Abstract

We have calculated intensity spectra and variances for waves emanating from a point source and propagating through extended three-dimensional random media by simulation. Spectra of the medium fluctuations considered were power-law, power-law with inner scale, and Gaussian spectra. The simulations covered the regimes of weak fluctuations and strong focusing, including the peak of the intensity variance and beyond. The intensity variances are substantially larger than both the corresponding results for plane-wave incidence and the theoretical calculations for point sources by other authors. Our simulation results agree reasonably closely with the results of laserpropagation experiments over kilometer-length paths in the atmosphere.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. D. Tappert, “The parabolic approximation method,” in Wave Propagation and Underwater Acoustics, J. B. Keller, J. S. Papadakis, eds. (Springer-Verlag, Berlin, 1977), pp. 224–287.
    [CrossRef]
  2. S. M. Flatté, F. D. Tappert, “Calculation of the effect of internal waves on oceanic sound transmission,” J. Acoust. Soc. Am. 58, 1151–1159 (1975).
    [CrossRef]
  3. D. J. Thomson, N. R. Chapman, “A wide-angle split-step algorithm for the parabolic equation,” J. Acoust. Soc. Am. 74, 1848–1854 (1983).
    [CrossRef]
  4. J. M. Martin, S. Flatté, “Intensity images and statistics from numerical simulation of plane wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
    [CrossRef] [PubMed]
  5. A. M. Whitman, M. J. Beran, “Two-scale solution for atmospheric scintillation from a point source,” J. Opt. Soc. Am A 5, 735–737 (1988).
    [CrossRef]
  6. 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]
  7. J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]
  8. D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
    [CrossRef]
  9. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, TT-68-50464 (National Technical Information Service, Springfield, Va., 1971).
  10. R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
    [CrossRef]
  11. S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).
  12. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).
  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. 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]
  15. R. J. Hill, “Theory of saturation of optical scintillation by strong turbulence: plane-wave variance and covariance and spherical wave covariance,” J. Opt. Soc. Am. 72, 212–222 (1982).
    [CrossRef]
  16. R. G. Frehlich, “Intensity covariance of a point source in a random medium with a Kolmogorov spectrum and an inner scale of turbulence,” J. Opt. Soc. Am. A 4, 360–366 (1987).
    [CrossRef]
  17. R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13, 953–961 (1978).
    [CrossRef]
  18. B. J. Uscinski, “Analytical solution of the fourth-moment equation and interpretation as a set of phase screens,” J. Opt. Soc. Am. A 2, 2077–2091 (1985).
    [CrossRef]
  19. V. R. Rumsey, “Scintillations due to a concentrated layer with a power-law turbulence spectrum,” Radio Sci. 10, 107–114 (1975).
    [CrossRef]
  20. D. P. Hinson, “Strong scintillations during atmospheric occultations: theoretical intensity spectra,” Radio Sci. 21, 257–270 (1986).
    [CrossRef]
  21. R. Frehlich, Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado 80309 (personal communication).
  22. K. S. Goshelashvily, V. I. Shishov, “Saturation of laser irradiance fluctuations beyond a turbulent layer,” Opt. Quantum Electron. 7, 524–536 (1975).
    [CrossRef]
  23. J. M. Martin, Voyager Microwave Scintillation Measurements of Solar Wind Plasma Parameters, Ph.D. dissertation (Stanford University, Stanford, Calif., 1985).
  24. G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979).
    [CrossRef]
  25. R. L. Phillips, L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71, 1440–1445 (1981).
    [CrossRef]
  26. R. G. Frehlich, Laser Propagation in Random Media, Ph.D. dissertation (University of California, San Diego, San Diego, Calif., 1982).
  27. A. M. Whitman, M. J. Beran, “Two-scale solution for atmospheric scintillation,” J. Opt. Soc. Am. A 2, 2133–2143 (1985).
    [CrossRef]
  28. 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]
  29. J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
    [CrossRef]
  30. S. Frankenthal, A. M. Whitman, M. J. Beran, “Two-scale solutions for intensity fluctuations in strong scattering,” J. Opt. Soc. Am. A 1, 585–597 (1984).
    [CrossRef]
  31. C. Macaskill, “An improved solution to the fourth moment equation for intensity fluctuations,” Proc. R. Soc. London Ser. A 386, 461–474 (1983).
    [CrossRef]
  32. A. Furutsu, “Intensity correlation functions of lightwaves in a turbulent medium: an exact version of the two-scale method,” Appl. Opt. 27, 2127–2144 (1988).
    [CrossRef] [PubMed]
  33. M. J. Beran, A. M. Whitman, “Effect of the turbulence inner scale on scintillation in the atmosphere,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1115, 2–10 (1989).
    [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. R. G. Frehlich, “Estimation of the parameters of the atmospheric turbulence spectrum using measurements of the spatial intensity covariance,” J. Opt. Soc. Am. A 5, 1–50 (1988).
  36. R. J. Hill, J. H. Churnside, “Observational challenges of strong scintillations of irradiance,” J. Opt. Soc. Am. A 3, 445–447 (1988).
    [CrossRef]
  37. G. R. Ochs, R. J. Hill, “Optical-scintillation method of measuring turbulence inner scale,” Appl. Opt. 24, 2430–2432 (1985).
    [CrossRef] [PubMed]

1988 (5)

A. M. Whitman, M. J. Beran, “Two-scale solution for atmospheric scintillation from a point source,” J. Opt. Soc. Am A 5, 735–737 (1988).
[CrossRef]

R. G. Frehlich, “Estimation of the parameters of the atmospheric turbulence spectrum using measurements of the spatial intensity covariance,” J. Opt. Soc. Am. A 5, 1–50 (1988).

R. J. Hill, J. H. Churnside, “Observational challenges of strong scintillations of irradiance,” J. Opt. Soc. Am. A 3, 445–447 (1988).
[CrossRef]

A. Furutsu, “Intensity correlation functions of lightwaves in a turbulent medium: an exact version of the two-scale method,” Appl. Opt. 27, 2127–2144 (1988).
[CrossRef] [PubMed]

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

1987 (3)

1986 (2)

D. P. Hinson, “Strong scintillations during atmospheric occultations: theoretical intensity spectra,” Radio Sci. 21, 257–270 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

1985 (3)

1984 (1)

1983 (4)

C. Macaskill, “An improved solution to the fourth moment equation for intensity fluctuations,” Proc. R. Soc. London Ser. A 386, 461–474 (1983).
[CrossRef]

D. J. Thomson, N. R. Chapman, “A wide-angle split-step algorithm for the parabolic equation,” J. Acoust. Soc. Am. 74, 1848–1854 (1983).
[CrossRef]

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

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]

1982 (2)

1981 (2)

1979 (2)

1978 (1)

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13, 953–961 (1978).
[CrossRef]

1976 (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

1975 (3)

S. M. Flatté, F. D. Tappert, “Calculation of the effect of internal waves on oceanic sound transmission,” J. Acoust. Soc. Am. 58, 1151–1159 (1975).
[CrossRef]

V. R. Rumsey, “Scintillations due to a concentrated layer with a power-law turbulence spectrum,” Radio Sci. 10, 107–114 (1975).
[CrossRef]

K. S. Goshelashvily, V. I. Shishov, “Saturation of laser irradiance fluctuations beyond a turbulent layer,” Opt. Quantum Electron. 7, 524–536 (1975).
[CrossRef]

Andrews, L. C.

Beran, M. J.

A. M. Whitman, M. J. Beran, “Two-scale solution for atmospheric scintillation from a point source,” J. Opt. Soc. Am A 5, 735–737 (1988).
[CrossRef]

A. M. Whitman, M. J. Beran, “Two-scale solution for atmospheric scintillation,” J. Opt. Soc. Am. A 2, 2133–2143 (1985).
[CrossRef]

S. Frankenthal, A. M. Whitman, M. J. Beran, “Two-scale solutions for intensity fluctuations in strong scattering,” J. Opt. Soc. Am. A 1, 585–597 (1984).
[CrossRef]

M. J. Beran, A. M. Whitman, “Effect of the turbulence inner scale on scintillation in the atmosphere,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1115, 2–10 (1989).
[CrossRef]

Chapman, N. R.

D. J. Thomson, N. R. Chapman, “A wide-angle split-step algorithm for the parabolic equation,” J. Acoust. Soc. Am. 74, 1848–1854 (1983).
[CrossRef]

Churnside, J. H.

R. J. Hill, J. H. Churnside, “Observational challenges of strong scintillations of irradiance,” J. Opt. Soc. Am. A 3, 445–447 (1988).
[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]

Clifford, S. F.

Codona, J. L.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

Coles, W. R.

Creamer, D. B.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

Dashen, R.

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

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).

Fante, R. L.

Feit, M. D.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Flatté, S.

Flatté, S. M.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

S. M. Flatté, F. D. Tappert, “Calculation of the effect of internal waves on oceanic sound transmission,” J. Acoust. Soc. Am. 58, 1151–1159 (1975).
[CrossRef]

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).

Fleck, J. A.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Frankenthal, S.

Frehlich, R.

R. Frehlich, Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado 80309 (personal communication).

Frehlich, R. G.

R. G. Frehlich, “Estimation of the parameters of the atmospheric turbulence spectrum using measurements of the spatial intensity covariance,” J. Opt. Soc. Am. A 5, 1–50 (1988).

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]

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

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

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]

R. G. Frehlich, Laser Propagation in Random Media, Ph.D. dissertation (University of California, San Diego, San Diego, Calif., 1982).

Furutsu, A.

Goshelashvily, K. S.

K. S. Goshelashvily, V. I. Shishov, “Saturation of laser irradiance fluctuations beyond a turbulent layer,” Opt. Quantum Electron. 7, 524–536 (1975).
[CrossRef]

Henyey, F. S.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

Hill, R. J.

Hinson, D. P.

D. P. Hinson, “Strong scintillations during atmospheric occultations: theoretical intensity spectra,” Radio Sci. 21, 257–270 (1986).
[CrossRef]

Knepp, D. L.

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

Macaskill, C.

C. Macaskill, “An improved solution to the fourth moment equation for intensity fluctuations,” Proc. R. Soc. London Ser. A 386, 461–474 (1983).
[CrossRef]

Martin, J. M.

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

J. M. Martin, Voyager Microwave Scintillation Measurements of Solar Wind Plasma Parameters, Ph.D. dissertation (Stanford University, Stanford, Calif., 1985).

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Munk, W. H.

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).

Ochs, G. R.

Parry, G.

Phillips, R. L.

Pusey, P. N.

Rumsey, V. R.

V. R. Rumsey, “Scintillations due to a concentrated layer with a power-law turbulence spectrum,” Radio Sci. 10, 107–114 (1975).
[CrossRef]

Shishov, V. I.

K. S. Goshelashvily, V. I. Shishov, “Saturation of laser irradiance fluctuations beyond a turbulent layer,” Opt. Quantum Electron. 7, 524–536 (1975).
[CrossRef]

Tappert, F. D.

S. M. Flatté, F. D. Tappert, “Calculation of the effect of internal waves on oceanic sound transmission,” J. Acoust. Soc. Am. 58, 1151–1159 (1975).
[CrossRef]

F. D. Tappert, “The parabolic approximation method,” in Wave Propagation and Underwater Acoustics, J. B. Keller, J. S. Papadakis, eds. (Springer-Verlag, Berlin, 1977), pp. 224–287.
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, TT-68-50464 (National Technical Information Service, Springfield, Va., 1971).

Thomson, D. J.

D. J. Thomson, N. R. Chapman, “A wide-angle split-step algorithm for the parabolic equation,” J. Acoust. Soc. Am. 74, 1848–1854 (1983).
[CrossRef]

Uscinski, B. J.

Wandzura, S. M.

Watson, K. M.

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).

Whitman, A. M.

A. M. Whitman, M. J. Beran, “Two-scale solution for atmospheric scintillation from a point source,” J. Opt. Soc. Am A 5, 735–737 (1988).
[CrossRef]

A. M. Whitman, M. J. Beran, “Two-scale solution for atmospheric scintillation,” J. Opt. Soc. Am. A 2, 2133–2143 (1985).
[CrossRef]

S. Frankenthal, A. M. Whitman, M. J. Beran, “Two-scale solutions for intensity fluctuations in strong scattering,” J. Opt. Soc. Am. A 1, 585–597 (1984).
[CrossRef]

M. J. Beran, A. M. Whitman, “Effect of the turbulence inner scale on scintillation in the atmosphere,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1115, 2–10 (1989).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).

Zachariasen, F.

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).

Appl. Opt. (3)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

J. Acoust. Soc. Am. (2)

S. M. Flatté, F. D. Tappert, “Calculation of the effect of internal waves on oceanic sound transmission,” J. Acoust. Soc. Am. 58, 1151–1159 (1975).
[CrossRef]

D. J. Thomson, N. R. Chapman, “A wide-angle split-step algorithm for the parabolic equation,” J. Acoust. Soc. Am. 74, 1848–1854 (1983).
[CrossRef]

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 A (1)

A. M. Whitman, M. J. Beran, “Two-scale solution for atmospheric scintillation from a point source,” J. Opt. Soc. Am A 5, 735–737 (1988).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (8)

Opt. Quantum Electron. (1)

K. S. Goshelashvily, V. I. Shishov, “Saturation of laser irradiance fluctuations beyond a turbulent layer,” Opt. Quantum Electron. 7, 524–536 (1975).
[CrossRef]

Proc. IEEE (1)

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

C. Macaskill, “An improved solution to the fourth moment equation for intensity fluctuations,” Proc. R. Soc. London Ser. A 386, 461–474 (1983).
[CrossRef]

Radio Sci. (4)

V. R. Rumsey, “Scintillations due to a concentrated layer with a power-law turbulence spectrum,” Radio Sci. 10, 107–114 (1975).
[CrossRef]

D. P. Hinson, “Strong scintillations during atmospheric occultations: theoretical intensity spectra,” Radio Sci. 21, 257–270 (1986).
[CrossRef]

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13, 953–961 (1978).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

Other (8)

F. D. Tappert, “The parabolic approximation method,” in Wave Propagation and Underwater Acoustics, J. B. Keller, J. S. Papadakis, eds. (Springer-Verlag, Berlin, 1977), pp. 224–287.
[CrossRef]

R. G. Frehlich, Laser Propagation in Random Media, Ph.D. dissertation (University of California, San Diego, San Diego, Calif., 1982).

J. M. Martin, Voyager Microwave Scintillation Measurements of Solar Wind Plasma Parameters, Ph.D. dissertation (Stanford University, Stanford, Calif., 1985).

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission through a Fluctuating Ocean (Cambridge U. Press, Cambridge, 1979).

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).

R. Frehlich, Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado 80309 (personal communication).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, TT-68-50464 (National Technical Information Service, Springfield, Va., 1971).

M. J. Beran, A. M. Whitman, “Effect of the turbulence inner scale on scintillation in the atmosphere,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1115, 2–10 (1989).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Intensity profiles through the center of a Gaussian beam with σ = 1 and Rf = 10: a, undiverged; b, diverged by a lens with x0 = 1.

Fig. 2
Fig. 2

a, Comparison of the U = 0.1 simulation (solid curve) with the Rytov calculation (dashed curve). Disagreement for κ > 2 is due to point-source structure artifacts. b, Comparison of intensity spectra generated by simulations (solid curves) versus low-frequency (long-dashed curves) and high-frequency (short-dashed curves) asymptotic theory of Frehlich19 for the point source, α = 1.7, and U = 0.1, 1.0, 10.0. For clarity, the U = 0.1 and U = 10.0 curves have been offset by − 10 and +10 dB, respectively. The normalization of wave number κ is such that κ = 1 corresponds to 1/Δ, where Δ is the grid sample size. Since normalized Rf = 10, then κ = 0.1 corresponds to 1/Rf.

Fig. 3
Fig. 3

Intensity variance σI2 versus U for pure power law α = 1.7. Point-source results are shown with error bars of ±1 σ, and plane-wave results are shown without error bars, as they are negligible. U is varied by changing the strength of medium fluctuations and keeping the geometry constant.

Fig. 4
Fig. 4

Intensity spectra for pure power law α = 1.7; point source (solid curves) versus plane wave (dashed curves) for a, U = 1.0; b, U = 5.0 (corresponding to peak value of σI2); c, U = 10.0.

Fig. 5
Fig. 5

Intensity variance for power-law medium with exponent α = 1.7 and inner scales of 0.17.Rf, 0.33Rf, and 0.50Rf. The error bars show 1σ uncertainties.

Fig. 6
Fig. 6

Intensity spectra for α = 1.7 and inner scales: a, 0.17Rf; b, 0.50Rf. The spectra for U = 1.0, 6.0 (dashed curves), and 15.0 (solid curves) are shown. The flattening of the spectra above κ ≈ 1 is due to point-source structure artifacts and does not reflect the true form of the intensity spectrum in that band.

Fig. 7
Fig. 7

Point-source intensity variance for a Gaussian medium with scale L = 1.6Rf. The error bars show 1σ uncertainties. Note the large values of intensity variance that are obtained owing to the intense focusing that is possible in such a smoothly varying medium.

Fig. 8
Fig. 8

Intensity spatial spectra for Gaussian medium with L = 1.6Rf at U = 1.0, 6.0, 15.0. The flattening of the spectra below a strength of ∼10−3 is a source-structure artifact, not the true spectral form.

Fig. 9
Fig. 9

Point-source intensity variance for power-law media with α = 1.7. The results for media with either pure power-law or inner-scale β = 0.5Rf media are shown. Simulation results (solid curves) are compared with the two-scale calculation of Whitman and Beran5 (dashed curves). This comparison yields more than a factor-of-2 difference for (β/Rf = 0.5. ζ k is a measure of the scattering strength related to σI2 as described in the text.

Fig. 10
Fig. 10

Intensity variance results from Fig. 5 rescaled as functions of the point-source wave-structure function Ds(Rf) for comparison with the measurements of Coles and Frehlich.6 The simulation results for β/Rf = 0.33, which would correspond to a measured inner scale r* of 0.74 cm, agree well with the data.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

2 i k x ψ + 2 ψ + 2 k 2 n 1 ψ = 0 ,
ψ ( x + δ , y , z ) = 1 ( exp ( i A δ ) { exp [ i θ ( y , z ) ] ψ ( x , y , z ) } ) ,
θ j = k x j δ / 2 x j + δ / 2 n 1 ( u , y , z ) d u .
W ( κ ) = 1 Δ 2 W ( κ / Δ ) ,
ψ ( 0 , y , z ) = exp ( ρ 2 2 σ 2 ) exp ( i ρ 2 2 x 0 2 ) ,
Im ( θ ) = { a 1 exp [ 1 2 ( ρ ρ e a 2 ) 2 ] if ρ ρ e a 1 if ρ > ρ e ,
[ n ( r ) n ( 0 ) ] 2 = C n 2 r α 1 .
Φ n ( κ ) = K ( α ) C n 2 κ α 2 ,
K ( α ) = Γ ( α + 1 ) 4 π 2 sin [ ( α 1 ) π 2 ] .
Φ n ( κ ) = K ( α ) C n 2 κ α 2 exp ( κ 2 β 2 / 4 )
Φ n ( κ ) = C n 2 exp ( κ 2 L 2 / 4 ) ,
σ I 2 I 2 I 2 1
U = 8 π k 2 0 R d x Φ n ( κ x = 0 , κ ; x ) sin 2 κ 2 L F 2 ( x ) 2 d 2 κ ,
L F ( x ) = { R f [ x R ( 1 x R ) ] 1 / 2 point source R f ( 1 x R ) 1 / 2 plane wave
σ 1 2 = 1.23 C n 2 k 7 / 6 R 11 / 6 .
ζ k = 1.23 k 7 / 11 ( C n 2 ) 6 / 11 R = 1.096 ( σ 1 2 ) 6 / 11 .
d ( ρ ) = 4 π k 2 ( 1 cos ρ · κ ) Φ n ( κ x = 0 , κ ) d 2 κ
D s ( ρ ) = 0 R d ( ρ x R ) d x .
ρ 0 = f ( α ) ( k 2 C n 2 R ) 1 / α .
D s ( R f ) = F ( α , β / R f ) U .
ζ k = 1.17 [ D s ( R f ) ] 6 / 11 .
Φ I h f ( κ ) = 1 ( 2 π ) 2 exp [ 0 R d ( ρ x R ) d x ] exp ( i ρ · κ ) d 2 ρ
Φ I l f ( κ ) = 8 π k 2 0 R ( R x ) 2 Φ n ( κ R x ) sin 2 [ κ 2 R f 2 ( R x ) 2 x ] × exp [ 0 R d [ κ R f 2 x h ( x , x ) ] d x ] d x ,
h ( x , x ) = { x ( x R ) / R x < x x ( x R ) / R x > x .
κ max = 10 1 / ( α + 2 ) ρ 0 1 , κ min = ρ 0 2 R f 2 .
R κ = 2 × 10 1 / ( α + 2 ) R f 2 ρ 0 2 = 2 × 10 1 / ( α + 2 ) [ U Γ ( 1 + α ) 2 α cos π α 4 Γ 2 ( 1 + α 2 ) ] 2 / α ,

Metrics