Abstract

Diffusion theory is widely used to describe photon migration in turbid media because it is simple and, in certain cases, can be accurate to within a few percent. However, it neglects the ballistic nature of photon propagation between successive scattering events and hence entirely breaks down for short times and distances, as well as for strong absorption. Here we generalize on the exact two-stream theory of one-dimensional photon migration to obtain a telegrapher equation that accounts for both diffusive and ballistic aspects of propagation in three-dimensional media. At long times and distances the standard diffusion theory is recovered, whereas at short times and distances we find improved predictions for such phenomena as pulse spreading, diffuse photon-density wave dispersion, transmission through a slab, and pulse reflection from a semi-infinite medium. Our theory should be useful for accurately characterizing turbid media, such as biological tissues, and may also aid in improving the spatial resolution of images made with diffuse light.

© 1997 Optical Society of America

Full Article  |  PDF Article

Corrections

D. J. Durian and J. Rudnick, "Photon migration at short times and distances and in cases of strong absorption: erratum," J. Opt. Soc. Am. A 14, 940-940 (1997)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-14-4-940

References

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    [CrossRef]
  9. B. J. Tromberg, L. O. Svaasand, T.-T. Tsay, and R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32, 607–616 (1993).
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  10. C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
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  11. W. E. Meador and W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and new improvement,” J. Atm. Sci. 37, 630–643 (1980).
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  12. C. Acquista, F. House, and J. Jafolla, “N-Stream approximations to radiative transfer,” J. Atm. Sci. 38, 1446–1451 (1981).
  13. M. Kac, “A stochastic model related to the telegrapher's equation,” Rocky Mount. J. Math. 4, 479–509 (1974).
    [CrossRef]
  14. D. D. Joseph and L. Preziosi, “Heat waves,” Rev. Mod. Phys. 61, 41–73 (1989).
    [CrossRef]
  15. J. M. Porra, G. H. Weiss, and J. Masoliver, “A diffusion model incorporating anisotropic properties,” Physica A 218, 229–236 (1995).
    [CrossRef]
  16. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
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  17. B. J. Berne and R. Pecora, Dynamic Light Scattering, with Applications to Chemistry, Biology, and Physics (Wiley, New York, 1976), pp. 83–88. This is not surprising, since in both cases the velocity autocorrelation function, from which the mean-squared displacement may be calculated, is an exponential. It is unlikely that the detailed shape of a spreading pulse is the same at very short times, however, since the Langevin case assumes a Maxwell distribution of particle speeds.
  18. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
    [CrossRef] [PubMed]
  19. J. M. Schmitt, A. Knuttel, and R. R. Knutson, “Interference of diffusive light waves,” J. Opt. Soc. Am. A 9, 1832–1843 (1992).
    [CrossRef] [PubMed]
  20. J. B. Fishkin and E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
    [CrossRef] [PubMed]
  21. J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiment,” Phys. Rev. E 53, 2307–2319 (1996).
    [CrossRef]
  22. J. Masoliver, J. M. Porra, and G. H. Weiss, “Solution to the telegrapher's equation in the presence of reflecting and partly reflecting boundaries,” Phys. Rev. E 48, 939–944 (1993).
    [CrossRef]
  23. 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]
  24. R. Aronson, “ Extrapolation distance for diffusion of light,”in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano and B. Chance, eds., (1993).Proc. SPIE 1888, 297–305(1993).
    [CrossRef]
  25. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  26. M. U. Vera and D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
    [CrossRef]
  27. D. J. Durian, “Penetration depth for diffusing-wave spectroscopy,” Appl. Opt. 34, 7100–7105 (1995).
    [CrossRef] [PubMed]
  28. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, 1959).
  29. G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
    [CrossRef]
  30. D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
    [CrossRef] [PubMed]
  31. M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
    [CrossRef]
  32. K. M. Yoo, F. Liu, and 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]
  33. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  34. I. J. D. Craig and A. M. Thompson, “Why Laplace transforms are difficult to invert numerically,” Comput. Phys. 8, 648–654 (1994).

1996 (2)

J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiment,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

M. U. Vera and D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

1995 (3)

D. J. Durian, “Penetration depth for diffusing-wave spectroscopy,” Appl. Opt. 34, 7100–7105 (1995).
[CrossRef] [PubMed]

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

J. M. Porra, G. H. Weiss, and J. Masoliver, “A diffusion model incorporating anisotropic properties,” Physica A 218, 229–236 (1995).
[CrossRef]

1994 (3)

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, and U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transf. 52, 713–727 (1994).
[CrossRef]

R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[CrossRef]

I. J. D. Craig and A. M. Thompson, “Why Laplace transforms are difficult to invert numerically,” Comput. Phys. 8, 648–654 (1994).

1993 (4)

R. Aronson, “ Extrapolation distance for diffusion of light,”in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano and B. Chance, eds., (1993).Proc. SPIE 1888, 297–305(1993).
[CrossRef]

J. Masoliver, J. M. Porra, and G. H. Weiss, “Solution to the telegrapher's equation in the presence of reflecting and partly reflecting boundaries,” Phys. Rev. E 48, 939–944 (1993).
[CrossRef]

J. B. Fishkin and E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[CrossRef] [PubMed]

B. J. Tromberg, L. O. Svaasand, T.-T. Tsay, and R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32, 607–616 (1993).
[CrossRef] [PubMed]

1992 (2)

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knuttel, and R. R. Knutson, “Interference of diffusive light waves,” J. Opt. Soc. Am. A 9, 1832–1843 (1992).
[CrossRef] [PubMed]

1991 (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]

1990 (1)

K. M. Yoo, F. Liu, and 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]

1989 (3)

1988 (2)

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

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

1987 (2)

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

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

1974 (1)

M. Kac, “A stochastic model related to the telegrapher's equation,” Rocky Mount. J. Math. 4, 479–509 (1974).
[CrossRef]

1941 (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Ackerson, B. J.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, and U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transf. 52, 713–727 (1994).
[CrossRef]

Alfano, R. R.

K. M. Yoo, F. Liu, and 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.

R. Aronson, “ Extrapolation distance for diffusion of light,”in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano and B. Chance, eds., (1993).Proc. SPIE 1888, 297–305(1993).
[CrossRef]

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

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]

Chance, B.

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

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

Craig, I. J. D.

I. J. D. Craig and A. M. Thompson, “Why Laplace transforms are difficult to invert numerically,” Comput. Phys. 8, 648–654 (1994).

Dorr-Nowkoorani, F.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, and U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transf. 52, 713–727 (1994).
[CrossRef]

Dougherty, R. L.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, and U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transf. 52, 713–727 (1994).
[CrossRef]

Durian, D. J.

M. U. Vera and D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

D. J. Durian, “Penetration depth for diffusing-wave spectroscopy,” Appl. Opt. 34, 7100–7105 (1995).
[CrossRef] [PubMed]

Fantini, S.

J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiment,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

Feng, T.-C.

Fishkin, J. B.

J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiment,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

J. B. Fishkin and E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[CrossRef] [PubMed]

Gratton, E.

J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiment,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

J. B. Fishkin and E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[CrossRef] [PubMed]

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Haskell, R. C.

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[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]

Ishimaru, I.

Joseph, D. D.

D. D. Joseph and L. Preziosi, “Heat waves,” Rev. Mod. Phys. 61, 41–73 (1989).
[CrossRef]

Kac, M.

M. Kac, “A stochastic model related to the telegrapher's equation,” Rocky Mount. J. Math. 4, 479–509 (1974).
[CrossRef]

Knutson, R. R.

Knuttel, A.

Liu, F.

K. M. Yoo, F. Liu, and 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]

Maret, G.

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

Masoliver, J.

J. M. Porra, G. H. Weiss, and J. Masoliver, “A diffusion model incorporating anisotropic properties,” Physica A 218, 229–236 (1995).
[CrossRef]

J. Masoliver, J. M. Porra, and G. H. Weiss, “Solution to the telegrapher's equation in the presence of reflecting and partly reflecting boundaries,” Phys. Rev. E 48, 939–944 (1993).
[CrossRef]

McAdams, M. S.

Nobbmann, U.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, and U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transf. 52, 713–727 (1994).
[CrossRef]

O’Leary, M. A.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Patterson, M. S.

Pine, D. J.

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]

Porra, J. M.

J. M. Porra, G. H. Weiss, and J. Masoliver, “A diffusion model incorporating anisotropic properties,” Physica A 218, 229–236 (1995).
[CrossRef]

J. Masoliver, J. M. Porra, and G. H. Weiss, “Solution to the telegrapher's equation in the presence of reflecting and partly reflecting boundaries,” Phys. Rev. E 48, 939–944 (1993).
[CrossRef]

Preziosi, L.

D. D. Joseph and L. Preziosi, “Heat waves,” Rev. Mod. Phys. 61, 41–73 (1989).
[CrossRef]

Reguigui, N. M.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, and U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transf. 52, 713–727 (1994).
[CrossRef]

Schmitt, J. M.

Stephen, M. J.

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

Svaasand, L. O.

Thompson, A. M.

I. J. D. Craig and A. M. Thompson, “Why Laplace transforms are difficult to invert numerically,” Comput. Phys. 8, 648–654 (1994).

Tromberg, B. J.

Tsay, T.-T.

van de Ven, M. J.

J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiment,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

Vera, M. U.

M. U. Vera and D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

Weiss, G. H.

J. M. Porra, G. H. Weiss, and J. Masoliver, “A diffusion model incorporating anisotropic properties,” Physica A 218, 229–236 (1995).
[CrossRef]

J. Masoliver, J. M. Porra, and G. H. Weiss, “Solution to the telegrapher's equation in the presence of reflecting and partly reflecting boundaries,” Phys. Rev. E 48, 939–944 (1993).
[CrossRef]

Weitz, D. A.

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]

Wilson, B. C.

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 65, 409–413 (1987).
[CrossRef]

Yodh, A. G.

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

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Yoo, K. M.

K. M. Yoo, F. Liu, and 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]

Zhu, J. X.

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]

Am. J. Phys. (1)

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

Appl. Opt. (4)

Astrophys. J. (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Comput. Phys. (1)

I. J. D. Craig and A. M. Thompson, “Why Laplace transforms are difficult to invert numerically,” Comput. Phys. 8, 648–654 (1994).

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

J. Quant. Spectrosc. Radiat. Transf. (1)

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, and U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transf. 52, 713–727 (1994).
[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 (1)

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

Phys. Rev. E (3)

M. U. Vera and D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiment,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

J. Masoliver, J. M. Porra, and G. H. Weiss, “Solution to the telegrapher's equation in the presence of reflecting and partly reflecting boundaries,” Phys. Rev. E 48, 939–944 (1993).
[CrossRef]

Phys. Rev. Lett. (3)

K. M. Yoo, F. Liu, and 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]

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

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Phys. Today (1)

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

Physica A (1)

J. M. Porra, G. H. Weiss, and J. Masoliver, “A diffusion model incorporating anisotropic properties,” Physica A 218, 229–236 (1995).
[CrossRef]

Proc. SPIE (1)

R. Aronson, “ Extrapolation distance for diffusion of light,”in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano and B. Chance, eds., (1993).Proc. SPIE 1888, 297–305(1993).
[CrossRef]

Rev. Mod. Phys. (1)

D. D. Joseph and L. Preziosi, “Heat waves,” Rev. Mod. Phys. 61, 41–73 (1989).
[CrossRef]

Rocky Mount. J. Math. (1)

M. Kac, “A stochastic model related to the telegrapher's equation,” Rocky Mount. J. Math. 4, 479–509 (1974).
[CrossRef]

Z. Phys. B (1)

G. Maret and 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 (9)

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, 1959).

W. E. Meador and W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and new improvement,” J. Atm. Sci. 37, 630–643 (1980).
[CrossRef]

C. Acquista, F. House, and J. Jafolla, “N-Stream approximations to radiative transfer,” J. Atm. Sci. 38, 1446–1451 (1981).

B. J. Berne and R. Pecora, Dynamic Light Scattering, with Applications to Chemistry, Biology, and Physics (Wiley, New York, 1976), pp. 83–88. This is not surprising, since in both cases the velocity autocorrelation function, from which the mean-squared displacement may be calculated, is an exponential. It is unlikely that the detailed shape of a spreading pulse is the same at very short times, however, since the Langevin case assumes a Maxwell distribution of particle speeds.

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

D. A. Weitz and D. J. Pine, in Dynamic Light Scattering: The Method and Some Applications, W. Brown, ed. (Clarendon, Oxford, 1993), pp. 652–720.

B. Chance and R. R. Alfano, eds., Optical Tomography, Photon Migration and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE 2389 (1995).

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

B. J. Ackerson, R. L. Dougherty, N. M. Reguigui, and U. Nobbmann, “Correlation transfer: application of radiative transfer solution methods to photon correlation problems,” J. Thermophys. Heat Transfer 6, 577–588 (1992).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Probability density versus radial distance for a pulse of photons released from r=0 in an infinite turbid medium with no absorption, for times as labeled. The simulation results in the upper plot agree very well with the telegrapher's equation predictions [Eq. (4.1)] in the middle plot. By contrast, the diffusion equation predictions [Eq. (2.2)] in the bottom plot violate causality and are an acceptable approximation only at asymptotically late times. The delta functions at r=t in the simulation and the telegrapher's results are not shown.

Fig. 2
Fig. 2

Probability density versus radial distance for a pulse of photons released from r = 0 in an infinite turbid medium with no absorption, for times as labeled. The heavy solid curves represent the telegrapher's-equation predictions [Eq. (4.1)]. The solid, long-dashed, short-dashed, and dotted curves represent random-walk simulation data with anisotropic scattering such that the average cosine of the scattering angles is g = 0, 0.5, 0.9, 0.99, respectively. The delta functions at r = t are not shown.

Fig. 3
Fig. 3

Probability density versus radial distance for the spreading of photons released from r = 0 in an infinite turbid medium with strong absorption, μa=1, for times as labeled. The simulation results in the upper plot agree very well with the telegrapher's-equation predictions [Eq. (4.1)] in the middle plot. The photons are almost entirely absorbed before the diffusion equation predictions [Eq. (2.2)], in the bottom plot, become acceptable at late times. The delta functions at r=t in the simulation and the telegrapher's results are not shown.

Fig. 4
Fig. 4

Decay constant for the amplitude of diffuse photon-density waves versus frequency, with absorption as labeled. The telegrapher's-equation prediction is given by Eq. (4.9), whereas the diffusion theory prediction is given by Eq. (4.8).

Fig. 5
Fig. 5

Phase velocity of diffuse photon-density waves versus frequency, with absorption as labeled. The telegrapher's-equation prediction is given by Eq. (4.9), with Vp=ω/kimag, whereas the diffusion theory prediction is given by Eq. (4.8). Note that the latter violates causality, rising above the speed of light, c=1, for high frequency and strong absorption.

Fig. 6
Fig. 6

Transmission probability versus slab thickness for photons migrating from zp=1, with wall reflectivity and absorption as labeled. The open circles denote simulation results with isotropic scattering, the solid curves denote the telegrapher's predictions [Eq. (5.7)], and the dashed curves denote the diffusion theory predictions, from the ω0 limit of Eq. (5.12).

Fig. 7
Fig. 7

Time-resolved reflectance for a short pulse backscattered from semi-infinite media with wall reflectivity and absorption as labeled; the first scattering event occurs at penetration depth zp=1. The solid curves denote simulation results, the dashed curves are from numerical inversion of the telegrapher's prediction [Eq. (5.15)], and the dotted curves denote the diffusion theory prediction [Eq. (5.14)]. The delta-function contribution to the telegrapher's prediction at t=zp is not shown.

Equations (32)

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2φ=(μa+1)D0φt+μa(μa+1)D0φ,
φ(r, t)=φ01+μa4πD0t3/2×exp(-μat)exp-r24D0t(1+μa),
It=Iz-μaI-μspbI+μspbI,
It=-Iz-μaI-μspbI+μspbI,
2(I+I)z2=2(I+I)t2+[2µa+(1-g)μs]×(I+I)t+μa[μa+(1-g)μs]×(I+I).
2φ=2φt2+2µa+1D0 φt+μaμa+1D0φ.
φ(r, t)=-φ0 exp[-(μa+γ)t]4πr(μa+γ)+t×r[I0(γt2-r2)Θ(t-r)],
φ(r, t)=φ0 γ exp[-(μa+γ)t]4πt2-r2(μa+γ)I1(γt2-r2)+γtt2-r2I2(γt2-r2).
φ(r, t)φ0 1+D0μa(4πD0t)3/2exp(-μat)exp-r24D0t.
φ(r, t)φ0(μa+γ)γ2Θ(t-r)8π+(μa+γ)δ(t-r)+δ(t-r)4πrexp(-μat).
r2(t)=6D0[t-D0+D0 exp(-t/D0)].
k2=μa(μa+1)D0+iωμa+1D0.
k2=μaμa+1D0-ω2+iω2µa+1D0.
kreal, imag=12α02+(β0ω)2±α02,
kreal, imag=12(α0-ω2)2+(β0ω)2±(α0-ω2)21/2,
0=-1+Rw1-Rw z+[(1-g)μs+μa]+t(I+I)z=0,
J(t)=1-Rw1+Rw(I+I)z=0.
ze=231+R21-R1,Rn=01(n+1)μnRw(μ)dμ,
0=zeD0nˆ·+1D0+μa+tφ(r, t)boundary,
J(t)=D0zeφ(r, t)boundary.
φ(z)=φ0D0α0[(1+D0μa)sinh(zα0)+zeα0cosh(zα0)]Bzpz<zp{(1+D0μa)sinh[(L-z)α0]+zeα0 cosh[(L-z)α0]}Tzpz>zp.
Tzp=(1+D0μa)sinh(zpα0)+zeα0 cosh(zpα0)[1+(ze2+D02)μa/D0]sinh(Lα0)+2zeα0 cosh(Lα0),
Bzp=(1+D0μa)sinh[(L-zp)α0]+zeα0 cosh[(L-zp)α0][1+(ze2+D02)μa/D0]sinh(Lα0)+2zeα0 cosh(Lα0).
φ¯(z, ω)=φ0α2(ω+μa)exp(-α|z-zp|)+A1 exp(αz)+A2 exp(-αz),
D0zeφ¯(L, ω)=φ0 [1+D0(ω+μa)]sinh(zpα)+zeα cosh(zpα)[1+(ze2+D02)(ω+μa)/D0]sinh(Lα)+2zeα cosh(Lα),
D0zeφ¯(0, ω)=φ0 [1+D0(ω+μa)]sinh[(L-zp)α]+zeα cosh[(L-zp)α][1+(ze2+D02)(ω+μa)/D0]sinh(Lα)+2zeα cosh(Lα).
D0zeφ¯(L, ω)
=φ0 sinh(zpα)+zeα cosh(zpα)(1+ze2α)sinh(Lα)+2zeα cosh(Lα),
D0zeφ¯(0, ω)
=φ0 sinh[(L-zp)α]+zeα cosh[(L-zp)α](1+ze2α)sinh(Lα)+2zeα cosh(Lα),
R(t)=zp1+μa4πD0t31/2 exp(-μat)exp-zp24D0t(1+μa).
R¯(ω)=[1+D0(ω+μa)]+zeα[1+(ze2+D02)(ω+μa)/D0]+2zeα×exp(-zpα),

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