Abstract

A comparison is presented of two different methods for polarized radiative transfer in coupled media consisting of two adjacent slabs with different refractive indices, each slab being a stratified medium with no change in optical properties except in the direction of stratification. One of the methods is based on solving the integro-differential radiative transfer equation for the two coupled slabs using the discrete ordinate approximation. The other method is based on probabilistic and statistical concepts and simulates the propagation of polarized light using the Monte Carlo approach. The emphasis is on non-Rayleigh scattering for particles in the Mie regime. Comparisons with benchmark results available for a slab with constant refractive index show that both methods reproduce these benchmark results when the refractive index is set to be the same in the two slabs. Computed results for test cases with coupling (different refractive indices in the two slabs) show that the two methods produce essentially identical results for identical input in terms of absorption and scattering coefficients and scattering phase matrices.

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2010 (4)

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, E. Sommersten, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions,” Dermatol. Surg.36, 1–8 (2010).
[CrossRef]

E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transfer111, 616–633 (2010).
[CrossRef]

J. K. Lotsberg and J. J. Stamnes, “Impact of particulate oceanic composition on the radiance and polarization of underwater and backscattered light,” Opt. Expr.18, 10432–10445 (2010).
[CrossRef]

A. A. Kokhanovsky, C. Cornet, M. Duan, C. Emde, I. L. Katsev, L. C-Labonnote, Q. Min, T. Nakajima, Y. Ota, and A.P. Prikhach, and others, “Benchmark results in vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer111, 1931–1946 (2010). .
[CrossRef]

2009 (2)

P. W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Expr.17, 2057–2079 (2009).
[CrossRef]

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions: a pilot study,” Skin Res. Technol.15, 330–337 (2009).
[CrossRef] [PubMed]

2008 (2)

K. Nielsen, L. Zhao, G. A. Ryzhikov, M. S. Biryulina, E. R. Sommersten, J. J. Stamnes, K. Stamnes, and J. Moan, “Retrieval of the physiological state of human skin from UV-VIS reflectance spectra: A feasibility study,” J. Photochem. Photobiol. B93, 23–31 (2008).
[CrossRef] [PubMed]

W. Li, K. Stamnes, R. Spurr, and J. J. Stamnes, “Simultaneous retrieval of aerosols and ocean properties: A classic inverse modeling approach. II. SeaWiFS case study for the Santa Barbara channel,” Int. J. Rem. Sens.29, 5689–5698 (2008).
[CrossRef]

2007 (2)

R. Spurr, K. Stamnes, H. Eide, W. Li, K. Zhang, and J. J. Stamnes, “Simultaneous retrieval of aerosol and ocean properties: A classic inverse modeling approach: I. Analytic Jacobians from the linearized CAO-DISORT model,” J. Quant. Spectrosc. Radiat. Transfer104, 428–449 (2007).
[CrossRef]

K. Hestenes, K. P. Nielsen, L. Zhao, J. J. Stamnes, and K. Stamnes, “Monte Carlo and discrete-ordinate simulations of spectral radiances in the coupled air-tissue system,” Appl. Opt.46, 2333–2350 (2007).
[CrossRef] [PubMed]

2004 (1)

K. P. Nielsen, L. Zhao, P. Juzenas, K. Stamnes, J. J. Stamnes, and J. Moan, “Reflectance spectra of pigmented and non-pigmented skin in the UV spectral region,” Photochem. Photobiol.80, 450–455 (2004).
[PubMed]

2003 (2)

2002 (1)

H. Ishimoto and K. Masuda, “A Monte Carlo approach for the calculation of polarized light,” J. Quant. Spectrosc. Radiat. Transfer72, 467–483 (2002).
[CrossRef]

2001 (1)

2000 (1)

F. M. Schulz and K. Stamnes, “Angular distribution of the Stokes vector in a plane parallel, vertically inhomogeneous medium in the vector discrete ordinate radiative transfer (VDISORT) model,” J. Quant. Spectrosc. Radiat. Transfer65, 609–620 (2000).
[CrossRef]

1999 (1)

F. M. Schulz, K. Stamnes, and F. Weng, “VDISORT: An improved and generalized discrete ordinate method for polarized vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer61, 105–122 (1999).
[CrossRef]

1998 (2)

A. A. Lacis, J. Chowdhary, M. I. Mishchenko, and B. Cairns, “Modeling errors in diffuse-sky radiation: vector vs. scalar treatment,” Geophys. Res. Lett.25, 135–138 (1998).
[CrossRef]

M. I. Mischenko and L. D. Travis, “Capabilities and Limitations of a Current FORTRAN Implementation of the T-Matrix Method for Randomly Oriented, Rotationally Symmetric Scatterers,” J. Quant. Spectrosc. Radiat. Transfer60, 309–324 (1998).
[CrossRef]

1997 (2)

H. R. Gordon, “Atmospheric correction of ocean color imagery in the Earth Observation System era,” J. Geophys. Res.102, 17081–17106 (1997).
[CrossRef]

M. Mishchenko and L. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res.102, 16989–7013 (1997).
[CrossRef]

1994 (2)

Z. Jin and K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt.33, 431–442 (1994).
[CrossRef] [PubMed]

M. I. Mishchenko, A.A. Lacis, and L. D. Travis, “Errors due to the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres,” J. Quant Spectrosc. Radiat Transfer51, 491–510 (1994).
[CrossRef]

1992 (3)

F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogenous, emitting and scattering atmosphere - I. Theory,” J. Quant. Spectrosc. Radiat. Transfer, 47, 19–33 (1992).
[CrossRef]

F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogenous, emitting and scattering atmosphere - II. Applications,” J. Quant. Spectrosc. Radiat. Transfer47, 35–42 (1992).
[CrossRef]

D. M. O’Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer48, 41–59 (1992).
[CrossRef]

1991 (1)

1989 (1)

C. N. Adams and G. W. Kattawar, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: Effect of interface refractive index on radiance and polarization,” Limm. Ocean.34, 1453–1472 (1989).
[CrossRef]

1988 (1)

1987 (2)

J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys.183, 371–391 (1987).

J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys.183, 371–391 (1987).

1984 (1)

K. Stamnes and P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer31, 273–282 (1984).
[CrossRef]

1983 (1)

J. W. Hovenier and C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys.128, 1–16 (1983).

1982 (1)

C. E. Siewert, “On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys.109, 195–200 (1982).

1981 (3)

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J.245, 1080–1086 (1981).
[CrossRef]

K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci.38, 387–399 (1981).
[CrossRef]

K. Stamnes and H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II. Intensity computations,” J. Atmos. Sci.38, 2696–2706 (1981).
[CrossRef]

1977 (1)

W. J. Wiscombe, “The delta-M method: Rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci.34, 1408–1422 (1977).
[CrossRef]

1974 (1)

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev.16, 527–610 (1974).
[CrossRef]

1973 (1)

K. N. Liou, “A numerical experiment on Chandrasekhar’s discrete-ordinate method for radiative transfer,” J. Atmos. Sci.30, 1303–1326 (1973).
[CrossRef]

1968 (1)

Adams, C. N.

C. N. Adams and G. W. Kattawar, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: Effect of interface refractive index on radiance and polarization,” Limm. Ocean.34, 1453–1472 (1989).
[CrossRef]

Biryulina, M.

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, E. Sommersten, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions,” Dermatol. Surg.36, 1–8 (2010).
[CrossRef]

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions: a pilot study,” Skin Res. Technol.15, 330–337 (2009).
[CrossRef] [PubMed]

Biryulina, M. S.

K. Nielsen, L. Zhao, G. A. Ryzhikov, M. S. Biryulina, E. R. Sommersten, J. J. Stamnes, K. Stamnes, and J. Moan, “Retrieval of the physiological state of human skin from UV-VIS reflectance spectra: A feasibility study,” J. Photochem. Photobiol. B93, 23–31 (2008).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1998).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics: 7th edition (Cambridge University, 2002).

Bosma, P. B.

J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys.183, 371–391 (1987).

J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys.183, 371–391 (1987).

Cairns, B.

A. A. Lacis, J. Chowdhary, M. I. Mishchenko, and B. Cairns, “Modeling errors in diffuse-sky radiation: vector vs. scalar treatment,” Geophys. Res. Lett.25, 135–138 (1998).
[CrossRef]

Castellana, F. S.

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, E. Sommersten, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions,” Dermatol. Surg.36, 1–8 (2010).
[CrossRef]

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions: a pilot study,” Skin Res. Technol.15, 330–337 (2009).
[CrossRef] [PubMed]

Chaikovskaya, L. I.

Chandrasekhar, S.

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

Chowdhary, J.

P. W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Expr.17, 2057–2079 (2009).
[CrossRef]

A. A. Lacis, J. Chowdhary, M. I. Mishchenko, and B. Cairns, “Modeling errors in diffuse-sky radiation: vector vs. scalar treatment,” Geophys. Res. Lett.25, 135–138 (1998).
[CrossRef]

C-Labonnote, L.

A. A. Kokhanovsky, C. Cornet, M. Duan, C. Emde, I. L. Katsev, L. C-Labonnote, Q. Min, T. Nakajima, Y. Ota, and A.P. Prikhach, and others, “Benchmark results in vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer111, 1931–1946 (2010). .
[CrossRef]

Conklin, P.

K. Stamnes and P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer31, 273–282 (1984).
[CrossRef]

Cornet, C.

A. A. Kokhanovsky, C. Cornet, M. Duan, C. Emde, I. L. Katsev, L. C-Labonnote, Q. Min, T. Nakajima, Y. Ota, and A.P. Prikhach, and others, “Benchmark results in vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer111, 1931–1946 (2010). .
[CrossRef]

Dale, H.

K. Stamnes and H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II. Intensity computations,” J. Atmos. Sci.38, 2696–2706 (1981).
[CrossRef]

Darbinjan, R. A.

G. I. Marchuk, G. A Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).
[CrossRef]

de Haan, J. F.

J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys.183, 371–391 (1987).

J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys.183, 371–391 (1987).

Duan, M.

A. A. Kokhanovsky, C. Cornet, M. Duan, C. Emde, I. L. Katsev, L. C-Labonnote, Q. Min, T. Nakajima, Y. Ota, and A.P. Prikhach, and others, “Benchmark results in vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer111, 1931–1946 (2010). .
[CrossRef]

Eide, H.

R. Spurr, K. Stamnes, H. Eide, W. Li, K. Zhang, and J. J. Stamnes, “Simultaneous retrieval of aerosol and ocean properties: A classic inverse modeling approach: I. Analytic Jacobians from the linearized CAO-DISORT model,” J. Quant. Spectrosc. Radiat. Transfer104, 428–449 (2007).
[CrossRef]

Elepov, B. S.

G. I. Marchuk, G. A Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).
[CrossRef]

Emde, C.

A. A. Kokhanovsky, C. Cornet, M. Duan, C. Emde, I. L. Katsev, L. C-Labonnote, Q. Min, T. Nakajima, Y. Ota, and A.P. Prikhach, and others, “Benchmark results in vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer111, 1931–1946 (2010). .
[CrossRef]

Gjerstad, K. I.

Gordon, H. R.

H. R. Gordon, “Atmospheric correction of ocean color imagery in the Earth Observation System era,” J. Geophys. Res.102, 17081–17106 (1997).
[CrossRef]

Hamre, B.

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D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions: a pilot study,” Skin Res. Technol.15, 330–337 (2009).
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K. P. Nielsen, L. Zhao, P. Juzenas, K. Stamnes, J. J. Stamnes, and J. Moan, “Reflectance spectra of pigmented and non-pigmented skin in the UV spectral region,” Photochem. Photobiol.80, 450–455 (2004).
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D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, E. Sommersten, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions,” Dermatol. Surg.36, 1–8 (2010).
[CrossRef]

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[CrossRef]

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions: a pilot study,” Skin Res. Technol.15, 330–337 (2009).
[CrossRef] [PubMed]

K. Nielsen, L. Zhao, G. A. Ryzhikov, M. S. Biryulina, E. R. Sommersten, J. J. Stamnes, K. Stamnes, and J. Moan, “Retrieval of the physiological state of human skin from UV-VIS reflectance spectra: A feasibility study,” J. Photochem. Photobiol. B93, 23–31 (2008).
[CrossRef] [PubMed]

W. Li, K. Stamnes, R. Spurr, and J. J. Stamnes, “Simultaneous retrieval of aerosols and ocean properties: A classic inverse modeling approach. II. SeaWiFS case study for the Santa Barbara channel,” Int. J. Rem. Sens.29, 5689–5698 (2008).
[CrossRef]

R. Spurr, K. Stamnes, H. Eide, W. Li, K. Zhang, and J. J. Stamnes, “Simultaneous retrieval of aerosol and ocean properties: A classic inverse modeling approach: I. Analytic Jacobians from the linearized CAO-DISORT model,” J. Quant. Spectrosc. Radiat. Transfer104, 428–449 (2007).
[CrossRef]

K. Hestenes, K. P. Nielsen, L. Zhao, J. J. Stamnes, and K. Stamnes, “Monte Carlo and discrete-ordinate simulations of spectral radiances in the coupled air-tissue system,” Appl. Opt.46, 2333–2350 (2007).
[CrossRef] [PubMed]

K. P. Nielsen, L. Zhao, P. Juzenas, K. Stamnes, J. J. Stamnes, and J. Moan, “Reflectance spectra of pigmented and non-pigmented skin in the UV spectral region,” Photochem. Photobiol.80, 450–455 (2004).
[PubMed]

Stamnes, K.

E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transfer111, 616–633 (2010).
[CrossRef]

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, E. Sommersten, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions,” Dermatol. Surg.36, 1–8 (2010).
[CrossRef]

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions: a pilot study,” Skin Res. Technol.15, 330–337 (2009).
[CrossRef] [PubMed]

K. Nielsen, L. Zhao, G. A. Ryzhikov, M. S. Biryulina, E. R. Sommersten, J. J. Stamnes, K. Stamnes, and J. Moan, “Retrieval of the physiological state of human skin from UV-VIS reflectance spectra: A feasibility study,” J. Photochem. Photobiol. B93, 23–31 (2008).
[CrossRef] [PubMed]

W. Li, K. Stamnes, R. Spurr, and J. J. Stamnes, “Simultaneous retrieval of aerosols and ocean properties: A classic inverse modeling approach. II. SeaWiFS case study for the Santa Barbara channel,” Int. J. Rem. Sens.29, 5689–5698 (2008).
[CrossRef]

R. Spurr, K. Stamnes, H. Eide, W. Li, K. Zhang, and J. J. Stamnes, “Simultaneous retrieval of aerosol and ocean properties: A classic inverse modeling approach: I. Analytic Jacobians from the linearized CAO-DISORT model,” J. Quant. Spectrosc. Radiat. Transfer104, 428–449 (2007).
[CrossRef]

K. Hestenes, K. P. Nielsen, L. Zhao, J. J. Stamnes, and K. Stamnes, “Monte Carlo and discrete-ordinate simulations of spectral radiances in the coupled air-tissue system,” Appl. Opt.46, 2333–2350 (2007).
[CrossRef] [PubMed]

K. P. Nielsen, L. Zhao, P. Juzenas, K. Stamnes, J. J. Stamnes, and J. Moan, “Reflectance spectra of pigmented and non-pigmented skin in the UV spectral region,” Photochem. Photobiol.80, 450–455 (2004).
[PubMed]

K. I. Gjerstad, J. J. Samnes, B. Hamre, J. K. Lotsberg, B. Yan, and K. Stamnes, “Monte Carlo and discrete-ordinate simulations of irradiances in the coupled atmosphere-ocean system,” Appl. Opt.42, 2609–2622 (2003).
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[CrossRef]

F. M. Schulz, K. Stamnes, and F. Weng, “VDISORT: An improved and generalized discrete ordinate method for polarized vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer61, 105–122 (1999).
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Swanson, D. L.

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, E. Sommersten, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions,” Dermatol. Surg.36, 1–8 (2010).
[CrossRef]

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions: a pilot study,” Skin Res. Technol.15, 330–337 (2009).
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M. Mishchenko and L. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res.102, 16989–7013 (1997).
[CrossRef]

Travis, L. D.

M. I. Mischenko and L. D. Travis, “Capabilities and Limitations of a Current FORTRAN Implementation of the T-Matrix Method for Randomly Oriented, Rotationally Symmetric Scatterers,” J. Quant. Spectrosc. Radiat. Transfer60, 309–324 (1998).
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[CrossRef]

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P. W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Expr.17, 2057–2079 (2009).
[CrossRef]

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Tynes, H. H.

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J. W. Hovenier and C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys.128, 1–16 (1983).

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Weng, F.

F. M. Schulz, K. Stamnes, and F. Weng, “VDISORT: An improved and generalized discrete ordinate method for polarized vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer61, 105–122 (1999).
[CrossRef]

F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogenous, emitting and scattering atmosphere - I. Theory,” J. Quant. Spectrosc. Radiat. Transfer, 47, 19–33 (1992).
[CrossRef]

F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogenous, emitting and scattering atmosphere - II. Applications,” J. Quant. Spectrosc. Radiat. Transfer47, 35–42 (1992).
[CrossRef]

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Wiscombe, W. J.

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P. W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Expr.17, 2057–2079 (2009).
[CrossRef]

Zhang, K.

R. Spurr, K. Stamnes, H. Eide, W. Li, K. Zhang, and J. J. Stamnes, “Simultaneous retrieval of aerosol and ocean properties: A classic inverse modeling approach: I. Analytic Jacobians from the linearized CAO-DISORT model,” J. Quant. Spectrosc. Radiat. Transfer104, 428–449 (2007).
[CrossRef]

Zhao, L.

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, E. Sommersten, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions,” Dermatol. Surg.36, 1–8 (2010).
[CrossRef]

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions: a pilot study,” Skin Res. Technol.15, 330–337 (2009).
[CrossRef] [PubMed]

K. Nielsen, L. Zhao, G. A. Ryzhikov, M. S. Biryulina, E. R. Sommersten, J. J. Stamnes, K. Stamnes, and J. Moan, “Retrieval of the physiological state of human skin from UV-VIS reflectance spectra: A feasibility study,” J. Photochem. Photobiol. B93, 23–31 (2008).
[CrossRef] [PubMed]

K. Hestenes, K. P. Nielsen, L. Zhao, J. J. Stamnes, and K. Stamnes, “Monte Carlo and discrete-ordinate simulations of spectral radiances in the coupled air-tissue system,” Appl. Opt.46, 2333–2350 (2007).
[CrossRef] [PubMed]

K. P. Nielsen, L. Zhao, P. Juzenas, K. Stamnes, J. J. Stamnes, and J. Moan, “Reflectance spectra of pigmented and non-pigmented skin in the UV spectral region,” Photochem. Photobiol.80, 450–455 (2004).
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Appl. Opt. (7)

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J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys.183, 371–391 (1987).

J. W. Hovenier and C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys.128, 1–16 (1983).

Astrophys. J. (1)

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[CrossRef]

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D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, E. Sommersten, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions,” Dermatol. Surg.36, 1–8 (2010).
[CrossRef]

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A. A. Lacis, J. Chowdhary, M. I. Mishchenko, and B. Cairns, “Modeling errors in diffuse-sky radiation: vector vs. scalar treatment,” Geophys. Res. Lett.25, 135–138 (1998).
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W. Li, K. Stamnes, R. Spurr, and J. J. Stamnes, “Simultaneous retrieval of aerosols and ocean properties: A classic inverse modeling approach. II. SeaWiFS case study for the Santa Barbara channel,” Int. J. Rem. Sens.29, 5689–5698 (2008).
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K. Stamnes and H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II. Intensity computations,” J. Atmos. Sci.38, 2696–2706 (1981).
[CrossRef]

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[CrossRef]

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H. R. Gordon, “Atmospheric correction of ocean color imagery in the Earth Observation System era,” J. Geophys. Res.102, 17081–17106 (1997).
[CrossRef]

M. Mishchenko and L. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res.102, 16989–7013 (1997).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Photochem. Photobiol. B (1)

K. Nielsen, L. Zhao, G. A. Ryzhikov, M. S. Biryulina, E. R. Sommersten, J. J. Stamnes, K. Stamnes, and J. Moan, “Retrieval of the physiological state of human skin from UV-VIS reflectance spectra: A feasibility study,” J. Photochem. Photobiol. B93, 23–31 (2008).
[CrossRef] [PubMed]

J. Quant Spectrosc. Radiat Transfer (1)

M. I. Mishchenko, A.A. Lacis, and L. D. Travis, “Errors due to the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres,” J. Quant Spectrosc. Radiat Transfer51, 491–510 (1994).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (11)

K. Stamnes and P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer31, 273–282 (1984).
[CrossRef]

R. Spurr, K. Stamnes, H. Eide, W. Li, K. Zhang, and J. J. Stamnes, “Simultaneous retrieval of aerosol and ocean properties: A classic inverse modeling approach: I. Analytic Jacobians from the linearized CAO-DISORT model,” J. Quant. Spectrosc. Radiat. Transfer104, 428–449 (2007).
[CrossRef]

E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transfer111, 616–633 (2010).
[CrossRef]

F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogenous, emitting and scattering atmosphere - I. Theory,” J. Quant. Spectrosc. Radiat. Transfer, 47, 19–33 (1992).
[CrossRef]

F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogenous, emitting and scattering atmosphere - II. Applications,” J. Quant. Spectrosc. Radiat. Transfer47, 35–42 (1992).
[CrossRef]

F. M. Schulz, K. Stamnes, and F. Weng, “VDISORT: An improved and generalized discrete ordinate method for polarized vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer61, 105–122 (1999).
[CrossRef]

F. M. Schulz and K. Stamnes, “Angular distribution of the Stokes vector in a plane parallel, vertically inhomogeneous medium in the vector discrete ordinate radiative transfer (VDISORT) model,” J. Quant. Spectrosc. Radiat. Transfer65, 609–620 (2000).
[CrossRef]

M. I. Mischenko and L. D. Travis, “Capabilities and Limitations of a Current FORTRAN Implementation of the T-Matrix Method for Randomly Oriented, Rotationally Symmetric Scatterers,” J. Quant. Spectrosc. Radiat. Transfer60, 309–324 (1998).
[CrossRef]

A. A. Kokhanovsky, C. Cornet, M. Duan, C. Emde, I. L. Katsev, L. C-Labonnote, Q. Min, T. Nakajima, Y. Ota, and A.P. Prikhach, and others, “Benchmark results in vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer111, 1931–1946 (2010). .
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H. Ishimoto and K. Masuda, “A Monte Carlo approach for the calculation of polarized light,” J. Quant. Spectrosc. Radiat. Transfer72, 467–483 (2002).
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[CrossRef]

Limm. Ocean. (1)

C. N. Adams and G. W. Kattawar, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: Effect of interface refractive index on radiance and polarization,” Limm. Ocean.34, 1453–1472 (1989).
[CrossRef]

Opt. Expr. (2)

J. K. Lotsberg and J. J. Stamnes, “Impact of particulate oceanic composition on the radiance and polarization of underwater and backscattered light,” Opt. Expr.18, 10432–10445 (2010).
[CrossRef]

P. W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Expr.17, 2057–2079 (2009).
[CrossRef]

Photochem. Photobiol. (1)

K. P. Nielsen, L. Zhao, P. Juzenas, K. Stamnes, J. J. Stamnes, and J. Moan, “Reflectance spectra of pigmented and non-pigmented skin in the UV spectral region,” Photochem. Photobiol.80, 450–455 (2004).
[PubMed]

Skin Res. Technol. (1)

D. L. Swanson, S. D. Laman, M. Biryulina, K. P. Nielsen, G. Ryzhikov, J. J. Stamnes, B. Hamre, L. Zhao, F. S. Castellana, and K. Stamnes, “Optical transfer diagnosis of pigmented lesions: a pilot study,” Skin Res. Technol.15, 330–337 (2009).
[CrossRef] [PubMed]

Space Sci. Rev. (1)

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev.16, 527–610 (1974).
[CrossRef]

Other (10)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

M. Born and E. Wolf, Principles of Optics: 7th edition (Cambridge University, 2002).

J. M. Wallace and P. V. Hobbs, Atmospheric Science: Introductory Survey (Academic, 1977).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1998).
[CrossRef]

C. Rodgers, Inverse Methods for Atmospheric Sounding (World Scientific, 2000).

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

J. K. Lotsberg, “Impact of particulate oceanic composition on the radiance and polarization of the natural light field,” PhD thesis, University of Bergen (2005).

G. I. Marchuk, G. A Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).
[CrossRef]

S. Jiang, “Radiative Transfer in the Coupled Atmosphere-Sea Ice-Ocean System with Applications in Remote Sensing,” PhD thesis, Stevens Institute of Technology (2003).

E. R. Sommersten, “CAO-VDISORT: A discrete ordinate method for polarized (vector) radiative transfer in a coupled system consisting of two media with different indices of refraction,” MSc thesis, University of Bergen (2005).

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Figures (9)

Fig. 1
Fig. 1

Left four panels: Comparison of reflected polarized radiation components computed by C-VDISORT (dashed curves) with benchmark results [45] (solid curves) for a homogeneous layer of non-absorbing aerosol particles, simulated by putting half of the aerosol particles (optical depth = 0.1631) in each slab, and setting the index of refraction to 1.0 in both slabs. From top to bottom: Stokes parameter I, Q, U, and Degree of linear polarization. Right four panels: Same as left four panels except that the comparison is between C-VDISORT (dashed curves) and C-PMC (solid curves). Red: Δϕ = 0°; green: Δϕ = 90°; blue: Δϕ = 180°. 200 discrete ordinate streams, 109 photons.

Fig. 2
Fig. 2

Similar results as in Fig. 1 but for the transmitted polarized radiation components.

Fig. 3
Fig. 3

Left three panels: Relative difference between reflected polarized radiation components computed by C-VDISORT and benchmark results for the results shown in Fig. 1. Right three panels: Relative difference between transmitted polarized radiation components computed by C-VDISORT and benchmark results for the results shown in Fig. 2. The color code is the same as in Fig. 1. Note that the U parameter computed by the C-VDISORT code is close to zero for both Δϕ = 0° and Δϕ = 180° (in agreement with the benchmark) which explains why the red and blue curves are missing for the U parameter.

Fig. 4
Fig. 4

Left four panels: Comparisons of reflected polarized radiation components computed by C-VDISORT (dashed curves) with benchmark results [45] (solid curves) for a homogeneous layer of non-absorbing cloud particles, simulated by setting the refractive index equal to 1.0 in both slabs, and putting half of the cloud particles (optical depth = 2.5) in each slab. From top to bottom: Stokes parameter I, Q, U and Degree of linear polarization. Right four panels: Same as left four panels except that the comparisons are between C-VDISORT (dashed curves) and C-PMC (solid curves). Red: Δϕ = 0°; green: Δϕ = 90°; blue: Δϕ = 180°. 200 discrete ordinate streams, 109 photons.

Fig. 5
Fig. 5

Results similar to those in Fig. 4, but for the transmitted polarized radiation components. 200 discrete ordinate streams, 109 photons.

Fig. 6
Fig. 6

Left three panels: Relative difference between reflected polarized radiation components computed by C-VDISORT and benchmark results for the cloud case shown in Fig. 4. Right three panels: Relative difference between transmitted polarized radiation components computed by C-VDISORT and benchmark results for the cloud case shown in Fig. 5. The color code is the same as in Fig. 1.

Fig. 7
Fig. 7

Comparisons between polarized radiation components computed with C-VDISORT (dashed curves) and C-PMC (solid curves) for the same physical situation as in Fig. 1 except that the refractive index was set equal 1.0 in the upper slab and equal to 1.338 in the lower slab. Upper four panels: just above interface. Lower four panels: just below the interface. Red: Δϕ = 45°; green: Δϕ = 90°; blue: Δϕ = 135°. 64 discrete ordinate streams in upper slab, 96 streams in lower slab, 109 photons.

Fig. 8
Fig. 8

Similar to Fig. 7 (aerosol particles with an optical depth of 0.1631 in each slab) except that we put non-absorbing aerosol particles of optical depth 0.3262 in the upper slab, and cloud-like particles with single-scattering albedo of 0.9, and optical depth of 5.0 in the lower slab. 96 discrete ordinate streams in upper slab, 128 streams in lower slab, 109 photons.

Fig. 9
Fig. 9

Upper four panels: Similar to Fig. 7, but at TOA for non-absorbing aerosol particles with an optical depth 0.1631 in each slab. Lower four panels: Similar to Fig. 8, but at TOA for non-absorbing aerosol particles with an optical depth of 0.3262 in the upper slab, and cloud particles with an optical depth of 5.0 and single-scattering albedo of 0.9 in the lower slab.

Equations (54)

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u d I ( τ , u , ϕ ) d τ = I ( τ , u , ϕ ) + Q ( τ , u , ϕ ) a ( τ ) 4 π 0 2 π d ϕ 1 1 d u M ( τ , u , ϕ ; u , ϕ ) I ( τ , u , ϕ ) .
x = 2 π r λ
n ( r ) = 1 2 π σ g 2 r exp [ ( ln r ln r g ) 2 / ( 2 σ g 2 ) ] , 0 n ( r ) d r = 1
ln r g = 0 ln r n ( r ) dr
σ g 2 = 0 ( ln r ln r g ) 2 n ( r ) d r .
r eff = 1 G r 1 r 2 r π r 2 n ( r ) d r = r g ( 1 + ν eff ) 5 / 2
ν eff = 1 G r eff 2 r 1 r 2 ( r r eff ) 2 π r 2 n ( r ) d r = exp [ σ g 2 ] 1 .
g = 1 4 π 4 π d ω p ( cos Θ ) cos Θ = 1 2 1 1 d ( cos Θ ) p ( cos Θ ) cos Θ
R | | = E | | r E | | i = m 1 cos θ t m 2 cos θ i m 1 cos θ t + m 2 cos θ i
T | | = E | | t E | | i = 2 m 1 cos θ i m 1 cos θ t + m 2 cos θ i
R = E r E i = m 1 cos θ i m 2 cos θ t m 1 cos θ i + m 2 cos θ t
T = E t E i = 2 m 1 cos θ i m 1 cos θ i + m 2 cos θ t
( I | | r I r U r V r ) = ( S 11 r 0 0 0 0 S 22 r 0 0 0 0 S 33 r S 34 r 0 0 S 43 r S 44 r ) ( I | | i I i U i V i )
S 11 r = R | | R | | * = | R | | | 2
S 22 r = R R * = | R | 2
S 33 r = S 44 r = Re ( R | | R * )
S 34 r = Im ( R | | R * )
S 43 r = S 34 r = Im ( R | | * R ) .
( I | | t I t U t V t ) = ( S 11 t 0 0 0 0 S 22 t 0 0 0 0 S 33 t S 34 t 0 0 S 43 t S 44 t ) ( I | | i I i U i V i )
S 11 t = | K t | 2 T | | T | | * = | K t | 2 | T | | | 2
S 22 t = | K t | 2 T T * = | K t | 2 | T | 2
S 33 t = S 44 t = | K t | 2 Re ( T | | T * )
S 34 t = | K t | 2 Im ( T | | T * )
S 43 t = S 34 t = | K t | 2 Im ( T | | * T ) .
| K t | 2 = m rel 3 μ t μ i
τ = j = 1 N d j k e , j = ln ρ
R = 1 2 [ sin 2 ( θ i θ t ) sin 2 ( θ i + θ t ) + tan 2 ( θ i θ t ) tan 2 ( θ i + θ t ) ] .
DIFF = { [ S C VDISORT S BENCH ] / | max { S BENCH } | } × 100
I S = R S ( ω ) I S
R S ( ω ) = [ 1 0 0 0 0 cos ( 2 ω ) sin ( 2 ω ) 0 0 sin ( 2 ω ) cos ( 2 ω ) 0 0 0 0 1 ]
F S ( Θ ) = [ a 1 b 1 0 0 b 1 a 2 0 0 0 0 a 3 b 2 0 0 b 2 a 4 ] .
M S ( Θ ) = R S ( Ψ ) F S ( Θ ) R S ( Φ )
I S sca = M S ( Θ ) I S inc = R S ( Ψ ) F S ( Θ ) R S ( Φ ) I S inc .
I S = D I
D = [ 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 ]
I = D 1 I S = D 1 R S ( ω ) I S = D 1 R S ( ω ) D I
R ( ω ) = D 1 R S ( ω ) D = [ cos 2 ω sin 2 ω 1 2 sin 2 ω 0 sin 2 ω cos 2 ω 1 2 sin ( 2 ω ) 0 sin ( 2 ω ) sin ( 2 ω ) cos 2 ω 0 0 0 0 1 ] .
M ( Θ ) = D 1 M S ( Θ ) D .
F ( Θ ) = D 1 F S ( Θ ) D = ( 1 2 ( a 1 + a 2 + 2 b 1 ) 1 2 ( a 1 a 2 ) 0 0 1 2 ( a 1 a 2 ) 1 2 ( a 1 + a 2 2 b 1 ) 0 0 0 0 a 3 b 2 0 0 b 2 a 4 ) .
F S ( Θ ) = 3 4 ( 1 ρ ) ( 1 + ρ / 2 ) [ 1 + ρ 1 ρ + cos 2 Θ sin 2 Θ 0 0 sin 2 Θ 1 + cos 2 Θ 0 0 0 0 2 cos Θ 0 0 0 0 2 4 ρ 1 ρ cos Θ ] .
F ( Θ ) = 3 2 ( 1 + 2 γ ) ( cos 2 Θ + γ sin 2 Θ γ 0 0 γ 1 0 0 0 0 ( 1 γ ) cos Θ 0 0 0 0 ( 1 3 γ ) cos Θ )
M ( u , ϕ ; u , ϕ ) = m = 0 2 N 1 { M m c ( u , u ) cos m ( ϕ ϕ ) + M m s ( u , u ) sin m ( ϕ ϕ ) } .
M m c ( u , u ) = A m ( u , u ) + D ˜ A m ( u , u ) D ˜
M m s ( u , u ) = A m ( u , u ) D ˜ D ˜ A m ( u , u )
A m ( u , u ) = = m 2 N 1 P m ( u ) Λ P m ( u )
Λ = ( α 1 β 1 0 0 β 1 α 2 0 0 0 0 α 3 β 2 0 0 β 2 α 4 )
a 1 ( Θ ) = = 0 2 N 1 α 1 P 0 , 0 ( cos Θ )
a 2 ( Θ ) + a 3 ( Θ ) = = 2 2 N 1 ( α 2 + α 3 ) P 2 , 2 ( cos Θ )
a 2 ( Θ ) a 3 ( Θ ) = = 2 2 N 1 ( α 2 α 3 ) P 2 , 2 ( cos Θ )
a 4 ( Θ ) = = 0 2 N 1 α 4 P 0 , 0 ( cos Θ )
b 1 ( Θ ) = = 2 2 N 1 β 1 P 0 , 2 ( cos Θ )
b 2 ( Θ ) = = 2 2 N 1 β 2 P 0 , 2 ( cos Θ ) .
P m ( u ) = ( P m , 0 ( u ) 0 0 0 0 P m , + ( u ) P m , ( u ) 0 0 P m , ( u ) P m , + ( u ) 0 0 0 0 P m , 0 ( u ) ) .
P m , ± ( u ) = 1 2 [ P m , 2 ( u ) ± P m , 2 ( u ) ] .

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