Abstract

Changes in the spectrum of light scattered by spatially random media are investigated theoretically within the accuracy of the first-order Rytov approximation. Such a phenomenon is analogous to that treated in recent studies on the correlation-induced spectral changes that are often referred to as the Wolf effect. In particular, the effects of multiple scattering on the spectrum of the scattered light are elucidated from comparison with the corresponding results derived previously under the first-order Born approximation. As a result, it is observed that the magnitude of the changes in the spectrum is enhanced in the forward-scattering directions owing to the effects of multiple scattering.

© 1995 Optical Society of America

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References

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  1. See, for example, E. Wolf, “Influence of source-correlations on spectra of radiated fields,” in International Trends in Optics, J. W. Goodman, ed. (Academic, New York, 1991), pp. 221–232, and references cited therein.
    [Crossref]
  2. E. Wolf, “Non-cosmological redshifts of spectral line,” Nature (London) 326, 363–365 (1987).
    [Crossref]
  3. J. V. Narlikar, “Noncosmological redshifts,” Space Sci. Rev. 50, 523–614 (1989).
    [Crossref]
  4. D. F. V. James, E. Wolf, “Determination of field correlations from spectral measurements with application to synthetic aperture imaging,” Radio Sci. 26, 1239–1243 (1991).
    [Crossref]
  5. E. Wolf, “Two inverse problems in spectroscopy with partially coherent sources and the scaling law,” J. Mod. Opt. 39, 9–20 (1992).
    [Crossref]
  6. T. Shirai, T. Asakura, “Spectral changes of light radiated by three-dimensional, anisotropic Gaussian Schell-model sources,” Opt. Commun. 105, 22–28 (1994).
    [Crossref]
  7. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
    [Crossref] [PubMed]
  8. E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989).
    [Crossref]
  9. J. T. Foley, E. Wolf, “Frequency shifts of spectral lines generated by scattering from space–time fluctuations,” Phys. Rev. A 40, 588–598 (1989).
    [Crossref] [PubMed]
  10. E. Wolf, “Correlation-induced Doppler-like frequency shifts of spectral lines,” Phys. Rev. Lett. 63, 2220–2223 (1989).
    [Crossref] [PubMed]
  11. D. F. V. James, M. P. Savedoff, E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Astrophys. J. 359, 67–71 (1990).
    [Crossref]
  12. D. F. V. James, E. Wolf, “Doppler-like frequency shifts generated by dynamic scattering,” Phys. Lett. A 146, 167–171 (1990).
    [Crossref]
  13. B. Cairns, E. Wolf, “Changes in the spectrum of light scattered by a moving diffuser plate,” J. Opt. Soc. Am. A 12, 1922–1928 (1991).
    [Crossref]
  14. A. Lagendijk, “Terrestrial redshifts from a diffuse light source,” Phys. Lett. A 147, 389–392 (1990).
    [Crossref]
  15. A. Lagendijk, “Comment on ‘Invariance of the spectrum of light on propagation’,” Phys. Rev. Lett. 65, 2082 (1990).
    [Crossref] [PubMed]
  16. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  17. See, for example, J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, Berlin, 1978).
    [Crossref]
  18. G. R. Heidbreder, “Multiple scattering and the method of Rytov,” J. Opt. Soc. Am. 57, 1477–1479 (1967).
    [Crossref]
  19. The applicability of the Born and Rytov approximations is shown clearly in some specific cases. See, for example, M. L. Oristaglio, “Accuracy of the Born and Rytov approximations for reflection and refraction at a plane interface,” J. Opt. Soc. Am. A 2, 1987–1993 (1985); F. C. Lin, M. A. Fiddy, “The Born–Rytov controversy. I. Comparing analytical and approximate expressions for the one-dimensional deterministic case,” J. Opt. Soc. Am. A 9, 1102–1110 (1992); “The Born–Rytov controversy. II. Applications to nonlinear and stochastic scattering problems in one-dimensional half-space media,” J. Opt. Soc. Am. A 10, 1971–1983 (1994).
    [Crossref]
  20. E. Wolf, J. T. Foley, “Scattering of electromagnetic fields of any state of coherence from space–time fluctuations,” Phys. Rev. A 40, 579–587 (1989).
    [Crossref] [PubMed]
  21. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 1.
  22. L. Mandel, E. Wolf, “Constitutive relations and the electromagnetic spectrum in a fluctuating medium,” Opt. Commun. 8, 95–99 (1973).
    [Crossref]
  23. See, for example, Chap. 17 of Ref. 16, and J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8. The treatment of the problem in these references is somewhat different from that in the present paper, mainly arising from the difference in the expression for the constitutive relation. However, both treatments are essentially the same as far as the condition regarding the depolarization effects is concerned.
  24. W. H. Carter, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
    [Crossref]
  25. B. Cairns, E. Wolf, “Comparison of the Born and the Rytov approximations for scattering on quasi-homogeneous media,” Opt. Commun. 74, 284–289 (1990).
    [Crossref]
  26. Equation (33) may be easily obtained with the help of the characteristic function for the sequence of the Gaussian random variables. See, for example, A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, Singapore, 1984), Sec. 8.3.
  27. The derivation of Eq. (45) is somewhat analogous to that of Eq. (43). See also Ref. 26.
  28. The present discussion of the applicability of the Born and Rytov approximations is a little bit qualitative. However, the same consequence has been given quantitatively in a recent paper. See T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
    [Crossref]
  29. To the authors’ knowledge, the relationship between the Rytov approximation and multiple scattering has rarely been discussed in most of the existing literature (Ref. 18 is an exception). However, from the discussion given in the present section, one may say that the Rytov solution describes some effects of multiple scattering.
  30. It was demonstrated in a recent paper that, unlike the widely recognized aspect, the Doppler effect yields not only the shift of the spectral lines but also the distortion of them, even in the situation in which one observes the spectrum of light produced by a uniformly moving source. See D. F. V. James, “The spectrum of radiation from a moving source,” Phys. Lett. A 140, 213–217 (1989).
    [Crossref]
  31. E. Wolf, T. Habashy, “Invisible bodies and uniqueness of the inverse scattering problem,” J. Mod. Opt. 40, 785–792 (1993).
    [Crossref]
  32. D. G. Fischer, E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11, 1128–1135 (1994).
    [Crossref]
  33. T. Shirai, T. Asakura, “Spectral changes of light scattered by spatially random media under the Rytov approximation,” in Meeting Digest, Topical Meeting of the International Commission for Optics, Frontiers in Information Optics, Kyoto, Japan, (Organizing Committees, Kyoto, 1994), p. 354.

1994 (2)

T. Shirai, T. Asakura, “Spectral changes of light radiated by three-dimensional, anisotropic Gaussian Schell-model sources,” Opt. Commun. 105, 22–28 (1994).
[Crossref]

D. G. Fischer, E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11, 1128–1135 (1994).
[Crossref]

1993 (2)

E. Wolf, T. Habashy, “Invisible bodies and uniqueness of the inverse scattering problem,” J. Mod. Opt. 40, 785–792 (1993).
[Crossref]

The present discussion of the applicability of the Born and Rytov approximations is a little bit qualitative. However, the same consequence has been given quantitatively in a recent paper. See T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[Crossref]

1992 (1)

E. Wolf, “Two inverse problems in spectroscopy with partially coherent sources and the scaling law,” J. Mod. Opt. 39, 9–20 (1992).
[Crossref]

1991 (2)

D. F. V. James, E. Wolf, “Determination of field correlations from spectral measurements with application to synthetic aperture imaging,” Radio Sci. 26, 1239–1243 (1991).
[Crossref]

B. Cairns, E. Wolf, “Changes in the spectrum of light scattered by a moving diffuser plate,” J. Opt. Soc. Am. A 12, 1922–1928 (1991).
[Crossref]

1990 (5)

A. Lagendijk, “Terrestrial redshifts from a diffuse light source,” Phys. Lett. A 147, 389–392 (1990).
[Crossref]

A. Lagendijk, “Comment on ‘Invariance of the spectrum of light on propagation’,” Phys. Rev. Lett. 65, 2082 (1990).
[Crossref] [PubMed]

D. F. V. James, M. P. Savedoff, E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Astrophys. J. 359, 67–71 (1990).
[Crossref]

D. F. V. James, E. Wolf, “Doppler-like frequency shifts generated by dynamic scattering,” Phys. Lett. A 146, 167–171 (1990).
[Crossref]

B. Cairns, E. Wolf, “Comparison of the Born and the Rytov approximations for scattering on quasi-homogeneous media,” Opt. Commun. 74, 284–289 (1990).
[Crossref]

1989 (6)

It was demonstrated in a recent paper that, unlike the widely recognized aspect, the Doppler effect yields not only the shift of the spectral lines but also the distortion of them, even in the situation in which one observes the spectrum of light produced by a uniformly moving source. See D. F. V. James, “The spectrum of radiation from a moving source,” Phys. Lett. A 140, 213–217 (1989).
[Crossref]

E. Wolf, J. T. Foley, “Scattering of electromagnetic fields of any state of coherence from space–time fluctuations,” Phys. Rev. A 40, 579–587 (1989).
[Crossref] [PubMed]

J. V. Narlikar, “Noncosmological redshifts,” Space Sci. Rev. 50, 523–614 (1989).
[Crossref]

E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989).
[Crossref]

J. T. Foley, E. Wolf, “Frequency shifts of spectral lines generated by scattering from space–time fluctuations,” Phys. Rev. A 40, 588–598 (1989).
[Crossref] [PubMed]

E. Wolf, “Correlation-induced Doppler-like frequency shifts of spectral lines,” Phys. Rev. Lett. 63, 2220–2223 (1989).
[Crossref] [PubMed]

1988 (1)

W. H. Carter, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[Crossref]

1987 (1)

E. Wolf, “Non-cosmological redshifts of spectral line,” Nature (London) 326, 363–365 (1987).
[Crossref]

1986 (1)

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[Crossref] [PubMed]

1985 (1)

1973 (1)

L. Mandel, E. Wolf, “Constitutive relations and the electromagnetic spectrum in a fluctuating medium,” Opt. Commun. 8, 95–99 (1973).
[Crossref]

1967 (1)

Asakura, T.

T. Shirai, T. Asakura, “Spectral changes of light radiated by three-dimensional, anisotropic Gaussian Schell-model sources,” Opt. Commun. 105, 22–28 (1994).
[Crossref]

T. Shirai, T. Asakura, “Spectral changes of light scattered by spatially random media under the Rytov approximation,” in Meeting Digest, Topical Meeting of the International Commission for Optics, Frontiers in Information Optics, Kyoto, Japan, (Organizing Committees, Kyoto, 1994), p. 354.

Cairns, B.

B. Cairns, E. Wolf, “Changes in the spectrum of light scattered by a moving diffuser plate,” J. Opt. Soc. Am. A 12, 1922–1928 (1991).
[Crossref]

B. Cairns, E. Wolf, “Comparison of the Born and the Rytov approximations for scattering on quasi-homogeneous media,” Opt. Commun. 74, 284–289 (1990).
[Crossref]

Carter, W. H.

W. H. Carter, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[Crossref]

Fischer, D. G.

Foley, J. T.

E. Wolf, J. T. Foley, “Scattering of electromagnetic fields of any state of coherence from space–time fluctuations,” Phys. Rev. A 40, 579–587 (1989).
[Crossref] [PubMed]

E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989).
[Crossref]

J. T. Foley, E. Wolf, “Frequency shifts of spectral lines generated by scattering from space–time fluctuations,” Phys. Rev. A 40, 588–598 (1989).
[Crossref] [PubMed]

Gori, F.

Groom, R. W.

The present discussion of the applicability of the Born and Rytov approximations is a little bit qualitative. However, the same consequence has been given quantitatively in a recent paper. See T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[Crossref]

Habashy, T.

E. Wolf, T. Habashy, “Invisible bodies and uniqueness of the inverse scattering problem,” J. Mod. Opt. 40, 785–792 (1993).
[Crossref]

Habashy, T. M.

The present discussion of the applicability of the Born and Rytov approximations is a little bit qualitative. However, the same consequence has been given quantitatively in a recent paper. See T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[Crossref]

Heidbreder, G. R.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

James, D. F. V.

D. F. V. James, E. Wolf, “Determination of field correlations from spectral measurements with application to synthetic aperture imaging,” Radio Sci. 26, 1239–1243 (1991).
[Crossref]

D. F. V. James, M. P. Savedoff, E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Astrophys. J. 359, 67–71 (1990).
[Crossref]

D. F. V. James, E. Wolf, “Doppler-like frequency shifts generated by dynamic scattering,” Phys. Lett. A 146, 167–171 (1990).
[Crossref]

It was demonstrated in a recent paper that, unlike the widely recognized aspect, the Doppler effect yields not only the shift of the spectral lines but also the distortion of them, even in the situation in which one observes the spectrum of light produced by a uniformly moving source. See D. F. V. James, “The spectrum of radiation from a moving source,” Phys. Lett. A 140, 213–217 (1989).
[Crossref]

Lagendijk, A.

A. Lagendijk, “Terrestrial redshifts from a diffuse light source,” Phys. Lett. A 147, 389–392 (1990).
[Crossref]

A. Lagendijk, “Comment on ‘Invariance of the spectrum of light on propagation’,” Phys. Rev. Lett. 65, 2082 (1990).
[Crossref] [PubMed]

Mandel, L.

L. Mandel, E. Wolf, “Constitutive relations and the electromagnetic spectrum in a fluctuating medium,” Opt. Commun. 8, 95–99 (1973).
[Crossref]

Narlikar, J. V.

J. V. Narlikar, “Noncosmological redshifts,” Space Sci. Rev. 50, 523–614 (1989).
[Crossref]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 1.

Oristaglio, M. L.

Papoulis, A.

Equation (33) may be easily obtained with the help of the characteristic function for the sequence of the Gaussian random variables. See, for example, A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, Singapore, 1984), Sec. 8.3.

Savedoff, M. P.

D. F. V. James, M. P. Savedoff, E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Astrophys. J. 359, 67–71 (1990).
[Crossref]

Shirai, T.

T. Shirai, T. Asakura, “Spectral changes of light radiated by three-dimensional, anisotropic Gaussian Schell-model sources,” Opt. Commun. 105, 22–28 (1994).
[Crossref]

T. Shirai, T. Asakura, “Spectral changes of light scattered by spatially random media under the Rytov approximation,” in Meeting Digest, Topical Meeting of the International Commission for Optics, Frontiers in Information Optics, Kyoto, Japan, (Organizing Committees, Kyoto, 1994), p. 354.

Spies, B. R.

The present discussion of the applicability of the Born and Rytov approximations is a little bit qualitative. However, the same consequence has been given quantitatively in a recent paper. See T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[Crossref]

Wolf, E.

D. G. Fischer, E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11, 1128–1135 (1994).
[Crossref]

E. Wolf, T. Habashy, “Invisible bodies and uniqueness of the inverse scattering problem,” J. Mod. Opt. 40, 785–792 (1993).
[Crossref]

E. Wolf, “Two inverse problems in spectroscopy with partially coherent sources and the scaling law,” J. Mod. Opt. 39, 9–20 (1992).
[Crossref]

D. F. V. James, E. Wolf, “Determination of field correlations from spectral measurements with application to synthetic aperture imaging,” Radio Sci. 26, 1239–1243 (1991).
[Crossref]

B. Cairns, E. Wolf, “Changes in the spectrum of light scattered by a moving diffuser plate,” J. Opt. Soc. Am. A 12, 1922–1928 (1991).
[Crossref]

D. F. V. James, E. Wolf, “Doppler-like frequency shifts generated by dynamic scattering,” Phys. Lett. A 146, 167–171 (1990).
[Crossref]

B. Cairns, E. Wolf, “Comparison of the Born and the Rytov approximations for scattering on quasi-homogeneous media,” Opt. Commun. 74, 284–289 (1990).
[Crossref]

D. F. V. James, M. P. Savedoff, E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Astrophys. J. 359, 67–71 (1990).
[Crossref]

J. T. Foley, E. Wolf, “Frequency shifts of spectral lines generated by scattering from space–time fluctuations,” Phys. Rev. A 40, 588–598 (1989).
[Crossref] [PubMed]

E. Wolf, “Correlation-induced Doppler-like frequency shifts of spectral lines,” Phys. Rev. Lett. 63, 2220–2223 (1989).
[Crossref] [PubMed]

E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989).
[Crossref]

E. Wolf, J. T. Foley, “Scattering of electromagnetic fields of any state of coherence from space–time fluctuations,” Phys. Rev. A 40, 579–587 (1989).
[Crossref] [PubMed]

E. Wolf, “Non-cosmological redshifts of spectral line,” Nature (London) 326, 363–365 (1987).
[Crossref]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[Crossref] [PubMed]

L. Mandel, E. Wolf, “Constitutive relations and the electromagnetic spectrum in a fluctuating medium,” Opt. Commun. 8, 95–99 (1973).
[Crossref]

See, for example, E. Wolf, “Influence of source-correlations on spectra of radiated fields,” in International Trends in Optics, J. W. Goodman, ed. (Academic, New York, 1991), pp. 221–232, and references cited therein.
[Crossref]

Astrophys. J. (1)

D. F. V. James, M. P. Savedoff, E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Astrophys. J. 359, 67–71 (1990).
[Crossref]

J. Geophys. Res. (1)

The present discussion of the applicability of the Born and Rytov approximations is a little bit qualitative. However, the same consequence has been given quantitatively in a recent paper. See T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[Crossref]

J. Mod. Opt. (2)

E. Wolf, T. Habashy, “Invisible bodies and uniqueness of the inverse scattering problem,” J. Mod. Opt. 40, 785–792 (1993).
[Crossref]

E. Wolf, “Two inverse problems in spectroscopy with partially coherent sources and the scaling law,” J. Mod. Opt. 39, 9–20 (1992).
[Crossref]

J. Opt. Soc. Am. (1)

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

Nature (London) (1)

E. Wolf, “Non-cosmological redshifts of spectral line,” Nature (London) 326, 363–365 (1987).
[Crossref]

Opt. Commun. (4)

T. Shirai, T. Asakura, “Spectral changes of light radiated by three-dimensional, anisotropic Gaussian Schell-model sources,” Opt. Commun. 105, 22–28 (1994).
[Crossref]

L. Mandel, E. Wolf, “Constitutive relations and the electromagnetic spectrum in a fluctuating medium,” Opt. Commun. 8, 95–99 (1973).
[Crossref]

W. H. Carter, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[Crossref]

B. Cairns, E. Wolf, “Comparison of the Born and the Rytov approximations for scattering on quasi-homogeneous media,” Opt. Commun. 74, 284–289 (1990).
[Crossref]

Phys. Lett. A (3)

D. F. V. James, E. Wolf, “Doppler-like frequency shifts generated by dynamic scattering,” Phys. Lett. A 146, 167–171 (1990).
[Crossref]

It was demonstrated in a recent paper that, unlike the widely recognized aspect, the Doppler effect yields not only the shift of the spectral lines but also the distortion of them, even in the situation in which one observes the spectrum of light produced by a uniformly moving source. See D. F. V. James, “The spectrum of radiation from a moving source,” Phys. Lett. A 140, 213–217 (1989).
[Crossref]

A. Lagendijk, “Terrestrial redshifts from a diffuse light source,” Phys. Lett. A 147, 389–392 (1990).
[Crossref]

Phys. Rev. A (2)

E. Wolf, J. T. Foley, “Scattering of electromagnetic fields of any state of coherence from space–time fluctuations,” Phys. Rev. A 40, 579–587 (1989).
[Crossref] [PubMed]

J. T. Foley, E. Wolf, “Frequency shifts of spectral lines generated by scattering from space–time fluctuations,” Phys. Rev. A 40, 588–598 (1989).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

E. Wolf, “Correlation-induced Doppler-like frequency shifts of spectral lines,” Phys. Rev. Lett. 63, 2220–2223 (1989).
[Crossref] [PubMed]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[Crossref] [PubMed]

A. Lagendijk, “Comment on ‘Invariance of the spectrum of light on propagation’,” Phys. Rev. Lett. 65, 2082 (1990).
[Crossref] [PubMed]

Radio Sci. (1)

D. F. V. James, E. Wolf, “Determination of field correlations from spectral measurements with application to synthetic aperture imaging,” Radio Sci. 26, 1239–1243 (1991).
[Crossref]

Space Sci. Rev. (1)

J. V. Narlikar, “Noncosmological redshifts,” Space Sci. Rev. 50, 523–614 (1989).
[Crossref]

Other (9)

See, for example, E. Wolf, “Influence of source-correlations on spectra of radiated fields,” in International Trends in Optics, J. W. Goodman, ed. (Academic, New York, 1991), pp. 221–232, and references cited therein.
[Crossref]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

See, for example, J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, Berlin, 1978).
[Crossref]

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 1.

T. Shirai, T. Asakura, “Spectral changes of light scattered by spatially random media under the Rytov approximation,” in Meeting Digest, Topical Meeting of the International Commission for Optics, Frontiers in Information Optics, Kyoto, Japan, (Organizing Committees, Kyoto, 1994), p. 354.

To the authors’ knowledge, the relationship between the Rytov approximation and multiple scattering has rarely been discussed in most of the existing literature (Ref. 18 is an exception). However, from the discussion given in the present section, one may say that the Rytov solution describes some effects of multiple scattering.

See, for example, Chap. 17 of Ref. 16, and J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8. The treatment of the problem in these references is somewhat different from that in the present paper, mainly arising from the difference in the expression for the constitutive relation. However, both treatments are essentially the same as far as the condition regarding the depolarization effects is concerned.

Equation (33) may be easily obtained with the help of the characteristic function for the sequence of the Gaussian random variables. See, for example, A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, Singapore, 1984), Sec. 8.3.

The derivation of Eq. (45) is somewhat analogous to that of Eq. (43). See also Ref. 26.

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Figures (5)

Fig. 1
Fig. 1

Geometry for analysis. A polychromatic plane wave of light is incident onto the scattering medium. The symbols s0 and s are unit vectors specifying directions of the incidence and the scattering, respectively.

Fig. 2
Fig. 2

Normalized spectral densities of the scattered light observed in the forward-scattering direction (θ = 0°) and at the angle θ = 20°. The dashed curve indicates the normalized spectral density of the incident light. These curves are plotted under the conditions that q = 1.0, (ω0/c)σ = 10.0, and the rms width of the incident spectrum is 0.1ω0.

Fig. 3
Fig. 3

Filter functions plotted against the relative frequency ω/ω0 for various angles θ of scattering obtained under (a) the first-order Born approximation and (b) the first-order Rytov approximation with q = 1.0 in the case of (ω0/c)σ = 10.0. These curves are normalized by the value at ω/ω0 = 1.0, so that the functions have the value of unity at ω/ω0 = 1.0.

Fig. 4
Fig. 4

Relative shift z of the peak spectrum of the scattered light plotted against the scattering angle θ for various values of q characterizing the degree of multiple scattering. These curves are plotted under the condition that the correlation width of the medium is (ω0/c)σ = 10.0 and the rms width of the incident spectrum is 0.1ω0. The dashed curve indicates the result obtained under the first-order Born approximation.

Fig. 5
Fig. 5

Spatial distributions of the spectral densities of the scattered monochromatic light for various values of q. These curves are drawn on the basis of Eq. (56) under the conditions that ω = ω0 and (ω0/c)σ = 10.0. The dashed curve indicates the result obtained under the first-order Born approximation.

Equations (64)

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× × E ( r , t ) + 1 c 2 2 E ( r , t ) t 2 = - 4 π c 2 2 P ( r , t ) t 2 ,
P ( r , t ) = 1 2 π 0 η ( r , t ; t ) E ( r , t - t ) d t .
E ˜ ( r , ω ) = 1 2 π - E ( r , t ) exp ( j ω t ) d t ,
P ˜ ( r , ω ) = 1 2 π - P ( r , t ) exp ( j ω t ) d t ,
η ^ ( r , t ; ω ) = 1 2 π 0 η ( r , t ; t ) exp ( j ω t ) d t ,
× × E ˜ ( r , ω ) - k 2 E ˜ ( r , ω ) = 4 π k 2 P ˜ ( r , ω ) ,
P ˜ ( r , ω ) = - η ¯ ( r , ω - ω ; ω ) E ˜ ( r , ω ) d ω ,
η ¯ ( r , ω ; ω ) = 1 2 π - η ^ ( r , t ; ω ) exp ( j ω t ) d t .
× × a = ( · a ) - 2 a ,
2 E ˜ ( r , ω ) + k 2 E ˜ ( r , ω ) = [ · E ˜ ( r , ω ) ] - 4 π k 2 P ˜ ( r , ω ) .
[ · E ˜ ( r , ω ) ] = - 4 π [ · P ˜ ( r , ω ) ] .
( 2 + k 2 ) E ˜ ( r , ω ) = - 4 π k 2 P ˜ ( r , ω ) .
( 2 + k 2 ) E ˜ ( r , ω ) = - 4 π k 2 P ˜ ( r , ω ) ,
P ˜ ( r , ω ) = - η ¯ ( r , ω - ω ; ω ) E ˜ ( r , ω ) d ω ,
η ^ ( r , t ; ω ) η ^ ( r ; ω ) ,
η ¯ ( r , ω ; ω ) = η ^ ( r ; ω ) δ ( ω ) .
P ˜ ( r , ω ) = η ^ ( r ; ω ) E ˜ ( r , ω ) .
( 2 + k 2 ) E ˜ ( r , ω ) = - 4 π k 2 η ^ ( r ; ω ) E ˜ ( r , ω ) ,
E ˜ ( i ) ( r , ω ) = a ( ω ) exp ( j k s 0 · r ) ,
S ( 0 ) ( r , ω ) = E ˜ ( i ) * ( r , ω ) E ˜ ( i ) ( r , ω ) ,
S ( 0 ) ( ω ) = a * ( ω ) a ( ω ) ,
E ˜ ( r , ω ) = E ˜ ( i ) ( r , ω ) exp [ ψ ( r , ω ) ] ,
( 2 + k 2 ) E ˜ ( i ) ( r , ω ) = 0 ,
( 2 + k 2 ) [ E ˜ ( i ) ( r , ω ) ψ ( r , ω ) ] = - [ ψ ( r , ω ) · ψ ( r , ω ) + 4 π k 2 η ^ ( r , ω ) ] E ˜ ( i ) ( r , ω ) .
ψ ( r , ω ) = 1 E ˜ ( i ) ( r , ω ) V G ( r , r ; ω ) [ ψ ( r , ω ) · ψ ( r , ω ) + 4 π k 2 η ^ ( r , ω ) ] E ˜ ( i ) ( r , ω ) d 3 r ,
G ( r , r ; ω ) = exp ( j k r - r ) 4 π r - r
ψ 1 ( r , ω ) = 4 π k 2 E ˜ ( i ) ( r , ω ) × V G ( r , r ; ω ) η ^ ( r , ω ) E ˜ ( i ) ( r , ω ) d 3 r .
E ˜ ( r , ω ) = E ˜ ( i ) ( r , ω ) × exp [ 4 π k 2 E ˜ ( i ) ( r , ω ) V G ( r , r ; ω ) η ^ ( r , ω ) E ˜ ( i ) ( r , ω ) d 3 r ] .
G ( r s , r ; ω ) ~ exp ( j k r ) 4 π r exp ( - j k s · r )
S T ( ) ( r , ω ) = E ˜ * ( r , ω ) E ˜ ( r , ω ) ,
S T ( ) ( r s , ω ) = a * ( ω ) a ( ω ) exp ( k 2 r { exp [ j k ( s 0 · r - r ) ] × V η ^ ( r , ω ) exp [ j k ( s - s 0 ) · r ] d 3 r + exp [ - j k ( s 0 · r - r ) ] × V η ^ ( r , ω ) exp [ - j k ( s - s 0 ) · r ] d 3 r } ) ,
S T ( ) ( r s , ω ) = S ( 0 ) ( ω ) exp { 2 k 2 r V η ^ ( r , ω ) × cos [ k ( s - s 0 ) · ( r - r ) ] d 3 r } ,
S T ( ) ( r , ω ) m = S ( 0 ) ( ω ) exp { 2 k 4 r 2 V V C ( r 1 , r 2 ; ω ) × cos [ k ( s - s 0 ) · ( r 1 - r ) ] × cos [ k ( s - s 0 ) · ( r 2 - r ) ] d 3 r 1 d 3 r 2 } ,
C ( r 1 , r 2 ; ω ) = η ^ * ( r 1 ; ω ) η ^ ( r 2 ; ω ) m
C ( r 1 , r 2 ; ω ) = I ( r 1 + r 2 2 ; ω ) g ( r 2 - r 1 ; ω ) ,
I ( r ; ω ) = η ^ ( r ; ω ) 2 m
g ( r 2 - r 1 ; ω ) = C ( r 1 , r 2 ; ω ) [ C ( r 1 , r 1 ; ω ) ] 1 / 2 [ C ( r 2 , r 2 ; ω ) ] 1 / 2
S T ( ) ( r , ω ) m = S ( 0 ) ( ω ) exp ( k 4 r 2 V d 3 r V d 3 r × I ( r ; ω ) g ( r ; ω ) { cos [ k ( s - s 0 ) · r ] + cos [ 2 k ( s - s 0 ) · r ] cos [ 2 k ( s - s 0 ) · r ] + sin [ 2 k ( s - s 0 ) · r ] sin [ 2 k ( s - s 0 ) · r ] } ) ,
r = r 2 - r 1 ,             r = r 1 + r 2 2 ,
S T ( ) ( r , ω ) m = S ( 0 ) ( ω ) exp { k 4 D r 2 V g ( r ; ω ) × exp [ - j k ( s - s 0 ) · r ] d 3 r } ,
D = V I ( r ; ω ) d 3 r
S ( ) ( r , ω ) m = S T ( ) ( r , ω ) m - S ( 0 ) ( ω ) .
S ( ) ( r , ω ) m = S ( 0 ) ( ω ) ( exp { k 4 D r 2 V g ( r ; ω ) × exp [ - j k ( s - s 0 ) · r ] d 3 r } - 1 ) ,
σ χ 2 < 0.2 ~ 0.5 ,
σ χ 2 = k 4 D 2 r 2 V g ( r ; ω ) exp [ - j k ( s - s 0 ) · r ] d 3 r ,
E ˜ ( r , ω ) = E ˜ ( i ) ( r , ω ) + E ˜ ( s ) ( r , ω ) ,
( 2 + k 2 ) E ˜ ( s ) ( r , ω ) = - 4 π k 2 η ^ ( r , ω ) E ˜ ( r , ω ) .
E ˜ ( i ) ( r , ω ) E ˜ ( s ) ( r , ω ) .
E ˜ ( s ) ( r , ω ) = 4 π k 2 V G ( r , r ; ω ) η ^ ( r , ω ) E ˜ ( i ) ( r , ω ) d 3 r ,
S ( ) ( r , ω ) m = S ( 0 ) ( ω ) k 4 D r 2 V g ( r ; ω ) × exp [ - j k ( s - s 0 ) · r ] d 3 r ,
k 4 D r 2 V g ( r ; ω ) exp [ - j k ( s - s 0 ) · r ] d 3 r 1 ,
g ( r ; ω ) = exp ( - r 2 2 σ 2 ) ,
S ( 0 ) ( ω ) = A exp [ - ( ω - ω 0 ) 2 2 Γ 0 2 ] ,
S ( ) ( θ , ω ) m = A exp [ - ( ω - ω 0 ) 2 2 Γ 0 2 ] × ( exp { D r 2 ( ω c ) 4 ( 2 π σ ) 3 × exp [ - 2 ( ω c σ ) 2 sin 2 ( θ 2 ) ] } - 1 ) ,
q = D r 2 ( ω 0 c ) 4 ( 2 π σ ) 3 .
S ( ) ( θ , ω ) m = A exp [ - ( ω - ω 0 ) 2 2 Γ 0 2 ] ( exp { q ( ω ω 0 ) 4 × exp [ - 2 ( ω x σ ) 2 sin 2 ( θ 2 ) ] } - 1 ) .
q ( ω ω 0 ) 4 exp [ - 2 ( ω c σ ) 2 sin 2 ( θ 2 ) ] 1.
1 2 q ( ω ω 0 ) 4 exp [ - 2 ( ω c σ ) 2 sin 2 ( θ 2 ) ] < 0.2 ~ 0.5.
F B ( ω , θ ) = ( ω ω 0 ) 4 exp { - 2 [ ( ω ω 0 ) 2 - 1 ] ( ω 0 c σ ) 2 × { sin 2 ( θ 2 ) }
F R ( ω , θ ) = exp { q ( ω ω 0 ) 4 exp [ - 2 ( ω c σ ) 2 sin 2 ( θ 2 ) ] } - 1 exp { q exp [ - 2 ( ω 0 c σ ) 2 sin 2 ( θ 2 ) ] } - 1
z = ω 0 - ω ω ,
F 1 = V g ( r ; ω ) cos [ k ( s - s 0 ) · r ] d 3 r ,
F 2 = V I ( r ; ω ) cos [ 2 k ( s - s 0 ) · r ] d 3 r ,
F 3 = V I ( r ; ω ) sin [ 2 k ( s - s 0 ) · r ] d 3 r .

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