Abstract

Within a gradient-index medium, the radiative rays propagate in curved paths, which makes polarized states change continuously and the solution to the radiative transfer be thus more complex and difficult. In this paper, an arbitrary multilayer model is developed to approximately simulate vector (polarized) radiative transfer in a gradient-index plane-parallel medium. The gradient-index medium is divided into an arbitrary number of sublayers, and each sublayer has a uniform refractive index and two virtual Fresnel’s interfaces where only transmission (refraction) is considered. Thus the polarization caused by the curved propagation of lights is approximated by that resulting from refraction on the interfaces. Radiative transfer with consideration of polarization caused by particle scattering and refraction (reflection) on the interfaces (surfaces) in the multilayer model is solved by the MC method. The grid independence of results obtained by the multilayer model for vector radiative transfer in gradient-index medium shows that the convergent solution of Stokes vector will be achieved provided that the sublayer number is large enough. The results for apparent emissivity of gradient-index medium and Stokes vector for two-layer medium are compared well with those in published literatures. Finally, we investigate polarized behaviors of radiative transfer in Rayleigh scattering slabs with linear and sinusoidal gradient indexes and present angular distributions of Stokes vector. Results show that total reflection inside the gradient-index medium resulting from the curved paths traveled by the photons affects greatly the angular distribution characteristics of Stokes vector.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Asllanaj and S. Fumeron, “Modified finite volume method applied to radiative transfer in 2D complex geometries and graded index media,” J. Quant. Spectrosc. Radiat. Transfer 111, 274–279 (2010).
    [CrossRef]
  2. O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, 1972).
  3. Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990).
  4. Y. T. Qiao, Graded Index Optics (Science, 1991).
  5. P. B. Abdallah and V. L. Dez, “Thermal emission of a semi-transparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 67, 185–198 (2000).
    [CrossRef]
  6. P. B. Abdallah and V. L. Dez, “Temperature field inside an absorbing-emitting semi-transparent slab at radiative equilibrium with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 65, 595–608 (2000).
    [CrossRef]
  7. P. B. Abdallah and V. L. Dez, “Radiative flux field inside an absorbing-emitting semi-transparent slab with variable spatial refractive index at radiative conductive coupling,” J. Quant. Spectrosc. Radiat. Transfer 67, 125–137 (2000).
    [CrossRef]
  8. P. B. Abdallah and V. L. Dez, “Thermal emission of a two-dimensional rectangular cavity with spatial affine refractive index,” J. Quant. Spectrosc. Radiat. Transfer 66, 555–569 (2000).
    [CrossRef]
  9. Y. Huang, X. L. Xia, and H.-P. Tan, “Radiative intensity solution and thermal emission analysis of a semitransparent medium layer with sinusoidal refractive index,” J. Quant. Spectrosc. Radiat. Transfer 74, 217–233 (2002).
    [CrossRef]
  10. Y. Huang, X. L. Xia, and H. P. Tan, “Temperature field of radiative equilibrium in a semitransparent slab with a linear refractive index and gray walls,” J. Quant. Spectrosc. Radiat. Transfer 74, 249–261 (2002).
    [CrossRef]
  11. Y. Huang, K. Y. Zhu, and J. Wang, “Temperature field of radiative equilibrium in a two-dimensional graded index medium with gray boundaries,” J. Quant. Spectrosc. Radiat. Transfer 110, 1013–1026 (2009).
    [CrossRef]
  12. L. H. Liu, H. P. Tan, and Q. Z. Yu, “Temperature distributions in an absorbing-emitting-scattering semitransparent slab with variable spatial refractive index,” Int. J. Heat Mass Transfer 46, 2917–2920 (2003).
    [CrossRef]
  13. Y. Huang, X. G. Liang, and X. L. Xia, “Monte Carlo simulation of radiative transfer in scattering, emitting, absorbing slab with gradient index,” J. Quant. Spectrosc. Radiat. Transfer 92, 111–120 (2005).
    [CrossRef]
  14. D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (2002).
    [CrossRef]
  15. L. H. Liu, “Finite element solution of radiative transfer across a slab with variable spatial refractive index,” Int. J. Heat Mass Transfer 48, 2260–2265 (2005).
    [CrossRef]
  16. L. H. Liu, “Finite volume method for radiation heat transfer in graded index medium,” J. Thermophys. Heat Transfer 20, 59–66 (2006).
    [CrossRef]
  17. N. A. Krishna and S. C. Mishra, “Discrete transfer method applied to radiative transfer in a variable refractive index semitransparent medium,” J. Quant. Spectrosc. Radiat. Transfer 102, 432–440 (2006).
    [CrossRef]
  18. J. M. Zhao and L. H. Liu, “Solution of radiative heat transfer in graded index media by least square spectral element method,” Int. J. Heat Mass Transfer 50, 2634–2642 (2007).
    [CrossRef]
  19. G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiple scattered light from haze and clouds,” Appl. Opt. 7, 1519–1527 (1968).
    [CrossRef]
  20. R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that in include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 41, 117–145 (1989).
    [CrossRef]
  21. K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
    [CrossRef]
  22. C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 64, 227–254 (2000).
    [CrossRef]
  23. 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. Transfer 107, 479–507 (2007).
    [CrossRef]
  24. 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. Transfer 117, 59–70 (2013).
    [CrossRef]
  25. 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. Transfer 72, 467–483 (2002).
    [CrossRef]
  26. X. D. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: a Monte Carlo study,” J. Biomed. Opt. 7, 279–290 (2002).
    [CrossRef]
  27. M. Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express 12, 6530–6539 (2004).
    [CrossRef]
  28. R. Vaillona, B. T. Wong, and M. P. Mengüç, “Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,” J. Quant. Spectrosc. Radiat. Transfer 84, 383–394 (2004).
    [CrossRef]
  29. 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. Remote Sens. 43, 1096–1101 (2005).
    [CrossRef]
  30. 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. Express 13, 4420–4438 (2005).
    [CrossRef]
  31. 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. Transfer 104, 277–287 (2007).
    [CrossRef]
  32. 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. Transfer 111, 174–186 (2010).
    [CrossRef]
  33. G. W. Kattawar, G. N. Plass, and J. A. Guinn, “Monte Carlo calculations of the polarization of radiation in the Earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3, 353–372 (1973).
    [CrossRef]
  34. 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, 1453–1472 (1989).
    [CrossRef]
  35. P. W. Zhai, Y. X. Hu, C. R. Trepte, and P. L. Lucker, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Express 17, 2057–2079 (2009).
    [CrossRef]
  36. P. W. Zhai, Y. X. Hu, J. Cdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transfer 111, 1025–1040 (2010).
    [CrossRef]
  37. P. W. Zhai, Y. X. Hu, D. B. Josset, C. R. Trepte, P. L. Lucker, and B. Lin, “Advanced angular interpolation in the vector radiative transfer for coupled atmosphere and ocean systems,” J. Quant. Spectrosc. Radiat. Transfer 115, 19–27 (2013).
    [CrossRef]
  38. X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhu, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China Ser. D 50, 442452 (2007).
    [CrossRef]
  39. X. Q. He, Y. Bai, Q. K. 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. Transfer 111, 1426–1448 (2010).
    [CrossRef]
  40. Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transfer 111, 878–894 (2010).
    [CrossRef]
  41. 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. Transfer 111, 616–633 (2010).
    [CrossRef]
  42. R. D. M. Garcia, “Fresnel boundary and interface conditions for polarized radiative transfer in a multilayer medium,” J. Quant. Spectrosc. Radiat. Transfer 113, 306–317 (2012).
    [CrossRef]
  43. R. D. M. Garcia, “Radiative transfer with polarization in a multi-layer medium subject to Fresnel boundary and interface conditions,” J. Quant. Spectrosc. Radiat. Transfer 115, 28–45 (2013).
    [CrossRef]
  44. S. Chandrasekhar, Radiative Transfer (Oxford University, 1950).
  45. C. W. Lau and K. M. Watson, “Radiation transport along curved ray paths,” J. Math. Phys. 11, 3125–3137 (1970).
    [CrossRef]
  46. J. M. Zhao, J. Y. Tan, and L. H. Liu, “On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media,” J. Quant. Spectrosc. Radiat. Transfer 113, 239–250 (2012).
    [CrossRef]
  47. M. F. Modest, Radiative Heat Transfer, 2nd ed. (Academic, 2003).
  48. M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, 1992).
  49. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  50. M. I. Mishchenko, Scattering, Absorption, and Emission of Light by Small Particles (NASA, 2002).
  51. 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, 400–412 (2001).
    [CrossRef]
  52. R. Green, Spherical Astronomy (Cambridge University, 1985).
  53. B. A. Whitney, “Monte Carlo radiative transfer,” Arxiv preprint arXiv: 1104.4990 (2011).

2013 (3)

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. Transfer 117, 59–70 (2013).
[CrossRef]

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

R. D. M. Garcia, “Radiative transfer with polarization in a multi-layer medium subject to Fresnel boundary and interface conditions,” J. Quant. Spectrosc. Radiat. Transfer 115, 28–45 (2013).
[CrossRef]

2012 (2)

J. M. Zhao, J. Y. Tan, and L. H. Liu, “On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media,” J. Quant. Spectrosc. Radiat. Transfer 113, 239–250 (2012).
[CrossRef]

R. D. M. Garcia, “Fresnel boundary and interface conditions for polarized radiative transfer in a multilayer medium,” J. Quant. Spectrosc. Radiat. Transfer 113, 306–317 (2012).
[CrossRef]

2010 (6)

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

X. Q. He, Y. Bai, Q. K. 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. Transfer 111, 1426–1448 (2010).
[CrossRef]

Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transfer 111, 878–894 (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. Transfer 111, 616–633 (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. Transfer 111, 174–186 (2010).
[CrossRef]

F. Asllanaj and S. Fumeron, “Modified finite volume method applied to radiative transfer in 2D complex geometries and graded index media,” J. Quant. Spectrosc. Radiat. Transfer 111, 274–279 (2010).
[CrossRef]

2009 (2)

Y. Huang, K. Y. Zhu, and J. Wang, “Temperature field of radiative equilibrium in a two-dimensional graded index medium with gray boundaries,” J. Quant. Spectrosc. Radiat. Transfer 110, 1013–1026 (2009).
[CrossRef]

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

2007 (4)

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhu, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China Ser. D 50, 442452 (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. Transfer 107, 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. Transfer 104, 277–287 (2007).
[CrossRef]

J. M. Zhao and L. H. Liu, “Solution of radiative heat transfer in graded index media by least square spectral element method,” Int. J. Heat Mass Transfer 50, 2634–2642 (2007).
[CrossRef]

2006 (2)

L. H. Liu, “Finite volume method for radiation heat transfer in graded index medium,” J. Thermophys. Heat Transfer 20, 59–66 (2006).
[CrossRef]

N. A. Krishna and S. C. Mishra, “Discrete transfer method applied to radiative transfer in a variable refractive index semitransparent medium,” J. Quant. Spectrosc. Radiat. Transfer 102, 432–440 (2006).
[CrossRef]

2005 (4)

Y. Huang, X. G. Liang, and X. L. Xia, “Monte Carlo simulation of radiative transfer in scattering, emitting, absorbing slab with gradient index,” J. Quant. Spectrosc. Radiat. Transfer 92, 111–120 (2005).
[CrossRef]

L. H. Liu, “Finite element solution of radiative transfer across a slab with variable spatial refractive index,” Int. J. Heat Mass Transfer 48, 2260–2265 (2005).
[CrossRef]

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. Remote Sens. 43, 1096–1101 (2005).
[CrossRef]

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. Express 13, 4420–4438 (2005).
[CrossRef]

2004 (2)

M. Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express 12, 6530–6539 (2004).
[CrossRef]

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

2003 (1)

L. H. Liu, H. P. Tan, and Q. Z. Yu, “Temperature distributions in an absorbing-emitting-scattering semitransparent slab with variable spatial refractive index,” Int. J. Heat Mass Transfer 46, 2917–2920 (2003).
[CrossRef]

2002 (5)

Y. Huang, X. L. Xia, and H.-P. Tan, “Radiative intensity solution and thermal emission analysis of a semitransparent medium layer with sinusoidal refractive index,” J. Quant. Spectrosc. Radiat. Transfer 74, 217–233 (2002).
[CrossRef]

Y. Huang, X. L. Xia, and H. P. Tan, “Temperature field of radiative equilibrium in a semitransparent slab with a linear refractive index and gray walls,” J. Quant. Spectrosc. Radiat. Transfer 74, 249–261 (2002).
[CrossRef]

D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (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. Transfer 72, 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, 279–290 (2002).
[CrossRef]

2001 (1)

2000 (5)

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

P. B. Abdallah and V. L. Dez, “Thermal emission of a semi-transparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 67, 185–198 (2000).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Temperature field inside an absorbing-emitting semi-transparent slab at radiative equilibrium with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 65, 595–608 (2000).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Radiative flux field inside an absorbing-emitting semi-transparent slab with variable spatial refractive index at radiative conductive coupling,” J. Quant. Spectrosc. Radiat. Transfer 67, 125–137 (2000).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Thermal emission of a two-dimensional rectangular cavity with spatial affine refractive index,” J. Quant. Spectrosc. Radiat. Transfer 66, 555–569 (2000).
[CrossRef]

1991 (1)

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

1989 (2)

G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere–ocean with scattering according to a Rayleigh phase matrix: Effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[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. Transfer 41, 117–145 (1989).
[CrossRef]

1973 (1)

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

1970 (1)

C. W. Lau and K. M. Watson, “Radiation transport along curved ray paths,” J. Math. Phys. 11, 3125–3137 (1970).
[CrossRef]

1968 (1)

Abdallah, P. B.

P. B. Abdallah and V. L. Dez, “Thermal emission of a semi-transparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 67, 185–198 (2000).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Temperature field inside an absorbing-emitting semi-transparent slab at radiative equilibrium with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 65, 595–608 (2000).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Thermal emission of a two-dimensional rectangular cavity with spatial affine refractive index,” J. Quant. Spectrosc. Radiat. Transfer 66, 555–569 (2000).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Radiative flux field inside an absorbing-emitting semi-transparent slab with variable spatial refractive index at radiative conductive coupling,” J. Quant. Spectrosc. Radiat. Transfer 67, 125–137 (2000).
[CrossRef]

Adams, C. N.

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

Asllanaj, F.

F. Asllanaj and S. Fumeron, “Modified finite volume method applied to radiative transfer in 2D complex geometries and graded index media,” J. Quant. Spectrosc. Radiat. Transfer 111, 274–279 (2010).
[CrossRef]

Bai, Y.

X. Q. He, Y. Bai, Q. K. 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. Transfer 111, 1426–1448 (2010).
[CrossRef]

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhu, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China Ser. D 50, 442452 (2007).
[CrossRef]

Brewster, M. Q.

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, 1992).

Cdhary, J.

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

Chaikovskaya, L. I.

Chandrasekhar, S.

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

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. Transfer 111, 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. Transfer 104, 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. Transfer 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. Remote Sens. 43, 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. Transfer 107, 479–507 (2007).
[CrossRef]

Dez, V. L.

D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (2002).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Temperature field inside an absorbing-emitting semi-transparent slab at radiative equilibrium with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 65, 595–608 (2000).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Thermal emission of a semi-transparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 67, 185–198 (2000).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Radiative flux field inside an absorbing-emitting semi-transparent slab with variable spatial refractive index at radiative conductive coupling,” J. Quant. Spectrosc. Radiat. Transfer 67, 125–137 (2000).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Thermal emission of a two-dimensional rectangular cavity with spatial affine refractive index,” J. Quant. Spectrosc. Radiat. Transfer 66, 555–569 (2000).
[CrossRef]

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. Remote Sens. 43, 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. Transfer 46, 413–423 (1991).
[CrossRef]

Fumeron, S.

F. Asllanaj and S. Fumeron, “Modified finite volume method applied to radiative transfer in 2D complex geometries and graded index media,” J. Quant. Spectrosc. Radiat. Transfer 111, 274–279 (2010).
[CrossRef]

Garcia, R. D. M.

R. D. M. Garcia, “Radiative transfer with polarization in a multi-layer medium subject to Fresnel boundary and interface conditions,” J. Quant. Spectrosc. Radiat. Transfer 115, 28–45 (2013).
[CrossRef]

R. D. M. Garcia, “Fresnel boundary and interface conditions for polarized radiative transfer in a multilayer medium,” J. Quant. Spectrosc. Radiat. Transfer 113, 306–317 (2012).
[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. Transfer 41, 117–145 (1989).
[CrossRef]

Gong, F.

X. Q. He, Y. Bai, Q. K. 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. Transfer 111, 1426–1448 (2010).
[CrossRef]

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhu, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China Ser. D 50, 442452 (2007).
[CrossRef]

Green, R.

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

Guinn, J. A.

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

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. Remote Sens. 43, 1096–1101 (2005).
[CrossRef]

He, X. Q.

X. Q. He, Y. Bai, Q. K. 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. Transfer 111, 1426–1448 (2010).
[CrossRef]

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhu, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China Ser. D 50, 442452 (2007).
[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. Transfer 107, 479–507 (2007).
[CrossRef]

Higurashi, A.

Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transfer 111, 878–894 (2010).
[CrossRef]

Hu, Y. X.

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

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

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

Huang, Y.

Y. Huang, K. Y. Zhu, and J. Wang, “Temperature field of radiative equilibrium in a two-dimensional graded index medium with gray boundaries,” J. Quant. Spectrosc. Radiat. Transfer 110, 1013–1026 (2009).
[CrossRef]

Y. Huang, X. G. Liang, and X. L. Xia, “Monte Carlo simulation of radiative transfer in scattering, emitting, absorbing slab with gradient index,” J. Quant. Spectrosc. Radiat. Transfer 92, 111–120 (2005).
[CrossRef]

Y. Huang, X. L. Xia, and H.-P. Tan, “Radiative intensity solution and thermal emission analysis of a semitransparent medium layer with sinusoidal refractive index,” J. Quant. Spectrosc. Radiat. Transfer 74, 217–233 (2002).
[CrossRef]

Y. Huang, X. L. Xia, and H. P. Tan, “Temperature field of radiative equilibrium in a semitransparent slab with a linear refractive index and gray walls,” J. Quant. Spectrosc. Radiat. Transfer 74, 249–261 (2002).
[CrossRef]

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. Transfer 72, 467–483 (2002).
[CrossRef]

Jacques, S. L.

Josset, D. B.

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

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

Katsev, I. L.

Kattawar, G. W.

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400–412 (2001).
[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, 1453–1472 (1989).
[CrossRef]

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

G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiple scattered light from haze and clouds,” Appl. Opt. 7, 1519–1527 (1968).
[CrossRef]

Kravtsov, Y. A.

Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990).

Krishna, N. A.

N. A. Krishna and S. C. Mishra, “Discrete transfer method applied to radiative transfer in a variable refractive index semitransparent medium,” J. Quant. Spectrosc. Radiat. Transfer 102, 432–440 (2006).
[CrossRef]

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. Transfer 111, 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. Transfer 107, 479–507 (2007).
[CrossRef]

Lau, C. W.

C. W. Lau and K. M. Watson, “Radiation transport along curved ray paths,” J. Math. Phys. 11, 3125–3137 (1970).
[CrossRef]

Lemonnier, D.

D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (2002).
[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. Transfer 107, 479–507 (2007).
[CrossRef]

Liang, X. G.

Y. Huang, X. G. Liang, and X. L. Xia, “Monte Carlo simulation of radiative transfer in scattering, emitting, absorbing slab with gradient index,” J. Quant. Spectrosc. Radiat. Transfer 92, 111–120 (2005).
[CrossRef]

Lin, B.

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

Liu, L. H.

J. M. Zhao, J. Y. Tan, and L. H. Liu, “On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media,” J. Quant. Spectrosc. Radiat. Transfer 113, 239–250 (2012).
[CrossRef]

J. M. Zhao and L. H. Liu, “Solution of radiative heat transfer in graded index media by least square spectral element method,” Int. J. Heat Mass Transfer 50, 2634–2642 (2007).
[CrossRef]

L. H. Liu, “Finite volume method for radiation heat transfer in graded index medium,” J. Thermophys. Heat Transfer 20, 59–66 (2006).
[CrossRef]

L. H. Liu, “Finite element solution of radiative transfer across a slab with variable spatial refractive index,” Int. J. Heat Mass Transfer 48, 2260–2265 (2005).
[CrossRef]

L. H. Liu, H. P. Tan, and Q. Z. Yu, “Temperature distributions in an absorbing-emitting-scattering semitransparent slab with variable spatial refractive index,” Int. J. Heat Mass Transfer 46, 2917–2920 (2003).
[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. Transfer 111, 616–633 (2010).
[CrossRef]

Lucker, P. L.

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

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

P. W. Zhai, Y. X. Hu, C. R. Trepte, and P. L. Lucker, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Express 17, 2057–2079 (2009).
[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. Transfer 72, 467–483 (2002).
[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. Transfer 104, 277–287 (2007).
[CrossRef]

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

Mishchenko, M. I.

M. I. Mishchenko, Scattering, Absorption, and Emission of Light by Small Particles (NASA, 2002).

Mishra, S. C.

N. A. Krishna and S. C. Mishra, “Discrete transfer method applied to radiative transfer in a variable refractive index semitransparent medium,” J. Quant. Spectrosc. Radiat. Transfer 102, 432–440 (2006).
[CrossRef]

Modest, M. F.

M. F. Modest, Radiative Heat Transfer, 2nd ed. (Academic, 2003).

Nakajima, T.

Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transfer 111, 878–894 (2010).
[CrossRef]

Orlov, Y. I.

Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990).

Ota, Y.

Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transfer 111, 878–894 (2010).
[CrossRef]

Pan, D. L.

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhu, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China Ser. D 50, 442452 (2007).
[CrossRef]

Plass, G. N.

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

G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiple scattered light from haze and clouds,” Appl. Opt. 7, 1519–1527 (1968).
[CrossRef]

Prahl, S. A.

Prikhach, A. S.

Qiao, Y. T.

Y. T. Qiao, Graded Index Optics (Science, 1991).

Ramella-Roman, J. C.

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. Transfer 107, 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. Transfer 64, 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. Transfer 41, 117–145 (1989).
[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. Transfer 111, 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. Transfer 111, 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. Transfer 111, 616–633 (2010).
[CrossRef]

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, 1972).

Stephens, G. L.

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

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. Transfer 104, 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. Transfer 111, 174–186 (2010).
[CrossRef]

Tan, H. P.

L. H. Liu, H. P. Tan, and Q. Z. Yu, “Temperature distributions in an absorbing-emitting-scattering semitransparent slab with variable spatial refractive index,” Int. J. Heat Mass Transfer 46, 2917–2920 (2003).
[CrossRef]

Y. Huang, X. L. Xia, and H. P. Tan, “Temperature field of radiative equilibrium in a semitransparent slab with a linear refractive index and gray walls,” J. Quant. Spectrosc. Radiat. Transfer 74, 249–261 (2002).
[CrossRef]

Tan, H.-P.

Y. Huang, X. L. Xia, and H.-P. Tan, “Radiative intensity solution and thermal emission analysis of a semitransparent medium layer with sinusoidal refractive index,” J. Quant. Spectrosc. Radiat. Transfer 74, 217–233 (2002).
[CrossRef]

Tan, J. Y.

J. M. Zhao, J. Y. Tan, and L. H. Liu, “On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media,” J. Quant. Spectrosc. Radiat. Transfer 113, 239–250 (2012).
[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. Transfer 107, 479–507 (2007).
[CrossRef]

Trepte, C. R.

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

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

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

Tynes, H. H.

Vaillona, R.

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

Van de Hulst, H. C.

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

Wang, J.

Y. Huang, K. Y. Zhu, and J. Wang, “Temperature field of radiative equilibrium in a two-dimensional graded index medium with gray boundaries,” J. Quant. Spectrosc. Radiat. Transfer 110, 1013–1026 (2009).
[CrossRef]

Wang, L. V.

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

Wang, X. D.

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

Watson, K. M.

C. W. Lau and K. M. Watson, “Radiation transport along curved ray paths,” J. Math. Phys. 11, 3125–3137 (1970).
[CrossRef]

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. Transfer 117, 59–70 (2013).
[CrossRef]

Whitney, B. A.

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

Wong, B. T.

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

Xia, X. L.

Y. Huang, X. G. Liang, and X. L. Xia, “Monte Carlo simulation of radiative transfer in scattering, emitting, absorbing slab with gradient index,” J. Quant. Spectrosc. Radiat. Transfer 92, 111–120 (2005).
[CrossRef]

Y. Huang, X. L. Xia, and H. P. Tan, “Temperature field of radiative equilibrium in a semitransparent slab with a linear refractive index and gray walls,” J. Quant. Spectrosc. Radiat. Transfer 74, 249–261 (2002).
[CrossRef]

Y. Huang, X. L. Xia, and H.-P. Tan, “Radiative intensity solution and thermal emission analysis of a semitransparent medium layer with sinusoidal refractive index,” J. Quant. Spectrosc. Radiat. Transfer 74, 217–233 (2002).
[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. Transfer 117, 59–70 (2013).
[CrossRef]

Xu, M.

Yokota, T.

Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transfer 111, 878–894 (2010).
[CrossRef]

Yu, Q. Z.

L. H. Liu, H. P. Tan, and Q. Z. Yu, “Temperature distributions in an absorbing-emitting-scattering semitransparent slab with variable spatial refractive index,” Int. J. Heat Mass Transfer 46, 2917–2920 (2003).
[CrossRef]

Zege, E. P.

Zhai, P. W.

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

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

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

Zhao, J. M.

J. M. Zhao, J. Y. Tan, and L. H. Liu, “On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media,” J. Quant. Spectrosc. Radiat. Transfer 113, 239–250 (2012).
[CrossRef]

J. M. Zhao and L. H. Liu, “Solution of radiative heat transfer in graded index media by least square spectral element method,” Int. J. Heat Mass Transfer 50, 2634–2642 (2007).
[CrossRef]

Zhu, K. Y.

Y. Huang, K. Y. Zhu, and J. Wang, “Temperature field of radiative equilibrium in a two-dimensional graded index medium with gray boundaries,” J. Quant. Spectrosc. Radiat. Transfer 110, 1013–1026 (2009).
[CrossRef]

Zhu, Q. K.

X. Q. He, Y. Bai, Q. K. 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. Transfer 111, 1426–1448 (2010).
[CrossRef]

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhu, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China Ser. D 50, 442452 (2007).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Geosci. Remote 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. Remote Sens. 43, 1096–1101 (2005).
[CrossRef]

Int. J. Heat Mass Transfer (3)

J. M. Zhao and L. H. Liu, “Solution of radiative heat transfer in graded index media by least square spectral element method,” Int. J. Heat Mass Transfer 50, 2634–2642 (2007).
[CrossRef]

L. H. Liu, H. P. Tan, and Q. Z. Yu, “Temperature distributions in an absorbing-emitting-scattering semitransparent slab with variable spatial refractive index,” Int. J. Heat Mass Transfer 46, 2917–2920 (2003).
[CrossRef]

L. H. Liu, “Finite element solution of radiative transfer across a slab with variable spatial refractive index,” Int. J. Heat Mass Transfer 48, 2260–2265 (2005).
[CrossRef]

J. Biomed. Opt. (1)

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

J. Math. Phys. (1)

C. W. Lau and K. M. Watson, “Radiation transport along curved ray paths,” J. Math. Phys. 11, 3125–3137 (1970).
[CrossRef]

J. Phys. Oceanogr. (1)

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

J. Quant. Spectrosc. Radiat. Transfer (28)

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

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

X. Q. He, Y. Bai, Q. K. 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. Transfer 111, 1426–1448 (2010).
[CrossRef]

Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transfer 111, 878–894 (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. Transfer 111, 616–633 (2010).
[CrossRef]

R. D. M. Garcia, “Fresnel boundary and interface conditions for polarized radiative transfer in a multilayer medium,” J. Quant. Spectrosc. Radiat. Transfer 113, 306–317 (2012).
[CrossRef]

R. D. M. Garcia, “Radiative transfer with polarization in a multi-layer medium subject to Fresnel boundary and interface conditions,” J. Quant. Spectrosc. Radiat. Transfer 115, 28–45 (2013).
[CrossRef]

N. A. Krishna and S. C. Mishra, “Discrete transfer method applied to radiative transfer in a variable refractive index semitransparent medium,” J. Quant. Spectrosc. Radiat. Transfer 102, 432–440 (2006).
[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. Transfer 104, 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. Transfer 111, 174–186 (2010).
[CrossRef]

Y. Huang, X. G. Liang, and X. L. Xia, “Monte Carlo simulation of radiative transfer in scattering, emitting, absorbing slab with gradient index,” J. Quant. Spectrosc. Radiat. Transfer 92, 111–120 (2005).
[CrossRef]

D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (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. Transfer 41, 117–145 (1989).
[CrossRef]

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

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 64, 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. Transfer 107, 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. Transfer 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. Transfer 72, 467–483 (2002).
[CrossRef]

F. Asllanaj and S. Fumeron, “Modified finite volume method applied to radiative transfer in 2D complex geometries and graded index media,” J. Quant. Spectrosc. Radiat. Transfer 111, 274–279 (2010).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Thermal emission of a semi-transparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 67, 185–198 (2000).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Temperature field inside an absorbing-emitting semi-transparent slab at radiative equilibrium with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 65, 595–608 (2000).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Radiative flux field inside an absorbing-emitting semi-transparent slab with variable spatial refractive index at radiative conductive coupling,” J. Quant. Spectrosc. Radiat. Transfer 67, 125–137 (2000).
[CrossRef]

P. B. Abdallah and V. L. Dez, “Thermal emission of a two-dimensional rectangular cavity with spatial affine refractive index,” J. Quant. Spectrosc. Radiat. Transfer 66, 555–569 (2000).
[CrossRef]

Y. Huang, X. L. Xia, and H.-P. Tan, “Radiative intensity solution and thermal emission analysis of a semitransparent medium layer with sinusoidal refractive index,” J. Quant. Spectrosc. Radiat. Transfer 74, 217–233 (2002).
[CrossRef]

Y. Huang, X. L. Xia, and H. P. Tan, “Temperature field of radiative equilibrium in a semitransparent slab with a linear refractive index and gray walls,” J. Quant. Spectrosc. Radiat. Transfer 74, 249–261 (2002).
[CrossRef]

Y. Huang, K. Y. Zhu, and J. Wang, “Temperature field of radiative equilibrium in a two-dimensional graded index medium with gray boundaries,” J. Quant. Spectrosc. Radiat. Transfer 110, 1013–1026 (2009).
[CrossRef]

J. M. Zhao, J. Y. Tan, and L. H. Liu, “On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media,” J. Quant. Spectrosc. Radiat. Transfer 113, 239–250 (2012).
[CrossRef]

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

J. Thermophys. Heat Transfer (1)

L. H. Liu, “Finite volume method for radiation heat transfer in graded index medium,” J. Thermophys. Heat Transfer 20, 59–66 (2006).
[CrossRef]

Limnol. Oceanogr. (1)

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

Opt. Express (3)

Sci. China Ser. D (1)

X. Q. He, D. L. Pan, Y. Bai, Q. K. Zhu, and F. Gong, “Vector radiative transfer numerical model of coupled ocean-atmosphere system using matrix-operator method,” Sci. China Ser. D 50, 442452 (2007).
[CrossRef]

Other (10)

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

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, 1972).

Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990).

Y. T. Qiao, Graded Index Optics (Science, 1991).

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

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

M. F. Modest, Radiative Heat Transfer, 2nd ed. (Academic, 2003).

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, 1992).

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

M. I. Mishchenko, Scattering, Absorption, and Emission of Light by Small Particles (NASA, 2002).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1.
Fig. 1.

Schematic of the multilayer approximation model of vector radiative transfer in gradient-index medium. The solid line denotes the incidence or refraction, and the dashed line denotes the reflection.

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.

Apparent directional emissivity just above the upper surface compared with that in [5]. (a) Isothermal medium and (b) nonisothermal medium with a linear distribution of temperature.

Fig. 4.
Fig. 4.

Apparent directional emissivity just above the upper surface and below the lower surface compared with those in [13], respectively. Two linear temperature distributions are considered in the medium with a linear refractive index of n(z)=1.80.6z/L. (a) T0/TL=0.8 and (b) T0/TL=1.2.

Fig. 5.
Fig. 5.

Apparent directional emissivity just above the upper surface compared with those in [13]. Three different values of albedo are considered, and the medium is isothermal with a sinusoidal refractive index of n(z)=1.2+0.6sin(πz/L).

Fig. 6.
Fig. 6.

Stokes vector just above the atmosphere–water interface for a collimated unpolarized incident beam. (a) I-component, (b) Q-component, (c) U-component, and (d) V-component.

Fig. 7.
Fig. 7.

Schematic of multilayer approximation model for vector radiative transfer in the gradient-index medium with an oblique incident polarized beam.

Fig. 8.
Fig. 8.

Distributions of Stokes vector just above the top surface of the medium. The gradient-index medium is divided into 5, 10, 30, and 50 sublayers, respectively, for the test of grid independence of results. (a) I-component, (b) Q-component, (c) U-component, and (d) V-component.

Fig. 9.
Fig. 9.

Angular distributions of Stokes vector just below the bottom surface of the medium at φ=90°. Two linear gradient refractive indexes of n(z)=1.2+0.6z/L and n(z)=1.80.6z/L are considered, respectively. (a) I-component, (b) Q-component, (c) U-component, and (d) V-component.

Fig. 10.
Fig. 10.

Angular distributions of Stokes vector just at the midpoint of the medium vat φ=90°. Two linear gradient refractive indexes of n(z)=1.2+0.6z/L and n(z)=1.80.6z/L are considered, respectively. (a) I-component, (b) Q-component, (c) U-component, and (d) V-component.

Fig. 11.
Fig. 11.

Angular distributions of Stokes vector just above the top surface of the medium at φ=90°. Two linear gradient refractive indexes of n(z)=1.2+0.6z/L and n(z)=1.80.6z/L are considered, respectively. (a) I-component, (b) Q-component, (c) U-component, and (d) V-component.

Fig. 12.
Fig. 12.

Angular distributions of Stokes vector just below the bottom surface of the medium at φ=90°. Two sinusoidal gradient refractive indexes of n(z)=1.2+0.6sin(πz/L) and n(z)=1.80.6sin(πz/L) are considered, respectively. (a) I-component, (b) Q-component, (c) U-component, and (d) V-component.

Fig. 13.
Fig. 13.

Angular distributions of Stokes vector at the midpoint of the medium at φ=90°. Two sinusoidal gradient refractive indexes of n(z)=1.2+0.6sin(πz/L) and n(z)=1.80.6sin(πz/L) are considered, respectively. (a) I-component, (b) Q-component, (c) U-component, and (d) V-component.

Fig. 14.
Fig. 14.

Angular distributions of Stokes vector just above the top surface of the medium at φ=90°. Two sinusoidal gradient refractive indexes of n(z)=1.2+0.6sin(πz/L) and n(z)=1.80.6sin(πz/L) are considered, respectively. (a) I-component, (b) Q-component, (c) U-component, and (d) V-component.

Equations (26)

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

Sref=RSi
Srefr=TSi,
R(θi)=12(r2+r2r2r200r2r2r2+r200002Re{rr*}2Im{rr*}002Im{rr*}2Re{rr*})
r=nicos(θi)nt2ni2sin2(θi)nicos(θi)+nt2ni2sin2(θi)r=nt2cos(θi)nint2ni2sin2(θi)nt2cos(θi)+nint2ni2sin2(θi),
T(θi)=12ntcos(θt)nicos(θi)(t2+t2t2t200t2t2t2+t200002Re{tt*}2Im{tt*}002Im{tt*}2Re{tt*}),
t=2nicos(θi)nicos(θi)+nt2ni2sin2(θi)t=2nintcos(θi)nt2cos(θi)+nint2ni2sin2(θi).
ρ=r2+r22.
ρ=II,
ρ=12(r2+r2)+Q2I(r2r2)=ρ+Q2I(r2r2).
Ss=L(πi2)M(Θ)L(i1)Si,
M(Θ)=(M1M200M2M10000M3M400M4M3),
L(ϕ)=(10000cos2ϕsin2ϕ00sin2ϕcos2ϕ00001).
Ss=M(Θ)L(i1)Si.
Is=IiM1(Θ)+QiM2(Θ)cos2i1UiM2(Θ)sin2i1Θ[0,π],i1[0,2π).
Is=IiM1(Θ)+Qi2+Ui2M2(Θ)cos(2i1+ϕ),
Is,Θ,max=IiM1(Θ)+Qi2+Ui2|M2(Θ)|Θ[0,π].
Is,max=max{IiM1(Θ)+Qi2+Ui2|M2(Θ)|}Θ[0,π].
Θ[0,π],i1[0,2π).
S=SI=(1Q/IU/IV/I).
Di,θ,φ=q=1NtSqN,
Si,θ,φ=I0|cosθ0||cosθ|dΩ·Di,θ,φ,
S(θ,φ)=j=1N4πκajdVjnj2Ib(Tj)DjsdAscosθdΩ,
εp=IpIb/2=(I+Q)/2Ib/2=I+QIb,
εv=IvIb/2=(IQ)/2Ib/2=IQIb,
ε=IIb,
M(Θ)=34(cos2Θ+1cos2Θ100cos2Θ1cos2Θ+100002cosΘ00002cosΘ).

Metrics