Abstract

Case’s method obtains solutions to the radiative transport equation as superpositions of elementary solutions when the specific intensity depends on one spatial variable. In this paper, we find elementary solutions when the specific intensity depends on three spatial variables in three-dimensional space. By using the reference frame whose z axis lies in the direction of the wave vector, the angular part of each elementary solution becomes the singular eigenfunction for the one-dimensional radiative transport equation. Thus, Case’s method is generalized.

© 2013 Optical Society of America

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References

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  1. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  2. J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).
  3. G. I. Marchuk and V. I. Lebedev, Numerical Methods in the Theory of Neutron Transport (Harwood Academic, 1986).
  4. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  5. S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
    [CrossRef]
  6. K. M. Case, “Elementary solutions of the transport equation and their applications,” Ann. Phys. 9, 1–23 (1960).
    [CrossRef]
  7. N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7, 2036–2045 (1966).
    [CrossRef]
  8. J. R. Mika, “Neutron transport with anisotropic scattering,” Nucl. Sci. Eng. 11, 415–427 (1961).
  9. M. M. R. Williams, “Diffusion length and criticality problems in two- and three-dimensional, one-speed neutron transport theory. I. Rectangular coordinates,” J. Math. Phys. 9, 1873–1885 (1968).
    [CrossRef]
  10. H. G. Kaper, “Elementary solutions of the reduced three-dimensional transport equation,” J. Math. Phys. 10, 286–297 (1969).
    [CrossRef]
  11. A. G. Gibbs, “Analytical solutions of the neutron transport equation in arbitrary convex geometry,” J. Math. Phys. 10, 875–890 (1969).
    [CrossRef]
  12. G. Garrettson and A. Leonard, “Green’s functions for multidimensional neutron transport in a slab,” J. Math. Phys. 11, 725–740 (1970).
    [CrossRef]
  13. K. M. Case and R. D. Hazeltine, “Three-dimensional linear transport theory,” J. Math. Phys. 11, 1126–1135 (1970).
    [CrossRef]
  14. C. J. Cannon, “An exact solution to the multi-dimensional line transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 13, 1011–1021 (1973).
    [CrossRef]
  15. G. C. Pomraning, “(Weakly) three-dimensional caseology,” Ann. Nucl. Energy 23, 413–427 (1996).
    [CrossRef]
  16. V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14, L13–L19 (2004).
    [CrossRef]
  17. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  18. A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92–98 (2003).
    [CrossRef]
  19. A. D. Kim, “Transport theory for light propagation in biological tissue,” J. Opt. Soc. Am. A 21, 820–827 (2004).
    [CrossRef]
  20. E. Inönü, “Orthogonality of a set of polynomials encountered in neutron-transport and radiative-transfer theories,” J. Math. Phys. 11, 568–577 (1970).
    [CrossRef]
  21. G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A 39, 115–137 (2006).
    [CrossRef]
  22. J. C. Schotland and V. A. Markel, “Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation,” Inverse Probl. Imaging 1, 181–188 (2007).
    [CrossRef]
  23. M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A 43, 065402 (2010).
    [CrossRef]
  24. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16, 5907–5925 (2008).
    [CrossRef]
  25. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, 1988).
  26. H. Hurwitz and P. F. Zweifel, “Slowing down of neutrons by hydrogenous moderators,” J. Appl. Phys. 26, 923–931 (1955).
    [CrossRef]

2010 (1)

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A 43, 065402 (2010).
[CrossRef]

2009 (1)

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

2008 (1)

2007 (1)

J. C. Schotland and V. A. Markel, “Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation,” Inverse Probl. Imaging 1, 181–188 (2007).
[CrossRef]

2006 (1)

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A 39, 115–137 (2006).
[CrossRef]

2004 (2)

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

A. D. Kim, “Transport theory for light propagation in biological tissue,” J. Opt. Soc. Am. A 21, 820–827 (2004).
[CrossRef]

2003 (1)

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

1996 (1)

G. C. Pomraning, “(Weakly) three-dimensional caseology,” Ann. Nucl. Energy 23, 413–427 (1996).
[CrossRef]

1973 (1)

C. J. Cannon, “An exact solution to the multi-dimensional line transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 13, 1011–1021 (1973).
[CrossRef]

1970 (3)

G. Garrettson and A. Leonard, “Green’s functions for multidimensional neutron transport in a slab,” J. Math. Phys. 11, 725–740 (1970).
[CrossRef]

K. M. Case and R. D. Hazeltine, “Three-dimensional linear transport theory,” J. Math. Phys. 11, 1126–1135 (1970).
[CrossRef]

E. Inönü, “Orthogonality of a set of polynomials encountered in neutron-transport and radiative-transfer theories,” J. Math. Phys. 11, 568–577 (1970).
[CrossRef]

1969 (2)

H. G. Kaper, “Elementary solutions of the reduced three-dimensional transport equation,” J. Math. Phys. 10, 286–297 (1969).
[CrossRef]

A. G. Gibbs, “Analytical solutions of the neutron transport equation in arbitrary convex geometry,” J. Math. Phys. 10, 875–890 (1969).
[CrossRef]

1968 (1)

M. M. R. Williams, “Diffusion length and criticality problems in two- and three-dimensional, one-speed neutron transport theory. I. Rectangular coordinates,” J. Math. Phys. 9, 1873–1885 (1968).
[CrossRef]

1966 (1)

N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7, 2036–2045 (1966).
[CrossRef]

1961 (1)

J. R. Mika, “Neutron transport with anisotropic scattering,” Nucl. Sci. Eng. 11, 415–427 (1961).

1960 (1)

K. M. Case, “Elementary solutions of the transport equation and their applications,” Ann. Phys. 9, 1–23 (1960).
[CrossRef]

1955 (1)

H. Hurwitz and P. F. Zweifel, “Slowing down of neutrons by hydrogenous moderators,” J. Appl. Phys. 26, 923–931 (1955).
[CrossRef]

1941 (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Arridge, S. R.

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

Cannon, C. J.

C. J. Cannon, “An exact solution to the multi-dimensional line transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 13, 1011–1021 (1973).
[CrossRef]

Case, K. M.

K. M. Case and R. D. Hazeltine, “Three-dimensional linear transport theory,” J. Math. Phys. 11, 1126–1135 (1970).
[CrossRef]

K. M. Case, “Elementary solutions of the transport equation and their applications,” Ann. Phys. 9, 1–23 (1960).
[CrossRef]

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

Duderstadt, J. J.

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

Foschum, F.

Garrettson, G.

G. Garrettson and A. Leonard, “Green’s functions for multidimensional neutron transport in a slab,” J. Math. Phys. 11, 725–740 (1970).
[CrossRef]

Gibbs, A. G.

A. G. Gibbs, “Analytical solutions of the neutron transport equation in arbitrary convex geometry,” J. Math. Phys. 10, 875–890 (1969).
[CrossRef]

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hazeltine, R. D.

K. M. Case and R. D. Hazeltine, “Three-dimensional linear transport theory,” J. Math. Phys. 11, 1126–1135 (1970).
[CrossRef]

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hurwitz, H.

H. Hurwitz and P. F. Zweifel, “Slowing down of neutrons by hydrogenous moderators,” J. Appl. Phys. 26, 923–931 (1955).
[CrossRef]

Inönü, E.

E. Inönü, “Orthogonality of a set of polynomials encountered in neutron-transport and radiative-transfer theories,” J. Math. Phys. 11, 568–577 (1970).
[CrossRef]

Kaper, H. G.

H. G. Kaper, “Elementary solutions of the reduced three-dimensional transport equation,” J. Math. Phys. 10, 286–297 (1969).
[CrossRef]

Keller, J. B.

Khersonskii, V. K.

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, 1988).

Kienle, A.

Kim, A. D.

Kušcer, I.

N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7, 2036–2045 (1966).
[CrossRef]

Lebedev, V. I.

G. I. Marchuk and V. I. Lebedev, Numerical Methods in the Theory of Neutron Transport (Harwood Academic, 1986).

Leonard, A.

G. Garrettson and A. Leonard, “Green’s functions for multidimensional neutron transport in a slab,” J. Math. Phys. 11, 725–740 (1970).
[CrossRef]

Machida, M.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A 43, 065402 (2010).
[CrossRef]

Marchuk, G. I.

G. I. Marchuk and V. I. Lebedev, Numerical Methods in the Theory of Neutron Transport (Harwood Academic, 1986).

Markel, V. A.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A 43, 065402 (2010).
[CrossRef]

J. C. Schotland and V. A. Markel, “Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation,” Inverse Probl. Imaging 1, 181–188 (2007).
[CrossRef]

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A 39, 115–137 (2006).
[CrossRef]

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

Martin, W. R.

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

McCormick, N. J.

N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7, 2036–2045 (1966).
[CrossRef]

Michels, R.

Mika, J. R.

J. R. Mika, “Neutron transport with anisotropic scattering,” Nucl. Sci. Eng. 11, 415–427 (1961).

Moskalev, A. N.

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, 1988).

Panasyuk, G.

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A 39, 115–137 (2006).
[CrossRef]

Panasyuk, G. Y.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A 43, 065402 (2010).
[CrossRef]

Pomraning, G. C.

G. C. Pomraning, “(Weakly) three-dimensional caseology,” Ann. Nucl. Energy 23, 413–427 (1996).
[CrossRef]

Schotland, J. C.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A 43, 065402 (2010).
[CrossRef]

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

J. C. Schotland and V. A. Markel, “Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation,” Inverse Probl. Imaging 1, 181–188 (2007).
[CrossRef]

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A 39, 115–137 (2006).
[CrossRef]

Varshalovich, D. A.

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, 1988).

Williams, M. M. R.

M. M. R. Williams, “Diffusion length and criticality problems in two- and three-dimensional, one-speed neutron transport theory. I. Rectangular coordinates,” J. Math. Phys. 9, 1873–1885 (1968).
[CrossRef]

Zweifel, P. F.

H. Hurwitz and P. F. Zweifel, “Slowing down of neutrons by hydrogenous moderators,” J. Appl. Phys. 26, 923–931 (1955).
[CrossRef]

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

Ann. Nucl. Energy (1)

G. C. Pomraning, “(Weakly) three-dimensional caseology,” Ann. Nucl. Energy 23, 413–427 (1996).
[CrossRef]

Ann. Phys. (1)

K. M. Case, “Elementary solutions of the transport equation and their applications,” Ann. Phys. 9, 1–23 (1960).
[CrossRef]

Astrophys. J. (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Inverse Probl. (2)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

Inverse Probl. Imaging (1)

J. C. Schotland and V. A. Markel, “Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation,” Inverse Probl. Imaging 1, 181–188 (2007).
[CrossRef]

J. Appl. Phys. (1)

H. Hurwitz and P. F. Zweifel, “Slowing down of neutrons by hydrogenous moderators,” J. Appl. Phys. 26, 923–931 (1955).
[CrossRef]

J. Math. Phys. (7)

E. Inönü, “Orthogonality of a set of polynomials encountered in neutron-transport and radiative-transfer theories,” J. Math. Phys. 11, 568–577 (1970).
[CrossRef]

N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7, 2036–2045 (1966).
[CrossRef]

M. M. R. Williams, “Diffusion length and criticality problems in two- and three-dimensional, one-speed neutron transport theory. I. Rectangular coordinates,” J. Math. Phys. 9, 1873–1885 (1968).
[CrossRef]

H. G. Kaper, “Elementary solutions of the reduced three-dimensional transport equation,” J. Math. Phys. 10, 286–297 (1969).
[CrossRef]

A. G. Gibbs, “Analytical solutions of the neutron transport equation in arbitrary convex geometry,” J. Math. Phys. 10, 875–890 (1969).
[CrossRef]

G. Garrettson and A. Leonard, “Green’s functions for multidimensional neutron transport in a slab,” J. Math. Phys. 11, 725–740 (1970).
[CrossRef]

K. M. Case and R. D. Hazeltine, “Three-dimensional linear transport theory,” J. Math. Phys. 11, 1126–1135 (1970).
[CrossRef]

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

J. Phys. A (2)

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A 39, 115–137 (2006).
[CrossRef]

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A 43, 065402 (2010).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

C. J. Cannon, “An exact solution to the multi-dimensional line transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 13, 1011–1021 (1973).
[CrossRef]

Nucl. Sci. Eng. (1)

J. R. Mika, “Neutron transport with anisotropic scattering,” Nucl. Sci. Eng. 11, 415–427 (1961).

Opt. Express (1)

Waves Random Media (1)

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

Other (4)

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, 1988).

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

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

G. I. Marchuk and V. I. Lebedev, Numerical Methods in the Theory of Neutron Transport (Harwood Academic, 1986).

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Figures (3)

Fig. 1.
Fig. 1.

Energy density U(z) for (a) the point source and (b) the source of length on the x axis.

Fig. 2.
Fig. 2.

Energy density Eq. (62) is plotted together with Eq. (66) and results from Monte Carlo simulation. The optical parameters (μa,μs,f1) are, from the top, (0.03cm1,100cm1,0), (0.03cm1,100cm1,0.3), and (0.3cm1,100cm1,0.3).

Fig. 3.
Fig. 3.

Energy density Eq. (68) is plotted. The optical parameters (μa,μs,f1) are, from the top, (0.03cm1,100cm1,0), (0.03cm1,100cm1,0.3), and (0.3cm1,100cm1,0.3). The blue line from Fig. 2 shows Eq. (62) for (μa,μs,f1)=(0.03cm1,100cm1,0.3).

Equations (81)

Equations on this page are rendered with MathJax. Learn more.

s^·I(r,s^)+(μa+μs)I(r,s^)=μsS2f(s^·s^)I(r,s^)ds^+S(r,s^),
f(s^·s^)=l=0Nm=llflYlm(s^)Ylm*(s^).
S2f(s^·s^)ds^=1.
s^·r˜I(r˜/μt,s^)+I(r˜/μt,s^)=cS2f(s^·s^)I(r˜/μt,s^)ds^+1μtS(r˜/μt,s^),
I˜(r˜,s^)=I(r˜/μt,s^),
s^·r˜I˜(r˜,s^)+I˜(r˜,s^)=cS2f(s^·s^)I˜(r˜,s^)ds^+1μtS(r˜/μt,s^).
s^·I(r,s^)+I(r,s^)=cS2f(s^·s^)I(r,s^)ds^.
f(s^·s^)=l=0Nm=llfl2l+14π(lm)!(l+m)!(1μ2)|m|/2×(1μ2)|m|/2plm(μ)plm(μ)eim(φφ).
Plm(μ)=(1)m(1μ2)|m|/2plm(μ).
(lm+1)pl+1m(μ)=(2l+1)μplm(μ)(l+m)pl1m(μ),
11plm(μ)plm(μ)dm(μ)=2(l+m)!(2l+1)(lm)!δll,
dm(μ)=(1μ2)|m|dμ.
p|m|m(μ)={2mm!2mm!form0,(1)m2|m|(|m|!)form<0.
k=1νk^,k^=(iνqQ(νq)),Q(νq)=1+(νq)2,
Iνm(r,s^;q)=Φνm(s^;k^)ek·r,
Φνm(s^;k^)=ϕm(ν,μ(k^))(1μ(k^)2)|m|/2eimφ(k^).
12πS2ϕm(ν,μ(k^))(1μ(k^)2)|m|ds^=11ϕm(ν,μ)dm(μ)=1.
(1μ(k^)ν)ϕm(ν,μ(k^))(1μ(k^)2)|m|/2eimφ(k^)=cS2f(s^(k^)·s^(k^))ϕm(ν,μ(k^))(1μ(k^)2)|m|/2eimφ(k^)ds^,
RHS=2πcΘ(N|m|)(1μ(k^)2)|m|/2eimφ(k^)×l=|m|Nfl2l+14π(lm)!(l+m)!×plm(μ(k^))11plm(μ)ϕm(ν,μ)dm(μ),
(νμ(k^))ϕm(ν,μ(k^))=2πcνΘ(N|m|)×l=|m|Nfl2l+14π(lm)!(l+m)!plm(μ(k^))hlm(ν),
hlm(ν)=11ϕm(ν,μ)plm(μ)dm(μ).
0|m|N.
σl=1cflΘ(Nl).
σlνhlm(ν)=11μϕm(ν,μ)plm(μ)dm(μ).
ν(2l+1)σlhlm(ν)(lm+1)hl+1m(ν)(l+m)hl1m(ν)=0
h|m|m(ν)=p|m|m
h|m|+1|m|(ν)=(2|m|+1)νσ|m|h|m||m|(ν).
hl|m|(ν)=(1)|m|(l|m|)!(l+|m|)!hl|m|(ν).
gm(ν,μ(k^))=l=|m|N(2l+1)fl(lm)!(l+m)!plm(μ(k^))hlm(ν).
ϕm(ν,μ(k^))=cν2Pgm(ν,μ(k^))νμ(k^)+λm(ν)(1ν2)|m|δ(νμ(k^)),
1=cν2P11gm(ν,μ)νμdm(μ)+11λm(ν)δ(νμ)dμ.
λm(ν)=1cν2P11gm(ν,μ)νμdm(μ).
Λm(z)=1cz211gm(z,μ)zμdm(μ),
Λm(ν)=0.
Φνm(s^;k^)=l=|m|clm(ν)Ylm(s^;k^).
clm(ν)1νl=|m|(S2μYlm(s^)Ylm*(s^)ds^)clm(ν)=cflΘ(Nl)clm(ν).
Bm|ψm(ν)=ν|ψm(ν),
Bllm=1σlσlS2μYlm(s^)Ylm*(s^)ds^=l2m2(4l21)σlσl1δl,l1+(l+1)2m2(4(l+1)21)σl+1σlδl,l+1,
l|ψm(ν)=1Zm(ν)σlclm(ν),
Φνm(s^;k^)=Zm(ν)l=|m|l|ψm(ν)σlYlm(s^;k^).
s^·G(r,s^;r0,s^0)+G(r,s^;r0,s^0)=cS2f(s^·s^)G(r,s^;r0,s^0)ds^+δ(rr0)δ(s^s^0).
S2μΦνm(s^;k^)[Φ˜νm(s^;k^)]*ds^=δνν,
{s^·G(r,s^;r0,s^0)+G(r,s^;r0,s^0)=cS2f(s^·s^)G(r,s^;r0,s^0)ds^,G(ρ,z0+0,s^;r0,s^0)G(ρ,z00,s^;r0,s^0)=1s^·z^δ(ρρ0)δ(s^s^0),
{G(ρ,z,s^;ρ0,z0,s^0)=m=NNR2[j=0M1aj+m(q)Ij+m(r,s^;q)+01Aνm(q)Iνm(r,s^;q)dν]dq(2π)2,z>z0,G(ρ,z,s^;ρ0,z0,s^0)=m=NNR2[j=0M1ajm(q)Ijm(r,s^;q)+01Aνm(q)Iνm(r,s^;q)dν]dq(2π)2,z<z0,.
aj±m(q)=eiq·ρ0e±Q(νjmq)z0/νjm[Φ˜j±m(s^0;k^)]*,Aνm(q)=eiq·ρ0eQ(νq)z0/ν[Φ˜νm(s^0;k^)]*.
G(ρ,z,s^;ρ0,z0,s^0)=±1(2π)2R2eiq·(ρρ0)m=NN{j=0M1Φj±m(s^;k^)[Φ˜j±m(s^0;k^)]*eQ(νjmq)|zz0|/νjm+01Φ±νm(s^;k^)[Φ˜±νm(s^0;k^)]*eQ(νq)|zz0|/νdν}dq,
G(r,s^;r0,s^0)=1(2π)2R2eiq·(ρρ0)ν>0m=NN1νQ(νq)Zm(ν)Φ±νm(s^;k^)[Φ±νm(s^0;k^)]*eQ(νq)|zz0|/νdq,
Φ˜±νm(s^;k^)=Φ±νm(s^;k^)[±νQ(νq)Zm(ν)]1.
G(z,s^;z0,s^0)=±m=NN{j=0M1Φj±m(s^;z^)[Φ˜j±m(s^0;z^)]*e|zz0|/νjm+01Φ±νm(s^;z^)[Φ˜±νm(s^0;z^)]*e|zz0|/νdν}.
G(z,s^;z0,s^0)=12πm=NN{j=0M11Njmϕm(±νjm,μ)ϕm(±νjm,μ0)(1μ2)|m|eim(φφ0)e|zz0|/νjm+011Nm(ν)ϕm(±ν,μ)ϕm(±ν,μ0)(1μ2)|m|eim(φφ0)e|zz0|/νdν},
Njm=Nm(νjm)=c2(νjm)2g(νjm,νjm)dΛm(z)dz|z=νjm,
Nm(ν)=νΛm+(ν)Λm(ν)(1ν2)|m|.
νjmZm(νjm)=2πNjm,νZm(ν)=2πNm(ν),
Φ˜νm(s^;k^)=Φνm(s^;k^)[2πQ(νq)Nm(ν)]1,
G(ρ,z,s^;ρ0,z0,s^0)=1(2π)3R2eiq·(ρρ0)m=NN{j=0M11Q(νjmq)NjmΦj±m(s^;k^)[Φj±m(s^0;k^)]*eQ(νjmq)|zz0|/νjm+011Q(νq)Nm(ν)Φ±νm(s^;k^)[Φ±νm(s^0;k^)]*eQ(νq)|zz0|/νdν}dq.
Φν0(s^;k^)=cν2P1νμ(k^)+λ0(ν)δ(νμ(k^))=ϕ0(ν,μ(k^)),
λ0(ν)=1cν2P111νμdμ=1cνtanh1ν.
Λ0(z)=1cz2111zμdμ=1cztanh11z=0.
G(ρ,z,s^;ρ0,z0,s^0)=1(2π)3R2eiq·(ρρ0){1Q(ν0q)N0ϕ0(±ν0,μ(k^))ϕ0*(±ν0,μ0(k^))eQ(ν0q)|zz0|/ν0+011Q(νq)N(ν)ϕ0(±ν,μ(k^))ϕ0*(±ν,μ0(k^))eQ(νq)|zz0|/νdν}dq.
U(z)=1vS2I(ρ=0,z,s^)ds^,
I(0,z,s^)=μt2SaR3×S2G(0,z,s^;ρ0,z0,s^0)×δ(ρ0)δ(z0)dρ0dz0ds^0,
U(z)μt2Sa=S2×S2G(0,z,s^;0,0,s^0)ds^ds^0=1vz[ez/ν0ν0N0+01ez/ννN(ν)dν],z>0.
Λ0(ν0)=1cν0211g0(ν0,μ)ν0μdμ.
G(r,s^;r0,s^0)=G0(r,s^;r0,s^0)+c4π(2π)3R3eik·(rr0)×1(1+ik·s^)(1+ik·s^0)[1c|k|tan1(|k|)]1dk,
G0(r,s^;r0,s^0)=e|rr0||rr0|2δ(s^rr0|rr0|)δ(s^s^0).
U(z)μt2Sa=e|z|vz2+2cπv0sin(kz)z(tan1k)2kctan1kdk.
I(0,z,s^)=μtSbR3×S2G(0,z,s^;x0,y0,z0,s^0)×Θ(μtx0)Θ(x0)δ(y0)δ(z0)dx0dy0dz0ds^0,
U(z)μtSb=R2S2×S2G(0,z,s^;ρ0,0,s^0)×Θ(μtx0)Θ(x0)δ(y0)ds^ds^0dρ0=1v0(0μtqJ0(t)dt)[eQ(ν0q)z/ν0Q(ν0q)N0+01eQ(νq)z/νQ(νq)N(ν)dν]dq,z>0,
U(z)μtSb=1v0μt[ex02+z2x02+z2+2cπ0sin(kx02+z2)x02+z2(tan1k)2kctan1kdk]dx0.
cosθk^=k^·z^=Q(νq),sinθk^=1cos2θk^=i|νq|
φk^={φq+πforν>0,φqforν<0,
μ(k^)=s^·k^=iνqsinθcos(φφq)+Q(νq)cosθ.
Ylm(s^;k^)=D(k^)Ylm(s^)=m=llDmml(φk^,θk^,0)Ylm(s^),
dmml(θk^)=dmml[iτ(νq)].
d000=1,
d001=1+x2,d011=i2|x|,d1±11=1±1+x22.
eimφ(k^)=(1μ(k^)2)|m|/2(1)m4π(2m+1)!(2m+1)!!×m=mmeimφk^dmmm(θk^)Ymm(s^),
clm(ν)=S2[cν2Pgm(ν,μ)νμ+λm(ν)(1ν2)|m|δ(νμ)]×(1μ2)|m|/2eimφYlm*(s^)ds^.
clm(ν)=2π2l+14π(lm)!(l+m)![cν2l=|m|Nfl(2l+1)×(lm)!(l+m)!hlm(ν)(1)mP11Plm(μ)Plm(μ)νμdμ+λm(ν)(1ν2)|m|/2Plm(ν)11δ(νμ)dμ].
clm(ν)=2π2l+14π(lm)!(l+m)!cν2l=|m|Nfl(2l+1)×(lm)!(l+m)!hlm(ν)(1)m2Qmax(l,l)m(ν)Pmin(l,l)m(ν),
clm(ν)=2π2l+14π(lm)!(l+m)![cν(1)m2l=|m|Nfl(2l+1)×(lm)!(l+m)!hlm(ν)×(iπPlm(ν)Plm(ν)0πPlm(eiθ)Plm(eiθ)νeiθieiθdθ)+λm(ν)(1ν2)|m|/2Plm(ν)].

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