Abstract

We study the propagation of light through a medium containing isotropic scattering and absorption centers. With a Monte Carlo simulation serving as the benchmark solution to the radiative transfer problem of light propagating through a turbid slab, we compare the transmission and reflection density computed from the telegrapher's equation, the diffusion equation, and multiple-flux theories such as the Kubelka-Munk and four-flux theories. Results are presented for both normally incident light and diffusely incident light. We find that we can always obtain very good results from the telegrapher's equation provided that two parameters that appear in the solution are set appropriately. We also find an interesting connection between certain solutions of the telegrapher's equation and solutions of the Kubelka-Munk and four-flux theories with a small modification to how the phenomenological parameters in those theories are traditionally related to the optical scattering and absorption coefficients of the slab. Finally, we briefly explore how well the theories can be extended to the case of anisotropic scattering by multiplying the scattering coefficient by a simple correction factor.

© 2007 Optical Society of America

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  3. S. H. Kong and J. D. Shore, "Modeling the impact of silver particle size and morphology on the covering power of photothermographic media," in Proceedings of the 30th International Congress of Imaging Science (The Society of Imaging Science and Technology, 2006) pp. 205-207.
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    [CrossRef]
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    [CrossRef]
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  19. W. Cai, M. Xu, M. Lax, and R. R. Alfano, "Diffusion coefficient depends on time, not on absorption," Opt. Lett. 27, 731-733 (2002).
  20. K. R. Naqvi, "On the diffusion coefficient of a photon migrating through a turbid medium: Fresh look from a broader perspective," arXiv.org e-print archive, cond-mat/0504429, June 9, 2005, http://arxiv.org/abs/cond-mat/0504429.
  21. R. C. Haskell, L. O. Svaasand, T. Tsay, T. Feng, M. S. MacAdams, and B. J. Tromberg, "Boundary conditions for the diffusion equation in radiative transfer," J. Opt. Soc. Am. A 11, 2727-2741 (1994).
  22. In the earliest two papers of Yang et al. [Refs. ], the authors make it clear that μ should be greater than or equal to 1, as we assume here, since μ physically corresponds to the factor of increase in the path length due to the scattering. In their later two papers [Refs. ], however, they are unclear on this point and, in fact, seem to implicitly assume that limits in which μ<1 are sensible to discuss. If we do not assume μ must be greater or equal to 1 but instead use the expression shown for s2>/=a2+as over the entire range of ad, it does significantly affect the quantitative results that we obtain for the transmission density for the RKM theory in Section . However, it does not change our basic conclusions concerning the quality of the agreement between the RKM theory and the Monte Carlo results.
  23. To verify the correctness of the Monte Carlo results, we have made comparisons between two different codes written independently by the two authors in different programming languages using different random number generators. Furthermore, the Monte Carlo results for no scattering (i.e., the rightmost data point in each figure) are found to be in excellent agreement with the exact results that can readily be calculated for that simpler case.
  24. L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
    [CrossRef]

2006 (1)

2005 (2)

2004 (2)

2002 (1)

2001 (1)

V. Gopal, S. A. Ramakrishna, A. K. Sood, and N. Kumar, "Photon transport in thin disordered slabs," Pramana, J. Phys. 56, 767-778 (2001).

1999 (1)

1998 (1)

1997 (1)

1994 (1)

1989 (1)

1981 (1)

1972 (1)

J. J. DePalma and J. Gasper, "Determining the optical properties of photographic emulsions by the Monte Carlo method," Photograph. Sci. Eng. 16, 181-191 (1972).

1941 (1)

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

1931 (1)

P. Kubelka and F. Munk, "Ein beitrag zur optik der farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Alfano, R. R.

Aronson, R.

Cai, W.

Carminati, R.

Chandrasekhar, S.

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

Corngold, N.

DePalma, J. J.

J. J. DePalma and J. Gasper, "Determining the optical properties of photographic emulsions by the Monte Carlo method," Photograph. Sci. Eng. 16, 181-191 (1972).

Durian, D. J.

Feng, T.

Gasper, J.

J. J. DePalma and J. Gasper, "Determining the optical properties of photographic emulsions by the Monte Carlo method," Photograph. Sci. Eng. 16, 181-191 (1972).

Gopal, V.

V. Gopal, S. A. Ramakrishna, A. K. Sood, and N. Kumar, "Photon transport in thin disordered slabs," Pramana, J. Phys. 56, 767-778 (2001).

Greenstein, J. L.

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

Greffet, J.-J.

Hanrahan, P.

H. W. Jensen, S. R. Marschner, M. Levoy, and P. Hanrahan, "A practical model for subsurface light transport," in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ACM, 2001), pp. 511-518.

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]

Ishimaru, A.

A. Ishimaru, "Diffusion of light in turbid material," Appl. Opt. 28, 2210-2215 (1989).

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

Jensen, H. W.

H. W. Jensen, S. R. Marschner, M. Levoy, and P. Hanrahan, "A practical model for subsurface light transport," in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ACM, 2001), pp. 511-518.

Kong, S. H.

S. H. Kong and J. D. Shore, "Modeling the impact of silver particle size and morphology on the covering power of photothermographic media," in Proceedings of the 30th International Congress of Imaging Science (The Society of Imaging Science and Technology, 2006) pp. 205-207.

Kruse, B.

Kubelka, P.

P. Kubelka and F. Munk, "Ein beitrag zur optik der farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Kumar, N.

V. Gopal, S. A. Ramakrishna, A. K. Sood, and N. Kumar, "Photon transport in thin disordered slabs," Pramana, J. Phys. 56, 767-778 (2001).

Lax, M.

Levoy, M.

H. W. Jensen, S. R. Marschner, M. Levoy, and P. Hanrahan, "A practical model for subsurface light transport," in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ACM, 2001), pp. 511-518.

MacAdams, M. S.

Marschner, S. R.

H. W. Jensen, S. R. Marschner, M. Levoy, and P. Hanrahan, "A practical model for subsurface light transport," in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ACM, 2001), pp. 511-518.

Miklavcic, S. J.

Munk, F.

P. Kubelka and F. Munk, "Ein beitrag zur optik der farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Naqvi, K. R.

K. R. Naqvi, "On the diffusion coefficient of a photon migrating through a turbid medium: Fresh look from a broader perspective," arXiv.org e-print archive, cond-mat/0504429, June 9, 2005, http://arxiv.org/abs/cond-mat/0504429.

Pierrat, R.

Ramakrishna, S. A.

V. Gopal, S. A. Ramakrishna, A. K. Sood, and N. Kumar, "Photon transport in thin disordered slabs," Pramana, J. Phys. 56, 767-778 (2001).

Rudnick, J.

Shore, J. D.

S. H. Kong and J. D. Shore, "Modeling the impact of silver particle size and morphology on the covering power of photothermographic media," in Proceedings of the 30th International Congress of Imaging Science (The Society of Imaging Science and Technology, 2006) pp. 205-207.

Sood, A. K.

V. Gopal, S. A. Ramakrishna, A. K. Sood, and N. Kumar, "Photon transport in thin disordered slabs," Pramana, J. Phys. 56, 767-778 (2001).

Svaasand, L. O.

Tromberg, B. J.

Tsay, T.

Völz, H. G.

H. G. Völz, Industrial Color Testing (VCH, 1995).

Waterman, P. C.

Xu, M.

Yang, L.

Appl. Opt. (1)

Astrophys. J. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Lett. (3)

Photograph. Sci. Eng. (1)

J. J. DePalma and J. Gasper, "Determining the optical properties of photographic emulsions by the Monte Carlo method," Photograph. Sci. Eng. 16, 181-191 (1972).

Pramana, J. Phys. (1)

V. Gopal, S. A. Ramakrishna, A. K. Sood, and N. Kumar, "Photon transport in thin disordered slabs," Pramana, J. Phys. 56, 767-778 (2001).

Z. Tech. Phys. (Leipzig) (1)

P. Kubelka and F. Munk, "Ein beitrag zur optik der farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Other (8)

H. G. Völz, Industrial Color Testing (VCH, 1995).

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

H. W. Jensen, S. R. Marschner, M. Levoy, and P. Hanrahan, "A practical model for subsurface light transport," in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ACM, 2001), pp. 511-518.

S. H. Kong and J. D. Shore, "Modeling the impact of silver particle size and morphology on the covering power of photothermographic media," in Proceedings of the 30th International Congress of Imaging Science (The Society of Imaging Science and Technology, 2006) pp. 205-207.

K. R. Naqvi, "On the diffusion coefficient of a photon migrating through a turbid medium: Fresh look from a broader perspective," arXiv.org e-print archive, cond-mat/0504429, June 9, 2005, http://arxiv.org/abs/cond-mat/0504429.

In the earliest two papers of Yang et al. [Refs. ], the authors make it clear that μ should be greater than or equal to 1, as we assume here, since μ physically corresponds to the factor of increase in the path length due to the scattering. In their later two papers [Refs. ], however, they are unclear on this point and, in fact, seem to implicitly assume that limits in which μ<1 are sensible to discuss. If we do not assume μ must be greater or equal to 1 but instead use the expression shown for s2>/=a2+as over the entire range of ad, it does significantly affect the quantitative results that we obtain for the transmission density for the RKM theory in Section . However, it does not change our basic conclusions concerning the quality of the agreement between the RKM theory and the Monte Carlo results.

To verify the correctness of the Monte Carlo results, we have made comparisons between two different codes written independently by the two authors in different programming languages using different random number generators. Furthermore, the Monte Carlo results for no scattering (i.e., the rightmost data point in each figure) are found to be in excellent agreement with the exact results that can readily be calculated for that simpler case.

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

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