Abstract

This paper is the first of two deriving the analytical solutions for light transport in infinite homogeneous tissue with an azimuth-dependent (m-dependent) anisotropic scattering kernel by two approaches, Case’s singular eigenfuncions expansion and Fourier transform, as well as proving the consistence of the two solutions. In this paper, Case’s method was applied and extended to the general m-dependent anisotropic scattering case. The explicit Green’s function of radiance distributions, which was regarded as the comparative standard for the equivalent solution via Fourier transform and inversion in our second accompanying paper, was expanded into a complete set of the discrete and continuous eigenfunctions. Considering that the two kinds of m-dependent Chandrasekhar orthogonal polynomials that play vital roles in these analytical solutions are very sensitive to the typical optical parameters of biological tissue as well as the degrees or orders, four numerical evaluation methods were benchmarked to find the stable, reliable and feasible numerical evaluation methods in high degrees and high orders.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. 18(1), 247–260 (2007).
    [Crossref]
  2. P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230(19), 7364–7383 (2011).
    [Crossref]
  3. A. D. Klose and E. W. Larsen, “Simplified spherical harmonics methods for modeling light transport in biological tissue,” in Biomedical Optics, Technical Digest (CD) (Optical Society of America, 2006), paper MH3.
    [Crossref]
  4. A. Liemert and A. Kienle, “Analytical solutions of the simplified spherical harmonics equations,” Opt. Lett. 35(20), 3507–3509 (2010).
    [Crossref] [PubMed]
  5. H. Zheng and W. Han, “On simplified spherical harmonics equations for the radiative transfer equation,” J. Math. Chem. 49(8), 1785–1797 (2011).
    [Crossref]
  6. A. Liemert and A. Kienle, “Comparison between radiative transfer theory and the simplified spherical harmonics approximation for a semi-infinite geometry,” Opt. Lett. 36(20), 4041–4043 (2011).
    [Crossref] [PubMed]
  7. M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “Diffusion approximation revisited,” J. Opt. Soc. Am. A 26(5), 1291–1300 (2009).
    [Crossref] [PubMed]
  8. A. Liemert and A. Kienle, “Light diffusion in a turbid cylinder. I. Homogeneous case,” Opt. Express 18(9), 9456–9473 (2010).
    [Crossref] [PubMed]
  9. A. Liemert and A. Kienle, “Light diffusion in a turbid cylinder. II. Layered case,” Opt. Express 18(9), 9266–9279 (2010).
    [Crossref] [PubMed]
  10. A. Kienle and A. Liemert, “Light diffusion in turbid media of different geometries in the steady-state, frequency, and time domains,” in Biomedical Optics and 3-D Imaging, OSA Technical Digest (CD) (Optical Society of America, 2010), paper BSuD41.
    [Crossref]
  11. C. Devaux and C. E. Siewert, “The FN method for radiative transfer problems without azimuthal symmetry,” Z. Angew. Math. Phys. 31(5), 592–604 (1980).
    [Crossref]
  12. M. Machida, “An FN method for the radiative transport equation in three dimensions,” J. Phys. A Math. Theor. 48(32), 325001 (2015).
    [Crossref]
  13. K. M. Case, “Elementary solutions of the transport equation and their applications,” Ann. Phys. 9(1), 1–23 (1960).
    [Crossref]
  14. N. J. Mccormick and N. Joseph, “One-speed neutron transport problems in plane geometry,” Univ. of Michigan Thesis, (1965).
  15. N. J. McCormick and I. Kuščer, “Half-space neutron transport with linearly anisotropic scattering,” J. Math. Phys. 6(12), 1939–1945 (1965).
    [Crossref]
  16. J. R. Mika, “Neutron transport with anisotropic scattering,” Nucl. Sci. Eng. 11(4), 415–427 (1961).
    [Crossref]
  17. G. Van den Eynde, “Neutron transport with anisotropic scattering theory and applications,” Thesis, (2005).
  18. N. V. Sultanov, “Elementary solution of the neutron transport equation with anisotropic scattering,” Sov. Atom Energ. 32(6), 539–546 (1972).
    [Crossref]
  19. A. D. Kim, “Transport theory for light propagation in tissues,” in Biomedical Topical Meeting, OSA Technical Digest (Optical Society of America, 2004), paper SD4.
  20. A. Liemert and A. Kienle, “Light transport in three-dimensional semi-infinite scattering media,” J. Opt. Soc. Am. A 29(7), 1475–1481 (2012).
    [Crossref] [PubMed]
  21. M. Machida, “Singular eigenfunctions for the three-dimensional radiative transport equation,” J. Opt. Soc. Am. A 31(1), 67–74 (2014).
    [Crossref] [PubMed]
  22. K. M. Case and R. D. Hazeltine, “Fourier transform methods in linear transport theory,” J. Math. Phys. 12(9), 1970–1980 (1971).
    [Crossref]
  23. B. D. Ganapol, “A consistent theory of neutral particle transport in an infinite medium,” Transp. Theory Stat. Phys. 29(1–2), 43–68 (2000).
    [Crossref]
  24. B. D. Ganapol, “Fourier transform transport solutions in spherical geometry,” Transp. Theory Stat. Phys. 32(5–7), 587–605 (2003).
    [Crossref]
  25. B. D. Ganapol, “The Fourier transform solution for the Green’s function of monoenergetic neutron transport theory,” in 2014 Hawaii University International Conferences on Science, Technology, Engineering, Math & Education (2014).
  26. B. D. Ganapol, “The infinite medium Green’s function of monoenergetic neutron transport theory via Fourier transform,” Nucl. Sci. Eng. 180(2), 224–246 (2015).
    [Crossref]
  27. Inönü and Erdal, “Orthogonality of a set of polynomials encountered in neutron transport and radiative transfer theories,” J. Math. Phys. 11(2), 568–577 (1970).
    [Crossref]
  28. K. M. Case, “Orthogonal polynomials from the viewpoint of scattering theory,” J. Math. Phys. 15(12), 2166–2174 (1974).
    [Crossref]
  29. K. M. Case, “Scattering theory, orthogonal polynomials, and the transport equation,” J. Math. Phys. 15(7), 974–983 (1974).
    [Crossref]
  30. R. D. M. Garcia and C. E. Siewert, “On discrete spectrum calculations in radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 42(5), 385–394 (1989).
    [Crossref]
  31. R. D. M. Garcia and C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transf. 43(3), 201–205 (1990).
    [Crossref]
  32. C. E. Siewert and N. J. Mccormick, “Some identities for Chandrasekhar polynomials,” J. Quant. Spectrosc. Radiat. Transf. 57(3), 399–404 (1997).
    [Crossref]
  33. B. D. Ganapol, “Chandrasekhar polynomials and the solution to the transport equation in an infinite medium,” J. Comput. Theor. Trans. 43(1–7), 433–473 (2014).
    [Crossref]
  34. M. Machida, “The Green’s function for the three-dimensional linear Boltzmann equation via Fourier transform,” J. Phys. A Math. Theor. 49(17), 175001 (2016).
    [Crossref]
  35. A. J. Welch and M. J. C. V. Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Springer Netherlands, 2011).
  36. Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989).
  37. N. I. Muskhelishvili, Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics (Dover Publications, 2011).
  38. D. Zwillinger and V. Moll, Table of Integrals, Series, and Products (Eighth Edition) (Academic Press, 2015).
  39. I. Kuščer and N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” J. Comput. Theor. Trans. 20(5–6), 351–381 (1974).
  40. N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7(11), 2036–2045 (1966).
    [Crossref]

2016 (1)

M. Machida, “The Green’s function for the three-dimensional linear Boltzmann equation via Fourier transform,” J. Phys. A Math. Theor. 49(17), 175001 (2016).
[Crossref]

2015 (2)

B. D. Ganapol, “The infinite medium Green’s function of monoenergetic neutron transport theory via Fourier transform,” Nucl. Sci. Eng. 180(2), 224–246 (2015).
[Crossref]

M. Machida, “An FN method for the radiative transport equation in three dimensions,” J. Phys. A Math. Theor. 48(32), 325001 (2015).
[Crossref]

2014 (2)

M. Machida, “Singular eigenfunctions for the three-dimensional radiative transport equation,” J. Opt. Soc. Am. A 31(1), 67–74 (2014).
[Crossref] [PubMed]

B. D. Ganapol, “Chandrasekhar polynomials and the solution to the transport equation in an infinite medium,” J. Comput. Theor. Trans. 43(1–7), 433–473 (2014).
[Crossref]

2012 (1)

2011 (3)

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230(19), 7364–7383 (2011).
[Crossref]

H. Zheng and W. Han, “On simplified spherical harmonics equations for the radiative transfer equation,” J. Math. Chem. 49(8), 1785–1797 (2011).
[Crossref]

A. Liemert and A. Kienle, “Comparison between radiative transfer theory and the simplified spherical harmonics approximation for a semi-infinite geometry,” Opt. Lett. 36(20), 4041–4043 (2011).
[Crossref] [PubMed]

2010 (3)

2009 (1)

2007 (1)

S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. 18(1), 247–260 (2007).
[Crossref]

2003 (1)

B. D. Ganapol, “Fourier transform transport solutions in spherical geometry,” Transp. Theory Stat. Phys. 32(5–7), 587–605 (2003).
[Crossref]

2000 (1)

B. D. Ganapol, “A consistent theory of neutral particle transport in an infinite medium,” Transp. Theory Stat. Phys. 29(1–2), 43–68 (2000).
[Crossref]

1997 (1)

C. E. Siewert and N. J. Mccormick, “Some identities for Chandrasekhar polynomials,” J. Quant. Spectrosc. Radiat. Transf. 57(3), 399–404 (1997).
[Crossref]

1990 (1)

R. D. M. Garcia and C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transf. 43(3), 201–205 (1990).
[Crossref]

1989 (1)

R. D. M. Garcia and C. E. Siewert, “On discrete spectrum calculations in radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 42(5), 385–394 (1989).
[Crossref]

1980 (1)

C. Devaux and C. E. Siewert, “The FN method for radiative transfer problems without azimuthal symmetry,” Z. Angew. Math. Phys. 31(5), 592–604 (1980).
[Crossref]

1974 (3)

K. M. Case, “Orthogonal polynomials from the viewpoint of scattering theory,” J. Math. Phys. 15(12), 2166–2174 (1974).
[Crossref]

K. M. Case, “Scattering theory, orthogonal polynomials, and the transport equation,” J. Math. Phys. 15(7), 974–983 (1974).
[Crossref]

I. Kuščer and N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” J. Comput. Theor. Trans. 20(5–6), 351–381 (1974).

1972 (1)

N. V. Sultanov, “Elementary solution of the neutron transport equation with anisotropic scattering,” Sov. Atom Energ. 32(6), 539–546 (1972).
[Crossref]

1971 (1)

K. M. Case and R. D. Hazeltine, “Fourier transform methods in linear transport theory,” J. Math. Phys. 12(9), 1970–1980 (1971).
[Crossref]

1970 (1)

Inönü and Erdal, “Orthogonality of a set of polynomials encountered in neutron transport and radiative transfer theories,” J. Math. Phys. 11(2), 568–577 (1970).
[Crossref]

1966 (1)

N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7(11), 2036–2045 (1966).
[Crossref]

1965 (1)

N. J. McCormick and I. Kuščer, “Half-space neutron transport with linearly anisotropic scattering,” J. Math. Phys. 6(12), 1939–1945 (1965).
[Crossref]

1961 (1)

J. R. Mika, “Neutron transport with anisotropic scattering,” Nucl. Sci. Eng. 11(4), 415–427 (1961).
[Crossref]

1960 (1)

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

Arridge, S. R.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230(19), 7364–7383 (2011).
[Crossref]

S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. 18(1), 247–260 (2007).
[Crossref]

Case, K. M.

K. M. Case, “Orthogonal polynomials from the viewpoint of scattering theory,” J. Math. Phys. 15(12), 2166–2174 (1974).
[Crossref]

K. M. Case, “Scattering theory, orthogonal polynomials, and the transport equation,” J. Math. Phys. 15(7), 974–983 (1974).
[Crossref]

K. M. Case and R. D. Hazeltine, “Fourier transform methods in linear transport theory,” J. Math. Phys. 12(9), 1970–1980 (1971).
[Crossref]

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

Devaux, C.

C. Devaux and C. E. Siewert, “The FN method for radiative transfer problems without azimuthal symmetry,” Z. Angew. Math. Phys. 31(5), 592–604 (1980).
[Crossref]

Erdal,

Inönü and Erdal, “Orthogonality of a set of polynomials encountered in neutron transport and radiative transfer theories,” J. Math. Phys. 11(2), 568–577 (1970).
[Crossref]

Ganapol, B. D.

B. D. Ganapol, “The infinite medium Green’s function of monoenergetic neutron transport theory via Fourier transform,” Nucl. Sci. Eng. 180(2), 224–246 (2015).
[Crossref]

B. D. Ganapol, “Chandrasekhar polynomials and the solution to the transport equation in an infinite medium,” J. Comput. Theor. Trans. 43(1–7), 433–473 (2014).
[Crossref]

B. D. Ganapol, “Fourier transform transport solutions in spherical geometry,” Transp. Theory Stat. Phys. 32(5–7), 587–605 (2003).
[Crossref]

B. D. Ganapol, “A consistent theory of neutral particle transport in an infinite medium,” Transp. Theory Stat. Phys. 29(1–2), 43–68 (2000).
[Crossref]

Garcia, R. D. M.

R. D. M. Garcia and C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transf. 43(3), 201–205 (1990).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “On discrete spectrum calculations in radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 42(5), 385–394 (1989).
[Crossref]

Han, W.

H. Zheng and W. Han, “On simplified spherical harmonics equations for the radiative transfer equation,” J. Math. Chem. 49(8), 1785–1797 (2011).
[Crossref]

Hazeltine, R. D.

K. M. Case and R. D. Hazeltine, “Fourier transform methods in linear transport theory,” J. Math. Phys. 12(9), 1970–1980 (1971).
[Crossref]

Inönü,

Inönü and Erdal, “Orthogonality of a set of polynomials encountered in neutron transport and radiative transfer theories,” J. Math. Phys. 11(2), 568–577 (1970).
[Crossref]

Kienle, A.

Kušcer, I.

I. Kuščer and N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” J. Comput. Theor. Trans. 20(5–6), 351–381 (1974).

N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7(11), 2036–2045 (1966).
[Crossref]

N. J. McCormick and I. Kuščer, “Half-space neutron transport with linearly anisotropic scattering,” J. Math. Phys. 6(12), 1939–1945 (1965).
[Crossref]

Liemert, A.

Machida, M.

M. Machida, “The Green’s function for the three-dimensional linear Boltzmann equation via Fourier transform,” J. Phys. A Math. Theor. 49(17), 175001 (2016).
[Crossref]

M. Machida, “An FN method for the radiative transport equation in three dimensions,” J. Phys. A Math. Theor. 48(32), 325001 (2015).
[Crossref]

M. Machida, “Singular eigenfunctions for the three-dimensional radiative transport equation,” J. Opt. Soc. Am. A 31(1), 67–74 (2014).
[Crossref] [PubMed]

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “Diffusion approximation revisited,” J. Opt. Soc. Am. A 26(5), 1291–1300 (2009).
[Crossref] [PubMed]

Markel, V. A.

Mccormick, N. J.

C. E. Siewert and N. J. Mccormick, “Some identities for Chandrasekhar polynomials,” J. Quant. Spectrosc. Radiat. Transf. 57(3), 399–404 (1997).
[Crossref]

I. Kuščer and N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” J. Comput. Theor. Trans. 20(5–6), 351–381 (1974).

N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7(11), 2036–2045 (1966).
[Crossref]

N. J. McCormick and I. Kuščer, “Half-space neutron transport with linearly anisotropic scattering,” J. Math. Phys. 6(12), 1939–1945 (1965).
[Crossref]

Mika, J. R.

J. R. Mika, “Neutron transport with anisotropic scattering,” Nucl. Sci. Eng. 11(4), 415–427 (1961).
[Crossref]

Mohan, P. S.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230(19), 7364–7383 (2011).
[Crossref]

Panasyuk, G. Y.

Pulkkinen, A.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230(19), 7364–7383 (2011).
[Crossref]

Schotland, J. C.

Schweiger, M.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230(19), 7364–7383 (2011).
[Crossref]

S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. 18(1), 247–260 (2007).
[Crossref]

Siewert, C. E.

C. E. Siewert and N. J. Mccormick, “Some identities for Chandrasekhar polynomials,” J. Quant. Spectrosc. Radiat. Transf. 57(3), 399–404 (1997).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transf. 43(3), 201–205 (1990).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “On discrete spectrum calculations in radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 42(5), 385–394 (1989).
[Crossref]

C. Devaux and C. E. Siewert, “The FN method for radiative transfer problems without azimuthal symmetry,” Z. Angew. Math. Phys. 31(5), 592–604 (1980).
[Crossref]

Sultanov, N. V.

N. V. Sultanov, “Elementary solution of the neutron transport equation with anisotropic scattering,” Sov. Atom Energ. 32(6), 539–546 (1972).
[Crossref]

Tarvainen, T.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230(19), 7364–7383 (2011).
[Crossref]

Wright, S.

S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. 18(1), 247–260 (2007).
[Crossref]

Zheng, H.

H. Zheng and W. Han, “On simplified spherical harmonics equations for the radiative transfer equation,” J. Math. Chem. 49(8), 1785–1797 (2011).
[Crossref]

Ann. Phys. (1)

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

J. Comput. Phys. (1)

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230(19), 7364–7383 (2011).
[Crossref]

J. Comput. Theor. Trans. (2)

B. D. Ganapol, “Chandrasekhar polynomials and the solution to the transport equation in an infinite medium,” J. Comput. Theor. Trans. 43(1–7), 433–473 (2014).
[Crossref]

I. Kuščer and N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” J. Comput. Theor. Trans. 20(5–6), 351–381 (1974).

J. Math. Chem. (1)

H. Zheng and W. Han, “On simplified spherical harmonics equations for the radiative transfer equation,” J. Math. Chem. 49(8), 1785–1797 (2011).
[Crossref]

J. Math. Phys. (6)

N. J. McCormick and I. Kuščer, “Half-space neutron transport with linearly anisotropic scattering,” J. Math. Phys. 6(12), 1939–1945 (1965).
[Crossref]

K. M. Case and R. D. Hazeltine, “Fourier transform methods in linear transport theory,” J. Math. Phys. 12(9), 1970–1980 (1971).
[Crossref]

N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7(11), 2036–2045 (1966).
[Crossref]

Inönü and Erdal, “Orthogonality of a set of polynomials encountered in neutron transport and radiative transfer theories,” J. Math. Phys. 11(2), 568–577 (1970).
[Crossref]

K. M. Case, “Orthogonal polynomials from the viewpoint of scattering theory,” J. Math. Phys. 15(12), 2166–2174 (1974).
[Crossref]

K. M. Case, “Scattering theory, orthogonal polynomials, and the transport equation,” J. Math. Phys. 15(7), 974–983 (1974).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Phys. A Math. Theor. (2)

M. Machida, “An FN method for the radiative transport equation in three dimensions,” J. Phys. A Math. Theor. 48(32), 325001 (2015).
[Crossref]

M. Machida, “The Green’s function for the three-dimensional linear Boltzmann equation via Fourier transform,” J. Phys. A Math. Theor. 49(17), 175001 (2016).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (3)

R. D. M. Garcia and C. E. Siewert, “On discrete spectrum calculations in radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 42(5), 385–394 (1989).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transf. 43(3), 201–205 (1990).
[Crossref]

C. E. Siewert and N. J. Mccormick, “Some identities for Chandrasekhar polynomials,” J. Quant. Spectrosc. Radiat. Transf. 57(3), 399–404 (1997).
[Crossref]

Meas. Sci. Technol. (1)

S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. 18(1), 247–260 (2007).
[Crossref]

Nucl. Sci. Eng. (2)

J. R. Mika, “Neutron transport with anisotropic scattering,” Nucl. Sci. Eng. 11(4), 415–427 (1961).
[Crossref]

B. D. Ganapol, “The infinite medium Green’s function of monoenergetic neutron transport theory via Fourier transform,” Nucl. Sci. Eng. 180(2), 224–246 (2015).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Sov. Atom Energ. (1)

N. V. Sultanov, “Elementary solution of the neutron transport equation with anisotropic scattering,” Sov. Atom Energ. 32(6), 539–546 (1972).
[Crossref]

Transp. Theory Stat. Phys. (2)

B. D. Ganapol, “A consistent theory of neutral particle transport in an infinite medium,” Transp. Theory Stat. Phys. 29(1–2), 43–68 (2000).
[Crossref]

B. D. Ganapol, “Fourier transform transport solutions in spherical geometry,” Transp. Theory Stat. Phys. 32(5–7), 587–605 (2003).
[Crossref]

Z. Angew. Math. Phys. (1)

C. Devaux and C. E. Siewert, “The FN method for radiative transfer problems without azimuthal symmetry,” Z. Angew. Math. Phys. 31(5), 592–604 (1980).
[Crossref]

Other (10)

N. J. Mccormick and N. Joseph, “One-speed neutron transport problems in plane geometry,” Univ. of Michigan Thesis, (1965).

A. Kienle and A. Liemert, “Light diffusion in turbid media of different geometries in the steady-state, frequency, and time domains,” in Biomedical Optics and 3-D Imaging, OSA Technical Digest (CD) (Optical Society of America, 2010), paper BSuD41.
[Crossref]

A. D. Klose and E. W. Larsen, “Simplified spherical harmonics methods for modeling light transport in biological tissue,” in Biomedical Optics, Technical Digest (CD) (Optical Society of America, 2006), paper MH3.
[Crossref]

B. D. Ganapol, “The Fourier transform solution for the Green’s function of monoenergetic neutron transport theory,” in 2014 Hawaii University International Conferences on Science, Technology, Engineering, Math & Education (2014).

A. D. Kim, “Transport theory for light propagation in tissues,” in Biomedical Topical Meeting, OSA Technical Digest (Optical Society of America, 2004), paper SD4.

G. Van den Eynde, “Neutron transport with anisotropic scattering theory and applications,” Thesis, (2005).

A. J. Welch and M. J. C. V. Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Springer Netherlands, 2011).

Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989).

N. I. Muskhelishvili, Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics (Dover Publications, 2011).

D. Zwillinger and V. Moll, Table of Integrals, Series, and Products (Eighth Edition) (Academic Press, 2015).

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Figures (9)

Fig. 1
Fig. 1 The 2-norms of the characteristic matrix vary with anisotropy factorg, the curves in each subfigure denote the 2-norms when m=0,5,10,15,20,25,30 from top to bottom
Fig. 2
Fig. 2 The distributions and variation tendency of the eigenvalues at some sampled positions, each curve in each subfigure corresponds to a certain sampled position
Fig. 3
Fig. 3 The result sequences and errors of the first and second kinds of Chandrasekhar polynomials as the degree (x-axis) increases when z = 0.48 by method (a) or (b).
Fig. 4
Fig. 4 The average errors of the first (left) and second (right) kinds of Chandrasekhar polynomials as z from 0 to 1 with step 0.2 and m from 0 to 30 with step 5 when N=599
Fig. 5
Fig. 5 The curves of the first (left) and second (right) kinds of Chandrasekhar polynomials as zfrom −1 to 1. g( l,m ) denotes g ^ l m ( z ), and ρ( l,m ) denotes ρ ^ l m ( z )
Fig. 6
Fig. 6 The average logarithmic absolute errors of the two sequences of the first (left) and second (right) kind of Chandrasekhar polynomials as z from 0 to 1 with step 0.2 and m from 0 to 30 with step 5 when N = 599 .
Fig. 7
Fig. 7 The average logarithmic residual errors of the first and second kind of Chandrasekhar polynomials as z from 0 to 1 with step 0.2 and m from 0 to 30 with step 5 when N = 599 .
Fig. 8
Fig. 8 The average logarithmic residual errors of the first and second kind of Chandrasekhar polynomials as z from 1.25 to 9.00 with step 0.25 and m from 0 to 30 with step 5 when N=251.
Fig. 9
Fig. 9 The logarithmic residual errors and absolute errors in method (d).

Equations (67)

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μ ψ( x,μ,φ; μ 0 , φ 0 ) x + μ t ψ( x,μ,φ; μ 0 , φ 0 )= μ s 0 2π 1 +1 p( cosγ )ψ( x, μ , φ ; μ 0 , φ 0 )d μ d φ +δ( x )δ( μ μ 0 )δ( φ φ 0 )
p( cosγ )= 1 4π l=0 L ( 2l+1 ) f l m=l l ( lm )! ( l+m )! P l m ( μ ) P l m ( μ ) e im( φ φ )
ψ( x,μ,φ; μ 0 , φ 0 )= 1 2π m= ψ m ( x,μ; μ 0 ) e imφ
μ ψ m ( x,μ; μ 0 ) x + μ t ψ m ( x,μ; μ 0 )= μ s 2 l=| m | L β l m P l m ( μ ) 1 +1 P l m ( μ ) ψ m ( x, μ ; μ 0 )d μ +δ( x )δ( μ μ 0 ) e im φ 0
ψ m ( x,μ )exp( μ t x/v ) ϕ m ( v,μ )
( 1 μ v ) ϕ m ( v,μ )= ϖ 2 l=| m | L β l m P l m ( μ ) 1 +1 P l m ( μ ) ϕ m ( v, μ )d μ
g l m ( v ) ( 1 ) m 1 +1 P l m ( μ ) ϕ m ( v,μ )dμ
ϕ m ( v,μ )= ϖv 2 g m ( v,μ ) vμ
1 1 ϕ m ( v,μ ) ( 1 μ 2 ) | m |/2 dμ =1
g m ( v,μ )= l=| m | L β l m g l m ( v ) P l m ( μ )
Λ m ( z )1 ϖz 2 1 1 g m ( z,μ ) zμ ( 1 μ 2 ) | m |/2 dμ
[ Λ m ( v ) ] ± =1 ϖv 2 P 1 1 g m ( v,μ ) vμ ( 1 μ 2 ) | m |/2 dμ ± iπϖv 2 g m ( v,v ) ( 1 v 2 ) | m |/2
ϕ m ( v,μ )= ϖv 2 P g m ( v,μ ) vμ + λ( v ) ( 1 v 2 ) | m |/2 δ( vμ )
λ( v )=1 ϖv 2 P 1 1 g m ( v,μ ) vμ ( 1 μ 2 ) | m |/2 dμ
[ Λ m ( v ) ] + + [ Λ m ( v ) ] =2λ( v )
[ Λ m ( v ) ] + [ Λ m ( v ) ] = λ 2 ( v )+ [ πϖv 2 g m ( v,v ) ( 1 v 2 ) | m |/2 ] 2
1 1 ϕ m ( v i m ,μ ) ϕ m ( v j m ,μ )μdμ = N 0 m ( v j m ) δ ij
1 1 ϕ m ( v,μ ) ϕ m ( v ,μ )μdμ = N m ( v )δ( v v )
1 1 ϕ m ( v j ,μ ) ϕ m ( v,μ )μdμ =0
N 0 m ( ± v j m )= ±ϖ v j m 2 [ ϖ ( v j m ) 2 l=| m | L β l m g l m ( v j m ) k=| m | L β k m g k m ( v j m ) γ lm δ m0 ( v j m ) 2 1 l=| m | L β l m [ g l m ( v j m ) ] 2 ( 1+ δ m0 ) v j m l=| m | L β l m g l m ( v j m ) k=0 [ l1 2 ] ( 2l4k1 ) g l2k1 m ( v j m ) ]
1 1 P l m ( μ ) P k m ( μ ) zμ dμ ={ 2 P l m ( z ) Q k m ( z ) lk 2 Q l m ( z ) P k m ( z ) l>k
d P l m ( x ) dx = 1+ δ m0 2 k=0 [ l1 2 ] ( 2l4k1 ) P l2k1 m ( x )
N 0 m ( ± v j m )=± 1 2 ϖ ( v j m ) 2 g m ( v j m , v j m ) [ ( v j m ) 2 1 ] | m |/2 d Λ m ( z ) dz | z= v j m
N m ( v )= v [ Λ m ( v ) ] + [ Λ m ( v ) ] ( 1 v 2 ) | m |
ψ m ( x,μ )= j=1 M a + m ϕ m ( v j m ,μ )exp( μ t x/ v j m ) + j=1 M a m ϕ m ( v j m ,μ )exp( μ t x/ v j m ) + 1 1 A m ( v ) ϕ m ( v,μ )exp( μ t x/v )dv
ψ m ( 0+,μ ) ψ m ( 0,μ )= 1 μ δ( x )δ( μ μ 0 ) e im φ 0
lim | x | ψ m ( x,μ )=0
ψ m ( x,μ; μ 0 )={ j=1 M a + m ϕ m ( v j m ,μ )exp( μ t x/ v j m ) + 0 1 A m ( v ) ϕ m ( v,μ )exp( μ t x/v )dv ( x>0 ) j=1 M a m ϕ m ( v j m ,μ )exp( μ t x/ v j m ) 1 0 A m ( v ) ϕ m ( v,μ )exp( μ t x/v )dv ( x<0 )
a j± m = 1 N 0 m ( ± v j m ) ϕ m ( ± v j m , μ 0 ) e im φ 0
A( v )= 1 N m ( v ) ϕ m ( v, μ 0 ) e im φ 0
ψ m ( x,μ; μ 0 )= e im φ 0 { j=1 M ϕ m ( v j m , μ 0 ) ϕ m ( v j m ,μ ) N 0 m ( v j m ) exp( μ t x/ v j m ) + 0 1 ϕ m ( v, μ 0 ) ϕ m ( v,μ ) N m ( v ) exp( μ t x/v )dv ( x>0 ) j=1 M ϕ m ( v j m , μ 0 ) ϕ m ( v j m ,μ ) N 0 m ( v j m ) exp( μ t x/ v j m ) 1 0 ϕ m ( v, μ 0 ) ϕ m ( v,μ ) N m ( v ) exp( μ t x/v )dv ( x<0 )
g l m ( z )= ( 1 ) m ( lm )! ( l+m )! g l m ( z )
g l m ( z )= ( 1 ) lm g l m ( z )
z h l g l m ( z )( lm+1 ) g l+1 m ( z )( l+m ) g l1 m ( z )=0
g ^ l m ( z )= ( lm )! ( l+m )! g l m ( z )
z h l g ^ l m ( z ) ( l+1 ) 2 m 2 g ^ l+1 m ( z ) l 2 m 2 g ^ l1 m ( z )=0
g ^ m m ( z )= ( 2m1 )!! ( 2m )! = ( 2m )! 2 m m!
z h l ρ l m ( z )( lm+1 ) ρ l+1 m ( z )( l+m ) ρ l1 m ( z )= ρ m+1 m ( z ) δ lm
ρ ^ l m ( z )= ( lm )! ( l+m )! ρ l m ( z )
z h l ρ ^ l m ( z ) ( l+1 ) 2 m 2 ρ ^ l+1 m ( z ) l 2 m 2 ρ ^ l1 m ( z )= 2m+1 ρ ^ m+1 m ( z ) δ lm
g l1 m ( z ) ρ l m ( z ) g l m ( z ) ρ l1 m ( z )= ( l+m1 )! ( lm )! ρ m+1 m ( z ) g m m ( z ) ( 2m )!
g ^ l1 m ( z ) ρ ^ l m ( z ) g ^ l m ( z ) ρ ^ l1 m ( z )= ( 2m+1 ) ρ ^ m+1 m ( z ) g ^ m m ( z ) l 2 m 2
g ^ l m ( z )= ( 2m )! g ^ m m ( z ) ( lm )!( l+m )! | A l1 m ( z ) |
ρ ^ l m ( z )= ( 2m+1 )! ρ ^ m+1 m ( z ) ( lm )!( l+m )! | A l1 m+1 ( z ) |
A l m ( z )=( z h m ( m+1 ) 2 m 2 ( m+1 ) 2 m 2 z h m+1 z h l1 l 2 m 2 l 2 m 2 z h l )
| A l m ( z ) |=( j=m l h j )| B l m +zI |
B l m =( 0 b m+1 m b m+1 m 0 b m+2 m b m+2 m 0 b l m b l m 0 )
g ^ l m ( z )= g ^ m m ( z ) j=0 lm1 h m+j ( z+ λ g,j m ) ( j+1 )( j+1+2m )
ρ ^ l m ( z )= ρ ^ m+1 m ( z ) j=1 lm1 h m+j ( z+ λ ρ,j m ) ( j+1 )( j+1+2m )
d ^ j m ( z )= z h j1 d ^ j1 m ( z ) ( j1 ) 2 m 2 d ^ j2 m ( z ) j 2 m 2
g ^ j m ( z )= z h j+1 g ^ j+1 m ( z ) ( j+2 ) 2 m 2 g ^ j+2 m ( z ) ( j+1 ) 2 m 2
g ^ j m ( z )= γ j m g ^ m+N m ( z )
γ j m ( z )= z h j+1 γ j+1 m ( z ) ( j+2 ) 2 m 2 γ j+2 m ( z ) ( j+1 ) 2 m 2
g ^ m m ( z )= γ m m ( z ) g ^ N m ( z )
g ^ j m ( z )= γ j m γ m m ( z ) g ^ m m ( z )
[ B m+N m ( z )zI ] G m+N m ( z )= G ¯ m+N+1 m ( z )
a j = h j g ^ j m ( z ),j=m,m+1,,m+N+1
[ B m+N m zI ] G m+N m ( z )=0
| B m+N m zI |=0
g ^ j m ( z i )= h m h j a j a m g ^ m m ( z i )
R 1,j m ( z ) g ^ j1 m ( z ) ρ ^ j m ( z ) g ^ j m ( z ) ρ ^ j1 m ( z ) 2m+1 ρ ^ m+1 m ( z ) g ^ m m ( z ) j 2 m 2
R 2,j m ( z )z h j ρ ^ j m ( z ) ( j+1 ) 2 m 2 ρ ^ j+1 m ( z ) j 2 m 2 ρ ^ j1 m ( z )
R 3,j m ( z )z h j g ^ j m ( z ) ( j+1 ) 2 m 2 g ^ j+1 m ( z ) j 2 m 2 g ^ j1 m ( z )
R 3,j m ( z ) d ^ j,1 m ( z ) d ^ j,2 m ( z )
R 4,j m ( z ) d ^ j,1 m ( z ) d ^ j,2 m ( z ) d ^ j,1 m ( z )
D i m ( z )log| R i,j m ( z ) |
D ¯ i m ( z )log{ R i,j m ( z ) [ R i,j m ( z ) ] T }

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