Abstract

We have carried out numerical simulations of waves traversing a three-dimensional random medium with Gaussian statistics and a power-law spectrum with inner-scale cutoff. The distributions of irradiance on the final observation screen provide the probability-density function (PDF) of irradiance. For both initially plane and initially spherical waves the simulation PDF’s in the strong-fluctuation regime lie between a K distribution and a log-normal-convolved-with-exponential distribution. We introduce a plot of the PDF of scaled log-normal irradiance, on which both the exponential and the lognormal PDF’s are universal curves and on which the PDF at both large and small irradiance is shown in detail. We have simulated a spherical-wave experiment, including aperture averaging, and find agreement between the simulated and the observed PDF’s.

© 1994 Optical Society of America

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Errata

Stanley M. Flatté, Charles Bracher, and Guang-Yu Wang, "Probability-density functions of irradiance for waves in atmospheric turbulence by numerical simulation: erratum," J. Opt. Soc. Am. A 12, 184-184 (1995)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-12-1-184

References

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  1. P. Deitz, N. Wright, “Saturation of scintillation magnitude in near-earth optical propagation,” J. Opt. Soc. Am. 59, 527–535 (1969).
    [CrossRef]
  2. A. Gurvich, M. Kallistratova, F. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. 20, 705–714 (1977).
    [CrossRef]
  3. J. Martin, S. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
    [CrossRef] [PubMed]
  4. J. Martin, S. Flatté, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990).
    [CrossRef]
  5. S. Flatté, C. Bracher, “Simulation of wave propagation in three-dimensional random media,” In Atmospheric, Volume and Surface Scattering and Propagation, A. Consortini, ed. (University of Florence, Florence, Italy, 1991), pp. 27–29.
  6. S. M. Flatté, G. Y. Wang, J. Martin, “Irradiance variance of optical waves through atmospheric turbulence by numerical simulation and comparison with experiment,” J. Opt. Soc. Am. A 10, 2363–2370 (1993).
    [CrossRef]
  7. V. Tatarskii, “Depolarization of light by turbulent atmospheric inhomogeneities,” Radiophys. Quantum Electron. 10, 987–988 (1967).
    [CrossRef]
  8. R. Dashen, G. Wang, S. M. Flatté, C. Bracher, “Moments of intensity and log intensity: new asymptotic results for waves in power-law media,” J. Opt. Soc. Am. A 10, 1233–1242 (1993).
    [CrossRef]
  9. E. Jakeman, P. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
    [CrossRef]
  10. E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
    [CrossRef]
  11. L. Andrews, R. Phillips, “I–K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2, 160–163 (1985).
    [CrossRef]
  12. L. Andrews, R. Phillips, “Mathematical genesis of the I–K distribution for random optical fields,” J. Opt. Soc. Am. A 3, 1912–1919 (1986).
    [CrossRef]
  13. J. Churnside, R. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
    [CrossRef]
  14. K. Furutsu, “Theory of irradiance distribution function in turbulent media—cluster approximation,” J. Math. Phys. 17, 1252–1263 (1976).
    [CrossRef]
  15. T. Ewart, “A model of the intensity probability distribution for wave propagation in random media,” J. Acoust. Soc. Am. 86, 1490–1498 (1989).
    [CrossRef]
  16. C. Macaskill, T. Ewart, “The probability distribution of intensity for acoustic propagation in a randomly varying ocean,” J. Acoust. Soc. Am. 76, 1466–1473 (1984).
    [CrossRef]
  17. T. E. Ewart, D. Percival, “Forward scattered waves in random media—the probability distribution of intensity,” J. Acoust. Soc. Am. 80, 1745–1753 (1986).
    [CrossRef]
  18. E. Jakeman, R. Tough, “Generalized K distribution: a statistical model for weak scattering,” J. Opt. Soc. Am. A 4, 1764–1772 (1987).
    [CrossRef]
  19. M. Teich, P. Diament, “Multiply stochastic representations for K distributions and their Poisson transforms,” J. Opt. Soc. Am. A 6, 80–91 (1989).
    [CrossRef]
  20. E. Rockower, “Quantum derivation of K-distributed noise for finite 〈N〉,” J. Opt. Soc. Am. A 5, 730–734 (1988).
    [CrossRef]
  21. R. Barakat, “Weak-scatter generalization of the K-density function with application to laser scattering in atmospheric turbulence,” J. Opt. Soc. Am. A 3, 401–409 (1986).
    [CrossRef]
  22. J. H. Churnside, S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
    [CrossRef]
  23. N. Ben-Yosef, E. Goldner, “Splitting-source model for the statistics of irradiance scintillations,” J. Opt. Soc. Am. A 5, 126–131 (1988).
    [CrossRef]
  24. R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media 2, 179–201 (1992).
    [CrossRef]
  25. M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
    [CrossRef]
  26. A. Consortini, F. Cochetti, J. H. Churnside, R. J. Hill, “Inner-scale effect on intensity variance measured for weak to strong atmospheric scintillation,” J. Opt. Soc. Am. A 10, 2354–2362 (1993).
    [CrossRef]
  27. V. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  28. A. Giovannini, L. V. Hove, “Negative binomial properties and clan structure in multiplicity distributions,” Acta Phys. Pol. B 19, 495–510 (1988).
  29. N. Yasuda, “The random-walk model of human migration,” Theor. Popul. Biol. 7, 156–167 (1975).
    [CrossRef] [PubMed]
  30. S. Ong, P. Lee, “The non-central negative binomial distribution,” Biomed. J. 21, 611–627 (1979).
  31. E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 (1982).
    [CrossRef]
  32. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  33. K. Goshelashvily, V. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).
  34. R. G. Frehlich, “Intensity covariance of a point source in a random medium with a Kolmogorov spectrum and an inner scale of turbulence,” J. Opt. Soc. Am. A 4, 360–366 (1987).
    [CrossRef]

1993

1992

R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media 2, 179–201 (1992).
[CrossRef]

1990

1989

T. Ewart, “A model of the intensity probability distribution for wave propagation in random media,” J. Acoust. Soc. Am. 86, 1490–1498 (1989).
[CrossRef]

M. Teich, P. Diament, “Multiply stochastic representations for K distributions and their Poisson transforms,” J. Opt. Soc. Am. A 6, 80–91 (1989).
[CrossRef]

1988

1987

1986

1985

1984

C. Macaskill, T. Ewart, “The probability distribution of intensity for acoustic propagation in a randomly varying ocean,” J. Acoust. Soc. Am. 76, 1466–1473 (1984).
[CrossRef]

1982

1980

E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
[CrossRef]

1979

S. Ong, P. Lee, “The non-central negative binomial distribution,” Biomed. J. 21, 611–627 (1979).

1978

E. Jakeman, P. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

1977

A. Gurvich, M. Kallistratova, F. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. 20, 705–714 (1977).
[CrossRef]

1976

K. Furutsu, “Theory of irradiance distribution function in turbulent media—cluster approximation,” J. Math. Phys. 17, 1252–1263 (1976).
[CrossRef]

1975

N. Yasuda, “The random-walk model of human migration,” Theor. Popul. Biol. 7, 156–167 (1975).
[CrossRef] [PubMed]

1974

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

K. Goshelashvily, V. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).

1969

1967

V. Tatarskii, “Depolarization of light by turbulent atmospheric inhomogeneities,” Radiophys. Quantum Electron. 10, 987–988 (1967).
[CrossRef]

Andrews, L.

Barakat, R.

Ben-Yosef, N.

Bracher, C.

R. Dashen, G. Wang, S. M. Flatté, C. Bracher, “Moments of intensity and log intensity: new asymptotic results for waves in power-law media,” J. Opt. Soc. Am. A 10, 1233–1242 (1993).
[CrossRef]

S. Flatté, C. Bracher, “Simulation of wave propagation in three-dimensional random media,” In Atmospheric, Volume and Surface Scattering and Propagation, A. Consortini, ed. (University of Florence, Florence, Italy, 1991), pp. 27–29.

Churnside, J.

Churnside, J. H.

Clifford, S. F.

Cochetti, F.

Consortini, A.

Dashen, R.

Deitz, P.

Diament, P.

Ewart, T.

T. Ewart, “A model of the intensity probability distribution for wave propagation in random media,” J. Acoust. Soc. Am. 86, 1490–1498 (1989).
[CrossRef]

C. Macaskill, T. Ewart, “The probability distribution of intensity for acoustic propagation in a randomly varying ocean,” J. Acoust. Soc. Am. 76, 1466–1473 (1984).
[CrossRef]

Ewart, T. E.

T. E. Ewart, D. Percival, “Forward scattered waves in random media—the probability distribution of intensity,” J. Acoust. Soc. Am. 80, 1745–1753 (1986).
[CrossRef]

Flatté, S.

J. Martin, S. Flatté, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990).
[CrossRef]

J. Martin, S. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
[CrossRef] [PubMed]

S. Flatté, C. Bracher, “Simulation of wave propagation in three-dimensional random media,” In Atmospheric, Volume and Surface Scattering and Propagation, A. Consortini, ed. (University of Florence, Florence, Italy, 1991), pp. 27–29.

Flatté, S. M.

Frehlich, R. G.

Furutsu, K.

K. Furutsu, “Theory of irradiance distribution function in turbulent media—cluster approximation,” J. Math. Phys. 17, 1252–1263 (1976).
[CrossRef]

Giovannini, A.

A. Giovannini, L. V. Hove, “Negative binomial properties and clan structure in multiplicity distributions,” Acta Phys. Pol. B 19, 495–510 (1988).

Goldner, E.

Goshelashvily, K.

K. Goshelashvily, V. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).

Gracheva, M.

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

Gurvich, A.

A. Gurvich, M. Kallistratova, F. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. 20, 705–714 (1977).
[CrossRef]

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

Hill, R.

Hill, R. J.

A. Consortini, F. Cochetti, J. H. Churnside, R. J. Hill, “Inner-scale effect on intensity variance measured for weak to strong atmospheric scintillation,” J. Opt. Soc. Am. A 10, 2354–2362 (1993).
[CrossRef]

R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media 2, 179–201 (1992).
[CrossRef]

Hove, L. V.

A. Giovannini, L. V. Hove, “Negative binomial properties and clan structure in multiplicity distributions,” Acta Phys. Pol. B 19, 495–510 (1988).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Jakeman, E.

E. Jakeman, R. Tough, “Generalized K distribution: a statistical model for weak scattering,” J. Opt. Soc. Am. A 4, 1764–1772 (1987).
[CrossRef]

E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 (1982).
[CrossRef]

E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
[CrossRef]

E. Jakeman, P. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

Kallistratova, M.

A. Gurvich, M. Kallistratova, F. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. 20, 705–714 (1977).
[CrossRef]

Khrupin, A.

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

Lee, P.

S. Ong, P. Lee, “The non-central negative binomial distribution,” Biomed. J. 21, 611–627 (1979).

Lomadze, S.

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

Macaskill, C.

C. Macaskill, T. Ewart, “The probability distribution of intensity for acoustic propagation in a randomly varying ocean,” J. Acoust. Soc. Am. 76, 1466–1473 (1984).
[CrossRef]

Martin, J.

Martvel, F.

A. Gurvich, M. Kallistratova, F. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. 20, 705–714 (1977).
[CrossRef]

Ong, S.

S. Ong, P. Lee, “The non-central negative binomial distribution,” Biomed. J. 21, 611–627 (1979).

Percival, D.

T. E. Ewart, D. Percival, “Forward scattered waves in random media—the probability distribution of intensity,” J. Acoust. Soc. Am. 80, 1745–1753 (1986).
[CrossRef]

Phillips, R.

Pokasov, V.

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

Pusey, P.

E. Jakeman, P. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

Rockower, E.

Shishov, V.

K. Goshelashvily, V. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).

Tatarskii, V.

V. Tatarskii, “Depolarization of light by turbulent atmospheric inhomogeneities,” Radiophys. Quantum Electron. 10, 987–988 (1967).
[CrossRef]

V. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Teich, M.

Tough, R.

Wang, G.

Wang, G. Y.

Wright, N.

Yasuda, N.

N. Yasuda, “The random-walk model of human migration,” Theor. Popul. Biol. 7, 156–167 (1975).
[CrossRef] [PubMed]

Acta Phys. Pol. B

A. Giovannini, L. V. Hove, “Negative binomial properties and clan structure in multiplicity distributions,” Acta Phys. Pol. B 19, 495–510 (1988).

Appl. Opt.

Biomed. J.

S. Ong, P. Lee, “The non-central negative binomial distribution,” Biomed. J. 21, 611–627 (1979).

J. Acoust. Soc. Am.

T. Ewart, “A model of the intensity probability distribution for wave propagation in random media,” J. Acoust. Soc. Am. 86, 1490–1498 (1989).
[CrossRef]

C. Macaskill, T. Ewart, “The probability distribution of intensity for acoustic propagation in a randomly varying ocean,” J. Acoust. Soc. Am. 76, 1466–1473 (1984).
[CrossRef]

T. E. Ewart, D. Percival, “Forward scattered waves in random media—the probability distribution of intensity,” J. Acoust. Soc. Am. 80, 1745–1753 (1986).
[CrossRef]

J. Math. Phys.

K. Furutsu, “Theory of irradiance distribution function in turbulent media—cluster approximation,” J. Math. Phys. 17, 1252–1263 (1976).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Martin, S. Flatté, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990).
[CrossRef]

S. M. Flatté, G. Y. Wang, J. Martin, “Irradiance variance of optical waves through atmospheric turbulence by numerical simulation and comparison with experiment,” J. Opt. Soc. Am. A 10, 2363–2370 (1993).
[CrossRef]

R. Dashen, G. Wang, S. M. Flatté, C. Bracher, “Moments of intensity and log intensity: new asymptotic results for waves in power-law media,” J. Opt. Soc. Am. A 10, 1233–1242 (1993).
[CrossRef]

E. Jakeman, R. Tough, “Generalized K distribution: a statistical model for weak scattering,” J. Opt. Soc. Am. A 4, 1764–1772 (1987).
[CrossRef]

M. Teich, P. Diament, “Multiply stochastic representations for K distributions and their Poisson transforms,” J. Opt. Soc. Am. A 6, 80–91 (1989).
[CrossRef]

E. Rockower, “Quantum derivation of K-distributed noise for finite 〈N〉,” J. Opt. Soc. Am. A 5, 730–734 (1988).
[CrossRef]

R. Barakat, “Weak-scatter generalization of the K-density function with application to laser scattering in atmospheric turbulence,” J. Opt. Soc. Am. A 3, 401–409 (1986).
[CrossRef]

J. H. Churnside, S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
[CrossRef]

N. Ben-Yosef, E. Goldner, “Splitting-source model for the statistics of irradiance scintillations,” J. Opt. Soc. Am. A 5, 126–131 (1988).
[CrossRef]

R. G. Frehlich, “Intensity covariance of a point source in a random medium with a Kolmogorov spectrum and an inner scale of turbulence,” J. Opt. Soc. Am. A 4, 360–366 (1987).
[CrossRef]

L. Andrews, R. Phillips, “I–K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2, 160–163 (1985).
[CrossRef]

L. Andrews, R. Phillips, “Mathematical genesis of the I–K distribution for random optical fields,” J. Opt. Soc. Am. A 3, 1912–1919 (1986).
[CrossRef]

J. Churnside, R. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
[CrossRef]

A. Consortini, F. Cochetti, J. H. Churnside, R. J. Hill, “Inner-scale effect on intensity variance measured for weak to strong atmospheric scintillation,” J. Opt. Soc. Am. A 10, 2354–2362 (1993).
[CrossRef]

J. Phys. A

E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
[CrossRef]

Phys. Rev. Lett.

E. Jakeman, P. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

Radiophys. Quantum Electron.

V. Tatarskii, “Depolarization of light by turbulent atmospheric inhomogeneities,” Radiophys. Quantum Electron. 10, 987–988 (1967).
[CrossRef]

A. Gurvich, M. Kallistratova, F. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. 20, 705–714 (1977).
[CrossRef]

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

Sov. Phys. JETP

K. Goshelashvily, V. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).

Theor. Popul. Biol.

N. Yasuda, “The random-walk model of human migration,” Theor. Popul. Biol. 7, 156–167 (1975).
[CrossRef] [PubMed]

Waves Random Media

R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media 2, 179–201 (1992).
[CrossRef]

Other

S. Flatté, C. Bracher, “Simulation of wave propagation in three-dimensional random media,” In Atmospheric, Volume and Surface Scattering and Propagation, A. Consortini, ed. (University of Florence, Florence, Italy, 1991), pp. 27–29.

V. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

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Figures (15)

Fig. 1
Fig. 1

First four moments of ln I for our plane-wave simulations. Symbols: triangles (light solid curves), l0 = 0; squares (light medium-dashed curves), l0 = 0.5Rf; diamonds (light long-dashed curves), l0 = Rf. Curves: heavy solid, locus of points reachable by a log-normal PDF; heavy long-dashed, K distribution; heavy short-dashed, log-normal ⊗ exponential distribution. Asterisks represent the moments of an exponential PDF. Values of β02 are 0.01, 0.03, 0.06, 0.1, 0.3, 0.5, 0.75, 1, 2, 4, 5, 10, 15, 20, and 25.

Fig. 2
Fig. 2

Same as Fig. 1, except that the scales have been magnified to show the strong-fluctuation regime in more detail. Only cases having β02 ≥ 1 are shown. One can obtain the values of β02 from the list in the caption for Fig. 1 and by counting in from the entry of each curve.

Fig. 3
Fig. 3

Same data as in Fig. 2(a); the ordinate as given in the label skews the data in such a way as to make them more visible.

Fig. 4
Fig. 4

Scaled PDF’s of In irradiance for plane-wave simulations at small β02. Curves: solid, log-normal; long-dashed, exponential. Symbols define values of β02: circles, 0.01; crosses, 0.10; triangles, 0.5; squares, 0.75; diamonds, 1.0. The insets make more visible the data near abscissa = 0. (a) l0 = 0, (b) l0 = 0.5Rf, (c) l0 = Rf.

Fig. 5
Fig. 5

Scaled PDF’s as in Fig. 4, except at intermediate β02. Symbols define values of β02: circles, 2.0; crosses, 3.0; triangles, 4.0; squares, 5.0. (a) l0 = 0, (b) l0 = 0.5Rf, (c) l0 = Rf.

Fig. 6
Fig. 6

Scaled PDF’s as in Fig. 4, except at large β02. Symbols define values of β02: circles, 10.0; crosses, 15.0; triangles, 20.0; squares, 25.0. (a) l0 = 0, (b) l0 = 0.5Rf, (c) l0 = Rf.

Fig. 7
Fig. 7

Scaled PDF’s of log-normal irradiance for plane-wave simulations at β02 = 25 and different l0: squares, simulation; curves: solid, log-normal; long-dashed, exponential; short-dashed, log-normal ⊗ exponential; medium-dashed, K distribution. Each of the parameters of the log-normal ⊗ exponential and the K distribution is set so that its value of the variance of I is the same as that of the simulation. The simulations lie closest to the K distribution. Some moments for these PDF’s are given in Tables 13. (a) l0 = 0, (b) l0 = 0.5Rf, (c) l0 = Rf.

Fig. 8
Fig. 8

PDF’s of irradiance for a point source with aperture averaging on a diameter of 1 mm. Circles are data from the experiment of Churnside and Hill13; squares are our numerical simulation with the same β02 as for the experiment and with an inner scale such that our aperture-averaged simulation has the same 〈I2〉 as does the experiment. Curves: solid, log-normal; long-dashed, exponential; short-dashed, log-normal ⊗ exponential; medium-dashed, K distribution. The distributions with a parameter have been set to have the same value of 〈I2〉 as for the experiment. Both the experimental data and the simulations lie closest to the log-normal ⊗ exponential PDF. Some moments for these PDF’s are given in Tables 4 and 5. (a) β02 = 23 and l0 = 0.5Rf, (b) β02 = 36 and l0 = 0.4Rf.

Fig. 9
Fig. 9

Scaled PDF’s of log-normal irradiance for a point source with aperture averaging on a diameter of 1 mm. Circles are data from the experiment of Churnside and Hill13; squares are our numerical simulation with the same β02 as for the experiment and with an inner scale such that our aperture-averaged simulation has the same 〈I2〉 as does the experiment. Curves: solid, log-normal; long-dashed, exponential. Scaling values of ln I mean and rms for the experimental data are chosen equal to those of the simulation. Some moments for these PDF’s are given in Tables 4 and 5. (a) β02 = 23 and l0 = 0.5Rf, (b) β02 = 36 and l0 = 0.4Rf.

Fig. 10
Fig. 10

Scaled PDF’s of log-normal irradiance for a point source. Squares, numerical simulations before aperture averaging. Curves: solid, log-normal; long-dashed, exponential; short-dashed, log-normal ⊗ exponential; medium-dashed, K distribution. The distributions with a parameter have been set to have the same value of 〈I2〉 as for the simulation. For the analytic functions, scaling values are determined by each function. These non-aperture-averaged simulations lie closest to the log-normal ⊗ exponential PDF. Some moments for these analytical PDF’s are given in Tables 4 and 5. (a) β02 = 23 and l0 = 0.5Rf, (b) β02 = 36 and l0 = 0.4Rf.

Fig. 11
Fig. 11

Variance-preserving plot (solid curve) of the irradiance spectrum for our numerical simulation with incident plane wave, β02 = 25, and l0 = 0. The dashed curves are the low- and high-wave-number analytic approximations described in the text. The length ρo is defined by D(ρo) = 1.

Fig. 12
Fig. 12

Variance-preserving plots of log-normal irradiance spectra for plane-wave simulations. The slight rises at the high-wave-number ends of the curves are artifacts of the finite grid lengths in the simulations, and they are associated with slight inaccuracies at very small irradiances, not at irradiance spikes. Numbers defining the different curves are values of β02. (a) l0 = 0; (b) l0 = 0.5Rf; (c) l0 = Rf.

Fig. 13
Fig. 13

Moments of ln I for simulations with β02= 25 and l0 = 0. The Fresnel lengths are set at a fixed ratio to the screen width, so that all simulations are the same, except for the resolutions at short distances. That is, the screen size is proportional to Rf and is 4096 × 4096 at Rf = 60. Each symbol represents a moment of ln I: circle, mean; cross, variance; triangle, skewness; square, kurtosis. The moments are unchanged within 1% between Rf values of 30 and 60.

Fig. 14
Fig. 14

PDF’s of scaled ln I for simulations with β02 = 25 and l0 = 0. The screen sizes are 4096 × Rf/60. The Fresnel lengths are 7.5, 15, 30, and 60 for the circle, cross, triangle, and square, respectively. The PDF’s are indistinguishable from one another.

Fig. 15
Fig. 15

Moment integrands for a simulation with β02 = 25 and l0 = 0. The curves represent the integrands of different moments: solid, zero (the PDF); long dashed, second; medium dashed, third; short dashed, fourth. The extent of the simulation sampling shows that very little contribution to these lower moments is lost as a result of finite data, (a) ln I; (b) I. For (b) the zeroth-, second-, and third-moment integrands are multiplied by 5 so that they are more visible.

Tables (5)

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Table 1 Moments for Plane-Wave Simulations at β02 = 25 and l0 = 0 and Moments of Two Analytic Distributions with the Same σI as the Simulation

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Table 2 Moments for Plane-Wave Simulations at β02 = 25 and l0 = 0.5Rf and Moments of Two Analytic Distributions with the Same σI as the Simulation

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Table 3 Moments for Plane-Wave Simulations at β02 = 25 and l0 = Rf and Moments of Two Analytic Distributions with the Same σI as the Simulation

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Table 4 Moments for Point-Source Initial Conditions from the Experiment and from a Simulation with β02 = 23 and l0 = 0.5Rfa

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Table 5 Moments for Point-Source Initial Conditions from the Experiment and from a Simulation with β02 = 36 and l0 = 0.4 Rfa

Equations (14)

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Φ n ( κ ) = K ( α ) C n 2 κ α 2 F ( κ l 0 ) ,
β 0 2 = 1.23 C n 2 k 7 / 6 L 11 / 6 ;
σ x 2 = ( x x ) 2 ,
skewness = ( x x ) 3 ,
kurtosis = ( x x ) 4 3 ( x x ) 2 2 ,
D ( x ) = 2.372 β 0 2 R f 5 / 3 x 5 / 3 ,
ρ 0 = 0.6 ( β o 2 ) 3 / 5 R f ,
S H ( κ ) = 1 ( 2 π ) 0 exp [ D ( x ) ] J 0 ( κ x ) x d x ,
S L ( κ ) = 8 π ( 0.8139 β 0 2 ) R f 5 / 3 κ 11 / 5 × 0 1 d z sin 2 [ κ 2 R f 2 2 ( 1 z ) ] × exp [ 2.372 β 0 2 1 + g ] 0 1 d z [ κ R f h ( z , z ) ] 5 / 3 ] ,
h ( z , z ) = { 1 z for z > z 1 z for z < z ,
g ( 1 + g ) 7 / 5 = 0.4916 β 0 2 / 5 .
ρ 0 = ρ 0 b ,
b = 2 0 exp ( u 5 / 3 ) J 0 ( b u ) u d u 0 exp ( u 5 / 3 ) J 2 ( b u ) u 2 d u ,
χ n = χ n P ( χ ) d χ .

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