Abstract

The intensity fluctuations of incoherent flat-topped Gaussian beams are evaluated when such sources are used in weakly turbulent horizontal atmospheric links. The formulation is developed for a detector having a response time much longer than the source coherence time. The flat-topped Gaussian profile is obtained by superposing many Gaussian beams, then the incoherence is introduced through delta correlation in space. The scintillation index of the incoherent flat-topped Gaussian beams is found to be smaller than the scintillation index of the corresponding incoherent Gaussian beams at the same link length, source size, and wavelength. When compared with the coherent counterparts, the intensity fluctuations of the incoherent flat-topped Gaussian beams are much smaller, yielding the same value only at the spherical wave limit, as expected. Transmitter aperture averaging is a special case of our solution.

© 2007 Optical Society of America

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References

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  1. F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
    [CrossRef]
  2. C. Palma and V. Bagini, "Expansions of general beams in Gaussian beams," Opt. Commun. 116, 1-7 (1995).
    [CrossRef]
  3. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, and M. Santarsiero, "Propagation of axially symmetric flattened Gaussian beams," J. Opt. Soc. Am. A 13, 1385-1394 (1996).
    [CrossRef]
  4. S. A. Amarande, "Beam propagation factor and the kurtosis parameter of flattened Gaussian beams," Opt. Commun. 129, 311-317 (1996).
    [CrossRef]
  5. M. Santarsiero and R. Borghi, "Correspondence between super-Gaussian and flattened Gaussian beams," J. Opt. Soc. Am. A 16, 188-190 (1999).
    [CrossRef]
  6. B. Lü and S. Lou, "General propagation equation of flattened Gaussian beams," J. Opt. Soc. Am. A 17, 2001-2004 (2000).
    [CrossRef]
  7. A. A. Tovar, "Propagation of flat-topped multi-Gaussian laser beams," J. Opt. Soc. Am. A 18, 1897-1904 (2001).
    [CrossRef]
  8. Y. Li, "Light beams with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002).
    [CrossRef]
  9. Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
    [CrossRef]
  10. X. L. Jil and B. Lu, "Propagation of a flattened Gaussian beam through multi-apertured optical ABCD systems," Optik 114, 394-400 (2003).
    [CrossRef]
  11. Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A: Pure Appl. Opt. 6, 390-395 (2004).
    [CrossRef]
  12. N. Zhou, G. Zeng, and L. Hu, "Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems," Opt. Commun. 240, 299-306 (2004).
    [CrossRef]
  13. M. Shen and S. Wang, "Decentered elliptical flattened Gaussian beam," Opt. Commun. 240, 245-252 (2004).
    [CrossRef]
  14. D. Ge, Y. Cai, and Q. Lin, "Partially coherent flat-topped beam and its propagation," Appl. Opt. 43, 4732-4738 (2004).
    [CrossRef] [PubMed]
  15. Y. Cai and Q. Lin, "A partially coherent elliptical flattened Gaussian beam and its propagation," J. Opt. A: Pure and Appl. Opt. 6, 1061-1066 (2004).
    [CrossRef]
  16. H. T. Eyyuboǧlu, Ç. Arpali, and Y. Baykal, "Flat topped beams and their characteristics in turbulent media," Opt. Express 14, 4196-4207 (2006).
    [CrossRef] [PubMed]
  17. Y. Cai, "Propagation of various flat-top beams in a turbulent atmosphere," J. Opt. A: Pure and Appl. Opt. 8, 537-545 (2006).
    [CrossRef]
  18. Y. Baykal and H. T. Eyyuboǧlu, "Scintillation index of flat-topped-Gaussian beams," Appl. Opt. 45, 3793-3797 (2006).
    [CrossRef] [PubMed]
  19. D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B-1-62150B-10 (2006).
  20. Y. Baykal, "Formulation of correlations for general-type beams in atmospheric turbulence," J. Opt. Soc. Am. A 23, 889-893 (2006).
    [CrossRef]
  21. Ç. Arpali, C. Yazicioǧlu, H. T. Eyyuboǧlu, S. A. Arpali, and Y. Baykal, "Simulator for general-type beam propagation in turbulent atmosphere," Opt. Express 14, 8918-8928 (2006).
    [CrossRef] [PubMed]
  22. J. Zhang and Y. Li, "Atmospherically turbulent effects on partially coherent flat-topped Gaussian beam," in Optical Technologies for Atmospheric, and Environmental Studies, D. Lu and G. G. Matvienko, eds., Proc. SPIE 5832, 48-55 (2005).
    [CrossRef]
  23. R. L. Fante, "Intensity fluctuations of an optical wave in a turbulent medium. Effect of source coherence," Optica Acta 28, 1203-1207 (1981).
    [CrossRef]
  24. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).
  25. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, 2000).
  26. Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
    [CrossRef]
  27. O. Korotkova, "Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation," J. Opt. A: Pure Appl. Opt. 8, 30-37 (2006).
    [CrossRef]

2006 (7)

H. T. Eyyuboǧlu, Ç. Arpali, and Y. Baykal, "Flat topped beams and their characteristics in turbulent media," Opt. Express 14, 4196-4207 (2006).
[CrossRef] [PubMed]

Y. Cai, "Propagation of various flat-top beams in a turbulent atmosphere," J. Opt. A: Pure and Appl. Opt. 8, 537-545 (2006).
[CrossRef]

Y. Baykal and H. T. Eyyuboǧlu, "Scintillation index of flat-topped-Gaussian beams," Appl. Opt. 45, 3793-3797 (2006).
[CrossRef] [PubMed]

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B-1-62150B-10 (2006).

Y. Baykal, "Formulation of correlations for general-type beams in atmospheric turbulence," J. Opt. Soc. Am. A 23, 889-893 (2006).
[CrossRef]

Ç. Arpali, C. Yazicioǧlu, H. T. Eyyuboǧlu, S. A. Arpali, and Y. Baykal, "Simulator for general-type beam propagation in turbulent atmosphere," Opt. Express 14, 8918-8928 (2006).
[CrossRef] [PubMed]

O. Korotkova, "Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation," J. Opt. A: Pure Appl. Opt. 8, 30-37 (2006).
[CrossRef]

2005 (1)

J. Zhang and Y. Li, "Atmospherically turbulent effects on partially coherent flat-topped Gaussian beam," in Optical Technologies for Atmospheric, and Environmental Studies, D. Lu and G. G. Matvienko, eds., Proc. SPIE 5832, 48-55 (2005).
[CrossRef]

2004 (5)

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A: Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

N. Zhou, G. Zeng, and L. Hu, "Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems," Opt. Commun. 240, 299-306 (2004).
[CrossRef]

M. Shen and S. Wang, "Decentered elliptical flattened Gaussian beam," Opt. Commun. 240, 245-252 (2004).
[CrossRef]

D. Ge, Y. Cai, and Q. Lin, "Partially coherent flat-topped beam and its propagation," Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, "A partially coherent elliptical flattened Gaussian beam and its propagation," J. Opt. A: Pure and Appl. Opt. 6, 1061-1066 (2004).
[CrossRef]

2003 (1)

X. L. Jil and B. Lu, "Propagation of a flattened Gaussian beam through multi-apertured optical ABCD systems," Optik 114, 394-400 (2003).
[CrossRef]

2002 (2)

Y. Li, "Light beams with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002).
[CrossRef]

Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
[CrossRef]

2001 (1)

2000 (2)

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, 2000).

B. Lü and S. Lou, "General propagation equation of flattened Gaussian beams," J. Opt. Soc. Am. A 17, 2001-2004 (2000).
[CrossRef]

1999 (1)

1997 (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).

1996 (2)

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, and M. Santarsiero, "Propagation of axially symmetric flattened Gaussian beams," J. Opt. Soc. Am. A 13, 1385-1394 (1996).
[CrossRef]

S. A. Amarande, "Beam propagation factor and the kurtosis parameter of flattened Gaussian beams," Opt. Commun. 129, 311-317 (1996).
[CrossRef]

1995 (1)

C. Palma and V. Bagini, "Expansions of general beams in Gaussian beams," Opt. Commun. 116, 1-7 (1995).
[CrossRef]

1994 (1)

F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
[CrossRef]

1983 (1)

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
[CrossRef]

1981 (1)

R. L. Fante, "Intensity fluctuations of an optical wave in a turbulent medium. Effect of source coherence," Optica Acta 28, 1203-1207 (1981).
[CrossRef]

Amarande, S. A.

S. A. Amarande, "Beam propagation factor and the kurtosis parameter of flattened Gaussian beams," Opt. Commun. 129, 311-317 (1996).
[CrossRef]

Andrews, L. C.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B-1-62150B-10 (2006).

Arpali, Ç.

Arpali, S. A.

Bagini, V.

Baykal, Y.

Borghi, R.

Cai, Y.

Y. Cai, "Propagation of various flat-top beams in a turbulent atmosphere," J. Opt. A: Pure and Appl. Opt. 8, 537-545 (2006).
[CrossRef]

Y. Cai and Q. Lin, "A partially coherent elliptical flattened Gaussian beam and its propagation," J. Opt. A: Pure and Appl. Opt. 6, 1061-1066 (2004).
[CrossRef]

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A: Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

D. Ge, Y. Cai, and Q. Lin, "Partially coherent flat-topped beam and its propagation," Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

Cowan, D. C.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B-1-62150B-10 (2006).

Eyyuboglu, H. T.

Fante, R. L.

R. L. Fante, "Intensity fluctuations of an optical wave in a turbulent medium. Effect of source coherence," Optica Acta 28, 1203-1207 (1981).
[CrossRef]

Ge, D.

Gori, F.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, 2000).

Hu, L.

N. Zhou, G. Zeng, and L. Hu, "Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems," Opt. Commun. 240, 299-306 (2004).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).

Jil, X. L.

X. L. Jil and B. Lu, "Propagation of a flattened Gaussian beam through multi-apertured optical ABCD systems," Optik 114, 394-400 (2003).
[CrossRef]

Korotkova, O.

O. Korotkova, "Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation," J. Opt. A: Pure Appl. Opt. 8, 30-37 (2006).
[CrossRef]

Li, Y.

J. Zhang and Y. Li, "Atmospherically turbulent effects on partially coherent flat-topped Gaussian beam," in Optical Technologies for Atmospheric, and Environmental Studies, D. Lu and G. G. Matvienko, eds., Proc. SPIE 5832, 48-55 (2005).
[CrossRef]

Y. Li, "Light beams with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002).
[CrossRef]

Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
[CrossRef]

Lin, Q.

D. Ge, Y. Cai, and Q. Lin, "Partially coherent flat-topped beam and its propagation," Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A: Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

Y. Cai and Q. Lin, "A partially coherent elliptical flattened Gaussian beam and its propagation," J. Opt. A: Pure and Appl. Opt. 6, 1061-1066 (2004).
[CrossRef]

Lou, S.

Lu, B.

X. L. Jil and B. Lu, "Propagation of a flattened Gaussian beam through multi-apertured optical ABCD systems," Optik 114, 394-400 (2003).
[CrossRef]

Lü, B.

Pacileo, A. M.

Palma, C.

C. Palma and V. Bagini, "Expansions of general beams in Gaussian beams," Opt. Commun. 116, 1-7 (1995).
[CrossRef]

Plonus, M. A.

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
[CrossRef]

Recolons, J.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B-1-62150B-10 (2006).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, 2000).

Santarsiero, M.

Shen, M.

M. Shen and S. Wang, "Decentered elliptical flattened Gaussian beam," Opt. Commun. 240, 245-252 (2004).
[CrossRef]

Tovar, A. A.

Wang, S.

M. Shen and S. Wang, "Decentered elliptical flattened Gaussian beam," Opt. Commun. 240, 245-252 (2004).
[CrossRef]

Wang, S. J.

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
[CrossRef]

Yazicioglu, C.

Young, C. Y.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B-1-62150B-10 (2006).

Zeng, G.

N. Zhou, G. Zeng, and L. Hu, "Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems," Opt. Commun. 240, 299-306 (2004).
[CrossRef]

Zhang, J.

J. Zhang and Y. Li, "Atmospherically turbulent effects on partially coherent flat-topped Gaussian beam," in Optical Technologies for Atmospheric, and Environmental Studies, D. Lu and G. G. Matvienko, eds., Proc. SPIE 5832, 48-55 (2005).
[CrossRef]

Zhou, N.

N. Zhou, G. Zeng, and L. Hu, "Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems," Opt. Commun. 240, 299-306 (2004).
[CrossRef]

Appl. Opt. (2)

J. Opt. A: Pure and Appl. Opt. (2)

Y. Cai and Q. Lin, "A partially coherent elliptical flattened Gaussian beam and its propagation," J. Opt. A: Pure and Appl. Opt. 6, 1061-1066 (2004).
[CrossRef]

Y. Cai, "Propagation of various flat-top beams in a turbulent atmosphere," J. Opt. A: Pure and Appl. Opt. 8, 537-545 (2006).
[CrossRef]

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

O. Korotkova, "Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation," J. Opt. A: Pure Appl. Opt. 8, 30-37 (2006).
[CrossRef]

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A: Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

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

Opt. Commun. (6)

S. A. Amarande, "Beam propagation factor and the kurtosis parameter of flattened Gaussian beams," Opt. Commun. 129, 311-317 (1996).
[CrossRef]

F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
[CrossRef]

C. Palma and V. Bagini, "Expansions of general beams in Gaussian beams," Opt. Commun. 116, 1-7 (1995).
[CrossRef]

N. Zhou, G. Zeng, and L. Hu, "Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems," Opt. Commun. 240, 299-306 (2004).
[CrossRef]

M. Shen and S. Wang, "Decentered elliptical flattened Gaussian beam," Opt. Commun. 240, 245-252 (2004).
[CrossRef]

Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Optica Acta (1)

R. L. Fante, "Intensity fluctuations of an optical wave in a turbulent medium. Effect of source coherence," Optica Acta 28, 1203-1207 (1981).
[CrossRef]

Optik (1)

X. L. Jil and B. Lu, "Propagation of a flattened Gaussian beam through multi-apertured optical ABCD systems," Optik 114, 394-400 (2003).
[CrossRef]

Proc. SPIE (2)

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B-1-62150B-10 (2006).

J. Zhang and Y. Li, "Atmospherically turbulent effects on partially coherent flat-topped Gaussian beam," in Optical Technologies for Atmospheric, and Environmental Studies, D. Lu and G. G. Matvienko, eds., Proc. SPIE 5832, 48-55 (2005).
[CrossRef]

Radio Sci. (1)

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
[CrossRef]

Other (2)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, 2000).

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Figures (8)

Fig. 1
Fig. 1

Fig. 1a.The source field distribution of flat-topped beam for N = 1 and 15.

Fig. 2
Fig. 2

Fig. 1b.The normalized receiver plane intensity level of flat-topped beam at N = 1 and 15 versus the link length.

Fig. 3
Fig. 3

Fig. 1c.The scintillation index of incoherent flat-topped Gaussian beam versus the link length.

Fig. 4
Fig. 4

The scintillation index of incoherent flat-topped Gaussian beams versus the source size.

Fig. 5
Fig. 5

The scintillation index of incoherent flat-topped Gaussian beams versus N.

Fig. 6
Fig. 6

The scintillation index of incoherent flat-topped Gaussian beams with different source sizes versus the wavelength.

Fig. 7
Fig. 7

The scintillation index of incoherent Gaussian beams with different source sizes versus the wavelength.

Fig. 8
Fig. 8

Comparison of the intensity fluctuations of coherent and incoherent flat-topped Gaussian beams.

Equations (23)

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u d ( s ) = n = 1 N ( 1 ) n 1 N ( N n ) exp ( n | s | 2 2 α s 2 ) .
u ( s ) = u d ( s ) u r ( s ) ,
u ( p ) = exp ( i k L ) λ i L d 2 s u ( s ) × exp [ i k 2 L | p s | 2 + ψ ( s , p ) ] .
I ( p , L ) = u ( p ) u * ( p ) = 1 ( λ L ) 2 d 2 s 1 d 2 s 2 Γ 2 s ( s 1 , s 2 ) × exp { i k 2 L [ | p s 1 | 2 | p s 2 | 2 ] } × exp [ ψ ( s 1 , p ) + ψ * ( s 2 , p ) ] ,
Γ 2 s ( s 1 , s 2 ) = u ( s 1 ) u * ( s 2 ) s = u d ( s 1 ) u d * ( s 2 ) u r ( s 1 ) u r * ( s 2 ) s = u d ( s 1 ) u d * ( s 2 ) u r ( s 1 ) u r * ( s 2 ) s = u d ( s 1 ) u d * ( s 2 ) exp [ i θ ( s 1 s 2 ) ] s ,
Γ 2 s ( s 1 , s 2 ) = λ 2 I [ ( s 1 + s 2 ) / 2 ] δ ( s 1 s 2 ) ,
Γ 2 s ( s 1 , s 2 ) = λ 2 [ n = 1 N ( 1 ) n 1 N ( N n ) exp ( n | s 1 + s 2 2 | 2 2 α s 2 ) ] × [ n = 1 N ( 1 ) n 1 N ( N n ) exp ( n | s 1 + s 2 2 | 2 2 α s 2 ) ] × δ ( s 1 s 2 ) .
I ( p , L ) = u ( p ) u * ( p ) ,
I ( p ) = 1 ( λ L ) 2 d 2 s 1 d 2 s 2 Γ 2 s ( s 1 , s 2 ) × exp { i k 2 L [ | p s 1 | 2 | p s 2 | 2 ] } × exp [ ψ ( s 1 , p ) + ψ * ( s 2 , p ) ] .
I ( p , L ) = 1 L 2 d 2 s 1 d 2 s 2 [ n = 1 N ( 1 ) n 1 N ( N n ) × exp ( n | s 1 + s 2 | 2 8 α s 2 ) ] [ n = 1 N ( 1 ) n 1 N ( N n ) × exp ( n | s 1 + s 2 | 2 8 α s 2 ) ] δ ( s 1 s 2 ) × exp { i k 2 L [ | p s 1 | 2 | p s 2 | 2 ] } × exp [ ρ 0 2 | s 1 s 2 | 2 ] ,
I ( p , L ) = 1 ( L N ) 2 n = 1 N n = 1 N ( 1 ) n + n ( N n ) ( N n ) × d 2 s 2 exp [ ( n + n ) | s 2 | 2 2 α s 2 ] .
I ( p , L ) = 2 π α s 2 ( L N ) 2 n = 1 N n = 1 N ( 1 ) n + n ( n + n ) ( N n ) ( N n ) .
I 2 ( p , L ) = 1 ( λ L ) 4 d 2 s 1 d 2 s 2 × d 2 s 3 d 2 s 4 Γ 2 s ( s 1 , s 2 ) Γ 2 s ( s 3 , s 4 ) × exp { i k 2 L [ | p s 1 | 2 | p s 2 | 2 + | p s 3 | 2 | p s 4 | 2 ] } × exp [ ψ ( s 1 , p ) + ψ * ( s 2 , p ) + ψ ( s 3 , p ) + ψ * ( s 4 , p ) ] ,
I 2 ( p , L ) = 1 ( L N ) 4 n = 1 N n = 1 N m = 1 N m = 1 N ( 1 ) n + n + m + m ( N n ) × ( N n ) ( N m ) ( N m ) d 2 s 2 d 2 s 4 × exp [ ( n + n ) | s 2 | 2 + ( m + m ) | s 4 | 2 2 α s 2 ] × exp [ ψ ( s 2 , p ) + ψ * ( s 2 , p ) + ψ ( s 4 , p ) + ψ * ( s 4 , p ) ] .
I 2 ( p , L ) = 1 ( L N ) 4 n = 1 N n = 1 N m = 1 N m = 1 N ( 1 ) n + n + m + m ( N n ) × ( N n ) ( N m ) ( N m ) d 2 s 2 d 2 s 4 × exp [ ( n + n ) | s 2 | 2 2 α s 2 ( m + m ) | s 4 | 2 2 α s 2 ] × exp [ 2 χ ( s 2 , p ) + 2 χ ( s 4 , p ) ] .
I 2 ( p , L ) = 1 ( L N ) 4 n = 1 N n = 1 N m = 1 N m = 1 N ( 1 ) n + n + m + m ( N n ) × ( N n ) ( N m ) ( N m ) d 2 s 2 d 2 s 4 × exp [ ( n + n ) | s 2 | 2 2 α s 2 ( m + m ) | s 4 | 2 2 α s 2 ] × exp [ 2 χ ( s 2 , p ) + 2 χ ( s 4 , p ) ] .
I 2 ( p , L ) I ( p , L ) 2 + 4 π 2 σ χ 2 ( L N ) 4 n = 1 N n = 1 N m = 1 N m = 1 N ( 1 ) n + n + m + m × ( N n ) ( N n ) ( N m ) ( N m ) × 4 α s 4 ( n + n ) ( m + m ) + 2 α s 2 ρ 0 2 ( m + m + n + n ) ,
exp [ 2 χ ( s 2 , p ) ] exp [ 2 χ ( s 4 , p ) ] 1 + 4 B χ ( s 2 , s 4 ) .
B χ ( s 1 , s 2 ) = [ χ ( s 1 , 0 ) χ ( s 1 , 0 ) ] × [ χ ( s 2 , 0 ) χ ( s 2 , 0 ) ] .
B χ ( s 1 , s 2 ) = σ χ 2 exp ( | s 1 s 2 | 2 ρ 0 2 ) ,
σ I 2 = I 2 ( p , L ) I ( p , L ) 2 I ( p , L ) 2 .
σ I 2 = 4 σ χ 2 Z ( ρ 0 ) Z ( ρ 0 = ) ,
Z ( ρ 0 ) = n = 1 N n = 1 N m = 1 N m = 1 N ( 1 ) r ( N n ) ( N n ) ( N m ) ( N m ) × 1 ( n + n ) ( m + m ) + 2 α s 2 ρ 0 2 r ,

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