Abstract

I derive a transport equation for the time correlation function in transmission and reflexion and inside a turbid medium. This equation goes beyond the diffusion approximation that is at the root of the well-established diffusing-wave spectroscopy technique. It takes into account all the transport regimes from ballistic to diffusive and the relaxation in direction at each scattering event. The derivation is based on a generalized form of the Bethe–Salpeter equation coupled to a generalized form of the scattering operator. The method presented can be easily adapted to compute the correlation function in systems with several time scales encountered, for example, in biology and polymer physics. The obtained equation is easily solvable numerically using a Monte Carlo scheme.

© 2008 Optical Society of America

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  1. P. Sebbah, Waves and Imaging through Complex Media (Kluwer Academic, 2001).
  2. B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, 1976).
  3. V. A. Bloomfield and T. K. Lim, “Quasi-elastic light scattering,” Methods Enzymol. 48, 415-494 (1978).
    [CrossRef] [PubMed]
  4. G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B: Condens. Matter 65, 409-413 (1987).
    [CrossRef]
  5. D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134-1137 (1988).
    [CrossRef] [PubMed]
  6. F. Scheffold, W. Hartk, G. Maret, and E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations of light,” Phys. Rev. B 56, 10942-10952 (1998).
    [CrossRef]
  7. D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252, 686-688 (1991).
    [CrossRef] [PubMed]
  8. P. D. Kaplan, A. G. Yodh, and D. J. Pine, “Diffusion and structure in dense binary suspensions,” Phys. Rev. Lett. 68, 393-396 (1992).
    [CrossRef] [PubMed]
  9. T. G. Mason and D. A. Weitz, “Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250-1253 (1995).
    [CrossRef] [PubMed]
  10. F. Scheffold and P. Schurtenberger, “Light scatteringprobes of viscoelastic fluids and solids,” Soft Mater. 1, 139-165 (2003).
    [CrossRef]
  11. B. Weber, C. Burger, M. T. Wyss, V. G. K. Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20, 2664-2670 (2004).
    [CrossRef] [PubMed]
  12. F. C. MacKintosh and S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383-2406 (1989).
    [CrossRef]
  13. L. F. Rojas-Ochoa, D. Lacoste, R. Lenke, P. Schurtenberger, and F. Scheffold, “Depolarization of backscattered linearly polarized light,” J. Opt. Soc. Am. A 21, 1799-1804 (2004).
    [CrossRef]
  14. J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948-3959 (1991).
    [CrossRef] [PubMed]
  15. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532-2538 (1995).
    [CrossRef]
  16. P.-A. Lemieux, M. U. Vera, and D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498-4514 (1998).
    [CrossRef]
  17. R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusion-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
    [CrossRef] [PubMed]
  18. R. Pierrat, N. Ben Braham, L. F. Rojas-Ochoa, F. Scheffold, and R. Carminati, “The influence of the scattering anisotropy parameter on diffuse reflection of light,” Opt. Commun. 281, 18-22 (2008).
    [CrossRef]
  19. G. Popescu and A. Dogariu, “Dynamic light scattering in subdiffusive regime,” Appl. Opt. 40, 4215-4221 (2001).
    [CrossRef]
  20. R. Pierrat, J.-J. Greffet, R. Elaloufi, and R. Carminati, “Spatial coherence in strongly scattering media,” J. Opt. Soc. Am. A 22, 2329-2337 (2005).
    [CrossRef]
  21. Y. N. Barabanenkov and V. M. Finkel'berg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587-591 (1968).
  22. S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.
  23. L. Tsang, K.-H. Ding, and B. Wen, “Dense media radiative transfer theory for dense discrete random media with particles of multiple sizes and permitivites,” Prog. Electromagn. Res. 6, 181-230 (1992).
  24. L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
    [CrossRef]
  25. Y. N. Barabanenkov, L. M. Zurk, and M. Y. Barabanenkov, “Single scattering and diffusion approximations for modified radiative transfer theory of wave multiple scattering in dense media near resonance,” Prog. Electromagn. Res. 15, 27-61 (1997).
    [CrossRef]
  26. L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000), Vol. 2, Chap. 7.
    [CrossRef]
  27. L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000), Vol. 3, Chap. 7.
    [CrossRef]
  28. F. Jaillon, S. E. Skipetrov, J. Li, G. Dietsche, G. Maret, and T. Gisler, “Diffusing-wave spectroscopy from head-like tissue phantoms: influence of a non-scattering layer,” Opt. Express 14, 10181-10194 (2001).
    [CrossRef]
  29. L. Le Goff, C. Amblard, F. Furst, and E. M. Furst, “Motor-driven dynamics in actin-myosin networks,” Phys. Rev. Lett. 88, 018101 (2001).
    [CrossRef]
  30. L. A. Apresyan and Y. A. Kravtsov, Radiation Transfer--Statistical and Wave Aspects (Gordon & Breach, 1996).
  31. U. Frish, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A.T.Bharuch-Reid, ed. (Academic, 1968), Vol. 1, pp. 75-198.
  32. F. Dyson, “The radiation theories of Tomonaga, Schwinger and Feynmann,” Phys. Rev. 75, 486-502 (1949).
    [CrossRef]
  33. F. Dyson, “The s matrix in quantum electrodynamics,” Phys. Rev. 75, 1736-1755 (1949).
    [CrossRef]
  34. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256-1259 (1968).
    [CrossRef]
  35. S. Chandrasekhar, Radiative Transfer (Dover, 1950).
  36. A. F. Molisch and B. P. Oehry, Radiation Trapping in Atomic Vapours (Clarendon, 1998).
  37. J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods (Chapman & Hall, 1964).
    [CrossRef]
  38. G. S. Fishman, Monte Carlo Concepts, Algorithms and Applications (Springer Verlag, 1996).

2008 (1)

R. Pierrat, N. Ben Braham, L. F. Rojas-Ochoa, F. Scheffold, and R. Carminati, “The influence of the scattering anisotropy parameter on diffuse reflection of light,” Opt. Commun. 281, 18-22 (2008).
[CrossRef]

2005 (1)

2004 (3)

R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusion-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef] [PubMed]

B. Weber, C. Burger, M. T. Wyss, V. G. K. Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20, 2664-2670 (2004).
[CrossRef] [PubMed]

L. F. Rojas-Ochoa, D. Lacoste, R. Lenke, P. Schurtenberger, and F. Scheffold, “Depolarization of backscattered linearly polarized light,” J. Opt. Soc. Am. A 21, 1799-1804 (2004).
[CrossRef]

2003 (1)

F. Scheffold and P. Schurtenberger, “Light scatteringprobes of viscoelastic fluids and solids,” Soft Mater. 1, 139-165 (2003).
[CrossRef]

2001 (3)

1998 (2)

P.-A. Lemieux, M. U. Vera, and D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498-4514 (1998).
[CrossRef]

F. Scheffold, W. Hartk, G. Maret, and E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations of light,” Phys. Rev. B 56, 10942-10952 (1998).
[CrossRef]

1997 (1)

Y. N. Barabanenkov, L. M. Zurk, and M. Y. Barabanenkov, “Single scattering and diffusion approximations for modified radiative transfer theory of wave multiple scattering in dense media near resonance,” Prog. Electromagn. Res. 15, 27-61 (1997).
[CrossRef]

1996 (1)

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

1995 (2)

T. G. Mason and D. A. Weitz, “Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250-1253 (1995).
[CrossRef] [PubMed]

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532-2538 (1995).
[CrossRef]

1992 (2)

P. D. Kaplan, A. G. Yodh, and D. J. Pine, “Diffusion and structure in dense binary suspensions,” Phys. Rev. Lett. 68, 393-396 (1992).
[CrossRef] [PubMed]

L. Tsang, K.-H. Ding, and B. Wen, “Dense media radiative transfer theory for dense discrete random media with particles of multiple sizes and permitivites,” Prog. Electromagn. Res. 6, 181-230 (1992).

1991 (2)

D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252, 686-688 (1991).
[CrossRef] [PubMed]

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

1989 (1)

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

1988 (1)

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

1987 (1)

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B: Condens. Matter 65, 409-413 (1987).
[CrossRef]

1978 (1)

V. A. Bloomfield and T. K. Lim, “Quasi-elastic light scattering,” Methods Enzymol. 48, 415-494 (1978).
[CrossRef] [PubMed]

1968 (2)

Y. N. Barabanenkov and V. M. Finkel'berg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587-591 (1968).

A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256-1259 (1968).
[CrossRef]

1949 (2)

F. Dyson, “The radiation theories of Tomonaga, Schwinger and Feynmann,” Phys. Rev. 75, 486-502 (1949).
[CrossRef]

F. Dyson, “The s matrix in quantum electrodynamics,” Phys. Rev. 75, 1736-1755 (1949).
[CrossRef]

Amblard, C.

L. Le Goff, C. Amblard, F. Furst, and E. M. Furst, “Motor-driven dynamics in actin-myosin networks,” Phys. Rev. Lett. 88, 018101 (2001).
[CrossRef]

Apresyan, L. A.

L. A. Apresyan and Y. A. Kravtsov, Radiation Transfer--Statistical and Wave Aspects (Gordon & Breach, 1996).

Aronson, R.

Barabanenkov, M. Y.

Y. N. Barabanenkov, L. M. Zurk, and M. Y. Barabanenkov, “Single scattering and diffusion approximations for modified radiative transfer theory of wave multiple scattering in dense media near resonance,” Prog. Electromagn. Res. 15, 27-61 (1997).
[CrossRef]

Barabanenkov, Y. N.

Y. N. Barabanenkov, L. M. Zurk, and M. Y. Barabanenkov, “Single scattering and diffusion approximations for modified radiative transfer theory of wave multiple scattering in dense media near resonance,” Prog. Electromagn. Res. 15, 27-61 (1997).
[CrossRef]

Y. N. Barabanenkov and V. M. Finkel'berg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587-591 (1968).

Ben Braham, N.

R. Pierrat, N. Ben Braham, L. F. Rojas-Ochoa, F. Scheffold, and R. Carminati, “The influence of the scattering anisotropy parameter on diffuse reflection of light,” Opt. Commun. 281, 18-22 (2008).
[CrossRef]

Berne, B. J.

B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, 1976).

Bloomfield, V. A.

V. A. Bloomfield and T. K. Lim, “Quasi-elastic light scattering,” Methods Enzymol. 48, 415-494 (1978).
[CrossRef] [PubMed]

Buck, A.

B. Weber, C. Burger, M. T. Wyss, V. G. K. Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20, 2664-2670 (2004).
[CrossRef] [PubMed]

Burger, C.

B. Weber, C. Burger, M. T. Wyss, V. G. K. Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20, 2664-2670 (2004).
[CrossRef] [PubMed]

Carminati, R.

R. Pierrat, N. Ben Braham, L. F. Rojas-Ochoa, F. Scheffold, and R. Carminati, “The influence of the scattering anisotropy parameter on diffuse reflection of light,” Opt. Commun. 281, 18-22 (2008).
[CrossRef]

R. Pierrat, J.-J. Greffet, R. Elaloufi, and R. Carminati, “Spatial coherence in strongly scattering media,” J. Opt. Soc. Am. A 22, 2329-2337 (2005).
[CrossRef]

R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusion-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef] [PubMed]

Chaikin, P. M.

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

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1950).

Dietsche, G.

Ding, K.-H.

L. Tsang, K.-H. Ding, and B. Wen, “Dense media radiative transfer theory for dense discrete random media with particles of multiple sizes and permitivites,” Prog. Electromagn. Res. 6, 181-230 (1992).

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000), Vol. 2, Chap. 7.
[CrossRef]

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000), Vol. 3, Chap. 7.
[CrossRef]

Dogariu, A.

Durian, D. J.

P.-A. Lemieux, M. U. Vera, and D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498-4514 (1998).
[CrossRef]

D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252, 686-688 (1991).
[CrossRef] [PubMed]

Dyson, F.

F. Dyson, “The radiation theories of Tomonaga, Schwinger and Feynmann,” Phys. Rev. 75, 486-502 (1949).
[CrossRef]

F. Dyson, “The s matrix in quantum electrodynamics,” Phys. Rev. 75, 1736-1755 (1949).
[CrossRef]

Elaloufi, R.

R. Pierrat, J.-J. Greffet, R. Elaloufi, and R. Carminati, “Spatial coherence in strongly scattering media,” J. Opt. Soc. Am. A 22, 2329-2337 (2005).
[CrossRef]

R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusion-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef] [PubMed]

Finkel'berg, V. M.

Y. N. Barabanenkov and V. M. Finkel'berg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587-591 (1968).

Fishman, G. S.

G. S. Fishman, Monte Carlo Concepts, Algorithms and Applications (Springer Verlag, 1996).

Frish, U.

U. Frish, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A.T.Bharuch-Reid, ed. (Academic, 1968), Vol. 1, pp. 75-198.

Furst, E. M.

L. Le Goff, C. Amblard, F. Furst, and E. M. Furst, “Motor-driven dynamics in actin-myosin networks,” Phys. Rev. Lett. 88, 018101 (2001).
[CrossRef]

Furst, F.

L. Le Goff, C. Amblard, F. Furst, and E. M. Furst, “Motor-driven dynamics in actin-myosin networks,” Phys. Rev. Lett. 88, 018101 (2001).
[CrossRef]

Gisler, T.

Greffet, J.-J.

R. Pierrat, J.-J. Greffet, R. Elaloufi, and R. Carminati, “Spatial coherence in strongly scattering media,” J. Opt. Soc. Am. A 22, 2329-2337 (2005).
[CrossRef]

R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusion-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef] [PubMed]

Hammersley, J. M.

J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods (Chapman & Hall, 1964).
[CrossRef]

Handscomb, D. C.

J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods (Chapman & Hall, 1964).
[CrossRef]

Hartk, W.

F. Scheffold, W. Hartk, G. Maret, and E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations of light,” Phys. Rev. B 56, 10942-10952 (1998).
[CrossRef]

Herbolzheimer, E.

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

Jaillon, F.

John, S.

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

Kaplan, P. D.

P. D. Kaplan, A. G. Yodh, and D. J. Pine, “Diffusion and structure in dense binary suspensions,” Phys. Rev. Lett. 68, 393-396 (1992).
[CrossRef] [PubMed]

Keller, J. B.

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000), Vol. 2, Chap. 7.
[CrossRef]

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000), Vol. 3, Chap. 7.
[CrossRef]

Kravtsov, Y. A.

L. A. Apresyan and Y. A. Kravtsov, Radiation Transfer--Statistical and Wave Aspects (Gordon & Breach, 1996).

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

Lacoste, D.

Le Goff, L.

L. Le Goff, C. Amblard, F. Furst, and E. M. Furst, “Motor-driven dynamics in actin-myosin networks,” Phys. Rev. Lett. 88, 018101 (2001).
[CrossRef]

Lemieux, P.-A.

P.-A. Lemieux, M. U. Vera, and D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498-4514 (1998).
[CrossRef]

Lenke, R.

Li, J.

Lim, T. K.

V. A. Bloomfield and T. K. Lim, “Quasi-elastic light scattering,” Methods Enzymol. 48, 415-494 (1978).
[CrossRef] [PubMed]

MacKintosh, F. C.

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

Maret, G.

F. Jaillon, S. E. Skipetrov, J. Li, G. Dietsche, G. Maret, and T. Gisler, “Diffusing-wave spectroscopy from head-like tissue phantoms: influence of a non-scattering layer,” Opt. Express 14, 10181-10194 (2001).
[CrossRef]

F. Scheffold, W. Hartk, G. Maret, and E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations of light,” Phys. Rev. B 56, 10942-10952 (1998).
[CrossRef]

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B: Condens. Matter 65, 409-413 (1987).
[CrossRef]

Mason, T. G.

T. G. Mason and D. A. Weitz, “Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250-1253 (1995).
[CrossRef] [PubMed]

Matijevic, E.

F. Scheffold, W. Hartk, G. Maret, and E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations of light,” Phys. Rev. B 56, 10942-10952 (1998).
[CrossRef]

Molisch, A. F.

A. F. Molisch and B. P. Oehry, Radiation Trapping in Atomic Vapours (Clarendon, 1998).

Oehry, B. P.

A. F. Molisch and B. P. Oehry, Radiation Trapping in Atomic Vapours (Clarendon, 1998).

Papanicolaou, G.

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

Pecora, R.

B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, 1976).

Pierrat, R.

R. Pierrat, N. Ben Braham, L. F. Rojas-Ochoa, F. Scheffold, and R. Carminati, “The influence of the scattering anisotropy parameter on diffuse reflection of light,” Opt. Commun. 281, 18-22 (2008).
[CrossRef]

R. Pierrat, J.-J. Greffet, R. Elaloufi, and R. Carminati, “Spatial coherence in strongly scattering media,” J. Opt. Soc. Am. A 22, 2329-2337 (2005).
[CrossRef]

Pine, D. J.

P. D. Kaplan, A. G. Yodh, and D. J. Pine, “Diffusion and structure in dense binary suspensions,” Phys. Rev. Lett. 68, 393-396 (1992).
[CrossRef] [PubMed]

D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252, 686-688 (1991).
[CrossRef] [PubMed]

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

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

Popescu, G.

Rojas-Ochoa, L. F.

R. Pierrat, N. Ben Braham, L. F. Rojas-Ochoa, F. Scheffold, and R. Carminati, “The influence of the scattering anisotropy parameter on diffuse reflection of light,” Opt. Commun. 281, 18-22 (2008).
[CrossRef]

L. F. Rojas-Ochoa, D. Lacoste, R. Lenke, P. Schurtenberger, and F. Scheffold, “Depolarization of backscattered linearly polarized light,” J. Opt. Soc. Am. A 21, 1799-1804 (2004).
[CrossRef]

Rytov, S. M.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

Ryzhik, L.

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

Scheffold, F.

R. Pierrat, N. Ben Braham, L. F. Rojas-Ochoa, F. Scheffold, and R. Carminati, “The influence of the scattering anisotropy parameter on diffuse reflection of light,” Opt. Commun. 281, 18-22 (2008).
[CrossRef]

L. F. Rojas-Ochoa, D. Lacoste, R. Lenke, P. Schurtenberger, and F. Scheffold, “Depolarization of backscattered linearly polarized light,” J. Opt. Soc. Am. A 21, 1799-1804 (2004).
[CrossRef]

B. Weber, C. Burger, M. T. Wyss, V. G. K. Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20, 2664-2670 (2004).
[CrossRef] [PubMed]

F. Scheffold and P. Schurtenberger, “Light scatteringprobes of viscoelastic fluids and solids,” Soft Mater. 1, 139-165 (2003).
[CrossRef]

F. Scheffold, W. Hartk, G. Maret, and E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations of light,” Phys. Rev. B 56, 10942-10952 (1998).
[CrossRef]

Schulthess, V. G. K.

B. Weber, C. Burger, M. T. Wyss, V. G. K. Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20, 2664-2670 (2004).
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Schurtenberger, P.

Sebbah, P.

P. Sebbah, Waves and Imaging through Complex Media (Kluwer Academic, 2001).

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Tatarskii, V. I.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

Tsang, L.

L. Tsang, K.-H. Ding, and B. Wen, “Dense media radiative transfer theory for dense discrete random media with particles of multiple sizes and permitivites,” Prog. Electromagn. Res. 6, 181-230 (1992).

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000), Vol. 3, Chap. 7.
[CrossRef]

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000), Vol. 2, Chap. 7.
[CrossRef]

Vera, M. U.

P.-A. Lemieux, M. U. Vera, and D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498-4514 (1998).
[CrossRef]

Walther, A.

Weber, B.

B. Weber, C. Burger, M. T. Wyss, V. G. K. Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20, 2664-2670 (2004).
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Weitz, D. A.

T. G. Mason and D. A. Weitz, “Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250-1253 (1995).
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D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252, 686-688 (1991).
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J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948-3959 (1991).
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D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134-1137 (1988).
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Wen, B.

L. Tsang, K.-H. Ding, and B. Wen, “Dense media radiative transfer theory for dense discrete random media with particles of multiple sizes and permitivites,” Prog. Electromagn. Res. 6, 181-230 (1992).

Wolf, P. E.

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B: Condens. Matter 65, 409-413 (1987).
[CrossRef]

Wyss, M. T.

B. Weber, C. Burger, M. T. Wyss, V. G. K. Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20, 2664-2670 (2004).
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Yodh, A. G.

P. D. Kaplan, A. G. Yodh, and D. J. Pine, “Diffusion and structure in dense binary suspensions,” Phys. Rev. Lett. 68, 393-396 (1992).
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J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948-3959 (1991).
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Zurk, L. M.

Y. N. Barabanenkov, L. M. Zurk, and M. Y. Barabanenkov, “Single scattering and diffusion approximations for modified radiative transfer theory of wave multiple scattering in dense media near resonance,” Prog. Electromagn. Res. 15, 27-61 (1997).
[CrossRef]

Appl. Opt. (1)

Eur. J. Neurosci. (1)

B. Weber, C. Burger, M. T. Wyss, V. G. K. Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20, 2664-2670 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Methods Enzymol. (1)

V. A. Bloomfield and T. K. Lim, “Quasi-elastic light scattering,” Methods Enzymol. 48, 415-494 (1978).
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Opt. Commun. (1)

R. Pierrat, N. Ben Braham, L. F. Rojas-Ochoa, F. Scheffold, and R. Carminati, “The influence of the scattering anisotropy parameter on diffuse reflection of light,” Opt. Commun. 281, 18-22 (2008).
[CrossRef]

Opt. Express (1)

Phys. Rev. (2)

F. Dyson, “The radiation theories of Tomonaga, Schwinger and Feynmann,” Phys. Rev. 75, 486-502 (1949).
[CrossRef]

F. Dyson, “The s matrix in quantum electrodynamics,” Phys. Rev. 75, 1736-1755 (1949).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. B (2)

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

F. Scheffold, W. Hartk, G. Maret, and E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations of light,” Phys. Rev. B 56, 10942-10952 (1998).
[CrossRef]

Phys. Rev. E (1)

P.-A. Lemieux, M. U. Vera, and D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498-4514 (1998).
[CrossRef]

Phys. Rev. Lett. (5)

R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusion-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef] [PubMed]

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

P. D. Kaplan, A. G. Yodh, and D. J. Pine, “Diffusion and structure in dense binary suspensions,” Phys. Rev. Lett. 68, 393-396 (1992).
[CrossRef] [PubMed]

T. G. Mason and D. A. Weitz, “Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250-1253 (1995).
[CrossRef] [PubMed]

L. Le Goff, C. Amblard, F. Furst, and E. M. Furst, “Motor-driven dynamics in actin-myosin networks,” Phys. Rev. Lett. 88, 018101 (2001).
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Prog. Electromagn. Res. (2)

L. Tsang, K.-H. Ding, and B. Wen, “Dense media radiative transfer theory for dense discrete random media with particles of multiple sizes and permitivites,” Prog. Electromagn. Res. 6, 181-230 (1992).

Y. N. Barabanenkov, L. M. Zurk, and M. Y. Barabanenkov, “Single scattering and diffusion approximations for modified radiative transfer theory of wave multiple scattering in dense media near resonance,” Prog. Electromagn. Res. 15, 27-61 (1997).
[CrossRef]

Science (1)

D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252, 686-688 (1991).
[CrossRef] [PubMed]

Soft Mater. (1)

F. Scheffold and P. Schurtenberger, “Light scatteringprobes of viscoelastic fluids and solids,” Soft Mater. 1, 139-165 (2003).
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Sov. Phys. JETP (1)

Y. N. Barabanenkov and V. M. Finkel'berg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587-591 (1968).

Wave Motion (1)

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Z. Phys. B: Condens. Matter (1)

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B: Condens. Matter 65, 409-413 (1987).
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P. Sebbah, Waves and Imaging through Complex Media (Kluwer Academic, 2001).

B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, 1976).

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000), Vol. 2, Chap. 7.
[CrossRef]

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000), Vol. 3, Chap. 7.
[CrossRef]

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

L. A. Apresyan and Y. A. Kravtsov, Radiation Transfer--Statistical and Wave Aspects (Gordon & Breach, 1996).

U. Frish, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A.T.Bharuch-Reid, ed. (Academic, 1968), Vol. 1, pp. 75-198.

S. Chandrasekhar, Radiative Transfer (Dover, 1950).

A. F. Molisch and B. P. Oehry, Radiation Trapping in Atomic Vapours (Clarendon, 1998).

J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods (Chapman & Hall, 1964).
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G. S. Fishman, Monte Carlo Concepts, Algorithms and Applications (Springer Verlag, 1996).

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Equations (34)

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E ( r 1 , t 1 ) E * ( r 2 , t 2 ) = G ( r 1 , r 1 , t 1 , t 1 ) G * ( r 2 , r 2 , t 2 , t 2 ) K m ( r 1 , ρ 1 , r 2 , ρ 2 , t 1 , τ 1 , t 2 , τ 2 ) E ( ρ 1 , τ 1 ) E * ( ρ 2 , τ 2 ) d 3 r 1 d 3 r 2 d 3 ρ 1 d 3 ρ 2 d t 1 d t 2 d τ 1 d τ 2 ,
G ( r , t ) = G ( k , ω ) exp [ i k r i ω t ] d 3 k 8 π 3 d ω 2 π ,
M m ( r , t ) = M m ( k , ω ) exp [ i k r i ω t ] d 3 k 8 π 3 d ω 2 π ,
K m ( r 1 , ρ 1 , r 2 , ρ 2 , t 1 , τ 1 , t 2 , τ 2 ) = K m ( k 1 , κ 1 , k 2 , κ 2 , ω 1 , Ω 1 , ω 2 , Ω 2 ) exp [ i k 1 r 1 i κ 1 ρ 1 i k 2 r 2 + i κ 2 ρ 2 ] exp [ i ω 1 t 1 + i Ω 1 τ 1 + i ω 2 t 2 i Ω 2 τ 2 ] d 3 k 1 8 π 3 d 3 κ 1 8 π 3 d 3 k 2 8 π 3 d 3 κ 2 8 π 3 d ω 1 2 π d ω 2 2 π d Ω 1 2 π d Ω 2 2 π ,
t m ( r 1 , r 2 , t 1 , t 2 ) = t m ( k 1 , k 2 , ω 1 , ω 2 ) exp [ i k 1 r 1 i k 2 r 2 i ω 1 t 1 + i ω 2 t 2 ] d 3 k 1 8 π 3 d 3 k 2 8 π 3 d ω 1 2 π d ω 2 2 π .
κ 1 , Ω 1 k 1 , ω 1 , κ 2 , Ω 2 k 2 , ω 2 ,
K m ( k 1 , κ 1 , k 2 , κ 2 , ω 1 , Ω 1 , ω 2 , Ω 2 ) = 8 π 3 δ ( k 1 κ 1 k 2 + κ 2 ) 2 π δ ( ω 1 Ω 1 ω 2 + Ω 2 ) K ̃ m ( k 1 , κ 1 , k 2 , κ 2 , ω 1 , Ω 1 , ω 2 , Ω 2 ) .
G ( k , ω ) = 1 ω 2 c 0 2 k 2 M m ( k , ω ) = 1 k eff 2 ( ω ) k 2 ,
[ 2 ω Ω c 0 2 + 2 k q + M m ( k + q 2 , Ω + ω 2 ) M m * ( k q 2 , Ω ω 2 ) ] f ( q , k , ω , Ω ) = [ G ( k + q 2 , Ω + ω 2 ) G * ( k q 2 , Ω ω 2 ) ] K ̃ m ( k + q 2 , κ + q 2 , k q 2 , κ q 2 , Ω + ω 2 , Ω + ω 2 , Ω ω 2 , Ω ω 2 ) f ( q , κ , ω , Ω ) d 3 κ 8 π 3 d Ω 2 π ,
f ( q , k , ω , Ω ) = E ( k + q 2 , Ω + ω 2 ) E * ( k q 2 , Ω ω 2 )
E sca ( k , ω ) = G 0 ( k , ω ) t m v ( k , k , ω , ω ) E inc ( k , ω ) d 3 k 8 π 3 d ω 2 π ,
E sca ( k , ω ) = G 0 ( k , ω + k v ) t m v ( k , k , ω + k v , ω + k v ) E inc ( k , ω ) d 3 k 8 π 3 d ω 2 π ,
E sca ( k , ω ) = G 0 ( k , ω ) t ( k , k , ω ) E inc ( k , ω ) d 3 k 8 π 3 .
t m v ( k , k , ω , ω ) = 2 π t ( k , k , ω k v ) δ [ ω ω ( k k ) v ] .
M m ( r r , t t ) = i = 1 N t m v ( r r i , r r i , t t i , t t i ) P ( r i , t i ) g ( v ) d 3 r i d t i d v ,
g ( v ) d v = 1 .
M m ( k , ω ) = lim T ρ T t m v ( k , k , ω , ω ) g ( v ) d v ,
M m ( k , ω ) = ρ t ( k , k , ω ) = M ( k , ω ) ,
K ̃ m ( k 1 , κ 1 , k 2 , κ 2 , ω 1 , Ω 1 , ω 2 , Ω 2 ) = lim T ρ T t m v ( k 1 , κ 1 , ω 1 , Ω 1 ) t m v * ( k 2 , κ 2 , ω 2 , Ω 2 ) g ( v ) d v .
K ̃ m ( k + q 2 , κ + q 2 , k q 2 , κ q 2 , Ω + ω 2 , Ω + ω 2 , Ω ω 2 , Ω ω 2 ) = ρ t ( k + q 2 , κ + q 2 , Ω + ω 2 ) t * ( k q 2 , κ q 2 , Ω ω 2 ) 1 k κ g ̃ ( Ω Ω k κ ) ,
g ̃ ( w ) = δ [ w w v w ] g ( v ) d v ,
K ̃ m = K ̃ k κ g ̃ ( Ω Ω k κ ) ,
[ 2 ω Ω c 0 2 + 2 k q + M ( k , Ω ) M * ( k , Ω ) ] f ( q , k , ω , Ω ) = [ G ( k , Ω ) G * ( k , Ω ) ] K ̃ ( k , κ , k , κ , Ω ) k κ g ̃ ( Ω Ω k κ ) f ( q , κ , ω , Ω ) d 3 κ 8 π 3 d Ω .
G ( k , ω ) = PV [ 1 k r 2 k 2 ] i π δ ( k r 2 k 2 ) ,
f ( q , k , ω , Ω ) = δ ( k r k ) I ( q , u , ω , Ω ) .
[ i ω Ω c 0 2 k r + i u q I [ M ( k r u , Ω ) ] k r ] I ( q , u , ω , Ω ) = 1 16 π 2 K ̃ ( k r u , k r u , k r u , k r u , Ω ) k r u u g ̃ ( Ω Ω k r u u ) I ( q , u , ω , Ω ) d u d Ω .
μ e ( Ω ) = I [ M ( k r u , Ω ) ] k r ,
μ s ( Ω ) p ( u , u , Ω ) = K ̃ ( k r u , k r u , k r u , k r u , Ω ) 4 π .
[ 1 c t + u r + μ e ] I ( r , u , t , Ω ) = μ s 4 π p ( u , u ) k r u u g ̃ ( Ω Ω k r u u ) I ( r , u , t , Ω ) d u d Ω .
g 1 ( r , t , τ ) = E ( r , t + τ 2 ) E * ( r , t τ 2 ) = k r 2 4 π I ( r , u , t , τ ) d u ,
[ 1 c t + u r + μ e ] g 1 ( r , u , t , τ ) = μ s 4 π p ( u , u ) g ͌ ( k r u u τ ) g 1 ( r , u , t , τ ) d u ,
g ( v ) = 1 [ σ 2 π ] 3 exp [ v 2 2 σ 2 ] ,
g ͌ ( k r u u τ ) = exp [ σ 2 k r 2 u u 2 τ 2 ] .
g ͌ ( k r u u τ ) = exp [ 2 D B k r 2 u u 2 τ ] .

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