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

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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]
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    [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]
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    [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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