Abstract

Reconstruction of the wind profile from the statistics of intensity fluctuations of an optical beam propagating in a turbulent atmosphere is considered. The equations for the spatiotemporal correlation function and the spectrum of weak intensity fluctuations of a Gaussian beam are obtained. The algorithms of wind profile retrieval from the spatiotemporal intensity spectrum are described and the results of end-to-end computer experiments on wind profiling based on the developed algorithms are presented. It is shown that the developed algorithms allow retrieval of the wind profile from the turbulent optical beam intensity fluctuations with acceptable accuracy in many practically feasible laser measurements set up in the atmosphere.

© 2007 Optical Society of America

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References

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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]
  2. A. Rossa, F. Roddier, and J. Vernin, "Detection of atmospheric turbulent layers by spatiotemporal and spatioangular correlation measurements of stellar-light scintillation," J. Opt. Soc. Am. 64, 1000-1004 (1974).
    [CrossRef]
  3. F. Roddier, "The effects of atmospheric turbulence on optical astronomy," Prog. Opt. 19, 281-319 (1981).
    [CrossRef]
  4. J. L. Caccia, M. Azouit, and J. Vernin, "Wind and CN2 profiling by single-star scintillation analysis," Appl. Opt. 26, 1288-1294 (1987).
    [CrossRef] [PubMed]
  5. V. A. Kliickers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, and J. C. Dainty, "Profiling of atmospheric turbulence strength and velocity using a generalized SCIDAR technique," Astron. Astrophys., Suppl. Ser. 130, 141-155 (1998).
    [CrossRef]
  6. R. Johnston, C. Dainty, N. Wooder, and R. Lane, "Generalized scintillation detection and ranging results obtained by use of a modified inversion technique," Appl. Opt. 41, 6768-6772 (2002).
    [CrossRef] [PubMed]
  7. 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]
  8. 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]
  9. 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]
  10. V. A. Banakh and V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech, 1987), p. 185.
  11. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (IPTS, 1971).
  12. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. Wave Propagation through Random Media (Springer, 1989), Vol. 4.
  13. E. A. Monastyrny, G. Ya. Patrushev, and V. V. Pokasov, "Temporal characteristics of the optical wave amplitude logarithm with fluctuating wind," Opt. Spectrosc. 56, 44-47 (1984) (in Russian).
  14. M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, "Wave propagation theories in random media based on the path-integral approach," Progress in Optics, Vol. XXXII, E.Wolf, ed. (North-Holland, 1993), p. 203.
    [CrossRef]
  15. H. A. Panofsky and J. A. Dutton, Atmospheric Turbulence: Models and Methods for Engineering Applications (Wiley-Interscience, 1983), p. 397.
  16. 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]
  17. V. Ya. Arsenin, Methods of Mathematical Physics and Special Functions (Nauka, 1984) (in Russian), p. 383.
  18. 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]
  19. V. P. Kandidov, "Monte Carlo method in nonlinear statistical optics," Usp. Fiz. Nauk 166, 1309-1338 (1996) (in Russian). This journal is translated into English with the title "Physics-Uspekhi (Advances in Physical Sciences)."
    [CrossRef]
  20. 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 (1998).
    [CrossRef]

2006

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]

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]

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]

1998

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

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 (1998).
[CrossRef]

1996

V. P. Kandidov, "Monte Carlo method in nonlinear statistical optics," Usp. Fiz. Nauk 166, 1309-1338 (1996) (in Russian). This journal is translated into English with the title "Physics-Uspekhi (Advances in Physical Sciences)."
[CrossRef]

1987

1984

E. A. Monastyrny, G. Ya. Patrushev, and V. V. Pokasov, "Temporal characteristics of the optical wave amplitude logarithm with fluctuating wind," Opt. Spectrosc. 56, 44-47 (1984) (in Russian).

1981

1979

1974

Adcock, M. J.

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

Arsenin, V. Ya.

V. Ya. Arsenin, Methods of Mathematical Physics and Special Functions (Nauka, 1984) (in Russian), p. 383.

Avila, R.

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]

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]

Azouit, M.

Banakh, V. A.

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]

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]

V. A. Banakh and V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech, 1987), p. 185.

Caccia, J. L.

Carrasco, E.

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]

Charnotskii, M. I.

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

Cruz, D. X.

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]

Daigne, G.

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]

Dainty, C.

Dainty, J. C.

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

Dutton, J. A.

H. A. Panofsky and J. A. Dutton, Atmospheric Turbulence: Models and Methods for Engineering Applications (Wiley-Interscience, 1983), p. 397.

Falits, A. V.

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]

Flatte, S. M.

Fuensalida, J. J.

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]

Garcia-Lorenzo, B.

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]

Gozani, J.

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

Ibanez, F.

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]

Johnston, R.

Kandidov, V. P.

V. P. Kandidov, "Monte Carlo method in nonlinear statistical optics," Usp. Fiz. Nauk 166, 1309-1338 (1996) (in Russian). This journal is translated into English with the title "Physics-Uspekhi (Advances in Physical Sciences)."
[CrossRef]

Kliickers, V. A.

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

Kravtsov, Yu. A.

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

Lane, R.

Lawrence, S.

Martin, J. M.

Mironov, V. L.

Monastyrny, E. A.

E. A. Monastyrny, G. Ya. Patrushev, and V. V. Pokasov, "Temporal characteristics of the optical wave amplitude logarithm with fluctuating wind," Opt. Spectrosc. 56, 44-47 (1984) (in Russian).

Munro, I.

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

Nicholls, T. W.

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

Ochs, G. R.

Panofsky, H. A.

H. A. Panofsky and J. A. Dutton, Atmospheric Turbulence: Models and Methods for Engineering Applications (Wiley-Interscience, 1983), p. 397.

Patrushev, G. Ya.

E. A. Monastyrny, G. Ya. Patrushev, and V. V. Pokasov, "Temporal characteristics of the optical wave amplitude logarithm with fluctuating wind," Opt. Spectrosc. 56, 44-47 (1984) (in Russian).

Pokasov, V. V.

E. A. Monastyrny, G. Ya. Patrushev, and V. V. Pokasov, "Temporal characteristics of the optical wave amplitude logarithm with fluctuating wind," Opt. Spectrosc. 56, 44-47 (1984) (in Russian).

Prieur, J.-L.

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]

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]

Roddier, F.

Rossa, A.

Rytov, S. M.

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

Tatarskii, V. I.

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

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

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

Vernin, J.

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]

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]

J. L. Caccia, M. Azouit, and J. Vernin, "Wind and CN2 profiling by single-star scintillation analysis," Appl. Opt. 26, 1288-1294 (1987).
[CrossRef] [PubMed]

A. Rossa, F. Roddier, and J. Vernin, "Detection of atmospheric turbulent layers by spatiotemporal and spatioangular correlation measurements of stellar-light scintillation," J. Opt. Soc. Am. 64, 1000-1004 (1974).
[CrossRef]

Wang, T.-I.

Wooder, N.

Wooder, N. J.

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

Zavorotny, V. U.

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

Appl. Opt.

Astron. Astrophys., Suppl. Ser.

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

J. Opt. Soc. Am.

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. Spectrosc.

E. A. Monastyrny, G. Ya. Patrushev, and V. V. Pokasov, "Temporal characteristics of the optical wave amplitude logarithm with fluctuating wind," Opt. Spectrosc. 56, 44-47 (1984) (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]

Prog. Opt.

F. Roddier, "The effects of atmospheric turbulence on optical astronomy," Prog. Opt. 19, 281-319 (1981).
[CrossRef]

Publ. Astron. Soc. Pac.

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]

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]

Usp. Fiz. Nauk

V. P. Kandidov, "Monte Carlo method in nonlinear statistical optics," Usp. Fiz. Nauk 166, 1309-1338 (1996) (in Russian). This journal is translated into English with the title "Physics-Uspekhi (Advances in Physical Sciences)."
[CrossRef]

Other

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

H. A. Panofsky and J. A. Dutton, Atmospheric Turbulence: Models and Methods for Engineering Applications (Wiley-Interscience, 1983), p. 397.

V. Ya. Arsenin, Methods of Mathematical Physics and Special Functions (Nauka, 1984) (in Russian), p. 383.

V. A. Banakh and V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech, 1987), p. 185.

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

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

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Figures (14)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Receiving point arrangement: 4, screen; 5, receiving point; 6, video camera; 7, filter; 7a, lens; 8, interface.

Fig. 3
Fig. 3

Component parts of the numerical scheme.

Fig. 4
Fig. 4

Wind profile simulation scheme.

Fig. 5
Fig. 5

Plot of the transverse-to-path-wind velocity normalized to the ratio of the spatial resolution to the temporal one ( Δ r Δ t ) as function of path length: C n 2 = 4 × 10 17 m 2 3 , λ = 0.5 × 10 6 m , Δ r = 5 × 10 3 m , N r = 256 , N t = 128 ; ◇, ●, ∎ are the wind profile data restored from the intensity fluctuations of plane, spherical waves, and Gaussian beam, respectively; 엯 is the initial wind profile.

Fig. 6
Fig. 6

Spectrum of plane-wave intensity. Turbulence is imitated by five phase screens.

Fig. 7
Fig. 7

Plane-wave spectrum variation along the ray. Squares connected by the solid curve represent the numerical data, and the dashed curve is calculated by Eq. (20).

Fig. 8
Fig. 8

Initial (circles) and reconstructed (crosses) profiles for the components of the velocity vector. Turbulence is imitated by five phase screens.

Fig. 9
Fig. 9

Initial (circles) and reconstructed (crosses) profiles for the components of the velocity vector. Turbulence is imitated by ten phase screens.

Fig. 10
Fig. 10

Spatiotemporal intensity spectrum for a focused Gaussian beam: x = 1.05 km , F = 2.1 km , C n 2 = 4 × 10 17 m 2 3 . Turbulence is imitated by seven phase screens.

Fig. 11
Fig. 11

Focused beam spectrum variation along the strip. Squares connected by the solid curve represent the numerical data, and the dashed curve is calculated by Eq. (15).

Fig. 12
Fig. 12

Initial (solid curve with circles) and reconstructed profiles from the focused Gaussian beam intensity fluctuations (dotted curve with crosses) for the components of the velocity vector. Turbulence is imitated by seven phase screens.

Fig. 13
Fig. 13

Spatiotemporal spectrum of the intensity of the collimated Gaussian beam propagating in a turbulent atmosphere, where C n 2 changes along the path ten times.

Fig. 14
Fig. 14

Initial (solid curve with circles) and reconstructed (dotted curve with crosses) velocity vector component profiles for a Gaussian beam propagating in a turbulent atmosphere with the structural characteristics parameter varying along the path.

Equations (40)

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

K I ( R , ρ , τ ) = I ( R + ρ 2 ; 0 ) I ( R ρ 2 ; τ ) I ( R + ρ 2 ; 0 ) I ( R ρ 2 ; τ )
F ̃ I ( q , ω ) = 1 ( 2 π ) 3 d ρ d τ K I ( R , ρ , τ ) exp ( i q ρ + i ω τ )
2 i k U ( x , ρ ) x + Δ ρ U ( x , ρ ) + 2 k 2 ϵ 1 ( x , ρ , t ) U ( x , ρ ) = 0
U ( 0 , ρ ) = U 0 ( ρ ) ,
U ( x , ρ , τ ) = k 2 π i x d ρ U 0 ( ρ ) exp { i k 2 x ( ρ ρ ) 2 } × 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 x x ) ρ + x x ρ + j = 1 N 1 ν j ( x x ) S j ; τ ) } ,
ϵ 1 ( x , ρ ; τ ) = ϵ 1 ( x , ρ V ( x ) τ ; 0 ) ,
U 0 ( ρ ) = U 0 exp ( ρ 2 2 a 2 i k ρ 2 2 F ) .
I ( x , ρ 1 ; 0 ) I ( x , ρ 2 ; τ ) = U ( x , ρ 1 ; 0 ) U * ( x , ρ 1 ; 0 ) U ( x , ρ 2 ; τ ) U * ( x , ρ 2 ; τ ) = ( k U 0 2 π x ) 4 d ρ 1 d ρ 4 exp { k 2 x Ω ( ρ 1 2 + ρ 2 2 + ρ 3 2 + ρ 4 2 ) + i k 2 x ( 1 x F ) ( ρ 1 2 ρ 2 2 + ρ 3 2 ρ 4 2 ) i k x ρ 1 ( ρ 1 ρ 2 ) i k x ρ 2 ( ρ 3 ρ 4 ) } lim N ( k 2 π x ) 4 ( N 1 ) × d S 1 , 1 d S 2 , 1 d S 3 , 1 d S 4 , 1 d S 1 , N 1 d S 2 , N 1 d S 3 , N 1 d S 4 , N 1 × exp { i k 2 x j = 1 N 1 ( S 1 , j 2 S 2 , j 2 + S 3 , j 2 S 4 , j 2 ) } exp { i k x 2 0 1 d ξ ( ϵ ( ρ 1 , ρ 1 , S 1 , j ) ) ϵ ( ρ 2 , ρ 1 , S 2 , j ) + ϵ ( ρ 3 , ρ 2 V ( ξ ) τ ξ , S 3 , j ) ϵ ( ρ 4 , ρ 2 V ( ξ ) τ ξ , S 4 , j ) } ,
Ω = k a 2 x , ϵ ( ρ , ρ , S i , j ) ϵ 1 ( x ξ , ( 1 ξ ) ρ + ξ ρ + j = 1 N 1 ν j ( ξ ) S i , j ) .
[ ϵ 1 ( x , ρ i , r l , a j ; t 1 ) ϵ 1 ( x , ρ k , r m , b j ; t 2 ) ] 2 = 2 π d κ C n 2 ( ξ ) Φ ϵ ( κ ) [ 1 exp { i κ [ ( ρ i ρ κ ) ( 1 ξ ) + ( r l r m ) ξ + j = 1 N 1 ν j ( ξ ) ( a i b j ) + V ( x ) ( t 1 t 2 ) ] } ] ,
K I ( R , ρ , τ ) = π k 2 x U 0 4 Ω 4 2 g 4 exp { k Ω 2 x g 2 ( ρ 2 + 4 R 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 { t n κ 2 + q n κ } ,
t 1 = t 3 * = x k ( 1 ξ ) [ i ξ + Ω g 2 ( 1 ξ ) ( 1 + i Ω ( 1 x F ) ) ] ,
t 2 = t 4 = x Ω k g 2 ( 1 ξ ) 2 ,
q 1 = q 3 * = [ i ξ + Ω g 2 ( 1 ξ ) ( 1 + i Ω ( 1 x F ) ) ] ρ ,
q 2 = q 4 * = 2 Ω g 2 ( 1 ξ ) R + i ρ ( ξ + Ω 2 g 2 ( 1 ξ ) ( 1 x F ) ) ,
g 2 = 1 + Ω 2 ( 1 x F ) 2 .
K I ( R , ρ , τ ) = 2 π k 2 x U 0 4 d κ Φ ϵ ( κ ) e i κ ρ 0 1 d ξ C n 2 ( ξ ) e i κ V ( ξ ) τ sin 2 ( x 2 k ( 1 ξ ) κ 2 ) .
K I ( R , ρ , τ ) = 2 π k 2 x U 0 4 Ω 4 d κ Φ ϵ ( κ ) 0 1 d ξ C n 2 ( ξ ) e i κ V ( ξ ) τ + i κ ρ ξ sin 2 ( x 2 k ξ ( 1 ξ ) κ 2 ) .
f I ( R , ρ , τ ) = 2 g 4 K I ( R , ρ , τ ) π k 2 x U 0 4 Ω 4 exp { k Ω 2 x g 2 ( ρ 2 + 4 R 2 ) } ,
D I ( R , ρ , τ ) = 4 i 0 1 d ξ d κ C n 2 ( ξ ) Φ ϵ ( κ ) e i κ V ( ξ ) τ exp { Ω g 2 ( 1 ξ ) 2 κ 2 x k + i γ ( ξ ) ( 1 ξ ) ρ κ } × sinh ( 2 Ω g 2 ( 1 ξ ) R κ ) sin ( γ ( ξ ) ( 1 ξ ) 2 κ 2 x k ) ,
D ̃ I ( R , q , ω ) = 0 1 d ξ d κ C n 2 ( ξ ) Φ ϵ ( κ ) exp { x k Ω g 2 ( 1 ξ ) 2 κ 2 } sinh ( 2 Ω g 2 ( 1 ξ ) R κ ) sin ( γ ( ξ ) ( 1 ξ ) 2 κ 2 x k ) δ ( ω + κ V ( ξ ) ) δ ( q γ ( ξ ) ( 1 ξ ) κ ) = 0 1 d ξ C n 2 ( ξ ) ( γ ( ξ ) ( 1 ξ ) ) 2 Φ ϵ ( q γ ( ξ ) ( 1 ξ ) ) exp { x k Ω q 2 g 2 γ 2 ( ξ ) } sinh ( 2 Ω g 2 γ ( ξ ) R q ) sin ( q 2 γ ( ξ ) x k ) δ ( ω + q V ( ξ ) γ ( ξ ) ( 1 ξ ) ) ,
f ( β , q , α ) = D ̃ I [ ( β q ) e i , q e i , α q ] Φ ϵ ( q e i ) = 0 1 d ξ P ( ξ ) δ ( α V i ( ξ ) ( 1 ξ ) γ ( ξ ) ) exp { x Ω k g 2 q 2 γ 2 ( ξ ) } sin x q 2 k γ ( ξ ) ,
P ( ξ ) = C n 2 ( ξ ) [ ( 1 ξ ) γ ( ξ ) ] 5 3 sinh ( 2 Ω g 2 β γ ( ξ ) ) ,
g ( β , η , α ) = 0 2 q f ( β , q , α ) exp ( i x k η q 2 ) d q .
g ( β , η , α ) = k x 0 1 d ξ P ( ξ ) γ ( ξ ) δ ( α V I ( ξ ) ( 1 ξ ) γ ( ξ ) ) 1 [ 1 γ 2 ( ξ ) ( η + i Ω g 2 γ 2 ( ξ ) ) 2 ] .
η 2 = 1 γ 2 ( ξ ) ( 1 Ω 2 g 4 γ 2 ( ξ ) ) .
α V i ( ξ ) ( 1 ξ ) γ ( ξ ) = 0 .
F ̃ I ( q , ω ) = 2 π k 2 x U 0 4 Φ ϵ ( q ) 0 1 d ξ C n 2 ( ξ ) sin 2 ( x 2 k ( 1 ξ ) q 2 ) δ ( ω + q V ( ξ ) ) .
f ( q , ω ) = F ̃ I ( q i e i , ω ) 2 π k 2 x U 0 4 Φ ϵ ( q ) = 0 1 d ξ C n 2 ( ξ ) sin 2 ( x 2 k ( 1 ξ ) q i 2 ) δ ( ω + q i V i ( ξ ) ) .
δ ( φ ( ξ ) ) = j δ ( ξ ξ j ) ] φ ( ξ j ) ,
f ( q i , α q i ) = j C n 2 ( ξ j ) sin 2 ( x 2 k ( 1 ξ j ) q i 2 ) 1 q i V i ( ξ ) ξ = ξ j .
g i ( η 2 , α ) = 0 d q i q i 2 f ( q i , α q i ) cos ( η 2 q i 2 ) .
g i ( η 2 , α ) = j C n 2 ( ξ j ) V i ( ξ ) ξ j π 8 [ 2 δ ( η 2 ) δ ( η 2 + x k ( 1 ξ j ) ) δ ( η 2 x k ( 1 ξ j ) ) ] .
F ̃ I ( q , ω ) = 2 π k 2 x U 0 4 Ω 4 0 1 d ξ Φ ϵ ( q ξ ) C n 2 ( ξ ) ξ 2 sin 2 ( x 2 k ξ 1 ( 1 ξ ) q 2 ) δ ( ω + q V ( ξ ) ξ ) ,
f ( q , ω ) = j ξ j 5 3 C n 2 ( ξ j ) q i ξ ( V i ( ξ ) ξ ) ξ = ξ j sin 2 ( x 2 k ξ j 1 ( 1 ξ j ) q i 2 ) ,
g i ( η 2 , α ) = 0 q i 2 f ( q i , α q i ) cos ( η 2 q i 2 ) d q i = j ξ j 5 3 C n 2 ( ξ j ) ξ ( V i ( ξ ) ξ ) ξ = ξ j π 8 [ 2 δ ( η 2 ) δ ( η 2 + x k ξ j 1 ( 1 ξ j ) ) δ ( η 2 x k ξ j 1 ( 1 ξ j ) ) ] ,
ω q i = V i ( ξ ) ξ
Δ V 2 4 x 2 k 2 R 0 2 T 0 2 + ( T 0 2 + V max 2 R 0 2 ) ( q 0 2 + V max 2 ω 0 2 )
Δ x 1.35 k ( 4 x 2 k 2 R 0 2 + q 0 2 + V max 2 ω 0 2 )
π k 2 x U 0 4 Φ ϵ ( q 0 ) C n 2 ( ξ ) = F N ( q 0 ) ,

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