Abstract

Approximate expressions are derived for the probability density functions of the i.f. signal magnitudes from optical heterodyne detection systems operating in the presence of clear air turbulence. The effects of log-normal amplitude fluctuations and Gaussian phase perturbations, in addition to local oscillator shot noise, are considered for both passive receivers and those employing active tilt-tracking systems to eliminate angle-of-arrival fluctuations. In Part 2, experimental results are presented that verify the density functions developed here.

© 1978 Optical Society of America

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References

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  1. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  2. I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965).
    [CrossRef]
  3. R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 6.
  4. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
    [CrossRef]
  5. David M. Chase, J. Opt. Soc. Am. 56, 33 (1966).
    [CrossRef]
  6. D. L. Fried, “Effects of Atmospheric Turbulence on Static and Tracking Optical Heterodyne Receivers/Average Antenna Gain and Antenna Gain Vairation,” Optical Science Consultants Technical Report TR-027 (August1971).
  7. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
    [CrossRef]
  8. M. C. Teich, S. Rosenberg, Appl. Opt. 12, 2616 (1973).
    [CrossRef] [PubMed]
  9. J. H. Churnside, C. M. McIntyre, “Distribution of Optical Heterodyne Detection Systems in the Presence of Clear Air Turbulence. 2: Experiment,” accompanying paper. Appl. Opt. 17, 2148 (1978).
    [CrossRef] [PubMed]
  10. R. L. Lawrence, J. W. Stronbehn, Proc. IEEE 58, 1523 (1970).
    [CrossRef]
  11. W. K. Pratt, Laser Communications Systems (Wiley, New York, 1969).
  12. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).
  13. D. A. DeWolf, J. Opt. Soc. Am. 63, 171 (1973).
    [CrossRef]
  14. D. L. Fried, R. A. Schmeltzer, Appl. Opt. 6, 1729 (1967).
    [CrossRef] [PubMed]
  15. P. Diament, M. C. Teich, J. Opt. Soc. Am. 60, 1489 (1970).
    [CrossRef]
  16. G. K. Born, R. Bogenberger, K. D. Erben, F. Frank, F. Mohr, G. Sepp, Appl. Opt. 14, 2857 (1975).
    [CrossRef] [PubMed]

1978 (1)

1975 (1)

1973 (2)

1970 (2)

R. L. Lawrence, J. W. Stronbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

P. Diament, M. C. Teich, J. Opt. Soc. Am. 60, 1489 (1970).
[CrossRef]

1967 (2)

1966 (2)

1965 (2)

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
[CrossRef]

Bogenberger, R.

Born, G. K.

Chabot, A.

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

Chase, David M.

Churnside, J. H.

DeWolf, D. A.

Diament, P.

Erben, K. D.

Frank, F.

Fried, D. L.

D. L. Fried, R. A. Schmeltzer, Appl. Opt. 6, 1729 (1967).
[CrossRef] [PubMed]

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
[CrossRef]

D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
[CrossRef]

D. L. Fried, “Effects of Atmospheric Turbulence on Static and Tracking Optical Heterodyne Receivers/Average Antenna Gain and Antenna Gain Vairation,” Optical Science Consultants Technical Report TR-027 (August1971).

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 6.

Goldstein, I.

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

Karp, S.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 6.

Lawrence, R. L.

R. L. Lawrence, J. W. Stronbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

McIntyre, C. M.

Miles, P. A.

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

Mohr, F.

Pratt, W. K.

W. K. Pratt, Laser Communications Systems (Wiley, New York, 1969).

Rosenberg, S.

Schmeltzer, R. A.

Sepp, G.

Stronbehn, J. W.

R. L. Lawrence, J. W. Stronbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).

Teich, M. C.

Appl. Opt. (4)

J. Opt. Soc. Am. (5)

Proc. IEEE (3)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

R. L. Lawrence, J. W. Stronbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

Other (4)

W. K. Pratt, Laser Communications Systems (Wiley, New York, 1969).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 6.

D. L. Fried, “Effects of Atmospheric Turbulence on Static and Tracking Optical Heterodyne Receivers/Average Antenna Gain and Antenna Gain Vairation,” Optical Science Consultants Technical Report TR-027 (August1971).

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Figures (5)

Fig. 1
Fig. 1

Block diagram of a generalized optical heterodyne receiver.

Fig. 2
Fig. 2

Probability density function p(I) vs normalized rms signal current I for static receiver (S), tracking receiver (T), and quiescent atmosphere (O) cases with D/r0 = 0.5.

Fig. 3
Fig. 3

Same as Fig. 2 with D/r0 = 1.0 and σχ = 0.3. Dashed line neglects first-order correction term.

Fig. 4
Fig. 4

Same as Fig. 2 with D/r0 = 1.0 and σχ = 0.8.

Fig. 5
Fig. 5

Same as Fig. 2 with D/r0 = 3.4.

Equations (65)

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W ( x ) = { 1 , if | x | ½ D , 0 , if | x | > ½ D .
E o ( x ) = E o = A o exp [ i ( 2 π f o t + ϕ o ) ]  ,
E s ( x ) = A s ( x ) exp { i [ 2 π f s t + ϕ s ( x ) ] } ,
i = 1 2 d x W ( x ) η [ E s ( x ) + E o ] * [ E s ( x ) + E o ]  .
i = η ( π / 8 ) D 2 A o 2 + η A o d x W ( x ) A s ( x ) × cos [ 2 π Δ f t + ϕ s ( x ) ϕ o ]  .
i s = η A o A s Z d x W ( x ) cos [ 2 π Δ f t ϕ o + ϕ s ( x ) ]  .
i n = ( 2 i e B ) 1 / 2 = ( π η e B ) 1 / 2 1 2 D A o  ,
i s = 2 1 / 2 γ Z π R 2 d x W ( x ) cos [ 2 π Δ f t ϕ o + ϕ s ( x ) ]  ,
ϕ s ( x ) Σ k = 1 6 a k F k ( x )
F 1 ( x ) ( π R 2 ) 1 / 2
F 2 ( x ) ( π R 4 / 4 ) 1 / 2 x ,
F 3 ( x ) ( π R 4 / 4 ) 1 / 2 y ,
F 4 ( x ) ( π R 6 / 12 ) 1 / 2 ( x 2 + y 2 1 2 R 2 )
F 5 ( x ) ( π R 6 / 6 ) 1 / 2 ( x 2 y 2 )  ,
F 6 ( x ) ( π R 6 / 24 ) 1 / 2 x y ,
a k = d x W ( x ) ϕ s ( x ) F k ( x )  .
i s = 2 1 / 2 γ Z / π R 2 d x W ( x ) cos [ 2 π Δ f t ϕ o + k = 1 6 a k F k ( x ) ] = 2 1 / 2 γ Z / π R 2 { cos ( 2 π Δ f t ϕ o + a 1 F 1 ) × d x W ( x ) cos [ k = 2 6 a k F k ( x ) ] sin ( 2 π Δ f t ϕ o + a 1 F 1 ) d x W ( x ) sin [ k = 2 6 a k F k ( x ) ] }  .
Δ 2 = ( 4 / π R 2 ) ( a 2 2 + a 3 2 )  ,
d x W ( x ) cos [ a 2 F 2 ( x ) + a 3 F 3 ( x ) ] = 2 π R 2 J 1 ( Δ ) / Δ
d x W ( x ) sin [ a 2 F 2 ( x ) + a 3 F 3 ( x ) ] = 0 ,
I = 2 γ Z | J 1 ( Δ ) / Δ |  ,
α 2 = ( 12 / π R 2 ) a 4 2 ,
β 2 = ( 6 / π R 2 ) ( a 5 2 + a 6 2 )  ,
C ( α , β ) = 0 1 cos ( α t ) J 0 ( β t ) d t ,
S ( α , β ) = 0 1 sin ( α t ) J 0 ( β t ) d t ,
d x W ( x ) cos [ Σ k = 4 6 a k F k ( x ) ] = π R 2 [ cos ( 1 2 α ) C ( α , β ) + sin ( 1 2 α ) S ( α , β ) ]
d x W ( x ) sin [ Σ k = 4 6 a k F k ( x ) ] = π R 2 [ sin ( 1 2 α ) C ( α , β ) cos ( 1 2 α ) S ( α , β ) ]
I = γ Z [ C 2 ( α , β ) + S 2 ( α , β ) ] 1 / 2 .
p ( I ) = d X d Δ p ( I | X, Δ ) p ( X ) p ( Δ )  ,
p ( I | X , Δ ) = 1 ( 2 π ) 1 / 2 exp { 1 2 [ I 2 γ Z J 1 ( Δ ) / Δ ] 2 } .
p ( X ) = 1 ( 2 π ) 1 / 2 σ χ exp [ ( X + σ χ 2 ) 2 2 σ χ 2 ]  .
p ( Δ ) = Δ σ Δ 2 exp ( Δ 2 2 σ Δ 2 )  .
p ( I ) = 1 2 π σ Δ 2 σ χ d X d Δ exp [ f ( X , Δ ) ]  ,
f ( X , Δ ) = 1 2 [ I 2 γ J 1 ( Δ ) / Δ ] 2 1 2 σ χ 2 ( X + σ χ 2 ) 2 Δ 2 2 σ Δ 2 + ln Δ  .
( f X ) | X 0  ,  Δ 0 = [ I 2 γ Z 0 J 1 ( Δ 0 ) / Δ 0 ] 2 γ Z 0 J 1 ( Δ 0 ) / Δ 0 + 1 σ χ 2 ( X 0 + σ χ 2 ) = 0 ,
( f Δ ) | X 0  ,  Δ 0 = [ I 2 γ Z 0 J 1 ( Δ 0 ) / Δ 0 ] 2 γ Z 0 J 1 ( Δ 0 ) / Δ 0 × [ J 0 ( Δ 0 ) / Δ 0 2 J 1 ( Δ 0 ) / Δ 0 2 ] Δ 0 σ Δ 2 + 1 Δ 0 = 0 ,
B = | 2 f X 2 2 f X Δ 2 f X Δ 2 f Δ 2 | ( X 0  ,  Δ 0 )
2 f X 2 = [ I 4 γ Z J 1 ( Δ ) / Δ ] ) 2 γ Z J 1 ( Δ ) / Δ 1 σ χ 2 ,
2 f X Δ = [ I 4 γ Z J 1 ( Δ ) / Δ ] 2 γ Z [ J 0 ( Δ ) / Δ 2 J 1 ( Δ ) / Δ 2 ]  ,
2 f Δ 2 = [ I 2 γ Z J 1 ( Δ ) / Δ ] 2 γ Z [ 6 J 1 ( Δ ) / Δ 3 J 1 ( Δ ) / Δ 3 J 0 ( Δ ) / Δ 2 ] 4 γ 2 Z 2 [ J 0 ( Δ ) / Δ 2 J 1 ( Δ ) / Δ 2 ] 2 1 σ Δ 2 1 Δ 2 .
p ( I ) = Δ 0 B 1 / 2 σ Δ 2 σ χ exp [ 1 2 [ I 2 γ Z 0 J 1 ( Δ 0 ) / Δ 0 ] 2 1 2 σ χ 2 ( X 0 + σ χ 2 ) 2 Δ 0 2 2 σ Δ 2 ]  .
p ( I ) = d X d α d β p ( I | X , α , β ) p ( X ) p ( α ) p ( β )  .
p ( I | X , α , β ) = 1 ( 2 π ) 1 / 2 exp { 1 2 [ I γ Z ( C 2 + S 2 ) 1 / 2 ] 2 } .
p ( α ) = 1 ( 2 π ) 1 / 2 σ α exp α 2 2 σ α 2 ,
p ( β ) = 2 β σ α 2 exp β 2 σ α 2 ,
σ α 2 = 12 a 4 2 π R 6 .
p ( I ) = 2 ( 2 π ) 3 / 2 σ χ σ α 3 d X d α d β exp [ f ( X , α , β ) ]  ,
f ( X , α , β ) = 1 2 [ I γ Z ( C 2 + S 2 ) 1 / 2 ] 2 1 2 σ χ 2 ( X + σ χ 2 ) 2 α 2 2 σ α 2 β 2 σ α 2 + ln β .
( f X ) | ( X 0 ,  α 0 , β 0 ) = [ I γ Z 0 ( C 2 + S 2 ) 1 / 2 ] γ Z 0 ( C 2 + S 2 ) 1 / 2 1 σ χ 2 ( X 0 + σ χ 2 ) = 0 ,
( f α ) | ( X 0 , α 0 , β 0 ) = [ I γ Z 0 ( C 2 + S 2 ) 1 / 2 ] × γ Z 0 C C α + S S α ( C 2 + S 2 ) 1 / 2 α 0 σ α 2 = 0 ,
( f β ) | ( X 0 , α 0 , β 0 ) = [ I γ Z 0 ( C 2 + S 2 ) 1 / 2 ] × γ Z 0 C C β + S S β ( C 2 + S 2 ) 1 / 2 2 β 0 σ α 2 + 1 β 0 = 0.
( f α ) | ( X 0 , 0 , β 0 ) = 0 ,
[ I γ Z 0 C ( 0 , β 0 ) ] γ Z 0 C ( 0 , β 0 ) 1 σ χ 2 ( X 0 + σ χ 2 ) = 0 ,
[ I γ Z 0 C ( 0 , β 0 ) ] γ Z 0 ( C β ) ( 0 , β 0 ) 2 β 0 σ α 2 + 1 β 0 = 0.
B = | 2 f X 2 0 2 f X β 0 2 f α 2 0 2 f X β 0 2 f β 2 | ( X 0 , 0 , β 0 )
( 2 f X 2 ) = ( I 2 γ Z C ) γ Z C 1 σ χ 2 ,
( 2 f X β ) = ( I 2 γ Z C ) γ Z ( C β ) ,
( 2 f α 2 ) = ( I γ Z C ) γ Z [ 2 C α 2 + ( S α ) 2 C ] 1 σ α 2 ,
( 2 f β 2 ) = ( I γ Z C ) γ Z ( 2 C β 2 ) γ 2 Z 2 ( 2 C β 2 ) C 2 σ α 2 1 β 0 2 ,
p ( I ) = 2 β 0 σ χ σ α 3 ( B ) 1 / 2 exp { 1 2 [ I γ Z 0 C ( 0 , β 0 ) ] 2 1 2 σ χ 2 ( X 0 + σ χ 2 ) 2 α 0 2 2 σ α 2 β 0 2 σ α 2 }  .
σ Δ 2 = k 2 D 2 4 θ 2 = D s ( D ) 4 = 0.73 k 2 C n 2 L D 5 / 3 = 1.72 ( D / r o ) 5 / 3 ,
σ α 2 = ( 12 / π R 2 ) a 4 2 = 0.253 ( D / r o ) 5 / 3 .
p ( I ) = 0 d X d Δ 1 ( 2 π ) 1 / 2 × exp { 1 2 [ I 2 γ Z | J 1 ( Δ ) / Δ | ] 2 } p ( X ) p ( Δ ) 0 d X d Δ 1 ( 2 π ) 1 / 2 exp { 1 2 [ I 2 γ Z J 1 ( Δ ) / Δ ] 2 } p ( X ) p ( Δ ) 3.83 7.02 d X d Δ p ( X ) p ( Δ ) 1 ( 2 π ) 1 / 2 × ( exp { 1 2 [ I 2 γ Z J 1 ( Δ ) / Δ ] 2 } exp { 1 2 [ I + 2 γ Z J 1 ( Δ ) / Δ ] 2 } )
p ( I ) = 1 ( 2 π ) 1 / 2 σ χ exp [ 1 2 ( I γ Z 0 ) 2 ] exp [ ( X 0 + σ χ 2 ) 2 2 σ χ 2 ] [ 1 σ χ 2 ( I 2 γ Z 0 ) γ Z 0 ] 1 / 2 ,
( I γ Z 0 ) γ Z 0 + 1 σ χ 2 ( X 0 + σ χ 2 ) = 0.

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