Abstract

A simple theoretical model involving only a single sample parameter, the depolarization ratio ρ for linearly polarized normally incident and normally scattered light, is developed to describe the angular intensity and all other polarization-dependent properties of diffuse transmission through multiple-scattering media. Initial experimental results that tend to support the theory are presented. Results for diffuse reflection are also described.

© 1993 Optical Society of America

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  6. M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
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  7. M. J. Stephen, G. Cwillich, “Rayleigh scattering and weak localization: effects of polarization,” Phys. Rev. B 34, 7564–7572 (1986).
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  8. 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).
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  12. M. Rosenbluh, M. Hoshen, I. Freund, M. Kaveh, “Time evolution of universal optical fluctuations,” Phys. Rev. Lett. 58, 2754–2757 (1987).
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  19. I. Freund, E. Barkai, “Third-order polarization correlations in highly random media,” J. Opt. Soc. Am. A 8, 1559–1567 (1991).
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  20. I. Freund, “Polarization correlations in two-dimensional random media,” Waves Random Media 1, 245–263 (1991).
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  21. I. Freund, “Image reconstruction through multiple scattering media,” Opt. Commun. 86, 216–227 (1991).
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    [CrossRef]
  25. R. Barakat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
    [CrossRef]
  26. S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply-scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
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  29. P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
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  31. A. Z. Genack, N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66, 2064–2067 (1991).
    [CrossRef] [PubMed]
  32. E. Akkermans, “Propagation d’ondes dans les milieux desordennés,” Ph.D. dissertation (University of Grenoble, Grenoble, France, 1986).
  33. 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]
  34. M. P. van Albada, A. Lagendijk, “Vector character of light in weak localization: spatial anisotropy in coherent back-scattering from a random medium,” Phys. Rev. B 36, 2353–2356 (1987).
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  39. Y. Kuga, “Third- and fourth-order iterative solutions for the vector radiative transfer equation,” J. Opt. Soc. Am. A 8, 1580–1586 (1991).
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  40. I. Freund, “Stokes vector reconstruction,” Opt. Lett. 15, 1425–1427 (1990).
    [CrossRef] [PubMed]
  41. I. Freund, M. Kaveh, “Comment on ‘Polarization memory of multiply scattered light,” Phys. Rev. B 45, 8162–8164 (1992).
    [CrossRef]
  42. 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]
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    [CrossRef] [PubMed]
  49. M. Abramowits, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964).
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    [CrossRef]
  54. 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]
  55. 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);Erratum, Phys. Rev. B 41, 9540 (1990).
    [CrossRef]
  56. I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
    [CrossRef] [PubMed]
  57. J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
    [CrossRef] [PubMed]
  58. M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
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    [CrossRef] [PubMed]
  61. J. A. Sánchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
    [CrossRef]
  62. R. M. A. Azzam, N. M. Bashra, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  63. C. Brosseau, R. Barakat, “Jones and Mueller polarization matrices for random media,” Opt. Commun. 84, 127–132 (1991).
    [CrossRef]

1992 (6)

C. J. Solomon, J. C. Dainty, “Use of polarization in double passage imaging through a random screen,” Opt. Commun. 87, 207–211 (1992).
[CrossRef]

I. Freund, “Time-reversal symmetry and image reconstruction through multiple-scattering media,” J. Opt. Soc. Am. A 9, 456–463 (1992).
[CrossRef]

I. I. Tarhan, G. H. Watson, “Polarization microstatistics of laser speckle,” Phys. Rev. A 45, 6013–6018 (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, D. Eliyahu, “Surface correlations in multiple-scattering media,” Phys. Rev. A 45, 6133–6134 (1992).
[CrossRef] [PubMed]

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

1991 (15)

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

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

Y. Kuga, “Third- and fourth-order iterative solutions for the vector radiative transfer equation,” J. Opt. Soc. Am. A 8, 1580–1586 (1991).
[CrossRef]

J. A. Sánchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
[CrossRef]

C. Brosseau, R. Barakat, “Jones and Mueller polarization matrices for random media,” Opt. Commun. 84, 127–132 (1991).
[CrossRef]

S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply-scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
[CrossRef] [PubMed]

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

A. Z. Genack, N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66, 2064–2067 (1991).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, “Time reversal symmetry of multiply scattered speckle patterns,” Opt. Commun. 82, 362–369 (1991).
[CrossRef]

I. Freund, M. Kaveh, R. Berkovits, M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1991).
[CrossRef]

I. Freund, “Optical intensity fluctuations in multiply scattering media,” Opt. Commun. 81, 251–257 (1991).
[CrossRef]

I. Freund, E. Barkai, “Third-order polarization correlations in highly random media,” J. Opt. Soc. Am. A 8, 1559–1567 (1991).
[CrossRef]

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

I. Freund, “Image reconstruction through multiple scattering media,” Opt. Commun. 86, 216–227 (1991).
[CrossRef]

I. Freund, “Diffractometry through multiple scattering media,” Opt. Commun. 87, 5–8 (1991).
[CrossRef]

1990 (4)

R. Berkovits, M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
[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);Erratum, Phys. Rev. B 41, 9540 (1990).
[CrossRef]

I. Freund, “Stokes vector reconstruction,” Opt. Lett. 15, 1425–1427 (1990).
[CrossRef] [PubMed]

M. Nieto-Vesperinas, J. A. Sánchez-Gil, A. J. Sant, J. C. Dainty, “Light transmission from a randomly rough dielectric diffuser: theoretical and experimental results,” Opt. Lett. 15, 1261–1263 (1990).
[CrossRef] [PubMed]

1989 (6)

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]

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]

Y. Kuga, A. Ishimaru, Q. Ma, “The second-order multiple scattering theory for the vector radiative transfer equation,” Radio Sci. 25, 247–252 (1989).
[CrossRef]

Y. Kuga, A. Ishimaru, “Backscattering enhancement by randomly distributed very large particles,” Appl. Opt. 28, 2165–2169 (1989).
[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]

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

1988 (6)

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]

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, (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]

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]

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]

I. Freund, “Antilocalization of light,” Phys. Rev. A 37, 1007–1008 (1988).
[CrossRef] [PubMed]

1987 (3)

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

M. P. van Albada, A. Lagendijk, “Vector character of light in weak localization: spatial anisotropy in coherent back-scattering from a random medium,” Phys. Rev. B 36, 2353–2356 (1987).
[CrossRef]

1986 (4)

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

1985 (4)

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]

P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
[CrossRef]

R. Barakat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
[CrossRef]

1984 (2)

S. John, “Electomagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984).
[CrossRef]

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

1982 (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, “Propagation d’ondes dans les milieux desordennés,” Ph.D. dissertation (University of Grenoble, Grenoble, France, 1986).

Anderson, P. W.

P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
[CrossRef]

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]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashra, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Barakat, R.

C. Brosseau, R. Barakat, “Jones and Mueller polarization matrices for random media,” Opt. Commun. 84, 127–132 (1991).
[CrossRef]

R. Barakat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
[CrossRef]

Barkai, E.

Bashra, N. M.

R. M. A. Azzam, N. M. Bashra, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Berkovits, R.

I. Freund, M. Kaveh, R. Berkovits, M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1991).
[CrossRef]

R. Berkovits, M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
[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);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]

Born, M.

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

Brosseau, C.

C. Brosseau, R. Barakat, “Jones and Mueller polarization matrices for random media,” Opt. Commun. 84, 127–132 (1991).
[CrossRef]

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]

Cohen, S. M.

S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply-scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
[CrossRef] [PubMed]

Cwillich, G.

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]

Edrei, I.

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]

Eliyahu, D.

I. Freund, D. Eliyahu, “Surface correlations in multiple-scattering media,” Phys. Rev. A 45, 6133–6134 (1992).
[CrossRef] [PubMed]

S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply-scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
[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]

Feschbach, H.

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

Freund, I.

I. Freund, D. Eliyahu, “Surface correlations in multiple-scattering media,” Phys. Rev. A 45, 6133–6134 (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, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

I. Freund, “Time-reversal symmetry and image reconstruction through multiple-scattering media,” J. Opt. Soc. Am. A 9, 456–463 (1992).
[CrossRef]

I. Freund, E. Barkai, “Third-order polarization correlations in highly random media,” J. Opt. Soc. Am. A 8, 1559–1567 (1991).
[CrossRef]

I. Freund, M. Rosenbluh, “Time reversal symmetry of multiply scattered speckle patterns,” Opt. Commun. 82, 362–369 (1991).
[CrossRef]

I. Freund, M. Kaveh, R. Berkovits, M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1991).
[CrossRef]

S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply-scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
[CrossRef] [PubMed]

I. Freund, “Optical intensity fluctuations in multiply scattering media,” Opt. Commun. 81, 251–257 (1991).
[CrossRef]

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

I. Freund, “Image reconstruction through multiple scattering media,” Opt. Commun. 86, 216–227 (1991).
[CrossRef]

I. Freund, “Diffractometry through multiple scattering media,” Opt. Commun. 87, 5–8 (1991).
[CrossRef]

I. Freund, “Stokes vector reconstruction,” Opt. Lett. 15, 1425–1427 (1990).
[CrossRef] [PubMed]

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);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, “Antilocalization of light,” Phys. Rev. A 37, 1007–1008 (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. 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]

Garcia, N.

A. Z. Genack, N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66, 2064–2067 (1991).
[CrossRef] [PubMed]

Genack, A. Z.

A. Z. Genack, N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66, 2064–2067 (1991).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

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

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.

John, S.

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (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]

S. John, “Electomagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984).
[CrossRef]

Kaveh, M.

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

S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply-scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
[CrossRef] [PubMed]

I. Freund, M. Kaveh, R. Berkovits, M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1991).
[CrossRef]

R. Berkovits, M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
[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]

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]

Kong, J. A.

L. J. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Kuga, Y.

Lagendijk, A.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

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, “Vector character of light in weak localization: spatial anisotropy in coherent back-scattering from a random medium,” Phys. Rev. B 36, 2353–2356 (1987).
[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]

Lap-Tung, R.

Ma, Q.

Y. Kuga, A. Ishimaru, Q. Ma, “The second-order multiple scattering theory for the vector radiative transfer equation,” Radio Sci. 25, 247–252 (1989).
[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, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (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]

Morse, P. M.

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

Nieto-Vesperinas, M.

Pine, D. J.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive 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]

Rosenbluh, M.

I. Freund, M. Rosenbluh, “Time reversal symmetry of multiply scattered speckle patterns,” Opt. Commun. 82, 362–369 (1991).
[CrossRef]

I. Freund, M. Kaveh, R. Berkovits, M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1991).
[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]

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]

Sánchez-Gil, J. A.

Sant, A. J.

Shapiro, B.

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

Shin, R. T.

L. J. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Solomon, C. J.

C. J. Solomon, J. C. Dainty, “Use of polarization in double passage imaging through a random screen,” Opt. Commun. 87, 207–211 (1992).
[CrossRef]

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, (1988).
[CrossRef]

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

Tarhan, I. I.

I. I. Tarhan, G. H. Watson, “Polarization microstatistics of laser speckle,” Phys. Rev. A 45, 6013–6018 (1992).
[CrossRef] [PubMed]

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]

Tip, A.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

Tsang, L. J.

L. J. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

van Albada, M. P.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

M. P. van Albada, A. Lagendijk, “Vector character of light in weak localization: spatial anisotropy in coherent back-scattering from a random medium,” Phys. Rev. B 36, 2353–2356 (1987).
[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]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1964).

van Tiggelen, B. A.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[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.

I. I. Tarhan, G. H. Watson, “Polarization microstatistics of laser speckle,” Phys. Rev. A 45, 6013–6018 (1992).
[CrossRef] [PubMed]

Weitz, D. A.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive 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 (Pergamon, 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]

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 diffusive 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]

Appl. Opt. (2)

Europhys. Lett. (1)

R. Berkovits, M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
[CrossRef]

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

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]

Opt. Acta (1)

R. Barakat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
[CrossRef]

Opt. Commun. (6)

I. Freund, “Image reconstruction through multiple scattering media,” Opt. Commun. 86, 216–227 (1991).
[CrossRef]

I. Freund, “Diffractometry through multiple scattering media,” Opt. Commun. 87, 5–8 (1991).
[CrossRef]

C. J. Solomon, J. C. Dainty, “Use of polarization in double passage imaging through a random screen,” Opt. Commun. 87, 207–211 (1992).
[CrossRef]

I. Freund, “Optical intensity fluctuations in multiply scattering media,” Opt. Commun. 81, 251–257 (1991).
[CrossRef]

I. Freund, M. Rosenbluh, “Time reversal symmetry of multiply scattered speckle patterns,” Opt. Commun. 82, 362–369 (1991).
[CrossRef]

C. Brosseau, R. Barakat, “Jones and Mueller polarization matrices for random media,” Opt. Commun. 84, 127–132 (1991).
[CrossRef]

Opt. Lett. (2)

Philos. Mag. B (1)

P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
[CrossRef]

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 (6)

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

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

I. Freund, D. Eliyahu, “Surface correlations in multiple-scattering media,” Phys. Rev. A 45, 6133–6134 (1992).
[CrossRef] [PubMed]

S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply-scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
[CrossRef] [PubMed]

I. I. Tarhan, G. H. Watson, “Polarization microstatistics of laser speckle,” Phys. Rev. A 45, 6013–6018 (1992).
[CrossRef] [PubMed]

I. Freund, “Antilocalization of light,” Phys. Rev. A 37, 1007–1008 (1988).
[CrossRef] [PubMed]

Phys. Rev. B (10)

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]

I. Freund, M. Kaveh, R. Berkovits, M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1991).
[CrossRef]

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

M. P. van Albada, A. Lagendijk, “Vector character of light in weak localization: spatial anisotropy in coherent back-scattering from a random medium,” Phys. Rev. B 36, 2353–2356 (1987).
[CrossRef]

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

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (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]

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);Erratum, Phys. Rev. B 41, 9540 (1990).
[CrossRef]

Phys. Rev. Lett. (11)

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

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

A. Z. Genack, N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66, 2064–2067 (1991).
[CrossRef] [PubMed]

S. John, “Electomagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984).
[CrossRef]

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]

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. 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]

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]

Phys. Today (1)

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

Radio Sci. (1)

Y. Kuga, A. Ishimaru, Q. Ma, “The second-order multiple scattering theory for the vector radiative transfer equation,” Radio Sci. 25, 247–252 (1989).
[CrossRef]

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 (12)

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

L. J. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

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

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1964).

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

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

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 (Pergamon, New York, 1959).

Buehler, 41 Waukegan Road, Lake Bluff, Ill. 60044, Gamma Micropolish (No. 40-6365-006).

White paraffin oil, measured refractive index n633 nm= 1.460.

R. M. A. Azzam, N. M. Bashra, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

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Figures (15)

Fig. 1
Fig. 1

(a) Experimental geometry. The angle of incidence α is measured from the normal to the input face, and the angle of scattering θ is measured from the normal to the output face. A planar geometry is used, with the incident and the scattered k vectors lying in the yz plane. The incident laser field is denoted by (t), the field on the input face of the sample by (u), the field at the output face of the sample by (v), and the field at the distant detector by (w). (b) Coordinate systems xyz and rθϕ used in the theory of Appendix A. As shown in (a), the +z axis is the outgoing normal to the output face and the x and y axes are parallel to the sample surfaces. The scattering angles are measured in this coordinate system with the incident k vector direction given by θi, ϕi and the scattered k vector direction by θs, ϕs.

Fig. 2
Fig. 2

Intensity in transmission versus scattering angle θ for linear x-polarized (ψ = 0°) normally incident (α = 0°) light for sample 1. The directly measured depolarization ratio is ρ = 1. ■ IV, vertically polarized output; ● IH, horizontally polarized output. The solid curves are a fit (using n = 1.37) of Eqs. (14), and the dashed curves (which largely overlap the solid curves) are the predictions of the model of Eqs. (19) and (20) below.

Fig. 3
Fig. 3

Intensity in transmission versus scattering angle for sample 1 under the same conditions as in Fig. 2, except that the angle of incidence is α = 60°.

Fig. 4
Fig. 4

(a) δI(L, C) [Eq. (16c)] and (b) δI(C, L) [Eq. (16b)] versus scattering angle θ for sample 1 in reflection for an angle of incidence α = 60°. Both data sets are normalized by the average peak intensity at θ = 0°, ½[I+(0) + I(0)]. The fact that these data are within experimental error nearly zero everywhere demonstrates that the polarization correlators Γ, Γ′, δ, and δ′ are all pure real, as is required by the average rotational symmetry of the random medium.

Fig. 5
Fig. 5

Intensity in transmission versus scattering angle θ for sample 2 (ρ = 0.45, n = 1.39) under the same conditions as in Fig. 2 These yield ρ′ = 0.67 for this sample.

Fig. 6
Fig. 6

Intensity in transmission versus scattering angle θ for sample 3 (ρ = 0.31, n = 1.39) under the same conditions as in Fig. 2, except that the incident light is y polarized (ψ = 90°), Eqs. (17). These data yield ρ′ = 0.51 for this sample.

Fig. 7
Fig. 7

(a) δI(L, L) [Eq. (16a)] and (b) δI(C, C) [Eq. (16d)] versus scattering angle θ for sample 1 (ρ = 1) in transmission for an angle of incidence α = 60°. Both data sets are normalized by the average peak intensity at θ = 0°, ½[I+(0) + I(0)]. The fact that these data are within experimental error nearly zero everywhere demonstrates that when ρ = 1 the polarization correlators Γ, Γ′, δ, and δ′ all vanish, as expected.

Fig. 8
Fig. 8

(a), (b) δI(L, L) [Eq. (16a)] versus scattering angle θ for sample 2 (ρ = 0.45) in transmission, (a), angle of incidence α = 20°; (b), α = 60°; (c) δI(C, C) [Eq. (16d)], α = 60°. The solid curves are theoretical fits to the data that yield from (a) and (b) (Γ′ + δ′)/(Γ + δ) = 0.5 and from (c) (Γ′ − δ′)/(Γ − δ) ≪ 1. The dashed curves are the predictions of the model of Eqs. (19) and (20). Since the data in (c) depend on the ratio of the differences of the correlation parameters, they are especially sensitive to small errors.

Fig. 9
Fig. 9

I+ for sample 2 [see Eqs. (16) and (A9)]. (a), (b) ■ I+(L, L), ● I(L, L); (c) ■ I+(C, C), ● I(C, C); α as in Fig. 8. The dashed curves are the predictions of the model of Eqs. (19) and (20).

Fig. 10
Fig. 10

Intensity in reflection for sample 1 versus scattering angle θ for an angle of incidence α = 60° with linearly polarized incident light: (a) input x polarized (ψ = 0°), ■ output copolarized (IV), ● output cross polarized (IH); (b) input y polarized (ψ = 90°), ● output copolarized (IH), ■ output cross polarized (IV). As is discussed in Section 7, these data illustrate the importance of single scattering in reflection.

Fig. 11
Fig. 11

Intensity in reflection for sample 1 versus scattering angle θ for normal incidence (α = 0°) for linearly polarized incident light: (a) input x polarized (ψ = 0°), ■ output copolarized (IV), ● output cross polarized (IH); (b) input y polarized (ψ = 90°), ● output copolarized (IH), ■ output cross polarized (IV). The dashed curves are the predictions of Eq. (10a) together with the model for reflection in Eqs. (21).

Fig. 12
Fig. 12

Coordinates used in Appendix B.

Fig. 13
Fig. 13

Comparison of (a) intensities I (solid curve) and I′ (dashed curve) with (b), (c) components Ijist (solid curves) and I j i s t (dashed curves) for a sample thickness L + 2Δl = 4l, where l is the mean free path and the Milne theory value Δ = 0.71 is used. The solid curves are the exact results [Eq. (B14a)] for the point-scatterer model, while the dashed curves are based on our simplifying approximations [Eq. (B14b)].

Fig. 14
Fig. 14

(a) Intensities I and I′ and (b), (c) components Ijist and I j i s t , as in Fig. 13, except that L + 2Δl= 6 · l.

Fig. 15
Fig. 15

(a) Intensities I and I′ and (b), (c) components Ijist and I j i s t , as in Fig. 13, except that L + 2Δl = 10 · l. Here the dashed and the solid curves overlap completely, and components xxyy and xyyx are of negligible amplitude.

Equations (82)

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δ I i δ I j = | E i | E j | 2 ,
υ j ( R ) = i d 2 r u i ( r ) E i j ( r , R ) ,
υ j ( R ) | υ l ( R ) = i k d 2 r d 2 r u i ( r ) * u k ( r ) × E i j ( r , R ) | E k l ( r , R ) .
E i j ( r , R ) | E k l ( r , R ) = δ ( r r ) F ( R ) | F ( R ) × D ( | R r | ) E i j | E k l ,
υ j ( R ) | υ l ( R ) = F ( R ) | F ( R ) i k u i u k E i j | E k l .
u = | E z z | 2 ,
ρ = | E x y | 2 = | E y x | 2 ,
ρ = | E x z | 2 = | E z x | 2 = | E y z | 2 = | E z y | 2 ,
Γ = E x x | E y y ,
Γ = E x x | E z z = E y y | E z z ,
δ = E x y | E y x ,
δ = E x z | E z x = E y z | E z y .
| Γ + δ | = 1 ρ
S ( θ ) = N cos 2 θ Σ d 2 ζ F ( R ) | F ( R + ζ ) | F ( R ) | 2 exp ( i κ · ζ ) ,
F ( R ) | F ( R + ζ ) | F ( R ) | 2 = 1 1 + Δ ln 2 [ Δ sin ( k ζ ) k ζ + j ( k ζ ) ] ,
j ( u ) = 0 1 d x x J 0 ( u x ) 1 + 1 x 2 ,
F ( R ) | F ( R + ζ ) | F ( R ) | 2 = 2 1 + 2 Δ [ Δ sin ( k ζ ) k ζ + J 1 ( k ζ ) k ζ ] .
S R ( θ ) = 2 1 + 2 Δ ( Δ cos θ + cos 2 θ 1 + cos θ )
S T ( θ ) = ( Δ cos θ + cos 2 θ ) / ( 1 + Δ ) .
υ x = E x x cos ψ + E y x sin ψ ,
υ y = E y y sin ψ + E x y cos ψ ,
υ z = E x z cos ψ + E y z sin ψ .
J 11 = w x | w x = cos 2 ψ + ρ sin 2 ψ ,
J 22 = w y | w y = ( sin 2 ψ + ρ cos 2 ψ ) cos 2 θ + ρ sin 2 θ ,
J 12 = w x | w y = 1 / 2 ( Γ + δ ) sin ( 2 ψ ) cos θ ,
Φ = 1 ρ ( ρ ρ ) sin 2 θ 1 + ρ + ( ρ ρ ) sin 2 θ .
I V ( θ ) = K S ( θ ) ,
I H ( θ ) = K S ( θ ) ( ρ cos 2 θ + ρ sin 2 θ ) ,
Δ ( R ) = 2 ( 1 + R ) [ 1 ( 3 / 16 ) ( 1 R ) ] 3 ( 1 R ) [ 1 ( 2 / 3 ) ( 1 ln 2 ) ( 1 R ) ] .
δ I ( L , L ) = K S ( θ ) [ Re ( Γ + δ ) cos α cos θ Re ( Γ + δ ) sin α sin θ ] .
δ I ( C , L ) = K S ( θ ) [ Im ( Γ δ ) cos α cos θ Im ( Γ δ ) sin α sin θ ] .
δ I ( L , C ) = K S ( θ ) [ Im ( Γ + δ ) cos α cos θ Im ( Γ + δ ) sin α sin θ ] ,
δ I ( C , C ) = K S ( θ ) [ Re ( Γ δ ) cos α cos θ Re ( Γ δ ) sin α sin θ ] ,
I V ( θ ) = K S ( θ ) ρ ,
I H ( θ ) = K S ( θ ) ( cos 2 θ + ρ sin 2 θ ) .
I + ( L , L ) + I ( L , L ) = K S ( θ ) [ 1 + cos 2 θ + 1 / 2 ( 1 + cos 2 α + u sin 2 α ) sin 2 θ ] ,
Γ = 1 ρ 1 + ρ ,
δ = ρ ( 1 ρ 1 + ρ ) .
ρ ρ = ρ ( 1 ρ ) .
Γ Γ = Γ ( 1 Γ ) .
δ = ρ Γ .
1 u = ρ ( 1 ρ ) .
u = 1 ,
ρ = ρ ,
Γ = Γ = 1 ρ ,
δ = δ = 0 ,
R ( θ , ϕ ) = [ θ ̂ · x ̂ θ ̂ · ŷ θ ̂ · ϕ ̂ · x ̂ ϕ ̂ · ŷ ϕ ̂ · r ̂ · x ̂ r ̂ · ŷ r ̂ · ] = [ cos θ cos ϕ cos θ sin ϕ sin θ sin ϕ cos ϕ 0 sin θ cos ϕ sin θ sin ϕ cos θ ] .
u = R 1 ( θ i , ϕ i ) t ,
w = R ( θ s , ϕ s ) v .
w = F ( θ i , ϕ i ; θ s , ϕ s ) t ,
F = S ( θ s ) R ( θ s , ϕ s ) A ( θ s , ϕ s ) E T R T ( θ i , ϕ i ) ,
F 11 / S ( θ s ) = ( cos θ i ) ( cos ϕ i ) [ ( cos θ s ) ( cos ϕ s ) E x x + ( cos θ s ) ( sin ϕ s ) E x y ( sin θ s ) E x z ] ( sin ϕ i ) [ ( cos θ s ) ( cos ϕ s ) E y x + ( cos θ s ) ( sin ϕ s ) E y y ( sin θ s ) E y z ] + ( sin θ i ) ( cos ϕ i ) [ ( cos θ s ) ( cos ϕ s ) E z x + ( cos θ s ) ( sin ϕ s ) E z y ( sin θ s ) E z z ] , F 12 / S ( θ s ) = ( cos θ i ) ( sin ϕ i ) [ ( cos θ s ) ( cos ϕ s ) E x x + ( cos θ s ) ( sin ϕ s ) E x y ( sin θ s ) E x z ] + cos ϕ i [ ( cos θ s ) ( cos ϕ s ) E y x + ( cos θ s ) ( sin ϕ s ) E y y ( sin θ s ) E y z ] + ( sin θ i ) ( sin ϕ i ) [ ( cos θ s ) ( cos ϕ s ) E z x + ( cos θ s ) ( sin ϕ s ) E z y ( sin θ s ) E z z ] , F 21 / S ( θ s ) = ( cos θ i ) ( cos ϕ i ) [ ( sin ϕ s ) E x x + ( cos ϕ s ) E x y ] sin ϕ i [ ( sin ϕ s ) E y x + ( cos ϕ s ) E y y ] + ( sin θ i ) ( cos ϕ i ) × [ ( sin ϕ s ) E z x + ( cos ϕ s ) E z y ] , F 22 / S ( θ s ) = ( cos θ i ) ( sin ϕ i ) [ ( sin ϕ s ) E x x + ( cos ϕ s ) E x y ] + cos ϕ i [ ( sin ϕ s ) E y x + ( cos ϕ s ) E y y ] + ( sin θ i ) ( sin ϕ i ) × [ ( sin ϕ s ) E z x + ( cos ϕ s ) E z y ] .
w = F ( θ , ϕ ; θ , ϕ + π ) t .
I = S ( θ ) { [ ( cos 4 θ ) + u ( sin 4 θ ) + ρ ( cos 2 θ ) + ρ × [ 2 ( sin 2 θ ) ( cos 2 θ ) + ( sin 2 θ ) ] + 2 Re ( Γ + δ ) ( sin 2 θ ) ( cos 2 θ ) ] + 2 [ ( sin 4 ϕ ) ( sin 2 ϕ ) ] [ 1 ρ Re ( Γ + δ ) ] × [ ( cos 2 θ ) ( cos 4 θ ) ] } .
w ± ( inc , scatt ) = G scatt F ( θ , π / 2 ; α , π / 2 ) t inc ,
G L = 1 / 2 [ 1 ± 1 ± 1 1 ] .
G C = 1 / 2 [ 1 ± i 1 ± i ] .
I ± ( inc , scatt ) = ( 1 / 2 ) K S ( θ ) { [ 1 + ρ ( cos 2 α ) + ρ ( sin 2 α ) ] + [ ρ + ( cos 2 α ) + ρ ( sin 2 α ) ] ( cos 2 θ ) + [ ρ [ 1 + ( cos 2 α ) ] + u ( sin 2 α ) ] ( sin 2 θ ) 2 Re ( Γ + δ ) ( sin α ) ( cos α ) ( sin θ ) × ( cos θ ) } ± 1 / 2 δ I ( inc , scatt ) ,
M = B F F B 1 ,
B = [ 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ] ,
M 11 = 1 / 2 ( | F 11 | 2 + | F 12 | 2 + | F 21 | 2 + | F 22 | 2 ) , M 12 = 1 / 2 ( | F 11 | 2 | F 12 | 2 + | F 21 | 2 | F 22 | 2 ) , M 13 = F 12 | F 11 + F 21 | F 22 , M 21 = 1 / 2 ( | F 11 | 2 + | F 12 | 2 | F 21 | 2 | F 22 | 2 ) , M 22 = 1 / 2 ( | F 11 | 2 | F 12 | 2 | F 21 | 2 + | F 22 | 2 ) , M 23 = F 12 | F 11 F 21 | F 22 , M 31 = F 21 | F 11 + F 12 | F 22 , M 32 = F 21 | F 11 F 12 | F 22 , M 33 = F 11 | F 22 + F 21 | F 12 , M 11 = F 11 | F 22 F 21 | F 12 .
[ I s Q s U s V s ] = [ I 1 Q 1 0 0 ] + [ I 2 cos ( 2 ϕ ) Q 2 cos ( 2 ϕ ) U sin ( 2 ϕ ) V sin ( 2 ϕ ) ] ,
I 1 = 1 / 2 S ( θ ) [ ( 1 + ρ ) ( 1 + cos 2 θ ) + 2 ρ sin 2 θ ] , Q 1 = 1 / 2 S ( θ ) ( 1 + ρ ρ ) sin 2 θ , I 2 = 1 / 2 S ( θ ) ( 1 ρ ) sin 2 θ , Q 2 = 1 / 2 S ( θ ) ( 1 ρ ) ( 1 + cos 2 θ ) , U = S ( θ ) ( 1 ρ ) cos θ , V = 0 .
Φ = | λ + λ | / ( λ + + λ ) = [ 1 4 det ( J s ) [ tr ( J s ) ] 2 ] 1 / 2 ,
υ j ( R ) = i G j i ( r , R ) u j ( r ) ,
G j i ( r , R ) = A j i G ( r , R )
G I ( r , R ) = exp ( i k | R r | ) | R r | .
G B ( R , P ) = exp [ Z / ( 2 l cos θ ) + i k | P R | ] | P R | ,
G P ( r , R ) = exp [ i φ ( r , R ) ] ,
υ j ( n ) ( R ) = i A j i ( n ) exp [ i φ ( r , R ) ] u i ( r ) ,
A j i ( n ) = [ m = n 1 A ( θ m , ϕ m ) ] j i .
υ j ( R ) = n υ j ( n ) D n ( r , R ) .
υ a ( P ) = d 3 R j G a j B ( R , P ) υ j ( R ) .
G B ( R , P ) = exp [ Z / ( 2 l cos θ ) + i k P ̂ · R ] | P | ,
υ a ( θ , ϕ ) | υ b ( θ , ϕ ) = j , s i , t d 3 R exp ( Z l cos θ ) × A a j ( θ , ϕ ) A b s ( θ , ϕ ) n D n ( r , R ) × A j i ( n ) | A s t ( n ) u i | u t ,
A x x ( n ) | A x x ( n ) = 1 + 2 ( 0.7 ) n 1 2 + ( 0.7 ) n 1 , A x y ( n ) | A x y ( n ) = 1 ( 0.7 ) n 1 2 + ( 0.7 ) n 1 , A x y ( n ) | A y x ( n ) = 3 2 ( 0.7 ) n 1 ( 0.5 ) n 1 2 + ( 0.7 ) n 1 , A x x ( n ) | A y y ( n ) = 3 2 ( 0.7 ) n 1 + ( 0.5 ) n 1 2 + ( 0.7 ) n 1 .
υ a ( θ , ϕ ) | υ b ( θ , ϕ ) = j , s i , t S ( θ ) A a j ( θ , ϕ ) A b s ( θ , ϕ ) E j i T | E s t T u i | u t .
I j i s t = d Z exp ( Z l cos θ ) n D n ( r , R ) A j i ( n ) | A s t ( n )
I j i s t = S ( θ ) E j i T | E s t T .
D n ( z = L l , Z ) = 1 π L m = 1 sin ( m π ( L l ) L + 2 Δ l ) × sin ( m π ( L + Δ l Z ) L + 2 Δ l ) × exp [ n 3 ( m π ( L + 2 Δ l ) / l ) 2 ] ,
D n ( z = L l , Z = Δ l ) = D n ( z = L l , Z = L + Δ l ) = 0
D n = 0 ( z = L l , Z ) = δ ( Z L + l ) .

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