Abstract

In this paper we propose the fast, but the accurate algorithm for numerical modeling of light fields in the turbid media slab. For the numerical solution of the radiative transfer equation (RTE) it is required its discretization based on the elimination of the solution anisotropic part and the replacement of the scattering integral by a finite sum. The solution regular part is determined numerically. A good choice of the method of the solution anisotropic part elimination determines the high convergence of the algorithm in the mean square metric. The method of synthetic iterations can be used to improve the convergence in the uniform metric. A significant increase in the solution accuracy with the use of synthetic iterations allows applying the two-stream approximation for the regular part determination. This approach permits to generalize the proposed method in the case of an arbitrary 3D geometry of the medium.

© 2015 Optical Society of America

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References

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  1. F. Hanson and I. Bendall, “Off-axis laser beam imaging and characterization with two cameras,” Appl. Opt. 52(22), 5342–5347 (2013).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  6. V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Convergence acceleration of radiative transfer equation solution at strongly anisotropic scattering,” in Light Scattering Reviews 5: Single Light Scattering and Radiative Transfer, A.A. Kokhanovsky eds. (Springer Praxis Books, 2010), pp.147–204.
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    [Crossref]
  8. W. E. Meador and W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: A unified description of existing methods and a new improvement,” J. Atmos. Sci. 37(3), 630–643 (1980).
    [Crossref]
  9. A. B. Davis and A. Marshak, “Multiple scattering in clouds, insights from three-dimensional diffusion/P_1 theory,” Nucl. Sci. Eng. 137, 251–288 (2001).

2013 (1)

2012 (1)

V. P. Budak, D. S. Efremenko, and O. V. Shagalov, “Efficiency of algorithm for solution of vector radiative transfer equation in turbid medium slab,” J. Phys. Conf. Ser. 369, 012021 (2012).
[Crossref]

2010 (1)

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

2008 (1)

N. Roy and F. Reid, “Off-axis laser detection model in coastal areas,” Opt. Eng. 47(8), 086002 (2008).
[Crossref]

2003 (1)

J.-P. Cariou, “Off-axis detection of pulsed laser beams: simulation and measurements in the lower atmosphere,” Proc. SPIE 5086, 129–138 (2003).

2002 (1)

M. L. Adams and E. W. Larsen, “Fast iterative methods for discrete-ordinates particle transport calculations,” Prog. Nucl. Energy 40(1), 3–159 (2002).
[Crossref]

2001 (1)

A. B. Davis and A. Marshak, “Multiple scattering in clouds, insights from three-dimensional diffusion/P_1 theory,” Nucl. Sci. Eng. 137, 251–288 (2001).

1980 (1)

W. E. Meador and W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: A unified description of existing methods and a new improvement,” J. Atmos. Sci. 37(3), 630–643 (1980).
[Crossref]

Adams, M. L.

M. L. Adams and E. W. Larsen, “Fast iterative methods for discrete-ordinates particle transport calculations,” Prog. Nucl. Energy 40(1), 3–159 (2002).
[Crossref]

Bendall, I.

Budak, V. P.

V. P. Budak, D. S. Efremenko, and O. V. Shagalov, “Efficiency of algorithm for solution of vector radiative transfer equation in turbid medium slab,” J. Phys. Conf. Ser. 369, 012021 (2012).
[Crossref]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Cariou, J.-P.

J.-P. Cariou, “Off-axis detection of pulsed laser beams: simulation and measurements in the lower atmosphere,” Proc. SPIE 5086, 129–138 (2003).

C-Labonnote, L.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Cornet, C.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Davis, A. B.

A. B. Davis and A. Marshak, “Multiple scattering in clouds, insights from three-dimensional diffusion/P_1 theory,” Nucl. Sci. Eng. 137, 251–288 (2001).

Duan, M.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Efremenko, D. S.

V. P. Budak, D. S. Efremenko, and O. V. Shagalov, “Efficiency of algorithm for solution of vector radiative transfer equation in turbid medium slab,” J. Phys. Conf. Ser. 369, 012021 (2012).
[Crossref]

Emde, C.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Hanson, F.

Katsev, I. L.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Klyukov, D. A.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Kokhanovsky, A. A.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Korkin, S. V.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Larsen, E. W.

M. L. Adams and E. W. Larsen, “Fast iterative methods for discrete-ordinates particle transport calculations,” Prog. Nucl. Energy 40(1), 3–159 (2002).
[Crossref]

Marshak, A.

A. B. Davis and A. Marshak, “Multiple scattering in clouds, insights from three-dimensional diffusion/P_1 theory,” Nucl. Sci. Eng. 137, 251–288 (2001).

Mayer, B.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Meador, W. E.

W. E. Meador and W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: A unified description of existing methods and a new improvement,” J. Atmos. Sci. 37(3), 630–643 (1980).
[Crossref]

Min, Q.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Nakajima, T.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Ota, Y.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Prikhach, A. S.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Reid, F.

N. Roy and F. Reid, “Off-axis laser detection model in coastal areas,” Opt. Eng. 47(8), 086002 (2008).
[Crossref]

Roy, N.

N. Roy and F. Reid, “Off-axis laser detection model in coastal areas,” Opt. Eng. 47(8), 086002 (2008).
[Crossref]

Rozanov, V. V.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Shagalov, O. V.

V. P. Budak, D. S. Efremenko, and O. V. Shagalov, “Efficiency of algorithm for solution of vector radiative transfer equation in turbid medium slab,” J. Phys. Conf. Ser. 369, 012021 (2012).
[Crossref]

Weaver, W. R.

W. E. Meador and W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: A unified description of existing methods and a new improvement,” J. Atmos. Sci. 37(3), 630–643 (1980).
[Crossref]

Yokota, T.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Zege, E. P.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Appl. Opt. (1)

J. Atmos. Sci. (1)

W. E. Meador and W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: A unified description of existing methods and a new improvement,” J. Atmos. Sci. 37(3), 630–643 (1980).
[Crossref]

J. Phys. Conf. Ser. (1)

V. P. Budak, D. S. Efremenko, and O. V. Shagalov, “Efficiency of algorithm for solution of vector radiative transfer equation in turbid medium slab,” J. Phys. Conf. Ser. 369, 012021 (2012).
[Crossref]

J. Quantum Spectrosc. Radiat. Transf. (1)

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quantum Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Nucl. Sci. Eng. (1)

A. B. Davis and A. Marshak, “Multiple scattering in clouds, insights from three-dimensional diffusion/P_1 theory,” Nucl. Sci. Eng. 137, 251–288 (2001).

Opt. Eng. (1)

N. Roy and F. Reid, “Off-axis laser detection model in coastal areas,” Opt. Eng. 47(8), 086002 (2008).
[Crossref]

Proc. SPIE (1)

J.-P. Cariou, “Off-axis detection of pulsed laser beams: simulation and measurements in the lower atmosphere,” Proc. SPIE 5086, 129–138 (2003).

Prog. Nucl. Energy (1)

M. L. Adams and E. W. Larsen, “Fast iterative methods for discrete-ordinates particle transport calculations,” Prog. Nucl. Energy 40(1), 3–159 (2002).
[Crossref]

Other (1)

V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Convergence acceleration of radiative transfer equation solution at strongly anisotropic scattering,” in Light Scattering Reviews 5: Single Light Scattering and Radiative Transfer, A.A. Kokhanovsky eds. (Springer Praxis Books, 2010), pp.147–204.

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Figures (2)

Fig. 1
Fig. 1 Normalized radiance angular distribution inside the slab.
Fig. 2
Fig. 2 Comparison of synthetic iteration (SI) with MDOM for the radiation reflected from the slab.

Equations (52)

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{ μ L r (τ, l ^ ) τ + L r (τ, l ^ )= Λ 4π x( l ^ , l ^ ) L r (τ, l ^ )d l ^ +Δ(τ, l ^ ); L r (τ, l ^ ) | τ=0,μ>0 =0; L r (τ, l ^ ) | τ= τ 0 ,μ<0 = L a (τ, l ^ ),
Δ(τ,μ,φ)=μ d L a (τ,μ,φ) dτ L a (τ,μ,φ)+ Λ 4π x( l ^ , l ^ ) L a (τ, μ , φ )d l ^ .
L a (τ,μ,φ)= e τ/ μ 0 δ( l ^ l ^ 0 )+ L ˜ a (τ,μ,φ),
Δ(τ,μ,φ)=μ d L ˜ a (τ,μ,φ) dτ L ˜ a (τ,μ,φ)+ Λ 4π x( l ^ , l ^ ) L ˜ a (τ, μ , φ )d l ^ + Λ 4π e τ/ μ 0 x( l ^ 0 , l ^ ).
L r (τ, μ i ± ,φ)= m=0 M (2 δ 0,m )cos(mφ) C m (τ, μ i ± ) , C ± (τ) C m (τ, μ i ± ),
d C (τ) dτ = B C (τ)+ M 1 Δ (τ), B M 1 ( 1 A W ),
C (0)+ P (0, τ 0 ) C ( τ 0 )= 0 τ 0 P (0,τ) M 1 Δ (τ, μ 0 )dτ
S U 1 C (0)+ H U 1 C ( τ 0 )= S 0 τ 0 e Γ t U 1 M 1 Δ (t, μ 0 )dt ,
e B τ 0 = U e Γ τ 0 U 1 ,
[ C (0) C + ( τ 0 ) ]=[ F F + ]+[ R T T R ][ C + (0) C ( τ 0 ) ]
L ˜ a (τ,μ,φ)= k=0 m=k k 2k+1 4π Z ˜ k (τ) Q k m ( μ 0 ) Q k m (μ) e imφ ,
μ d L ˜ a (τ,μ,φ) dτ = μ μ 0 k=0 K m=k k 2k+1 4π ( Λ x k e τ/ μ 0 d k Z ˜ k (τ) ) Q k m ( μ 0 ) Q k m (μ) e imφ .
Δ m (τ,μ)=( 1 μ μ 0 ) k=m K 2k+1 4π ( Λ x k e τ/ μ 0 d k Z ˜ k (τ) ) Q k m ( μ 0 ) Q k m (μ) .
Δ m (τ)=( 1 M / μ 0 ) k=m K 2k+1 4π ( Λ x k e τ/ μ 0 d k Z ˜ k (τ) ) Q k m ( μ 0 ) Q k m .
J = k=m K 2k+1 4π Q k m ( μ 0 )( (1Λ x k ) J k J 0 ) U 1 ( 1 μ 0 M 1 ) Q k m ,
J k diag( 1 γ i μ 0 d k )( H e d k τ 0 / μ 0 S ), J 0 diag( 1 γ i μ 0 1 )( H e τ 0 / μ 0 S ).
C(μ)= k=0 N 2k+1 2 c k P k (μ) .
P k (μ)= i=1 N+1 P k ( μ i ) P N+1 (μ) (μ μ i ) P N+1 ( μ i ) ,
C(μ)= i=1 N+1 P N+1 (μ) (μ μ i ) P N+1 ( μ i ) k=0 N 2k+1 2 c k P k ( μ i ) = i=1 N+1 C( μ i ) P N+1 (μ) (μ μ i ) P N+1 ( μ i ) ,
U 1 C ( τ ) = e Γ τ U 1 C ( 0 ) + e Γ τ 0 τ e Γ t U 1 M 1 Δ ( t , μ 0 ) d t ,
U 1 C ( τ ) = e Γ ( τ 0 τ ) U 1 C ( τ 0 ) e Γ τ τ τ 0 e Γ t U 1 M 1 Δ ( t , μ 0 ) d t .
U 1 C (τ)=[ e Γ ( τ 0 τ) u 11 C ( τ 0 )+ e Γ ( τ 0 τ) u 12 C + ( τ 0 ) e Γ + τ u 21 C (0) ]+ e Γ τ [ ( τ τ 0 e Γ t U 1 M 1 Δ (t, μ 0 )dt ) ( 0 τ e Γ t U 1 M 1 Δ (t, μ 0 )dt ) + ].
U 1 C (τ)=[ e Γ ( τ 0 τ) 0 0 e Γ + τ ] G ( τ 0 )+ F (τ),
F (τ) k=m K 2k+1 4π Q k m ( μ 0 )diag( d k e d k τ/ μ 0 γ i μ 0 d k e τ/ μ 0 γ i μ 0 1 ) U 1 ( 1 μ 0 M 1 ) Q k m k=m K diag( d k e d k τ/ μ 0 γ μ 0 d k e τ/ μ 0 γ i μ 0 1 ) f k ,
G ( τ 0 )[ u 11 C ( τ 0 )+ u 12 C + ( τ 0 ) u 21 C (0) ][ ( F ( τ 0 ) ) ( F (0) ) + ].
L(τ, l ^ )={ e τ/ μ 0 δ( l ^ l ^ 0 )+ Λ 4πμ 0 τ e (τt) /μ x( l ^ , l ^ )L(t, l ^ )d l ^ dt ,μ0; Λ 4πμ τ τ 0 e (τt) /μ x( l ^ , l ^ )L(t, l ^ )d l ^ dt ,μ<0.
L (1) (τ, l ^ )={ Λ 4πμ 0 τ e (τt) /μ x( l ^ , l ^ )( L a (t, l ^ )+ L r (t, l ^ ) )d l ^ dt ,μ0; Λ 4πμ τ τ 0 e (τt) /μ x( l ^ , l ^ )( L a (t, l ^ )+ L r (t, l ^ ) )d l ^ dt ,μ<0.
L(τ,μ,φ)= L a (τ,μ,φ)+ L r (τ,μ,φ)= k=0 K 2k+1 4π Z k (τ) P k (ν) + m=0 M (2 δ 0m ) C m (τ,μ)cos(mφ) ,
x( l ^ , l ^ )L(t, l ^ )d l ^ S a (τ, l ^ )+ S r (τ, l ^ )= k=0 K (2k+1) x k Z k (t) P k ( l ^ l ^ 0 ) +2π m=M M (2 δ 0m )cos(mφ) k=0 K (2k+1) x k Q k m (μ) 1 1 C m (t, μ ) Q k m ( μ )d μ .
L + a ( τ 0 )= Λ 4πμ 0 τ 0 e ( τ 0 t) /μ x( l ^ , l ^ ) L a (t, l ^ )d l ^ dt = Λ μ 0 4π k=0 K (2k+1) x k μ 0 μ d k ( e d k τ 0 / μ 0 e τ 0 /μ ) P k ( l ^ l ^ 0 ) ,
L a (0)= Λ 4πμ 0 τ 0 e t/μ x( l ^ , l ^ ) L a (t, l ^ )d l ^ dt = Λ μ 0 4π k=0 K (2k+1) x k μ 0 μ d k ( 1 e (1/μ d k / μ 0 ) τ 0 ) P k ( l ^ l ^ 0 ) .
S m (τ)[ S m (τ) S + m (τ) ]= M 1 A W [ C (τ) C + (τ) ]= M 1 A W C (τ),
S m (τ)= X { [ e Γ ( τ 0 τ) 0 0 e Γ + τ ] G ( τ 0 )+ F (τ) }, X M 1 A W U ,
[ L ˜ (0) L ˜ + ( τ 0 ) ]= m=0 M (2 δ 0m )cos(mφ)diag( 1 , e τ 0 / μ i ) 0 τ 0 diag( e t/ μ i ) S m (t)dt .
0 τ 0 diag( e t/ μ i ) S m (t)dt = 0 τ 0 diag( e t/ μ i ) X [ e Γ ( τ 0 t) 0 0 e Γ + t ]dt G ( τ 0 )+ 0 τ 0 diag( e t/ μ i ) X F (t)dt .
diag( e t/ μ i ) X [ e Γ ( τ 0 t) 0 0 e Γ + t ]=( X [ e (1/ μ i γ j )t ] )[ e Γ τ 0 0 0 1 ],
diag( e t/ μ i ) X F (t)= k=m K X [ e t/ μ i ( d k e d k t/ μ 0 γ j μ 0 d k e t/ μ 0 γ j μ 0 1 ) ] f k ,
[ L ˜ (0) L ˜ + ( τ 0 ) ]= m=0 M (2 δ 0m )cos(mφ)( P m (1) ( τ 0 )+ k=m K P k (2) ( τ 0 ) ) ,
P m (1) ( τ 0 )=( X [ e τ 0 / μ i + e γ j + τ 0 1/ μ i + γ j + e (1/ μ i + + γ j + ) τ 0 1 1/ μ i + + γ j + e (1/ μ i + + γ j + ) τ 0 1 1/ μ i + + γ j + e τ 0 / μ i + e γ j + τ 0 1/ μ i + γ j + ] ) G ( τ 0 ),
P k (2) ( τ 0 )= μ 0 ( X [ d k ( e ( μ 0 / μ i d k ) τ 0 / μ 0 1 ) ( γ j μ 0 d k ) ( μ 0 / μ i d k ) + e ( μ 0 / μ i 1) τ 0 / μ 0 1 ( γ j μ 0 1)( μ 0 / μ i 1) d k ( e d k τ 0 / μ 0 e τ 0 / μ i ) ( γ j μ 0 d k ) ( μ 0 / μ i d k ) e τ 0 / μ 0 e τ 0 / μ i ( γ j μ 0 1)( μ 0 / μ i 1) ] ) f k , Γ ± =[ γ j ± ].
{ Ω μ L ˜ (τ, l ^ , l ^ 0 ) τ d l ^ + Ω L ˜ (τ, l ^ , l ^ 0 )d l ^ = Λ 4π Ω x( l ^ , l ^ ) L ˜ (τ, l ^ , l ^ 0 )d l ^ d l ^ + Ω Δ(τ, l ^ , l ^ 0 )d l ^ , Ω + μ L ˜ (τ, l ^ , l ^ 0 ) τ d l ^ + Ω + L ˜ (τ, l ^ , l ^ 0 )d l ^ = Λ 4π Ω + x( l ^ , l ^ ) L ˜ (τ, l ^ , l ^ 0 )d l ^ d l ^ + Ω + Δ(τ, l ^ , l ^ 0 )d l ^ ,
E ˜ (τ) Ω ± L ˜ (τ,μ)d l ^ , Δ (τ) Ω ± Δ(τ, l ^ , l ^ 0 )d l ^ , μ = 1 2π Ω ± μd l ^ ,
β c 1 4π Ω ± x( l ^ , l ^ )d l ^ | l ^ Ω ± , β o 1 4π Ω ± x( l ^ , l ^ )d l ^ | l ^ Ω
M d E (τ) dτ = A E (τ)+ Δ (τ),
E (τ)[ E ˜ (τ) E ˜ (τ) ], M diag( μ , μ ), A =[ 1Λ β c Λ β o Λ β o 1Λ β c ] 1 Λ[ β c β o β o β c ], Δ (τ)=[ Δ (τ) Δ (τ) ].
E ˜ (τ) | τ=0 =0, E ˜ (τ) | τ= τ 0 = Ω L a ( τ 0 ,μ,φ)d l ^ E a ( τ 0 )
[ u 11 e γ 1 τ 0 u 12 u 21 e γ 2 τ 0 u 22 ][ E ˜ (0) E ˜ ( τ 0 ) ]= S (0, τ 0 )+[ e γ 1 τ 0 u 11 e γ 2 τ 0 u 21 ] E a ,
E (τ)= U ( e Γ τ U 1 [ E ˜ (0) 0 ]+ e Γ τ S (0,τ) ), E (τ)= U ( e Γ (τ τ 0 ) U 1 [ E a E ˜ ( τ 0 ) ] e Γ τ S (τ, τ 0 ) ),
L(τ, l ^ )= L a (τ, l ^ 0 , l ^ )+ 1 2π E ˜ (τ)θ(μ)+ 1 2π E ˜ (τ)θ(μ),θ(μ)={ 1,μ0, 0,μ<0.
L ˜ ( τ 0 ,μ)= Λ e τ 0 /μ 4πμ [ 0 τ 0 ( E ˜ (τ)+ E ˜ (τ) ) e t/μ dt +l(μ) 0 τ 0 ( E ˜ (τ) E ˜ (τ) ) e t/μ dt ],
L ˜ (0,μ)= Λ 4πμ [ 0 τ 0 ( E ˜ (τ)+ E ˜ (τ) ) e t/μ dt +l(μ) 0 τ 0 ( E ˜ (τ) E ˜ (τ) ) e t/μ dt ],
l(μ)= j=1 (K+1)/2 (4j1) x 2j1 P 2j1 (μ) α j , α j 0 1 P 2j1 (μ)dμ , α j+1 = 2j1 2(j+1) α j , α 1 =0.5.

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