Abstract

The characteristics of the transient and polarization must be considered for a complete and correct description of short-pulse laser transfer in a scattering medium. A Monte Carlo (MC) method combined with a time shift and superposition principle is developed to simulate transient vector (polarized) radiative transfer in a scattering medium. The transient vector radiative transfer matrix (TVRTM) is defined to describe the transient polarization behavior of short-pulse laser propagating in the scattering medium. According to the definition of reflectivity, a new criterion of reflection at Fresnel surface is presented. In order to improve the computational efficiency and accuracy, a time shift and superposition principle is applied to the MC model for transient vector radiative transfer. The results for transient scalar radiative transfer and steady-state vector radiative transfer are compared with those in published literatures, respectively, and an excellent agreement between them is observed, which validates the correctness of the present model. Finally, transient radiative transfer is simulated considering the polarization effect of short-pulse laser in a scattering medium, and the distributions of Stokes vector in angular and temporal space are presented.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2013 (1)

F. Xu, R. A. West, and A. B. Davis, “A hybrid method for modeling polarized radiative transfer in a spherical-shell planetary atmosphere,” J. Quant. Spectrosc. Radiat. Transf.117, 59–70 (2013).
[CrossRef]

2011 (4)

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

J. M. Wang and C. Y. Wu, “Second-order-accurate discrete ordinates solutions of transient radiative transfer in a scattering slab with variable refractive index,” Int. Commun. Heat Mass Transf.38(9), 1213–1218 (2011).
[CrossRef]

M. Akamatsu and Z. X. Guo, “Ultrafast radiative heat transfer in three-dimensional highly-scattering media subjected to pulse train irradiation,” Numer. Heat Tranf. Anal. Appl.59, 653–671 (2011).

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 Transfer53(19-20), 3799–3806 (2010).
[CrossRef]

C. Cornet, L. C. Labonnote, and F. Szczap, “Three-dimensional polarized Monte Carlo atmospheric radiative transfer model (3DMCPOL): 3D effects on polarized visible reflectances of a cirrus cloud,” J. Quant. Spectrosc. Radiat. Transf.111(1), 174–186 (2010).
[CrossRef]

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

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 medium with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf.111(4), 616–633 (2010).
[CrossRef]

2009 (1)

C. Y. Wu, “Monte Carlo simulation of transient radiative transfer in a medium with a variable refractive index,” Int. J. Heat Mass Transfer52(19-20), 4151–4159 (2009).
[CrossRef]

2008 (1)

E. A. Sergeeva and A. I. Korytin, “Theoretical and experimental study of blurring of a femtosecond laser pulse in a turbid medium,” Radiophys. Quantum Electron.51(4), 301–314 (2008).
[CrossRef]

2007 (3)

R. Singh, S. C. Mishra, N. K. Roy, N. S. Shekhawat, and K. Mitra, “An insight into the modeling of short-pulse laser transport through a participating medium,” Numer Heat Tranf. B-Fundam.52(4), 373–385 (2007).
[CrossRef]

J. Lenoble, M. Herman, J. L. Deuze, B. Lafrance, R. Santer, and D. Tanre, “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]

J. N. Swamy, C. Crofcheck, and M. P. Mengüç, “A Monte Carlo ray tracing study of polarized light propagation in liquid foams,” J. Quant. Spectrosc. Radiat. Transf.104(2), 277–287 (2007).
[CrossRef]

2006 (2)

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006).
[CrossRef]

M. Sakami and A. Dogariu, “Polarized light-pulse transport through scattering media,” J. Opt. Soc. Am. A23(3), 664–670 (2006).
[CrossRef] [PubMed]

2005 (2)

J. C. Ramella-Roman, S. A. Prahl, and S. L. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express13(12), 4420–4438 (2005).
[CrossRef] [PubMed]

C. Davis, C. Emde, and R. Harwood, “A 3-D polarized reversed monte carlo radiative transfer model for millimeter and submillimeter passive remote sensing in cloudy atmospheres,” IEEE Trans. Geosci. Rem. Sens.43(5), 1096–1101 (2005).
[CrossRef]

2004 (3)

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

J. C. Chai, P. F. Hsu, and Y. C. Lam, “Three-dimensional transient radiative transfer modeling using the finite-volume method,” J. Quant. Spectrosc. Radiat. Transf.86(3), 299–313 (2004).
[CrossRef]

M. Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express12(26), 6530–6539 (2004).
[CrossRef] [PubMed]

2003 (3)

J. C. Chai, “One-dimensional transient radiation heat transfer modeling using a finite-volume method,” Numer Heat Tranf. B-Fundam.44(2), 187–208 (2003).
[CrossRef]

X. D. 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, C. S. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Tranf. Anal. Appl.44, 183–197 (2003).

2002 (3)

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 169–179 (2002).
[CrossRef]

H. Ishimoto and K. Masuda, “A Monte Carlo approach for the calculation of polarized light: application to an incident narrow beam,” J. Quant. Spectrosc. Radiat. Transf.72(4), 467–483 (2002).
[CrossRef]

X. D. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: A Monte Carlo study,” J. Biomed. Opt.7(3), 279–290 (2002).
[CrossRef] [PubMed]

2001 (4)

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]

A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Polarized pulse waves in random discrete scatterers,” Appl. Opt.40(30), 5495–5502 (2001).
[CrossRef] [PubMed]

P. F. Hsu, “Effects of multiple scattering and reflective boundary on the transient radiative transfer process,” Int. J. Therm. Sci.40(6), 539–549 (2001).
[CrossRef]

S. H. Wu and C. Y. Wu, “Time-resolved spatial distribution of scattered radiative energy in a two-dimensional cylindrical medium with a large mean free path for scattering,” Int. J. Heat Mass Transfer44(14), 2611–2619 (2001).
[CrossRef]

2000 (3)

C. Y. Wu and S. H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer43(11), 2009–2020 (2000).
[CrossRef]

Z. X. Guo, S. Kumar, and K. C. San, “Multidimensional Monte Carlo simulation of short-pulse laser transport in scattering media,” J. Thermophys. Heat Transf.14, 504–511 (2000).

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]

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 (2)

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

G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere–ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr.34(8), 1453–1472 (1989).
[CrossRef]

1986 (1)

K. Masuda and T. Takashima, “Computational accuracy of radiation emerging from the ocean surface in the model atmosphere–ocean system,” Pap. Meteorol. Geophys.37(1), 1–13 (1986).
[CrossRef]

1968 (1)

Adams, C. N.

G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere–ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr.34(8), 1453–1472 (1989).
[CrossRef]

Akamatsu, M.

M. Akamatsu and Z. X. Guo, “Ultrafast radiative heat transfer in three-dimensional highly-scattering media subjected to pulse train irradiation,” Numer. Heat Tranf. Anal. Appl.59, 653–671 (2011).

Budak, Y. P.

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

Chai, J. C.

J. C. Chai, P. F. Hsu, and Y. C. Lam, “Three-dimensional transient radiative transfer modeling using the finite-volume method,” J. Quant. Spectrosc. Radiat. Transf.86(3), 299–313 (2004).
[CrossRef]

J. C. Chai, “One-dimensional transient radiation heat transfer modeling using a finite-volume method,” Numer Heat Tranf. B-Fundam.44(2), 187–208 (2003).
[CrossRef]

Chaikovskaya, L. I.

Chowdhary, J.

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

Chugh, P.

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006).
[CrossRef]

Cornet, C.

C. Cornet, L. C. Labonnote, and F. Szczap, “Three-dimensional polarized Monte Carlo atmospheric radiative transfer model (3DMCPOL): 3D effects on polarized visible reflectances of a cirrus cloud,” J. Quant. Spectrosc. Radiat. Transf.111(1), 174–186 (2010).
[CrossRef]

Crofcheck, C.

J. N. Swamy, C. Crofcheck, and M. P. Mengüç, “A Monte Carlo ray tracing study of polarized light propagation in liquid foams,” J. Quant. Spectrosc. Radiat. Transf.104(2), 277–287 (2007).
[CrossRef]

Davis, A. B.

F. Xu, R. A. West, and A. B. Davis, “A hybrid method for modeling polarized radiative transfer in a spherical-shell planetary atmosphere,” J. Quant. Spectrosc. Radiat. Transf.117, 59–70 (2013).
[CrossRef]

Davis, C.

C. Davis, C. Emde, and R. Harwood, “A 3-D polarized reversed monte carlo radiative transfer model for millimeter and submillimeter passive remote sensing in cloudy atmospheres,” IEEE Trans. Geosci. Rem. Sens.43(5), 1096–1101 (2005).
[CrossRef]

Deuze, J. L.

J. Lenoble, M. Herman, J. L. Deuze, B. Lafrance, R. Santer, and D. Tanre, “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.

Emde, C.

C. Davis, C. Emde, and R. Harwood, “A 3-D polarized reversed monte carlo radiative transfer model for millimeter and submillimeter passive remote sensing in cloudy atmospheres,” IEEE Trans. Geosci. Rem. Sens.43(5), 1096–1101 (2005).
[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 in include polarization effects,” J. Quant. Spectrosc. Radiat. Transf.41(2), 117–145 (1989).
[CrossRef]

Guo, Z. X.

M. Akamatsu and Z. X. Guo, “Ultrafast radiative heat transfer in three-dimensional highly-scattering media subjected to pulse train irradiation,” Numer. Heat Tranf. Anal. Appl.59, 653–671 (2011).

Z. X. Guo, S. Kumar, and K. C. San, “Multidimensional Monte Carlo simulation of short-pulse laser transport in scattering media,” J. Thermophys. Heat Transf.14, 504–511 (2000).

Harwood, R.

C. Davis, C. Emde, and R. Harwood, “A 3-D polarized reversed monte carlo radiative transfer model for millimeter and submillimeter passive remote sensing in cloudy atmospheres,” IEEE Trans. Geosci. Rem. Sens.43(5), 1096–1101 (2005).
[CrossRef]

Herman, M.

J. Lenoble, M. Herman, J. L. Deuze, B. Lafrance, R. Santer, and D. Tanre, “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.

J. C. Chai, P. F. Hsu, and Y. C. Lam, “Three-dimensional transient radiative transfer modeling using the finite-volume method,” J. Quant. Spectrosc. Radiat. Transf.86(3), 299–313 (2004).
[CrossRef]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 169–179 (2002).
[CrossRef]

P. F. Hsu, “Effects of multiple scattering and reflective boundary on the transient radiative transfer process,” Int. J. Therm. Sci.40(6), 539–549 (2001).
[CrossRef]

Hu, Y. X.

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

Ilyushin, Y. A.

Y. A. Ilyushin and Y. 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.

Ishimoto, H.

H. Ishimoto and K. Masuda, “A Monte Carlo approach for the calculation of polarized light: application to an incident narrow beam,” J. Quant. Spectrosc. Radiat. Transf.72(4), 467–483 (2002).
[CrossRef]

Jacques, S. L.

Jaruwatanadilok, S.

Josset, D. B.

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

Katsev, I. L.

Kattawar, G. W.

Korytin, A. I.

E. A. Sergeeva and A. I. Korytin, “Theoretical and experimental study of blurring of a femtosecond laser pulse in a turbid medium,” Radiophys. Quantum Electron.51(4), 301–314 (2008).
[CrossRef]

Kuga, Y.

Kumar, P.

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006).
[CrossRef]

Kumar, S.

Z. X. Guo, S. Kumar, and K. C. San, “Multidimensional Monte Carlo simulation of short-pulse laser transport in scattering media,” J. Thermophys. Heat Transf.14, 504–511 (2000).

Labonnote, L. C.

C. Cornet, L. C. Labonnote, and F. Szczap, “Three-dimensional polarized Monte Carlo atmospheric radiative transfer model (3DMCPOL): 3D effects on polarized visible reflectances of a cirrus cloud,” J. Quant. Spectrosc. Radiat. Transf.111(1), 174–186 (2010).
[CrossRef]

Lafrance, B.

J. Lenoble, M. Herman, J. L. Deuze, B. Lafrance, R. Santer, and D. Tanre, “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]

Lam, Y. C.

J. C. Chai, P. F. Hsu, and Y. C. Lam, “Three-dimensional transient radiative transfer modeling using the finite-volume method,” J. Quant. Spectrosc. Radiat. Transf.86(3), 299–313 (2004).
[CrossRef]

Lenoble, J.

J. Lenoble, M. Herman, J. L. Deuze, B. Lafrance, R. Santer, and D. Tanre, “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]

Lotsberg, J. K.

E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two medium with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf.111(4), 616–633 (2010).
[CrossRef]

Lucker, P. L.

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

Mahanta, P.

P. Rath, C. S. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Tranf. Anal. Appl.44, 183–197 (2003).

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]

Masuda, K.

H. Ishimoto and K. Masuda, “A Monte Carlo approach for the calculation of polarized light: application to an incident narrow beam,” J. Quant. Spectrosc. Radiat. Transf.72(4), 467–483 (2002).
[CrossRef]

K. Masuda and T. Takashima, “Computational accuracy of radiation emerging from the ocean surface in the model atmosphere–ocean system,” Pap. Meteorol. Geophys.37(1), 1–13 (1986).
[CrossRef]

Mengüç, M. P.

J. N. Swamy, C. Crofcheck, and M. P. Mengüç, “A Monte Carlo ray tracing study of polarized light propagation in liquid foams,” J. Quant. Spectrosc. Radiat. Transf.104(2), 277–287 (2007).
[CrossRef]

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

Mishra, C. S.

P. Rath, C. S. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Tranf. Anal. Appl.44, 183–197 (2003).

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]

R. Singh, S. C. Mishra, N. K. Roy, N. S. Shekhawat, and K. Mitra, “An insight into the modeling of short-pulse laser transport through a participating medium,” Numer Heat Tranf. B-Fundam.52(4), 373–385 (2007).
[CrossRef]

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006).
[CrossRef]

Mitra, K.

R. Singh, S. C. Mishra, N. K. Roy, N. S. Shekhawat, and K. Mitra, “An insight into the modeling of short-pulse laser transport through a participating medium,” Numer Heat Tranf. B-Fundam.52(4), 373–385 (2007).
[CrossRef]

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006).
[CrossRef]

P. Rath, C. S. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Tranf. Anal. Appl.44, 183–197 (2003).

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 169–179 (2002).
[CrossRef]

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]

Plass, G. N.

Prahl, S. A.

Prikhach, A. S.

Ramella-Roman, J. C.

Rath, P.

P. Rath, C. S. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Tranf. Anal. Appl.44, 183–197 (2003).

Roy, N. K.

R. Singh, S. C. Mishra, N. K. Roy, N. S. Shekhawat, and K. Mitra, “An insight into the modeling of short-pulse laser transport through a participating medium,” Numer Heat Tranf. B-Fundam.52(4), 373–385 (2007).
[CrossRef]

Saha, U. K.

P. Rath, C. S. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Tranf. Anal. Appl.44, 183–197 (2003).

Sakami, M.

M. Sakami and A. Dogariu, “Polarized light-pulse transport through scattering media,” J. Opt. Soc. Am. A23(3), 664–670 (2006).
[CrossRef] [PubMed]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 169–179 (2002).
[CrossRef]

San, K. C.

Z. X. Guo, S. Kumar, and K. C. San, “Multidimensional Monte Carlo simulation of short-pulse laser transport in scattering media,” J. Thermophys. Heat Transf.14, 504–511 (2000).

Santer, R.

J. Lenoble, M. Herman, J. L. Deuze, B. Lafrance, R. Santer, and D. Tanre, “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]

Sergeeva, E. A.

E. A. Sergeeva and A. I. Korytin, “Theoretical and experimental study of blurring of a femtosecond laser pulse in a turbid medium,” Radiophys. Quantum Electron.51(4), 301–314 (2008).
[CrossRef]

Shekhawat, N. S.

R. Singh, S. C. Mishra, N. K. Roy, N. S. Shekhawat, and K. Mitra, “An insight into the modeling of short-pulse laser transport through a participating medium,” Numer Heat Tranf. B-Fundam.52(4), 373–385 (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 in include polarization effects,” J. Quant. Spectrosc. Radiat. Transf.41(2), 117–145 (1989).
[CrossRef]

Singh, R.

R. Singh, S. C. Mishra, N. K. Roy, N. S. Shekhawat, and K. Mitra, “An insight into the modeling of short-pulse laser transport through a participating medium,” Numer Heat Tranf. B-Fundam.52(4), 373–385 (2007).
[CrossRef]

Sommersten, E. R.

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 medium with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf.111(4), 616–633 (2010).
[CrossRef]

Stamnes, J. J.

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 medium with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf.111(4), 616–633 (2010).
[CrossRef]

Stamnes, K.

E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two medium with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf.111(4), 616–633 (2010).
[CrossRef]

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]

Sun, C. W.

X. D. 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]

Swamy, J. N.

J. N. Swamy, C. Crofcheck, and M. P. Mengüç, “A Monte Carlo ray tracing study of polarized light propagation in liquid foams,” J. Quant. Spectrosc. Radiat. Transf.104(2), 277–287 (2007).
[CrossRef]

Szczap, F.

C. Cornet, L. C. Labonnote, and F. Szczap, “Three-dimensional polarized Monte Carlo atmospheric radiative transfer model (3DMCPOL): 3D effects on polarized visible reflectances of a cirrus cloud,” J. Quant. Spectrosc. Radiat. Transf.111(1), 174–186 (2010).
[CrossRef]

Takashima, T.

K. Masuda and T. Takashima, “Computational accuracy of radiation emerging from the ocean surface in the model atmosphere–ocean system,” Pap. Meteorol. Geophys.37(1), 1–13 (1986).
[CrossRef]

Tanre, D.

J. Lenoble, M. Herman, J. L. Deuze, B. Lafrance, R. Santer, and D. Tanre, “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]

Trepte, C. R.

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

Tynes, H. H.

Vaillon, R.

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

Wang, J. M.

J. M. Wang and C. Y. Wu, “Second-order-accurate discrete ordinates solutions of transient radiative transfer in a scattering slab with variable refractive index,” Int. Commun. Heat Mass Transf.38(9), 1213–1218 (2011).
[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 Transfer53(19-20), 3799–3806 (2010).
[CrossRef]

Wang, L. V.

X. D. 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]

X. D. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: A Monte Carlo study,” J. Biomed. Opt.7(3), 279–290 (2002).
[CrossRef] [PubMed]

Wang, X. D.

X. D. 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]

X. D. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: A Monte Carlo study,” J. Biomed. Opt.7(3), 279–290 (2002).
[CrossRef] [PubMed]

West, R. A.

F. Xu, R. A. West, and A. B. Davis, “A hybrid method for modeling polarized radiative transfer in a spherical-shell planetary atmosphere,” J. Quant. Spectrosc. Radiat. Transf.117, 59–70 (2013).
[CrossRef]

Wong, B. T.

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

Wu, C. Y.

J. M. Wang and C. Y. Wu, “Second-order-accurate discrete ordinates solutions of transient radiative transfer in a scattering slab with variable refractive index,” Int. Commun. Heat Mass Transf.38(9), 1213–1218 (2011).
[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 Transfer53(19-20), 3799–3806 (2010).
[CrossRef]

C. Y. Wu, “Monte Carlo simulation of transient radiative transfer in a medium with a variable refractive index,” Int. J. Heat Mass Transfer52(19-20), 4151–4159 (2009).
[CrossRef]

S. H. Wu and C. Y. Wu, “Time-resolved spatial distribution of scattered radiative energy in a two-dimensional cylindrical medium with a large mean free path for scattering,” Int. J. Heat Mass Transfer44(14), 2611–2619 (2001).
[CrossRef]

C. Y. Wu and S. H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer43(11), 2009–2020 (2000).
[CrossRef]

Wu, S. H.

S. H. Wu and C. Y. Wu, “Time-resolved spatial distribution of scattered radiative energy in a two-dimensional cylindrical medium with a large mean free path for scattering,” Int. J. Heat Mass Transfer44(14), 2611–2619 (2001).
[CrossRef]

C. Y. Wu and S. H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer43(11), 2009–2020 (2000).
[CrossRef]

Xu, F.

F. Xu, R. A. West, and A. B. Davis, “A hybrid method for modeling polarized radiative transfer in a spherical-shell planetary atmosphere,” J. Quant. Spectrosc. Radiat. Transf.117, 59–70 (2013).
[CrossRef]

Xu, M.

Yang, C. C.

X. D. 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]

Zege, E. P.

Zhai, P. W.

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

Appl. Opt. (3)

Comput. Phys. Commun. (1)

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

IEEE Trans. Geosci. Rem. Sens. (1)

C. Davis, C. Emde, and R. Harwood, “A 3-D polarized reversed monte carlo radiative transfer model for millimeter and submillimeter passive remote sensing in cloudy atmospheres,” IEEE Trans. Geosci. Rem. Sens.43(5), 1096–1101 (2005).
[CrossRef]

Int. Commun. Heat Mass Transf. (2)

J. M. Wang and C. Y. Wu, “Second-order-accurate discrete ordinates solutions of transient radiative transfer in a scattering slab with variable refractive index,” Int. Commun. Heat Mass Transf.38(9), 1213–1218 (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]

Int. J. Heat Mass Transfer (5)

C. Y. Wu and S. H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer43(11), 2009–2020 (2000).
[CrossRef]

S. H. Wu and C. Y. Wu, “Time-resolved spatial distribution of scattered radiative energy in a two-dimensional cylindrical medium with a large mean free path for scattering,” Int. J. Heat Mass Transfer44(14), 2611–2619 (2001).
[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 Transfer53(19-20), 3799–3806 (2010).
[CrossRef]

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006).
[CrossRef]

C. Y. Wu, “Monte Carlo simulation of transient radiative transfer in a medium with a variable refractive index,” Int. J. Heat Mass Transfer52(19-20), 4151–4159 (2009).
[CrossRef]

Int. J. Therm. Sci. (1)

P. F. Hsu, “Effects of multiple scattering and reflective boundary on the transient radiative transfer process,” Int. J. Therm. Sci.40(6), 539–549 (2001).
[CrossRef]

J. Biomed. Opt. (2)

X. D. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: A Monte Carlo study,” J. Biomed. Opt.7(3), 279–290 (2002).
[CrossRef] [PubMed]

X. D. 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 (1)

J. Quant. Spectrosc. Radiat. Transf. (13)

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

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 medium with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf.111(4), 616–633 (2010).
[CrossRef]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 169–179 (2002).
[CrossRef]

R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that in 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]

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]

J. Lenoble, M. Herman, J. L. Deuze, B. Lafrance, R. Santer, and D. Tanre, “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]

F. Xu, R. A. West, and A. B. Davis, “A hybrid method for modeling polarized radiative transfer in a spherical-shell planetary atmosphere,” J. Quant. Spectrosc. Radiat. Transf.117, 59–70 (2013).
[CrossRef]

H. Ishimoto and K. Masuda, “A Monte Carlo approach for the calculation of polarized light: application to an incident narrow beam,” J. Quant. Spectrosc. Radiat. Transf.72(4), 467–483 (2002).
[CrossRef]

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

J. C. Chai, P. F. Hsu, and Y. C. Lam, “Three-dimensional transient radiative transfer modeling using the finite-volume method,” J. Quant. Spectrosc. Radiat. Transf.86(3), 299–313 (2004).
[CrossRef]

J. N. Swamy, C. Crofcheck, and M. P. Mengüç, “A Monte Carlo ray tracing study of polarized light propagation in liquid foams,” J. Quant. Spectrosc. Radiat. Transf.104(2), 277–287 (2007).
[CrossRef]

C. Cornet, L. C. Labonnote, and F. Szczap, “Three-dimensional polarized Monte Carlo atmospheric radiative transfer model (3DMCPOL): 3D effects on polarized visible reflectances of a cirrus cloud,” J. Quant. Spectrosc. Radiat. Transf.111(1), 174–186 (2010).
[CrossRef]

J. Thermophys. Heat Transf. (1)

Z. X. Guo, S. Kumar, and K. C. San, “Multidimensional Monte Carlo simulation of short-pulse laser transport in scattering media,” J. Thermophys. Heat Transf.14, 504–511 (2000).

Limnol. Oceanogr. (1)

G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere–ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr.34(8), 1453–1472 (1989).
[CrossRef]

Numer Heat Tranf. B-Fundam. (2)

R. Singh, S. C. Mishra, N. K. Roy, N. S. Shekhawat, and K. Mitra, “An insight into the modeling of short-pulse laser transport through a participating medium,” Numer Heat Tranf. B-Fundam.52(4), 373–385 (2007).
[CrossRef]

J. C. Chai, “One-dimensional transient radiation heat transfer modeling using a finite-volume method,” Numer Heat Tranf. B-Fundam.44(2), 187–208 (2003).
[CrossRef]

Numer. Heat Tranf. Anal. Appl. (2)

P. Rath, C. S. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Tranf. Anal. Appl.44, 183–197 (2003).

M. Akamatsu and Z. X. Guo, “Ultrafast radiative heat transfer in three-dimensional highly-scattering media subjected to pulse train irradiation,” Numer. Heat Tranf. Anal. Appl.59, 653–671 (2011).

Opt. Express (2)

Pap. Meteorol. Geophys. (1)

K. Masuda and T. Takashima, “Computational accuracy of radiation emerging from the ocean surface in the model atmosphere–ocean system,” Pap. Meteorol. Geophys.37(1), 1–13 (1986).
[CrossRef]

Radiophys. Quantum Electron. (1)

E. A. Sergeeva and A. I. Korytin, “Theoretical and experimental study of blurring of a femtosecond laser pulse in a turbid medium,” Radiophys. Quantum Electron.51(4), 301–314 (2008).
[CrossRef]

Other (5)

S. Chandrasekhar, Radiative Transfer (Oxford University, 1950).

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

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

R. Green, Spherical Astronomy (Cambridge University, 1985).

B. A. Whitney, “Monte Carlo radiative transfer,” Arxiv preprint arXiv: 1104.4990 (2011).

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Figures (18)

Fig. 1
Fig. 1

Schematic of square pulse chain

Fig. 2
Fig. 2

Geometry of Scattering plane and Meridian planes. The photon’s direction of propagation before and after scattering is (θ1,φ1) and (θ2,φ2) respectively.

Fig. 3
Fig. 3

Schematic of the geometry of the model.

Fig. 4
Fig. 4

Comparison of transmittance and reflectance signals with respect to dimensionless time for different optical thickness: (a) τ = 0.1; (b) τ = 2.0.

Fig. 5
Fig. 5

Comparison of computational accuracy between MCM and MCM_TSS.

Fig. 6
Fig. 6

Schematic of the coupled atmosphere-water system with an oblique incident unpolarized beam.

Fig. 7
Fig. 7

Stokes vector elements just above the atmosphere–water interface for a collimated unpolarized incident beam.

Fig. 8
Fig. 8

Distributions of Stokes vector just above the atmosphere–water interface varying with time and direction.

Fig. 9
Fig. 9

Distributions of Stokes vector on the surface A1+ varying with time and direction for n = 1.44.

Fig. 10
Fig. 10

Distributions of Stokes vector on the surface A2+ varying with time and direction for n = 1.44.

Fig. 11
Fig. 11

Angular distributions of Stokes vector on the surface A1+ at three different moments for n = 1.44.

Fig. 12
Fig. 12

Time-resolved Stokes vector on the surface A1+ at three different directions for n = 1.44.

Fig. 13
Fig. 13

Angular distributions of Stokes vector on the surface A2+ at three different moments for n = 1.44.

Fig. 14
Fig. 14

Time-resolved Stokes vector on the surface A2+ at three different directions for n = 1.44.

Fig. 15
Fig. 15

Angular distributions of Stokes vector on the surface A2+ at three different moments for n = 1.0.

Fig. 16
Fig. 16

Time-resolved Stokes vector on the surface A2+ at three different directions for n = 1.0.

Fig. 17
Fig. 17

Time-resolved Stokes vector on the surface A2+ at μ = 0.707 direction with scattering albedos of 0.5 and 1 for n = 1.0.

Fig. 18
Fig. 18

Time-resolved Stokes vector on the surface A2+ at μ = 0.707 direction with optical thicknesses of 1 and 5 for n = 1.0.

Tables (1)

Tables Icon

Table 1 Comparison of computation time by MCM and MCM_TSS

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

1 c S(r,Ω) t S(r,Ω)= κ e S(r,Ω)+ κ a I b (r)+ 4π Z(r, Ω'Ω)S(r,Ω')dΩ,
S= ( I,Q,U,V ) T ,
t=(p1+ R t ) t p +( p1 ) t d ,
L= ln(1 R L ) κ ,
{ x'=x+Lsinθcosφ, y'=y+Lsinθsinφ, z'=z+Lcosθ,
t L = L c = nL c 0 ,
t'=t+ t L .
t * = c 0 κ 0 t,
S s =L(π i 2 )M(Θ)L( i 1 ) S i ,
M(Θ)=( M 1 M 2 0 0 M 2 M 1 0 0 0 0 M 3 M 4 0 0 M 4 M 3 ),
L(ϕ)=( 1 0 0 0 0 cos2ϕ sin2ϕ 0 0 sin2ϕ cos2ϕ 0 0 0 0 1 ).
S * =S/I = ( 1,Q/I ,U/ I,V/I ) T .
S ref =R S i ,
S tra =T S i ,
R=( ρ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ),
T=( 1ρ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ),
θ r =arccos( 1 R θ ), φ r =2π R φ ,
R( θ i )= 1 2 ( r 2 + r 2 r 2 r 2 0 0 r 2 r 2 r 2 + r 2 0 0 0 0 2Re{ r r * } 2Im{ r r * } 0 0 2Im{ r r * } 2Re{ r r * } ),
r = n i cos( θ i ) n t 2 n i 2 sin 2 ( θ i ) n i cos( θ i )+ n t 2 n i 2 sin 2 ( θ i ) , r = n t 2 cos( θ i ) n i n t 2 n i 2 sin 2 ( θ i ) n t 2 cos( θ i )+ n i n t 2 n i 2 sin 2 ( θ i ) ,
T( θ i )= 1 2 n t cos( θ t ) n i cos( θ i ) ( t 2 + t 2 t 2 t 2 0 0 t 2 t 2 t 2 + t 2 0 0 0 0 2Re{ t t * } 2Im{ t t * } 0 0 2Im{ t t * } 2Re{ t t * } ),
t = 2 n i cos( θ i ) n i cos( θ i )+ n t 2 n i 2 sin 2 ( θ i ) , t = 2 n i n t cos( θ i ) n t 2 cos( θ i )+ n i n t 2 n i 2 sin 2 ( θ i ) .
θ r = θ i , n t sin θ t = n i sin θ i , φ r = φ t = φ i ,
ρ= r 2 + r 2 2 .
ρ'= I' I ,
ρ'= 1 2 ( r 2 + r 2 )+ Q 2I ( r 2 r 2 )=ρ+ Q 2I ( r 2 r 2 ).
D i,t,θ,φ = q=1 N t S q / Δt N/ t p ,
S i,t,θ,φ = I 0 | cos θ 0 | | cosθ |dΩ D i,t,θ,φ ,
S i,t,θ,φ = 1 min( N Δt ,int( t/ Δt ) ) j=1 min( N Δt ,int( t/ Δt ) ) S i,( int( t/ Δt )j )Δt,θ,φ ' .
R( t )= 2π I A 1 ,t,θ,φ | cosθ |dΩ I 0 | cos θ 0 | , T( t )= 2π I A 2 ,t,θ,φ | cosθ |dΩ I 0 | cos θ 0 | .

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