Abstract

The results of a Monte Carlo calculation of the probability distribution of the far-field irradiance produced by a thin turbulent layer placed in front of a finite-aperture transmitter are discussed. These results clearly show that Rice-Nakagami and other well-known distributions such as the lognormal distribution are inapplicable to this problem.

© 1982 Optical Society of America

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References

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  1. R. L. Fante, Proc. IEEE 68, 1424 (1980).
    [CrossRef]
  2. E. J. Fremouw, R. C. Livingston, D. A. Miller, J. Atmos. Terr. Phys. 42, 717 (1980).
    [CrossRef]
  3. D. L. Knepp, G. C. Valley, Radio Sci. 13, 59 (1978).
    [CrossRef]
  4. G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, Opt. Commun. 22, 195 (1977).
    [CrossRef]
  5. R. P. Mercier, Proc. Cambridge Philos. Soc. 62, 382 (1962).
    [CrossRef]
  6. S. O. Rice, “Mathematical Analysis of Random Noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, Ed. (Dover, New York, 1954), pp. 133–162.
  7. M. Nakagami, “The m-Distribution—A General Formula of Intensity Distribution of Rapid Fading,” in Statistical Methods in Radio Propagation, W. C. Hoffman, Ed. (Pergamon, New York, 1960), pp. 3–36.
  8. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975), pp. 9–75.
    [CrossRef]
  9. J. W. Strohbehn, T. Wang, J. P. Speck, Radio Sci. 10, 59 (1975).
    [CrossRef]
  10. W. P. Brown, “Computer Simulation of Adaptive Optical Systems,” Hughes Research Laboratories Report, contract N60921-74-C-0249 (1975).
  11. R. L. Fante, Proc. IEEE 63, 1669 (1975).
    [CrossRef]
  12. G. Parry, J. G. Walker, J. Opt. Soc. Am. 70, 1157 (1980).
    [CrossRef]
  13. L. Bissonette, P. Wizinowich, Appl. Opt. 18, 1590 (1979).
    [CrossRef]
  14. C. R. Reddi, K. K. Moorthy, B. V. K. Murthy, Radio Sci. 15, 1001 (1980).
    [CrossRef]

1980

R. L. Fante, Proc. IEEE 68, 1424 (1980).
[CrossRef]

E. J. Fremouw, R. C. Livingston, D. A. Miller, J. Atmos. Terr. Phys. 42, 717 (1980).
[CrossRef]

G. Parry, J. G. Walker, J. Opt. Soc. Am. 70, 1157 (1980).
[CrossRef]

C. R. Reddi, K. K. Moorthy, B. V. K. Murthy, Radio Sci. 15, 1001 (1980).
[CrossRef]

1979

1978

D. L. Knepp, G. C. Valley, Radio Sci. 13, 59 (1978).
[CrossRef]

1977

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, Opt. Commun. 22, 195 (1977).
[CrossRef]

1975

J. W. Strohbehn, T. Wang, J. P. Speck, Radio Sci. 10, 59 (1975).
[CrossRef]

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[CrossRef]

1962

R. P. Mercier, Proc. Cambridge Philos. Soc. 62, 382 (1962).
[CrossRef]

Bissonette, L.

Brown, W. P.

W. P. Brown, “Computer Simulation of Adaptive Optical Systems,” Hughes Research Laboratories Report, contract N60921-74-C-0249 (1975).

Fante, R. L.

R. L. Fante, Proc. IEEE 68, 1424 (1980).
[CrossRef]

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[CrossRef]

Fremouw, E. J.

E. J. Fremouw, R. C. Livingston, D. A. Miller, J. Atmos. Terr. Phys. 42, 717 (1980).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975), pp. 9–75.
[CrossRef]

Jakeman, E.

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, Opt. Commun. 22, 195 (1977).
[CrossRef]

Knepp, D. L.

D. L. Knepp, G. C. Valley, Radio Sci. 13, 59 (1978).
[CrossRef]

Livingston, R. C.

E. J. Fremouw, R. C. Livingston, D. A. Miller, J. Atmos. Terr. Phys. 42, 717 (1980).
[CrossRef]

McWhirter, J. G.

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, Opt. Commun. 22, 195 (1977).
[CrossRef]

Mercier, R. P.

R. P. Mercier, Proc. Cambridge Philos. Soc. 62, 382 (1962).
[CrossRef]

Miller, D. A.

E. J. Fremouw, R. C. Livingston, D. A. Miller, J. Atmos. Terr. Phys. 42, 717 (1980).
[CrossRef]

Moorthy, K. K.

C. R. Reddi, K. K. Moorthy, B. V. K. Murthy, Radio Sci. 15, 1001 (1980).
[CrossRef]

Murthy, B. V. K.

C. R. Reddi, K. K. Moorthy, B. V. K. Murthy, Radio Sci. 15, 1001 (1980).
[CrossRef]

Nakagami, M.

M. Nakagami, “The m-Distribution—A General Formula of Intensity Distribution of Rapid Fading,” in Statistical Methods in Radio Propagation, W. C. Hoffman, Ed. (Pergamon, New York, 1960), pp. 3–36.

Parry, G.

G. Parry, J. G. Walker, J. Opt. Soc. Am. 70, 1157 (1980).
[CrossRef]

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, Opt. Commun. 22, 195 (1977).
[CrossRef]

Pusey, P. N.

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, Opt. Commun. 22, 195 (1977).
[CrossRef]

Reddi, C. R.

C. R. Reddi, K. K. Moorthy, B. V. K. Murthy, Radio Sci. 15, 1001 (1980).
[CrossRef]

Rice, S. O.

S. O. Rice, “Mathematical Analysis of Random Noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, Ed. (Dover, New York, 1954), pp. 133–162.

Speck, J. P.

J. W. Strohbehn, T. Wang, J. P. Speck, Radio Sci. 10, 59 (1975).
[CrossRef]

Strohbehn, J. W.

J. W. Strohbehn, T. Wang, J. P. Speck, Radio Sci. 10, 59 (1975).
[CrossRef]

Valley, G. C.

D. L. Knepp, G. C. Valley, Radio Sci. 13, 59 (1978).
[CrossRef]

Walker, J. G.

Wang, T.

J. W. Strohbehn, T. Wang, J. P. Speck, Radio Sci. 10, 59 (1975).
[CrossRef]

Wizinowich, P.

Appl. Opt.

J. Atmos. Terr. Phys.

E. J. Fremouw, R. C. Livingston, D. A. Miller, J. Atmos. Terr. Phys. 42, 717 (1980).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, Opt. Commun. 22, 195 (1977).
[CrossRef]

Proc. Cambridge Philos. Soc.

R. P. Mercier, Proc. Cambridge Philos. Soc. 62, 382 (1962).
[CrossRef]

Proc. IEEE

R. L. Fante, Proc. IEEE 68, 1424 (1980).
[CrossRef]

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[CrossRef]

Radio Sci.

C. R. Reddi, K. K. Moorthy, B. V. K. Murthy, Radio Sci. 15, 1001 (1980).
[CrossRef]

D. L. Knepp, G. C. Valley, Radio Sci. 13, 59 (1978).
[CrossRef]

J. W. Strohbehn, T. Wang, J. P. Speck, Radio Sci. 10, 59 (1975).
[CrossRef]

Other

W. P. Brown, “Computer Simulation of Adaptive Optical Systems,” Hughes Research Laboratories Report, contract N60921-74-C-0249 (1975).

S. O. Rice, “Mathematical Analysis of Random Noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, Ed. (Dover, New York, 1954), pp. 133–162.

M. Nakagami, “The m-Distribution—A General Formula of Intensity Distribution of Rapid Fading,” in Statistical Methods in Radio Propagation, W. C. Hoffman, Ed. (Pergamon, New York, 1960), pp. 3–36.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975), pp. 9–75.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Irradiance distributions on standard probability paper for D/L0 = 22.

Fig. 2
Fig. 2

Irradiance distributions on standard probability paper for D/L0 = 10.5.

Fig. 3
Fig. 3

Irradiance distributions on standard probability paper for D/L0 = 3.5.

Fig. 4
Fig. 4

Irradiance distributions on log-probability paper for D/L0 = 10.5.

Fig. 5
Fig. 5

Irradiance distributions on log-probability paper for D/L0 = 3.5.

Fig. 6
Fig. 6

Rice-Nakagami probability distributions calculated for the mean and standard deviations of the data in Fig. 5.

Tables (1)

Tables Icon

Table I Strehl Ratio and Irradiance Fluctuation Standard Deviation Obtained for Various Aperture Sizes and Phase-Fluctuation Strengths

Equations (8)

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I ( r , z ) = ( k 2 π z ) 2 d r d r u ( r ) u * ( r ) × exp [ i k ( r - r ) · r z + i ϕ ( r ) - i ϕ ( r ) ] ,
u ( r ) = u 0 exp ( - r 2 R 2 ) ,             0 r R D / 2 = 0 ,             R < r .
S ϕ ( K ) = 5 L 0 2 σ ϕ 2 6 π ( 1 + K 2 L 0 2 ) 11 / 6 ,
C ϕ ( ρ ) = 5 ( ρ / 2 L 0 ) 5 / 6 σ ϕ 2 K 5 / 6 ( ρ / L 0 ) 3 Γ ( 11 / 6 ) .
C ϕ ( 0 ) = d K S ϕ ( K ) = σ ϕ 2 .
N = m = 0 N β m ( 1 - β ) N - m N ! ( N - m ) ! m ! = N β .
N 2 = m = 0 N m 2 β m ( 1 - β ) N - m N ! ( N - m ) ! m ! = N ( N - 1 ) β 2 + N β .
fractional error = ( N 2 - N 2 N 2 ) 1 / 2 = ( 1 - β N β ) 1 / 2 .

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