Abstract

We study light propagation in a random medium governed by the radiative transport equation. We present a theory for the transport equation with an inhomogeneous absorption coefficient. We obtain an analytical expression for the specific intensity in a uniform absorbing and scattering medium containing a point absorber. Using that result we derive a self-consistent system of integral equations to study a collection of point absorbers. We show numerical results that demonstrate the use of this theory.

© 2006 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1996).
  2. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  3. K. M. Case, 'On boundary value problems of linear transport theory,' in Proceedings of the Symposium in Applied Mathematics, Vol. 1, R.Bellman, G.Birkhoff, and I.Abu-Shumays, eds. (American Mathematical Society, 1969), pp. 17-36.
  4. L. L. Foldy, 'The multiple scattering of waves,' Phys. Rev. 67, 107-119 (1945).
    [CrossRef]
  5. M. Lax, 'Multiple scattering of waves II. The effective field in dense systems,' Phys. Rev. 85, 261-269 (1952).
    [CrossRef]
  6. L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2001).
  7. L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
    [CrossRef]
  8. A. D. Kim, 'Transport theory for light propagation in biological tissue,' J. Opt. Soc. Am. A 43, 555-563 (2004).
  9. V. A. Markel, 'Modified spherical harmonics method for solving the radiative transport equation,' Waves Random Complex Media 14, L13-L19 (2004).
  10. A. D. Kim, 'A boundary integral method to compute Green's functions for the radiative transport equation,' Waves Random Complex Media 15, 17-42 (2005).
    [CrossRef]
  11. L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, 1985).
    [CrossRef]

2005 (1)

A. D. Kim, 'A boundary integral method to compute Green's functions for the radiative transport equation,' Waves Random Complex Media 15, 17-42 (2005).
[CrossRef]

2004 (2)

A. D. Kim, 'Transport theory for light propagation in biological tissue,' J. Opt. Soc. Am. A 43, 555-563 (2004).

V. A. Markel, 'Modified spherical harmonics method for solving the radiative transport equation,' Waves Random Complex Media 14, L13-L19 (2004).

1952 (1)

M. Lax, 'Multiple scattering of waves II. The effective field in dense systems,' Phys. Rev. 85, 261-269 (1952).
[CrossRef]

1945 (1)

L. L. Foldy, 'The multiple scattering of waves,' Phys. Rev. 67, 107-119 (1945).
[CrossRef]

Case, K. M.

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

K. M. Case, 'On boundary value problems of linear transport theory,' in Proceedings of the Symposium in Applied Mathematics, Vol. 1, R.Bellman, G.Birkhoff, and I.Abu-Shumays, eds. (American Mathematical Society, 1969), pp. 17-36.

Delves, L. M.

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, 1985).
[CrossRef]

Foldy, L. L.

L. L. Foldy, 'The multiple scattering of waves,' Phys. Rev. 67, 107-119 (1945).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1996).

Kim, A. D.

A. D. Kim, 'A boundary integral method to compute Green's functions for the radiative transport equation,' Waves Random Complex Media 15, 17-42 (2005).
[CrossRef]

A. D. Kim, 'Transport theory for light propagation in biological tissue,' J. Opt. Soc. Am. A 43, 555-563 (2004).

Kong, J. A.

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2001).

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Lax, M.

M. Lax, 'Multiple scattering of waves II. The effective field in dense systems,' Phys. Rev. 85, 261-269 (1952).
[CrossRef]

Markel, V. A.

V. A. Markel, 'Modified spherical harmonics method for solving the radiative transport equation,' Waves Random Complex Media 14, L13-L19 (2004).

Mohamed, J. L.

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, 1985).
[CrossRef]

Tsang, L.

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2001).

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Zweifel, P. F.

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

J. Opt. Soc. Am. A (1)

A. D. Kim, 'Transport theory for light propagation in biological tissue,' J. Opt. Soc. Am. A 43, 555-563 (2004).

Phys. Rev. (2)

L. L. Foldy, 'The multiple scattering of waves,' Phys. Rev. 67, 107-119 (1945).
[CrossRef]

M. Lax, 'Multiple scattering of waves II. The effective field in dense systems,' Phys. Rev. 85, 261-269 (1952).
[CrossRef]

Waves Random Complex Media (2)

V. A. Markel, 'Modified spherical harmonics method for solving the radiative transport equation,' Waves Random Complex Media 14, L13-L19 (2004).

A. D. Kim, 'A boundary integral method to compute Green's functions for the radiative transport equation,' Waves Random Complex Media 15, 17-42 (2005).
[CrossRef]

Other (6)

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, 1985).
[CrossRef]

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2001).

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1996).

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

K. M. Case, 'On boundary value problems of linear transport theory,' in Proceedings of the Symposium in Applied Mathematics, Vol. 1, R.Bellman, G.Birkhoff, and I.Abu-Shumays, eds. (American Mathematical Society, 1969), pp. 17-36.

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Figures (3)

Fig. 1
Fig. 1

Sketch of the problem used for the numerical example. Two point absorbers are located between the planar source at z = 0 and the detector plane at z = z d .

Fig. 2
Fig. 2

(Color online) Average scattered intensity due to the two point absorbers. The optical properties of the medium are μ a 0 = 0.034 mm 1 , μ s 0 = 6.11 mm 1 and g = 0.70 . The positions of the two point absorbers are r 1 = ( 1.0 , 1.0 , 4.0 ) mm and r 2 = ( 2.0 , 2.0 , 4.1 ) mm . Both point absorbers have absorption cross section σ a = 0.25 mm 2 . The detector planes are located at (a) z d = 5 mm , (b) 6 mm , (c) 7 mm , (d) 8 mm , (e) 9 mm , (f) 10 mm .

Fig. 3
Fig. 3

(Color online) Same as Fig. 2, but with r 2 = ( 1.0 , 1.0 , 4.1 ) mm . The detector planes are located at (a) z d = 5 mm , (b) 6 mm , (c) 7 mm .

Equations (54)

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ω Ψ + μ a Ψ + μ s L Ψ = S ,
L Ψ = Ψ ( r , ω ) S 2 f ( ω ω ) Ψ ( r , ω ) d 2 ω .
Ψ ( r , ω ) = 0 , r D , ω n ̂ ( r ) > 0 ,
Ψ ( r , ω ) = D S 2 G ( r , ω ; r , ω ) S ( r , ω ) d 2 ω d 3 r .
Ψ ̂ ( q , z , ω ) = R 2 Ψ ( ρ , z , ω ) exp ( i q ρ ) d 2 ρ ,
ϑ z Ψ ̂ + i q Ω Ψ ̂ + μ a Ψ ̂ + μ s L Ψ ̂ = 0 ,
λ ϑ V + i q Ω V + μ a V + μ s L V = 0 .
[ λ j ( q ) λ k ( q ) ] S 2 V j ( q , ω ) V k ( q , ω ) ϑ d 2 ω = 0 , for j k .
S 2 V j 2 ( q , ω ) ϑ d 2 ω = { 1 j > 0 , + 1 j < 0 } .
G ( r , ω ; r , ω ) = 1 ( 2 π ) 2 R 2 G ̂ ( z , ω , z , ω ; q ) exp [ i q ( ρ ρ ) ] d 2 ρ .
ϑ z G ̂ + i q Ω G ̂ + μ a G ̂ + μ s L G ̂ = δ ( z z ) δ ( ω ω ) .
ϑ G ̂ ( z + 0 , ω , z , ω ; q ) ϑ G ̂ ( z 0 , ω , z , ω ; q ) = δ ( ω ω ) .
G ̂ ( z , ω , z , ω ; q ) = j g j ( z , ω , z ; q ) V j ( q , ω ) .
j [ z g j ( z , ω , z ; q ) λ j ( q ) g j ( z , ω , z ; q ) ] ϑ V j ( q , ω ) = δ ( z z ) δ ( ω ω ) .
z g j ( z , ω , z ; q ) λ j ( q ) g j ( z , ω , z ; q ) = sgn ( j ) V j ( q , ω ) δ ( z z ) .
z C j ( z , z ; q ) λ j ( q ) C j ( z , z ; q ) = δ ( z z ) .
C j ( z + 0 , z ; q ) C j ( z 0 , z ; q ) = 1 .
C j ( z , z ; q ) = { exp [ λ j ( q ) ( z z ) ] , z < z , j > 0 , 0 , z < z , j < 0 , 0 , z > z , j > 0 , + exp [ λ j ( q ) ( z z ) ] , z > z , j < 0 } .
G ̂ ( z , ω , z , ω ; q ) = { j > 0 exp [ λ j ( q ) ( z z ) ] V j ( q , ω ) V j ( q , ω ) , z < z , j < 0 exp [ λ j ( q ) ( z z ) ] V j ( q , ω ) V j ( q , ω ) , z > z } .
μ a ( r ) = μ a 0 + δ μ a ( r ) .
ω Ψ + μ a 0 Ψ + μ s L Ψ = S δ μ a Ψ .
Ψ ( r , ω ) = Ψ i ( r , ω ) D S 2 G 0 ( r , ω ; r , ω ) δ μ a ( r ) Ψ ( r , ω ) d 2 ω d 3 r .
Ψ i ( ω , r ) = D S 2 G 0 ( r , ω ; r , ω ) S ( r , ω ) d 2 ω d 3 r .
Ψ s ( r , ω ) = D S 2 G 0 ( r , ω ; r , ω ) δ μ a ( r ) Ψ ( r , ω ) d 2 ω d 3 r ,
G ( r , ω ; r , ω ) = G 0 ( r , ω ; r , ω ) D S 2 G 0 ( r , ω ; r , ω ) × δ μ a ( r ) G ( r , ω ; r , ω ) d 2 ω d 3 r .
G = G 0 G 0 δ μ a G .
G = G 0 G 0 δ μ a G 0 + G 0 δ μ a G 0 δ μ a G 0 G 0 δ μ a G 0 δ μ a G 0 δ μ a G 0 + .
G G 0 G 0 δ μ a G 0 .
G = G 0 + G 0 ( δ μ a + δ μ a G 0 δ μ a + ) G 0 = G 0 + G 0 T G 0 .
Ψ s ( r , ω ) = D S 2 D S 2 G 0 ( r , ω ; r , ω ) T ( r , ω ; r , ω ) × Ψ i ( r , ω ) d 2 ω d 3 r d 2 ω d 3 r .
Ψ ( r , ω ) = S 0 S 2 G 0 ( r , ω ; r 0 , ω ) d 2 ω .
Ψ s ( r , ω ) = σ a S 2 G 0 ( r , ω ; r 0 , ω ) Ψ i ( r 0 , ω ) d 2 ω .
T ( r , ω ; r , ω ) = σ a δ ( r r 0 ) δ ( ω ω ) δ ( r r 0 ) .
Ψ s ( r , ω ) = σ n j S 2 G 0 ( r , ω ; r j , ω ) Ψ i ( r j , ω ) d 2 ω ,
T j Ψ i ( r , ω ) ,
Ψ = Ψ i + j = 1 N T j Ψ E .
Ψ E ( r j , ω ) = Ψ i ( r j , ω ) + k j T k Ψ i ( r j , ω ) + .
Ψ E ( r j , ω ) = Ψ i ( r j , ω ) + k j T k Ψ i ( r j , ω ) + k j l k T k T l Ψ i ( r j , ω ) + k j l k m l T k T l T m Ψ i ( r j , ω ) + .
Ψ E ( r j , ω ) = Ψ i ( r j , ω ) + k j T k ( Ψ i + l k T l Ψ i + l k m l T l T m Ψ i + ) ( r j , ω ) .
Ψ E ( r j , ω ) = Ψ i ( r j , ω ) + k j T k Ψ E ( r j , ω ) .
Ψ E ( r 1 , ω ) σ a 2 S 2 G 0 ( r 1 , ω ; r 2 , ω ) Ψ E ( r 2 , ω ) d 2 ω = Ψ i ( r 1 , ω ) ,
Ψ E ( r 2 , ω ) σ a 1 S 2 G 0 ( r 2 , ω ; r 1 , ω ) Ψ E ( r 1 , ω ) d 2 ω = Ψ i ( r 2 , ω ) .
Ψ s ( r , ω ) = σ a 1 S 2 G 0 ( r , ω ; r 1 , ω ) Ψ E ( r 1 , ω ) d 2 ω σ a 2 S 2 G 0 ( r , ω ; r 2 , ω ) Ψ E ( r 2 , ω ) d 2 ω .
f ( ω ω ) = 1 4 π 1 g 2 ( 1 + g 2 2 g ω ω ) 3 2 ,
U ( x , y ) = S 2 Ψ ( ω , x , y , z d ) d 2 ω
S ( r , ω ) = F δ ( z ) δ ( ω z ̂ ) .
Ψ i ( r , ω ) = F R 2 G 0 ( r , ω ; ρ , 0 , z ̂ ) d ρ .
ω = [ ( 1 ϑ 2 ) 1 2 cos φ , ( 1 ϑ 2 ) 1 2 sin φ , ϑ ] .
L M V ( ϑ m , φ m ; q ) = V ( ϑ m , φ n ; q ) π M n = 1 2 M m = 1 M p ( ϑ m , ϑ m , φ n φ n ) V ( ϑ m , φ n ; q ) w m .
p ( ϑ , ϑ , φ φ ) = f { ϑ ϑ + [ ( 1 ϑ 2 ) ( 1 ϑ 2 ) ] 1 2 cos ( φ φ ) } .
λ ϑ m V m n ( q ) + i ( 1 ϑ m 2 ) 1 2 ( q x cos φ n + q y sin φ n ) V m n ( q ) + μ a V m n ( q ) + μ s L M V m n ( q ) = 0 ,
m = 1 , , M , n = 1 , , 2 M .
γ j ( q ) = π M n = 1 2 M m = 1 M V j , m n 2 ( q ) ϑ m w m .
Re [ λ M 2 ( q ) ] < < Re [ λ 1 ( q ) ] < Re [ λ + 1 ( q ) ] < < Re [ λ + M 2 ( q ) ] .

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