Abstract

We calculated the probability distribution function (PDF) from simulations for an initially spherical wave propagated through homogeneous atmospheric turbulence. The onset of strong scintillation was calculated. By onset of strong scintillation, we mean conditions of weak scintillation changing to the condition of strong focusing. In addition, one case in the saturation regime was calculated. The simulations’ PDF’s are compared with several heuristic models of the PDF and are seen to progress from close to log normal for the cases of weakest scintillation to close to the log normally modulated exponential PDF (LNME PDF) for the cases of strong scintillation. The simulations’ PDF’s are not in agreement with the K PDF for any of the calculated cases. The best agreement was obtained in comparison with Beckmann’s PDF [P. Beckmann, Probability in Communication Engineering (Harcourt, Brace, & World, 1967)]. Beckmann’s PDF varies from being the log-normal PDF for weak scintillation to being the LNME PDF for strong scintillation and progresses further to the theoretically expected exponential PDF in the limit of saturated scintillation. We recommend that simulation be used to predict the irradiance PDF for plane and diverged waves in homogeneous turbulence in preference to using heuristic models. More complicated propagation cases remain in the domain of heuristic argumentation.

© 1997 Optical Society of America

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References

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  1. A. Consortini, F. Cochetti, J. H. Churnside, R. J. Hill, “Inner-scale effect on irradiance variance measured for weak-to-strong atmospheric scintillation,” J. Opt. Soc. Am. A 10, 2354–2362 (1993).
    [CrossRef]
  2. S. M. Flatté, C. Bracher, G.-Y. Wang, “Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation,” J. Opt. Soc. Am. A 11, 2080–2092 (1994).
    [CrossRef]
  3. R. J. Hill, R. G. Frehlich, “Onset of strong scintillation with application to remote sensing of turbulence inner scale,” Appl. Opt. 35, 986–997 (1996).
    [CrossRef] [PubMed]
  4. E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
    [CrossRef]
  5. E. Jakeman, P. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
    [CrossRef]
  6. J. H. Churnside, R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
    [CrossRef]
  7. P. Beckmann, Probability in Communication Engineering (Harcourt, Brace & World, New York, 1967).
  8. 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]
  9. G. Ya. Patrushev, O. A. Rubtsova, “Probability density of the intensity and flux fluctuations of optical radiation propagating and reflecting in the turbulent atmosphere,” Atmos. Oceanic Opt. 6, 760–769 (1993).
  10. R. J. Hill, R. G. Frehlich, W. D. Otto, “The probability distribution of irradiance scintillation,” , NOAA Environmental Research Laboratories, Boulder, Colorado (1996). Available from the National Technical Information Service, 5285 Port Royal Road, Springfield, Va. 22161.
  11. A. M. Obukhov, “The structure of the temperature field in turbulent flow,” Izv. Akad. Nauk. SSSR Ser. Geogr. Geofiz. 13, 58–69 (1949).
  12. R. J. Hill, S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. 68, 892–899 (1978).
    [CrossRef]
  13. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971).
  14. R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (;1978).
  15. R. J. Hill, “Comparison of scintillation methods for measuring the inner scale of turbulence,” Appl. Opt. 27, 2187–2193 (1988).
    [CrossRef] [PubMed]
  16. S. M. Flatté, G. 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]
  17. C. A. Davis, D. L. Walters, “Atmospheric inner-scale effects on normalized irradiance variance,” Appl. Opt. 33, 8406–8411 (1994).
    [CrossRef] [PubMed]
  18. E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azar, “Optical measurements of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
    [CrossRef]
  19. V. Thiermann, H. Grassl, “The measurement of turbulent surface-layer fluxes by use of bichromatic scintillation,” Boundary-Layer Meteorol. 58, 367–389 (1992).
  20. R. J. Hill, G. R. Ochs, “Fine calibration of large-aperture optical scintillometers and an optical estimate of the inner scale of turbulence,” Appl. Opt. 17, 3608–3612 (1978).
    [CrossRef] [PubMed]
  21. G. R. Ochs, R. J. Hill, “Optical-scintillation method of measuring turbulence inner scale,” Appl. Opt. 24, 2430–2432 (1985).
    [CrossRef] [PubMed]
  22. R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
    [CrossRef]
  23. R. J. Hill, G. R. Ochs, “Inner-scale dependence of scintillation variances measured in weak scintillation,” J. Opt. Soc. Am. A 9, 1406–1411 (1992).
    [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. Teich, P. Diament, “Multiply stochastic representations for K distributions and their Poisson transforms,” J. Opt. Soc. Am. A 6, 80–91 (1989).
    [CrossRef]
  26. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [CrossRef]
  27. E. R. Milyutin, Yu. I. Yaremenko, “Distribution of intensity fluctuation in optical radiation propagating in a turbulent atmosphere,” Radiotekh. Elektron. 25, 2272–2278 (1980). [English translation: Radio Eng. Electron. Phys.25, 1–5 (1980).]
  28. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Applied Mathematics Series 55, U.S. Government Printing Office, Washington, D.C., 1964).
  29. M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Zh. Eksp. Teor. Fiz. 67, 2035–2046 (1974). [English translation: Sov. Phys. JETP40, 1011–1016 (1974).]
  30. R. J. Hill, R. A. Bohlander, S. F. Clifford, R. W. McMillan, J. T. Priestley, W. P. Schoenfeld, “Turbulence-induced millimeter-wave scintillation compared with micrometeorological measurements,” IEEE Trans. Geosci. Remote Sens. 26, 330–342 (1988).
    [CrossRef]
  31. J. H. Churnside, R. G. Frehlich, “Experimental evaluation of log-normally modulated Rician and IK models of optical scintillation in the atmosphere,” J. Opt. Soc. Am. A 6, 1760–1766 (1989).
    [CrossRef]
  32. C. G. Little, “A diffraction theory of the scintillation of stars on optical and radio wavelengths,” Mon. Not. R. Astron. Soc. 111, 289–302 (1951).
  33. W. A. Coles, J. P. Filice, R. G. Frehlich, M. Yadlowsky, “Simulation of wave propagation in three-dimensional ran dom media,” Appl. Opt. 34, 2089–2101 (1995).
    [CrossRef] [PubMed]
  34. 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]
  35. E. Jakeman, R. Tough, “Generalized K distribution: a statistical model for weak scattering,” J. Opt. Soc. Am. A 4, 1764–1772 (1987).
    [CrossRef]
  36. 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]
  37. 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]

1996

1995

1994

1993

1992

R. J. Hill, G. R. Ochs, “Inner-scale dependence of scintillation variances measured in weak scintillation,” J. Opt. Soc. Am. A 9, 1406–1411 (1992).
[CrossRef]

V. Thiermann, H. Grassl, “The measurement of turbulent surface-layer fluxes by use of bichromatic scintillation,” Boundary-Layer Meteorol. 58, 367–389 (1992).

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[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]

1989

1988

R. J. Hill, “Comparison of scintillation methods for measuring the inner scale of turbulence,” Appl. Opt. 27, 2187–2193 (1988).
[CrossRef] [PubMed]

R. J. Hill, R. A. Bohlander, S. F. Clifford, R. W. McMillan, J. T. Priestley, W. P. Schoenfeld, “Turbulence-induced millimeter-wave scintillation compared with micrometeorological measurements,” IEEE Trans. Geosci. Remote Sens. 26, 330–342 (1988).
[CrossRef]

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azar, “Optical measurements of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
[CrossRef]

1987

1986

1985

1980

E. R. Milyutin, Yu. I. Yaremenko, “Distribution of intensity fluctuation in optical radiation propagating in a turbulent atmosphere,” Radiotekh. Elektron. 25, 2272–2278 (1980). [English translation: Radio Eng. Electron. Phys.25, 1–5 (1980).]

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

1978

1975

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

1974

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Zh. Eksp. Teor. Fiz. 67, 2035–2046 (1974). [English translation: Sov. Phys. JETP40, 1011–1016 (1974).]

1951

C. G. Little, “A diffraction theory of the scintillation of stars on optical and radio wavelengths,” Mon. Not. R. Astron. Soc. 111, 289–302 (1951).

1949

A. M. Obukhov, “The structure of the temperature field in turbulent flow,” Izv. Akad. Nauk. SSSR Ser. Geogr. Geofiz. 13, 58–69 (1949).

Andrews, L.

Azar, Z.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azar, “Optical measurements of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
[CrossRef]

Azoulay, E.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azar, “Optical measurements of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
[CrossRef]

Barakat, R.

Beckmann, P.

P. Beckmann, Probability in Communication Engineering (Harcourt, Brace & World, New York, 1967).

Bohlander, R. A.

R. J. Hill, R. A. Bohlander, S. F. Clifford, R. W. McMillan, J. T. Priestley, W. P. Schoenfeld, “Turbulence-induced millimeter-wave scintillation compared with micrometeorological measurements,” IEEE Trans. Geosci. Remote Sens. 26, 330–342 (1988).
[CrossRef]

Bracher, C.

Churnside, J. H.

Clifford, S. F.

Cochetti, F.

Coles, W. A.

Consortini, A.

Davis, C. A.

Diament, P.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Filice, J. P.

Flatté, S. M.

Frehlich, R.

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

Frehlich, R. G.

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Zh. Eksp. Teor. Fiz. 67, 2035–2046 (1974). [English translation: Sov. Phys. JETP40, 1011–1016 (1974).]

Grassl, H.

V. Thiermann, H. Grassl, “The measurement of turbulent surface-layer fluxes by use of bichromatic scintillation,” Boundary-Layer Meteorol. 58, 367–389 (1992).

Gurvich, A. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Zh. Eksp. Teor. Fiz. 67, 2035–2046 (1974). [English translation: Sov. Phys. JETP40, 1011–1016 (1974).]

Hill, R. J.

R. J. Hill, R. G. Frehlich, “Onset of strong scintillation with application to remote sensing of turbulence inner scale,” Appl. Opt. 35, 986–997 (1996).
[CrossRef] [PubMed]

A. Consortini, F. Cochetti, J. H. Churnside, R. J. Hill, “Inner-scale effect on irradiance 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]

R. J. Hill, G. R. Ochs, “Inner-scale dependence of scintillation variances measured in weak scintillation,” J. Opt. Soc. Am. A 9, 1406–1411 (1992).
[CrossRef]

R. J. Hill, R. A. Bohlander, S. F. Clifford, R. W. McMillan, J. T. Priestley, W. P. Schoenfeld, “Turbulence-induced millimeter-wave scintillation compared with micrometeorological measurements,” IEEE Trans. Geosci. Remote Sens. 26, 330–342 (1988).
[CrossRef]

R. J. Hill, “Comparison of scintillation methods for measuring the inner scale of turbulence,” Appl. Opt. 27, 2187–2193 (1988).
[CrossRef] [PubMed]

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

G. R. Ochs, R. J. Hill, “Optical-scintillation method of measuring turbulence inner scale,” Appl. Opt. 24, 2430–2432 (1985).
[CrossRef] [PubMed]

R. J. Hill, S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. 68, 892–899 (1978).
[CrossRef]

R. J. Hill, G. R. Ochs, “Fine calibration of large-aperture optical scintillometers and an optical estimate of the inner scale of turbulence,” Appl. Opt. 17, 3608–3612 (1978).
[CrossRef] [PubMed]

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (;1978).

R. J. Hill, R. G. Frehlich, W. D. Otto, “The probability distribution of irradiance scintillation,” , NOAA Environmental Research Laboratories, Boulder, Colorado (1996). Available from the National Technical Information Service, 5285 Port Royal Road, Springfield, Va. 22161.

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, “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]

Jetter, A.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azar, “Optical measurements of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
[CrossRef]

Kashkarov, S. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Zh. Eksp. Teor. Fiz. 67, 2035–2046 (1974). [English translation: Sov. Phys. JETP40, 1011–1016 (1974).]

Kohnle, A.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azar, “Optical measurements of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
[CrossRef]

Little, C. G.

C. G. Little, “A diffraction theory of the scintillation of stars on optical and radio wavelengths,” Mon. Not. R. Astron. Soc. 111, 289–302 (1951).

Martin, J.

McMillan, R. W.

R. J. Hill, R. A. Bohlander, S. F. Clifford, R. W. McMillan, J. T. Priestley, W. P. Schoenfeld, “Turbulence-induced millimeter-wave scintillation compared with micrometeorological measurements,” IEEE Trans. Geosci. Remote Sens. 26, 330–342 (1988).
[CrossRef]

Milyutin, E. R.

E. R. Milyutin, Yu. I. Yaremenko, “Distribution of intensity fluctuation in optical radiation propagating in a turbulent atmosphere,” Radiotekh. Elektron. 25, 2272–2278 (1980). [English translation: Radio Eng. Electron. Phys.25, 1–5 (1980).]

Obukhov, A. M.

A. M. Obukhov, “The structure of the temperature field in turbulent flow,” Izv. Akad. Nauk. SSSR Ser. Geogr. Geofiz. 13, 58–69 (1949).

Ochs, G. R.

Otto, W. D.

R. J. Hill, R. G. Frehlich, W. D. Otto, “The probability distribution of irradiance scintillation,” , NOAA Environmental Research Laboratories, Boulder, Colorado (1996). Available from the National Technical Information Service, 5285 Port Royal Road, Springfield, Va. 22161.

Patrushev, G. Ya.

G. Ya. Patrushev, O. A. Rubtsova, “Probability density of the intensity and flux fluctuations of optical radiation propagating and reflecting in the turbulent atmosphere,” Atmos. Oceanic Opt. 6, 760–769 (1993).

Phillips, R.

Pokasov, V. V.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Zh. Eksp. Teor. Fiz. 67, 2035–2046 (1974). [English translation: Sov. Phys. JETP40, 1011–1016 (1974).]

Priestley, J. T.

R. J. Hill, R. A. Bohlander, S. F. Clifford, R. W. McMillan, J. T. Priestley, W. P. Schoenfeld, “Turbulence-induced millimeter-wave scintillation compared with micrometeorological measurements,” IEEE Trans. Geosci. Remote Sens. 26, 330–342 (1988).
[CrossRef]

Pusey, P.

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

Rubtsova, O. A.

G. Ya. Patrushev, O. A. Rubtsova, “Probability density of the intensity and flux fluctuations of optical radiation propagating and reflecting in the turbulent atmosphere,” Atmos. Oceanic Opt. 6, 760–769 (1993).

Schoenfeld, W. P.

R. J. Hill, R. A. Bohlander, S. F. Clifford, R. W. McMillan, J. T. Priestley, W. P. Schoenfeld, “Turbulence-induced millimeter-wave scintillation compared with micrometeorological measurements,” IEEE Trans. Geosci. Remote Sens. 26, 330–342 (1988).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971).

Teich, M.

Thiermann, V.

V. Thiermann, H. Grassl, “The measurement of turbulent surface-layer fluxes by use of bichromatic scintillation,” Boundary-Layer Meteorol. 58, 367–389 (1992).

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azar, “Optical measurements of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
[CrossRef]

Tough, R.

Walters, D. L.

Wang, G.

Wang, G.-Y.

Yadlowsky, M.

Yaremenko, Yu. I.

E. R. Milyutin, Yu. I. Yaremenko, “Distribution of intensity fluctuation in optical radiation propagating in a turbulent atmosphere,” Radiotekh. Elektron. 25, 2272–2278 (1980). [English translation: Radio Eng. Electron. Phys.25, 1–5 (1980).]

Appl. Opt.

Atmos. Oceanic Opt.

G. Ya. Patrushev, O. A. Rubtsova, “Probability density of the intensity and flux fluctuations of optical radiation propagating and reflecting in the turbulent atmosphere,” Atmos. Oceanic Opt. 6, 760–769 (1993).

Boundary-Layer Meteorol.

V. Thiermann, H. Grassl, “The measurement of turbulent surface-layer fluxes by use of bichromatic scintillation,” Boundary-Layer Meteorol. 58, 367–389 (1992).

IEEE Trans. Geosci. Remote Sens.

R. J. Hill, R. A. Bohlander, S. F. Clifford, R. W. McMillan, J. T. Priestley, W. P. Schoenfeld, “Turbulence-induced millimeter-wave scintillation compared with micrometeorological measurements,” IEEE Trans. Geosci. Remote Sens. 26, 330–342 (1988).
[CrossRef]

Izv. Akad. Nauk. SSSR Ser. Geogr. Geofiz.

A. M. Obukhov, “The structure of the temperature field in turbulent flow,” Izv. Akad. Nauk. SSSR Ser. Geogr. Geofiz. 13, 58–69 (1949).

J. Atmos. Sci.

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

J. Fluid Mech.

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (;1978).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

S. M. Flatté, G. 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. J. Hill, G. R. Ochs, “Inner-scale dependence of scintillation variances measured in weak scintillation,” J. Opt. Soc. Am. A 9, 1406–1411 (1992).
[CrossRef]

S. M. Flatté, C. Bracher, G.-Y. Wang, “Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation,” J. Opt. Soc. Am. A 11, 2080–2092 (1994).
[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]

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]

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

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

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

J. H. Churnside, R. G. Frehlich, “Experimental evaluation of log-normally modulated Rician and IK models of optical scintillation in the atmosphere,” J. Opt. Soc. Am. A 6, 1760–1766 (1989).
[CrossRef]

A. Consortini, F. Cochetti, J. H. Churnside, R. J. Hill, “Inner-scale effect on irradiance 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]

J. Phys. D

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azar, “Optical measurements of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
[CrossRef]

Mon. Not. R. Astron. Soc.

C. G. Little, “A diffraction theory of the scintillation of stars on optical and radio wavelengths,” Mon. Not. R. Astron. Soc. 111, 289–302 (1951).

Phys. Rev. Lett.

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

Proc. IEEE

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Radiotekh. Elektron.

E. R. Milyutin, Yu. I. Yaremenko, “Distribution of intensity fluctuation in optical radiation propagating in a turbulent atmosphere,” Radiotekh. Elektron. 25, 2272–2278 (1980). [English translation: Radio Eng. Electron. Phys.25, 1–5 (1980).]

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]

Zh. Eksp. Teor. Fiz.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Zh. Eksp. Teor. Fiz. 67, 2035–2046 (1974). [English translation: Sov. Phys. JETP40, 1011–1016 (1974).]

Other

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Applied Mathematics Series 55, U.S. Government Printing Office, Washington, D.C., 1964).

R. J. Hill, R. G. Frehlich, W. D. Otto, “The probability distribution of irradiance scintillation,” , NOAA Environmental Research Laboratories, Boulder, Colorado (1996). Available from the National Technical Information Service, 5285 Port Royal Road, Springfield, Va. 22161.

P. Beckmann, Probability in Communication Engineering (Harcourt, Brace & World, New York, 1967).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971).

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Fig. 2
Fig. 2

PDF of ln I scaled by σ for our case of weakest scintillation. (a) l0/RF=0.0 and (b) l0/RF=1.0. The simulation, lognormal, and Beckmann’s PDFs are shown. Curve symbols are given in Fig. 1. The inset figures give details near the maxima.

Fig. 3
Fig. 3

PDF of ln I scaled by σ for near-unity variance. (a) l0/RF=0.0 and (b) l0/RF=1.0. Curve symbols are given in Fig. 1. The K PDF is not shown because it is indistinguishable from the LNME PDF. The inset figures give details near the maxima.

Fig. 4
Fig. 4

PDF of ln I scaled by σ for stronger scintillation than Figs. 2 and 3. (a) l0/RF=0.0 and (b) l0/RF=1.0. Curve symbols are given in Fig. 1. The LNME PDF is not shown because it is nearly indistinguishable from Beckmann’s PDF. The inset figures give details near the maxima.

Fig. 5
Fig. 5

PDF of ln I scaled by σ for our case of strongest scintillation in the strong focusing regime. (a) l0/RF=0.0 and (b) l0/RF=1.0. Curve symbols are given in Fig. 1. The LNME PDF is not shown in (a) because it is indistinguishable from Beckmann’s PDF, and Beckmann’s PDF is not shown in (b) for reasons given in the text. The inset figures give details near the maxima.

Fig. 6
Fig. 6

PDF of ln I scaled by σ for the regime of saturation of scintillation, with l0/RF=0.8. Curve symbols are given in Fig. 1. Beckmann’s PDF is not shown for reasons given in the text. The inset figure gives details near the maxima.

Equations (27)

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P(I)=2Γ(y) y(y+1)/2I(y-1)/2Ky-1[2(Iy)1/2],
y=2/(I2-2),
P(I)=12πσz 0 dzz2 exp-Iz-ln z+12 σz222σz2,
Φn(κ)=0.033 Cn2κ-11/3f(κl0).
σRytov2=β02σ˜2(l0/RF),
β020.496 k7/6L11/6Cn2,
σ˜2(l0/RF)10.501du0dxx-8/3f(xl0/RF)×sin2[x2u(1-u)/2],
σRytov2=σI2=σln I2=-2ln I,
σI2=(I-1)2,
ln I,
σln I2=[(ln I-ln I)2].
P3(x|y¯, σx, σy, )=dy P1(x|y, σx, )×P2(y|y¯, σy, ),
limσy0P3(x|y¯, σx, )=P1(x|y¯, σx, ).
E1(g; y, σx, )dx g(x) P1(x|y, σx, ),
E3(g; y¯, σx, σy, )
dx g(x) P3(x|y¯, σx, σy, ),
=dydx g(x) P1(x|y, σx, )P2(x|y¯, σy, ),
=dy E1(g; y, σx, ) P2(y|y¯, σy, ),
=E2[E1(g; y, σx, ); y¯, σy, ],
PRN(I)=(r+1)z-1 exp-r-(r+1) IzI0×4r(r+1) Iz1/2,
PLN(z)=12πσzz exp-ln z+12 σz22/2σz2,
PB(I|r, σz2)=0dzPRN(I|z, r)PLN(z|σz2).
I2B=exp(σz2)(r2+4r+2)/(r+1)2,
I-1/2B=π1/2(r+1)1/2 exp(-r/2)×I0(r/2)exp(3σz2/8),
exp(ln IB)=[r/(r+1)]×exp-12 σz2+E1(r),
E1(r)rdt t-1 exp(-t).
σσln I2,

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