Abstract

A heuristic model of irradiance fluctuations for a propagating optical wave in a weakly inhomogeneous medium is developed under the assumption that small-scale irradiance fluctuations are modulated by large-scale irradiance fluctuations of the wave. The upper bound for small turbulent cells is defined by the smallest cell size between the Fresnel zone and the transverse spatial coherence radius of the optical wave. A lower bound for large turbulent cells is defined by the largest cell size between the Fresnel zone and the scattering disk. In moderate-to-strong irradiance fluctuations, cell sizes between those defined by the spatial coherence radius and the scattering disk are eliminated through spatial-frequency filtering as a consequence of the propagation process. The resulting scintillation index from this theory has the form σI2=σx2+σy2+σx2σy2, where σx2 denotes large-scale scintillation and σy2 denotes small-scale scintillation. By means of a modification of the Rytov method that incorporates an amplitude spatial-frequency filter function under strong-fluctuation conditions, tractable expressions are developed for the scintillation index of a plane wave and a spherical wave that are valid under moderate-to-strong irradiance fluctuations. In many cases the models also compare well with conventional results in weak-fluctuation regimes. Inner-scale effects are taken into account by use of a modified atmospheric spectrum that exhibits a bump at large spatial frequencies. Quantitative values predicted by these models agree well with experimental and simulation data previously published. In addition to the scintillation index, expressions are also developed for the irradiance covariance function of a plane wave and a spherical wave, both of which have the form BI(ρ)=Bx(ρ)+By(ρ)+Bx(ρ)By(ρ), where Bx(ρ) is the covariance function associated with large-scale fluctuations and By(ρ) is the covariance function associated with small-scale fluctuations. In strong turbulence the derived covariance shows the characteristic two-scale behavior, in which the correlation length is determined by the spatial coherence radius of the field and the width of the long residual correlation tail is determined by the scattering disk.

© 1999 Optical Society of America

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References

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  1. V. I. Tatarskii, Wave Propagation in a Turbulent Medium, translated from Russian by R. A. Silverman (McGraw-Hill, New York, 1961).
  2. L. A. Chernov, Wave Propagation in a Random Medium, translated from Russian by R. A. Silverman (McGraw-Hill, New York, 1960).
  3. M. E. Gracheva, A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 8, 717–724 (1965).
  4. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971).
  5. J. Dunphy, J. Kerr, “Scintillation measurements for large integrated-path turbulence,” J. Opt. Soc. Am. 63, 981–986 (1973).
    [CrossRef]
  6. M. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Sov. Phys. JETP 40, 1011–1016 (1974).
  7. H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. 64, 59–67 (1974).
    [CrossRef]
  8. S. F. Clifford, G. R. Ochs, R. S. Lawrence, “Saturation of optical scintillation by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
    [CrossRef]
  9. R. J. Hill, 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).
    [CrossRef]
  10. R. G. Frehlich, S. M. Wandzura, R. J. Hill, “Log-amplitude covariance for waves propagating through very strong turbulence,” J. Opt. Soc. Am. A 4, 2158–2161 (1987).
    [CrossRef]
  11. 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]
  12. K. S. Gochelashvili, V. I. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).
  13. R. L. Fante, “Inner-scale size effect on the scintillations of light in the turbulent atmosphere,” J. Opt. Soc. Am. 73, 277–281 (1983).
    [CrossRef]
  14. 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); erratum, 4, 1324 (1987).
    [CrossRef]
  15. G. Parry, “Measurements of atmospheric turbulence-induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
    [CrossRef]
  16. R. L. Phillips, L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71, 1440–1445 (1981).
    [CrossRef]
  17. W. R. Coles, R. G. Frehlich, “Simultaneous measurements of angular scattering and intensity scintillation in the atmosphere,” J. Opt. Soc. Am. 72, 1042–1048 (1982).
    [CrossRef]
  18. A. Consortini, R. 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]
  19. J. M. Martin, S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
    [CrossRef] [PubMed]
  20. J. M. Martin, S. M. Flatté, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990).
    [CrossRef]
  21. 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]
  22. S. M. Flatté, C. Bracher, G.-Y. Wang, “Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulations,” J. Opt. Soc. Am. A 11, 2080–2092 (1994).
    [CrossRef]
  23. R. J. Hill, R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14, 1530–1540 (1997).
    [CrossRef]
  24. V. I. Tatarskii, V. U. Zavorotnyi, “Wave propagation in random media with fluctuating turbulent parameters,” J. Opt. Soc. Am. A 2, 2069–2076 (1985).
    [CrossRef]
  25. 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]
  26. L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
    [CrossRef]
  27. L. C. Andrews, R. L. Phillips, Laser Propagation through Random Media (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1998).
  28. W. B. Miller, J. C. Ricklin, L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11, 2719–2726 (1994).
    [CrossRef]
  29. E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
    [CrossRef]
  30. E. Jakeman, P. N. Pusey, “The significance of K-distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
    [CrossRef]
  31. E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
    [CrossRef]
  32. L. C. Andrews, R. L. 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]
  33. L. C. Andrews, R. L. Phillips, “Mathematical genesis of the I–K distribution for random optical fields,” J. Opt. Soc. Am. A 3, 1912–1919 (1986).
    [CrossRef]
  34. 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]
  35. 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]
  36. 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]
  37. A. M. Prokhorov, F. V. Bunkin, K. S. Goshelashvili, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
    [CrossRef]
  38. C. Y. Young, L. C. Andrews, “Effects of a modified spectral model on the spatial coherence of a laser beam,” Waves Random Media 4, 385–397 (1994).
    [CrossRef]
  39. B. J. Rickett, W. A. Coles, G. Bourgois, “Slow scintillation in the interstellar medium,” Astron. Astrophys. 134, 390–395 (1984).
  40. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1998) [formerly published as 2nd ed (McGraw-Hill, New York, 1992)].
  41. J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 3.
  42. R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
    [CrossRef]

1997 (1)

1996 (1)

1994 (3)

1993 (2)

1992 (1)

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

1990 (1)

1989 (1)

1988 (1)

1987 (4)

1986 (1)

1985 (2)

1984 (1)

B. J. Rickett, W. A. Coles, G. Bourgois, “Slow scintillation in the interstellar medium,” Astron. Astrophys. 134, 390–395 (1984).

1983 (1)

1982 (1)

1981 (3)

1980 (1)

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

1978 (2)

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

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]

1976 (1)

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

1975 (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Goshelashvili, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

1974 (4)

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

M. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Sov. Phys. JETP 40, 1011–1016 (1974).

H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. 64, 59–67 (1974).
[CrossRef]

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

1973 (1)

1970 (1)

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

1965 (1)

M. E. Gracheva, A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 8, 717–724 (1965).

Andrews, L. C.

W. B. Miller, J. C. Ricklin, L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11, 2719–2726 (1994).
[CrossRef]

C. Y. Young, L. C. Andrews, “Effects of a modified spectral model on the spatial coherence of a laser beam,” Waves Random Media 4, 385–397 (1994).
[CrossRef]

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

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

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

L. C. Andrews, R. L. Phillips, Laser Propagation through Random Media (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1998).

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1998) [formerly published as 2nd ed (McGraw-Hill, New York, 1992)].

Bourgois, G.

B. J. Rickett, W. A. Coles, G. Bourgois, “Slow scintillation in the interstellar medium,” Astron. Astrophys. 134, 390–395 (1984).

Bracher, C.

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Goshelashvili, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium, translated from Russian by R. A. Silverman (McGraw-Hill, New York, 1960).

Churnside, J. H.

Clifford, S. F.

Cochetti, R.

Coles, W. A.

B. J. Rickett, W. A. Coles, G. Bourgois, “Slow scintillation in the interstellar medium,” Astron. Astrophys. 134, 390–395 (1984).

Coles, W. R.

Consortini, A.

Dunphy, J.

Fante, R. L.

Flatté, S. M.

Frehlich, R. G.

Gochelashvili, K. S.

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

Goshelashvili, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Goshelashvili, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Gracheva, M.

M. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Sov. Phys. JETP 40, 1011–1016 (1974).

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 8, 717–724 (1965).

Gurvich, A. S.

M. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Sov. Phys. JETP 40, 1011–1016 (1974).

M. E. Gracheva, A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 8, 717–724 (1965).

Hill, R. J.

Jakeman, E.

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

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

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

Kashkarov, S. S.

M. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Sov. Phys. JETP 40, 1011–1016 (1974).

Kerr, J.

Lawrence, R. S.

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

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

Martin, J.

Martin, J. M.

Miller, W. B.

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]

Phillips, R. L.

Pokasov, V. V.

M. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Sov. Phys. JETP 40, 1011–1016 (1974).

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Goshelashvili, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Pusey, P. N.

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

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

Rickett, B. J.

B. J. Rickett, W. A. Coles, G. Bourgois, “Slow scintillation in the interstellar medium,” Astron. Astrophys. 134, 390–395 (1984).

Ricklin, J. C.

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Goshelashvili, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

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

Strohbehn, J. W.

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

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 3.

Tatarskii, V. I.

V. I. Tatarskii, V. U. Zavorotnyi, “Wave propagation in random media with fluctuating turbulent parameters,” J. Opt. Soc. Am. A 2, 2069–2076 (1985).
[CrossRef]

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

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

Wandzura, S. M.

Wang, G.

Wang, G.-Y.

Young, C. Y.

C. Y. Young, L. C. Andrews, “Effects of a modified spectral model on the spatial coherence of a laser beam,” Waves Random Media 4, 385–397 (1994).
[CrossRef]

Yura, H. T.

Zavorotnyi, V. U.

Appl. Opt. (2)

Astron. Astrophys. (1)

B. J. Rickett, W. A. Coles, G. Bourgois, “Slow scintillation in the interstellar medium,” Astron. Astrophys. 134, 390–395 (1984).

IEEE Trans. Antennas Propag. (1)

E. Jakeman, 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, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 8, 717–724 (1965).

J. Mod. Opt. (1)

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

J. Opt. Soc. Am. (8)

J. Opt. Soc. Am. A (14)

A. Consortini, R. 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. 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); erratum, 4, 1324 (1987).
[CrossRef]

R. G. Frehlich, S. M. Wandzura, R. J. Hill, “Log-amplitude covariance for waves propagating through very strong turbulence,” J. Opt. Soc. Am. A 4, 2158–2161 (1987).
[CrossRef]

W. B. Miller, J. C. Ricklin, L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11, 2719–2726 (1994).
[CrossRef]

J. M. Martin, S. M. 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. 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]

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

R. J. Hill, R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14, 1530–1540 (1997).
[CrossRef]

V. I. Tatarskii, V. U. Zavorotnyi, “Wave propagation in random media with fluctuating turbulent parameters,” J. Opt. Soc. Am. A 2, 2069–2076 (1985).
[CrossRef]

L. C. Andrews, R. L. 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. C. Andrews, R. L. 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]

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]

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]

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]

Phys. Rev. Lett. (1)

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

Proc. IEEE (2)

A. M. Prokhorov, F. V. Bunkin, K. S. Goshelashvili, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

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

Sov. Phys. JETP (2)

M. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations for strong fluctuations of light in a turbulent medium,” Sov. Phys. JETP 40, 1011–1016 (1974).

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

Waves Random Media (1)

C. Y. Young, L. C. Andrews, “Effects of a modified spectral model on the spatial coherence of a laser beam,” Waves Random Media 4, 385–397 (1994).
[CrossRef]

Other (6)

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1998) [formerly published as 2nd ed (McGraw-Hill, New York, 1992)].

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 3.

L. C. Andrews, R. L. Phillips, Laser Propagation through Random Media (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1998).

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

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

L. A. Chernov, Wave Propagation in a Random Medium, translated from Russian by R. A. Silverman (McGraw-Hill, New York, 1960).

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Figures (9)

Fig. 1
Fig. 1

Relative positions of scale sizes in (a) weak and (b) strong turbulence.

Fig. 2
Fig. 2

The solid curves depict the scintillation index for the plane-wave and spherical-wave models in the absence of inner-scale effects. The dashed curves correspond to the asymptotic theory valid for the saturation regime, and the dotted curve is the small-scale scintillation for the spherical-wave model.

Fig. 3
Fig. 3

Scintillation index of a plane wave with inner-scale effects for λ=0.488 µm and Cn2=5×10-13 m-2/3.

Fig. 4
Fig. 4

Same as Fig. 3, but for a spherical wave.

Fig. 5
Fig. 5

Scintillation index of a spherical wave with two values of inner scale, λ=0.488 µm, and a 1-order-of-magnitude change in Cn2.

Fig. 6
Fig. 6

Scintillation index as a function of inner scale with β0=5.48 for the spherical-wave case and σ1=5.48 for the plane-wave case. The dashed curves represent simulation data taken from Ref. 21, and the solid curves are based on the theory presented here.

Fig. 7
Fig. 7

Normalized plane-wave covariance for various strengths of turbulence.

Fig. 8
Fig. 8

Normalized spherical-wave covariance for various strengths of turbulence.

Fig. 9
Fig. 9

Spatial scales of irradiance fluctuations normalized by the Fresnel zone L/k and plotted as a function of the strength of turbulence. The emergence of two scales occurs at the onset of strong fluctuations.

Equations (91)

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σI2=I2I2-1,
σ12=1.23Cn2k7/6L11/6,
Φn(κ)=0.033Cn2κ-11/3f(κl0),
f(κl0)=exp(-κ2/κl2)[1+1.802(κ/κl)-0.254(κ/κl)7/6],κl=3.3/l0.
I2=x2y2=(1+σx2)(1+σy2),
σI2=(1+σx2)(1+σy2)-1=σx2+σy2+σx2σy2.
U(r, L)=U0(r, L)exp[ψ(r, L)],
U(r, L)=U0(r, L)exp[ψx(r, L)+ψy(r, L)],
Φn(κ)=0.033Cn2κ-11/3G(κ, l0),
G(κ, l0)=Gx(κ, l0)+Gy(κ)=f(κl0)exp-κ2κx2+κ11/3(κ2+κy2)11/6.
l1ρ0(spatialcoherenceradius),
l2L/k(Fresnelzonesize),
l3L/kρ0(scatteringdisk),
κ11ρ0,κ2kL1/2,κ3kρ0L.
κxκ2(weakfluctuations)κ3(strongfluctuations),
κyκ2(weakfluctuations)κ1(strongfluctuations).
Lκ12kLkρ02,Lκ22k1,Lκ32kkρ02L.
Lklx2=1κx2=c1Lk+c2Lkρ02,
1ly2=κy2=c3L/k+c4ρ02,
σI2=exp(σln I2)-1σln I2,σ121,
σln I2=8π2k20L0κΦn(κ)1-cosκ2zkdκdz=1.06σ12010η-11/6(1-cos ηξ)dηdξ.
σI2σ12=0.847Lkρ025/6,σ121,
σI21+0.86σ14/5=1+0.919kρ02L1/3,σ121.
σx2=exp(σln x2)-1,σy2=exp(σln y2)-1,
σI2=exp(σln x2+σln y2)-1.
σln x2=8π2k20L0κΦn(κ)Gx(κ)1-cosκ2zkdκdz1.06σ12Lk7/601ξ20κ4/3 exp(-κ2/κx2)dκdξ0.15σ12ηx7/6,
σln y2=1.06σ12010(η+ηy)-11/6(1-cos ηξ)dηdξ1.272σ12ηy-5/6,ηy1,
ηx=1c1+c2L/kρ021/c1,L/kρ021kρ02/c2L,L/kρ021,
ηy=c3+c4L/kρ02c3,L/kρ021c4L/kρ02,L/kρ021.
σI22 exp(σln x2)-11+0.24σ14/51c27/6,σ121,
ηx=31+L/kρ023,L/kρ0213kρ02/L,L/kρ021,
ηy=3+1.7L/kρ023,L/kρ0211.7L/kρ02,L/kρ021.
σln x2=0.54σ12(1+1.22σ112/5)7/60.54σ12,σ1210.43/σ14/5,σ121,
σln y2=0.509σ12(1+0.69σ112/5)5/60.509σ12,σ121ln 2,σ121,
σI2=exp0.54σ12(1+1.22σ112/5)7/6+0.509σ12(1+0.69σ112/5)5/6-1,0σ12<.
σI2=exp(σln I2)-1σln I2,σ121,
σln I2=8π2k20L0κΦn(κ)1-cosκ2kz(1-z/L)×dκdz=1.06σ12010η-11/6{1-cos[ηξ(1-ξ)]}dηdξ,
σ121.
σI2=0.4σ12,σ1211+2.73σ14/5,σ121.
σln x2=1.06σ12010η-11/6×exp(-η/ηx){1-cos[ηξ(1-ξ)]}dηdξ0.015σ12ηx7/6,
σln y2=1.06σ12010(η+ηy)-11/6×{1-cos[ηξ(1-ξ)]}dηdξ1.272σ12ηy-5/6,
ηx=81+0.137L/kρ028,L/kρ02158.4kρ02/L,L/kρ021,
ηy=8+1.7L/kρ028,L/kρ0211.7L/kρ02,L/kρ021.
σln x2=0.17σ12(1+0.167σ112/5)7/60.17σ12,σ1211.37/σ14/5,σ121,
σln y2=0.225σ12(1+0.259σ112/5)5/60.225σ12,σ121ln 2,σ121.
σI2=exp0.17σ12(1+0.167σ112/5)7/6+0.225σ12(1+0.259σ112/5)5/6-1,0σ12<.
σI2(L)3.86σ12(1+1/Ql2)11/12sin116tan-1 Ql+1.507(1+Ql2)1/4sin43tan-1 Ql-0.273(1+Ql2)7/24×sin54tan-1 Ql-3.50Ql-5/6,σ12<1,
σI2(L)3.86σ120.40(1+9/Ql2)11/12sin116tan-1Ql3+2.610(9+Ql2)1/4sin43tan-1Ql3-0.518(9+Ql2)7/24×sin54tan-1Ql3-3.50Ql-5/6,σ12<1,
σI21+2.39(σ12Ql7/6)1/6,σ12Ql7/6100(planewave)1+7.65(σ12Ql7/6)1/6,σ12Ql7/6100(sphericalwave).
σln x2=1.06σ12010η-11/6 exp(-η/Ql-η/ηx)×[1+1.802(η/Ql)1/2-0.254(η/Ql)7/12]×(1-cos ηξ)dηdξ0.15σ12ηx7/6(l0),
ηx(l0)=ηxQlηx+Ql1+1.753ηxηx+Ql1/2-0.252ηxηx+Ql7/126/7.
ηx,p=31+0.49L/kρ02=31+0.50σ12Ql1/6
(planewave),
ηx,s=81+0.068L/kρ02=81+0.069σ12Ql1/6
(sphericalwave),
σln x,p2(l0)=0.15σ12ηx,pQlηx,p+Ql7/6×1+1.753ηx,pηx,p+Ql1/2-0.252ηx,pηx,p+Ql7/12
(planewave),
σln x,s2(l0)=0.015σ12ηx,sQlηx,s+Ql7/6×1+1.753ηx,sηx,s+Ql1/2-0.252ηx,sηx,s+Ql7/12
(sphericalwave).
σI2=expσln x,p2(l0)+0.509σ12(1+0.69σ112/5)5/6-1
(planewave),
σI2=expσln x,s2(l0)+0.225σ12(1+0.259σ112/5)5/6-1
(sphericalwave).
BI(ρ)=Bx(ρ)+By(ρ)+Bx(ρ)By(ρ),
BI(ρ)=exp[Bln I(ρ)]-1=exp[Bln x(ρ)+Bln y(ρ)]-1,
Bln x(ρ)=1.06σ12010η-11/6 exp(-η/ηx)J0(ρkη/L)×(1-cos ηξ)dηdξ0.15σ12ηx7/6 1F176; 1;-kρ2ηx4L,
Bln y(ρ)=1.06σ12010(η+ηy)-11/6J0(ρkη/L)×(1-cos ηξ)dηdξ1.265σ12kρ2Lηy5/12K5/6kρ2ηyL1/2,
Bln x(ρ)=1.06σ12010η-11/6 exp(-η/ηx)J0(ρξkη/L)×{1-cos[ηξ(1-ξ)]}dηdξ0.015σ12ηx7/6×3F376,32, 2;72, 3, 1;-kρ2ηx4L,
Bln y(ρ)=1.06σ12010(η+ηy)-11/6J0(ρξkη/L)×{1-cos[ηξ(1-ξ)]}dηdξ1.272σ12ηy-5/6 1F212;16,32;kρ2ηy4L-0.889σ12(kρ2/L)5/6 1F243;116,73;kρ2ηy4L,
BI(ρ)=expσln x2 1F176; 1;-kρ2ηx4L+0.994σln y2kρ2ηyL5/12K5/6kρ2ηyL1/2-1,
BI(ρ)=expσln x2 3F376,32, 2;72, 3, 1;-kρ2ηx4L+σln y2 1F212;16,32;kρ2ηy4L-0.70σln y2kρ2ηyL5/6×1F243;116,73;kρ2ηy4L-1,
BI(ρ)C0hf(ρ)+C1lf(ρ)+C1hf(ρ)=C0hf(ρ)+Aσ14/5[b1(ρ/l3)+b2(ρ/ρ0)],σ121,
C0hf(ρ)=By(ρ),
C1lf(ρ)=Bx(ρ)Bln x(ρ),
C1hf(ρ)=Bx(ρ)By(ρ)Bln x(ρ)By(ρ),
b1(ρ/l3)=1F176; 1;-3ρ24l32,
b2(ρ/ρ0)=1F176; 1;-3(ρ/ρ0)24(l3/ρ0)2×exp0.86ρρ05/6K5/61.30ρρ0-1.
b1(ρ/l3)=0.73601ξ20u4/3J0(0.658ρu/l3)×exp[-u5/3ξ5/3(1-58ξ)]dudξ.
C0hf(ρ)=exp[-D(ρ)]=exp[-2(ρ/ρ0)5/3],σ12,
By(ρ)=exp0.86ρρ05/6K5/61.30ρρ0-1exp[-2(ρ/ρ0)5/3]+exp[-2(ρ/ρ0)5/3]-1exp[-2(ρ/ρ0)5/3],ρρ0,σ12.
bI(ρ)=Bx(ρ)+By(ρ)+Bx(ρ)By(ρ)σx2+σy2+σx2σy2.
bI(ρ)=1-2.37(kρ2/L)5/6+1.82(kρ2/L)+,l0ρL/k(planewave)1-2.22(kρ2/L)5/6+1.97(kρ2/L)+,l0ρL/k(sphericalwave).
bI(ρ)=1-2.36(kρ2/L)5/6+1.71(kρ2/L)+,l0ρL/k(planewave)1-2.2(kρ2/L)5/6+1.71(kρ2/L)+,l0ρL/k(sphericalwave).
σI2(L)=1+32π2k2L010κΦn(κ)sin2Lκ22kh(ξ, ξ)×exp-01DLκkh(τ, ξ)dτdκdξ,
h(τ, ξ)=τ(1-αξ),τ<ξξ(1-ατ),τ>ξ,
01DLκkh(τ, ξ)dτ=1.02σ12Ql1/6Lκ2kξ21-23ξ,
σI,pl2(L)=1+0.982σ12Ql7/6×01ξ2[1+1.02σ12Ql7/6ξ2(1-23ξ)]7/6dξ+1.802Γ(5/3)Γ(7/6)×01ξ2[1+1.02σ12Ql7/6ξ2(1-23ξ)]5/3dξ-0.254Γ(7/4)Γ(7/6)×01ξ2[1+1.02σ12Ql7/6ξ2(1-23ξ)]7/4dξ.
σI,pl2(L)=1+2.39(σ12Ql7/6)1/6,σ12Ql7/6100.
01DLκkh(τ, ξ)dτ=0.34σ12Ql1/6Lκ2kξ2(1-ξ)2,
σI,sph2(L)=1+0.982σ12Ql7/6×01ξ2(1-ξ)2[1+0.34σ12Ql7/6ξ2(1-ξ)2]7/6dξ+1.802Γ(5/3)Γ(7/6)×01ξ2(1-ξ)2[1+0.34σ12Ql7/6ξ2(1-ξ)2]5/3dξ-0.254Γ(7/4)Γ(7/6)×01ξ2(1-ξ)2[1+0.34σ12Ql7/6ξ2(1-ξ)2]7/4dξ.
σI,sph2(L)=1+7.65(σ12Ql7/6)1/6,σ12Ql7/6100.

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