Abstract

In highly scattering media, light energy fluence rate distributions can be described by diffusion theory. Boundary conditions, appropriate to the diffusion approximation, are derived for surfaces where reflection of diffuse light occurs. Both outer surfaces and interfaces separating media with different indices of refraction can be treated. The diffusion equation together with its boundary conditions is solved using the finite element method. This numerical method allows much freedom of geometry.

© 1988 Optical Society of America

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References

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  1. K. M. Case, P. R. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, MA, 1967), p. 9.
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1: Single Scattering and Transport Theory (Academic, New York, 1978), p. 157.
  3. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).
  4. K. Furutsu, “Diffusion Equation Derived from Space-Time Transport Equation,” J. Opt. Soc. Am. 70, 360 (1980).
    [CrossRef]
  5. H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas and Applications (Academic, New York, 1980).
  6. R. A. J. Groenhuis, H. A. Ferwerda, J. J. ten Bosch, “Scattering and Absorption of Turbid Materials Determined from Reflection Measurements. 1: Theory,” Appl. Opt. 22, 2456 (1983).
    [CrossRef] [PubMed]
  7. I. Fried, Numerical Solutions of Differential Equations (Academic, New York, 1979).

1983 (1)

1980 (1)

Case, K. M.

K. M. Case, P. R. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, MA, 1967), p. 9.

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Ferwerda, H. A.

Fried, I.

I. Fried, Numerical Solutions of Differential Equations (Academic, New York, 1979).

Furutsu, K.

Groenhuis, R. A. J.

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1: Single Scattering and Transport Theory (Academic, New York, 1978), p. 157.

ten Bosch, J. J.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas and Applications (Academic, New York, 1980).

Zweifel, P. R.

K. M. Case, P. R. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, MA, 1967), p. 9.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Other (5)

H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas and Applications (Academic, New York, 1980).

K. M. Case, P. R. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, MA, 1967), p. 9.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1: Single Scattering and Transport Theory (Academic, New York, 1978), p. 157.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

I. Fried, Numerical Solutions of Differential Equations (Academic, New York, 1979).

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Figures (4)

Fig. 1
Fig. 1

Diffuse reflection at a medium–vacuum boundary. The symbols are explained in the text.

Fig. 2
Fig. 2

Reflection function for a medium(n = 1.5)–vacuum boundary as a function of the cosine of the angle of incidence.

Fig. 3
Fig. 3

Vector n ˆ 12 normal to the boundary surface S separating medium(1) and medium(2).

Fig. 4
Fig. 4

Reflection function for a medium(1)/medium(2) boundary [n(1) = 1.2, n(2) = 1.6] for light in medium(1).

Equations (32)

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s ˆ · L ( r , s ˆ ) + ρ σ t L ( r , s ˆ ) = ( ρ σ t / 4 π ) ( 4 π ) p ( s ˆ , s ˆ ) L ( r , s ˆ ) d Ω + ε ( r , s ˆ ) ,
L ( r , s ˆ ) L r i ( r , s ˆ ) + L d ( r , s ˆ ) .
s ˆ · L d ( r , s ˆ ) + ρ σ t L d ( r , s ˆ ) = ( ρ σ t / 4 π ) ( 4 π ) p ( s ˆ , s ˆ ) L d ( r , s ˆ ) d Ω + ε r i ( r , s ˆ ) ,
ε r i ( r , s ˆ ) ( ρ σ t / 4 π ) ( 4 π ) p ( s ˆ , s ˆ ) L r i ( r , s ˆ ) d Ω .
L d ( r , s ˆ ) ( 1 / 4 π ) Ψ ( r ) + ( 3 / 4 π ) F ( r ) · s ˆ ,
Ψ ( r ) ( 4 π ) L d ( r , s ˆ ) d Ω is the radiant energy fluence rate and F d ( r ) ( 4 π ) L d ( r , s ˆ ) s ˆ d Ω is the net energy flux vector .
· F ( r ) = ρ σ a Ψ ( r ) + ρ σ s ( 4 π ) L r i ( r , s ˆ ) d Ω .
Ψ ( r ) = ( 3 / 4 π ) ρ σ t r F ( r ) + ( 3 / 4 π ) ( 4 π ) ε r i ( r , s ˆ ) s ˆ d Ω ,
g [ ( 4 π ) p ( s ˆ , s ˆ ) s ˆ · s ˆ d Ω ] / [ ( 4 π ) p ( s ˆ , s ˆ ) d Ω ] .
2 Ψ ( r ) κ d 2 Ψ ( r ) = 3 ρ σ s ρ σ t r ( 4 π ) L r i ( r , s ˆ ) d Ω + ( 3 / 4 π ) · ( 4 π ) ε r i ( r , s ˆ ) s ˆ d Ω Q ( r ) ,
S m n ˆ i . s ˆ L d ( r , s ˆ ) d S S υ n ˆ i · s ˆ L d ( r , s ˆ ) d S = 0 ,
n ˆ i · s ˆ > 0 n ˆ i · s ˆ L d ( r , s ˆ ) d Ω = n ˆ i · s ˆ < 0 R ( s ˆ ) n ˆ i · s ˆ L d ( r , s ˆ ) d Ω [ r o n S υ ] .
R ( s ˆ ) = ( 1 / 2 ) [ n m cos θ υ n υ cos θ n m cos θ υ + n υ cos θ ] 2 + ( 1 / 2 ) [ n m cos θ n υ cos θ υ n m cos θ + n υ cos θ υ ] 2 ,
n ˆ i · s ˆ > 0 n ˆ i · s ˆ L d ( r , s ˆ ) d Ω n ˆ i · s ˆ < 0 R ( s ˆ ) n ˆ i · s ˆ L d ( r , s ˆ ) d Ω = 0 [ r on S ] .
R ( s ˆ ) [ 1 , 0 cos θ cos θ c , R 0 , cos θ c cos θ 1 .
Ψ ( r ) + 2 A F ( r ) · n ˆ i = 0 [ r on S ] ,
Ψ ( r ) A h n ˆ i · Ψ ( r ) + 2 A n ˆ i · Q 1 ( r ) = 0 [ r on S ] ,
Q 1 ( r ) ( 1 / ρ σ t r ( 4 π ) ε r i ( r , s ˆ ) s ˆ d Ω .
n ˆ 12 · s ˆ > 0 n ˆ 12 · s ˆ L d ( 1 ) d Ω = n ˆ 12 · s ˆ < 0 R ( 1 ) n ˆ 12 · s ˆ L d ( 1 ) d Ω + n ˆ 12 · s ˆ > 0 T ( 2 ) n ˆ 12 · s ˆ L d ( 2 ) d Ω ,
n ˆ 12 · s ˆ < 0 n ˆ 12 · s ˆ L d ( 2 ) d Ω = n ˆ 12 · s ˆ > 0 R ( 2 ) n ˆ 12 · s ˆ L d ( 2 ) d Ω + n ˆ 12 · s ˆ < 0 T ( 1 ) n ˆ 12 · s ˆ L d ( 1 ) d Ω ,
T ( 1 ) = 1 R ( 1 ) ,
T ( 2 ) = 1 R ( 2 ) .
R ( 1 ) ( s ˆ ) exp ( b | cos θ | ) ,
b 2 ln { ( 1 / 2 ) [ n ( 1 ) cos θ 2 n ( 2 ) / 2 n ( 1 ) cos θ 2 + n ( 2 ) / 2 ] 2 + ( 1 / 2 ) [ n ( 1 ) / 2 n ( 2 ) cos θ 2 n ( 1 ) / 2 + n ( 2 ) cos θ 2 ] 2 } ,
n ˆ 12 · F ( 1 ) ( r ) = n ˆ 12 · F ( 2 ) ( r ) [ r on S ] .
n ˆ 12 · Ψ ( 1 ) = 1 h ( 1 ) [ B 1 B Ψ ( 1 ) + B 2 B Ψ ( 2 ) + 2 n ˆ 12 · Q 1 ( 1 ) ] ,
n ˆ 12 · Ψ ( 2 ) = 1 h ( 2 ) [ B 1 B Ψ ( 1 ) + B 2 B Ψ ( 2 ) + 2 n ˆ 12 · Q 1 ( 2 ) ] ,
B 1 = b 3 2 b 2 e b + 2 b e b 2 b , B 2 = b 3 ( 1 R 0 ) ( cos 2 θ c 1 ) , B = b 3 ( R 0 cos 3 θ c R 0 cos 3 θ c ) 3 b 2 e b + 6 b e b 6 e b + 6 .
Φ ( r j ) [ 1 for i = j , 0 for i j .
Ψ ( r ) i = 1 N ψ i Φ i ( r ) .
V Φ k i = 1 N ψ i 2 Φ i d V + k d 2 V Φ k i = 1 N ψ 1 Φ i d V = V Φ k Q d V [ k = 1 , 2 n ] .
S i = 1 N ψ i Φ k Φ i · n ˆ i d S + V i = 1 N ψ i ( Φ k Φ i + k d 2 Φ k Φ i ) d V = V Φ k Q d V [ k = 1 , 2 N ] .

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