Abstract

The concept of photon diffusion on the velocity sphere and the non-Euclidean diffusion equation (NED) are introduced to describe photon migration in highly forward-scattering random media. The NED covers the ballistic, transient, and developed diffusion modes of photon migration in random media. An approximate analytic solution to the NED is presented.

© 1996 Optical Society of America

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References

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  1. B. Chance, R. R. Alfano, eds., Biomedical Optics Conference, Proc. SPIE2387 (1995).
  2. A. Ya. Polishchuk, S. Gutman, R. R. Alfano, presented at the Optical Society of America Annual Meeting, Portland, Oregon, Sept. 10 (1995); “Photon density modes beyond the diffusion approximation: scalar wave-diffusion equation,” submitted to J. Opt. Soc. Am. A.
  3. R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
  4. Sh. Feng, F. Zeng, B. Chance, Proc. SPIE 1888, 78 (1993).
    [CrossRef]
  5. J. Tassendorf, Phys. Rev. A. 35, 872 (1987).
    [CrossRef]
  6. L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
    [CrossRef] [PubMed]
  7. J. A. Moon, J. Reintjes, Opt. Lett. 19, 521 (1994).
    [CrossRef] [PubMed]
  8. J. S. Shotland, J. C. Haselgrove, J. S. Leigh, Appl. Opt. 32, 448 (1993).
    [CrossRef]
  9. M. S. Patterson, S. Andersson-Engeis, B. S. Wilson, E. K. Osei, Appl. Opt. 34, 22 (1995).
    [CrossRef] [PubMed]
  10. A. Ya. Polishchuk, R. R. Alfano, Opt. Lett. 20, 1937 (1995).
    [CrossRef] [PubMed]
  11. H. Risken, The Fokker-Planck Equation (Springer-Verlag, Berlin, 1984).
    [CrossRef]
  12. A. Ya. Polishchuk, M. Zevalos, F. Liu, R. R. Alfano, “A generalization of Fermat principle for photons in random media: mean-square curvature of the most favorable paths and photon diffusion on the velocity sphere,”Phys. Rev. E. (to be published).
  13. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

1995 (2)

1994 (2)

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
[CrossRef] [PubMed]

J. A. Moon, J. Reintjes, Opt. Lett. 19, 521 (1994).
[CrossRef] [PubMed]

1993 (2)

1987 (1)

J. Tassendorf, Phys. Rev. A. 35, 872 (1987).
[CrossRef]

Alfano, R. R.

A. Ya. Polishchuk, R. R. Alfano, Opt. Lett. 20, 1937 (1995).
[CrossRef] [PubMed]

A. Ya. Polishchuk, M. Zevalos, F. Liu, R. R. Alfano, “A generalization of Fermat principle for photons in random media: mean-square curvature of the most favorable paths and photon diffusion on the velocity sphere,”Phys. Rev. E. (to be published).

A. Ya. Polishchuk, S. Gutman, R. R. Alfano, presented at the Optical Society of America Annual Meeting, Portland, Oregon, Sept. 10 (1995); “Photon density modes beyond the diffusion approximation: scalar wave-diffusion equation,” submitted to J. Opt. Soc. Am. A.

Andersson-Engeis, S.

Chance, B.

Sh. Feng, F. Zeng, B. Chance, Proc. SPIE 1888, 78 (1993).
[CrossRef]

Feld, M. S.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
[CrossRef] [PubMed]

Feng, Sh.

Sh. Feng, F. Zeng, B. Chance, Proc. SPIE 1888, 78 (1993).
[CrossRef]

Feynman, R. P.

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

Gutman, S.

A. Ya. Polishchuk, S. Gutman, R. R. Alfano, presented at the Optical Society of America Annual Meeting, Portland, Oregon, Sept. 10 (1995); “Photon density modes beyond the diffusion approximation: scalar wave-diffusion equation,” submitted to J. Opt. Soc. Am. A.

Haselgrove, J. C.

Hibbs, A. R.

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

Ishimaru, A.

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

Itzkan, I.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
[CrossRef] [PubMed]

Leigh, J. S.

Liu, F.

A. Ya. Polishchuk, M. Zevalos, F. Liu, R. R. Alfano, “A generalization of Fermat principle for photons in random media: mean-square curvature of the most favorable paths and photon diffusion on the velocity sphere,”Phys. Rev. E. (to be published).

Moon, J. A.

Osei, E. K.

Patterson, M. S.

Perelman, L. T.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
[CrossRef] [PubMed]

Polishchuk, A. Ya.

A. Ya. Polishchuk, R. R. Alfano, Opt. Lett. 20, 1937 (1995).
[CrossRef] [PubMed]

A. Ya. Polishchuk, M. Zevalos, F. Liu, R. R. Alfano, “A generalization of Fermat principle for photons in random media: mean-square curvature of the most favorable paths and photon diffusion on the velocity sphere,”Phys. Rev. E. (to be published).

A. Ya. Polishchuk, S. Gutman, R. R. Alfano, presented at the Optical Society of America Annual Meeting, Portland, Oregon, Sept. 10 (1995); “Photon density modes beyond the diffusion approximation: scalar wave-diffusion equation,” submitted to J. Opt. Soc. Am. A.

Reintjes, J.

Risken, H.

H. Risken, The Fokker-Planck Equation (Springer-Verlag, Berlin, 1984).
[CrossRef]

Shotland, J. S.

Tassendorf, J.

J. Tassendorf, Phys. Rev. A. 35, 872 (1987).
[CrossRef]

Wilson, B. S.

Wu, J.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
[CrossRef] [PubMed]

Zeng, F.

Sh. Feng, F. Zeng, B. Chance, Proc. SPIE 1888, 78 (1993).
[CrossRef]

Zevalos, M.

A. Ya. Polishchuk, M. Zevalos, F. Liu, R. R. Alfano, “A generalization of Fermat principle for photons in random media: mean-square curvature of the most favorable paths and photon diffusion on the velocity sphere,”Phys. Rev. E. (to be published).

Appl. Opt. (2)

Opt. Lett. (2)

Phys. Rev. A. (1)

J. Tassendorf, Phys. Rev. A. 35, 872 (1987).
[CrossRef]

Phys. Rev. Lett. (1)

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
[CrossRef] [PubMed]

Proc. SPIE (1)

Sh. Feng, F. Zeng, B. Chance, Proc. SPIE 1888, 78 (1993).
[CrossRef]

Other (6)

H. Risken, The Fokker-Planck Equation (Springer-Verlag, Berlin, 1984).
[CrossRef]

A. Ya. Polishchuk, M. Zevalos, F. Liu, R. R. Alfano, “A generalization of Fermat principle for photons in random media: mean-square curvature of the most favorable paths and photon diffusion on the velocity sphere,”Phys. Rev. E. (to be published).

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

B. Chance, R. R. Alfano, eds., Biomedical Optics Conference, Proc. SPIE2387 (1995).

A. Ya. Polishchuk, S. Gutman, R. R. Alfano, presented at the Optical Society of America Annual Meeting, Portland, Oregon, Sept. 10 (1995); “Photon density modes beyond the diffusion approximation: scalar wave-diffusion equation,” submitted to J. Opt. Soc. Am. A.

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

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Figures (1)

Fig. 1
Fig. 1

Temporal evolution of the photon number density N, defined by Eqs. (4), (7), and (9), at distances of (a) 5 lt and (b) 15 lt from the collimated source in the longitudinal (L) and the transverse (T) directions. Diffusion approximation is also shown (D). lt = 1 mm, medium index of refraction 1.33. The incident pulse contains 100 photons. The inset in (a) shows the coordinate system used in the derivation of the collisional integral.

Equations (11)

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n t + s r n = I c = ρ σ 0 c d s f ( s , s ) [ n ( s ) n ( s ) ] ,
I c = ρ σ 0 c d x d y cos [ θ ( x 2 + y 2 ) ] f ( x 2 + y 2 ) × [ n ( 0 , 0 ) n ( x , y ) ] .
I c = ρ c Δ 2 n ( x , y ) | x , y = 0 d x d y 4 cos θ ( x 2 + y 2 ) σ ( x 2 + y 2 ) .
I c = D s Δ s n ,
n t + s r n D s Δ s n = 0 .
N ( t , r ) = 1 ( 4 π ) 3 / 2 1 det Δ i j × exp [ 1 4 Δ i j 1 ( r r c ) i ( r r c ) j ] .
δ F δ t D s Δ s F = δ ( t t ) δ ( s s 0 ) .
F ( t , s | t , s 0 ) = l = 0 2 l + 1 4 π × exp [ D s l ( l + 1 ) ( t t ) c 2 ] P i [ cos ( S S ˆ 0 ) ] ,
r c ( t ) = r ( t ) = 0 t d t d s F ( t , s | 0 , s 0 ) s = s 0 ( l t / c ) × [ 1 exp ( c t / l t ) ] .
s i ( t ) s j ( t ) = d s d s · s i · F ( t , s | t , s ) · s j · F ( t , s | 0 , s 0 ) = exp [ 2 D s c 2 ( t t ) ] × { exp ( 6 D s c 2 t ) ( s 0 ) i ( s 0 ) j + 1 3 [ 1 exp ( 6 D s c 2 t ) ] δ i j } .
Δ i j ( t ) = 1 2 0 t d t 0 t d t [ s i ( t ) s i ( t ) ] × [ s j ( t ) s j ( t ) ] = l t 2 2 δ i j ( 2 3 τ f 1 + 1 9 f 3 ) + l t 2 2 c 2 ( s 0 ) i ( s 0 ) j ( f 1 1 3 f 3 f 1 2 ) .

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