Abstract

Within the Markov approximation, we introduce a novel class of random media which can produce a scattered field with optical lattice patterns. It is shown that the array dimension, lobes intensity profile, and the periodicity of the optical lattice can be flexibly controlled by altering the correlation parameters of scattering potential of the random medium. In addition, a new method for designing random media is proposed. It is shown that the convolution of any two legitimate degrees of potential correlation can lead to a new degree of potential correlation corresponding to a new scattered intensity distribution. An example of a novel family of random media is cited to demonstrate the result.

© 2017 Optical Society of America

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References

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  2. 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).
    [PubMed]
  3. D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32(24), 3483–3485 (2007).
    [PubMed]
  4. 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).
    [PubMed]
  5. T. Wang, Z. Jiang, X. Ji, and D. Zhao, “Can different media generate scattered field with identical spectral coherence,” Opt. Commun. 363, 134–137 (2016).
  6. X. Du and D. Zhao, “Rotationally symmetric scattering from anisotropic media,” Phys. Lett. A 375, 1269–1273 (2011).
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    [PubMed]
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    [PubMed]
  12. Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010).
  13. X. Wang, Z. Liu, and K. Huang, “Weak scattering of a random electromagnetic source of circular frames upon a deterministic medium,” J. Opt. Soc. Am. B 34(8), 1755–1764 (2017).
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    [PubMed]
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  16. O. Korotkova, “Design of weak scattering media for controllable light scattering,” Opt. Lett. 40(2), 284–287 (2015).
    [PubMed]
  17. O. Korotkova, “Can a sphere scatter light producing rectangular intensity patterns?” Opt. Lett. 40(8), 1709–1712 (2015).
    [PubMed]
  18. G. Zheng, D. Ye, X. Peng, M. Song, and Q. Zhao, “Tunable scattering intensity with prescribed weak media,” Opt. Express 24(21), 24169–24178 (2016).
    [PubMed]
  19. J. Li and O. Korotkova, “Random medium model for cusping of plane waves,” Opt. Lett. 42(17), 3251–3254 (2017).
    [PubMed]
  20. J. Li and O. Korotkova, “Deterministic mode representation of random stationary media for scattering problems,” J. Opt. Soc. Am. A 34(6), 1021–1028 (2017).
  21. Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40(23), 5662–5665 (2015).
    [PubMed]
  22. L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
    [PubMed]
  23. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  24. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [PubMed]
  25. H. Wu, X. Pan, Z. Zhu, X. Ji, and T. Wang, “Reciprocity relations of an electromagnetic light wave on scattering from a quasi-homogeneous anisotropic medium,” Opt. Express 25(10), 11297–11305 (2017).
    [PubMed]
  26. O. Korotkova and Z. Mei, “Convolution of degrees of coherence,” Opt. Lett. 40(13), 3073–3076 (2015).
    [PubMed]
  27. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [PubMed]
  28. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [PubMed]
  29. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [PubMed]
  30. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
    [PubMed]
  31. J. Li and O. Korotkova, “Scattering of light from a stationary nonuniformly correlated medium,” Opt. Lett. 41(11), 2616–2619 (2016).
    [PubMed]

2017 (4)

2016 (3)

2015 (5)

2014 (1)

2013 (2)

2012 (3)

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

C. Ding, Y. Cai, and Y. Zhang, “Scattering of partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 35, 384–386 (2012).

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[PubMed]

2011 (2)

X. Du and D. Zhao, “Rotationally symmetric scattering from anisotropic media,” Phys. Lett. A 375, 1269–1273 (2011).

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[PubMed]

2010 (3)

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

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010).

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

2009 (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).
[PubMed]

2008 (1)

S. Sahin and O. Korotkova, “Scattering of scalar light field from collection of particles,” Phys. Rev. A 78(6), 063815 (2008).

2007 (2)

2006 (1)

1997 (1)

1989 (1)

Cai, Y.

C. Ding, Y. Cai, and Y. Zhang, “Scattering of partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 35, 384–386 (2012).

Chang, L.

Ding, C.

C. Ding, Y. Cai, and Y. Zhang, “Scattering of partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 35, 384–386 (2012).

Du, X.

X. Du and D. Zhao, “Rotationally symmetric scattering from anisotropic media,” Phys. Lett. A 375, 1269–1273 (2011).

Fischer, D. G.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

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).
[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).
[PubMed]

Foley, J. T.

Gori, F.

Huang, K.

Ji, X.

H. Wu, X. Pan, Z. Zhu, X. Ji, and T. Wang, “Reciprocity relations of an electromagnetic light wave on scattering from a quasi-homogeneous anisotropic medium,” Opt. Express 25(10), 11297–11305 (2017).
[PubMed]

T. Wang, Z. Jiang, X. Ji, and D. Zhao, “Can different media generate scattered field with identical spectral coherence,” Opt. Commun. 363, 134–137 (2016).

Jiang, Z.

T. Wang, Z. Jiang, X. Ji, and D. Zhao, “Can different media generate scattered field with identical spectral coherence,” Opt. Commun. 363, 134–137 (2016).

Korotkova, O.

J. Li and O. Korotkova, “Deterministic mode representation of random stationary media for scattering problems,” J. Opt. Soc. Am. A 34(6), 1021–1028 (2017).

J. Li and O. Korotkova, “Random medium model for cusping of plane waves,” Opt. Lett. 42(17), 3251–3254 (2017).
[PubMed]

J. Li and O. Korotkova, “Scattering of light from a stationary nonuniformly correlated medium,” Opt. Lett. 41(11), 2616–2619 (2016).
[PubMed]

O. Korotkova and Z. Mei, “Convolution of degrees of coherence,” Opt. Lett. 40(13), 3073–3076 (2015).
[PubMed]

O. Korotkova, “Design of weak scattering media for controllable light scattering,” Opt. Lett. 40(2), 284–287 (2015).
[PubMed]

O. Korotkova, “Can a sphere scatter light producing rectangular intensity patterns?” Opt. Lett. 40(8), 1709–1712 (2015).
[PubMed]

Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40(23), 5662–5665 (2015).
[PubMed]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[PubMed]

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[PubMed]

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010).

S. Sahin and O. Korotkova, “Scattering of scalar light field from collection of particles,” Phys. Rev. A 78(6), 063815 (2008).

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32(24), 3483–3485 (2007).
[PubMed]

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).
[PubMed]

Lajunen, H.

Li, J.

Liu, Z.

Ma, L.

Mao, Y.

Mei, Z.

Pan, X.

Peng, X.

Ponomarenko, S. A.

Saastamoinen, T.

Sahin, S.

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[PubMed]

S. Sahin and O. Korotkova, “Scattering of scalar light field from collection of particles,” Phys. Rev. A 78(6), 063815 (2008).

Santarsiero, M.

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).
[PubMed]

Song, M.

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), 033836 (2010).

van Dijk, T.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

Visser, T. D.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[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).
[PubMed]

Wang, T.

H. Wu, X. Pan, Z. Zhu, X. Ji, and T. Wang, “Reciprocity relations of an electromagnetic light wave on scattering from a quasi-homogeneous anisotropic medium,” Opt. Express 25(10), 11297–11305 (2017).
[PubMed]

T. Wang, Z. Jiang, X. Ji, and D. Zhao, “Can different media generate scattered field with identical spectral coherence,” Opt. Commun. 363, 134–137 (2016).

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

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

Wang, X.

Wolf, E.

Wu, H.

Ye, D.

Zhang, Y.

C. Ding, Y. Cai, and Y. Zhang, “Scattering of partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 35, 384–386 (2012).

Zhao, D.

T. Wang, Z. Jiang, X. Ji, and D. Zhao, “Can different media generate scattered field with identical spectral coherence,” Opt. Commun. 363, 134–137 (2016).

Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40(23), 5662–5665 (2015).
[PubMed]

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

X. Du and D. Zhao, “Rotationally symmetric scattering from anisotropic media,” Phys. Lett. A 375, 1269–1273 (2011).

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

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32(24), 3483–3485 (2007).
[PubMed]

Zhao, Q.

Zheng, G.

Zhu, Z.

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

T. Wang, Z. Jiang, X. Ji, and D. Zhao, “Can different media generate scattered field with identical spectral coherence,” Opt. Commun. 363, 134–137 (2016).

Opt. Express (3)

Opt. Lett. (14)

J. Li and O. Korotkova, “Random medium model for cusping of plane waves,” Opt. Lett. 42(17), 3251–3254 (2017).
[PubMed]

O. Korotkova and Z. Mei, “Convolution of degrees of coherence,” Opt. Lett. 40(13), 3073–3076 (2015).
[PubMed]

Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40(23), 5662–5665 (2015).
[PubMed]

J. Li and O. Korotkova, “Scattering of light from a stationary nonuniformly correlated medium,” Opt. Lett. 41(11), 2616–2619 (2016).
[PubMed]

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32(24), 3483–3485 (2007).
[PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[PubMed]

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

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[PubMed]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[PubMed]

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[PubMed]

L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
[PubMed]

O. Korotkova, “Design of weak scattering media for controllable light scattering,” Opt. Lett. 40(2), 284–287 (2015).
[PubMed]

O. Korotkova, “Can a sphere scatter light producing rectangular intensity patterns?” Opt. Lett. 40(8), 1709–1712 (2015).
[PubMed]

Phys. Lett. A (2)

X. Du and D. Zhao, “Rotationally symmetric scattering from anisotropic media,” Phys. Lett. A 375, 1269–1273 (2011).

C. Ding, Y. Cai, and Y. Zhang, “Scattering of partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 35, 384–386 (2012).

Phys. Rev. A (2)

S. Sahin and O. Korotkova, “Scattering of scalar light field from collection of particles,” Phys. Rev. A 78(6), 063815 (2008).

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010).

Phys. Rev. Lett. (2)

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

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).
[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).

Other (1)

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

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Figures (8)

Fig. 1
Fig. 1 Degree of the scattering potential correlation varying with k ρ d along x (left columns) and y (right columns) directions, respectively. k= 10 7 , (a-b) M=N=1 (blue curve), M=7, N=6 (red curve), and M=11, N=10 (green curve); (c-d) R x = R y =0.25 (blue curve), R x = R y =0.5 (red curve), and R x = R y =1 (green curve); (e-f) k μ x =k μ y =15 (blue curve), k μ x =k μ y =10 (red curve), and k μ x =k μ y =5 (green curve).
Fig. 2
Fig. 2 Spectral density distribution in far-zone scattered field for M=4, N=6, R x = R y =1, and k μ x =k μ y =20. The other parameters are A=1, kσ=30.
Fig. 3
Fig. 3 Optical lattice patterns with different dimensions and elliptical lobes in far-zone scattered field. The parameters are k μ x =20, k μ y =10, R x = R y =1, (a) M=6, N=1, and (b) M=6, N=5.
Fig. 4
Fig. 4 Rectangular and circular intensity distribution patterns in far-zone scattered field. The parameters are k μ x =k μ y =20, R x = R y =0.25, (a) M=4,N=6, and (b) M=N=1.
Fig. 5
Fig. 5 First column is the degree of potential correlation corresponding to Eq. (15); the second column is the degree of potential correlation corresponding to Eq. (21); the third column is their convolution.
Fig. 6
Fig. 6 Optical lattice patterns with adjustable lobes intensity calculated from Eq. (26). The medium with M=N=5, k μ x =k μ y =20, R x = R y =1, k δ x =k δ y =4, kσ=30. (a) L=40, (b) L=10, (c) L=1, (d) L=1 with k δ x =k δ y =15.
Fig. 7
Fig. 7 First row is circular Gaussian degree of potential correlation, the second row is rectangular multi-Gaussian degree of potential correlation, and the third row is their convolution.
Fig. 8
Fig. 8 Far-zone scattered fields calculated from Eq. (26). The medium with M=N=1, k μ x =k μ y =8, R x = R y =1, L=40. (a) k δ x =k δ y =5, (b) k δ x =25, k δ y =8, (c) k δ x =8, k δ y =25, and (d) k δ x =k δ y =25.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

C F ( r 1 , r 2 ;ω )= F ( r 1 ,ω )F( r 2 ,ω ) .
C F ( r 1 , r 2 ;ω )= H 0 * ( r 1 ,u,ω ) H 0 ( r 2 ,u,ω )p( u,ω ) d 3 u,
H 0 ( r ,u;ω )=τ( r )exp(i r u),
C F ( r 1 , r 2 ;ω )= τ ( r 1 )τ( r 2 ) μ F ( r 2 r 1 ,ω ),
μ F ( r 2 r 1 ,ω )= D p( u,ω )exp[ iu( r 2 r 1 ) ] d 3 u.
C F ( r 1 , r 2 ;ω )= τ 2 ( r 1 + r 2 2 ,ω ) μ F ( r 2 r 1 ,ω ).
S ( s ) ( rs,ω )= 1 r 2 S ( i ) ( ω ) I ˜ F ( 0,ω ) μ ˜ F [ k( s s 0 ),ω ],
μ ˜ F [ k( s s 0 ),ω ]= D μ F ( r 2 r 1 ,ω )exp [ ik( s s 0 )( r 2 r 1 ) ] d 3 r 1 d 3 r 2
I ˜ F ( 0,ω )= D τ 2 ( r ) d 3 r
τ F ( r ,ω )=Aexp( r 2 4 σ 2 ),
I ˜ F ( 0,ω )=A ( 2π ) 3 2 σ 3 .
μ F ( r 2 r 1 ,ω )= μ F ( ρ 2 ρ 1 ,ω )δ( z 2 z 1 ),
S ( s ) ( rs,ω )= 1 r 2 A ( 2π ) 3 2 σ 3 S ( i ) ( ω ) μ ˜ F [ K ρ ,ω ],
μ ˜ F ( K ρ ,ω )= D μ F ( ρ 2 ρ 1 ,ω )exp [ i K ρ ( ρ 2 ρ 1 ) ] d 2 ρ 1 d 2 ρ 2
μ F ( ρ 2 ρ 1 ,ω )= 1 N×M exp[ ( x 2 x 1 ) 2 2 μ x 2 ]exp[ ( y 2 y 1 ) 2 2 μ y 2 ] n=P P cos[ 2πn( x 2 x 1 ) R x μ x ] × m=Q Q cos[ 2πm( y 2 y 1 ) R y μ y ] ,
S ( s ) ( rs,ω )= S ( i ) ( ω )A σ 3 ( 2π ) 3 μ x μ y 2MN r 2 exp( k 2 μ x 2 s x 2 2 )exp( k 2 μ y 2 s y 2 2 ) × n=P P cosh( 2πnk s x μ x R x ) exp( 2 π 2 n 2 R x 2 ) × n=Q Q cosh( 2πnk s y μ y R y ) exp( 2 π 2 n 2 R y 2 ).
μ c ( ρ 2 ρ 1 ,ω )= μ F1 ( ρ 2 ρ 1 ,ω ) μ F2 ( ρ 2 ρ 1 ,ω ),
p c ( u ρ ,ω )= p 1 ( u ρ ,ω ) p 2 ( u ρ ,ω ),
p j ( u ρ ,ω )= - + μ j ( ρ 2 ρ 1 ,ω ) exp[ i u ρ ( ρ 2 ρ 1 ) ] d 2 ρ 1 d 2 ρ 2 ,
S c ( s ) ( rs,ω )= 1 r 2 S ( i ) ( ω ) I ˜ F ( 0,ω ) p 1 ( K ρ ) p 2 ( K ρ ).
μ F2 ( ρ 2 ρ 1 ,ω )= 1 C L x C L y l=1 L x ( 1 ) l1 l ( L x l )exp( ( x 2 x 1 ) 2 2l δ x 2 ) × l=1 L y ( 1 ) l1 l ( L y l )exp( ( y 2 y 1 ) 2 2l δ y 2 )
C L x = l=1 L x ( 1 ) l1 l ( L x l )
C L y = l=1 L y ( 1 ) l1 l ( L y l ).
μ c ( ρ 2 ρ 1 ,ω )= 1 N×M exp[ ( x 2 x 1 ) 2 2 μ x 2 ]exp[ ( y 2 y 1 ) 2 2 μ y 2 ] n=P P cos[ 2πn( x 2 x 1 ) R x μ x ] × m=Q Q cos[ 2πm( y 2 y 1 ) R y μ y ] 1 C L x C L y l=1 L x ( 1 ) l1 l ( L x l )exp( ( x 2 x 1 ) 2 2l δ x 2 ) × l=1 L y ( 1 ) l1 l ( L y l )exp( ( y 2 y 1 ) 2 2l δ y 2 ).
p c ( K ρ ,ω )= p 1 ( K ρ ,ω ) p 2 ( K ρ ,ω ) = μ x μ y δ x δ y 2MN C L x C L y exp( k 2 μ x 2 s x 2 2 )exp( k 2 μ y 2 s y 2 2 ) × n=P P cosh( 2πnk s x μ x R x ) exp( 2 π 2 n 2 R x 2 ) × m=Q Q cosh( 2πmk s y μ y R y ) exp( 2 π 2 m 2 R y 2 ) × l=1 L x ( 1 ) l1 l ( L x l )exp[ l s x 2 k 2 δ x 2 2 ] × l=1 L y ( 1 ) l1 l ( L y l )exp[ l s y 2 k 2 δ y 2 2 ].
S c ( s ) ( rs,ω )= S ( i ) ( ω )A σ 3 ( 2π ) 3 μ x μ y δ x δ y 2MN C L x C L y r 2 exp( k 2 μ x 2 s x 2 2 )exp( k 2 μ y 2 s y 2 2 ) × n=P P cosh( 2πnk s x μ x R x ) exp( 2 π 2 n 2 R x 2 ) × m=Q Q cosh( 2πmk s y μ y R y ) exp( 2 π 2 m 2 R y 2 ) × l=1 L x ( 1 ) l1 l ( L x l )exp[ l s x 2 k 2 δ x 2 2 ] × l=1 L y ( 1 ) l1 l ( L y l )exp[ l s y 2 k 2 δ y 2 2 ].
μ c ( ρ 2 ρ 1 ,ω )= μ F1 ( ρ 2 ρ 1 ,ω ) μ F2 ( ρ 2 ρ 1 ,ω )··· μ Fn ( ρ 2 ρ 1 ,ω ).
p c ( u ρ ,ω )= p 1 ( u ρ ,ω ) p 2 ( u ρ ,ω )··· p n ( u ρ ,ω ).

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