Abstract

The diffraction theory of optical scintillations has so far failed to describe the propagation of light over paths where the integrated amount of refractive-index turbulence is sufficient to cause saturation of the scintillations. We present a simple, physically based elaboration of the first-order perturbation theory and compare it with observations. Our theory reproduces in detail the observed saturation curve and the observed spatial covariance of the scintillations. In particular, we show why the fine-scale structure of scintillations persists deep into the saturation regime.

© 1974 Optical Society of America

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References

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  1. C. G. Little, Mon. Not. R. Astron. Soc. 111, 289 (1951).
  2. V. I. Tatarskii, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman (McGraw–Hill, New York, 1961).
  3. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, translated from the Russian (originally published in 1967) (National Technical Information Service, Springfield, Va., 1971).
  4. D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
    [CrossRef]
  5. R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
    [CrossRef]
  6. V. I. Tatarskii and M. E. Gerstenshtein, Zh. Eksp. Teor. Fiz. 44, 676 (1963) [Sov. Phys.-JETP 17, 458 (1963)].
  7. V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 46, 1399 (1964) [Sov. Phys.-JETP 19, 946 (1964)].
  8. V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 49, 1581 (1965) [Sov. Phys.-JETP 22, 1083 (1966)].
  9. D. A. de Wolf, Radio Sci. 2, 1379 (1967).
  10. D. A. de Wolf, J. Opt. Soc. Am. 58, 461 (1968).
    [CrossRef]
  11. D. A. de Wolf, J. Opt. Soc. Am. 59, 1455 (1969).
    [CrossRef]
  12. W. P. Brown, in Proceedings of the Symposium on Modern Optics, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1967), p. 717.
  13. W. P. Brown, IEEE Trans. Antennas Propag. 15, 81 (1967).
    [CrossRef]
  14. M. E. Gracheva and A. S. Gurvich, Sov. Radiophys. 8, 511 (1955).
    [CrossRef]
  15. G. R. Ochs and R. S. Lawrence, J. Opt. Soc. Am. 59, 226 (1969).
    [CrossRef]
  16. M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, Radiophys. and Quantum Electron. 13, 40 (1970).
    [CrossRef]
  17. J. R. Kerr, J. Opt. Soc. Am. 62, 1040 (1972).
    [CrossRef]
  18. W. P. Brown, J. Opt. Soc. Am. 62, 45 (1972).
    [CrossRef]
  19. W. P. Brown, J. Opt. Soc. Am. 62, 966 (1972).
    [CrossRef]
  20. D. A. de Wolf, J. Opt. Soc. Am. 63, 171 (1973).
    [CrossRef]
  21. A. T. Young, J. Opt. Soc. Am. 60, 1495 (1970).
    [CrossRef]
  22. G. R. Ochs and S. F. Clifford, J. Opt. Soc. Am. 62, 728A (1972).
  23. S. F. Clifford, J. Opt. Soc. Am. 63, 471A (1973).
  24. P. M. Livingston, Appl. Opt. 11, 684 (1972).
    [CrossRef] [PubMed]
  25. R. S. Lawrence, G. R. Ochs, and S. F. Clifford, J. Opt. Soc. Am. 60, 826 (1970).
    [CrossRef]
  26. R. F. Lutomirski and H. T. Yura, Appl. Opt. 10, 1652 (1971).
    [CrossRef] [PubMed]
  27. D. L. Fried, J. Opt. Soc. 56, 1372 (1966).
    [CrossRef]
  28. H. T. Yura, J. Opt. Soc. Am. 63, 567 (1973).
    [CrossRef]
  29. R. F. Lutomirski and H. T. Yura, J. Opt. Soc. Am. 61, 482 (1971).
    [CrossRef]

1973 (3)

1972 (5)

1971 (2)

1970 (4)

R. S. Lawrence, G. R. Ochs, and S. F. Clifford, J. Opt. Soc. Am. 60, 826 (1970).
[CrossRef]

A. T. Young, J. Opt. Soc. Am. 60, 1495 (1970).
[CrossRef]

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, Radiophys. and Quantum Electron. 13, 40 (1970).
[CrossRef]

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

1969 (2)

1968 (1)

1967 (3)

D. A. de Wolf, Radio Sci. 2, 1379 (1967).

D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
[CrossRef]

W. P. Brown, IEEE Trans. Antennas Propag. 15, 81 (1967).
[CrossRef]

1966 (1)

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

1965 (1)

V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 49, 1581 (1965) [Sov. Phys.-JETP 22, 1083 (1966)].

1964 (1)

V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 46, 1399 (1964) [Sov. Phys.-JETP 19, 946 (1964)].

1963 (1)

V. I. Tatarskii and M. E. Gerstenshtein, Zh. Eksp. Teor. Fiz. 44, 676 (1963) [Sov. Phys.-JETP 17, 458 (1963)].

1955 (1)

M. E. Gracheva and A. S. Gurvich, Sov. Radiophys. 8, 511 (1955).
[CrossRef]

1951 (1)

C. G. Little, Mon. Not. R. Astron. Soc. 111, 289 (1951).

Brown, W. P.

W. P. Brown, J. Opt. Soc. Am. 62, 45 (1972).
[CrossRef]

W. P. Brown, J. Opt. Soc. Am. 62, 966 (1972).
[CrossRef]

W. P. Brown, IEEE Trans. Antennas Propag. 15, 81 (1967).
[CrossRef]

W. P. Brown, in Proceedings of the Symposium on Modern Optics, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1967), p. 717.

Clifford, S. F.

S. F. Clifford, J. Opt. Soc. Am. 63, 471A (1973).

G. R. Ochs and S. F. Clifford, J. Opt. Soc. Am. 62, 728A (1972).

R. S. Lawrence, G. R. Ochs, and S. F. Clifford, J. Opt. Soc. Am. 60, 826 (1970).
[CrossRef]

de Wolf, D. A.

Fried, D. L.

D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
[CrossRef]

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

Gerstenshtein, M. E.

V. I. Tatarskii and M. E. Gerstenshtein, Zh. Eksp. Teor. Fiz. 44, 676 (1963) [Sov. Phys.-JETP 17, 458 (1963)].

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, Radiophys. and Quantum Electron. 13, 40 (1970).
[CrossRef]

M. E. Gracheva and A. S. Gurvich, Sov. Radiophys. 8, 511 (1955).
[CrossRef]

Gurvich, A. S.

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, Radiophys. and Quantum Electron. 13, 40 (1970).
[CrossRef]

M. E. Gracheva and A. S. Gurvich, Sov. Radiophys. 8, 511 (1955).
[CrossRef]

Kallistratova, M. A.

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, Radiophys. and Quantum Electron. 13, 40 (1970).
[CrossRef]

Kerr, J. R.

Lawrence, R. S.

Little, C. G.

C. G. Little, Mon. Not. R. Astron. Soc. 111, 289 (1951).

Livingston, P. M.

Lutomirski, R. F.

Ochs, G. R.

Strohbehn, J. W.

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

Tatarskii, V. I.

V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 49, 1581 (1965) [Sov. Phys.-JETP 22, 1083 (1966)].

V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 46, 1399 (1964) [Sov. Phys.-JETP 19, 946 (1964)].

V. I. Tatarskii and M. E. Gerstenshtein, Zh. Eksp. Teor. Fiz. 44, 676 (1963) [Sov. Phys.-JETP 17, 458 (1963)].

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman (McGraw–Hill, New York, 1961).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, translated from the Russian (originally published in 1967) (National Technical Information Service, Springfield, Va., 1971).

Young, A. T.

Yura, H. T.

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

W. P. Brown, IEEE Trans. Antennas Propag. 15, 81 (1967).
[CrossRef]

J. Opt. Soc. (1)

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

J. Opt. Soc. Am. (14)

Mon. Not. R. Astron. Soc. (1)

C. G. Little, Mon. Not. R. Astron. Soc. 111, 289 (1951).

Proc. IEEE (1)

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

Radio Sci. (1)

D. A. de Wolf, Radio Sci. 2, 1379 (1967).

Radiophys. and Quantum Electron. (1)

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, Radiophys. and Quantum Electron. 13, 40 (1970).
[CrossRef]

Sov. Radiophys. (1)

M. E. Gracheva and A. S. Gurvich, Sov. Radiophys. 8, 511 (1955).
[CrossRef]

Zh. Eksp. Teor. Fiz. (3)

V. I. Tatarskii and M. E. Gerstenshtein, Zh. Eksp. Teor. Fiz. 44, 676 (1963) [Sov. Phys.-JETP 17, 458 (1963)].

V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 46, 1399 (1964) [Sov. Phys.-JETP 19, 946 (1964)].

V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 49, 1581 (1965) [Sov. Phys.-JETP 22, 1083 (1966)].

Other (3)

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman (McGraw–Hill, New York, 1961).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, translated from the Russian (originally published in 1967) (National Technical Information Service, Springfield, Va., 1971).

W. P. Brown, in Proceedings of the Symposium on Modern Optics, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1967), p. 717.

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Figures (8)

Fig. 1
Fig. 1

The variation of the square root of the log-amplitude variance at each of four path lengths compared to the refractive-index structure parameter CN for a 24-h period. All paths greater than 50 m show evidence of saturation.

Fig. 2
Fig. 2

Plots of the observed log-amplitude variance σχ2 and the prediction σt2 from the first-order theory. These data were taken simultaneously at 50 m (top left), 250 m (center), 500 m (top right), and 1000 m (bottom). The solid curve is our theoretical prediction from Eq. (13). The dashed line is a plot of Eq. (1).

Fig. 3
Fig. 3

Examples of direct recordings of covariance function Cχ(ρn) and spacing in Fresnel zones √(λL) taken on the 1000-m path.

Fig. 4
Fig. 4

Covariance functions taken at 500 m under progressively stronger (a–c) turbulent conditions. These smoothed curves represent averages of the raw data.

Fig. 5
Fig. 5

Four covariance functions taken at 1000 m under progressively stronger (a–d) turbulent conditions. These smoothed curves represent averages of the raw data.

Fig. 6
Fig. 6

The geometry of the simple eddy model used to explain the salient points of the analysis. The source at S illuminates the eddy of diameter 2l at z and produces scintillations that are observed at position L.

Fig. 7
Fig. 7

The propagation geometry showing the incident perturbed field u(r′), the scattering at z and the ultimate field variables at z = L.

Fig. 8
Fig. 8

Theoretical covariance curves determined from Eq. (13). Each curve results from a different amount of integrated turbulence σt2.

Tables (1)

Tables Icon

Table I A comparison of integrated-path refractive turbulence σt2 as determined from temperature measurements and from fitting Eq. (13) of our present theory to the observations of Figs. 4 and 5. Here α is the rms discrepancy in Cχ between the observations and the present theory.

Equations (15)

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σ χ 2 = 0.124 k 7 / 6 L 11 / 6 C N 2 ,             L l 0 2 / λ ,
C χ ( ϱ ) = 1 2 T - T T d t χ ( r + ϱ , t ) χ ( r , t ) / σ χ 2 ,
l ( z / L ) 1 2 [ 1 - ( z / L ) ] 1 2 · ( λ L ) 1 2 .
I ( p ) = [ 1 + cos χ 0 2 λ s 0 ] 2 A A exp [ i k ( s 1 - s 2 ) + ψ ( s 1 ) + ψ * ( s 2 ) + ψ ( r 1 ) + ψ * ( r 2 ) ] u 0 ( r 1 ) u 0 * ( r 2 ) d 2 r 1 d 2 r 2 .
H ( K p ) = 1 ( 2 π ) 2 d 2 ϱ e - i K p · ϱ I ( ϱ ) T .
I ( ϱ ) T = [ k 2 π ( L - z ) ] 2 × d 2 ξ exp { i k ( ϱ · ξ ) ( L - z ) } M ST ( ξ , z ) M ST ( ξ , L - z ) × d 2 η exp { - i k ( η · ξ ) ( L - z ) } u 0 * ( η - ξ / 2 ) u 0 ( η + ξ / 2 ) .
H N ( K p ) = M ST ( K p ( L - z ) / k , z ) M ST × ( K p ( L - z ) / k , L - z ) .
M ( ρ , z ) = exp { - 4 π 2 k 2 0 z d z × 0 d κ κ ϕ n ( κ ) [ 1 - J 0 ( κ ρ z z ) ] } ,
M ( ρ , z ) = exp { - 4 π 2 k 2 0 z d z ( z / z ) 2 × 0 d K p K p ϕ n ( K p z / z ) [ 1 - J 0 ( K p ρ ) ] } .
M ST ( ρ , z ) = exp { - 4 π 2 k 2 0 z d z ( z / z ) 2 × γ K d K p K p ϕ n ( K p z / z ) [ 1 - J 0 ( K p ρ ) ] } .
Φ n ( K ) = 0.033 C N 2 K - 11 / 3 ,             L 0 - 1 K l 0 - 1 ,
M ST ( ρ , z ) = exp { - 0.05 π 2 k 2 C N 2 z ρ 5 / 3 × 0.35 K ρ d ξ ξ - 8 / 3 [ 1 - J 0 ( ξ ) ] } .
C χ ( ρ ) = 0.132 π 2 k 2 L C N 2 0 1 d u 0 d K × K - 8 / 3 J 0 ( K ρ u ) M ST ( K L u ( 1 - u ) / k , L ) × sin 2 [ K 2 u ( 1 - u ) L 2 k ] .
C χ ( ρ n , σ t 2 ) = 2.95 σ t 2 0 1 d u [ u ( 1 - u ) ] 5 / 6 0 d y sin 2 y y 11 / 6 × exp { - σ t 2 [ u ( 1 - u ) ] 5 / 6 f ( y ) } J 0 [ ( 4 π y u 1 - u ) 1 2 ρ n ] ,
f ( y ) = 7.02 y 5 / 6 0.7 y d ξ ξ - 8 / 3 [ 1 - J 0 ( ξ ) ] .