Abstract

The relationship between the properties of the far-zone field and the characteristics of the scattering medium for an electromagnetic light wave on scattering from a quasi-homogeneous anisotropic medium is investigated. It is shown that the spectral density and the spectral degree of coherence of the scattered field can be factorized as a product of two parts, the one is dependent on the polarization of the incident field, and the other is dependent on the characteristics of the medium. The medium-dependent part displays two reciprocity relations, i.e. the normalized spectral density of the scattered field is proportional to the Fourier transform of the normalized correlation coefficient of the scattering potential, and the spectral degree of coherence of the scattered field is proportional to the Fourier transform of the strength of scattering potential. An example of Gaussian-correlated, quasi-homogeneous, anisotropic medium is discussed to illustrate these reciprocity relations.

© 2017 Optical Society of America

1. Introduction

Over the past few years, the theory of weak scattering, which discussed the relationship between the structure characteristics of a scattering medium and the statistical optical properties of its scattered field, attracted substantial attentions due to its potential applications in areas such as remote sensing, detection, and medical diagnosis. Since the seminal work made by Wolf and his collaborators that the spectrum of a polychromatic light wave may change as it is scattered from a random medium [1], numerous papers have been published to discuss the far-zone property of light waves on scattering from various media, including the random medium, the deterministic medium, and the collection of particles [2–13]. It is shown from these publications that there is a close relationship between the properties of the far-zone scattered field and the characteristics of the scattering medium (for a review of this work, please see [14]). These results have been applied to the inverse scattering problem, that is, the determination of the structure information of an unknown medium from measurement of its scattered field (see, for examples [15–19]).

Among all the discussions, a general and important class of random media, i.e. the quasi-homogeneous medium, is frequently mentioned. A medium can be regarded as quasi-homogeneous medium when the strength of the scattering potential SF(r,ω) varies much more slowly with position r than the normalized correlation coefficient μF(r1,r2,ω)=μF(r2r1,ω) varies with the position difference r2r1 [20]. The far-zone properties of light waves on scattering from a quasi-homogeneous medium are discussed extensively (see, for examples [21–26]). It is found that there are two reciprocity relations between the properties of the scattered field and the characteristics of the scattering medium [3, 23]. More recently, the reciprocity relations of the third-order correlation between intensity fluctuations of light waves on scattering have been discussed by Chang and Li and the reciprocity relations of light waves from Young’s pinholes on scattering have been discussed by Yu and Li [27, 28]. These researches may have potential applications in the reconstruction of structure information of a medium from the measurement of the scattered field. However, to the best of our knowledge, almost all the discussions on the reciprocity relations of light wave scattering are based on the incidence of scalar light waves (one exception is Ref [5].). In this manuscript, this discussion may be generalized to the scattering of electromagnetic light waves from anisotropic medium. The relationship between the characteristics of the quasi-homogeneous anisotropic medium and the properties of the scattered field will be discussed and two reciprocity relations for the scattering of electromagnetic light wave will be presented. An example of electromagnetic light wave on scattering from a Gaussian-correlated, quasi-homogeneous, anisotropic medium will be discussed to illustrate these relations.

2. Theory

Let us start by considering a spatially coherent electromagnetic plane light wave, with a propagation direction specified by a real unit vector s0, is incident on a statistically stationary random scattering medium occupying a finite domain D (see Fig. 1). The property of the incident field at a pair of points r1 and r2 within the area of the scattering medium can be described by its cross-spectral density matrix, which is defined as [29]

W(i)(r1,r2,s0,ω)E(i)(r1,s0,ω)E(i)(r2,s0,ω),
where ω denotes the frequency, E(i)(r,s0,ω) is the Hermitian adjoint matrix of the incident field Ε(i)(r,s0,ω), and the angular brackets denote the average, taken over the statistically ensemble of realizations of the incident field. The incident field Ε(i)(r,s0,ω) within the area of the scattering medium can be expressed as [30, 31]
E(i)(r,s0,ω)=[Ex(i)(r,s0,ω),Ey(i)(r,s0,ω),0],
where
Ej(i)(r,s0,ω)=aj(ω)exp(iks0r)(j=x,y),
is the Cartesian coordinate component of the incident field along the j-th axis, aj(ω) is a random function, and k=ω/c is the free-space wave number with c being the speed of light in vacuum. Assume that the interaction between the incident wave and the scattering medium is sufficiently weak, so that the scattering process can be treated within the accuracy of the first-order Born approximation [32]. Under these circumstances, the far-zone scattered field at a point specified by a position vector rs(s2=1) may be expressed in the matrix form [30]
E(s)(rs,s0,ω)=exp(ikr)rDF(r,ω)[Ex(i)(r,ω)Ey(i)(r,ω)Ez(i)(r,ω)]T[1sx2sxsysxszsysx1sy2syszszsxszsy1sz2]exp(iksr)d3r,
where “T” denotes the transpose operation of a matrix, F(r,ω) is the scattering potential of the scatterer, and
sx=sinθcosϕ,sy=sinθsinϕ,sz=cosθ,
are the three Cartesian coordinate components of the unit vector of the scattering direction s with θ being the polar angle made by the scattering direction and the z axis and ϕ being the azimuthal angle made by the perpendicular projection of the scattering direction s in the x-y plane and the x axis (as shown in Fig. 1).

 figure: Fig. 1

Fig. 1 Illustration of notations.

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The property of the scattered field may likewise be represented by an ensemble of realizations,E(s)(rs,s0,ω), whose cross-spectral density matrix may be represented in a similar form as the incident field, i.e.,

W(s)(rs1,rs2,s0,ω)E(s)(rs1,s0,ω)E(s)(rs2,s0,ω),
where the angular brackets “” denote the ensemble average. On substituting from Eq. (4) into Eq. (6), and after some calculations, we obtain for the cross-spectral density matrix the expression
W(s)(rs1,rs2,s0,ω)=A(s1,s2)r2DCF(r1,r2,ω)exp{ik[(s2s0)r2(s1s0)r1]}d3r1d3r2,
where
A(s1,s2)=[1s1x2s1xs1ys1xs1zs1ys1x1s1y2s1ys1zs1zs1xs1zs1y1s1z2]T[Sx(ω)000Sy(ω)0000][1s2x2s2xs2ys2xs2zs2ys2x1s2y2s2ys2zs2zs2xs2zs2y1s2z2],
here, Sj(ω)=aj(ω)aj(ω),(j=x,y) represents the spectrum of the incident field along the j-th axis, and
CF(r1,r2,ω)=F(r1,ω)F(r2,ω)
is the two-point correlation function of the scattering potentials with the “” denoting the complex conjugate.

The far-zone scattered spectral density can be obtained from its cross-spectral density matrix [i.e., Eq. (7)], by the following formula [31]

S(s)(rs,s0,ω)=Tr[W(s)(rs,rs,s0,ω)],
where “Tr” denotes the trace of the matrix. And the spectral degree of coherence of the scattered field can also be derived from Eq. (7) by using the following definition [31]

μ(s)(rs1,rs2,s0,ω)=Tr[W(s)(rs1,rs2,s0,ω)]Tr[W(s)(rs1,rs1,s0,ω)]Tr[W(s)(rs2,rs2,s0,ω)].

For the simplicity of following discussion, we rewrite the correlation function of the scattering potential as its strength of the scattering potential and its normalized correlation coefficient of the scattering potential [29], i.e.,

CF(r1,r2,ω)=[SF(r1,ω)SF(r2,ω)]1/2μF(r1,r2,ω),
where SF(r,ω)=CF(r,r,ω) is the strength of the scattering potential at location of r, and μF(r1,r2,ω) is the normalized correlation coefficient of the scattering potential that depends on two spatial position variables r1and r2. We will now assume that the scattering medium is spatially quasi-homogeneous and anisotropic. In this case, the correlation function of the scattering potentials of the scattering medium can be expressed as
CF(r1,r2,ω)=SFx(Rsx+,ω)SFy(Rsy+,ω)SFz(Rsz+,ω),×μFx(Rsx,ω)μFy(Rsy,ω)μFz(Rsz,ω),
where
Rsj+=(r1j+r2j)/2Rsj=r2jr1j(j=x,y,z).
It should be noted that there is other definition of the correlation function of a quasi-homogeneous anisotropic medium (see, for examples, Refs [13]. and [25]), which presents the scattering potential of the medium by a 3×3 diagonal matrix. For the convenience of discussion, we assume that the property of the scattering medium is independent on the vibration direction of the incident field. In this case, Eq. (13) is reasonable enough to describe the property of the scattering medium. On substituting from Eq. (13) together with Eq. (14) into Eq. (7), and after manipulating the Fourier transform, the cross-spectral density matrix of scattered field in the far-zone of a quasi-homogeneous anisotropic medium can be represented by the form
W(s)(rs1,rs2,s0,ω)=A(s1,s2)r2S˜Fx(Ksx+,ω)S˜Fy(Ksy+,ω)S˜Fz(Ksz+,ω)×μ˜Fx(Ksx,ω)μ˜Fy(Ksy,ω)μ˜Fz(Ksz,ω),
where
S˜Fj(Ksj+,ω)=DSFj(Rsj+,ω)exp(iKsj+Rsj+)dRsj+
and
μ˜Fj(Ksj,ω)=DμFj(Rsj,ω)exp(iKsjRsj)dRsj
are the Fourier transform of the strength function SFj and the Fourier transform of the normalized correlation coefficient μFj, respectively, with

Ksj+=k(s2js1j),Ksj=k[(s1j+s2j)2s0j](j=x,y,z).

The spectral density and the spectral degree of coherence of the scattered field in the far-zone can be obtained by substituting from Eq. (15) into Eqs. (10) and (11), respectively, with the following forms

S(s)(rs,s0,ω)=Tr[A(s,s)]S˜F(0,ω)r2×μ˜Fx[k(sxs0x),ω]μ˜Fy[k(sys0y),ω]μ˜Fz[k(szs0z),ω],
and
μ(s)(rs1,rs2,s0,ω)=Tr[A(s1,s2)]Tr[A(s1,s1)]Tr[A(s2,s2)]1S˜F(0,ω)×S˜Fx(Ksx+,ω)S˜Fy(Ksy+,ω)S˜Fz(Ksz+,ω)μ˜Fx[k(s1xs0x),ω]μ˜Fy[k(s1ys0y),ω]μ˜Fz[k(s1zs0z),ω].×μ˜Fx(Ksx,ω)μ˜Fy(Ksy,ω)μ˜Fz(Ksz,ω)μ˜Fx[k(s2xs0x),ω]μ˜Fy[k(s2ys0y),ω]μ˜Fz[k(s2zs0z),ω]
For a quasi-homogeneous medium, the normalized correlation coefficient of the scattering potential μF(r,ω) is a fast function of its spatial argument r, so from the property of the Fourier transform we know that the Fourier transform of the correlation function μ˜F(K,ω) is a slow function of K. Then we can, without loss of reasonability, make the following approximations [29]
μ˜Fj[k(s1js0j),ω]μ˜Fj[k(s2js0j),ω]μ˜Fj{k[(s1j+s2j)2s0j],ω}
Based on the approximation of Eq. (21), the spectral degree of coherence of the far-zone scattered field can be simplified as
μ(s)(rs1,rs2,s0,ω)=Tr[A(s1,s2)]Tr[A(s1,s1)]Tr[A(s2,s2)]S˜Fx(Ksx+,ω)S˜Fy(Ksy+,ω)S˜Fz(Ksz+,ω)S˜F(0,ω).
As shown in Eqs. (19) and (22), when an electromagnetic light wave is scattered from a quasi-homogeneous anisotropic medium, the spectral density and the spectral degree of coherence can be expressed as a product of two factors, the one is dependent on the polarization of the incident field, and the other is dependent on the correlation function of the scattering medium, and

  • 1. The medium-depended spectral density is proportional to Fourier transform of the normalized correlation coefficient of the scattering potential, and
  • 2. The medium-depended spectral degree of coherence is proportional to the Fourier transform of the strength of the scattering potential.

3. An example

In this part, an example of an electromagnetic plane light wave on scattering from a quasi-homogeneous anisotropic medium will be discussed to illustrate the reciprocity relations that obtained in the above section. Assume that both the strength function and the normalized correlation coefficient of the quasi-homogeneous anisotropic medium obey the Gaussian distributions [22], i.e.,

SF(Rs+,ω)=C0exp[Rsx+22σsx2]exp[Rsy+22σsy2]exp[Rsz+22σsz2],
and
μF(Rs,ω)=exp[Rsx22σμx2]exp[Rsy22σμy2]exp[Rsz22σμz2],
where C0 is a positive constant, σs and σμ denote the effective length and the effective correlation length of the scattering potential of the medium, respectively. Substitution of Eqs. (23) and (24) into Eqs. (16) and (17), respectively, one can obtain the Fourier transform of the strength function and the Fourier transform of the normalized correlation coefficient, with the following forms
S˜F[k(s2s1),ω]=(2π)3/2C0σsxσsyσsz×exp{k22[(s2xs1x)2σsx2+(s2ys1y)2σsy2+(s2zs1z)2σsz2]},
and
μ˜F[k(s1+s22s0),ω]=(2π)3/2σμxσμyσμz.×exp{k22[(s1x+s2x2s0x)2σμx2+(s1y+s2y2s0y)2σμy2+(s1z+s2z2s0z)2σμz2]}
On substituting from Eqs. (25) and (26) into Eqs. (19) and (22), after performing tedious but straightforward calculations, we obtain the following expressions for the spectral density and for the spectral degree of coherence of the scattered field in the far-zone, i.e.,
S(s)(rs,s0,ω)=Sx(ω)(1sx2)+Sy(ω)(1sy2)r2(2π)3C0σsxσsyσszσμxσμyσμz×exp{k22[(sxs0x)2σμx2+(sys0y)2σμy2+(szs0z)2σμz2]}
and

μ(s)(rs1,rs2,s0,ω)={Sx(ω)[(1s1x2)(1s2x2)+s1xs2x(s1ys2y+s1zs2z)]Sx(ω)(1s1x2)+Sy(ω)(1s1y2)Sx(ω)(1s2x2)+Sy(ω)(1s2y2).+Sy(ω)[(1s1y2)(1s2y2)+s1ys2y(s1xs2x+s1zs2z)]Sx(ω)(1s1x2)+Sy(ω)(1s1y2)Sx(ω)(1s2x2)+Sy(ω)(1s2y2)}×exp{k22[(s2xs1x)2σsx2+(s2ys1y)2σsy2+(s2zs1z)2σsz2]}

In the following, some numerical results will be presented to show the reciprocity relations of an electromagnetic light waves on scattering from a Gaussian-correlated, quasi-homogeneous, anisotropic medium. In Fig. 2, the normalized spectral density of an electromagnetic light waves on scattering from two different Gaussian-correlated, quasi-homogeneous, anisotropic media is plotted. Figure 2(a) presents the normalized spectral density produced by a quasi-homogeneous medium with isotropic strength and with anisotropic correlation coefficient, while Fig. 2(b) presents the normalized spectral density produced by a quasi-homogeneous medium with anisotropic strength and with isotropic correlation coefficient. It is shown from Fig. 2 that the distribution of the spectral density is isotropic for a scattering medium with isotropic correlation coefficient, and is anisotropic for a scattering medium with anisotropic correlation coefficient, i.e. the normalized spectral density of the scattered field is governed by the correlation coefficient of the scattering potential of the medium. This phenomenon can be explained by the first reciprocity theorem (see also Eq. (19)), i.e. the scattered spectral density is independent on the strength of the scattering potential of the medium.

 figure: Fig. 2

Fig. 2 Normalized spectral density (S(s)(rs,s0,ω)/S(s)(rs0,s0,ω)) of the far-zone scattered field. The parameters for calculations are as follows: (a)Sx(ω)=Sy(ω)=1, σsx=σsy=σsz=20λ,σμx=1λ,σμy=2λ,σμz=3λ; (b)Sx(ω)=Sy(ω)=1,σsx=10λ, σsy=20λ,σsz=30λ,σμx=σμy=σμz=1λ.

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In Fig. 3, the spectral degree of coherence of the far-zone scattered field is plotted. In Fig. 3(a), we choose the scattering medium with isotropic strength and with anisotropic correlation coefficient, while in Fig. 3(b), we choose the scattering medium with anisotropic strength and with isotropic correlation coefficient. It is shown from Fig. 3 that the distribution of spectral degree of coherence of the scattered field is isotropic when the strength of the scattering potential of the medium is isotropic, and is anisotropic when the strength of the scattering potential of the medium is anisotropic, i.e. the scattered spectral degree of coherence is governed by the strength of the scattering potential of the medium. This phenomenon can be explained by the second reciprocity theorem (see also Eq. (22)), i.e. the scattered spectral degree of coherence is independent on the correlation coefficient of the scattering potential of the medium.

 figure: Fig. 3

Fig. 3 Spectral degree of coherence (μ(s)(rs0,rs,s0,ω)) of the far-zone scattered field. The parameters are chosen as follows: (a)Sx(ω)=Sy(ω)=1,σsx=σsy=σsz=1λ,σμx=1λ, σμy=2λ,σμz=3λ;(b)Sx(ω)=Sy(ω)=1,σsx=1λ,σsy=2λ,σsz=4λ, σμx=σμy=σμz=1λ.

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4. Conclusion

In summary, the scattering behaviors of an electromagnetic light wave from a quasi-homogeneous, anisotropic medium have been discussed within the accuracy of the first-order Born approximation. It is shown that there are two factors that play roles in the scattered field, the one is the polarization of the incident field, and the other is the properties of the scattering medium, and the medium-dependent factors shows two reciprocity relations between the properties of the scattered field and the characteristics of the scattering medium. As an example, the scattering of an electromagnetic light wave from a Gaussian-correlated, quasi-homogeneous, anisotropic medium has been discussed. These results may have potential applications in inverse scattering problem. For example, the anisotropic information of the scattering medium may be determined from the measurement of the scattered field.

Funding

This research was supported by the National Natural Science Foundation of China (NSFC) under Grants 11404231, 61475105.

References and links

1. E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989). [CrossRef]  

2. A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23(17), 1340–1342 (1998). [CrossRef]   [PubMed]  

3. T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23(7), 1631–1638 (2006). [CrossRef]   [PubMed]  

4. S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78(6), 063815 (2008). [CrossRef]  

5. Y. Xin, Y. Chen, Q. Zhao, and M. Zhou, “Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media,” Opt. Commun. 278(2), 247–252 (2007). [CrossRef]  

6. S. Sahin and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett. 34(12), 1762–1764 (2009). [CrossRef]   [PubMed]  

7. J. Li and L. Chang, “Spectral shifts and spectral switches of light generated by scattering of arbitrary coherent waves from a quasi-homogeneous media,” Opt. Express 23(13), 16602–16616 (2015). [CrossRef]   [PubMed]  

8. S. Sahin, G. Gbur, and O. Korotkova, “Scattering of light from particles with semisoft boundaries,” Opt. Lett. 36(20), 3957–3959 (2011). [CrossRef]   [PubMed]  

9. O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056609 (2007). [CrossRef]   [PubMed]  

10. X. Du and D. Zhao, “Scattering of light by a system of anisotropic particles,” Opt. Lett. 35(10), 1518–1520 (2010). [CrossRef]   [PubMed]  

11. Z. Mei and O. Korotkova, “Random light scattering by collections of ellipsoids,” Opt. Express 20(28), 29296–29307 (2012). [CrossRef]   [PubMed]  

12. C. Ding, Y. Cai, Y. Zhang, and L. Pan, “Scattering of a partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 376(42), 2697–2702 (2012). [CrossRef]  

13. J. Li, P. Wu, and L. Chang, “Condition for invariant spectrum of an electromagnetic wave scattered from an anisotropic random media,” Opt. Express 23(17), 22123–22133 (2015). [CrossRef]   [PubMed]  

14. D. Zhao and T. Wang, “Direct and inverse problems in the theory of light scattering,” Prog. Opt. 57, 261–308 (2012). [CrossRef]  

15. D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11(11), 1128–1135 (1994). [CrossRef]  

16. G. Gbur and E. Wolf, “Determination of density correlation functions from scattering of polychromatic light,” Opt. Commun. 168(168), 39–45 (1999). [CrossRef]  

17. M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009). [CrossRef]   [PubMed]  

18. T. Wang and D. Zhao, “Determination of pair-structure factor of scattering potential of a collection of particles,” Opt. Lett. 35(3), 318–320 (2010). [CrossRef]   [PubMed]  

19. J. Li, “Determination of correlation function of scattering potential of random medium by Gauss vortex beam,” Opt. Commun. 308(11), 164–168 (2013). [CrossRef]  

20. W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67(2), 85–90 (1988). [CrossRef]  

21. D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133(1), 17–21 (1997). [CrossRef]  

22. X. Du and D. Zhao, “Scattering of light by Gaussian-correlated quasi-homogeneous anisotropic media,” Opt. Lett. 35(3), 384–386 (2010). [CrossRef]   [PubMed]  

23. X. Du and D. Zhao, “Reciprocity relations for scattering from quasi-homogeneous anisotropic media,” Opt. Commun. 284(16), 3808–3810 (2011). [CrossRef]  

24. Y. Xin, Y. He, Y. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett. 35(23), 4000–4002 (2010). [CrossRef]   [PubMed]  

25. J. Li, F. Chen, and L. Chang, “Correlation between intensity fluctuations of electromagnetic waves scattered from a spatially quasi-homogeneous, anisotropic medium,” Opt. Express 24(21), 24274–24286 (2016). [CrossRef]   [PubMed]  

26. D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294(294), 43–48 (2013). [CrossRef]  

27. L. Chang and J. Li, “Reciprocal relations for third-order correlation between intensity fluctuations of light scattered from a quasi-homogeneous medium,” IEEE Photonics J. 9(2), 6100507 (2017). [CrossRef]  

28. J. Yu and J. Li, “Reciprocal relations for light from Young’s pinholes scattering upon a quasi-homogeneous medium,” Laser Phys. Lett. 14(5), 056003 (2017). [CrossRef]  

29. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

30. Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 193–200 (2010). [CrossRef]  

31. T. Wang and D. Zhao, “Scattering theory of stochastic electromagnetic light waves,” Opt. Lett. 35(14), 2412–2414 (2010). [CrossRef]   [PubMed]  

32. M. Born and E. Wolf, 32. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

References

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  • |

  1. E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989).
    [Crossref]
  2. A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23(17), 1340–1342 (1998).
    [Crossref] [PubMed]
  3. T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23(7), 1631–1638 (2006).
    [Crossref] [PubMed]
  4. S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78(6), 063815 (2008).
    [Crossref]
  5. Y. Xin, Y. Chen, Q. Zhao, and M. Zhou, “Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media,” Opt. Commun. 278(2), 247–252 (2007).
    [Crossref]
  6. S. Sahin and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett. 34(12), 1762–1764 (2009).
    [Crossref] [PubMed]
  7. J. Li and L. Chang, “Spectral shifts and spectral switches of light generated by scattering of arbitrary coherent waves from a quasi-homogeneous media,” Opt. Express 23(13), 16602–16616 (2015).
    [Crossref] [PubMed]
  8. S. Sahin, G. Gbur, and O. Korotkova, “Scattering of light from particles with semisoft boundaries,” Opt. Lett. 36(20), 3957–3959 (2011).
    [Crossref] [PubMed]
  9. O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056609 (2007).
    [Crossref] [PubMed]
  10. X. Du and D. Zhao, “Scattering of light by a system of anisotropic particles,” Opt. Lett. 35(10), 1518–1520 (2010).
    [Crossref] [PubMed]
  11. Z. Mei and O. Korotkova, “Random light scattering by collections of ellipsoids,” Opt. Express 20(28), 29296–29307 (2012).
    [Crossref] [PubMed]
  12. C. Ding, Y. Cai, Y. Zhang, and L. Pan, “Scattering of a partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 376(42), 2697–2702 (2012).
    [Crossref]
  13. J. Li, P. Wu, and L. Chang, “Condition for invariant spectrum of an electromagnetic wave scattered from an anisotropic random media,” Opt. Express 23(17), 22123–22133 (2015).
    [Crossref] [PubMed]
  14. D. Zhao and T. Wang, “Direct and inverse problems in the theory of light scattering,” Prog. Opt. 57, 261–308 (2012).
    [Crossref]
  15. D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11(11), 1128–1135 (1994).
    [Crossref]
  16. G. Gbur and E. Wolf, “Determination of density correlation functions from scattering of polychromatic light,” Opt. Commun. 168(168), 39–45 (1999).
    [Crossref]
  17. M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
    [Crossref] [PubMed]
  18. T. Wang and D. Zhao, “Determination of pair-structure factor of scattering potential of a collection of particles,” Opt. Lett. 35(3), 318–320 (2010).
    [Crossref] [PubMed]
  19. J. Li, “Determination of correlation function of scattering potential of random medium by Gauss vortex beam,” Opt. Commun. 308(11), 164–168 (2013).
    [Crossref]
  20. W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67(2), 85–90 (1988).
    [Crossref]
  21. D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133(1), 17–21 (1997).
    [Crossref]
  22. X. Du and D. Zhao, “Scattering of light by Gaussian-correlated quasi-homogeneous anisotropic media,” Opt. Lett. 35(3), 384–386 (2010).
    [Crossref] [PubMed]
  23. X. Du and D. Zhao, “Reciprocity relations for scattering from quasi-homogeneous anisotropic media,” Opt. Commun. 284(16), 3808–3810 (2011).
    [Crossref]
  24. Y. Xin, Y. He, Y. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett. 35(23), 4000–4002 (2010).
    [Crossref] [PubMed]
  25. J. Li, F. Chen, and L. Chang, “Correlation between intensity fluctuations of electromagnetic waves scattered from a spatially quasi-homogeneous, anisotropic medium,” Opt. Express 24(21), 24274–24286 (2016).
    [Crossref] [PubMed]
  26. D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294(294), 43–48 (2013).
    [Crossref]
  27. L. Chang and J. Li, “Reciprocal relations for third-order correlation between intensity fluctuations of light scattered from a quasi-homogeneous medium,” IEEE Photonics J. 9(2), 6100507 (2017).
    [Crossref]
  28. J. Yu and J. Li, “Reciprocal relations for light from Young’s pinholes scattering upon a quasi-homogeneous medium,” Laser Phys. Lett. 14(5), 056003 (2017).
    [Crossref]
  29. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  30. Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 193–200 (2010).
    [Crossref]
  31. T. Wang and D. Zhao, “Scattering theory of stochastic electromagnetic light waves,” Opt. Lett. 35(14), 2412–2414 (2010).
    [Crossref] [PubMed]
  32. M. Born and E. Wolf, 32. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

2017 (2)

L. Chang and J. Li, “Reciprocal relations for third-order correlation between intensity fluctuations of light scattered from a quasi-homogeneous medium,” IEEE Photonics J. 9(2), 6100507 (2017).
[Crossref]

J. Yu and J. Li, “Reciprocal relations for light from Young’s pinholes scattering upon a quasi-homogeneous medium,” Laser Phys. Lett. 14(5), 056003 (2017).
[Crossref]

2016 (1)

2015 (2)

2013 (2)

D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294(294), 43–48 (2013).
[Crossref]

J. Li, “Determination of correlation function of scattering potential of random medium by Gauss vortex beam,” Opt. Commun. 308(11), 164–168 (2013).
[Crossref]

2012 (3)

D. Zhao and T. Wang, “Direct and inverse problems in the theory of light scattering,” Prog. Opt. 57, 261–308 (2012).
[Crossref]

Z. Mei and O. Korotkova, “Random light scattering by collections of ellipsoids,” Opt. Express 20(28), 29296–29307 (2012).
[Crossref] [PubMed]

C. Ding, Y. Cai, Y. Zhang, and L. Pan, “Scattering of a partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 376(42), 2697–2702 (2012).
[Crossref]

2011 (2)

S. Sahin, G. Gbur, and O. Korotkova, “Scattering of light from particles with semisoft boundaries,” Opt. Lett. 36(20), 3957–3959 (2011).
[Crossref] [PubMed]

X. Du and D. Zhao, “Reciprocity relations for scattering from quasi-homogeneous anisotropic media,” Opt. Commun. 284(16), 3808–3810 (2011).
[Crossref]

2010 (6)

2009 (2)

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett. 34(12), 1762–1764 (2009).
[Crossref] [PubMed]

2008 (1)

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78(6), 063815 (2008).
[Crossref]

2007 (2)

Y. Xin, Y. Chen, Q. Zhao, and M. Zhou, “Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media,” Opt. Commun. 278(2), 247–252 (2007).
[Crossref]

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056609 (2007).
[Crossref] [PubMed]

2006 (1)

1999 (1)

G. Gbur and E. Wolf, “Determination of density correlation functions from scattering of polychromatic light,” Opt. Commun. 168(168), 39–45 (1999).
[Crossref]

1998 (1)

1997 (1)

D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133(1), 17–21 (1997).
[Crossref]

1994 (1)

1989 (1)

1988 (1)

W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67(2), 85–90 (1988).
[Crossref]

Cai, Y.

C. Ding, Y. Cai, Y. Zhang, and L. Pan, “Scattering of a partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 376(42), 2697–2702 (2012).
[Crossref]

Carter, W. H.

W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67(2), 85–90 (1988).
[Crossref]

Chang, L.

Chen, F.

Chen, Y.

Y. Xin, Y. He, Y. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett. 35(23), 4000–4002 (2010).
[Crossref] [PubMed]

Y. Xin, Y. Chen, Q. Zhao, and M. Zhou, “Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media,” Opt. Commun. 278(2), 247–252 (2007).
[Crossref]

Ding, C.

C. Ding, Y. Cai, Y. Zhang, and L. Pan, “Scattering of a partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 376(42), 2697–2702 (2012).
[Crossref]

Dogariu, A.

Du, X.

Fischer, D. G.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
[Crossref] [PubMed]

T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23(7), 1631–1638 (2006).
[Crossref] [PubMed]

D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133(1), 17–21 (1997).
[Crossref]

D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11(11), 1128–1135 (1994).
[Crossref]

Foley, J. T.

Gbur, G.

S. Sahin, G. Gbur, and O. Korotkova, “Scattering of light from particles with semisoft boundaries,” Opt. Lett. 36(20), 3957–3959 (2011).
[Crossref] [PubMed]

G. Gbur and E. Wolf, “Determination of density correlation functions from scattering of polychromatic light,” Opt. Commun. 168(168), 39–45 (1999).
[Crossref]

Gori, F.

He, Y.

Korotkova, O.

Z. Mei and O. Korotkova, “Random light scattering by collections of ellipsoids,” Opt. Express 20(28), 29296–29307 (2012).
[Crossref] [PubMed]

S. Sahin, G. Gbur, and O. Korotkova, “Scattering of light from particles with semisoft boundaries,” Opt. Lett. 36(20), 3957–3959 (2011).
[Crossref] [PubMed]

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 193–200 (2010).
[Crossref]

S. Sahin and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett. 34(12), 1762–1764 (2009).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78(6), 063815 (2008).
[Crossref]

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056609 (2007).
[Crossref] [PubMed]

Kuebel, D.

D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294(294), 43–48 (2013).
[Crossref]

Lahiri, M.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
[Crossref] [PubMed]

Li, J.

Mei, Z.

Pan, L.

C. Ding, Y. Cai, Y. Zhang, and L. Pan, “Scattering of a partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 376(42), 2697–2702 (2012).
[Crossref]

Sahin, S.

Shirai, T.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
[Crossref] [PubMed]

Tong, Z.

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 193–200 (2010).
[Crossref]

Visser, T. D.

D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294(294), 43–48 (2013).
[Crossref]

T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23(7), 1631–1638 (2006).
[Crossref] [PubMed]

Wang, T.

Wolf, E.

D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294(294), 43–48 (2013).
[Crossref]

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
[Crossref] [PubMed]

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056609 (2007).
[Crossref] [PubMed]

T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23(7), 1631–1638 (2006).
[Crossref] [PubMed]

G. Gbur and E. Wolf, “Determination of density correlation functions from scattering of polychromatic light,” Opt. Commun. 168(168), 39–45 (1999).
[Crossref]

A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23(17), 1340–1342 (1998).
[Crossref] [PubMed]

D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133(1), 17–21 (1997).
[Crossref]

D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11(11), 1128–1135 (1994).
[Crossref]

E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989).
[Crossref]

W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67(2), 85–90 (1988).
[Crossref]

Wu, P.

Xin, Y.

Y. Xin, Y. He, Y. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett. 35(23), 4000–4002 (2010).
[Crossref] [PubMed]

Y. Xin, Y. Chen, Q. Zhao, and M. Zhou, “Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media,” Opt. Commun. 278(2), 247–252 (2007).
[Crossref]

Yu, J.

J. Yu and J. Li, “Reciprocal relations for light from Young’s pinholes scattering upon a quasi-homogeneous medium,” Laser Phys. Lett. 14(5), 056003 (2017).
[Crossref]

Zhang, Y.

C. Ding, Y. Cai, Y. Zhang, and L. Pan, “Scattering of a partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 376(42), 2697–2702 (2012).
[Crossref]

Zhao, D.

Zhao, Q.

Y. Xin, Y. Chen, Q. Zhao, and M. Zhou, “Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media,” Opt. Commun. 278(2), 247–252 (2007).
[Crossref]

Zhou, M.

Y. Xin, Y. Chen, Q. Zhao, and M. Zhou, “Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media,” Opt. Commun. 278(2), 247–252 (2007).
[Crossref]

IEEE Photonics J. (1)

L. Chang and J. Li, “Reciprocal relations for third-order correlation between intensity fluctuations of light scattered from a quasi-homogeneous medium,” IEEE Photonics J. 9(2), 6100507 (2017).
[Crossref]

J. Opt. Soc. Am. A (3)

Laser Phys. Lett. (1)

J. Yu and J. Li, “Reciprocal relations for light from Young’s pinholes scattering upon a quasi-homogeneous medium,” Laser Phys. Lett. 14(5), 056003 (2017).
[Crossref]

Opt. Commun. (7)

J. Li, “Determination of correlation function of scattering potential of random medium by Gauss vortex beam,” Opt. Commun. 308(11), 164–168 (2013).
[Crossref]

W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67(2), 85–90 (1988).
[Crossref]

D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133(1), 17–21 (1997).
[Crossref]

X. Du and D. Zhao, “Reciprocity relations for scattering from quasi-homogeneous anisotropic media,” Opt. Commun. 284(16), 3808–3810 (2011).
[Crossref]

D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294(294), 43–48 (2013).
[Crossref]

G. Gbur and E. Wolf, “Determination of density correlation functions from scattering of polychromatic light,” Opt. Commun. 168(168), 39–45 (1999).
[Crossref]

Y. Xin, Y. Chen, Q. Zhao, and M. Zhou, “Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media,” Opt. Commun. 278(2), 247–252 (2007).
[Crossref]

Opt. Express (4)

Opt. Lett. (8)

Phys. Lett. A (1)

C. Ding, Y. Cai, Y. Zhang, and L. Pan, “Scattering of a partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 376(42), 2697–2702 (2012).
[Crossref]

Phys. Rev. A (2)

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78(6), 063815 (2008).
[Crossref]

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 193–200 (2010).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056609 (2007).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102(12), 123901 (2009).
[Crossref] [PubMed]

Prog. Opt. (1)

D. Zhao and T. Wang, “Direct and inverse problems in the theory of light scattering,” Prog. Opt. 57, 261–308 (2012).
[Crossref]

Other (2)

M. Born and E. Wolf, 32. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

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Figures (3)

Fig. 1
Fig. 1 Illustration of notations.
Fig. 2
Fig. 2 Normalized spectral density ( S (s) (rs, s 0 ,ω)/ S (s) (r s 0 , s 0 ,ω) ) of the far-zone scattered field. The parameters for calculations are as follows: (a) S x (ω)= S y (ω)=1, σ sx = σ sy = σ sz =20λ, σ μx =1λ, σ μy =2λ, σ μz =3λ; (b) S x (ω)= S y (ω)=1, σ sx =10λ, σ sy =20λ, σ sz =30λ, σ μx = σ μy = σ μz =1λ.
Fig. 3
Fig. 3 Spectral degree of coherence ( μ (s) (r s 0 ,rs, s 0 ,ω) ) of the far-zone scattered field. The parameters are chosen as follows: (a) S x (ω)= S y (ω)=1, σ sx = σ sy = σ sz =1λ, σ μx =1λ, σ μy =2λ, σ μz =3λ; (b) S x (ω)= S y (ω)=1, σ sx =1λ, σ sy =2λ, σ sz =4λ, σ μx = σ μy = σ μz =1λ.

Equations (28)

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W (i) ( r 1 , r 2 , s 0 ,ω) E (i) ( r 1 , s 0 ,ω) E (i) ( r 2 , s 0 ,ω) ,
E (i) ( r , s 0 ,ω)=[ E x (i) ( r , s 0 ,ω), E y (i) ( r , s 0 ,ω), 0 ],
E j (i) ( r , s 0 ,ω)= a j (ω)exp(ik s 0 r ) (j=x,y),
E (s) (rs, s 0 ,ω)= exp(ikr) r D F( r ,ω) [ E x (i) ( r ,ω) E y (i) ( r ,ω) E z (i) ( r ,ω) ] T [ 1 s x 2 s x s y s x s z s y s x 1 s y 2 s y s z s z s x s z s y 1 s z 2 ] exp(iks r ) d 3 r ,
s x =sinθcosϕ, s y =sinθsinϕ, s z =cosθ,
W (s) (r s 1 ,r s 2 , s 0 ,ω) E (s) (r s 1 , s 0 ,ω) E (s) (r s 2 , s 0 ,ω) ,
W (s) (r s 1 ,r s 2 , s 0 ,ω)= A( s 1 , s 2 ) r 2 D C F ( r 1 , r 2 ,ω)exp{ ik[ ( s 2 s 0 ) r 2 ( s 1 s 0 ) r 1 ] } d 3 r 1 d 3 r 2 ,
A( s 1 , s 2 )= [ 1 s 1x 2 s 1x s 1y s 1x s 1z s 1y s 1x 1 s 1y 2 s 1y s 1z s 1z s 1x s 1z s 1y 1 s 1z 2 ] T [ S x (ω) 0 0 0 S y (ω) 0 0 0 0 ][ 1 s 2x 2 s 2x s 2y s 2x s 2z s 2y s 2x 1 s 2y 2 s 2y s 2z s 2z s 2x s 2z s 2y 1 s 2z 2 ],
C F ( r 1 , r 2 ,ω)= F ( r 1 ,ω)F( r 2 ,ω)
S (s) ( rs, s 0 ,ω )=Tr[ W (s) (rs,rs, s 0 ,ω) ],
μ (s) ( r s 1 ,r s 2 , s 0 ,ω )= Tr[ W (s) (r s 1 ,r s 2 , s 0 ,ω) ] Tr[ W (s) (r s 1 ,r s 1 , s 0 ,ω) ] Tr[ W (s) (r s 2 ,r s 2 , s 0 ,ω) ] .
C F ( r 1 , r 2 ,ω)= [ S F ( r 1 ,ω) S F ( r 2 ,ω) ] 1/2 μ F ( r 1 , r 2 ,ω),
C F ( r 1 , r 2 ,ω)= S F x ( R sx + ,ω) S F y ( R sy + ,ω) S F z ( R sz + ,ω), × μ F x ( R sx ,ω) μ F y ( R sy ,ω) μ F z ( R sz ,ω),
R sj + =( r 1j + r 2j )/2 R sj = r 2j r 1j (j=x,y,z).
W (s) (r s 1 ,r s 2 , s 0 ,ω)= A( s 1 , s 2 ) r 2 S ˜ F x ( K sx + ,ω) S ˜ F y ( K sy + ,ω) S ˜ F z ( K sz + ,ω) × μ ˜ F x ( K sx ,ω) μ ˜ F y ( K sy ,ω) μ ˜ F z ( K sz ,ω),
S ˜ F j ( K sj + ,ω)= D S F j ( R sj + ,ω)exp(i K sj + R sj + )d R sj +
μ ˜ F j ( K sj ,ω)= D μ F j ( R sj ,ω)exp(i K sj R sj )d R sj
K sj + =k( s 2j s 1j ), K sj =k[ ( s 1j + s 2j ) 2 s 0j ] (j=x,y,z).
S (s) ( rs, s 0 ,ω )= Tr[A( s,s )] S ˜ F (0,ω) r 2 × μ ˜ F x [ k( s x s 0x ),ω ] μ ˜ F y [ k( s y s 0y ),ω ] μ ˜ F z [ k( s z s 0z ),ω ],
μ (s) ( r s 1 ,r s 2 , s 0 ,ω )= Tr[A( s 1 , s 2 )] Tr[A( s 1 , s 1 )] Tr[A( s 2 , s 2 )] 1 S ˜ F (0,ω) × S ˜ F x ( K sx + ,ω) S ˜ F y ( K sy + ,ω) S ˜ F z ( K sz + ,ω) μ ˜ F x [ k( s 1x s 0x ),ω ] μ ˜ F y [ k( s 1y s 0y ),ω ] μ ˜ F z [ k( s 1z s 0z ),ω ] . × μ ˜ F x ( K sx ,ω) μ ˜ F y ( K sy ,ω) μ ˜ F z ( K sz ,ω) μ ˜ F x [ k( s 2x s 0x ),ω ] μ ˜ F y [ k( s 2y s 0y ),ω ] μ ˜ F z [ k( s 2z s 0z ),ω ]
μ ˜ F j [ k( s 1j s 0j ),ω ] μ ˜ F j [ k( s 2j s 0j ),ω ] μ ˜ F j { k[ ( s 1j + s 2j ) 2 s 0j ],ω }
μ (s) ( r s 1 ,r s 2 , s 0 ,ω )= Tr[A( s 1 , s 2 )] Tr[A( s 1 , s 1 )] Tr[A( s 2 , s 2 )] S ˜ F x ( K sx + ,ω) S ˜ F y ( K sy + ,ω) S ˜ F z ( K sz + ,ω) S ˜ F (0,ω) .
S F ( R s + ,ω)= C 0 exp[ R sx + 2 2 σ sx 2 ]exp[ R sy + 2 2 σ sy 2 ]exp[ R sz + 2 2 σ sz 2 ],
μ F ( R s ,ω)=exp[ R sx 2 2 σ μx 2 ]exp[ R sy 2 2 σ μy 2 ]exp[ R sz 2 2 σ μz 2 ],
S ˜ F [ k( s 2 s 1 ),ω ]= (2π) 3/2 C 0 σ sx σ sy σ sz ×exp{ k 2 2 [ ( s 2x s 1x ) 2 σ sx 2 + ( s 2y s 1y ) 2 σ sy 2 + ( s 2z s 1z ) 2 σ sz 2 ] },
μ ˜ F [ k( s 1 + s 2 2 s 0 ),ω ]= (2π) 3/2 σ μx σ μy σ μz . ×exp{ k 2 2 [ ( s 1x + s 2x 2 s 0x ) 2 σ μx 2 + ( s 1y + s 2y 2 s 0y ) 2 σ μy 2 + ( s 1z + s 2z 2 s 0z ) 2 σ μz 2 ] }
S (s) (rs, s 0 ,ω)= S x (ω)(1 s x 2 )+ S y (ω)(1 s y 2 ) r 2 (2π) 3 C 0 σ sx σ sy σ sz σ μx σ μy σ μz ×exp{ k 2 2 [ ( s x s 0x ) 2 σ μx 2 + ( s y s 0y ) 2 σ μy 2 + ( s z s 0z ) 2 σ μz 2 ] }
μ (s) (r s 1 ,r s 2 , s 0 ,ω)={ S x (ω)[ (1 s 1x 2 )(1 s 2x 2 )+ s 1x s 2x ( s 1y s 2y + s 1z s 2z ) ] S x (ω)(1 s 1x 2 )+ S y (ω)(1 s 1y 2 ) S x (ω)(1 s 2x 2 )+ S y (ω)(1 s 2y 2 ) . + S y (ω)[ (1 s 1y 2 )(1 s 2y 2 )+ s 1y s 2y ( s 1x s 2x + s 1z s 2z ) ] S x (ω)(1 s 1x 2 )+ S y (ω)(1 s 1y 2 ) S x (ω)(1 s 2x 2 )+ S y (ω)(1 s 2y 2 ) } ×exp{ k 2 2 [ ( s 2x s 1x ) 2 σ sx 2 + ( s 2y s 1y ) 2 σ sy 2 + ( s 2z s 1z ) 2 σ sz 2 ] }

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