentum. In order to obtain an expression for the angular momentum carried by light we will use paraxial approximation within classical electromagnetic theory. It is found that the resulting value can be divided naturally in two parts: a polarization contribution and a spatial contribution. By analogy, those two terms are also called spin part and orbital part of the angular momentum. In order to interpret this result we will consider two beam families, Laguerre-Gauss modes and Gauss Schell-model family. Several procedures used to obtain beams with angular momentum are also mentioned. Finally, different methods used to observe and measure the angular momentum transported by light beams are briefly explained.
1
Introduction
Everybody knows that light carries energy and linear momentum. We are used to think that we can send information or cut metal sheets using the energy transported by light beams. Also devices such as solar sails and laser cooling traps are not so unfamiliar, and they are based on the linear momentum carried by light. On the other hand there is not such a clear relation between light and angular momentum. But it should come to no surprise that light transports angular momentum too. In fact, this relation was established long time ago. It seems that Maxwell based his identification of light as an electromagnetic wave on the rotational properties of the luminiferous aether, and not too later Poynting associated angular momentum with circular polarization, as noted in Ref. 1. The mechanical effects of angular momentum carried by circularly polarized light were later demonstrated by Beth with a very nice experiment.2 Even in the classical textbook by Jackson3 we can find a definition of the density of angular momentum of an electromagnetic field, together with a very illustrative exercise. Nevertheless, the angular momentum of light is not such a well-known concept, and we will try to clarify several aspects in this lesson. First of all it would be good to identify in what kind of situations we could notice angular momentum. The first experiment coming to our mind could be a lever and fulcrum device, creating angular momentum by transferring linear momentum from a light beam to the lever. But this is a somewhat trivial case. If we think harder circularly polarized light beams, mentioned in the previous paragraph, could come to our mind. In such circularly polarized light there is "something", the electric field in this case, that "rotates" inside the beam. And something like that should be related with angular momentum. In fact, as we said, the relation was demonstrated experimentally by Beth. 2 We could call intrinsic or spin part to that contribution to the angular momentum, because polarization can exist by itself with no dependence on the spatial coordinates. Finally we could think about an extrinsic or orbital part of angular momentum: in some beams there is an explicit rotational contribution
207
208
Figure 1. Spiraling phase, exp( — ilrji), 1 = 3.
Figure 2. General astigmatic beam from the Gauss Schell-model family.
when the field, or the cross spectral density function if we are dealing with partially coherent beams, is expressed in terms of the spatial coordinates. Something like that happens to some types of optical vortices. For example to beams with spiraling phase terms (azimuthal angular dependence, shown in Figure 1), such as LaguerreGauss beams.4 It is also the case of general astigmatic beams, such as the Gauss Schell-model beam shown in Figure 2. 5 ' 6 This orbital part is perhaps the most striking and interesting contribution to the angular momentum. There are several interesting and comprehensive papers on the subject of angular momentum of light. To name just a few, we would recommend reading Ref. 4, where the essentials within the paraxial approximation are introduced, Ref. 7, that develops the quantum mechanical approach, the review paper appeared in Progress in Optics,8 and the special issue on Atoms and angular momentum of light, appeared in the Journal of Optics B: Quantum and Semiclassical Optics in 2002, 9 that brings us to the front edge of the subject. In those works many more references can be found, covering this field from the experimental results to the deep
209
theoretical point of view. In this lesson we will concentrate in the paraxial approach to the problem, working within classical electromagnetic theory. Although this approach simplifies some aspects, it brings out most of the interesting features of the subject. In Section 2 the basics of the paraxial approach are introduced, while the case of angular momentum in partially coherent beams is contemplated in Section 3. Section 4 considers several examples within Laguerre-Gauss and Gauss Schell-model beam families. Finally, Section 5 is devoted to the experimental problem of how to detect and measure the angular momentum of light. 2
Angular momentum in paraxial approximation
The first step is to define what we intend to calculate. If we are interested in angular momentum we know from the usual definitions in mechanics that we need to start with linear momentum. Recalling classical electromagnetic theory, linear momentum is related to the Poynting vector of the field S, that is obtained from the electric field E and the magnetic field B. In free space and SI units, S = c2e0ExB,
(1)
where c is the speed of light and SQ is the permittivity of free space. From the Pointing vector we could obtain the energy flux and the linear momentum flux. But instead of using fluxes we will work with densities, following Ref.4 . Doing so we find out that the linear momentum density is given by 7 > = zl s = £ o E x B . c
(2)
Now we can obtain the angular momentum density as the cross product of the radius vector and the linear momentum density,
M = r xP- — r x S = £ 0 r x (E x B).
(3)
At this point it should be noted that the choice between using fluxes or densities is not trivial. I can be shown1 that the rigorous and natural way to attack the problem would be to use fluxes. Otherwise there does not appear to be a simple way to separate angular momentum in its spin and orbital parts. In our case we are going to use a paraxial approximation, and this separation problem does not exist . The second step is to find an approach to problem, or how to find an adequate way to introduce the electric and magnetic fields in Eq. (3). We are going to follow the approach used in Refs. 10 and 4. If we have light propagating in the z direction we would be interested in the non trivial part of the angular momentum, that is, its longitudinal z component. Since M. oc r x E we have to accept some non-transversal component in our light. Otherwise we cannot have a longitudinal component in M. In the usual simplified paraxial approach that component is usually neglected. To include it in our calculations we can consider as usually a light beam propagating in z direction with a slowly varying amplitude in transversal direction. But instead of starting from the electric or magnetic fields we could use the vector potential A
210
and the scalar potential (p,
!/,z)e-'(""-^e;
(4)
where AQ(X, y, z) and if>o(x,y,z) are the slowly varying amplitudes, ui is the frequency, k = 27T/A, A being the wavelength, and e = ax + /?y is a complex unit vector that represents polarization. For example, e = x for linear polarization along x, and e = (x + iy)/\/2 for circular polarization to the left. We can use Lorentz gauge and obtain the electric and magnetic fields from A and
where the overbar indicates time average, * indicates complex conjugation, Vj_ stands for the transverse components of the gradient and instead of using AQ we have introduced a new quantity if) defined using A0(x,y,z) = --
— i/j(x,y,z).
(6)
Looking at Eq. (5) we find that |^|2 gives the irradiance, the z component of S, so that <j> can be identified with what we usually call amplitude. Once we have the Poynting vector S we can calculate what we are looking for, the (time averaged) longitudinal component of the angular momentum. Using Eq. (5) and cylindrical coordinates we have, o2 L —SI16 i ^ o 2 v *• = c3l» =kc
r
^
/
f-i
or
We have found the result we were looking for. Now from this equation we can conclude that there are two sources for angular momentum. The first term in Mz depends on polarization, and represents the intrinsic or spin part of angular momentum. As expected, this term is zero for linearly polarized light and has opposite signs for left and right circularly polarized light. The second term comes from an azimuthal dependence of the field, and represents the extrinsic or orbital part of angular momentum. As we anticipated before, we have found that within paraxial approximation angular momentum splits clearly in spin and orbital components. We could move forward and calculate the angular momentum flux through a given z plane. Such value can be obtained from the integral J = c 11 M dxdy. •)J
Using Eq. (7) and similar expressions for the other two components we find
(8)
211
Jy = -(-{x) + z(u)),
(10)
Jz = -((xv) - (yu)) + -s3;
s3 = 2Ilm(a*/3),
(11)
where we have introduced the notation used in the beam characterization formalism based on the irradiance second-order moments,11"15 / is the beam power and 83 is the fourth stokes parameter. Within the second order moments formalism beam power and irradiance second order moments are defined as / = j! Szdxdy=
ff\^\2dxdy,
(x) = j I x\^dxdy,
(12) (13)
<•>=57 (15) (16) Definitions for (y) and (v) are symmetric to those given for (x) and (u). In the previous expressions, ( x ) , (y), (w), (v) are the so-called first-order irradiance moments of the beam. They represent the beam center of gravity ((x), (y)) and the mean direction of the beam propagation ( ( u ) , (v)). With that in mind Eqs. (9) and (10) come to no surprise. A beam that forms an angle with the z direction or does not pass through x = y = 0 will introduce angular momentum along the transversal directions. But these are trivial contributions to the angular momentum. If (x) = (j/) = (w) = (v) — 0, something that can be accomplished with a suitable translation and rotation of the Cartesian coordinate system, the only term that remains is the non-trivial longitudinal angular momentum component Jz. As expected, this component of the angular momentum flux is divided in two terms. The intrinsic or spin term,
depends on the Stokes parameter 83 and therefore it is proportional to the amount of circular polarization of the beam. On the other hand, the extrinsic or orbital term, Jf = -c ((xv) - (yu)) ,
(18)
is expressed as the difference of two second-order beam moments that, for coherent fields, are related to the spatial structure of the phase. The interesting point is that we have found a link between the angular momentum problem and the beam characterization formalism based on the irradiance moments. Such characterization relies on the so-called beam matrix p.11-13.14 The beam matrix P is a real 4 x 4 symmetric positive definite matrix that contains all
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the second-order moments of the irradiance. Therefore, it has only ten independent elements. It is not too difficult to relate nine of these elements with spatial characteristics of the beam such as beam widths, divergences and curvature radii. Equation (18) allows us to identify the remaining term, (xv) — (yu), the antisymmetric part of one of the submatrices of the beam matrix, with the orbital angular momentum transported by the beam. This identification with angular momentum closes the physical interpretation of all the elements of the beam matrix P. 3
Partially coherent beams
Although Eq. (18) was derived only for totally coherent fields, we could think about using it also for general partially coherent beams. For such beams the beam matrix P is also defined, but in this case the second-order moments have to be defined starting from the cross-spectral density function or from the Wigner distribution of the beam. Therefore Eq. (18) can be used, but we need to justify that it has the adequate physical meaning. The plausibility of Eq. (18) for partially coherent beams can be rigorously justified by means of the so-called Mercer expansion of partially coherent beams.16 The cross-spectral density function of partially coherent beams can be expanded as an incoherent superposition of cross-spectral density functions associated to coherent fields: n(ri,r2),
(19)
Now we can apply Eq. (18) to each coherent component F n . But since each one of this Jz contributions is mutually incoherent, we can add them to obtain the total orbital angular momentum. It is found that Eq. (18) is recovered in this way,17 but now its use for partially coherent beams is justified. 4
Examples and sources of beams with angular momentum
The most typical type of beams with angular momentum are the Laguerre-Gauss modes. These beams can have orbital angular momentum if their azimuthal index / is not zero. In such case they have a spiral phase (Figure 1), and the angular momentum flux transported by the beam is given by Jz =-l(jj
(20)
We could say that Laguerre-Gauss modes carry an angular momentum of ITi per photon (note that this a way of speaking, since we have used just classical electromagnetic theory). The orbital angular momentum contribution has to be added to the spin (polarization) term, giving different distributions for the total angular momentum, as shown in the example given in Figure 3. Gauss Schell-model beams form another family of beams that can transport angular momentum. Gauss Schell-model beams are defined as the most general beams with Gaussian profile both in irradiance and transversal spatial coherence. These beams are characterized by their beam width matrix 07, their transversal
213
Figure 3. Irradiance profile and angular momentum contributions rpj. for the Laguerre-Gauss mode LGj (radial index p — 0; azimuthal index I = 1). Linear polarization (33 = 0) and circular polarizations (to the left, 53 = 1; to the right, 83 = —1) have been considered.
coherence width matrix ag, their curvature radius matrix R and their twisted phase parameter r.14 For them the orbital angular momentum is given by6 J* = ^ {tr [(R-V2- - a] R-1) J] - 2rtr(
(21)
where J is a 2 x 2 matrix, J = ((0, 1), (—1, 0)). An example of Gauss Schell-model beam with angular momentum is presented in Figure 2. The beam in this figure presents a twisted irradiance profile, but it should be pointed out that there are Gauss Schell beams transporting orbital angular momentum and having a rotationally symmetric irradiance profile at the same time.6'14 Another interesting topic is how to create beams with orbital angular momentum. In some cases lasers can oscillate naturally or can be forced to oscillate in a Laguerre-Gauss mode. But in other cases beams with angular momentum have to be created starting from other types of beam. Astigmatic mode converters,18'19 spiral phase plates,20 and holograms21 have been proposed and used. An astigmatic mode converted composed of two cylindrical lenses is schematized in Figure 4.
214
Figure 4. Astigmatic mode converter. Two cylindrical lenses can be used to convert a HermiteGauss mode HGio into a Laguerre-Gauss mode LGj.
5
Angular momentum measurements
Now that we have established that beams can transport angular momentum it would be interesting to know how to measure it. We can follow two approaches. In the first one we can try to measure in a direct way the mechanical effects of the angular momentum. In the second one, that we can call beam characterization approach, optical elements are used to transform angular momentum in other properties of light, so that indirect methods are used in the measurement. In the following we will consider both methods. The most direct approach when we want to quantify angular momentum in a light beam is to try to measure its mechanical effect. The problem is that we are talking about a very small effect. So we have to consider two types of experimental setups: a) those based on very sensitive and well designed macroscopic devices; and b) experiments based on the movement of small particles, where small forces produce big effects. We also have to choose what type of interaction we want between the light beam and the mechanical system, because there are mechanical systems that respond only to spin angular momentum (polarization), others that respond only to orbital angular momentum and systems that respond to both of them. This is schematized in Figure 5, where torsional pendulums are considered. Torsional pendulums are one of the most sensitive mechanical devices. If an absorbing plate is suspended from a thin wire and a light beam is directed to the plate, a fraction of light will be absorbed by the plate. That means that the fraction of angular momentum that is not carried away by the reflected beam will be transferred to the absorbing plate. In this case both spin and orbital angular momentum are transferred to the plate. On the contrary, if a birefringent plate is suspended instead of an absorbing plate we will just detect the spin angular momentum. The birefringent
215
IHHH
tlttttt
ttttttt
birrefringent plate
absorbing plate
astigmatic system
Figure 5. Transfer of the angular momentum of light to a plate suspended from a torsional pendulum. An absorbing plate can be used to measure the spin and orbital components, a + I while birefringent plates and astigmatic optical systems are only sensitive to the spin a or orbital / components, respectively.
plate modifies the polarization state of the beam, changing the 83 parameter of the beam. Angular momentum has to be preserved, and therefore some angular momentum has to be transferred to the birefringent plate. Finally, astigmatic optical systems change the amount of orbital momentum transported by light beams. In this way if an astigmatic optical system is suspended from the torsional pendulum, the orbital angular momentum component could be transferred to the pendulum and detected directly. A cylindrical lens could be the simplest astigmatic optical system. The first direct measurement of the mechanical effects of a light beam was done by R. Beth in 1936.2 He made a really neat and nice experiment. In his work R. Beth measured the spin angular momentum of light using a torsional pendulum with a birefringent plate. His experiment gave not only the qualitative behavior, but also quantitative results that matched the theory. Basically, he suspended a half-wave plate in a torsional pendulum between two quarter-wave plates. When linearly polarized light propagates up through the first quarter-wave plate it is converted into circularly polarized light. Then its polarization state is reversed by the halfwave plate that forms the torsional pendulum. After that light passes through the second quarter-wave plate and a mirror sends the beam down again. In its second pass the beam goes down through the same changes but with reversed sign if we think in terms of propagation direction. The result is that the angular momentum transferred to the half-wave plate is four times what would have been transferred to an ideal absorbing plate illuminated by a circularly polarized beam. Reference2 is also worth reading just because the way it is written. In his paper R. Beth explains step by step the theory, all the precautions taken in the apparatus design, how the measurement were carried out and checked and how the measurements compare with the theory. Of course the fact that it was a successful experiment helped a lot in order to have a nice paper. In any case, to have an idea about the quality of the experimental work, we have to mention that the torsional pendulum used by R. Beth remains as one of the most sensitive torque measuring devices ever made. 22 A second measurement of the spin component of the angular momentum was made by P. J. Allen in 1966.23 He used 1 W of circularly polarized microwave radiation and a thin dipole suspended a distance A/4 away from the closed end of a circular waveguide. When circularly polarized radiation hits the dipole it
216
starts to rotate as a result of the transfer of angular momentum. He found that the dipole rotation rate / was proportional to the incident power, and that the reflected radiation also changed its frequency, following the law AF = — 2f. Analogous experiments trying to measure the orbital component of the angular momentum using a torsional pendulum have been attempted by M. Beijersbergen in his doctoral thesis.24 Unfortunately the experiment was not successful, and probably for this reason the results have not been published in a regular journal. Nevertheless it is a very interesting work, and the chapter devoted to this measurements in Ref.24 is worth reading. They tried to follow an experimental design parallel to that used by R. Beth,2 employing a cylindrical lens suspended from a torsional pendulum and a mirror to obtain a double pass device. Even though they used all the available modern technology they were unable to separate the effects due to the torque coming from orbital angular momentum from those resulting from unintentional torques. The lack of rotational symmetry of the system used by M. Beijersbergen, compared to that used by R. Beth, appears to be the main cause of sensitivity to all kinds of non-ideal conditions in the experiment. Cylindrical lenses are needed and they are asymmetric by construction. Also, angular momentum is only transferred in the second pass through the cylindrical lens, not in both passes,18 increasing the asymmetry of the setup. As it was mentioned before, another approach is to use very small particles. When small particles are used they can be trapped by the light beam, and at the same time angular momentum can be transferred from the beam to the particle. In this way experimental evidence of the existence of orbital angular momentum was obtained by He et al.25 They created a beam with angular momentum using a hologram. Other interesting results are those presented in Refs. 26 and 27 , where it is shown that orbital angular momentum is mechanically equivalent to spin angular momentum, and they can be added or subtracted. There have been, of course, much more recent results, since these experiments take us to the field of optical tweezers and optical spanners, with very interesting and innovative developments.28 The last approach we are going to consider in the measurement of angular momentum is the beam characterization method. Instead of a direct measure of the mechanical effects we can transform angular momentum into other properties of light that could be easier to measure. This is quite clear in the case of the spin part of angular momentum. Polarization of fully polarized light can be linear, circular or elliptical. The customary method to find out the state of polarization of this type of light is to use linear polarizers and retarder plates. Any method based on the mechanical effects of angular momentum of light would be rather unusual. Something similar happens with the orbital component of angular momentum of light. As we saw in Sections 2 (Eq. (18)) and 3, the orbital angular momentum transported by a beam can be obtained from the second order moments that appear in its beam matrix. An experiment based this type of measurements is described in Ref. 29. The authors used the setup shown in Figure 6 in order to quantify the orbital angular momentum carried by the doughnut beam produced by a TEA COs laser. In a first set of measurements no cylindrical lens was used. The squared beam widths of the beam at different z planes were measured and fitted to parabolas. From those fitting parabolas eight elements of the beam matrix were obtained, while
217
ruler
Figure 6. Setup used in Ref.29 in order to measure the orbital angular momentum transported by the beam produced by a TEA COg laser.
the other two remained coupled. In order to remove the coupling a cylindrical lens was used in a second set of measurements. It was found that the beam transported an orbital angular momentum lz ~ —H per photon and it was a nearly pure LG^"1 mode. Acknowledgment s I would like to thank G. Neme§, 3. Movilla, F. Encinas-Sanz and A. E. Siegman for their collaboration and for many useful discussions with them. I am also indebted to P. M. Mejias, R. Martinez-Herrero and G. Piquero for their interest and support. This work was partially financed by the Ettore Majorana Foundation and Centre for Scientific Culture and by the Ministerio de Ciencia y Tecnologi'a of Spain under Project BFM 2001-1356, within the framework of EUREKA project EU-2359. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
S. M. Harriett, J. Opt. B: Quantum Semidass. Opt. 4, (2002). S7 (2002). R. Beth, Phys. Rev. 50, 115 (1936). 3. D. Jackson, Classical electrodynamics (John Wiley & Sons, New York, 1962). L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw and J.P. Woerdman, Phys. Rev. A 45, 8185 (1992). J. A. Arnaud and H. Kogelnik, Appl. Opt. 8, 1687 (1969). J. Serna and J. M. Movilla, Opt. Lett. 26, (2001). 405 (2001). S. J. van-Enk and G. Nienhuis, Opt. Commun. 94, 147 (1992). L. Allen, M.J. Padgett and M. Babiker, Prog. Opt. 39, 291 (1999). Special issue: Atoms and angular momentum of light, J. Opt. B: Quantum Semidass. Opt. 4, Si (2002). H. A. Haus, Waves and fields in optoelectronics (Prentice Hall, Englewood Cliffs, N. J., 1984) M. J. Bastiaans, /. Opt. Soc. Am. 69, 1710 (1979). S. Lavi, R. Prochaska and E. Keren, Appl. Opt. 27, 3696 (1988). J. Serna, R. Martinez-Herrero and P.M. Mejias, J. Opt. Soc. Am. A 8, 1094 (1991). G. Neme§ and J. Serna, in Proceedings of the 4th Workshop on Laser Beam and Optics Characterization, Adolf Giesen and Michel Morin, eds., 92 (IFSW, Stuttgart, 1997).
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15. ISO 11146:1999 (International Organization for Standardization, Geneva, 1999). 16. E. Wolf, J. Opt. Soc. Am. 72, 343 (1982). 17. J. Serna and J. M. Movilla, Opt. Lett. 26, (2001). 405 (2001). 18. M.W. Beijersbergen, L. Allen, H.E.L.O. van der Veen and J.P. Woerdman, Opt. Commun. 96, 123 (1993). 19. H. Laabs, C. Gao and H. Weber, J. Mod. Opt. 46, (1999). 709 (1999). 20. M.W. Beijersbergen, R.P.C. Coerwinkel, M. Kristensen and J.P. Woerdman, Opt. Commun. 94, 321 (1994). 21. H. He, N.R. Heckenberg and H. Rubinsztein-Dunlop, J. Mod. Optics 42, 127 (1995). 22. G. T. Gillies and R. C. Ritter, Rev. Sci. Instrum. 64, 283 (1993). 23. P. J. Allen, Am. J. Phys. 34, 1185 (1966). 24. M. W. Beijersbergen, Phase singularities in optical beams (Ph. D. thesis, Leiden University, Leiden, 1996). 25. H. He, M. E. J. Friese, N. R. Heckenberg and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995). 26. M.E.J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N.R. Heckenberg, Phys. Rev. A 54, (1996). 1593 (1996). 27. N. B. Simpson, K. Dholakia, L. Allen and M. J. Padgett, Opt. Lett. 22, 52 (1997). 28. K. Dholakia, G. Spalding and M. MacDonald, Phys. World October 2002, 31. 29. J. Serna, F. Encinas-Sanz and G. Nemes, J. Opt. Soc. Am. A 18, 1726 (2001).
EXPERIMENTAL ASPECTS IN BEAM CHARACTERIZATION ALBERTO SONA Department of Physics, University oflnsubria Via Valleggio 11 - 22100 Como (Italy) E-mail: [email protected]
Abstract Beam characterization is the pre-requisite of any research exploiting light beams, especially in cases involving laser beams. One can rely on the beam parameters provided by the manufacturer but often they are inadequate and/or not sufficient for the experimental data analysis. The full characterization of a laser beam can require the determination of many parameters (about ten for a generic beam); however for symmetrical beams the significant ones can reduce to only to a few. The characterization can be performed with the accuracy requested by the application and limited to the relevant parameters. The main parameters of interest will be defined and the measurement procedures and equipment will be discussed. The ISO standards consider the following parameters mainly of interest for industrial applications: 1) Beam widths, divergence angle and beam propagation ratio. 2) Power, energy density distribution 3) Parameters for stigmatic and simple astigmatic beams 4) .Parameters for general astigmatic beams 5) Geometrical laser beams classification and propagation 6) Power, energy and temporal characteristics 7) Beam positional stability 8) Beam polarization 9) Spectral characteristics 10) Shape of a laser wavefront: Phase distribution All the above points will be briefly discussed as regards the experimental problems involved. Special attention will be given to the methods for measuring the intensity distribution and to the related instrumentation to derive the Beam propagation ratio, the Beam Quality factor M2 or the Beam Parameters Product. Examples of the parameters relevance for specific applications will be given. Depending on the spectral range, specific detectors are used: CCD cameras with detector arrays in the visible and near infrared, thermocameras with a single detector and scanning system for the medium and far IR. The major problems in data collection and processing will be discussed. Another new and not yet fully investigated area is the characterization of laser beam by wavefront measuring instruments. One possible approach is the use of selfreferencing interferometers such as the point diffraction interferometers. Alternatively wavefront gradient measuring instruments can be used such as the Hartmann-Shack sensors.
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Wavefront intensity and phase joint distributions can now be measured at the same time. This can provide in addition new methods to derive the modal content. A short review of the experimental problems in this area still looking for a practical solution will be given.
References 1. Proceedings of the 6yh Intenational Workshop on Laser Beam and Optics Characterization: LBOC6 Munich, 18-20 June 2001
SEMINARS
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APPLICATIONS OF THE TIME-RESOLVED INTEGRAL-GEOMETRIC METHODS FOR THE COMPOSITE MATERIALS DIAGNOSTICS ABOUTRAB A. ALIVERDIEV Institute of Physics Daghestan Scientific Centre of Russian Academy of Sciences (DSC HAS) M. Yaragskogo 94 - 367003 Makhachkala (Russia) E-mail: [email protected]
Abstract Here we present some specific application of non-destructive diagnostics in the framework of chronotomography. We designated key difficulties, bound with strongly restricted number of angular projections having a place in actual physical examinations, and have considered series of mathematical approaches (including based on Hartley transformation with as a base function), allowing to overcome them. We consider also the direct Radon transformation in time-space square in application to precision velocimetry.
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THE ELECTRODYNAMICS OF PROCESSES HAVING PLACED IN THE VOLUME OF A MONOCHROMATIC COHERENT POLARIZED RADIATION BEAM ALEXANDER A. ALIVERDIEV, ABOUTRAB A. ALIVERDIEV, and A. A. AMIROV Institute of Physics Daghestan Scientific Centre of Russian Academy of Sciences (DSC RAS) M. Yaragskogo 94 - 367003 Makhachkala (Russia) E-mail: [email protected]
Abstract Here we present the analysis of processes happening in a volume of a monochromatic polarized radiation beam. This analysis is carried out by the means of the time-space function of the projections of a dot vector electrodynamic charge, propellented on a spiral, on two crossly perpendicular planes. All quantum properties (including of a spin presence) are explained. There is obtained, that the magnification of intensity of monochromatic polarized radiation has a limit, which depends on a radiation wavelength. The formula for the calculation of a fine structure constant is obtained.
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NATURALLY GENERATED BEAMS BY QUASI-OPTICAL PHENOMENA
MARIA BRANESCU National Institute for R&D on Material Physics, P. O. Box MG 7, Bucharest, Romania, E-mail: [email protected]
The paper describes the specific quasi-optical phenomena involved in the generation of the celestial and biological radiation. It examines the generation of the free space and guided biological beams in the presence of the ambient celestial primordial isotropic background. Josephson and Kerr mechanisms provide an explanation of the concept of bio-coherence as a novel possibility of electromagnetic communication.
1
Introduction
The goal of this paper is to present a description of different aspects and phenomena that occurs in the quasi-optical spectrum (QOS) involved in the generation of the celestial and the biological radiation. The living systems detect and use all the biosphere ambient conditions: temperature, geomagnetic field, Earth atmosphere and celestial radiation. Nature uses in an adequate mode the specific phenomena and properties of each spectrum range, i.e. visible, infrared, microwave, etc. The model of the QOS coherent dynamics was introduced by H. Frohlich1 in the 70' and developed by Wu and Austin2, E. Del Giudice et a/3'4, C. W. Smith5, F. A. Popp6, and others. Frohlich has proposed coherent dynamics as the fundamental driving force of living processes. The QOS radiation has long wavelengths (sub-mm to mm), corresponding to little energy per quantum and high medium penetrability7. The biological cells use these properties in the molecular transport and high sensitivity communication. Two specific properties of the living systems made possible the occurrence of Josephson-like phenomena, at the ambient temperature. The living systems are relatively stable but far from equilibrium and have specific dielectric properties involving very high electric fields. Using long-range coherent Josephson-like phenomena, the living system surpasses the disadvantage of the QOS small signal to noise ratio. Experimentally it was proven the generation of the high coherent radiation by the metabolic dynamic processes of the living systems4. The Josephson-like generated beam propagates in the free space or is guided inside the living cells. In the living cells, inside the cytoplasm, the beams are guided by the filamentary cytoskeleton formations8. The living cell cytoplasm presents birefringence in the existing electric field3. This fieldinduced anisorropy makes possible the occurrence of Kerr-like phenomena, known from nonlinear physics of lasers9' 10. Kerr-like mechanisms assure the guided transport of the high coherent microwave through the hostile watery medium of the cytoplasm3'8. The celestial primordial background (CPB) presents a wide spectrum of a weak microwave isotropic radiation. Almost all the spectrum of the CPB is situated in the QOS range". As a development of the Frohlich model, in this work we hypothesize that any
227
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weak, microwave mode of the CPB may be tuned, via Josephson-like phenomena12, and used in the coherent dynamics of the living processes. Section 2 presents the concepts, the specific properties, and the propagation of the QOS free space and guided beams13'14. It also discusses the detection techniques for the QOS radiation7. Section 3 deals with the QOS celestial radiation sources, continuum or line spectra, and their astronomical importance11' 15~18. Section 4 treats the Josephson-like1 generating mechanism of the high spatial and temporal coherence beams in the living systems. Special attention is drawn to Kerr3' 8 and Davydov3' 19 mechanisms observed in the propagation of the guided biological beams. Section 5 examines a possible implication of the CPB in the quantum macroscopic Josephson-like phenomena observed in the living matter. The conclusions are given in the last section.
2 QOS Beam Concept and Propagation. Specific Properties and Detection Techniques of the QOS Radiation The QOS comprises wavelength roughly in the sub-mm and mm range (0.1 mm -10 cm). It represents the boundary between the high frequency electronics and very far infrared optics7. The specific characteristics of the QOS radiation come from this boundary, where kT » hv. This is situated at a frequency of about 6 THz or a wavelength of 50 urn. It corresponds to the ambient temperature of about 300 K, or 27° C, and quantum energy of 4 x 10"21 J. This quantum energy has about the value of the environmental thermal background. In other words, the signal becomes comparable to the thermal ambient noise. At frequencies below 6 THz, the wave properties predominate over the quantum properties. The frequencies greater than 6 THz are associated with quanta. Several consequences come from this boundary: - Transversal (spatial) coherence and symmetry properties of the QOS beams justify the concept's application of the Ideal Gaussian Beam in the QOS. The beam definition contains the condition of paraxiality (60 < 20°), and the condition w0/A, > 10, where 60 is the half angle divergence and w0 is the waist half diameter. Paraxial distributions with Wo/A, between 1 and 10 can be called quasi-beams, and distributions with w0/A < 1 and/or no paraxiality can be called non-beams. Free QOS beams can be characterized using the beam propagation parameter, M2, as an extension of the beam concept from the optical spectrum. For symmetrical beams as the stigmatic and the aligned simple astigmatic ones, the M2 parameter is useful to characterize antenna directivities, to calculate the minimum beam spot size after the focusing optics, or to design an appropriately large aperture optical system, avoiding beam truncation. The general astigmatic beams are more complicated, but they can be measured too, and can be transformed into more symmetrical beams14. - In certain materials the refractive index, n, and the permeability, u, can be less than one20. - QOS borrows techniques and components from the visible, the infrared, and the microwave spectrum. These can be applied in the QOS, taking into account such phenomena, as diffraction and dispersion, and the corresponding limitations7. - In the QOS some natural mechanisms to generate high spatially and temporally coherent radiation can occur: astronomic masers and Josephson-like biological generation.
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- The radiation in QOS has a small energy per quantum, close to the resonance value of many molecular and ionic degrees of freedom. Therefore, a wide variety of photon and phonon absorption and emission phenomena are situated in this spectral range. The richness of these phenomena determines or is involved in a multitude of events, as: the radiation sources in radio astronomy; the dynamics of the atmosphere; the structure of the condensed matter; the dynamic structure of the living matter. - The radiation is less scattered in the QOS than in the other spectral domains (visible or infrared). Therefore, in the QOS we can see far away in time and space through dust, fog and haze, with Earth based radio telescopes. QOS represents the cosmic and Earth's atmospheric windows18 (see Figure 1). - The QOS wavelength is sufficiently short for an excellent spatial resolution and reduced antenna size, but sufficiently long for precision pointing of the beam and acquisition data time for the radar. In the QOS the receivers have a wide input frequency bandwidth. The long output time constant and the greater period of the signal provide a greater processing time and therefore more sensibility than in the optical spectrum. The QOS reception techniques, based on very large baseline interferometers with greater collecting area, may provide a better spatial and angular resolution power than in the optical spectrum7'18. - The superconductor devices, based on quantum macroscopic Josephson phenomena, hold the ultimate quantum limit in detection of the QOS radiation. The superconductorisolator-superconductor (SIS) devices have some of the highest-order nonlinearities known in nature. Some can generate high order harmonics, of about 103. Josephson parametric amplifiers work well at cryogenic temperature. These have only one degree of freedom and reach the lowest quantum noise known in the electronic devices. The Josephson generated QOS radiation has high spatial and temporal coherence21.
3
QOS Celestial Sources
Radio-astronomy represents a good tool for studying the Universe on a much larger scale than the optical astronomy can. It answers fundamental cosmological questions, such as how big and how old the Universe is. 3.1
QOS Celestial Sources with Continuum Spectrum
The radiation produced by thermal mechanisms is unpolarized. This radiation obeys the Rayleigh-Jeans law: the radiative flux density of a cosmic object at a wavelength A in the radio spectrum at absolute temperature T is proportional to kT /)C, where k is the Boltzmann's constant. Cosmic primordial background, the relic part of the Big Bang (BB) event, with isotropic continuum spectrum, is generated by thermal mechanism. It has the maximum at 2.7 K. The radiation started to travel approximately one million years after the BB event". Almost all of the CPB energy is situated in the QOS range, because of the red shift of the radiation. The CPB represents the strongest evidence of the existence of the BB event. The directional variation of the CPB temperature is only 30 uK, so the cosmos isotropy is a good approach. The radiation produced by non-thermal mechanisms. The spectral flux density emitted by the radio sources, via non-thermal mechanism, is proportional to A,*, where x is a positive number, of about 0.3-1.3, depending on the generating mechanism.
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Synchrotron radiation is produced by very low concentrations of electrons and weak magnetic field. The electrons velocity is comparable to the speed of light, so the particles are relativistic. The received radiation is linearly polarized. The other effect of high speed is the spread of the spectrum, which results from the "beaming" of the emission in the forward direction11. The "beaming" means that the observer only sees sharp pulses of emission at shower instant when a gyrating electron happens to be directed towards the observer. Synchrotron radiation covers a wide frequency band, since the spectrum of a short pulse always contains a wide range of frequencies. The spectrum of radio emission produced by the synchrotron process has the intensity proportional to X", where x is between 0.2 and 1.2. It is a valuable indicator of the sizes of small but powerful sources. Galactic radio emission contains synchrotron radiation. The "shock waves". Large explosive events, such as solar eruptions or supernovae, present radio emission. The most powerful types of solar radio emission are the bursts associated with solar flares. The non-thermal mechanism18 emits the quiet sun radiation. 3.2
QOS Coherent Celestial Sources with Line Spectra
The various possible "pumping" mechanisms of the QOS coherent celestial sources include infrared radiation, collisions or chemical reactions18. Stellar and interstellar absorption and emission lines. These spectra belong mainly to the QOS. The most important is the CO molecule 2.6 mm line, induced by collisions with hydrogen molecules that are about 10,000 times more abundant. The widespread presence of CO is particularly valuable as an indicator of the principal distribution of molecular hydrogen and the cold molecular gas cloud, which are the seats of the star formation17. Celestial Microwave amplification by stimulated emission of radiation (Maser) Sources represent the brightest celestial radio-sources from small volume of gas. They have very narrow lines with extremely high intensity16. The most important masers are HO, H2O and SiO. 4
QOS Biological Sources
Active biological systems have three main properties1. First, they are relatively stable but far from equilibrium, pointing to existence of metastable states. Second, they exhibit a non-trivial order that leads to coherence and high stability of wave propagation inside the biological tissues. Third, they have extraordinary dielectric properties. The metabolic activity maintains high electric fields in the membranes and in the strongly electrical polarizable constituents. 4.1
Biological Josephson-like Phenomenology
Most living system operate at about 300 K ambient temperature, where the signal is comparable with environmental noise. It is well known in electronics that the systems, which use coherent detection, can operate at a signal to noise far below unity. The coherence decreases with the increasing number of degrees of freedom. Exploiting the dielectric properties of the electrically polarizable constituents from the active biomaterial, non-equilibrium phase transitions could be established. Frohlich proved that the quantum macroscopic long-range coherence and order in space and time might exist in the living systems.
231
Frohlich1 assumed emission and absorption energy transfer between two normal modes of the vibration system with participation of the heat bath. The Frohlich system behaves like a special type of non-linear system. The non-linearity can arise in electrically strong polarizable constituents as proteins, cytoskeleton polymers, including the living cell membrane, subjected to a high electric field (107 V/cm). The non-linear interaction of the Frohlich system with the heat bath induces the energy exchange between the spectral components. The energy is transferred between the wide spectrum polarized waves and the heat bath via long-range interactions. The cooperative phenomena create the spacetime coherent regions within the living systems. These represent the same type of correlation as found in the Josephson-like phenomenology. Quantum macroscopic coherent states of Bose-like type may occur at the ambient temperature. Frohlich's model introduces the so called "non-equilibrium superconductivity". In Ref. 4 the experimental evidences of coherent radiation and Josephson-like phenomena occurring in the living matter are reported. In the presence of the magnetic field, Josephson-like phenomena generate weak highly coherent radio frequency radiation outside the living cells. 4.2
Quasi-Optical Kerr Phenomena and Biologically Guided Beams
The dominant component of the heat-bath model is water, which represents more than 70% of the living matter. It strongly absorbs in the microwave spectrum, representing a cytoplasm "hostile medium" for the polarized microwaves. The polarized waves have been introduced as a model for order at very long distance1. The long-range communication uses the cytoskeleton as a waveguide structure, to penetrate through the cytoplasm. The cytoskeleton continuously changes its shape and size with time, cell activity, and external stimuli. Its main components are polymers of proteins with high values of the permanent dipole moments3'8. These components are: tubulin actin (with the diameter of about 25 nm) in the microtubules, intermediary filament actin (with the diameter of about 10 nm) and actin (with the diameter of about 6 nm) in the microfilaments9. Biological media show birefringence at rather low values of the existent electric field (104 V/m). That is a necessary condition for Kerr phenomena and the associated selffocusing and self-trapping effects, known from nonlinear optics9'10. The filamentation of the electromagnetic field produces strong field gradients on the filament boundaries. As a consequence, frequency-selective forces appear. When resonance occurs between the field frequency and a molecular mode, condensation of selected molecules can occur at the interface of the electric field filaments. The field intensity remains constant along the filament, giving rise to a complete transparency of the medium. These filaments behave as optical waveguides, where the beam propagates without divergence. The biological system has loosely bound networks of microtubules, microfilaments and microtrabeculae immersed in cytoplasma, which fills up the interstitial space3. The far infrared Raman spectroscopy is used to put in evidence the long-range coherence, or the dynamical order in living cells3. 4.3 Soliton Mechanism Davydov19 proposed a possible mechanism for energy transfer in proteins' a-helix. According to this mechanism some local excited vibrational states behave as solitons, called the Davydov solitons. They can exist and propagate only in one-dimensional
232
systems. The velocity of a soliton is always lower than the sound velocity, so it does not emit photons. This partially explains the great stability of the solitons and the fact that they do not change their form with time3. This mechanism can exist in microscopic, mesoscopic, and / or macroscopic systems supporting waves.
Galactic center Galactic pole Atmospheric absorption noise
300
100
m
10
m
i
100
10
Wavelength Figure 1 . Antenna sky noise temperature as a function of frequency and angle from zenith18.
5
CPB - Ambient Biosphere Condition
The ultimate threshold for magnetic effects is the linkage of the elementary magnetic flux 0 = h/2e = 2.07 x 10" Wb (or quantum of magnetic flux) with the cross-sectional area of a living cell, considered an area of coherence. The devices based on Josephson-like phenomena (superconductor quantum interference devices) can detect magnetic quanta. Josephson and SIS detectors can detect quanta in the QOS. These Josephson phenomena devices represent the ultimate limit on quantum sensitivity devices known to date. The living organisms may also detect and use the magnetic flux quanta4. That fact constitutes the strongest evidence that Josephson-like phenomena occur in the living matter. Living cells thus posses the mechanism capable to detect and use the QOS radiation naturally produced by celestial sources or by another living cell. The CPB peak flux density11 is situated at a wavelength of about 1.9 mm or 160 GHz, corresponding to quantum energy of about 6.34 x 10'4 eV.
233
5.1 Biological Josephson Magnetic Field Quantization in the CPB Radiation Field In the presence of the CPB microwave field, the Josephson-like multi-photonic generation processes may occur. The magnitude of the steps in the magnetic field quantization, observed experimentally in the voltage-current characteristics4, is proportional to the energy of the QOS quanta. We hypothesize that some of the observed steps may occur as a consequence of the ambient CPB radiation and the geomagnetic field, when the energy of their quanta is tuned to the specific physiological conditions of the living cells. The Josephson-like generation observed in Ref. 4 at about 107 Hz, means that a phonon generation of the relaxation process had taken place. Further investigations must be done to study if the Josephson photon22 generation in the recombination process at 109 Hz does exist. It is known that the superconductor junctions generate weak, high spatially and temporally coherent, QOS beams. The challenge for the new investigations is to detect and measure the free space, highly coherent, weak QOS beams, produced by biological quantum macroscopic Josephson-like phenomena. 5.2
CPB in the Biological Josephson Parametric Amplification
Biological Josephson parametric amplification can explain tuned effects of weak external electromagnetic fields by biological systems. Amplification, in which two of the coupled modes drive the third, may even be several orders of magnitude. The amplification is conditioned by the "pumping" signal, the excitation of the system. Therefore, the biological effect depends on the excitation, i.e. on the internal state of the system. The Frohlich model introduces the thermal environment as the heat-bath. We consider that the CPB represents the second heat-bath used in the parametric Josephson amplification as the second pumping wave source. The CPB may be thus involved in the Frohlich mechanism of the coherent dynamics of the living processes. 5.3 CPB is an Important Ambient Condition on the Life Occurrence and Development The water, the major constituent of the living systems, can be itself a coherent system. The water molecules possess a permanent electric dipole moment. So, it becomes macroscopically polarized in the presence of a dynamic coherent excitation. The spatial extent of the coherent domains in water is within an order of magnitude the same as that of a typical living cell23. Thus a water structure may be formatted and memorized. We consider the ambient temperature water surrounded by natural sunlight and the isotropic weak microwave CPB radiation, a nest for the life occurrence and development. The Sun's infrared radiation is partially involved in the ambient heat of Earth. The sunlight represents the source of energy for the photosynthesis mechanism, and is involved also in the sense of the vision. The ultraviolet Sun's radiation is involved in creating protective mechanism for the living tissues. We hypothesize that the CPB radiation is involved in the biologic Josephson-like phenomenology. This would open the way to understand why and how external electromagnetic fields could interfere with the fundamental processes of the living cells.
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6
Conclusions
We conclude by noting some ideas and remarks: - QOS may establish links between unconnected fields as astrophysics, biophysics, microwave electronics, nonlinear optics and physics of superconducting materials. These were developed as different scientific fields but a similarity of some QOS phenomena (for ex. Josephson-like or maser phenomena) is manifested in different fields. - At the beginning of life on Earth, the presence of the CPB and the geomagnetic field has created the conditions for occurrence of a biological mechanism, which could detect and use these fields. This is the Josephson-like biological mechanism, the unique one capable to detect both the CPB and the geomagnetic field with quantum sensibility. - Beside the geomagnetic field and the sunlight, we must consider as an ambient condition for the occurrence and development of the biosphere, the isotropic, always present CPB. Acknowledgments I thank the organizing committee and especially Prof. Massimo Santarsiero, Dipartimento di Fisica - Universita Roma Tre, and Prof. Sergio Martelluci, Engineering Faculty, University "Torr Vergata", Rome, for the possibility to attend the ISQE-35 Erice School on "Free and Guided Optical Beams", and to present this work. References 1. H. Frohlich in Biological Coherence and Response to External Stimuli, ed. H. Frohlich (Springer Verlag, Berlin 1988). 2. T. M. Wu and S. Austin, Phys. Lett. 73A, 266 (1979). 3. E. Del Giudice, S. Doglia and M. Milani, Physica Scripta 26, 232 (1982). 4. E. Del Giudice, S. Doglia, M. Milani, C. W. Smith and G. Vitiello, Physica Scripta 40, 786 (1989). 5. C. W. Smith, Frontier Perspectives 7, 9 (1998). 6. F. A. Popp in Bioelectrodynamics and Communication, eds. M. W. Ho, F. A. Popp and U. Warnke (World Scientific, Singapore 1994). 7. P. F. Goldsmith, Quasioptical Systems: Gaussian Beam Quasioptical Propagation and Applications (Chapman & Hall, IEEE Press, Piscataway 1998), chs. 1, 10, 11. 8. E. Del Giudice, S. Doglia, M. Milani, Physics Letters, 90A, 104 (1982). 9. E. Siegman, Lasers (University Science Books, Mill Valley 1986), 379-382. 10. G. Nemes, Introduction to Nonlinear Optics (in Romanian), (Ed. Acad., Bucuresti 1972), ch. 3. 11. Ryden in Introduction to cosmology, ed. A. Black (Addison Wesley, San Francisco, CA 2003), chs. 2,9. 12. M. Branescu, Proc. SPIE4762, 388 (2002). 13. H. Martin and J. W. Bowen, IEEE Trans. Microwave Theory Tech., MTT-41, 1676 (1993). 14. M. Branescu and G. Nemes in Proc. 7-th ISRAMT, eds. C. C. Penalosa and B. S. Rawat (CEDMA, Malaga 1999). 15. Th. De Graauw in New Directions in Terahertz Technology, eds. J. M. Chamberlain and R. E. Miles (Kluwer Academic, Norwell, MA 1997).
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16. M. Elitzur in Proc. Millimeter Wave Summer School, eds. W. F. Wall (Kluwer Academic, Norwell, MA 1999). 17. F. Wall in Proc. Millimeter Wave Summer School, eds. W. F. Wall (Kluwer Academic, Norwell, MA 1999). 18. J. D. Kraus, Radio Astronomy (Cygnus-Quasar Books, Powell, OH 1986), chs. 7, 8. 19. Davydov, Physica Scripta 20, 387 (1979). 20. G. R. Tremblay and A. Boivin, Appl. Opt. 5, 249 (1966). 21. K. K. Likharev, Dynamics of Josephson Junctions and Circuits, (Gordon and Breach Science, Amsterdam 1986), chs.10-13. 22. W. Eisenmenger and A. Dayem, Phys. Rev. Lett., 5, 125 (1967). 23. G. J. Hyland in What is life?, eds. H. P. Durr, F. A. Popp, W. Schommers (World Scientific, New Jersey 2002), ch. 12.
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LASER AND SATELLITE REMOTE SENSING OF THE OCEAN LUCA FIORANI ENEA, FIS-LAS, Via Fermi 45, 00044 Frascati, Italy The physical, chemical and biological processes taking place in the ocean are of primary importance for the climatic equilibrium of our planet. This explains why the more recent techniques are applied in the hydrographic studies. In this paper, we review briefly one laser system and three satellite radiometers routinely used for the study of the phytoplankton dynamics. Their data are compared and, as a consequence, the accuracy on the measurement of relevant seawater parameters is assessed.
1
Introduction
More than 40 years after the discovery of the laser (conceived by Schawlow and Townes in December 1958 and realized by Maiman in July I960)', laser remote sensors2 and, in particular, hydrospheric lidars3 are mature systems. Also Earth observation satellites4 are now reliable instruments. In particular, ocean color radiometers5 provided us with an unprecedented insight into the phytoplankton dynamics. Unfortunately, their data are not exempt from uncertainties. From one hand, atmospheric corrections6 are necessary to obtain water-leaving radiances, removing from the radiometer measurements the contributions of air molecules and aerosols, which represent up to the 90% of the total in the visible bands. From the other, the bio-optical algorithms7, i.e. the set of semi-empirical equations used to calculate biogeochemical properties from the water-leaving radiances in the visible bands, are usually calibrated with in situ measurements, not representing all the provinces of the world ocean. This could explain why regional models may misestimate primary production: e.g. in the Southern Ocean, some calculations suffered of the "Antarctic paradox"8 (primary production insufficient to support the population of grazers). The important consequences of those studies in decision making, as fishing quota setting, justifies the efforts to assess ocean color data sensed from space borne radiometers with the "sea truth" provided by in situ measurements. In this framework, the ENEA lidar fluorosensor9 (ELF), operational aboard the research vessel (RV) Italica during four oceanographic campaigns, gathered a large amount of data useful for the validation and/or the calibration of space borne radiometers.
la) Figure 1. a) Main picture: RV Italica; ELF and the ancillary instruments are housed in a container (inside the circle). Left insert: external view of the container; note on the left of the container the box accommodating the mirror for water surface observation. Right insert: internal view of the container; ELF and the spectrofluorometer are visible behind and on the left of the operator, respectively, b) ELF (left) and the author (right) during the MIPOT oceanographic campaign.
237
238
2 Instruments and methods ELF (fig. 1) excites water with a laser pulse (at 355 nm) and observes the Raman backscattering of water (at 404 nm) and the fluorescence of chlorophyll-a (at 680 nm). The absolute concentration of that pigment is directly proportional to the fluorescence-toRaman ratio (also known as concentration released in Raman units). The Raman units are then converted into mg m~3, i.e. the absolute concentration is retrieved from the fluorescence-to-Raman ratio, by calibration against conventional analysis on the same water. ELF consists mainly of a frequency-tripled Nd:YAG laser (transmitter) and of a Cassegrain telescope coupled to detectors (receiver): the laser emits a pulse to the sea surface and the telescope collects the backscattered light at different wavelengths. The radiation is then directed through a fiber bundle to spectral bandpass filters and is detected with photomultiplier tubes. Other instruments provide ancillary measurements: lamp spectrofluorometer, pulsed amplitude fluorometer, solar radiance detector and global positioning system. The specifications of ELF are summarized in table 1. The first ocean color radiometer was CZCS10, operated on the Nimbus-7 satellite from 1978 to 1986. Four new ocean color radiometers, SeaWiFS11, two MODIS sensors12 and MERIS13, were launched in 1 August 1997, 18 December 1999, 4 May 2002 and 1 March 2002, respectively. The first, aboard OrbView-2, is a joint venture of Orbital Sciences Corporation and the National Aeronautics and Space Administration (NASA). The two MODIS sensors, integrated on Terra and Aqua, respectively, are part of the NASA's Earth Observing System (EOS), a program including six spacecrafts: Aqua, Aura, ICESat, Landsat 7, SORCE and Terra. MERIS is one of the instruments aboard ENVISAT, the more important Earth observing satellite of the European Space Agency (ESA). The modern ocean color radiometers have been designed mainly for the observation of the phytoplankton biochemistry: in particular, they are expected to provide surface chlorophyll-a concentration over the range 0.05 to 50 mg m"3. The main characteristics of SeaWiFS, MODIS and MERIS are summarized in table 2. Transmitter
Receiver
Electronics
Table 1. Specifications of ELF.
Laser Wavelength Pulse energy Pulse duration Pulse repetition rate Telescope Clear aperture Focal length Center wavelengths Bandwidths Detectors Gate width Dynamic range Bus Central Processing Unit
Nd:YAG 355 nm 3mJ 10ns 10 Hz Cassegrain 0.4m 1.65m 404, 450, 585, 680 nm 5 nm FWHM Photomultipliers 100ns 15 bit ISA -VME mixed VME embedded 486
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Radiometer Scan Width Scan Coverage Nadir Resolution
SeaWiFS ±580.3*, ±45°.0" 2,800 kmM, 500 km" 1.13kma,4.5km»
MODIS ±55° 2,330 km 0.25 kmc, 0.5 km", 1 km8 36 (405- 14,385 nm) 0.903 s» No 12 bits 24 February 2000
Bands 8 (41 2- 865 nm) Scan Period 0.167sh Tilt 20°, 0°, +20° Digitization 10 bits 4 September 1997 Acquisition a Local area coverage. b Global area coverage. c Bands 1 - 2 i Bands3-7 e Bands 8-36 ' Full resolution. a Reduced resolution. h From the period of the continuously rotating scan mirror. ' From the period of the CCD frame. i For MODIS aboard Terra; 24 June 2002 for MODIS aboard Aqua.
MERIS ±34°.25 1,150km 0.3 km', 1.2kms 1 5 (41 2.5- 900 nm) 0.044 S' No 12 bits 22 March 2002
Table 2. Main characteristics of SeaWiFS, MODIS and MERIS.
3
Results and discussion
ELF participated to three Italian expeditions in Antarctica (13th, 15th and 16th in 1997-98, 2000 and 2001, respectively) and to the MIPOT (Mediterranean Sea, Indian and Pacific Oceans Transect) oceanographic campaign (2001-02). Only during the latter both SeaWiFS and MODIS were operational: this is why we will describe it here as a case study of the validation of radiometers by lidar. An example of SeaWiFS calibration by ELF will be also given, exploiting the data of the 13th expedition. CL (line), Cs ( + ) and CM f» 10.OOp
01
1.00
E a. o _p _c O
0.10 -
0.01
0
2000 4000 6000 8000 Lidar measurement number
10000
Figure 2. Surface chlorophyll-a concentration measured by ELF (CL), SeaWiFS (Cs) and MODIS (CM).
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During MIPOT the RV Italica covered the Italy - New Zealand and New Zealand - Italy transects and provided in situ data for 69 days continuously (13 November - 18 December 2001, 28 February - 1 April 2002, respectively) in marine zones among the less studied (Red Sea, Indian Ocean). The temporal and spatial resolutions of ELF, SeaWiFS and MODIS are very different: while a laser pulse is emitted every 0.1 s and the beam footprint on the water surface has a diameter of about 10 cm, a satellite swath is acquired in some minutes and a radiometer pixel sizes around 1 km. The temporal and spatial resolution of the spaceborne sensors degrades further passing from raw to processed imagery. In this study, Level 3 (L3) daily Standard Mapped Image (SMI) products were used both for SeaWiFS and MODIS and only non questionable pixels were retained. L3 is the highest processing level for chlorophyll-a and ensure the best accuracy, 1 day is the finer temporal resolution for L3 files and SMI products of SeaWiFS and MODIS are readily comparable (one SeaWiFS pixel contain exactly four MODIS pixels). As far as depth is concerned, lidar and radiometers probe a similar layer of the order of magnitude of 10 m. Some processing was necessary to compare ELF, SeaWiFS and MODIS data. At first, the four MODIS pixels falling in a SeaWiFS pixel were averaged, so that the two radiometers had the same temporal and spatial resolution (1 day, about 9 km * 9 km, respectively): from now on, this average will be simply called "MODIS pixel". Secondly, all the ELF measurements falling in a SeaWiFS pixel were averaged, thus representing a nearly straight track (length: — 5-10 km, width: -0.1 m) acquired in about 10 - 20 minutes: in the following, we will name this track "ELF granule". Eventually, the measurements carried out by lidar and radiometers were considered concurrent if the in situ track was acquired in the same day as the remote sensed pixel. The remaining dissimilarity in temporal and spatial resolutions can be hardly avoided and can be responsible for part of the disagreement between SeaWiFS and ELF. The concurrent measurements of surface chlorophyll-a concentration carried out by ELF (CL), SeaWiFS (Cs) and MODIS (CM) are shown in fig. 2. In many occasions, ELF was able to carry out measurements when SeaWiFS and MODIS were not: during the entire cruise, 9086 ELF granules were collected and only 1318 of them were concurrent to at least one radiometer pixel (784 with a SeaWiFS pixel and 695 with a MODIS pixel). This partial failure of the space borne sensors is probably due to the cloud coverage lasting for many days, especially in the Indian Ocean, and to the 10-day shut down of MODIS in March 2002. Fig. 2 suggests that ELF performed better in the detection of steep peeks, probably because the residual higher spatial resolution of the ELF granule with respect to the SeaWiFS and MODIS pixel. CL is high (about 1 mg m"3 or more) in 7 cases [during the Italy - New Zealand transect: in coastal waters of the Mediterranean Sea (lidar measurement number - 100), in the Suez Channel (lidar measurement number - 700), in coastal waters of the Red Sea (lidar measurement number ~ 1300), in the Hobart Harbour (lidar measurement number - 4000) and in coastal waters near New Zealand (lidar measurement number - 4600); during the New Zealand - Italy transect: in coastal waters of the Red Sea (lidar measurement number - 7900) and in coastal waters of the Mediterranean Sea (lidar measurement number - 9100)] while Cs and CM, apart from isolated points, are high in 2 cases [during the Italy - New Zealand transect: in coastal waters of the Red Sea (lidar measurement number - 1300); during the New Zealand Italy transect: in coastal waters of the Pacific Ocean (lidar measurement number ~ 5400)].
241
• Southern Ocean
u
, 1-
- ••• • Ross Sea
..-•, ••-
o
•-«•'
a | ]! 0.5 Q.
£•
s
£
^N^-
' \
£ n
-0.5
-0.25
0
logio(Chlorophyll-a [mg m^])
m Oct-97 Nov-97 Dec-97 Jan-98 Feb-98 Mar-9
0.25
a)
Period
b)
Figure 3. a) SeaWiFS bio-optical algorithms: standard and ELF-calibrated (during the 16* and 13* expeditions). The band ratio is the ratio of the remote sensing reflectance measured at 490 and 555 nm. b) Primary production estimated with the ELF-calibrated SeaWiFS bio-optical algorithm in the Southern Ocean (longitude: 157°.5 - 180°, latitude: -61°.875 - -78°.75) and in the Ross Sea (longitude: 157°.5 - 180°, latitude: 72°.07 - -78°.75).
Although Cs and CM are in general higher than CL, the overall agreement between lidar and radiometers is good (of the order of 25% in average): all the sensors show strong concentrations in the Mediterranean Sea, the Red Sea and the Pacific Ocean, especially in the Italy - New Zealand transect. A way to improve our understanding of the phytoplankton dynamics is to calibrate the bio-optical algorithms used by satellite radiometers with the measurements performed by in situ instruments. This approach has been undertaken for SeaWiFS and ELF14 up to obtain new estimates of chlorophyll-a concentration and primary production in the Southern Ocean15. The chlorophyll-a results (fig. 3a) show that, given a band ratio (the ratio of the remote sensing reflectance measured at 490 and 555 nm) measured by SeaWiFS, the standard bio-optical algorithm overestimates the low concentrations (lower than about 1.1 - 1.25 mg m"3) and underestimates the high concentrations (higher than about 1.1 - 1.25 mg m"3). Note that this behavior could explain why the "Antarctic paradox" arose, indicating that ELF could help in solving it. Also in the Southern Ocean, the discrepancy between lidar and radiometer is of the order of 25%, but the behavior is different (in the temperate waters crossed during MIPOT, SeaWiFS and MODIS tended to overestimate in all the range of concentrations). The ELF-calibrated bio-optical algorithms are similar for the 16th and 13th expeditions, suggesting the need of a regional bio-optical algorithm for the Southern Ocean. Examples of primary production estimates provided by such an algorithm are shown in fig. 3b. It is interesting to observe how fast is the increase of primary production near Antarctica (Ross Sea) during spring.
4
Conclusions
The remote sensing data provided by laser systems and satellite radiometers can be combined, thus merging their advantages (accuracy of the laser systems and coverage of the satellite radiometers). Moreover, the in situ instruments can provide information even when the sea is screened by clouds.
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The comparison of the imagery provided by SeaWiFS and MODIS with the measurements performed by ELF suggests that the final accuracy in the chlorophyll-a concentration is of the order of 25%. With this encouraging finding in mind, the SeaWiFS bio-optical algorithm has been calibrated in the Southern Ocean with the ELF data. The results indicate that this new regional algorithm could give better estimates of primary production in the Antarctic seawaters.
Acknowledgements The author is deeply grateful to R. Barbini, F. Colao, L. De Dominicis, R. Fantoni and A. Palucci for the fundamental contribution and the constant encouragement. A special acknowledgment is addressed to S. Fonda Umani et al and V. Saggiomo et al. for kindly providing unpublished data. This work has been supported by the Italian Antarctic Research Program (PNRA), Technology Sector, 5bl Lidar Fluorosensor and 11-5 Palucc5 Projects (for the periods 1996-1998 and 1999-2001, respectively). The author would like to thank the SeaWiFS Project (Code 970.2) and the Distributed Active Archive Center (Code 902) at the Goddard Space Flight Center, Greenbelt, MD 20771, for the production and distribution of these data, respectively. These activities are sponsored by NASA's Mission to Planet Earth Program.
References 1. HECHT, J. 1985. Laser Pioneers. San Diego: Academic Press, 312 pp. 2. MEASURES, R. M. 1992. Laser Remote Sensing. Malabar: Krieger Publishing Company, 510pp. 3. GRANT, W. B. 1995. Lidar for atmospheric and hydrospheric studies, in Tunable Laser Applications. Duarte, F. J., ed. New York: Marcel Dekker, 313 pp. 4. COMMITTEE ON EARTH OBSERVATION SATELLITES. 2002. Earth Observation Handbook. Paris: European Space Agency, 167 pp. 5. JOINT I. & GROOM S. B. 2000. Estimation of phytoplankton production from space: current status and future potential of satellite remote sensing. Journal of Experimental Marine Biology and Ecology, 250, 233-255. 6. FIORANI, L., MATTEI, S. & VETRELLA, S. 1998. Laser methods for the atmospheric correction of marine radiance data sensed from satellite. Proceedings ofSPIE, 3496, 176-187. 7. O'REILLY, J. E., MARITORENA, S., MITCHELL, B. G., SIEGEL, D. A., CARDER, K. L., GARVER, S. A., KAHRU, M. & MCCLAIN, C. 1998. Ocean color chlorophyll algorithms for SeaWiFS. Journal of Geophysical Research C, 103, 24937-24953.
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8. ARRIGO, K. R., WORTHEN, D. L., SCHNELL, A. & LIZOTTE, M. P. 1998. Primary production in Southern Ocean waters. Journal of Geophysical Research C, 103, 15587-15600.
9. BARBINI, R., COLAO, F., FANTONI, R., FIORANI, L. & PALUCCI, A. 2001. Remote sensing of the Southern Ocean: techniques and results. Journal of Optoelectronics and Advanced Materials, 3, 817-830. 10. HoviS, W.A. 1980. Nimbus-7 coastal zone color scanner: system description and initial imagery. Science, 210, 60-63.
11. HOOKER, S. B., ESAIAS, W. E., FELDMAN, G. C., GREGG, W. W. & MCCLAIN, C. R. 1992. An overview of SeaWiFS and ocean color. SeaWiFS Technical Report Series NASA Technical Memorandum 104566- Vol. 1. HOOKER, S. B. & FIRESTONE, E. R., eds. Greenbelt: NASA, 25 pp.
12. ESAIAS, W. E., ABBOTT, M. R., BARTON, L, BROWN, O. B., CAMPBELL, J. W., CARDER, K. L., CLARK, D. K., EVANS, R. H., HOGE, F. E., GORDON, H. R., BALCH, W. M., LETELIER, R. & MINNETT, P. J. 1998. An overview of MODIS capabilities for ocean science observations. IEEE Transactions on Geoscience and Remote Sensing, 36, 1250-1265. 13. HUOT, J.-P., TAIT, H., RAST, M., DELWART, S., BEZY, J.-L. & LEVRINI, G. 2002. The optical imaging instruments and their applications: AATSR and MERIS. ESA Bulletin, 106, 56-66. 14. BARBINI, R., COLAO, F., FANTONI, R., FIORANI, L. & PALUCCI, A. Lidar fluorosensor calibration of the SeaWiFS chlorophyll algorithm in the Ross Sea. International Journal of Remote Sensing, in press. 15. BARBINI, R., COLAO, F., FANTONI, R., FIORANI, L. & PALUCCI, A. Remote sensed primary production in the western Ross Sea: results of in situ tuned models. Antarctic Science, in press.
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TIME-RESOLVED SPECTROSCOPY OF SEMICONDUCTOR PHOTODETECTORS B. PURA, J. STRZESZEWSKI, A. TADEUSIAK, and Z. WRZESINSK1 Faculty of Physics, Warsaw University of Technology Koszykowa 75 - 00661 Warsaw (Poland) E-mail: [email protected]
Abstract A variety of different semiconductor detectors from wide-band gap materials, Si, GaAs, low temperature grown GaAs, and InP were tested for fast response to X-ray pulses and for sensitivity to shallow penetrating red and blue light from LEDs. All detectors are of planar MSM structure, The detectors are intended for dosimetry of subnanosecond X-ray pulses from hot plasmas. The devices .made from InP and the most defected GaAs are the fastest and mostly promising for the X-ray diagnostics. Recently we also design a new system for femtosecond timeresolved spectroscopy. In preliminary investigations we observe for about 100 fs light excitation the electrical response in the range of 10-100 ps.
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POLYMER WAVEGUIDES FOR OPTICAL MODULATORS B. PURA, J. STRZESZEWSKI, A. TADEUSIAK, and Z. WRZESINSKI Faculty of Physics, Warsaw University of Technology Koszykowa 75 - 00661 Warsaw (Poland) E-mail: [email protected]
Abstract Nonlinear optical materials are expected to be active elements for optical communications and fotonics. In particular, a large number of organic compounds have been already reported as a promising materials for waveguides and devices. Several nonlinear optical polymers containing azometine side-chain groups have been obtained. In this communication we present our results of nonlinear properties of polyarylates and preparation of strip waveguides from these polymers. A patterned sequence of polymer films with different refractive index was deposited on silicon wafer in special design to produce optical switchers and modulators.
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ACCELERATION OF ELECTRONS IN FREE SPACE BY A TM01 LASER BEAM
CHARLES VARIN, MICHEL PICHE Centre d'optique photonique et laser, Universite Laval Sainte-Foy, Quebec, Canada E-mail: [email protected]
In this paper we describe how high-energy electrons are accelerated in22 free space by the longitudinal electric field of a TM0i laser beam. For intensities reaching 10 W/cm2, numerical calculations show that a synchronous acceleration is achieved for certain values of the phase of the laser field, leading to multi-GeV energy gains.
1
Introduction
The future of accelerator-based particle physics depends on the development of new acceleration techniques in order to build experimental facilities producing particle beams of higher energy, with reduced size and lower cost. Laser-driven accelerators are considered as a potential future direction to meet that goal. Despite many proposals since the last four decades there is still no laser-based acceleration scheme that meets the requirements of particle physics. In this paper, we point out that the electromagnetic field of a TM0i laser beam possesses very interesting features that could be exploited to accelerate electrons. The basic properties of that particular laser beam are presented. Compared to actual linear accelerator facilities, numerical calculations show that the proposed laser-based acceleration scheme would significantly increase the energy gain per unit length.
2
Overview of the TMoi laser beam
The TM0i laser beam is a solution of the paraxial wave equation. For a wave propagation along the z axis, the lowest-order solution for a transverse magnetic (TM) laser beam (e.g. with Bz = 0) in circular cylindrical coordinates (r, d, z) exhibits a ring-shaped intensity profile1 with a zero on the axis (r = 0). It should be noted that the beam shape does not depend upon 9, the azimuthal angle (see Fig. 1) and that the field is radially polarized. The field components of the beam can be written as :
E = Re B =
+
R
e
+ R
"
249
e
*
,
0)
250
To satisfy Maxwell's equations, the electromagnetic field of the TMoi beam has only three nonzero components : Er, EZ, and Be(Eg the paraxial approximation, one finds :
E=En
Bf, and B^ are zero). Within
(3)
(4) (5)
where :
"f; 2
(6)
Z0 = kw 0 I 2,
(7)
Az =
(8)
z — Zf (Zf is the position of the beam waist),
+ Az2 / z ,
(9)
R(z) = Az + zl I Az ,
(10)
= tan
(11)
kr2
(12)
1R(z) iff
= tan"
(13)
1 - r 2 / w 2 (z) 1/2
F(r, z) =
1 -
(14)
The amplitude of the magnetic field Bg and the longitudinal electric field EZ are related to the amplitude of the transverse electric field EQ with Bg = E0 I c and Ez ~ 0.74 A E0 IW0, where c is the speed of light in vacuum, A is the wavelength, and WQ is the beam spot size at focus (beam waist). The intensity of the beam at focus is
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proportional r
-
to
r
2
exp(-2r
9
9
/ WQ).
It
reaches
its
maximum
value at
WQ I V2 so that:
(15)
and EO
= J 2 T]Q /max , where T]Q is the intrinsic impedance of free space (12071Q). At the center of the beam, the longitudinal electric field Ez is the only non-
vanishing field component. We will show that it can possibly be used to synchronously accelerate electrons along the propagation axis of the beam.
Longitudinal Field E Transverse Field E
0,0
0,5
1,0
1,5
2,0
2,5
Normalized radial position ( r I wa)
Figure 1. Electric field (left) and intensity profile (right) of a TM0i laser beam at beam waist (z=z )• 3
Relativistic electron dynamics
Within the framework of classical electrodynamics, the Lorentz force2 is used. Namely, one can write the following equation of motion for charged particles : dp
(-
-
~\
-£- = q\E + v X B)
(16)
The momentum and energy of a particle of charge q and velocity V are defined as p = ymv and W - ymc2, with y
= 1 / \1 - v
I c . Then, the energy
2
gain is AVT = (7 - y0 }mc . If an electron (q = - 1.602x10"" C) is injected at the center of a TM0i laser beam (r = 0) with a velocity close to the speed of light (v., = C) and with Vr
dt
— vg
= 0,
~ 0, the Lorentz equation simplifies to :
(17)
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dt
= o,
(18)
- = qE(r = 0).
(19)
dt
Following Eq. (4), the longitudinal electric field along the z axis is:
E(r = 0) =
- kz + 2 arctan(Az
1 + Az2 / z02
(20)
where Az = z — z , , z , is the position of the beam waist, ZQ
2
= kwQ I 2 is
the Rayleigh distance, W0 is the beam waist, and (j)^ is the initial phase of the field.
4
Numerical results for electrons propagating along the beam axis
Typical numerical results obtained from equation (19) — for an electron with a given initial energy WQ = 2 GeV moving along the propagation axis of a strongly focused TM0i laser beam ( "H> » 10 A. and /„,„ > 1020 W/cm2) — are shown in figures 2 and 3.
4000-1 2000-
1022 W/cm2
0^-2000«
40-
S-
04
•1 - °: <*
-80-
bo
u W
0,4-, 0.0 0,20,0-0,2-0,4 0,0
0,5
1,0
1,5
2,0
1,0
1,5
2,0
102' W/cm2 0,5
0,5
Field initial phase (in units of n) Figure 2. Phase (
The top curve of Fig. 2, showing energy gain as a function of the phase of the laser beam, clearly exhibits phase domains that produce positive energy gains. These regions correspond to positions in a field cycle for which the electron can be synchronously accelerated by the longitudinal electric field of the beam. Since laser
253
beams propagating in free space have phase velocities higher than c, the speed of light in vacuo, such phase matching is generally believed to be impossible. In fact, the longitudinal electric field of the TM0i beam has a phase velocity equal to c, plus a weak correction due to the Gouy phase shift (2 n}. As a result, the phase slippage experienced by an ultra-relativistic electron is such that its trajectory is contained within a half-cycle of the field, for the last part of its transit. This behaviour is clearly seen in Fig. 3; when fa = 0.62^, electrons are decelerated just before focus, and then they experience a violent acceleration beyond focus. The acceleration benefits also from the fact that the longitudinal field falls off as (z - z/)"2 away from focus; this fallout is much slower than the Gaussian envelope of the transverse electromagnetic field components. The results are very similar to those obtained previously with another type of TM0i laser beam3 for which the longitudinal electric field falls off as l/(z - zj). We recently, found that this beam had a radial field amplitude falling as 1 / r away from center, leading to a divergence of its total power. =0.1471
A = 0.0
0,996
0,998
1 ,(KX)
1,002
1,004
0,996
Position along the z axis
0,998
1,000
1,002
1,004
Position along the z axis
4000 <(0=0.62jL
3000-
-1500-2000 • 0,996
0,998
1,000
1,002
1,004
0,996
Position along the z axis
0,998
1,000
1,002
Position along the z axis
7000-
1.66 it
6000>" 50002, 4000-
C
•g 3000 cm & 2000•1000•20000,996
0,998
1,000
1,002
Position along the z axis
1,004
0,996
0,998
1,000
1,002
1,004
Position along the z axis
Figure 3. Detailed dynamics of 2-GeV electrons along the z axis (meters) of a TM0i laser beam. The laser wavelength is 0.8 ^m, the beam waist is at z = 1 m (the interaction begins at z = 0), and the beam waist is 10 Um. The maximum beam intensity is 1022 W/cm2. The electric field initial phase ^ is given on each graph.
254
5
Conclusion
It has been numerically shown that ultra-relativistic electrons could possibly be accelerated to multi-GeV energies by the longitudinal electric field of an intense laser beam (/^ « 1022 W/cm2). Acknowledgements We would like to thank the Natural Sciences and Engineering Research Council (Canada), Les fonds pour la formation de chercheurs et 1'aide a la recherche (Quebec), and the Canadian Institute for Photonic Innovations for their financial support.
References 1 2 3
A.E. Siegman, Lasers (University Science Books, 1986). D.J. Jackson, Classical Electrodynamics, Third edition (Wiley, New York 1999) C. Varin, M. Piche; Ultrafast Phenomena XIII, Springer Series in Chemical Physics; © Springer-Verlag, Berlin Heidelberg 2003; 164-166 (2003).
