Abstract

A discontinuous finite element method (DFEM) with unstructured meshes is extended for solving the polarized radiative transfer problem in participating media. Two-dimensional media with irregular geometries and exposure to external irradiation are considered. The computational domain is divided into unstructured triangular meshes to handle the inclined and curved boundaries. The radiative intensities are assumed to be discontinuous across the inner-element boundaries of the adjacent elements. By utilizing a numerical flux with an up-winding scheme, all the discrete elements are connected and this guarantees the continuity of the computational domain. The correctness of the DFEM model for solving the polarized radiative transfer is first verified by comparing the DFEM solutions with published data of polarized radiative flux scenarios. Then the DFEM is applied to solve the polarized radiative transfer problems in irregular media, and the Stokes vector variables are analyzed.

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

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  32. C. H. Wang, Y. Y. Feng, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in a two-dimensional scattering medium considering the polarization effect,” Opt. Express 25(13), 14621–14634 (2017).
    [Crossref]
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2018 (5)

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, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonnotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transfer 209, 19–44 (2018).
[Crossref]

Z. Chen, B. Liu, S. Wang, and E. Liu, “Polarization-modulated three-dimensional imaging using a large-aperture electro-optic modulator,” Appl. Opt. 57(27), 7750–7757 (2018).
[Crossref]

C. H. Wang and H. Y. Liu, “Discontinuous Galerkin finite element method for radiative heat transfer in two-dimensional media with inner obstacles,” Numer. Heat Transfer, Part A 73(11), 806–822 (2018).
[Crossref]

Y. Y. Feng and C. H. Wang, “Discontinuous finite element method with a local numerical flux scheme for radiative transfer with strong inhomogeneity,” Int. J. Heat Mass Transfer 126, 783–795 (2018).
[Crossref]

2017 (6)

2016 (1)

C. Aghanajafi and A. Abjadpour, “Discrete ordinates method applied to radiative transfer equation in complex geometries meshed by structured and unstructured grids,” J. Braz. Soc. Mech. Sci. Eng. 38(3), 1007–1019 (2016).
[Crossref]

2015 (3)

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - Phase A,” J. Quant. Spectrosc. Radiat. Transfer 164, 8–36 (2015).
[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. Transfer 152, 114–126 (2015).
[Crossref]

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, and G. Videen, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transfer 153, 131–143 (2015).
[Crossref]

2014 (1)

2013 (1)

2012 (1)

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

2011 (1)

R. D. M. Garcia and C. E. Siewert, “A simplified implementation of the discrete-ordinates method for a class of problems in radiative transfer with polarization,” J. Quant. Spectrosc. Radiat. Transfer 112(18), 2801–2813 (2011).
[Crossref]

2010 (3)

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C. Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 111(12-13), 1931–1946 (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. Transfer 111(4), 616–633 (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 with a rough interface,” J. Quant. Spectrosc. Radiat. Transfer 111(7-8), 1025–1040 (2010).
[Crossref]

2009 (1)

2007 (1)

C. Bordier, C. Andraud, E. Charron, and J. Lafait, “Radiative transfer model with polarization effects applied to organic matter,” Phys. B 394(2), 301–305 (2007).
[Crossref]

2006 (1)

P. Talukdar, “Discrete transfer method with the concept of blocked-off region for irregular geometries,” J. Quant. Spectrosc. Radiat. Transfer 98(2), 238–248 (2006).
[Crossref]

2004 (2)

J. B. Breckinridge and B. R. Oppenheimer, “Polarization effects in reflecting coronagraphs for white-light applications in astronomy,” Astrophys. J. 600(2), 1091–1098 (2004).
[Crossref]

R. Vaillona, 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. Transfer 84(4), 383–394 (2004).
[Crossref]

2001 (1)

2000 (2)

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

M. Sakami and A. Charette, “Application of a modified discrete ordinates method to two-dimensional enclosures of irregular geometry,” J. Quant. Spectrosc. Radiat. Transfer 64(3), 275–298 (2000).
[Crossref]

1989 (1)

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

Abjadpour, A.

C. Aghanajafi and A. Abjadpour, “Discrete ordinates method applied to radiative transfer equation in complex geometries meshed by structured and unstructured grids,” J. Braz. Soc. Mech. Sci. Eng. 38(3), 1007–1019 (2016).
[Crossref]

Aghanajafi, C.

C. Aghanajafi and A. Abjadpour, “Discrete ordinates method applied to radiative transfer equation in complex geometries meshed by structured and unstructured grids,” J. Braz. Soc. Mech. Sci. Eng. 38(3), 1007–1019 (2016).
[Crossref]

Andraud, C.

C. Bordier, C. Andraud, E. Charron, and J. Lafait, “Radiative transfer model with polarization effects applied to organic matter,” Phys. B 394(2), 301–305 (2007).
[Crossref]

Barlakas, V.

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonnotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transfer 209, 19–44 (2018).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - Phase A,” J. Quant. Spectrosc. Radiat. Transfer 164, 8–36 (2015).
[Crossref]

Bordier, C.

C. Bordier, C. Andraud, E. Charron, and J. Lafait, “Radiative transfer model with polarization effects applied to organic matter,” Phys. B 394(2), 301–305 (2007).
[Crossref]

Boss, E.

Brasselet, S.

H. B. de Aguiar, S. Gigan, and S. Brasselet, “Polarization recovery through scattering media,” Sci. Adv. 3(9), e1600743 (2017).
[Crossref]

Breckinridge, J. B.

J. B. Breckinridge and B. R. Oppenheimer, “Polarization effects in reflecting coronagraphs for white-light applications in astronomy,” Astrophys. J. 600(2), 1091–1098 (2004).
[Crossref]

Brown, A. J.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, and G. Videen, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transfer 153, 131–143 (2015).
[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]

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

Budak, V. P.

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

Byrne, S.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, and G. Videen, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transfer 153, 131–143 (2015).
[Crossref]

Chami, M.

Chandrasekhar, S.

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

Charette, A.

M. Sakami and A. Charette, “Application of a modified discrete ordinates method to two-dimensional enclosures of irregular geometry,” J. Quant. Spectrosc. Radiat. Transfer 64(3), 275–298 (2000).
[Crossref]

Charron, E.

C. Bordier, C. Andraud, E. Charron, and J. Lafait, “Radiative transfer model with polarization effects applied to organic matter,” Phys. B 394(2), 301–305 (2007).
[Crossref]

Chen, Z.

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 with a rough interface,” J. Quant. Spectrosc. Radiat. Transfer 111(7-8), 1025–1040 (2010).
[Crossref]

Cohen, D.

Colaprete, A.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, and G. Videen, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transfer 153, 131–143 (2015).
[Crossref]

Cornet, C.

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonnotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transfer 209, 19–44 (2018).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - Phase A,” J. Quant. Spectrosc. Radiat. Transfer 164, 8–36 (2015).
[Crossref]

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

de Aguiar, H. B.

H. B. de Aguiar, S. Gigan, and S. Brasselet, “Polarization recovery through scattering media,” Sci. Adv. 3(9), e1600743 (2017).
[Crossref]

Dilligeard, E.

Duan, M. Z.

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

Emde, C.

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonnotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transfer 209, 19–44 (2018).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - Phase A,” J. Quant. Spectrosc. Radiat. Transfer 164, 8–36 (2015).
[Crossref]

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

Evans, F.

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonnotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transfer 209, 19–44 (2018).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - Phase A,” J. Quant. Spectrosc. Radiat. Transfer 164, 8–36 (2015).
[Crossref]

Feng, Y. Y.

Y. Y. Feng and C. H. Wang, “Discontinuous finite element method with a local numerical flux scheme for radiative transfer with strong inhomogeneity,” Int. J. Heat Mass Transfer 126, 783–795 (2018).
[Crossref]

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

Franz, B. A.

Garcia, R. D. M.

R. D. M. Garcia and C. E. Siewert, “A simplified implementation of the discrete-ordinates method for a class of problems in radiative transfer with polarization,” J. Quant. Spectrosc. Radiat. Transfer 112(18), 2801–2813 (2011).
[Crossref]

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

Gigan, S.

H. B. de Aguiar, S. Gigan, and S. Brasselet, “Polarization recovery through scattering media,” Sci. Adv. 3(9), e1600743 (2017).
[Crossref]

Hesthaven, J. S.

J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (Springer, 2007).

Hu, Y.

Huang, Y.

Y. Huang, G. D. Shi, and K. Y. Zhu, “Backward and forward Monte Carlo method in polarized radiative transfer,” Astrophys. J. 847(1), L9 (2017).
[Crossref]

Josset, D. B.

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 with a rough interface,” J. Quant. Spectrosc. Radiat. Transfer 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.

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

Kliger, D. S.

D. S. Kliger and J. W. Lewis, Polarized Light in Optics and Spectroscopy (Elsevier, 2012).

Klyukov, D. A.

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

Kokhanovsky, A. A.

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

Kong, J. A.

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

Korkin, S.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - Phase A,” J. Quant. Spectrosc. Radiat. Transfer 164, 8–36 (2015).
[Crossref]

Korkin, S. V.

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

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - Phase A,” J. Quant. Spectrosc. Radiat. Transfer 164, 8–36 (2015).
[Crossref]

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

Labonnotte, L. C.

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonnotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transfer 209, 19–44 (2018).
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Lafait, J.

C. Bordier, C. Andraud, E. Charron, and J. Lafait, “Radiative transfer model with polarization effects applied to organic matter,” Phys. B 394(2), 301–305 (2007).
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Lewis, J. W.

D. S. Kliger and J. W. Lewis, Polarized Light in Optics and Spectroscopy (Elsevier, 2012).

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B. Q. Li, Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer (Springer, 2006).

Liu, B.

Liu, E.

Liu, H. Y.

C. H. Wang and H. Y. Liu, “Discontinuous Galerkin finite element method for radiative heat transfer in two-dimensional media with inner obstacles,” Numer. Heat Transfer, Part A 73(11), 806–822 (2018).
[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. Transfer 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]

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. Transfer 111(4), 616–633 (2010).
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Lucker, P. L.

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 with a rough interface,” J. Quant. Spectrosc. Radiat. Transfer 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]

Lyapustin, A.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - Phase A,” J. Quant. Spectrosc. Radiat. Transfer 164, 8–36 (2015).
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Macke, A.

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonnotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transfer 209, 19–44 (2018).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - Phase A,” J. Quant. Spectrosc. Radiat. Transfer 164, 8–36 (2015).
[Crossref]

Mayer, B.

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonnotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transfer 209, 19–44 (2018).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - Phase A,” J. Quant. Spectrosc. Radiat. Transfer 164, 8–36 (2015).
[Crossref]

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

Mengüç, M. P.

R. Vaillona, 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. Transfer 84(4), 383–394 (2004).
[Crossref]

Michaels, T. I.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, and G. Videen, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transfer 153, 131–143 (2015).
[Crossref]

Min, Q.

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

Nakajima, T.

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

J. B. Breckinridge and B. R. Oppenheimer, “Polarization effects in reflecting coronagraphs for white-light applications in astronomy,” Astrophys. J. 600(2), 1091–1098 (2004).
[Crossref]

Ota, Y.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - Phase A,” J. Quant. Spectrosc. Radiat. Transfer 164, 8–36 (2015).
[Crossref]

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

Prikhach, A. S.

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

Rozanov, V. V.

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

Sakami, M.

M. Sakami and A. Charette, “Application of a modified discrete ordinates method to two-dimensional enclosures of irregular geometry,” J. Quant. Spectrosc. Radiat. Transfer 64(3), 275–298 (2000).
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Santer, R.

Shi, G. D.

Y. Huang, G. D. Shi, and K. Y. Zhu, “Backward and forward Monte Carlo method in polarized radiative transfer,” Astrophys. J. 847(1), L9 (2017).
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Shin, R. T.

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

Siewert, C. E.

R. D. M. Garcia and C. E. Siewert, “A simplified implementation of the discrete-ordinates method for a class of problems in radiative transfer with polarization,” J. Quant. Spectrosc. Radiat. Transfer 112(18), 2801–2813 (2011).
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C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 64(3), 227–254 (2000).
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R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 41(2), 117–145 (1989).
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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]

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. Transfer 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]

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. Transfer 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]

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

Stamnes, S.

Sun, W.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, and G. Videen, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transfer 153, 131–143 (2015).
[Crossref]

Talukdar, P.

P. Talukdar, “Discrete transfer method with the concept of blocked-off region for irregular geometries,” J. Quant. Spectrosc. Radiat. Transfer 98(2), 238–248 (2006).
[Crossref]

Tan, H. P.

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. Transfer 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]

Titus, T. N.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, and G. Videen, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transfer 153, 131–143 (2015).
[Crossref]

Trepte, C. R.

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 with a rough interface,” J. Quant. Spectrosc. Radiat. Transfer 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]

Tsang, L.

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

Vaillona, R.

R. Vaillona, 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. Transfer 84(4), 383–394 (2004).
[Crossref]

Videen, G.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, and G. Videen, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transfer 153, 131–143 (2015).
[Crossref]

Wang, C. H.

Y. Y. Feng and C. H. Wang, “Discontinuous finite element method with a local numerical flux scheme for radiative transfer with strong inhomogeneity,” Int. J. Heat Mass Transfer 126, 783–795 (2018).
[Crossref]

C. H. Wang and H. Y. Liu, “Discontinuous Galerkin finite element method for radiative heat transfer in two-dimensional media with inner obstacles,” Numer. Heat Transfer, Part A 73(11), 806–822 (2018).
[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).
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C. H. Wang, Y. Y. Feng, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in a two-dimensional scattering medium considering the polarization effect,” Opt. Express 25(13), 14621–14634 (2017).
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C. H. Wang, H. L. Yi, and H. P. Tan, “Discontinuous finite element method for vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 189, 383–397 (2017).
[Crossref]

Wang, S.

Wang, Z.

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonnotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transfer 209, 19–44 (2018).
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Warburton, T.

J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (Springer, 2007).

Wendisch, M.

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonnotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transfer 209, 19–44 (2018).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - Phase A,” J. Quant. Spectrosc. Radiat. Transfer 164, 8–36 (2015).
[Crossref]

Werdell, J.

Winker, D. M.

Wolff, M. J.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, and G. Videen, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transfer 153, 131–143 (2015).
[Crossref]

Wong, B. T.

R. Vaillona, 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. Transfer 84(4), 383–394 (2004).
[Crossref]

Xie, Y.

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

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.

Yokota, T.

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

Zege, E. P.

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

Zhai, P. W.

Zhang, Y.

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. Transfer 152, 114–126 (2015).
[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. 847(1), L9 (2017).
[Crossref]

Appl. Opt. (2)

Astrophys. J. (2)

Y. Huang, G. D. Shi, and K. Y. Zhu, “Backward and forward Monte Carlo method in polarized radiative transfer,” Astrophys. J. 847(1), L9 (2017).
[Crossref]

J. B. Breckinridge and B. R. Oppenheimer, “Polarization effects in reflecting coronagraphs for white-light applications in astronomy,” Astrophys. J. 600(2), 1091–1098 (2004).
[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]

Int. J. Heat Mass Transfer (1)

Y. Y. Feng and C. H. Wang, “Discontinuous finite element method with a local numerical flux scheme for radiative transfer with strong inhomogeneity,” Int. J. Heat Mass Transfer 126, 783–795 (2018).
[Crossref]

J. Braz. Soc. Mech. Sci. Eng. (1)

C. Aghanajafi and A. Abjadpour, “Discrete ordinates method applied to radiative transfer equation in complex geometries meshed by structured and unstructured grids,” J. Braz. Soc. Mech. Sci. Eng. 38(3), 1007–1019 (2016).
[Crossref]

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

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

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. Transfer 152, 114–126 (2015).
[Crossref]

P. Talukdar, “Discrete transfer method with the concept of blocked-off region for irregular geometries,” J. Quant. Spectrosc. Radiat. Transfer 98(2), 238–248 (2006).
[Crossref]

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

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

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - Phase A,” J. Quant. Spectrosc. Radiat. Transfer 164, 8–36 (2015).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonnotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transfer 209, 19–44 (2018).
[Crossref]

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, and G. Videen, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transfer 153, 131–143 (2015).
[Crossref]

A. J. Brown and Y. Xie, “Symmetry relations revealed in Mueller matrix hemispherical maps,” J. Quant. Spectrosc. Radiat. Transfer 113(8), 644–651 (2012).
[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 with a rough interface,” J. Quant. Spectrosc. Radiat. Transfer 111(7-8), 1025–1040 (2010).
[Crossref]

M. Sakami and A. Charette, “Application of a modified discrete ordinates method to two-dimensional enclosures of irregular geometry,” J. Quant. Spectrosc. Radiat. Transfer 64(3), 275–298 (2000).
[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. Transfer 111(4), 616–633 (2010).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “A simplified implementation of the discrete-ordinates method for a class of problems in radiative transfer with polarization,” J. Quant. Spectrosc. Radiat. Transfer 112(18), 2801–2813 (2011).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 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. Transfer 64(3), 227–254 (2000).
[Crossref]

R. Vaillona, 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. Transfer 84(4), 383–394 (2004).
[Crossref]

Numer. Heat Transfer, Part A (1)

C. H. Wang and H. Y. Liu, “Discontinuous Galerkin finite element method for radiative heat transfer in two-dimensional media with inner obstacles,” Numer. Heat Transfer, Part A 73(11), 806–822 (2018).
[Crossref]

Opt. Express (5)

Phys. B (1)

C. Bordier, C. Andraud, E. Charron, and J. Lafait, “Radiative transfer model with polarization effects applied to organic matter,” Phys. B 394(2), 301–305 (2007).
[Crossref]

Sci. Adv. (1)

H. B. de Aguiar, S. Gigan, and S. Brasselet, “Polarization recovery through scattering media,” Sci. Adv. 3(9), e1600743 (2017).
[Crossref]

Other (5)

B. Q. Li, Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer (Springer, 2006).

J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (Springer, 2007).

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

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

D. S. Kliger and J. W. Lewis, Polarized Light in Optics and Spectroscopy (Elsevier, 2012).

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Figures (13)

Fig. 1.
Fig. 1. Sketch of (a) the discrete element, its unit outward normal vectors and neighboring elements, and (b) element nodes and boundaries
Fig. 2.
Fig. 2. Physical model of the two-dimensional medium irradiated by an external collimated illumination and the spatial discretization
Fig. 3.
Fig. 3. Dimensionless polarized radiative flux distributions along the boundaries: (a) right boundary, (b) top boundary.
Fig. 4.
Fig. 4. Physical model of the two-dimensional trapezoidal medium irradiated by an external collimated illumination and the spatial discretization
Fig. 5.
Fig. 5. Stokes vector radiative flux on the bottom boundary of the trapezoidal medium
Fig. 6.
Fig. 6. Stokes vector radiative flux along the slanted and top boundaries of the trapezoidal medium
Fig. 7.
Fig. 7. Physical model of the two-dimensional trapezoidal medium irradiated by an external collimated illumination and the spatial discretization
Fig. 8.
Fig. 8. (a) Stokes vector radiative flux on the bottom boundary, and (b) artificial temperature distribution inside the semi-circle medium
Fig. 9.
Fig. 9. Angular distributions of the reflected Stokes vector components at location A/(x, y) = (0.0, 0.0) of the semicircle medium with an inner obstacle.
Fig. 10.
Fig. 10. Angular distributions of Stokes vector components at location B/(x, y) = (0.0, 1.0) for the semicircle medium with an inner obstacle.
Fig. 11.
Fig. 11. Physical model of the square medium with two obstacles and the spatial discretization.
Fig. 12.
Fig. 12. Angular distributions of the Stokes vector components at (a) location C/(x, y) = (0.3, 1.0) and (b) location D/(x, y) = (0.7, 1.0) of the square medium with two obstacles.
Fig. 13.
Fig. 13. Polarized incident radiation of Stokes vector components for the medium with two obstacles and exposed to an external irradiation on the bottom boundary.

Equations (16)

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ΩI(r,Ω)+β(r)I(r,Ω)=S(r,Ω),
β=(ββββ),
S(r,Ω)=κa(r)Ib(r)+κs4π4πZ(Ω,Ω)I(r,Ω)dΩ,
I=Ic+Id,
ΩIc(r,Ω)=β(r)Ic(r,Ω),
Ic(r,Ω)=I0exp(βs)δ(ΩΩ0),
ΩId(r,Ω)+β(r)Id(r,Ω)=Sd(r,Ω),
Sd(r,Ω) = κs4π4πZ(ΩΩ)Id(r,Ω)dΩ+κs4πZ(Ω0Ω)Ic(r,Ω0),
I(rw,Ω) = RsId(rw,Ω)+1πnwΩn>0RdId(rw,Ω)|nwΩ|dΩ,
ΩmIm(r,Ωm)+β(r)Im(r,Ωm)=Sm(r,Ωm),m = 1,2,3,,Nθ×Nφ,
Imi=1Ne=3Iimϕi,
Smi=1Ne=3Simϕi,
KmUm=Hm,
Um=(I1mQ1mU1mV1mI2mQ2mU2mV2mI3mQ3mU3mV3m),
Kjim=ΩmAeϕiϕjdA+AeϕiϕjβdA+l=1Nb=3max(0,nlΩm)ΓlϕiϕjdΓl,i,j=1,2,3,
Hjkm=AeSikmϕiϕjdA+l=1Nb=3max(0,nlΩm)ΓlIikm,+ϕiϕjdΓl,i=1,2,3;k=1,2,3,4,

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