Abstract

This paper discusses some of the present knowledge of the mathematical techniques used to describe light diffusion in turbid material such as tissues. Attention will be paid to the usefulness and limitations of various techniques. First, we review the transport theory, radiance, radiant energy fluence rate, phase functions, boundary conditions, and measurement techniques. We then discuss the first-order solution, multiple scattering, diffusion approximation, and their limitations. The plane wave, spherical wave, beam wave, and pulse wave excitations are discussed followed by a brief review of the surface scattering effects due to rough interfaces.

© 1989 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  2. J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986).
  3. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).
  4. V. V. Varadan, V. K. Varadan, Eds., Multiple Scattering of Waves in Random Rough Surfaces (Pennsylvania State U., University Park, 1987).
  5. W. M. Star, J. P. A. Marijnissen, “New Trends in Photobiology (Invited Review) Light Dosimetry: Status and Prospects,” J. Photochem. Photobiol. B Biol. 1, 149 (1987).
    [CrossRef]
  6. A. Ishimaru, C. Yeh, “Matrix Representations of the Vector Radiative-Transfer Theory for Randomly Distributed Nonspherical Particles,” J. Opt. Soc. Am. A 1, 359–364 (1984).
    [CrossRef]
  7. A. Ishimaru, Y. Kuga, R. L.-T. Cheung, K. Shimizu, “Scattering and Diffusion of a Beam Wave in Randomly Distributed Scatterers,” J. Opt. Soc. Am. 73, 131–136 (1983).
    [CrossRef]
  8. A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. 68, 1045–1050 (1978).
    [CrossRef]
  9. K. Furutsu, “Diffusion Equation Derived from a Space-Time Transport Equation,” J. Opt. Soc. Am. 70, 360–366 (1980).
    [CrossRef]
  10. K. Shimizu, A. Ishimaru, “Experimental Test of the Reduced Velocity of Light in a Diffuse Medium,” Opt. Lett 5, 205–207 (1980).
    [CrossRef] [PubMed]
  11. Y. Kuga, A. Ishimaru, A. P. Bruckner, “Experiments on Picosecond Pulse Propagation in a Diffuse Medium,” J. Opt. Soc. Am. 73, 1812–1815 (1983).
    [CrossRef]
  12. A. Ishimaru, L. Tsang, “Backscattering Enhancement of Random Discrete Scatterers of Moderate Sizes,” J. Opt. Soc. Am. A 5, 228 (1988).
    [CrossRef]
  13. P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical Coherent Backscattering by Random Media,” J. Phys. France 49, 63 (1988).
    [CrossRef]
  14. J. A. DeSanto, G. S. Brown, “Analytical Techniques for Multiple Scattering from Rough Surfaces,” Prog. Opt. 23, 3–62 (1986).

1988

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical Coherent Backscattering by Random Media,” J. Phys. France 49, 63 (1988).
[CrossRef]

A. Ishimaru, L. Tsang, “Backscattering Enhancement of Random Discrete Scatterers of Moderate Sizes,” J. Opt. Soc. Am. A 5, 228 (1988).
[CrossRef]

1987

W. M. Star, J. P. A. Marijnissen, “New Trends in Photobiology (Invited Review) Light Dosimetry: Status and Prospects,” J. Photochem. Photobiol. B Biol. 1, 149 (1987).
[CrossRef]

1986

J. A. DeSanto, G. S. Brown, “Analytical Techniques for Multiple Scattering from Rough Surfaces,” Prog. Opt. 23, 3–62 (1986).

1984

1983

1980

K. Shimizu, A. Ishimaru, “Experimental Test of the Reduced Velocity of Light in a Diffuse Medium,” Opt. Lett 5, 205–207 (1980).
[CrossRef] [PubMed]

K. Furutsu, “Diffusion Equation Derived from a Space-Time Transport Equation,” J. Opt. Soc. Am. 70, 360–366 (1980).
[CrossRef]

1978

Akkermans, E.

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical Coherent Backscattering by Random Media,” J. Phys. France 49, 63 (1988).
[CrossRef]

Brown, G. S.

J. A. DeSanto, G. S. Brown, “Analytical Techniques for Multiple Scattering from Rough Surfaces,” Prog. Opt. 23, 3–62 (1986).

Bruckner, A. P.

Cheung, R. L.-T.

DeSanto, J. A.

J. A. DeSanto, G. S. Brown, “Analytical Techniques for Multiple Scattering from Rough Surfaces,” Prog. Opt. 23, 3–62 (1986).

Furutsu, K.

Ishimaru, A.

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986).

Kuga, Y.

Maret, G.

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical Coherent Backscattering by Random Media,” J. Phys. France 49, 63 (1988).
[CrossRef]

Marijnissen, J. P. A.

W. M. Star, J. P. A. Marijnissen, “New Trends in Photobiology (Invited Review) Light Dosimetry: Status and Prospects,” J. Photochem. Photobiol. B Biol. 1, 149 (1987).
[CrossRef]

Maynard, R.

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical Coherent Backscattering by Random Media,” J. Phys. France 49, 63 (1988).
[CrossRef]

Shimizu, K.

A. Ishimaru, Y. Kuga, R. L.-T. Cheung, K. Shimizu, “Scattering and Diffusion of a Beam Wave in Randomly Distributed Scatterers,” J. Opt. Soc. Am. 73, 131–136 (1983).
[CrossRef]

K. Shimizu, A. Ishimaru, “Experimental Test of the Reduced Velocity of Light in a Diffuse Medium,” Opt. Lett 5, 205–207 (1980).
[CrossRef] [PubMed]

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Star, W. M.

W. M. Star, J. P. A. Marijnissen, “New Trends in Photobiology (Invited Review) Light Dosimetry: Status and Prospects,” J. Photochem. Photobiol. B Biol. 1, 149 (1987).
[CrossRef]

Tsang, L.

Wolf, P. E.

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical Coherent Backscattering by Random Media,” J. Phys. France 49, 63 (1988).
[CrossRef]

Yeh, C.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Photochem. Photobiol. B Biol.

W. M. Star, J. P. A. Marijnissen, “New Trends in Photobiology (Invited Review) Light Dosimetry: Status and Prospects,” J. Photochem. Photobiol. B Biol. 1, 149 (1987).
[CrossRef]

J. Phys. France

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical Coherent Backscattering by Random Media,” J. Phys. France 49, 63 (1988).
[CrossRef]

Opt. Lett

K. Shimizu, A. Ishimaru, “Experimental Test of the Reduced Velocity of Light in a Diffuse Medium,” Opt. Lett 5, 205–207 (1980).
[CrossRef] [PubMed]

Prog. Opt.

J. A. DeSanto, G. S. Brown, “Analytical Techniques for Multiple Scattering from Rough Surfaces,” Prog. Opt. 23, 3–62 (1986).

Other

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

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

V. V. Varadan, V. K. Varadan, Eds., Multiple Scattering of Waves in Random Rough Surfaces (Pennsylvania State U., University Park, 1987).

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Figures (6)

Fig. 1
Fig. 1

Boundary conditions for the radiance at the interface between two homogeneous mediums.

Fig. 2
Fig. 2

(a) Isotropic detector; (b) detector with FOV.

Fig. 3
Fig. 3

Discrete ordinates method.

Fig. 4
Fig. 4

Boundary conditions for diffuse intensity.

Fig. 5
Fig. 5

Pulse propagation in a turbid medium.

Fig. 6
Fig. 6

Rough surface scattering.

Equations (28)

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d d s I ( r , s ˆ ) = γ t I ( r , s ˆ ) + γ t 4 π p ( s ˆ , s ˆ ) I ( r ¯ , s ˆ ) d ω ,
ψ ( r ) = 4 π I ( r , s ˆ ) d ω .
F ( r ) = 4 π I ( r , s ˆ ) s ˆ d ω ,
F + γ a ψ = 0 ,
p ( θ ) = W 0 [ 1 g 2 ) ( 1 + g 2 2 g cos θ ) 3 / 2 ,
W 0 = 1 4 π 4 π p ( θ ) d ω , g = 4 π p ( θ ) cos θ d ω 4 π p ( θ ) d ω .
I = I c + I d .
I c = F 0 δ ( ω ˆ ω ˆ 0 ) exp ( τ ) ,
τ = 0 s γ t d s .
n 1 sin θ 1 = n 2 sin θ 2 .
I r = R p I i ,
R p = | R | 2 .
I t = n 2 3 cos θ 2 n 1 3 cos θ 1 | T | 2 I i ,
I t ( s ) = I t ( o ) exp ( τ ) .
I d ( s ˆ ) = 0
P r = 4 π A ( s ˆ ) I ( s ˆ ) d ω .
P r = A 0 4 π I ( s ˆ ) d ω = A 0 ψ .
P r = A 0 2 π I ( s ˆ ) cos θ d ω .
P r = A 0 Δ ω I ( s ˆ ) .
P r = A 0 F 0 exp ( τ ) + A 0 Δ ω I d .
d d τ [ I 1 I 2 ] + [ S 11 S 12 S 21 S 22 ] [ I 1 I 2 ] = [ B 1 B 2 ] exp ( t )
I d = n = 0 I n = 1 4 π ( ψ d + 3 F d · s ˆ ) + .
( 2 κ 2 ) ψ d = Q ,
ψ t = ψ c + ψ d , ψ c = F 0 exp ( τ ) .
ψ d + h n ψ d + 2 ( n ˆ · s ˆ 0 ) Q 1 = 0 , h = 2 3 γ tr ,
( 2 3 υ 2 2 t 2 1 D t + 3 γ a γ tr ) ψ d ( t ) = 0 ,
D = υ 3 ( γ a + γ tr ) .
σ > λ / ( 8 cos θ i ) ,

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