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.

© 2019 Optical Society of America

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References

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2016 (3)

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)

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)

2005 (1)

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)

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]

1999 (1)

1998 (1)

1997 (4)

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, “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)

1984 (1)

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)

1971 (1)

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.

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]

Alonso, A. G.

Amrani, B.

Andraud, C.

Arancibia-Bulnes, C. A.

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.

Briton, J.

Budak, V. P.

Bugliaro, L.

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]

Buras-Schnell, R.

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]

Chandrasekhar, S.

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

Chauvet, O.

de la Osa, R. A.

Dowling, T.

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]

El Haber, F.

Eldridge, J. I.

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]

Emde, C.

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]

Feld, M. S.

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]

Ference, R. J.

Froyer, G.

Fukshansky, L.

Gasteiger, J.

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]

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]

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]

González, F.

Gouesbet, G.

Granqvist, C.-G.

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]

Gréhan, G.

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]

Guo, S.

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]

Haka, A. S.

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]

Hamann, U.

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]

Hébert, M.

Hecht, E.

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

Hersch, R. D.

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.

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]

Kylling, J.

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]

Laaksonen, K.

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]

Letoulouzan, J.-N.

Li, S.-Y.

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]

Maheu, B.

Matarrese, R.

Mayer, B.

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]

Mazauric, S.

McGee, S. A.

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]

Mcnicholas, H. J.

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

Miklavcic, S. J.

Mirkovic, J.

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]

Mishchenko, M. I.

Moreno, F.

Mudgett, P.

Munk, F.

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

Nieminen, R.

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]

Niklasson, G. A.

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]

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.

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]

Platnick, S.

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

Preuss, L. E.

Puisto, S.

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]

Richards, L.

Richter, B.

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]

Rocquefelte, X.

Rostedt, N.

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]

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]

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[Crossref]

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[Crossref]

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Figures (8)

Fig. 1.
Fig. 1. Sketch of the system and the fluxes. A finitely thick plane-parallel light-scattering medium (thicknessh) placed in optical contact with an arbitrary substrate. Incident light I, as well as the reflected (RI) and transmitted light (TI), are considered to be partially collimated.
Fig. 2.
Fig. 2. Probability allocation for (a) a diffuse beam and (b) a collimated beam.
Fig. 3.
Fig. 3. Absolute difference (in percentage points) between simulated and theoretical results of the collimated transmittance, as a function of Nl and iPi (top axis). Parameters: α=0.4mm1, β=(12α)/3mm1, h=1mm, n1=n3=n4=1.0, n2=1.8, τc=1, and f=0.5.
Fig. 4.
Fig. 4. Theoretical and simulated values comparison of Rc, Rd, Tc, and Td as a function of the thickness with increasing extinction (ζ=α+β with α=10β) for different discretizations, i.e., different iPi, and number of beams (NT). Parameters: h=1mm, n1=n3=n4=1.0, n2=1.5, τc=1, and f=0.5.
Fig. 5.
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=1mm1, h=1mm, n1=n4=1.0, n2=1.5, n3=1.0, τc=1, and f=0.5.
Fig. 6.
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=0.5mm1, β=0.005mm1, h=1mm, n1=1.0, n2=1.41 (refractive index of a mammal tissue [32]), τc=1.0, and n3=1.5, n4=1 for the first and n3=n4=1.41 second case, respectively, NT=106 beams.
Fig. 7.
Fig. 7. Theoretical (solid lines) and simulated (markers) results of the total absorptance A as well as the film’s Af and substrate’s As contribution to the absorptance for a low-absorbing substrate, shown in the figure at the top (τc=0.8), and for a high-absorbing substrate, figure at the bottom (τc=0.2). Parameters: α=1mm1, h=1mm, n1=n4=1.0, n2=1.3, n3=1.5, and f=0.5.
Fig. 8.
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 Δt. The total number of incident beams utilized in this simulation is NT=5×107.

Tables (3)

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Table 1. Probabilities Correspondencea

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Algorithm 1. Four-Flux Monte Carlo

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Table 2. System’s Propertiesa

Equations (81)

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dIcdz=(α+β)Ic,
dJcdz=(α+β)Jc,
dIddz=ξ[(1σd)α(JdId)βId]+σcαIc+(1σc)αJc,
dJddz=ξ[βJd+(1σd)α(JdId)]σcαJc(1σc)αIc,
Ic(z)=c1eζz,
Jc(z)=c2eζz,
Jd(z)=c3cosh(Az)+c4sinh(Az)+c5eζz+c6eζz,
Id(z)=c7cosh(Az)+c8sinh(Az)+c9eζz+c10eζz,
A=ξβ[β+2α(1σd)],
B=α[ξβσc+ξα(1σd)+ζσc],
C=α{ξ[α(1σd)+β(1σc)]ζ(1σc)},
D=ξ[β+α(1σd)],
E=ξα(1σd)=Dξβ,
U(z)=Ic(z)+Id(z)+Jc(z)+Jd(z).
R=fRc+Rd=f(Rc+Rcd)+(1f)Rdd,
T=fTc+Td=f(Tc+Tcd)+(1f)Tdd,
Rc=rc12+(1rc12)2Rsce2ζh1rc12Rsce2ζh,
Rcd=(1rd21)(1rc12)eζh[C0+C1eζh+C2eζh](A2ζ2)(1rc12Rsce2ζh)DEN,
Rdd=rd12(1rd21)(1rd12)DEN×[ARsdcosh(Ah)+(ERsdD)sinh(Ah)],
Tc=(1rc23)(1rc34)τc(1rc12)eζh(1rc23rc34τc2)(1rc12Rsce2ζh),
Tcd=[(1rc12)(1rd23)(1rd34)τdeζh]×D1cosh(Ah)+D2sinh(Ah)+D3eζh+D4eζh(1rd32rd34τd2)(1rc12Rsceζh)(A2ζ2)DEN,
Tdd=(1rd12)(1rd23)(1rd34)Aτd(1rd32rd34τd2)DEN,
DEN=A(Rsdrd211)cosh(Ah)+[E(Rsd+rd21)D(1+rd21Rsd)]sinh(Ah).
Rsc=rc23(12rc34τc2)+rc34τc21rc23rc34τc2,
Rsd=rd23+(1rd23rd32)rd34τd21rd32rd34τd2,
A=1RT.
A=Af+As=Af+fAsc+(1f)Asd,
Asc=(1rc12)[D1cosh(Ah)+D2sinh(Ah)+D3eζh+D4eζh](1rc12Rsce2ζh)(A2ζ2)DEN+[(1rc23)(1τc)+rc31τc(1τcrc23+rc23τc)](1rc12eζh)(1rc23rc34τc2)(1rc12Rsce2ζh),
Asd=[(1rd23)(1τd)+rd34τd(1τdrd23τd)](1rd12)A(1rd32rd34τd2)DEN.
Rg=rd23=RsdA=K2+2KS=Sb,Tg=(1rd23)τdD=K+S,K=ξβE=S,S=ξα(1σd)X=h,
Nl=(ξ+1)ζhξασdhiPi,
iPi=PKcol+PKdif+PSbackcol+PSbackdif+PSforwardcol.
iPi0.01
Is(z)=Ic(z)+Jc(z)=(αhN)(1rc12)fI1rc12Rsce2ζh[eζz+Rsce2ζheζz],
d2(Jd+Id)dz2=ξ2β[β+2(1σd)α](Jd+Id)ξ[β+2(1σd)α]α(Jc+Ic)+(12σc)αζ(Jc+Ic),
d2(JdId)dz2=ξ2β[β+2(1σd)α](Jd+Id)+ξβ(12σc)α(JcIc)αζ[JcIc].
d2Jddz2+A2Jd=Bc2eζz+Cc1eζz,
d2Iddz2+A2Id=Cc2eζz+Bc1eζz.
c5=Bc2A2ζ2,
c6=Cc1A2ζ2,
c9=Cc2A2ζ2,
c10=Bc1A2ζ2.
0=sinh(Az)[ξ(β+α(1σd))c4+ξ(1σd)αc8Ac3]+cosh(Az)[ξ(β+α(1σd))c3+ξ(1σd)αc7Ac4]+eζz[ξ(β+α(1σd))c5+ξ(1σd)αc9σcαc2ζc5]+eζz[ξ(β+α(1σd))c6+ξ(1σd)αc10(1σc)αc1ζc6].
c7=Dc3Ac4E,
c8=Dc4Ac3E.
Ic(0)=(1rc12)fI+rc12Jc(0),
Id(0)=(1rd12)(1f)Ird21Jd(0),
Jc(h)=RscIc(h),
Jd(h)=RsdId(h).
c1=(1rc12)fI1rc12Rsce2ζh,
c2=(1rc12)fIRsce2ζh1rc12Rsce2ζh.
c3=σ1fI+σ2(1f)I,
c4=σ3fI+σ4(1f)I,
σ1={[A[(C+BRsc)Rsd(B+CRsc)]eζh]+[ARsdcosh(Ah)+(ERsdD)sinh(Ah)]×[(Crd21B)Rsce2ζh+(Brd21C)]}×(1rc12)[(1rc12Rsce2ζh)(A2ζ2)DEN]1,
σ2=(rd121)[ARsdcosh(Ah)+(ERsdD)sinh(Ah)]DEN,
σ3={[(Drd21E)[(C+BRsc)Rsd(B+CRsc)]eζh][(ERsdD)cosh(Ah)+ARsdsinh(Ah)]×[(Crd21B)Rsce2ζh+(Brd21C)]}×(1rc12)[(1rc12Rsce2ζh)(A2ζ2)DEN]1,
σ4=(1rd12)[(ERsdD)cosh(Ah)+ARsdsinh(Ah)]DEN.
Rc=rc12fI+(1rc12)Jc(0)I.
Rd=rd12(1f)I+(1rd21)Jd(0)I.
C0=A[(C+BRsc)Rsd(B+CRsc)],
C1=[A(BRsdC)cosh(Ah)+[B(EDRsd)+C(ERsdD)]sinh(Ah)],
C2=Rsc[A(CRsdB)cosh(Ah)+[C(EDRsd)+B(ERsdD)]sinh(Ah)].
Tc=TscIc(h)I=[(1rc23)(1rc34)τc(1rc23rc34τc2)]Ic(h)I,
Td=TsdId(h)I=[(1rd23)(1rd34)1rd32rd34τd2]Id(h)I.
D1=A[(rd21CB)+Rsc(rd21BC)],
D2=(Erd21D)(C+BRsc)(Drd21E)(B+CRsc),
D3=A(Brd21C),
D4=ARsc(Crd21B).
As=AdId(h)+AcIc(h)I,
Ad=1RsdTsd=(1rd23)(1τd)+rd34τd(1τdrd23τd)1rd32rd34τd2,
Ac=1RscTsc=(1rc23)(1τc)+rc34τc(1τcrc23τc)1rc23rc34τc2.
ξ(z)=01I(z,μ)dμ01μI(z,μ)dμ.
α=ηVCsca,
β=ηVCabs,
σc=01p(μ)dμ11p(μ)dμ,
rs(θi)=|njcosθinicosθjnjcosθi+nicosθj|2
rp=|njcosθjnicosθinjcosθj+nicosθi|2
rcij=12(rs+rp),
rdij=120π/2[rs(α)+rp(α)]sin(2α)dα.
τ=exp(0lα(z)dz),
τd=τcξ.