Abstract

Transient (time-dependent) polarized radiative transfer in a scattering medium exposed to an external collimated beam illumination is conducted based on the time-dependent polarized radiative transfer theory. The transient term, which persists the nanosecond order time and cannot be ignored for the time-dependent radiative transfer problems induced by a short-pulsed beam, is considered as well as the polarization effect of the radiation. A discontinuous finite element method (DFEM) is developed for the transient vector radiative transfer problem and the derivation of the discrete form of the governing equation is presented. The correctness of the developed DFEM is first verified by comparing the DFEM solutions with the results from the literature. The DFEM is then applied to study the transient polarized radiative transfer induced by a pulsed beam. The time-dependent Stokes vector components are calculated, plotted and analyzed as functions of the axis coordinate and discrete direction. Effects of the diffuse/specular boundary and the incident beam polarization state with respect to the Stokes vector components are further analyzed for cases of different boundary reflection modes and incident beam illuminations.

© 2017 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative transfer (Dover, 1960).
  2. D. S. Kliger and J. W. Lewis, Polarized light in optics and spectroscopy (Elsevier, 2012).
  3. M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18(4), 948–960 (2001).
    [Crossref] [PubMed]
  4. A. J. Brown, “Spectral bluing induced by small particles under the Mie and Rayleigh regimes,” Icarus 239, 85–95 (2014).
    [Crossref]
  5. A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
    [Crossref]
  6. S. A. Kartazayeva, X. Ni, and R. R. Alfano, “Backscattering target detection in a turbid medium by use of circularly and linearly polarized light,” Opt. Lett. 30(10), 1168–1170 (2005).
    [Crossref] [PubMed]
  7. R. D. M. Garcia and C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36(5), 401–423 (1986).
    [Crossref]
  8. 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]
  9. K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
    [Crossref]
  10. J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earth’s atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
    [Crossref]
  11. 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]
  12. 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]
  13. 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]
  14. 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]
  15. X. Q. He, Y. Bai, Q. Zhu, 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]
  16. 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]
  17. T. Yun, N. Zeng, W. Li, D. Li, X. Jiang, and H. Ma, “Monte Carlo simulation of polarized photon scattering in anisotropic media,” Opt. Express 17(19), 16590–16602 (2009).
    [Crossref] [PubMed]
  18. 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. Transf. 111(12), 1931–1946 (2010).
    [Crossref]
  19. 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. Transf. 164, 8–36 (2015).
    [Crossref]
  20. S. Kumar and K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transf. 33, 187–294 (1999).
    [Crossref]
  21. M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
    [Crossref]
  22. B. Su and G. L. Olson, “An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium,” Ann. Nucl. Energy 24(13), 1035–1055 (1997).
    [Crossref]
  23. J. V. P. De Oliveira, A. V. Cardona, M. T. Vilhena, and R. C. Barros, “A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering,” J. Quant. Spectrosc. Radiat. Transf. 73(1), 55–62 (2002).
    [Crossref]
  24. Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123(3), 466–475 (2001).
    [Crossref]
  25. K. M. Katika and L. Pilon, “Modified method of characteristics in transient radiation transfer,” J. Quant. Spectrosc. Radiat. Transf. 98(2), 220–237 (2006).
    [Crossref]
  26. S. C. Mishra, R. Muthukumaran, and S. Maruyama, “The finite volume method approach to the collapsed dimension method in analyzing steady/transient radiative transfer problems in participating media,” Int. Commun. Heat Mass Transf. 38(3), 291–297 (2011).
    [Crossref]
  27. Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
    [Crossref]
  28. Z. X. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 159–168 (2002).
    [Crossref]
  29. 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]
  30. J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer 53(19), 3799–3806 (2010).
    [Crossref]
  31. P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
    [Crossref]
  32. 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]
  33. A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Polarized pulse waves in random discrete scatterers,” Appl. Opt. 40(30), 5495–5502 (2001).
    [Crossref] [PubMed]
  34. 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]
  35. M. Sakami and A. Dogariu, “Polarized light-pulse transport through scattering media,” J. Opt. Soc. Am. A 23(3), 664–670 (2006).
    [Crossref] [PubMed]
  36. 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]
  37. H. Yi, X. Ben, and H. Tan, “Transient radiative transfer in a scattering slab considering polarization,” Opt. Express 21(22), 26693–26713 (2013).
    [Crossref] [PubMed]
  38. W. H. Reed and T. Hill, “Triangular mesh methods for the neutron transport equation,” Los. Alamos Rep. LA-UR-73-479 (1973).
  39. J. S. Hesthaven and T. Warburton, Nodal discontinuous Galerkin methods: algorithms, analysis, and applications (Springer Science & Business Media, 2007).
  40. X. Cui and B. Q. Li, “Discontinuous finite element solution of 2-D radiative transfer with and without axisymmetry,” J. Quant. Spectrosc. Radiat. Transf. 96(3), 383–407 (2005).
    [Crossref]
  41. L. H. Liu and L. J. Liu, “Discontinuous finite element method for radiative heat transfer in semitransparent graded index medium,” J. Quant. Spectrosc. Radiat. Transf. 105(3), 377–387 (2007).
    [Crossref]
  42. P. D. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation,” Commun. Pure Appl. Math. 7(1), 159–193 (1954).
    [Crossref]

2017 (2)

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. 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]

2016 (1)

M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
[Crossref]

2015 (2)

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. Transf. 164, 8–36 (2015).
[Crossref]

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

2014 (2)

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

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]

2013 (2)

2011 (2)

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]

S. C. Mishra, R. Muthukumaran, and S. Maruyama, “The finite volume method approach to the collapsed dimension method in analyzing steady/transient radiative transfer problems in participating media,” Int. Commun. Heat Mass Transf. 38(3), 291–297 (2011).
[Crossref]

2010 (4)

J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer 53(19), 3799–3806 (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]

X. Q. He, Y. Bai, Q. Zhu, 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]

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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

2009 (1)

2008 (1)

Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
[Crossref]

2007 (2)

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earth’s atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[Crossref]

L. H. Liu and L. J. Liu, “Discontinuous finite element method for radiative heat transfer in semitransparent graded index medium,” J. Quant. Spectrosc. Radiat. Transf. 105(3), 377–387 (2007).
[Crossref]

2006 (2)

M. Sakami and A. Dogariu, “Polarized light-pulse transport through scattering media,” J. Opt. Soc. Am. A 23(3), 664–670 (2006).
[Crossref] [PubMed]

K. M. Katika and L. Pilon, “Modified method of characteristics in transient radiation transfer,” J. Quant. Spectrosc. Radiat. Transf. 98(2), 220–237 (2006).
[Crossref]

2005 (2)

X. Cui and B. Q. Li, “Discontinuous finite element solution of 2-D radiative transfer with and without axisymmetry,” J. Quant. Spectrosc. Radiat. Transf. 96(3), 383–407 (2005).
[Crossref]

S. A. Kartazayeva, X. Ni, and R. R. Alfano, “Backscattering target detection in a turbid medium by use of circularly and linearly polarized light,” Opt. Lett. 30(10), 1168–1170 (2005).
[Crossref] [PubMed]

2003 (2)

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]

P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
[Crossref]

2002 (2)

Z. X. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 159–168 (2002).
[Crossref]

J. V. P. De Oliveira, A. V. Cardona, M. T. Vilhena, and R. C. Barros, “A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering,” J. Quant. Spectrosc. Radiat. Transf. 73(1), 55–62 (2002).
[Crossref]

2001 (4)

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]

1999 (1)

S. Kumar and K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transf. 33, 187–294 (1999).
[Crossref]

1997 (1)

B. Su and G. L. Olson, “An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium,” Ann. Nucl. Energy 24(13), 1035–1055 (1997).
[Crossref]

1991 (1)

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[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. Transf. 41(2), 117–145 (1989).
[Crossref]

1986 (1)

R. D. M. Garcia and C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36(5), 401–423 (1986).
[Crossref]

1954 (1)

P. D. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation,” Commun. Pure Appl. Math. 7(1), 159–193 (1954).
[Crossref]

Aber, J.

Z. X. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 159–168 (2002).
[Crossref]

Alfano, R. R.

Bai, Y.

X. Q. He, Y. Bai, Q. Zhu, 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]

Barlakas, V.

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. Transf. 164, 8–36 (2015).
[Crossref]

Barros, R. C.

J. V. P. De Oliveira, A. V. Cardona, M. T. Vilhena, and R. C. Barros, “A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering,” J. Quant. Spectrosc. Radiat. Transf. 73(1), 55–62 (2002).
[Crossref]

Ben, X.

Brown, A. J.

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

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

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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Buividas, R.

M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
[Crossref]

Byrne, S.

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

Cardona, A. V.

J. V. P. De Oliveira, A. V. Cardona, M. T. Vilhena, and R. C. Barros, “A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering,” J. Quant. Spectrosc. Radiat. Transf. 73(1), 55–62 (2002).
[Crossref]

Chaikovskaya, L. I.

Cheng, Q.

Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
[Crossref]

C-Labonnote, 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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Cohen, D.

Colaprete, A.

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

Cornet, 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. Transf. 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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Cui, X.

X. Cui and B. Q. Li, “Discontinuous finite element solution of 2-D radiative transfer with and without axisymmetry,” J. Quant. Spectrosc. Radiat. Transf. 96(3), 383–407 (2005).
[Crossref]

De Oliveira, J. V. P.

J. V. P. De Oliveira, A. V. Cardona, M. T. Vilhena, and R. C. Barros, “A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering,” J. Quant. Spectrosc. Radiat. Transf. 73(1), 55–62 (2002).
[Crossref]

Deuzé, J. L.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earth’s atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[Crossref]

Dogariu, A.

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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Emde, 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. Transf. 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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Evans, F.

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. Transf. 164, 8–36 (2015).
[Crossref]

Evans, K. F.

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[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]

R. D. M. Garcia and C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36(5), 401–423 (1986).
[Crossref]

Garetz, B. A.

Z. X. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 159–168 (2002).
[Crossref]

Gong, F.

X. Q. He, Y. Bai, Q. Zhu, 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]

Grund, C. J.

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

Guo, Z. X.

Z. X. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 159–168 (2002).
[Crossref]

Hasegawa, S.

M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
[Crossref]

Hayasaki, Y.

M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
[Crossref]

He, X. Q.

X. Q. He, Y. Bai, Q. Zhu, 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]

Herman, M.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earth’s atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[Crossref]

Hsu, P. F.

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123(3), 466–475 (2001).
[Crossref]

Huang, D. X.

Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
[Crossref]

Huang, Z. F.

Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
[Crossref]

Ilyushin, Y. A.

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.

Jiang, X.

Juodkazis, S.

M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
[Crossref]

Kartazayeva, S. A.

Katika, K. M.

K. M. Katika and L. Pilon, “Modified method of characteristics in transient radiation transfer,” J. Quant. Spectrosc. Radiat. Transf. 98(2), 220–237 (2006).
[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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

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]

Kattawar, G. W.

Keller, J. B.

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. Transf. 111(12), 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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

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. Transf. 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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Kuga, Y.

Kumar, S.

Z. X. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 159–168 (2002).
[Crossref]

S. Kumar and K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transf. 33, 187–294 (1999).
[Crossref]

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. Transf. 164, 8–36 (2015).
[Crossref]

Lafrance, B.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earth’s atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[Crossref]

Lax, P. D.

P. D. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation,” Commun. Pure Appl. Math. 7(1), 159–193 (1954).
[Crossref]

Lenoble, J.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earth’s atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[Crossref]

Li, B. Q.

X. Cui and B. Q. Li, “Discontinuous finite element solution of 2-D radiative transfer with and without axisymmetry,” J. Quant. Spectrosc. Radiat. Transf. 96(3), 383–407 (2005).
[Crossref]

Li, D.

Li, W.

Liu, L. H.

L. H. Liu and L. J. Liu, “Discontinuous finite element method for radiative heat transfer in semitransparent graded index medium,” J. Quant. Spectrosc. Radiat. Transf. 105(3), 377–387 (2007).
[Crossref]

Liu, L. J.

L. H. Liu and L. J. Liu, “Discontinuous finite element method for radiative heat transfer in semitransparent graded index medium,” J. Quant. Spectrosc. Radiat. Transf. 105(3), 377–387 (2007).
[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]

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. Transf. 164, 8–36 (2015).
[Crossref]

Ma, H.

Macke, 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. Transf. 164, 8–36 (2015).
[Crossref]

Mahanta, P.

P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
[Crossref]

Malinauskas, M.

M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
[Crossref]

Maruyama, S.

S. C. Mishra, R. Muthukumaran, and S. Maruyama, “The finite volume method approach to the collapsed dimension method in analyzing steady/transient radiative transfer problems in participating media,” Int. Commun. Heat Mass Transf. 38(3), 291–297 (2011).
[Crossref]

Mayer, B.

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. Transf. 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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Michaels, T. I.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Mishra, S. C.

S. C. Mishra, R. Muthukumaran, and S. Maruyama, “The finite volume method approach to the collapsed dimension method in analyzing steady/transient radiative transfer problems in participating media,” Int. Commun. Heat Mass Transf. 38(3), 291–297 (2011).
[Crossref]

P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
[Crossref]

Mitra, K.

P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
[Crossref]

S. Kumar and K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transf. 33, 187–294 (1999).
[Crossref]

Mizeikis, V.

M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
[Crossref]

Moscoso, M.

Muthukumaran, R.

S. C. Mishra, R. Muthukumaran, and S. Maruyama, “The finite volume method approach to the collapsed dimension method in analyzing steady/transient radiative transfer problems in participating media,” Int. Commun. Heat Mass Transf. 38(3), 291–297 (2011).
[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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Ni, X.

Olson, G. L.

B. Su and G. L. Olson, “An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium,” Ann. Nucl. Energy 24(13), 1035–1055 (1997).
[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. Transf. 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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Papanicolaou, G.

Pilon, L.

K. M. Katika and L. Pilon, “Modified method of characteristics in transient radiation transfer,” J. Quant. Spectrosc. Radiat. Transf. 98(2), 220–237 (2006).
[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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

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]

Rath, P.

P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
[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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Saha, U. K.

P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
[Crossref]

Sakami, M.

Santer, R.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earth’s atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[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]

R. D. M. Garcia and C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36(5), 401–423 (1986).
[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.

Stephens, G. L.

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[Crossref]

Su, B.

B. Su and G. L. Olson, “An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium,” Ann. Nucl. Energy 24(13), 1035–1055 (1997).
[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]

Sun, W.

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

Tan, H.

Tan, H. P.

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]

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, Z. M.

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123(3), 466–475 (2001).
[Crossref]

Tanikawa, T.

Tanré, D.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earth’s atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[Crossref]

Titus, T. N.

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

Tynes, H. H.

Videen, G.

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

Vilhena, M. T.

J. V. P. De Oliveira, A. V. Cardona, M. T. Vilhena, and R. C. Barros, “A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering,” J. Quant. Spectrosc. Radiat. Transf. 73(1), 55–62 (2002).
[Crossref]

Wang, C. H.

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, 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]

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, J. M.

J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer 53(19), 3799–3806 (2010).
[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]

Wendisch, M.

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. Transf. 164, 8–36 (2015).
[Crossref]

Wolff, M. J.

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

Wu, C. Y.

J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer 53(19), 3799–3806 (2010).
[Crossref]

Xie, M.

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]

Yi, H.

Yi, H. L.

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, 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]

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]

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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Yu, Y. L.

Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
[Crossref]

Yun, T.

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. Transf. 111(12), 1931–1946 (2010).
[Crossref]

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]

Zeng, N.

Zhang, Y.

Zhou, H. C.

Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
[Crossref]

Zhu, Q.

X. Q. He, Y. Bai, Q. Zhu, 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]

Žukauskas, A.

M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
[Crossref]

Adv. Heat Transf. (1)

S. Kumar and K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transf. 33, 187–294 (1999).
[Crossref]

Ann. Nucl. Energy (1)

B. Su and G. L. Olson, “An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium,” Ann. Nucl. Energy 24(13), 1035–1055 (1997).
[Crossref]

Appl. Opt. (3)

ASME J. Heat Transfer (2)

Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
[Crossref]

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123(3), 466–475 (2001).
[Crossref]

Commun. Pure Appl. Math. (1)

P. D. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation,” Commun. Pure Appl. Math. 7(1), 159–193 (1954).
[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. Commun. Heat Mass Transf. (1)

S. C. Mishra, R. Muthukumaran, and S. Maruyama, “The finite volume method approach to the collapsed dimension method in analyzing steady/transient radiative transfer problems in participating media,” Int. Commun. Heat Mass Transf. 38(3), 291–297 (2011).
[Crossref]

Int. J. Heat Mass Transfer (2)

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. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer 53(19), 3799–3806 (2010).
[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. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transf. (16)

X. Cui and B. Q. Li, “Discontinuous finite element solution of 2-D radiative transfer with and without axisymmetry,” J. Quant. Spectrosc. Radiat. Transf. 96(3), 383–407 (2005).
[Crossref]

L. H. Liu and L. J. Liu, “Discontinuous finite element method for radiative heat transfer in semitransparent graded index medium,” J. Quant. Spectrosc. Radiat. Transf. 105(3), 377–387 (2007).
[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]

X. Q. He, Y. Bai, Q. Zhu, 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]

Z. X. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 159–168 (2002).
[Crossref]

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

R. D. M. Garcia and C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36(5), 401–423 (1986).
[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]

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[Crossref]

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earth’s atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[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]

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]

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. Transf. 111(12), 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. Transf. 164, 8–36 (2015).
[Crossref]

K. M. Katika and L. Pilon, “Modified method of characteristics in transient radiation transfer,” J. Quant. Spectrosc. Radiat. Transf. 98(2), 220–237 (2006).
[Crossref]

J. V. P. De Oliveira, A. V. Cardona, M. T. Vilhena, and R. C. Barros, “A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering,” J. Quant. Spectrosc. Radiat. Transf. 73(1), 55–62 (2002).
[Crossref]

Light Sci. Appl. (1)

M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
[Crossref]

Numer. Heat Transf. A (1)

P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Other (4)

W. H. Reed and T. Hill, “Triangular mesh methods for the neutron transport equation,” Los. Alamos Rep. LA-UR-73-479 (1973).

J. S. Hesthaven and T. Warburton, Nodal discontinuous Galerkin methods: algorithms, analysis, and applications (Springer Science & Business Media, 2007).

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 (15)

Fig. 1
Fig. 1 Schematic of the one-dimensional medium exposed to a square pulsed beam illumination.
Fig. 2
Fig. 2 Sketch of elements, element boundaries and the radiation values on the boundaries
Fig. 3
Fig. 3 Reflectance and transmittance obtained by FVM and DFEM for a scalar radiative transfer problem.
Fig. 4
Fig. 4 The four unique elements in the scattering phase matrix.
Fig. 5
Fig. 5 The transient DFEM solutions and the quasi-steady solutions of the Stokes vector components at z/L = 0.5 for the atmosphere with τ = 1.0, ω = 0.99 and ρd = 0.1.
Fig. 6
Fig. 6 Stokes vector component distributions at locations (a) z/L = 0, (b) z/L = 0.5 and (c) z/L = 1.0 for the atmosphere with a diffuse boundary and exposed to a collimated beam illumination.
Fig. 7
Fig. 7 Time-resolved Stokes vector components for different discrete directions at locations (a) z/L = 0 and (b) z/L = 1.0 for the atmosphere with a diffuse boundary and exposed to a collimated beam illumination.
Fig. 8
Fig. 8 Radiative flux distributions at different time for the atmosphere with a diffuse boundary and exposed to a collimated beam illumination.
Fig. 9
Fig. 9 Stokes vector component distributions at locations (a) z/L = 0, (b) z/L = 0.5 and (c) z/L = 1.0 for the atmosphere with a diffuse boundary and exposed to a short-pulsed beam illumination.
Fig. 10
Fig. 10 Time-resolved Stokes vector components for different discrete directions at locations (a) z/L = 0 and (b) z/L = 1.0 for the atmosphere with a diffuse boundary and exposed to a short-pulsed beam illumination.
Fig. 11
Fig. 11 Radiative flux distributions at different time for the atmosphere with a diffuse boundary and exposed to a short-pulsed beam illumination.
Fig. 12
Fig. 12 Stokes vector component distributions at positions (a) z/L = 0, (b) z/L = 0.5 and (c) z/L = 1.0 for the atmosphere with a specular boundary and exposed to a short-pulsed beam illumination.
Fig. 13
Fig. 13 Comparisons of (a) reflected Stokes vector components for μ = −0.4187 and (b) transmitted Stokes vector components for μ = 0.4187 for the atmosphere with diffuse/specular boundary.
Fig. 14
Fig. 14 Comparisons of time-resolved Stokes vector components at locations (a) z/L = 0 and (b) z/L = 1.0 for the atmosphere with a diffuse boundary and exposed to a linear/circular polarized beam illumination.
Fig. 15
Fig. 15 Radiative fluxes for Stokes vector components at different time for the atmosphere with a diffuse boundary and exposed to circular polarized beam illumination.

Tables (1)

Tables Icon

Table 1 The scattering angles of the selected discrete directions for the considered case with laser incident direction μ0 = 0.2.

Equations (24)

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1 c 0 I(z,Ω,t) t +ΩI(z,Ω,t)+βI(z,Ω,t)=S(z,Ω,t),
S(z,Ω,t)= κ s 4π 4π Z( Ω Ω)I(r, Ω ,t) d Ω ,
I( z w ,Ω,t)= R s I( z w , Ω ,t)+ 1 π n w Ω n >0 R d I( z w , Ω ,t)| n w Ω |d Ω ,
I(t)= I 0 [H(t)H(t t p )],
I= I c + I d ,
I c (z,t)= I 0 exp(βz), z c t z c + t p .
S(r,Ω,t)= κ s 4π [ Z(z, Ω 0 Ω) I c (z, Ω 0 ,t)+ 4π Z(z, Ω Ω) I d (z, Ω ,t)d Ω ].
I(z,Ω, t * ) t * =F[I(z,Ω, t * )],
F[(z,Ω, t * )]=LΩI(z,Ω, t * )LβI(z,Ω, t * )+LS(z,Ω, t * ).
I k I k1 Δ t * = 1 2 F[ I k ]+ 1 2 F[ I k1 ].
Ω I k (z,Ω)+ β ˜ I k (z,Ω)= S ˜ k (z,Ω),
β ˜ = 2 LΔ t * +β,
S ˜ k (z,Ω)= S k (z,Ω)+ S k1 (z,Ω)Ω I k1 (z,Ω)(β 2 LΔ t * ) I k1 (z,Ω),
Ω n I n (z, Ω n )+ β ˜ I n (z, Ω n )= S ˜ n (z, Ω n ),
< I n , Ω n W > e + ([ I n ] n e ,W) e +< β ˜ I n ,W > e =< S ˜ n ,W > e ,
<f,g > e = e fg dV, (f,g) e = e fg dA,
[ I n ]=Ω I ¯ n +| Ω n | I n n K ,
I ¯ n = 1 2 ( I n + I + n ), I n = 1 2 ( I n I + n ),
< I n , Ω n W > e + 1 2 ( ( Ω n n e +| Ω n |) I + n ,W ) e +< β ˜ I n ,W > E =< S ˜ n ,W > e 1 2 ( ( Ω n n e +| Ω n |) I n ,W ) e .
I n = i I i n ϕ i ,
K n I n = H n ,
K ji n = Ω n < ϕ i , ϕ j > e + 1 2 Ω n ( n e ϕ i , ϕ j ) e + 1 2 | Ω n | ( ϕ i , ϕ j ) e +< β ˜ ϕ i , ϕ j > e ,
H j = i=1 S ˜ i n < ϕ i , ϕ j > K 1 2 Ω n i=1 I i, n ( n K ϕ i , ϕ j ) e + 1 2 | Ω n | i=1 I i, n ( ϕ i , ϕ j ) e .
Z=( P 1 P 2 0 0 P 2 P 1 0 0 0 0 P 3 P 4 0 0 P 4 P 3 ),

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