Abstract

Novel analytical expressions for the cross-spectral density function of a Gaussian Schell-model pulsed (GSMP) beam propagating through atmospheric turbulence are derived. Based on the cross-spectral density function, the average spectral density and the spectral degree of coherence of a GSMP beam in atmospheric turbulence are in turn examined. The dependence of the spectral degree of coherence on the turbulence strength measured by the atmospheric spatial coherence length is calculated numerically and analyzed in depth. The results obtained are useful for applications involving spatially and spectrally partially coherent pulsed beams propagating through atmospheric turbulence.

© 2011 OSA

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  1. G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
    [CrossRef]
  2. C. Chen, H. Yang, X. Feng, and H. Wang, “Optimization criterion for initial coherence degree of lasers in free-space optical links through atmospheric turbulence,” Opt. Lett. 34(4), 419–421 (2009).
    [CrossRef] [PubMed]
  3. K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011).
    [CrossRef]
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chaps. 4 and 5.
  5. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005).
    [CrossRef] [PubMed]
  6. L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
    [CrossRef] [PubMed]
  7. C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
    [CrossRef]
  8. R. L. Fante, “Two-position, two-frequency mutual-coherence function in turbulence,” J. Opt. Soc. Am. 71(12), 1446–1451 (1981).
    [CrossRef]
  9. C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009).
    [CrossRef]
  10. C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
    [CrossRef]
  11. H. Mao and D. Zhao, “Second-order intensity-moment characteristics for broadband partially coherent flat-topped beams in atmospheric turbulence,” Opt. Express 18(2), 1741–1755 (2010).
    [CrossRef] [PubMed]
  12. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 4 and 7.
  13. Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. 283(7), 1209–1212 (2010).
    [CrossRef]
  14. G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
    [CrossRef]
  15. C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002).
    [CrossRef]
  16. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007), Chap. 3.

2011 (3)

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011).
[CrossRef]

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[CrossRef]

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[CrossRef]

2010 (2)

2009 (2)

C. Chen, H. Yang, X. Feng, and H. Wang, “Optimization criterion for initial coherence degree of lasers in free-space optical links through atmospheric turbulence,” Opt. Lett. 34(4), 419–421 (2009).
[CrossRef] [PubMed]

C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009).
[CrossRef]

2008 (1)

C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
[CrossRef]

2007 (1)

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[CrossRef]

2005 (1)

2003 (1)

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[CrossRef] [PubMed]

2002 (1)

C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002).
[CrossRef]

1981 (1)

Berman, G. P.

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[CrossRef]

Bishop, A. R.

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[CrossRef]

Chen, C.

Chen, H.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[CrossRef] [PubMed]

Chernobrod, B. M.

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[CrossRef]

Dang, A.

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[CrossRef]

Ding, C.

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[CrossRef]

C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009).
[CrossRef]

C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
[CrossRef]

Drexler, K.

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011).
[CrossRef]

Fante, R. L.

Feng, X.

Gbur, G.

Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. 283(7), 1209–1212 (2010).
[CrossRef]

Gorshkov, V. N.

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[CrossRef]

Gu, Y.

Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. 283(7), 1209–1212 (2010).
[CrossRef]

Guo, H.

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[CrossRef]

Lajunen, H.

Li, X.

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[CrossRef]

Lin, Q.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[CrossRef] [PubMed]

Lü, B.

C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009).
[CrossRef]

C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
[CrossRef]

Luo, B.

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[CrossRef]

Mao, H.

Nguyen, D. C.

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[CrossRef]

Pan, L.

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[CrossRef]

C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009).
[CrossRef]

Roggemann, M.

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011).
[CrossRef]

Tervo, J.

Vahimaa, P.

Voelz, D.

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011).
[CrossRef]

Wang, H.

Wang, L. G.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[CrossRef] [PubMed]

Wu, G.

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[CrossRef]

Yang, H.

Young, C. Y.

C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002).
[CrossRef]

Yu, S.

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[CrossRef]

Yuan, X.

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[CrossRef]

Zhao, D.

Zhao, Z.

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[CrossRef]

Zhu, S. Y.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[CrossRef] [PubMed]

Chin. Phys. Lett. (1)

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[CrossRef]

J. Opt. (1)

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009).
[CrossRef]

C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[CrossRef]

Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. 283(7), 1209–1212 (2010).
[CrossRef]

Opt. Eng. (1)

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

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

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[CrossRef] [PubMed]

Proc. SPIE (1)

C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002).
[CrossRef]

Other (3)

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007), Chap. 3.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 4 and 7.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chaps. 4 and 5.

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Figures (3)

Fig. 1
Fig. 1

Dependence relationships between |μ(ρ 1,ρ 2,L,ω,ω)| and the wavelength λ with different combinations of ρ 0 and σ 0.

Fig. 2
Fig. 2

|μ(ρ,ρ,L,ω 0,ω 1)| as a function of the wavelength separation δλ with ρ 0 = ∞, 3cm, 2cm and 1cm.

Fig. 3
Fig. 3

|μ(ρ 1,ρ 2,L,ω 0,ω 1)| in terms of the relative atmospheric spatial coherence length ρ 0/[w 0 2∙Δ(L,ω 0)]1/2.

Equations (24)

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Γ ( 0 ) ( r 1 , r 2 , t 1 , t 2 ) = Γ 0 exp [ R ( r 1 , r 2 ) T ( t 1 , t 2 ) ] ,
R ( r 1 , r 2 ) = r 1 2 + r 2 2 w 0 2 + | r 1 r 2 | 2 2 σ 0 2 ,
T ( t 1 , t 2 ) = t 1 2 + t 2 2 2 T 0 2 + ( t 1 t 2 ) 2 2 T c 2 + i ω 0 ( t 1 t 2 ) ,
W ( 0 ) ( r 1 , r 2 , ω 1 , ω 2 ) = W 0 exp [ R ( r 1 , r 2 ) F ( ω 1 , ω 2 ) ] ,
F ( ω 1 , ω 2 ) = ( ω 1 ω 0 ) 2 + ( ω 2 ω 0 ) 2 2 Ω 0 2 + ( ω 1 ω 2 ) 2 2 Ω c 2 ,
W ( ρ 1 , z 1 , ρ 2 , z 2 , ω 1 , ω 2 ) = ω 1 ω 2 4 π 2 z 1 z 2 c 2 exp [ i ( ω 2 z 2 ω 1 z 1 ) / c ] × d 2 r 1 d 2 r 2 W ( 0 ) ( r 1 , r 2 , ω 1 , ω 2 ) × exp [ i ω 2 | ρ 2 r 2 | 2 2 c z 2 i ω 1 | ρ 1 r 1 | 2 2 c z 1 ] × exp [ ψ * ( r 1 , ρ 1 , z 1 , ω 1 ) + ψ ( r 2 , ρ 2 , z 2 , ω 2 ) ] m ,
exp [ ψ * ( r 1 , ρ 1 , L , ω 1 ) + ψ ( r 2 , ρ 2 , L , ω 2 ) ] m Γ 2 exp [ | ρ 1 ρ 2 | 2 + ( ρ 1 ρ 2 ) ( r 1 r 2 ) + | r 1 r 2 | 2 ρ 0 2 ] ,
exp ( p 2 x 2 ± q x ) d x = exp ( q 2 4 p 2 ) π p ,           ( Re { p 2 } > 0 ) ,
W ( ρ 1 , ρ 2 , L , ω 1 , ω 2 ) = W 0 Γ 2 g 1 g 2 W b exp [ F ( ω 1 , ω 2 ) ] exp ( | ρ 1 ρ 2 | 2 ρ 0 2 ) × exp { 1 W b 2 [ g 1 2 g 2 2 R ( ρ 1 , ρ 2 ) + X ( ρ 1 , ρ 2 ) w 0 2 w c 2 + Q 1 ( ρ 1 , ρ 2 ) + Q 2 ( ρ 1 , ρ 2 ) ρ 0 2 ] } × exp { i W b 2 [ a g 1 g 2 ( g 1 g 2 ) R ( ρ 1 , ρ 2 ) u ( g 1 , g 2 ) Y ( ρ 1 , ρ 2 ) + Q 3 ( ρ 1 , ρ 2 ) ρ 0 2 ] } × exp [ i η ( g 1 , g 2 ) ] exp [ i ( ω 2 ω 1 ) L / c ] ,
a = 1 w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 ,
X ( ρ 1 , ρ 2 ) = g 1 2 ρ 1 2 + g 2 2 ρ 2 2 w 0 2 + | g 1 ρ 1 g 2 ρ 2 | 2 2 σ 0 2 ,
Y ( ρ 1 , ρ 2 ) = g 1 ρ 1 2 g 2 ρ 2 2 w 0 2 w c 2 ,
1 w c 2 = 1 w 0 2 + 1 σ 0 2 + 2 ρ 0 2 ,
W b 2 = u ( g 1 , g 1 ) u ( g 2 , g 2 ) + ( g 1 g 2 ) 2 4 ( 1 σ 0 2 + 2 ρ 0 2 ) 2 ,
u ( g 1 , g 2 ) = 1 w 0 2 w c 2 + g 1 g 2 ,
Q 1 ( ρ 1 , ρ 2 ) = g 1 g 2 2 ( 1 ρ 0 4 + 1 σ 0 2 w 0 2 ) ( ρ 1 2 + ρ 2 2 ) { ( g 1 + g 2 ) 2 w 0 2 ( 1 2 ρ 0 2 + 1 w 0 2 ) + 5 g 1 g 2 2 ρ 0 2 w 0 2 + ( g 1 2 + g 2 2 ) [ 1 2 σ 0 2 w 0 2 1 2 ρ 0 2 ( 1 2 σ 0 2 + 1 ρ 0 2 ) ] } ρ 1 ρ 2 + [ 2 g 1 2 g 2 2 + 1 ρ 0 2 ( 1 4 σ 0 2 + 1 w 0 2 ) g 1 g 2 1 2 w c 2 ρ 0 2 w 0 4 ] | ρ 1 ρ 2 | 2 ,
Q 2 ( ρ 1 , ρ 2 ) = g 1 2 ρ 2 2 + g 2 2 ρ 1 2 8 ρ 0 2 σ 0 2 + [ ( 22 w 0 2 1 σ 0 2 ) 1 8 ρ 0 2 1 w 0 4 ] ( g 1 2 ρ 1 2 + g 2 2 ρ 2 2 ) ( 1 ρ 0 2 + 1 w 0 2 ) | g 1 ρ 2 + g 2 ρ 1 | 2 4 ρ 0 2 | g 1 ρ 1 + g 2 ρ 2 | 2 4 ρ 0 4 + 3 X ( ρ 1 , ρ 2 ) w 0 2 ,
Q 3 ( ρ 1 , ρ 2 ) = [ 1 w c 2 ( g 1 2 g 2 g 1 g 2 2 g 1 w 0 4 ) + g 1 g 2 4 ρ 0 2 ( g 1 g 2 1 w 0 4 ) g 1 g 2 2 w 0 2 ] ρ 1 2 + [ 1 w c 2 ( g 1 2 g 2 g 1 g 2 2 + g 2 w 0 4 ) + g 1 g 2 4 ρ 0 2 ( g 1 g 2 1 w 0 4 ) + g 1 2 g 2 w 0 2 ] ρ 2 2 + [ 1 w c 2 ( 2 g 1 2 g 2 + 2 g 1 g 2 2 + g 1 g 2 w 0 4 ) g 1 g 2 2 ρ 0 2 ( g 1 g 2 1 w 0 4 ) g 1 g 2 ( g 1 g 2 ) w 0 2 ] ρ 1 ρ 2 ,
η ( g 1 , g 2 ) = arctan [ a ( g 1 g 2 ) / u ( g 1 , g 2 ) ] ,       ρ j = | ρ j | ,       g j = ω j 2 c L ,       ( j = 1 ,  2 ) .
S ( ρ , L , ω ) = W 0 Δ ( L , ω ) exp [ 2 ρ 2 w 0 2 Δ ( L , ω ) ] exp [ ( ω ω 0 ) 2 Ω 0 2 ] ,
Δ ( L , ω ) = 1 + ( 2 c L w 0 w c ω ) 2 = 1 + 4 c 2 L 2 w 0 2 ω 2 ( 1 w 0 2 + 1 σ 0 2 + 2 ρ 0 2 ) .
μ ( ρ 1 , ρ 2 , L , ω 1 , ω 2 ) = W ( ρ 1 , ρ 2 , L , ω 1 , ω 2 ) S ( ρ 1 , L , ω 1 ) S ( ρ 2 , L , ω 2 ) .
μ ( ρ 1 , ρ 2 , L , ω 1 , ω 2 ) = g 1 g 2 Γ 2 Δ ( L , ω 1 ) Δ ( L , ω 2 ) W b × exp ( | ρ 1 ρ 2 | 2 ρ 0 2 ) exp [ ( ω 1 ω 2 ) 2 2 Ω c 2 ] × exp { 1 W b 2 [ g 1 2 g 2 2 R ( ρ 1 , ρ 2 ) + X ( ρ 1 , ρ 2 ) w 0 2 w c 2 + Q 1 ( ρ 1 , ρ 2 ) + Q 2 ( ρ 1 , ρ 2 ) ρ 0 2 ] + V ( ρ 1 , ρ 2 ) w 0 2 } × exp { i W b 2 [ a g 1 g 2 ( g 1 g 2 ) R ( ρ 1 , ρ 2 ) u ( g 1 , g 2 ) Y ( ρ 1 , ρ 2 ) + ρ 0 2 Q 3 ( ρ 1 , ρ 2 ) ] } × exp [ i η ( g 1 , g 2 ) ] exp [ i ( ω 2 ω 1 ) L / c ] ,
V ( ρ 1 , ρ 2 ) = ρ 1 2 Δ ( L , ω 1 ) + ρ 2 2 Δ ( L , ω 2 ) .

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