Abstract

Using the angular spectrum representation of plane waves, we investigate the scattering of multi-Gaussian Schell-model (MGSM) beams from a random medium within the accuracy of the first-order Born approximation. The far-zone properties, including the normalized spectral density and the spectral degree of coherence, are discussed. It is shown that the normalized spectral density and the spectral degree of coherence are influenced by the boundary of the beam profile (i.e., M), the transverse beam width, the correlation width of the source, and the properties of the scatterer.

© 2013 Optical Society of America

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References

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  1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
  2. D. Zhao and T. Wang, “Direct and inverse problem in the theory of light scattering,” Prog. Opt.57, 262–308 (2012).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  18. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  19. M. Born and E. Wolf, Principles of Optics, 7th ed (Cambridge University Press, 1999).

2013 (3)

Y. Zhang and D. Zhao, “Scattering of Hermite-Gaussian beams on Gaussian Schell-model random media,” Opt. Commun.300, 38–44 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun.305, 57–65 (2013).
[CrossRef]

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt.15(2), 025705 (2013).
[CrossRef]

2012 (3)

2011 (1)

2010 (5)

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]

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

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]

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A82(3), 033836 (2010).
[CrossRef]

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).
[CrossRef] [PubMed]

2007 (1)

2006 (1)

1998 (1)

P. S. Carney and E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun.155(1–3), 1–6 (1998).
[CrossRef]

1988 (1)

Cai, Y.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun.305, 57–65 (2013).
[CrossRef]

C. Ding, Y. Cai, O. Korotkova, Y. Zhang, and L. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett.36(4), 517–519 (2011).
[CrossRef] [PubMed]

Carney, P. S.

P. S. Carney and E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun.155(1–3), 1–6 (1998).
[CrossRef]

Chen, Y.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun.305, 57–65 (2013).
[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]

Ding, C.

Du, X.

Eyyuboglu, H. T.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun.305, 57–65 (2013).
[CrossRef]

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).
[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. A23(7), 1631–1638 (2006).
[CrossRef] [PubMed]

He, Y.

Jannson, J.

Jannson, T.

Korotkova, O.

Li, J.

Liu, X.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun.305, 57–65 (2013).
[CrossRef]

Mei, Z.

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt.15(2), 025705 (2013).
[CrossRef]

Pan, L.

Qu, J.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun.305, 57–65 (2013).
[CrossRef]

Sahin, S.

Shchepakina, E.

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt.15(2), 025705 (2013).
[CrossRef]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A29(10), 2159–2164 (2012).
[CrossRef] [PubMed]

Tong, Z.

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A82(3), 033836 (2010).
[CrossRef]

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).
[CrossRef] [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).
[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. A23(7), 1631–1638 (2006).
[CrossRef] [PubMed]

Wang, F.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun.305, 57–65 (2013).
[CrossRef]

Wang, T.

D. Zhao and T. Wang, “Direct and inverse problem in the theory of light scattering,” Prog. Opt.57, 262–308 (2012).

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]

Wolf, E.

Xin, Y.

Yuan, Y.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun.305, 57–65 (2013).
[CrossRef]

Zhang, Y.

Zhao, D.

J. Opt. (1)

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt.15(2), 025705 (2013).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

P. S. Carney and E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun.155(1–3), 1–6 (1998).
[CrossRef]

Y. Zhang and D. Zhao, “Scattering of Hermite-Gaussian beams on Gaussian Schell-model random media,” Opt. Commun.300, 38–44 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun.305, 57–65 (2013).
[CrossRef]

Opt. Lett. (7)

Phys. Rev. A (1)

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A82(3), 033836 (2010).
[CrossRef]

Phys. Rev. Lett. (1)

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).
[CrossRef] [PubMed]

Prog. Opt. (1)

D. Zhao and T. Wang, “Direct and inverse problem in the theory of light scattering,” Prog. Opt.57, 262–308 (2012).

Other (3)

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

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

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

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Figures (5)

Fig. 1
Fig. 1

The far-zone scattered field for selected values of M as a function of s x . The other parameters are chosen as follows: σ = 15 λ , δ = 5 λ , σ I = 25 λ , σ μ = 10 λ . (a) Normalized spectral density. (b) The spectral degree of coherence for two directions of scattering s and incidence s 0 .

Fig. 2
Fig. 2

The far-zone scattered field for selected values of the transverse beam width of the source σ as a function of s x . The other parameters are chosen as follows: δ = 5 λ , σ I = 25 λ , σ μ = 10 λ , M = 4 . (a) Normalized spectral density. (b) The spectral degree of coherence for two directions of scattering s and incidence s 0 .

Fig. 3
Fig. 3

The far-zone scattered field for selected values of the correlation width of the source δ as a function of s x . The other parameters are chosen as follows: σ = 15 λ , σ I = 25 λ , σ μ = 10 λ , M = 4 . (a) Normalized spectral density. (b) The spectral degree of coherence for two directions of scattering s and incidence s 0 .

Fig. 4
Fig. 4

The far-zone scattered field for selected values of the effective radius of the scatterer σ I as a function of s x . The other parameters are chosen as follows: σ = 15 λ , δ = 5 λ , σ μ = 10 λ , M = 4 . (a) Normalized spectral density. (b) The spectral degree of coherence for two directions of scattering s and incidence s 0 .

Fig. 5
Fig. 5

The far-zone scattered field for selected values of the correlation length of the scatterer σ μ as a function of s x . The other parameters are chosen as follows: σ = 15 λ , δ = 5 λ , σ I = 25 λ , M = 4 . (a) Normalized spectral density. (b) The spectral degree of coherence for two directions of scattering s and incidence s 0 .

Equations (32)

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U (i) (r,ω)= p 2 + q 2 1 a(p,q,ω) exp(ik s 0 r)dpdq,
U (s) (r,ω)= e ikr r p 2 + q 2 1 a(p,q,ω) f(s, s 0 ,ω)dpdq,
f(s, s 0 ,ω)= D F( r ,ω)exp[ ik( s s 0 ) r ] d 3 r ,
W (s) (r s 1 ,r s 2 ,ω)= U (s) (r s 1 ,ω) U (s) (r s 2 ,ω) ,
W (s) (r s 1 ,r s 2 ,ω)= 1 r 2 p 1 2 + q 1 2 1 p 2 2 + q 2 2 1 A( s 01 , s 02 ,ω) C ˜ F [ k( s 1 s 01 ),k( s 2 s 02 ),ω ]d p 1 d q 1 d p 2 d q 2 ,
A( s 01 , s 02 ,ω)= a ( p 1 , q 1 ,ω)a( p 2 , q 2 ,ω)
C ˜ F [ k( s 1 s 01 ),k( s 2 s 02 ),ω ]= f ( s 1 , s 01 ,ω)f( s 2 , s 02 ,ω) = D D C F ( r 1 , r 2 ,ω) exp[ ik( s 1 s 01 ) r 1 ik( s 2 s 02 ) r 2 ] d 3 r 1 d 3 r 2
W (0) ( ρ 1 , ρ 2 ,ω)=exp( ρ 1 2 + ρ 2 2 4 σ 2 ) 1 C 0 m=1 M ( M m ) ( 1 ) m1 m exp( | ρ 2 ρ 1 | 2 2m δ 2 ).
A( s 01 , s 02 ,ω)= ( k 2π ) 4 + W (0) ( ρ 1 , ρ 2 ,ω) exp[ ik( s 02 ρ 2 s 01 ρ 1 ) ] d 2 ρ 1 d 2 ρ 2 .
A( s 01 , s 02 ,ω)= k 4 σ 2 4 π 2 1 C 0 m=1 M ( M m ) ( 1 ) m1 m σ eff 2 ×exp{ k 2 2 [ ( s 01 s 02 ) 2 σ 2 + ( s 01 + s 02 ) 2 σ eff 2 4 ] },
1 σ eff 2 = 1 4 σ 2 + 1 m δ 2 .
C F ( r 1 , r 2 ,ω)= A 0 exp( r 1 2 + r 2 2 4 σ I 2 )exp( | r 2 r 1 | 2 2 σ μ 2 ),
C ˜ F [ k( s 1 s 01 ),k( s 2 s 02 ),ω ]=64 π 3 A 0 σ I 6 σ μ 3 ( σ μ 2 +4 σ I 2 ) 3 2 ×exp{ k 2 σ I 2 2 [ ( s 1 s 01 )( s 2 s 02 ) ] 2 } ×exp{ k 2 σ I 2 σ μ 2 2( σ μ 2 +4 σ I 2 ) [ ( s 1 s 01 )+( s 2 s 02 ) ] 2 }.
W (s) (r s 1 ,r s 2 ,ω)= 16π k 4 σ 2 A 0 σ I 6 σ μ 3 ( σ μ 2 +4 σ I 2 ) 3 2 C 0 r 2 m=1 M ( M m ) ( 1 ) m1 m σ eff 2 p 1 2 + q 1 2 1 p 2 2 + q 2 2 1 exp[ k 2 σ 2 2 ( p 2 2 2 p 2 p 1 + p 1 2 + q 2 2 2 q 2 q 1 + q 1 2 )] ×exp[ k 2 σ eff 2 8 ( p 2 2 +2 p 2 p 1 + p 1 2 + q 2 2 +2 q 2 q 1 + q 1 2 )] ×exp{ k 2 σ I 2 2 [ ( s 1x s 2x + p 2 ) 2 2( s 1x s 2x + p 2 ) p 1 + ( s 1y s 2y + q 2 ) 2 2( s 1y s 2y + q 2 ) q 1 + ( s 1z s 2z + 1 p 2 2 q 2 2 ) 2 2( s 1z s 2z + 1 p 2 2 q 2 2 ) 1 p 1 2 q 1 2 + 1 ] } ×exp{ k 2 σ I 2 σ μ 2 2( σ μ 2 +4 σ I 2 ) [ ( s 1x + s 2x p 2 ) 2 2( s 1x + s 2x p 2 ) p 1 + ( s 1y + s 2y q 2 ) 2 2( s 1y + s 2y q 2 ) q 1 + ( s 1z + s 2z 1 p 2 2 q 2 2 ) 2 2( s 1z + s 2z 1 p 2 2 q 2 2 ) 1 p 1 2 q 1 2 + 1 ] }d p 1 d q 1 d p 2 d q 2 .
W (s) (r s 1 ,r s 2 ,ω)== 16π k 4 σ 2 A 0 σ I 6 σ μ 3 ( σ μ 2 +4 σ I 2 ) 3 2 C 0 r 2 m=1 M ( M m ) ( 1 ) m1 m σ eff 2 p 1 2 + q 1 2 <<1 p 2 2 + q 2 2 1 exp[ k 2 σ 2 2 ( p 2 2 2 p 2 p 1 + p 1 2 + q 2 2 2 q 2 q 1 + q 1 2 )] ×exp[ k 2 σ eff 2 8 ( p 2 2 +2 p 2 p 1 + p 1 2 + q 2 2 +2 q 2 q 1 + q 1 2 )] ×exp{ k 2 σ I 2 2 [ ( s 1x s 2x + p 2 ) 2 2( s 1x s 2x + p 2 ) p 1 + ( s 1y s 2y + q 2 ) 2 2( s 1y s 2y + q 2 ) q 1 + ( s 1z s 2z + 1 p 2 2 q 2 2 ) 2 2( s 1z s 2z + 1 p 2 2 q 2 2 )+( s 1z s 2z + 1 p 2 2 q 2 2 )( p 1 2 + q 1 2 )+1]} ×exp{ k 2 σ I 2 σ μ 2 2( σ μ 2 +4 σ I 2 ) [ ( s 1x + s 2x p 2 ) 2 2( s 1x + s 2x p 2 ) p 1 + ( s 1y + s 2y q 2 ) 2 2( s 1y + s 2y q 2 ) q 1 + ( s 1z + s 2z 1 p 2 2 q 2 2 ) 2 2( s 1z + s 2z 1 p 2 2 q 2 2 )+( s 1z + s 2z 1 p 2 2 q 2 2 )( p 1 2 + q 1 2 ) + 1 ] }d p 1 d q 1 d p 2 d q 2 .
W (s) (r s 1 ,r s 2 ,ω)= 16π k 4 σ 2 A 0 σ I 6 σ μ 3 ( σ μ 2 +4 σ I 2 ) 3 2 C 0 r 2 m=1 M ( M m ) ( 1 ) m1 m σ eff 2 p 2 2 + q 2 2 1 exp[ k 2 σ 2 2 ( p 2 2 2 p 2 p 1 + p 1 2 + q 2 2 2 q 2 q 1 + q 1 2 )] ×exp[ k 2 σ eff 2 8 ( p 2 2 +2 p 2 p 1 + p 1 2 + q 2 2 +2 q 2 q 1 + q 1 2 )] ×exp{ k 2 σ I 2 2 [ ( s 1x s 2x + p 2 ) 2 2( s 1x s 2x + p 2 ) p 1 + ( s 1y s 2y + q 2 ) 2 2( s 1y s 2y + q 2 ) q 1 + ( s 1z s 2z + 1 p 2 2 q 2 2 ) 2 2( s 1z s 2z + 1 p 2 2 q 2 2 )+( s 1z s 2z + 1 p 2 2 q 2 2 )( p 1 2 + q 1 2 )+1]} ×exp{ k 2 σ I 2 σ μ 2 2( σ μ 2 +4 σ I 2 ) [ ( s 1x + s 2x p 2 ) 2 2( s 1x + s 2x p 2 ) p 1 + ( s 1y + s 2y q 2 ) 2 2( s 1y + s 2y q 2 ) q 1 + ( s 1z + s 2z 1 p 2 2 q 2 2 ) 2 2( s 1z + s 2z 1 p 2 2 q 2 2 )+( s 1z + s 2z 1 p 2 2 q 2 2 )( p 1 2 + q 1 2 ) + 1 ] }d p 1 d q 1 d p 2 d q 2 .
W (s) (r s 1 ,r s 2 ,ω)= 16 π 2 k 4 σ 2 A 0 σ I 6 σ μ 3 ( σ μ 2 +4 σ I 2 ) 3 2 C 0 r 2 m=1 M ( M m ) ( 1 ) m1 m σ eff 2 p 2 2 + q 2 2 1 1 a exp( b 2 + c 2 4a )exp[ k 2 2 ( σ 2 + σ eff 2 4 )( p 2 2 + q 2 2 )] ×exp{ k 2 σ I 2 2 [ ( s 1x s 2x + p 2 ) 2 + ( s 1y s 2y + q 2 ) 2 + ( s 1z s 2z + 1 p 2 2 q 2 2 ) 2 2( s 1z s 2z + 1 p 2 2 q 2 2 )+1]} ×exp{ k 2 σ I 2 σ μ 2 2( σ μ 2 +4 σ I 2 ) [ ( s 1x + s 2x p 2 ) 2 + ( s 1y + s 2y q 2 ) 2 + ( s 1z + s 2z 1 p 2 2 q 2 2 ) 2 2( s 1z + s 2z 1 p 2 2 q 2 2 )+ 1 ] }d p 2 d q 2 ,
a= k 2 σ 2 2 + k 2 σ eff 2 8 + k 2 σ I 2 2 ( s 1z s 2z + 1 p 2 2 q 2 2 )+ k 2 σ I 2 σ μ 2 2( σ μ 2 +4 σ I 2 ) ( s 1z + s 2z 1 p 2 2 q 2 2 ),
b= k 2 σ 2 p 2 k 2 σ eff 2 p 2 4 + k 2 σ I 2 ( s 1x s 2x + p 2 )+ k 2 σ I 2 σ μ 2 σ μ 2 +4 σ I 2 ( s 1x + s 2x p 2 ),
c= k 2 σ 2 q 2 k 2 σ eff 2 q 2 4 + k 2 σ I 2 ( s 1y s 2y + q 2 )+ k 2 σ I 2 σ μ 2 σ μ 2 +4 σ I 2 ( s 1y + s 2y q 2 ).
S (s) (rs,ω)= 16 π 2 k 4 σ 2 A 0 σ I 6 σ μ 3 ( σ μ 2 +4 σ I 2 ) 3 2 C 0 r 2 m=1 M ( M m ) ( 1 ) m1 m σ eff 2 p 2 2 + q 2 2 1 1 a exp( b 2 + c 2 4 a )exp[ k 2 2 ( σ 2 + σ eff 2 4 )( p 2 2 + q 2 2 )] ×exp[ k 2 σ I 2 (1 1 p 2 2 q 2 2 )] ×exp[ k 2 σ I 2 σ μ 2 2( σ μ 2 +4 σ I 2 ) (64 s x p 2 4 s y q 2 4 s z 1 p 2 2 q 2 2 4 s z +2 1 p 2 2 q 2 2 )]d p 2 d q 2 ,
a = k 2 σ 2 2 + k 2 σ eff 2 8 + k 2 σ I 2 2 1 p 2 2 q 2 2 + k 2 σ I 2 σ μ 2 2( σ μ 2 +4 σ I 2 ) ( 2 s z 1 p 2 2 q 2 2 ),
b = k 2 σ 2 p 2 k 2 σ eff 2 p 2 4 + k 2 σ I 2 p 2 + k 2 σ I 2 σ μ 2 σ μ 2 +4 σ I 2 ( 2 s x p 2 ),
c = k 2 σ 2 q 2 k 2 σ eff 2 q 2 4 + k 2 σ I 2 q 2 + k 2 σ I 2 σ μ 2 σ μ 2 +4 σ I 2 ( 2 s y q 2 ).
μ (s) (r s 1 ,r s 2 ,ω)= W (s) (r s 1 ,r s 2 ,ω) S (s) (r s 1 ,ω) S (s) (r s 2 ,ω) ,
μ (s) (r s 1 ,r s 2 ,ω)= m=1 M ( M m ) ( 1 ) m1 m σ eff 2 p 2 2 + q 2 2 1 1 a exp( b 2 + c 2 4a ) ×exp[ k 2 2 ( σ 2 + σ eff 2 4 )( p 2 2 + q 2 2 )] ×exp{ k 2 σ I 2 2 [ ( s 1x s 2x + p 2 ) 2 + ( s 1y s 2y + q 2 ) 2 + ( s 1z s 2z + 1 p 2 2 q 2 2 ) 2 2( s 1z s 2z + 1 p 2 2 q 2 2 )+1]} ×exp{ k 2 σ I 2 σ μ 2 2( σ μ 2 +4 σ I 2 ) [ ( s 1x + s 2x p 2 ) 2 + ( s 1y + s 2y q 2 ) 2 + ( s 1z + s 2z 1 p 2 2 q 2 2 ) 2 2( s 1z + s 2z 1 p 2 2 q 2 2 )+ 1 ] }d p 2 d q 2 / { m=1 M ( M m ) ( 1 ) m1 m σ eff 2 p 2 2 + q 2 2 1 1 a 1 exp( b 1 2 + c 1 2 4 a 1 ) ×exp[ k 2 2 ( σ 2 + σ eff 2 4 )( p 2 2 + q 2 2 )]exp[ k 2 σ I 2 (1 1 p 2 2 q 2 2 )] ×exp[ k 2 σ I 2 σ μ 2 2( σ μ 2 +4 σ I 2 ) (64 s 1x p 2 4 s 1y q 2 4 s 1z 1 p 2 2 q 2 2 4 s 1z +2 1 p 2 2 q 2 2 )]d p 2 d q 2 } 1 2 / { m=1 M ( M m ) ( 1 ) m1 m σ eff 2 p 2 2 + q 2 2 1 1 a 2 exp( b 2 2 + c 2 2 4 a 2 ) ×exp[ k 2 2 ( σ 2 + σ eff 2 4 )( p 2 2 + q 2 2 )]exp[ k 2 σ I 2 (1 1 p 2 2 q 2 2 )] ×exp[ k 2 σ I 2 σ μ 2 2( σ μ 2 +4 σ I 2 ) (64 s 2x p 2 4 s 2y q 2 4 s 2z 1 p 2 2 q 2 2 4 s 2z +2 1 p 2 2 q 2 2 )]d p 2 d q 2 } 1 2 ,
a 1 = k 2 σ 2 2 + k 2 σ eff 2 8 + k 2 σ I 2 2 1 p 2 2 q 2 2 + k 2 σ I 2 σ μ 2 2( σ μ 2 +4 σ I 2 ) ( 2 s 1z 1 p 2 2 q 2 2 ),
b 1 = k 2 σ 2 p 2 k 2 σ eff 2 p 2 4 + k 2 σ I 2 p 2 + k 2 σ I 2 σ μ 2 σ μ 2 +4 σ I 2 ( 2 s 1x p 2 ),
c 1 = k 2 σ 2 q 2 k 2 σ eff 2 q 2 4 + k 2 σ I 2 q 2 + k 2 σ I 2 σ μ 2 σ μ 2 +4 σ I 2 ( 2 s 1y q 2 ),
a 2 = k 2 σ 2 2 + k 2 σ eff 2 8 + k 2 σ I 2 2 1 p 2 2 q 2 2 + k 2 σ I 2 σ μ 2 2( σ μ 2 +4 σ I 2 ) ( 2 s 2z 1 p 2 2 q 2 2 ),
b 2 = k 2 σ 2 p 2 k 2 σ eff 2 p 2 4 + k 2 σ I 2 p 2 + k 2 σ I 2 σ μ 2 σ μ 2 +4 σ I 2 ( 2 s 2x p 2 ),
c 2 = k 2 σ 2 q 2 k 2 σ eff 2 q 2 4 + k 2 σ I 2 q 2 + k 2 σ I 2 σ μ 2 σ μ 2 +4 σ I 2 ( 2 s 2y q 2 ).

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