Abstract

We calculate the radiance of a light beam propagating in a uniformly scattering and absorbing slab and determine the point-spread function. We do this by solving numerically the governing radiative transport equation by use of plane-wave mode expansions. When scattering is sharply peaked in the forward direction and it becomes difficult to solve the radiative transport equation, we replace it with either the Fokker–Planck or the Leakeas–Larsen equation. We also solve these equations by using plane-wave mode expansions. Numerical results show that these two equations agree with the radiative transport equation for large anisotropy factors. The agreement improves as the optical thickness increases.

© 2004 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1996).
  3. M. Moscoso, J. B. Keller, G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948–960 (2001).
    [CrossRef]
  4. A. D. Kim, M. Moscoso, “Radiative transfer computations for optical beams,” J. Comput. Phys. 185, 50–60 (2003).
    [CrossRef]
  5. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).
  6. G. C. Pomraning, “The Fokker–Planck operator as an asymptotic limit,” Math. Models Meth. Appl. Sci. 2, 21–36 (1992).
    [CrossRef]
  7. E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413–423 (1999).
    [CrossRef]
  8. C. L. Leakeas, E. W. Larsen, “Generalized Fokker–Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236–250 (2001).
  9. A. D. Kim, J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92–98 (2003).
    [CrossRef]
  10. A. D. Kim, M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Comput. (USA) 23, 2075–2095 (2002).
  11. A. Hamilton, “Uncorrelated modes of the nonlinear power spectrum,” Mon. Not. R. Astron. Soc.312, 257–284 (2000), http://casa.colorado.edu/∼ajsh/FFTLog/ .
    [CrossRef]
  12. A. E. Siegman, “Quasi-fast Hankel transform,” Opt. Lett. 1, 13–15 (1977).
    [CrossRef]
  13. W.-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  14. J. E. Morel, “An improved Fokker–Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131–136 (1985).
  15. S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence on HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–334 (1987).
  16. A. Dunn, R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2, 898–905 (1996).
    [CrossRef]
  17. J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, D. Shen, T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical–tissue diagnostics,” Appl. Opt. 37, 3586–3593 (1998).
    [CrossRef]
  18. M. Hammer, D. Schweitzer, B. Michel, E. Thamm, A. Kolb, “Single scattering by red blood cells,” Appl. Opt. 37, 7410–7418 (1998).
    [CrossRef]

2003 (2)

A. D. Kim, M. Moscoso, “Radiative transfer computations for optical beams,” J. Comput. Phys. 185, 50–60 (2003).
[CrossRef]

A. D. Kim, J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92–98 (2003).
[CrossRef]

2002 (1)

A. D. Kim, M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Comput. (USA) 23, 2075–2095 (2002).

2001 (2)

C. L. Leakeas, E. W. Larsen, “Generalized Fokker–Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236–250 (2001).

M. Moscoso, J. B. Keller, G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948–960 (2001).
[CrossRef]

1999 (1)

E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413–423 (1999).
[CrossRef]

1998 (2)

1996 (1)

A. Dunn, R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2, 898–905 (1996).
[CrossRef]

1992 (1)

G. C. Pomraning, “The Fokker–Planck operator as an asymptotic limit,” Math. Models Meth. Appl. Sci. 2, 21–36 (1992).
[CrossRef]

1990 (1)

W.-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1987 (1)

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence on HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–334 (1987).

1985 (1)

J. E. Morel, “An improved Fokker–Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131–136 (1985).

1977 (1)

Alter, C. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence on HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–334 (1987).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Cheong, W.-F.

W.-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Dunn, A.

A. Dunn, R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2, 898–905 (1996).
[CrossRef]

Eick, A. A.

Freyer, J. P.

Hammer, M.

Hielscher, A. H.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1996).

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).

Jacques, S. L.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence on HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–334 (1987).

Johnson, T. M.

Katsev, I. L.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).

Keller, J. B.

Kim, A. D.

A. D. Kim, J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92–98 (2003).
[CrossRef]

A. D. Kim, M. Moscoso, “Radiative transfer computations for optical beams,” J. Comput. Phys. 185, 50–60 (2003).
[CrossRef]

A. D. Kim, M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Comput. (USA) 23, 2075–2095 (2002).

Kolb, A.

Larsen, E. W.

C. L. Leakeas, E. W. Larsen, “Generalized Fokker–Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236–250 (2001).

E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413–423 (1999).
[CrossRef]

Leakeas, C. L.

C. L. Leakeas, E. W. Larsen, “Generalized Fokker–Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236–250 (2001).

Michel, B.

Morel, J. E.

J. E. Morel, “An improved Fokker–Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131–136 (1985).

Moscoso, M.

A. D. Kim, M. Moscoso, “Radiative transfer computations for optical beams,” J. Comput. Phys. 185, 50–60 (2003).
[CrossRef]

A. D. Kim, M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Comput. (USA) 23, 2075–2095 (2002).

M. Moscoso, J. B. Keller, G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948–960 (2001).
[CrossRef]

Mourant, J. R.

Papanicolaou, G.

Pomraning, G. C.

G. C. Pomraning, “The Fokker–Planck operator as an asymptotic limit,” Math. Models Meth. Appl. Sci. 2, 21–36 (1992).
[CrossRef]

Prahl, S. A.

W.-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence on HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–334 (1987).

Richards-Kortum, R.

A. Dunn, R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2, 898–905 (1996).
[CrossRef]

Schweitzer, D.

Shen, D.

Siegman, A. E.

Thamm, E.

Welch, A. J.

W.-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Zege, E. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

W.-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Dunn, R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2, 898–905 (1996).
[CrossRef]

J. Comput. Phys. (1)

A. D. Kim, M. Moscoso, “Radiative transfer computations for optical beams,” J. Comput. Phys. 185, 50–60 (2003).
[CrossRef]

J. Opt. Soc. Am. A (2)

Lasers Life Sci. (1)

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence on HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–334 (1987).

Math. Models Meth. Appl. Sci. (1)

G. C. Pomraning, “The Fokker–Planck operator as an asymptotic limit,” Math. Models Meth. Appl. Sci. 2, 21–36 (1992).
[CrossRef]

Nucl. Sci. Eng. (2)

C. L. Leakeas, E. W. Larsen, “Generalized Fokker–Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236–250 (2001).

J. E. Morel, “An improved Fokker–Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131–136 (1985).

Opt. Lett. (1)

Prog. Nucl. Energy (1)

E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413–423 (1999).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Sci. Comput. (USA) (1)

A. D. Kim, M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Comput. (USA) 23, 2075–2095 (2002).

Other (4)

A. Hamilton, “Uncorrelated modes of the nonlinear power spectrum,” Mon. Not. R. Astron. Soc.312, 257–284 (2000), http://casa.colorado.edu/∼ajsh/FFTLog/ .
[CrossRef]

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1996).

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).

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Figures (8)

Fig. 1
Fig. 1

Sketch of a pencil beam incident normally on a slab with the coordinates indicated.

Fig. 2
Fig. 2

Point-spread function for the radiative transport equation by use of plane-wave mode expansions (solid curve), Monte Carlo simulations (circles), and the Chebyshev spectral method (dashed curve). Scattering is governed by the Henyey–Greenstein phase function with g = 0.2 . The slab thickness is z 0 = 5 l 0 with Σ a / Σ s = 0.01 . The beam width is W 0 = 0.5 l 0 .

Fig. 3
Fig. 3

Point-spread function for the Fokker–Planck (dotted curves) and Leakeas–Larsen (dashed curves) equations compared with Monte Carlo simulations of the radiative transport equation (solid curves). Scattering is governed by the exponential phase function with = 0.1 so g = 0.9 . The slab thickness is z 0 = 1 l 0 (top), z 0 = 5 l 0 (middle) and z 0 = 10 l 0 (bottom) with Σ a / Σ s = 0.01 . The beam width is W 0 = 0.5 l 0 .

Fig. 4
Fig. 4

Error of the Fokker–Planck E FP (circles) and Leakeas–Larsen E LL (triangles) equations for ρ = 0.2 l 0 and z 0 = 1 l 0 with Σ a / Σ s = 0.01 . The beam width is W 0 = 0.5 l 0 .

Fig. 5
Fig. 5

Peak flux at beam center as a function of slab thickness for the radiative transport, Fokker–Planck and Leakeas–Larsen equations.

Fig. 6
Fig. 6

Beam width at half-peak flux as a function of slab thickness for the radiative transport, Fokker–Planck and Leakeas–Larsen equations. All parameters are the same as for Fig. 3.

Fig. 7
Fig. 7

Eigenvalue spectrum of the scattering operator L for the radiative transport (circles), Fokker–Planck (squares) and Leakeas–Larsen (triangles) equations. We do not show the first eigenvalue ν 0 = 0 .

Fig. 8
Fig. 8

Point-spread function for a slab of thickness z 0 = 5 l 0 due to a very narrow beam W 0 = 0.05 l 0 . All other parameters are the same as for Fig. 3.

Equations (52)

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μ z Ψ + ( 1 - μ 2 ) 1 / 2 ( cos   ϕ   x Ψ + sin   ϕ   y Ψ ) + Σ a Ψ
= Σ s L Ψ .
L Ψ ( μ ,   ϕ ,   x ,   y ,   z ) = - Ψ + - π + π - 1 + 1 f ( μ ,   ϕ ; μ ,   ϕ )× Ψ ( μ ,   ϕ ,   x ,   y ,   z ) d μ d ϕ .
Ψ ( μ ,   ϕ ,   x ,   y ,   0 ) = ( 2 π ) - 1 exp [ - ( x 2 + y 2 ) / W 0 2 ] δ ( μ - 1 ) ,
0 < μ 1 , - π ϕ < π .
Ψ ( μ ,   ϕ ,   x ,   y ,   z 0 ) = 0 ,
- 1 μ < 0 , - π ϕ < π .
Ψ ˆ ( μ ,   ϕ ,   k ,   φ ¯ ,   z ) = Ψ ( μ ,   ϕ ,   x ,   y ,   z )× exp [ - ik ( x   cos   φ ¯ + y   sin   φ ¯ ) ] d x d y .
μ z Ψ ˆ + ik ( 1 - μ 2 ) 1 / 2   cos ( φ ¯ - ϕ ) Ψ ˆ + Σ a Ψ ˆ = L Ψ ˆ ,
Ψ ˆ ( μ ,   ϕ ,   k ,   φ ¯ ,   0 ) = W 0 2 ( 4 π ) - 1   exp ( - W 0 2 k 2 / 4 ) δ ( μ - 1 ) , 0 < μ 1 , - π ϕ < π ,
Ψ ˆ ( μ ,   ϕ ,   k ,   φ ¯ ,   z 0 ) = 0 ,
- 1 μ < 0 , - π ϕ < π .
Ψ ˆ ( μ ,   φ ,   k ,   z ) = exp ( λ z ) V ( μ ,   φ ) .
λ μ V + ik ( 1 - μ 2 ) 1 / 2       cos   φ V + Σ a V = Σ s LV .
LV ( μ m ,   φ n ) - V ( μ m ,   φ n ) + m = 1 M n = 1 N f ( μ m ,   φ n ;   μ m ,   φ n )× V ( μ m ,   φ n ) w m Δ φ .
λ μ m V ( μ m ,   φ n ) + ik ( 1 - μ m 2 ) 1 / 2       cos   φ n V ( μ m ,   φ n )
+ Σ a V ( μ m ,   φ n ) = LV ( μ m ,   φ n ) ,
m = 1 , , M , n = 1 , , N .
Re [ λ - P / 2 ] < < Re [ λ - 1 ] < Re [ λ + 1 ] < < Re [ λ + P / 2 ] .
Ψ ˆ ( μ m ,   φ n ,   k ,   z ) = p = 1 P / 2 { c p   exp [ λ p ( z - z 0 ) ] V p ( μ m ,   φ n ) + c - p       exp ( - λ p z ) V p ( μ M - m + 1 ,   φ n ) } .
p = 1 P / 2 [ c p   exp ( - λ p z 0 ) V p ( μ m ,   φ n ) + c - p V p ( μ M - m + 1 ,   φ n ) ]
= ( W 0 2 / 2 ) G ( μ m ) exp ( - W 0 2 k 2 / 4 )
m = M / 2 + 1 , , M , n = 1 , , N ,
p = 1 P / 2 [ c p V p ( μ m ,   φ n ) + c - p   exp ( - λ p z 0 ) V p ( μ M - m + 1 ,   φ n ) ]
= 0 m = 1 , , M / 2 , n = 1 , , N .
1 2 π   δ ( μ - 1 ) G ( μ m ) = ( 2 π 3 μ w 2 ) - 1 / 2×   exp [ - ( μ m - 1 ) 2 / 2 μ w 2 ] ,
m = M / 2 + 1 , , M .
Ψ ( μ ,   ϕ ,   x ,   y ,   z ) = ( 2 π ) - 2 0 - π π Ψ ˆ ( μ ,   φ ¯ - ϕ ,   k ,   z )×  exp [ ik ( x   cos   φ ¯ + y   sin   φ ¯ ) ] d φ k ¯ d k .
F ( x ,   y ;   z 0 ) = - π π 0 1 Ψ ( μ ,   ϕ ,   x ,   y ,   z 0 ) μ d μ d ϕ .
F ( ρ ;   z 0 ) = ( 2 π ) - 1 0 F ˆ ( k ;   z 0 ) J 0 ( k ρ ) k d k .
F ˆ ( k ;   z 0 ) = - π + π 0 + 1 Ψ ˆ ( μ ,   φ ,   k ,   z 0 ) μ d μ d φ .
L Ψ = 1 2   ( 1 - g ) Δ Ψ .
Δ Ψ = μ ( 1 - μ 2 )   Ψ μ + 1 1 - μ 2 2 Ψ ϕ 2 ,
g = 2 π - 1 + 1 f ( cos   Θ ) cos   Θ d ( cos   Θ ) ,
cos   Θ = μ μ + ( 1 - μ 2 ) 1 / 2 ( 1 - μ 2 ) 1 / 2 cos ( ϕ - ϕ ) .
L Ψ = α Δ [ I - β Δ ] - 1 Ψ .
D m + 1 / 2 = D m - 1 / 2 - 2 μ m w m , m = 1 , , M ,
Δ V ( μ m ,   φ n )
1 w m       D m + 1 / 2 V ( μ m + 1 ,   φ n ) - V ( μ m ,   φ n ) μ m + 1 - μ m - D m - 1 / 2 V ( μ m ,   φ n ) - V ( μ m - 1 ,   φ n ) μ m - μ m - 1 + 1 1 - μ m 2  
×  V ( μ m ,   φ n + 1 ) - 2 V ( μ m ,   φ n ) + V ( μ m ,   φ n - 1 ) ( Δ φ ) 2 .
 
V ( μ ,   φ ) = V ( μ ,   φ + 2 π ) .
λ μ m V ( μ m ,   φ n ) + ik ( 1 - μ m 2 ) 1 / 2       cos   φ n V ( μ m ,   φ n ) + Σ a V ( μ m ,   φ n ) = 1 2   Σ s ( 1 - g ) Δ V ( μ m ,   φ n ) ,
m = 1 , , M , n = 1 , , N .
V ( μ m ,   φ n ) = ( I - β Δ ) W ( μ m ,   φ n ) .
λ μ m ( I - β Δ ) W ( μ m ,   φ n ) + Σ a ( I - β Δ ) W ( μ m ,   φ n )
= α Δ W ( μ m ,   φ n ) , m = 1 , , M , n = 1 , , N .
f ( cos   Θ ) = 1 4 π       1 - g 2 ( 1 + g 2 - 2 g   cos   Θ ) 3 / 2 ,
f ( cos   Θ ) = 1 2 π       exp [ - ( 1 - cos   Θ ) / ] 1 - exp ( - 2 / ) .
g = coth   - 1 - .
E FP , LL ( ρ ) | F FP , LL ( ρ ;   z 0 ) - F RT ( ρ ;   z 0 ) | F RT ( ρ ;   z 0 ) .
LY nm ( μ ,   ϕ ) = - ν n Y nm ( μ ,   ϕ ) .

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