Abstract

To allow the determination of scattering and absorption parameters of a turbid material from reflection measurements the relation of these parameters to the reflection has been described by two theoretical approaches. One approach is based on the diffusion theory which has been extended to include anisotropic scattering. This results in a reflection formula in which the scattering and absorption are described by one parameter each. As a second more general approach a Monte Carlo model is applied. Comparison of the results indicates the range of values of the scattering and absorption parameters where the computationally fast diffusion approach is applicable.

© 1983 Optical Society of America

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  1. J. Langerholc, Appl. Opt. 21, 1593 (1982).
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  5. S. Chandrasekhar, Radiative Transfer (Oxford U. P., New York, 1960).
  6. V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand-Reinhold, Princeton, N.J., 1963).
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  8. R. L. Fante, J. Opt. Soc. Am. 71, 460 (1981).
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  9. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
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  13. E. D. Cashwell, C. J. Everett, Monte Carlo Method for Random Walk Problems (Pergamon, London, 1959).
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  15. H. C. van de Hulst, Multiple Light Scattering, Vol. 2 (Academic, New York, 1980).
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1983

1982

1981

1980

1979

1978

1977

A. N. Witt, Astrophys. J. Suppl. 35, 1 (1977).
[CrossRef]

A. Ishimaru, Proc. IEEE 65, 1030 (1977).
[CrossRef]

1976

Anderson, D. E.

Carter, L. L.

L. L. Carter, H. G. Horak, M. T. Sandford, J. Comput. Phys. 26, 119 (1978).
[CrossRef]

Case, K. M.

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

Cashwell, E. D.

E. D. Cashwell, C. J. Everett, Monte Carlo Method for Random Walk Problems (Pergamon, London, 1959).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. P., New York, 1960).

Egan, W. G.

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

Everett, C. J.

E. D. Cashwell, C. J. Everett, Monte Carlo Method for Random Walk Problems (Pergamon, London, 1959).

Fante, R. L.

Ferwerda, H. A.

Furutsu, K.

Groenhuis, R. A. J.

R. A. J. Groenhuis, J. J. Ten Bosch, H. A. Ferwerda, Appl. Opt. 22, 2463 (1983).
[CrossRef] [PubMed]

R. A. J. Groenhuis, in Technical Digest, Topical Meeting on Optical Phenomena Peculiar to Matter of Small Dimensions (Optical Society of America, Washington, D.C., 1980), paper WB-6.

Hilgeman, T. W.

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

Horak, H. G.

L. L. Carter, H. G. Horak, M. T. Sandford, J. Comput. Phys. 26, 119 (1978).
[CrossRef]

Ishimaru, A.

A. Ishimaru, J. Opt. Soc. Am. 68, 1045 (1978).
[CrossRef]

A. Ishimaru, Proc. IEEE 65, 1030 (1977).
[CrossRef]

L. Reynolds, C. Johnson, A. Ishimaru, Appl. Opt. 15, 2059 (1976).
[CrossRef] [PubMed]

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

Johnson, C.

Langerholc, J.

Lee, J.-S.

Meador, W. E.

Meier, R. R.

Reynolds, L.

Sandford, M. T.

L. L. Carter, H. G. Horak, M. T. Sandford, J. Comput. Phys. 26, 119 (1978).
[CrossRef]

Sobolev, V. V.

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand-Reinhold, Princeton, N.J., 1963).

Ten Bosch, J. J.

Twersky, V.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering, Vol. 2 (Academic, New York, 1980).

Weaver, W. R.

Witt, A. N.

A. N. Witt, Astrophys. J. Suppl. 35, 1 (1977).
[CrossRef]

Wolf, E.

E. Wolf, Phys. Rev. D 13, 869 (1976).
[CrossRef]

Zweifel, P. F.

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

Appl. Opt.

Astrophys. J. Suppl.

A. N. Witt, Astrophys. J. Suppl. 35, 1 (1977).
[CrossRef]

J. Comput. Phys.

L. L. Carter, H. G. Horak, M. T. Sandford, J. Comput. Phys. 26, 119 (1978).
[CrossRef]

J. Opt. Soc. Am.

Phys. Rev. D

E. Wolf, Phys. Rev. D 13, 869 (1976).
[CrossRef]

Proc. IEEE

A. Ishimaru, Proc. IEEE 65, 1030 (1977).
[CrossRef]

Other

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

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

E. D. Cashwell, C. J. Everett, Monte Carlo Method for Random Walk Problems (Pergamon, London, 1959).

R. A. J. Groenhuis, in Technical Digest, Topical Meeting on Optical Phenomena Peculiar to Matter of Small Dimensions (Optical Society of America, Washington, D.C., 1980), paper WB-6.

H. C. van de Hulst, Multiple Light Scattering, Vol. 2 (Academic, New York, 1980).

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

S. Chandrasekhar, Radiative Transfer (Oxford U. P., New York, 1960).

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand-Reinhold, Princeton, N.J., 1963).

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Figures (5)

Fig. 1
Fig. 1

Geometry to measure the reflected light as a function of the distance to a small incident beam: 3-D representation and cross section through the z axis.

Fig. 2
Fig. 2

Relative radiance R is calculated as a function of the distance r (Fig. 1) between source and detector using the diffusion theory (full lines) and the Monte Carlo method (symbols). Several values of ρσs, the linear scattering coefficient, are used. For clarity, symbols and full lines are used instead of histograms.

Fig. 3
Fig. 3

Relative radiance R for several values of g, the average cosine of the scattering angle (see Fig. 2).

Fig. 4
Fig. 4

Relative radiance R for several values of ρσa, the linear absorption coefficient (see Fig. 2).

Fig. 5
Fig. 5

Average proportional deviation of the relative radiance values calculated with the diffusion theory and those calculated with the Monte Carlo method mapped as a function of ρ σ s = ρ σ s ( 1 g ) and ρ σ a / ρ σ s . Three areas are indicated: deviation < 10%; 10% < deviation < 25%; and deviation > 25%.

Equations (32)

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( s r ) L ( r , s ) = ρ σ t L ( r , s ) + ρ σ t 4 π 4 π p ( s , s ) L ( r , s ) d ω ,
p ( μ ) = ω 0 ( 1 g 2 ) ( 1 + g 2 2 g μ ) 3 / 2 ,
p ( μ ) = ω 0 [ ( 1 g ) + 4 g δ ( μ 1 ) ] .
( s r ) L ( r , s ) = ρ σ t r L ( r , s ) + ρ σ s 4 π 4 π L ( r , s ) d ω ,
σ t r = σ s + σ a
L inc ( r , s ) = E 0 u ( r r f ) δ ( μ 1 ) ,
L ( r , s ) = L r i ( r , z ) + L d ( r , s ) ,
d d z L r i ( r , z ) = ρ σ t r L r i ( r , z ) .
L r i ( r , z ) = E 0 u ( r r f ) δ ( μ 1 ) exp ( ρ σ t r z ) .
( s r ) L d ( r , s ) = ρ σ t r L d ( r , s ) + 1 4 ρ σ s E 0 u ( r r f ) exp ( ρ σ t r z ) + 1 4 π ρ σ s 4 π L d ( r , s ) d ω .
L d ( r , s ) = U d ( r ) + 3 4 π F d ( r ) s ,
s r U d ( r ) + 3 4 π ( s r ) [ s F d ( r ) ] = ρ σ a U d ( r ) 3 4 π ρ σ t r [ F d ( r ) s ] + ¼ ρ σ s E 0 u ( r r f ) exp ( ρ σ t r z ) .
r F d ( r ) = 4 π ρ σ a U d ( r ) + π ρ σ s E 0 u ( r r f ) exp ( ρ σ t r z ) .
r U d ( r ) = 3 4 π ρ σ t r F d ( r ) .
( r 2 ρ σ a D 1 ) U d ( r ) = 1 4 ρ σ s D 1 E 0 u ( r r f ) exp ( ρ σ t r z ) ,
2 π , μ > 0 L d ( r , s ) ( s z ) d ω = 0 at z = 0 .
U d ( r ) h z U d ( r ) = 0 at z = 0 ,
2 π , μ > 0 L d ( r , s ) ( s z ) d ω = r d 2 π , μ < 0 L d ( r , s ) ( s z ) d ω .
r d 1.4399 n 2 + 0.7099 n 1 + 0.6681 + 0.0636 n ,
U d ( r ) 2 D z U d ( r ) = r d [ U d ( r ) + 2 D z U d ( r ) ]
U d ( r ) h z U d ( r ) = 0 at z = 0 .
U d ( r ) + h z U d ( r ) = 0 at z = d ,
0 d exp ( ρ σ t r z ) sin ( k i z + γ i ) d z = z i [ k i 2 + ( ρ σ t r ) 2 ] 1 ,
z i = sin γ i [ ρ σ t r + exp ( ρ σ t r d ) ( k i sin k i d ρ σ t r cos k i d ) ] + cos γ i [ k i exp ( ρ σ t r d ) ( ρ σ t r sin k i d + k i cos k i d ) ] .
R ( r ) = 1 2 π ( 1 r s ) 2 π r d R d ( r ) d r ( m 2 sr 1 ) ,
8 ρ σ s D r f 2 i = 1 Γ i z i R i ( r ) ( k i 2 + ρ 2 σ t r 2 ) λ i 2 ,
r f 2 / 2 r r f I 1 ( λ i r f ) K 1 ( λ i r ) r r f ,
k i [ ( sin γ i ) / 2 + k i D cos γ i ] 2 k i d + sin ( 2 γ i ) sin [ 2 ( k i d + γ i ) ] ,
( k i 2 + ρ σ a D 1 ) 1 / 2 ,
τ = ln R N / ρ σ s ,
ϕ = 2 π R N , μ = [ ( 1 g 2 ) ( 1 g 2 ) 2 ( 1 g + 2 gRN ) 2 ] [ 2 g ] 1 .
P = w p ( μ 0 ) d ω 0 exp [ ( ρ σ s + a ) z 0 ] ,

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