Abstract

On the basis of the extended Huygens–Fresnel principle, the scattering of partially coherent Gaussian–Schell-model (GSM) beams from a diffuse target in slant double-passage atmospheric turbulence is studied and compared with that of fully coherent Gaussian beams. Using the cross-spectral density function of the GSM beams, we derive the expressions of the mutual coherence function, angle-of-arrival fluctuation, and covariance and variance of the intensity of the scattered field, taking into account the fluctuations of both the log-amplitude and phase. The numerical results are presented, and the influences of the wavelength, propagation distance, and waist radius on scattering properties are discussed. The perturbation region of the normalized intensity variance of the partially coherent GSM beam is smaller than that of the fully coherent Gaussian beam at the middle turbulence level. The normalized intensity variance of long-distance beam propagation is smaller than that of beam propagation along a short distance.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. E. Bennett, “Specular reflectance of aluminized ground glass and the height distribution of surface irregularities,” J. Opt. Soc. Am. 53, 1389–1393 (1963).
    [CrossRef]
  2. J. O. Porfens, “Relation between the height distribution of a rough surface and the reflection at normal incidence,” J. Opt. Soc. Am. 53, 1394–1399 (1963).
    [CrossRef]
  3. L. I. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle patterns,” J. Opt. Soc. Am. 55, 247–253 (1965).
    [CrossRef]
  4. D. P. Greenwood, “The scattering from curved rough surfaces of an EM wave transmitted through a turbulent medium,” IEEE Trans. Antennas Propag. 20, 19–24 (1972).
    [CrossRef]
  5. R. A. Sparague, “Surface roughness measurement using white light speckle,” Appl. Opt. 11, 2811–2817 (1972).
    [CrossRef]
  6. C. N. Kurtz, “Transmitted characteristics of surface and the design of nearly band-limited binary diffusers,” J. Opt. Soc. Am. 62, 982–989 (1972).
    [CrossRef]
  7. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972).
    [CrossRef] [PubMed]
  8. S. F. Clifford and H. T. Yura, “Equivalence of two theories of strong optical scintillation,” J. Opt. Soc. Am. 64, 1641–1644(1974).
    [CrossRef]
  9. M. G. Miller, A. M. Schneiderman, and P. F. Kellen, “Second-order statistics of laser speckle pattern,” J. Opt. Soc. Am. 65, 779–785 (1975).
    [CrossRef]
  10. P. H. Deitz, “Image information by means of speckle-pattern processing,” J. Opt. Soc. Am. 65, 279–285 (1975).
    [CrossRef]
  11. D. E. Estes and R. Boucher, “Temporal and spatial intensity-interferometer imaging through a random medium,” J. Opt. Soc. Am. 65, 760–768 (1975).
    [CrossRef]
  12. M. J. Lahart and A. S. Maratay, “Image speckle patterns of weak diffusers,” J. Opt. Soc. Am. 65, 769–778 (1975).
    [CrossRef]
  13. J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
    [CrossRef]
  14. M. H. Lee, J. F. Holmes, and J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 66, 1164–1172 (1976).
    [CrossRef]
  15. R. Fante, “Multiple-frequency mutual coherence functions for a beam in a random medium,” IEEE Trans. Antennas Propag. 26, 621–623 (1978).
    [CrossRef]
  16. J. F. Holmes, H. L. Myung, and J. R. Kerr, “Effect of the log-amplitude covariance function on the statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 70, 355–360 (1980).
    [CrossRef]
  17. C. M. McIntyre, J. R. Kerr, and M. H. Lee, “Enhanced variance of irradiance from target glint,” Appl. Opt. 18, 3211–3212(1979).
    [CrossRef] [PubMed]
  18. Yu. A. Kravtsov and A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
    [CrossRef]
  19. J. Wu and A. D. Boardman, “Coherence length of a Gauss–Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
    [CrossRef]
  20. O. Korotrova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109(2002).
    [CrossRef]
  21. 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]
  22. W. H. Liu and J. Wu, “Effect of multiple Gauss–Schell beams through strong turbulence on log-intensity variance,” J. Appl. Opt. 26, 25–28 (2005) (in Chinese).
  23. L. C. Andrews, R. L. Phillips, and W. B. Miller, “Mutual coherence function for a double-passage retroreflected optical wave in atmospheric turbulence,” Appl. Opt. 36, 698–670 (1997).
    [CrossRef] [PubMed]
  24. Ya. A. Ilyushin, “Impact of the plasma fluctuations in the Martian ionosphere on the performance of the synthetic aperture ground-penetrating radar,” Planet. Space Sci. 57, 1458–1466 (2009).
    [CrossRef]
  25. H. Y. Wei, Z. S. Wu, and H. Peng, “Scattering from a diffuse target in the slant atmospheric turbulence,” Acta. Phys. Sin. 57, 7666–7672 (2008) (in Chinese).
  26. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. 19, 1794–1802 (2002).
    [CrossRef]
  27. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658(1971).
    [CrossRef] [PubMed]
  28. H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta. 22, 523–535 (1975).
    [CrossRef]
  29. ITU-R, Document3J/31-E, “On propagation data and prediction methods required for the design of space-to-Earth and Earth-to-space optical communication systems,” in Proceedings of the Radio Communication Study Group Meeting (International Telecommunication Union, 2001), pp. 618–619.
  30. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  31. R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1524 (1970).
    [CrossRef]
  32. H. Y. Wei and Z. S. Wu, “Study on the effect of laser beam propagation on the slant path through atmospheric turbulence,” J. Electromagn. Waves Appl. 22, 787–802 (2008).
    [CrossRef]

2009 (1)

Ya. A. Ilyushin, “Impact of the plasma fluctuations in the Martian ionosphere on the performance of the synthetic aperture ground-penetrating radar,” Planet. Space Sci. 57, 1458–1466 (2009).
[CrossRef]

2008 (2)

H. Y. Wei, Z. S. Wu, and H. Peng, “Scattering from a diffuse target in the slant atmospheric turbulence,” Acta. Phys. Sin. 57, 7666–7672 (2008) (in Chinese).

H. Y. Wei and Z. S. Wu, “Study on the effect of laser beam propagation on the slant path through atmospheric turbulence,” J. Electromagn. Waves Appl. 22, 787–802 (2008).
[CrossRef]

2005 (1)

W. H. Liu and J. Wu, “Effect of multiple Gauss–Schell beams through strong turbulence on log-intensity variance,” J. Appl. Opt. 26, 25–28 (2005) (in Chinese).

2002 (2)

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. 19, 1794–1802 (2002).
[CrossRef]

O. Korotrova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109(2002).
[CrossRef]

1997 (1)

1991 (1)

J. Wu and A. D. Boardman, “Coherence length of a Gauss–Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[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]

1982 (1)

Yu. A. Kravtsov and A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
[CrossRef]

1980 (1)

1979 (1)

1978 (1)

R. Fante, “Multiple-frequency mutual coherence functions for a beam in a random medium,” IEEE Trans. Antennas Propag. 26, 621–623 (1978).
[CrossRef]

1976 (1)

1975 (6)

1974 (1)

1972 (4)

1971 (1)

1970 (1)

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1524 (1970).
[CrossRef]

1965 (1)

1963 (2)

Andrews, L. C.

O. Korotrova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109(2002).
[CrossRef]

L. C. Andrews, R. L. Phillips, and W. B. Miller, “Mutual coherence function for a double-passage retroreflected optical wave in atmospheric turbulence,” Appl. Opt. 36, 698–670 (1997).
[CrossRef] [PubMed]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Baykal, Y.

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]

Bennett, H. E.

Boardman, A. D.

J. Wu and A. D. Boardman, “Coherence length of a Gauss–Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[CrossRef]

Boucher, R.

Clifford, S. F.

Davidson, F. M.

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. 19, 1794–1802 (2002).
[CrossRef]

Deitz, P. H.

Estes, D. E.

Fante, R.

R. Fante, “Multiple-frequency mutual coherence functions for a beam in a random medium,” IEEE Trans. Antennas Propag. 26, 621–623 (1978).
[CrossRef]

Goldfischer, L. I.

Goodman, J. W.

J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[CrossRef]

Greenwood, D. P.

D. P. Greenwood, “The scattering from curved rough surfaces of an EM wave transmitted through a turbulent medium,” IEEE Trans. Antennas Propag. 20, 19–24 (1972).
[CrossRef]

Holmes, J. F.

Ilyushin, Ya. A.

Ya. A. Ilyushin, “Impact of the plasma fluctuations in the Martian ionosphere on the performance of the synthetic aperture ground-penetrating radar,” Planet. Space Sci. 57, 1458–1466 (2009).
[CrossRef]

Kellen, P. F.

Kerr, J. R.

Korotrova, O.

O. Korotrova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109(2002).
[CrossRef]

Kravtsov, Yu. A.

Yu. A. Kravtsov and A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
[CrossRef]

Kurtz, C. N.

Lahart, M. J.

Lawrence, R. S.

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1524 (1970).
[CrossRef]

Lee, M. H.

Liu, W. H.

W. H. Liu and J. Wu, “Effect of multiple Gauss–Schell beams through strong turbulence on log-intensity variance,” J. Appl. Opt. 26, 25–28 (2005) (in Chinese).

Lutomirski, R. F.

Maratay, A. S.

McIntyre, C. M.

Miller, M. G.

Miller, W. B.

Myung, H. L.

Pedersen, H. M.

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta. 22, 523–535 (1975).
[CrossRef]

Peng, H.

H. Y. Wei, Z. S. Wu, and H. Peng, “Scattering from a diffuse target in the slant atmospheric turbulence,” Acta. Phys. Sin. 57, 7666–7672 (2008) (in Chinese).

Phillips, R. L.

O. Korotrova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109(2002).
[CrossRef]

L. C. Andrews, R. L. Phillips, and W. B. Miller, “Mutual coherence function for a double-passage retroreflected optical wave in atmospheric turbulence,” Appl. Opt. 36, 698–670 (1997).
[CrossRef] [PubMed]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

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]

Porfens, J. O.

Ricklin, J. C.

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. 19, 1794–1802 (2002).
[CrossRef]

Saichev, A. I.

Yu. A. Kravtsov and A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
[CrossRef]

Schneiderman, A. M.

Sparague, R. A.

Strohbehn, J. W.

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1524 (1970).
[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]

Wei, H. Y.

H. Y. Wei and Z. S. Wu, “Study on the effect of laser beam propagation on the slant path through atmospheric turbulence,” J. Electromagn. Waves Appl. 22, 787–802 (2008).
[CrossRef]

H. Y. Wei, Z. S. Wu, and H. Peng, “Scattering from a diffuse target in the slant atmospheric turbulence,” Acta. Phys. Sin. 57, 7666–7672 (2008) (in Chinese).

Wu, J.

W. H. Liu and J. Wu, “Effect of multiple Gauss–Schell beams through strong turbulence on log-intensity variance,” J. Appl. Opt. 26, 25–28 (2005) (in Chinese).

J. Wu and A. D. Boardman, “Coherence length of a Gauss–Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[CrossRef]

Wu, Z. S.

H. Y. Wei, Z. S. Wu, and H. Peng, “Scattering from a diffuse target in the slant atmospheric turbulence,” Acta. Phys. Sin. 57, 7666–7672 (2008) (in Chinese).

H. Y. Wei and Z. S. Wu, “Study on the effect of laser beam propagation on the slant path through atmospheric turbulence,” J. Electromagn. Waves Appl. 22, 787–802 (2008).
[CrossRef]

Yura, H. T.

Acta. Phys. Sin. (1)

H. Y. Wei, Z. S. Wu, and H. Peng, “Scattering from a diffuse target in the slant atmospheric turbulence,” Acta. Phys. Sin. 57, 7666–7672 (2008) (in Chinese).

Appl. Opt. (5)

IEEE Trans. Antennas Propag. (2)

D. P. Greenwood, “The scattering from curved rough surfaces of an EM wave transmitted through a turbulent medium,” IEEE Trans. Antennas Propag. 20, 19–24 (1972).
[CrossRef]

R. Fante, “Multiple-frequency mutual coherence functions for a beam in a random medium,” IEEE Trans. Antennas Propag. 26, 621–623 (1978).
[CrossRef]

J. Appl. Opt. (1)

W. H. Liu and J. Wu, “Effect of multiple Gauss–Schell beams through strong turbulence on log-intensity variance,” J. Appl. Opt. 26, 25–28 (2005) (in Chinese).

J. Electromagn. Waves Appl. (1)

H. Y. Wei and Z. S. Wu, “Study on the effect of laser beam propagation on the slant path through atmospheric turbulence,” J. Electromagn. Waves Appl. 22, 787–802 (2008).
[CrossRef]

J. Mod. Opt. (1)

J. Wu and A. D. Boardman, “Coherence length of a Gauss–Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[CrossRef]

J. Opt. Soc. Am. (12)

J. F. Holmes, H. L. Myung, and J. R. Kerr, “Effect of the log-amplitude covariance function on the statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 70, 355–360 (1980).
[CrossRef]

M. H. Lee, J. F. Holmes, and J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 66, 1164–1172 (1976).
[CrossRef]

S. F. Clifford and H. T. Yura, “Equivalence of two theories of strong optical scintillation,” J. Opt. Soc. Am. 64, 1641–1644(1974).
[CrossRef]

M. G. Miller, A. M. Schneiderman, and P. F. Kellen, “Second-order statistics of laser speckle pattern,” J. Opt. Soc. Am. 65, 779–785 (1975).
[CrossRef]

P. H. Deitz, “Image information by means of speckle-pattern processing,” J. Opt. Soc. Am. 65, 279–285 (1975).
[CrossRef]

D. E. Estes and R. Boucher, “Temporal and spatial intensity-interferometer imaging through a random medium,” J. Opt. Soc. Am. 65, 760–768 (1975).
[CrossRef]

M. J. Lahart and A. S. Maratay, “Image speckle patterns of weak diffusers,” J. Opt. Soc. Am. 65, 769–778 (1975).
[CrossRef]

C. N. Kurtz, “Transmitted characteristics of surface and the design of nearly band-limited binary diffusers,” J. Opt. Soc. Am. 62, 982–989 (1972).
[CrossRef]

H. E. Bennett, “Specular reflectance of aluminized ground glass and the height distribution of surface irregularities,” J. Opt. Soc. Am. 53, 1389–1393 (1963).
[CrossRef]

J. O. Porfens, “Relation between the height distribution of a rough surface and the reflection at normal incidence,” J. Opt. Soc. Am. 53, 1394–1399 (1963).
[CrossRef]

L. I. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle patterns,” J. Opt. Soc. Am. 55, 247–253 (1965).
[CrossRef]

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. 19, 1794–1802 (2002).
[CrossRef]

Opt. Acta. (1)

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta. 22, 523–535 (1975).
[CrossRef]

Opt. Commun. (1)

J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[CrossRef]

Planet. Space Sci. (1)

Ya. A. Ilyushin, “Impact of the plasma fluctuations in the Martian ionosphere on the performance of the synthetic aperture ground-penetrating radar,” Planet. Space Sci. 57, 1458–1466 (2009).
[CrossRef]

Proc. IEEE (1)

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1524 (1970).
[CrossRef]

Proc. SPIE (1)

O. Korotrova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109(2002).
[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]

Sov. Phys. Usp. (1)

Yu. A. Kravtsov and A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
[CrossRef]

Other (2)

ITU-R, Document3J/31-E, “On propagation data and prediction methods required for the design of space-to-Earth and Earth-to-space optical communication systems,” in Proceedings of the Radio Communication Study Group Meeting (International Telecommunication Union, 2001), pp. 618–619.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Geometry of the laser beam propagating in the slant double-passage atmospheric turbulence.

Fig. 2
Fig. 2

MCF of the collimated beams as a function of variable p for different waist radii.

Fig. 3
Fig. 3

MCF of the convergent beams as a function of variable p for different waist radii and wavelengths.

Fig. 4
Fig. 4

MCF of the divergent beams as a function of variable p for different waist radii.

Fig. 5
Fig. 5

Average intensity of the divergent, convergent, collimated beams at any receiving point as a function of distance L for different waist radii.

Fig. 6
Fig. 6

Angle-of-arrival fluctuation of the collimated and convergent partially coherent GSM and the fully coherent Gaussian beams as a function of variable p.

Fig. 7
Fig. 7

Normalized intensity variance of the partially coherent GSM beams and the fully coherent Gaussian beams as a function of the log-amplitude variance.

Equations (48)

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

u 0 ( r ) = u ¯ 0 exp [ r 2 W 0 2 i k r 2 2 F 0 ] ,
u ˜ 0 ( r ) = u 0 ( r ) exp [ i φ d ( r ) ] ,
u i ( ρ ) = k e i k L 2 π i L d r u ˜ 0 ( r ) exp [ i k | ρ r | 2 2 L + ψ 1 ( ρ , r ) ] ,
ψ 1 ( ρ , r ) = χ ( ρ , r ) + i S ( ρ , r ) .
u r ( p ) = k e i k L 2 π i L d ρ u s ( ρ ) exp [ i k | p ρ | 2 2 L + ψ 2 ( p , ρ ) ] ,
u s ( ρ ) = u i ( ρ ) T ( ρ ) ,
T d ( ρ 1 ) T d * ( ρ 2 ) = λ 2 R d π δ ( ρ 1 ρ 2 ) ,
T d ( ρ 1 ) T d * ( ρ 2 ) T d ( ρ 3 ) T d * ( ρ 4 ) = ( λ 2 R d π ) 2 [ δ ( ρ 1 ρ 2 ) δ ( ρ 3 ρ 4 ) + δ ( ρ 1 ρ 4 ) δ ( ρ 3 ρ 2 ) ] ,
u s ( ρ 1 ) u s * ( ρ 2 ) = λ 2 R d π I i ( ρ 2 ) δ ( ρ 1 ρ 2 ) ,
u s ( ρ 1 ) u s * ( ρ 2 ) u s ( ρ 3 ) u s * ( ρ 4 ) = ( λ 2 R d π ) 2 [ I i ( ρ 1 ) I i ( ρ 3 ) δ ( ρ 1 ρ 3 ) δ ( ρ 3 ρ 4 ) + I i ( ρ 4 ) I i ( ρ 2 ) δ ( ρ 1 ρ 4 ) δ ( ρ 3 ρ 2 ) ] ,
Γ ( p 1 , p 2 ) = u r ( p 1 ) u r * ( p 2 ) = ( k 2 π L ) 2 d ρ 1 d ρ 2 u s ( ρ 1 ) u s * ( ρ 2 ) × exp [ i k 2 L ( | p 1 ρ 1 | 2 | p 2 ρ 2 | 2 ) ] exp [ ψ 2 ( p 1 , ρ 1 ) + ψ 2 * ( p 2 , ρ 2 ) ] .
exp [ ψ ( p 1 , ρ 1 ) + ψ * ( p 2 , ρ 2 ) + + ψ ( p 2 n 1 , ρ 2 n 1 ) + ψ * ( p 2 n , ρ 2 n ) ] = exp [ 1 2 i = 1 2 n 1 j = i + 1 2 n ( 1 ) i + j + 1 D s ( i , j ) + 2 i = 1 2 n 2 j = 1 n C x ( i , 2 j + 1 ) ] ,
Γ ( p 1 , p 2 ) = ( k 2 π L ) 2 d ρ 1 d ρ 2 u s ( ρ 1 ) u s * ( ρ 2 ) × exp [ i k 2 L ( | p 1 ρ 1 | 2 | p 2 ρ 2 | 2 ) ] exp ( 1 2 D s 12 + 2 C χ 13 ) ,
C χ 13 = 0.132 π 2 L k 2 0 1 C n 2 ( t L ) d t 0 d u u 8 / 3 sin 2 ( u 2 t ( 1 t ) L k ) J 0 ( u | ( p 1 p 3 ) t + ( 1 t ) ( ρ 1 ρ 2 ) | ) ,
D S 12 = 2.92 k 2 L 0 1 d t C n 2 ( t L ) | ( 1 t ) ( ρ 1 ρ 2 ) | 5 / 3 = 2 ( ρ 1 ρ 2 ρ T ) 5 / 3 ,
Γ d ( p 1 , p 2 ) = u r ( p 1 ) u r * ( p 2 ) = R d π L 2 d ρ I i ( ρ ) exp { i k 2 L [ p 1 2 p 2 2 2 ρ · p ] ( p ρ T ) 5 / 3 + 2 C χ 13 } ,
W ( ρ , ρ ) = I i ( ρ ) = U i ( ρ ) U i * ( ρ ) = 1 ( λ L ) 2 d r 1 d r 2 W 0 ( r 1 , r 2 ) exp [ ψ ( ρ , r 1 ) + ψ * ( ρ , r 2 ) ] exp { i k 2 L [ ( ρ r 1 ) 2 ( ρ r 2 ) 2 ] } ,
W 0 ( r 1 , r 2 ) = u ˜ 0 ( r 1 ) u ˜ 0 * ( r 2 ) = u 0 ( r 1 ) u 0 * ( r 2 ) × exp [ i φ d 1 ( r 1 ) ] exp [ i φ d 2 ( r 2 ) ] = u 0 ( r 1 ) u 0 * ( r 2 ) exp [ ( r 1 r 2 ) 2 2 σ μ 2 ] .
r S = ( r 1 + r 2 ) / 2 , r = r 1 r 2 .
W 0 ( r S , r ) = | u ¯ 0 | 2 exp { 1 W 0 2 [ 1 2 ( r 2 + 4 r S 2 ) ] i k 2 F 0 ( 2 r · r S ) r 2 2 σ μ 2 } .
exp [ ψ ( ρ , r 1 ) + ψ * ( ρ , r 2 ) ] = exp ( 1 2 D S + 2 C χ 13 ) exp [ r 2 ρ T 2 + 2 C χ 13 ] ,
W ( ρ , ρ ) = I i ( ρ ) = | u ¯ 0 | 2 ( λ L ) 2 d r d r S exp ( 2 r S 2 W 0 2 ) exp [ i k r S · r F 0 + i k r S · r L ] × exp [ r 2 2 W 0 2 r 2 2 σ μ 2 r 2 ρ T 2 i k ρ · r L + 2 C χ 13 ] .
I i ( ρ ) = W 0 2 | u ¯ 0 | 2 W ζ 2 ( L ) exp [ 2 ρ 2 W ζ 2 ( L ) + 2 C χ 13 ] ,
Γ d ( p 1 , p 2 ) = 1 π L 2 W 0 2 | u ¯ 0 | 2 W ζ 2 ( L ) exp [ ( p ρ T ) 5 / 3 + 4 C χ 13 + i k ( p 1 2 p 2 2 ) 2 L ] × d ρ J 0 ( k L ρ p ) exp [ 2 ρ 2 W ζ 2 ( L ) ] .
I ( p ) = Γ d ( p , p ) = R d π L 2 W 0 2 | u ¯ 0 | 2 W ζ 2 ( L ) d ρ exp [ 2 ρ 2 W ζ 2 ( L ) ] .
exp [ 1 2 D ( p 1 , p 2 , L ) ] = | Γ ( p 1 , p 2 , L ) | [ Γ ( p 1 , p 1 , L ) Γ ( p 2 , p 2 , L ) ] 1 / 2 .
D ( p 1 , p 2 , L ) = [ 2 ( p / ρ T ) 5 / 3 8 C χ 13 i k p 2 / L ] 2 ln { d ρ J 0 ( k p ρ / L ) exp ( 2 ρ 2 / W ζ 2 ( L ) ) d ρ exp ( 2 ρ 2 / W ζ 2 ( L ) ) } .
α 2 = D ( p 1 , p 2 , L ) ( k w ζ ) 2 = [ 2 ( p / ρ T ) 5 / 3 8 C χ 13 i k p 2 / L ] / ( k w ζ ) 2 2 ( k w ζ ) 2 · ln { d ρ J 0 ( k p ρ / L ) exp ( 2 ρ 2 / W ζ 2 ( L ) ) d ρ exp ( 2 ρ 2 / W ζ 2 ( L ) ) } .
D ( p 1 , p 2 , L ) = { 2 [ 1 4 W 0 2 + ( W 0 k 2 L ) 2 ( 1 L F 0 ) 2 ] p 2 i k p 2 L + 4 ( p ρ T ) 5 / 3 8 C χ 13 } .
α 2 = D ( p 1 , p 2 , L ) ( k W e ) 2 = { 2 [ 1 4 W 0 2 + ( W 0 k 2 L ) 2 ( 1 L F 0 ) 2 ] p 2 i k p 2 L + 4 ( p ρ T ) 5 / 3 8 C χ 13 } / ( k W e ) 2 ,
C I ( p 1 , p 2 ) = I ( p 1 ) I ( p 2 ) I ( p 1 ) I ( p 2 ) ,
B I ( p 1 , p 2 ) = I ( p 1 ) I ( p 2 ) = u r ( p 1 ) u r * ( p 1 ) u r ( p 2 ) u r * ( p 2 ) = ( k 2 π L ) 4 d ρ 1 d ρ 2 d ρ 3 d ρ 4 u s ( ρ 1 ) u s * ( ρ 2 ) u s ( ρ 3 ) u s * ( ρ 4 ) exp { i k 2 L [ | p 1 ρ 1 | 2 | p 1 ρ 2 | 2 | p 2 ρ 3 | 2 | p 2 ρ 4 | 2 ] } × exp [ ψ 2 ( p 1 , ρ 1 ) + ψ 2 ( p 1 , ρ 2 ) + ψ 2 ( p 2 , ρ 3 ) + ψ 2 ( p 2 , ρ 4 ) ] ,
H ( p 1 , p 2 ; ρ 1 , ρ 2 , ρ 3 , ρ 4 ) = exp [ ψ 2 ( p 1 , ρ 1 ) + ψ 2 * ( p 1 , ρ 2 ) + ψ 2 ( p 2 , ρ 3 ) + ψ 2 * ( p 2 , ρ 4 ) ] = exp { 1 2 [ D s 12 D s 13 + D s 14 + D s 23 D s 24 + D s 34 ] + 2 C χ 13 + 2 C χ 24 } .
B I ( p 1 , p 2 ) = I ( p 1 ) I ( p 2 ) = ( R d π L 2 ) 2 d ρ 1 d ρ 2 I i ( ρ 1 ) I i ( ρ 2 ) H 1 + ( R d π L 2 ) 2 d ρ 1 d ρ 2 I i ( ρ 1 ) I i ( ρ 2 ) exp ( i k ρ · p L ) H 2 ,
H 1 = H | ρ 1 = ρ 2 ; ρ 3 = ρ 4 ; p 1 = p 2 = exp [ 4 C χ ( p , ρ ) ] ,
H 2 = exp { ρ T 5 / 3 [ 2 p 5 / 3 + 2 ρ 5 / 3 8 3 0 1 d t | p t + ( 1 t ) ρ | 5 / 3 8 3 0 1 d t | p t ( 1 t ) ρ | 5 / 3 ] + 2 C χ ( p , ρ ) + 2 C χ ( p , ρ ) } ,
C I ( p 1 , p 2 ) = C I 1 ( p 1 , p 2 ) + C I 2 ( p 1 , p 2 ) ,
C I 1 = ( R d π L 2 ) 2 d ρ 1 d ρ 2 I i ( ρ 1 ) I i ( ρ 2 ) H 1 I ( p 1 ) I ( p 2 ) ,
C I 2 = ( R d π L 2 ) 2 d ρ 1 d ρ 2 I i ( ρ 1 ) I i ( ρ 2 ) exp ( i k ρ · p L ) H 2 .
C I 2 = ( R d π L 2 · w 0 2 | u ¯ 0 | 2 w ζ 2 ( L ) ) 2 exp [ 4 C χ ( p , ρ ) ] d ρ d r exp ( ρ 2 + 4 r 2 w ζ 2 ( L ) + i k ρ · p L ) · H 2 ( p , ρ ) ,
C I 1 = I 2 · { exp [ 8 C χ ( p , ρ ) ] 1 } .
C I ( p ) = I 2 · { exp [ 8 C χ ( p , ρ ) ] 1 } + ( R d π L 2 · w 0 2 | u ¯ 0 | 2 w ζ 2 ( L ) ) 2 exp [ 4 C χ ( p , ρ ) ] × d ρ d r exp ( ρ 2 + 4 r 2 w ζ 2 ( L ) + i k ρ · p L ) · H 2 ( p , ρ ) .
σ I 2 = I 2 · { 2 exp [ 8 C χ ( ρ , 0 ) ] 1 } .
σ I 2 = 2 exp [ 8 C χ ( ρ , 0 ) ] 1 .
C χ ( ρ , 0 ) = 2.95 σ T 2 0 1 d u ( u ( 1 u ) ) 5 / 6 0 d y sin 2 y y 11 / 6 × exp { σ T 2 [ ( u ( 1 u ) ) 5 / 6 ] f ( y ) } J 0 [ ( 4 π y u 1 u ) 1 / 2 ρ λ L ] ,
σ T 2 = 0.124 k 7 / 6 L 11 / 6 C n 2 < 0.3 ,
f ( y ) = 7.02 y 5 / 6 0.7 y d x · x 8 / 3 [ 1 J 0 ( x ) ] ,
σ χ 2 = 0.36 ( σ T 2 ) 2 / 5 + 0.42 α 5 / 3 .

Metrics