Abstract

Forward-peaked and large-angle scattering approximations of the radiative transport equation give rise to generalized Fokker–Planck equations whose main feature is the replacement of the integral scattering operator with differential operators in the direction-space variables. Using the PN method, an appraisal of generalized Fokker–Planck equations due to González-Rodríguez and Kim [Appl. Opt. 47, 2599–2609 (2008)], Leakeas and Larsen [Nucl. Sci. Eng. 137, 236–250 (2001), and J. Opt. Soc. Am. A 20, 92–98 (2003)], and Pomraning [Math. Models Meth. Appl. Sci. 2, 21–36 (1992)] is carried out by computing the relative error between the backscattered and transmitted surface flux predicted by the generalized Fokker–Planck equations and the transport equation with Henyey–Greenstein phase function for anisotropies ranging from 0 to 1. Generalized Fokker–Planck equations whose scattering operators incorporate large-angle scattering and possess eigenvalues similar to the integral scattering operator with Henyey–Greenstein phase function are found to minimize the relative error in the limit of unit anisotropy.

© 2009 Optical Society of America

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References

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  1. B. Davison, Neutron Transport Theory (Oxford University, 1957).
  2. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  3. A. D. Kim and M. Moscoso, “Light transport in two-layer tissues,” J. Biomed. Opt. 10, 034015 (2005).
    [CrossRef] [PubMed]
  4. K. Ren and G. Bal, “Generalized diffusion model in optical tomography with clear layers,” J. Opt. Soc. Am. A 20, 2355-2364 (2003).
    [CrossRef]
  5. W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166-2185 (1990).
    [CrossRef]
  6. G. C. Pomraning, “The Fokker-Planck equation as an asymptotic limit,” Math. Models Methods 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 and E. W. Larsen, “Generalized Fokker-Planck approximations of particle transport with forward-peaked scattering,” Nucl. Sci. Eng. 137, 236-250 (2001).
  9. A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92-98 (2003).
    [CrossRef]
  10. P. González-Rodríguez and A. D. Kim, “Light propagation in tissues with forward-peaked and large angle scattering,” Appl. Opt. 47, 2599-2609 (2008).
    [CrossRef] [PubMed]
  11. E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).
  12. J. E. Morel, “An improved Fokker-Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131-136 (1985).
  13. V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14, L13-L19 (2004).
    [CrossRef]
  14. K. G. Phillips and C. Lancellotti, “A universal numerical treatment of radiative transport equations with differential and integral scattering operators,” Proc. SPIE , 6864, 68640Z (2008).
    [CrossRef]
  15. J. K. Barton, T. J. Pfefer, A. J. Welch, D. J. Smithies, J. S. Nelson, and M. J. C. van Gemert, “Optical Monte Carlo modeling of a true port wine stain anatomy,” Opt. Express 2, 391-396(1998).
    [CrossRef] [PubMed]
  16. A. J. Welch and M. J. C. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum, 1995).
  17. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, 2006).
  18. G. C. Pomraning, “A non-Gaussian treatment of radiation pencil beams,” Nucl. Sci. Eng. 127, 182-198 (1997).
  19. K. M. Yoo, F. Liu, and R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. 64, 2647-2650 (1990).
    [CrossRef] [PubMed]
  20. E. W. Larsen and J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75-81 (1974).
    [CrossRef]
  21. K. G. Phillips, “Numerical solution of linear transport equations with scattering operators of integral and differential type,” PhD thesis (CUNY Graduate Center, 2008).
  22. A. D. Kim and M. Moscoso, “Backscattering of beams by forward-peaked scattering media,” Opt. Lett. 29, 74-76(2004).
    [CrossRef] [PubMed]
  23. K. G. Phillips, M. Xu, S. K. Gayen, and R. R. Alfano, “Time-resolved ring structure of backscattered circularly polarized beams from forward scattering media,” Opt. Express 13, 7954-7969 (2005).
    [CrossRef] [PubMed]

2008

K. G. Phillips and C. Lancellotti, “A universal numerical treatment of radiative transport equations with differential and integral scattering operators,” Proc. SPIE , 6864, 68640Z (2008).
[CrossRef]

P. González-Rodríguez and A. D. Kim, “Light propagation in tissues with forward-peaked and large angle scattering,” Appl. Opt. 47, 2599-2609 (2008).
[CrossRef] [PubMed]

2005

2004

A. D. Kim and M. Moscoso, “Backscattering of beams by forward-peaked scattering media,” Opt. Lett. 29, 74-76(2004).
[CrossRef] [PubMed]

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14, L13-L19 (2004).
[CrossRef]

2003

2001

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

1999

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

1998

1997

G. C. Pomraning, “A non-Gaussian treatment of radiation pencil beams,” Nucl. Sci. Eng. 127, 182-198 (1997).

1992

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

1990

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

K. M. Yoo, F. Liu, and R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. 64, 2647-2650 (1990).
[CrossRef] [PubMed]

1985

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

1974

E. W. Larsen and J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75-81 (1974).
[CrossRef]

Alfano, R. R.

K. G. Phillips, M. Xu, S. K. Gayen, and R. R. Alfano, “Time-resolved ring structure of backscattered circularly polarized beams from forward scattering media,” Opt. Express 13, 7954-7969 (2005).
[CrossRef] [PubMed]

K. M. Yoo, F. Liu, and R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. 64, 2647-2650 (1990).
[CrossRef] [PubMed]

Bal, G.

Barton, J. K.

Canuto, C.

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, 2006).

Case, K. M.

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

Cheong, W. F.

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

Davison, B.

B. Davison, Neutron Transport Theory (Oxford University, 1957).

Gayen, S. K.

González-Rodríguez, P.

Hussaini, M. Y.

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, 2006).

Keller, J. B.

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

E. W. Larsen and J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75-81 (1974).
[CrossRef]

Kim, A. D.

Lancellotti, C.

K. G. Phillips and C. Lancellotti, “A universal numerical treatment of radiative transport equations with differential and integral scattering operators,” Proc. SPIE , 6864, 68640Z (2008).
[CrossRef]

Larsen, E. W.

C. L. Leakeas and E. W. Larsen, “Generalized Fokker-Planck approximations of particle transport with 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]

E. W. Larsen and J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75-81 (1974).
[CrossRef]

Leakeas, C. L.

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

Lewis, E. E.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).

Liu, F.

K. M. Yoo, F. Liu, and R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. 64, 2647-2650 (1990).
[CrossRef] [PubMed]

Markel, V. A.

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14, L13-L19 (2004).
[CrossRef]

Miller, W. F.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).

Morel, J. E.

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

Moscoso, M.

Nelson, J. S.

Pfefer, T. J.

Phillips, K. G.

K. G. Phillips and C. Lancellotti, “A universal numerical treatment of radiative transport equations with differential and integral scattering operators,” Proc. SPIE , 6864, 68640Z (2008).
[CrossRef]

K. G. Phillips, M. Xu, S. K. Gayen, and R. R. Alfano, “Time-resolved ring structure of backscattered circularly polarized beams from forward scattering media,” Opt. Express 13, 7954-7969 (2005).
[CrossRef] [PubMed]

K. G. Phillips, “Numerical solution of linear transport equations with scattering operators of integral and differential type,” PhD thesis (CUNY Graduate Center, 2008).

Pomraning, G. C.

G. C. Pomraning, “A non-Gaussian treatment of radiation pencil beams,” Nucl. Sci. Eng. 127, 182-198 (1997).

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

Prahl, S. A.

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

Quarteroni, A.

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, 2006).

Ren, K.

Smithies, D. J.

van Gemert, M. J. C.

Welch, A. J.

J. K. Barton, T. J. Pfefer, A. J. Welch, D. J. Smithies, J. S. Nelson, and M. J. C. van Gemert, “Optical Monte Carlo modeling of a true port wine stain anatomy,” Opt. Express 2, 391-396(1998).
[CrossRef] [PubMed]

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

A. J. Welch and M. J. C. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum, 1995).

Xu, M.

Yoo, K. M.

K. M. Yoo, F. Liu, and R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. 64, 2647-2650 (1990).
[CrossRef] [PubMed]

Zang, T. A.

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, 2006).

Zweifel, P. F.

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

Appl. Opt.

IEEE J. Quantum Electron.

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

J. Biomed. Opt.

A. D. Kim and M. Moscoso, “Light transport in two-layer tissues,” J. Biomed. Opt. 10, 034015 (2005).
[CrossRef] [PubMed]

J. Math. Phys.

E. W. Larsen and J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75-81 (1974).
[CrossRef]

J. Opt. Soc. Am. A

Math. Models Methods Appl. Sci.

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

Nucl. Sci. Eng.

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

G. C. Pomraning, “A non-Gaussian treatment of radiation pencil beams,” Nucl. Sci. Eng. 127, 182-198 (1997).

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

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

K. M. Yoo, F. Liu, and R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. 64, 2647-2650 (1990).
[CrossRef] [PubMed]

Proc. SPIE

K. G. Phillips and C. Lancellotti, “A universal numerical treatment of radiative transport equations with differential and integral scattering operators,” Proc. SPIE , 6864, 68640Z (2008).
[CrossRef]

Prog. Nucl. Energy

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

Waves Random Media

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14, L13-L19 (2004).
[CrossRef]

Other

K. G. Phillips, “Numerical solution of linear transport equations with scattering operators of integral and differential type,” PhD thesis (CUNY Graduate Center, 2008).

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).

B. Davison, Neutron Transport Theory (Oxford University, 1957).

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

A. J. Welch and M. J. C. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum, 1995).

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, 2006).

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Figures (11)

Fig. 1
Fig. 1

Backscattered radiance, ψ r , for various values of anisotropy factor with material properties σ s = 100 [ cm 1 ] , σ a = 1 [ cm 1 ] , L = 0.1 [ cm ] = 10 [ l s ] , N = 60 Legendre polynomials. Scattering was determined by the Henyey–Greenstein phase function.

Fig. 2
Fig. 2

Transmitted radiance, ψ t , for various values of anisotropy factor with material properties σ s = 100 [ cm 1 ] , σ a = 1 [ cm 1 ] , L = 0.1 [ cm ] = 10 [ l s ] , N = 60 Legendre polynomials. Scattering was determined by the Henyey–Greenstein phase function.

Fig. 3
Fig. 3

Comparison of the upward (in bold) and downward flux for various values of anisotropy factor. Material properties σ s = 100 [ cm 1 ] , σ a = 1 [ cm 1 ] , L = 0.1 [ cm ] = 10 [ l s ] , N = 60 Legendre polynomials.

Fig. 4
Fig. 4

Comparison of the upward (in bold) and downward flux for various values of scattering coefficient. Material properties g = 0.9 , σ a = 1 [ cm 1 ] , L = 0.1 [ cm ] = 10 [ l s ] , N = 60 Legendre polynomials.

Fig. 5
Fig. 5

Relative error of the backscattered current, J ( 0 ) , over the full range of anisotropy factor, with optical properties σ s = 100 [ cm 1 ] , σ a = 1 [ cm 1 ] . Slab thickness L = 0.1 [ cm ] . N = 60 Legendre polynomials.

Fig. 6
Fig. 6

Relative error of the transmitted current, J + ( L ) , over the full range of anisotropy factor, with optical properties σ s = 100 [ cm 1 ] , σ a = 1 [ cm 1 ] . Slab thickness L = 0.1 [ cm ] . N = 60 Legendre polynomials.

Fig. 7
Fig. 7

Relative error of the backscattered current, J ( 0 ) , over the partial range of anisotropy factor g [ 0.9 , 1 ] , with optical properties σ s = 100 [ cm 1 ] , σ a = 1 [ cm 1 ] . Slab thickness L = 0.1 [ cm ] . N = 60 Legendre polynomials.

Fig. 8
Fig. 8

Relative error of the transmitted current, J + ( L ) , over the partial range of anisotropy factor g [ 0.9 , 1 ] , with optical properties σ s = 100 [ cm 1 ] , σ a = 1 [ cm 1 ] . Slab thickness L = 0.1 [ cm ] . N = 60 Legendre polynomials.

Fig. 9
Fig. 9

Comparison of the 9th eigenvalue of the GFPE to the RTE with Henyey–Greenstein phase function.

Fig. 10
Fig. 10

Comparison of the 29th eigenvalue of the GFPE to the RTE with Henyey–Greenstein phase function.

Fig. 11
Fig. 11

Comparison of the 59th eigenvalue of the GFPE to the RTE with Henyey–Greenstein phase function.

Equations (60)

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( ω · r + σ a ) ψ ( ω , r ) = σ s S 2 f ( ω · ω ) ψ ( ω , r ) d ω σ s ψ ( ω , r ) ,
ψ ( ω , r b ) = h ( ω , r b ) , ω · n ( r b ) < 0 , r b D .
( μ z + σ a ) ψ ( μ , z ) = σ s S 2 f ( ω · ω ) ψ ( ω · k ^ , z ) d ω σ s ψ ( μ , z ) ,
ψ ( μ , z = 0 ) = h ( μ ) , μ [ 0 , 1 ] ,
ψ ( μ , z = L ) = 0 , μ [ 1 , 0 ] .
σ s S 2 f ( ω · ω ) Y n m ( ω ) d ω σ s Y n m ( ω ) = σ s ( 1 f ^ n ) Y n m ( ω ) ,
f ^ n = 2 π 1 1 f ( ω · ω ) P n ( ω · ω ) d ( ω · ω ) .
ω · ψ ( ω , r ) + σ a ψ ( ω , r ) = σ s ( 1 g ) 2 Δ ω ψ ( ω , r ) .
λ n = σ s ( 1 g ) 2 [ n ( n + 1 ) ] .
F [ n ( n + 1 ) ] Y n m λ n .
F ( Δ ω ) Y n m ( ω ) = α Δ ω ( 1 β Δ ω ) 1 Y n m ( ω ) ,
= α [ n ( n + 1 ) ] ( 1 + β [ n ( n + 1 ) ] ) 1 Y n m ( ω ) .
α = σ s ( 1 g ) 2 g , β = α σ s .
( ω · r + σ a ) ψ ( ω , r ) = α Δ ω ( 1 β Δ ω ) 1 ψ ( ω , r ) ,
λ n = α [ n ( n + 1 ) ] 1 + β [ n ( n + 1 ) ] .
( 1 β Δ ω ) 1 δ ( ω · ω 1 ) 2 π = n = 0 2 n + 1 4 π P n ( ω · ω ) 1 + β n ( n + 1 ) = G ( ω · ω )
( 1 β Δ ω ) 1 ψ ( ω ) = S 2 G ( ω · ω ) ψ ( ω ) d ω .
σ s S 2 f ( ω · ω ) Y n m ( ω ) d ω σ s Y n m ( ω ) = σ s ( 1 g n ) Y n m ( ω ) .
σ s S 2 f ( ω · ω ) Y n m ( ω ) d ω σ s Y n m ( ω ) = σ ϵ ( 1 ( 1 ϵ ) n ) Y n m ( ω ) .
S 2 f ( ω · ω ) Y n m ( ω ) d ω σ s Y n m ( ω ) = ( σ ( 1 + ϵ ) [ n ] ϵ σ 2 [ n ( n + 1 ) ] + O ( ϵ 2 ) ) Y n m ( ω ) .
L 3 / 2 [ Y n m ( ω ) ] 1 4 π 2 S 2 Y n m ( ω ) ( 1 ω · ω ) 3 / 2 d ω = n Y n m ( ω ) .
( ω · r + σ a ) ψ ( ω , r ) = σ s ( 1 g ) ( 2 g ) L 3 / 2 [ ψ ( t , ω , r ) ] σ s ( 1 g ) 2 2 Δ ω ψ ( t , ω , r ) .
λ n = σ s ( 1 g ) ( 2 g ) [ n ] + σ s ( 1 g ) 2 2 [ n ( n + 1 ) ] .
f ( ω · ω ) m = 0 M a m δ ( 2 m ) ( ω ω ) + n = 0 N 2 n + 1 4 π b n P n ( ω · ω ) .
f ( ω · ω ) a 0 δ ( ω ω ) + a 1 δ ( ω ω ) + 1 4 π [ b 0 P 0 ( ω · ω ) + 3 b 1 P 1 ( ω · ω ) ] .
L FPE [ ψ ] = σ s [ ( 1 a 0 ) a 1 Δ ω ] ψ + σ s 4 π S 2 [ b 0 P 0 ( ω · ω ) + 3 b 1 P 1 ( ω · ω ) ] ψ ( ω ) d ω .
λ n = σ s [ ( 1 a 0 ) + a 1 n ( n + 1 ) b 0 δ n , 0 b 1 δ n , 1 ] ,
a 0 = 2 f ^ 2 f ^ 3 ,
a 1 = ( f ^ 2 f ^ 3 ) / 6 ,
b 0 = 1 2 f ^ 2 + f ^ 3 ,
b 1 = f ^ 1 5 f ^ 2 / 3 + 2 f ^ 3 / 3.
L gFPEd [ ψ ] = σ s [ ( 1 a 0 ) a 1 Δ ω ( 1 a 2 Δ ω ) 1 ] ψ + σ s 4 π S 2 [ b 0 P 0 ( ω · ω ) + 3 b 1 P 1 ( ω · ω ) ] ψ ( ω ) d ω ,
λ n = σ s [ ( 1 a 0 ) a 1 n ( n + 1 ) 1 + a 2 n ( n + 1 ) b 0 δ n , 0 b 1 δ n , 1 ] .
( μ z + σ a ) ψ ( μ , z ) = L [ ψ ( μ , z ) ] , ( μ , z ) [ 1 , 1 ] × [ 0 , L ] ,
ψ ( μ , z = 0 ) = h ( μ ) , μ [ 0 , 1 ] ,
ψ ( μ , z = L ) = 0 , μ [ 1 , 0 ] .
ψ ( μ , z ) = ϕ ( μ ) e γ z .
( μ γ + σ a ) ϕ ( μ ) = L [ ϕ ( μ ) ] , μ [ 1 , 1 ] .
L [ ϕ ( ω · k ^ ) ] = L [ n = 0 m = n + n ϕ ^ n Y n m ( ω ) Y n m * ( k ^ ) ] ,
ϕ ^ n = 2 π 1 1 ϕ ( μ ) P n ( μ ) d μ .
L [ Y n m ] = λ n Y n m ,
( γ μ + σ a ) n = 0 2 n + 1 4 π ϕ ^ n P n ( μ ) = n = 0 2 n + 1 4 π λ n ϕ ^ n P n ( μ ) .
ϕ ( μ ) n = 0 N 1 2 n + 1 4 π ϕ ^ n P n ( μ ) .
γ 2 k + 1 [ k ϕ ^ k 1 + ( k + 1 ) ϕ ^ k + 1 ] + ( σ a λ k ) ϕ ^ k = 0 for     k = 0 , 1 , , N 1.
( γ A + D ) ϕ ^ = 0 or A 1 D ϕ ^ = γ ϕ ^ ,
A = ( 0 1 0 0 · 0 1 / 3 0 2 / 3 0 · 0 0 2 / 5 0 3 / 5 · 0 · · · · · 0 · · k / ( 2 k + 1 ) 0 ( k + 1 ) / ( 2 k + 1 ) · · · · · · · · · · · 0 N / ( 2 N 1 ) 0 0 0 0 N / ( 2 N + 1 ) 0 ) ,
D = diag ( σ a λ 0 , σ a λ 1 , , σ a λ N 1 ) .
ψ ( μ , z ) = l = 1 N / 2 c l ϕ l + ( μ ) e γ l + ( z L ) + l = 1 N / 2 d l ϕ l ( μ ) e γ l z .
ϕ l ± ( μ ) n = 0 N 1 2 n + 1 4 π ϕ ^ l n ± P n ( μ ) .
ψ ( μ i , z = 0 ) = l = 1 N / 2 c l e γ l + L ϕ l + ( μ i ) + l = 1 N / 2 d l ϕ l ( μ i ) = h ( μ i ) , μ i [ 0 , 1 ] ,
ψ ( μ i , z = L ) = l = 1 N / 2 c l ϕ l + ( μ i ) + l = 1 N / 2 d l e γ l L ϕ l ( μ i ) = 0 , μ i [ 1 , 0 ] .
ψ r ( μ i ) ψ ( μ i , z = 0 ) = l = 1 N / 2 c l e γ l + L ϕ l + ( μ i ) + l = 1 N / 2 d l ϕ l ( μ i ) , μ i [ 1 , 0 ] ,
ψ t ( μ i ) ψ ( μ i , z = L ) = l = 1 N / 2 c l ϕ l + ( μ i ) + l = 1 N / 2 d l e γ l L ϕ l ( μ i ) , μ i [ 0 , 1 ] .
J + ( z ) = 0 1 μ ψ ( μ , z ) d μ ,
J ( z ) = 1 0 | μ | ψ ( μ , z ) d μ .
J + ( 0 ) = J ( 0 ) + J + ( L ) + σ a 0 L 1 + 1 ψ ( μ , z ) d μ d z .
h ( μ ) = B W 2 π exp ( ( μ 1 ) 2 / 2 W 2 ) , μ [ 0 , 1 ] .
J + ( z ) J + ( 0 ) = e σ a z .
E = | J H G ( 0 ) J G F P E ( 0 ) | J H G ( 0 ) ,
E + = | J H G + ( L ) J G F P E + ( L ) | J H G + ( L ) .

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