Abstract

The statistical properties associated with the intensity fluctuations of a He-Ne laser beam propagating through extended clear-air turbulence over a homogeneous range were measured and compared with existing theoretical models. The statistics were determined by measuring the first five normalized moments of the beam intensity over varying distances from 183 m to 3 km. The beam geometry was approximately that of a spherical wave. Calculated values of the refractive-index-structure parameter C<sub>n</sub> <sup>2</sup> varied from a low of approximately 10<sup>-14</sup> m<sup>-2/3</sup> to a high value of the order of 10<sup>-10</sup> m<sup>-2/3</sup>. Thus the data obtained correspond to many conditions of turbulence from weak to strong turbulence in the saturation regime. We found that the data generally support the lognormal model under conditions of weak turbulence and the negative-exponential distribution in the superstrong-turbulence regime. The K distribution fits some of, but not all, the data in the saturation regime. For conditions of strong turbulence before the superstrong regime, none of the above statistical models fits the data well. This is due primarily to the fact that a looping effect appeared in the data whereby the third- and higher-order normalized moments appear to follow a looped curve in the vicinity of their maximum values when they are plotted as functions of the second normalized moment. None of the theoretical curves for the normalized moments of the above-named distributions exhibits this looping effect (since they are essentially one-parameter distributions), and therefore they have limited applicability with respect to our data.

© 1981 Optical Society of America

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  1. V. I. Tatarskii, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman (McGraw-Hill, New York, 1961); The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  2. G. R. Ochs, R. R. Bergman, and J. R. Snyder, "Laser-beam scintillation over horizontal paths from 5.5 to 145 kilometers," J. Opt. Soc. Am. 59, 231–234 (1969).
  3. 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. Uchebn. Zaved. Radiofiz. 13, 55–60 (1970).
  4. R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," Proc. IEEE 58, 1523–1545 (1970).
  5. D. A. DeWolf, "Strong irradiance fluctuations in turbulent air: plane waves," J. Opt. Soc. Am. 62, 966–971 (1972).
  6. J. R. Dunphy and J. R. Kerr, "Scintillation measurements for large integrated-path turbulence," J. Opt. Soc. Am. 63, 981–986 (1973).
  7. D. A. DeWolf, "Waves in turbulent air: a phenomenological model," Proc. IEEE 62, 1523–1529 (1974).
  8. 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).
  9. E. Jakeman and P. N. Pusey, "A model for non-Rayleigh sea echo," IEEE Trans. Antenn. and Prop. AP-24, 806–814 (1976).
  10. A. Ishimaru, "Theory and applications of wave propagation and scattering in random media," Proc. IEEE 65, 1030–1061 (1977).
  11. J. K. Jao and M. Elbaum, "First-order statistics of a non-Rayleigh fading signal and its detection," Proc. IEEE 66 781–789 (1978).
  12. G. Parry and P. N. Pusey, "K distributions in atmospheric propagation of laser light," J. Opt. Soc. Am. 69, 796–798 (1979).
  13. G. Parry, "Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam," Opt. Acta 28, 715–728 (1981).
  14. D. L. Fried, "Propagation of a spherical wave in a turbulent medium," J. Opt. Soc. Am. 53, 175–180 (1967).
  15. A. Ishimaru, Wave Propagation and Scattering Media (Academic, New York, 1978).
  16. S. F. Clifford, "The classical theory of wave propagation in a turbulent medium," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978).
  17. 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).
  18. R. A. Elliot, J. R. Dunphy, and J. R. Kerr, "Statistical tests of distribution 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 WAH-1.
  19. 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).
  20. I. G. Yakushkin, "Moments of field propagating in randomly inhomogeneous medium in the limit of saturated fluctuations," Izv. Vyssh. Uchelon. Zaved. Radiofiz. 21, 1194–1201 (1978).
  21. T.-I. Wang and J. W. Strohbehn, "Perturbed lognormal distribution of irradiance fluctuations," J. Opt. Soc. Am. 64, 994–1004 (1974).
  22. 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).
  23. L. R. Bissonnette and P. L. Wizinowich, "Probability distribution of turbulence irradiance in a saturation regime," Appl. Opt. 18, 1590–1599 (1979).
  24. R. J. Hill and S. F. Clifford, "Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation," J. Opt. Soc. Am. 68, 892–899 (1978).
  25. S. F. Clifford and R. J. Hill, "Relation between irradiance and log-amplitude variance for optical scintillation described by the K-distribution," J. Opt; Soc. Am. 71, 112–114 (1981).
  26. R. J. Hill and S. F. Clifford, "Theory of saturation of optical scintillation by strong turbulence for arbitrary refractive-index spectra," J. Opt. Soc. Am. 71, 675–686 (1981).
  27. B. J. Uscinski, The Elements of Wave Propagation in Random Media (McGraw-Hill, New York, 1977).

1981 (3)

1979 (2)

1978 (3)

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).

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

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

1976 (1)

E. Jakeman and P. N. Pusey, "A model for non-Rayleigh sea echo," IEEE Trans. Antenn. and Prop. AP-24, 806–814 (1976).

1975 (1)

1974 (3)

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).

T.-I. Wang and J. W. Strohbehn, "Perturbed lognormal distribution of irradiance fluctuations," J. Opt. Soc. Am. 64, 994–1004 (1974).

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).

1973 (1)

1972 (1)

1970 (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. Uchebn. Zaved. Radiofiz. 13, 55–60 (1970).

1969 (1)

1967 (1)

D. L. Fried, "Propagation of a spherical wave in a turbulent medium," J. Opt. Soc. Am. 53, 175–180 (1967).

Bergman, R. R.

Bissonnette, L. R.

Clifford, S. F.

Davidson, F.

DeWolf, D. A.

D. A. DeWolf, "Strong irradiance fluctuations in turbulent air: plane waves," J. Opt. Soc. Am. 62, 966–971 (1972).

D. A. DeWolf, "Waves in turbulent air: a phenomenological model," Proc. IEEE 62, 1523–1529 (1974).

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).

R. A. Elliot, J. R. Dunphy, and J. R. Kerr, "Statistical tests of distribution 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 WAH-1.

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).

Elliot, R. A.

R. A. Elliot, J. R. Dunphy, and J. R. Kerr, "Statistical tests of distribution 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 WAH-1.

Fried, D. L.

D. L. Fried, "Propagation of a spherical wave in a turbulent medium," J. Opt. Soc. Am. 53, 175–180 (1967).

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. Uchebn. Zaved. Radiofiz. 13, 55–60 (1970).

Gurvich, 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).

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. Uchebn. Zaved. Radiofiz. 13, 55–60 (1970).

Hill, R. J.

Ishimaru, A.

A. Ishimaru, "Theory and applications of wave propagation and scattering in random media," Proc. IEEE 65, 1030–1061 (1977).

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

Jakeman, E.

E. Jakeman and P. N. Pusey, "A model for non-Rayleigh sea echo," IEEE Trans. Antenn. and Prop. AP-24, 806–814 (1976).

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).

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. Uchebn. 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).

R. A. Elliot, J. R. Dunphy, and J. R. Kerr, "Statistical tests of distribution 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 WAH-1.

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).

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

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).

Ochs, G. R.

Parry, G.

G. Parry, "Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam," Opt. Acta 28, 715–728 (1981).

G. Parry and P. N. Pusey, "K distributions in atmospheric propagation of laser light," J. Opt. Soc. Am. 69, 796–798 (1979).

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).

E. Jakeman and P. N. Pusey, "A model for non-Rayleigh sea echo," IEEE Trans. Antenn. and Prop. AP-24, 806–814 (1976).

Snyder, J. R.

Strohbehn, J. W.

T.-I. Wang and J. W. Strohbehn, "Perturbed lognormal distribution of irradiance fluctuations," J. Opt. Soc. Am. 64, 994–1004 (1974).

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

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman (McGraw-Hill, New York, 1961); The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Uscinski, B. J.

B. J. Uscinski, The Elements of Wave Propagation in Random Media (McGraw-Hill, New York, 1977).

Wang, T.-I.

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. Uchelon. Zaved. Radiofiz. 21, 1194–1201 (1978).

Appl. Opt. (1)

IEEE Trans. Antenn. and Prop. (1)

E. Jakeman and P. N. Pusey, "A model for non-Rayleigh sea echo," IEEE Trans. Antenn. and Prop. AP-24, 806–814 (1976).

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (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. Uchebn. Zaved. Radiofiz. 13, 55–60 (1970).

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. Uchelon. Zaved. Radiofiz. (1)

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

J. Opt. Soc. Am. (11)

T.-I. Wang and J. W. Strohbehn, "Perturbed lognormal distribution of irradiance fluctuations," J. Opt. Soc. Am. 64, 994–1004 (1974).

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).

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

S. F. Clifford and R. J. Hill, "Relation between irradiance and log-amplitude variance for optical scintillation described by the K-distribution," J. Opt; Soc. Am. 71, 112–114 (1981).

R. J. Hill and S. F. Clifford, "Theory of saturation of optical scintillation by strong turbulence for arbitrary refractive-index spectra," J. Opt. Soc. Am. 71, 675–686 (1981).

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).

G. Parry and P. N. Pusey, "K distributions in atmospheric propagation of laser light," J. Opt. Soc. Am. 69, 796–798 (1979).

D. L. Fried, "Propagation of a spherical wave in a turbulent medium," J. Opt. Soc. Am. 53, 175–180 (1967).

G. R. Ochs, R. R. Bergman, and J. R. Snyder, "Laser-beam scintillation over horizontal paths from 5.5 to 145 kilometers," J. Opt. Soc. Am. 59, 231–234 (1969).

D. A. DeWolf, "Strong irradiance fluctuations in turbulent air: plane waves," J. Opt. Soc. Am. 62, 966–971 (1972).

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

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).

Opt. Acta (1)

G. Parry, "Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam," Opt. Acta 28, 715–728 (1981).

Other (9)

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

S. F. Clifford, "The classical theory of wave propagation in a turbulent medium," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978).

R. A. Elliot, J. R. Dunphy, and J. R. Kerr, "Statistical tests of distribution 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 WAH-1.

D. A. DeWolf, "Waves in turbulent air: a phenomenological model," Proc. IEEE 62, 1523–1529 (1974).

A. Ishimaru, "Theory and applications of wave propagation and scattering in random media," Proc. IEEE 65, 1030–1061 (1977).

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

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman (McGraw-Hill, New York, 1961); The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

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

B. J. Uscinski, The Elements of Wave Propagation in Random Media (McGraw-Hill, New York, 1977).

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