Abstract

Time-dependent polarized radiative transfer in an atmosphere-ocean system exposed to external illumination is numerically investigated. The specular reflection and transmission effects based on the relative refractive index between the atmosphere and water are considered. A modified Monte Carlo (MMC) algorithm combined with time shift and superposition principle, which significantly improves the computational efficiency of the traditional Monte Carlo (TMC) method, is developed to simulate the time-dependent polarized radiative transfer process. The accuracy and computational superiority of the MMC for polarized radiative transfer in the atmosphere-ocean system are validated, and the time-resolved polarized radiative signals are discussed.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method

Peng-Wang Zhai, Yongxiang Hu, Charles R. Trepte, and Patricia L. Lucker
Opt. Express 17(4) 2057-2079 (2009)

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2019 (2)

D. Ramon, F. Steinmetz, D. Jolivet, M. Compiegne, and R. Frouin, “Modeling polarized radiative transfer in the ocean-atmosphere system with the GPU-accelerated SMART-G Monte Carlo code,” J. Quant. Spectrosc. Radiat. Transf. 222, 89–107 (2019).
[Crossref]

Y. A. Ilyushin, “Transient polarized radiative transfer in cloud layers: numerical simulation of imaging lidar returns,” J. Opt. Soc. Am. A 36(4), 540–548 (2019).
[Crossref] [PubMed]

2018 (3)

C. H. Wang, L. Qu, Y. Zhang, and H. L. Yi, “Three-dimensional polarized radiative transfer simulation using the discontinuous finite element method,” J. Quant. Spectrosc. Radiat. Transf. 208, 108–124 (2018).
[Crossref]

R. Tapimo, H. T. T. Kamdem, and D. Yemele, “A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere,” Astrophys. Space Sci. 363(3), 52 (2018).
[Crossref]

C. Emde and B. Mayer, “Errors induced by the neglect of polarization in radiance calculations for three-dimensional cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 218, 151–160 (2018).
[Crossref]

2017 (5)

2016 (2)

Y. Huang, G. D. Shi, and K. Y. Zhu, “Backward and forward Monte Carlo method in polarized radiative transfer,” Astrophys. J. 820(1), 9 (2016).
[Crossref]

C. H. Wang, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in two dimensional graded index medium by Monte Carlo method combined with the time shift and superposition principle,” Numer. Heat Transfer. Part A 69(6), 574–588 (2016).
[Crossref]

2015 (2)

C. H. Wang, Q. Ai, H. L. Yi, and H. P. Tan, “Transient radiative transfer in a graded index medium with specularly reflecting surfaces,” Numer. Heat Transfer. Part A 67(11), 1232–1252 (2015).

J. M. Zhao, J. Y. Tan, and L. H. Liu, “Monte Carlo method for polarized radiative transfer in gradient-index media,” J. Quant. Spectrosc. Radiat. Transf. 152, 114–126 (2015).
[Crossref]

2014 (3)

H. L. Yi, C. H. Wang, and H. P. Tan, “Transient radiative transfer in a complex refracting medium by a modified Monte Carlo simulation,” Int. J. Heat Mass Transfer 79, 437–449 (2014).
[Crossref]

A. J. Brown, “Equivalence relations and symmetries for laboratory, LIDAR, and planetary Müeller matrix scattering geometries,” J. Opt. Soc. Am. A 31(12), 2789–2794 (2014).
[Crossref] [PubMed]

A. J. Brown, “Spectral bluing induced by small particles under the Mie and Rayleigh regimes,” Icarus 239, 85–95 (2014).
[Crossref]

2013 (2)

P. W. Zhai, X. Y. Hu, D. B. Josset, C. R. Trepte, P. L. Lucker, and B. Lin, “Advanced angular interpolation in the vector radiative transfer for coupled atmosphere and ocean systems,” J. Quant. Spectrosc. Radiat. Transf. 115, 19–27 (2013).
[Crossref]

D. Cohen, S. Stamnes, T. Tanikawa, E. R. Sommersten, J. J. Stamnes, J. K. Lotsberg, and K. Stamnes, “Comparison of discrete ordinate and Monte Carlo simulations of polarized radiative transfer in two coupled slabs with different refractive indices,” Opt. Express 21(8), 9592–9614 (2013).
[Crossref] [PubMed]

2012 (1)

A. J. Brown and Y. Xie, “Symmetry relations revealed in Mueller matrix hemispherical maps,” J. Quant. Spectrosc. Radiat. Transf. 113(8), 644–651 (2012).
[Crossref]

2011 (1)

Y. A. Ilyushin and V. P. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182(4), 940–945 (2011).
[Crossref]

2010 (3)

X. Q. He, Y. Bai, Q. K. Zhou, and F. Gong, “A vector radiative transfer model of coupled ocean-atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectrosc. Radiat. Transf. 111(10), 1426–1448 (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. Transf. 111(4), 616–633 (2010).
[Crossref]

P. W. Zhai, Y. X. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transf. 111(7–8), 1025–1040 (2010).
[Crossref]

2009 (1)

2007 (1)

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhou, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China. Ser. D Earth Sci. 50(3), 442–452 (2007).
[Crossref]

2006 (1)

2004 (2)

Q. L. Min and M. Z. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 87(3–4), 243–259 (2004).
[Crossref]

R. Vaillon, B. T. Wong, and M. P. Mengüç, “Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,” J. Quant. Spectrosc. Radiat. Transf. 84(4), 383–394 (2004).
[Crossref]

2003 (1)

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[Crossref] [PubMed]

2002 (1)

R. Elaloufi, R. Carminati, and J. J. Greffet, “Time-dependent transport through scattering media: from radiative transfer to diffusion,” J. Opt. A 4(5), S103–S108 (2002).
[Crossref]

2001 (2)

2000 (1)

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64(3), 227–254 (2000).
[Crossref]

1994 (1)

M. I. Mishchenko, A. A. Lacis, and L. D. Travis, “Errors induced by the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 51(3), 491–510 (1994).
[Crossref]

1989 (2)

G. W. Kattawar and C. N. Adams, “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,” Limnol. Oceanogr. 34(8), 1453–1472 (1989).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 41(2), 117–145 (1989).
[Crossref]

1973 (1)

G. W. Kattawar, G. N. Plass, and J. A. Guinn, “Monte Carlo calculations of the polarization of radiation in the Earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3(4), 353–372 (1973).
[Crossref]

Adams, C. N.

G. W. Kattawar and C. N. Adams, “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,” Limnol. Oceanogr. 34(8), 1453–1472 (1989).
[Crossref]

Ai, Q.

C. H. Wang, Q. Ai, H. L. Yi, and H. P. Tan, “Transient radiative transfer in a graded index medium with specularly reflecting surfaces,” Numer. Heat Transfer. Part A 67(11), 1232–1252 (2015).

Bai, Y.

X. Q. He, Y. Bai, Q. K. Zhou, and F. Gong, “A vector radiative transfer model of coupled ocean-atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectrosc. Radiat. Transf. 111(10), 1426–1448 (2010).
[Crossref]

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhou, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China. Ser. D Earth Sci. 50(3), 442–452 (2007).
[Crossref]

Boss, E.

Brown, A. J.

A. J. Brown, “Equivalence relations and symmetries for laboratory, LIDAR, and planetary Müeller matrix scattering geometries,” J. Opt. Soc. Am. A 31(12), 2789–2794 (2014).
[Crossref] [PubMed]

A. J. Brown, “Spectral bluing induced by small particles under the Mie and Rayleigh regimes,” Icarus 239, 85–95 (2014).
[Crossref]

A. J. Brown and Y. Xie, “Symmetry relations revealed in Mueller matrix hemispherical maps,” J. Quant. Spectrosc. Radiat. Transf. 113(8), 644–651 (2012).
[Crossref]

Budak, V. P.

Y. A. Ilyushin and V. P. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182(4), 940–945 (2011).
[Crossref]

Carminati, R.

R. Elaloufi, R. Carminati, and J. J. Greffet, “Time-dependent transport through scattering media: from radiative transfer to diffusion,” J. Opt. A 4(5), S103–S108 (2002).
[Crossref]

Chaikovskaya, L. I.

Chowdhary, J.

P. W. Zhai, Y. X. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transf. 111(7–8), 1025–1040 (2010).
[Crossref]

Cohen, D.

Compiegne, M.

D. Ramon, F. Steinmetz, D. Jolivet, M. Compiegne, and R. Frouin, “Modeling polarized radiative transfer in the ocean-atmosphere system with the GPU-accelerated SMART-G Monte Carlo code,” J. Quant. Spectrosc. Radiat. Transf. 222, 89–107 (2019).
[Crossref]

Dogariu, A.

Duan, M. Z.

Q. L. Min and M. Z. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 87(3–4), 243–259 (2004).
[Crossref]

Elaloufi, R.

R. Elaloufi, R. Carminati, and J. J. Greffet, “Time-dependent transport through scattering media: from radiative transfer to diffusion,” J. Opt. A 4(5), S103–S108 (2002).
[Crossref]

Emde, C.

C. Emde and B. Mayer, “Errors induced by the neglect of polarization in radiance calculations for three-dimensional cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 218, 151–160 (2018).
[Crossref]

Feng, Y. Y.

Franz, B. A.

Frouin, R.

D. Ramon, F. Steinmetz, D. Jolivet, M. Compiegne, and R. Frouin, “Modeling polarized radiative transfer in the ocean-atmosphere system with the GPU-accelerated SMART-G Monte Carlo code,” J. Quant. Spectrosc. Radiat. Transf. 222, 89–107 (2019).
[Crossref]

Garcia, R. D. M.

R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 41(2), 117–145 (1989).
[Crossref]

Gong, F.

X. Q. He, Y. Bai, Q. K. Zhou, and F. Gong, “A vector radiative transfer model of coupled ocean-atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectrosc. Radiat. Transf. 111(10), 1426–1448 (2010).
[Crossref]

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhou, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China. Ser. D Earth Sci. 50(3), 442–452 (2007).
[Crossref]

Greffet, J. J.

R. Elaloufi, R. Carminati, and J. J. Greffet, “Time-dependent transport through scattering media: from radiative transfer to diffusion,” J. Opt. A 4(5), S103–S108 (2002).
[Crossref]

Guinn, J. A.

G. W. Kattawar, G. N. Plass, and J. A. Guinn, “Monte Carlo calculations of the polarization of radiation in the Earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3(4), 353–372 (1973).
[Crossref]

He, X. Q.

X. Q. He, Y. Bai, Q. K. Zhou, and F. Gong, “A vector radiative transfer model of coupled ocean-atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectrosc. Radiat. Transf. 111(10), 1426–1448 (2010).
[Crossref]

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhou, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China. Ser. D Earth Sci. 50(3), 442–452 (2007).
[Crossref]

Hu, X. Y.

P. W. Zhai, X. Y. Hu, D. B. Josset, C. R. Trepte, P. L. Lucker, and B. Lin, “Advanced angular interpolation in the vector radiative transfer for coupled atmosphere and ocean systems,” J. Quant. Spectrosc. Radiat. Transf. 115, 19–27 (2013).
[Crossref]

Hu, Y.

Hu, Y. X.

P. W. Zhai, Y. X. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transf. 111(7–8), 1025–1040 (2010).
[Crossref]

Huang, Y.

Y. Huang, G. D. Shi, and K. Y. Zhu, “Backward and forward Monte Carlo method in polarized radiative transfer,” Astrophys. J. 820(1), 9 (2016).
[Crossref]

Ilyushin, Y. A.

Y. A. Ilyushin, “Transient polarized radiative transfer in cloud layers: numerical simulation of imaging lidar returns,” J. Opt. Soc. Am. A 36(4), 540–548 (2019).
[Crossref] [PubMed]

Y. A. Ilyushin and V. P. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182(4), 940–945 (2011).
[Crossref]

Ishimaru, A.

Jaruwatanadilok, S.

Jolivet, D.

D. Ramon, F. Steinmetz, D. Jolivet, M. Compiegne, and R. Frouin, “Modeling polarized radiative transfer in the ocean-atmosphere system with the GPU-accelerated SMART-G Monte Carlo code,” J. Quant. Spectrosc. Radiat. Transf. 222, 89–107 (2019).
[Crossref]

Josset, D. B.

P. W. Zhai, X. Y. Hu, D. B. Josset, C. R. Trepte, P. L. Lucker, and B. Lin, “Advanced angular interpolation in the vector radiative transfer for coupled atmosphere and ocean systems,” J. Quant. Spectrosc. Radiat. Transf. 115, 19–27 (2013).
[Crossref]

P. W. Zhai, Y. X. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transf. 111(7–8), 1025–1040 (2010).
[Crossref]

Kamdem, H. T. T.

R. Tapimo, H. T. T. Kamdem, and D. Yemele, “A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere,” Astrophys. Space Sci. 363(3), 52 (2018).
[Crossref]

Katsev, I. L.

Kattawar, G. W.

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40(3), 400–412 (2001).
[Crossref] [PubMed]

G. W. Kattawar and C. N. Adams, “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,” Limnol. Oceanogr. 34(8), 1453–1472 (1989).
[Crossref]

G. W. Kattawar, G. N. Plass, and J. A. Guinn, “Monte Carlo calculations of the polarization of radiation in the Earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3(4), 353–372 (1973).
[Crossref]

Kuga, Y.

Lacis, A. A.

M. I. Mishchenko, A. A. Lacis, and L. D. Travis, “Errors induced by the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 51(3), 491–510 (1994).
[Crossref]

Lin, B.

P. W. Zhai, X. Y. Hu, D. B. Josset, C. R. Trepte, P. L. Lucker, and B. Lin, “Advanced angular interpolation in the vector radiative transfer for coupled atmosphere and ocean systems,” J. Quant. Spectrosc. Radiat. Transf. 115, 19–27 (2013).
[Crossref]

Liu, L. H.

J. M. Zhao, J. Y. Tan, and L. H. Liu, “Monte Carlo method for polarized radiative transfer in gradient-index media,” J. Quant. Spectrosc. Radiat. Transf. 152, 114–126 (2015).
[Crossref]

Lotsberg, J. K.

D. Cohen, S. Stamnes, T. Tanikawa, E. R. Sommersten, J. J. Stamnes, J. K. Lotsberg, and K. Stamnes, “Comparison of discrete ordinate and Monte Carlo simulations of polarized radiative transfer in two coupled slabs with different refractive indices,” Opt. Express 21(8), 9592–9614 (2013).
[Crossref] [PubMed]

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. Transf. 111(4), 616–633 (2010).
[Crossref]

Lucker, P. L.

P. W. Zhai, X. Y. Hu, D. B. Josset, C. R. Trepte, P. L. Lucker, and B. Lin, “Advanced angular interpolation in the vector radiative transfer for coupled atmosphere and ocean systems,” J. Quant. Spectrosc. Radiat. Transf. 115, 19–27 (2013).
[Crossref]

P. W. Zhai, Y. X. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transf. 111(7–8), 1025–1040 (2010).
[Crossref]

P. W. Zhai, Y. Hu, C. R. Trepte, and P. L. Lucker, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Express 17(4), 2057–2079 (2009).
[Crossref] [PubMed]

Mayer, B.

C. Emde and B. Mayer, “Errors induced by the neglect of polarization in radiance calculations for three-dimensional cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 218, 151–160 (2018).
[Crossref]

Mengüç, M. P.

R. Vaillon, B. T. Wong, and M. P. Mengüç, “Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,” J. Quant. Spectrosc. Radiat. Transf. 84(4), 383–394 (2004).
[Crossref]

Min, Q. L.

Q. L. Min and M. Z. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 87(3–4), 243–259 (2004).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, A. A. Lacis, and L. D. Travis, “Errors induced by the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 51(3), 491–510 (1994).
[Crossref]

Pan, D. L.

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhou, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China. Ser. D Earth Sci. 50(3), 442–452 (2007).
[Crossref]

Plass, G. N.

G. W. Kattawar, G. N. Plass, and J. A. Guinn, “Monte Carlo calculations of the polarization of radiation in the Earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3(4), 353–372 (1973).
[Crossref]

Prikhach, A. S.

Qu, L.

C. H. Wang, L. Qu, Y. Zhang, and H. L. Yi, “Three-dimensional polarized radiative transfer simulation using the discontinuous finite element method,” J. Quant. Spectrosc. Radiat. Transf. 208, 108–124 (2018).
[Crossref]

Ramon, D.

D. Ramon, F. Steinmetz, D. Jolivet, M. Compiegne, and R. Frouin, “Modeling polarized radiative transfer in the ocean-atmosphere system with the GPU-accelerated SMART-G Monte Carlo code,” J. Quant. Spectrosc. Radiat. Transf. 222, 89–107 (2019).
[Crossref]

Sakami, M.

Shi, G. D.

Y. Huang, G. D. Shi, and K. Y. Zhu, “Backward and forward Monte Carlo method in polarized radiative transfer,” Astrophys. J. 820(1), 9 (2016).
[Crossref]

Siewert, C. E.

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64(3), 227–254 (2000).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 41(2), 117–145 (1989).
[Crossref]

Sommersten, E. R.

D. Cohen, S. Stamnes, T. Tanikawa, E. R. Sommersten, J. J. Stamnes, J. K. Lotsberg, and K. Stamnes, “Comparison of discrete ordinate and Monte Carlo simulations of polarized radiative transfer in two coupled slabs with different refractive indices,” Opt. Express 21(8), 9592–9614 (2013).
[Crossref] [PubMed]

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. Transf. 111(4), 616–633 (2010).
[Crossref]

Stamnes, J. J.

D. Cohen, S. Stamnes, T. Tanikawa, E. R. Sommersten, J. J. Stamnes, J. K. Lotsberg, and K. Stamnes, “Comparison of discrete ordinate and Monte Carlo simulations of polarized radiative transfer in two coupled slabs with different refractive indices,” Opt. Express 21(8), 9592–9614 (2013).
[Crossref] [PubMed]

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. Transf. 111(4), 616–633 (2010).
[Crossref]

Stamnes, K.

D. Cohen, S. Stamnes, T. Tanikawa, E. R. Sommersten, J. J. Stamnes, J. K. Lotsberg, and K. Stamnes, “Comparison of discrete ordinate and Monte Carlo simulations of polarized radiative transfer in two coupled slabs with different refractive indices,” Opt. Express 21(8), 9592–9614 (2013).
[Crossref] [PubMed]

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. Transf. 111(4), 616–633 (2010).
[Crossref]

Stamnes, S.

Steinmetz, F.

D. Ramon, F. Steinmetz, D. Jolivet, M. Compiegne, and R. Frouin, “Modeling polarized radiative transfer in the ocean-atmosphere system with the GPU-accelerated SMART-G Monte Carlo code,” J. Quant. Spectrosc. Radiat. Transf. 222, 89–107 (2019).
[Crossref]

Sun, C. W.

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[Crossref] [PubMed]

Tan, H. P.

C. H. Wang, H. L. Yi, and H. P. Tan, “Transient polarized radiative transfer analysis in a scattering medium by a discontinuous finite element method,” Opt. Express 25(7), 7418–7442 (2017).
[Crossref] [PubMed]

C. H. Wang, H. L. Yi, and H. P. Tan, “Discontinuous finite element method for vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 189, 383–397 (2017).
[Crossref]

C. H. Wang, Y. Y. Feng, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient/time-dependent radiative transfer in a two-dimensional scattering medium considering the polarization effect,” Opt. Express 25(13), 14621–14634 (2017).
[Crossref] [PubMed]

C. H. Wang, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in two dimensional graded index medium by Monte Carlo method combined with the time shift and superposition principle,” Numer. Heat Transfer. Part A 69(6), 574–588 (2016).
[Crossref]

C. H. Wang, Q. Ai, H. L. Yi, and H. P. Tan, “Transient radiative transfer in a graded index medium with specularly reflecting surfaces,” Numer. Heat Transfer. Part A 67(11), 1232–1252 (2015).

H. L. Yi, C. H. Wang, and H. P. Tan, “Transient radiative transfer in a complex refracting medium by a modified Monte Carlo simulation,” Int. J. Heat Mass Transfer 79, 437–449 (2014).
[Crossref]

Tan, J. Y.

J. M. Zhao, J. Y. Tan, and L. H. Liu, “Monte Carlo method for polarized radiative transfer in gradient-index media,” J. Quant. Spectrosc. Radiat. Transf. 152, 114–126 (2015).
[Crossref]

Tanikawa, T.

Tapimo, R.

R. Tapimo, H. T. T. Kamdem, and D. Yemele, “A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere,” Astrophys. Space Sci. 363(3), 52 (2018).
[Crossref]

Travis, L. D.

M. I. Mishchenko, A. A. Lacis, and L. D. Travis, “Errors induced by the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 51(3), 491–510 (1994).
[Crossref]

Trepte, C. R.

P. W. Zhai, X. Y. Hu, D. B. Josset, C. R. Trepte, P. L. Lucker, and B. Lin, “Advanced angular interpolation in the vector radiative transfer for coupled atmosphere and ocean systems,” J. Quant. Spectrosc. Radiat. Transf. 115, 19–27 (2013).
[Crossref]

P. W. Zhai, Y. X. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transf. 111(7–8), 1025–1040 (2010).
[Crossref]

P. W. Zhai, Y. Hu, C. R. Trepte, and P. L. Lucker, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Express 17(4), 2057–2079 (2009).
[Crossref] [PubMed]

Tynes, H. H.

Vaillon, R.

R. Vaillon, B. T. Wong, and M. P. Mengüç, “Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,” J. Quant. Spectrosc. Radiat. Transf. 84(4), 383–394 (2004).
[Crossref]

Wang, C. H.

C. H. Wang, L. Qu, Y. Zhang, and H. L. Yi, “Three-dimensional polarized radiative transfer simulation using the discontinuous finite element method,” J. Quant. Spectrosc. Radiat. Transf. 208, 108–124 (2018).
[Crossref]

C. H. Wang, H. L. Yi, and H. P. Tan, “Discontinuous finite element method for vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 189, 383–397 (2017).
[Crossref]

C. H. Wang, H. L. Yi, and H. P. Tan, “Transient polarized radiative transfer analysis in a scattering medium by a discontinuous finite element method,” Opt. Express 25(7), 7418–7442 (2017).
[Crossref] [PubMed]

C. H. Wang, Y. Y. Feng, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient/time-dependent radiative transfer in a two-dimensional scattering medium considering the polarization effect,” Opt. Express 25(13), 14621–14634 (2017).
[Crossref] [PubMed]

C. H. Wang, Y. Zhang, H. L. Yi, and M. Xie, “Analysis of transient radiative transfer induced by an incident short-pulsed laser in a graded-index medium with Fresnel boundaries,” Appl. Opt. 56(7), 1861–1871 (2017).
[Crossref] [PubMed]

C. H. Wang, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in two dimensional graded index medium by Monte Carlo method combined with the time shift and superposition principle,” Numer. Heat Transfer. Part A 69(6), 574–588 (2016).
[Crossref]

C. H. Wang, Q. Ai, H. L. Yi, and H. P. Tan, “Transient radiative transfer in a graded index medium with specularly reflecting surfaces,” Numer. Heat Transfer. Part A 67(11), 1232–1252 (2015).

H. L. Yi, C. H. Wang, and H. P. Tan, “Transient radiative transfer in a complex refracting medium by a modified Monte Carlo simulation,” Int. J. Heat Mass Transfer 79, 437–449 (2014).
[Crossref]

Wang, L. V.

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[Crossref] [PubMed]

Wang, X.

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[Crossref] [PubMed]

Werdell, J.

Winker, D. M.

Wong, B. T.

R. Vaillon, B. T. Wong, and M. P. Mengüç, “Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,” J. Quant. Spectrosc. Radiat. Transf. 84(4), 383–394 (2004).
[Crossref]

Xie, M.

Xie, Y.

A. J. Brown and Y. Xie, “Symmetry relations revealed in Mueller matrix hemispherical maps,” J. Quant. Spectrosc. Radiat. Transf. 113(8), 644–651 (2012).
[Crossref]

Yang, C. C.

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[Crossref] [PubMed]

Yemele, D.

R. Tapimo, H. T. T. Kamdem, and D. Yemele, “A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere,” Astrophys. Space Sci. 363(3), 52 (2018).
[Crossref]

Yi, H. L.

C. H. Wang, L. Qu, Y. Zhang, and H. L. Yi, “Three-dimensional polarized radiative transfer simulation using the discontinuous finite element method,” J. Quant. Spectrosc. Radiat. Transf. 208, 108–124 (2018).
[Crossref]

C. H. Wang, H. L. Yi, and H. P. Tan, “Discontinuous finite element method for vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 189, 383–397 (2017).
[Crossref]

C. H. Wang, H. L. Yi, and H. P. Tan, “Transient polarized radiative transfer analysis in a scattering medium by a discontinuous finite element method,” Opt. Express 25(7), 7418–7442 (2017).
[Crossref] [PubMed]

C. H. Wang, Y. Zhang, H. L. Yi, and M. Xie, “Analysis of transient radiative transfer induced by an incident short-pulsed laser in a graded-index medium with Fresnel boundaries,” Appl. Opt. 56(7), 1861–1871 (2017).
[Crossref] [PubMed]

C. H. Wang, Y. Y. Feng, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient/time-dependent radiative transfer in a two-dimensional scattering medium considering the polarization effect,” Opt. Express 25(13), 14621–14634 (2017).
[Crossref] [PubMed]

C. H. Wang, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in two dimensional graded index medium by Monte Carlo method combined with the time shift and superposition principle,” Numer. Heat Transfer. Part A 69(6), 574–588 (2016).
[Crossref]

C. H. Wang, Q. Ai, H. L. Yi, and H. P. Tan, “Transient radiative transfer in a graded index medium with specularly reflecting surfaces,” Numer. Heat Transfer. Part A 67(11), 1232–1252 (2015).

H. L. Yi, C. H. Wang, and H. P. Tan, “Transient radiative transfer in a complex refracting medium by a modified Monte Carlo simulation,” Int. J. Heat Mass Transfer 79, 437–449 (2014).
[Crossref]

Zege, E. P.

Zhai, P. W.

P. W. Zhai, Y. Hu, D. M. Winker, B. A. Franz, J. Werdell, and E. Boss, “Vector radiative transfer model for coupled atmosphere and ocean systems including inelastic sources in ocean waters,” Opt. Express 25(8), A223–A239 (2017).
[Crossref] [PubMed]

P. W. Zhai, X. Y. Hu, D. B. Josset, C. R. Trepte, P. L. Lucker, and B. Lin, “Advanced angular interpolation in the vector radiative transfer for coupled atmosphere and ocean systems,” J. Quant. Spectrosc. Radiat. Transf. 115, 19–27 (2013).
[Crossref]

P. W. Zhai, Y. X. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transf. 111(7–8), 1025–1040 (2010).
[Crossref]

P. W. Zhai, Y. Hu, C. R. Trepte, and P. L. Lucker, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Express 17(4), 2057–2079 (2009).
[Crossref] [PubMed]

Zhang, Y.

C. H. Wang, L. Qu, Y. Zhang, and H. L. Yi, “Three-dimensional polarized radiative transfer simulation using the discontinuous finite element method,” J. Quant. Spectrosc. Radiat. Transf. 208, 108–124 (2018).
[Crossref]

C. H. Wang, Y. Zhang, H. L. Yi, and M. Xie, “Analysis of transient radiative transfer induced by an incident short-pulsed laser in a graded-index medium with Fresnel boundaries,” Appl. Opt. 56(7), 1861–1871 (2017).
[Crossref] [PubMed]

C. H. Wang, Y. Y. Feng, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient/time-dependent radiative transfer in a two-dimensional scattering medium considering the polarization effect,” Opt. Express 25(13), 14621–14634 (2017).
[Crossref] [PubMed]

C. H. Wang, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in two dimensional graded index medium by Monte Carlo method combined with the time shift and superposition principle,” Numer. Heat Transfer. Part A 69(6), 574–588 (2016).
[Crossref]

Zhao, J. M.

J. M. Zhao, J. Y. Tan, and L. H. Liu, “Monte Carlo method for polarized radiative transfer in gradient-index media,” J. Quant. Spectrosc. Radiat. Transf. 152, 114–126 (2015).
[Crossref]

Zhou, Q. K.

X. Q. He, Y. Bai, Q. K. Zhou, and F. Gong, “A vector radiative transfer model of coupled ocean-atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectrosc. Radiat. Transf. 111(10), 1426–1448 (2010).
[Crossref]

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhou, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China. Ser. D Earth Sci. 50(3), 442–452 (2007).
[Crossref]

Zhu, K. Y.

Y. Huang, G. D. Shi, and K. Y. Zhu, “Backward and forward Monte Carlo method in polarized radiative transfer,” Astrophys. J. 820(1), 9 (2016).
[Crossref]

Appl. Opt. (3)

Astrophys. J. (1)

Y. Huang, G. D. Shi, and K. Y. Zhu, “Backward and forward Monte Carlo method in polarized radiative transfer,” Astrophys. J. 820(1), 9 (2016).
[Crossref]

Astrophys. Space Sci. (1)

R. Tapimo, H. T. T. Kamdem, and D. Yemele, “A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere,” Astrophys. Space Sci. 363(3), 52 (2018).
[Crossref]

Comput. Phys. Commun. (1)

Y. A. Ilyushin and V. P. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182(4), 940–945 (2011).
[Crossref]

Icarus (1)

A. J. Brown, “Spectral bluing induced by small particles under the Mie and Rayleigh regimes,” Icarus 239, 85–95 (2014).
[Crossref]

Int. J. Heat Mass Transfer (1)

H. L. Yi, C. H. Wang, and H. P. Tan, “Transient radiative transfer in a complex refracting medium by a modified Monte Carlo simulation,” Int. J. Heat Mass Transfer 79, 437–449 (2014).
[Crossref]

J. Biomed. Opt. (1)

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[Crossref] [PubMed]

J. Opt. A (1)

R. Elaloufi, R. Carminati, and J. J. Greffet, “Time-dependent transport through scattering media: from radiative transfer to diffusion,” J. Opt. A 4(5), S103–S108 (2002).
[Crossref]

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

J. Phys. Oceanogr. (1)

G. W. Kattawar, G. N. Plass, and J. A. Guinn, “Monte Carlo calculations of the polarization of radiation in the Earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3(4), 353–372 (1973).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (15)

M. I. Mishchenko, A. A. Lacis, and L. D. Travis, “Errors induced by the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 51(3), 491–510 (1994).
[Crossref]

C. Emde and B. Mayer, “Errors induced by the neglect of polarization in radiance calculations for three-dimensional cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 218, 151–160 (2018).
[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. Transf. 111(4), 616–633 (2010).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 41(2), 117–145 (1989).
[Crossref]

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64(3), 227–254 (2000).
[Crossref]

Q. L. Min and M. Z. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 87(3–4), 243–259 (2004).
[Crossref]

C. H. Wang, H. L. Yi, and H. P. Tan, “Discontinuous finite element method for vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 189, 383–397 (2017).
[Crossref]

C. H. Wang, L. Qu, Y. Zhang, and H. L. Yi, “Three-dimensional polarized radiative transfer simulation using the discontinuous finite element method,” J. Quant. Spectrosc. Radiat. Transf. 208, 108–124 (2018).
[Crossref]

R. Vaillon, B. T. Wong, and M. P. Mengüç, “Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,” J. Quant. Spectrosc. Radiat. Transf. 84(4), 383–394 (2004).
[Crossref]

J. M. Zhao, J. Y. Tan, and L. H. Liu, “Monte Carlo method for polarized radiative transfer in gradient-index media,” J. Quant. Spectrosc. Radiat. Transf. 152, 114–126 (2015).
[Crossref]

X. Q. He, Y. Bai, Q. K. Zhou, and F. Gong, “A vector radiative transfer model of coupled ocean-atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectrosc. Radiat. Transf. 111(10), 1426–1448 (2010).
[Crossref]

D. Ramon, F. Steinmetz, D. Jolivet, M. Compiegne, and R. Frouin, “Modeling polarized radiative transfer in the ocean-atmosphere system with the GPU-accelerated SMART-G Monte Carlo code,” J. Quant. Spectrosc. Radiat. Transf. 222, 89–107 (2019).
[Crossref]

P. W. Zhai, Y. X. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transf. 111(7–8), 1025–1040 (2010).
[Crossref]

P. W. Zhai, X. Y. Hu, D. B. Josset, C. R. Trepte, P. L. Lucker, and B. Lin, “Advanced angular interpolation in the vector radiative transfer for coupled atmosphere and ocean systems,” J. Quant. Spectrosc. Radiat. Transf. 115, 19–27 (2013).
[Crossref]

A. J. Brown and Y. Xie, “Symmetry relations revealed in Mueller matrix hemispherical maps,” J. Quant. Spectrosc. Radiat. Transf. 113(8), 644–651 (2012).
[Crossref]

Limnol. Oceanogr. (1)

G. W. Kattawar and C. N. Adams, “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,” Limnol. Oceanogr. 34(8), 1453–1472 (1989).
[Crossref]

Numer. Heat Transfer. Part A (2)

C. H. Wang, Q. Ai, H. L. Yi, and H. P. Tan, “Transient radiative transfer in a graded index medium with specularly reflecting surfaces,” Numer. Heat Transfer. Part A 67(11), 1232–1252 (2015).

C. H. Wang, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in two dimensional graded index medium by Monte Carlo method combined with the time shift and superposition principle,” Numer. Heat Transfer. Part A 69(6), 574–588 (2016).
[Crossref]

Opt. Express (5)

Sci. China. Ser. D Earth Sci. (1)

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhou, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China. Ser. D Earth Sci. 50(3), 442–452 (2007).
[Crossref]

Other (5)

S. Chandrasekhar, Radiative Transfer (Oxford, 1960).

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

H. C. Hulst and H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

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

G. C. Pomraning, The Equations of Radiation Hydrodynamics (Dover, 1973).

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Figures (10)

Fig. 1
Fig. 1 The incident direction, scattering direction, and rotation angle in the scattering event.
Fig. 2
Fig. 2 Schematic of the atmosphere-ocean system exposed to an external illumination.
Fig. 3
Fig. 3 Stokes vector components just below the interface obtained by different algorithms.
Fig. 4
Fig. 4 Comparisons of Stokes vector components obtained by MMC at a long time against the DOM results.
Fig. 5
Fig. 5 Zenith distributions of Stokes vector components at different time moments for the location just below the atmosphere-ocean interface.
Fig. 6
Fig. 6 Stokes vector contour varying with time and direction for the location just below the atmosphere-ocean interface.
Fig. 7
Fig. 7 Zenith distributions of Stokes vector components at different time moments for the location just above the atmosphere-ocean interface.
Fig. 8
Fig. 8 Stokes vector contour varying with time and direction for the location just above the atmosphere-ocean interface.
Fig. 9
Fig. 9 Zenith distributions of Stokes vector components at different time moments for the location at the top of the atmosphere.
Fig. 10
Fig. 10 Stokes vector contour varying with time and direction for the location at the top of the atmosphere.

Equations (19)

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I= ( I,Q,U,V ) T ,
t 0 = R t t p ,
L= ln(1 R L ) β ,
{ x'=x+Lsinθcosφ y'=y+Lsinθsinφ z'=z+Lcosθ t =t+ nL c 0 ,
I( Ω s )=Z( Ω i Ω s )I( Ω i ),
Z( Ω i Ω s )=(π ψ s )P( Ω i , Ω s )( ψ i ),
(ψ)=( 1 0 0 0 0 cos2ψ sin2ψ 0 0 sin2ψ cos2ψ 0 0 0 0 1 ).
P(Θ)= 3 4 ( cos 2 Θ+1 sin 2 Θ 0 0 sin 2 Θ cos 2 Θ+1 0 0 0 0 2cosΘ 0 0 0 0 2cosΘ ),
I * = I I = ( 1, Q I , U I , V I ) T ,
R(θ)= 1 2 ( r h 2 + r v 2 r h 2 r v 2 0 0 r h 2 r v 2 r h 2 + r v 2 0 0 0 0 2Re( r v r h * ) 2Im( r h r v * ) 0 0 2Im( r v r h * ) 2Re( r v r h * ) ),
r v (θ)= cosθ n 2 sin θ 2 cosθ+ n 2 sin θ 2 ,
r h (θ)= n 2 cosθ n 2 sin θ 2 n 2 cosθ+ n 2 sin θ 2 ,
T(θ)= n 2 cosϑ cosθ ( t h 2 + t v 2 t h 2 t v 2 0 0 t h 2 t v 2 t h 2 + t v 2 0 0 0 0 2Re( t v t h * ) 2Im( t h t v * ) 0 0 2Im( t v t h * ) 2Re( t v t h * ) ),
t v (θ)= 2cosθ cosθ+ n 2 sin 2 θ ,
t h (θ)= 2ncosθ n 2 cosθ+ n 2 sin 2 θ .
M i,Ω,P = j=1 N t I j * / Δt N/ t p ,
I i,Ω,P = I 0 | cos θ 0 | | cosθ |dΩ M i,Ω,P ,
I i,Ω,P Δ = I 0 | cos θ 0 | | cosθ |dΩ M i,Ω,P Δ = I 0 | cos θ 0 | | cosθ |dΩ j=1 N t I j * N Δ ,
I m,Ω,P = 1 min(m,M) k=1 min(m,M) I ( mk+1 ),Ω,P Δ .

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