Abstract

A generalized Huygens–Fresnel integral, valid for optical wave propagation through random inhomogeneities in the presence of any complex optical system characterized by an ABCD ray matrix, is used to derive a general expression for the mutual coherence function (MCF) associated with a Gaussian-beam wave in the weak-fluctuation regime. The mean irradiance obtained from this expression shows excellent agreement with all known asymptotic relations. By introducing a pair of effective beam parameters Θt and Λt that account for additional diffraction on the receiving aperture, resulting from turbulence, the normalized MCF and the related degree of coherence are formally extended into the regime of strong fluctuations. Results for the normalized MCF from this heuristic approach compare well with numerical calculations obtained directly from the formal solution of the parabolic equation. Also, the implied spatial coherence length from this analysis in moderate-to-strong-fluctuation regimes generally agrees more closely with numerical solutions of the parabolic equation than do previous approximate solutions. All calculations are based on the modified von Kármán spectrum for direct comparison with established results.

© 1994 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  25. H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCDoptical systems,” J. Opt. Soc. Am. 10, 316–323 (1993).
    [CrossRef]

1993 (4)

W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
[CrossRef]

L. C. Andrews, W. B. Miller, J. C. Ricklin, “Geometrical representation of Gaussian beams propagating through complex paraxial optical systems,” Appl. Opt. 32, 5918–5929 (1993).
[CrossRef] [PubMed]

L. C. Andrews, S. Vester, C. E. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40, 931–938 (1993).
[CrossRef]

H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCDoptical systems,” J. Opt. Soc. Am. 10, 316–323 (1993).
[CrossRef]

1989 (1)

1987 (1)

1983 (1)

M. S. Belen’kii, V. L. Mironov, “Phase fluctuations when focusing light in a turbulent atmosphere,” Radiophys. Quantum Electron. 12, 1096–1101 (1983).
[CrossRef]

1980 (2)

M. S. Belen’kii, V. L. Mironov, “Mean diffracted rays of an optical beam in a turbulent medium,” J. Opt. Soc. Am. 70, 159–163 (1980).
[CrossRef]

M. S. Belen’kii, V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 595–597 (1980).
[CrossRef]

1977 (1)

M. S. Belen’kii, A. I. Kon, V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287–290 (1977).
[CrossRef]

1976 (1)

1975 (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

1974 (1)

1972 (1)

1971 (4)

1970 (1)

1967 (1)

I. Z. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967).
[CrossRef]

Andrews, L. C.

W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
[CrossRef]

L. C. Andrews, W. B. Miller, J. C. Ricklin, “Geometrical representation of Gaussian beams propagating through complex paraxial optical systems,” Appl. Opt. 32, 5918–5929 (1993).
[CrossRef] [PubMed]

L. C. Andrews, S. Vester, C. E. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40, 931–938 (1993).
[CrossRef]

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (McGraw-Hill, New York, 1992).

Belen’kii, M. S.

M. S. Belen’kii, V. L. Mironov, “Phase fluctuations when focusing light in a turbulent atmosphere,” Radiophys. Quantum Electron. 12, 1096–1101 (1983).
[CrossRef]

M. S. Belen’kii, V. L. Mironov, “Mean diffracted rays of an optical beam in a turbulent medium,” J. Opt. Soc. Am. 70, 159–163 (1980).
[CrossRef]

M. S. Belen’kii, V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 595–597 (1980).
[CrossRef]

M. S. Belen’kii, A. I. Kon, V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287–290 (1977).
[CrossRef]

Beran, M. J.

Brown, W. P.

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Fante, R. L.

R. L. Fante, “Mutual coherence function and frequency of a laser beam propagating through atmospheric turbulence,” J. Opt. Soc. Am. 64, 592–598 (1974).
[CrossRef]

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, New York, 1985).
[CrossRef]

Feizulin, I. Z.

I. Z. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967).
[CrossRef]

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Grum, T. P.

H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCDoptical systems,” J. Opt. Soc. Am. 10, 316–323 (1993).
[CrossRef]

Hanson, S. G.

Ho, T. L.

Ishimaru, A.

A. Ishimaru, “The beam wave case and remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 5.
[CrossRef]

Kon, A. I.

M. S. Belen’kii, A. I. Kon, V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287–290 (1977).
[CrossRef]

Kravtsov, Yu. A.

I. Z. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967).
[CrossRef]

Lutomirski, R. F.

Lutomirski, R. T.

Miller, W. B.

Mironov, V. L.

M. S. Belen’kii, V. L. Mironov, “Phase fluctuations when focusing light in a turbulent atmosphere,” Radiophys. Quantum Electron. 12, 1096–1101 (1983).
[CrossRef]

M. S. Belen’kii, V. L. Mironov, “Mean diffracted rays of an optical beam in a turbulent medium,” J. Opt. Soc. Am. 70, 159–163 (1980).
[CrossRef]

M. S. Belen’kii, V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 595–597 (1980).
[CrossRef]

M. S. Belen’kii, A. I. Kon, V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287–290 (1977).
[CrossRef]

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Richardson, C. E.

L. C. Andrews, S. Vester, C. E. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40, 931–938 (1993).
[CrossRef]

Ricklin, J. C.

Seigman, A. E.

A. E. Seigman, Lasers (University Science Books, Mill Valley, Calif., 1986).

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Tavis, M. T.

Vester, S.

L. C. Andrews, S. Vester, C. E. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40, 931–938 (1993).
[CrossRef]

Whitman, A. M.

Yura, H. T.

Appl. Opt. (4)

J. Mod. Opt. (1)

L. C. Andrews, S. Vester, C. E. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40, 931–938 (1993).
[CrossRef]

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (3)

Proc. IEEE (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Radiophys. Quantum Electron. (2)

M. S. Belen’kii, V. L. Mironov, “Phase fluctuations when focusing light in a turbulent atmosphere,” Radiophys. Quantum Electron. 12, 1096–1101 (1983).
[CrossRef]

I. Z. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967).
[CrossRef]

Sov. J. Quantum Electron. (2)

M. S. Belen’kii, A. I. Kon, V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287–290 (1977).
[CrossRef]

M. S. Belen’kii, V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 595–597 (1980).
[CrossRef]

Other (5)

A. Ishimaru, “The beam wave case and remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 5.
[CrossRef]

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, New York, 1985).
[CrossRef]

A. E. Seigman, Lasers (University Science Books, Mill Valley, Calif., 1986).

J. C. Ricklin, “Optical turbulence effects on Gaussian beam wave propagation in the unstable surface boundary layer,” M.S. thesis (New Mexico State University, Las Cruces, N.M., 1990).

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (McGraw-Hill, New York, 1992).

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Figures (5)

Fig. 1
Fig. 1

Propagation geometry of a point source located at distance L to the left of a thin Gaussian lens of real focal length f and transmission radius W0.

Fig. 2
Fig. 2

MCF normalized by its on-axis value shown as a function of the ratio (2/L)1/2 for a collimated beam (Ω = 1) in both weak- (q = 0.1) and strong- (q = 10) fluctuation regimes. The dashed curves are obtained from Eq. (49), and the solid curves represent numerical calculations based on the parabolic-equation method.

Fig. 3
Fig. 3

Ratio of coherence lengths as a function of the Fresnel ratio Ω and turbulence strengths q = 1 and q = 100. The solid curves are numerical calculations based on the parabolic-equation method, and the dashed and the dotted curves are obtained from Eqs. (60) and (58), respectively.

Fig. 4
Fig. 4

Ratio of coherence lengths predicted by Eq. (60) (solid curve), Eq. (58) (dashed curve), and weak-fluctuation theory (dotted curve) as a function of the Fresnel ratio Ω for a propagating beam that is initially convergent. Numerical calculations based on the parabolic equation are denoted by filled triangles. The assumed beam conditions are W0 = 1 cm, Ωf = 1, and λ = 1.06 μm.

Fig. 5
Fig. 5

Comparison of normalized coherence length as a function of the Rytov variance σ12 for a collimated beam, a focused beam, an unbounded plane wave, and a spherical wave. The results are based on Eq. (60), with l0 = 0 and κ0 = 0.

Equations (64)

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U 0 ( r , 0 ) = exp ( r 2 / W 0 2 i k r 2 / 2 R 0 ) ,
U 0 ( r , L ) = exp ( i k L ) Ω 0 + i Ω exp [ r 2 Ω 0 + i Ω ( 1 W 0 2 + i k 2 R 0 ) ] = ( Θ i Λ ) exp ( i k L ) exp ( r 2 W 2 i k r 2 2 R ) ,
Ω 0 = 1 L R 0 , Ω = 2 L k W 0 2 ,
W = W 0 ( Ω 0 2 + Ω 2 ) 1 / 2 ,
R = L ( Ω 0 2 + Ω 2 ) Ω 0 ( 1 Ω 0 ) Ω 2 .
Θ = Ω 0 Ω 0 2 + Ω 2 = 1 + L R ,
Λ = Ω Ω 0 2 + Ω 2 = 2 L k W 2 .
Φ n ( κ ) = 0.033 C n 2 exp ( κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) 11 / 6 ,
σ 1 2 = 1.23 C n 2 k 7 / 6 L 11 / 6 , q = L k ρ p 2 ,
Q m = L κ m 2 k , Q 0 = L κ 0 2 k .
L < z i , q Λ 1
Λ Q m 1 , q Λ 1 ,
Λ Q m 1
L z i , q Λ 1 .
U ( r , L ) = i k 2 π B exp ( i k L ) d 2 s U i ( s ) exp [ ψ ( s , r ) ] × exp [ i k 2 B ( A s 2 2 s r + D r 2 ) ] ,
[ A B C D ] = [ 1 η 0 1 ] [ 1 0 1 f + i 2 k W 0 2 1 ] [ 1 L 0 1 ] = [ 1 η f + i 2 η k W 0 2 L + η ( 1 L f + i 2 L k W 0 2 ) 1 f + i 2 k W 0 2 1 L f + i 2 L k W 0 2 ] .
Γ 2 ( r 1 , r 2 , L ) = Γ 2 0 ( r 1 , r 2 , L ) exp [ ψ ( 0 , r 1 ) + ψ * ( 0 , r 2 ) ] ,
exp [ ψ ( 0 , r 1 ) + ψ * ( 0 , r 2 ) = exp { 4 π 2 k 2 0 L 0 κ Φ n ( κ ) × [ 1 exp ( κ 2 β i / k ) J 0 ( | α r ρ + 2 i α i r | κ ) ] d κ d η } ,
α = α r i α i = B ( η ) / B ( L ) ,
β = β r i β i = α ( L η ) ,
Γ 1 ( r 1 , r 2 , L ) = Γ 2 0 ( r 1 , r 2 , L ) × exp ( 4 π 2 k 2 L 0 1 0 κ Φ n ( κ ) × { 1 exp ( Λ L ξ 2 κ 2 / k ) J 0 [ | ( 1 Θ ξ ) ρ + 2 i Λ ξ r | κ ] } d κ d ξ ) ,
Γ 2 0 ( r 1 , r 2 , L ) = W 0 2 W 2 exp ( 2 r 2 W 2 ρ 2 2 W 2 i k R ρ r ) .
Γ 2 ( r 1 , r 2 , L ) = Γ 2 0 ( r 1 , r 2 , L ) × exp [ σ r 2 ( r 1 , L ) + σ r 2 ( r 2 , L ) H ] × exp [ ½ Δ ( r 1 , r 2 , L ) ] ,
σ r 2 ( r , L ) = 2 π 2 k 2 L 0 1 0 κ Φ n ( κ ) × exp ( Λ L ξ 2 κ 2 / k ) [ I 0 ( 2 Λ r ξ κ ) 1 ] d κ d ξ ,
H = 4 π 2 k 2 L 0 1 0 κ Φ n ( κ ) × [ 1 exp ( Λ L ξ 2 κ 2 / k ) ] d κ d ξ ,
Δ ( r 1 , r 2 , L ) = 4 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( Λ L ξ 2 κ 2 / k ) × { I 0 ( 2 Λ r 1 ξ κ ) + I 0 ( 2 Λ r 2 ξ κ ) 2 J 0 [ | ( 1 Θ ξ ) ρ + 2 i Λ ξ r | κ ] } d κ d ξ .
γ ( r 1 , r 2 , L ) = | Γ 2 ( r 1 , r 2 , L ) | [ Γ 2 ( r 1 , r 1 , L ) Γ 2 ( r 2 , r 2 , L ) ] 1 / 2 ,
γ ( r 1 , r 2 , L ) = exp [ ½ D ( r 1 , r 2 , L ) ] ,
I ( r , L ) = Γ 2 ( r , r , L ) = W 0 2 W 2 exp ( 2 r 2 / W 2 ) exp [ 2 σ r 2 ( r , L ) H ] ,
σ r 2 ( r , L ) = [ 0.982 σ 1 2 Λ Q m 1 / 6 F 2 1 ( 1 / 6 , 3 / 2 ; 5 / 2 ; Λ Q m ) 1.270 σ 1 2 Λ Q 0 1 / 6 ] r 2 W 2 0.982 σ 1 2 Λ 5 / 6 [ ( Λ Q m 1 + 0.52 Λ Q m ) 1 / 6 1.293 ( Λ Q 0 ) 1 / 6 ] r 2 W 2 , r < W ,
H = 3.537 σ 1 2 Q m 5 / 6 [ F 2 1 ( 5 / 6 , 1 / 2 ; 3 / 2 ; Λ Q m ) 1 ] 1.271 σ 1 2 Λ Q 0 1 / 6 3.537 σ 1 2 Λ 5 / 6 × [ ( 1 + 0.31 Λ Q m ) 5 / 6 1 ( Λ Q m ) 5 / 6 0.359 ( Λ Q 0 ) 1 / 6 ] ,
I ( r , L ) = W 0 2 W 1 2 exp ( 2 r 2 / W 2 2 ) , r > W ,
W 1 = W { 1 + 3.537 σ 1 2 Λ 5 / 6 × [ ( 1 + 0.31 Λ Q m ) 5 / 6 1 ( Λ Q m ) 5 / 6 0.359 ( Λ Q 0 ) 1 / 6 ] } 1 / 2 ,
W 2 = W { 1 + 0.982 σ 1 2 Λ 5 / 6 × [ ( Λ Q m 1 + 0.52 Λ Q m ) 1 / 6 1.293 ( Λ Q 0 ) 1 / 6 ] } 1 / 2 .
W 1 = W 2 = W [ 1 + 0.982 σ 1 2 Λ ( Q m 1 / 6 1.293 Q 0 1 / 6 ) ] 1 / 2 , Λ Q m 1 ,
W 1 = W { 1 + 1.327 σ 1 2 Λ 5 / 6 [ 1 0.957 ( Λ Q 0 ) 1 / 6 ] } 1 / 2 , Λ Q m 1 ,
W 2 = W { 1 + 1.105 σ 1 2 Λ 5 / 6 [ 1 1.170 ( Λ Q 0 ) 1 / 6 ] } 1 / 2 , Λ Q m 1 .
I ( 0 , L ) = W 0 2 W 2 e H 1 Ω 0 2 + Ω 2 ( 1 H ) , H 1 .
R t = L ( 1 + 4 q Λ / 3 ) Θ + 2 q Λ ,
W t = W ( 1 + 4 q Λ / 3 ) 1 / 2 .
Θ t = 1 + L R t = Θ 2 q Λ / 3 1 + 4 q Λ / 3 ,
Λ t = 2 L k W t 2 = Λ 1 + 4 q Λ / 3 ,
Γ 2 ( ρ , L ) = W 0 2 W 1 2 exp [ 1 4 Λ ( k ρ 2 L ) 1 2 d ( ρ , L ) ] ,
d ( ρ , L ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( Λ L ξ 2 κ 2 / k ) × { 1 J 0 [ ( 1 Θ ξ ) κ ρ ] } d κ d ξ .
Γ 2 ( ρ , L ) Γ 2 ( 0 , L ) = exp [ 1 4 Λ t ( k ρ 2 L ) 1 2 d t ( ρ , L ) ] ,
d t ( ρ , L ) = 0.491 σ 1 2 Q m 1 / 6 { 1 1 Θ t [ ( 1 + 0.11 Λ t Q m + 0.029 Q m k ρ 2 / L ) 1 / 6 Θ t 3 ( 1 + 0.11 Λ t Q m + 0.029 Θ t 2 Q m k ρ 2 / L ) 1 / 6 ] 0.715 ( 1 + Θ t + Θ t 2 ) ( κ 0 l 0 ) 1 / 3 } ( k ρ 2 L ) ,
Γ 2 ( ρ , L ) Γ 2 ( 0 , L ) = exp ( { 1 4 Λ t + 1 3 q ( 1 + Θ t + Θ t 2 ) × [ 1 0.715 ( κ 0 l 0 ) 1 / 3 ] } ( k ρ 2 L ) ) , ρ l 0 ,
Γ 2 ( ρ , L ) Γ 2 ( 0 , L ) = exp { 0.41 C n 2 k 2 L l 0 1 / 3 ρ 2 [ 1 0.715 ( κ 0 l 0 ) 1 / 3 ] } , L z i ,
Γ 2 ( ρ , L ) Γ 2 ( 0 , L ) = exp { [ 1 4 Λ t + 0.268 q 5 / 6 ( 1 + Θ t + Θ t 2 ) × ( κ 0 l 0 ) 1 / 3 ] ( k ρ 2 L ) 3 8 q 5 / 6 a t ( k ρ 2 L ) 5 / 6 } , ρ l 0 ,
a t = { 1 Θ t 8 / 3 1 Θ t Θ t 0 1 + | Θ t | 8 / 3 1 Θ t Θ t < 0 .
D ( ρ , L ) = 4 σ r 2 ( ρ / 2 , L ) + d ( ρ , L ) .
D ( ρ , L ) = 1.093 C n 2 k 2 L l 0 1 / 3 ρ 2 g t ( ρ , L ) ,
g t ( ρ , L ) = Λ t 2 ( 1 + 0.52 Λ t Q m ) 1 / 6 0.715 ( 1 + Θ t + Θ t 2 + Λ t 2 ) ( κ 0 l 0 ) 1 / 3 + 1 1 Θ t [ ( 1 + 0.11 Λ t Q m + ρ 2 / l 0 2 ) 1 / 6 Θ t 3 ( 1 + 0.11 Λ t Q m + Θ t 2 ρ 2 / l 0 2 ) 1 / 6 ] .
g t ( ρ , L ) = [ ( 1 + ρ 2 / l 0 2 ) 1 / 6 + ( 1 + ρ 2 / 4 l 0 2 ) 1 / 6 ] , q Λ 1 .
D ( ρ , L ) = ( 1 + Θ t + Θ t 2 + Λ t 2 ) ( ρ / ρ p ) 2 , ρ l 0 ,
D ( ρ , L ) = 0.888 σ 1 2 [ a t ( k ρ 2 L ) 5 / 6 + 0.618 Λ t 11 / 6 ( k ρ 2 L ) ] 3 4 ( a t + 0.618 Λ t 11 / 6 ) ( ρ / ρ p ) 5 / 3 , ρ l 0 ,
ρ 0 2 ρ p 2 = 3 + 4 q Λ 1 + Θ + Θ 2 + Λ 2 + 3 Λ / 4 q + q Λ ,
ρ 0 2 ρ p 2 = 3 + 4 q Λ 1 + Θ + Θ 2 + Λ 2 + q Λ , q Λ .
ρ 0 2 ρ p 2 = 3 1 + Θ t + Θ t 2 + Λ t 2 , ρ 0 l 0 ,
ρ 0 2 ρ p 2 = [ 8 3 ( a t + 0.618 Λ t 11 / 6 ) ] 6 / 5 , ρ 0 l 0 ,
Ω f = 2 R 0 k W 0 2 .
Γ ( r 1 , r 2 , L ) = W 0 2 8 π d 2 u × exp ( ρ 2 2 W b 2 1 8 W 2 u 2 + Θ 2 Λ u ρ + i u r G ) ,
W b = W 0 | Ω f | ( 1 + Ω f 2 ) 1 / 2 ,
G = 4 π 2 k 2 L 0 1 0 κ Φ n ( κ ) × { 1 J 0 [ κ | ρ ( L ξ / k ) u | ] } d κ d ξ .

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