Abstract

The problem of wind profile reconstruction from scintillations of an optical wave scattered off a rough surface in a telescope focus plane is considered. Both the expression for the spatiotemporal correlation function and the algorithm of cross-wind velocity and direction profiles reconstruction based on the spatiotemporal spectrum of intensity of an optical wave scattered by a diffuse target in a turbulent atmosphere are presented. Computer simulations performed under conditions of weak optical turbulence show wind profiles reconstruction by the developed algorithm.

© 2007 Optical Society of America

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  1. T.-I. Wang, G. R. Ochs, and S. Lawrence, "Wind measurements by the temporal cross-correlation of the optical scintillations," Appl. Opt. 20, 4073-4081 (1981).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  7. R. Avila, E. Carrasco, F. Ibanez, J. Vernin, J.-L. Prieur, and D. X. Cruz, "Generalized SCIDAR measurements at San Pedro Martin. II Wind profile statistics," Publ. Astron. Soc. Pac. 118, 503-515 (2006).
    [CrossRef]
  8. B. Garcia-Lorenzo and J. J. Fuensalida, "Processing of turbulent layer wind speed with generalized SCIDAR through wavelet analysis," Mon. Not. R. Astron. Soc. 372, 1483-1495 (2006).
    [CrossRef]
  9. M. Yu and M. A. Vorontsov, "Remote sensing of atmospheric wind profiles using spatio-temporal intensity information and wavefront sensor information," Proc. SPIE 5998, 599808 (2005).
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    [PubMed]
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  12. J. F. Holmes, F. Amzajerdian, V. S. Rao Gudimetla, and J. M. Hunt, "Remote crosswind measurement using speckle-turbulence interaction and optical heterodyne detection," J. Opt. Soc. Am. A 2, P104 (1985).
  13. M. H. Lee, J. F. Holmes, and J. R. Kerr, "Statistics of speckle propagation through the turbulent atmosphere," J. Opt. Soc. Am. A 66, 1164-1172 (1976).
    [CrossRef]
  14. J. F. Holmes, M. H. Lee, 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. A 70, 355-360 (1980).
    [CrossRef]
  15. V. S. Rao Gudimetla and J. F. Holmes, "Two-point joint density function of the intensity for a laser speckle pattern after propagation through the turbulent atmosphere," J. Opt. Soc. Am. A 3, P118 (1986).
  16. V. A. Banakh, D. S. Rytchkov, V. V. Zhmylevski, and V. V. Morozov, "Mean power of the partially spatially coherent laser beam backscattered by the atmospheric layer," Atmos. Oceanic Opt. 20, 953-958 (2007).
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    [CrossRef] [PubMed]
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    [CrossRef]
  19. V. A. Banakh and V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, 1987).
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    [CrossRef]
  21. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (IPTS, 1971).
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  23. V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk . 54, 626-633 (1983) (in Russian).
  24. V. A. Banakh and V. M. Buldakov, "Impact of the initial spatial coherency of the light beam on the intensity fluctuations," Opt. Spektrosk. 55, 707-711 (1983) (in Russian).
  25. C. M. McIntyre, M. H. Lee, and J. H. Churnside, "Statistics of irradiance scattered from a diffuse target containing multiple glints," J. Opt. Soc. Am. A 70, 1084-1095 (1980).
    [CrossRef]
  26. H. A. Panofsky and J. A. Dutton, Atmospheric Turbulence: Models and Methods for Engineering Applications (Wiley, 1983).
  27. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. Wave Propagation through Random Media (Springer, 1989), Vol. 4.
  28. V. A. Banakh and V. L. Mironov, "Phase approximation of the Huygens-Kirchhoff method in problems of space-limited optical-beam propagation in the turbulent atmosphere," Opt. Lett. 4, 259-261 (1979).
    [CrossRef] [PubMed]
  29. V. A. Banakh, "Modeling of image of laser illuminated scattering layer in a turbulent atmosphere," Atmos. Oceanic Opt. 20, 303-307 (2007).
  30. J. M. Martin and S. M. Flatte, "Intensity images and statistics from numerical simulation of wave propagation in 3-D random media," Appl. Opt. 27, 2111-2126 (1988).
    [CrossRef] [PubMed]
  31. V. P. Kandidov, "Monte Carlo method in nonlinear statistical optics," Usp. Fiz. Nauk 166, 1309-1338 (1996) (in Russian).
    [CrossRef]
  32. V. A. Banakh, I. N. Smalikho, and Ch. Werner, "Numerical simulation of effect of refractive turbulence on the statistics of a coherent lidar return in the atmosphere," Appl. Opt. 39, 5403-5414 (2000).
    [CrossRef]
  33. V. A. Banakh and A. V. Falits, "Turbulent statistics of laser beam intensity on ground-to-satellite optical link," Proc. SPIE 4678, 132-143 (2001).
    [CrossRef]
  34. V. A. Banakh and I. N. Smalikho, "Laser beam propagation along extended vertical and slant paths in the turbulent atmosphere," Atmos. Oceanic Opt. 6, 211-286 (1993).
  35. V. H. Rumsey, "Scintillations due to a concentrated layer with a power-law turbulence spectrum," Radio Sci. 10, 107-114 (1975).
    [CrossRef]
  36. V. A. Banakh and D. A. Marakasov, "Wind profiling based on the optical beam intensity statistics in a turbulent atmosphere," J. Opt. Soc. Am. A 24, 3245-3254 (2007).
    [CrossRef]

2007

V. A. Banakh, D. S. Rytchkov, V. V. Zhmylevski, and V. V. Morozov, "Mean power of the partially spatially coherent laser beam backscattered by the atmospheric layer," Atmos. Oceanic Opt. 20, 953-958 (2007).

V. A. Banakh and D. A. Marakasov, "Wind velocity profile reconstruction from intensity fluctuations of a plane wave propagating in a turbulent atmosphere," Opt. Lett. 32, 2236-2238 (2007).
[CrossRef] [PubMed]

V. A. Banakh, "Modeling of image of laser illuminated scattering layer in a turbulent atmosphere," Atmos. Oceanic Opt. 20, 303-307 (2007).

V. A. Banakh and D. A. Marakasov, "Wind profiling based on the optical beam intensity statistics in a turbulent atmosphere," J. Opt. Soc. Am. A 24, 3245-3254 (2007).
[CrossRef]

2006

R. Avila, E. Carrasco, F. Ibanez, J. Vernin, J.-L. Prieur, and D. X. Cruz, "Generalized SCIDAR measurements at San Pedro Martin. II Wind profile statistics," Publ. Astron. Soc. Pac. 118, 503-515 (2006).
[CrossRef]

B. Garcia-Lorenzo and J. J. Fuensalida, "Processing of turbulent layer wind speed with generalized SCIDAR through wavelet analysis," Mon. Not. R. Astron. Soc. 372, 1483-1495 (2006).
[CrossRef]

2005

M. Yu and M. A. Vorontsov, "Remote sensing of atmospheric wind profiles using spatio-temporal intensity information and wavefront sensor information," Proc. SPIE 5998, 599808 (2005).

2004

J.-L. Prieur, R. Avila, G. Daigne, and J. Vernin, "Automatic determination of wind profiles with generalized SCIDAR," Publ. Astron. Soc. Pac. 116, 778-789 (2004).
[CrossRef]

2002

2001

V. A. Banakh and A. V. Falits, "Turbulent statistics of laser beam intensity on ground-to-satellite optical link," Proc. SPIE 4678, 132-143 (2001).
[CrossRef]

2000

1998

V. A. Kluckers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, and J. C. Dainty, "Profiling of atmospheric turbulence strength and velocity using generalized SCIDAR technique," Astron. Astrophys. , Suppl. Ser. 130, 141-155 (1998).

1996

V. P. Kandidov, "Monte Carlo method in nonlinear statistical optics," Usp. Fiz. Nauk 166, 1309-1338 (1996) (in Russian).
[CrossRef]

1993

V. A. Banakh and I. N. Smalikho, "Laser beam propagation along extended vertical and slant paths in the turbulent atmosphere," Atmos. Oceanic Opt. 6, 211-286 (1993).

1988

1987

1986

V. S. Rao Gudimetla and J. F. Holmes, "Two-point joint density function of the intensity for a laser speckle pattern after propagation through the turbulent atmosphere," J. Opt. Soc. Am. A 3, P118 (1986).

1985

J. F. Holmes, F. Amzajerdian, V. S. Rao Gudimetla, and J. M. Hunt, "Remote crosswind measurement using speckle-turbulence interaction and optical heterodyne detection," J. Opt. Soc. Am. A 2, P104 (1985).

1984

1983

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk . 54, 626-633 (1983) (in Russian).

V. A. Banakh and V. M. Buldakov, "Impact of the initial spatial coherency of the light beam on the intensity fluctuations," Opt. Spektrosk. 55, 707-711 (1983) (in Russian).

1981

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Strong fluctuations of noncoherent radiation in a turbulent atmosphere," Izv. Vyssh. Uchebn. Zaved. , Radiofiz. 24, 703-708 (1981) (in Russian).

T.-I. Wang, G. R. Ochs, and S. Lawrence, "Wind measurements by the temporal cross-correlation of the optical scintillations," Appl. Opt. 20, 4073-4081 (1981).
[CrossRef] [PubMed]

1980

C. M. McIntyre, M. H. Lee, and J. H. Churnside, "Statistics of irradiance scattered from a diffuse target containing multiple glints," J. Opt. Soc. Am. A 70, 1084-1095 (1980).
[CrossRef]

J. F. Holmes, M. H. Lee, and M. E. Fossey, "Remote crosswind measurement utilizing the interaction of a target-induced speckle field with the turbulent atmosphere," J. Opt. Soc. Am. A 70, 1586 (1980).

J. F. Holmes, M. H. Lee, 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. A 70, 355-360 (1980).
[CrossRef]

1979

1976

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

1975

S. F. Clifford, G. R. Ochs, and T.-I. Wang, "Optical wind sensing by observing the scintillations of a random scene," Appl. Opt. 14, 2844-2850 (1975).
[PubMed]

V. H. Rumsey, "Scintillations due to a concentrated layer with a power-law turbulence spectrum," Radio Sci. 10, 107-114 (1975).
[CrossRef]

1974

Appl. Opt.

Astron. Astrophys.

V. A. Kluckers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, and J. C. Dainty, "Profiling of atmospheric turbulence strength and velocity using generalized SCIDAR technique," Astron. Astrophys. , Suppl. Ser. 130, 141-155 (1998).

Atmos. Oceanic Opt.

V. A. Banakh and I. N. Smalikho, "Laser beam propagation along extended vertical and slant paths in the turbulent atmosphere," Atmos. Oceanic Opt. 6, 211-286 (1993).

V. A. Banakh, D. S. Rytchkov, V. V. Zhmylevski, and V. V. Morozov, "Mean power of the partially spatially coherent laser beam backscattered by the atmospheric layer," Atmos. Oceanic Opt. 20, 953-958 (2007).

V. A. Banakh, "Modeling of image of laser illuminated scattering layer in a turbulent atmosphere," Atmos. Oceanic Opt. 20, 303-307 (2007).

Izv. Vyssh. Uchebn. Zaved.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Strong fluctuations of noncoherent radiation in a turbulent atmosphere," Izv. Vyssh. Uchebn. Zaved. , Radiofiz. 24, 703-708 (1981) (in Russian).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. F. Holmes, M. H. Lee, and M. E. Fossey, "Remote crosswind measurement utilizing the interaction of a target-induced speckle field with the turbulent atmosphere," J. Opt. Soc. Am. A 70, 1586 (1980).

J. F. Holmes, F. Amzajerdian, V. S. Rao Gudimetla, and J. M. Hunt, "Remote crosswind measurement using speckle-turbulence interaction and optical heterodyne detection," J. Opt. Soc. Am. A 2, P104 (1985).

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

J. F. Holmes, M. H. Lee, 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. A 70, 355-360 (1980).
[CrossRef]

V. S. Rao Gudimetla and J. F. Holmes, "Two-point joint density function of the intensity for a laser speckle pattern after propagation through the turbulent atmosphere," J. Opt. Soc. Am. A 3, P118 (1986).

V. P. Aksenov, V. A. Banakh, and V. L. Mironov, "Fluctuations of retroreflected laser radiation in a turbulent atmosphere," J. Opt. Soc. Am. A 1, 263-274 (1984).
[CrossRef]

C. M. McIntyre, M. H. Lee, and J. H. Churnside, "Statistics of irradiance scattered from a diffuse target containing multiple glints," J. Opt. Soc. Am. A 70, 1084-1095 (1980).
[CrossRef]

V. A. Banakh and D. A. Marakasov, "Wind profiling based on the optical beam intensity statistics in a turbulent atmosphere," J. Opt. Soc. Am. A 24, 3245-3254 (2007).
[CrossRef]

Mon. Not. R. Astron. Soc.

B. Garcia-Lorenzo and J. J. Fuensalida, "Processing of turbulent layer wind speed with generalized SCIDAR through wavelet analysis," Mon. Not. R. Astron. Soc. 372, 1483-1495 (2006).
[CrossRef]

Opt. Lett.

Opt. Spektrosk

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk . 54, 626-633 (1983) (in Russian).

Opt. Spektrosk.

V. A. Banakh and V. M. Buldakov, "Impact of the initial spatial coherency of the light beam on the intensity fluctuations," Opt. Spektrosk. 55, 707-711 (1983) (in Russian).

Proc. SPIE

V. A. Banakh and A. V. Falits, "Turbulent statistics of laser beam intensity on ground-to-satellite optical link," Proc. SPIE 4678, 132-143 (2001).
[CrossRef]

M. Yu and M. A. Vorontsov, "Remote sensing of atmospheric wind profiles using spatio-temporal intensity information and wavefront sensor information," Proc. SPIE 5998, 599808 (2005).

Publ. Astron. Soc. Pac.

J.-L. Prieur, R. Avila, G. Daigne, and J. Vernin, "Automatic determination of wind profiles with generalized SCIDAR," Publ. Astron. Soc. Pac. 116, 778-789 (2004).
[CrossRef]

R. Avila, E. Carrasco, F. Ibanez, J. Vernin, J.-L. Prieur, and D. X. Cruz, "Generalized SCIDAR measurements at San Pedro Martin. II Wind profile statistics," Publ. Astron. Soc. Pac. 118, 503-515 (2006).
[CrossRef]

Radio Sci.

V. H. Rumsey, "Scintillations due to a concentrated layer with a power-law turbulence spectrum," Radio Sci. 10, 107-114 (1975).
[CrossRef]

Usp. Fiz. Nauk

V. P. Kandidov, "Monte Carlo method in nonlinear statistical optics," Usp. Fiz. Nauk 166, 1309-1338 (1996) (in Russian).
[CrossRef]

Other

H. A. Panofsky and J. A. Dutton, Atmospheric Turbulence: Models and Methods for Engineering Applications (Wiley, 1983).

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. Wave Propagation through Random Media (Springer, 1989), Vol. 4.

V. A. Banakh and V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, 1987).

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, "Wave propagation theories in random media based on the path-integral approach," in Progress in Optics, E. Wolf, ed., (North-Holland, 1993), Vol. XXXII, pp. 203.
[CrossRef]

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (IPTS, 1971).

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Figures (8)

Fig. 1
Fig. 1

Spatiotemporal spectrum variations along the distinguished ray. φ 0 = 0.1 , q = 200 m 1 ; □, x = 2000   m ; ○, x = 1000   m ; +, x = 500   m .

Fig. 2
Fig. 2

Scheme of wind profile measurer. a, Collocated receiver and transmitter; b, spaced receiver and transmitter.

Fig. 3
Fig. 3

Components of the spatiotemporal intensity spectrum for the collimated illuminating Gaussian beam affected by a single phase screen near the transmitter and retroreflected by a diffuse target at x = 1   km . a, Z-component; b, Y-component.

Fig. 4
Fig. 4

Z-components of the spatiotemporal spectrum of intensity of a collimated laser beam, passed through a moving random phase screen near the transmitter and retroreflected by a diffuse target at x = 1   km , for different strengths of phase fluctuations. a, σ I 2 = 0.01 ; b, σ I 2 = 0.1 ; c, σ I 2 = 0.25 ; d, σ I 2 = 0.5 .

Fig. 5
Fig. 5

Components of the spatiotemporal intensity spectrum of a defocused illuminating beam affected by five phase screens and retroreflected by a diffuse target at x = 1   km . a, Z-component; b, Y-component.

Fig. 6
Fig. 6

Initial profiles (solid curves with circles) and reconstruction results (dashed curves with crosses) for defocused illuminating beam retroreflected by a diffuse target at x = 1   km .

Fig. 7
Fig. 7

Components of the spatiotemporal intensity spectrum of a defocused illuminating beam affected by seven phase screens and retroreflected by a diffuse target at x = 1.4   km . a, Z-component; b, Y-component.

Fig. 8
Fig. 8

Initial profiles (solid curves with circles) and reconstruction results (dashed curves with crosses) for defocused illuminating beam retroreflected by a diffuse target at x = 1.4   km .

Equations (88)

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

U ± ( x , ρ , τ ) = k U 0 2 π i x d r   exp { ± i φ 0 f ( r ) + i k 2 x ( r ρ + V τ ) 2 } ,
I ± ( x , ρ ; τ ) = | U ± ( x , ρ , τ ) | 2 ,
K I ( ρ 1 , ρ 2 ; τ ) = I ( x , ρ 1 ; 0 ) I ( x , ρ 2 ; τ ) I ( x , ρ 1 ; 0 ) × I ( x , ρ 2 ; τ ) = 1 4 [ ( I + ( x , ρ 1 ; 0 ) I ( x , ρ 1 ; 0 ) ) × ( I + ( x , ρ 2 ; τ ) I ( x , ρ 2 ; τ ) ) ] .
F ˜ I ( x , κ ; ω ) = d τ d ρ 1 d ρ 2 e i ω τ i κ ρ 1 + i κ ρ 2 K I ( ρ 1 , ρ 2 ; τ ) = π 2 δ ( ω + κ V ) | d ρ e i κ ρ ( I + ( x , ρ ; 0 ) I ( x , ρ ; 0 ) ) | 2 .
I ± ( x , ρ ; 0 ) | U 0 | 2 ± k φ 0 π x | U 0 | 2 d r f ( r ) cos ( k 2 x ( ρ r ) 2 ) ,
F ˜ I ( x , κ ; ω ) = 8 φ 0 2 | U 0 | 4 δ ( ω + κ v ) sin 2 ( x 2 k κ 2 ) × | d r e i κ r f ( r ) | 2 .
κ 2 = 2 π n k x .
U 0 ( ρ ) = U 0   exp { ρ 2 2 a 2 i k ρ 2 2 F } ,
U R ( 0 , ρ ) = d r d r d ρ U 0 ( ρ ) V ( r , r ) G ( x , r ; 0 , ρ ) × G ( x , r ; 0 , ρ ) ,
G ( x , r ; 0 , ρ ) = k e i k x 2 π i x G R ( x , r ; 0 , ρ ) × exp { i k 2 x ( r ρ ) 2 }
G R ( x , r ; 0 , ρ ) = lim N ( k 2 π i x ) N 1 d S 1   …   d S N 1 × exp { i k 2 x j = 1 N 1 S j 2 + i k 2 0 x d x × ε 1 ( x , ( 1 ξ ) ρ + ξ r + j = 1 N 1 ν j ( ξ ) S j ) } ,
ξ = x x ,
ν j ( ξ ) = sin ( j π ξ ) 2 N   sin ( j π 2 N ) .
U ( l , ρ ) = k 2 π i l   exp { i k l + i k ρ 2 2 l } d ρ T ( ρ ) U R ( x 0 , ρ ) × exp { i k l ρ ρ + i k 2 ( 1 l 1 F t ) ρ 2 } ,
T ( ρ ) = T 0   exp { ρ 2 2 a t 2 } ,
I ( F t , ρ ) = U ( F t , ρ ) U * ( F t , ρ ) = ( k | T 0 | 2 π F t ) 2 d r 1 , 2 d r 1 , 2 d ρ 1 , 2 d ρ 1 , 2 U 0 ( ρ 1 ) × U 0 * ( ρ 2 ) V ( r 1 , r 1 ) V * ( r 2 , r 2 ) × G ( x , r 1 ; 0 , ρ 1 ) G * ( x , r 2 ; 0 , ρ 2 ) × G ( x , r 1 ; 0 , ρ 1 ) G * ( x , r 2 ; 0 , ρ 2 ) × exp { ρ 1 2 + ρ 2 2 2 a t 2 i k F t ρ ( ρ 1 ρ 2 ) } .
I ( F t , ρ 1 , 0 ) I ( F t , ρ 2 , τ )
= ( k | T 0 | 2 π F t ) 4 ( k 2 π x ) 8 d r 1 4 d r 1 4 d ρ 1 4 d ρ 1 4 U 0 ( ρ 1 ) × U 0 * ( ρ 2 ) U 0 ( ρ 3 ) U 0 * ( ρ 4 ) V ( r 1 , r 1 ) V * ( r 2 , r 2 ) × V ( r 3 , r 3 ) V * ( r 4 , r 4 )
× { j = 1 4 ρ j 2 2 a t 2 i k F t ( ρ 1 ( ρ 1 ρ 2 ) + ρ 2 ( ρ 3 ρ 4 ) ) } × exp { i k 2 x [ ( r 1 ρ 1 ) 2 ( r 2 ρ 2 ) 2 + ( r 1 ρ 1 ) 2
( r 2 ρ 2 ) 2 + ( r 3 ρ 3 ) 2 ( r 4 ρ 4 ) 2 + ( r 3 ρ 3 ) 2 ( r 4 ρ 4 ) 2 ] } lim N ( k 2 π x ) 8 ( N 1 )
× d a 1   …   N 1 d b 1   …   N 1 d c 1   …   N 1 d e 1   …   N 1 d f 1   …   N 1 × d g 1   …   N 1 d h 1   …   N 1 d t 1   …   N 1
× exp { i k 2 x j = 1 N 1 ( a j 2 + b j 2 c j 2 e j 2 + f j 2 + g j 2 h j 2 t j 2 ) + i k 2 0 x d x [ ε 1 ( x , ρ 1 , r 1 , a j ; 0 ) ε 1 ( x , ρ 2 , r 1 , c j ; 0 ) + ε 1 ( x , ρ 1 , r 1 , b j ; 0 ) ε 1 ( x , ρ 2 , r 2 , e j ; 0 ) + ε 1 ( x , ρ 3 , r 3 , f j ; τ ) ε 1 ( x , ρ 4 , r 4 , h j ; τ ) + ε 1 ( x , ρ 3 , r 3 , g j ; τ ) ε 1 ( x , ρ 4 , r 4 , t j ; τ ) ] } ,
K I ( ρ 1 , ρ 2 ; τ ) = I ( F t , ρ 1 , 0 ) I ( F t , ρ 2 , τ ) I ( F t , ρ 1 , 0 ) I ( F t , ρ 2 , τ ) ,
V ( r 1 , r 1 ) V * ( r 2 , r 2 ) V ( r 3 , r 3 ) V * ( r 4 , r 4 ) = V ( r 1 , r 1 ) V * ( r 2 , r 2 ) V ( r 3 , r 3 ) V * ( r 4 , r 4 ) ,
V ( r i , r i ) V * ( r l , r l ) = 4 π k 2 | A ( r i ) | 2 δ ( r i r l ) δ ( r i r i ) × δ ( r l r l ) .
A ( r i ) = A 0   exp { r i 2 2 a r 2 } ,
ε 1 ( x , ρ ; τ ) = ε 1 ( x , ρ V ( x ) τ ; 0 ) ,
e i φ = e ( 1 / 2 ) φ 2 ,
e ( 1 / 2 ) φ 2 1 1 2 φ 2 .
K I ( ρ 1 , ρ 2 ; τ ) = 2 A   exp { D ( 2 R 2 + ρ 2 / 2 ) } d κ 0 x d x × Φ ε ( x , κ ) e i κ V ( x ) τ { exp ( Re   t 1 κ 2 + i   Im   γ 1 κ ρ ) [ cosh ( 2   Re   γ 1 κ R ) cosh ( Re   γ 1 κ ρ + i   Im   t 1 κ 2 ) ] + exp ( Re   t 2 κ 2 + i   Im   γ 2 κ ρ ) × [ cosh ( 2   Re   γ 2 κ R ) cosh ( Re   γ 2 κ ρ + i   Im   t 2 κ 2 ) ] + 2   Re [ exp ( t 1 + t 2 * 2 κ 2 + γ 1 γ 2 * 2 κ ρ ) cosh ( ( γ 1 + γ 2 * ) κ R ) exp ( t 1 + t 2 2 κ 2 + γ 1 + γ 2 2 κ ρ ) × cosh ( ( γ 1 γ 2 ) κ R ) ] } .
R = ρ 1 + ρ 2 2 ,     ρ = ρ 1 ρ 2 ,
A = k 2 | k 2 a d A 0 T 0 U 0 x 2 F t z z t | 4 ,
1 a d 2 = 1 a r 2 + k 2 x 2 a t 2 | z t 2 | + k 2 x 2 a 2 | z 2 | ,
D = k 2 a d 2 F t 2 a t 2 | z t | 2 ( 1 a r 2 + k 2 x 2 a 2 | z 2 | ) ,
t 1 = i x k ξ ( 1 ξ ) ( 1 ξ ) 2 z + a d 2 2 ( i ξ + k x 1 ξ z ) 2 ,
t 2 = i x k ξ ( 1 ξ ) ( 1 ξ ) 2 z t + a d 2 2 ( i ξ + k x 1 ξ z t ) 2 ,
z = 1 a 2 i k ( 1 x 1 F ) ,     z t = 1 a t 2 i k x ,
γ 1 = a d 2 k 2 a t 2 x F t | z t | 2 ( i ξ + k x 1 ξ z ) ,
γ 2 = k ( 1 ξ ) F t z t a d 2 k 2 a t 2 x F t | z t | 2 ( i ξ + k x 1 ξ z t ) .
F I ( ρ , R ; τ ) = B exp { D ( 2 R 2 + ρ 2 / 2 ) } d κ 0 x d x × exp { β j κ 2 + ν j κ ρ + δ j κ R } × Φ ε ( x , κ ) e i κ V ( x ) τ ,
d R d ρ F I ( ρ , R ; τ ) e i q ρ = π B D 2 d κ 0 x d x × exp { ( β j δ j 2 8 D ) κ 2 + ( ν j κ + i q ) 2 2 D } Φ ε ( x , κ ) e i κ V ( x ) τ .
Re { } = Re   η j ( κ + Im   ν j 2 D   Re   η j q ) 2 + q 2 ( Im   ν j ) 2 2 D   Re   η j 4 D 2   Re   η j ,
d κ Φ ε ( x , κ ) e i κ V ( x ) τ   exp { η j κ 2 + i ν j D q κ } π η j Φ ε ( x , Im   ν j 2 D   Re   η j q ) e i ( ( τ   Im   ν j ) / ( 2 D   Re   η j ) ) q V ( x ) × exp { ν j 2 q 2 4 η j D 2 } .
d R d ρ F I ( ρ , R ; τ ) e i q ρ = π 3 B D 2 0 x d x Φ ε ( x , Im   ν j 2 D   Re   η j q ) η j × e i ( ( τ   Im   ν j ) / ( 2 D   Re   η j ) ) q V ( x ) × exp { q 2 ( ν j 2 4 η j D 2 + 1 2 D ) } .
g ( α , q ) = 1 2 π d τ d R d ρ F I ( ρ , R ; τ ) e i q ρ + i ω τ = π 3 B D 2 0 x d x 1 η j q Φ ε ( x , Im 2 D   Re   η j q ) × δ ( α Im   ν j 2 D   Re   η j V i ( x ) ) × exp { q 2 ( ν j 2 4 η j D 2 + 1 2 D ) } .
f 0 ( α , p ) = 0 g 0 ( α , q ) e i p q 2 d q = 0 x d x C n 2 ( x ) 1 η j δ ( α Im   ν j 2 D   Re   η j V i ( x ) ) ν j 2 4 η j D 2 + 1 2 D i p .
f 0 ( α , p ) = 0 x d x C n 2 ( x ) j = 1 6 k j η j δ ( α Im   Γ j V i ( x ) 2 D   Re   η j ) Γ j 4 η j D 2 + 1 2 D i p .
( 1 ) k 1 = 2 ,
η 1 = Re   t 1 ( Re   γ 1 ) 2 2 D + ( Im   γ 1 ) 2 2 D = Re   t 1 Re ( γ 1 2 ) 2 D ,
Γ 1 = i   Im   γ 1 .
( 2 ) k 2 = 2 , η 2 = Re   t 2 Re ( γ 2 2 ) 2 D ,
Γ 2 = i   Im   γ 2 .
( 3 ) k 3 = 1 , η 3 = t 1 γ 1 2 2 D , Γ 3 = γ 1 .
p 3 = Im ( 1 2 ( 1 + 2 t 1 γ 1 2 D ) ) ,
( 4 ) k 4 = 1 , η 4 = η 3 * , Γ 4 = γ 1 * .
( 5 ) k 5 = 1 , η 5 = t 2 γ 2 2 2 D , Γ 5 = γ 2 .
p 5 = Im ( 1 2 ( 1 + 2 D t 2 γ 2 2 ) ) ,
( 6 ) k 6 = 1 , η 6 = η 5 * , Γ 6 = γ 2 * .
K I ( ρ 1 , ρ 2 ; τ ) K I ( ρ , τ ) = 4 A d κ 0 x d x Φ ε ( x , κ ) × e i κ V ( x ) τ i ξ ( L / F t ) κ ρ ( ( ξ 2 a t 2 ) / 2 ) κ 2 × sin 2 ( L κ 2 2 k ξ ( 1 ξ ) ) ,
g ( α , q ) = 1 ( 2 π ) 3 d τ d ρ K I ( ρ , τ ) e i q ρ i ω τ = 4 A 0 x d x Φ ε ( x , q F t x ξ ) F t 2 x 2 ξ 2 q   exp ( q 2 F t 2 a t 2 2 x 2 ) × sin 2 ( F t 2 q 2 2 k x 1 ξ ξ ) δ ( α F t V i ( x ) x ξ ) .
g 0 ( α , q ) = q 2 g ( α , q ) 4 A Φ 0 ( q )   exp ( q 2 F t 2 a t 2 2 x 2 )
Φ ε ( x , q F t x ξ ) = C n 2 ( x ) Φ 0 ( q ) ( F t x ξ ) 11 / 3
g 0 ( α , q ) = 0 x d x C n 2 ( x ) ( x ξ F t ) 5 / 3 q sin 2 ( F t 2 q 2 2 k x 1 ξ ξ ) × δ ( α F t V i ( x ) x ξ ) .
0 g 0 ( α , q ) cos ( p q 2 ) d q = π 8 0 x d x C n 2 ( x ) ( x ξ F t ) 5 / 3 × δ ( α F t V i ( x ) x ξ ) × [ 2 δ ( p ) δ ( p + F t 2 2 k x 1 ξ ξ ) δ ( p F t 2 2 k x 1 ξ ξ ) ] .
I ( F t , ρ ) = d r d ρ 1 , 2 U 0 ( ρ 1 ) U 0 * ( ρ 2 ) G ( x , r ; 0 , ρ 1 ) × G * ( x , r ; 0 , ρ 2 ) ( k A 0 2 π F t ) 2 d ρ 1 , 2 T ( ρ 1 ) T ( ρ 2 ) × exp { i k F t ρ ( ρ 1 ρ 2 ) } G ( x , r ; 0 , ρ 1 ) × G * ( x , r ; 0 , ρ 2 ) = d r I I ( x , r ) I T ( x , r , ρ ) .
I ( F t , ρ ) I ( F t , ρ , r ) = d r I I ( x , r ) I T ( x , r , ρ ) = d r I I ( x , r ) I T ( x , r + ρ x / F t ) .
I ( F t , ρ ) = ( 2 π ) 2 d κ I ˜ I ( x , κ ) I ˜ T ( x , κ ) exp ( i κ ρ x / F t ) ,
I ˜ I ( x , κ ) = ( 2 π ) 2 d r I I ( x , r ) e i κ r ,
I ˜ T ( x , κ ) = ( 2 π ) 2 d r I T ( x , r , ρ ) exp [ i κ ( r + ρ x / F t ) ] ,
I ( F t , ρ ) d r I I 2 ( x , r ) I T 0 ( x , r , ρ ) ,
I I ( x , r ) = | d ρ U 0 ( ρ ) G ( x , r ; 0 , ρ ) | 2 ,
I T 0 ( x , r , ρ ) = B   exp { ( r + x F t ρ ) 2 a t d 2 } ,
I I ( x , r , t ) I I ( x , r , t ) + δ I I ( x , r , t ) ,
I I 2 ( x , r , t ) I I ( x , r , t ) 2 + 2 I I ( x , r , t ) δ I I ( x , r , t ) + δ I I 2 ( x , r , t ) ,
I I 2 ( x , r 1 ; t ) I I 2 ( x , r 2 ; t + τ )
I I ( x , r 1 , t ) 2 I I ( x , r 2 ; t + τ ) 2 + 2 I I ( x , r 1 , t ) × I I ( x , r 2 ; t + τ ) ( I I ( x , r 1 , t ) δ I I ( x , r 2 ; t + τ ) + I I ( x , r 2 ; t + τ ) δ I I ( x , r 1 , t ) ) + I I ( x , r 1 , t ) 2 δ I I 2 ( x , r 2 ; t + τ ) I I ( x , r 2 ; t + τ ) 2 × δ I I 2 ( x , r 1 , t ) + 4 I I ( x , r 1 , t ) I I ( x , r 2 ; t + τ ) × δ I I ( x , r 1 , t ) δ I I ( x , r 2 ; t + τ ) .
I I 2 ( x , r 1 ; t ) I I 2 ( x , r 2 ; t + τ ) = K ( r 1 , r 2 ) + 4 I I ( x , r 1 , t ) × I I ( x , r 2 ; t + τ ) × δ I I ( x , r 1 ; t ) δ I I ( x , r 2 ; t + τ ) , = K ( r 1 , r 2 ) + 4 I I ( x , r 1 , t ) × I I ( x , r 2 ; t + τ ) K I ( r 1 , r 2 , τ ) .
I I ( x , r 1 ; t ) I I ( x , r 2 ; t ) exp { k Ω x g 2 ( r 1 2 + r 2 2 ) } ,
K I ( ρ 1 , ρ 2 , τ ) = d r 1 d r 2 [ I I 2 ( x , r 1 ; t ) I I 2 ( x , r 2 ; t + τ ) × I T 0 ( x , r 1 , ρ 1 ) I T 0 ( x , r 2 , ρ 2 ) I I 2 ( x , r 1 ) I I 2 ( x , r 2 ) I T 0 ( x , r 1 , ρ 1 ) × I T 0 ( x , r 2 , ρ 2 ) ] , exp { ρ 2 + 4 R 2 2 a F 2 } 0 1 d ξ d κ C n 2 ( ξ ) Φ ε ( κ ) × e i κ V ( ξ ) τ n = 1 4 ( 1 ) n exp { c n κ 2 + Z n κ } ,
a F 2 = F t 2 x 2 ( a t d 2 + x g 2 2 k Ω ) ,
c 1 = γ ( ξ ) ( x g 2 k Ω   Re   γ ( ξ ) γ ( ξ ) / ( 2 a t d 2 + k Ω x g 2 ) ) ,
c 3 = c 1 * , c 2 , 4 = Re   c 1 ,
Z n = x / F t 1 + 2 k Ω a t d 2 x g 2 q n ,
q 1 = q 3 * = γ ( ξ ) ρ ,
q 2 = q 4 * = 2   Re   γ ( ξ ) R + i   Im   γ ( ξ ) ρ ,
γ ( ξ ) = i ξ + Ω g 2 ( 1 ξ ) ( 1 + i Ω ( 1 x F ) ) .
p = ± F t 2 g 2 3 k Ω x χ ( ξ ) 1 + χ 2 ( ξ ) , χ ( ξ ) = Im   γ ( ξ ) 3   Re   γ ( ξ ) ,
α = ω / q = F t x Im   γ ( ξ ) | γ ( ξ ) | 2 + 2 ( Re   γ ( ξ ) ) 2 V i ( ξ ) ,

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