Abstract

We study the deviation from diffusion theory that occurs in the dynamic transport of light through thin scattering slabs. Solving numerically the time-dependent radiative transfer equation, we obtain the decay time and the effective diffusion coefficient Deff. We observe a nondiffusive behavior for systems whose thickness L is smaller than 8ltr, where ltr is the transport mean free path. We introduce a simple model that yields the position of the transition between the diffusive and the nondiffusive regimes. The size dependence of Deff in the nondiffusive region is strongly affected by internal reflections. We show that the reduction of ∼50% of Deff that was observed experimentally [Phys. Rev. Lett. 79, 4369 (1997)] can be reproduced by the radiative transfer approach. We demonstrate that the radiative transfer equation is an appropriate tool for studying dynamic light transport in thin scattering systems when coherent effects play no significant role.

© 2004 Optical Society of America

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References

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  5. S. K. Gayen, R. R. Alfano, “Biomedical imaging techniques,” Opt. Photon. News 7, 17–22 (1996).
    [CrossRef]
  6. L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
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  7. M. R. Hee, J. A. Izatt, J. M. Jacobson, J. G. Fujimoto, E. A. Swanson, “Femtosecond transillumination optical coherence tomography,” Opt. Lett. 18, 950–952 (1993).
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  8. 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]
  9. D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
    [CrossRef] [PubMed]
  10. A. Mandelis, “Diffusion waves and their uses,” Phys. Today 53, 29–34 (2000).
    [CrossRef]
  11. A. Majumdar, “Microscale heat conduction in dielectric thin films,” J. Heat Transfer 115, 7–16 (1993).
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    [CrossRef]
  13. M. Xu, W. Cai, M. Lax, R. R. Alfano, “Photon migration in turbid media using a cumulant approximation to radiative transfer,” Phys. Rev. E 65, 066609 (2002).
    [CrossRef]
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    [CrossRef] [PubMed]
  15. R. H. J. Kop, P. de Vries, R. Sprik, A. Lagendijk, “Observation of anomalous transport of strongly multiple-scattered light in thin disordered slabs,” Phys. Rev. Lett. 79, 4369–4372 (1997).
    [CrossRef]
  16. Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, P. Sheng, “Wave transport in random media: The ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
    [CrossRef]
  17. I. Freund, M. Kaveh, M. Rosenbluh, “Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation,” Phys. Rev. Lett. 60, 1130–1133 (1988).
    [CrossRef] [PubMed]
  18. K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length resolved dynamic light scattering in highly scattering random media: The transition to diffusing-wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
    [CrossRef]
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  22. D. J. Durian, J. Rudnick, “Photon migration at short times and distances and in cases of strong absorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).
  23. X. Zhang, Z. Q. Zhang, “Wave transport through thin slabs of random media with internal reflection: Ballistic to diffusive transition,” Phys. Rev. E 66, 016612 (2002).
  24. J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, C. W. Rella, “Static and dynamic transport of light close to the Anderson localization transition,” Phys. Rev. E 63, 046613 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  30. R. Graaff, J. J. Ten Bosch, “Diffusion coefficient in photon diffusion theory,” Opt. Lett. 25, 43–45 (2000).
  31. R. Graaff, K. Rinzema, “Practical improvements on photon diffusion theory: application to isotropic scattering,” Phys. Med. Biol. 46, 3043–3050 (2001).
    [PubMed]
  32. R. Elaloufi, R. Carminati, J.-J. Greffet, “Definition of the diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 20, 678–685 (2003).
  33. J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
    [PubMed]

2003 (1)

2002 (3)

M. Xu, W. Cai, M. Lax, R. R. Alfano, “Photon migration in turbid media using a cumulant approximation to radiative transfer,” Phys. Rev. E 65, 066609 (2002).
[CrossRef]

R. Elaloufi, R. Carminati, J.-J. Greffet, “Time-dependent transport through scattering media: From radiative transfer to diffusion,” J. Opt. A, Pure Appl. Opt. 4, S103–S108 (2002).

X. Zhang, Z. Q. Zhang, “Wave transport through thin slabs of random media with internal reflection: Ballistic to diffusive transition,” Phys. Rev. E 66, 016612 (2002).

2001 (2)

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, C. W. Rella, “Static and dynamic transport of light close to the Anderson localization transition,” Phys. Rev. E 63, 046613 (2001).
[CrossRef]

R. Graaff, K. Rinzema, “Practical improvements on photon diffusion theory: application to isotropic scattering,” Phys. Med. Biol. 46, 3043–3050 (2001).
[PubMed]

2000 (2)

A. Mandelis, “Diffusion waves and their uses,” Phys. Today 53, 29–34 (2000).
[CrossRef]

R. Graaff, J. J. Ten Bosch, “Diffusion coefficient in photon diffusion theory,” Opt. Lett. 25, 43–45 (2000).

1999 (4)

R. Aronson, N. Corngold, “Photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A 16, 1066–1071 (1999).
[CrossRef]

K. Mitra, S. Kumar, “Development and comparison of models for light-pulse transport through scattering-absorbing media,” Appl. Opt. 38, 188–196 (1999).

G. Chen, “Phonon wave heat conduction in thin films and superlattices,” J. Heat Transfer 121, 945–953 (1999).
[CrossRef]

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, P. Sheng, “Wave transport in random media: The ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

1998 (2)

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length resolved dynamic light scattering in highly scattering random media: The transition to diffusing-wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

A. D. Kim, A. Ishimaru, “Optical diffusion of continuous wave, pulsed and density waves in scattering media and comparisons with radiative transfer,” Appl. Opt. 37, 5313–5319 (1998).

1997 (2)

D. J. Durian, J. Rudnick, “Photon migration at short times and distances and in cases of strong absorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).

R. H. J. Kop, P. de Vries, R. Sprik, A. Lagendijk, “Observation of anomalous transport of strongly multiple-scattered light in thin disordered slabs,” Phys. Rev. Lett. 79, 4369–4372 (1997).
[CrossRef]

1996 (1)

S. K. Gayen, R. R. Alfano, “Biomedical imaging techniques,” Opt. Photon. News 7, 17–22 (1996).
[CrossRef]

1995 (2)

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[CrossRef]

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

1993 (2)

1991 (2)

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
[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).
[PubMed]

1990 (1)

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

1988 (2)

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

I. Freund, M. Kaveh, M. Rosenbluh, “Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation,” Phys. Rev. Lett. 60, 1130–1133 (1988).
[CrossRef] [PubMed]

1987 (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]

Alfano, R. R.

M. Xu, W. Cai, M. Lax, R. R. Alfano, “Photon migration in turbid media using a cumulant approximation to radiative transfer,” Phys. Rev. E 65, 066609 (2002).
[CrossRef]

S. K. Gayen, R. R. Alfano, “Biomedical imaging techniques,” Opt. Photon. News 7, 17–22 (1996).
[CrossRef]

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Aronson, R.

Bizheva, K. K.

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length resolved dynamic light scattering in highly scattering random media: The transition to diffusing-wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

Boas, D. A.

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length resolved dynamic light scattering in highly scattering random media: The transition to diffusing-wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

Cai, W.

M. Xu, W. Cai, M. Lax, R. R. Alfano, “Photon migration in turbid media using a cumulant approximation to radiative transfer,” Phys. Rev. E 65, 066609 (2002).
[CrossRef]

Carminati, R.

R. Elaloufi, R. Carminati, J.-J. Greffet, “Definition of the diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 20, 678–685 (2003).

R. Elaloufi, R. Carminati, J.-J. Greffet, “Time-dependent transport through scattering media: From radiative transfer to diffusion,” J. Opt. A, Pure Appl. Opt. 4, S103–S108 (2002).

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Chaikin, P. M.

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

Chance, B.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Chen, G.

G. Chen, “Phonon wave heat conduction in thin films and superlattices,” J. Heat Transfer 121, 945–953 (1999).
[CrossRef]

Corngold, N.

de Vries, P.

R. H. J. Kop, P. de Vries, R. Sprik, A. Lagendijk, “Observation of anomalous transport of strongly multiple-scattered light in thin disordered slabs,” Phys. Rev. Lett. 79, 4369–4372 (1997).
[CrossRef]

Durian, D. J.

Elaloufi, R.

R. Elaloufi, R. Carminati, J.-J. Greffet, “Definition of the diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 20, 678–685 (2003).

R. Elaloufi, R. Carminati, J.-J. Greffet, “Time-dependent transport through scattering media: From radiative transfer to diffusion,” J. Opt. A, Pure Appl. Opt. 4, S103–S108 (2002).

Freund, I.

I. Freund, M. Kaveh, M. Rosenbluh, “Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation,” Phys. Rev. Lett. 60, 1130–1133 (1988).
[CrossRef] [PubMed]

Fujimoto, J. G.

Gayen, S. K.

S. K. Gayen, R. R. Alfano, “Biomedical imaging techniques,” Opt. Photon. News 7, 17–22 (1996).
[CrossRef]

Gomez Rivas, J.

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, C. W. Rella, “Static and dynamic transport of light close to the Anderson localization transition,” Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Graaff, R.

R. Graaff, K. Rinzema, “Practical improvements on photon diffusion theory: application to isotropic scattering,” Phys. Med. Biol. 46, 3043–3050 (2001).
[PubMed]

R. Graaff, J. J. Ten Bosch, “Diffusion coefficient in photon diffusion theory,” Opt. Lett. 25, 43–45 (2000).

Greffet, J.-J.

R. Elaloufi, R. Carminati, J.-J. Greffet, “Definition of the diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 20, 678–685 (2003).

R. Elaloufi, R. Carminati, J.-J. Greffet, “Time-dependent transport through scattering media: From radiative transfer to diffusion,” J. Opt. A, Pure Appl. Opt. 4, S103–S108 (2002).

Hee, M. R.

Herbolzheimer, E.

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

Ho, P. P.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Ishimaru, A.

Izatt, J. A.

Jacobson, J. M.

Jones, I. P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, P. Sheng, “Wave transport in random media: The ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

Kaveh, M.

I. Freund, M. Kaveh, M. Rosenbluh, “Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation,” Phys. Rev. Lett. 60, 1130–1133 (1988).
[CrossRef] [PubMed]

Kim, A. D.

Kop, R. H. J.

R. H. J. Kop, P. de Vries, R. Sprik, A. Lagendijk, “Observation of anomalous transport of strongly multiple-scattered light in thin disordered slabs,” Phys. Rev. Lett. 79, 4369–4372 (1997).
[CrossRef]

Kumar, S.

Lagendijk, A.

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, C. W. Rella, “Static and dynamic transport of light close to the Anderson localization transition,” Phys. Rev. E 63, 046613 (2001).
[CrossRef]

R. H. J. Kop, P. de Vries, R. Sprik, A. Lagendijk, “Observation of anomalous transport of strongly multiple-scattered light in thin disordered slabs,” Phys. Rev. Lett. 79, 4369–4372 (1997).
[CrossRef]

Lax, M.

M. Xu, W. Cai, M. Lax, R. R. Alfano, “Photon migration in turbid media using a cumulant approximation to radiative transfer,” Phys. Rev. E 65, 066609 (2002).
[CrossRef]

Liu, C.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Liu, F.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Majumdar, A.

A. Majumdar, “Microscale heat conduction in dielectric thin films,” J. Heat Transfer 115, 7–16 (1993).
[CrossRef]

Mandelis, A.

A. Mandelis, “Diffusion waves and their uses,” Phys. Today 53, 29–34 (2000).
[CrossRef]

Maret, G.

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]

Mitra, K.

Noordam, L. D.

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, C. W. Rella, “Static and dynamic transport of light close to the Anderson localization transition,” Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Page, J. H.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, P. Sheng, “Wave transport in random media: The ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

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).
[PubMed]

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

Rella, C. W.

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, C. W. Rella, “Static and dynamic transport of light close to the Anderson localization transition,” Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Rinzema, K.

R. Graaff, K. Rinzema, “Practical improvements on photon diffusion theory: application to isotropic scattering,” Phys. Med. Biol. 46, 3043–3050 (2001).
[PubMed]

Rosenbluh, M.

I. Freund, M. Kaveh, M. Rosenbluh, “Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation,” Phys. Rev. Lett. 60, 1130–1133 (1988).
[CrossRef] [PubMed]

Rudnick, J.

Schriemer, H. P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, P. Sheng, “Wave transport in random media: The ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

Sheng, P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, P. Sheng, “Wave transport in random media: The ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic, New York, 1995).

Siegel, A. M.

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length resolved dynamic light scattering in highly scattering random media: The transition to diffusing-wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

Sprik, R.

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, C. W. Rella, “Static and dynamic transport of light close to the Anderson localization transition,” Phys. Rev. E 63, 046613 (2001).
[CrossRef]

R. H. J. Kop, P. de Vries, R. Sprik, A. Lagendijk, “Observation of anomalous transport of strongly multiple-scattered light in thin disordered slabs,” Phys. Rev. Lett. 79, 4369–4372 (1997).
[CrossRef]

Stamnes, K.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.

Swanson, E. A.

Ten Bosch, J. J.

Thomas, G. E.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.

Waitz, D. A.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, P. Sheng, “Wave transport in random media: The ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

Wang, L.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
[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).
[PubMed]

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

Wolf, P. E.

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]

Xu, M.

M. Xu, W. Cai, M. Lax, R. R. Alfano, “Photon migration in turbid media using a cumulant approximation to radiative transfer,” Phys. Rev. E 65, 066609 (2002).
[CrossRef]

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

Yoo, K. M.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Zhang, G.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Zhang, X.

X. Zhang, Z. Q. Zhang, “Wave transport through thin slabs of random media with internal reflection: Ballistic to diffusive transition,” Phys. Rev. E 66, 016612 (2002).

Zhang, Z. Q.

X. Zhang, Z. Q. Zhang, “Wave transport through thin slabs of random media with internal reflection: Ballistic to diffusive transition,” Phys. Rev. E 66, 016612 (2002).

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, P. Sheng, “Wave transport in random media: The ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

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).
[PubMed]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Appl. Opt. (2)

J. Heat Transfer (2)

A. Majumdar, “Microscale heat conduction in dielectric thin films,” J. Heat Transfer 115, 7–16 (1993).
[CrossRef]

G. Chen, “Phonon wave heat conduction in thin films and superlattices,” J. Heat Transfer 121, 945–953 (1999).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

R. Elaloufi, R. Carminati, J.-J. Greffet, “Time-dependent transport through scattering media: From radiative transfer to diffusion,” J. Opt. A, Pure Appl. Opt. 4, S103–S108 (2002).

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

Opt. Lett. (2)

Opt. Photon. News (1)

S. K. Gayen, R. R. Alfano, “Biomedical imaging techniques,” Opt. Photon. News 7, 17–22 (1996).
[CrossRef]

Phys. Med. Biol. (1)

R. Graaff, K. Rinzema, “Practical improvements on photon diffusion theory: application to isotropic scattering,” Phys. Med. Biol. 46, 3043–3050 (2001).
[PubMed]

Phys. Rev. A (1)

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

Phys. Rev. E (5)

X. Zhang, Z. Q. Zhang, “Wave transport through thin slabs of random media with internal reflection: Ballistic to diffusive transition,” Phys. Rev. E 66, 016612 (2002).

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, C. W. Rella, “Static and dynamic transport of light close to the Anderson localization transition,” Phys. Rev. E 63, 046613 (2001).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Inverse decay time versus slab thickness L for different values of the medium effective refractive index n2 (n1=n3=1 for the half-spaces z<0 and z>L). The medium parameters are ls=0.95 μm, la=46.5 μm (albedo ω0=0.98). (a) g=0, (b) g=0.4. Phase function: Henyey–Greenstein.

Fig. 2
Fig. 2

Effective diffusion coefficient Deff versus slab thickness L. The medium parameters are the same as in Fig. 1(a) with g=0. (a) Deff/L2 versus L; this quantity becomes independent of L for Lltr. (b) Deff versus L.

Fig. 3
Fig. 3

Inverse decay time and diffusion coefficient for a slab with parameters similar to those in Ref. 15. The slab contains TiO2 particles illuminated at λ=780 nm. g=0.27, ls=0.65 μm, la=200 μm (albedo ω0=0.997). The effective index of the slab is n=1.39. (a) τ-1 versus L, (b) diffusion coefficient D as defined in Ref. 15 normalized by its asymptotic value D0; solid curve, R¯=0. The inset shows the results obtained for different values of R¯. Phase function: Mie scattering.

Fig. 4
Fig. 4

Dispersion relations [Re(s) versus k] in an infinite medium for the solutions of the RTE and of the diffusion equation; s and k are in dimensionless units, the reference length scale being L*=(μs+μa)-1 and the reference time scale being t*=L*/v. The medium parameters are ltr=0.95 μm, ω0=0.995 (albedo). (a) s* versus k* for g=0. The numerical solution obtained from the RTE is compared with an analytical result valid for isotropic scattering and k*<πω0/2 and with the solution obtained from the diffusion approximation. (b) s* versus k* for g=0.5; no analytical solution can be found in this case. Phase function: Henyey–Greenstein.

Fig. 5
Fig. 5

(a) Decay time versus slab thickness L obtained from the analytical model Eq. (16). (b) Effective diffusion coefficient for slab thickness L obtained from the analytical model Eq. (17). The medium parameters are ltr=0.95 μm, ω0=0.995.

Equations (20)

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1vI(z, μ, t)t+μ I(z, μ, t)z
=-(μs+μa)I(z, μ, t)+μs2-1+1p(0)(μ, μ)I(z, μ, t)dμ,
μ I(z, μ, ω)z=-μs+μa-i ωvI(z, μ, ω)+μs2-1+1p(0)(μ, μ)I(z, μ, ω)dμ,
I(z, μ, ω)=Ib+(z, ω)δ(μ-1)+Ib-(z, ω)δ(μ+1)+Id(τ, μ, ω),
dIb±(z, ω)dz=-α(ω)Ib±(z, ω)
μ Id(z, μ, ω)z=-α(ω)Id(z, μ, ω)+μs2-1+1p(0)(μ, μ)Id(z, μ, ω)dμ+S(z, μ, ω)
Ib+(z, ω)=T12(μ=1)I0(ω)exp[-α(ω)z]Γ,
Ib-(z, ω)=T12(μ=1)I0(ω)exp[-α(ω)×(2L-z)]R23(μ=1)Γ,
S(z, μ, ω)=μs2 p(0)(μ, 1)Ib+(z, ω)+μs2 p(0)(μ, -1)Ib-(z, ω).
Id(z=0, μ, ω)=R21(μ)Id(z=0, -μ, ω),
forμ>0
Id(z=L, μ, ω)=R23(|μ|)Id(z=L, -μ, ω),
forμ<0.
Deff=L2π21τ-μav.
D=Leff2π21τ-μav.
s=-(μa+μs)v+kvtan(k/μs),for|k|<πμs/2,
u(r, t)t-D2u(r, t)+μavu(r, t)=0,
s=-μav-k2D.
τ=(μa+μs)v-kvtan(k/μs)-1.
Deff=L2π2 μsv-Lvπ tan(π/μsL).

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