Abstract

The scintillation (power fluctuations due to causes other than beam wandering) of laser beams directed upward through the atmosphere is analyzed as a function of turbulence profile, zenith angle, beam waist at the transmitter, and wave-front curvature at the transmitter. Turbulence at the tropopause significantly reduces transmitter- aperture-averaging effects for collimated beams. Defocused beams have greater scintillation (measured by Cl, the log-amplitude variance) than collimated beams, for the same beam waist at the transmitter, and over certain ranges of defocusing may scintillate more than a point source. The analysis also shows that focused beams may have significantly less scintillation than collimated beams with the same initial beam waist. Because this occurs only for focusing at or above the tropopause, all of the scintillation may be considered as averaged at the focal region, so that far-field receivers receive flux that has been averaged over all of the turbulence.

© 1973 Optical Society of America

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References

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  1. P. J. Titterton, Appl. Opt. 12, 423 (1973).
    [Crossref] [PubMed]
  2. D. L. Fried and J. D. Cloud, J. Opt. Soc. Am. 56, 1667 (1966).
    [Crossref]
  3. R. A. Schmeltzer, Q. Appl. Math. 24, 339 (1967).
  4. A. Ishimaru, Proc. IEEE 57, 407 (1969).
    [Crossref]
  5. R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
    [Crossref]
  6. D. L. Fried and J. B. Seidman, J. Opt. Soc. Am. 57, 181 (1967).
    [Crossref]
  7. D. L. Fried, J. Opt. Soc. Am. 57, 980 (1967).
    [Crossref]
  8. J. R. Kerr and J. P. Dunphy, J. Opt. Soc. Am. 63, 1 (1973).
    [Crossref]
  9. J. L. Bufton, P. O. Minott, M. W. Fitzmaurice, and P. J. Titterton, J. Opt. Soc. Am. 62, 1068 (1972).
    [Crossref]
  10. J. L. Bufton (private communication).
  11. C. E. Coulman, Sol. Phys. 7, 122 (1969).
    [Crossref]
  12. Yu. A. Volkov, V. P. Kukharets, and L. R. Tsvang, Izv. Acad. Sci. USSR Atmos Ocean. Phys. 4, 591 (1968).
  13. J. Wyngaard, Y. Izumi, and S. A. Collins, J. Opt. Soc. Am. 61, 1646 (1971).
    [Crossref]
  14. J. L. Bufton, Master’s thesis, University of Maryland (1970).
  15. R. E. Hufnagel, as cited in Ref. 11.
  16. E. Brookner, Appl. Opt. 10, 1533 (1971).
  17. H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
    [Crossref]
  18. P. J. Titterton and J. P. Speck, Appl. Opt. 12, 425 (1973).
    [Crossref] [PubMed]

1973 (3)

1972 (1)

1971 (2)

1969 (3)

C. E. Coulman, Sol. Phys. 7, 122 (1969).
[Crossref]

A. Ishimaru, Proc. IEEE 57, 407 (1969).
[Crossref]

R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
[Crossref]

1968 (1)

Yu. A. Volkov, V. P. Kukharets, and L. R. Tsvang, Izv. Acad. Sci. USSR Atmos Ocean. Phys. 4, 591 (1968).

1967 (3)

1966 (2)

Brookner, E.

E. Brookner, Appl. Opt. 10, 1533 (1971).

Bufton, J. L.

J. L. Bufton, P. O. Minott, M. W. Fitzmaurice, and P. J. Titterton, J. Opt. Soc. Am. 62, 1068 (1972).
[Crossref]

J. L. Bufton, Master’s thesis, University of Maryland (1970).

J. L. Bufton (private communication).

Cloud, J. D.

Collins, S. A.

Coulman, C. E.

C. E. Coulman, Sol. Phys. 7, 122 (1969).
[Crossref]

Dunphy, J. P.

Fitzmaurice, M. W.

Fried, D. L.

Harp, J. C.

R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
[Crossref]

Hufnagel, R. E.

R. E. Hufnagel, as cited in Ref. 11.

Ishimaru, A.

A. Ishimaru, Proc. IEEE 57, 407 (1969).
[Crossref]

Izumi, Y.

Kerr, J. R.

Kogelnik, H.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[Crossref]

Kukharets, V. P.

Yu. A. Volkov, V. P. Kukharets, and L. R. Tsvang, Izv. Acad. Sci. USSR Atmos Ocean. Phys. 4, 591 (1968).

Lee, R. W.

R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
[Crossref]

Li, T.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[Crossref]

Minott, P. O.

Schmeltzer, R. A.

R. A. Schmeltzer, Q. Appl. Math. 24, 339 (1967).

Seidman, J. B.

Speck, J. P.

Titterton, P. J.

Tsvang, L. R.

Yu. A. Volkov, V. P. Kukharets, and L. R. Tsvang, Izv. Acad. Sci. USSR Atmos Ocean. Phys. 4, 591 (1968).

Volkov, Yu. A.

Yu. A. Volkov, V. P. Kukharets, and L. R. Tsvang, Izv. Acad. Sci. USSR Atmos Ocean. Phys. 4, 591 (1968).

Wyngaard, J.

Appl. Opt. (3)

Izv. Acad. Sci. USSR Atmos Ocean. Phys. (1)

Yu. A. Volkov, V. P. Kukharets, and L. R. Tsvang, Izv. Acad. Sci. USSR Atmos Ocean. Phys. 4, 591 (1968).

J. Opt. Soc. Am. (6)

Proc. IEEE (3)

A. Ishimaru, Proc. IEEE 57, 407 (1969).
[Crossref]

R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
[Crossref]

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[Crossref]

Q. Appl. Math. (1)

R. A. Schmeltzer, Q. Appl. Math. 24, 339 (1967).

Sol. Phys. (1)

C. E. Coulman, Sol. Phys. 7, 122 (1969).
[Crossref]

Other (3)

J. L. Bufton, Master’s thesis, University of Maryland (1970).

R. E. Hufnagel, as cited in Ref. 11.

J. L. Bufton (private communication).

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Figures (7)

F. 1
F. 1

Ratio of beam-wave to point-source log-amplitude variance vs wave-front curvature at the transmitter, for defocused conditions, with λ = 6328 Å, f(h) = h−1/3eh/1000, θ = 45°, H = 30 km, and Cls = 0.413.

F. 2
F. 2

Ratio of beam-wave to point-source log-amplitude variance vs wave-front curvature at the transmitter, for defocused conditions with λ = 6328 Å, f(h) = h−4/3, θ = 45°, H = 30 km, and Cls = 0.00363.

F. 3
F. 3

Ratio of beam-wave to point-source log-amplitude variance vs wave-front curvature at the transmitter, for defocused conditions, with the same parameters as Fig. 2, except f(h) = h−4/3 +0.001 exp{−[h−1.5(104)]/103}2 and Cls = 0.049 79.

F. 4
F. 4

Ratio of beam-wave to point-source log-amplitude variance vs wave-front curvature at the transmitter, for focused conditions, with λ = 5320 Å, θ = 45°, H = 35 000 km, f(h) = h−4/3 +0.002 43 exp{−[h−1.5(104)]/103}2, and Cls = 0.25.

F. 5
F. 5

Beam-waist-size effects on the ratio of beam-wave to point-source log-amplitude variance vs wave-front curvature at the transmitter, for focused conditions, with λ = 6328 Å, θ = 45°, H = 30 km, f(h) = h−4/3+0.002 43 exp{[h−1.5(104)]/103}2, and Cls = 0.115.

F. 6
F. 6

Tropopause-altitude effects on the ratio of beam-wave to point-source log-amplitude variance vs wave-front curvature at the transmitter for focused conditions, with λ = 6238 Å, θ = 45°, H = 30 km, w = 6 cm, and f(h) = h−4/3+0.002 43 exp{−[hhp]/103}2.

F. 7
F. 7

Zenith-angle effects on the ratio of beam-wave to point-source log-amplitude variance vs wave-front curvature at the transmitter for focused conditions, with λ = 6328 Å, H = 30 km, w = 6 cm, and f(h) = h−4/3+0.002 43 exp{−[h−1.5 × 104]/103}2.

Tables (1)

Tables Icon

Table I Behavior of integrand of Eq. (11) for focused beams at the tropopause.

Equations (39)

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C l ( l l ¯ ) 2 av ,
l 1 2 ln ( I / ) , I = instantaneous irradiance , = average irradiance .
σ I 2 = e 4 C l 1
the ripple = ( σ I 2 ) 1 2 × 100 % .
C l ( 0 ) = 8.16 8 π k 2 Re { 0 S d Z C N 2 ( Z ) 0 d σ σ 11 / 6 ( exp [ σ 2 Re γ ( S , Z ) ] exp [ σ 2 γ ( S , Z ) ] ) }
γ ( S , Z ) = 2 S Z i k [ Z i k α 2 S i k α 2 ]
1 α 2 = 1 α 0 2 i k R ,
U = A exp { ( r 2 2 α 0 2 i k r 2 2 R ) }
= A exp { ( r 2 w 2 i k r 2 2 R ) }
R > 0 defocused beam , R < 0 focused beam , R = collimated beam .
α 2 = α 0 2 ( 1 + i k α 0 2 / R ) 1 + ( k α 0 2 / R ) 2 .
C l s ( 0 ) = 5.6086 ( 10 14 ) K 0 k 7 / 6 ( sec θ ) 11 / 6 0 H f ( h ) ( 1 h H ) 5 / 6 h 5 / 6 d h .
C N 2 ( h ) = 10 13 K 0 f ( h ) ,
C l ( 0 ) C l s ( 0 ) = 3.8637 [ 0 H f ( h ) ( 1 h H ) 5 / 6 h 5 / 6 d h ] 1 [ 0 H f ( h ) ( H h ) 5 / 6 { ( M 2 + N 2 P 2 ) 5 / 12 × cos 5 6 ( tan 1 ( N M ) ) ( M P ) 5 / 6 } d h ]
M = ( H h cos θ ) A ( 1 + ( A R ) 2 ) ,
N = { h cos θ ( 1 + ( A R ) 2 ) + A 2 R } × { H cos ( 1 + ( A R ) 2 ) + A 2 R } + A 2 ,
P = { H cos θ ( 1 + ( A R ) 2 ) + A 2 R } 2 + A 2 ,
A = π w 2 λ .
f ( h ) = h 4 3 + 0.001 exp { ( h 1.5 × 10 4 10 3 ) 2 } ,
f ( h ) = h 4 3 + { 0.003 5 0.002 43 0.001 68 } exp { ( h 1.5 × 10 4 10 3 ) 2 } ,
f ( h ) = h 4 3 + 0.002 43 exp { h h p 10 3 } 2
f ( h ) = h 4 / 3 + 0.001 exp { h 1.5 × 10 4 10 3 } 2
f ( h ) = h 4 / 3 + 0.002 43 exp { ( h 1.5 × 10 4 10 3 ) 2 } ,
J ( H h ) 5 / 6 f ( h ) { ( M 2 + N 2 P 2 ) 5 / 12 cos ( 5 6 tan 1 N M ) ( M P ) 5 / 6 } .
N M = Z S ( S Z ) A ( 1 + A 2 R 2 ) ( S + Z S Z ) A R + A 3 ( S Z ) R 2 ( 1 + A 2 / R 2 ) + A ( S Z ) ( 1 + A 2 / R 2 )
Z = h / cos θ and S = H cos θ .
N M = Z A ( 1 + A 2 R 2 ) A R + A S .
d d R ( N M ) = 0 = 2 Z A R 3 + A R 2 .
N M = Z A A 4 Z .
R = 2 Z
A = 2 Z or Z = π w 2 2 λ .
w 2 = w 0 2 [ 1 + ( λ Z π w 0 2 ) 2 ] ,
R = Z [ 1 + ( π w 0 2 λ Z ) 2 ]
R = 2 π w 0 2 λ = π w 2 λ = R min .
N M = 2 Z A ( 1 + A 2 R 2 ) 3 A R + A Z .
| R | = 2 π w 2 2 λ = π w 2 λ = A .
( M 2 + N 2 P 2 ) 5 / 12 = ( 4 α 2 2 α + 1 8 α 2 4 α + 1 ) 5 / 6 , N M = 4 α 2 3 α + 1 α ,
( M P ) 5 / 6 = ( α 8 α 2 4 α + 1 ) 5 / 6 .
( M 2 + N 2 P 2 ) 5 / 12 cos 5 6 tan 1 ( N M ) ( M P ) 5 / 6