Abstract

Quantitative error analyses for the simulation of wave propagation in three-dimensional random media, when narrow angular scattering is assumed, are presented for plane-wave and spherical-wave geometry. This includes the errors that result from finite grid size, finite simulation dimensions, and the separation of the two-dimensional screens along the propagation direction. Simple error scalings are determined for power-law spectra of the random refractive indices of the media. The effects of a finite inner scale are also considered. The spatial spectra of the intensity errors are calculated and compared with the spatial spectra of intensity. The numerical requirements for a simulation of given accuracy are determined for realizations of the field. The numerical requirements for accurate estimation of higher moments of the field are less stringent.

© 1995 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  4. R. G. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
    [CrossRef] [PubMed]
  5. R. G. Frehlich, “Effects of refractive turbulence on coherent laser radar,” Appl. Opt. 32, 2122–2139 (1993).
    [CrossRef] [PubMed]
  6. W. A. Coles, J. P. Filice, “Dynamic spectra of interplanetary scintillations,” Nature (London) 312, 251–254 (1985).
    [CrossRef]
  7. R. Buckley, “Diffraction by a random phase-changing screen: a numerical experiment,” J. Atmos. Terr. Phys. 37, 1431–1446 (1975).
    [CrossRef]
  8. C. L. Rino, “On the application of phase screen models to the interpretation of ionospheric scintillation data,” Radio Sci. 17, 855–867 (1982).
    [CrossRef]
  9. D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
    [CrossRef]
  10. C. H. Liu, S. J. Franke, “Experimental and theoretical studies of ionospheric irregularities using scintillation techniques,” Radio Sci. 21, 363–374 (1986).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  20. R. G. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
    [CrossRef]
  21. V. I. Klyatskin, V. I. Tatarskii, “A new method of successive approximations in the problem of the propagation of waves in a medium having random large-scale inhomogeneities,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1400–1415 (1971) [Radio-phys. Quantum Electron. 14, 1100–1111 (1971)].
  22. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  23. V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978) [Sov. Phys. JETP 48, 27–31 (1978)].
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    [CrossRef]
  25. 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]
  26. R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
    [CrossRef]
  27. W. A. Coles, R. G. Frehlich, B. J. Rickett, J. L. Codona, “Refractive scintillation in the interstellar medium,” Astrophys. J. 315, 666–674 (1987).
    [CrossRef]
  28. R. Narayan, W. B. Hubbard, “Theory of anisotropic refractive scintillation: application to stellar occultations by Neptune,” Astrophys. J. 325, 503–518 (1988).
    [CrossRef]
  29. S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission Through a Fluctuating Ocean, Cambridge Monographs on Mechanics and Applied Mathematics (Cambridge U. Press, Cambridge, 1979).
  30. R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  31. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  32. 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]
  33. M. Spivack, “Accuracy of the moments from simulation of waves in random media,” J. Opt. Soc. Am. A 7, 790–793 (1990).
    [CrossRef]
  34. N. Ben-Yosef, E. Goldner, “Sample size influence on optical scintillation analysis. 1: Analytical treatment of the higher-order irradiance moments,” Appl. Opt. 27, 2167–2177 (1988).
    [CrossRef] [PubMed]
  35. R. G. Frehlich, J. H. Churnside, “Statistical properties of estimates of the moments of laser scintillation,” J. Mod. Opt. 36, 1645–1659 (1989).
    [CrossRef]

1994 (2)

1993 (2)

1992 (2)

R. G. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

1991 (1)

1990 (3)

1989 (2)

R. G. Frehlich, J. H. Churnside, “Statistical properties of estimates of the moments of laser scintillation,” J. Mod. Opt. 36, 1645–1659 (1989).
[CrossRef]

M. Spivack, B. J. Uscinski, “The split-step solution in random wave propagation,” J. Comput. Appl. Math. 27, 349–361 (1989).
[CrossRef]

1988 (3)

1987 (2)

W. A. Coles, R. G. Frehlich, B. J. Rickett, J. L. Codona, “Refractive scintillation in the interstellar medium,” Astrophys. J. 315, 666–674 (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]

1986 (2)

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]

C. H. Liu, S. J. Franke, “Experimental and theoretical studies of ionospheric irregularities using scintillation techniques,” Radio Sci. 21, 363–374 (1986).
[CrossRef]

1985 (1)

W. A. Coles, J. P. Filice, “Dynamic spectra of interplanetary scintillations,” Nature (London) 312, 251–254 (1985).
[CrossRef]

1984 (2)

C. L. Rino, J. Owen, “Numerical simulations of intensity scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
[CrossRef]

C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” Inst. Math. Appl. J. Appl. Math. 33, 1–15 (1984).
[CrossRef]

1983 (2)

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 (1)

C. L. Rino, “On the application of phase screen models to the interpretation of ionospheric scintillation data,” Radio Sci. 17, 855–867 (1982).
[CrossRef]

1979 (1)

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

1978 (1)

V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978) [Sov. Phys. JETP 48, 27–31 (1978)].

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 (2)

R. Buckley, “Diffraction by a random phase-changing screen: a numerical experiment,” J. Atmos. Terr. Phys. 37, 1431–1446 (1975).
[CrossRef]

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

1971 (1)

V. I. Klyatskin, V. I. Tatarskii, “A new method of successive approximations in the problem of the propagation of waves in a medium having random large-scale inhomogeneities,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1400–1415 (1971) [Radio-phys. Quantum Electron. 14, 1100–1111 (1971)].

Ben-Yosef, N.

Brasher, C.

Buckley, R.

R. Buckley, “Diffraction by a random phase-changing screen: a numerical experiment,” J. Atmos. Terr. Phys. 37, 1431–1446 (1975).
[CrossRef]

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Churnside, J. H.

R. G. Frehlich, J. H. Churnside, “Statistical properties of estimates of the moments of laser scintillation,” J. Mod. Opt. 36, 1645–1659 (1989).
[CrossRef]

Codona, J. L.

W. A. Coles, R. G. Frehlich, B. J. Rickett, J. L. Codona, “Refractive scintillation in the interstellar medium,” Astrophys. J. 315, 666–674 (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]

Coles, W. A.

W. A. Coles, R. G. Frehlich, B. J. Rickett, J. L. Codona, “Refractive scintillation in the interstellar medium,” Astrophys. J. 315, 666–674 (1987).
[CrossRef]

W. A. Coles, J. P. Filice, “Dynamic spectra of interplanetary scintillations,” Nature (London) 312, 251–254 (1985).
[CrossRef]

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]

Dainty, J. C.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[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 Monographs on Mechanics and Applied Mathematics (Cambridge U. Press, Cambridge, 1979).

Ewart, T. E.

C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” Inst. Math. Appl. J. Appl. Math. 33, 1–15 (1984).
[CrossRef]

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]

Filice, J. P.

W. A. Coles, J. P. Filice, “Dynamic spectra of interplanetary scintillations,” Nature (London) 312, 251–254 (1985).
[CrossRef]

J. P. Filice, “Studies of the microscale density fluctuations in the solar wind using interplanetary scintillations,” Ph.D. dissertation (University of California, San Diego, Calif., 1984).

Flatté, S. M.

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]

Franke, S. J.

C. H. Liu, S. J. Franke, “Experimental and theoretical studies of ionospheric irregularities using scintillation techniques,” Radio Sci. 21, 363–374 (1986).
[CrossRef]

Frehlich, R. G.

R. G. Frehlich, “Effects of global intermittency on wave propagation in random media,” Appl. Opt. 33, 5764–5769 (1994).
[CrossRef] [PubMed]

R. G. Frehlich, “Effects of refractive turbulence on coherent laser radar,” Appl. Opt. 32, 2122–2139 (1993).
[CrossRef] [PubMed]

R. G. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

R. G. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
[CrossRef] [PubMed]

R. G. Frehlich, J. H. Churnside, “Statistical properties of estimates of the moments of laser scintillation,” J. Mod. Opt. 36, 1645–1659 (1989).
[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]

W. A. Coles, R. G. Frehlich, B. J. Rickett, J. L. Codona, “Refractive scintillation in the interstellar medium,” Astrophys. J. 315, 666–674 (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]

Glindemann, A.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Goldner, E.

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]

Hubbard, W. B.

R. Narayan, W. B. Hubbard, “Theory of anisotropic refractive scintillation: application to stellar occultations by Neptune,” Astrophys. J. 325, 503–518 (1988).
[CrossRef]

Kavaya, M. J.

Klyatskin, V. I.

V. I. Klyatskin, V. I. Tatarskii, “A new method of successive approximations in the problem of the propagation of waves in a medium having random large-scale inhomogeneities,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1400–1415 (1971) [Radio-phys. Quantum Electron. 14, 1100–1111 (1971)].

Knepp, D. L.

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

Lane, R. G.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Liu, C. H.

C. H. Liu, S. J. Franke, “Experimental and theoretical studies of ionospheric irregularities using scintillation techniques,” Radio Sci. 21, 363–374 (1986).
[CrossRef]

Macaskill, C.

C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” Inst. Math. Appl. J. Appl. Math. 33, 1–15 (1984).
[CrossRef]

Martin, J.

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]

J. Martin, “Simulation of wave propagation in random: theory and applications,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993).

Martin, J. M.

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 Monographs on Mechanics and Applied Mathematics (Cambridge U. Press, Cambridge, 1979).

Narayan, R.

R. Narayan, W. B. Hubbard, “Theory of anisotropic refractive scintillation: application to stellar occultations by Neptune,” Astrophys. J. 325, 503–518 (1988).
[CrossRef]

Owen, J.

C. L. Rino, J. Owen, “Numerical simulations of intensity scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
[CrossRef]

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Rickett, B. J.

W. A. Coles, R. G. Frehlich, B. J. Rickett, J. L. Codona, “Refractive scintillation in the interstellar medium,” Astrophys. J. 315, 666–674 (1987).
[CrossRef]

Rino, C. L.

C. L. Rino, J. Owen, “Numerical simulations of intensity scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
[CrossRef]

C. L. Rino, “On the application of phase screen models to the interpretation of ionospheric scintillation data,” Radio Sci. 17, 855–867 (1982).
[CrossRef]

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Spivack, M.

M. Spivack, “Accuracy of the moments from simulation of waves in random media,” J. Opt. Soc. Am. A 7, 790–793 (1990).
[CrossRef]

M. Spivack, B. J. Uscinski, “The split-step solution in random wave propagation,” J. Comput. Appl. Math. 27, 349–361 (1989).
[CrossRef]

Tatarskii, V. I.

V. I. Klyatskin, V. I. Tatarskii, “A new method of successive approximations in the problem of the propagation of waves in a medium having random large-scale inhomogeneities,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1400–1415 (1971) [Radio-phys. Quantum Electron. 14, 1100–1111 (1971)].

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

Uscinski, B. J.

M. Spivack, B. J. Uscinski, “The split-step solution in random wave propagation,” J. Comput. Appl. Math. 27, 349–361 (1989).
[CrossRef]

Wang, G.

Watson, K. M.

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission Through a Fluctuating Ocean, Cambridge Monographs on Mechanics and Applied Mathematics (Cambridge U. Press, Cambridge, 1979).

Zachariasen, F.

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission Through a Fluctuating Ocean, Cambridge Monographs on Mechanics and Applied Mathematics (Cambridge U. Press, Cambridge, 1979).

Zavorotnyi, V. U.

V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978) [Sov. Phys. JETP 48, 27–31 (1978)].

Appl. Opt. (5)

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]

Astrophys. J. (2)

W. A. Coles, R. G. Frehlich, B. J. Rickett, J. L. Codona, “Refractive scintillation in the interstellar medium,” Astrophys. J. 315, 666–674 (1987).
[CrossRef]

R. Narayan, W. B. Hubbard, “Theory of anisotropic refractive scintillation: application to stellar occultations by Neptune,” Astrophys. J. 325, 503–518 (1988).
[CrossRef]

Inst. Math. Appl. J. Appl. Math. (1)

C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” Inst. Math. Appl. J. Appl. Math. 33, 1–15 (1984).
[CrossRef]

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

V. I. Klyatskin, V. I. Tatarskii, “A new method of successive approximations in the problem of the propagation of waves in a medium having random large-scale inhomogeneities,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1400–1415 (1971) [Radio-phys. Quantum Electron. 14, 1100–1111 (1971)].

J. Atmos. Sci. (1)

R. G. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

J. Atmos. Terr. Phys. (1)

R. Buckley, “Diffraction by a random phase-changing screen: a numerical experiment,” J. Atmos. Terr. Phys. 37, 1431–1446 (1975).
[CrossRef]

J. Comput. Appl. Math. (1)

M. Spivack, B. J. Uscinski, “The split-step solution in random wave propagation,” J. Comput. Appl. Math. 27, 349–361 (1989).
[CrossRef]

J. Math. Phys. (1)

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

J. Mod. Opt. (1)

R. G. Frehlich, J. H. Churnside, “Statistical properties of estimates of the moments of laser scintillation,” J. Mod. Opt. 36, 1645–1659 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Nature (London) (1)

W. A. Coles, J. P. Filice, “Dynamic spectra of interplanetary scintillations,” Nature (London) 312, 251–254 (1985).
[CrossRef]

Opt. Eng. (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Proc. IEEE (2)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

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

Radio Sci. (4)

C. H. Liu, S. J. Franke, “Experimental and theoretical studies of ionospheric irregularities using scintillation techniques,” Radio Sci. 21, 363–374 (1986).
[CrossRef]

C. L. Rino, J. Owen, “Numerical simulations of intensity scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
[CrossRef]

C. L. Rino, “On the application of phase screen models to the interpretation of ionospheric scintillation data,” Radio Sci. 17, 855–867 (1982).
[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]

Waves Random Media (1)

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978) [Sov. Phys. JETP 48, 27–31 (1978)].

Other (4)

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

S. M. Flatté, R. Dashen, W. H. Munk, K. M. Watson, F. Zachariasen, Sound Transmission Through a Fluctuating Ocean, Cambridge Monographs on Mechanics and Applied Mathematics (Cambridge U. Press, Cambridge, 1979).

J. P. Filice, “Studies of the microscale density fluctuations in the solar wind using interplanetary scintillations,” Ph.D. dissertation (University of California, San Diego, Calif., 1984).

J. Martin, “Simulation of wave propagation in random: theory and applications,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993).

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Figures (12)

Fig. 1
Fig. 1

Intensity error irms that is due to transverse sampling Δs for a plane wave that is incident upon a thin screen. The phase spectrum for the calculations in the top graph is a pure power-law spectrum with exponent α = 1, and the phase spectrum in the middle graph has α = 5/3. The phase spectrum used in the lower panel is an atmospheric spectrum Eq. (2) with inner scale l0 = 5Δsref. The turbulence levels for these spectra are described by the Born variance with zero-inner-scale Eq. (15). These are mb2 = 0.1 (○), 1.0 (□), and 10 (⋄). The chi-square, best-fit, power law [irms = 0.671(Δs/s0)0.514 for α = 1 and irms = 0.382(Δs/s0)0.875 for α = 5/3] is plotted as a straight line on the top and middle graphs. The estimated rms intensity mdec for mb2 = 10 is shown as a dashed line.

Fig. 2
Fig. 2

Spatial spectra of intensity (●) and intensity error that is due to transverse sampling (open symbols) for a plane-wave phase-screen simulation (N = 2048, Δs = rf/64). The phase spectrum was a pure Kolmogorov power law with mb2 = 10. The intensity-error spectra were calculated for Δs = rf/32 (○) and Δs = rf/16 (□).

Fig. 3
Fig. 3

Intensity variance m2 (top graph) and the windowing error Δm (bottom graph) for a plane-wave phase-screen simulation as a function of θ0R/L. The phase spectrum was a pure Kolmogorov power law with mb2 = 1. The results of the simulation are marked with (●). The chi-square, best-fit, power-law models [m2 = 0.8858 − 6.73(θ0R/L)2.36 and Δm = 2.59(θ0R/L)1.18] are drawn as solid curves, and the theoretical model Eqs. (19) and (20) for δ = 0.65 are drawn as dashed curves.

Fig. 4
Fig. 4

Normalized intensity error that is due to longitudinal sampling Δz for a plane-wave extended-medium simulation (N = 512, Δs = rf/16, NS = 128). The refractive-index spectra were pure power-law spectra with α = 1 (top graph) and α = 5/3 (middle graph) and an atmospheric spectrum Eq. (2) with l0 = 5Δs (bottom graph). The turbulence levels (for zero inner scale) are marked mb2 = 0.1 (○), 1.0 (□), and 10 (⋄). The normalized rms intensity m/mb for mb2 = 10 is shown as a dashed line.

Fig. 5
Fig. 5

Spatial spectra of the intensity (●) and of the intensity error that is due to longitudinal sampling Δz (open symbols) for a plane-wave extended-medium simulation (N = 512, Δs = rf/16, NS = 128). The refractive-index spectrum was a pure Kolmogorov power-law spectrum with Born variance mb2 = 1. The error spectra were calculated for NS = 64 screens (○) and NS = 32 screens (□).

Fig. 6
Fig. 6

Intensity error irms that is due to transverse sampling Δs for a plane-wave extended-medium simulation (N = 512, Δs = rf/16, NS = 128). The refractive-index spectra were pure power-law spectra with α = 1 (top graph), α = 5/3 (middle graph), and an atmospheric spectrum Eq. (2) with inner scale l0 = 5Δsref (bottom graph). The turbulence level for these spectra are described by the Born variance with zero-inner-scale Eq. (23). These are mb2 = 0.1 (○), 1.0 (□), and 10 (⋄). The chi-square, best-fit power law from the thin-screen simulations (Fig. 1) is plotted as a straight line in the top and middle graphs. The estimated rms intensity mdec for mb2 = 10 is shown as a dashed line.

Fig. 7
Fig. 7

Spatial spectra of intensity (●) and of the intensity error that is due to transverse sampling (open symbols) for a plane-wave extended-medium simulation (N = 512, Δs = rf/16, NS = 128). The refractive-index spectrum was a pure Kolmogorov power-law spectrum with Born variance mb2 = 1. The intensity-error spectra were calculated for Δs = rf/8 (○) and Δs = rf/4 (□).

Fig. 8
Fig. 8

Normalized variance m2 for plane-wave and point-source extended-medium simulations as functions of θ0R/L. The phase spectrum was a pure Kolmogorov power-law spectrum with mb2 = 1. The results of the simulation are represented by the solid circles. The chi-square, best-fit, power-law models [m2 = 0.938 − 1.25(θ0R/L)2.15 for a plane wave and m2 = 1.08 − 4.65(θ0R/L)2.03 for a point source] are drawn as solid curves, and the theoretical expressions are drawn as dashed curves. For the plane-wave case, the theoretical model is given by Eq. (28) with δ = 0.65. For the point-source case, the theoretical model is given by Eq. (40) with δ = 0.75.

Fig. 9
Fig. 9

Normalized intensity error that is due to longitudinal sampling Δz for a point-source extended-medium simulation (N = 512, Δs = rf/16, NS = 128). The refractive-index spectra were pure power-law spectra with α = 1 (top graph) and 5/3 (middle graph) and an atmospheric spectrum Eq. (2) with l0 = 5Δs (bottom graph). The turbulence levels (for zero inner scale) are mb2 = 0.1 (○), 1.0 (□), and 10 (⋄). The estimated rms intensity m/mb for mb2 = 10 is shown as a dashed line.

Fig. 10
Fig. 10

Spatial spectra of the intensity (●) and of the intensity error that is due to longitudinal sampling Δz (open symbols) for a point-source extended-medium simulation (N = 512, Δs = rf/16, NS = 128). The refractive-index spectrum was a pure Kolmogorov power-law spectrum with Born variance mb2 = 1. The error spectra were calculated for NS = 64 screens (○) and NS = 32 screens (□).

Fig. 11
Fig. 11

Intensity error irms that is due to transverse sampling Δs for a point-source simulation (N = 512, Δs = rf/16, NS = 128). The refractive-index spectra were pure power-law spectra with α = 1 (top graph), α = 5/3 (middle graph), and an atmospheric spectrum Eq. (2) with l0 = 5Δsref (bottom graph). The turbulence levels for these spectra are described by the Born variance with zero-inner-scale Eq. (35). These levels are mb2 = 0.1 (○), 1.0 (□), and 10 (⋄). The chi-square, best-fit, power law from the thin-screen simulation (Fig. 1) is plotted as a solid line in the top and middle graphs. The estimated rms intensity mdec for mb2 = 10 is shown as a dashed curve.

Fig. 12
Fig. 12

Spatial spectra of intensity (●) and intensity error that is due to transverse sampling (open symbols) for a point-source extended-medium simulation (N = 512, Δs = rf/16, NS = 128). The refractive-index spectrum was a pure Kolmogorov power-law spectrum with Born variance mb2 = 1. The intensity error spectra were calculated for Δs = rf/8 (○) and Δs = rf/4 (□).

Equations (41)

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Φ n ( q ) = A ( α ) C n 2 q α 2 , A ( α ) = Γ ( α + 1 ) sin [ ( α 1 ) π / 2 ] / 4 π 2 .
Φ n ( q ) = 0 . 0330054 C n 2 q 11 / 3 f ( q l 0 ) , f ( x ) = ( 1 + a 1 x + a 2 x 2 + a 3 x 3 ) exp ( x ) .
f z ( s , z ) = j 2 k t 2 f ( s , z ) + j k n ( s , z ) f ( s , z ) ,
Γ 2 ( s , R ) = exp [ 1 2 D P ( s , R ) ] ,
D P ( s , R ) = 0 R d z D ( s , z ) ,
D ( s , z ) = 4 π k 2 d 2 q [ 1 cos ( q · s ) Φ n ( q , q z = 0 , z ) .
D S ( s , R ) = 0 R d z D ( s z / R , z ) .
f [ s , ( n + 1 ) Δ z ] = F t 1 ( F t { f ( s , n Δ z ) exp [ j ϕ n ( s ) ] } × exp [ j q 2 Δ z / ( 2 k ) ] ) .
ϕ ( s , z ) = k Δ z / 2 + Δ z / 2 d z n ( s , z ) .
Φ ϕ ( q x , q y , z ) = 2 π k 2 Δ z Φ n ( q x , q y , q z = 0 , z ) .
σ p q 2 = | ϕ p q | 2 = 4 π 2 N x N y Δ x Δ y Φ ϕ ( 2 π p / L x , 2 π q / L y ) .
S ϕ ( q ) = 1 L x L y | 0 L x 0 L y ϕ ( s ) exp ( j q · s ) d 2 s | 2 = ( Δ x Δ y ) 2 L x L y | m = 0 N x n = 0 N y ϕ m n × exp [ j ( q x m Δ x + q y n Δ y ) ] | 2 .
S ϕ ( 2 π p / L x , 2 π q / L y ) = ( Δ x Δ y ) 2 L x L y | ϕ p q t | 2 4 π 2 = Δ x Δ y N x N y | ϕ p q t | 2 4 π 2 ,
Φ ϕ ( q ) = T q α 2 f ( q l 0 ) ,
m b 2 = 4 π T Γ ( 1 α / 2 ) cos ( απ / 4 ) r f α / α = K 1 ( α ) D TS ( r f ) ,
K 1 ( α ) = 2 α Γ ( 1 + α / 2 ) cos ( απ 4 ) ,
D TS ( s ) = ( s / s 0 ) α = 4 π T Γ ( 1 α / 2 ) α Γ ( 1 + α / 2 ) ( s / 2 ) α .
m dec 2 = Var ( I ref + i ) = m ref 2 + i rms 2 + Cov ( I ref , i ) ,
Δ m 2 = 2 δπ/ L x 2 δπ/ L x d q x 2 δπ/ L y 2 δπ/ L y d q y Φ I ( q x , q y ) ,
Φ I ( q ) = 4 Φ ϕ ( q ) sin 2 [ q 2 R / ( 2 k ) ] exp [ D TS ( q R / k ) ] .
Δ m 2 m b 2 = α ( 2 πδ r f / L ) 4 α W ( α ) π cos ( πα / 4 ) Γ ( 1 α / 2 ) ,
W ( α ) = 0 1 d x 0 1 d y ( x 2 + y 2 ) 1 α / 2 ,
m b 2 = K 2 ( α ) D P ( r f ) ,
K 2 ( α ) = 2 α + 1 Γ ( 1 + α / 2 ) cos ( απ / 4 ) / ( 2 + α ) ,
D P ( s ) = ( s / s 0 ) α = B 1 ( α ) k 2 C n 2 R s α ,
B 1 ( α ) = 2 1 α Γ ( α ) Γ ( 1 α / 2 ) sin [ ( α 1 ) π / 2 ] Γ ( 1 + α / 2 ) .
Φ I ( q ) = 8 π k 2 Φ n ( q ) 0 R d z sin 2 [ q 2 ( R z ) / ( 2 k ) ] × exp { D P [ q k ( R z ) ( 1 + α z / R ) / ( 1 + α ) ] } .
Δ m 2 = α 2 α + 2 Γ ( 1 + α / 2 ) D P ( r f ) 2 π Γ ( 1 α / 2 ) × 0 2 πδθ 0 R / L d u x 0 2 πδθ 0 R / L d u y 0 1 d t × u α 2 sin 2 [ t u 2 s 0 2 / ( 2 r f 2 ) ] × exp [ u α t α ( 1 + α α t ) / ( 1 + α ) ] .
Δ m 2 m b 2 = α ( 2 + α ) ( 2 πδ r f / L ) 4 α W ( α ) 6 π Γ ( 1 α / 2 ) cos ( απ / 4 ) .
F ( s , r , t ) = f s ( η , r ) exp [ j ( k r ω t ) ] / r .
f s r = j 2 k r 2 ( 2 f s η x 2 + 2 f s η y 2 ) + j k n ( r η x , r η x , r ) f s .
f s t ( β , r ) = f s t ( β , r 0 ) exp [ j β 2 ( 1 / r 0 1 / r ) / ( 2 k ) ] .
Φ ϕ ( β , r ) = 2 π k 2 r Δ r / 2 r + Δ r / 2 z 2 Φ n ( β / z , q z = 0 ) d z .
Φ ϕ ( β , r ) = 2 π k 2 Δ r Φ n ( β / r , q z = 0 ) / r 2 .
m b 2 = K 3 ( α ) D S ( r f ) ,
K 3 ( α ) = 2 α Γ 3 ( 1 + α / 2 ) cos ( απ / 4 ) / Γ ( α + 1 ) ,
D S ( s ) = ( s / s 0 ) α = B 2 ( α ) k 2 C n 2 R s α ,
B 2 ( α ) = 2 1 α Γ ( α ) Γ ( 1 α / 2 ) sin [ ( α 1 ) π / 2 ] [ ( α + 1 ) Γ ( 1 + α / 2 ) ] .
Φ I ( q ) = 8 π k 2 0 R d z Φ n ( q R / z ) sin 2 [ q 2 z ( R z ) / ( 2 k ) ] × exp { D S [ q ( R z ) / k ] } ,
Δ m 2 = 2 α + 2 α 2 ( α + 1 ) Γ ( 1 + α / 2 ) D S ( r f ) 2 π Γ ( 1 α / 2 ) × 0 2 πδθ 0 R / L d u x 0 2 πδθ 0 R / L d u y 0 1 d t × t α u α 2 sin 2 [ ( 1 1 / t ) u 2 s 0 2 / ( 2 r f 2 ) ] × exp [ u α ( 1 t ) α ] .
Δ m 2 m b 2 = 2 Γ ( 1 + α ) ( 2 πδ r f / L ) 4 α W ( α ) ( α 1 ) Γ ( 1 α / 2 ) Γ 2 ( 1 + α / 2 ) cos ( απ / 4 ) .

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