Abstract

All the existing Rytov method solutions in atmospheric turbulence deal with coherent sources. In this paper we introduce the spatial partial coherence of a beam wave source to Rytov’s method and evaluate the intensity covariance and the scintillation index due to a spatially partially coherent beam wave source. The advantage of this solution is that in the calculation of the scintillations, the need for the use of the quadratic approximation for the medium structure functions is eliminated. The disadvantage is that, since it is a weak-fluctuation solution, we cannot extend the results to the incoherent source limit when (weak) turbulence is present.

© 1985 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.
  2. J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere, Vol. 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
    [CrossRef]
  3. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  4. R. A. Schmeltzer, “Means, variances and covariances for laser beam propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).
  5. A. Ishimaru, “Fluctuations of a beam wave propagation through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
    [CrossRef]
  6. H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. 64, 59–67 (1974).
    [CrossRef]
  7. S. F. Clifford, G. R. Ochs, R. S. Lawrence, “Saturation of optical scintillations by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
    [CrossRef]
  8. J. C. Leader, “Beam intensity fluctuations in atmospheric turbulence,” J. Opt. Soc. Am. 71, 542–558 (1981).
    [CrossRef]
  9. J. C. Leader, “Intensity fluctuations resulting from partially coherent light propagating through atmospheric turbulence,” J. Opt. Soc. Am. 69, 73–84 (1979).
    [CrossRef]
  10. Y. Baykal, M. A. Plonus, S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Sci. 18, 551–556 (1983).
    [CrossRef]
  11. Y. Baykal, “Coherence and turbulence effects on the intensity scintillations at optical frequencies,” Ph.D. dissertation (Northwest University, Evanston, Ill., 1982).
  12. H. M. Pederson, “Theory of speckle dependence of surface roughness,” J. Opt. Soc. Am. 66, 1204–1210 (1976).
    [CrossRef]
  13. L. Mandel, E. Wolf, “Coherence of optical fields,”Rev. Mod. Phys. 37, 231–287 (1965).
    [CrossRef]
  14. Z. I. Feizulin, Y. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967).
    [CrossRef]
  15. A. Ishimaru, “Fluctuations of a focused beam wave for atmospheric turbulence probing,” Proc. IEEE 57, 407–414 (1969).
    [CrossRef]
  16. R. F. Lutomirski, H. T. Yura, “Aperture-averaging factor of a fluctuating light signal,” J. Opt. Soc. Am. 59, 1247–1248 (1969).
    [CrossRef]
  17. M. H. Lee, J. F. Holmes, J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 66, 1164–1172 (1976).
    [CrossRef]
  18. R. L. Fante, “Intensity fluctuations of an optical wave in a turbulent medium. Effect of source coherence,” Opt. Acta 28, 1203–1207 (1981).
    [CrossRef]
  19. J. F. Holmes, M. H. Lee, J. R. Kerr, “Effect of the log-amplitude covariance function on the statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 70, 355–360 (1980).
    [CrossRef]
  20. P. A. Pincus, M. E. Fossey, J. F. Holmes, J. R. Kerr, “Speckle propagation through turbulence: Experimental,” J. Opt. Soc. Am. 68, 760–762 (1978).
    [CrossRef]
  21. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]

1983 (1)

Y. Baykal, M. A. Plonus, S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Sci. 18, 551–556 (1983).
[CrossRef]

1981 (2)

J. C. Leader, “Beam intensity fluctuations in atmospheric turbulence,” J. Opt. Soc. Am. 71, 542–558 (1981).
[CrossRef]

R. L. Fante, “Intensity fluctuations of an optical wave in a turbulent medium. Effect of source coherence,” Opt. Acta 28, 1203–1207 (1981).
[CrossRef]

1980 (1)

1979 (1)

1978 (1)

1976 (2)

1974 (2)

1969 (3)

A. Ishimaru, “Fluctuations of a beam wave propagation through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

A. Ishimaru, “Fluctuations of a focused beam wave for atmospheric turbulence probing,” Proc. IEEE 57, 407–414 (1969).
[CrossRef]

R. F. Lutomirski, H. T. Yura, “Aperture-averaging factor of a fluctuating light signal,” J. Opt. Soc. Am. 59, 1247–1248 (1969).
[CrossRef]

1967 (2)

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

R. A. Schmeltzer, “Means, variances and covariances for laser beam propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).

1966 (1)

1965 (1)

L. Mandel, E. Wolf, “Coherence of optical fields,”Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Baykal, Y.

Y. Baykal, M. A. Plonus, S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Sci. 18, 551–556 (1983).
[CrossRef]

Y. Baykal, “Coherence and turbulence effects on the intensity scintillations at optical frequencies,” Ph.D. dissertation (Northwest University, Evanston, Ill., 1982).

Clifford, S. F.

Fante, R. L.

R. L. Fante, “Intensity fluctuations of an optical wave in a turbulent medium. Effect of source coherence,” Opt. Acta 28, 1203–1207 (1981).
[CrossRef]

Feizulin, Z. I.

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

Fossey, M. E.

Fried, D. L.

Holmes, J. F.

Ishimaru, A.

A. Ishimaru, “Fluctuations of a focused beam wave for atmospheric turbulence probing,” Proc. IEEE 57, 407–414 (1969).
[CrossRef]

A. Ishimaru, “Fluctuations of a beam wave propagation through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

Kerr, J. R.

Kravtsov, Y.

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

Lawrence, R. S.

Leader, J. C.

Lee, M. H.

Lutomirski, R. F.

Mandel, L.

L. Mandel, E. Wolf, “Coherence of optical fields,”Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Ochs, G. R.

Pederson, H. M.

Pincus, P. A.

Plonus, M. A.

Y. Baykal, M. A. Plonus, S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Sci. 18, 551–556 (1983).
[CrossRef]

Schmeltzer, R. A.

R. A. Schmeltzer, “Means, variances and covariances for laser beam propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Wang, S. J.

Y. Baykal, M. A. Plonus, S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Sci. 18, 551–556 (1983).
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, “Coherence of optical fields,”Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Yura, H. T.

J. Opt. Soc. Am. (10)

H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. 64, 59–67 (1974).
[CrossRef]

S. F. Clifford, G. R. Ochs, R. S. Lawrence, “Saturation of optical scintillations by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
[CrossRef]

J. C. Leader, “Beam intensity fluctuations in atmospheric turbulence,” J. Opt. Soc. Am. 71, 542–558 (1981).
[CrossRef]

J. C. Leader, “Intensity fluctuations resulting from partially coherent light propagating through atmospheric turbulence,” J. Opt. Soc. Am. 69, 73–84 (1979).
[CrossRef]

H. M. Pederson, “Theory of speckle dependence of surface roughness,” J. Opt. Soc. Am. 66, 1204–1210 (1976).
[CrossRef]

R. F. Lutomirski, H. T. Yura, “Aperture-averaging factor of a fluctuating light signal,” J. Opt. Soc. Am. 59, 1247–1248 (1969).
[CrossRef]

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

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

P. A. Pincus, M. E. Fossey, J. F. Holmes, J. R. Kerr, “Speckle propagation through turbulence: Experimental,” J. Opt. Soc. Am. 68, 760–762 (1978).
[CrossRef]

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
[CrossRef]

Opt. Acta (1)

R. L. Fante, “Intensity fluctuations of an optical wave in a turbulent medium. Effect of source coherence,” Opt. Acta 28, 1203–1207 (1981).
[CrossRef]

Proc. IEEE (1)

A. Ishimaru, “Fluctuations of a focused beam wave for atmospheric turbulence probing,” Proc. IEEE 57, 407–414 (1969).
[CrossRef]

Q. Appl. Math. (1)

R. A. Schmeltzer, “Means, variances and covariances for laser beam propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).

Radio Sci. (2)

A. Ishimaru, “Fluctuations of a beam wave propagation through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

Y. Baykal, M. A. Plonus, S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Sci. 18, 551–556 (1983).
[CrossRef]

Radiophys. Quantum Electron. (1)

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

Rev. Mod. Phys. (1)

L. Mandel, E. Wolf, “Coherence of optical fields,”Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Other (4)

Y. Baykal, “Coherence and turbulence effects on the intensity scintillations at optical frequencies,” Ph.D. dissertation (Northwest University, Evanston, Ill., 1982).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere, Vol. 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
[CrossRef]

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

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Figures (12)

Fig. 1
Fig. 1

The scintillation index, m2, versus the normalized source size, α s / λ L, for various ρs in vacuum.

Fig. 2
Fig. 2

m2 versus α s / λ L for different ρs in vacuum.

Fig. 3
Fig. 3

m2 versus α s / λ L for different ρs for σa2 = 0.5 in vacuum.

Fig. 4
Fig. 4

Variation of the scintillations for constant ζ = αss in vacuum.

Fig. 5
Fig. 5

Variation of the scintillations for more incoherent sources (ζ flarge) in vacuum.

Fig. 6
Fig. 6

Variation of the scintillations for constant ζ and σa2 = 3 in vacuum.

Fig. 7
Fig. 7

m2 versus α s / λ L for constant ρs in turbulence (Cn2 = 3.5 × 10−16).

Fig. 8
Fig. 8

m2 versus α s / λ L for constant ρs in stronger turbulence (Cn2 = 10−15).

Fig. 9
Fig. 9

m2 versus ( α s / λ L ) for constant ρs and σa2 = 0.5 in turbulence.

Fig. 10
Fig. 10

Variation of the scintillations for constant ζ in turbulence.

Fig. 11
Fig. 11

m2 versus the path length L for a coherent source in turbulence.

Fig. 12
Fig. 12

m2 versus L for a partially coherent source in turbulence.

Equations (36)

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u ( r ) = u 0 ( r ) exp { ψ ( r ) } ,
ψ ( r ) = k 2 2 π u 0 ( r ) υ n 1 ( r ) u 0 ( r ) exp ( i k | r r | ) | r r | d 3 r
u 0 ( p , z ) = k e ikz A s 2 π i z d 2 s exp [ ( 1 2 α s 2 i k 2 F ) s 2 + i ϕ ( s ) ] × exp [ i k 2 z ( p s ) 2 ] ,
u ( s ) = A s exp [ ( 1 2 α s 2 + i k 2 F ) s 2 ]
ϕ = a + b · s ,
u 0 ( p , z ) = A s 1 + i α z exp [ ikz k α 2 p 2 ( 1 + i α z ) ] × exp { i [ a + b · p ( 1 + i α z ) ( z / 2 k ) b 2 ( 1 + i α z ) ] } ,
α = 1 k α s 2 + i F .
Γ 4 = u 0 ( r 1 ) u 0 * ( r 1 ) u 0 ( r 2 ) u 0 * ( r 2 ) exp [ ψ ( r 1 ) + ψ * ( r 1 ) + ψ ( r 2 ) + ψ * ( r 2 ) ] s , m ,
Γ 4 Γ 4 υ Γ 4 t ,
Γ 4 υ = u 0 ( p 1 , L ) u 0 * ( p 1 , L ) u 0 ( p 2 , L ) u 0 * ( p 2 , L ) s
Γ 4 t exp [ ψ ( p 1 , L ) + ψ * ( p 1 , L ) + ψ ( p 2 , L ) + ψ * ( p 2 , L ) ] s , m
Γ 4 υ = ( 1 λ L ) 4 d 2 s 1 d 2 s 2 d 2 s 3 d 2 s 4 Γ 4 s × exp { i k 2 L [ ( p 1 s 1 ) 2 ( p 1 s 2 ) 2 + ( p 2 s 3 ) 2 ( p 2 s 4 ) 2 ] } ,
Γ 4 s = A s 4 Γ 4 Φ exp [ 1 2 α s 2 ( s 1 2 + s 2 2 + s 3 2 + s 4 2 ) i k 2 F ( s 1 2 s 2 2 + s 3 2 s 4 2 ) ] ,
Γ 4 Φ exp { i [ ϕ ( s 1 ) ϕ ( s 2 ) + ϕ ( s 3 ) ϕ ( s 4 ) ] } .
Γ 4 Φ = exp { 1 4 ρ s 2 [ ( s 1 s 2 ) 2 + ( s 3 s 4 ) 2 ] } + exp { 1 4 ρ s 2 [ ( s 1 s 4 ) 2 + ( s 3 s 2 ) 2 ] } + exp { 4 σ a 2 1 4 ρ s 2 [ ( s 1 + s 3 ) 2 + ( s 2 + s 2 ) 2 ] } 2 exp [ 2 σ a 2 1 4 ρ s 2 ( s 1 2 + s 2 2 + s 3 2 + s 4 2 ) ] ,
Γ 4 υ = 2 ( k A s α s ) 4 L 4 [ 1 α s 2 + 1 ρ s 2 + ( A α s ) 2 ] × exp { ( k L ) 2 ( p 1 2 + p 2 2 ) [ 1 α s 2 + 1 ρ s 2 + ( A α s ) 2 ] } + ( k A s L ) 4 exp ( 4 σ a 2 ) [ ( 1 α s 2 + 1 ρ s 2 ) 2 + A 2 ] [ ( 1 α s 2 ) 2 + A 2 ] × exp { ( k L ) 2 ( p 1 2 + p 2 2 ) 2 [ 1 α s 2 + ( A α s ) 2 ] } × exp { ( k L ) 2 ( p 1 2 + p 2 2 ) 2 [ ( 1 α s 2 + 1 ρ s 2 ) + A 2 ( 1 α s 2 + 1 ρ s 2 ) ] } 2 ( k A s L ) 4 exp ( 2 σ a 2 ) [ ( 1 α s 2 + 1 2 ρ s 2 ) 2 + A 2 ] 2 × exp { ( k L ) 2 ( p 1 2 + p 2 2 ) ( 1 α s 2 + 1 2 ρ s 2 + A 2 1 α s 2 + 1 2 ρ s 2 ) } ,
Γ 4 t = exp [ ½ D ψ ( p 1 , p 1 ) ½ D ψ ( p 2 , p 2 ) + 4 C χ ( p 1 , p 2 ) ] ,
C I ( p 1 , p 2 ) I ( p 1 ) I ( p 2 ) I ( p 1 ) I ( p 2 ) 1 .
I ( p ) = u 0 ( p ) u 0 * ( p ) s exp [ ψ ( p ) + ψ * ( p ) ] s , m ,
I ( p ) = ( k α s A s L ) 2 1 α s 2 + 1 ρ s 2 + ( α s A ) 2 exp [ ( k L ) 2 p 2 1 α s 2 + 1 ρ s 2 + ( α s A ) 2 ] × exp [ ½ D ψ ( p , p ) ] ,
C I ( p 1 , p 2 ) = ( 2 + exp ( 4 σ a 2 ) [ 1 α s 2 ( 1 α s 2 + 1 ρ s 2 ) + A 2 ] 2 [ ( 1 α s 2 + 1 ρ s 2 ) 2 + A 2 ] [ ( 1 α s 2 ) 2 + A 2 ] × exp { ( k L ) 2 ( p 1 p 2 ) 2 2 [ 1 s 2 + ( A α s ) 2 ] ( k L ) 2 ( p 1 + p 2 ) 2 2 [ ( 1 α s 2 + 1 ρ s 2 ) + A 2 ( 1 α s 2 + 1 ρ s 2 ) ] + ( k L ) 2 ( p 1 2 + p 2 2 ) 1 α s 2 + 1 ρ s 2 + ( A α s ) 2 } 2 exp ( 2 σ a 2 ) [ 1 α s 2 ( 1 α s 2 + 1 ρ s 2 ) + A 2 ] 2 [ ( 1 α s 2 + 1 2 ρ s 2 ) 2 + A 2 ] 2 × exp { ( k L ) 2 ( p 1 2 + p 2 2 ) 1 2 ρ s 2 + A 2 α s 4 2 ρ s 2 + α s 2 [ 1 α s 2 + 1 ρ s 2 + ( A α s ) 2 ] [ 1 α s 2 + 1 2 ρ s 2 + A 2 1 α s 2 + 1 2 ρ s 2 ] } ) exp [ 4 C χ ( p 1 , p 2 ) ] 1 .
m 2 = { 2 + exp ( 4 σ a 2 ) [ 1 α s 2 ( 1 α s 2 + 1 ρ s 2 ) + A 2 ] 2 [ ( 1 α s 2 + 1 ρ s 2 ) 2 + A 2 ] [ ( 1 α s 2 ) 2 + A 2 ] 2 exp ( 2 σ a 2 ) [ 1 α s 2 ( 1 α s 2 + 1 ρ s 2 ) + A 2 ] 2 [ ( 1 α s 2 + 1 2 ρ s 2 ) 2 + A 2 ] 2 } × exp [ 4 σ χ ( p . c . b . ) 2 ] 1 ,
σ χ ( p . c . b . ) 2 C χ ( 0 , 0 )
m 2 | σ a 2 = 0 ρ s = exp ( 4 σ χ 2 ) 1 ,
σ χ 2 = σ χ ( p . c . b . ) 2 | σ a 2 = 0 ρ s
B χ ( P x 1 , P y 1 , P x 2 , P y 2 , L ) χ ( P x 1 , P y 1 , L ) χ ( P x 2 , P y 2 , L )
B χ ( P x 1 , P y 1 , P x 2 , P y 2 , L ) = 2 π 2 Re { 0 L d η 0 κ d κ [ J 0 ( κ P ) | H 1 | 2 + J 0 ( κ Q ) H 2 ] Φ n ( κ ) } ,
Q = γ [ ( P x 1 P x 2 ) 2 + ( P y 1 P y 2 ) 2 ] 1 / 2 , P = [ ( γ P x 1 γ * P x 2 ) 2 + ( γ P y 1 γ * P y 2 ) 2 ] 1 / 2 , γ = 1 + i α η 1 + i α L , α = 1 k α s 2 + i F , H 2 = k 2 exp [ i γ ( L η ) k κ 2 ] , | H 1 | 2 = k 2 exp ( β 2 κ 2 ) , β 2 = ( L η ) 2 ξ 1 , ξ 1 = ρ s 2 L 2 [ k 2 α s 2 ( 1 L F ) 2 + L 2 α s 2 ] 1 ρ s 2 L 2 [ k 2 α s 2 ( 1 L F ) 2 + L 2 α s 2 ] 2 .
C χ ( P 1 , P 2 ) = B χ ( P 1 , P 2 ) .
B s ( P x 1 , P y 1 , P x 2 , P y 2 , L ) S ( P x 1 , P y 1 , L ) S ( P x 2 , P y 2 , L ) = 2 π 2 Re { 0 L d η 0 κ d κ [ J 0 ( κ P ) | H 1 | 2 J 0 ( κ Q ) H 2 ] Φ n ( κ ) } ,
B χ s ( P x 1 , P y 1 , P x 2 , P y 2 , L ) χ ( P x 1 , P y 1 , L ) S ( P x 2 , P y 2 , L ) = 2 π 2 Im { 0 L d η 0 κ d κ [ J 0 ( κ Q ) H 2 J 0 ( κ P ) | H 1 | 2 ] Φ n ( κ ) } ,
D χ ( P x 1 , P y 1 , P x 2 , P y 2 , L ) [ χ ( P x 1 , P y 1 , L ) χ ( P x 2 , P y 2 , L ) ] 2 = 4 π 2 Re [ 0 L d η 0 κ d κ ( { 1 2 J 0 [ 2 i κ P 1 Im ( γ ) + 1 2 J 0 ( 2 i κ P 2 Im ( γ ) ] } | H 1 | 2 + [ 1 J 0 ( κ P ) ] H 2 ) Φ n ( κ ) ] ,
P m = ( P x m 2 + P y m 2 ) 2 , m = 1 , 2 , D s ( P x 1 , P y 1 , P x 2 , P y 2 , L ) [ S ( P x 1 , P y 1 , L ) S ( P x 2 , P y 2 , L ) ] 2 = 4 π 2 Re [ 0 L d η 0 κ d κ ( { 1 2 J 0 [ 2 i κ P 1 Im ( γ ) ] + 1 2 J 0 [ 2 i κ P 2 Im ( γ ) ] } | H 1 | 2 [ 1 J 0 ( κ Q ) ] H 2 ) Φ n ( κ ) ] ,
D ψ = D χ + D s ,
D χ s ( P x 1 , P y 1 , P x 2 , P y 2 , L ) [ χ ( P x 1 , P y 1 , L ) χ ( P x 2 , P y 2 , L ) ] × [ S ( P x 1 , P y 1 , L ) S ( P x 2 , P y 2 , L ) ] = 4 π 2 Im { 0 L d η 0 κ d κ [ 1 J 0 ( π Q ) ] H 2 Φ n ( κ ) } .
σ χ ( p . c . b . ) 2 = 2.176 C n 2 k 7 / 6 L 11 / 6 Re { 0 1 d t [ i ( 1 + i α L t ) ( 1 t ) ( 1 + i α L ) + k L κ m 2 ] 5 / 6 [ ξ 1 L k ( 1 t ) 2 + k L κ m 2 ] 5 / 6 } ,

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