Abstract

The problem of narrow-beam propagation in a medium with highly anisotropic scattering is considered within radiative transfer theory. A novel solution of the radiative transfer equation within small-angle approximation, accounting for the path length spread, is presented. A new stable numerical algorithm for the simulation of a narrow beam is developed and applied to a narrow beam in a two-dimensional scattering medium. A truly three-dimensional version of the proposed approach is formulated.

© 2011 Optical Society of America

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References

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    [CrossRef]
  11. A. S. Kompaneets, “Multiple scattering of fast electrons and α-particles in heavy elements,” Sov. Phys. JETP 15, 235-243(1945).
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    [CrossRef]
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  17. V. P. Budak and Y. A. Ilyushin, “Development of the small angle approximation of the radiative transfer theory taking into account the photon path distribution function,” Atmos. Oceanic Opt. 23, 181-185 (2010).
    [CrossRef]
  18. K. van Wijk, M. M. Haney, and J. A. Scales, “1D energy transport in a strongly scattering laboratory model,” Phys. Rev. E 69, 036611 (2004).
    [CrossRef]

2010 (1)

V. P. Budak and Y. A. Ilyushin, “Development of the small angle approximation of the radiative transfer theory taking into account the photon path distribution function,” Atmos. Oceanic Opt. 23, 181-185 (2010).
[CrossRef]

2008 (1)

V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 109, 220-234 (2008).
[CrossRef]

2006 (1)

E. A. Sergeeva, M. Y. Kirillin, and A. V. Priezzhev, “Propagation of a femtosecond pulse in a scattering medium: theoretical analysis and numerical simulation,” Quantum Electron. 36, 1023-1031 (2006).
[CrossRef]

2005 (1)

V. P. Budak and A. V. Kozelskii, “Accuracy and applicability domain of the small angle approximation,” Atmos. Oceanic Opt. 18, 32-37 (2005).

2004 (1)

K. van Wijk, M. M. Haney, and J. A. Scales, “1D energy transport in a strongly scattering laboratory model,” Phys. Rev. E 69, 036611 (2004).
[CrossRef]

1999 (1)

1995 (1)

1989 (1)

1970 (1)

1968 (1)

W. M. Irvine, “Diffuse reflection and transmission by cloud and dust layers,” J. Quant. Spectrosc. Radiat. Transfer 8, 471-485(1968).
[CrossRef]

1963 (1)

1945 (1)

A. S. Kompaneets, “Multiple scattering of fast electrons and α-particles in heavy elements,” Sov. Phys. JETP 15, 235-243(1945).

1940 (1)

S. Goudsmit and J. L. Saunderson, “Multiple scattering of electrons,” Phys. Rev. 57, 24-29 (1940).
[CrossRef]

1929 (1)

W. Bothe, “Die streuabsorption der elektronenstrahlen,” Zeit. Phys. 54, 161-178 (1929).
[CrossRef]

1922 (1)

G. Wentzel, “Zur theorie der streuung von β-strahlen,” Ann. Physik 374, 335-368 (1922).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Alexandrov, M. D.

Bothe, W.

W. Bothe, “Die streuabsorption der elektronenstrahlen,” Zeit. Phys. 54, 161-178 (1929).
[CrossRef]

Budak, V. P.

V. P. Budak and Y. A. Ilyushin, “Development of the small angle approximation of the radiative transfer theory taking into account the photon path distribution function,” Atmos. Oceanic Opt. 23, 181-185 (2010).
[CrossRef]

V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 109, 220-234 (2008).
[CrossRef]

V. P. Budak and A. V. Kozelskii, “Accuracy and applicability domain of the small angle approximation,” Atmos. Oceanic Opt. 18, 32-37 (2005).

Chandrasekhar, S.

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

Dave, J. V.

Duntley, S.

Gazdag, J.

Goudsmit, S.

S. Goudsmit and J. L. Saunderson, “Multiple scattering of electrons,” Phys. Rev. 57, 24-29 (1940).
[CrossRef]

Haney, M. M.

K. van Wijk, M. M. Haney, and J. A. Scales, “1D energy transport in a strongly scattering laboratory model,” Phys. Rev. E 69, 036611 (2004).
[CrossRef]

Ilyushin, Y. A.

V. P. Budak and Y. A. Ilyushin, “Development of the small angle approximation of the radiative transfer theory taking into account the photon path distribution function,” Atmos. Oceanic Opt. 23, 181-185 (2010).
[CrossRef]

Irvine, W. M.

W. M. Irvine, “Diffuse reflection and transmission by cloud and dust layers,” J. Quant. Spectrosc. Radiat. Transfer 8, 471-485(1968).
[CrossRef]

Ito, S.

Kirillin, M. Y.

E. A. Sergeeva, M. Y. Kirillin, and A. V. Priezzhev, “Propagation of a femtosecond pulse in a scattering medium: theoretical analysis and numerical simulation,” Quantum Electron. 36, 1023-1031 (2006).
[CrossRef]

Kompaneets, A. S.

A. S. Kompaneets, “Multiple scattering of fast electrons and α-particles in heavy elements,” Sov. Phys. JETP 15, 235-243(1945).

Korkin, S. V.

V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 109, 220-234 (2008).
[CrossRef]

Kozelskii, A. V.

V. P. Budak and A. V. Kozelskii, “Accuracy and applicability domain of the small angle approximation,” Atmos. Oceanic Opt. 18, 32-37 (2005).

McLean, J. W.

Morton, K. W.

R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems (Interscience, 1967).

Oguchi, T.

Priezzhev, A. V.

E. A. Sergeeva, M. Y. Kirillin, and A. V. Priezzhev, “Propagation of a femtosecond pulse in a scattering medium: theoretical analysis and numerical simulation,” Quantum Electron. 36, 1023-1031 (2006).
[CrossRef]

Remizovich, V. S.

Richtmyer, R. D.

R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems (Interscience, 1967).

Saunderson, J. L.

S. Goudsmit and J. L. Saunderson, “Multiple scattering of electrons,” Phys. Rev. 57, 24-29 (1940).
[CrossRef]

Scales, J. A.

K. van Wijk, M. M. Haney, and J. A. Scales, “1D energy transport in a strongly scattering laboratory model,” Phys. Rev. E 69, 036611 (2004).
[CrossRef]

Sergeeva, E. A.

E. A. Sergeeva, M. Y. Kirillin, and A. V. Priezzhev, “Propagation of a femtosecond pulse in a scattering medium: theoretical analysis and numerical simulation,” Quantum Electron. 36, 1023-1031 (2006).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

van Wijk, K.

K. van Wijk, M. M. Haney, and J. A. Scales, “1D energy transport in a strongly scattering laboratory model,” Phys. Rev. E 69, 036611 (2004).
[CrossRef]

Walker, R. E.

Wentzel, G.

G. Wentzel, “Zur theorie der streuung von β-strahlen,” Ann. Physik 374, 335-368 (1922).
[CrossRef]

Ann. Physik (1)

G. Wentzel, “Zur theorie der streuung von β-strahlen,” Ann. Physik 374, 335-368 (1922).
[CrossRef]

Appl. Opt. (2)

Atmos. Oceanic Opt. (2)

V. P. Budak and Y. A. Ilyushin, “Development of the small angle approximation of the radiative transfer theory taking into account the photon path distribution function,” Atmos. Oceanic Opt. 23, 181-185 (2010).
[CrossRef]

V. P. Budak and A. V. Kozelskii, “Accuracy and applicability domain of the small angle approximation,” Atmos. Oceanic Opt. 18, 32-37 (2005).

J. Opt. Soc. Am. (1)

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

J. Quant. Spectrosc. Radiat. Transfer (2)

W. M. Irvine, “Diffuse reflection and transmission by cloud and dust layers,” J. Quant. Spectrosc. Radiat. Transfer 8, 471-485(1968).
[CrossRef]

V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 109, 220-234 (2008).
[CrossRef]

Phys. Rev. (1)

S. Goudsmit and J. L. Saunderson, “Multiple scattering of electrons,” Phys. Rev. 57, 24-29 (1940).
[CrossRef]

Phys. Rev. E (1)

K. van Wijk, M. M. Haney, and J. A. Scales, “1D energy transport in a strongly scattering laboratory model,” Phys. Rev. E 69, 036611 (2004).
[CrossRef]

Quantum Electron. (1)

E. A. Sergeeva, M. Y. Kirillin, and A. V. Priezzhev, “Propagation of a femtosecond pulse in a scattering medium: theoretical analysis and numerical simulation,” Quantum Electron. 36, 1023-1031 (2006).
[CrossRef]

Sov. Phys. JETP (1)

A. S. Kompaneets, “Multiple scattering of fast electrons and α-particles in heavy elements,” Sov. Phys. JETP 15, 235-243(1945).

Zeit. Phys. (1)

W. Bothe, “Die streuabsorption der elektronenstrahlen,” Zeit. Phys. 54, 161-178 (1929).
[CrossRef]

Other (3)

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

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems (Interscience, 1967).

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Figures (4)

Fig. 1
Fig. 1

To the evaluation of Eq. (15).

Fig. 2
Fig. 2

Total irradiance C 0 ( x , y ) (arbitrary units) of the point unidirectional source in a 2D scattering medium. Λ = 1 , g = 0.99 . Optical depths x are labeled by the numbers.

Fig. 3
Fig. 3

Angular distributions of the outgoing radiance L 2 + L D . Λ = 0.99 , g = 0.9 , X = 1 , x = 0 . y = 0 (solid curve), y = 0.05 (dashed curve).

Fig. 4
Fig. 4

Angular distributions of the outgoing radiance L 2 + L D . Λ = 0.99 , g = 0.9 , X = 1 , x = 0 . y = 0.75 (solid curve), y = 1.5 (dashed curve).

Equations (40)

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( Ω · ) L = L + Λ 4 π L ( r , Ω ) x ( Ω , Ω ) d Ω + f ( r ) ,
z L + 1 μ z μ x x L + 1 μ z μ y y L = 1 μ z L + 1 μ z Λ 4 π L ( r , Ω ) x ( Ω , Ω ) d Ω + 1 μ z f ( r ) ,
1 μ z 1 + ( 1 μ z ) + ( 1 μ z ) 2 + + ( 1 μ z ) n + o ( ( 1 μ z ) n ) μ z .
z L + μ z μ x x L + μ z μ y y L = μ z L + μ z Λ 4 π L ( r , Ω ) x ( Ω , Ω ) d Ω + μ z f ( r ) ,
f s ( r ) = ( Ω · ) L a + L a Λ 4 π L a ( r , Ω ) x ( Ω , Ω ) d Ω + f ( r ) .
f s ( r ) = ( 1 μ z μ z ) ( ( Ω · ) L a + L a Λ 4 π L a ( r , Ω ) x ( Ω , Ω ) d Ω + f ( r ) ) ,
L = n = 0 m = n n c n m Y n m ( θ , ϕ ) ,
x ( Ω , Ω ) = 4 π n = 0 m = n n x n Y n m ( θ , ϕ ) Y n m ( θ , ϕ ) .
( n · ) L = L + Λ 2 π L ( x , y , ϕ ) x ( ϕ , ϕ ) d ϕ ,
L ( x , y , ϕ ) = 1 2 π exp ( i k y y ) L y ( x , k y ) d k y = 1 2 π exp ( i k y y ) n = C n ( x , k y ) e i n ϕ d k y .
x ( ϕ , ϕ ) = n = x n ( r ) e i n ( ϕ ϕ ) .
1 2 ( k y C n 1 ( x , k y ) k y C n + 1 ( x , k y ) + C n 1 ( x , k y ) x + C n + 1 ( x , k y ) x ) = ( 1 Λ x n ) C n ( x , k y ) .
x C n ( x , k y ) k y n C n ( x , k y ) = ( 1 Λ x n ) C n ( x , k y ) .
k y C ( p , q , k y ) q = ( 1 Λ x p q ) C ( p , q , k y ) ,
C ( p , q , k y ) = exp { 1 k y q 0 q ( 1 Λ x p q ) d q }
x ( ϕ ) = 1 a 2 1 a cos ϕ ,
x n = 1 2 π 0 2 π exp ( i n ϕ ) x ( ϕ ) d ϕ = g | n | ,
L ( 0 , y , ϕ ) = θ ( y ) δ ( ϕ ) ,
θ ( y ) = { 1 , y > 0 0 , y < 0 .
L + ( 0 , y , ϕ ) = δ ( ϕ ) / 2 = 1 2 n = 1 2 π exp ( i n ϕ ) ,
L ( 0 , y , ϕ ) = { δ ( ϕ ) / 2 , y > 0 δ ( ϕ ) / 2 , y < 0 .
L + ( x , y , ϕ ) = δ ( ϕ ) / 2 = 1 2 n = 1 2 π exp ( ( 1 Λ x n ) x + i n ϕ ) .
L ( 0 , y , ϕ ) = 1 2 π exp ( i k y y ) d k y i k y n = 1 2 π exp ( i n ϕ ) ,
1 k y q 0 q ( 1 Λ x p q ) d q = Λ g k y x k y ln g x Λ k y ln g .
1 k y q 0 q ( 1 Λ x p q ) d q 1 2 k y Λ x 2 ln g ( 1 Λ ) x .
1 2 π C 0 ( x , k y ) exp ( i k y y ) d k y = 1 2 π 2 exp ( ( 1 Λ ) x ) arctan ( 2 y x 2 Λ ln g ) .
1 2 π ( L + + L ) d ϕ = exp ( ( 1 Λ x n ) x ) ( 1 4 π + 1 2 π 2 arctan ( 2 y x 2 Λ log g ) ) .
L ( 0 , y , ϕ ) = δ ( ϕ ) δ ( y ) = 1 2 π exp ( i k y y ) d k y ,
C 0 ( x , y ) = 2 x 2 Λ ln g π ( Λ 2 x 4 ln g 2 + 4 y 2 ) exp ( x ( Λ 1 ) ) .
L 0 ( x , y , ϕ ) = δ ( ϕ ) δ ( y ) exp ( x ) = 1 2 π exp ( i k y y ) C n ( 0 ) ( x , k y ) exp ( i n ϕ ) d k y ,
( n · ) L 1 = L 1 + Λ 2 π ( L 0 ( x , y , ϕ ) + L 1 ( x , y , ϕ ) ) x ( ϕ , ϕ ) d ϕ .
x L 1 + μ x μ y y L 1 = μ x L 1 + μ x Λ 2 π ( L 0 ( x , y , ϕ ) + L 1 ( x , y , ϕ ) ) x ( ϕ , ϕ ) d ϕ ,
x C ( 1 ) + μ ^ x μ ^ y i k y C ( 1 ) = μ ^ x C ( 1 ) + μ ^ x Λ x ^ C ( 1 ) + μ ^ x Λ x ^ C ( 0 ) ,
x C ( 2 ) + μ ^ x + μ ^ x x C ( 1 ) + μ ^ x + μ ^ y i k y ( C ( 1 ) + C ( 2 ) ) = μ ^ x + ( C ( 1 ) + C ( 2 ) ) + μ ^ x + Λ x ^ ( C ( 0 ) + C ( 1 ) + C ( 2 ) ) ,
L 2 ( X , ϕ ) = M ( ϕ ) L 1 ( X , ϕ ) ,
exp ( a ( 1 + cos ϕ ) ) = exp ( a ) m = ( 1 ) m I m ( a ) exp ( i m ϕ ) ,
C k ( 2 ) = exp ( a ) j ( 1 ) j + k I j + k ( a ) C j ( 1 ) ,
L D = L 1 L 2
f ( ϕ ) d ϕ m = 0 N 1 a m f ( ϕ m ) ,
A ^ i , j ( DO ) = 1 N m x m e i m ( ϕ i ϕ j ) .

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