Abstract

Wave propagation in a weak random medium has generally been understood to give rise to lognormal field statistics, and first Born approximation of log amplitude and phase. Yet no derivation of the field probability distribution seems to exist; only an inference. The moments of the field (not just the amplitude) are derived from moment equations, and from these it is shown that extra terms must be identified with phase and amplitude in the customary representation of the field.

© 1978 Optical Society of America

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  1. A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shishov, "Laser irradiance propagation in turbulent media," Proc. IEEE 63, 790–811 (1975).
  2. R. L. Fante, "Electromagnetic beam propagation in turbulent media," Proc. IEEE 63, 1669–1692 (1975).
  3. Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, and V. I. Tatarski, "Status of the theory of propagation of waves in a randomly inhomogeneous medium," Sov. Phys. Uspekhi, 13, 551–680 (1971).
  4. P. G. Bergmann, "Propagation of radiation in a medium with random inhomogeneities," Phys. Rev. 70, 486–492 (1946).
  5. C. L. Pekeris, "Note on the scattering of radiation in an inhomogeneous medium," Phys. Rev. 71, 268–269 (1947).
  6. J. B. Keller, "Wave propagation in random media," in Proc. Symp. in Applied Mathematics, edited by G. Birkhoff, R. Bellman, and C. C. Lin, (American Mathematical Society, 1962, Providence, R.I.), Vol. 13, pp. 227–246.
  7. H. G. Booker and W. E. Gordon, "A theory of radio scattering in the troposphere," Proc. IRE 38, 401–412 (1950).
  8. L. Chernov, Wave Propagation in a Random Medium (McGraw- Hill, New York, 1960) (also Dover, 1967).
  9. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961) (also Dover, 1967).
  10. H. Bremmer, "Random volume scattering," J. Res. Nat. Bur. Stand. Sect. D: 68, 967–981 (1964).
  11. D. L. Fried, "Optical resolution through a randomly inhomogeneous medium for very long and very short exposures," J. Opt. Soc. Am. 56, 1372–1379 (1966).
  12. V. I. Tatarski, "Strong amplitude fluctuations of a wave propagating in a medium with weak random inhomogeneities," Izv. Vyssh. Ucheb. Zaved. Radiofiz., 10, 48–58 (1967).
  13. This calculation is given in D. A. de Wolf, "Strong irradiance fluctuations in turbulent air: plane waves", J. Opt. Soc. Am. 63, 171–179 (1973), in Secs. I and II. Sections III and IV of this work were based on a saturation-regime approximation that was later shown to be incorrect, hence are not correct. However, Secs. I and II for weak scattering, and Sec. V for very strong scattering (not realizable in optical turbulence) do not depend on the invalidated saturation-regime approximation.
  14. V. I. Tatarski, The Effect of the Turbulent Atmosphere on Wave Propagation (U.S. Dept. of Commerce, Springfield, Va., 1971), (see Chap. 5 for historic review of Russian contributions).
  15. W. P. Brown, Jr., "Moment equations for waves propagated in random media," J. Opt. Soc. Am., 62, 45–54 (1972).
  16. K. Furutsu, "Statistical theory of wave propagation in a random medium and the irradiance distribution function," J. Opt. Soc. Am. 62, 240–254 (1972).
  17. H. Bremmer, "General remarks concerning theories dealing with scattering and diffraction in random media," Radio Sci. 8, 511–534 (1973).
  18. W. P. Brown, Jr., "Propagation in random media-cumulative effect of weak inhomogeneities," IEEE Trans. Antennas Propag. AP-15, 81–89 (1967).
  19. D. A. de Wolf, "Multiple scattering in a random continuum," Radio Sci. 2, 1379–1392 (1967) [and erratum, Radio Sci. 3, 308 (1968)].
  20. A. S. Gurvich and V. I. Tatarski, "Coherence and intensity fluctuations of light in the turbulent atmosphere," Radio Sci. 10, 3–14 (1975).
  21. R. L. Fante, "Electric field spectrum and intensity covariance of a wave in a random medium," Radio Sci. 10, 77–85 (1975).
  22. K. Gochelashvili and V. Shishov, "Saturated intensity fluctuations of laser radiation in a turbulent medium," (in Russian), Zh. Eksp. Teor. Fiz. 66, 1237–1247 (1974).
  23. W. C. Hoffmann, "Wave propagation in a general random continuous medium," in Proc. Symp. in Applied Mathematics, edited by R. Bellman, (American Mathematical Society, Providence, R.I., 1964), Vol. 16, pp. 117–144.
  24. Reference 3, sections 3b (and 3c) can direct the reader to the pertinent literature.
  25. H. Bremmer, "A mathematical connecting scattering and diffraction phenomena, including Bragg-type interferences", TH-Rep. 71-E-26 (Technische Hogeschool Eindhoven), 64 pp., Dec. 1971.
  26. R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," Proc. IEEE 58, 1523–1545 (1970) (tabular summaries of some of these equations).
  27. Reference 14, sections 40–49, contains plane-wave and sphericalwave derivations by Tatarski and co-workers of these formulas.
  28. R. Barakat, "Sums of independent lognormally distributed random variables," J. Opt. Soc. Am. 66, 211–216 (1976).
  29. J. W. Strohbehn, T-I. Wang, and J. P. Speck, "On the probability distribution of line-of-sight fluctuations of optical signals," Radio Sci. 10, 59–70 (1975).

1976

1975

J. W. Strohbehn, T-I. Wang, and J. P. Speck, "On the probability distribution of line-of-sight fluctuations of optical signals," Radio Sci. 10, 59–70 (1975).

A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shishov, "Laser irradiance propagation in turbulent media," Proc. IEEE 63, 790–811 (1975).

R. L. Fante, "Electromagnetic beam propagation in turbulent media," Proc. IEEE 63, 1669–1692 (1975).

A. S. Gurvich and V. I. Tatarski, "Coherence and intensity fluctuations of light in the turbulent atmosphere," Radio Sci. 10, 3–14 (1975).

R. L. Fante, "Electric field spectrum and intensity covariance of a wave in a random medium," Radio Sci. 10, 77–85 (1975).

1974

K. Gochelashvili and V. Shishov, "Saturated intensity fluctuations of laser radiation in a turbulent medium," (in Russian), Zh. Eksp. Teor. Fiz. 66, 1237–1247 (1974).

1973

This calculation is given in D. A. de Wolf, "Strong irradiance fluctuations in turbulent air: plane waves", J. Opt. Soc. Am. 63, 171–179 (1973), in Secs. I and II. Sections III and IV of this work were based on a saturation-regime approximation that was later shown to be incorrect, hence are not correct. However, Secs. I and II for weak scattering, and Sec. V for very strong scattering (not realizable in optical turbulence) do not depend on the invalidated saturation-regime approximation.

H. Bremmer, "General remarks concerning theories dealing with scattering and diffraction in random media," Radio Sci. 8, 511–534 (1973).

1972

W. P. Brown, Jr., "Moment equations for waves propagated in random media," J. Opt. Soc. Am., 62, 45–54 (1972).

K. Furutsu, "Statistical theory of wave propagation in a random medium and the irradiance distribution function," J. Opt. Soc. Am. 62, 240–254 (1972).

1971

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, and V. I. Tatarski, "Status of the theory of propagation of waves in a randomly inhomogeneous medium," Sov. Phys. Uspekhi, 13, 551–680 (1971).

1970

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

1967

V. I. Tatarski, "Strong amplitude fluctuations of a wave propagating in a medium with weak random inhomogeneities," Izv. Vyssh. Ucheb. Zaved. Radiofiz., 10, 48–58 (1967).

W. P. Brown, Jr., "Propagation in random media-cumulative effect of weak inhomogeneities," IEEE Trans. Antennas Propag. AP-15, 81–89 (1967).

D. A. de Wolf, "Multiple scattering in a random continuum," Radio Sci. 2, 1379–1392 (1967) [and erratum, Radio Sci. 3, 308 (1968)].

1966

1964

H. Bremmer, "Random volume scattering," J. Res. Nat. Bur. Stand. Sect. D: 68, 967–981 (1964).

W. C. Hoffmann, "Wave propagation in a general random continuous medium," in Proc. Symp. in Applied Mathematics, edited by R. Bellman, (American Mathematical Society, Providence, R.I., 1964), Vol. 16, pp. 117–144.

1950

H. G. Booker and W. E. Gordon, "A theory of radio scattering in the troposphere," Proc. IRE 38, 401–412 (1950).

1947

C. L. Pekeris, "Note on the scattering of radiation in an inhomogeneous medium," Phys. Rev. 71, 268–269 (1947).

1946

P. G. Bergmann, "Propagation of radiation in a medium with random inhomogeneities," Phys. Rev. 70, 486–492 (1946).

Barabanenkov, Y. N.

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, and V. I. Tatarski, "Status of the theory of propagation of waves in a randomly inhomogeneous medium," Sov. Phys. Uspekhi, 13, 551–680 (1971).

Barakat, R.

Bergmann, P. G.

P. G. Bergmann, "Propagation of radiation in a medium with random inhomogeneities," Phys. Rev. 70, 486–492 (1946).

Booker, H. G.

H. G. Booker and W. E. Gordon, "A theory of radio scattering in the troposphere," Proc. IRE 38, 401–412 (1950).

Bremmer, H.

H. Bremmer, "A mathematical connecting scattering and diffraction phenomena, including Bragg-type interferences", TH-Rep. 71-E-26 (Technische Hogeschool Eindhoven), 64 pp., Dec. 1971.

H. Bremmer, "General remarks concerning theories dealing with scattering and diffraction in random media," Radio Sci. 8, 511–534 (1973).

H. Bremmer, "Random volume scattering," J. Res. Nat. Bur. Stand. Sect. D: 68, 967–981 (1964).

Brown, Jr., W. P.

W. P. Brown, Jr., "Moment equations for waves propagated in random media," J. Opt. Soc. Am., 62, 45–54 (1972).

W. P. Brown, Jr., "Propagation in random media-cumulative effect of weak inhomogeneities," IEEE Trans. Antennas Propag. AP-15, 81–89 (1967).

Bunkin, F.

A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shishov, "Laser irradiance propagation in turbulent media," Proc. IEEE 63, 790–811 (1975).

Chernov, L.

L. Chernov, Wave Propagation in a Random Medium (McGraw- Hill, New York, 1960) (also Dover, 1967).

de Wolf, D. A.

This calculation is given in D. A. de Wolf, "Strong irradiance fluctuations in turbulent air: plane waves", J. Opt. Soc. Am. 63, 171–179 (1973), in Secs. I and II. Sections III and IV of this work were based on a saturation-regime approximation that was later shown to be incorrect, hence are not correct. However, Secs. I and II for weak scattering, and Sec. V for very strong scattering (not realizable in optical turbulence) do not depend on the invalidated saturation-regime approximation.

D. A. de Wolf, "Multiple scattering in a random continuum," Radio Sci. 2, 1379–1392 (1967) [and erratum, Radio Sci. 3, 308 (1968)].

Fante, R. L.

R. L. Fante, "Electromagnetic beam propagation in turbulent media," Proc. IEEE 63, 1669–1692 (1975).

R. L. Fante, "Electric field spectrum and intensity covariance of a wave in a random medium," Radio Sci. 10, 77–85 (1975).

Fried, D. L.

Furutsu, K.

Gochelashvili, K.

K. Gochelashvili and V. Shishov, "Saturated intensity fluctuations of laser radiation in a turbulent medium," (in Russian), Zh. Eksp. Teor. Fiz. 66, 1237–1247 (1974).

Gochelashvily, K.

A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shishov, "Laser irradiance propagation in turbulent media," Proc. IEEE 63, 790–811 (1975).

Gordon, W. E.

H. G. Booker and W. E. Gordon, "A theory of radio scattering in the troposphere," Proc. IRE 38, 401–412 (1950).

Gurvich, A. S.

A. S. Gurvich and V. I. Tatarski, "Coherence and intensity fluctuations of light in the turbulent atmosphere," Radio Sci. 10, 3–14 (1975).

Hoffmann, W. C.

W. C. Hoffmann, "Wave propagation in a general random continuous medium," in Proc. Symp. in Applied Mathematics, edited by R. Bellman, (American Mathematical Society, Providence, R.I., 1964), Vol. 16, pp. 117–144.

Keller, J. B.

J. B. Keller, "Wave propagation in random media," in Proc. Symp. in Applied Mathematics, edited by G. Birkhoff, R. Bellman, and C. C. Lin, (American Mathematical Society, 1962, Providence, R.I.), Vol. 13, pp. 227–246.

Kravtsov, Y. A.

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, and V. I. Tatarski, "Status of the theory of propagation of waves in a randomly inhomogeneous medium," Sov. Phys. Uspekhi, 13, 551–680 (1971).

Lawrence, R. S.

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

Pekeris, C. L.

C. L. Pekeris, "Note on the scattering of radiation in an inhomogeneous medium," Phys. Rev. 71, 268–269 (1947).

Prokhorov, A.

A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shishov, "Laser irradiance propagation in turbulent media," Proc. IEEE 63, 790–811 (1975).

Rytov, S. M.

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, and V. I. Tatarski, "Status of the theory of propagation of waves in a randomly inhomogeneous medium," Sov. Phys. Uspekhi, 13, 551–680 (1971).

Shishov, V.

A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shishov, "Laser irradiance propagation in turbulent media," Proc. IEEE 63, 790–811 (1975).

K. Gochelashvili and V. Shishov, "Saturated intensity fluctuations of laser radiation in a turbulent medium," (in Russian), Zh. Eksp. Teor. Fiz. 66, 1237–1247 (1974).

Strohbehn, J. W.

J. W. Strohbehn, T-I. Wang, and J. P. Speck, "On the probability distribution of line-of-sight fluctuations of optical signals," Radio Sci. 10, 59–70 (1975).

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

Tatarski, V. I.

A. S. Gurvich and V. I. Tatarski, "Coherence and intensity fluctuations of light in the turbulent atmosphere," Radio Sci. 10, 3–14 (1975).

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, and V. I. Tatarski, "Status of the theory of propagation of waves in a randomly inhomogeneous medium," Sov. Phys. Uspekhi, 13, 551–680 (1971).

V. I. Tatarski, "Strong amplitude fluctuations of a wave propagating in a medium with weak random inhomogeneities," Izv. Vyssh. Ucheb. Zaved. Radiofiz., 10, 48–58 (1967).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961) (also Dover, 1967).

V. I. Tatarski, The Effect of the Turbulent Atmosphere on Wave Propagation (U.S. Dept. of Commerce, Springfield, Va., 1971), (see Chap. 5 for historic review of Russian contributions).

IEEE Trans. Antennas Propag.

W. P. Brown, Jr., "Propagation in random media-cumulative effect of weak inhomogeneities," IEEE Trans. Antennas Propag. AP-15, 81–89 (1967).

Izv. Vyssh. Ucheb. Zaved. Radiofiz.

V. I. Tatarski, "Strong amplitude fluctuations of a wave propagating in a medium with weak random inhomogeneities," Izv. Vyssh. Ucheb. Zaved. Radiofiz., 10, 48–58 (1967).

J. Opt. Soc. Am.

J. Res. Nat. Bur. Stand. Sect. D:

H. Bremmer, "Random volume scattering," J. Res. Nat. Bur. Stand. Sect. D: 68, 967–981 (1964).

Phys. Rev.

P. G. Bergmann, "Propagation of radiation in a medium with random inhomogeneities," Phys. Rev. 70, 486–492 (1946).

C. L. Pekeris, "Note on the scattering of radiation in an inhomogeneous medium," Phys. Rev. 71, 268–269 (1947).

Radio Sci.

H. Bremmer, "General remarks concerning theories dealing with scattering and diffraction in random media," Radio Sci. 8, 511–534 (1973).

D. A. de Wolf, "Multiple scattering in a random continuum," Radio Sci. 2, 1379–1392 (1967) [and erratum, Radio Sci. 3, 308 (1968)].

A. S. Gurvich and V. I. Tatarski, "Coherence and intensity fluctuations of light in the turbulent atmosphere," Radio Sci. 10, 3–14 (1975).

R. L. Fante, "Electric field spectrum and intensity covariance of a wave in a random medium," Radio Sci. 10, 77–85 (1975).

J. W. Strohbehn, T-I. Wang, and J. P. Speck, "On the probability distribution of line-of-sight fluctuations of optical signals," Radio Sci. 10, 59–70 (1975).

Sov. Phys. Uspekhi

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, and V. I. Tatarski, "Status of the theory of propagation of waves in a randomly inhomogeneous medium," Sov. Phys. Uspekhi, 13, 551–680 (1971).

Zh. Eksp. Teor. Fiz.

K. Gochelashvili and V. Shishov, "Saturated intensity fluctuations of laser radiation in a turbulent medium," (in Russian), Zh. Eksp. Teor. Fiz. 66, 1237–1247 (1974).

Other

W. C. Hoffmann, "Wave propagation in a general random continuous medium," in Proc. Symp. in Applied Mathematics, edited by R. Bellman, (American Mathematical Society, Providence, R.I., 1964), Vol. 16, pp. 117–144.

Reference 3, sections 3b (and 3c) can direct the reader to the pertinent literature.

H. Bremmer, "A mathematical connecting scattering and diffraction phenomena, including Bragg-type interferences", TH-Rep. 71-E-26 (Technische Hogeschool Eindhoven), 64 pp., Dec. 1971.

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

Reference 14, sections 40–49, contains plane-wave and sphericalwave derivations by Tatarski and co-workers of these formulas.

A. Prokhorov, F. Bunkin, K. Gochelashvily, and V. Shishov, "Laser irradiance propagation in turbulent media," Proc. IEEE 63, 790–811 (1975).

R. L. Fante, "Electromagnetic beam propagation in turbulent media," Proc. IEEE 63, 1669–1692 (1975).

J. B. Keller, "Wave propagation in random media," in Proc. Symp. in Applied Mathematics, edited by G. Birkhoff, R. Bellman, and C. C. Lin, (American Mathematical Society, 1962, Providence, R.I.), Vol. 13, pp. 227–246.

H. G. Booker and W. E. Gordon, "A theory of radio scattering in the troposphere," Proc. IRE 38, 401–412 (1950).

L. Chernov, Wave Propagation in a Random Medium (McGraw- Hill, New York, 1960) (also Dover, 1967).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961) (also Dover, 1967).

This calculation is given in D. A. de Wolf, "Strong irradiance fluctuations in turbulent air: plane waves", J. Opt. Soc. Am. 63, 171–179 (1973), in Secs. I and II. Sections III and IV of this work were based on a saturation-regime approximation that was later shown to be incorrect, hence are not correct. However, Secs. I and II for weak scattering, and Sec. V for very strong scattering (not realizable in optical turbulence) do not depend on the invalidated saturation-regime approximation.

V. I. Tatarski, The Effect of the Turbulent Atmosphere on Wave Propagation (U.S. Dept. of Commerce, Springfield, Va., 1971), (see Chap. 5 for historic review of Russian contributions).

W. P. Brown, Jr., "Moment equations for waves propagated in random media," J. Opt. Soc. Am., 62, 45–54 (1972).

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