Abstract

We revisit the definition of the diffusion coefficient for light transport in scattering and absorbing media. From an asymptotic analysis of the transport equation, we present a novel derivation of the diffusion coefficient, which is restricted neither to low absorption nor to a situation in which the specific intensity is quasi-isotropic. Our result agrees with previous expressions of the diffusion coefficient in the appropriate limit. Using numerical simulations, we discuss the implications of the proper choice of the diffusion coefficient for time-dependent transport.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
    [CrossRef]
  2. S. K. Gayen, R. R. Alfano, “Biomedical imaging techniques,” Opt. Photon. News, July1996, pp. 17–22.
  3. A. Mandelis, “Diffusion waves and their uses,” Phys. Today 53, 29–34 (2000).
    [CrossRef]
  4. P Sebbah, ed., Waves and Imaging through Complex Media (Kluwer Academic, Dordrecht, The Netherlands2001).
  5. S Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  6. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  7. A Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, Piscataway, N.J.1997).
  8. R. Aronson, N. Corngold, “Photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A 16, 1066–1071 (1999).
    [CrossRef]
  9. K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
    [CrossRef]
  10. M. Bassani, F. Martelli, G. Zaccanti, D. Contini, “Independence of the diffusion coefficient from absorption: experimental and numerical evidence,” Opt. Lett. 22, 853–855 (1997).
    [CrossRef] [PubMed]
  11. T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
    [CrossRef]
  12. D. J. Durian, “The diffusion coefficient depends on absorption,” Opt. Lett. 23, 1502–1504 (1998).
    [CrossRef]
  13. R. Graaff, J. J. Ten Bosch, “Diffusion coefficient in photon diffusion theory,” Opt. Lett. 25, 43–45 (2000).
    [CrossRef]
  14. G. E. Thomas, K Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.
  15. A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143 (1996).
    [CrossRef]
  16. F. Liu, K. M. Yoo, R. R. Alfano, “Transmitted photon intensity through biological tissues within various time windows,” Opt. Lett. 19, 740–742 (1994).
    [CrossRef] [PubMed]
  17. D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusive-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
    [CrossRef] [PubMed]
  18. D. A. Boas, L. E. Campbell, A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
    [CrossRef] [PubMed]
  19. 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]
  20. 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).
    [CrossRef]
  21. M Lax, V Nayaranamurti, R. C. Fulton, “Classical diffusive photon transport in a slab,” in Proceedings of the Symposium on Laser Optics of Condensed Matter, J. L. Birman, H. Z. Cummins, A. A. Kaplyanskii, eds. (Plenum, New York, 1987), pp. 229–235.
  22. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [CrossRef]
  23. 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]
  24. B. A. van Tiggelen, A. Lagendijk, M. P. van Albada, A. Tip, “Speed of light in random media,” Phys. Rev. B 45, 12233–12243 (1992).
    [CrossRef]
  25. Y. Kuga, A. Ishimaru, D. Rice, “Velocity of coherent and incoherent electromagnetic waves in a dense strongly scattering medium,” Phys. Rev. B 48, 13155–13158 (1993).
    [CrossRef]
  26. H. P. Schriemer, M. L. Cowan, J. H. Page, P. Sheng, Z. Liu, D. A. Weitz, “Energy velocity of diffusing waves in strongly scattering media,” Phys. Rev. Lett. 79, 3166–3169 (1997).
    [CrossRef]
  27. 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]
  28. D. J. Durian, J. Rudnick, “Photon migration at short times and distances and in case of strong absorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).
    [CrossRef]
  29. 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).
    [CrossRef]
  30. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 90.

2002

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).
[CrossRef]

2000

1999

1998

1997

M. Bassani, F. Martelli, G. Zaccanti, D. Contini, “Independence of the diffusion coefficient from absorption: experimental and numerical evidence,” Opt. Lett. 22, 853–855 (1997).
[CrossRef] [PubMed]

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

T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
[CrossRef]

H. P. Schriemer, M. L. Cowan, J. H. Page, P. Sheng, Z. Liu, D. A. Weitz, “Energy velocity of diffusing waves in strongly scattering media,” Phys. Rev. Lett. 79, 3166–3169 (1997).
[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]

1996

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143 (1996).
[CrossRef]

S. K. Gayen, R. R. Alfano, “Biomedical imaging techniques,” Opt. Photon. News, July1996, pp. 17–22.

1995

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

D. A. Boas, L. E. Campbell, A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

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

1994

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

F. Liu, K. M. Yoo, R. R. Alfano, “Transmitted photon intensity through biological tissues within various time windows,” Opt. Lett. 19, 740–742 (1994).
[CrossRef] [PubMed]

1993

Y. Kuga, A. Ishimaru, D. Rice, “Velocity of coherent and incoherent electromagnetic waves in a dense strongly scattering medium,” Phys. Rev. B 48, 13155–13158 (1993).
[CrossRef]

1992

B. A. van Tiggelen, A. Lagendijk, M. P. van Albada, A. Tip, “Speed of light in random media,” Phys. Rev. B 45, 12233–12243 (1992).
[CrossRef]

1991

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]

1988

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

Alfano, R. R.

Aronson, R.

Bassani, M.

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]

T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
[CrossRef]

D. A. Boas, L. E. Campbell, A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

Campbell, L. E.

D. A. Boas, L. E. Campbell, A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

Carminati, R.

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).
[CrossRef]

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, “Diffusive-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Chance, B.

Chandrasekhar, S

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

Contini, D.

Corngold, N.

Cowan, M. L.

H. P. Schriemer, M. L. Cowan, J. H. Page, P. Sheng, Z. Liu, D. A. Weitz, “Energy velocity of diffusing waves in strongly scattering media,” Phys. Rev. Lett. 79, 3166–3169 (1997).
[CrossRef]

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]

Durduran, T.

Durian, D. J.

Elaloufi, R.

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).
[CrossRef]

Fulton, R. C.

M Lax, V Nayaranamurti, R. C. Fulton, “Classical diffusive photon transport in a slab,” in Proceedings of the Symposium on Laser Optics of Condensed Matter, J. L. Birman, H. Z. Cummins, A. A. Kaplyanskii, eds. (Plenum, New York, 1987), pp. 229–235.

Furutsu, K.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Gayen, S. K.

S. K. Gayen, R. R. Alfano, “Biomedical imaging techniques,” Opt. Photon. News, July1996, pp. 17–22.

Graaff, R.

Greffet, J.-J.

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).
[CrossRef]

Herbolzheimer, E.

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

Ishimaru, A

A Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, Piscataway, N.J.1997).

Ishimaru, A.

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).
[CrossRef]

Y. Kuga, A. Ishimaru, D. Rice, “Velocity of coherent and incoherent electromagnetic waves in a dense strongly scattering medium,” Phys. Rev. B 48, 13155–13158 (1993).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 90.

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]

Kuga, Y.

Y. Kuga, A. Ishimaru, D. Rice, “Velocity of coherent and incoherent electromagnetic waves in a dense strongly scattering medium,” Phys. Rev. B 48, 13155–13158 (1993).
[CrossRef]

Lagendijk, A.

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]

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143 (1996).
[CrossRef]

B. A. van Tiggelen, A. Lagendijk, M. P. van Albada, A. Tip, “Speed of light in random media,” Phys. Rev. B 45, 12233–12243 (1992).
[CrossRef]

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]

Lax, M

M Lax, V Nayaranamurti, R. C. Fulton, “Classical diffusive photon transport in a slab,” in Proceedings of the Symposium on Laser Optics of Condensed Matter, J. L. Birman, H. Z. Cummins, A. A. Kaplyanskii, eds. (Plenum, New York, 1987), pp. 229–235.

Liu, F.

Liu, Z.

H. P. Schriemer, M. L. Cowan, J. H. Page, P. Sheng, Z. Liu, D. A. Weitz, “Energy velocity of diffusing waves in strongly scattering media,” Phys. Rev. Lett. 79, 3166–3169 (1997).
[CrossRef]

Mandelis, A.

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

Martelli, F.

Nayaranamurti, V

M Lax, V Nayaranamurti, R. C. Fulton, “Classical diffusive photon transport in a slab,” in Proceedings of the Symposium on Laser Optics of Condensed Matter, J. L. Birman, H. Z. Cummins, A. A. Kaplyanskii, eds. (Plenum, New York, 1987), pp. 229–235.

Page, J. H.

H. P. Schriemer, M. L. Cowan, J. H. Page, P. Sheng, Z. Liu, D. A. Weitz, “Energy velocity of diffusing waves in strongly scattering media,” Phys. Rev. Lett. 79, 3166–3169 (1997).
[CrossRef]

Pine, D. J.

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

Rice, D.

Y. Kuga, A. Ishimaru, D. Rice, “Velocity of coherent and incoherent electromagnetic waves in a dense strongly scattering medium,” Phys. Rev. B 48, 13155–13158 (1993).
[CrossRef]

Rudnick, J.

Schriemer, H. P.

H. P. Schriemer, M. L. Cowan, J. H. Page, P. Sheng, Z. Liu, D. A. Weitz, “Energy velocity of diffusing waves in strongly scattering media,” Phys. Rev. Lett. 79, 3166–3169 (1997).
[CrossRef]

Sheng, P.

H. P. Schriemer, M. L. Cowan, J. H. Page, P. Sheng, Z. Liu, D. A. Weitz, “Energy velocity of diffusing waves in strongly scattering media,” Phys. Rev. Lett. 79, 3166–3169 (1997).
[CrossRef]

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.

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.

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.

Tip, A.

B. A. van Tiggelen, A. Lagendijk, M. P. van Albada, A. Tip, “Speed of light in random media,” Phys. Rev. B 45, 12233–12243 (1992).
[CrossRef]

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]

van Albada, M. P.

B. A. van Tiggelen, A. Lagendijk, M. P. van Albada, A. Tip, “Speed of light in random media,” Phys. Rev. B 45, 12233–12243 (1992).
[CrossRef]

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]

van Tiggelen, B. A.

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143 (1996).
[CrossRef]

B. A. van Tiggelen, A. Lagendijk, M. P. van Albada, A. Tip, “Speed of light in random media,” Phys. Rev. B 45, 12233–12243 (1992).
[CrossRef]

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]

Weitz, D. A.

H. P. Schriemer, M. L. Cowan, J. H. Page, P. Sheng, Z. Liu, D. A. Weitz, “Energy velocity of diffusing waves in strongly scattering media,” Phys. Rev. Lett. 79, 3166–3169 (1997).
[CrossRef]

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

Yamada, Y.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Yodh, A.

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

Yodh, A. G.

T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
[CrossRef]

D. A. Boas, L. E. Campbell, A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

Yoo, K. M.

Zaccanti, G.

Zweifel, P. F.

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

Appl. Opt.

J. Opt. A Pure Appl. Opt.

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).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Opt. Photon. News

S. K. Gayen, R. R. Alfano, “Biomedical imaging techniques,” Opt. Photon. News, July1996, pp. 17–22.

Phys. Rep.

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143 (1996).
[CrossRef]

Phys. Rev. B

B. A. van Tiggelen, A. Lagendijk, M. P. van Albada, A. Tip, “Speed of light in random media,” Phys. Rev. B 45, 12233–12243 (1992).
[CrossRef]

Y. Kuga, A. Ishimaru, D. Rice, “Velocity of coherent and incoherent electromagnetic waves in a dense strongly scattering medium,” Phys. Rev. B 48, 13155–13158 (1993).
[CrossRef]

Phys. Rev. E

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

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]

Phys. Rev. Lett.

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

D. A. Boas, L. E. Campbell, A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

H. P. Schriemer, M. L. Cowan, J. H. Page, P. Sheng, Z. Liu, D. A. Weitz, “Energy velocity of diffusing waves in strongly scattering media,” Phys. Rev. Lett. 79, 3166–3169 (1997).
[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]

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]

Phys. Today

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

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

Other

P Sebbah, ed., Waves and Imaging through Complex Media (Kluwer Academic, Dordrecht, The Netherlands2001).

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

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

A Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, Piscataway, N.J.1997).

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

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 90.

M Lax, V Nayaranamurti, R. C. Fulton, “Classical diffusive photon transport in a slab,” in Proceedings of the Symposium on Laser Optics of Condensed Matter, J. L. Birman, H. Z. Cummins, A. A. Kaplyanskii, eds. (Plenum, New York, 1987), pp. 229–235.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

(a) Angular behavior (polar diagram with θ in degrees) of the fundamental mode g 0 + ( μ = cos   θ ) for different values of the albedo ω 0 . l tr = 0.95   μ m and g = 0.5 . (b) Variation of g 0 versus absorption for two values of the anisotropy factor g. l sca = 1   μ m . Phase function: Henyey–Greenstein.

Fig. 2
Fig. 2

Dependence on absorption [(a) and (b)] and scattering anisotropy [(c) and (d)] of the diffusion coefficients obtained from three different approaches: the P 1 approximation D P 1 , the random-walk approach in a nonabsorbing medium D st , and the asymptotic approach D as . D as ( 0 ) in (a) is a first-order approximation of D as valid for isotropic scattering and low absorption. (a) l tr = 0.95   μ m , g = 0 ; (b) l tr = 0.95   μ m , g = 0.5 ; (c) l tr = 0.95   μ m , l abs = 0.1   mm ; (d) l sca = 1   μ m , ω 0 = 0.85 . Phase function: Henyey–Greenstein.

Fig. 3
Fig. 3

Diffuse-transmission impulse response through a slab of width L = 15   μ m , with l tr = 0.95   μ m and l abs = 5.4   μ m . (a) g = 0 , (b) g = 0.5 . Phase function: Henyey–Greenstein. The energy velocity is assumed to be a constant: v = 3.10 8   m   s - 1 . The long-time decay is correctly predicted by diffusion theory only when the diffusion coefficient given by Eq. (9) is used.

Tables (1)

Tables Icon

Table 1 Numerical Values of the First Three Eigenvalues k 0 , k 1 , and k 2 Obtained in Seeking Solutions of Eq. (1) of the Form I ( z ,   μ ) = g ± ( μ ) exp ( kz ) for Four Cases a

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

μ   I ( z ,   μ ) z = - ( μ a + μ s ) I ( z ,   μ ) + μ s 2 - 1 + 1 p ( 0 ) ( μ ,   μ ) I ( z ,   μ ) d μ ,
q z ( z ) = - 2 π μ a + μ s z - 1 + 1 μ 2 I ( z ,   μ ) d μ .
I ( z ,   μ ) = c 0 + g 0 + ( μ ) exp ( - k 0 z ) + c 0 - g 0 - ( μ ) exp ( k 0 z ) ,
q z ( z ) = 2 π μ a + μ s   k 0 [ c 0 + g 0 exp ( - k 0 z ) - c 0 - g 0 exp ( k 0 z ) ] ,
g 0 = - 1 + 1 μ 2 g 0 ± ( μ ) d μ ,
U ( z ) = 2 π c   [ c 0 + exp ( - k 0 z ) + c 0 - exp ( k 0 z ) ] .
q z ( z ) = - g 0 μ a + μ s   v   U ( z ) z .
D = g 0   v μ a + μ s .
D = g 0 μ a + μ s .
q z ( z ) / z = - μ a vU ( z ) .
- 1 μ a + μ s   g 0   2 U ( z ) z 2 = - μ a U ( z ) .
1 v I ( z ,   μ ,   t ) t + μ   I ( z ,   μ ,   t ) z
= - ( μ a + μ s ) I ( z ,   μ ,   t ) + μ s 2 - 1 + 1 p ( 0 ) ( μ ,   μ ) I ( z ,   μ ,   t ) d μ .
T ( t ) = H ( t ) D d exp ( - μ a vt ) m = 1 π m d sin π mL d × exp - π 2 m 2 D t d 2 ,
- 1 + 1 μ 2 I ( z ,   μ ) z d μ
= - ( μ a + μ s ) - 1 + 1 μ I ( z ,   μ ) d μ + μ s 2 - 1 + 1 μ p ( 0 ) ( μ ,   μ ) I ( z ,   μ ) d μ d μ .
z - 1 + 1 μ 2 I ( z ,   μ ) d μ = - ( μ a + μ s ) - 1 + 1 μ I ( z ,   μ ) d μ + μ s g - 1 + 1 μ I ( z ,   μ ) d μ .
q z ( z ) = - 2 π μ a + μ s ( 1 - g ) z - 1 + 1 μ 2 I ( z ,   μ ) d μ ,
p ( u u ) = n = 0 + a n P n ( u u ) ,
P n ( u u ) = P n ( μ ) P n ( μ ) + 2 m = 1 n ( n - m ) ! ( n + m ) !   P n m ( μ ) P n m ( μ ) × cos ( ϕ - ϕ ) ,
p ( 0 ) ( μ ,   μ ) = n = 0 + a n P n ( μ ) P n ( μ ) .
- 1 + 1 μ p ( 0 ) ( μ ,   μ ) d μ = n = 0 + a n P n ( μ ) - 1 + 1 μ P n ( μ ) d μ = n = 0 + a n P n ( μ ) × - 1 + 1 μ P n ( μ ) P 0 ( μ ) d μ ,
- 1 + 1 μ p ( 0 ) ( μ ,   μ ) d μ = 2 3   a 1 P 1 ( μ ) = 2 g μ ,

Metrics