Abstract

Using a Gaussian weighting function for the receiver aperture, we obtain a closed-form representation for the receiver-aperture averaging effect for the intensity fluctuation of a beam wave in the turbulent atmosphere. It is shown that, unlike for the plane-wave case, the power scintillations do not always decrease when the receiver aperture is increased. The reasons are that (1) the intensity fluctuations on the axis for a coherent beam-wave source are smaller than these off the axis and (2) the averaging effect cannot show up when the total beam is within a coherent patch (i.e., the coherence length is larger than the beamwidth).

© 1983 Optical Society of America

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References

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  1. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propation (National Technical Information Service, Springfield, Va., 1971).
  2. F. X. Kneizys and et al., Atmospheric Transmittance/Radiation: Computer Codelowtran 5, AFGL-TR-80-0067, NTIS #ADA 088-215 (National Technical Information Service, Springfield, Va., 1980).
  3. J. C. Leader, “Intensity fluctuations resulting from partially coherent light propagating through atmospheric turbulence,” J. Opt. Soc. Am. 69, 73–84 (1979).
    [Crossref]
  4. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979).
    [Crossref]
  5. C. F. Ouyang, M. A. Plonus, Y. Baykal, and S. J. Wang, “Transmitter and receiver aperture averaging effects for the intensity fluctuations of a beam wave in the turbulent atmosphere,” (Northwestern University, Evanston, Ill., July1982).
  6. H. M. Pederson, “Theory of speckle dependence on surface roughness,” J. Opt. Soc. Am. 66, 1204–1210 (1976).
    [Crossref]
  7. We define the random phase of the source field8 as ϕ= a+ b· s, where a and b are random numbers with zero mean, i.e., 〈a〉 = 〈b〉 = 0 and σa2 = 〈a2〉, ρs= 1/〈b2〉. As ρs→ ∞, σa2→ 0; we consider the source coherent; if ρs→ 0 or/and σa2→ ∞, it is incoherent.
  8. S. M. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70, 745–747 (1980).
    [Crossref]
  9. A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogenous medium,” Radio Sci. 4, 295–305 (1969).
    [Crossref]
  10. R. L. Fante, “Intensity fluctuations of an optical wave in a turbulent medium. Effect of source coherence,” Opt. Acta 28, 1203–1207 (1981).
    [Crossref]
  11. E. M. Kuznetsova, “Fluctuations in radiation,” Opt. Spectrosk. (USSR) 48, 170–172 (1980).
  12. A. G. Arutyanyan, S. A. Akhmanov, Yu. D. Golyaev, V. G. Tunkin, and A. S. Chirkin, “Spatial field and intensity correlation functions of laser radiation,” Sov. Phys. JETP 37, 764–771 (1973).
  13. R. L. Fante, “Two-source sperical wave structure functions in atmospheric turbulence,” J. Opt. Sec. Am. 66, 74 (1974).
    [Crossref]
  14. Z. I. Feizulin and Y. Krautsov, “Broadening of a laser beam in a turbulent medium,” Radio Phys. Quantum Electron. 10, 33–35 (1967).
    [Crossref]

1981 (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]

1980 (2)

E. M. Kuznetsova, “Fluctuations in radiation,” Opt. Spectrosk. (USSR) 48, 170–172 (1980).

S. M. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70, 745–747 (1980).
[Crossref]

1979 (2)

1976 (1)

1974 (1)

R. L. Fante, “Two-source sperical wave structure functions in atmospheric turbulence,” J. Opt. Sec. Am. 66, 74 (1974).
[Crossref]

1973 (1)

A. G. Arutyanyan, S. A. Akhmanov, Yu. D. Golyaev, V. G. Tunkin, and A. S. Chirkin, “Spatial field and intensity correlation functions of laser radiation,” Sov. Phys. JETP 37, 764–771 (1973).

1969 (1)

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

1967 (1)

Z. I. Feizulin and Y. Krautsov, “Broadening of a laser beam in a turbulent medium,” Radio Phys. Quantum Electron. 10, 33–35 (1967).
[Crossref]

Akhmanov, S. A.

A. G. Arutyanyan, S. A. Akhmanov, Yu. D. Golyaev, V. G. Tunkin, and A. S. Chirkin, “Spatial field and intensity correlation functions of laser radiation,” Sov. Phys. JETP 37, 764–771 (1973).

Arutyanyan, A. G.

A. G. Arutyanyan, S. A. Akhmanov, Yu. D. Golyaev, V. G. Tunkin, and A. S. Chirkin, “Spatial field and intensity correlation functions of laser radiation,” Sov. Phys. JETP 37, 764–771 (1973).

Baykal, Y.

C. F. Ouyang, M. A. Plonus, Y. Baykal, and S. J. Wang, “Transmitter and receiver aperture averaging effects for the intensity fluctuations of a beam wave in the turbulent atmosphere,” (Northwestern University, Evanston, Ill., July1982).

Chirkin, A. S.

A. G. Arutyanyan, S. A. Akhmanov, Yu. D. Golyaev, V. G. Tunkin, and A. S. Chirkin, “Spatial field and intensity correlation functions of laser radiation,” Sov. Phys. JETP 37, 764–771 (1973).

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]

R. L. Fante, “Two-source sperical wave structure functions in atmospheric turbulence,” J. Opt. Sec. Am. 66, 74 (1974).
[Crossref]

Feizulin, Z. I.

Z. I. Feizulin and Y. Krautsov, “Broadening of a laser beam in a turbulent medium,” Radio Phys. Quantum Electron. 10, 33–35 (1967).
[Crossref]

Golyaev, Yu. D.

A. G. Arutyanyan, S. A. Akhmanov, Yu. D. Golyaev, V. G. Tunkin, and A. S. Chirkin, “Spatial field and intensity correlation functions of laser radiation,” Sov. Phys. JETP 37, 764–771 (1973).

Ishimaru, A.

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

Kneizys, F. X.

F. X. Kneizys and et al., Atmospheric Transmittance/Radiation: Computer Codelowtran 5, AFGL-TR-80-0067, NTIS #ADA 088-215 (National Technical Information Service, Springfield, Va., 1980).

Krautsov, Y.

Z. I. Feizulin and Y. Krautsov, “Broadening of a laser beam in a turbulent medium,” Radio Phys. Quantum Electron. 10, 33–35 (1967).
[Crossref]

Kuznetsova, E. M.

E. M. Kuznetsova, “Fluctuations in radiation,” Opt. Spectrosk. (USSR) 48, 170–172 (1980).

Leader, J. C.

Ouyang, C. F.

C. F. Ouyang, M. A. Plonus, Y. Baykal, and S. J. Wang, “Transmitter and receiver aperture averaging effects for the intensity fluctuations of a beam wave in the turbulent atmosphere,” (Northwestern University, Evanston, Ill., July1982).

Pederson, H. M.

Plonus, M. A.

S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979).
[Crossref]

C. F. Ouyang, M. A. Plonus, Y. Baykal, and S. J. Wang, “Transmitter and receiver aperture averaging effects for the intensity fluctuations of a beam wave in the turbulent atmosphere,” (Northwestern University, Evanston, Ill., July1982).

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propation (National Technical Information Service, Springfield, Va., 1971).

Tunkin, V. G.

A. G. Arutyanyan, S. A. Akhmanov, Yu. D. Golyaev, V. G. Tunkin, and A. S. Chirkin, “Spatial field and intensity correlation functions of laser radiation,” Sov. Phys. JETP 37, 764–771 (1973).

Wandzura, S. M.

Wang, S. C. H.

Wang, S. J.

C. F. Ouyang, M. A. Plonus, Y. Baykal, and S. J. Wang, “Transmitter and receiver aperture averaging effects for the intensity fluctuations of a beam wave in the turbulent atmosphere,” (Northwestern University, Evanston, Ill., July1982).

J. Opt. Sec. Am. (1)

R. L. Fante, “Two-source sperical wave structure functions in atmospheric turbulence,” J. Opt. Sec. Am. 66, 74 (1974).
[Crossref]

J. Opt. Soc. Am. (4)

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]

Opt. Spectrosk. (USSR) (1)

E. M. Kuznetsova, “Fluctuations in radiation,” Opt. Spectrosk. (USSR) 48, 170–172 (1980).

Radio Phys. Quantum Electron. (1)

Z. I. Feizulin and Y. Krautsov, “Broadening of a laser beam in a turbulent medium,” Radio Phys. Quantum Electron. 10, 33–35 (1967).
[Crossref]

Radio Sci. (1)

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

Sov. Phys. JETP (1)

A. G. Arutyanyan, S. A. Akhmanov, Yu. D. Golyaev, V. G. Tunkin, and A. S. Chirkin, “Spatial field and intensity correlation functions of laser radiation,” Sov. Phys. JETP 37, 764–771 (1973).

Other (4)

We define the random phase of the source field8 as ϕ= a+ b· s, where a and b are random numbers with zero mean, i.e., 〈a〉 = 〈b〉 = 0 and σa2 = 〈a2〉, ρs= 1/〈b2〉. As ρs→ ∞, σa2→ 0; we consider the source coherent; if ρs→ 0 or/and σa2→ ∞, it is incoherent.

C. F. Ouyang, M. A. Plonus, Y. Baykal, and S. J. Wang, “Transmitter and receiver aperture averaging effects for the intensity fluctuations of a beam wave in the turbulent atmosphere,” (Northwestern University, Evanston, Ill., July1982).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propation (National Technical Information Service, Springfield, Va., 1971).

F. X. Kneizys and et al., Atmospheric Transmittance/Radiation: Computer Codelowtran 5, AFGL-TR-80-0067, NTIS #ADA 088-215 (National Technical Information Service, Springfield, Va., 1980).

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Figures (7)

Fig. 1
Fig. 1

The averaging factor G(R), calculated for a unit step-function receiver (dashed line, by numerical integration) and for Gaussian-function receiver (solid line) for a coherent source (ρs → ∞); αs = 2 cm, L = 3 km, and λ = 1 μm.

Fig. 2
Fig. 2

The normalized intensity fluctuation m2(p)/m2(0) off the axis for a coherent source; αs = 2 cm, L = 3 km, λ = 1 μm, Cn2 = 10−16 m−2/3.

Fig. 3
Fig. 3

The normalized intensity fluctuation off the axis for an incoherent source in a turbulence-free medium (Cn2 = 10−30 m−2/3); αs = 2 cm, L = 3 km, λ = 1 μm.

Fig. 4
Fig. 4

Power scintillation for a coherent source in a turbulent medium when αs = 2 cm, L = 3 km, and λ = 1 μm.

Fig. 5
Fig. 5

Averaging factor G(R) for a coherent source in a turbulent medium when αs = 2 cm, L = 3 km, λ = 1 μm.

Fig. 6
Fig. 6

Averaging factor G(R) for an incoherent source in a turbulent medium when αs = 2 cm, L = 3 km, λ = 1 μm.

Fig. 7
Fig. 7

Averaging factor G(R) for an incoherent source in a turbulence-free medium when αs = 2 cm, L = 3 km, λ = 1 μm.

Equations (24)

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I ( L , p ) = ( k α s 2 L ) 2 A s 2 1 4 α s 2 + 1 4 ρ s 2 + 1 ρ 0 2 + [ k α s 2 L ( 1 - L F ) ] 2 × exp { - p 2 ( k α s 2 L ) 2 α s 2 [ 1 + ( 2 α s ρ 0 ) 2 + ( α s ρ s ) 2 + ( k α s 2 L ) 2 ( 1 - L F ) 2 ] } ,
P ( L ) = - I ( L , p ) exp [ - ( p 2 / R 2 ) ] d 2 p = π A s 2 ( k α s 2 L ) 2 [ 1 4 α s 2 + 1 4 ρ s 2 + 1 ρ 0 2 + ( α s A 2 ) 2 ] { 1 R 2 + ( k α s 2 L ) 2 α s 2 [ 1 + ( α s ρ s ) 2 + ( 2 α s ρ 0 ) 2 + ( α s 2 A ) 2 ] } ,
B I ( p c , p d , L ) = I 1 ( L , p 1 ) I 2 ( L , p 2 ) = 4 ( k A s 2 L ) 4 exp ( 4 σ χ s 2 ) × [ α s 2 2 B D G exp ( - H 1 p c 2 - F 1 p d 2 ) + α s 2 2 B K N exp ( - H 2 p c 2 - F 2 p d 2 ) + exp ( - 4 σ a 2 ) T U D W exp ( - H 3 p c 2 - F 3 p d 2 ) - exp ( - 2 σ a 2 ) 2 X Y Z Q exp ( - H 4 p c 2 - F 4 p d 2 ) ] ,
H 1 = H 2 = k 2 B L 2 , H 3 = k 2 U L 2 , H 4 = k 2 Y L 2 , F 1 = 2 ( 1 ρ 0 2 - 1 ρ χ 2 ) - 1 D ( 1 ρ 0 2 - 1 ρ χ 2 ) 2 + [ 1 D ( A - 4 ρ χ S 2 ) ( 1 ρ 0 2 - 1 ρ χ 2 ) + k L + 2 ρ χ S 2 ] 2 4 G , F 2 = 2 ( 1 ρ 0 2 - 1 ρ χ 2 ) - 1 K ( 1 ρ 0 2 - 1 ρ χ 2 ) 2 + [ 1 K ( A - 4 ρ χ S 2 ) ( 1 ρ 0 2 - 1 ρ χ 2 ) + k L + 2 ρ χ S 2 ] 2 4 N , F 3 = 2 ( 1 ρ 0 2 - 1 ρ χ 2 ) - 1 D ( 1 ρ 0 2 - 1 ρ χ 2 ) 2 + [ 1 D ( A - 4 ρ χ S 2 ) ( 1 ρ 0 2 - 1 ρ χ 2 ) + k L + 2 ρ χ S 2 ] 2 4 W , F 4 = 2 ( 1 ρ 0 2 - 1 ρ χ 2 ) - 1 Q ( 1 ρ 0 2 - 1 ρ χ 2 ) 2 + 1 Q ( A - 4 ρ χ S 2 ) ( 1 ρ 0 2 - 1 ρ χ 2 ) + k L + 2 ρ χ S 2 ] 4 Z , B = 1 2 α s 2 + 1 2 ρ s 2 + α s 2 A 2 2 + 4 ρ 0 2 , D = 1 2 α s 2 + 2 ρ 0 2 - 2 ρ χ 2 , G = D + 1 2 ρ s 2 + ( A - 4 ρ χ S 2 ) 2 4 D , K = D + 1 2 ρ s 2 , N = D + ( A - 4 ρ χ S 2 ) 2 4 K , T = 2 α s 2 + 2 ρ s 2 , U = T 4 + A 2 T + 4 ρ 0 2 = A 2 T + 1 2 ρ s 2 + 1 2 α s 2 + 4 ρ 0 2 , W = G - 1 2 ρ s 2 X = 1 2 α s 2 + 1 4 ρ s 2 , Y = X + A 2 4 X + 4 ρ 0 2 , Q = D + 1 4 ρ s 2 , Z = Q + ( A - 4 ρ χ S 2 ) 2 4 Q , σ χ s 2 = 0.124 k 7 / 6 C n 2 L 11 / 6 , 1 ρ 0 2 = ( 0.546 C n 2 k 2 L ) 6 / 5 , 1 ρ χ 2 = 0.425 C n 2 k 13 / 6 L 5 / 6 , 1 ρ χ S 2 = 0.114 C n 2 k 13 / 6 L 5 / 6 , p c = 1 2 ( p 1 + p 2 ) , p d = p 1 - p 2 ,
m p 2 ( P - P ) 2 P 2 = Σ I 1 I 2 exp ( - p 1 2 R 2 - p 2 2 R 2 ) d 2 p 1 d 2 p 2 P 2 - 1 - B I ( p c , p d , L ) exp ( - 2 p c 2 R 2 - p d 2 2 R 2 ) d 2 p 1 d 2 p 2 P - 1.
m P 2 = 4 exp ( 4 σ χ s 2 ) α s 4 [ 1 4 α s 2 + 1 4 ρ s 2 + 1 ρ 0 2 + ( α s A 2 ) 2 ] 2 × [ 1 R 2 + ( k α s 2 L ) 2 α s 2 [ 1 + ( α s ρ s ) 2 + ( 2 α s ρ 0 ) 2 + ( α s 2 A ) 2 ] ] 2 × [ α s 2 2 B D G ( H 1 + 2 R 2 ) ( F 1 + 1 2 R 2 ) + α s 2 2 B K N ( H 2 + 2 R 2 ) ( F 2 + 1 2 R 2 ) + exp ( - 4 σ a 2 ) T U D G ( H 3 + 2 R 2 ) ( F 3 + 1 2 R 2 ) - exp ( - 2 σ a 2 ) 2 X Y Z Q ( H 4 + 2 R 2 ) ( F 4 + 1 2 R 2 ) ] - 1.
G ( R ) = m P 2 m 2 ( p ) p = 0 = σ P 2 / P 2 σ I 2 ( p ) / I ( p ) 2 p = 0 .
m 2 ( p ) = 4 exp ( 4 σ χ s 2 ) α s 4 [ 1 4 α s 2 + 1 4 ρ s 2 + 1 ρ 0 2 + ( α s A 2 ) 2 ] 2 × exp [ 2 ( k L ) 2 ( A 2 α s 2 ) + 1 α s 2 + 1 ρ s 2 + 4 ρ 0 2 ] p 2 × [ α s 2 2 B D G exp ( - k 2 B L 2 p 2 ) + α s 2 2 B K N exp ( - k 2 B L 2 p 2 ) + exp ( - 4 σ a 2 ) T U D W exp ( - k 2 U L 2 p 2 ) - exp ( - 2 σ a 2 ) 2 X Y Z Q exp ( - k 2 Y L 2 p 2 ) ] - 1.
m 2 ( p ) p = 0 = 4 exp ( 4 σ χ s 2 ) α s 4 [ 1 4 α s 2 + 1 4 ρ s 2 + 1 ρ 0 2 + ( α s A 2 ) 2 ] 2 × [ α s 2 2 B D G + α s 2 2 B K N + exp ( - 4 σ a 2 ) T U D W - exp ( - 2 σ a 2 ) 2 X Y Z Q ] - 1.
4 exp ( 4 σ χ s 2 ) α s 4 [ 1 4 α s 2 + 1 4 ρ s 2 + 1 ρ 0 2 + ( α s A 2 ) 2 ] 2 [ 1 R 2 + ( k α s 2 L ) 2 α s 2 [ 1 + ( α s ρ s ) 2 + ( 2 α s ρ 0 ) 2 + ( α s 2 A ) 2 ] ] 2 G ( R ) = 4 exp ( 4 σ χ s 2 ) α s 4 [ 1 4 α s 2 + 1 4 ρ s 2 + 1 ρ 0 2 + ( α s A 2 ) 2 ] 2 [ α s 2 2 B D G + α s 2 2 B K N + exp ( - 4 σ a 2 ) T U D W - exp ( - 2 σ a 2 ) 2 X Y Z Q ] - 1 × [ α s 2 2 B D G ( H 1 + 2 R 2 ) ( F 1 + 1 2 R 2 ) + α s 2 2 B K N ( H 2 + 2 R 2 ) ( F 2 + 1 2 R 2 ) + exp ( - 4 σ a 2 ) T U D G ( H 3 + 2 R 2 ) ( F 3 + 1 2 R 2 ) - exp ( - 2 σ a 2 ) 2 X Y Z Q ( H 4 + 2 R 2 ) ( F 4 + 1 2 R 2 ) ] - 1. 1
u ( L , p ) = e i k L i λ L Σ d 2 s u s ( s ) × exp [ i k 2 L ( p - s ) 2 ] exp [ ψ ( s , p ) ] ,
I ( p 1 ) I ( p 2 ) = ( 1 λ L ) 4 d 2 s 1 d 2 s 2 d 2 s 3 d 2 s 4 × Γ 4 s ( s 1 , s 2 , s 3 , s 4 ) F 4 ( s 1 , s 2 , s 3 , s 4 ; p 1 , p 2 ) × exp { - i k L [ p 1 · ( s 1 - s 2 ) + p 2 · ( s 3 - s 4 ) ] + i k 2 L ( s 1 2 - s 2 2 + s 3 2 - s 4 2 ) } ,
Γ 4 s ( s 1 , s 2 , s 3 , s 4 ) u s ( s 1 ) u s * ( s 2 ) u s ( s 3 ) u s * ( s 4 ) s
F 4 ( s 1 , s 2 , s 3 , s 4 ; p 1 , p 2 ) exp [ ψ ( s 1 , p 1 ) + ψ * ( s 2 , p 1 ) + ψ ( s 3 , p 2 ) + ψ * ( s 4 , p 2 ) ] m
F 4 = exp [ - 1 2 D ψ ( s 1 - s 2 , 0 ) - 1 2 D ψ ( s 1 - s 4 , p d ) - 1 2 D ψ ( s 2 - s 3 , p d ) - 1 2 D ψ ( s 3 - s 4 , 0 ) + 1 2 D ψ ( s 2 - s 4 , p d ) + 1 2 D ψ ( s 1 - s 3 , p d ) + 2 B χ ( s 2 - s 4 , p d ) + 2 B χ ( s 1 - s 3 , p d ) + i D χ S ( s 2 - s 4 , p d ) - i D χ S ( s 1 - s 3 , p d ) ] ,
D ψ ( s i - s j , p d ) ψ ( s i , p 1 ) - ψ ( s j , p 2 ) 2 ,
D χ S ( s i - s j , p d ) χ ( s i , p 1 ) - χ ( s j , p 2 ) S ( s i , p 1 ) - S ( s j , p 2 ) ,
B χ ( s i - s j , p d ) χ ( s i , p 1 ) χ ( s j , p 2 ) - χ 2 = σ χ s 2 - 1 2 D χ ( s i - s j , p d ) ,
σ χ s 2 = 0.124 k 7 / 6 C n 2 L 11 / 6 .
1 2 D ψ ( s d , p d ) = 1 ρ 0 2 ( s d 2 + s d · p d + p d 2 ) ,
B χ ( s d , p d ) = σ χ s 2 - 1 2 ( 1 ρ 0 2 - 1 ρ χ 2 ) ( s d 2 + p d · s d + p d 2 ) ,
D χ S ( s d , p d ) = 1 ρ χ S 2 ( s d 2 + p d · s d + p d 2 ) .
U s d ( s ) = A s exp [ - ( 1 2 α s 2 + i k 2 F ) s 2 ] ,
Γ s 4 ( s 1 , s 2 , s 3 , s 4 ) = A s 4 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 ) ] × ( 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 4 ) 2 ] } - 2 exp [ - 2 σ a 2 - 1 4 ρ s 2 ( s 1 2 + s 2 2 + s 3 2 + s 4 2 ) ] ) ,