Abstract

Luminescent waveguides (LWs) occur in a wide range of applications, from solar concentrators to doped fiber amplifiers. Here we report a comprehensive analysis of escape-cone losses in LWs, which are losses associated with internal rays making an angle less than the critical angle with a waveguide surface. For applications such as luminescent solar concentrators, escape-cone losses often dominate all others. A statistical treatment of escape-cone losses is given accounting for photoselection, photon polarization, and the Fresnel relations, and the model is used to analyze light absorption and propagation in waveguides with isotropic and orientationally aligned luminophores. The results are then compared to experimental measurements performed on a fluorescent dye-doped poly(methyl methacrylate) waveguide.

© 2013 Optical Society of America

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2012 (1)

M. G. Debije and P. P. C. Verbunt, “Thirty years of luminescent solar concentrator research: solar energy for the built environment,” Adv. Energy Mater. 2, 12–35 (2012).
[CrossRef]

2010 (6)

2009 (1)

2008 (1)

2007 (1)

Z. Chen and T. M. Swager, “Synthesis and characterization of fluorescent acenequinones as dyes for guest–host liquid crystal displays,” Org. Lett. 9, 997–1000 (2007).
[CrossRef]

2006 (1)

X. Zhang, R. Yamaguchi, K. Moriyama, M. Kadowaki, T. Kobayashi, T. Ishi-i, T. Thiemann, and S. Mataka, “Highly dichroic benzo-2,1,3-thiadiazole dyes containing five linearly π-conjugated aromatic residues, with fluorescent emission ranging from green to red, in a liquid crystal guest-host system,” J. Mater. Chem. 16, 736–740 (2006).
[CrossRef]

2000 (1)

M. I. Barnik, V. G. Vasil’chenko, S. V. Golovkin, A. M. Medvedkov, A. S. Solov’ev, and S. G. Yudin, “Scintillation properties of materials based on liquid crystals in static and dynamic states,” Instrum. Exp. Tech. 43, 602–611 (2000).
[CrossRef]

1995 (1)

V. V. Popov and V. N. Yakimenko, “State of the art of prospects for investigations of luminescent solar concentrators,” J. Appl. Spectrosc. 62, 573–577 (1995).
[CrossRef]

1981 (1)

1979 (1)

1976 (1)

1973 (1)

G. Baur, A. Stieb, and G. Meier, “Polarized fluorescence of dyes oriented in room temperature nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 22, 261–269 (1973).
[CrossRef]

1970 (1)

R. H. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2, 883–888 (1970).
[CrossRef]

1969 (1)

G. J. Keil, “Radiance amplification by a fluorescence radiation converter,” Appl. Phys. 40, 3544–3547 (1969).
[CrossRef]

1964 (1)

1949 (1)

Baldo, M. A.

Barnham, K. W. J.

Barnik, M. I.

M. I. Barnik, V. G. Vasil’chenko, S. V. Golovkin, A. M. Medvedkov, A. S. Solov’ev, and S. G. Yudin, “Scintillation properties of materials based on liquid crystals in static and dynamic states,” Instrum. Exp. Tech. 43, 602–611 (2000).
[CrossRef]

Bastiaansen, C. W. M.

P. P. C. Verbunt, C. W. M. Bastiaansen, D. J. Broer, and M. G. Debije, “The effect of dyes aligned by liquid crystals on luminescent solar concentrator performance,” presented at the 24th European Photovoltaic Solar Energy Conference, Hamburg, 21–25 September 2009.

Batchelder, J. S.

Baur, G.

G. Baur, A. Stieb, and G. Meier, “Polarized fluorescence of dyes oriented in room temperature nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 22, 261–269 (1973).
[CrossRef]

Bende, E. E.

Beyler, A. P.

Bose, R.

Broer, D. J.

P. P. C. Verbunt, C. W. M. Bastiaansen, D. J. Broer, and M. G. Debije, “The effect of dyes aligned by liquid crystals on luminescent solar concentrator performance,” presented at the 24th European Photovoltaic Solar Energy Conference, Hamburg, 21–25 September 2009.

Büchtemann, A.

Budel, T.

Burgers, A. R.

Chatten, A. J.

Chen, Z.

Z. Chen and T. M. Swager, “Synthesis and characterization of fluorescent acenequinones as dyes for guest–host liquid crystal displays,” Org. Lett. 9, 997–1000 (2007).
[CrossRef]

Cheng, Y. Y.

Clady, R. G. C. R.

Cole, T.

Coles, H.

H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photonics 4, 676–685 (2010).
[CrossRef]

De Mello Donegá, C.

Debije, M. G.

M. G. Debije and P. P. C. Verbunt, “Thirty years of luminescent solar concentrator research: solar energy for the built environment,” Adv. Energy Mater. 2, 12–35 (2012).
[CrossRef]

M. G. Debije, “Solar energy collectors with tunable transmission,” Adv. Funct. Mater. 20, 1498–1502 (2010).
[CrossRef]

P. P. C. Verbunt and M. G. Debije, “Progress in luminescent solar concentrator research: solar energy for the built environment,” in World Renewable Energy Congress (Linköping University Electronic Press, 2011), p. 2751–2758.

P. P. C. Verbunt, C. W. M. Bastiaansen, D. J. Broer, and M. G. Debije, “The effect of dyes aligned by liquid crystals on luminescent solar concentrator performance,” presented at the 24th European Photovoltaic Solar Energy Conference, Hamburg, 21–25 September 2009.

Farrell, D. J.

Golovkin, S. V.

M. I. Barnik, V. G. Vasil’chenko, S. V. Golovkin, A. M. Medvedkov, A. S. Solov’ev, and S. G. Yudin, “Scintillation properties of materials based on liquid crystals in static and dynamic states,” Instrum. Exp. Tech. 43, 602–611 (2000).
[CrossRef]

Ishi-i, T.

X. Zhang, R. Yamaguchi, K. Moriyama, M. Kadowaki, T. Kobayashi, T. Ishi-i, T. Thiemann, and S. Mataka, “Highly dichroic benzo-2,1,3-thiadiazole dyes containing five linearly π-conjugated aromatic residues, with fluorescent emission ranging from green to red, in a liquid crystal guest-host system,” J. Mater. Chem. 16, 736–740 (2006).
[CrossRef]

Johnson, B. L.

S. McDowall, B. L. Johnson, and D. L. Patrick, “Simulations of luminescent solar concentrators: effects of polarization and fluorophore alignment,” J. Appl. Phys. 108, 053508 (2010).
[CrossRef]

Jones, R. C.

Kadowaki, M.

X. Zhang, R. Yamaguchi, K. Moriyama, M. Kadowaki, T. Kobayashi, T. Ishi-i, T. Thiemann, and S. Mataka, “Highly dichroic benzo-2,1,3-thiadiazole dyes containing five linearly π-conjugated aromatic residues, with fluorescent emission ranging from green to red, in a liquid crystal guest-host system,” J. Mater. Chem. 16, 736–740 (2006).
[CrossRef]

Keil, G. J.

G. J. Keil, “Radiance amplification by a fluorescence radiation converter,” Appl. Phys. 40, 3544–3547 (1969).
[CrossRef]

Kennedy, M.

Kim, H.

Kobayashi, T.

X. Zhang, R. Yamaguchi, K. Moriyama, M. Kadowaki, T. Kobayashi, T. Ishi-i, T. Thiemann, and S. Mataka, “Highly dichroic benzo-2,1,3-thiadiazole dyes containing five linearly π-conjugated aromatic residues, with fluorescent emission ranging from green to red, in a liquid crystal guest-host system,” J. Mater. Chem. 16, 736–740 (2006).
[CrossRef]

Koester, C. J.

Koole, R.

Lakowicz, J. R.

J. R. Lakowicz, Principles of Fluorescence Spectroscopy, 3rd ed. (Springer, 2006).

Lambe, J.

Lehmberg, R. H.

R. H. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2, 883–888 (1970).
[CrossRef]

MacQueen, R. W.

Mataka, S.

X. Zhang, R. Yamaguchi, K. Moriyama, M. Kadowaki, T. Kobayashi, T. Ishi-i, T. Thiemann, and S. Mataka, “Highly dichroic benzo-2,1,3-thiadiazole dyes containing five linearly π-conjugated aromatic residues, with fluorescent emission ranging from green to red, in a liquid crystal guest-host system,” J. Mater. Chem. 16, 736–740 (2006).
[CrossRef]

McCormack, S. J.

McDowall, S.

S. McDowall, B. L. Johnson, and D. L. Patrick, “Simulations of luminescent solar concentrators: effects of polarization and fluorophore alignment,” J. Appl. Phys. 108, 053508 (2010).
[CrossRef]

Medvedkov, A. M.

M. I. Barnik, V. G. Vasil’chenko, S. V. Golovkin, A. M. Medvedkov, A. S. Solov’ev, and S. G. Yudin, “Scintillation properties of materials based on liquid crystals in static and dynamic states,” Instrum. Exp. Tech. 43, 602–611 (2000).
[CrossRef]

Meier, G.

G. Baur, A. Stieb, and G. Meier, “Polarized fluorescence of dyes oriented in room temperature nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 22, 261–269 (1973).
[CrossRef]

Meijerink, A.

Meyer, A.

Meyer, T.

Moriyama, K.

X. Zhang, R. Yamaguchi, K. Moriyama, M. Kadowaki, T. Kobayashi, T. Ishi-i, T. Thiemann, and S. Mataka, “Highly dichroic benzo-2,1,3-thiadiazole dyes containing five linearly π-conjugated aromatic residues, with fluorescent emission ranging from green to red, in a liquid crystal guest-host system,” J. Mater. Chem. 16, 736–740 (2006).
[CrossRef]

Morris, S.

H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photonics 4, 676–685 (2010).
[CrossRef]

Mulder, C. L.

Patrick, D. L.

S. McDowall, B. L. Johnson, and D. L. Patrick, “Simulations of luminescent solar concentrators: effects of polarization and fluorophore alignment,” J. Appl. Phys. 108, 053508 (2010).
[CrossRef]

Popov, V. V.

V. V. Popov and V. N. Yakimenko, “State of the art of prospects for investigations of luminescent solar concentrators,” J. Appl. Spectrosc. 62, 573–577 (1995).
[CrossRef]

Quilitz, J.

Reusswig, P. D.

Richards, B. S.

Rotschild, C.

Schmidt, T. W.

Shurcliff, W. A.

Slooff, L. H.

Snitzer, E.

Solov’ev, A. S.

M. I. Barnik, V. G. Vasil’chenko, S. V. Golovkin, A. M. Medvedkov, A. S. Solov’ev, and S. G. Yudin, “Scintillation properties of materials based on liquid crystals in static and dynamic states,” Instrum. Exp. Tech. 43, 602–611 (2000).
[CrossRef]

Stieb, A.

G. Baur, A. Stieb, and G. Meier, “Polarized fluorescence of dyes oriented in room temperature nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 22, 261–269 (1973).
[CrossRef]

Swager, T. M.

Z. Chen and T. M. Swager, “Synthesis and characterization of fluorescent acenequinones as dyes for guest–host liquid crystal displays,” Org. Lett. 9, 997–1000 (2007).
[CrossRef]

Thiemann, T.

X. Zhang, R. Yamaguchi, K. Moriyama, M. Kadowaki, T. Kobayashi, T. Ishi-i, T. Thiemann, and S. Mataka, “Highly dichroic benzo-2,1,3-thiadiazole dyes containing five linearly π-conjugated aromatic residues, with fluorescent emission ranging from green to red, in a liquid crystal guest-host system,” J. Mater. Chem. 16, 736–740 (2006).
[CrossRef]

van Sark, W. G. J. H. M.

Vanmaekelbergh, D.

Vasil’chenko, V. G.

M. I. Barnik, V. G. Vasil’chenko, S. V. Golovkin, A. M. Medvedkov, A. S. Solov’ev, and S. G. Yudin, “Scintillation properties of materials based on liquid crystals in static and dynamic states,” Instrum. Exp. Tech. 43, 602–611 (2000).
[CrossRef]

Velázquez, A. M.

Verbunt, P. P. C.

M. G. Debije and P. P. C. Verbunt, “Thirty years of luminescent solar concentrator research: solar energy for the built environment,” Adv. Energy Mater. 2, 12–35 (2012).
[CrossRef]

P. P. C. Verbunt and M. G. Debije, “Progress in luminescent solar concentrator research: solar energy for the built environment,” in World Renewable Energy Congress (Linköping University Electronic Press, 2011), p. 2751–2758.

P. P. C. Verbunt, C. W. M. Bastiaansen, D. J. Broer, and M. G. Debije, “The effect of dyes aligned by liquid crystals on luminescent solar concentrator performance,” presented at the 24th European Photovoltaic Solar Energy Conference, Hamburg, 21–25 September 2009.

Weber, W. H.

Wilson, L. R.

Yakimenko, V. N.

V. V. Popov and V. N. Yakimenko, “State of the art of prospects for investigations of luminescent solar concentrators,” J. Appl. Spectrosc. 62, 573–577 (1995).
[CrossRef]

Yamaguchi, R.

X. Zhang, R. Yamaguchi, K. Moriyama, M. Kadowaki, T. Kobayashi, T. Ishi-i, T. Thiemann, and S. Mataka, “Highly dichroic benzo-2,1,3-thiadiazole dyes containing five linearly π-conjugated aromatic residues, with fluorescent emission ranging from green to red, in a liquid crystal guest-host system,” J. Mater. Chem. 16, 736–740 (2006).
[CrossRef]

Yudin, S. G.

M. I. Barnik, V. G. Vasil’chenko, S. V. Golovkin, A. M. Medvedkov, A. S. Solov’ev, and S. G. Yudin, “Scintillation properties of materials based on liquid crystals in static and dynamic states,” Instrum. Exp. Tech. 43, 602–611 (2000).
[CrossRef]

Zewail, A. H.

Zhang, X.

X. Zhang, R. Yamaguchi, K. Moriyama, M. Kadowaki, T. Kobayashi, T. Ishi-i, T. Thiemann, and S. Mataka, “Highly dichroic benzo-2,1,3-thiadiazole dyes containing five linearly π-conjugated aromatic residues, with fluorescent emission ranging from green to red, in a liquid crystal guest-host system,” J. Mater. Chem. 16, 736–740 (2006).
[CrossRef]

Adv. Energy Mater. (1)

M. G. Debije and P. P. C. Verbunt, “Thirty years of luminescent solar concentrator research: solar energy for the built environment,” Adv. Energy Mater. 2, 12–35 (2012).
[CrossRef]

Adv. Funct. Mater. (1)

M. G. Debije, “Solar energy collectors with tunable transmission,” Adv. Funct. Mater. 20, 1498–1502 (2010).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. (1)

G. J. Keil, “Radiance amplification by a fluorescence radiation converter,” Appl. Phys. 40, 3544–3547 (1969).
[CrossRef]

Instrum. Exp. Tech. (1)

M. I. Barnik, V. G. Vasil’chenko, S. V. Golovkin, A. M. Medvedkov, A. S. Solov’ev, and S. G. Yudin, “Scintillation properties of materials based on liquid crystals in static and dynamic states,” Instrum. Exp. Tech. 43, 602–611 (2000).
[CrossRef]

J. Appl. Phys. (1)

S. McDowall, B. L. Johnson, and D. L. Patrick, “Simulations of luminescent solar concentrators: effects of polarization and fluorophore alignment,” J. Appl. Phys. 108, 053508 (2010).
[CrossRef]

J. Appl. Spectrosc. (1)

V. V. Popov and V. N. Yakimenko, “State of the art of prospects for investigations of luminescent solar concentrators,” J. Appl. Spectrosc. 62, 573–577 (1995).
[CrossRef]

J. Mater. Chem. (1)

X. Zhang, R. Yamaguchi, K. Moriyama, M. Kadowaki, T. Kobayashi, T. Ishi-i, T. Thiemann, and S. Mataka, “Highly dichroic benzo-2,1,3-thiadiazole dyes containing five linearly π-conjugated aromatic residues, with fluorescent emission ranging from green to red, in a liquid crystal guest-host system,” J. Mater. Chem. 16, 736–740 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

Mol. Cryst. Liq. Cryst. (1)

G. Baur, A. Stieb, and G. Meier, “Polarized fluorescence of dyes oriented in room temperature nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 22, 261–269 (1973).
[CrossRef]

Nat. Photonics (1)

H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photonics 4, 676–685 (2010).
[CrossRef]

Opt. Express (4)

Org. Lett. (1)

Z. Chen and T. M. Swager, “Synthesis and characterization of fluorescent acenequinones as dyes for guest–host liquid crystal displays,” Org. Lett. 9, 997–1000 (2007).
[CrossRef]

Phys. Rev. A (1)

R. H. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2, 883–888 (1970).
[CrossRef]

Other (3)

P. P. C. Verbunt and M. G. Debije, “Progress in luminescent solar concentrator research: solar energy for the built environment,” in World Renewable Energy Congress (Linköping University Electronic Press, 2011), p. 2751–2758.

P. P. C. Verbunt, C. W. M. Bastiaansen, D. J. Broer, and M. G. Debije, “The effect of dyes aligned by liquid crystals on luminescent solar concentrator performance,” presented at the 24th European Photovoltaic Solar Energy Conference, Hamburg, 21–25 September 2009.

J. R. Lakowicz, Principles of Fluorescence Spectroscopy, 3rd ed. (Springer, 2006).

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Figures (10)

Fig. 1.
Fig. 1.

Typical geometry of a LW: incoming light is absorbed by a lumninophore and when emitted may be emitted in a direction resulting in TIR (dashed red ray) or in a direction resulting in loss out the escape cone (solid red ray). When used as an LSC, photovoltaic cells are placed around the waveguide edge to convert captured light into electricity.

Fig. 2.
Fig. 2.

Successive absorption/emission events for LWs containing isotropic (triangles) and oriented (circles) luminophores. The prediction of Eq. (1) is shown by the dashed lines, based on a PMMA waveguide cladded by air, neglecting self-absorption and accounting for Fresnel relations. Triangles plus dashed line: simplistic assumption given by Eq. (1); triangles plus solid line: present model assuming isotropically oriented luminophores; circles plus solid lines: present model assuming oriented luminophores with P2=0.5,0.6,,1.0. (a) Proportion of absorbed light lost out the escape cone at successive absorption/emission events. (b) Proportion of light remaining within the LW at successive absorption/emission events.

Fig. 3.
Fig. 3.

Order parameters of photoselected dye molecules, by generation, for an LW with randomly oriented luminophores and n1=1.49, n0=1, β=0.

Fig. 4.
Fig. 4.

Description of the computational geometry when θemθesc. (a) Case θemθesc<σ<θem. (b) Case θem<σ<θem<θesc.

Fig. 5.
Fig. 5.

Geometry involved in the computation of |bd|. (a) Case σ<θesc. (b) Case σ>θesc.

Fig. 6.
Fig. 6.

Description of the computational geometry when θem<θesc. (a) Case θescθem<σ<θem+θesc. (b) Case θem<θesc,0<σ<θescθem.

Fig. 7.
Fig. 7.

Probability of escape conditional on emission from μem. Solid curve: including Fresnel relations; dashed curve: neglecting Fresnel relations.

Fig. 8.
Fig. 8.

Experimental setup: collimated monochromatic light is polarized at angle α and illuminates a rectangular window on the fluorescent PMMA block. Emitted light escaping out the circular window is collected by a fiber optic terminated by a cosine corrector.

Fig. 9.
Fig. 9.

Linearized experimental data. The straight dashed blue line is the linear regression line to this data, and the shaded region is the 95% confidence interval.

Fig. 10.
Fig. 10.

Experimental data points, least squares fit I(α) (dashed blue curve), scaled theoretical model (solid purple curve), and 95% confidence region (shaded).

Tables (1)

Tables Icon

Table 1. Proportion of Light Lost Out of the Escape Cone at Successive Absorption/Emission Events for Various Degrees of Luminophore Alignmenta

Equations (44)

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

P(esc)=11(n0/n1)2.
vabs=vabs(θph)=(sinθph,0,cosθph).
pabs=pabs(θph,αp)=cosαp(cosθph,0,sinθph)+sinαp(0,1,0).
μem=μem(θμe,φμe)=(sinθμecosφμe,sinθμesinφμe,cosθμe).
P(esc|pabs,vabs)=S2P(esc|μem)P(μem|pabs,vabs)dA(μem).
P(esc|μem)=S2P(esc|vem)P(vem|μem)dA(vem),
vem(σ,ρ;μem)=sinσ[cosρ(μem×kμem×k)×μem+sinρμem×kμem×k]+cosσμem.
P(vem|μem)dA(vem)=sin2σS2(1(v·μem)2)dA(v)sinσdσdρ=38πsin3σdσdρ.
P(esc|μem)=38π0π(ππP(esc|vem(σ,ρ;μem))dρ)sin3σdσ.
R+(θem,σ)=arccos(cosθesccosσcosθemsinθemsinσ).
pem(σ,ρ;μem)=vem×(vem×μem)vem×(vem×μem).
pem=(pem·s)s+(pem·(vem×s))(vem×s).
Rs(vem)(pem·s)2+Rp(vem)(pem·(vem×s))2,
Rs(v)=(h(v)n1/n0h(v)+n1/n0)2,Rp(v)=(h(v)n0/n1h(v)+n0/n1)2,
h(v)=1v·k1(n1/n0)2(1(v·k)2).
P(esc|μem)=34π0πR+(θem,σ)R+(θem,σ)[1Rs(vem)(pem·s)2Rp(vem)(pem·(vem×s))2]dρsin3σdσ.
P(esc|μem)=(13332cosθesc+132cos(3θesc))332(cosθesccos(3θesc))cos(2θem)=0.2170.125cos(2θem)whenn0=1andn1=1.49.
P(μem|pabs,vabs)=q2πSP(μabs(τ)|pabs,vabs)ds(τ).
P(μabs(τ)|pabs,vabs)=P(absorption|μabs(τ),pabs,vabs)S2P(absorption|μ,pabs,vabs)dA(μ).
P(μabs(τ)|pabs,vabs)=(μabs(τ)·pabs)2S2(μ·pabs)2dA(μ)=34π(μabs(τ)·pabs)2.
P(μem|pabs,vabs)=3q8π2S(μabs(τ)·pabs)2ds(τ).
μabs(τ)=(sinβcosτ,sinβsinτ,cosβ)
S(μabs(τ)·pabs)2ds(τ)=02π(cosρsinβcosτ+sinρcosβ)2dτ=π(cos2ρsin2β+2sin2ρcos2β)=π[(1(μem·pabs)2)sin2β+2(μem·pabs)2cos2β],
P(μem|pabs,vabs)=3q8π[(1(μem·pabs)2)sin2β+2(μem·pabs)2cos2β].
μem·pabs=cosαp(cosθphcosφμesinθμesinθphcosθμe)+sinαpsinφμesinθμe.
14π12π02π0π02πP(esc|vabs(θph,φph),pabs(θph,φph,α))dαsinθphdθphdφph=0.2587,
a[7995,9187]andc[38746,39610].
8591x+39178±37550.013+0.025(x0.052)2.
P(esc|α=0°)P(esc|α=90°)=0.692.
P(esc|α)=0.0104cos(2α)+0.051,for whichP(esc|α=0°)P(esc|α=90°)=0.658.
γ=argmini=1N(γ(0.0104xi+0.051)I^i)2=775864,
μabs=μemhμ(θ)sinθdθdφ=ec2P2(cosθ)sinθ2π0πec2P2(cosθ)sinθdθdθdφ,
P(μ(θ,φ)|v0,p(π,α))=(μ(θ,φ)·p(π,α))2hμ(θ)sinθdθdφ02π0π(μ(θ,φ)·p(π,α))2hμ(θ)sinθdθdφ,
P(μ(θ,φ)|v0,randomp)=12π02πP(μ(θ,φ)|v0,p(π,α))dα.
P(v|μ)=38π(1(μ·v)2),
P(v|1)P(v|first emission event)=S2P(v|μ)P(μ|v0,randomp)dA(μ).
P(esc|first emission event)=202π0θescP(v(θ,φ)|1)sinθdθdφ.
P(v,p|n)=f(θ,α)dαdθdφ02πθescπθesc02πf(θ,α)dαdθdφ,
f(θ,α)=possibleμP(v,p|μ)P(μ|n)ds(μ);
μposs(τ;θ,α)=cos(τ)v(θ)+sin(τ)p(θ,α).
P(v(θ),p(θ,α)|μ(τ))=12π2(1(v(θ)·μ(τ;θ,α))2).
P(μ|n+1)=02πθescπθesc02πP(μ|v(θ),p(θ,α))P(v,p|n)sinθdαdθdφ,
P(μ(θμ,φμ)|v(θ),p(θ,α))=(μ(θμ,φμ)·p(θ,α))2hμ(θμ)sin(θμ)dθμdφμ02π0π(μ(θ,φ)·p(θ,α))2hμ(θ)sin(θ)dθdφ.
P(v|n+1)P(v|(n+1)st emission event)=S2P(v|μ)P(μ|n+1)dA(μ),P(v|μ)=38π(1(μ·v)2),P(esc|(n+1)st emission event)=202π0θescP(v(θ,φ)|n+1)sinθdθdφ.

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