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Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media
Optics Communications 278 (2007) 247–252 www.elsevier.com/locate/optcom
Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media Yu Xin *, YanRu Chen, Qi Zhao, MuChun Zhou Department of Optical Engineering, Nanjing University of Science and Technology, 200 Xiao Ling Wei, Nanjing 210094, PR China Received 31 January 2007; received in revised form 11 June 2007; accepted 15 June 2007
Abstract We study the coherence properties of the field generated by beam radiated from quasi-homogeneous (QH) electromagnetic source scattering on QH media. Formulas for the spectral density and spectral degree of coherence of the three dimensional scattered field are derived. The results show under assumption that the diagonal correlation coefficients of the source are proportional to each other, the far field of the scattered light satisfy two reciprocity relations analogous to that in the scalar case, that, the spectral density is proportional to the convolution of the spectral density of the source and the spatial Fourier transform of the correlation coefficient of the scattering potential; the spectral degree of coherence is proportional to the convolution of the diagonal correlation coefficients and the strength of the scattering potential. Ó 2007 Elsevier B.V. All rights reserved. PACS: 03.50.z; 42.25.Kb; 42.25.Fx; 42.25.Dd Keywords: Coherence; Scattering; Random media
1. Introduction Light scattering by random media is now of intensive interest in various field [1–3]. A general and important class of model random media is the quasi-homogeneous (QH) media [4]. This kind of random media has a property that the strength of the scattering potential S ðF Þ ðr; xÞ varies much more slowly with position r at a particular frequency x than the correlation coefficient lðF Þ ðr1 ; r2 ; xÞ lðF Þ ðr2 r1 ; xÞ with position r 0 = r2 r1. In the framework of scalar scattering of light, an interesting result analogous to that in the radiation situation [5] was obtained by Visser et al. [6], that within the accuracy of the first-order Born approximation, for the far field of scattered light generated by light from a QH scalar source *
Corresponding author. Tel.: +86 2584315435. E-mail address: [email protected] (Y. Xin).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.06.030
scattering by QH random media, the angular distribution of spectral density and the spectral degree of coherence form two reciprocity relations. Recently, the QH scalar source was generalized by Korotkova et al. [7] to the electromagnetic case and it was shown that for the radiated far field, analogous reciprocity relations to that of the scalar case were satisfied by the spectral density and spectral degree of coherence. Theoretically, it is desirable to know the coherence properties of the scattered electromagnetic field generated by light from a QH electromagnetic source scattering by a QH random media, however, this problem has not been studied yet. In the present paper, we discuss the coherence property of the scattered field of electromagnetic beam from a QH source scattering from a QH random media. We first study the coherence property of scattered field of a plane wave, next, we consider the case of beam scattering as a more general problem, at last, we make a conclusion.
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Y. Xin et al. / Optics Communications 278 (2007) 247–252
2. Electromagnetic plane wave scattering on a QH media Consider a linearly polarized plane electromagnetic wave EðiÞ ðr; tÞ ¼ EðiÞ ðr; xÞ expðixtÞ incident on a QH ðiÞ ðiÞ ðiÞ random media. EðiÞ ðr; xÞ ¼ ½EðiÞ a ; E b , E a ; Eb are two components of the incident electric field along two mutually orthogonal direction ai ; bi perpendicular to the direction of wave propagation. For the sake of simplicity, we assume that both components have unit amplitude: ðiÞ
EðiÞ a ðr; xÞ ¼ E b ðr; xÞ ¼ expðiks0 rÞ;
ð1Þ
where r is the position vector, and s0 is the direction of propagation, k = x/c is the wave number, x is the frequency, c is the speed of light in vacuum. Within the accuracy of the first-order Born approximation, the amplitude of the scattered field is given by [8]: Z 3 0 ðsÞ ðiÞ 0 0 0 E ðrs; xÞ ¼ s s F ðr ; xÞE ðr ; xÞGðr; r ; xÞ d r ; D
ð2Þ where superscript (s) denotes quantities pertaining to scattered field, s is the unit vector along a typical scattering path, D is the domain the scatterer occupies. F ðr0 ; xÞ is the scattering potential of the random media. Gðr; r0 ; xÞ is the free space outgoing Green’s function of the Helmholz operator, in the far zone of the scatter, a asymptotic approximation may be used for the Green’s function: expðikrÞ exp½iks r0 : ð3Þ r The correlation function of the scattering potential is defined as: Gðrs; r0 ; xÞ ¼
C ðF Þ ðr1 ; r2 ; xÞ ¼ hF ðr1 ; xÞF ðr2 ; xÞi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ S ðF Þ ðr1 ; xÞS ðF Þ ðr2 ; xÞgðF Þ ðr1 ; r2 ; xÞ:
C
ðr1 ; r2 ; xÞ S
ðF Þ
ðF Þ
½ðr1 þ r2 Þ=2; xg ðr2 r1 ; xÞ:
x ¼ ai ;
y ¼ bi ;
z ¼ s0 :
ð8Þ
The scattered electromagnetic field is three dimensional, and its three components may be calculated as: Z 3 0 ðsÞ 0 0 Ex ¼ Ax ð/Þ F ðr0 ; xÞEðiÞ x ðr ; xÞGðr; r ; xÞ d r ; D Z 3 0 0 0 EyðsÞ ¼ Ay ð/Þ F ðr0 ; xÞEðiÞ ð9Þ y ðr ; xÞGðr; r ; xÞ d r ; D Z 3 0 0 0 EzðsÞ ¼ Az ð/Þ F ðr0 ; xÞEðiÞ y ðr ; xÞGðr; r ; xÞ d r ; D
where / is the angle s makes with s0. The geometrical factors Ai ði ¼ x; y; zÞ in front of the integrals in Eq. (9) have the form: Ax ð/Þ ¼ 1; Ay ð/Þ ¼ cos2 /;
ð10Þ
Az ð/Þ ¼ sin / cos /: Based on the unified theory of coherence and polarization [9], the coherence properties of the scattered field can be represented by a third-order matrix in terms of the three components of the scattered electric field: ðsÞ
ð4Þ
ð5Þ
On substituting Eq. (5) into Eq. (4) we obtain for the correlation function: ðF Þ
we also assume that the cartesian coordinate is chosen as:
WðsÞ ½W ij ðr1 ; r2 ; xÞ
In Eq. (4), the angle brackets denote average taken over a statistical ensemble of realizations of the scatterer, the asterisk denotes complex conjugate. S ðF Þ ðr; xÞ ¼ C ðF Þ ðr; r; xÞ is the strength of the scattering potential, gðF Þ ðr1 ; r2 ; xÞ is the correlation coefficient of the scattering potential. For a QH random media, the correlation coefficient depends on r1 ; r2 only through the difference r2 r1, and the strength of the scattering potential varies much more slowly with its spatial argument than the correlation coefficient with its spatial argument. In other words, over the effective with of |g(F)| the function S(F) is essentially constant. Consequently, over regions of the scatterer for which jgðF Þ ðr02 r01 ; xÞj is appreciable, it is reasonable to make the approximation: S ðF Þ ðr1 ; xÞ S ðF Þ ðr2 ; xÞ S ðF Þ ½ðr1 þ r2 Þ=2; x:
ðsÞ
Let ½EðsÞ a ; Eb be the two components of scattered electric field vector along two mutually orthogonal direction as , bs perpendicular to the scattering path s. as , bs are chosen in the following way: s s0 ; ai ¼ as ¼ js s0 j ð7Þ bi ¼ s 0 ai ; bs ¼ s as ;
ð6Þ
ðsÞ
ðsÞ
¼ ½hEi ðr1 ; xÞEj ðr2 ; xÞi
ði; j ¼ x; y; zÞ:
ð11Þ
In Eq. (11) the superscript (s) represents quantities pertainðsÞ ing to the scattered field, Ei ði ¼ x; y; zÞ denote the cartesian components of the typical member of the statistical ensemble of the scattered electric field, and the angle brackets denote the average taken on a statistical ensemble of realizations of the electric filed, the asterisk denotes complex conjugation. The x was included in equations above to indicate that we treat the problem in space-frequency domain and the equations hold for a single frequency component of the field, but to avoid unnecessary cluttering of the formulas, we henceforth omit x in various notations. On substituting Eq. (9) into Eq. (11) and using Eqs. (1), (3) we obtain: exp½ikðr2 r1 Þ ðsÞ W ij ðr1 s1 ; r2 s2 Þ ¼ Ai ð/1 ÞAj ð/2 Þ r1 r 2 Z Z C ðF Þ ðr01 ; r02 Þ exp½iks0 ðr02 r01 Þ D
D
exp½ikðs1 r01 s2 r02 Þ d3 r01 d3 r02
ð12Þ
Y. Xin et al. / Optics Communications 278 (2007) 247–252
here, s1 and s2 are two unit vectors along two scattering path, /1 and /2 are angles s1 and s2 make with s0. r01 and r02 are position vectors of points in the region of scatterer. As a matter of convenience, we make changes of the spatial arguments: r02 r01 ¼ R S;
ð13Þ
r02 þ r01 ¼ 2Rþ S;
substituting Eq. (13) into Eq. (12) and using Eq. (6) we obtain: W
ðsÞ ij ðr 1 s1 ; r2 s2 Þ
Z Z exp½ikðr2 r1 Þ ðF Þ S ðF Þ ðRþ ðR S Þg SÞ r1 r 2 D D þ exp½ikðs0 ðs1 þ s2 Þ=2Þ R S exp½ikðs1 s2 Þ RS
¼ Ai ð/1 ÞAj ð/2 Þ
3 þ d3 R S d RS :
ðsÞ
In Eqs. (18) and (19) we have suppressed the dependence of various quantities on the frequency x. As we have noted that the correlation function of the scattering potential of a QH media is a fast function of its spatial argument, so from the property of the Fourier transform we know that the spatial Fourier transform of the correlation function is a slow function, then we can, with no loss of reasonability, make approximations: ~gðF Þ ½kðs0 s1 Þ ¼ ~gðF Þ ½kðs0 s2 Þ ¼ ~gðF Þ ½kðs0 ðs1 þ s2 Þ=2Þ
exp½ikðr2 r1 Þ e ðF Þ S ½kðs1 s2 Þ r1 r2 ~ gðF Þ ½kðs0 ðs1 þ s2 Þ=2Þ; ð15Þ
W ij ðr1 s1 ; r2 s2 Þ ¼ Ai1 Aj2
gðF Þ are the three dimensional Fourier transwhere e S ðF Þ , ~ form of the strength and correlation coefficient of the scattering potential of the QH media. In Eq. (15) we have suppressed the arguments /1 , /2 and make use of Ai1 , Aj2 to represent the geometrical factors. According to the definition by Korotkova [10], the spectral density and spectral degree of coherence of the three dimensional random scattered field are given by: S ðsÞ ðrÞ ¼ TrWðsÞ ðr; rÞ;
ð16Þ ðsÞ
TrW ðr1 ; r2 Þ gðsÞ ðr1 ; r2 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; TrWðsÞ ðr1 ; r1 Þ TrWðsÞ ðr2 ; r2 Þ
ð17Þ
where Tr denotes the trace of a matrix. On substituting Eq. (15) into Eqs. (16), (17) we obtain for the scattered field the spectral density: 1 ðF Þ S ðsÞ ðrsÞ ¼ 2 e ð18Þ S ð0ÞgðF Þ ½kðs0 sÞðA2x þ A2y þ A2z Þ r and the spectral degree of coherence: ~ gðF Þ ½kðs0 ðs1 þ s2 Þ=2Þ g ðr1 s1 ; r2 s2 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ gðF Þ ½kðs0 s2 Þ gðF Þ ½kðs0 s1 Þ~ e ðF Þ ½kðs1 s2 Þ S Að/1 ; /2 Þ exp½ikðr2 r1 Þ; e S ðF Þ ð0Þ ð19Þ ðsÞ
where Ax1 Ax2 þ Ay1 Ay2 þ Az1 Az2 Að/1 ; /2 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 ðAx1 þ A2y1 þ A2z1 ÞðA2x2 þ A2y2 þ A2z2 Þ
ð20Þ
ð21Þ
then the first factor on the right-hand side of Eq. (19) can be approximated by unity, and Eq. (19) reduces to: gðsÞ ðr1 s1 ; r2 s2 Þ ¼ Að/1 ; /2 Þ
ð14Þ
In Eq. (14), the integral extends over the whole region of the scatterer and it is a six dimensional Fourier transform of the strength and correlation coefficient of the scattering potential. Making integration of the right-hand side of Eq. (14) results into the formula:
249
e S ðF Þ ½kðs1 s2 Þ exp½ikðr2 r1 Þ: e S ðF Þ ð0Þ ð22Þ
From Eqs. (18) and (22) we can see there are two reciprocities analogous to that in the scalar case, i.e., the spectral density of scattered electromagnetic field in the far zone of the scatterer, is proportional to spatial Fourier transform of the correlation coefficient of the scattering potential and the spectral degree of coherence of the scatterer field is proportional to the normalized spatial Fourier transform of the strength of the scattering potential of the media. 3. Beam radiated from a QH uniformly polarized electromagnetic source scattering on a QH media Recently, the concepts of QH source were generalized from scalar case to the electromagnetic case [7], and conditions [11] for such a source to generate electromagnetic beam was derived. By assuming that the source is uniformly polarized, it was verified that the far field of the beam it generates supports two reciprocity relations that is analogous to that in the scalar case. According to the unified theory of coherence, the cross-spectral density matrix of the secondary planar source which generate a electromagnetic beam is ðQÞ
ðQÞ
ðQÞ
W ij ðr1 ; r2 Þ ¼ hEi ðr1 ÞEj ðr2 Þi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðQÞ ðQÞ ¼ S i ðr1 Þ S ðQÞ ðr2 Þlij ðr1 ; r2 Þ
ði; j ¼ x; yÞ: ð23Þ
In Eq. (23), Ei , Ej are two components of the electric field along two mutually orthogonal directions x, y. The angle brackets denote average taken over the statistical ensemble of realizations of the electric field, and the asterisk denotes complex conjugation. The superscript (Q) denotes quantities pertaining to the source, r1 , r2 are the position vectors of two ðQÞ ðQÞ points in the source plane. S i ðrÞ ¼ W ii ðr; rÞ ði ¼ x; yÞ is the spectral density of one component of the electric field, ðQÞ lij ðr1 ; r2 Þ ði; j ¼ x; yÞ is the correlation coefficient between two components of the electric field in the source plane.
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Y. Xin et al. / Optics Communications 278 (2007) 247–252
For the QH electromagnetic source, approximations analogous to the scalar case are made: ðQÞ
ðQÞ
ðQÞ
S i ðr1 Þ S i ðr2 Þ S i ½ðr1 þ r2 Þ=2; ðQÞ
ð24Þ
ðQÞ
lij ðr1 ; r2 Þ lij ðr2 r1 Þ:
Uniform polarization of the source means that the spectral ðQÞ are proportional density of the two components S ðQÞ x , Sy to each other, i.e., ðQÞ S ðQÞ y ðrÞ ¼ aS x ðrÞ;
ðQÞ lðQÞ xy ðr; rÞ ¼ lxy :
ð26Þ
On substituting Eqs. (24), (25) into Eq. (23) we obtain: W
ðQÞ
¼ aij S
½ðr1 þ
ðQÞ r2 Þ=2lij ðr2
r1 Þ;
S
ðrÞ ¼
and aij ¼
S ðQÞ x ðrÞ
8 1 > < 1þa pffiffi a
> : 1þa a
1þa
þ
S ðQÞ y ðrÞ
ð29Þ
when i ¼ j ¼ y:
We use the cartesian coordinate that was defined in the previous section and assume that the beam propagates along the z axis, the cross-spectral density matrix of the far field generated by the source is: ðiÞ
2
W ij ðr1 u1 ; r2 u2 Þ ¼ ð2pkÞ cos h1 cos h2 exp½ikðr2 r1 Þ e ðQÞ W ; ij ðku1 ; ku2 Þ r1 r 2 where
Z Z
1 ð2pÞ
4
Q
ð30Þ
D
Q
exp½ikðu1 r01 u2 r02 Þ d2 r01 d2 r02
ð31Þ
is the four dimensional spatial Fourier transform of ðQÞ W ij ðr1 ; r2 Þ. In Eq. (30), u1 , u2 are unit vectors along r1 , r2 , h1 , h2 are angles u1 and u2 make with the positive z direction. It is convenient to make changes of variables: 0 0 R Q ¼ r2 r1 ;
ð32Þ
0 0 Rþ Q ¼ ðr2 þ r1 Þ=2:
D
G ðr1 s1 ; r01 ÞGðr2 s2 ; r02 Þ d3 r01 d3 r02 ;
On substituting Eq. (27) into Eq. (31) and using Eq. (32) we obtain: Z Z 1 ðQÞ e ðQÞ W ðku ; ku Þ ¼ a S ðQÞ ðRþ 1 2 ij ij R Þlij ðRQ Þ 4 ð2pÞ Q Q exp½iku1 ðRþ Q RQ =2Þ=r1 2 2 þ exp½iku2 ðRþ Q þ RQ =2Þ=r 2 d RQ d RQ
ð33Þ
ð35Þ
where
ðiÞ ij
8 ðiÞ > W > < ij ¼ W ðiÞ iy > > : ðiÞ W yj
when i; j ¼ x; y; when j ¼ z;
ð36Þ
when i ¼ z:
If it is assumed that the scatterer is located in the far zone of the source, it is reasonable before substitute Eq. (34) into Eq. (35) to make some approximations. The denominator r1r2 in front of the integral in Eq. (34) may be approximated by r1r2 R2 and the two denominator in the two exponentials under the integral on the right-hand side of Eq. (34) may be approximated by R, R = |R| is the distance from the origin (located in the source plane) to the region of the scatterer. R is the position vector of the scatter. We combine the exponential term exp½ikðr02 r01 Þ with the correlation function of the scatterer to obtain a modified correlation function: C ðF Þ ðr01 ; r02 Þ ¼ C ðF Þ ðr01 ; r02 Þ exp½ikðr02 r01 Þ:
ðQÞ
W ij ðr01 ; r02 Þ
ð34Þ
From the discussion in the previous section, we can see that the elements of the cross-spectral density matrix of the scattered field may be presented as: Z Z ðsÞ ðiÞ W ij ðr1 s1 ; r2 s2 Þ ¼ Ai ð/1 ÞAj ð/2 Þ W ij ðr01 ; r02 ÞC ðF Þ ðr01 ; r02 Þ
W
when i 6¼ j;
Q
2 2 þ exp½ikr2 u2 ðRþ Q þ QQ =2Þ=r2 d RQ d RQ :
ð28Þ
when i ¼ j ¼ x;
e ðQÞ W ij ðku1 ; ku2 Þ ¼
Q
ð27Þ
where S(Q) is the spectral density of the electromagnetic field of the source: ðQÞ
ðiÞ
W ij ðr1 u1 ; r2 u2 Þ 2 k exp½ikðr2 r1 Þ ¼ aij cos h1 cos h2 2p r1 r2 Z Z ðQÞ þ S ðQÞ ðRþ Q Þlij ðRQ Þ exp½ikr 1 u1 ðRQ RQ =2Þ=r1
ð25Þ
where a is a quantity depending only on frequency, and the correlation coefficient lðQÞ xy ðr; rÞ is independent of position:
ðQÞ ij ðr1 ; r2 Þ
hence, Eq. (30) becomes:
ð37Þ
Naturally, the combination results in a modified correlation coefficient: gðF Þ ðr01 ; r02 Þ ¼ gðF Þ ðr01 ; r02 Þ exp½ikðr02 r01 Þ:
ð38Þ
By passing we note that the modified correlation coefficient gðF Þ is still a fast function of its spatial argument and the correlation function C(F) is still a product of a slow function and a fast function. The factor cos h1 cos h2 in Eq. (34) can be approximated by cos2 h, where h is the angle R makes with the positive z direction. On substituting Eq. (34) into Eq. (35) we obtain the formula for the elements of the cross-spectral density matrix of the scattered filed: exp½ikðr2 r1 Þ ðsÞ W ij ðr1 s1 ; r2 s2 Þ ¼ Ai ð/1 ÞAj ð/2 Þ cos2 haij R2 r 1 r 2 Z Z Z Z Q ðF Þ S ðQÞ ðRþ ðRþ gðF Þ ðR R Þlij ðRQ ÞS S Þ SÞ 2 þ 3 3 þ expðikUÞ d2 R Q d RQ d RS d RS ;
ð39Þ
Y. Xin et al. / Optics Communications 278 (2007) 247–252
where U is the phase factor and given by the formula: U
þ 0 ¼ ðRþ Q RQ =2Þ=R r2 ðRQ þ RQ =2Þ=R Rþ S ðs2 s1 Þ RS ðs1 s2 Þ ¼ Rþ S ½ðs2 s1 Þ þ RQ =R RS ½ðs1 þ s2 Þ=2
r01
þ Rþ Q =R: ð40Þ
By integrating over
Rþ S
and
R S
we obtain:
exp½ikðr2 r1 Þ R2 r 1 r 2 M ij ½s2 s1 N ½ðs2 þ s1 Þ=2; ð41Þ
ðsÞ
W ij ðr1 s1 ; r2 s2 Þ ¼ aij Ai ð/1 ÞAj ð/2 Þ cos2 h
where M ij ½s2 s1 ¼
Z Q
N ½ðs1 þ s2 Þ=2 ¼
ðQÞ e ðF Þ fk½ðs2 lij ðR Q ÞS
Z S
ðQÞ
ðRþ gðF Þ fk½ðs1 Q Þ~
3 þ þ s2 Þ=2 þ Rþ Q =Rg d RQ :
Q
ð42Þ Here the meaning of the subscripts ij pertaining to quantities of the source is understood from Eq. (36). e S ðF Þ and ~gðF Þ are three dimensional spatial Fourier transform of the strength of the scattering potential S(F) and the modified correlation coefficient of the scattering potential gðF Þ , respectively. It is easy to see that Mij and N have the form of a convolution. After making substitution of Eq. (41) into Eq. (16), we get the spectral density of the scattered field: 2 k cos2 h S ðsÞ ðrsÞ ¼ LðsÞN ðsÞ; ð43Þ 2p ½1 þ aR2 r2 where L¼
A2x M xx ð0Þ
þ
aðA2y
þ
A2z ÞM yy ð0Þ:
ð44Þ
On substituting Eq. (41) into Eq. (17) we obtain the spectral degree of coherence of the scattered field: N ½ðs1 þ s2 Þ=2 g ðr1 s1 ; r2 s2 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi exp½ikðr2 r1 Þ N ðs1 Þ N ðs2 Þ L12 pffiffiffiffiffipffiffiffiffiffi ; L1 L2 ðsÞ
ð45Þ
where L12 ¼ Ax1 Ax2 M xx ðs2 s1 Þ þ aðAy1 Ay2 þ Az1 Az2 ÞM yy ðs2 s1 Þ; L1 ¼ A2x1 M xx ð0Þ þ aðA2y1 þ A2z1 ÞM yy ð0Þ; L2 ¼ A2x2 M xx ð0Þ þ aðA2y2 þ A2z2 ÞM yy ð0Þ: ð46Þ Since the modified correlation coefficient of the scattering potential gðF Þ is a fast function of its spatial argument and its Fourier transform is a slow function, we can deduce from Eq. (42) that N is a slow function of its argument and the approximation can be made: N ðs1 Þ N ðs2 Þ N ½ðs1 þ s2 Þ=2:
As a consequence, the first factor on the right-hand side of Eq. (45) can be approximated by unity and Eq. (45) reduce to: L12 gðsÞ ðr1 s1 ; r2 s2 Þ ¼ exp½ikðr2 r1 Þ pffiffiffiffiffipffiffiffiffiffi : L1 L2
ð47Þ
ð48Þ
From Eqs. (43) and (48) we can find that the scattered electromagnetic field do not support the reciprocity relations like that holding for the scalar light scattering, because in ðQÞ general, the diagonal correlation coefficients lðQÞ xx and lyy do not have definite relation to each other. However, if we assume that the two diagonal correlation coefficients are proportional to each other, i.e., ðQÞ lðQÞ yy ðrÞ ¼ blxx ðrÞ;
3 s1 Þ þ R Q =Rg d RQ ;
251
ð49Þ
where b is a quantity depending only on frequency, then, on substituting Eq. (49) into Eqs. (44) and (46), we obtain: L ¼ ½A2x þ abðA2y þ A2z ÞM xx ð0Þ; L12 ¼ ½Ax1 Ax2 þ abðAy1 Ay2 þ Az1 Az2 ÞM xx ðs2 s1 Þ; L1 ¼ ½A2x1 þ abðA2y1 þ A2z1 ÞM xx ð0Þ
ð50Þ
L2 ¼ ½A2x2 þ abðA2y2 þ A2z2 ÞM xx ð0Þ: Consequently, the spectral density and spectral degree of coherence of the scattered field reduce to: 2 k cos2 h S ðsÞ ðrsÞ ¼ KM xx ð0ÞN ðsÞ; ð51Þ 2p ½1 þ aR2 r2 M xx ðs2 s1 Þ ; ð52Þ gðsÞ ðr1 s1 ; r2 s2 Þ ¼ exp½ikðr2 r1 ÞK0 M xx ð0Þ where K ¼ A2x þ abðA2y þ A2z Þ;
ð53Þ
Ax1 Ax2 þ abðAy1 Ay2 þ Az1 Az2 Þ K0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 Ax1 þ abðA2y1 þ A2z1 Þ A2x2 þ abðA2y2 þ A2z2 Þ
ð54Þ
Now from Eqs. (51) and (52), it comes into focus that there holds two reciprocity relations for the spectral density and spectral degree of coherence of the scattered field generated by a electromagnetic beam from a QH uniformly polarized source scattering by a QH random media: The spectral density of the scattered field is proportional to the convolution of the spectral density S(Q) of the source and the spatial Fourier transform of the modified correlation coefficient gðF Þ of the scatter; the spectral degree of coherence of the scattered field is proportional to the convoluðQÞ tion of the diagonal correlation coefficients lii ði ¼ x; yÞ of the source and the spatial Fourier transform of the strength of the scattering potential. 4. Conclusion We have studied the coherence properties of scattered electromagnetic field from a QH random media. For the simple case of linearly polarized electromagnetic plane wave scattering, it is found that the scattered far field
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satisfy two reciprocity relations analogous to that in the scalar wave scattering. For a beam radiated from QH electromagnetic source scattering on a QH random media, the far field do not always satisfy the reciprocity relations. However, the reciprocity relations come into existence when the two diagonal correlation coefficients are proportional to each other: the spectral density of the scattered field is related to a convolution of the spectral density of the source and the spatial Fourier transform of the modified correlation coefficient of the scatterer; the spectral degree of coherence is related to a convolution of the diagonal correlation coefficients and the Fourier transform of the scattering strength of the scatterer. In most occasions, electromagnetic waves are utilized rather than a scalar wave, so the results of our research may find some applications in fields as target detection and imaging. Acknowledgments This work was supported by the National Natural Science Foundation of China (50176020). The authors
gratefully acknowledge Prof. Emil Wolf and Dr. Olga Korotkova for sending some of their papers to us. References [1] G. Jarry, E. Steimer, V. Damaschini, M. Epifanie, M. Jurczak, R. Kaiser, Appl. Opt. 37 (1998) 7357. [2] B.G. Hoover, L. Deslauriers, S.M. Grannell, R.E. Ahmed, D.S. Dilworth, B.D. Athey, E.N. Leith, Phys. Rev. E 65 (2002) 026614. [3] T.A. Leskova, A.A. Maradudin, J.M. Lopez, Phys. Rev. E 71 (2005) 036606. [4] R.A. Silverman, Proc. Cambridge Philos. Soc. 54 (1958) 530. [5] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995. [6] T.D. Visser, D.G. Fischer, E. Wolf, J. Opt. Soc. Am. A 23 (2006) 1631. [7] O. Korotkova, B.G. Hoover, V.L. Gamiz, E. Wolf, J. Opt. Soc. Am. A 22 (2005) 2547. [8] L. Tsang, J.A. Kong, K.H. Ding, Scattering of Electromagnetic Waves: Theory and Applications, Wiley, 2000. [9] E. Wolf, Phys. Lett. A 312 (2003) 263. [10] O. Korotkova, E. Wolf, J. Opt. Soc. Am. A 21 (2004) 2382. [11] O. Korotkova, M. Salem, E. Wolf, Opt. Lett. 29 (2004) 1173.
Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media Yu Xin *, YanRu Chen, Qi Zhao, MuChun Zhou Department of Optical Engineering, Nanjing University of Science and Technology, 200 Xiao Ling Wei, Nanjing 210094, PR China Received 31 January 2007; received in revised form 11 June 2007; accepted 15 June 2007
Abstract We study the coherence properties of the field generated by beam radiated from quasi-homogeneous (QH) electromagnetic source scattering on QH media. Formulas for the spectral density and spectral degree of coherence of the three dimensional scattered field are derived. The results show under assumption that the diagonal correlation coefficients of the source are proportional to each other, the far field of the scattered light satisfy two reciprocity relations analogous to that in the scalar case, that, the spectral density is proportional to the convolution of the spectral density of the source and the spatial Fourier transform of the correlation coefficient of the scattering potential; the spectral degree of coherence is proportional to the convolution of the diagonal correlation coefficients and the strength of the scattering potential. Ó 2007 Elsevier B.V. All rights reserved. PACS: 03.50.z; 42.25.Kb; 42.25.Fx; 42.25.Dd Keywords: Coherence; Scattering; Random media
1. Introduction Light scattering by random media is now of intensive interest in various field [1–3]. A general and important class of model random media is the quasi-homogeneous (QH) media [4]. This kind of random media has a property that the strength of the scattering potential S ðF Þ ðr; xÞ varies much more slowly with position r at a particular frequency x than the correlation coefficient lðF Þ ðr1 ; r2 ; xÞ lðF Þ ðr2 r1 ; xÞ with position r 0 = r2 r1. In the framework of scalar scattering of light, an interesting result analogous to that in the radiation situation [5] was obtained by Visser et al. [6], that within the accuracy of the first-order Born approximation, for the far field of scattered light generated by light from a QH scalar source *
Corresponding author. Tel.: +86 2584315435. E-mail address: [email protected] (Y. Xin).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.06.030
scattering by QH random media, the angular distribution of spectral density and the spectral degree of coherence form two reciprocity relations. Recently, the QH scalar source was generalized by Korotkova et al. [7] to the electromagnetic case and it was shown that for the radiated far field, analogous reciprocity relations to that of the scalar case were satisfied by the spectral density and spectral degree of coherence. Theoretically, it is desirable to know the coherence properties of the scattered electromagnetic field generated by light from a QH electromagnetic source scattering by a QH random media, however, this problem has not been studied yet. In the present paper, we discuss the coherence property of the scattered field of electromagnetic beam from a QH source scattering from a QH random media. We first study the coherence property of scattered field of a plane wave, next, we consider the case of beam scattering as a more general problem, at last, we make a conclusion.
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2. Electromagnetic plane wave scattering on a QH media Consider a linearly polarized plane electromagnetic wave EðiÞ ðr; tÞ ¼ EðiÞ ðr; xÞ expðixtÞ incident on a QH ðiÞ ðiÞ ðiÞ random media. EðiÞ ðr; xÞ ¼ ½EðiÞ a ; E b , E a ; Eb are two components of the incident electric field along two mutually orthogonal direction ai ; bi perpendicular to the direction of wave propagation. For the sake of simplicity, we assume that both components have unit amplitude: ðiÞ
EðiÞ a ðr; xÞ ¼ E b ðr; xÞ ¼ expðiks0 rÞ;
ð1Þ
where r is the position vector, and s0 is the direction of propagation, k = x/c is the wave number, x is the frequency, c is the speed of light in vacuum. Within the accuracy of the first-order Born approximation, the amplitude of the scattered field is given by [8]: Z 3 0 ðsÞ ðiÞ 0 0 0 E ðrs; xÞ ¼ s s F ðr ; xÞE ðr ; xÞGðr; r ; xÞ d r ; D
ð2Þ where superscript (s) denotes quantities pertaining to scattered field, s is the unit vector along a typical scattering path, D is the domain the scatterer occupies. F ðr0 ; xÞ is the scattering potential of the random media. Gðr; r0 ; xÞ is the free space outgoing Green’s function of the Helmholz operator, in the far zone of the scatter, a asymptotic approximation may be used for the Green’s function: expðikrÞ exp½iks r0 : ð3Þ r The correlation function of the scattering potential is defined as: Gðrs; r0 ; xÞ ¼
C ðF Þ ðr1 ; r2 ; xÞ ¼ hF ðr1 ; xÞF ðr2 ; xÞi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ S ðF Þ ðr1 ; xÞS ðF Þ ðr2 ; xÞgðF Þ ðr1 ; r2 ; xÞ:
C
ðr1 ; r2 ; xÞ S
ðF Þ
ðF Þ
½ðr1 þ r2 Þ=2; xg ðr2 r1 ; xÞ:
x ¼ ai ;
y ¼ bi ;
z ¼ s0 :
ð8Þ
The scattered electromagnetic field is three dimensional, and its three components may be calculated as: Z 3 0 ðsÞ 0 0 Ex ¼ Ax ð/Þ F ðr0 ; xÞEðiÞ x ðr ; xÞGðr; r ; xÞ d r ; D Z 3 0 0 0 EyðsÞ ¼ Ay ð/Þ F ðr0 ; xÞEðiÞ ð9Þ y ðr ; xÞGðr; r ; xÞ d r ; D Z 3 0 0 0 EzðsÞ ¼ Az ð/Þ F ðr0 ; xÞEðiÞ y ðr ; xÞGðr; r ; xÞ d r ; D
where / is the angle s makes with s0. The geometrical factors Ai ði ¼ x; y; zÞ in front of the integrals in Eq. (9) have the form: Ax ð/Þ ¼ 1; Ay ð/Þ ¼ cos2 /;
ð10Þ
Az ð/Þ ¼ sin / cos /: Based on the unified theory of coherence and polarization [9], the coherence properties of the scattered field can be represented by a third-order matrix in terms of the three components of the scattered electric field: ðsÞ
ð4Þ
ð5Þ
On substituting Eq. (5) into Eq. (4) we obtain for the correlation function: ðF Þ
we also assume that the cartesian coordinate is chosen as:
WðsÞ ½W ij ðr1 ; r2 ; xÞ
In Eq. (4), the angle brackets denote average taken over a statistical ensemble of realizations of the scatterer, the asterisk denotes complex conjugate. S ðF Þ ðr; xÞ ¼ C ðF Þ ðr; r; xÞ is the strength of the scattering potential, gðF Þ ðr1 ; r2 ; xÞ is the correlation coefficient of the scattering potential. For a QH random media, the correlation coefficient depends on r1 ; r2 only through the difference r2 r1, and the strength of the scattering potential varies much more slowly with its spatial argument than the correlation coefficient with its spatial argument. In other words, over the effective with of |g(F)| the function S(F) is essentially constant. Consequently, over regions of the scatterer for which jgðF Þ ðr02 r01 ; xÞj is appreciable, it is reasonable to make the approximation: S ðF Þ ðr1 ; xÞ S ðF Þ ðr2 ; xÞ S ðF Þ ½ðr1 þ r2 Þ=2; x:
ðsÞ
Let ½EðsÞ a ; Eb be the two components of scattered electric field vector along two mutually orthogonal direction as , bs perpendicular to the scattering path s. as , bs are chosen in the following way: s s0 ; ai ¼ as ¼ js s0 j ð7Þ bi ¼ s 0 ai ; bs ¼ s as ;
ð6Þ
ðsÞ
ðsÞ
¼ ½hEi ðr1 ; xÞEj ðr2 ; xÞi
ði; j ¼ x; y; zÞ:
ð11Þ
In Eq. (11) the superscript (s) represents quantities pertainðsÞ ing to the scattered field, Ei ði ¼ x; y; zÞ denote the cartesian components of the typical member of the statistical ensemble of the scattered electric field, and the angle brackets denote the average taken on a statistical ensemble of realizations of the electric filed, the asterisk denotes complex conjugation. The x was included in equations above to indicate that we treat the problem in space-frequency domain and the equations hold for a single frequency component of the field, but to avoid unnecessary cluttering of the formulas, we henceforth omit x in various notations. On substituting Eq. (9) into Eq. (11) and using Eqs. (1), (3) we obtain: exp½ikðr2 r1 Þ ðsÞ W ij ðr1 s1 ; r2 s2 Þ ¼ Ai ð/1 ÞAj ð/2 Þ r1 r 2 Z Z C ðF Þ ðr01 ; r02 Þ exp½iks0 ðr02 r01 Þ D
D
exp½ikðs1 r01 s2 r02 Þ d3 r01 d3 r02
ð12Þ
Y. Xin et al. / Optics Communications 278 (2007) 247–252
here, s1 and s2 are two unit vectors along two scattering path, /1 and /2 are angles s1 and s2 make with s0. r01 and r02 are position vectors of points in the region of scatterer. As a matter of convenience, we make changes of the spatial arguments: r02 r01 ¼ R S;
ð13Þ
r02 þ r01 ¼ 2Rþ S;
substituting Eq. (13) into Eq. (12) and using Eq. (6) we obtain: W
ðsÞ ij ðr 1 s1 ; r2 s2 Þ
Z Z exp½ikðr2 r1 Þ ðF Þ S ðF Þ ðRþ ðR S Þg SÞ r1 r 2 D D þ exp½ikðs0 ðs1 þ s2 Þ=2Þ R S exp½ikðs1 s2 Þ RS
¼ Ai ð/1 ÞAj ð/2 Þ
3 þ d3 R S d RS :
ðsÞ
In Eqs. (18) and (19) we have suppressed the dependence of various quantities on the frequency x. As we have noted that the correlation function of the scattering potential of a QH media is a fast function of its spatial argument, so from the property of the Fourier transform we know that the spatial Fourier transform of the correlation function is a slow function, then we can, with no loss of reasonability, make approximations: ~gðF Þ ½kðs0 s1 Þ ¼ ~gðF Þ ½kðs0 s2 Þ ¼ ~gðF Þ ½kðs0 ðs1 þ s2 Þ=2Þ
exp½ikðr2 r1 Þ e ðF Þ S ½kðs1 s2 Þ r1 r2 ~ gðF Þ ½kðs0 ðs1 þ s2 Þ=2Þ; ð15Þ
W ij ðr1 s1 ; r2 s2 Þ ¼ Ai1 Aj2
gðF Þ are the three dimensional Fourier transwhere e S ðF Þ , ~ form of the strength and correlation coefficient of the scattering potential of the QH media. In Eq. (15) we have suppressed the arguments /1 , /2 and make use of Ai1 , Aj2 to represent the geometrical factors. According to the definition by Korotkova [10], the spectral density and spectral degree of coherence of the three dimensional random scattered field are given by: S ðsÞ ðrÞ ¼ TrWðsÞ ðr; rÞ;
ð16Þ ðsÞ
TrW ðr1 ; r2 Þ gðsÞ ðr1 ; r2 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; TrWðsÞ ðr1 ; r1 Þ TrWðsÞ ðr2 ; r2 Þ
ð17Þ
where Tr denotes the trace of a matrix. On substituting Eq. (15) into Eqs. (16), (17) we obtain for the scattered field the spectral density: 1 ðF Þ S ðsÞ ðrsÞ ¼ 2 e ð18Þ S ð0ÞgðF Þ ½kðs0 sÞðA2x þ A2y þ A2z Þ r and the spectral degree of coherence: ~ gðF Þ ½kðs0 ðs1 þ s2 Þ=2Þ g ðr1 s1 ; r2 s2 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ gðF Þ ½kðs0 s2 Þ gðF Þ ½kðs0 s1 Þ~ e ðF Þ ½kðs1 s2 Þ S Að/1 ; /2 Þ exp½ikðr2 r1 Þ; e S ðF Þ ð0Þ ð19Þ ðsÞ
where Ax1 Ax2 þ Ay1 Ay2 þ Az1 Az2 Að/1 ; /2 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 ðAx1 þ A2y1 þ A2z1 ÞðA2x2 þ A2y2 þ A2z2 Þ
ð20Þ
ð21Þ
then the first factor on the right-hand side of Eq. (19) can be approximated by unity, and Eq. (19) reduces to: gðsÞ ðr1 s1 ; r2 s2 Þ ¼ Að/1 ; /2 Þ
ð14Þ
In Eq. (14), the integral extends over the whole region of the scatterer and it is a six dimensional Fourier transform of the strength and correlation coefficient of the scattering potential. Making integration of the right-hand side of Eq. (14) results into the formula:
249
e S ðF Þ ½kðs1 s2 Þ exp½ikðr2 r1 Þ: e S ðF Þ ð0Þ ð22Þ
From Eqs. (18) and (22) we can see there are two reciprocities analogous to that in the scalar case, i.e., the spectral density of scattered electromagnetic field in the far zone of the scatterer, is proportional to spatial Fourier transform of the correlation coefficient of the scattering potential and the spectral degree of coherence of the scatterer field is proportional to the normalized spatial Fourier transform of the strength of the scattering potential of the media. 3. Beam radiated from a QH uniformly polarized electromagnetic source scattering on a QH media Recently, the concepts of QH source were generalized from scalar case to the electromagnetic case [7], and conditions [11] for such a source to generate electromagnetic beam was derived. By assuming that the source is uniformly polarized, it was verified that the far field of the beam it generates supports two reciprocity relations that is analogous to that in the scalar case. According to the unified theory of coherence, the cross-spectral density matrix of the secondary planar source which generate a electromagnetic beam is ðQÞ
ðQÞ
ðQÞ
W ij ðr1 ; r2 Þ ¼ hEi ðr1 ÞEj ðr2 Þi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðQÞ ðQÞ ¼ S i ðr1 Þ S ðQÞ ðr2 Þlij ðr1 ; r2 Þ
ði; j ¼ x; yÞ: ð23Þ
In Eq. (23), Ei , Ej are two components of the electric field along two mutually orthogonal directions x, y. The angle brackets denote average taken over the statistical ensemble of realizations of the electric field, and the asterisk denotes complex conjugation. The superscript (Q) denotes quantities pertaining to the source, r1 , r2 are the position vectors of two ðQÞ ðQÞ points in the source plane. S i ðrÞ ¼ W ii ðr; rÞ ði ¼ x; yÞ is the spectral density of one component of the electric field, ðQÞ lij ðr1 ; r2 Þ ði; j ¼ x; yÞ is the correlation coefficient between two components of the electric field in the source plane.
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Y. Xin et al. / Optics Communications 278 (2007) 247–252
For the QH electromagnetic source, approximations analogous to the scalar case are made: ðQÞ
ðQÞ
ðQÞ
S i ðr1 Þ S i ðr2 Þ S i ½ðr1 þ r2 Þ=2; ðQÞ
ð24Þ
ðQÞ
lij ðr1 ; r2 Þ lij ðr2 r1 Þ:
Uniform polarization of the source means that the spectral ðQÞ are proportional density of the two components S ðQÞ x , Sy to each other, i.e., ðQÞ S ðQÞ y ðrÞ ¼ aS x ðrÞ;
ðQÞ lðQÞ xy ðr; rÞ ¼ lxy :
ð26Þ
On substituting Eqs. (24), (25) into Eq. (23) we obtain: W
ðQÞ
¼ aij S
½ðr1 þ
ðQÞ r2 Þ=2lij ðr2
r1 Þ;
S
ðrÞ ¼
and aij ¼
S ðQÞ x ðrÞ
8 1 > < 1þa pffiffi a
> : 1þa a
1þa
þ
S ðQÞ y ðrÞ
ð29Þ
when i ¼ j ¼ y:
We use the cartesian coordinate that was defined in the previous section and assume that the beam propagates along the z axis, the cross-spectral density matrix of the far field generated by the source is: ðiÞ
2
W ij ðr1 u1 ; r2 u2 Þ ¼ ð2pkÞ cos h1 cos h2 exp½ikðr2 r1 Þ e ðQÞ W ; ij ðku1 ; ku2 Þ r1 r 2 where
Z Z
1 ð2pÞ
4
Q
ð30Þ
D
Q
exp½ikðu1 r01 u2 r02 Þ d2 r01 d2 r02
ð31Þ
is the four dimensional spatial Fourier transform of ðQÞ W ij ðr1 ; r2 Þ. In Eq. (30), u1 , u2 are unit vectors along r1 , r2 , h1 , h2 are angles u1 and u2 make with the positive z direction. It is convenient to make changes of variables: 0 0 R Q ¼ r2 r1 ;
ð32Þ
0 0 Rþ Q ¼ ðr2 þ r1 Þ=2:
D
G ðr1 s1 ; r01 ÞGðr2 s2 ; r02 Þ d3 r01 d3 r02 ;
On substituting Eq. (27) into Eq. (31) and using Eq. (32) we obtain: Z Z 1 ðQÞ e ðQÞ W ðku ; ku Þ ¼ a S ðQÞ ðRþ 1 2 ij ij R Þlij ðRQ Þ 4 ð2pÞ Q Q exp½iku1 ðRþ Q RQ =2Þ=r1 2 2 þ exp½iku2 ðRþ Q þ RQ =2Þ=r 2 d RQ d RQ
ð33Þ
ð35Þ
where
ðiÞ ij
8 ðiÞ > W > < ij ¼ W ðiÞ iy > > : ðiÞ W yj
when i; j ¼ x; y; when j ¼ z;
ð36Þ
when i ¼ z:
If it is assumed that the scatterer is located in the far zone of the source, it is reasonable before substitute Eq. (34) into Eq. (35) to make some approximations. The denominator r1r2 in front of the integral in Eq. (34) may be approximated by r1r2 R2 and the two denominator in the two exponentials under the integral on the right-hand side of Eq. (34) may be approximated by R, R = |R| is the distance from the origin (located in the source plane) to the region of the scatterer. R is the position vector of the scatter. We combine the exponential term exp½ikðr02 r01 Þ with the correlation function of the scatterer to obtain a modified correlation function: C ðF Þ ðr01 ; r02 Þ ¼ C ðF Þ ðr01 ; r02 Þ exp½ikðr02 r01 Þ:
ðQÞ
W ij ðr01 ; r02 Þ
ð34Þ
From the discussion in the previous section, we can see that the elements of the cross-spectral density matrix of the scattered field may be presented as: Z Z ðsÞ ðiÞ W ij ðr1 s1 ; r2 s2 Þ ¼ Ai ð/1 ÞAj ð/2 Þ W ij ðr01 ; r02 ÞC ðF Þ ðr01 ; r02 Þ
W
when i 6¼ j;
Q
2 2 þ exp½ikr2 u2 ðRþ Q þ QQ =2Þ=r2 d RQ d RQ :
ð28Þ
when i ¼ j ¼ x;
e ðQÞ W ij ðku1 ; ku2 Þ ¼
Q
ð27Þ
where S(Q) is the spectral density of the electromagnetic field of the source: ðQÞ
ðiÞ
W ij ðr1 u1 ; r2 u2 Þ 2 k exp½ikðr2 r1 Þ ¼ aij cos h1 cos h2 2p r1 r2 Z Z ðQÞ þ S ðQÞ ðRþ Q Þlij ðRQ Þ exp½ikr 1 u1 ðRQ RQ =2Þ=r1
ð25Þ
where a is a quantity depending only on frequency, and the correlation coefficient lðQÞ xy ðr; rÞ is independent of position:
ðQÞ ij ðr1 ; r2 Þ
hence, Eq. (30) becomes:
ð37Þ
Naturally, the combination results in a modified correlation coefficient: gðF Þ ðr01 ; r02 Þ ¼ gðF Þ ðr01 ; r02 Þ exp½ikðr02 r01 Þ:
ð38Þ
By passing we note that the modified correlation coefficient gðF Þ is still a fast function of its spatial argument and the correlation function C(F) is still a product of a slow function and a fast function. The factor cos h1 cos h2 in Eq. (34) can be approximated by cos2 h, where h is the angle R makes with the positive z direction. On substituting Eq. (34) into Eq. (35) we obtain the formula for the elements of the cross-spectral density matrix of the scattered filed: exp½ikðr2 r1 Þ ðsÞ W ij ðr1 s1 ; r2 s2 Þ ¼ Ai ð/1 ÞAj ð/2 Þ cos2 haij R2 r 1 r 2 Z Z Z Z Q ðF Þ S ðQÞ ðRþ ðRþ gðF Þ ðR R Þlij ðRQ ÞS S Þ SÞ 2 þ 3 3 þ expðikUÞ d2 R Q d RQ d RS d RS ;
ð39Þ
Y. Xin et al. / Optics Communications 278 (2007) 247–252
where U is the phase factor and given by the formula: U
þ 0 ¼ ðRþ Q RQ =2Þ=R r2 ðRQ þ RQ =2Þ=R Rþ S ðs2 s1 Þ RS ðs1 s2 Þ ¼ Rþ S ½ðs2 s1 Þ þ RQ =R RS ½ðs1 þ s2 Þ=2
r01
þ Rþ Q =R: ð40Þ
By integrating over
Rþ S
and
R S
we obtain:
exp½ikðr2 r1 Þ R2 r 1 r 2 M ij ½s2 s1 N ½ðs2 þ s1 Þ=2; ð41Þ
ðsÞ
W ij ðr1 s1 ; r2 s2 Þ ¼ aij Ai ð/1 ÞAj ð/2 Þ cos2 h
where M ij ½s2 s1 ¼
Z Q
N ½ðs1 þ s2 Þ=2 ¼
ðQÞ e ðF Þ fk½ðs2 lij ðR Q ÞS
Z S
ðQÞ
ðRþ gðF Þ fk½ðs1 Q Þ~
3 þ þ s2 Þ=2 þ Rþ Q =Rg d RQ :
Q
ð42Þ Here the meaning of the subscripts ij pertaining to quantities of the source is understood from Eq. (36). e S ðF Þ and ~gðF Þ are three dimensional spatial Fourier transform of the strength of the scattering potential S(F) and the modified correlation coefficient of the scattering potential gðF Þ , respectively. It is easy to see that Mij and N have the form of a convolution. After making substitution of Eq. (41) into Eq. (16), we get the spectral density of the scattered field: 2 k cos2 h S ðsÞ ðrsÞ ¼ LðsÞN ðsÞ; ð43Þ 2p ½1 þ aR2 r2 where L¼
A2x M xx ð0Þ
þ
aðA2y
þ
A2z ÞM yy ð0Þ:
ð44Þ
On substituting Eq. (41) into Eq. (17) we obtain the spectral degree of coherence of the scattered field: N ½ðs1 þ s2 Þ=2 g ðr1 s1 ; r2 s2 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi exp½ikðr2 r1 Þ N ðs1 Þ N ðs2 Þ L12 pffiffiffiffiffipffiffiffiffiffi ; L1 L2 ðsÞ
ð45Þ
where L12 ¼ Ax1 Ax2 M xx ðs2 s1 Þ þ aðAy1 Ay2 þ Az1 Az2 ÞM yy ðs2 s1 Þ; L1 ¼ A2x1 M xx ð0Þ þ aðA2y1 þ A2z1 ÞM yy ð0Þ; L2 ¼ A2x2 M xx ð0Þ þ aðA2y2 þ A2z2 ÞM yy ð0Þ: ð46Þ Since the modified correlation coefficient of the scattering potential gðF Þ is a fast function of its spatial argument and its Fourier transform is a slow function, we can deduce from Eq. (42) that N is a slow function of its argument and the approximation can be made: N ðs1 Þ N ðs2 Þ N ½ðs1 þ s2 Þ=2:
As a consequence, the first factor on the right-hand side of Eq. (45) can be approximated by unity and Eq. (45) reduce to: L12 gðsÞ ðr1 s1 ; r2 s2 Þ ¼ exp½ikðr2 r1 Þ pffiffiffiffiffipffiffiffiffiffi : L1 L2
ð47Þ
ð48Þ
From Eqs. (43) and (48) we can find that the scattered electromagnetic field do not support the reciprocity relations like that holding for the scalar light scattering, because in ðQÞ general, the diagonal correlation coefficients lðQÞ xx and lyy do not have definite relation to each other. However, if we assume that the two diagonal correlation coefficients are proportional to each other, i.e., ðQÞ lðQÞ yy ðrÞ ¼ blxx ðrÞ;
3 s1 Þ þ R Q =Rg d RQ ;
251
ð49Þ
where b is a quantity depending only on frequency, then, on substituting Eq. (49) into Eqs. (44) and (46), we obtain: L ¼ ½A2x þ abðA2y þ A2z ÞM xx ð0Þ; L12 ¼ ½Ax1 Ax2 þ abðAy1 Ay2 þ Az1 Az2 ÞM xx ðs2 s1 Þ; L1 ¼ ½A2x1 þ abðA2y1 þ A2z1 ÞM xx ð0Þ
ð50Þ
L2 ¼ ½A2x2 þ abðA2y2 þ A2z2 ÞM xx ð0Þ: Consequently, the spectral density and spectral degree of coherence of the scattered field reduce to: 2 k cos2 h S ðsÞ ðrsÞ ¼ KM xx ð0ÞN ðsÞ; ð51Þ 2p ½1 þ aR2 r2 M xx ðs2 s1 Þ ; ð52Þ gðsÞ ðr1 s1 ; r2 s2 Þ ¼ exp½ikðr2 r1 ÞK0 M xx ð0Þ where K ¼ A2x þ abðA2y þ A2z Þ;
ð53Þ
Ax1 Ax2 þ abðAy1 Ay2 þ Az1 Az2 Þ K0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 Ax1 þ abðA2y1 þ A2z1 Þ A2x2 þ abðA2y2 þ A2z2 Þ
ð54Þ
Now from Eqs. (51) and (52), it comes into focus that there holds two reciprocity relations for the spectral density and spectral degree of coherence of the scattered field generated by a electromagnetic beam from a QH uniformly polarized source scattering by a QH random media: The spectral density of the scattered field is proportional to the convolution of the spectral density S(Q) of the source and the spatial Fourier transform of the modified correlation coefficient gðF Þ of the scatter; the spectral degree of coherence of the scattered field is proportional to the convoluðQÞ tion of the diagonal correlation coefficients lii ði ¼ x; yÞ of the source and the spatial Fourier transform of the strength of the scattering potential. 4. Conclusion We have studied the coherence properties of scattered electromagnetic field from a QH random media. For the simple case of linearly polarized electromagnetic plane wave scattering, it is found that the scattered far field
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satisfy two reciprocity relations analogous to that in the scalar wave scattering. For a beam radiated from QH electromagnetic source scattering on a QH random media, the far field do not always satisfy the reciprocity relations. However, the reciprocity relations come into existence when the two diagonal correlation coefficients are proportional to each other: the spectral density of the scattered field is related to a convolution of the spectral density of the source and the spatial Fourier transform of the modified correlation coefficient of the scatterer; the spectral degree of coherence is related to a convolution of the diagonal correlation coefficients and the Fourier transform of the scattering strength of the scatterer. In most occasions, electromagnetic waves are utilized rather than a scalar wave, so the results of our research may find some applications in fields as target detection and imaging. Acknowledgments This work was supported by the National Natural Science Foundation of China (50176020). The authors
gratefully acknowledge Prof. Emil Wolf and Dr. Olga Korotkova for sending some of their papers to us. References [1] G. Jarry, E. Steimer, V. Damaschini, M. Epifanie, M. Jurczak, R. Kaiser, Appl. Opt. 37 (1998) 7357. [2] B.G. Hoover, L. Deslauriers, S.M. Grannell, R.E. Ahmed, D.S. Dilworth, B.D. Athey, E.N. Leith, Phys. Rev. E 65 (2002) 026614. [3] T.A. Leskova, A.A. Maradudin, J.M. Lopez, Phys. Rev. E 71 (2005) 036606. [4] R.A. Silverman, Proc. Cambridge Philos. Soc. 54 (1958) 530. [5] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995. [6] T.D. Visser, D.G. Fischer, E. Wolf, J. Opt. Soc. Am. A 23 (2006) 1631. [7] O. Korotkova, B.G. Hoover, V.L. Gamiz, E. Wolf, J. Opt. Soc. Am. A 22 (2005) 2547. [8] L. Tsang, J.A. Kong, K.H. Ding, Scattering of Electromagnetic Waves: Theory and Applications, Wiley, 2000. [9] E. Wolf, Phys. Lett. A 312 (2003) 263. [10] O. Korotkova, E. Wolf, J. Opt. Soc. Am. A 21 (2004) 2382. [11] O. Korotkova, M. Salem, E. Wolf, Opt. Lett. 29 (2004) 1173.