Abstract

We present a unified derivation of the photon diffusion coefficient for both steady-state and time-dependent transport in disordered absorbing media. The derivation is based on a modal analysis of the time-dependent radiative transfer equation. This approach confirms that the dynamic diffusion coefficient is given by the random-walk result D=cl*3, where l* is the transport mean free path and c is the energy velocity, independent of the level of absorption. It also shows that the diffusion coefficient for steady-state transport, often used in biomedical optics, depends on absorption, in agreement with recent theoretical and experimental works. These two results resolve a recurrent controversy in light propagation and imaging in scattering media.

© 2006 Optical Society of America

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).
  2. P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic, 1995).
  3. P.Sebbah, ed., Waves and Imaging through Complex Media (Kluwer Academic, 2001).
  4. A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48(3), 34-40 (1995).
    [CrossRef]
  5. S. K. Gayen and R. R. Alfano, "Biomedical imaging techniques," Opt. Photon. News, July 1996, pp. 17-22.
  6. A. Mandelis, "Diffusion waves and their uses," Phys. Today 53, 29-34 (2000).
    [CrossRef]
  7. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  8. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  9. K. Furutsu and Y. Yamada, "Diffusion approximation for a dissipative random medium and the applications," Phys. Rev. E 50, 3634-3640 (1994).
    [CrossRef]
  10. T. Durduran, A. G. Yodh, B. Chance, and D. A. Boas, "Does the photon-diffusion coefficient depend on absorption?" J. Opt. Soc. Am. A 14, 3358-3365 (1997).
    [CrossRef]
  11. M. Bassani, F. Martelli, G. Zaccanti, and D. Contini, "Independence of the diffusion coefficient from absorption: experimental and numerical evidence," Opt. Lett. 22, 853-855 (1997).
    [CrossRef] [PubMed]
  12. W. Cai, M. Xu, M. Lax, and R. R. Alfano, "Diffusion coefficient depends on time, not on absorption," Opt. Lett. 27, 731-733 (2002).
    [CrossRef]
  13. D. J. Durian, "The diffusion coefficient depends on absorption," Opt. Lett. 23, 1502-1504 (1998).
    [CrossRef]
  14. R. Aronson and N. Corngold, "Photon diffusion coefficient in an absorbing medium," J. Opt. Soc. Am. A 16, 1066-1071 (1999).
    [CrossRef]
  15. R. Graaff and J. J. Ten Bosch, "Diffusion coefficient in photon diffusion theory," Opt. Lett. 25, 43-45 (2000).
    [CrossRef]
  16. R. Graaff and K. Rinzema, "Practical improvements on photon diffusion theory: application to isotropic scattering," Phys. Med. Biol. 46, 3043-3050 (2001).
    [CrossRef] [PubMed]
  17. R. Elaloufi, R. Carminati, and J.-J. Greffet, "Definition of the diffusion coefficient in scattering and absorbing media," J. Opt. Soc. Am. A 20, 678-685 (2003). In this reference, the analytical expression of the steady-state diffusion is established. Numerical simulations are also used to show that the long-time behavior of pulse transmission through absorbing slabs is better described by using this coefficient rather than the P1 expression. The good agreement with the RTE result that is observed is due to the fact that the long-time decay in these calculations is dominated by the term exp(−μact). It is clear from the present study that the dynamic diffusion coefficient, independent of absorption, should be used in order to obtain a perfect agreement.
    [CrossRef]
  18. I. Kuscer and N. J. McCormick, "Some analytical results for radiative transfer in thick atmospheres," Transp. Theory Stat. Phys. 20, 351-381 (1991).
    [CrossRef]
  19. J. Ripoll, D. Yessayan, G. Zacharakis, and V. Ntziachristos, "Experimental determination of photon propagation in highly scattering and absorbing media," J. Opt. Soc. Am. A 22, 546-551 (2005).
    [CrossRef]
  20. A. Lagendijk and B. A. van Tiggelen, "Resonant multiple scattering of light," Phys. Rep. 270, 143-216 (1996).
    [CrossRef]
  21. L. A. Apresyan and Yu. A. Kravtsov, Radiative Transfer: Statistical and Wave Aspects (Gordon & Breach,1996).
  22. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.
  23. G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, 1999), Chap. 8.
    [CrossRef]
  24. R. Elaloufi, R. Carminati, and 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]
  25. R. Elaloufi, R. Carminati, and J.-J. Greffet, "Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach," J. Opt. Soc. Am. A 21, 1430-1437 (2004).
    [CrossRef]
  26. A. M. Sedletskii, Fourier Transforms and Approximations (Gordon & Breach, 2000).
  27. L. Sirovich, Techniques of Asymptotic Analysis (Springer-Verlag, 1971).
  28. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Pt. 2, Sect. 8.4.
  29. Equation is a homogeneous Fredholm integral equation of the second kind, whose kernel (the phase function) is real and nonsingular. The procedure described in Ref. shows that, under these two conditions on the kernel only, α(μ) is a constant function. This result is valid for any type of phase function.

2005 (1)

2004 (1)

2003 (1)

2002 (2)

R. Elaloufi, R. Carminati, and 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]

W. Cai, M. Xu, M. Lax, and R. R. Alfano, "Diffusion coefficient depends on time, not on absorption," Opt. Lett. 27, 731-733 (2002).
[CrossRef]

2001 (2)

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

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

2000 (3)

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

R. Graaff and J. J. Ten Bosch, "Diffusion coefficient in photon diffusion theory," Opt. Lett. 25, 43-45 (2000).
[CrossRef]

A. M. Sedletskii, Fourier Transforms and Approximations (Gordon & Breach, 2000).

1999 (2)

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

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

1998 (1)

1997 (3)

1996 (3)

S. K. Gayen and R. R. Alfano, "Biomedical imaging techniques," Opt. Photon. News, July 1996, pp. 17-22.

A. Lagendijk and B. A. van Tiggelen, "Resonant multiple scattering of light," Phys. Rep. 270, 143-216 (1996).
[CrossRef]

L. A. Apresyan and Yu. A. Kravtsov, Radiative Transfer: Statistical and Wave Aspects (Gordon & Breach,1996).

1995 (2)

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

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48(3), 34-40 (1995).
[CrossRef]

1994 (1)

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

1991 (1)

I. Kuscer and N. J. McCormick, "Some analytical results for radiative transfer in thick atmospheres," Transp. Theory Stat. Phys. 20, 351-381 (1991).
[CrossRef]

1989 (1)

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

1971 (1)

L. Sirovich, Techniques of Asymptotic Analysis (Springer-Verlag, 1971).

1967 (1)

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

1960 (1)

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

1953 (1)

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Pt. 2, Sect. 8.4.

Alfano, R. R.

W. Cai, M. Xu, M. Lax, and R. R. Alfano, "Diffusion coefficient depends on time, not on absorption," Opt. Lett. 27, 731-733 (2002).
[CrossRef]

S. K. Gayen and R. R. Alfano, "Biomedical imaging techniques," Opt. Photon. News, July 1996, pp. 17-22.

Apresyan, L. A.

L. A. Apresyan and Yu. A. Kravtsov, Radiative Transfer: Statistical and Wave Aspects (Gordon & Breach,1996).

Aronson, R.

Bassani, M.

Boas, D. A.

Bosch, J. J.

Cai, W.

Carminati, R.

Case, K. M.

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

Chance, B.

Chandrasekhar, S.

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

Contini, D.

Corngold, N.

Durduran, T.

Durian, D. J.

Elaloufi, R.

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Pt. 2, Sect. 8.4.

Furutsu, K.

K. Furutsu and 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 and R. R. Alfano, "Biomedical imaging techniques," Opt. Photon. News, July 1996, pp. 17-22.

Graaff, R.

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

R. Graaff and J. J. Ten Bosch, "Diffusion coefficient in photon diffusion theory," Opt. Lett. 25, 43-45 (2000).
[CrossRef]

Greffet, J.-J.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

Kravtsov, Yu. A.

L. A. Apresyan and Yu. A. Kravtsov, Radiative Transfer: Statistical and Wave Aspects (Gordon & Breach,1996).

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

Kuscer, I.

I. Kuscer and N. J. McCormick, "Some analytical results for radiative transfer in thick atmospheres," Transp. Theory Stat. Phys. 20, 351-381 (1991).
[CrossRef]

Lagendijk, A.

A. Lagendijk and B. A. van Tiggelen, "Resonant multiple scattering of light," Phys. Rep. 270, 143-216 (1996).
[CrossRef]

Lax, M.

Mandelis, A.

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

Martelli, F.

McCormick, N. J.

I. Kuscer and N. J. McCormick, "Some analytical results for radiative transfer in thick atmospheres," Transp. Theory Stat. Phys. 20, 351-381 (1991).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Pt. 2, Sect. 8.4.

Ntziachristos, V.

Rinzema, K.

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

Ripoll, J.

Rytov, S. M.

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

Sedletskii, A. M.

A. M. Sedletskii, Fourier Transforms and Approximations (Gordon & Breach, 2000).

Sheng, P.

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

Sirovich, L.

L. Sirovich, Techniques of Asymptotic Analysis (Springer-Verlag, 1971).

Stamnes, K.

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

Tatarskii, V. I.

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

Thomas, G. E.

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

van Tiggelen, B. A.

A. Lagendijk and B. A. van Tiggelen, "Resonant multiple scattering of light," Phys. Rep. 270, 143-216 (1996).
[CrossRef]

Xu, M.

Yamada, Y.

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

Yessayan, D.

Yodh, A.

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48(3), 34-40 (1995).
[CrossRef]

Yodh, A. G.

Zaccanti, G.

Zacharakis, G.

Zweifel, P. F.

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

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

R. Elaloufi, R. Carminati, and 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 (5)

R. Elaloufi, R. Carminati, and J.-J. Greffet, "Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach," J. Opt. Soc. Am. A 21, 1430-1437 (2004).
[CrossRef]

R. Elaloufi, R. Carminati, and J.-J. Greffet, "Definition of the diffusion coefficient in scattering and absorbing media," J. Opt. Soc. Am. A 20, 678-685 (2003). In this reference, the analytical expression of the steady-state diffusion is established. Numerical simulations are also used to show that the long-time behavior of pulse transmission through absorbing slabs is better described by using this coefficient rather than the P1 expression. The good agreement with the RTE result that is observed is due to the fact that the long-time decay in these calculations is dominated by the term exp(−μact). It is clear from the present study that the dynamic diffusion coefficient, independent of absorption, should be used in order to obtain a perfect agreement.
[CrossRef]

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

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

J. Ripoll, D. Yessayan, G. Zacharakis, and V. Ntziachristos, "Experimental determination of photon propagation in highly scattering and absorbing media," J. Opt. Soc. Am. A 22, 546-551 (2005).
[CrossRef]

Opt. Lett. (4)

Phys. Med. Biol. (1)

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

Phys. Rep. (1)

A. Lagendijk and B. A. van Tiggelen, "Resonant multiple scattering of light," Phys. Rep. 270, 143-216 (1996).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Today (2)

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

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48(3), 34-40 (1995).
[CrossRef]

Transp. Theory Stat. Phys. (1)

I. Kuscer and N. J. McCormick, "Some analytical results for radiative transfer in thick atmospheres," Transp. Theory Stat. Phys. 20, 351-381 (1991).
[CrossRef]

Other (13)

A. M. Sedletskii, Fourier Transforms and Approximations (Gordon & Breach, 2000).

L. Sirovich, Techniques of Asymptotic Analysis (Springer-Verlag, 1971).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Pt. 2, Sect. 8.4.

Equation is a homogeneous Fredholm integral equation of the second kind, whose kernel (the phase function) is real and nonsingular. The procedure described in Ref. shows that, under these two conditions on the kernel only, α(μ) is a constant function. This result is valid for any type of phase function.

S. K. Gayen and R. R. Alfano, "Biomedical imaging techniques," Opt. Photon. News, July 1996, pp. 17-22.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

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

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

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

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

L. A. Apresyan and Yu. A. Kravtsov, Radiative Transfer: Statistical and Wave Aspects (Gordon & Breach,1996).

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

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

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Equations (15)

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

1 c 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 ,
I ( z , μ , t ) = + I ̃ ( k , μ , t ) exp ( i k z ) d k .
I ̃ ( k , μ , t ) = g k s ( μ ) exp ( s t ) .
s ( k ) c g k s ( μ ) + i k μ g k s ( μ ) = ( μ a + μ s ) g k s ( μ ) + μ s 2 1 + 1 p ( 0 ) ( μ , μ ) g k s ( μ ) d μ .
s ( k ) c G k s 0 + i k G k s 1 = μ a G k s 0 ,
ϕ k s = μ a c + s ( k ) k 2 ( i k U k s ) .
D k s = μ a c s ( k ) k 2 .
D s t = μ a k 0 2 c .
s 0 ( k ) = ( μ a + μ s ) c + k c tan ( k μ s ) for k < π μ s 2 .
s ( k ) c G k s 1 + i k G k s 2 = [ μ a + μ s ( 1 g ) ] G k s 1 .
s 0 ( k ) μ a c D d y n k 2 when k 0 .
g k s ( μ ) α ( μ ) + β ( μ ) k + γ ( μ ) k 2 when k 0 ,
D d y n = c μ s ( 1 g ) 1 1 μ 2 α ( μ ) d μ 1 1 α ( μ ) d μ .
α ( μ ) = 1 2 1 1 p ( 0 ) ( μ , μ ) α ( μ ) d μ .
D d y n = c 3 μ s ( 1 g ) = 1 3 c l * .

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