Abstract

Within the validity of the first-order Born approximation, expressions are derived for the correlation between intensity fluctuations (CIF) of an electromagnetic plane wave scattered from a spatially quasi-homogeneous (QH), anisotropic medium. Upon establishing the correlation matrix of the scattering potential of the medium, we show that the CIF is the summation of Fourier transforms of the strengths and normalized correlation coefficients (NCCs) of the scattering potential matrix. Numerical results reveal that the CIF is susceptible to the effective width and correlation length of the medium, and degree of polarization of the incident electromagnetic wave. Our study not only extends the current knowledge of the CIF of a scattered field but also provides an important reference to the study of high-order intensity correlations of light scattered from a spatially anisotropic medium.

© 2016 Optical Society of America

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References

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  1. F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
    [Crossref] [PubMed]
  2. S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modeling of classical ghost images obtained using scattered light,” New J. Phys. 9(8), 285 (2007).
    [Crossref]
  3. Q. Wang, “Theoretical study on the statistical properties of phase difference between two interfering speckle fields,” Opt. Commun. 284(22), 5233–5239 (2011).
    [Crossref]
  4. K. W. Chan, “Role of photon statistics of light source in ghost imaging,” Opt. Lett. 37(13), 2739–2741 (2012).
    [Crossref] [PubMed]
  5. B. I. Erkmen, “Computational ghost imaging for remote sensing,” J. Opt. Soc. Am. A 29(5), 782–789 (2012).
    [Crossref] [PubMed]
  6. T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
    [Crossref]
  7. H. Liu, D. Guan, L. Li, S. Zhang, and J. Xiong, “The impact of light polarization on imaging visibility of Nth-order intensity correlation with thermal light,” Opt. Commun. 283(3), 405–408 (2010).
    [Crossref]
  8. S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10(5), 055001 (2008).
    [Crossref]
  9. Y. Xin, Y. Chen, Q. Zhao, and M. Zhou, “Effect of cross-polarization of electromagnetic source on the degree of polarization of generated beam,” Opt. Commun. 281(8), 1954–1957 (2008).
    [Crossref]
  10. O. Korotkova, “Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation,” J. Opt. A, Pure Appl. Opt. 8(1), 30–37 (2006).
    [Crossref]
  11. S. Sahin, O. Korotkova, G. Zhang, and J. Pu, “Free-space propagation of the spectral degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 11(8), 085703 (2009).
    [Crossref]
  12. D. Kuebel, “Properties of the degree of cross-polarization in the space-time domain,” Opt. Commun. 282(17), 3397–3401 (2009).
    [Crossref]
  13. A. Al-Qasimi, M. Lahiri, D. Kuebel, D. F. V. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown-Twiss effect,” Opt. Express 18(16), 17124–17129 (2010).
    [Crossref] [PubMed]
  14. M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
    [Crossref] [PubMed]
  15. 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]
  16. H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A 28(6), 1139–1144 (2011).
    [Crossref] [PubMed]
  17. J. Li, Y. Qin, and S. Zhou, “Fourth-order correlation statistics of an electromagnetic plane wave scattered by a quasi-homogeneous medium,” J. Opt. 13(11), 115702 (2011).
    [Crossref]
  18. G. Zhang and Z. Wu, “Fluctuation correlation of the scattered intensity from two-dimensional rough surfaces,” Opt. Express 20(2), 1491–1502 (2012).
    [Crossref] [PubMed]
  19. H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence,” J. Opt. Soc. Am. A 29(2), 169–173 (2012).
    [Crossref] [PubMed]
  20. Y. Baykal, H. T. Eyyuboğlu, C. Z. Çil, Y. Cai, and O. Korotkova, “Intensity fluctuations of partially coherent cos Gaussian and cosh Gaussian beams in atmospheric turbulence,” J. Opt. 13(5), 055709 (2011).
    [Crossref]
  21. H. M. Boots, “Light scattering in an optically anisotropic medium,” J. Opt. Soc. Am. A 11(9), 2539–2544 (1994).
    [Crossref]
  22. H. Stark and T. C. Lubensky, “Multiple light scattering in nematic liquid crystals,” Phys. Rev. Lett. 77(11), 2229–2232 (1996).
    [Crossref] [PubMed]
  23. K. Khairy, J. Foo, and J. Howard, “Shapes of red blood cells: comparison of 3D confocal images with the bilayer-couple model,” Cell. Mol. Bioeng. 1(2-3), 173–181 (2008).
    [Crossref] [PubMed]
  24. T. Wang, Z. Jiang, X. Ji, and D. Zhao, “Spectrum of an electromagnetic light wave on scattering from an anisotropic semisoft boundary medium,” J. Opt. Soc. Am. A 33(4), 625–629 (2016).
    [Crossref] [PubMed]
  25. S. A. Ponomarenko and A. V. Shchegrov, “Spectral changes of light produced by scattering from disordered anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(3), 3310–3313 (1999).
    [Crossref] [PubMed]
  26. J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered electromagnetic field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
    [Crossref]
  27. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  28. X. Du and D. Zhao, “Spectral shifts produced by scattering from rotational quasi-homogeneous anisotropic media,” Opt. Lett. 36(24), 4749–4751 (2011).
    [Crossref] [PubMed]
  29. X. Du and D. Zhao, “Frequency shifts of spectral lines induced by scattering from a rotational anisotropic particle,” Opt. Commun. 285(6), 934–936 (2012).
    [Crossref]

2016 (1)

2012 (6)

2011 (5)

Q. Wang, “Theoretical study on the statistical properties of phase difference between two interfering speckle fields,” Opt. Commun. 284(22), 5233–5239 (2011).
[Crossref]

J. Li, Y. Qin, and S. Zhou, “Fourth-order correlation statistics of an electromagnetic plane wave scattered by a quasi-homogeneous medium,” J. Opt. 13(11), 115702 (2011).
[Crossref]

Y. Baykal, H. T. Eyyuboğlu, C. Z. Çil, Y. Cai, and O. Korotkova, “Intensity fluctuations of partially coherent cos Gaussian and cosh Gaussian beams in atmospheric turbulence,” J. Opt. 13(5), 055709 (2011).
[Crossref]

H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A 28(6), 1139–1144 (2011).
[Crossref] [PubMed]

X. Du and D. Zhao, “Spectral shifts produced by scattering from rotational quasi-homogeneous anisotropic media,” Opt. Lett. 36(24), 4749–4751 (2011).
[Crossref] [PubMed]

2010 (3)

2009 (2)

S. Sahin, O. Korotkova, G. Zhang, and J. Pu, “Free-space propagation of the spectral degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 11(8), 085703 (2009).
[Crossref]

D. Kuebel, “Properties of the degree of cross-polarization in the space-time domain,” Opt. Commun. 282(17), 3397–3401 (2009).
[Crossref]

2008 (3)

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10(5), 055001 (2008).
[Crossref]

Y. Xin, Y. Chen, Q. Zhao, and M. Zhou, “Effect of cross-polarization of electromagnetic source on the degree of polarization of generated beam,” Opt. Commun. 281(8), 1954–1957 (2008).
[Crossref]

K. Khairy, J. Foo, and J. Howard, “Shapes of red blood cells: comparison of 3D confocal images with the bilayer-couple model,” Cell. Mol. Bioeng. 1(2-3), 173–181 (2008).
[Crossref] [PubMed]

2007 (2)

T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
[Crossref]

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modeling of classical ghost images obtained using scattered light,” New J. Phys. 9(8), 285 (2007).
[Crossref]

2006 (1)

O. Korotkova, “Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation,” J. Opt. A, Pure Appl. Opt. 8(1), 30–37 (2006).
[Crossref]

2005 (1)

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

1999 (1)

S. A. Ponomarenko and A. V. Shchegrov, “Spectral changes of light produced by scattering from disordered anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(3), 3310–3313 (1999).
[Crossref] [PubMed]

1996 (1)

H. Stark and T. C. Lubensky, “Multiple light scattering in nematic liquid crystals,” Phys. Rev. Lett. 77(11), 2229–2232 (1996).
[Crossref] [PubMed]

1994 (1)

1987 (1)

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
[Crossref] [PubMed]

Al-Qasimi, A.

Aruldoss, C.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modeling of classical ghost images obtained using scattered light,” New J. Phys. 9(8), 285 (2007).
[Crossref]

Bache, M.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

Baykal, Y.

H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence,” J. Opt. Soc. Am. A 29(2), 169–173 (2012).
[Crossref] [PubMed]

Y. Baykal, H. T. Eyyuboğlu, C. Z. Çil, Y. Cai, and O. Korotkova, “Intensity fluctuations of partially coherent cos Gaussian and cosh Gaussian beams in atmospheric turbulence,” J. Opt. 13(5), 055709 (2011).
[Crossref]

Boots, H. M.

Brambilla, E.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

Cai, Y.

Y. Baykal, H. T. Eyyuboğlu, C. Z. Çil, Y. Cai, and O. Korotkova, “Intensity fluctuations of partially coherent cos Gaussian and cosh Gaussian beams in atmospheric turbulence,” J. Opt. 13(5), 055709 (2011).
[Crossref]

Castelletto, S.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modeling of classical ghost images obtained using scattered light,” New J. Phys. 9(8), 285 (2007).
[Crossref]

Chan, K. W.

Chen, F.

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered electromagnetic field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

Chen, J.

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered electromagnetic field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

Chen, Y.

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered electromagnetic field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

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, “Effect of cross-polarization of electromagnetic source on the degree of polarization of generated beam,” Opt. Commun. 281(8), 1954–1957 (2008).
[Crossref]

Çil, C. Z.

Y. Baykal, H. T. Eyyuboğlu, C. Z. Çil, Y. Cai, and O. Korotkova, “Intensity fluctuations of partially coherent cos Gaussian and cosh Gaussian beams in atmospheric turbulence,” J. Opt. 13(5), 055709 (2011).
[Crossref]

Crosby, S.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modeling of classical ghost images obtained using scattered light,” New J. Phys. 9(8), 285 (2007).
[Crossref]

Cwilich, G.

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
[Crossref] [PubMed]

Du, X.

X. Du and D. Zhao, “Frequency shifts of spectral lines induced by scattering from a rotational anisotropic particle,” Opt. Commun. 285(6), 934–936 (2012).
[Crossref]

X. Du and D. Zhao, “Spectral shifts produced by scattering from rotational quasi-homogeneous anisotropic media,” Opt. Lett. 36(24), 4749–4751 (2011).
[Crossref] [PubMed]

Erkmen, B. I.

Eyyuboglu, H. T.

Y. Baykal, H. T. Eyyuboğlu, C. Z. Çil, Y. Cai, and O. Korotkova, “Intensity fluctuations of partially coherent cos Gaussian and cosh Gaussian beams in atmospheric turbulence,” J. Opt. 13(5), 055709 (2011).
[Crossref]

Ferri, F.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

Foo, J.

K. Khairy, J. Foo, and J. Howard, “Shapes of red blood cells: comparison of 3D confocal images with the bilayer-couple model,” Cell. Mol. Bioeng. 1(2-3), 173–181 (2008).
[Crossref] [PubMed]

Gatti, A.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

Gerçekcioglu, H.

Guan, D.

H. Liu, D. Guan, L. Li, S. Zhang, and J. Xiong, “The impact of light polarization on imaging visibility of Nth-order intensity correlation with thermal light,” Opt. Commun. 283(3), 405–408 (2010).
[Crossref]

He, Y.

Howard, J.

K. Khairy, J. Foo, and J. Howard, “Shapes of red blood cells: comparison of 3D confocal images with the bilayer-couple model,” Cell. Mol. Bioeng. 1(2-3), 173–181 (2008).
[Crossref] [PubMed]

Jacks, H. C.

James, D. F. V.

A. Al-Qasimi, M. Lahiri, D. Kuebel, D. F. V. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown-Twiss effect,” Opt. Express 18(16), 17124–17129 (2010).
[Crossref] [PubMed]

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10(5), 055001 (2008).
[Crossref]

Ji, X.

Jiang, Z.

Khairy, K.

K. Khairy, J. Foo, and J. Howard, “Shapes of red blood cells: comparison of 3D confocal images with the bilayer-couple model,” Cell. Mol. Bioeng. 1(2-3), 173–181 (2008).
[Crossref] [PubMed]

Korotkova, O.

H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A 28(6), 1139–1144 (2011).
[Crossref] [PubMed]

Y. Baykal, H. T. Eyyuboğlu, C. Z. Çil, Y. Cai, and O. Korotkova, “Intensity fluctuations of partially coherent cos Gaussian and cosh Gaussian beams in atmospheric turbulence,” J. Opt. 13(5), 055709 (2011).
[Crossref]

S. Sahin, O. Korotkova, G. Zhang, and J. Pu, “Free-space propagation of the spectral degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 11(8), 085703 (2009).
[Crossref]

O. Korotkova, “Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation,” J. Opt. A, Pure Appl. Opt. 8(1), 30–37 (2006).
[Crossref]

Kuebel, D.

Lahiri, M.

Li, J.

J. Li, Y. Qin, and S. Zhou, “Fourth-order correlation statistics of an electromagnetic plane wave scattered by a quasi-homogeneous medium,” J. Opt. 13(11), 115702 (2011).
[Crossref]

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]

Li, L.

H. Liu, D. Guan, L. Li, S. Zhang, and J. Xiong, “The impact of light polarization on imaging visibility of Nth-order intensity correlation with thermal light,” Opt. Commun. 283(3), 405–408 (2010).
[Crossref]

Liu, H.

H. Liu, D. Guan, L. Li, S. Zhang, and J. Xiong, “The impact of light polarization on imaging visibility of Nth-order intensity correlation with thermal light,” Opt. Commun. 283(3), 405–408 (2010).
[Crossref]

Lubensky, T. C.

H. Stark and T. C. Lubensky, “Multiple light scattering in nematic liquid crystals,” Phys. Rev. Lett. 77(11), 2229–2232 (1996).
[Crossref] [PubMed]

Lugiato, L. A.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

Magatti, D.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

Ponomarenko, S. A.

S. A. Ponomarenko and A. V. Shchegrov, “Spectral changes of light produced by scattering from disordered anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(3), 3310–3313 (1999).
[Crossref] [PubMed]

Pu, J.

S. Sahin, O. Korotkova, G. Zhang, and J. Pu, “Free-space propagation of the spectral degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 11(8), 085703 (2009).
[Crossref]

Qin, Y.

J. Li, Y. Qin, and S. Zhou, “Fourth-order correlation statistics of an electromagnetic plane wave scattered by a quasi-homogeneous medium,” J. Opt. 13(11), 115702 (2011).
[Crossref]

Roberts, A.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modeling of classical ghost images obtained using scattered light,” New J. Phys. 9(8), 285 (2007).
[Crossref]

Sahin, S.

S. Sahin, O. Korotkova, G. Zhang, and J. Pu, “Free-space propagation of the spectral degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 11(8), 085703 (2009).
[Crossref]

Scholten, R. E.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modeling of classical ghost images obtained using scattered light,” New J. Phys. 9(8), 285 (2007).
[Crossref]

Shchegrov, A. V.

S. A. Ponomarenko and A. V. Shchegrov, “Spectral changes of light produced by scattering from disordered anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(3), 3310–3313 (1999).
[Crossref] [PubMed]

Shirai, T.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10(5), 055001 (2008).
[Crossref]

T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
[Crossref]

Stark, H.

H. Stark and T. C. Lubensky, “Multiple light scattering in nematic liquid crystals,” Phys. Rev. Lett. 77(11), 2229–2232 (1996).
[Crossref] [PubMed]

Stephen, M. J.

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
[Crossref] [PubMed]

Volkov, S. N.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10(5), 055001 (2008).
[Crossref]

Wang, Q.

Q. Wang, “Theoretical study on the statistical properties of phase difference between two interfering speckle fields,” Opt. Commun. 284(22), 5233–5239 (2011).
[Crossref]

Wang, T.

Wang, Y.

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered electromagnetic field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

Wolf, E.

A. Al-Qasimi, M. Lahiri, D. Kuebel, D. F. V. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown-Twiss effect,” Opt. Express 18(16), 17124–17129 (2010).
[Crossref] [PubMed]

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10(5), 055001 (2008).
[Crossref]

T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
[Crossref]

Wu, Z.

Xin, Y.

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered electromagnetic field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

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, “Effect of cross-polarization of electromagnetic source on the degree of polarization of generated beam,” Opt. Commun. 281(8), 1954–1957 (2008).
[Crossref]

Xiong, J.

H. Liu, D. Guan, L. Li, S. Zhang, and J. Xiong, “The impact of light polarization on imaging visibility of Nth-order intensity correlation with thermal light,” Opt. Commun. 283(3), 405–408 (2010).
[Crossref]

Zhang, G.

G. Zhang and Z. Wu, “Fluctuation correlation of the scattered intensity from two-dimensional rough surfaces,” Opt. Express 20(2), 1491–1502 (2012).
[Crossref] [PubMed]

S. Sahin, O. Korotkova, G. Zhang, and J. Pu, “Free-space propagation of the spectral degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 11(8), 085703 (2009).
[Crossref]

Zhang, S.

H. Liu, D. Guan, L. Li, S. Zhang, and J. Xiong, “The impact of light polarization on imaging visibility of Nth-order intensity correlation with thermal light,” Opt. Commun. 283(3), 405–408 (2010).
[Crossref]

Zhao, D.

Zhao, Q.

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered electromagnetic field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

Y. Xin, Y. Chen, Q. Zhao, and M. Zhou, “Effect of cross-polarization of electromagnetic source on the degree of polarization of generated beam,” Opt. Commun. 281(8), 1954–1957 (2008).
[Crossref]

Zhou, M.

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered electromagnetic field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

Y. Xin, Y. Chen, Q. Zhao, and M. Zhou, “Effect of cross-polarization of electromagnetic source on the degree of polarization of generated beam,” Opt. Commun. 281(8), 1954–1957 (2008).
[Crossref]

Zhou, S.

J. Li, Y. Qin, and S. Zhou, “Fourth-order correlation statistics of an electromagnetic plane wave scattered by a quasi-homogeneous medium,” J. Opt. 13(11), 115702 (2011).
[Crossref]

Cell. Mol. Bioeng. (1)

K. Khairy, J. Foo, and J. Howard, “Shapes of red blood cells: comparison of 3D confocal images with the bilayer-couple model,” Cell. Mol. Bioeng. 1(2-3), 173–181 (2008).
[Crossref] [PubMed]

J. Opt. (2)

J. Li, Y. Qin, and S. Zhou, “Fourth-order correlation statistics of an electromagnetic plane wave scattered by a quasi-homogeneous medium,” J. Opt. 13(11), 115702 (2011).
[Crossref]

Y. Baykal, H. T. Eyyuboğlu, C. Z. Çil, Y. Cai, and O. Korotkova, “Intensity fluctuations of partially coherent cos Gaussian and cosh Gaussian beams in atmospheric turbulence,” J. Opt. 13(5), 055709 (2011).
[Crossref]

J. Opt. A, Pure Appl. Opt. (3)

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10(5), 055001 (2008).
[Crossref]

O. Korotkova, “Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation,” J. Opt. A, Pure Appl. Opt. 8(1), 30–37 (2006).
[Crossref]

S. Sahin, O. Korotkova, G. Zhang, and J. Pu, “Free-space propagation of the spectral degree of cross-polarization of stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 11(8), 085703 (2009).
[Crossref]

J. Opt. Soc. Am. A (5)

New J. Phys. (1)

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modeling of classical ghost images obtained using scattered light,” New J. Phys. 9(8), 285 (2007).
[Crossref]

Opt. Commun. (7)

Q. Wang, “Theoretical study on the statistical properties of phase difference between two interfering speckle fields,” Opt. Commun. 284(22), 5233–5239 (2011).
[Crossref]

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered electromagnetic field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

D. Kuebel, “Properties of the degree of cross-polarization in the space-time domain,” Opt. Commun. 282(17), 3397–3401 (2009).
[Crossref]

Y. Xin, Y. Chen, Q. Zhao, and M. Zhou, “Effect of cross-polarization of electromagnetic source on the degree of polarization of generated beam,” Opt. Commun. 281(8), 1954–1957 (2008).
[Crossref]

T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
[Crossref]

H. Liu, D. Guan, L. Li, S. Zhang, and J. Xiong, “The impact of light polarization on imaging visibility of Nth-order intensity correlation with thermal light,” Opt. Commun. 283(3), 405–408 (2010).
[Crossref]

X. Du and D. Zhao, “Frequency shifts of spectral lines induced by scattering from a rotational anisotropic particle,” Opt. Commun. 285(6), 934–936 (2012).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

S. A. Ponomarenko and A. V. Shchegrov, “Spectral changes of light produced by scattering from disordered anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(3), 3310–3313 (1999).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987).
[Crossref] [PubMed]

H. Stark and T. C. Lubensky, “Multiple light scattering in nematic liquid crystals,” Phys. Rev. Lett. 77(11), 2229–2232 (1996).
[Crossref] [PubMed]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

Other (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram for the scattering of an electromagnetic wave from a spatially QH, anisotropic medium that occupies the spatial volume D. s 0 is the unit vector that represents the propagation direction of the incident wave, s is the unit vector that accounts for the propagation direction of the scattered wave, θ is the scattering angle. U y ( i ) denotes the incident field component along the y axis. U x ( s ) , U y ( s ) , U z ( s ) are the 3D components of the scattered field. U α ( s ) and U β ( s ) are the mutually perpendicular field components that remain orthogonal to s. The sign “•” means that the field component is perpendicular to the y-z plane.

Fig. 2
Fig. 2

Dependence of the CIF of the scattered field on the effective widths σ, σxy, σyz, and σxz of the QH, anisotropic medium. The unpolarized plane wave on incidence is considered, i.e. S x ( i ) ( ω )= S y ( i ) ( ω ), the uniform parameters are chosen as δ = 0.3λ, δxy = δyz = 0.2λ, δxz = 0.1λ. (a) σyz = 3λ, σxy = σxz = 4λ, (b) σ = 5λ, σyz = 3λ, σxz = 4λ, (c) σ = 5λ, σxy = σxz = 4λ, (d) σ = 5λ, σxy = 4λ, σyz = 3λ.

Fig. 3
Fig. 3

Dependence of the CIF of the scattered field on the correlation lengths δ, δxy, δyz, and δxz of the QH, anisotropic medium. The unpolarized incident plane wave on incidence is considered, i.e., S x ( i ) ( ω )= S y ( i ) ( ω ), the uniform parameters are chosen as σ = 5λ, σxy = σxz = 4λ, σyz = 3λ. (a) δxy = δyz = 0.2λ, δxz = 0.1λ, (b) δ = 0.3λ, δxz = 0.1λ, δyz = 0.2λ, (c) δ = 0.3λ, δxy = 0.2λ, δxz = 0.1λ, (d) δ = 0.3λ, δxy = δyz = 0.2λ.

Fig. 4
Fig. 4

Effects of the polarization of the incident plane wave on the CIF distributions of the scattered field for different correlation lengths of the medium. Linear polarization along x axis: S y ( i ) ( ω )=0; unpolarized wave: S x ( i ) ( ω )= S y ( i ) ( ω ); linear polarization along y axis: S x ( i ) ( ω )=0. (a) δ = 0.1λ, (b) δ = 0.6λ, (c) δxy = 0.2λ, (d) δxy = 0.6λ. Other numerical parameters are the same as those in Fig. 3.

Equations (27)

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U j ( i ) ( r',ω )= a j ( ω )exp( ik s 0 r' ), ( j=x, y ),
U x ( s ) ( rs,ω )= D F x ( r',ω ) U x ( i ) ( r',ω )G( rs,r',ω ) d 3 r',
U y ( s ) ( rs,ω )= cos 2 θ D F y ( r',ω ) U y ( i ) ( r',ω )G( rs,r',ω ) d 3 r',
U z ( s ) ( rs,ω )=sinθcosθ D F z ( r',ω ) U y ( i ) ( r',ω )G( rs,r',ω ) d 3 r',
F( r',ω )=( F x ( r',ω ) 0 0 0 F y ( r',ω ) 0 0 0 F z ( r',ω ) ).
G( rs,r',ω ) exp( ikr ) r exp( iksr' ).
U α ( s ) ( rs,ω )= U x ( s ) ( rs,ω ) e x ,
U β ( s ) ( rs,ω )= U y ( s ) ( rs,ω ) e y + U z ( s ) ( rs,ω ) e z ,
U α ( s ) ( rs,ω )×s= U β ( s ) ( rs,ω ), U β ( s ) ( rs,ω )×s= U α ( s ) ( rs,ω ), U α ( s ) ( rs,ω )× U β ( s ) ( rs,ω )=s.
CI F ( s ) ( r 1 s 1 , r 2 s 2 )= Δ I ( s ) ( r 1 s 1 )Δ I ( s ) ( r 2 s 2 ) = I ( s ) ( r 1 s 1 ) I ( s ) ( r 2 s 2 ) I ( s ) ( r 1 s 1 ) I ( s ) ( r 2 s 2 ) .
CI F ( s ) ( r 1 s 1 , r 2 s 2 )= [ U α ( s )* ( r 1 s 1 ) U β ( s ) ( r 1 s 1 ) ][ U α ( s ) ( r 1 s 1 ) U β ( s ) ( r 1 s 1 ) ][ U α ( s )* ( r 2 s 2 ) U β ( s ) ( r 2 s 2 ) ][ U α ( s ) ( r 2 s 2 ) U β ( s ) ( r 2 s 2 ) ] U α ( s )* ( r 1 s 1 ) U α ( s ) ( r 1 s 1 )+ U β ( s )* ( r 1 s 1 ) U β ( s ) ( r 1 s 1 ) U α ( s )* ( r 2 s 2 ) U α ( s ) ( r 2 s 2 )+ U β ( s )* ( r 2 s 2 ) U β ( s ) ( r 2 s 2 ) = U α ( s )* ( r 1 s 1 ) U α ( s ) ( r 1 s 1 ) U α ( s )* ( r 2 s 2 ) U α ( s ) ( r 2 s 2 ) + U α ( s )* ( r 1 s 1 ) U α ( s ) ( r 1 s 1 ) U β ( s )* ( r 2 s 2 ) U β ( s ) ( r 2 s 2 ) + U α ( s )* ( r 2 s 2 ) U α ( s ) ( r 2 s 2 ) U β ( s )* ( r 1 s 1 ) U β ( s ) ( r 1 s 1 ) + U β ( s )* ( r 1 s 1 ) U β ( s ) ( r 1 s 1 ) U β ( s )* ( r 2 s 2 ) U β ( s ) ( r 2 s 2 ) U α ( s )* ( r 1 s 1 ) U α ( s ) ( r 1 s 1 ) U α ( s )* ( r 2 s 2 ) U α ( s ) ( r 2 s 2 ) U α ( s )* ( r 1 s 1 ) U α ( s ) ( r 1 s 1 ) U β ( s )* ( r 2 s 2 ) U β ( s ) ( r 2 s 2 ) U α ( s )* ( r 2 s 2 ) U α ( s ) ( r 2 s 2 ) U β ( s )* ( r 1 s 1 ) U β ( s ) ( r 1 s 1 ) U β ( s )* ( r 1 s 1 ) U β ( s ) ( r 1 s 1 ) U β ( s )* ( r 2 s 2 ) U β ( s ) ( r 2 s 2 ) ,
CI F ( s ) ( r 1 s 1 , r 2 s 2 )= ( S x ( i ) ( ω ) r 1 r 2 ) 2 D D D D [ F x * ( r 1 ' ) F x ( r 2 ' ) F x * ( r 3 ' ) F x ( r 4 ' ) F x * ( r 1 ' ) F x ( r 2 ' ) F x * ( r 3 ' ) F x ( r 4 ' ) ] ×exp[ ik( s 1 s 0 ) r 1 'ik( s 1 s 0 ) r 2 '+ik( s 2 s 0 ) r 3 'ik( s 2 s 0 ) r 4 ' ] d 3 r 1 ' d 3 r 2 ' d 3 r 3 ' d 3 r 4 ' + S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 cos 4 θ 2 D D D D [ F x * ( r 1 ' ) F x ( r 2 ' ) F y * ( r 3 ' ) F y ( r 4 ' ) F x * ( r 1 ' ) F x ( r 2 ' ) F y * ( r 3 ' ) F y ( r 4 ' ) ] ×exp[ ik( s 1 s 0 ) r 1 'ik( s 1 s 0 ) r 2 '+ik( s 2 s 0 ) r 3 'ik( s 2 s 0 ) r 4 ' ] d 3 r 1 ' d 3 r 2 ' d 3 r 3 ' d 3 r 4 ' + S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 sin 2 θ 2 cos 2 θ 2 D D D D [ F x * ( r 1 ' ) F x ( r 2 ' ) F z * ( r 3 ' ) F z ( r 4 ' ) F x * ( r 1 ' ) F x ( r 2 ' ) F z * ( r 3 ' ) F z ( r 4 ' ) ] ×exp[ ik( s 1 s 0 ) r 1 'ik( s 1 s 0 ) r 2 '+ik( s 2 s 0 ) r 3 'ik( s 2 s 0 ) r 4 ' ] d 3 r 1 ' d 3 r 2 ' d 3 r 3 ' d 3 r 4 ' + S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 cos 4 θ 1 D D D D [ F x * ( r 1 ' ) F x ( r 2 ' ) F y * ( r 3 ' ) F y ( r 4 ' ) F x * ( r 1 ' ) F x ( r 2 ' ) F y * ( r 3 ' ) F y ( r 4 ' ) ] ×exp[ ik( s 2 s 0 ) r 1 'ik( s 2 s 0 ) r 2 '+ik( s 1 s 0 ) r 3 'ik( s 1 s 0 ) r 4 ' ] d 3 r 1 ' d 3 r 2 ' d 3 r 3 ' d 3 r 4 ' + S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 sin 2 θ 1 cos 2 θ 1 D D D D [ F x * ( r 1 ' ) F x ( r 2 ' ) F z * ( r 3 ' ) F z ( r 4 ' ) F x * ( r 1 ' ) F x ( r 2 ' ) F z * ( r 3 ' ) F z ( r 4 ' ) ] ×exp[ ik( s 2 s 0 ) r 1 'ik( s 2 s 0 ) r 2 '+ik( s 1 s 0 ) r 3 'ik( s 1 s 0 ) r 4 ' ] d 3 r 1 ' d 3 r 2 ' d 3 r 3 ' d 3 r 4 ' + ( S y ( i ) ( ω ) r 1 r 2 ) 2 cos 4 θ 1 cos 4 θ 2 D D D D [ F y * ( r 1 ' ) F y ( r 2 ' ) F y * ( r 3 ' ) F y ( r 4 ' ) F y * ( r 1 ' ) F y ( r 2 ' ) F y * ( r 3 ' ) F y ( r 4 ' ) ] ×exp[ ik( s 1 s 0 ) r 1 'ik( s 1 s 0 ) r 2 '+ik( s 2 s 0 ) r 3 'ik( s 2 s 0 ) r 4 ' ] d 3 r 1 ' d 3 r 2 ' d 3 r 3 ' d 3 r 4 ' + ( S y ( i ) ( ω ) r 1 r 2 ) 2 cos 4 θ 1 sin 2 θ 2 cos 2 θ 2 D D D D [ F y * ( r 1 ' ) F y ( r 2 ' ) F z * ( r 3 ' ) F z ( r 4 ' ) F y * ( r 1 ' ) F y ( r 2 ' ) F z * ( r 3 ' ) F z ( r 4 ' ) ] ×exp[ ik( s 1 s 0 ) r 1 'ik( s 1 s 0 ) r 2 '+ik( s 2 s 0 ) r 3 'ik( s 2 s 0 ) r 4 ' ] d 3 r 1 ' d 3 r 2 ' d 3 r 3 ' d 3 r 4 ' + ( S y ( i ) ( ω ) r 1 r 2 ) 2 cos 4 θ 2 sin 2 θ 1 cos 2 θ 1 D D D D [ F z * ( r 1 ' ) F z ( r 2 ' ) F y * ( r 3 ' ) F y ( r 4 ' ) F z * ( r 1 ' ) F z ( r 2 ' ) F y * ( r 3 ' ) F y ( r 4 ' ) ] ×exp[ ik( s 1 s 0 ) r 1 'ik( s 1 s 0 ) r 2 '+ik( s 2 s 0 ) r 3 'ik( s 2 s 0 ) r 4 ' ] d 3 r 1 ' d 3 r 2 ' d 3 r 3 ' d 3 r 4 ' + ( S y ( i ) ( ω ) r 1 r 2 ) 2 sin 2 θ 1 cos 2 θ 1 sin 2 θ 2 cos 2 θ 2 D D D D [ F z * ( r 1 ' ) F z ( r 2 ' ) F z * ( r 3 ' ) F z ( r 4 ' ) F z * ( r 1 ' ) F z ( r 2 ' ) F z * ( r 3 ' ) F z ( r 4 ' ) ] ×exp[ ik( s 1 s 0 ) r 1 'ik( s 1 s 0 ) r 2 '+ik( s 2 s 0 ) r 3 'ik( s 2 s 0 ) r 4 ' ] d 3 r 1 ' d 3 r 2 ' d 3 r 3 ' d 3 r 4 ',
C ij ( F ) ( r 1 ', r 2 ',ω )= F i * ( r 1 ',ω ) F j ( r 2 ',ω ) , ( i, j = x, y, z ).
C ij ( F ) ( r 1 ', r 2 ',ω ) S ij ( F ) ( r 1 '+ r 2 ' 2 ,ω ) η ij ( F ) ( r 1 ' r 2 ',ω ), ( i, j = x, y, z ).
F i * ( r 1 ' ) F j ( r 2 ' ) F u * ( r 3 ' ) F v ( r 4 ' ) = F i * ( r 1 ' ) F j ( r 2 ' ) F u * ( r 3 ' ) F v ( r 4 ' ) + F i * ( r 1 ' ) F v ( r 4 ' ) F u * ( r 3 ' ) F j ( r 2 ' ) , ( i, j, u, v = x, y, z ).
R s + = r 1 '+ r 4 ', R s = r 1 ' r 4 ', U s + = r 3 '+ r 2 ', U s = r 3 ' r 2 '.
CI F ( s ) ( r 1 s 1 , r 2 s 2 )= 1 4 ( S x ( i ) ( ω ) r 1 r 2 ) 2 D D D D S xx ( F ) ( R s + 2 ) η xx ( F ) ( R s ) S xx ( F ) ( U s + 2 ) η xx ( F ) ( U s ) ×exp[ ik R s + 2 ( s 1 s 2 )+ik R s ( s 1 + s 2 2 s 0 )ik U s + 2 ( s 1 s 2 )+ik U s ( s 1 + s 2 2 s 0 ) ] d 3 R s + d 3 R s d 3 U s + d 3 U s + 1 4 S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 cos 4 θ 2 D D D D S xy ( F ) ( R s + 2 ) η xy ( F ) ( R s ) S yx ( F ) ( U s + 2 ) η yx ( F ) ( U s ) ×exp[ ik R s + 2 ( s 1 s 2 )+ik R s ( s 1 + s 2 2 s 0 )ik U s + 2 ( s 1 s 2 )+ik U s ( s 1 + s 2 2 s 0 ) ] d 3 R s + d 3 R s d 3 U s + d 3 U s + 1 4 S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 sin 2 θ 2 cos 2 θ 2 D D D D S xz ( F ) ( R s + 2 ) η xz ( F ) ( R s ) S zx ( F ) ( U s + 2 ) η zx ( F ) ( U s ) ×exp[ ik R s + 2 ( s 1 s 2 )+ik R s ( s 1 + s 2 2 s 0 )ik U s + 2 ( s 1 s 2 )+ik U s ( s 1 + s 2 2 s 0 ) ] d 3 R s + d 3 R s d 3 U s + d 3 U s + 1 4 S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 cos 4 θ 1 D D D D S xy ( F ) ( R s + 2 ) η xy ( F ) ( R s ) S yx ( F ) ( U s + 2 ) η yx ( F ) ( U s ) ×exp[ ik R s + 2 ( s 2 s 1 )+ik R s ( s 1 + s 2 2 s 0 )ik U s + 2 ( s 2 s 1 )+ik U s ( s 1 + s 2 2 s 0 ) ] d 3 R s + d 3 R s d 3 U s + d 3 U s + 1 4 S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 sin 2 θ 1 cos 2 θ 1 D D D D S xz ( F ) ( R s + 2 ) η xz ( F ) ( R s ) S zx ( F ) ( U s + 2 ) η zx ( F ) ( U s ) + 1 4 ( S y ( i ) ( ω ) r 1 r 2 ) 2 cos 4 θ 1 cos 4 θ 2 D D D D S yy ( F ) ( R s + 2 ) η yy ( F ) ( R s ) S yy ( F ) ( U s + 2 ) η yy ( F ) ( U s ) ×exp[ ik R s + 2 ( s 2 s 1 )+ik R s ( s 1 + s 2 2 s 0 )ik U s + 2 ( s 2 s 1 )+ik U s ( s 1 + s 2 2 s 0 ) ] d 3 R s + d 3 R s d 3 U s + d 3 U s + 1 4 ( S y ( i ) ( ω ) r 1 r 2 ) 2 cos 4 θ 1 sin 2 θ 2 cos 2 θ 2 D D D D S yz ( F ) ( R s + 2 ) η yz ( F ) ( R s ) S zy ( F ) ( U s + 2 ) η zy ( F ) ( U s ) ×exp[ ik R s + 2 ( s 1 s 2 )+ik R s ( s 1 + s 2 2 s 0 )ik U s + 2 ( s 1 s 2 )+ik U s ( s 1 + s 2 2 s 0 ) ] d 3 R s + d 3 R s d 3 U s + d 3 U s + 1 4 ( S y ( i ) ( ω ) r 1 r 2 ) 2 cos 4 θ 2 sin 2 θ 1 cos 2 θ 1 D D D D S zy ( F ) ( R s + 2 ) η zy ( F ) ( R s ) S yz ( F ) ( U s + 2 ) η yz ( F ) ( U s ) ×exp[ ik R s + 2 ( s 1 s 2 )+ik R s ( s 1 + s 2 2 s 0 )ik U s + 2 ( s 1 s 2 )+ik U s ( s 1 + s 2 2 s 0 ) ] d 3 R s + d 3 R s d 3 U s + d 3 U s + 1 4 ( S y ( i ) ( ω ) r 1 r 2 ) 2 sin 2 θ 1 cos 2 θ 1 sin 2 θ 2 cos 2 θ 2 D D D D S zz ( F ) ( R s + 2 ) η zz ( F ) ( R s ) S zz ( F ) ( U s + 2 ) η zz ( F ) ( U s ) ×exp[ ik R s + 2 ( s 1 s 2 )+ik R s ( s 1 + s 2 2 s 0 )ik U s + 2 ( s 1 s 2 )+ik U s ( s 1 + s 2 2 s 0 ) ] d 3 R s + d 3 R s d 3 U s + d 3 U s .
CI F ( s ) ( r 1 s 1 , r 2 s 2 )= 1 4 ( S x ( i ) ( ω ) r 1 r 2 ) 2 S ˜ xx ( F ) [ k( s 2 s 1 ) ] S ˜ xx ( F ) [ k( s 1 s 2 ) ] { η ˜ xx ( F ) [ k( s 0 s 1 + s 2 2 ) ] } 2 + 1 4 S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 cos 4 θ 2 S ˜ xy ( F ) [ k( s 2 s 1 ) ] η ˜ xy ( F ) [ k( s 0 s 1 + s 2 2 ) ] S ˜ yx ( F ) [ k( s 1 s 2 ) ] η ˜ yx ( F ) [ k( s 0 s 1 + s 2 2 ) ] + 1 4 S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 sin 2 θ 2 cos 2 θ 2 S ˜ xz ( F ) [ k( s 2 s 1 ) ] η ˜ xz ( F ) [ k( s 0 s 1 + s 2 2 ) ] S ˜ zx ( F ) [ k( s 1 s 2 ) ] η ˜ zx ( F ) [ k( s 0 s 1 + s 2 2 ) ] + 1 4 S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 cos 4 θ 1 S ˜ xy ( F ) [ k( s 1 s 2 ) ] η ˜ xy ( F ) [ k( s 0 s 1 + s 2 2 ) ] S ˜ yx ( F ) [ k( s 2 s 1 ) ] η ˜ yx ( F ) [ k( s 0 s 1 + s 2 2 ) ] + 1 4 S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 sin 2 θ 1 cos 2 θ 1 S ˜ xz ( F ) [ k( s 1 s 2 ) ] η ˜ xz ( F ) [ k( s 0 s 1 + s 2 2 ) ] S ˜ zx ( F ) [ k( s 2 s 1 ) ] η ˜ zx ( F ) [ k( s 0 s 1 + s 2 2 ) ] + 1 4 ( S y ( i ) ( ω ) r 1 r 2 ) 2 cos 4 θ 1 cos 4 θ 2 S ˜ yy ( F ) [ k( s 2 s 1 ) ] S ˜ yy ( F ) [ k( s 1 s 2 ) ] { η ˜ yy ( F ) [ k( s 0 s 1 + s 2 2 ) ] } 2 + 1 4 ( S y ( i ) ( ω ) r 1 r 2 ) 2 cos 4 θ 1 sin 2 θ 2 cos 2 θ 2 S ˜ yz ( F ) [ k( s 2 s 1 ) ] η ˜ yz ( F ) [ k( s 0 s 1 + s 2 2 ) ] S ˜ zy ( F ) [ k( s 1 s 2 ) ] η ˜ zy ( F ) [ k( s 0 s 1 + s 2 2 ) ] + 1 4 ( S y ( i ) ( ω ) r 1 r 2 ) 2 cos 4 θ 2 sin 2 θ 1 cos 2 θ 1 S ˜ yz ( F ) [ k( s 1 s 2 ) ] η ˜ yz ( F ) [ k( s 0 s 1 + s 2 2 ) ] S ˜ zy ( F ) [ k( s 2 s 1 ) ] η ˜ zy ( F ) [ k( s 0 s 1 + s 2 2 ) ] + 1 4 ( S y ( i ) ( ω ) r 1 r 2 ) 2 sin 2 θ 1 cos 2 θ 1 sin 2 θ 1 cos 2 θ 2 S ˜ zz ( F ) [ k( s 2 s 1 ) ] S ˜ zz ( F ) [ k( s 1 s 2 ) ] { η ˜ zz ( F ) [ k( s 0 s 1 + s 2 2 ) ] } 2 ,
S ˜ ij ( F ) ( ks )= D S ij ( F ) ( R s + )exp( ik R s + s ) d 3 R s + ,
η ˜ ij ( F ) ( ks )= D η ij ( F ) ( R s )exp( ik R s s ) d 3 R s ,
S ˜ ij ( F ) [ k( s 2 s 1 ) ]= S ˜ ji ( F ) [ k( s 2 s 1 ) ]= { S ˜ ij ( F ) [ k( s 1 s 2 ) ] } .
{ η ˜ ij ( F ) [ k( s 1 + s 2 2 s 0 ) ] } * { η ˜ ij ( F ) [ k( s 0 s 1 + s 2 2 ) ] } * = η ˜ ji ( F ) [ k( s 1 + s 2 2 s 0 ) ].
CI F ( s ) ( r 1 s 1 , r 2 s 2 )= 1 4 ( S x ( i ) ( ω ) r 1 r 2 ) 2 | S ˜ xx ( F ) [ k( s 2 s 1 ) ] | 2 { η ˜ xx ( F ) [ k( s 0 s 1 + s 2 2 ) ] } 2 + 1 4 S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 ( cos 4 θ 1 + cos 4 θ 2 ) | S ˜ xy ( F ) [ k( s 2 s 1 ) ] | 2 | η ˜ xy ( F ) [ k( s 0 s 1 + s 2 2 ) ] | 2 + 1 4 S x ( i ) ( ω ) S y ( i ) ( ω ) ( r 1 r 2 ) 2 ( sin 2 θ 1 cos 2 θ 1 + sin 2 θ 2 cos 2 θ 2 ) | S ˜ xz ( F ) [ k( s 2 s 1 ) ] | 2 | η ˜ xz ( F ) [ k( s 0 s 1 + s 2 2 ) ] | 2 + 1 4 ( S y ( i ) ( ω ) r 1 r 2 ) 2 { cos 4 θ 1 cos 4 θ 2 | S ˜ yy ( F ) [ k( s 2 s 1 ) ] | 2 { η ˜ yy ( F ) [ k( s 0 s 1 + s 2 2 ) ] } 2 + 1 4 cos 2 θ 1 cos 2 θ 2 ( cos 2 θ 1 sin 2 θ 2 + cos 2 θ 2 sin 2 θ 1 ) | S ˜ yz ( F ) [ k( s 2 s 1 ) ] | 2 | η ˜ yz ( F ) [ k( s 0 s 1 + s 2 2 ) ] | 2 + 1 4 sin 2 θ 1 cos 2 θ 1 sin 2 θ 2 cos 2 θ 2 | S ˜ zz ( F ) [ k( s 2 s 1 ) ] | 2 { η ˜ zz ( F ) [ k( s 0 s 1 + s 2 2 ) ] } 2 .
S ˜ ij ( F ) [ k( s 2 s 1 ) ]= S ˜ ( F ) [ k( s 2 s 1 ) ], η ˜ ij ( F ) [ k( s 0 s 1 + s 2 2 ) ]= η ˜ ( F ) [ k( s 0 s 1 + s 2 2 ) ], ( i, j=x, y, z ).
CI F ( s ) ( r 1 s 1 , r 2 s 2 )= 1 r 1 2 r 2 2 [ S x ( i ) ( ω )+ cos 2 θ 1 S y ( i ) ( ω ) ][ S x ( i ) ( ω )+ cos 2 θ 2 S y ( i ) ( ω ) ] | S ˜ ( F ) [ k( s 2 s 1 ) ] | 2 { η ˜ ( F ) [ k( s 0 s 1 + s 2 2 ) ] } 2 .
S ij ( F ) ( R s + )= A ( 2π σ ij 2 ) 3/2 exp[ ( R s + ) 2 2 σ ij 2 ], η ij ( F ) ( R s )= B ( 2π δ ij 2 ) 3/2 exp[ ( R s ) 2 2 δ ij 2 ], ( i , j=x, y, z ),
CI F ( s ) ( θ,-θ,r )= 1 4 ( S x ( i ) ( ω ) r 2 ) 2 A 2 B 2 exp[ 4 k 2 σ xx 2 sin 2 θ k 2 δ xx 2 ( cosθ1 ) 2 ] + S x ( i ) ( ω ) S y ( i ) ( ω ) 2 r 4 A 2 B 2 cos 4 θexp[ 4 k 2 σ xy 2 sin 2 θ k 2 δ xy 2 ( cosθ1 ) 2 ] + S x ( i ) ( ω ) S y ( i ) ( ω ) 2 r 4 A 2 B 2 sin 2 θ cos 2 θexp[ 4 k 2 σ xz 2 sin 2 θ k 2 δ xz 2 ( cosθ1 ) 2 ] + 1 4 ( S y ( i ) ( ω ) r 2 ) 2 A 2 B 2 { cos 8 θexp[ 4 k 2 σ yy 2 sin 2 θ k 2 δ yy 2 ( cosθ1 ) 2 ] + 1 2 cos 6 θ sin 2 θexp[ 4 k 2 σ yz 2 sin 2 θ k 2 δ yz 2 ( cosθ1 ) 2 ] + 1 4 cos 4 θ sin 4 θexp[ 4 k 2 σ zz 2 sin 2 θ k 2 δ zz 2 ( cosθ1 ) 2 ] }.

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