Abstract

Surface reflections in multiple-scattering media are treated theoretically for the coherent backscattering of light, the optical memory effect, time correlations in dynamical systems, and frequency correlations in static systems. A finite number of surface reflections and a finite separation between the surface of the scattering medium and the reflecting surface are included in the theory and are found to induce a variety of new effects. Simple, closed-form expressions are given for one, two, and three dimensions. The theory is found to be in good accord with recent experiments on the coherent backscattering of light.

© 1994 Optical Society of America

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.
  2. Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
    [CrossRef]
  3. M. P. van Albada, A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
    [CrossRef] [PubMed]
  4. P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
    [CrossRef] [PubMed]
  5. E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media; analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
    [CrossRef] [PubMed]
  6. S. Etemad, R. Thompson, M. J. Andrejco, “Weak localization of photons: universal fluctuations and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
    [CrossRef] [PubMed]
  7. M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
    [CrossRef] [PubMed]
  8. M. J. Stephen, G. Cwillich, “Rayleigh scattering and weak localization: effects of polarization,” Phys. Rev. B 34, 7564–7572 (1986).
    [CrossRef]
  9. E. Akkermans, “Propagation d’ondes dans les milieux desordennés,” Ph.D. dissertation (University of Grenoble, Grenoble, France, 1986).
  10. F. C. MacKintosh, S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and non-parity conserving media,” Phys. Rev. B 37, 884–897 (1988).
    [CrossRef]
  11. E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
    [CrossRef]
  12. S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
    [CrossRef] [PubMed]
  13. I. Freund, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
    [CrossRef] [PubMed]
  14. I. Freund, M. Rosenbluh, R. Berkovits, “Geometric scaling of the optical memory effect in coherent-wave propagation through random media,” Phys. Rev. B 39, 12,403–12,406 (1989).
    [CrossRef]
  15. G. Maret, P. E. Wolf, “Multiple light scattering from disordered media: the effect of brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
    [CrossRef]
  16. M. Kaveh, M. Rosenbluh, I. Freund, “Speckle patterns permit direct observation of phase breaking,” Nature (London) 326, 778–780 (1987).
    [CrossRef]
  17. M. Rosenbluh, M. Hoshen, I. Freund, M. Kaveh, “Time evolution of universal optical fluctuations,” Phys. Rev. Lett. 58, 2754–2757 (1987).
    [CrossRef] [PubMed]
  18. M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
    [CrossRef]
  19. D. J. Pine, D. A. Weitz, P. M. Chaiken, E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
    [CrossRef] [PubMed]
  20. D. N. Qu, J. C. Dainty, “Polarization dependence of dynamic light scattering by dense disordered media,” Opt. Lett. 13, 1066–1068 (1988).
    [CrossRef] [PubMed]
  21. F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
    [CrossRef]
  22. I. Freund, M. Kaveh, “Comment on ‘Polarization memory of multiply scattered light,’” Phys. Rev. B 45, 8162–8164 (1992).
    [CrossRef]
  23. B. Shapiro, “Large intensity fluctuations for wave propagation in random media,” Phys. Rev. Lett. 57, 2168–2171 (1986).
    [CrossRef] [PubMed]
  24. M. J. Stephen, G. Cwillich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59, 285–288 (1987).
    [CrossRef] [PubMed]
  25. A. Z. Genack, “Optical transmission in disordered media,” Phys. Rev. Lett. 58, 2043–2046 (1987).
    [CrossRef] [PubMed]
  26. I. Edrei, M. Kaveh, “Wavelength dependence of static intensity correlation functions,” Phys. Rev. B 38, 950–953 (1988).
    [CrossRef]
  27. A. Z. Genack, J. M. Drake, “Relationship between optical intensity, fluctuations, and pulse propagation in random media,” Europhys. Lett. 11, 331–336 (1990).
    [CrossRef]
  28. A. Lagendijk, R. Vreeker, P. De Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
    [CrossRef]
  29. I. Freund, R. Berkovits, “Surface reflections and optical transport through random media: coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B 41, 496–503 (1990); in the definition of xfollowing Eq. (8) in this reference, xshould be multiplied by 2 [Erratum, Phys. Rev. B 41, 9540 (1990).]
    [CrossRef]
  30. J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffuse light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
    [CrossRef] [PubMed]
  31. D. Ben-Avraham, H. Taitelbaum, G. H. Weiss, “Boundary conditions for a model of photon migration in turbid media,” Lasers Life Sci. 4, 1–8 (1991).
  32. I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
    [CrossRef] [PubMed]
  33. Th. M. Nieuwenhuizen, J. M. Luck, “The skin layer in diffusive media,” Phys. Rev. E 48, 569–588 (1993).
    [CrossRef]
  34. M. Ospeck, S. Fraden, “Boundary conditions and the weak localization of light,” Phys. Rev. E 49, 4578–4589 (1994).
    [CrossRef]
  35. I. Freund, M. Rosenbluh, R. Berkovits, M. Kaveh, “Coherent backscattering of light in a quasi-two-dimensional system,” Phys. Rev. Lett. 61, 1214–1217 (1988).
    [CrossRef] [PubMed]
  36. I. Freund, “Polarization correlations in two-dimensional random media,” Waves Random Media 1, 245–263 (1991).
    [CrossRef]
  37. P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 2.
  38. P. M. Saulnier, G. H. Watson, “Role of surface reflectivity in coherent backscattering measurements,” Opt. Lett. 17, 946–948 (1992).
    [CrossRef] [PubMed]
  39. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1959).
  40. R. Berkowits, M. Kaveh, “Weak localization of photons in a disordered two-dimensional medium,” J. Phys. C 20, L181–L185 (1987).
    [CrossRef]
  41. G. Cwillich, M. J. Stephen, “Rayleigh scattering and weak localization: geometric effects and fluctuations,” Phys. Rev. B 35, 6517–6520 (1987).
    [CrossRef]
  42. M. Abramowits, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964).

1994 (1)

M. Ospeck, S. Fraden, “Boundary conditions and the weak localization of light,” Phys. Rev. E 49, 4578–4589 (1994).
[CrossRef]

1993 (1)

Th. M. Nieuwenhuizen, J. M. Luck, “The skin layer in diffusive media,” Phys. Rev. E 48, 569–588 (1993).
[CrossRef]

1992 (3)

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

I. Freund, M. Kaveh, “Comment on ‘Polarization memory of multiply scattered light,’” Phys. Rev. B 45, 8162–8164 (1992).
[CrossRef]

P. M. Saulnier, G. H. Watson, “Role of surface reflectivity in coherent backscattering measurements,” Opt. Lett. 17, 946–948 (1992).
[CrossRef] [PubMed]

1991 (3)

I. Freund, “Polarization correlations in two-dimensional random media,” Waves Random Media 1, 245–263 (1991).
[CrossRef]

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffuse light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

D. Ben-Avraham, H. Taitelbaum, G. H. Weiss, “Boundary conditions for a model of photon migration in turbid media,” Lasers Life Sci. 4, 1–8 (1991).

1990 (2)

I. Freund, R. Berkovits, “Surface reflections and optical transport through random media: coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B 41, 496–503 (1990); in the definition of xfollowing Eq. (8) in this reference, xshould be multiplied by 2 [Erratum, Phys. Rev. B 41, 9540 (1990).]
[CrossRef]

A. Z. Genack, J. M. Drake, “Relationship between optical intensity, fluctuations, and pulse propagation in random media,” Europhys. Lett. 11, 331–336 (1990).
[CrossRef]

1989 (3)

A. Lagendijk, R. Vreeker, P. De Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

I. Freund, M. Rosenbluh, R. Berkovits, “Geometric scaling of the optical memory effect in coherent-wave propagation through random media,” Phys. Rev. B 39, 12,403–12,406 (1989).
[CrossRef]

1988 (9)

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
[CrossRef]

D. J. Pine, D. A. Weitz, P. M. Chaiken, E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

D. N. Qu, J. C. Dainty, “Polarization dependence of dynamic light scattering by dense disordered media,” Opt. Lett. 13, 1066–1068 (1988).
[CrossRef] [PubMed]

F. C. MacKintosh, S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and non-parity conserving media,” Phys. Rev. B 37, 884–897 (1988).
[CrossRef]

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

I. Edrei, M. Kaveh, “Wavelength dependence of static intensity correlation functions,” Phys. Rev. B 38, 950–953 (1988).
[CrossRef]

I. Freund, M. Rosenbluh, R. Berkovits, M. Kaveh, “Coherent backscattering of light in a quasi-two-dimensional system,” Phys. Rev. Lett. 61, 1214–1217 (1988).
[CrossRef] [PubMed]

1987 (7)

M. J. Stephen, G. Cwillich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59, 285–288 (1987).
[CrossRef] [PubMed]

A. Z. Genack, “Optical transmission in disordered media,” Phys. Rev. Lett. 58, 2043–2046 (1987).
[CrossRef] [PubMed]

G. Maret, P. E. Wolf, “Multiple light scattering from disordered media: the effect of brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

M. Kaveh, M. Rosenbluh, I. Freund, “Speckle patterns permit direct observation of phase breaking,” Nature (London) 326, 778–780 (1987).
[CrossRef]

M. Rosenbluh, M. Hoshen, I. Freund, M. Kaveh, “Time evolution of universal optical fluctuations,” Phys. Rev. Lett. 58, 2754–2757 (1987).
[CrossRef] [PubMed]

R. Berkowits, M. Kaveh, “Weak localization of photons in a disordered two-dimensional medium,” J. Phys. C 20, L181–L185 (1987).
[CrossRef]

G. Cwillich, M. J. Stephen, “Rayleigh scattering and weak localization: geometric effects and fluctuations,” Phys. Rev. B 35, 6517–6520 (1987).
[CrossRef]

1986 (5)

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media; analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

S. Etemad, R. Thompson, M. J. Andrejco, “Weak localization of photons: universal fluctuations and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

M. J. Stephen, G. Cwillich, “Rayleigh scattering and weak localization: effects of polarization,” Phys. Rev. B 34, 7564–7572 (1986).
[CrossRef]

B. Shapiro, “Large intensity fluctuations for wave propagation in random media,” Phys. Rev. Lett. 57, 2168–2171 (1986).
[CrossRef] [PubMed]

1985 (2)

M. P. van Albada, A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

1984 (1)

Abramowits, M.

M. Abramowits, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964).

Akkermans, E.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media; analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

E. Akkermans, “Propagation d’ondes dans les milieux desordennés,” Ph.D. dissertation (University of Grenoble, Grenoble, France, 1986).

Andrejco, M. J.

S. Etemad, R. Thompson, M. J. Andrejco, “Weak localization of photons: universal fluctuations and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

Ben-Avraham, D.

D. Ben-Avraham, H. Taitelbaum, G. H. Weiss, “Boundary conditions for a model of photon migration in turbid media,” Lasers Life Sci. 4, 1–8 (1991).

Berkovits, R.

I. Freund, R. Berkovits, “Surface reflections and optical transport through random media: coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B 41, 496–503 (1990); in the definition of xfollowing Eq. (8) in this reference, xshould be multiplied by 2 [Erratum, Phys. Rev. B 41, 9540 (1990).]
[CrossRef]

I. Freund, M. Rosenbluh, R. Berkovits, “Geometric scaling of the optical memory effect in coherent-wave propagation through random media,” Phys. Rev. B 39, 12,403–12,406 (1989).
[CrossRef]

I. Freund, M. Rosenbluh, R. Berkovits, M. Kaveh, “Coherent backscattering of light in a quasi-two-dimensional system,” Phys. Rev. Lett. 61, 1214–1217 (1988).
[CrossRef] [PubMed]

Berkowits, R.

R. Berkowits, M. Kaveh, “Weak localization of photons in a disordered two-dimensional medium,” J. Phys. C 20, L181–L185 (1987).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1959).

Chaiken, P. M.

D. J. Pine, D. A. Weitz, P. M. Chaiken, E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Cwillich, G.

M. J. Stephen, G. Cwillich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59, 285–288 (1987).
[CrossRef] [PubMed]

G. Cwillich, M. J. Stephen, “Rayleigh scattering and weak localization: geometric effects and fluctuations,” Phys. Rev. B 35, 6517–6520 (1987).
[CrossRef]

M. J. Stephen, G. Cwillich, “Rayleigh scattering and weak localization: effects of polarization,” Phys. Rev. B 34, 7564–7572 (1986).
[CrossRef]

Dainty, J. C.

De Vries, P.

A. Lagendijk, R. Vreeker, P. De Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

Drake, J. M.

A. Z. Genack, J. M. Drake, “Relationship between optical intensity, fluctuations, and pulse propagation in random media,” Europhys. Lett. 11, 331–336 (1990).
[CrossRef]

Edrei, I.

I. Edrei, M. Kaveh, “Wavelength dependence of static intensity correlation functions,” Phys. Rev. B 38, 950–953 (1988).
[CrossRef]

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

Etemad, S.

S. Etemad, R. Thompson, M. J. Andrejco, “Weak localization of photons: universal fluctuations and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

Feng, S.

I. Freund, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Feschbach, H.

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 2.

Fraden, S.

M. Ospeck, S. Fraden, “Boundary conditions and the weak localization of light,” Phys. Rev. E 49, 4578–4589 (1994).
[CrossRef]

Freund, I.

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

I. Freund, M. Kaveh, “Comment on ‘Polarization memory of multiply scattered light,’” Phys. Rev. B 45, 8162–8164 (1992).
[CrossRef]

I. Freund, “Polarization correlations in two-dimensional random media,” Waves Random Media 1, 245–263 (1991).
[CrossRef]

I. Freund, R. Berkovits, “Surface reflections and optical transport through random media: coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B 41, 496–503 (1990); in the definition of xfollowing Eq. (8) in this reference, xshould be multiplied by 2 [Erratum, Phys. Rev. B 41, 9540 (1990).]
[CrossRef]

I. Freund, M. Rosenbluh, R. Berkovits, “Geometric scaling of the optical memory effect in coherent-wave propagation through random media,” Phys. Rev. B 39, 12,403–12,406 (1989).
[CrossRef]

I. Freund, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, R. Berkovits, M. Kaveh, “Coherent backscattering of light in a quasi-two-dimensional system,” Phys. Rev. Lett. 61, 1214–1217 (1988).
[CrossRef] [PubMed]

M. Kaveh, M. Rosenbluh, I. Freund, “Speckle patterns permit direct observation of phase breaking,” Nature (London) 326, 778–780 (1987).
[CrossRef]

M. Rosenbluh, M. Hoshen, I. Freund, M. Kaveh, “Time evolution of universal optical fluctuations,” Phys. Rev. Lett. 58, 2754–2757 (1987).
[CrossRef] [PubMed]

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

Genack, A. Z.

A. Z. Genack, J. M. Drake, “Relationship between optical intensity, fluctuations, and pulse propagation in random media,” Europhys. Lett. 11, 331–336 (1990).
[CrossRef]

A. Z. Genack, “Optical transmission in disordered media,” Phys. Rev. Lett. 58, 2043–2046 (1987).
[CrossRef] [PubMed]

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, P. M. Chaiken, E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Hoshen, M.

M. Rosenbluh, M. Hoshen, I. Freund, M. Kaveh, “Time evolution of universal optical fluctuations,” Phys. Rev. Lett. 58, 2754–2757 (1987).
[CrossRef] [PubMed]

Ishimaru, A.

Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

John, S.

F. C. MacKintosh, S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and non-parity conserving media,” Phys. Rev. B 37, 884–897 (1988).
[CrossRef]

Kane, C.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Kaveh, M.

I. Freund, M. Kaveh, “Comment on ‘Polarization memory of multiply scattered light,’” Phys. Rev. B 45, 8162–8164 (1992).
[CrossRef]

I. Edrei, M. Kaveh, “Wavelength dependence of static intensity correlation functions,” Phys. Rev. B 38, 950–953 (1988).
[CrossRef]

I. Freund, M. Rosenbluh, R. Berkovits, M. Kaveh, “Coherent backscattering of light in a quasi-two-dimensional system,” Phys. Rev. Lett. 61, 1214–1217 (1988).
[CrossRef] [PubMed]

R. Berkowits, M. Kaveh, “Weak localization of photons in a disordered two-dimensional medium,” J. Phys. C 20, L181–L185 (1987).
[CrossRef]

M. Rosenbluh, M. Hoshen, I. Freund, M. Kaveh, “Time evolution of universal optical fluctuations,” Phys. Rev. Lett. 58, 2754–2757 (1987).
[CrossRef] [PubMed]

M. Kaveh, M. Rosenbluh, I. Freund, “Speckle patterns permit direct observation of phase breaking,” Nature (London) 326, 778–780 (1987).
[CrossRef]

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

Kuga, Y.

Lagendijk, A.

A. Lagendijk, R. Vreeker, P. De Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

M. P. van Albada, A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

Lee, P. A.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Luck, J. M.

Th. M. Nieuwenhuizen, J. M. Luck, “The skin layer in diffusive media,” Phys. Rev. E 48, 569–588 (1993).
[CrossRef]

MacKintosh, F. C.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

F. C. MacKintosh, S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and non-parity conserving media,” Phys. Rev. B 37, 884–897 (1988).
[CrossRef]

Maret, G.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

G. Maret, P. E. Wolf, “Multiple light scattering from disordered media: the effect of brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Maynard, R.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media; analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

Morse, P. M.

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 2.

Nieuwenhuizen, Th. M.

Th. M. Nieuwenhuizen, J. M. Luck, “The skin layer in diffusive media,” Phys. Rev. E 48, 569–588 (1993).
[CrossRef]

Ospeck, M.

M. Ospeck, S. Fraden, “Boundary conditions and the weak localization of light,” Phys. Rev. E 49, 4578–4589 (1994).
[CrossRef]

Pine, D. J.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffuse light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

D. J. Pine, D. A. Weitz, P. M. Chaiken, E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Qu, D. N.

Rosenbluh, M.

I. Freund, M. Rosenbluh, R. Berkovits, “Geometric scaling of the optical memory effect in coherent-wave propagation through random media,” Phys. Rev. B 39, 12,403–12,406 (1989).
[CrossRef]

I. Freund, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, R. Berkovits, M. Kaveh, “Coherent backscattering of light in a quasi-two-dimensional system,” Phys. Rev. Lett. 61, 1214–1217 (1988).
[CrossRef] [PubMed]

M. Kaveh, M. Rosenbluh, I. Freund, “Speckle patterns permit direct observation of phase breaking,” Nature (London) 326, 778–780 (1987).
[CrossRef]

M. Rosenbluh, M. Hoshen, I. Freund, M. Kaveh, “Time evolution of universal optical fluctuations,” Phys. Rev. Lett. 58, 2754–2757 (1987).
[CrossRef] [PubMed]

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

Saulnier, P. M.

Shapiro, B.

B. Shapiro, “Large intensity fluctuations for wave propagation in random media,” Phys. Rev. Lett. 57, 2168–2171 (1986).
[CrossRef] [PubMed]

Stegun, I. A.

M. Abramowits, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964).

Stephen, M. J.

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
[CrossRef]

M. J. Stephen, G. Cwillich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59, 285–288 (1987).
[CrossRef] [PubMed]

G. Cwillich, M. J. Stephen, “Rayleigh scattering and weak localization: geometric effects and fluctuations,” Phys. Rev. B 35, 6517–6520 (1987).
[CrossRef]

M. J. Stephen, G. Cwillich, “Rayleigh scattering and weak localization: effects of polarization,” Phys. Rev. B 34, 7564–7572 (1986).
[CrossRef]

Stone, A. D.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Taitelbaum, H.

D. Ben-Avraham, H. Taitelbaum, G. H. Weiss, “Boundary conditions for a model of photon migration in turbid media,” Lasers Life Sci. 4, 1–8 (1991).

Thompson, R.

S. Etemad, R. Thompson, M. J. Andrejco, “Weak localization of photons: universal fluctuations and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

van Albada, M. P.

M. P. van Albada, A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

Vreeker, R.

A. Lagendijk, R. Vreeker, P. De Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

Watson, G. H.

Weiss, G. H.

D. Ben-Avraham, H. Taitelbaum, G. H. Weiss, “Boundary conditions for a model of photon migration in turbid media,” Lasers Life Sci. 4, 1–8 (1991).

Weitz, D. A.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffuse light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

D. J. Pine, D. A. Weitz, P. M. Chaiken, E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1959).

Wolf, P. E.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

G. Maret, P. E. Wolf, “Multiple light scattering from disordered media: the effect of brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media; analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Zhu, J. X.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffuse light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Europhys. Lett. (1)

A. Z. Genack, J. M. Drake, “Relationship between optical intensity, fluctuations, and pulse propagation in random media,” Europhys. Lett. 11, 331–336 (1990).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. (Paris) (1)

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

J. Phys. C (1)

R. Berkowits, M. Kaveh, “Weak localization of photons in a disordered two-dimensional medium,” J. Phys. C 20, L181–L185 (1987).
[CrossRef]

Lasers Life Sci. (1)

D. Ben-Avraham, H. Taitelbaum, G. H. Weiss, “Boundary conditions for a model of photon migration in turbid media,” Lasers Life Sci. 4, 1–8 (1991).

Nature (London) (1)

M. Kaveh, M. Rosenbluh, I. Freund, “Speckle patterns permit direct observation of phase breaking,” Nature (London) 326, 778–780 (1987).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

A. Lagendijk, R. Vreeker, P. De Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

Phys. Rev. A (2)

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffuse light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

Phys. Rev. B (9)

G. Cwillich, M. J. Stephen, “Rayleigh scattering and weak localization: geometric effects and fluctuations,” Phys. Rev. B 35, 6517–6520 (1987).
[CrossRef]

I. Edrei, M. Kaveh, “Wavelength dependence of static intensity correlation functions,” Phys. Rev. B 38, 950–953 (1988).
[CrossRef]

I. Freund, R. Berkovits, “Surface reflections and optical transport through random media: coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B 41, 496–503 (1990); in the definition of xfollowing Eq. (8) in this reference, xshould be multiplied by 2 [Erratum, Phys. Rev. B 41, 9540 (1990).]
[CrossRef]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

I. Freund, M. Kaveh, “Comment on ‘Polarization memory of multiply scattered light,’” Phys. Rev. B 45, 8162–8164 (1992).
[CrossRef]

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
[CrossRef]

I. Freund, M. Rosenbluh, R. Berkovits, “Geometric scaling of the optical memory effect in coherent-wave propagation through random media,” Phys. Rev. B 39, 12,403–12,406 (1989).
[CrossRef]

M. J. Stephen, G. Cwillich, “Rayleigh scattering and weak localization: effects of polarization,” Phys. Rev. B 34, 7564–7572 (1986).
[CrossRef]

F. C. MacKintosh, S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and non-parity conserving media,” Phys. Rev. B 37, 884–897 (1988).
[CrossRef]

Phys. Rev. E (2)

Th. M. Nieuwenhuizen, J. M. Luck, “The skin layer in diffusive media,” Phys. Rev. E 48, 569–588 (1993).
[CrossRef]

M. Ospeck, S. Fraden, “Boundary conditions and the weak localization of light,” Phys. Rev. E 49, 4578–4589 (1994).
[CrossRef]

Phys. Rev. Lett. (13)

I. Freund, M. Rosenbluh, R. Berkovits, M. Kaveh, “Coherent backscattering of light in a quasi-two-dimensional system,” Phys. Rev. Lett. 61, 1214–1217 (1988).
[CrossRef] [PubMed]

B. Shapiro, “Large intensity fluctuations for wave propagation in random media,” Phys. Rev. Lett. 57, 2168–2171 (1986).
[CrossRef] [PubMed]

M. J. Stephen, G. Cwillich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59, 285–288 (1987).
[CrossRef] [PubMed]

A. Z. Genack, “Optical transmission in disordered media,” Phys. Rev. Lett. 58, 2043–2046 (1987).
[CrossRef] [PubMed]

M. P. van Albada, A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media; analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

S. Etemad, R. Thompson, M. J. Andrejco, “Weak localization of photons: universal fluctuations and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, P. M. Chaiken, E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

M. Rosenbluh, M. Hoshen, I. Freund, M. Kaveh, “Time evolution of universal optical fluctuations,” Phys. Rev. Lett. 58, 2754–2757 (1987).
[CrossRef] [PubMed]

Waves Random Media (1)

I. Freund, “Polarization correlations in two-dimensional random media,” Waves Random Media 1, 245–263 (1991).
[CrossRef]

Z. Phys. B (1)

G. Maret, P. E. Wolf, “Multiple light scattering from disordered media: the effect of brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

Other (5)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

E. Akkermans, “Propagation d’ondes dans les milieux desordennés,” Ph.D. dissertation (University of Grenoble, Grenoble, France, 1986).

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 2.

M. Abramowits, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964).

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1959).

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Figures (4)

Fig. 1
Fig. 1

Relative half-width W of CB as a function of average surface reflectivity. The open squares are the computer data of Ospeck and Fraden.34 The curves are based on the theoretical expressions in Eq. (2): dotted curve, Eq. (2a), Lagendijk et al.28; dashed curve, Eq. (2c), Freund32; solid curve, Eq. (2b), Freund and Berkovits.29

Fig. 2
Fig. 2

CB shape function γ versus Ql* for n2 = 1.5, n3 = 1.33: (a) 3D, R = 0.43; (b) 2D, R = 0.25. Dotted curves, γ0, no surface reflections, Eq. (1); dotted–dashed curves, γ, surface reflections for N = ∞, Eq. (16b) with W = 0; dashed curves, γ1, surface reflection for N = 1, Eq. (16a) with W = 50l*; solid curves, γ, surface reflections for N = ∞, Eq. (16b) with W = 50l*.

Fig. 3
Fig. 3

Time correlation function Γ versus t/〈t1〉 for 3D, R = 0.43. This correlation function is independent of W. Dotted curve, no surface reflections, Eq. (23); dashed curve, surface reflections for N = 1, Eq. (22a); solid curve, surface reflections for N = ∞, Eq. (22c).

Fig. 4
Fig. 4

Intensity–intensity frequency correlation function Ci versus ωτ00 for n2 = 1.5, n3 = 1.33: (a) 3D, R = 0.43; (b) 2D, R = 0.25; (c) 1D, R = 0.5. Dotted curves, C0i, no surface reflections, Eqs. (29b); dotted–dashed curves, Ci, surface reflections for N = ∞, Eq. (31b) with W = 0; dashed curves, C1i, surface reflections for N = 1, Eq. (31a) with W = 50l*; solid curves, Ci, surface reflections for N = ∞, Eq. (31b) with W = 50l*.

Equations (64)

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γ 0 ( Q ) = ( 1 1 + 2 Δ 0 ) [ 1 1 + ( Q l * ) 2 ] × [ 1 + 1 - exp ( - 2 Δ 0 Q l * ) Q l * ] ,
γ L . V . DV ( Q ) = 1 1 + 2 ( + Δ 0 ) [ 1 1 + ( Q l * ) 2 ] × { 1 + 1 Q l * [ 1 + ( Q l * - 1 Q l * + 1 ) exp ( - 2 Δ 0 Q l * ) ] } ,
γ F . B ( Q ) = ( 1 - R ) γ 0 ( Q ) 1 - R γ 0 ( Q ) ,
γ F ( Q ) = [ 1 1 + 2 Δ ( 1 + R 1 - R ) ] [ 1 1 + ( Q l * ) 2 ] × { 1 + 1 - exp [ - 2 Δ 0 ( 1 + R 1 - R ) Q l * ] Q l * } .
R = Δ Ω d R ( θ ) S ( θ ) d Ω d Δ Ω d S ( θ ) d Ω d ,
S ( θ ) = 0 d z ρ ( z ) exp ( - z / l cos θ ) ,
ρ ( z ) = 1 + Δ ( R ) - exp ( - z / l * ) .
S R ( θ ) = Δ ( R ) cos θ + cos 2 ( θ ) η + cos θ .
Δ ( R ) = Δ 0 ( 1 + R 1 - R )
( A 0 - B 0 ) R 2 + ( A 0 + B 0 - A + B ) R - ( A + B ) = 0 ,
A = Δ Ω d R ( θ ) A ( θ ) d Ω d ,             B = Δ Ω d R ( θ ) B ( θ ) d Ω d ,
A 0 = Δ Ω d A ( θ ) d Ω d ,             B 0 = Δ Ω d B ( θ ) d Ω d ,
R d = R 1 + ( 1 - R 1 ) ( 1 - n 3 1 - d ) ,
Δ ( R ) = Δ 0 [ 2 ( 1 - R 1 ) - 1 n 3 d - 1 - 1 ] .
γ ( Q ) = d d - 1 x P ( x ) exp ( i Q · x ) ,
γ 0 ( Q ) = M diff ( Q ) = d d - 1 x P diff ( x ) exp ( i Q · x ) ,
P 1 ( x ) = d d - 1 u Π 1 ( u ) P diff ( x - u ) ,
Π 1 ( u ) = d d - 1 x P diff ( x ) φ ( u - x ) .
M n ( Q ) = d d - 1 x P n ( x ) exp ( i Q · x ) ,
M 1 ( Q ) = M diff ( Q ) [ Ψ ( Q ) M diff ( Q ) ] ,
Ψ ( Q ) = d d - 1 w φ ( w ) exp ( i Q · w ) .
M n ( Q ) = M diff ( Q ) [ Ψ ( Q ) M diff ( Q ) ] n .
P ( x ) = P 0 ( x ) + R P 1 ( x ) + R 2 P 2 ( x ) + ,
γ N ( Q ) = γ ( Q ) f N ( Q ) ,
γ ( Q ) = ( 1 - R ) γ 0 ( Q ) 1 - R Ψ ( Q ) γ 0 ( Q ) ,
f N ( Q ) = 1 - [ R Ψ ( Q ) γ 0 ( Q ) ] N + 1 1 - R N + 1 .
Ψ ( Q ) = Ψ 3 D ( Q ) = N 3 D u min u max φ ( u ) J 0 ( Q u ) u d u ,
φ ( u ) = φ θ ( θ 3 ) | d u d θ 3 | - 1 .
φ ( u ) = [ 1 + ( u 2 W ) 2 ] - 3 / 2 .
Ψ 3 D ( Q ) = N 3 D x min x max x J 0 ( 2 Q W x ) ( 1 + x 2 ) 3 / 2 d x ,
Ψ ( Q ) = Ψ 2 D ( Q ) = N 2 D x min x max cos ( 2 Q W x ) ( 1 + x 2 ) 3 / 2 d x ,
F N ( Q ) = F N ( Q ) = F ( Q ) f N 2 ( Q ) ,
F ( Q ) = ( 1 - R ) 2 F 0 ( Q ) [ 1 - R Ψ ( Q ) F 0 ( Q ) ] 2 ,
Γ N ( t ) = Γ ( t ) g N 2 ( t ) ,
g N ( t ) = 1 - [ R Γ 0 ( t ) ] N + 1 1 - R N + 1 ,
Γ ( t ) = ( 1 - R ) 2 Γ 0 ( t ) [ 1 - R Γ 0 ( t ) ] 2 .
Γ 0 ( t ) = exp [ - ( 1 + Δ 0 ) 2 d t / t 1 ] ,
C e ( ω ) = E ( ω 0 ) E * ( ω 0 + ω )
C e ( ω ) = σ ( t ) exp ( i ω t ) d t .
C i ( ω ) = C e ( ω ) 2 .
C N e ( ω ) = C e ( ω ) h N ( ω ) ,
C e ( ω ) = ( 1 - R ) C 0 e ( ω ) 1 - R Ξ ( ω ) C 0 e ( ω ) ,
h N ( ω ) = 1 - [ R Ξ ( ω ) C 0 e ( ω ) ] N + 1 1 - R N + 1 ,
Ξ ( ω ) = χ ( t ) exp ( i ω t ) d t ,
χ ( t ) d t = φ Ω ( Ω ) d Ω d ,
Ξ ( ω ) = Ξ d ( ω ) = Δ Ω d φ Ω ( Ω ) exp ( i ω τ 0 / cos θ ) d Ω d .
C 0 e ( ω ) = C 0 i ( ω ) exp [ i ζ ( ω ) ] ,
C 0 i ( ω ) = exp [ - 2 ζ ( ω ) ] ,
ζ ( ω ) = ζ d ( ω ) = d τ 00 ( ω ) ,
τ 00 ( ω ) = 2 l * ω / c .
Ξ ( ω ) = Ξ re ( ω ) + i Ξ im ( ω ) = α ( ω ) exp [ i β ( ω ) ] ,
C N i = C i H N ,
C i = ( 1 - R ) 2 C 0 i 1 + R 2 α 2 C 0 i - 2 R α C 0 i cos ( ζ + B ) ,
H N = 1 + ( R 2 α 2 C 0 i ) N + 1 - 2 ( R α C 0 i ) N + 1 cos [ ( N + 1 ) ( ζ + β ) ] ( 1 - R N + 1 ) 2 .
Ξ re ( ω ) = Ξ 0 ( ω ) [ Ξ ( v max ) - Ξ ( v min ) ] ,
Ξ im ( ω ) = Ξ 0 ( ω ) [ Ξ ( v max ) - Ξ ( v min ) ] ,
v min = ω τ 0 [ 1 - ( 1 / n 2 ) 2 ] - 1 / 2 ,
v max = ω τ 0 [ 1 - ( n 3 / n 2 ) 2 ] - 1 / 2 ,
Ξ 2 D 0 ( ω ) = - n 2 ω τ 0 ( n 2 2 - 1 - n 2 2 - n 3 2 ) - 1 ,
Ξ 2 D ( v ) = cos v v + s i ( v ) ,
Ξ 2 D ( v ) = sin v v - c i ( v ) .
Ξ 3 D 0 ( ω ) = - ( n 2 ω τ 0 ) 2 ( n 3 2 - 1 ) - 1 ,
Ξ 3 D ( v ) = cos ( v ) v 2 - sin ( v ) v + c i ( v ) ,
Ξ 3 D ( v ) = sin ( v ) v 2 + cos ( v ) v + s i ( v ) .

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