Abstract

We have studied the effect of residual log-amplitude fluctuations and imperfect phase compensation on the characteristics of a phase-compensated optical link operating through atmospheric turbulence.

© 1976 Optical Society of America

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References

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  1. M. Yellin, J. Opt. Soc. Am. 65, 1211 (1975).
  2. J. Hardy, J. Opt. Soc. Am. 65, 1212 (1975).
  3. P. Cone and J. Feinleib, J. Opt. Soc. Am,  65, 1212 (1975).
  4. J. Pearson, W. Brown, S. Kokorowski, T. O’Meara, and M. Pedinoff, J. Opt. Soc. Am. 65, 1212 (1975).
  5. D. Fried and H. Yura, J. Opt. Soc. Am. 62, 600 (1972).
    [CrossRef]
  6. Z. Feizulin and Y. Kravtsov, Radiophys. Quantum Electron. 10, 33 (1967).
    [CrossRef]
  7. R. Fante, J. Opt. Soc. Am,  66, 74 (1976).
    [CrossRef]
  8. R. Fante, J. Opt. Soc. Am. 65, 548 (1975).
    [CrossRef]
  9. That is ρ0/λx)1/2=(2.5 k7/6 Cn2 x11/6)-3/5=[6(σ12)3/5]-1≪1, since σ12≫1 in strong turbulence.
  10. V. Banakh, G. Krekov, V. Mironov, S. Khmelevstov, and S. Tsvik, J. Opt. Soc. Am. 64, 516 (1974).
    [CrossRef]
  11. A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shisov, Proc. IEEE 63, 790 (1975).
    [CrossRef]
  12. In Eq. (13) we have replaced the phase structure function dS by the wave structure function d1, because the two are approximately equal for |ρ1 − ρ2| > (λx)1/2.
  13. R. Hudgin, J. Opt. Soc. Am. 65, 1211 (1975).
  14. L. Bradley and M. Cheifetz, J. Opt. Soc. Am. 65, 1212 (1975).
  15. J. Shapiro (private communication, 1975).
  16. R. Lawrence and J. Strohbehn, Proc. IEEE 58, 1523 (1970).
    [CrossRef]

1976 (1)

R. Fante, J. Opt. Soc. Am,  66, 74 (1976).
[CrossRef]

1975 (8)

R. Fante, J. Opt. Soc. Am. 65, 548 (1975).
[CrossRef]

M. Yellin, J. Opt. Soc. Am. 65, 1211 (1975).

J. Hardy, J. Opt. Soc. Am. 65, 1212 (1975).

P. Cone and J. Feinleib, J. Opt. Soc. Am,  65, 1212 (1975).

J. Pearson, W. Brown, S. Kokorowski, T. O’Meara, and M. Pedinoff, J. Opt. Soc. Am. 65, 1212 (1975).

A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shisov, Proc. IEEE 63, 790 (1975).
[CrossRef]

R. Hudgin, J. Opt. Soc. Am. 65, 1211 (1975).

L. Bradley and M. Cheifetz, J. Opt. Soc. Am. 65, 1212 (1975).

1974 (1)

1972 (1)

1970 (1)

R. Lawrence and J. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

1967 (1)

Z. Feizulin and Y. Kravtsov, Radiophys. Quantum Electron. 10, 33 (1967).
[CrossRef]

Banakh, V.

Bradley, L.

L. Bradley and M. Cheifetz, J. Opt. Soc. Am. 65, 1212 (1975).

Brown, W.

J. Pearson, W. Brown, S. Kokorowski, T. O’Meara, and M. Pedinoff, J. Opt. Soc. Am. 65, 1212 (1975).

Bunkin, F.

A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shisov, Proc. IEEE 63, 790 (1975).
[CrossRef]

Cheifetz, M.

L. Bradley and M. Cheifetz, J. Opt. Soc. Am. 65, 1212 (1975).

Cone, P.

P. Cone and J. Feinleib, J. Opt. Soc. Am,  65, 1212 (1975).

Fante, R.

R. Fante, J. Opt. Soc. Am,  66, 74 (1976).
[CrossRef]

R. Fante, J. Opt. Soc. Am. 65, 548 (1975).
[CrossRef]

Feinleib, J.

P. Cone and J. Feinleib, J. Opt. Soc. Am,  65, 1212 (1975).

Feizulin, Z.

Z. Feizulin and Y. Kravtsov, Radiophys. Quantum Electron. 10, 33 (1967).
[CrossRef]

Fried, D.

Gochelashvily, K.

A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shisov, Proc. IEEE 63, 790 (1975).
[CrossRef]

Hardy, J.

J. Hardy, J. Opt. Soc. Am. 65, 1212 (1975).

Hudgin, R.

R. Hudgin, J. Opt. Soc. Am. 65, 1211 (1975).

Khmelevstov, S.

Kokorowski, S.

J. Pearson, W. Brown, S. Kokorowski, T. O’Meara, and M. Pedinoff, J. Opt. Soc. Am. 65, 1212 (1975).

Kravtsov, Y.

Z. Feizulin and Y. Kravtsov, Radiophys. Quantum Electron. 10, 33 (1967).
[CrossRef]

Krekov, G.

Lawrence, R.

R. Lawrence and J. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

Mironov, V.

O’Meara, T.

J. Pearson, W. Brown, S. Kokorowski, T. O’Meara, and M. Pedinoff, J. Opt. Soc. Am. 65, 1212 (1975).

Pearson, J.

J. Pearson, W. Brown, S. Kokorowski, T. O’Meara, and M. Pedinoff, J. Opt. Soc. Am. 65, 1212 (1975).

Pedinoff, M.

J. Pearson, W. Brown, S. Kokorowski, T. O’Meara, and M. Pedinoff, J. Opt. Soc. Am. 65, 1212 (1975).

Prokhorov, A.

A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shisov, Proc. IEEE 63, 790 (1975).
[CrossRef]

Shapiro, J.

J. Shapiro (private communication, 1975).

Shisov, V.

A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shisov, Proc. IEEE 63, 790 (1975).
[CrossRef]

Strohbehn, J.

R. Lawrence and J. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

Tsvik, S.

Yellin, M.

M. Yellin, J. Opt. Soc. Am. 65, 1211 (1975).

Yura, H.

J. Opt. Soc. Am (2)

P. Cone and J. Feinleib, J. Opt. Soc. Am,  65, 1212 (1975).

R. Fante, J. Opt. Soc. Am,  66, 74 (1976).
[CrossRef]

J. Opt. Soc. Am. (8)

R. Fante, J. Opt. Soc. Am. 65, 548 (1975).
[CrossRef]

M. Yellin, J. Opt. Soc. Am. 65, 1211 (1975).

J. Hardy, J. Opt. Soc. Am. 65, 1212 (1975).

J. Pearson, W. Brown, S. Kokorowski, T. O’Meara, and M. Pedinoff, J. Opt. Soc. Am. 65, 1212 (1975).

D. Fried and H. Yura, J. Opt. Soc. Am. 62, 600 (1972).
[CrossRef]

V. Banakh, G. Krekov, V. Mironov, S. Khmelevstov, and S. Tsvik, J. Opt. Soc. Am. 64, 516 (1974).
[CrossRef]

R. Hudgin, J. Opt. Soc. Am. 65, 1211 (1975).

L. Bradley and M. Cheifetz, J. Opt. Soc. Am. 65, 1212 (1975).

Proc. IEEE (2)

A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shisov, Proc. IEEE 63, 790 (1975).
[CrossRef]

R. Lawrence and J. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

Radiophys. Quantum Electron. (1)

Z. Feizulin and Y. Kravtsov, Radiophys. Quantum Electron. 10, 33 (1967).
[CrossRef]

Other (3)

That is ρ0/λx)1/2=(2.5 k7/6 Cn2 x11/6)-3/5=[6(σ12)3/5]-1≪1, since σ12≫1 in strong turbulence.

In Eq. (13) we have replaced the phase structure function dS by the wave structure function d1, because the two are approximately equal for |ρ1 − ρ2| > (λx)1/2.

J. Shapiro (private communication, 1975).

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Equations (43)

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u ( ρ ) = k 2 π i x - d 2 ρ 1 u 0 ( ρ 1 ) × exp ( i k 2 x ( ρ - ρ 1 ) 2 + χ ( ρ , ρ 1 ) + i S ( ρ , ρ 1 ) ) ,
u 0 ( ρ 1 ) = ũ 0 ( ρ 1 ) exp [ - i ( k / 2 x ) ρ 1 2 - i S ( 0 , ρ 1 ) ] ,
u c ( ρ ) = ( k 2 π i x ) - d 2 ρ 1 μ ˜ 0 ( ρ 1 ) × exp ( - i k x ρ · ρ 1 + i k 2 x ρ 2 + χ ( ρ , ρ 1 ) ) ,
I ( x , ρ ) c u c ( ρ ) u c * ( ρ ) = ( k 2 π x ) 2 = d 2 ρ 1 - d 2 ρ 2 ũ 0 ( ρ 1 ) ũ 0 ( ρ 2 ) × exp ( - i k x ρ · ( ρ 1 - ρ 2 ) - 1 2 d x ( 0 , ρ 1 - ρ 2 ) ) ,
d x ( ρ - ρ , ρ 1 - ρ 2 ) = [ χ ( ρ , ρ 1 ) - χ ( ρ , ρ 2 ) ] 2
I 2 ( x , ρ ) c = ( k 2 π x ) 4 e 4 σ x 2 - d 2 ρ 1 d 2 ρ 2 d 2 ρ 3 d 2 ρ 1 ũ 0 ( ρ 4 ) ũ 0 ( ρ 2 ) ũ 0 ( ρ 3 ) ũ 0 ( ρ 4 ) exp ( - i k x ρ · ( ρ 1 - ρ 2 + ρ 3 - ρ 4 ) - 1 2 [ d χ ( 0 , ρ 1 - ρ 2 ) + d χ ( 0 , ρ 1 - ρ 3 ) + d χ ( 0 , ρ 1 - ρ 4 ) + d χ ( 0 , ρ 2 - ρ 3 ) + d χ ( 0 , ρ 2 - ρ 4 ) + d χ ( 0 , ρ 3 - ρ 4 ) ] ) ,
d χ ( 0 , ρ 1 - ρ 2 ) { 0.545 k 2 C n 2 x ρ 1 - ρ 2 5 / 3 for ρ 1 - ρ 2 ( λ x ) 1 / 2 , 2 σ χ 2 for ρ 1 - ρ 2 ( λ x ) 1 / 2 .
d χ ( 0 , ρ 1 - ρ 2 ) = 2 [ σ χ 2 - δ χ ( ρ 1 ) δ χ ( ρ 2 ) ] 2 σ χ 2
I ( x , ρ ) c exp ( - σ χ 2 ) I 0 ( ρ ) ,
ρ 1 - ρ 2 ( λ x ) 1 / 2 ,             ρ 1 - ρ 3 ( λ x ) 1 / 2 ,             etc . ,
I 2 ( x , ρ ) c exp ( - 2 σ χ 2 ) I 0 2 ( ρ )
σ c 2 ( I 2 ) c - I c 2 ) / I c 2 F 0 ,
σ u 2 = ( I 2 u - I u 2 ) / I u 2 F 1.
I 2 ( x , ρ ) u = ( k 2 π x ) 4 - d 2 ρ 1 d 2 ρ 2 d 2 ρ 3 d 2 ρ 4 ũ 0 ( ρ 1 ) ũ 0 ( ρ 2 ) ũ 0 ( ρ 3 ) ũ 0 ( ρ 4 ) exp ( - i k x ρ · ( ρ 1 - ρ 2 + ρ 3 - ρ 4 ) - 1 2 [ d 1 ( 0 , ρ 1 - ρ 2 ) + d 1 ( 0 , ρ 1 - ρ 4 ) + d 1 ( 0 , ρ 2 - ρ 3 ) + d 1 ( 0 , ρ 3 - ρ 4 ) - d 1 ( 0 , ρ 2 - ρ 4 ) - d 1 ( 0 , ρ 1 - ρ 3 ) ] ) ,
I ( x , 0 ) c I 0 ( 0 ) / [ 1 + 11.7 F ( σ χ 2 ) 6 / 5 ] ,
I 2 ( x , 0 ) c I 0 2 ( 0 ) exp ( 4 σ χ 2 ) / [ 1 + 23.4 F ( σ χ 2 ) 6 / 5 ] 3 ,
σ c 2 [ exp ( 4 σ χ 2 ) - 1 ] - 46.8 F ( σ χ 2 ) 6 / 5 exp ( 4 σ χ 2 ) ,
exp ( - 1.2 σ χ 2 ) I 0 ( 0 ) I ( x , 0 ) c I 0 ( 0 ) ,
exp ( - 3.2 σ χ 2 ) I 0 2 ( 0 ) I 2 ( x , 0 ) c exp ( 4 σ χ 2 ) I 0 ( 0 ) ,
exp ( 4 σ χ 2 ) - 1 σ c 2 0.
u 0 ( ρ 1 ) = ũ 0 ( ρ 1 ) exp [ - i ( k / 2 x ) ρ 1 2 - i S 0 ( ρ 1 ) ] ,
u c ( ρ ) = ( k 2 π i x ) e i ( k / 2 x ) ρ 2 - d 2 ρ 1 ũ 0 ( ρ 1 ) × exp ( i k x ρ · ρ 1 + χ ( 0 , ρ 1 ) + i [ S ( 0 , ρ 1 ) - S 0 ( ρ 1 ) ] ) .
I ( x , 0 ) c ( k 2 π x ) 2 - d 2 ρ 1 - d 2 ρ 2 ũ 0 ( ρ 1 ) ũ 0 ( ρ 2 ) × exp [ - 1 2 d χ ( 0 , ρ 1 - ρ 2 ) - 1 2 d Δ S ( ρ 1 - ρ 2 ) ] ,
d Δ S ( ρ 1 - ρ 2 ) 4.78 ( ρ 0 ω 0 ) 5 / 3 1 d ξ ξ 8 / 3 [ 1 - J 0 ( ξ ω 0 ρ 1 - ρ 2 ) ] × 0 1 d t t 5 / 3 cos 2 ( α ξ 2 ( 1 - t ) 2 t ) ,
d Δ S ( ρ 1 - ρ 2 ) { 1 ( ω 0 ρ 0 ) 5 / 3 [ A ( ω 0 ρ 1 - ρ 2 ) 5 / 3 - B ( ω 0 ρ 1 - ρ 2 ) 2 ] for ρ 1 - ρ 2 < ω 0 - 1 , C / ( ω 0 ρ 0 ) 5 / 3 for ρ 1 - ρ 2 ω 0 - 1 ,
d Δ S ( ρ 1 - ρ 2 ) C / ( ω 0 ρ 0 ) 5 / 3
d Δ S 1 ,
d Δ S ( ρ 1 - ρ 2 ) { ( Q / ρ 0 2 ) ρ 1 - ρ 2 2 for ρ 1 - ρ 2 ω 0 - 1 , Q / ( ω 0 ρ 0 ) 2 for ρ 1 - ρ 2 > ω 0 - 1 .
I ( x , 0 ) c I 0 ( 0 ) exp ( - σ χ 2 - Q 2 ω 0 2 ρ 0 2 ) [ exp ( - 1 4 a 2 ω 0 2 ) + exp ( Q / 2 ω 0 2 ρ 0 2 ) - exp ( - 1 / 4 a 2 ω 0 2 ) 1 + 2 Q a 2 / ρ 0 2 ] ,
I ( x , 0 ) c I 0 ( 0 ) exp ( - σ χ 2 ) ( 1 + ( 2 Q a 2 / ρ 0 2 ) exp ( - 1 / 4 a 2 ω 0 2 ) 1 + 2 Q a 2 / ρ 0 2 ) I 0 ( 0 ) exp ( - σ χ 2 ) .
I ( x , 0 ) c I 0 ( 0 ) exp ( - σ χ 2 ) 1 + 2 Q a 2 / ρ 0 2 .
I 2 ( x , 0 ) c ( k 2 π x ) 4 exp ( 4 σ χ 2 ) - d 2 ρ 1 d 2 ρ 2 d 2 ρ 3 d 2 ρ 4 ũ 0 ( ρ 1 ) ũ 0 ( ρ 2 ) ũ 0 ( ρ 3 ) ũ 0 ( ρ 4 ) × exp [ - 1 2 d χ ( 0 , ρ 1 - ρ 2 ) - 1 2 d χ ( 0 , ρ 1 - ρ 3 ) - 1 2 d χ ( 0 , ρ 1 - ρ 4 ) - 1 2 d χ ( 0 , ρ 2 - ρ 3 ) - 1 2 d χ ( 0 , ρ 2 - ρ 4 ) - 1 2 d χ ( 0 , ρ 3 - ρ 4 ) - 1 2 d Δ S ( ρ 1 - ρ 2 ) - 1 2 d Δ S ( ρ 1 - ρ 4 ) + 1 2 d Δ S ( ρ 1 - ρ 3 ) - 1 2 d Δ S ( ρ 2 - ρ 3 ) - 1 2 d Δ S ( ρ 3 - ρ 4 ) + 1 2 d Δ S ( ρ 2 - ρ 4 ) ] .
I 2 ( x , 0 ) c exp ( - 2 σ χ 2 ) I 0 ( 0 ) ,
S ( 0 , ρ 1 ) = - s ( ω ) exp ( i ω · ρ 1 ) d 2 ω ,
S 0 ( ρ 1 ) = - s 0 ( ω ) exp ( i ω · ρ ) d 2 ω ,
[ Δ S ( ρ 1 ) - Δ S ( ρ 2 ) ] 2 = ω > ω 0 d 2 ω ω > ω 0 d 2 ω s ( ω ) s ( ω ) × { exp [ i ( ω + ω ) · ρ 1 ] + exp [ i ( ω + ω ) · ρ 2 ] - exp [ i ( ω · ρ 1 + ω · ρ 2 ) ] - exp [ i ( ω · ρ 1 + ω · ρ 2 ) ] } .
G ( ω ) = ( 1 2 π ) 2 - d 2 ρ g ( ρ ) exp ( - i ρ · ω )
g ( ρ 1 - ρ 2 ) S ( ρ 1 ) S ( ρ 2 )
[ Δ S ( ρ 1 ) - Δ S ( ρ 2 ) ] 2 = 2 ω > ω 0 d 2 ω G ( ω ) [ 1 - exp { i ω · ( ρ 1 - ρ 2 ) } ] = 4 π ω 0 ω d ω G ( ω ) [ 1 - J 0 ( ω ρ 1 - ρ 2 ) ] .
g ( ρ 1 - ρ 2 ) = 2 π k 2 - d 2 κ ( 0.033 C n 2 κ 11 / 3 ) × 0 x d η exp ( i η x κ · ( ρ 1 - ρ 2 ) ) cos 2 ( κ 2 η ( x - η ) 2 k x ) .
G ( ω ) = 0.207 k 2 C n 2 x ω 11 / 3 0 1 d t t 5 / 3 cos 2 ( ω 2 x 2 k 1 - t t ) .
[ Δ S ( ρ 1 ) - Δ S ( ρ 2 ) ] 2 = 2.6 k 2 C n 2 x × ω 0 s d ω ω 8 / 3 [ 1 - J 0 ( ω ρ 1 - ρ 2 ) ] × 0 1 d t t 5 / 3 cos 2 ( ω 2 x ( 1 - t ) 2 k t ) .
d Δ S ( ρ 1 - ρ 2 ) = 4.78 ( ω 0 ρ 0 ) 5 / 3 1 d ξ ξ 8 / 3 [ 1 - J 0 ( ξ ω 0 ρ 1 - ρ 2 ) ] × 0 1 d t t 5 / 3 cos 2 ( α ξ 2 ( 1 - t ) 2 t ) ,