Abstract

The problem of fluctuations of the laser radiation field retroreflected from a target in a turbulent atmosphere is formulated based on the equations of statistical moments of the location Green function. The asymptotic methods for solving the above equations were used to investigate spatial coherence, mean value, and normalized variance and covariance of retroreflected radiation intensity in the case of weak and strong turbulence. The reflections of plane and spherical waves from a mirror and a diffuse target were considered. It is ascertained that, in some cases, the scales of the mutual coherence function of the retroreflected field were statistically inhomogeneous. The effect of amplification of fluctuation intensity is the strongest for a point reflector. The level of the residual covariance strongly depends on the field at the transmitting aperture, the type of the target, and its dimensions.

© 1984 Optical Society of America

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References

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  1. V. A. Banakh, V. L. Mironov, presented at Second All-Union Meeting on Atmospheric Optics, Tomsk, USSR, 1980.
  2. K. S. Gochelashvili, V. I. Shishov, Kvant. Elektron. 8, 1953–1956 (1981).
  3. V. P. Aksenov, V. L. Mironov, J. Opt. Soc. Am. 69, 1609–1614 (1979).
    [CrossRef]
  4. S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Introduction to Statistical Radiophysics, Part II (Nauka, Moscow, 1978).
  5. K. S. Gochelashvili, V. I. Shishov, “Waves in randomly inhomogeneous media,” in Results in Science and Technology. Radiophysics. Physical Foundations of Electronics (Acoustic. T.I.M. VINITI, 1981).
  6. V. I. Gel’fgat, Akust. Zh. 22, 123–124 (1976).
  7. V. P. Aksenov, V. A. Banakh, V. L. Mironov, VINITI, May21, 1981.
  8. J. F. Molyneux, J. Opt. Soc. Am. 61, 369–377 (1971).
    [CrossRef]
  9. I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 18, 1660–1666 (1975).
  10. I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 21, 1194–1201 (1978).
  11. A. K. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Laser Radiation in a Turbulent Atmosphere, V. I. Tatarskii, ed. (Nauka, Moscow, 1976).
  12. M. H. Lee, J. F. Holmes, R. Kerr, J. Opt. Soc. Am. 66, 1164–1172 (1976).
    [CrossRef]
  13. M. G. Miller, A. M. Schneiderman, P. F. Killer, J. Opt. Soc. Am. 65, 779–785 (1975).
    [CrossRef]
  14. M. Born, E. Wolf, Foundations of Optics (Nauka, Moscow, 1973).
  15. A. G. Vinogradov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 1584–1586 (1974).
  16. A. G. Vinogradov, Yu. A. Kravtsov, V. I. Tatarskii, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 1064–1070 (1973).
  17. M. S. Belen’kii, V. L. Mironov, Kvant. Elektron. 1, 2253–2263 (1974).
  18. V. P. Aksenov, V. L. Mironov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 22, 141–149 (1979).
  19. A. B. Krupnik, A. I. Saichev, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 24, 1234–1238 (1981).
  20. A. G. Vinogradov, Yu. A. Kravtsov, in “VIth All-Union Symposium on Diffraction and Propagation of Waves (Erevan, Moscow, 1973), pp. 294–296.
  21. G. Ya. Patrushev, Kvant. Elektron. 5, 2342–2347 (1978).
  22. V. A. Banakh, V. M. Buldakov, V. L. Mironov, Opt. Spektrosk. 54, 1054–1059 (1983).
  23. V. L. Mironov, Laser Beam Propagation in a Turbulent Atmosphere (Nauka, Novosibirsk, 1981).
  24. V. U. Zavorotnii, V. I. Klyatskin, V. I. Tatarskii, Zh. Eksp. Theor. Fiz. 73, 481–497 (1977).
  25. K. S. Gochelashvili, V. I. Shishov, Zh. Eksp. Teor. Fiz. 74, 1974–1978 (1978).

1983 (1)

V. A. Banakh, V. M. Buldakov, V. L. Mironov, Opt. Spektrosk. 54, 1054–1059 (1983).

1981 (3)

A. B. Krupnik, A. I. Saichev, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 24, 1234–1238 (1981).

K. S. Gochelashvili, V. I. Shishov, Kvant. Elektron. 8, 1953–1956 (1981).

V. P. Aksenov, V. A. Banakh, V. L. Mironov, VINITI, May21, 1981.

1979 (2)

V. P. Aksenov, V. L. Mironov, J. Opt. Soc. Am. 69, 1609–1614 (1979).
[CrossRef]

V. P. Aksenov, V. L. Mironov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 22, 141–149 (1979).

1978 (3)

K. S. Gochelashvili, V. I. Shishov, Zh. Eksp. Teor. Fiz. 74, 1974–1978 (1978).

G. Ya. Patrushev, Kvant. Elektron. 5, 2342–2347 (1978).

I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 21, 1194–1201 (1978).

1977 (1)

V. U. Zavorotnii, V. I. Klyatskin, V. I. Tatarskii, Zh. Eksp. Theor. Fiz. 73, 481–497 (1977).

1976 (2)

1975 (2)

M. G. Miller, A. M. Schneiderman, P. F. Killer, J. Opt. Soc. Am. 65, 779–785 (1975).
[CrossRef]

I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 18, 1660–1666 (1975).

1974 (2)

M. S. Belen’kii, V. L. Mironov, Kvant. Elektron. 1, 2253–2263 (1974).

A. G. Vinogradov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 1584–1586 (1974).

1973 (1)

A. G. Vinogradov, Yu. A. Kravtsov, V. I. Tatarskii, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 1064–1070 (1973).

1971 (1)

Aksenov, V. P.

V. P. Aksenov, V. A. Banakh, V. L. Mironov, VINITI, May21, 1981.

V. P. Aksenov, V. L. Mironov, J. Opt. Soc. Am. 69, 1609–1614 (1979).
[CrossRef]

V. P. Aksenov, V. L. Mironov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 22, 141–149 (1979).

Banakh, V. A.

V. A. Banakh, V. M. Buldakov, V. L. Mironov, Opt. Spektrosk. 54, 1054–1059 (1983).

V. P. Aksenov, V. A. Banakh, V. L. Mironov, VINITI, May21, 1981.

V. A. Banakh, V. L. Mironov, presented at Second All-Union Meeting on Atmospheric Optics, Tomsk, USSR, 1980.

Belen’kii, M. S.

M. S. Belen’kii, V. L. Mironov, Kvant. Elektron. 1, 2253–2263 (1974).

Born, M.

M. Born, E. Wolf, Foundations of Optics (Nauka, Moscow, 1973).

Buldakov, V. M.

V. A. Banakh, V. M. Buldakov, V. L. Mironov, Opt. Spektrosk. 54, 1054–1059 (1983).

Gel’fgat, V. I.

V. I. Gel’fgat, Akust. Zh. 22, 123–124 (1976).

Gochelashvili, K. S.

K. S. Gochelashvili, V. I. Shishov, Kvant. Elektron. 8, 1953–1956 (1981).

K. S. Gochelashvili, V. I. Shishov, Zh. Eksp. Teor. Fiz. 74, 1974–1978 (1978).

K. S. Gochelashvili, V. I. Shishov, “Waves in randomly inhomogeneous media,” in Results in Science and Technology. Radiophysics. Physical Foundations of Electronics (Acoustic. T.I.M. VINITI, 1981).

Gurvich, A. K.

A. K. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Laser Radiation in a Turbulent Atmosphere, V. I. Tatarskii, ed. (Nauka, Moscow, 1976).

Holmes, J. F.

Kerr, R.

Khmelevtsov, S. S.

A. K. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Laser Radiation in a Turbulent Atmosphere, V. I. Tatarskii, ed. (Nauka, Moscow, 1976).

Killer, P. F.

Klyatskin, V. I.

V. U. Zavorotnii, V. I. Klyatskin, V. I. Tatarskii, Zh. Eksp. Theor. Fiz. 73, 481–497 (1977).

Kon, A. I.

A. K. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Laser Radiation in a Turbulent Atmosphere, V. I. Tatarskii, ed. (Nauka, Moscow, 1976).

Kravtsov, Yu. A.

A. G. Vinogradov, Yu. A. Kravtsov, V. I. Tatarskii, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 1064–1070 (1973).

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Introduction to Statistical Radiophysics, Part II (Nauka, Moscow, 1978).

A. G. Vinogradov, Yu. A. Kravtsov, in “VIth All-Union Symposium on Diffraction and Propagation of Waves (Erevan, Moscow, 1973), pp. 294–296.

Krupnik, A. B.

A. B. Krupnik, A. I. Saichev, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 24, 1234–1238 (1981).

Lee, M. H.

Miller, M. G.

Mironov, V. L.

V. A. Banakh, V. M. Buldakov, V. L. Mironov, Opt. Spektrosk. 54, 1054–1059 (1983).

V. P. Aksenov, V. A. Banakh, V. L. Mironov, VINITI, May21, 1981.

V. P. Aksenov, V. L. Mironov, J. Opt. Soc. Am. 69, 1609–1614 (1979).
[CrossRef]

V. P. Aksenov, V. L. Mironov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 22, 141–149 (1979).

M. S. Belen’kii, V. L. Mironov, Kvant. Elektron. 1, 2253–2263 (1974).

A. K. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Laser Radiation in a Turbulent Atmosphere, V. I. Tatarskii, ed. (Nauka, Moscow, 1976).

V. A. Banakh, V. L. Mironov, presented at Second All-Union Meeting on Atmospheric Optics, Tomsk, USSR, 1980.

V. L. Mironov, Laser Beam Propagation in a Turbulent Atmosphere (Nauka, Novosibirsk, 1981).

Molyneux, J. F.

Patrushev, G. Ya.

G. Ya. Patrushev, Kvant. Elektron. 5, 2342–2347 (1978).

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Introduction to Statistical Radiophysics, Part II (Nauka, Moscow, 1978).

Saichev, A. I.

A. B. Krupnik, A. I. Saichev, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 24, 1234–1238 (1981).

Schneiderman, A. M.

Shishov, V. I.

K. S. Gochelashvili, V. I. Shishov, Kvant. Elektron. 8, 1953–1956 (1981).

K. S. Gochelashvili, V. I. Shishov, Zh. Eksp. Teor. Fiz. 74, 1974–1978 (1978).

K. S. Gochelashvili, V. I. Shishov, “Waves in randomly inhomogeneous media,” in Results in Science and Technology. Radiophysics. Physical Foundations of Electronics (Acoustic. T.I.M. VINITI, 1981).

Tatarskii, V. I.

V. U. Zavorotnii, V. I. Klyatskin, V. I. Tatarskii, Zh. Eksp. Theor. Fiz. 73, 481–497 (1977).

A. G. Vinogradov, Yu. A. Kravtsov, V. I. Tatarskii, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 1064–1070 (1973).

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Introduction to Statistical Radiophysics, Part II (Nauka, Moscow, 1978).

Vinogradov, A. G.

A. G. Vinogradov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 1584–1586 (1974).

A. G. Vinogradov, Yu. A. Kravtsov, V. I. Tatarskii, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 1064–1070 (1973).

A. G. Vinogradov, Yu. A. Kravtsov, in “VIth All-Union Symposium on Diffraction and Propagation of Waves (Erevan, Moscow, 1973), pp. 294–296.

Wolf, E.

M. Born, E. Wolf, Foundations of Optics (Nauka, Moscow, 1973).

Yakushkin, I. G.

I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 21, 1194–1201 (1978).

I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 18, 1660–1666 (1975).

Zavorotnii, V. U.

V. U. Zavorotnii, V. I. Klyatskin, V. I. Tatarskii, Zh. Eksp. Theor. Fiz. 73, 481–497 (1977).

Akust. Zh. (1)

V. I. Gel’fgat, Akust. Zh. 22, 123–124 (1976).

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (6)

I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 18, 1660–1666 (1975).

I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 21, 1194–1201 (1978).

A. G. Vinogradov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 1584–1586 (1974).

A. G. Vinogradov, Yu. A. Kravtsov, V. I. Tatarskii, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 1064–1070 (1973).

V. P. Aksenov, V. L. Mironov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 22, 141–149 (1979).

A. B. Krupnik, A. I. Saichev, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 24, 1234–1238 (1981).

J. Opt. Soc. Am. (4)

Kvant. Elektron. (3)

K. S. Gochelashvili, V. I. Shishov, Kvant. Elektron. 8, 1953–1956 (1981).

M. S. Belen’kii, V. L. Mironov, Kvant. Elektron. 1, 2253–2263 (1974).

G. Ya. Patrushev, Kvant. Elektron. 5, 2342–2347 (1978).

Opt. Spektrosk. (1)

V. A. Banakh, V. M. Buldakov, V. L. Mironov, Opt. Spektrosk. 54, 1054–1059 (1983).

VINITI (1)

V. P. Aksenov, V. A. Banakh, V. L. Mironov, VINITI, May21, 1981.

Zh. Eksp. Teor. Fiz. (1)

K. S. Gochelashvili, V. I. Shishov, Zh. Eksp. Teor. Fiz. 74, 1974–1978 (1978).

Zh. Eksp. Theor. Fiz. (1)

V. U. Zavorotnii, V. I. Klyatskin, V. I. Tatarskii, Zh. Eksp. Theor. Fiz. 73, 481–497 (1977).

Other (7)

A. K. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Laser Radiation in a Turbulent Atmosphere, V. I. Tatarskii, ed. (Nauka, Moscow, 1976).

V. A. Banakh, V. L. Mironov, presented at Second All-Union Meeting on Atmospheric Optics, Tomsk, USSR, 1980.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Introduction to Statistical Radiophysics, Part II (Nauka, Moscow, 1978).

K. S. Gochelashvili, V. I. Shishov, “Waves in randomly inhomogeneous media,” in Results in Science and Technology. Radiophysics. Physical Foundations of Electronics (Acoustic. T.I.M. VINITI, 1981).

V. L. Mironov, Laser Beam Propagation in a Turbulent Atmosphere (Nauka, Novosibirsk, 1981).

A. G. Vinogradov, Yu. A. Kravtsov, in “VIth All-Union Symposium on Diffraction and Propagation of Waves (Erevan, Moscow, 1973), pp. 294–296.

M. Born, E. Wolf, Foundations of Optics (Nauka, Moscow, 1973).

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Figures (7)

Fig. 1
Fig. 1

(a) The modulus of the complex coherence degree of the reflected spherical wave D s ( L / k ) = 0.1. 1 is the point reflector (Ω2 = 10−3); the same curve describes the coherence of the spherical wave at a single passage through a path of length L. 2 and 3 are infinite reflectors (Ωr = 103): 2, ρ1, ρ 2 L / k(the coherence of a spherical wave at a single passage through a path of length 2L is described by the same curve); 3, R = 0 or R = ρ/2. (b) The modulus of a complex spatial-coherence degree of the reflected plane wave, D s ( L / k ) = 0.1. 1 is the point reflector (Ω2 = 10−3); 2 is the infinite reflector (Ω2 = 103); 3 is the plane wave at a single passage through a path of length 2L.

Fig. 2
Fig. 2

Dependence of the degree of coherence of the reflected spherical wave [ Ω D s - 6 / 5 ( L / k )] on the position of the center of gravity of observation points [ D s ( L / k ) 1]. 1, 2, 3: infinite reflector [ Ω r D s 6 / 5 ( L / k )] 4, 5, 6: point reflector [ Ω r D s ( - 6 / 5 ( L / k )]; 1, 4: |R| ≫ ρs (|ρ1|, |ρ2| ≫ ρs); 2,5: R = ρ/2; 3,6: R =0.

Fig. 3
Fig. 3

Normalized variance of the weak intensity fluctuations as a function of the receiver position. 1–4: mirror reflector; 5, 6: diffuse target (Ωr = 1). In this case the value σI,T2/β02 is plotted along the axis of ordinates, where σI,T2 = σR,I2 − 1. 1, 3, 5: Ω = 10−3; 2, 4, 6: Ω = 103; 1, 2: Ωr = 10−3; 3, 4: Ωr. = 103.

Fig. 4
Fig. 4

Saturation level of the normalized variance of the strong intensity fluctuations as a function of the distance to the center of the receiver. 1, a spherical wave [ Ω D s - 6 / 5 ( L / k )] and a point reflector [ Ω r D s - 6 / 5 ( L / k )]; 2, a spherical wave and an infinite mirror reflector [ Ω r D s 6 / 5 ( L / k )]; 3, a plane wave [ Ω D s 6 / 5 ( L / k )] and a point reflector.

Fig. 5
Fig. 5

Normalized variance of the strictly backward-reflected wave intensity fluctuations as a function of the parameter D s ( L / k ). 1, 2: spherical wave; 3, 4: plane wave; 1, 3: point reflector; 2, 4: infinite plane mirror; 5, 6: diffuse target (Ωr = 1); 5: spherical wave; 6: plane wave; 7: σR,I = 1.

Fig. 6
Fig. 6

Spatial correlation coefficient of the reflected wave weak intensity fluctuations R = 0. 1–4: mirror reflector; 1, 3: Ω = 10−3; 2, 4: Ω = 103; 1, 2: Ωr = 10−3; 3, 4: Ω r = 103; 5, 6: diffuse target, β02 = 0.1; 5: Ωr = 1, Ω = 10−3, or Ω = 103; 6: Ωr = 103, Ω = 10−1 or Ω = 10.

Fig. 7
Fig. 7

Correlation coefficient of strong intensity fluctuations of the reflected spherical wave depending on the way of separating the observation points. 1, 2, 3: point reflector; 4, 5, 6: infinite mirror plane; 1,4: |R| ≫ ρs. (|ρ1|, |ρ2| ≫ ρs); 2, 5: R = ρ/2; 3, 6: R = 0.

Equations (85)

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u R ( x 0 , ρ ) = d 2 t d 2 r u 0 ( t ) K ( r ) G ( x , x 0 ; r , r ; ρ , t ) ,
- 2 i k G x + ( Δ ρ + Δ t ) G + k 2 [ 1 ( x , t ) + 1 ( x , ρ ) ] G = 0
G ( x , x ; r , τ ; ρ , t ) = δ ( r - ρ ) δ ( τ - t ) .
Γ R , 2 n ( x 0 , ρ 2 n ) = u R ( x 0 , ρ 1 ) u R * ( x 0 , ρ 2 ) u R ( x 0 , ρ 2 n - 1 ) × u R ( x 0 , ρ 2 n - 1 )
Γ R , 2 n ( x 0 , ρ 2 n ) = d 8 t 2 n , d 8 r 2 n u 2 n ( x 0 , t 2 n ) K 2 n ( r 2 n ) × G 2 n ( x , x 0 ; r 2 n , r 2 n ; ρ 2 n , t 2 n ) ,
G 2 n ( x , x 0 ; r 2 n , τ 2 n ; ρ 2 n , t 2 n ) = j = 1 n G ( x , x 0 ; r 2 j - 1 , τ 2 j - 1 ; ρ 2 j - 1 , t 2 j - 1 ) × G * ( x , x 0 ; r 2 j , τ 2 j , ρ 2 j , t 2 j ) , K 2 n ( r 2 n ) = j = 1 n K ( r 2 j - 1 ) K * ( r 2 j ) , u 2 n ( x 0 , t 2 n ) = j = 1 n u ( t 2 j - 1 ) u * ( t 2 j ) .
σ R , I 2 ( x 0 , R ) = B R , I ( x 0 ; R , R ) / Γ R , 2 2 ( x 0 , R , R )
b R , I ( x 0 ; ρ 1 , ρ 2 ) = B R , I ( x 0 ; ρ 1 , ρ 2 ) / i = 1 , 2 B R , I 1 / 2 ( x 0 ; ρ i , ρ i )
B R , I ( x 0 ; ρ 1 , ρ 2 ) = Γ R , 4 ( x 0 ; ρ 1 , ρ 1 , ρ 2 , ρ 2 ) - Γ R , 2 ( x 0 ; ρ 1 , ρ 1 ) Γ R , 2 ( x 0 ; ρ 2 , ρ 2 ) .
1 ( x , ρ ) 1 ( x , ρ ) = 2 π δ ( x - x ) × d 2 χ Φ ( x , χ ) exp [ i χ ( ρ - ρ ) ]
ζ G 2 n + i 2 j = 1 2 n ( - 1 ) j + 1 ( Δ t j + Δ ρ j ) G 2 n - D n ( L / k ) f 2 n ( ρ 2 n , t 2 n ) G 2 n = 0 ,
f 2 n ( ρ 2 n , t 2 n ) = 4 3 [ j = 1 2 n - 1 i = 2 , j < i 2 n ( - 1 ) i + j ( ρ j - ρ i 5 / 3 + t j - t i 5 / 3 ) + j = 1 2 n i = 1 2 n ( - 1 ) i + j ρ j - t i 5 / 3 ] ,
G 2 n ( ζ , ζ ; r 2 n , τ 2 n ; ρ 2 n , t 2 n ) = ( L / k ) 4 n i = 1 n δ ( τ 2 j - 1 - t 2 j - 1 ) δ ( r 2 j - 1 - ρ 2 j - 1 ) × δ * ( τ 2 j - t 2 j ) δ * ( r 2 j - ρ 2 j ) .
G 2 n ( ζ , ζ 0 ; r 2 n , τ 2 n ; ρ 2 n , t 2 n ) = k = 1 n P ( r 2 k - 1 , ρ 2 k - 1 ; ζ - ζ 0 ) P ( r 2 k - 1 , t 2 k - 1 ; ζ - ζ 0 ) × P * ( r 2 k , ρ 2 k ; ζ - ζ 0 ) P * ( r 2 k , t 2 k ; ζ - ζ 0 ) + ( L / k ) 4 n × D s ( L / k ) ζ 0 ζ d ξ k = 1 n d 2 ρ 2 k - 1 d 2 ρ 2 k d 2 t 2 k - 1 × d 2 t 2 k P ( ρ 2 k - 1 , ρ 2 k - 1 , ζ - ξ ) P ( t 2 k - 1 , t 2 k - 1 ; ζ - ξ ) × P * ( ρ 2 k , ρ 2 k ; ζ - ξ ) P * ( t 2 k , t 2 k ; ζ - ξ ) f 2 n ( ρ 2 n , t 2 n ) × G 2 n ( ζ , ξ ; r 2 n , r 2 n ; ρ 2 n , t 2 n ) ,
u 0 ( t ) = u 0 exp ( - t 2 / 2 a 2 ) ,
K ( r ) = - exp ( - r 2 / 2 a r 2 )
K ( r ) ¯ = 0
K ( r 1 ) K * ( r 2 ) ¯ = K ( r 1 ) 2 δ ( r 1 - r 2 ) ,
u ( x , r 1 ) u * ( x , r 2 ) = β I ( x , r 1 ) δ ( r 1 - r 2 ) K ( r 1 ) 2 ,
K ( r 1 ) K * ( r 2 ) K ( r 3 ) K * ( r 4 ) ¯ = K ( r 1 ) K * ( r 2 ) ¯ · K ( r 3 ) K * ( r 4 ) ¯ + K ( r 1 ) K * ( r 4 ) ¯ · K ( r 2 ) K * ( r 3 ) ¯ = K ( r 1 ) 2 K ( r 3 ) 2 δ ( r 1 - r 2 ) δ ( r 3 - r 4 ) + K ( r 1 ) 2 K ( r 3 ) 2 δ ( r 1 - r 4 ) δ ( r 2 - r 3 ) .
γ R ( R , ρ ) = Γ R , 2 ( R , ρ ) Γ R , 2 1 / 2 ( R + ρ / 2 , 0 ) Γ R , 2 1 / 2 ( R - ρ / 2 , 0 ) ,
γ R ( ρ ) = 1 - ½ D s ( L / k ) ( ρ / L / k ) 5 / 3 + 0 [ D S 2 ( L / k ) ] ,
Γ R , 2 ( R , ρ ) = u 0 2 a r 2 L 2 Ω 2 exp ( i R f ρ f - 1 4 Ω r ρ f 2 ) × { 1 + 1 2 ( 1 u 0 Ω ) 4 [ B I , c Φ ( R + ρ / 2 ) + B I , c Φ * ( R - ρ / 2 ) ] - 1 2 D s ( L / k ) ρ f 5 / 3 + 0 ( D s 2 ( L / k ) } ,
ρ f = ρ k / L ,             R f = R k / L ,             ρ f = ρ f ,             R f = R f , Re B I , c Φ ( z ) = 7.92 D s ( L / k ) u 0 4 Ω 4 × 0 1 d ξ ( Re { [ - i ξ ( 1 - ξ ) ] 5 / 6 F 1 1 [ - , 1 ; i z 2 k / L × ( 1 - ξ ) / ( 4 ξ ) ] } - 2 - 5 / 3 Γ - 1 ( ¹¹ / ) [ z 2 ( 1 - ξ ) 2 k / L ] 5 / 6 )
Γ R , 2 ( R , ρ ) = u 0 2 Ω r k L exp ( i R f ρ f - 1 4 Ω R ρ f 2 ) × ( 1 + 3.96 D s ( L / k ) 0 1 d ξ j = 1 , 2 × { r j 5 / 6 F 1 1 ( - , 1 ; ( 1 - ξ ) 2 ( R j + i Ω r ρ f / 2 ) 2 4 r j - v j 5 / 6 × F 1 1 [ - , 1 ; - ( R j + i Ω r ρ f / 2 ) 2 / ( 4 v j ) ] } - 1 2 D s ( L / k ) ρ f 5 / 3 ) ,
γ R ( ρ ) = γ p ( ρ ) - 7.92 D s ( L / k ) × 0 1 d ξ Re { [ i ( 1 - ξ ) ] 5 / 6 [ F 1 1 ( - 5 6 , 1 ; i ρ f 2 4 ( 1 - ξ ) ) - 1 ] } ,
Γ R , 2 ( R , ρ ) = u 0 2 Ω 2 4 e i / 2 R f ρ f { γ s ( ρ ) + 1 2 ( 1 u 0 Ω ) 4 × [ B I , c Φ ( R + ρ / 2 ) + B I , c Φ ( R - ρ / 2 ) ] - 4 3 D s ( L / k ) × 0 1 d ξ [ R f ( 1 - ξ ) + ρ f / 2 5 / 3 + R f ( 1 - ξ ) - ρ f / 2 5 / 3 - ( R f + ρ f / 2 5 / 3 + R f - ρ f / 2 5 / 3 ) ( 1 - ξ ) 5 / 3 } ,
γ p ( ρ ) = 1 - 8 3 D s ( L / k ) ρ f 5 / 3
γ s ( ρ ) = 1 - D s ( L / k ) ρ f 5 / 3
Γ R , 2 ( R , ρ ) = u 0 2 ( Ω 2 ) 2 e i / 2 R f ρ f ( exp [ - D s ( L / k ) ρ f 5 / 3 ] + 0 { D s - 2 / 5 ( L / k ) exp [ - D s ( L / k ) ρ f 5 / 3 - D s - 1 ( L / k ) R f 5 / 3 ] } + exp { - 4 3 D s ( L / k ) × [ 0 1 d ξ ( R f ξ - ( R f + ρ f / 2 ) 5 / 3 + R f ξ - ( R - ρ f / 2 ) 5 / 3 ) ] } + 0 { D s - 2 / 5 ( L / k ) × exp [ - D s ( L / k ) ρ f 5 / 3 - D s ( L / k ) R f 5 / 3 ] } ) .
Γ R , 2 ( R , ρ ) = u 0 2 Ω r 2 Ω 2 e i R f ρ f ( exp [ - 1 2 D s ( L / k ) ρ f 5 / 3 ] + 0 { D s - 2 / 5 ( L / k ) exp [ - D s ( L / k ) ρ f 5 / 3 - D s - 1 ( L / k ) R f 5 / 3 ] } + exp { - 1 2 D s ( L / k ) × [ R f + ρ f / 2 5 / 3 + R f - ρ f / 2 5 / 3 ] } + 0 { D s - 2 / 5 ( L / k ) exp [ - D s ( L / k ) ρ f 5 / 3 - D s ( L / k ) R f 5 / 3 ] } ) .
ρ C R = ( 1 2 ) 3 / 2 ρ s ,             Ω R D s 6 / 5 ( L / k ) , ρ C R = ρ s ,             Ω r D s - 6 / 5 ( L / k ) .
ρ C R 1.42 ρ p , Ω r D s 6 / 5 ( L / k ) , ρ C R 2.15 ρ p , Ω r D s - 6 / 5 ( L / k ) ,
γ R ( ρ ) = e - ( ρ / ρ s ) 5 / 3 + e - 2 ( ρ / ( 2 ρ s ) ) 5 / 3 1 + e - ( ρ / ( 2 ρ s ) ) 5 / 3 ,
γ R ( ρ ) = e - 2 ( ρ / ρ s ) 5 / 3 + e - 16 / 3 ( ρ / ( 2 ρ s ) ) 5 / 3 1 + e - 2 ( ρ / ρ s ) 5 / 3 , Ω r D s - 6 / 5 ( L / k ) .
ρ C R 1.6 ρ p , Ω r D s 6 / 5 ( L / k ) , ρ C R 2.46 ρ p , Ω r D s - 6 / 5 ( L / k ) .
Γ R , 2 ( 0 ) = u 0 2 ( Ω 2 ) 2 { 2 + 2.6 [ D s ( L / k ) ] - 2 / 5 + 0 [ D s - 4 / 5 ( L / k ) ] } , Ω D s - 6 / 5 ( L / k ) , Ω r D s 6 / 5 ( L / k ) , Γ R , 2 ( 0 ) = u 0 2 Ω r 2 Ω { 2 + 2.6 [ D s ( L / k ) ] - 2 / 5 + 0 [ D s - 4 / 5 ( L / k ) ] } , Ω D s - 6 / 5 ( L / k ) , Ω r D s - 6 / 5 ( L / k ) .
γ R ( ρ ) e - 1 / 4 Ω r ρ f 2
Γ R , 2 ( R , ρ ) = u 0 2 ( { e - 8 / 3 D s ( L / k ) ρ f 5 / 3 + 0 [ ρ f 2 D s - 2 / 5 ( L / k ) e - D s ( L / k ) ρ f 5 / 3 ] } + 0 [ D s - 12 / 5 ( L / k ) e - D s - 1 ( L / k ) ρ f 5 / 3 ] ) , Ω r D s 6 / 5 ( L / k ) ,
Γ R , 2 ( R , ρ ) = u 0 2 Ω r 2 ( { e - 1 / 2 D s ( L / k ) ρ f 5 / 3 + 0 [ D s - 2 / 5 ( L / k ) e - D s ( L / k ) ρ f 5 / 3 - D s - 1 ( L / k ) R f 5 / 3 ] } + 0 [ D s - 12 / 5 ( L / k ) e - D s - 1 ( L / k ) ρ f 5 / 3 - D s - 1 ( L / k ) R f 5 / 3 ] ) , Ω r D s - 6 / 5 ( L / k ) .
Γ R , 2 ( 0 ) = u 0 2 a r 2 L 2 { 1 + 0.15 D s - 2 / 5 ( L / k ) + 0 [ D s - 4 / 5 ( L / k ) ] } , Ω r D s 6 / 5 ( L / k ) , Γ R , 2 ( 0 ) = u 0 2 a r 2 L 2 { 1 + 0 [ D s ( L / k ) ] } Ω r - 7 / 6 ] } ,             Ω > Ω r , Ω r D s 6 / 5 ( L / k ) .
σ R , I 2 ( R ) = σ I , c Φ 2 + σ I , b 2 + 2 B I , b ( R ) ,
B I , b ( R ) = 0.57 D s ( L / k ) 0 1 d ξ [ ( 1 - ξ 2 ) 5 / 6 F 1 1 × ( - , 1 ; i R f 2 ( 1 - ξ ) 1 + ξ ) - ( 1 - ξ ) 5 / 3 F 1 1 ( - , 1 ; i R f 2 2 ) ] .
σ R , I 2 ( R ) = σ I , b 2 ( 2 L ) + 2 B I , b ( R ) ,
σ I , b 2 ( 2 L ) = σ I , c Φ 2 ( 2 L ) = 4.1 D s ( L / k ) × 0 1 d ξ [ ξ ( 1 - ξ / 2 ) ] 5 / 6 = 1.6 D s ( L / k )
σ I , b 2 ( 2 L ) = σ I , n v 2 ( 2 L ) = β 0 2 × 0 1 d ξ [ ( 2 - ξ ) 5 / 6 + ξ 5 / 6 ] = 4 D s ( L / k )
B I , b ( R ) = B I , c Φ ( R ) = 7.92 D s ( L / k ) × Re 0 1 d ξ { [ i ξ ( 1 - ξ / 2 ) ] 5 / 6 F 1 1 ( - , 1 ; i R f 2 ( 1 - ξ ) 2 4 ξ ( 1 - ξ / 2 ) ) - Γ - 1 ( 11 / 6 ) [ R f ( 1 - ξ ) / 2 ] 5 / 3 }
B I , b ( R ) = 4.1 D s ( L / k ) 0 1 d ξ [ 1 - ( 1 - ξ ) 5 / 6 ] = 1.87 D s ( L / k )
σ R , I 2 ( R ) = 1 + 4 σ I , g 2 + 4 B I , g ( R ) .
B I , g ( R ) = 7.92 D s ( L / k ) 0 1 d ξ { Re [ p 5 / 6 × F 1 1 ( - 5 6 , 1 ; - R f 2 ( 1 - ξ ) 2 4 p ) ] - q 5 / 6 F 1 1 ( - 5 6 , 1 ; R f 2 ( 1 - ξ ) 2 4 q ) } , σ I , g 2 = B I , g ( 0 ) ,             p = q - i ξ ( 1 - ξ ) , q = 1 2 ξ 2 Ω r ,
σ R , I 2 ( R ) = 1 + 2 σ I , g 2 + 2 σ I , p 2 + 4 B I , p ( R ) ,
σ I , p 2 = 7.92 D s ( L / k ) × 0 1 d ξ { [ 1 2 Ω r + i ( 1 - ξ ) ] 5 / 6 - ( Ω r 2 ) 5 / 6 }
B I , p ( R ) = 7.92 D s ( L / k ) 0 1 d ξ [ s 5 / 6 F 1 1 ( - 5 6 , 1 ; - R f 2 ( 1 - ξ ) 2 4 s ) - t 5 / 6 F 1 1 ( - 5 6 , 1 ; - R f 2 ( 1 - ξ ) 2 4 t ) ] , s = 1 4 ( 1 + ξ 2 ) Ω r - i 2 ( 1 - ξ 2 ) , t = 1 4 ( 1 + ξ 2 ) Ω r - i 2 ( 1 - ξ ) 2 .
σ R , I 2 = 1 + 1.52 D s ( L / k ) + 0 [ D s 2 ( L / k ) ] ,             R L / k ,
σ R , I 2 = 1 + 076 D s ( L / k ) + 0 [ D s 2 ( L / k ) ] ,             R L / k .
σ R , I 2 ( 0 ) = 1 + 2 σ I , g 2 + 2 σ I , p 2 + 4 × 7.92 D s ( L / k ) × 0 1 d ξ ( s 5 / 6 - t 5 / 6 ) ,             σ R , I 2 = 1 + 2 σ I , g 2 + 2 σ I , p 2 ,
σ R , I 2 ( 0 ) = 1 + 1.56 D s ( L / k ) + 0 [ D s 2 ( L / k ) ] ,
σ R , I 2 = 1 + 1.02 D s ( L / k ) + 0 [ D s 2 ( L / k ) ] ,             R L / k .
σ R , I 2 = 1 + 0 [ D s ( L / k ) Ω r - 7 / 6 ] ,
σ R , I 2 ( R ) = 3 + 14 e - D s ( L / k ) R f 5 / 3 + 3 e - D s ( L / k ) R f 5 / 3 1 + e - 2 D s ( L / k ) R f 5 / 3 + 2 e - D s ( L / k ) R f 5 / 3 .
σ R , I 2 ( 0 ) = 5 + 44.2 D s - 2 / 5 ( L / k ) 1 + 2.6 D s - 2 / 5 ( L / k ) + 0 [ D s - 4 / 5 ( L / k ) ] ,             Ω r , Ω D s - 6 / 5 ( L / k ) , D s ( L / k ) 1 ,
σ R , I 2 ( 0 ) = 1 + 5.72 D s - 2 / 5 ( L / k ) 1 + 2.6 D s - 2 / 5 ( L / k ) + 0 [ D s - 4 / 5 ( L / k ) ] ,             Ω r - 1 , Ω D s - 6 / 5 ( L / k ) , D s ( L / k ) 1 ,
σ R , I 2 ( 0 ) = 3 + 12.2 D s - 2 / 5 ( L / k ) 1 + 0.76 D s - 2 / 5 ( L / k ) + 0 [ D s - 4 / 5 ( L / k ) ] ,             Ω r , Ω - 1 D s - 6 / 5 ( L / k ) , D s ( L / k ) 1 ,
σ R , I 2 ( 0 ) = 1 + 0.78 D s - 2 / 5 ( L / k ) + 0 [ D s - 4 / 5 ( L / k ) ] ,             Ω r , Ω D s 6 / 5 ( L / k ) , D s ( L / k ) 1.
σ R , I 2 ( 0 ) = 11 + 91 D s - 2 / 5 ( L / k ) 1 + 2.6 D s - 2 / 5 ( L / k ) + 0 [ D s - 4 / 5 ( L / k ) ] ,             Ω D s - 6 / 5 ( L / k ) , σ R , I 2 ( 0 ) = 7 + 25.1 D s - 2 / 5 ( L / k ) 1 + 0.76 D s - 2 / 5 ( L / k ) + 0 [ D s - 4 / 5 ( L / k ) ] ,             Ω D s 6 / 5 ( L / k ) .
σ R , I 2 ( 0 ) = 1 + 10.4 D s - 2 / 5 ( L / k ) + 0 [ D s - 4 / 5 ( L / k ) ] ,             Ω D s - 6 / 5 ( L / k ) , σ R , I 2 ( 0 ) = 1 + 4.96 D s - 2 / 5 ( L / k ) + 0 [ D s - 4 / 5 ( L / k ) ] ,             Ω D s 6 / 5 ( L / k ) .
b R , I = σ I , g 2 / ( 1 + 4 σ I , g 2 ) , b R , I = 2 σ I , g 2 / [ ( 1 + 4 σ I , g 2 ) B R , I ( 0 ) ] 1 / 2
b R , I ( ρ ) = e - 1 / 2 Ω r ρ f 2 .
b R , I ( R , ρ ) = 1 3 [ 1 + 2 e - 2 ( ρ / ρ s ) 5 / 3 ] ,
R ρ s ( ρ 1 , ρ 2 ρ s ) , b R , I ( R , ρ ) = 2 5 × 1 + 4 exp [ - 2 ( ρ / ρ s ) 5 / 3 ] 3 + 14 exp [ - 2 ( ρ / ρ s ) 5 / 3 ] + 3 exp [ - 4 ( ρ / ρ s ) 5 / 3 ] ,
b R , I ( R , ρ ) = [ 1 + 2 e - 2 ( ρ / ρ s ) 5 / 3 + 6 e - 2 ( ρ / 2 ρ s ) 5 / 3 + 8 e - ( ρ / ρ s ) 5 / 3 e - 2 ( ρ / 2 ρ s ) 5 / 3 + 3 e - 4 ( ρ / 2 ρ s ) 5 / 3 ] × [ 3 + 14 e - 2 ( ρ / 2 ρ s ) 5 / 3 + 3 e - 4 ( ρ / 2 ρ s ) 5 / 3 ] - 1 ,
b R , I ( R , ρ ) = e - 4 ( ρ / ρ s ) 5 / 3 ,             R ρ s ( ρ 1 , ρ 2 ρ s ) ,
b R , I ( R , ρ ) = 1 2 α 1 / 2 ( ρ ) [ 3 e - 4 ( ρ / ρ s ) 5 / 3 + e - 2 ( ρ / ρ s ) 5 / 3 ] ,             R = ρ / 2 ,
α ( ρ ) = 1 + e - 4 ( ρ / ρ s ) 5 / 3 + 2 e - 2 ( ρ / ρ s ) 5 / 3 , b R , I ( R , ρ ) = 1 α ( ρ / 2 ) [ e - 4 ( ρ / ρ s ) 5 / 3 + e - 4 ( ρ / ρ s ) 5 / 3 + 2 e - 2 ( ρ / ρ s ) 5 / 3 e - 16 / 3 ( ρ / ρ s ) 5 / 3 ] ,
b R , I ( R , ρ ) = b R , I ( ρ ) = exp [ - 4 8 3 ( ρ / ρ s ) 5 / 3 ] ,
b R , I ( R , ρ ) = 1 2 α 1 / 2 ( ρ ) [ 3 e - 1 / 2 Ω r ρ f 2 - 2 ( ρ / ρ s ) 5 / 3 + e - 2 ( ρ / ρ s ) 5 / 3 ] , R = ρ / 2 , b R , I ( R , ρ ) = 1 α ( ρ / 2 ) { exp [ - 1 2 Ω r ρ f 2 - 2 ( ρ / ρ s ) 5 / 3 ] + exp [ - 4 ( ρ 2 ρ s ) 5 / 3 ] + 2 exp [ - 1 2 Ω r ρ f 2 - ( ρ / ρ s ) 5 / 3 - 2 ( ρ 2 ρ s ) 5 / 3 ] } , R = 0 , b R , I ( ρ ) = exp [ - 1 2 Ω r ρ f 2 - 2 ( ρ / ρ s ) 5 / 3 ] ,             ρ 1 , ρ 2 ρ s .
ω ( I ) 1 I exp { - I I } ,
I R n ( x 0 , 0 ) = I 0 - n I n p n ( x , 0 ) I 0 δ p n ( x 0 , 0 ) = I 0 - n I n p 2 n ( x , 0 ) ,
I R n ( x 0 , R ) = I 0 - n I n p n ( x , 0 ) I 0 δ p n ( x 0 , R ) , R R K ,
I n p n ( x , 0 ) = I n p n ( x , R )
I R n ( x 0 , R ) = I 0 - n I n p n ( x , 0 ) 2 .
I n ( x , 0 ) = n ! { 1 + n ( n - 1 ) 2 1.3 [ D s ( L / k ) ] - 2 / 5 } × I n , D s ( L / k ) 1.
I R n ( x 0 , 0 ) = ( 2 n ) ! { 1 + n ( 2 n - 1 ) × 1.3 [ D s ( L / k ) ] - 2 / 5 } I R 0 n .
I R n ( x 0 , 0 ) = n ! I R ( x 0 , 0 ) n ,
I R n ( x 0 , R ) = ( n ! ) 2 { 1 + n ( n - 1 ) 1.3 [ D s ( L / k ) ] - 2 / 5 } I R , 0 n , R R b k ~ ρ s .

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