Physically meaningful Monte Carlo approach to the four-flux solution of a dense multilayered system

E. de la Hoz; R. Alcaraz de la Osa; D. Ortiz; J. M. Saiz; F. Moreno; F. González

doi:10.1364/JOSAA.36.000292

Physically meaningful Monte Carlo approach to the four-flux solution of a dense multilayered system

E. de la Hoz, R. Alcaraz de la Osa, D. Ortiz, J. M. Saiz, F. Moreno, and F. González

Journal of the Optical Society of America A
Vol. 36,
Issue 2,
pp. 292-304
(2019)
•https://doi.org/10.1364/JOSAA.36.000292

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E. de la Hoz, R. Alcaraz de la Osa, D. Ortiz, J. M. Saiz, F. Moreno, and F. González, "Physically meaningful Monte Carlo approach to the four-flux solution of a dense multilayered system," J. Opt. Soc. Am. A 36, 292-304 (2019)

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Related Topics
Table of Contents Category
- Physical Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: September 26, 2018
- Revised Manuscript: November 26, 2018
- Manuscript Accepted: January 6, 2019
- Published: February 1, 2019

Accessible

Open Access

Abstract

Due to the complexity of the radiative transfer equation, light transport problems are commonly solved using either models under restrictive assumptions, e.g., $N$ -flux models where infinite lateral extension is assumed, or numerical methods. While the latter can be applied to more general cases, it is difficult to relate their parameters to the physical properties of the systems under study. Hence in this contribution we present, first, a review of a four-flux formalism to study the light transport problem in a plane-parallel system together with a derivation of equations to evaluate the different contributions to the total absorptance and, second, as a complementary tool, a Monte Carlo algorithm with a direct correspondence between its inputs and the properties of the system. The combination of the four-flux model and the Monte Carlo approach provides (i) all convergence warranties since the formalism has been established as a limit and (ii) new added capabilities, i.e., both temporal (transient states) and spatial (arbitrarily inhomogeneous media) resolution. The support between the theoretical model and the numerical tool is reciprocal since the model is utilized to set a Monte Carlo discretization criterion, while the Monte Carlo approach is used to validate the aforementioned model. This reinforces the parallel approach used in this work. Furthermore, we provide some examples to show its capabilities and potential, e.g., the study of the temporal distribution of a delta-like pulse of light.

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References

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|
Year
|
Author
|
Publication

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[Crossref]
Y. Zhang and J. C. Tan, “Radiation transfer of models of massive star formation. I. Dependence on basic core properties,” Astrophys. J. 733, 55–75 (2011).
[Crossref]
C. Emde, R. Buras-Schnell, A. Kylling, B. Mayer, J. Gasteiger, U. Hamann, J. Kylling, B. Richter, C. Pause, T. Dowling, and L. Bugliaro, “The libRadtran software package for radiative transfer calculations (version 2.0.1),” Geosci. Model Dev. 9, 1647–1672 (2016).
[Crossref]
L. G. Sokoletsky, V. P. Budak, F. Shen, and A. A. Kokhanovsky, “Comparative analysis of radiative transfer approaches for calculation of plane transmittance and diffuse attenuation coefficient of plane-parallel light scattering layers,” Appl. Opt. 53, 459–468 (2014).
[Crossref]
J. Hovenier and C. Van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).
S. Platnick, “Vertical photon transport in cloud remote sensing problems,” J. Geophys. Res. Atmos. 105, 22919–22935 (2000).
[Crossref]
S. Y. Kotchenova, E. F. Vermote, R. Matarrese, and F. J. Klemm, “Validation of a vector version of the 6S radiative transfer code for atmospheric correction of satellite data. Part I: Path radiance,” Appl. Opt. 45, 6762–6774 (2006).
[Crossref]
S. Chandrasekhar, Radiative Transfer (Courier, 2013).
P. Mudgett and L. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[Crossref]
P. Kubelka and F. Munk, “An article on optics of paint layers,” Z. Tech. Phys. 12, 593–601 (1931).
L. Fukshansky and N. Kazarinova, “Extension of the Kubelka–Munk theory of light propagation in intensely scattering materials to fluorescent media,” J. Opt. Soc. Am. 70, 1101–1111 (1980).
[Crossref]
W. E. Vargas and G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
[Crossref]
L. Yang and S. J. Miklavcic, “Revised Kubelka–Munk theory. III. A general theory of light propagation in scattering and absorptive media,” J. Opt. Soc. Am. A 22, 1866–1873 (2005).
[Crossref]
R. A. de la Osa, A. G. Alonso, D. Ortiz, F. González, F. Moreno, and J. Saiz, “Extension of the Kubelka–Munk theory to an arbitrary substrate: a Monte Carlo approach,” J. Opt. Soc. Am. A 33, 2053–2060 (2016).
[Crossref]
V. D.-M. Ž. Barbarić-Mikočević and K. Itrić, “Kubelka–Munk theory in describing optical properties of paper (I),” Teh. Vjesn. 18, 117–124 (2011).
H. R. Kang, “Applications of color mixing models to electronic printing,” J. Electron. Imaging 3, 276–288 (1994).
[Crossref]
J. Beasley, J. Atkins, and F. Billmeyer, “Scattering and absorption of light in turbid media,” in ICES Electromagnetic Scattering (1967), Vol. 63, pp. 765–785.
A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.
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]
W. E. Vargas and G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997).
[Crossref]
W. E. Vargas, “Generalized four-flux radiative transfer model,” Appl. Opt. 37, 2615–2623 (1998).
[Crossref]
C. A. Arancibia-Bulnes and J. C. Ruiz-Suárez, “Average path-length parameter of diffuse light in scattering media,” Appl. Opt. 38, 1877–1883 (1999).
[Crossref]
W. E. Vargas and G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997).
[Crossref]
F. El Haber, X. Rocquefelte, C. Andraud, B. Amrani, S. Jobic, O. Chauvet, and G. Froyer, “Prediction of the transparency in the visible range of x-ray absorbing nanocomposites built upon the assembly of LaF3 or LaPO4 nanoparticles with poly (methyl methacrylate),” J. Opt. Soc. Am. B 29, 305–311 (2012).
[Crossref]
L. Wang, J. I. Eldridge, and S. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014).
[Crossref]
K. Laaksonen, S.-Y. Li, S. Puisto, N. Rostedt, T. Ala-Nissila, C.-G. Granqvist, R. Nieminen, and G. A. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Solar Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]
C. Rozé, T. Girasole, and A.-G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]
B. Maheu, J. Briton, and G. Gouesbet, “Four-flux model and a Monte Carlo code: comparisons between two simple, complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24(1989).
[Crossref]
L. Simonot, R. D. Hersch, M. Hébert, and S. Mazauric, “Multilayer four-flux matrix model accounting for directional-diffuse light transfers,” Appl. Opt. 55, 27–37 (2016).
[Crossref]
M. I. Mishchenko, “Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics,” Appl. Opt. 41, 7114–7134(2002).
[Crossref]
J. W. Tunnell, A. S. Haka, S. A. McGee, J. Mirkovic, and M. S. Feld, “Diagnostic tissue spectroscopy and its applications to gastrointestinal endoscopy,” Tech. Gastrointest. Endosc. 5, 65–73 (2003).
[Crossref]
F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989).
[Crossref]
W. E. Vargas and G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[Crossref]
C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, and B. Maheu, “Average crossing parameter and forward scattering ratio values in four-flux model for multiple scattering media,” Opt. Commun. 194, 251–263 (2001).
[Crossref]
E. Hecht, Optics, 5th ed. (Pearson Education, 2017).
H. J. Mcnicholas, “Absolute methods in reflectometry,” Bur. Stand. J. Res. 1, 29–73 (1928).
[Crossref]

2016 (3)

C. Emde, R. Buras-Schnell, A. Kylling, B. Mayer, J. Gasteiger, U. Hamann, J. Kylling, B. Richter, C. Pause, T. Dowling, and L. Bugliaro, “The libRadtran software package for radiative transfer calculations (version 2.0.1),” Geosci. Model Dev. 9, 1647–1672 (2016).
[Crossref]

L. Simonot, R. D. Hersch, M. Hébert, and S. Mazauric, “Multilayer four-flux matrix model accounting for directional-diffuse light transfers,” Appl. Opt. 55, 27–37 (2016).
[Crossref]

R. A. de la Osa, A. G. Alonso, D. Ortiz, F. González, F. Moreno, and J. Saiz, “Extension of the Kubelka–Munk theory to an arbitrary substrate: a Monte Carlo approach,” J. Opt. Soc. Am. A 33, 2053–2060 (2016).
[Crossref]

2014 (3)

L. G. Sokoletsky, V. P. Budak, F. Shen, and A. A. Kokhanovsky, “Comparative analysis of radiative transfer approaches for calculation of plane transmittance and diffuse attenuation coefficient of plane-parallel light scattering layers,” Appl. Opt. 53, 459–468 (2014).
[Crossref]

L. Wang, J. I. Eldridge, and S. Guo, “Comparison of different models for the determination of the absorption and scattering coefficients of thermal barrier coatings,” Acta Mater. 64, 402–410 (2014).
[Crossref]

K. Laaksonen, S.-Y. Li, S. Puisto, N. Rostedt, T. Ala-Nissila, C.-G. Granqvist, R. Nieminen, and G. A. Niklasson, “Nanoparticles of TiO2 and VO2 in dielectric media: conditions for low optical scattering, and comparison between effective medium and four-flux theories,” Solar Energy Mater. Sol. Cells 130, 132–137 (2014).
[Crossref]

2012 (1)

F. El Haber, X. Rocquefelte, C. Andraud, B. Amrani, S. Jobic, O. Chauvet, and G. Froyer, “Prediction of the transparency in the visible range of x-ray absorbing nanocomposites built upon the assembly of LaF3 or LaPO4 nanoparticles with poly (methyl methacrylate),” J. Opt. Soc. Am. B 29, 305–311 (2012).
[Crossref]

2011 (2)

Y. Zhang and J. C. Tan, “Radiation transfer of models of massive star formation. I. Dependence on basic core properties,” Astrophys. J. 733, 55–75 (2011).
[Crossref]

V. D.-M. Ž. Barbarić-Mikočević and K. Itrić, “Kubelka–Munk theory in describing optical properties of paper (I),” Teh. Vjesn. 18, 117–124 (2011).

2006 (1)

S. Y. Kotchenova, E. F. Vermote, R. Matarrese, and F. J. Klemm, “Validation of a vector version of the 6S radiative transfer code for atmospheric correction of satellite data. Part I: Path radiance,” Appl. Opt. 45, 6762–6774 (2006).
[Crossref]

2005 (1)

L. Yang and S. J. Miklavcic, “Revised Kubelka–Munk theory. III. A general theory of light propagation in scattering and absorptive media,” J. Opt. Soc. Am. A 22, 1866–1873 (2005).
[Crossref]

2003 (1)

J. W. Tunnell, A. S. Haka, S. A. McGee, J. Mirkovic, and M. S. Feld, “Diagnostic tissue spectroscopy and its applications to gastrointestinal endoscopy,” Tech. Gastrointest. Endosc. 5, 65–73 (2003).
[Crossref]

2002 (1)

M. I. Mishchenko, “Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics,” Appl. Opt. 41, 7114–7134(2002).
[Crossref]

2001 (2)

C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, and B. Maheu, “Average crossing parameter and forward scattering ratio values in four-flux model for multiple scattering media,” Opt. Commun. 194, 251–263 (2001).
[Crossref]

C. Rozé, T. Girasole, and A.-G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]

2000 (1)

S. Platnick, “Vertical photon transport in cloud remote sensing problems,” J. Geophys. Res. Atmos. 105, 22919–22935 (2000).
[Crossref]

W. E. Vargas and G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997).
[Crossref]

W. E. Vargas and G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997).
[Crossref]

1994 (1)

H. R. Kang, “Applications of color mixing models to electronic printing,” J. Electron. Imaging 3, 276–288 (1994).
[Crossref]

1989 (2)

F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989).
[Crossref]

B. Maheu, J. Briton, and G. Gouesbet, “Four-flux model and a Monte Carlo code: comparisons between two simple, complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24(1989).
[Crossref]

1984 (1)

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]

1983 (1)

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

1980 (1)

L. Fukshansky and N. Kazarinova, “Extension of the Kubelka–Munk theory of light propagation in intensely scattering materials to fluorescent media,” J. Opt. Soc. Am. 70, 1101–1111 (1980).
[Crossref]

1971 (1)

P. Mudgett and L. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[Crossref]

1931 (1)

P. Kubelka and F. Munk, “An article on optics of paint layers,” Z. Tech. Phys. 12, 593–601 (1931).

1928 (1)

H. J. Mcnicholas, “Absolute methods in reflectometry,” Bur. Stand. J. Res. 1, 29–73 (1928).
[Crossref]

1905 (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[Crossref]

Ala-Nissila, T.

Alonso, A. G.

Amrani, B.

Andraud, C.

Arancibia-Bulnes, C. A.

C. A. Arancibia-Bulnes and J. C. Ruiz-Suárez, “Average path-length parameter of diffuse light in scattering media,” Appl. Opt. 38, 1877–1883 (1999).
[Crossref]

Atkins, J.

J. Beasley, J. Atkins, and F. Billmeyer, “Scattering and absorption of light in turbid media,” in ICES Electromagnetic Scattering (1967), Vol. 63, pp. 765–785.

Barbaric-Mikocevic, V. D.-M. Ž.

V. D.-M. Ž. Barbarić-Mikočević and K. Itrić, “Kubelka–Munk theory in describing optical properties of paper (I),” Teh. Vjesn. 18, 117–124 (2011).

Beasley, J.

J. Beasley, J. Atkins, and F. Billmeyer, “Scattering and absorption of light in turbid media,” in ICES Electromagnetic Scattering (1967), Vol. 63, pp. 765–785.

Billmeyer, F.

J. Beasley, J. Atkins, and F. Billmeyer, “Scattering and absorption of light in turbid media,” in ICES Electromagnetic Scattering (1967), Vol. 63, pp. 765–785.

Bolin, F. P.

F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989).
[Crossref]

Briton, J.

Budak, V. P.

Bugliaro, L.

Buras-Schnell, R.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Courier, 2013).

Chauvet, O.

de la Osa, R. A.

Dowling, T.

El Haber, F.

Eldridge, J. I.

Emde, C.

Feld, M. S.

Ference, R. J.

F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989).
[Crossref]

Froyer, G.

Fukshansky, L.

Gasteiger, J.

Girasole, T.

C. Rozé, T. Girasole, and A.-G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]

González, F.

Gouesbet, G.

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]

Granqvist, C.-G.

Gréhan, G.

Guo, S.

Haka, A. S.

Hamann, U.

Hébert, M.

L. Simonot, R. D. Hersch, M. Hébert, and S. Mazauric, “Multilayer four-flux matrix model accounting for directional-diffuse light transfers,” Appl. Opt. 55, 27–37 (2016).
[Crossref]

Hecht, E.

E. Hecht, Optics, 5th ed. (Pearson Education, 2017).

Hersch, R. D.

L. Simonot, R. D. Hersch, M. Hébert, and S. Mazauric, “Multilayer four-flux matrix model accounting for directional-diffuse light transfers,” Appl. Opt. 55, 27–37 (2016).
[Crossref]

Hovenier, J.

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

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.

Itric, K.

V. D.-M. Ž. Barbarić-Mikočević and K. Itrić, “Kubelka–Munk theory in describing optical properties of paper (I),” Teh. Vjesn. 18, 117–124 (2011).

Jobic, S.

Kang, H. R.

H. R. Kang, “Applications of color mixing models to electronic printing,” J. Electron. Imaging 3, 276–288 (1994).
[Crossref]

Kazarinova, N.

Klemm, F. J.

Kokhanovsky, A. A.

Kotchenova, S. Y.

Kubelka, P.

P. Kubelka and F. Munk, “An article on optics of paint layers,” Z. Tech. Phys. 12, 593–601 (1931).

Kylling, A.

Kylling, J.

Laaksonen, K.

Letoulouzan, J.-N.

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]

Li, S.-Y.

Maheu, B.

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]

Matarrese, R.

Mayer, B.

Mazauric, S.

L. Simonot, R. D. Hersch, M. Hébert, and S. Mazauric, “Multilayer four-flux matrix model accounting for directional-diffuse light transfers,” Appl. Opt. 55, 27–37 (2016).
[Crossref]

McGee, S. A.

Mcnicholas, H. J.

H. J. Mcnicholas, “Absolute methods in reflectometry,” Bur. Stand. J. Res. 1, 29–73 (1928).
[Crossref]

Miklavcic, S. J.

L. Yang and S. J. Miklavcic, “Revised Kubelka–Munk theory. III. A general theory of light propagation in scattering and absorptive media,” J. Opt. Soc. Am. A 22, 1866–1873 (2005).
[Crossref]

Mirkovic, J.

Mishchenko, M. I.

Moreno, F.

Mudgett, P.

P. Mudgett and L. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[Crossref]

Munk, F.

P. Kubelka and F. Munk, “An article on optics of paint layers,” Z. Tech. Phys. 12, 593–601 (1931).

Nieminen, R.

Niklasson, G. A.

W. E. Vargas and G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997).
[Crossref]

W. E. Vargas and G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
[Crossref]

W. E. Vargas and G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[Crossref]

W. E. Vargas and G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997).
[Crossref]

Ortiz, D.

Pause, C.

Platnick, S.

S. Platnick, “Vertical photon transport in cloud remote sensing problems,” J. Geophys. Res. Atmos. 105, 22919–22935 (2000).
[Crossref]

Preuss, L. E.

F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989).
[Crossref]

Puisto, S.

Richards, L.

P. Mudgett and L. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[Crossref]

Richter, B.

Rocquefelte, X.

Rostedt, N.

Rozé, C.

C. Rozé, T. Girasole, and A.-G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]

Ruiz-Suárez, J. C.

C. A. Arancibia-Bulnes and J. C. Ruiz-Suárez, “Average path-length parameter of diffuse light in scattering media,” Appl. Opt. 38, 1877–1883 (1999).
[Crossref]

Saiz, J.

Schuster, A.

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[Crossref]

Shen, F.

Simonot, L.

L. Simonot, R. D. Hersch, M. Hébert, and S. Mazauric, “Multilayer four-flux matrix model accounting for directional-diffuse light transfers,” Appl. Opt. 55, 27–37 (2016).
[Crossref]

Sokoletsky, L. G.

Tafforin, A.-G.

C. Rozé, T. Girasole, and A.-G. Tafforin, “Multilayer four-flux model of scattering, emitting and absorbing media,” Atmos. Environ. 35, 5125–5130 (2001).
[Crossref]

Tan, J. C.

Y. Zhang and J. C. Tan, “Radiation transfer of models of massive star formation. I. Dependence on basic core properties,” Astrophys. J. 733, 55–75 (2011).
[Crossref]

Taylor, R. C.

F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989).
[Crossref]

Tunnell, J. W.

Van der Mee, C.

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

Vargas, W. E.

W. E. Vargas, “Generalized four-flux radiative transfer model,” Appl. Opt. 37, 2615–2623 (1998).
[Crossref]

W. E. Vargas and G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[Crossref]

W. E. Vargas and G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
[Crossref]

W. E. Vargas and G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997).
[Crossref]

W. E. Vargas and G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997).
[Crossref]

Vermote, E. F.

Wang, L.

Yang, L.

L. Yang and S. J. Miklavcic, “Revised Kubelka–Munk theory. III. A general theory of light propagation in scattering and absorptive media,” J. Opt. Soc. Am. A 22, 1866–1873 (2005).
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Fig. 1. Sketch of the system and the fluxes. A finitely thick plane-parallel light-scattering medium (

thickness \equiv h

) placed in optical contact with an arbitrary substrate. Incident light

I

, as well as the reflected (

R I

) and transmitted light (

T I

), are considered to be partially collimated.

Download Full Size | PPT Slide | PDF

Fig. 2. Probability allocation for (a) a diffuse beam and (b) a collimated beam.

Download Full Size | PPT Slide | PDF

Fig. 3. Absolute difference (in percentage points) between simulated and theoretical results of the collimated transmittance, as a function of

N_{l}

and

\sum_{i} P_{i}

(top axis). Parameters:

α = 0.4 {mm}^{- 1}

β = (1 - 2 α) / 3 {mm}^{- 1}

h = 1 mm

n_{1} = n_{3} = n_{4} = 1.0

n_{2} = 1.8

τ_{c} = 1

, and

f = 0.5

Download Full Size | PPT Slide | PDF

Fig. 4. Theoretical and simulated values comparison of

R_{c}

R_{d}

T_{c}

, and

T_{d}

as a function of the thickness with increasing extinction (

ζ = α + β

with

α = 10 β

) for different discretizations, i.e., different

\sum_{i} P_{i}

, and number of beams (

N T

). Parameters:

h = 1 mm

n_{1} = n_{3} = n_{4} = 1.0

n_{2} = 1.5

τ_{c} = 1

, and

f = 0.5

Download Full Size | PPT Slide | PDF

Fig. 5. Transverse energy density relative to the initial intensity

I

as a function of the medium’s depth for different values of the

β / α

ratio. The solid lines (markers) show the theoretical (simulated) results. Parameters:

α = 1 {mm}^{- 1}

h = 1 mm

n_{1} = n_{4} = 1.0

n_{2} = 1.5

n_{3} = 1.0

τ_{c} = 1

, and

f = 0.5

Download Full Size | PPT Slide | PDF

Fig. 6. Simulated (markers) and theoretical (solid lines) percentage of beams that have become diffuse at a given medium’s depth for different fractions of the initially collimated beams when reflections at the interface tissue/substrate are taken into account (solid lines) and when they are not (dotted lines). Parameters:

α = 0.5 {mm}^{- 1}

β = 0.005 {mm}^{- 1}

h = 1 mm

n_{1} = 1.0

n_{2} = 1.41

(refractive index of a mammal tissue [32]),

τ_{c} = 1.0

, and

n_{3} = 1.5

n_{4} = 1

for the first and

n_{3} = n_{4} = 1.41

second case, respectively,

N_{T} = 10^{6}

beams.

Download Full Size | PPT Slide | PDF

Fig. 7. Theoretical (solid lines) and simulated (markers) results of the total absorptance

A

as well as the film’s

A_{f}

and substrate’s

A_{s}

contribution to the absorptance for a low-absorbing substrate, shown in the figure at the top (

τ_{c} = 0.8

), and for a high-absorbing substrate, figure at the bottom (

τ_{c} = 0.2

). Parameters:

α = 1 {mm}^{- 1}

h = 1 mm

n_{1} = n_{4} = 1.0

n_{2} = 1.3

n_{3} = 1.5

, and

f = 0.5

Download Full Size | PPT Slide | PDF

Fig. 8. Response function of the system to a delta-like pulse. Histogram showing the ratio of collimated and diffuse beams that have been either reflected or transmitted as a function of time. The bin width is equal to a time element

Δ t

. The total number of incident beams utilized in this simulation is

N_{T} = 5 \times 10^{7}

Download Full Size | PPT Slide | PDF

Tables (3)

Table 1. Probabilities Correspondence^a

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Algorithm 1. Four-Flux Monte Carlo

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Table 2. System’s Properties^a

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Equations (81)

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\frac{d I_{c}}{d z} = - (α + β) I_{c},

\frac{d J_{c}}{d z} = (α + β) J_{c},

\frac{d I_{d}}{d z} = ξ [(1 - σ_{d}) α (J_{d} - I_{d}) - β I_{d}] + σ_{c} α I_{c} + (1 - σ_{c}) α J_{c},

\frac{d J_{d}}{d z} = ξ [β J_{d} + (1 - σ_{d}) α (J_{d} - I_{d})] - σ_{c} α J_{c} - (1 - σ_{c}) α I_{c},

I_{c} (z) = c_{1} e^{- ζ z},

J_{c} (z) = c_{2} e^{ζ z},

J_{d} (z) = c_{3} \cosh (A z) + c_{4} \sinh (A z) + c_{5} e^{ζ z} + c_{6} e^{- ζ z},

I_{d} (z) = c_{7} \cosh (A z) + c_{8} \sinh (A z) + c_{9} e^{ζ z} + c_{10} e^{- ζ z},

A = ξ \sqrt{β [β + 2 α (1 - σ_{d})]},

B = α [ξ β σ_{c} + ξ α (1 - σ_{d}) + ζ σ_{c}],

C = α {ξ [α (1 - σ_{d}) + β (1 - σ_{c})] - ζ (1 - σ_{c})},

D = ξ [β + α (1 - σ_{d})],

E = ξ α (1 - σ_{d}) = D - ξ β,

U (z) = I_{c} (z) + I_{d} (z) + J_{c} (z) + J_{d} (z) .

R = f R_{c} + R_{d} = f (R_{c} + R_{c d}) + (1 - f) R_{d d},

T = f T_{c} + T_{d} = f (T_{c} + T_{c d}) + (1 - f) T_{d d},

R_{c} = r_{c 12} + \frac{{(1 - r_{c 12})}^{2} R_{s c} e^{- 2 ζ h}}{1 - r_{c 12} R_{s c} e^{- 2 ζ h}},

R_{c d} = \frac{(1 - r_{d 21}) (1 - r_{c 12}) e^{- ζ h} [C_{0} + C_{1} e^{ζ h} + C_{2} e^{- ζ h}]}{(A^{2} - ζ^{2}) (1 - r_{c 12} R_{s c} e^{- 2 ζ h}) DEN},

R_{d d} = r_{d 12} - \frac{(1 - r_{d 21}) (1 - r_{d 12})}{DEN} \times [A R_{s d} \cosh (A h) + (E - R_{s d} D) \sinh (A h)],

T_{c} = \frac{(1 - r_{c 23}) (1 - r_{c 34}) τ_{c} (1 - r_{c 12}) e^{- ζ h}}{(1 - r_{c 23} r_{c 34} τ_{c}^{2}) (1 - r_{c 12} R_{s c} e^{- 2 ζ h})},

T_{c d} = [(1 - r_{c 12}) (1 - r_{d 23}) (1 - r_{d 34}) τ_{d} e^{- ζ h}] \times \frac{D_{1} \cosh (A h) + D_{2} \sinh (A h) + D_{3} e^{ζ h} + D_{4} e^{- ζ h}}{(1 - r_{d 32} r_{d 34} τ_{d}^{2}) (1 - r_{c 12} R_{s c} e^{- ζ h}) (A^{2} - ζ^{2}) DEN},

T_{d d} = - \frac{(1 - r_{d 12}) (1 - r_{d 23}) (1 - r_{d 34}) A τ_{d}}{(1 - r_{d 32} r_{d 34} τ_{d}^{2}) DEN},

DEN = A (R_{s d} r_{d 21} - 1) \cosh (A h) + [E (R_{s d} + r_{d 21}) - D (1 + r_{d 21} R_{s d})] \sinh (A h) .

R_{s c} = \frac{r_{c 23} (1 - 2 r_{c 34} τ_{c}^{2}) + r_{c 34} τ_{c}^{2}}{1 - r_{c 23} r_{c 34} τ_{c}^{2}},

R_{s d} = \frac{r_{d 23} + (1 - r_{d 23} - r_{d 32}) r_{d 34} τ_{d}^{2}}{1 - r_{d 32} r_{d 34} τ_{d}^{2}},

A = 1 - R - T .

A = A_{f} + A_{s} = A_{f} + f A_{s c} + (1 - f) A_{s d},

A_{s c} = \frac{(1 - r_{c 12}) [D_{1} \cosh (A h) + D_{2} \sinh (A h) + D_{3} e^{ζ h} + D_{4} e^{- ζ h}]}{(1 - r_{c 12} R_{s c} e^{- 2 ζ h}) (A^{2} - ζ^{2}) DEN} + \frac{[(1 - r_{c 23}) (1 - τ_{c}) + r_{c 31} τ_{c} (1 - τ_{c} - r_{c 23} + r_{c 23} τ_{c})] (1 - r_{c 12} e^{- ζ h})}{(1 - r_{c 23} r_{c 34} τ_{c}^{2}) (1 - r_{c 12} R_{s c} e^{- 2 ζ h})},

A_{s d} = - \frac{[(1 - r_{d 23}) (1 - τ_{d}) + r_{d 34} τ_{d} (1 - τ_{d} - r_{d 23} τ_{d})] (1 - r_{d 12}) A}{(1 - r_{d 32} r_{d 34} τ_{d}^{2}) DEN} .

\begin{array}{l} R_{g} = r_{d 23} = R_{s d} & A = \sqrt{K^{2} + 2 K S} = S b, \\ T_{g} = (1 - r_{d 23}) τ_{d} & D = \sqrt{K + S}, \\ K = ξ β & E = S, \\ S = ξ α (1 - σ_{d}) & X = h, \end{array}

N_{l} = \frac{(ξ + 1) ζ h - ξ α σ_{d} h}{\sum_{i} P_{i}},

\sum_{i} P_{i} = P K^{col} + P K^{dif} + P S_{back}^{col} + P S_{back}^{dif} + P S_{forward}^{col} .

\sum_{i} P_{i} \leq 0.01

I_{s} (z) = I_{c} (z) + J_{c} (z) = (\frac{α h}{N}) \frac{(1 - r_{c 12}) f I}{1 - r_{c 12} R_{s c} e^{- 2 ζ h}} [e^{- ζ z} + R_{s c} e^{- 2 ζ h} e^{ζ z}],

\frac{d^{2} (J_{d} + I_{d})}{d z^{2}} = ξ^{2} β [β + 2 (1 - σ_{d}) α] (J_{d} + I_{d}) - ξ [β + 2 (1 - σ_{d}) α] α (J_{c} + I_{c}) + (1 - 2 σ_{c}) α ζ (J_{c} + I_{c}),

\frac{d^{2} (J_{d} - I_{d})}{d z^{2}} = ξ^{2} β [β + 2 (1 - σ_{d}) α] (J_{d} + I_{d}) + ξ β (1 - 2 σ_{c}) α (J_{c} - I_{c}) - α ζ [J_{c} - I_{c}] .

- \frac{d^{2} J_{d}}{d z^{2}} + A^{2} J_{d} = B c_{2} e^{ζ z} + C c_{1} e^{- ζ z},

- \frac{d^{2} I_{d}}{d z^{2}} + A^{2} I_{d} = C c_{2} e^{ζ z} + B c_{1} e^{- ζ z} .

c_{5} = \frac{B c_{2}}{A^{2} - ζ^{2}},

c_{6} = \frac{C c_{1}}{A^{2} - ζ^{2}},

c_{9} = \frac{C c_{2}}{A^{2} - ζ^{2}},

c_{10} = \frac{B c_{1}}{A^{2} - ζ^{2}} .

0 = \sin h (A z) [ξ (β + α (1 - σ_{d})) c_{4} + ξ (1 - σ_{d}) α c_{8} - A c_{3}] + \cos h (A z) [ξ (β + α (1 - σ_{d})) c_{3} + ξ (1 - σ_{d}) α c_{7} - A c_{4}] + e^{ζ z} [ξ (β + α (1 - σ_{d})) c_{5} + ξ (1 - σ_{d}) α c_{9} - σ_{c} α c_{2} - ζ c_{5}] + e^{- ζ z} [ξ (β + α (1 - σ_{d})) c_{6} + ξ (1 - σ_{d}) α c_{10} - (1 - σ_{c}) α c_{1} - ζ c_{6}] .

c_{7} = \frac{D c_{3} - A c_{4}}{E},

c_{8} = \frac{D c_{4} - A c_{3}}{E} .

I_{c} (0) = (1 - r_{c 12}) f I + r_{c 12} J_{c} (0),

I_{d} (0) = (1 - r_{d 12}) (1 - f) I - r_{d 21} J_{d} (0),

J_{c} (h) = R_{s c} I_{c} (h),

J_{d} (h) = R_{s d} I_{d} (h) .

c_{1} = \frac{(1 - r_{c 12}) f I}{1 - r_{c 12} R_{s c} e^{- 2 ζ h}},

c_{2} = \frac{(1 - r_{c 12}) f I R_{s c} e^{- 2 ζ h}}{1 - r_{c 12} R_{s c} e^{- 2 ζ h}} .

c_{3} = σ_{1} f I + σ_{2} (1 - f) I,

c_{4} = σ_{3} f I + σ_{4} (1 - f) I,

σ_{1} = {[A [(C + B R_{s c}) - R_{s d} (B + C R_{s c})] e^{- ζ h}] + [A R_{s d} \cosh (A h) + (E - R_{s d} D) \sinh (A h)] \times [(C - r_{d 21} B) R_{s c} e^{- 2 ζ h} + (B - r_{d 21} C)]} \times (1 - r_{c 12}) {[(1 - r_{c 12} R_{s c} e^{- 2 ζ h}) (A^{2} - ζ^{2}) DEN]}^{- 1},

σ_{2} = \frac{(r_{d 12} - 1) [A R_{s d} \cosh (A h) + (E - R_{s d} D) \sinh (A h)]}{DEN},

σ_{3} = {[(D - r_{d 21} E) [(C + B R_{s c}) - R_{s d} (B + C R_{s c})] e^{- ζ h}] - [(E - R_{s d} D) \cosh (A h) + A R_{s d} \sinh (A h)] \times [(C - r_{d 21} B) R_{s c} e^{- 2 ζ h} + (B - r_{d 21} C)]} \times (1 - r_{c 12}) {[(1 - r_{c 12} R_{s c} e^{- 2 ζ h}) (A^{2} - ζ^{2}) DEN]}^{- 1},

σ_{4} = \frac{(1 - r_{d 12}) [(E - R_{s d} D) \cosh (A h) + A R_{s d} \sinh (A h)]}{DEN} .

R_{c} = \frac{r_{c 12} f I + (1 - r_{c 12}) J_{c} (0)}{I} .

R_{d} = \frac{r_{d 12} (1 - f) I + (1 - r_{d 21}) J_{d} (0)}{I} .

C_{0} = A [(C + B R_{s c}) - R_{s d} (B + C R_{s c})],

C_{1} = [A (B R_{s d} - C) \cos h (A h) + [B (E - D R_{s d}) + C (E R_{s d} - D)] \sinh (A h)],

C_{2} = R_{s c} [A (C R_{s d} - B) \cos h (A h) + [C (E - D R_{s d}) + B (E R_{s d} - D)] \sinh (A h)] .

T_{c} = T_{s c} \frac{I_{c} (h)}{I} = [\frac{(1 - r_{c 23}) (1 - r_{c 34}) τ_{c}}{(1 - r_{c 23} r_{c 34} τ_{c}^{2})}] \frac{I_{c} (h)}{I},

T_{d} = T_{s d} \frac{I_{d} (h)}{I} = [\frac{(1 - r_{d 23}) (1 - r_{d 34})}{1 - r_{d 32} r_{d 34} τ_{d}^{2}}] \frac{I_{d} (h)}{I} .

D_{1} = A [(r_{d 21} C - B) + R_{s c} (r_{d 21} B - C)],

D_{2} = (E - r_{d 21} D) (C + B R_{s c}) - (D - r_{d 21} E) (B + C R_{s c}),

D_{3} = A (B - r_{d 21} C),

D_{4} = A R_{s c} (C - r_{d 21} B) .

A_{s} = \frac{A_{d} I_{d} (h) + A_{c} I_{c} (h)}{I},

A_{d} = 1 - R_{s d} - T_{s d} = \frac{(1 - r_{d 23}) (1 - τ_{d}) + r_{d 34} τ_{d} (1 - τ_{d} - r_{d 23} τ_{d})}{1 - r_{d 32} r_{d 34} τ_{d}^{2}},

A_{c} = 1 - R_{s c} - T_{s c} = \frac{(1 - r_{c 23}) (1 - τ_{c}) + r_{c 34} τ_{c} (1 - τ_{c} - r_{c 23} τ_{c})}{1 - r_{c 23} r_{c 34} τ_{c}^{2}} .

ξ (z) = \frac{\int_{0}^{1} I (z, μ) d μ}{\int_{0}^{1} μ I (z, μ) d μ} .

α = \frac{η}{V} C_{sca},

β = \frac{η}{V} C_{abs},

σ_{c} = \frac{\int_{0}^{1} p (μ) d μ}{\int_{- 1}^{1} p (μ) d μ},

r_{s} (θ_{i}) = {| \frac{n_{j} \cos θ_{i} - n_{i} \cos θ_{j}}{n_{j} \cos θ_{i} + n_{i} \cos θ_{j}} |}^{2}

r_{p} = {| \frac{n_{j} \cos θ_{j} - n_{i} \cos θ_{i}}{n_{j} \cos θ_{j} + n_{i} \cos θ_{i}} |}^{2}

r_{c i j} = \frac{1}{2} (r_{s} + r_{p}),

r_{d i j} = \frac{1}{2} \int_{0}^{π / 2} [r_{s} (α) + r_{p} (α)] \sin (2 α) d α .

τ = \exp (- \int_{0}^{l} α (z) d z),

τ_{d} = τ_{c}^{ξ} .

Probability	Physical Parameter	Description
$P K^{col}$	$β d z$	Absorption probability for a collimated beam
$P K^{dif}$	$ξ β d z$	Absorption probability for a diffuse beam
$P R^{col}$	$r_{c i j}$	(Fresnel) Reflection probability for a collimated beam
$P R^{dif}$	$r_{d i j}$	(Fresnel) Reflection probability for a diffuse beam
$P S_{back}^{col}$	$α (1 - σ_{c}) d z$	Backscattering probability for a collimated beam
$P S_{back}^{dif}$	$ξ α (1 - σ_{d}) d z$	Backscattering probability for a diffuse beam
$P S_{forward}^{col}$	$α σ_{c} d z$	Forward scattering probability for a collimated beam

Probability	Physical Parameter	Description
$P K^{col}$	$β d z$	Absorption probability for a collimated beam
$P K^{dif}$	$ξ β d z$	Absorption probability for a diffuse beam
$P R^{col}$	$r_{c i j}$	(Fresnel) Reflection probability for a collimated beam
$P R^{dif}$	$r_{d i j}$	(Fresnel) Reflection probability for a diffuse beam
$P S_{back}^{col}$	$α (1 - σ_{c}) d z$	Backscattering probability for a collimated beam
$P S_{back}^{dif}$	$ξ α (1 - σ_{d}) d z$	Backscattering probability for a diffuse beam
$P S_{forward}^{col}$	$α σ_{c} d z$	Forward scattering probability for a collimated beam

$i$	$n$	$α ({mm}^{- 1})$	$β ({mm}^{- 1})$	$h (mm)$	$l$
1	1.5	0.1	0.1	1	502
2	1.8	1	1	1	502
3	1.5	0	0	2	1004

Abstract

References

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