Abstract

We present a model for calculating the angular distribution of light, including polarization effects from multilayered inhomogeneous media, with an index of refraction mismatch between layers. The model is based on the resolution of the radiative transfer equation by the discrete ordinate method. Comparisons with previous simpler models and examples of simulations are presented.

© 2008 Optical Society of America

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  1. K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387-399 (1981).
    [CrossRef]
  2. S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).
  3. A. Ishimaru, “Diffusion of light in turbid material,” Appl. Opt. 28, 2210-2215 (1989).
    [CrossRef] [PubMed]
  4. P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).
  5. B. Maheu, J. N. Letoulouzan, and G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz-Mie parameters,” Appl. Opt. 23, 3353-3362 (1984).
    [CrossRef] [PubMed]
  6. P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485-1501 (1971).
    [CrossRef] [PubMed]
  7. K. Stamnes and P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 31, 273-282 (1984).
    [CrossRef]
  8. K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502-2509 (1988).
    [CrossRef] [PubMed]
  9. Z. Jin and K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431-442 (1994).
    [CrossRef] [PubMed]
  10. A. Da Silva, C. Andraud, J. Lafait, T. Robin, and R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313-332 (2004).
    [CrossRef]
  11. A. Da Silva, C. Andraud, E. Charron, B. Stout, and J. Lafait, “Light scattering through layered media with strong interaction between scatterers: theory and experiments,” Physica B 338, 74-78 (2003).
    [CrossRef]
  12. M. I. Mishchenko, A. A. Lacis, and L. D. Travis, “Errors induced by the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 51, 491-510 (1994).
    [CrossRef]
  13. C. Bordier, C. Andraud, J. Lafait, and E. Charron, “Radiative transfer model with polarization effects applied to organic matter,” Physica B 394, 301-305 (2007).
    [CrossRef]
  14. F. M. Schulz, K. Stamnes, and F. Weng, “VDISORT: an improved and generalized discrete ordinate method for polarized (vector) radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 61, 105-122 (1999).
    [CrossRef]
  15. F. M. Schulz and K. Stamnes, “Angular distribution of the Stokes vector in a plane-parallel vertically inhomogeneous medium in the vector discrete ordinate radiative transfer (VDISORT) model,” J. Quant. Spectrosc. Radiat. Transf. 65, 609-620 (2000).
    [CrossRef]
  16. C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64, 227-254 (2000).
    [CrossRef]
  17. L. Roux, “Etude numérique de la diffusion multiple dans des milieux aléatoires. Modèles de transfert radiatif et electromagnétique,” Ph.D. thesis (Ecole Centrale, 1999).
  18. J. Caron, “Diffusion de la lumière dans les milieux stratifiés: prise en compte des interfaces rugueuses et des effets de polarisation,” Ph.D. thesis (University of Paris, 2003).
  19. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  20. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  21. G. Fardella, “Modélisation de l'émission thermique de rayonnement infrarouge par les milieux inhomogènes,” Ph.D. thesis (University of Paris, 1995).
  22. J. C. Auger, “Dependant light scattering in dense heterogeneous media,” Physica B 279, 21-24 (2000).
    [CrossRef]
  23. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).
  24. J. Caron, C. Andraud, and J. Lafait, “Radiative transfer calculations in multilayer systems with smooth or rough interfaces,” J. Mod. Opt. 51, 575-596 (2004).
    [CrossRef]
  25. J. W. Hovenier and C. V. M. Van der Mee, “Fundamental relationships relevant to the transfer of polarized light in scattering atmosphere,” Astron. Astrophys. 128, 1-16 (1983).
  26. C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080-1086 (1981).
    [CrossRef]
  27. V. H. Domke, “Fourier expansion of the phase matrix for Mie scattering,” Z. Meteorol. 25, 357-361 (1975).
  28. W. A. De Rooij and C. C. A. H. Van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237-248 (1984).
  29. P. Vestrucci and C. E. Siewert, “A numerical evaluation of an analytical representation of the components in a Fourier decomposition of the phase matrix for the scattering of polarized light,” J. Quant. Spectrosc. Radiat. Transf. 31, 177-183 (1984).
    [CrossRef]
  30. W. J. Wiscombe, “Improved Mie scattering algorithm,” Appl. Opt. 19, 1505-1509 (1980).
    [CrossRef] [PubMed]
  31. W. M. F. Wauben and J. W. Hovenier, “Polarized radiation of an atmosphere containing randomly-oriented spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47, 491-503 (1992).
    [CrossRef]
  32. F. Kuik, J. F. de Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47, 477-489 (1992).
    [CrossRef]
  33. J. E. Hansen and D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527-610 (1974).
    [CrossRef]

2007 (1)

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

2004 (2)

A. Da Silva, C. Andraud, J. Lafait, T. Robin, and R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313-332 (2004).
[CrossRef]

J. Caron, C. Andraud, and J. Lafait, “Radiative transfer calculations in multilayer systems with smooth or rough interfaces,” J. Mod. Opt. 51, 575-596 (2004).
[CrossRef]

2003 (1)

A. Da Silva, C. Andraud, E. Charron, B. Stout, and J. Lafait, “Light scattering through layered media with strong interaction between scatterers: theory and experiments,” Physica B 338, 74-78 (2003).
[CrossRef]

2000 (3)

F. M. Schulz and K. Stamnes, “Angular distribution of the Stokes vector in a plane-parallel vertically inhomogeneous medium in the vector discrete ordinate radiative transfer (VDISORT) model,” J. Quant. Spectrosc. Radiat. Transf. 65, 609-620 (2000).
[CrossRef]

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

J. C. Auger, “Dependant light scattering in dense heterogeneous media,” Physica B 279, 21-24 (2000).
[CrossRef]

1999 (1)

F. M. Schulz, K. Stamnes, and F. Weng, “VDISORT: an improved and generalized discrete ordinate method for polarized (vector) radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 61, 105-122 (1999).
[CrossRef]

1994 (2)

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

Z. Jin and K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431-442 (1994).
[CrossRef] [PubMed]

1992 (2)

W. M. F. Wauben and J. W. Hovenier, “Polarized radiation of an atmosphere containing randomly-oriented spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47, 491-503 (1992).
[CrossRef]

F. Kuik, J. F. de Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47, 477-489 (1992).
[CrossRef]

1989 (1)

1988 (1)

1984 (4)

B. Maheu, J. N. Letoulouzan, and G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz-Mie parameters,” Appl. Opt. 23, 3353-3362 (1984).
[CrossRef] [PubMed]

K. Stamnes and P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 31, 273-282 (1984).
[CrossRef]

W. A. De Rooij and C. C. A. H. Van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237-248 (1984).

P. Vestrucci and C. E. Siewert, “A numerical evaluation of an analytical representation of the components in a Fourier decomposition of the phase matrix for the scattering of polarized light,” J. Quant. Spectrosc. Radiat. Transf. 31, 177-183 (1984).
[CrossRef]

1983 (1)

J. W. Hovenier and C. V. M. Van der Mee, “Fundamental relationships relevant to the transfer of polarized light in scattering atmosphere,” Astron. Astrophys. 128, 1-16 (1983).

1981 (2)

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080-1086 (1981).
[CrossRef]

K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387-399 (1981).
[CrossRef]

1980 (1)

1975 (1)

V. H. Domke, “Fourier expansion of the phase matrix for Mie scattering,” Z. Meteorol. 25, 357-361 (1975).

1974 (1)

J. E. Hansen and D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527-610 (1974).
[CrossRef]

1971 (1)

1931 (1)

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Andraud, C.

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

J. Caron, C. Andraud, and J. Lafait, “Radiative transfer calculations in multilayer systems with smooth or rough interfaces,” J. Mod. Opt. 51, 575-596 (2004).
[CrossRef]

A. Da Silva, C. Andraud, J. Lafait, T. Robin, and R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313-332 (2004).
[CrossRef]

A. Da Silva, C. Andraud, E. Charron, B. Stout, and J. Lafait, “Light scattering through layered media with strong interaction between scatterers: theory and experiments,” Physica B 338, 74-78 (2003).
[CrossRef]

Auger, J. C.

J. C. Auger, “Dependant light scattering in dense heterogeneous media,” Physica B 279, 21-24 (2000).
[CrossRef]

Barrera, R. G.

A. Da Silva, C. Andraud, J. Lafait, T. Robin, and R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313-332 (2004).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bordier, C.

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

Caron, J.

J. Caron, C. Andraud, and J. Lafait, “Radiative transfer calculations in multilayer systems with smooth or rough interfaces,” J. Mod. Opt. 51, 575-596 (2004).
[CrossRef]

J. Caron, “Diffusion de la lumière dans les milieux stratifiés: prise en compte des interfaces rugueuses et des effets de polarisation,” Ph.D. thesis (University of Paris, 2003).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).

Charron, E.

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

A. Da Silva, C. Andraud, E. Charron, B. Stout, and J. Lafait, “Light scattering through layered media with strong interaction between scatterers: theory and experiments,” Physica B 338, 74-78 (2003).
[CrossRef]

Conklin, P.

K. Stamnes and P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 31, 273-282 (1984).
[CrossRef]

Da Silva, A.

A. Da Silva, C. Andraud, J. Lafait, T. Robin, and R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313-332 (2004).
[CrossRef]

A. Da Silva, C. Andraud, E. Charron, B. Stout, and J. Lafait, “Light scattering through layered media with strong interaction between scatterers: theory and experiments,” Physica B 338, 74-78 (2003).
[CrossRef]

de Haan, J. F.

F. Kuik, J. F. de Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47, 477-489 (1992).
[CrossRef]

De Rooij, W. A.

W. A. De Rooij and C. C. A. H. Van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237-248 (1984).

Domke, V. H.

V. H. Domke, “Fourier expansion of the phase matrix for Mie scattering,” Z. Meteorol. 25, 357-361 (1975).

Fardella, G.

G. Fardella, “Modélisation de l'émission thermique de rayonnement infrarouge par les milieux inhomogènes,” Ph.D. thesis (University of Paris, 1995).

Gouesbet, G.

Hansen, J. E.

J. E. Hansen and D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527-610 (1974).
[CrossRef]

Hovenier, J. W.

F. Kuik, J. F. de Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47, 477-489 (1992).
[CrossRef]

W. M. F. Wauben and J. W. Hovenier, “Polarized radiation of an atmosphere containing randomly-oriented spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47, 491-503 (1992).
[CrossRef]

J. W. Hovenier and C. V. M. Van der Mee, “Fundamental relationships relevant to the transfer of polarized light in scattering atmosphere,” Astron. Astrophys. 128, 1-16 (1983).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Ishimaru, A.

Jayaweera, K.

Jin, Z.

Kubelka, P.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Kuik, F.

F. Kuik, J. F. de Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47, 477-489 (1992).
[CrossRef]

Lacis, A. A.

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

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

Lafait, J.

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

J. Caron, C. Andraud, and J. Lafait, “Radiative transfer calculations in multilayer systems with smooth or rough interfaces,” J. Mod. Opt. 51, 575-596 (2004).
[CrossRef]

A. Da Silva, C. Andraud, J. Lafait, T. Robin, and R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313-332 (2004).
[CrossRef]

A. Da Silva, C. Andraud, E. Charron, B. Stout, and J. Lafait, “Light scattering through layered media with strong interaction between scatterers: theory and experiments,” Physica B 338, 74-78 (2003).
[CrossRef]

Letoulouzan, J. N.

Maheu, B.

Mishchenko, M. I.

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

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

Mudgett, P. S.

Munk, F.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Richards, L. W.

Robin, T.

A. Da Silva, C. Andraud, J. Lafait, T. Robin, and R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313-332 (2004).
[CrossRef]

Roux, L.

L. Roux, “Etude numérique de la diffusion multiple dans des milieux aléatoires. Modèles de transfert radiatif et electromagnétique,” Ph.D. thesis (Ecole Centrale, 1999).

Schulz, F. M.

F. M. Schulz and K. Stamnes, “Angular distribution of the Stokes vector in a plane-parallel vertically inhomogeneous medium in the vector discrete ordinate radiative transfer (VDISORT) model,” J. Quant. Spectrosc. Radiat. Transf. 65, 609-620 (2000).
[CrossRef]

F. M. Schulz, K. Stamnes, and F. Weng, “VDISORT: an improved and generalized discrete ordinate method for polarized (vector) radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 61, 105-122 (1999).
[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, 227-254 (2000).
[CrossRef]

P. Vestrucci and C. E. Siewert, “A numerical evaluation of an analytical representation of the components in a Fourier decomposition of the phase matrix for the scattering of polarized light,” J. Quant. Spectrosc. Radiat. Transf. 31, 177-183 (1984).
[CrossRef]

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080-1086 (1981).
[CrossRef]

Stamnes, K.

F. M. Schulz and K. Stamnes, “Angular distribution of the Stokes vector in a plane-parallel vertically inhomogeneous medium in the vector discrete ordinate radiative transfer (VDISORT) model,” J. Quant. Spectrosc. Radiat. Transf. 65, 609-620 (2000).
[CrossRef]

F. M. Schulz, K. Stamnes, and F. Weng, “VDISORT: an improved and generalized discrete ordinate method for polarized (vector) radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 61, 105-122 (1999).
[CrossRef]

Z. Jin and K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431-442 (1994).
[CrossRef] [PubMed]

K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502-2509 (1988).
[CrossRef] [PubMed]

K. Stamnes and P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 31, 273-282 (1984).
[CrossRef]

K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387-399 (1981).
[CrossRef]

Stout, B.

A. Da Silva, C. Andraud, E. Charron, B. Stout, and J. Lafait, “Light scattering through layered media with strong interaction between scatterers: theory and experiments,” Physica B 338, 74-78 (2003).
[CrossRef]

Swanson, R. A.

K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387-399 (1981).
[CrossRef]

Travis, D.

J. E. Hansen and D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527-610 (1974).
[CrossRef]

Travis, L. D.

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

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

Tsay, S.-C.

Van de Hulst, H. C.

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

Van der Mee, C. V. M.

J. W. Hovenier and C. V. M. Van der Mee, “Fundamental relationships relevant to the transfer of polarized light in scattering atmosphere,” Astron. Astrophys. 128, 1-16 (1983).

Van der Stap, C. C. A. H.

W. A. De Rooij and C. C. A. H. Van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237-248 (1984).

Vestrucci, P.

P. Vestrucci and C. E. Siewert, “A numerical evaluation of an analytical representation of the components in a Fourier decomposition of the phase matrix for the scattering of polarized light,” J. Quant. Spectrosc. Radiat. Transf. 31, 177-183 (1984).
[CrossRef]

Wauben, W. M. F.

W. M. F. Wauben and J. W. Hovenier, “Polarized radiation of an atmosphere containing randomly-oriented spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47, 491-503 (1992).
[CrossRef]

Weng, F.

F. M. Schulz, K. Stamnes, and F. Weng, “VDISORT: an improved and generalized discrete ordinate method for polarized (vector) radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 61, 105-122 (1999).
[CrossRef]

Wiscombe, W.

Wiscombe, W. J.

Appl. Opt. (6)

Astron. Astrophys. (2)

W. A. De Rooij and C. C. A. H. Van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237-248 (1984).

J. W. Hovenier and C. V. M. Van der Mee, “Fundamental relationships relevant to the transfer of polarized light in scattering atmosphere,” Astron. Astrophys. 128, 1-16 (1983).

Astrophys. J. (1)

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080-1086 (1981).
[CrossRef]

J. Atmos. Sci. (1)

K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387-399 (1981).
[CrossRef]

J. Mod. Opt. (2)

J. Caron, C. Andraud, and J. Lafait, “Radiative transfer calculations in multilayer systems with smooth or rough interfaces,” J. Mod. Opt. 51, 575-596 (2004).
[CrossRef]

A. Da Silva, C. Andraud, J. Lafait, T. Robin, and R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313-332 (2004).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (8)

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

F. M. Schulz, K. Stamnes, and F. Weng, “VDISORT: an improved and generalized discrete ordinate method for polarized (vector) radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 61, 105-122 (1999).
[CrossRef]

F. M. Schulz and K. Stamnes, “Angular distribution of the Stokes vector in a plane-parallel vertically inhomogeneous medium in the vector discrete ordinate radiative transfer (VDISORT) model,” J. Quant. Spectrosc. Radiat. Transf. 65, 609-620 (2000).
[CrossRef]

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

P. Vestrucci and C. E. Siewert, “A numerical evaluation of an analytical representation of the components in a Fourier decomposition of the phase matrix for the scattering of polarized light,” J. Quant. Spectrosc. Radiat. Transf. 31, 177-183 (1984).
[CrossRef]

K. Stamnes and P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 31, 273-282 (1984).
[CrossRef]

W. M. F. Wauben and J. W. Hovenier, “Polarized radiation of an atmosphere containing randomly-oriented spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47, 491-503 (1992).
[CrossRef]

F. Kuik, J. F. de Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47, 477-489 (1992).
[CrossRef]

Physica B (3)

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

A. Da Silva, C. Andraud, E. Charron, B. Stout, and J. Lafait, “Light scattering through layered media with strong interaction between scatterers: theory and experiments,” Physica B 338, 74-78 (2003).
[CrossRef]

J. C. Auger, “Dependant light scattering in dense heterogeneous media,” Physica B 279, 21-24 (2000).
[CrossRef]

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[CrossRef]

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

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

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Figures (13)

Fig. 1
Fig. 1

Geometry of the system.

Fig. 2
Fig. 2

Notations for polar angles.

Fig. 3
Fig. 3

Scattering angle.

Fig. 4
Fig. 4

Collimated fluxes between two layers.

Fig. 5
Fig. 5

Schematic representation of the fluxes at the interface between two layers.

Fig. 6
Fig. 6

Multilayer representation.

Fig. 7
Fig. 7

Stokes parameters I and Q as a function of the polar angle at the top and the bottom of the atmosphere for an unpolarized source and for an azimuth of φ = 0 . Wauben and Hovenier values, circles; our model, solid curves.

Fig. 8
Fig. 8

Stokes parameters I , Q , U , V as a function of the polar angle at the top and at the bottom of the atmosphere for an unpolarized source and for an azimuth of φ = 90 ° . Wauben and Hovenier values, circles; our model, solid curves.

Fig. 9
Fig. 9

Angular distribution of polarized scattered flux by a solid angle unit for an angle of incidence ( θ i = 0 ° , φ i = 0 ° ) for three types of particle distributions: gamma distribution (black curves), log-normal distribution (dotted curves), Dirac distribution (gray curves), and for two types of polarization: (a) pp polarization and (b) ps polarization.

Fig. 10
Fig. 10

Angular distribution of reflected scattered light by a dielectric slab containing scattering particles. Black curves are for r = 0.2 μ m , and gray curves are for r = 0.1 μ m without polarization (a), pp polarization (b), and ps polarization (c).

Fig. 11
Fig. 11

Angular distribution of reflected scattered light by a dielectric slab containing scattering particles. Black curves are for r = 2 μ m , and gray curves are for r = 0.2 μ m without polarization (a), pp polarization (solid curves), and ps polarization (dotted curves) (b).

Fig. 12
Fig. 12

Two configurations of the system: latex beads in water with n sph = 1.59 and n s = 1.33 .

Fig. 13
Fig. 13

Angular distribution of the polarized scattered flux by a solid angle unit for an angle of incidence ( θ i = 0 ° , φ i = 0 ° ) for two types of polarization: pp polarization (solid curves) and ps polarization (dotted curves). The gray curve refers to the configuration of Fig. 12a, and the black curve refers to the configuration of Fig. 12b.

Tables (4)

Tables Icon

Table 1 Characteristic Quantities of Model Scattering

Tables Icon

Table 2 Parameters and Some Characteristic Quantities of the Two Media

Tables Icon

Table 3 Parameters of the Two Media

Tables Icon

Table 4 Input Parameters of the System

Equations (85)

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BRDF = d I r I i × cos ( θ i ) × d ω i .
ρ ( θ i , θ r ) = BRDF ( θ i , θ r ) cos θ r .
τ ( θ i , θ t ) = BTDF ( θ i , θ t ) cos θ t .
I p = E p E p * ,
I s = E s E s * ,
U = E p E s * + E p * E s ,
V = i E p E s * E p * E s ,
I = D I ,
I = E p E p * + E s E s * ,
Q = E p E p * E s E s * ,
D = ( 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 ) .
( 1 0 0 0 ) .
( E p E s ) = A ( E p i E s i ) ,
( I sca Q sca U sca V sca ) = 1 k 2 R 2 ( P 11 P 12 0 0 P 12 P 22 0 0 0 0 P 33 P 34 0 0 P 34 P 44 ) ( I i Q i U i V i ) ,
α sca = η C sca , α ext = η C ext ,
C sca = 2 π k 2 n = 1 ( 2 n + 1 ) { b n 2 + a n 2 } ,
C ext = 2 π k 2 Re n = 1 ( 2 n + 1 ) ( b n + a n ) .
C ext = r min r max p ( r ) C ext ( r ) d r = i = 1 N r u i p ( r i ) C ext ( r i ) ,
C sca = r min r max p ( r ) C sca ( r ) d r = i = 1 N r u i p ( r i ) C sca ( r i ) .
cos Θ = cos θ cos θ + sin θ sin θ cos ( φ φ ) .
± d f ( z , μ , φ ) d z = α ext f ( z , μ , φ ) μ + α sca 4 π 0 2 π 1 1 M ( μ , φ , μ φ ) f ( z , μ , φ ) μ d μ d φ + S ( z , μ , φ ) ,
f ( z , μ , φ ) = I ( z , μ , φ ) μ ,
M ( θ , μ ; θ , φ ) = L ( π i 2 ) P ( Θ ) L ( i 1 ) ,
L ( i ) = ( 1 0 0 0 0 cos 2 i sin 2 i 0 0 sin 2 i cos 2 i 0 0 0 0 1 ) .
P ̃ ( Θ ) = 4 π k 2 C sca P ( Θ ) = ( P ̃ 11 ( Θ ) P ̃ 12 ( Θ ) 0 0 P ̃ 12 ( Θ ) P ̃ 22 ( Θ ) 0 0 0 0 P ̃ 33 ( Θ ) P ̃ 34 ( Θ ) 0 0 P ̃ 34 ( Θ ) P ̃ 44 ( Θ ) ) ,
1 2 1 1 P ̃ 11 ( Θ ) d ( cos Θ ) = 1 .
P ̃ 11 ( Θ ) = l = 0 β l P 0 , 0 l cos ( Θ ) ,
P ̃ 22 ( Θ ) + P ̃ 33 ( Θ ) = l = 2 ( α l + ξ l ) P 2 , 2 l cos ( Θ ) ,
P ̃ 22 ( Θ ) P ̃ 33 ( Θ ) = l = 2 ( α l ξ l ) P 2 , 2 l cos ( Θ ) ,
P ̃ 44 ( Θ ) = l = 0 δ l P 0 , 0 l cos ( Θ ) ,
P ̃ 12 ( Θ ) = l = 2 γ l P 0 , 2 l cos ( Θ ) ,
P ̃ 34 ( Θ ) = l = 2 ε l P 0 , 2 l cos ( Θ ) .
β l = ( l + 1 2 ) 1 1 d ( cos Θ ) P ̃ 11 ( Θ ) d 00 l l ( Θ ) = ( l + 1 2 ) j = 1 N Θ w j P ̃ 11 ( arccos μ j ) d 00 l l ( arccos μ j ) ,
P ̃ 11 ( Θ ) = 1 C sca r min r max p ( r ) C sca ( r ) P ̃ 11 ( r ; Θ ) d r .
r min r max p ( r ) C sca ( r ) P ̃ 11 ( r ; Θ ) d r = i = 1 N r u i p ( r i ) C sca ( r i ) P ̃ 11 ( r i ; Θ ) .
For layer n , we have ( F + ( n ) F ( n ) ) = I n M n 1 ( F + ( n 1 ) F ( n 1 ) ) ,
I n = ( 1 T n ( n 1 ) R ( n 1 ) n T n ( n 1 ) R n ( n 1 ) T n ( n 1 ) T n ( n 1 ) R ( n 1 ) n R n ( n 1 ) T n ( n 1 ) ) .
M n 1 = ( exp [ α ext ( z n 2 z n 1 ) ] 0 0 exp [ α ext ( z n 2 z n 1 ) ] ) .
R i j = ( r i j p 0 0 0 0 r i j s 0 0 0 0 Re ( r i j p r i j s * ) Im ( r i j p r i j s * ) 0 0 Im ( r i j p r i j s * ) Re ( r i j p r i j s * ) ) ,
T i j = n j cos θ j n i cos θ i ( t i j p 0 0 0 0 t i j s 0 0 0 0 Re ( t i j p r i j s * ) Im ( t i j p r i j s * ) 0 0 Im ( t i j p r i j s * ) Re ( t i j p r i j s * ) ) ,
S ( z , μ , φ ) = α sca 4 π M ( μ , φ , μ s , φ i ) F s + ( z , μ s , φ i ) μ s + α sca 4 π M ( μ , φ , μ s , φ i ) F s ( z , μ s , φ i ) μ s .
I ( z , μ , φ ) = m = 0 N leg I m c ( z , μ ) cos m ( φ φ i ) + m = 0 N leg I m s ( z , μ ) sin m ( φ φ i ) ,
M ( μ , φ ; μ , φ ) = m = 0 N leg M m c ( μ , μ ) cos m ( φ φ ) + m = 0 N leg M m s ( μ , μ ) sin m ( φ φ ) ,
S ( z , μ , φ ) = m = 0 N leg S m c ( z , μ ) cos m ( φ φ i ) + m = 0 N leg S m s ( z , μ ) sin m ( φ φ i ) ,
M m c = ( M 11 m c M 12 m c 0 0 M 21 m c M 22 m c 0 0 0 0 M 33 m c M 34 m c 0 0 M 43 m c M 44 m c ) ,
M m s = ( 0 0 M 13 m s M 14 m s 0 0 M 23 m s M 24 m s M 31 m s M 32 m s 0 0 M 41 m s M 42 m s 0 0 0 ) .
± d f m c ( z , μ ) d z = α ext f m c ( z , μ ) μ + α sca 4 1 1 M m c ( μ , μ ) f m c ( z , μ ) ( 1 + δ 0 m ) + M m s ( μ , μ ) f m s ( z , μ ) ( δ 0 m 1 ) μ d μ + α sca 4 π M m c ( μ , μ s ) F s + ( z , μ s ) μ s + α sca 4 π M m c ( μ , μ s ) F s ( z , μ s ) μ s ,
± d f m s ( z , μ ) d z = α ext f m c ( z , μ ) μ + α sca 4 1 1 M m c ( μ , μ ) f m s ( z , μ ) ( 1 δ 0 m ) + M m s ( μ , μ ) f m c ( z , μ ) ( 1 δ 0 m ) μ d μ + α sca 4 π M m s ( μ , μ s ) F s + ( z , μ s ) μ s + α sca 4 π M m s ( μ , μ s ) F s ( z , μ s ) μ s .
F j m = f m ( μ j ) Ω j ,
± d F i m c d z = α ext F i m c μ i + α sca 8 π Ω i j ( 1 + δ 0 m ) M m c ( μ i , μ j ) F j m c + ( δ 0 m 1 ) M m s ( μ i , μ j ) F j m s μ j + Ω i α sca 4 π M m c ( μ i , μ s ) F s + ( z , μ s ) μ s + Ω i α sca 4 π M m c ( μ i , μ s ) F s ( z , μ s ) μ s ,
± d F i m s d z = α ext F i m s μ i + α sca 8 π Ω i j ( 1 δ 0 m ) M m c ( μ i , μ j ) F j m s + ( 1 δ 0 m ) M m s ( μ i , μ j ) F j m c μ j + Ω i α sca 4 π M m s ( μ i , μ s ) F s + ( z , μ s ) μ s + Ω i α sca 4 π M m s ( μ i , μ s ) F s ( z , μ s ) μ s .
F i m s c = ( I i m s Q i m s U i m c V i m c ) , F i m c s = ( I i m c Q i m c U i m s V i m s ) .
± d F i m c s d z = α ext F i m c s μ i + α sca 8 π Ω i j A m c s ( μ i , μ j ) F j m c s μ j + Ω i α sca 4 π M m c s ( μ i , μ s ) F s + ( z , μ s ) μ s + Ω i α sca 4 π M m c s ( μ i , μ s ) F s ( z , μ s ) μ s ,
± d F i m s c d z = α ext F i m s c μ i + α sca 8 π Ω i j A m s c ( μ i , μ j ) F j m s c μ j + Ω i α sca 4 π M m s c ( μ i , μ s ) F s + ( z , μ s ) μ s + Ω i α sca 4 π M m s c ( μ i , μ s ) F s ( z , μ s ) μ s ,
M m c s ( μ , μ s ) = ( M m c 0 M m s 0 ) , M m s c ( μ , μ s ) = ( 0 M m s 0 M m c ) ,
A m s c ( μ i , μ j ) = ( ( 1 δ 0 m ) M m c ( 1 δ 0 m ) M m s ( δ 0 m 1 ) M m s ( 1 + δ 0 m ) M m c ) ,
A m c s ( μ i , μ j ) = ( ( 1 + δ 0 m ) M m c ( δ 0 m + 1 ) M m s ( 1 δ 0 m ) M m s ( 1 δ 0 m ) M m c ) ,
M 0 c ( μ , μ ) = 1 2 [ A 0 ( μ , μ ) + D A 0 ( μ , μ ) D ] ,
M m c ( μ , μ ) = A m ( μ , μ ) + D A m ( μ , μ ) D ,
M m s ( μ , μ ) = A m ( μ , μ ) D D A m ( μ , μ ) ,
A m ( μ , μ ) = l = m ( l m ) ! ( l m ) ! Π l m ( μ ) B l Π l m ( μ ) ,
B l = ( β l γ l 0 0 γ l α l 0 0 0 0 ζ l ε l 0 0 ε l δ l ) .
F i ( z ) = j = 1 p s A i j ( z ) C j + B i ( z ) ,
A i j ( z ) = a i j e λ J ( z Z s 1 ) for 1 j p s 2 ,
A i j ( z ) = a i j e λ J ( z ) , for p s 2 + 1 j p s .
F 1 u = T 21 F 2 u + R 12 F 1 d ,
F 2 d = T 12 F 1 d + R 21 F 2 u ,
R 12 ( 1 : 8 , 1 : 8 ) = ( 0 0 0 0 r 12 p 0 0 0 0 0 0 0 0 r 12 s 0 0 0 0 0 0 0 0 Re ( r 12 p r 12 s * ) Im ( r 12 p r 12 s * ) 0 0 0 0 0 0 Im ( r 12 p r 12 s * ) Re ( r 12 p r 12 s * ) r 12 p 0 0 0 0 0 0 0 0 r 12 s 0 0 0 0 0 0 0 0 Re ( r 12 p r 12 s * ) Im ( r 12 p r 12 s * ) 0 0 0 0 0 0 Im ( r 12 p r 12 s * ) Re ( r 12 p r 12 s * ) 0 0 0 0 ) ,
( F M u m ( Z M ) F M d m ( Z M 1 ) ) = Q 1 m ( M ) C j ( M ) + B 1 m ( M ) ,
( F M u m ( Z M 1 ) F M d m ( Z M ) ) = Q 2 m ( M ) C j ( M ) + B 2 m ( M ) ,
( F M 1 u m ( Z M 2 ) F M 1 d m ( Z M 1 ) ) = Q 2 m ( M 1 ) C j ( M 1 ) + B 2 m ( M 1 ) ,
Q 1 m ( M ) = ( A i j ( Z M ) A i j ( Z M ) A i j ( Z M 1 ) A i j ( Z M 1 ) ) , B 1 m ( M ) = ( B M u m ( Z M ) B M d m ( Z M 1 ) ) ,
Q 2 m ( M ) = ( A i j ( Z M 1 ) A i j ( Z M 1 ) A i j ( Z M ) A i j ( Z M ) ) , B 2 m ( M ) = ( B M u m ( Z M 1 ) B M d m ( Z M ) ) ,
Q 2 m ( M 1 ) = ( A i j ( Z M 2 ) A i j ( Z M 2 ) A i j ( Z M 1 ) A i j ( Z M 1 ) ) ,
B 2 m ( M 1 ) = ( B M u m ( Z M 2 ) B M d m ( Z M 1 ) ) .
Q 1 m ( M ) × C j ( M ) + B 1 m ( M ) = R ( M ) [ Q 2 m ( M ) × C j ( M ) + B 2 m ( M ) ] + T ( M ) [ Q 2 m ( M 1 ) × C j ( M 1 ) + B 2 m ( M 1 ) ] ,
C j ( M ) = R ( M ) B 2 m ( M ) B 1 m ( M ) + T ( M ) [ Q 2 m ( M 1 ) × C j ( M 1 ) + B 2 m ( M 1 ) ] Q 1 m ( M ) R Q 2 m ( M ) .
( F 1 u m ( Z 1 ) F 1 d m ( 0 ) ) = ( 0 R 1 , 2 R 1 , 0 0 ) ( F 1 u m ( 0 ) F 1 d m ( Z 1 ) ) + ( T 2 , 1 0 0 0 ) ( F 2 u m ( Z 1 ) F 2 d m ( Z 2 ) ) .
( F out m ( 0 ) F out m ( Z M ) ) = ( T 1 , 0 0 0 T M , 0 ) ( F 1 u m ( 0 ) F M d m ( Z M ) ) .
( I p ( 0 , μ , φ ) I s ( 0 , μ , φ ) U ( 0 , μ , φ ) V ( 0 , μ , φ ) ) = m = 0 N leg ( I p m c ( 0 , μ , φ ) I s m c ( 0 , μ , φ ) U m c ( 0 , μ , φ ) V m c ( 0 , μ , φ ) ) cos m φ + m = 1 N leg ( I p m s ( 0 , μ , φ ) I s m s ( 0 , μ , φ ) U m s ( 0 , μ , φ ) V m s ( 0 , μ , φ ) ) sin m φ .
( I p ( Z M , μ , φ ) I s ( Z M , μ , φ ) U ( Z M , μ , φ ) V ( Z M , μ , φ ) ) = m = 0 N leg ( I p m c ( Z M , μ , φ ) I s m c ( Z M , μ , φ ) U m c ( Z M , μ , φ ) V m c ( Z M , μ , φ ) ) cos m φ + m = 1 N leg ( I p m s ( Z M , μ , φ ) I s m s ( Z M , μ , φ ) U m s ( Z M , μ , φ ) V m s ( Z M , μ , φ ) ) sin m φ .
n ( r ) = constant × r ( 1 3 b ) b exp ( r a b ) .
n ( r ) = constant × r 1 exp ( ( ln ( r ) ln ( r g ) ) 2 2 ln 2 ( σ g ) ) .
n ( r ) = { f o r r = R 0 f o r r R .
0 n ( r ) d r = 1 .

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