Abstract

A universal model is proposed for the irradiance fluctuations of an optical beam propagating through atmospheric turbulence. When this model was compared with existing measured data, we found good qualitative and quantitative agreement, which suggests that this new theoretical model is applicable under all known conditions of turbulence. In the regime of weak scattering, the normalized moments of the distribution are essentially the same as those predicted by the lognormal model, although they show large deviations from lognormal statistics in the saturation regime. The limiting form of the universal model for conditions of super-strong turbulence is that of the negative-exponential distribution, but, for more moderate conditions of turbulence, the form is that of an exponential times an infinite series of Laguerre polynomials. The new distribution was derived under the assumption that the field irradiance consists of two principal components, each of which has an amplitude that is m distributed.

© 1982 Optical Society of America

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References

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  1. G. R. Ochs, R. R. Bergmen, and J. R. Snyder, “Laser-beam scintillation over horizontal paths from 5.5 to 145 kilometers,” J. Opt. Soc. Am. 59, 231–234 (1969).
    [Crossref]
  2. M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, “Measurements of the variance of ‘strong’ intensity fluctuations of laser radiation in the atmosphere,” Izv. Vyssh. Ucheb. Zaved. Radiofiz. 13, 55–60 (1970).
  3. J. R. Dunphy and J. R. Kerr, “Scintillation measurements for large integrated-path turbulence,” J. Opt. Soc. Am. 63, 981–986 (1973).
    [Crossref]
  4. G. Parry and P. N. Pusey, “K-distributions in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979).
    [Crossref]
  5. R. L. Phillips and L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71, 1440–1445 (1981).
    [Crossref]
  6. V. I. Tatarskii, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman (McGraw-Hill, New York, 1961).
  7. D. L. Fried, “Propagation of a spherical wave in a turbulent medium,” J. Opt. Soc. Am. 57, 175–180 (1967).
    [Crossref]
  8. A. Ishimaru, Wave Propagation and Scattering Media (Academic, New York, 1978).
  9. S. F. Clifford, G. R. Ochs, and R. S. Lawrence, “Saturation of optical scintillations by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
    [Crossref]
  10. A. S. Gurvich and V. I. Tatarskii, “Coherence and intensity fluctuations of light in the turbulent atmosphere,” Radio Sci. 10, 3–14 (1975).
    [Crossref]
  11. G. Parry, “Measurements of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
    [Crossref]
  12. M. E. Gracheva, A. S. Gurvich, S. O. Lowadze, Vl. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Izv. Vyssh, Uchebn. Zaved. Radiofiz. 17, 105–112 (1974).
  13. R. A. Elliot, J. R. Dunphy, and J. R. Kerr, “Statistical tests of distributional hypotheses applied to irradiance fluctuations,” in Digest of Topical Meeting on Optical Propagation through Turbulence, Rain, and Fog (Optical Society of America, Washington, D.C., 1977), paper WA4.
  14. T.-I. Wang and J. W. Strohbehn, “Perturbed log-normal distribution of irradiance fluctuations,” J. Opt. Soc. Am. 64, 994–1004 (1974).
    [Crossref]
  15. F. Davidson and A. Gonzalez-Del-Valle, “Measurements of three-parameter log-normally distributed optical-field irradiance fluctuations in a turbulent medium,” J. Opt. Soc. Am. 65, 655–663 (1975).
    [Crossref]
  16. E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
    [Crossref]
  17. E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
    [Crossref]
  18. L. R. Bissonnette and P. L. Wizinowich, “Probability distribution of turbulent irradiance in a saturation regime,” Appl. Opt. 18, 1590–1599 (1979).
    [Crossref] [PubMed]
  19. R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE,  58, 1523–1545 (1970).
    [Crossref]
  20. P. F. Miller, “The probability distribution of a wave at a very large depth within an extended region,” J. Phys A 11, 403–422 (1978).
    [Crossref]
  21. I. G. Yakushkin, “Moments of field propagating in randomly inhomogeneous medium in the limit of saturated fluctuations,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 21, 1194–1201 (1978).
  22. J. W. Strohbehn, T.-I. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci.10, 59–70 (1975).
  23. D. A. deWolf, “Waves in turbulent air: a phenomenological model,” Proc. IEEE 62, 1523–1529 (1974).
    [Crossref]
  24. J. K. Jao and M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
    [Crossref]
  25. R. L. Fante, “Optical beam propagation in turbulent media,” AFCRL-TR-75-0439, Phys. Sci. Res. Papers No. 640, Air Force Cambridge Res. Labs., Air Force Systems Command, USAF (U.S. Air Force Geophysical Laboratory, Bedford, Mass., 1975).
  26. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).
  27. M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” reprint from Statistical Methods of Radio Wave Propagation (Pergamon, Oxford, 1960).
  28. S. H. Lin, “Statistical behavior of a fading signal,” Bell Syst. Tech. J. 50, 3211–3270 (1971).
    [Crossref]
  29. I. S. Gradshteyn and I. W. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).
  30. A. Erdelyi and et al., Higher Transcendental Functions, Bateman Manuscript Project (McGraw-Hill, New York, 1953), Vols. 1 and 2.
  31. R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
    [Crossref]

1981 (2)

G. Parry, “Measurements of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[Crossref]

R. L. Phillips and L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71, 1440–1445 (1981).
[Crossref]

1980 (1)

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

1979 (3)

1978 (3)

J. K. Jao and M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
[Crossref]

P. F. Miller, “The probability distribution of a wave at a very large depth within an extended region,” J. Phys A 11, 403–422 (1978).
[Crossref]

I. G. Yakushkin, “Moments of field propagating in randomly inhomogeneous medium in the limit of saturated fluctuations,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 21, 1194–1201 (1978).

1976 (1)

E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[Crossref]

1975 (3)

J. W. Strohbehn, T.-I. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci.10, 59–70 (1975).

A. S. Gurvich and V. I. Tatarskii, “Coherence and intensity fluctuations of light in the turbulent atmosphere,” Radio Sci. 10, 3–14 (1975).
[Crossref]

F. Davidson and A. Gonzalez-Del-Valle, “Measurements of three-parameter log-normally distributed optical-field irradiance fluctuations in a turbulent medium,” J. Opt. Soc. Am. 65, 655–663 (1975).
[Crossref]

1974 (4)

M. E. Gracheva, A. S. Gurvich, S. O. Lowadze, Vl. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Izv. Vyssh, Uchebn. Zaved. Radiofiz. 17, 105–112 (1974).

D. A. deWolf, “Waves in turbulent air: a phenomenological model,” Proc. IEEE 62, 1523–1529 (1974).
[Crossref]

S. F. Clifford, G. R. Ochs, and R. S. Lawrence, “Saturation of optical scintillations by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
[Crossref]

T.-I. Wang and J. W. Strohbehn, “Perturbed log-normal distribution of irradiance fluctuations,” J. Opt. Soc. Am. 64, 994–1004 (1974).
[Crossref]

1973 (1)

1971 (1)

S. H. Lin, “Statistical behavior of a fading signal,” Bell Syst. Tech. J. 50, 3211–3270 (1971).
[Crossref]

1970 (2)

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE,  58, 1523–1545 (1970).
[Crossref]

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, “Measurements of the variance of ‘strong’ intensity fluctuations of laser radiation in the atmosphere,” Izv. Vyssh. Ucheb. Zaved. Radiofiz. 13, 55–60 (1970).

1969 (1)

1967 (1)

Andrews, L. C.

Bergmen, R. R.

Bissonnette, L. R.

Clifford, S. F.

Dashen, R.

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[Crossref]

Davidson, F.

deWolf, D. A.

D. A. deWolf, “Waves in turbulent air: a phenomenological model,” Proc. IEEE 62, 1523–1529 (1974).
[Crossref]

Dunphy, J. R.

J. R. Dunphy and J. R. Kerr, “Scintillation measurements for large integrated-path turbulence,” J. Opt. Soc. Am. 63, 981–986 (1973).
[Crossref]

R. A. Elliot, J. R. Dunphy, and J. R. Kerr, “Statistical tests of distributional hypotheses applied to irradiance fluctuations,” in Digest of Topical Meeting on Optical Propagation through Turbulence, Rain, and Fog (Optical Society of America, Washington, D.C., 1977), paper WA4.

Elbaum, M.

J. K. Jao and M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
[Crossref]

Elliot, R. A.

R. A. Elliot, J. R. Dunphy, and J. R. Kerr, “Statistical tests of distributional hypotheses applied to irradiance fluctuations,” in Digest of Topical Meeting on Optical Propagation through Turbulence, Rain, and Fog (Optical Society of America, Washington, D.C., 1977), paper WA4.

Erdelyi, A.

A. Erdelyi and et al., Higher Transcendental Functions, Bateman Manuscript Project (McGraw-Hill, New York, 1953), Vols. 1 and 2.

Fante, R. L.

R. L. Fante, “Optical beam propagation in turbulent media,” AFCRL-TR-75-0439, Phys. Sci. Res. Papers No. 640, Air Force Cambridge Res. Labs., Air Force Systems Command, USAF (U.S. Air Force Geophysical Laboratory, Bedford, Mass., 1975).

Fried, D. L.

Gonzalez-Del-Valle, A.

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, S. O. Lowadze, Vl. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Izv. Vyssh, Uchebn. Zaved. Radiofiz. 17, 105–112 (1974).

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, “Measurements of the variance of ‘strong’ intensity fluctuations of laser radiation in the atmosphere,” Izv. Vyssh. Ucheb. Zaved. Radiofiz. 13, 55–60 (1970).

Gradshteyn, I. S.

I. S. Gradshteyn and I. W. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Gurvich, A. S.

A. S. Gurvich and V. I. Tatarskii, “Coherence and intensity fluctuations of light in the turbulent atmosphere,” Radio Sci. 10, 3–14 (1975).
[Crossref]

M. E. Gracheva, A. S. Gurvich, S. O. Lowadze, Vl. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Izv. Vyssh, Uchebn. Zaved. Radiofiz. 17, 105–112 (1974).

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, “Measurements of the variance of ‘strong’ intensity fluctuations of laser radiation in the atmosphere,” Izv. Vyssh. Ucheb. Zaved. Radiofiz. 13, 55–60 (1970).

Ishimaru, A.

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

Jakeman, E.

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

E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[Crossref]

Jao, J. K.

J. K. Jao and M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
[Crossref]

Kallistratova, M. A.

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, “Measurements of the variance of ‘strong’ intensity fluctuations of laser radiation in the atmosphere,” Izv. Vyssh. Ucheb. Zaved. Radiofiz. 13, 55–60 (1970).

Kerr, J. R.

J. R. Dunphy and J. R. Kerr, “Scintillation measurements for large integrated-path turbulence,” J. Opt. Soc. Am. 63, 981–986 (1973).
[Crossref]

R. A. Elliot, J. R. Dunphy, and J. R. Kerr, “Statistical tests of distributional hypotheses applied to irradiance fluctuations,” in Digest of Topical Meeting on Optical Propagation through Turbulence, Rain, and Fog (Optical Society of America, Washington, D.C., 1977), paper WA4.

Khrupin, A. S.

M. E. Gracheva, A. S. Gurvich, S. O. Lowadze, Vl. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Izv. Vyssh, Uchebn. Zaved. Radiofiz. 17, 105–112 (1974).

Lawrence, R. S.

S. F. Clifford, G. R. Ochs, and R. S. Lawrence, “Saturation of optical scintillations by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
[Crossref]

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE,  58, 1523–1545 (1970).
[Crossref]

Lin, S. H.

S. H. Lin, “Statistical behavior of a fading signal,” Bell Syst. Tech. J. 50, 3211–3270 (1971).
[Crossref]

Lowadze, S. O.

M. E. Gracheva, A. S. Gurvich, S. O. Lowadze, Vl. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Izv. Vyssh, Uchebn. Zaved. Radiofiz. 17, 105–112 (1974).

Miller, P. F.

P. F. Miller, “The probability distribution of a wave at a very large depth within an extended region,” J. Phys A 11, 403–422 (1978).
[Crossref]

Nakagami, M.

M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” reprint from Statistical Methods of Radio Wave Propagation (Pergamon, Oxford, 1960).

Ochs, G. R.

Parry, G.

G. Parry, “Measurements of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[Crossref]

G. Parry and P. N. Pusey, “K-distributions in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979).
[Crossref]

Phillips, R. L.

Pokasov, Vl. V.

M. E. Gracheva, A. S. Gurvich, S. O. Lowadze, Vl. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Izv. Vyssh, Uchebn. Zaved. Radiofiz. 17, 105–112 (1974).

Pusey, P. N.

G. Parry and P. N. Pusey, “K-distributions in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979).
[Crossref]

E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[Crossref]

Ryzhik, I. W.

I. S. Gradshteyn and I. W. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Snyder, J. R.

Speck, J. P.

J. W. Strohbehn, T.-I. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci.10, 59–70 (1975).

Strohbehn, J. W.

J. W. Strohbehn, T.-I. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci.10, 59–70 (1975).

T.-I. Wang and J. W. Strohbehn, “Perturbed log-normal distribution of irradiance fluctuations,” J. Opt. Soc. Am. 64, 994–1004 (1974).
[Crossref]

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE,  58, 1523–1545 (1970).
[Crossref]

Tatarskii, V. I.

A. S. Gurvich and V. I. Tatarskii, “Coherence and intensity fluctuations of light in the turbulent atmosphere,” Radio Sci. 10, 3–14 (1975).
[Crossref]

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

Wang, T.-I.

J. W. Strohbehn, T.-I. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci.10, 59–70 (1975).

T.-I. Wang and J. W. Strohbehn, “Perturbed log-normal distribution of irradiance fluctuations,” J. Opt. Soc. Am. 64, 994–1004 (1974).
[Crossref]

Wizinowich, P. L.

Yakushkin, I. G.

I. G. Yakushkin, “Moments of field propagating in randomly inhomogeneous medium in the limit of saturated fluctuations,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 21, 1194–1201 (1978).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

S. H. Lin, “Statistical behavior of a fading signal,” Bell Syst. Tech. J. 50, 3211–3270 (1971).
[Crossref]

IEEE Trans. Antennas Propag. (1)

E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[Crossref]

Izv. Vyssh, Uchebn. Zaved. Radiofiz. (1)

M. E. Gracheva, A. S. Gurvich, S. O. Lowadze, Vl. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Izv. Vyssh, Uchebn. Zaved. Radiofiz. 17, 105–112 (1974).

Izv. Vyssh. Ucheb. Zaved. Radiofiz. (1)

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, “Measurements of the variance of ‘strong’ intensity fluctuations of laser radiation in the atmosphere,” Izv. Vyssh. Ucheb. Zaved. Radiofiz. 13, 55–60 (1970).

Izv. Vyssh. Uchebn. Zaved., Radiofiz. (1)

I. G. Yakushkin, “Moments of field propagating in randomly inhomogeneous medium in the limit of saturated fluctuations,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 21, 1194–1201 (1978).

J. Math. Phys. (1)

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[Crossref]

J. Opt. Soc. Am. (8)

J. Phys A (1)

P. F. Miller, “The probability distribution of a wave at a very large depth within an extended region,” J. Phys A 11, 403–422 (1978).
[Crossref]

J. Phys. A (1)

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

Opt. Acta (1)

G. Parry, “Measurements of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[Crossref]

Proc. IEEE (3)

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE,  58, 1523–1545 (1970).
[Crossref]

D. A. deWolf, “Waves in turbulent air: a phenomenological model,” Proc. IEEE 62, 1523–1529 (1974).
[Crossref]

J. K. Jao and M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
[Crossref]

Radio Sci. (2)

J. W. Strohbehn, T.-I. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci.10, 59–70 (1975).

A. S. Gurvich and V. I. Tatarskii, “Coherence and intensity fluctuations of light in the turbulent atmosphere,” Radio Sci. 10, 3–14 (1975).
[Crossref]

Other (8)

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

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

R. A. Elliot, J. R. Dunphy, and J. R. Kerr, “Statistical tests of distributional hypotheses applied to irradiance fluctuations,” in Digest of Topical Meeting on Optical Propagation through Turbulence, Rain, and Fog (Optical Society of America, Washington, D.C., 1977), paper WA4.

R. L. Fante, “Optical beam propagation in turbulent media,” AFCRL-TR-75-0439, Phys. Sci. Res. Papers No. 640, Air Force Cambridge Res. Labs., Air Force Systems Command, USAF (U.S. Air Force Geophysical Laboratory, Bedford, Mass., 1975).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” reprint from Statistical Methods of Radio Wave Propagation (Pergamon, Oxford, 1960).

I. S. Gradshteyn and I. W. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

A. Erdelyi and et al., Higher Transcendental Functions, Bateman Manuscript Project (McGraw-Hill, New York, 1953), Vols. 1 and 2.

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Figures (5)

Fig. 1
Fig. 1

Multiple-scattering model.

Fig. 2
Fig. 2

Parameters of the m distributions associated with the diffuse and specular components of the laser beam as a function of the normalized variance.

Fig. 3
Fig. 3

Measured values of the normalized third, fourth, and fifth moments obtained during March 1980 experiments and compared with values predicted by the universal model.

Fig. 4
Fig. 4

Measured values of the normalized third, fourth, and fifth moments obtained during August 1980 experiments and compared with values predicted by the universal model.

Fig. 5
Fig. 5

Measured values of the normalized third, fourth, and fifth moments obtained over long path lengths and compared with values from the lognormal and universal models.

Equations (36)

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U ( t ) = ( A e i θ + R e i ϕ ) e i ω t ,
p ( R ) = 2 m m R 2 m - 1 Γ ( m ) b m exp ( - m R 2 / b ) ,             R > 0 ,
p ( A ) = 2 M M A 2 M - 1 Γ ( M ) c M exp ( - M A 2 / c ) ,             A > 0 ,
I = U ( t ) 2 = A 2 + R 2 + 2 A R cos ( ϕ - θ ) .
p ( I ) = ½ 0 z J 0 ( I z ) J 0 ( A z ) J 0 ( R z ) d z ,
J 0 ( A z ) = 0 p ( A ) J 0 ( A z ) d A
J 0 ( R z ) = 0 p ( R ) J 0 ( R z ) d R .
J 0 ( A z ) = 2 M M Γ ( M ) c M 0 A 2 M - 1 exp ( - M A 2 / c ) J 0 ( A z ) d A = F 1 1 ( M ; 1 ; - c z 2 / 4 M ) ,
J 0 ( R z ) = F 1 1 ( m ; 1 ; - b z 2 / 4 m ) .
F 1 1 ( a ; b ; - x ) = e - x F 1 1 ( b - a ; b ; x )
J 0 ( A z ) J 0 ( R z ) = exp ( - b z 2 / 4 m ) F 1 1 ( M ; 1 ; - c z 2 / 4 M ) × F 1 1 ( 1 - m ; 1 ; b z 2 / 4 m ) .
F 1 1 ( a ; c ; p x ) F 1 1 ( a ; c ; q x ) = k = 0 ( a ) k ( p x ) k k ! ( c ) k F 3 2 ( a , 1 - c - k , - k ; c , 1 - a - k ; - q / p ) ,
( a ) k = Γ ( a + k ) Γ ( a ) ,             k = 0 , 1 , 2 ,
J 0 ( A z ) J 0 ( R z ) = e - b z 2 / 4 m k = 0 ( 1 - m ) k ( b z 2 / 4 m ) k ( k ! ) 2 × F 3 2 ( M , - k , - k ; 1 , m - k ; c m / b M ) .
0 e - x 2 x 2 n + 1 J 0 ( 2 x z ) d x = 1 / 2 n ! e - z L n ( z ) ,
p ( I ) = m b exp ( - mI / b ) k = 0 ( 1 - m ) k k ! × L k ( m I / b ) F 3 2 ( M , - k , - k ; 1 , m - k ; c m / b M ) .
p ( I ) = m Γ ( m ) b Γ ( M ) e - m I / b k = 0 ( - 1 ) k k ! L k ( m I / b ) × j = 0 k ( k j ) 2 Γ ( M + j ) ( r m / M ) j Γ ( m - k + j ) ,
( k j ) = k ! j ! ( k - j ) !
r = c / b .
p ( I ) = 1 b e - I / b .
I n = 0 I n p ( I ) d I ,             n = 1 , 2 ,
I n = ( b m ) n k = 0 n ( n k ) 2 × Γ ( M + k ) Γ ( m + n - k ) Γ ( M ) Γ ( m ) ( r m / M ) k .
I = b ( 1 + r ) ,
I n I n = 1 ( 1 + r ) n k = 0 n ( n k ) 2 μ n - k a k r k ,
μ k = R 2 k R 2 k = Γ ( m + k ) m k Γ ( m )
a k = A 2 k A 2 k = Γ ( M + k ) M k Γ ( M ) .
r = 2 - x + [ ( a 2 + μ 2 - 4 ) x + 4 - a 2 μ 2 ] 1 / 2 x - a 2 ,
I n = 0 I n p ( I ) d I ,             n = 1 , 2 , ,
0 e - s t t β L n ( t ) d t = Γ ( β + 1 ) s - ( β + 1 ) F ( - n , β + 1 ; 1 ; 1 / s ) ,
I n = ( b m ) n Γ ( m ) Γ ( M ) k = 0 ( - 1 ) k k ! n ! F ( - k , n + 1 ; 1 ; 1 ) × j = 0 k ( k j ) 2 Γ ( M + j ) Γ ( m + j - k ) ( r m / M ) j ,
F ( a , b ; c ; 1 ) = Γ ( c ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) ,
F ( - k , n + 1 ; 1 ; 1 ) = Γ ( k - n ) Γ ( k + 1 ) Γ ( - n ) = { ( - 1 ) k n ! / k ! ( n - k ) ! , k n 0 k > n ,
I n = ( b m ) n k = 0 n j = 0 k × ( n ! ) 2 Γ ( m ) Γ ( M + j ) ( r m / M ) j ( n - k ) ! ( j ! ) 2 [ ( k - j ) ! ] 2 Γ ( M ) Γ ( m + j - k ) .
I n = ( b m ) n j = 0 n ( n ! ) 2 Γ ( M + j ) ( r m / M ) j Γ ( M ) ( j ! ) 2 ( n - j ) ! × p = 0 n - j ( n - j ) ! Γ ( m ) ( n - p - j ) ! Γ ( m - p ) ( p ! ) 2 .
p = 0 n - j ( n - j ) ! Γ ( m ) ( n - p - j ) ! Γ ( m - p ) ( p ! ) 2 = p = 0 n - j ( 1 - m ) p ( - n + j ) p ( 1 ) p ( 1 ) p p ! = F ( 1 - m , - n + j ; 1 ; 1 ) = Γ ( m + n - j ) Γ ( m ) Γ ( n - j + 1 ) ,
I n = ( b m ) n j = 0 n ( n j ) 2 Γ ( M + j ) Γ ( m + n - j ) Γ ( M ) Γ ( m ) ( r m M ) j .