Abstract

An extended random medium is modeled by a set of 2-D thin Gaussian phase-changing screens with phase power spectral densities appropriate to the natural medium being modeled. Details of the algorithm and limitations on its application to experimental conditions are discussed, concentrating on power-law spectra describing refractive-index fluctuations of the neutral atmosphere. Inner and outer scale effects on intensity scintillation spectra and intensity variance are also included. Images of single realizations of the intensity field at the observing plane are presented, showing that under weak scattering the small-scale Fresnel length structure of the medium dominates the intensity scattering pattern. As the strength of scattering increases, caustics and interference fringes around focal regions begin to form. Finally, in still stronger scatter, the clustering of bright regions begins to reflect the large-scale structure of the medium. For plane waves incident on the medium, physically reasonable inner scales do not produce the large values of intensity variance observed in the focusing region during laser propagation experiments over kilometer paths in the atmosphere. Values as large as experimental observations have been produced in the simulations, but they require inner scales of the order of 10 cm. Inclusion of an outer scale depresses the low-frequency end of the intensity spectrum and reduces the maximum of the intensity variance. Increasing the steepness of the power law also slightly increases the maximum value of intensity variance.

© 1988 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. I. Tatarski, “The Effects of the Turbulent Atmosphere on Wave Propagation,” National Technical Information Service, TT-68-50464 (1971).
  2. J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, F. S. Henyey, “Solution for the Fourth Moment of Waves Propagating in Random Media,” Radio Sci. 21, 929 (1986).
    [CrossRef]
  3. R. L. Fante, “Electric Field Spectrum and Intensity Covariance of a Wave in a Random Medium,” Radio Sci. 10, 77 (1975).
    [CrossRef]
  4. C. Macaskill, “An Improved Solution to the Fourth Moment Equation for Intensity Fluctuations,” Proc. R. Soc. London Ser. A 386, 461 (1983).
    [CrossRef]
  5. M. Whitman, M. J. Beran, “Two-Scale Solution for Atmospheric Scintillation,” J. Opt. Soc. Am. A 2, 2133 (1985).
    [CrossRef]
  6. S. M. Flatte, F. D. Tappert, “Calculation of the Effect of Internal Waves on Oceanic Sound Transmission,” J. Acoust. Soc. Am. 58, 1151 (1975).
    [CrossRef]
  7. C. Macaskill, T. E. Ewart, “Computer Simulation of Two-Dimensional Random Wave Propagation,” IMA J. Appl. Math. 33, 1 (1984).
    [CrossRef]
  8. R. Buckley, “Diffraction by a Random Phase-Changing Screen: A Numerical Experiment,” J. Atmos. Terr. Phys. 37, 1431 (1975).
    [CrossRef]
  9. C. L. Rino, J. Owen, “Numerical Simulations of Intensity Scintillation Using the Power Law Phase Screen Model,” Radio Sci. 15, 41 (1980).
    [CrossRef]
  10. J. P. Filice, “Studies of the Microscale Density Fluctuations in the Solar Wind Using Interplanetary Scintillations,” U. California, San Diego, Ph.D. Thesis (1984).
  11. H. G. Booker, J. A. Ferguson, H. O. Vats, “Comparison between the Extended-Medium and the Phase-Screen Scintillation Theories,” J. Atmos. Terr. Phys. 47, 381 (1985).
    [CrossRef]
  12. V. A. Fock, “Theory of Radio-Wave Propagation in an Inhomogeneous Atmosphere for a Raised Source,” Bull. Acad. Sci. URSS Ser. Phys. 14, 70 (1950).
  13. S. M. Flatte, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission Through a Fluctuating Ocean (Cambridge U.P., London, 1979).
  14. R. Dashen, “Path Integrals for Waves in Random Media,” J. Math. Phys. 20, 894 (1979).
    [CrossRef]
  15. 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 (1985).
    [CrossRef]
  16. D. L. Knepp, “Multiple Phase-Screen Calculation of the Tem poral Behavior of Stochastic Waves,” Proc. IEEE 71, 722 (1983).
    [CrossRef]
  17. R. L. Fante, “Inner-Scale Size Effect on the Scintillations of Light in the Turbulent Atmosphere,” J. Opt. Soc. Am. 73, 277 (1983).
    [CrossRef]
  18. V. R. Rumsey, “Scintillations due to a Concentrated Layer with a Power-Law Turbulence Spectrum,” Radio Sci. 10, 107 (1975).
    [CrossRef]
  19. K. S. Goshelashvily, V. I. Shishov, “Saturated Fluctuations in the Laser Radiation Intensity in a Turbulent Medium,” Sov. Phys. JETP 39, 605 (1974).
  20. 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 (1987).
    [CrossRef]
  21. R. Esswein, S. M. Flatte, “Calculation of the Phase-Structure Function Density from Oceanic Internal Waves,” J. Acoust. Soc. Am. 70, 1387 (1981).
    [CrossRef]
  22. R. G. Frehlich, “Laser Propagation in Random Media,” U. California, San Diego, Ph.D. Thesis (1982).
  23. K. S. Goshelashvily, V. I. Shishov, “Saturation of Laser Irradiance Fluctuations beyond a Turbulent Layer,” Opt. Quantum Electron. 7, 524 (1975).
    [CrossRef]
  24. D. P. Hinson, “Strong Scintillations during Atmospheric Occupations: Theoretical Intensity Spectra,” Radio Sci. 21, 257 (1986).
    [CrossRef]
  25. R. Woo, J. W. Armstrong, “Spacecraft Radio Scattering Observations of the Power Spectrum of Electron Density Fluctuations in the Solar Wind,” J. Geophys. Res. 84, 7288 (1979).
    [CrossRef]
  26. 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 (1981).
    [CrossRef]
  27. 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 (1978).
    [CrossRef]
  28. 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 (1982).
    [CrossRef]
  29. W. R. Coles, R. G. Frehlich, “Simultaneous Measurements of Angular Scattering and Intensity Scintillation in the Atmosphere,” J. Opt. Soc. Am. 72, 1042 (1982).
    [CrossRef]
  30. R. L. Phillips, L. C. Andrews, “Measured Statistics of Laser-Light Scattering in Atmospheric Turbulence,” J. Opt. Soc. Am. 71, 1440 (1981).
    [CrossRef]
  31. J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).
  32. C. L. Rino, “A Power Law Phase Screen Model for Ionospheric Scintillation 1. Weak Scatter,” Radio Sci. 14, 1135 (1979).
    [CrossRef]
  33. B. J. Uscinski, C. Macaskill, M. Spivack, “Path Integrals for Wave Intensity Fluctuations in Random Media,” J. Sound Vib. 106, 509 (1986).
    [CrossRef]
  34. E. E. Salpeter, “Interplanetary Scintillations. I. Theory,” Astrophys. J. 147, 433 (1967).
    [CrossRef]
  35. S. Frankenthal, A. M. Whitman, M. J. Beran, “Two-Scale Solutions for Intensity Fluctuations in Strong Scattering,” J. Opt. Soc. Am. A 1, 585 (1984).
    [CrossRef]

1987 (1)

1986 (3)

B. J. Uscinski, C. Macaskill, M. Spivack, “Path Integrals for Wave Intensity Fluctuations in Random Media,” J. Sound Vib. 106, 509 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, F. S. Henyey, “Solution for the Fourth Moment of Waves Propagating in Random Media,” Radio Sci. 21, 929 (1986).
[CrossRef]

D. P. Hinson, “Strong Scintillations during Atmospheric Occupations: Theoretical Intensity Spectra,” Radio Sci. 21, 257 (1986).
[CrossRef]

1985 (3)

1984 (2)

S. Frankenthal, A. M. Whitman, M. J. Beran, “Two-Scale Solutions for Intensity Fluctuations in Strong Scattering,” J. Opt. Soc. Am. A 1, 585 (1984).
[CrossRef]

C. Macaskill, T. E. Ewart, “Computer Simulation of Two-Dimensional Random Wave Propagation,” IMA J. Appl. Math. 33, 1 (1984).
[CrossRef]

1983 (3)

C. Macaskill, “An Improved Solution to the Fourth Moment Equation for Intensity Fluctuations,” Proc. R. Soc. London Ser. A 386, 461 (1983).
[CrossRef]

D. L. Knepp, “Multiple Phase-Screen Calculation of the Tem poral Behavior of Stochastic Waves,” Proc. IEEE 71, 722 (1983).
[CrossRef]

R. L. Fante, “Inner-Scale Size Effect on the Scintillations of Light in the Turbulent Atmosphere,” J. Opt. Soc. Am. 73, 277 (1983).
[CrossRef]

1982 (2)

1981 (3)

1980 (1)

C. L. Rino, J. Owen, “Numerical Simulations of Intensity Scintillation Using the Power Law Phase Screen Model,” Radio Sci. 15, 41 (1980).
[CrossRef]

1979 (3)

R. Woo, J. W. Armstrong, “Spacecraft Radio Scattering Observations of the Power Spectrum of Electron Density Fluctuations in the Solar Wind,” J. Geophys. Res. 84, 7288 (1979).
[CrossRef]

C. L. Rino, “A Power Law Phase Screen Model for Ionospheric Scintillation 1. Weak Scatter,” Radio Sci. 14, 1135 (1979).
[CrossRef]

R. Dashen, “Path Integrals for Waves in Random Media,” J. Math. Phys. 20, 894 (1979).
[CrossRef]

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 (1978).
[CrossRef]

1975 (5)

K. S. Goshelashvily, V. I. Shishov, “Saturation of Laser Irradiance Fluctuations beyond a Turbulent Layer,” Opt. Quantum Electron. 7, 524 (1975).
[CrossRef]

V. R. Rumsey, “Scintillations due to a Concentrated Layer with a Power-Law Turbulence Spectrum,” Radio Sci. 10, 107 (1975).
[CrossRef]

S. M. Flatte, F. D. Tappert, “Calculation of the Effect of Internal Waves on Oceanic Sound Transmission,” J. Acoust. Soc. Am. 58, 1151 (1975).
[CrossRef]

R. Buckley, “Diffraction by a Random Phase-Changing Screen: A Numerical Experiment,” J. Atmos. Terr. Phys. 37, 1431 (1975).
[CrossRef]

R. L. Fante, “Electric Field Spectrum and Intensity Covariance of a Wave in a Random Medium,” Radio Sci. 10, 77 (1975).
[CrossRef]

1974 (1)

K. S. Goshelashvily, V. I. Shishov, “Saturated Fluctuations in the Laser Radiation Intensity in a Turbulent Medium,” Sov. Phys. JETP 39, 605 (1974).

1967 (1)

E. E. Salpeter, “Interplanetary Scintillations. I. Theory,” Astrophys. J. 147, 433 (1967).
[CrossRef]

1950 (1)

V. A. Fock, “Theory of Radio-Wave Propagation in an Inhomogeneous Atmosphere for a Raised Source,” Bull. Acad. Sci. URSS Ser. Phys. 14, 70 (1950).

Andrews, L. C.

Armstrong, J. W.

R. Woo, J. W. Armstrong, “Spacecraft Radio Scattering Observations of the Power Spectrum of Electron Density Fluctuations in the Solar Wind,” J. Geophys. Res. 84, 7288 (1979).
[CrossRef]

Beran, M. J.

Booker, H. G.

H. G. Booker, J. A. Ferguson, H. O. Vats, “Comparison between the Extended-Medium and the Phase-Screen Scintillation Theories,” J. Atmos. Terr. Phys. 47, 381 (1985).
[CrossRef]

Buckley, R.

R. Buckley, “Diffraction by a Random Phase-Changing Screen: A Numerical Experiment,” J. Atmos. Terr. Phys. 37, 1431 (1975).
[CrossRef]

Clifford, S. F.

Codona, J. L.

J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, F. S. Henyey, “Solution for the Fourth Moment of Waves Propagating in Random Media,” Radio Sci. 21, 929 (1986).
[CrossRef]

Coles, W. R.

Creamer, D. B.

J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, F. S. Henyey, “Solution for the Fourth Moment of Waves Propagating in Random Media,” Radio Sci. 21, 929 (1986).
[CrossRef]

Dashen, R.

R. Dashen, “Path Integrals for Waves in Random Media,” J. Math. Phys. 20, 894 (1979).
[CrossRef]

S. M. Flatte, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission Through a Fluctuating Ocean (Cambridge U.P., London, 1979).

Esswein, R.

R. Esswein, S. M. Flatte, “Calculation of the Phase-Structure Function Density from Oceanic Internal Waves,” J. Acoust. Soc. Am. 70, 1387 (1981).
[CrossRef]

Ewart, T. E.

C. Macaskill, T. E. Ewart, “Computer Simulation of Two-Dimensional Random Wave Propagation,” IMA J. Appl. Math. 33, 1 (1984).
[CrossRef]

Fante, R. L.

R. L. Fante, “Inner-Scale Size Effect on the Scintillations of Light in the Turbulent Atmosphere,” J. Opt. Soc. Am. 73, 277 (1983).
[CrossRef]

R. L. Fante, “Electric Field Spectrum and Intensity Covariance of a Wave in a Random Medium,” Radio Sci. 10, 77 (1975).
[CrossRef]

Ferguson, J. A.

H. G. Booker, J. A. Ferguson, H. O. Vats, “Comparison between the Extended-Medium and the Phase-Screen Scintillation Theories,” J. Atmos. Terr. Phys. 47, 381 (1985).
[CrossRef]

Filice, J. P.

J. P. Filice, “Studies of the Microscale Density Fluctuations in the Solar Wind Using Interplanetary Scintillations,” U. California, San Diego, Ph.D. Thesis (1984).

Flatte, S. M.

J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, F. S. Henyey, “Solution for the Fourth Moment of Waves Propagating in Random Media,” Radio Sci. 21, 929 (1986).
[CrossRef]

R. Esswein, S. M. Flatte, “Calculation of the Phase-Structure Function Density from Oceanic Internal Waves,” J. Acoust. Soc. Am. 70, 1387 (1981).
[CrossRef]

S. M. Flatte, F. D. Tappert, “Calculation of the Effect of Internal Waves on Oceanic Sound Transmission,” J. Acoust. Soc. Am. 58, 1151 (1975).
[CrossRef]

S. M. Flatte, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission Through a Fluctuating Ocean (Cambridge U.P., London, 1979).

Fock, V. A.

V. A. Fock, “Theory of Radio-Wave Propagation in an Inhomogeneous Atmosphere for a Raised Source,” Bull. Acad. Sci. URSS Ser. Phys. 14, 70 (1950).

Frankenthal, S.

Frehlich, R. G.

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 (1987).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, F. S. Henyey, “Solution for the Fourth Moment of Waves Propagating in Random Media,” Radio Sci. 21, 929 (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 (1982).
[CrossRef]

R. G. Frehlich, “Laser Propagation in Random Media,” U. California, San Diego, Ph.D. Thesis (1982).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).

Goshelashvily, K. S.

K. S. Goshelashvily, V. I. Shishov, “Saturation of Laser Irradiance Fluctuations beyond a Turbulent Layer,” Opt. Quantum Electron. 7, 524 (1975).
[CrossRef]

K. S. Goshelashvily, V. I. Shishov, “Saturated Fluctuations in the Laser Radiation Intensity in a Turbulent Medium,” Sov. Phys. JETP 39, 605 (1974).

Henyey, F. S.

J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, F. S. Henyey, “Solution for the Fourth Moment of Waves Propagating in Random Media,” Radio Sci. 21, 929 (1986).
[CrossRef]

Hill, R. J.

Hinson, D. P.

D. P. Hinson, “Strong Scintillations during Atmospheric Occupations: Theoretical Intensity Spectra,” Radio Sci. 21, 257 (1986).
[CrossRef]

Knepp, D. L.

D. L. Knepp, “Multiple Phase-Screen Calculation of the Tem poral Behavior of Stochastic Waves,” Proc. IEEE 71, 722 (1983).
[CrossRef]

Macaskill, C.

B. J. Uscinski, C. Macaskill, M. Spivack, “Path Integrals for Wave Intensity Fluctuations in Random Media,” J. Sound Vib. 106, 509 (1986).
[CrossRef]

C. Macaskill, T. E. Ewart, “Computer Simulation of Two-Dimensional Random Wave Propagation,” IMA J. Appl. Math. 33, 1 (1984).
[CrossRef]

C. Macaskill, “An Improved Solution to the Fourth Moment Equation for Intensity Fluctuations,” Proc. R. Soc. London Ser. A 386, 461 (1983).
[CrossRef]

Munk, W. H.

S. M. Flatte, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission Through a Fluctuating Ocean (Cambridge U.P., London, 1979).

Owen, J.

C. L. Rino, J. Owen, “Numerical Simulations of Intensity Scintillation Using the Power Law Phase Screen Model,” Radio Sci. 15, 41 (1980).
[CrossRef]

Phillips, R. L.

Rino, C. L.

C. L. Rino, J. Owen, “Numerical Simulations of Intensity Scintillation Using the Power Law Phase Screen Model,” Radio Sci. 15, 41 (1980).
[CrossRef]

C. L. Rino, “A Power Law Phase Screen Model for Ionospheric Scintillation 1. Weak Scatter,” Radio Sci. 14, 1135 (1979).
[CrossRef]

Rumsey, V. R.

V. R. Rumsey, “Scintillations due to a Concentrated Layer with a Power-Law Turbulence Spectrum,” Radio Sci. 10, 107 (1975).
[CrossRef]

Salpeter, E. E.

E. E. Salpeter, “Interplanetary Scintillations. I. Theory,” Astrophys. J. 147, 433 (1967).
[CrossRef]

Shishov, V. I.

K. S. Goshelashvily, V. I. Shishov, “Saturation of Laser Irradiance Fluctuations beyond a Turbulent Layer,” Opt. Quantum Electron. 7, 524 (1975).
[CrossRef]

K. S. Goshelashvily, V. I. Shishov, “Saturated Fluctuations in the Laser Radiation Intensity in a Turbulent Medium,” Sov. Phys. JETP 39, 605 (1974).

Spivack, M.

B. J. Uscinski, C. Macaskill, M. Spivack, “Path Integrals for Wave Intensity Fluctuations in Random Media,” J. Sound Vib. 106, 509 (1986).
[CrossRef]

Tappert, F. D.

S. M. Flatte, F. D. Tappert, “Calculation of the Effect of Internal Waves on Oceanic Sound Transmission,” J. Acoust. Soc. Am. 58, 1151 (1975).
[CrossRef]

Tatarski, V. I.

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

Uscinski, B. J.

B. J. Uscinski, C. Macaskill, M. Spivack, “Path Integrals for Wave Intensity Fluctuations in Random Media,” J. Sound Vib. 106, 509 (1986).
[CrossRef]

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 (1985).
[CrossRef]

Vats, H. O.

H. G. Booker, J. A. Ferguson, H. O. Vats, “Comparison between the Extended-Medium and the Phase-Screen Scintillation Theories,” J. Atmos. Terr. Phys. 47, 381 (1985).
[CrossRef]

Watson, K. M.

S. M. Flatte, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission Through a Fluctuating Ocean (Cambridge U.P., London, 1979).

Whitman, A. M.

Whitman, M.

Woo, R.

R. Woo, J. W. Armstrong, “Spacecraft Radio Scattering Observations of the Power Spectrum of Electron Density Fluctuations in the Solar Wind,” J. Geophys. Res. 84, 7288 (1979).
[CrossRef]

Zachariasen, F.

S. M. Flatte, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission Through a Fluctuating Ocean (Cambridge U.P., London, 1979).

Astrophys. J. (1)

E. E. Salpeter, “Interplanetary Scintillations. I. Theory,” Astrophys. J. 147, 433 (1967).
[CrossRef]

Bull. Acad. Sci. URSS Ser. Phys. (1)

V. A. Fock, “Theory of Radio-Wave Propagation in an Inhomogeneous Atmosphere for a Raised Source,” Bull. Acad. Sci. URSS Ser. Phys. 14, 70 (1950).

IMA J. Appl. Math. (1)

C. Macaskill, T. E. Ewart, “Computer Simulation of Two-Dimensional Random Wave Propagation,” IMA J. Appl. Math. 33, 1 (1984).
[CrossRef]

J. Acoust. Soc. Am. (2)

S. M. Flatte, F. D. Tappert, “Calculation of the Effect of Internal Waves on Oceanic Sound Transmission,” J. Acoust. Soc. Am. 58, 1151 (1975).
[CrossRef]

R. Esswein, S. M. Flatte, “Calculation of the Phase-Structure Function Density from Oceanic Internal Waves,” J. Acoust. Soc. Am. 70, 1387 (1981).
[CrossRef]

J. Atmos. Terr. Phys. (2)

H. G. Booker, J. A. Ferguson, H. O. Vats, “Comparison between the Extended-Medium and the Phase-Screen Scintillation Theories,” J. Atmos. Terr. Phys. 47, 381 (1985).
[CrossRef]

R. Buckley, “Diffraction by a Random Phase-Changing Screen: A Numerical Experiment,” J. Atmos. Terr. Phys. 37, 1431 (1975).
[CrossRef]

J. Geophys. Res. (1)

R. Woo, J. W. Armstrong, “Spacecraft Radio Scattering Observations of the Power Spectrum of Electron Density Fluctuations in the Solar Wind,” J. Geophys. Res. 84, 7288 (1979).
[CrossRef]

J. Math. Phys. (1)

R. Dashen, “Path Integrals for Waves in Random Media,” J. Math. Phys. 20, 894 (1979).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (4)

J. Sound Vib. (1)

B. J. Uscinski, C. Macaskill, M. Spivack, “Path Integrals for Wave Intensity Fluctuations in Random Media,” J. Sound Vib. 106, 509 (1986).
[CrossRef]

Opt. Quantum Electron. (1)

K. S. Goshelashvily, V. I. Shishov, “Saturation of Laser Irradiance Fluctuations beyond a Turbulent Layer,” Opt. Quantum Electron. 7, 524 (1975).
[CrossRef]

Proc. IEEE (1)

D. L. Knepp, “Multiple Phase-Screen Calculation of the Tem poral Behavior of Stochastic Waves,” Proc. IEEE 71, 722 (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 (1983).
[CrossRef]

Radio Sci. (7)

C. L. Rino, “A Power Law Phase Screen Model for Ionospheric Scintillation 1. Weak Scatter,” Radio Sci. 14, 1135 (1979).
[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 (1978).
[CrossRef]

D. P. Hinson, “Strong Scintillations during Atmospheric Occupations: Theoretical Intensity Spectra,” Radio Sci. 21, 257 (1986).
[CrossRef]

V. R. Rumsey, “Scintillations due to a Concentrated Layer with a Power-Law Turbulence Spectrum,” Radio Sci. 10, 107 (1975).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, F. S. Henyey, “Solution for the Fourth Moment of Waves Propagating in Random Media,” Radio Sci. 21, 929 (1986).
[CrossRef]

R. L. Fante, “Electric Field Spectrum and Intensity Covariance of a Wave in a Random Medium,” Radio Sci. 10, 77 (1975).
[CrossRef]

C. L. Rino, J. Owen, “Numerical Simulations of Intensity Scintillation Using the Power Law Phase Screen Model,” Radio Sci. 15, 41 (1980).
[CrossRef]

Sov. Phys. JETP (1)

K. S. Goshelashvily, V. I. Shishov, “Saturated Fluctuations in the Laser Radiation Intensity in a Turbulent Medium,” Sov. Phys. JETP 39, 605 (1974).

Other (5)

R. G. Frehlich, “Laser Propagation in Random Media,” U. California, San Diego, Ph.D. Thesis (1982).

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

J. P. Filice, “Studies of the Microscale Density Fluctuations in the Solar Wind Using Interplanetary Scintillations,” U. California, San Diego, Ph.D. Thesis (1984).

S. M. Flatte, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission Through a Fluctuating Ocean (Cambridge U.P., London, 1979).

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).

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 (19)

Fig. 1
Fig. 1

Schematic representation of Φ I (κ) in strong scattering conditions. κmax and κmin are wavenumber limits defining simulation spectral window requirements.

Fig. 2
Fig. 2

Comparison of simulations vs low-frequency (long dash) and high-frequency (short dash) asymptotic theory of Codona et al. for extended medium: α = 1.7, and U = 0.1, 1.0, and 10.0.

Fig. 3
Fig. 3

Comparison of single-screen simulation results with calculations of Hinson for α = 1.7 and U = 0.1, 1.0, and 10.0.

Fig. 4
Fig. 4

Sample realization of a phase screen for a pure power law, α = 1.7. Note the presence of both strong large-scale variation and small-scale filamentary structure.

Fig. 5
Fig. 5

Intensity statistics for pure power laws (α = 1.0, 1.7, and 2.0) for (a) intensity variance as a function of U; (b) intensity spectra for U = 1.0 describing conditions before the peak in σ I 2 ; (c) intensity spectra at the maximum σ I 2 , U = 5.0; and (d) intensity spectra for U = 10.0 in the declining region of (a). U is varied by changing strength of medium fluctuations, keeping geometry constant.

Fig. 6
Fig. 6

Images of intensity for one realization of the 3-D pure power-law medium with α = 1.7: (a) U = 0.01; (b) U = 1.0; (c) U = 5.0; and (d) U = 10.0. Intensity range is hard limited from 0 to 10, and the mean intensity is unity for all images.

Fig. 7
Fig. 7

Sample realization of a phase screen for a power law α = 1.7 and an inner scale β = 5.0. Large-scale variation remains, but smallscale fluctuations are suppressed.

Fig. 8
Fig. 8

Intensity statistics for a power law of 1.7 and inner scales of: β = 1.6, 5.0, and 15.0 corresponding to β/R f = 0.3,1, and 3: (a) intensity variance; (b) intensity spectra for U = 1.0; (c) U = 5.0; and (d) U = 10.0.

Fig. 9
Fig. 9

Intensity images of a single realization of a power-law medium with α = 1.7, R f = 5.0, and β = 5.0: (a) U = 1.0; (b) U = 5.0; and (c) U = 10.0.

Fig. 10
Fig. 10

Intensity images of a single realization of the same medium as in Fig. 9 but with β = 15.0: (a) U = 1.0; (b) U = 3.0 instead of 5.0, since the peak in σ I 2 occurs here; (c) U = 10.0; (d) a close-up of the intensity realization for U = 4.0 showing a remarkably clear set of fringes emanating from the bright point just to the left of center; and (e) a close-up of (c) showing checkerboard oscillations at the highest spatial frequencies.

Fig. 11
Fig. 11

Sample realization of a phase screen for a power-law α = 1.7 and an outer scale L0 = 15.0. Small-scale filamentary structure is similar to Fig. 4, but large-scale variation is suppressed.

Fig. 12
Fig. 12

Intensity statistics for a power-law medium with α = 1.7, outer scales of 5,15, and 50, and R f = 5.0: (a) intensity variance as a function of scattering strength U; (b) intensity spectra for U = 1.0; (c) U = 5.0; and (d) U = 10.0. Spectra for all strengths of scatter show a substantial effect at low frequencies from the outer scale.

Fig. 13
Fig. 13

Intensity images of a single realization of a power-law medium with α = 1.7, R f = 5.0, and L0 = 15.0: (a) U = 1.0, (b) U = 5.0, and (c) U = 10.0.

Fig. 14
Fig. 14

Sample realization of a phase screen for a (single-scale) Gaussian medium with scale L = 16. In this case both large- and small-scale structures are suppressed relative to the pure power law of Fig. 4

Fig. 15
Fig. 15

Intensity statistics for a Gaussian medium with varying scale size: (a) intensity variance as a function of scattering strength U for L0/R f of 1.6 and 0.8 showing the increase in peak value of σ I 2 when the scale size becomes greater than the Fresnel length and (b) intensity spectra for U = 1.0, 5.0, and 40.0. Note that essentially no aliasing is encountered even at U = 40.0 due to the narrow bandwidth of the Gaussian medium spectrum. κmin and κmax do separate but much more slowly than for a power-law medium.

Fig. 16
Fig. 16

Intensity images of a single realization of a (single-scale) Gaussian medium with L = 16.0, R f = 10.0: (a) U = 1.0; (b) U = 5.0 (the peak); (c) U = 10.0 (past the peak); and (d) U = 40.0. Intensity range is 0-5 for these images to enhance the detail.

Fig. 17
Fig. 17

Comparison of intensity standard deviation computed by WB (solid lines) with simulation results (dashed lines) for 3-D power-law spectrum with inner scale (γ K = 10) and pure power law (γ K = 0). The strength parameter σ is varied by change in propagation distance, not strength of fluctuations.

Fig. 18
Fig. 18

Comparison of simulation intensity variance results for an extended medium (continuous curve) with those of the equivalent thin screen model (symbols) of Booker et al. at U = 2,4, and 6. The medium spectrum has a Kolmogorov power law with inner scale: β/R f = 0.3,1.0, and 3.0 (boxes, pluses, and crosses, respectively).

Fig. 19
Fig. 19

Intensity spectra for extended medium simulation (solid curve) vs single equivalent screen (dashed curve) for U = 6 and β = 1.6.

Equations (29)

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

ϕ = ψ exp ( - i k x ) ,
2 i k x ψ + 2 ψ + 2 k 2 n 1 ψ = 0 ,
B 0 ( ρ 1 , ρ 2 ) = k 2 0 δ x n 1 ( x , ρ 1 ) n 1 ( x , p 2 ) d x d x = k 2 δ x A n ( ρ 1 - ρ 2 ) ,
A n ( ρ ) = 2 π - Φ n ( κ x = 0 , κ ) exp ( i κ · ρ ) d 2 κ .
Φ θ ( κ ) = 2 π k 2 δ x Φ n ( κ x = 0 , κ ) .
Ψ ( x , κ ) x = - i κ 2 2 k Ψ ( x , κ ) ,
Ψ ( x , κ ) = Ψ ( x , κ ) exp [ - i κ 2 ( x - x ) 2 k ] .
W ^ ( κ ^ ) = 1 Δ 2 W ( κ ^ / Δ ) ,
σ I 2 = I 2 I 2 - 1 ,
U = 4 π k 2 R 0 Φ n ( 0 , κ ) [ 1 - sin ( κ 2 R f 2 ) κ 2 R f 2 ] d κ ,
σ 1 2 = C ( α ) C n 2 k 2 - α / 2 R 1 + α / 2 ,
C ( α ) = 2 Γ ( 1 - α / 2 ) Γ ( 1 + α ) α ( 1 + α / 2 ) cos ( π α 4 ) sin [ ( α - 1 ) π 2 ] .
C 0 hf ( ρ ) = Γ 2 ( ρ , R ) Γ 2 * ( ρ , R ) = exp [ - D ( ρ ) ] ,
D ( ρ ) = 4 π k 2 R - [ 1 - cos ( ρ · k ) ] Φ n ( κ x , κ ) d 2 κ .
Φ I hf ( κ ) = 1 ( 2 π ) 2 - exp [ - D ( ρ ) ] exp ( - i ρ · κ ) d 2 ρ ,
ρ 0 2 Γ ( 2 α ) 2 π α ,
4 π k 2 R Φ n ( κ x = 0 , κ ) .
κ max = ρ 0 - 1 10 1 α + 2 .
Φ 1 lf ( κ ) = 4 π k 2 Φ k ( κ ) 0 R exp [ - 0 R d [ κ k h ( x , x ) ] d x ] × sin 2 [ κ 2 ( R - x ) 2 k ] d x ,
h ( x , x ) = { R - x x < x , R - x x > x ,
κ 0 = k R [ Γ ( 1 + α / 2 ) B ( α ) k 2 C n 2 R ] ,
R k = 2 · 10 1 α + 2 [ B ( α ) k 2 - α 2 C n 2 R 1 + α 2 Γ ( 1 + α 2 ) ] 2 α = 2 ( σ 1 2 ) 2 α 10 1 α + 2 [ 1 + α 2 2 α Γ ( 1 + α 2 ) cos π α 4 ] 2 α ,
Φ n ( k ) = 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 ) = K ( α ) C n 2 ( κ 2 + 1 / L 0 2 ) - α 2 - 1 .
Φ n ( κ ) = C n 2 exp ( - κ 2 L 2 / 4 ) ,
γ k = 6.6098 ( C 1 β 11 / 3 r f ) ,
σ 1 = 2.4377 N 11 / 12 r f 5 / 6 C 1 1 / 2 ,

Metrics