Abstract

We theoretically and numerically investigate the capability of disordered media to enhance the optical path length in dielectric slabs and augment their light absorption efficiency due to scattering. We first perform a series of Monte Carlo simulations of random walks to determine the path length distribution in weakly to strongly (single to multiple) scattering, non-absorbing dielectric slabs under normally incident light and derive analytical expressions for the path length enhancement in these two limits. Quite interestingly, while multiple scattering is expected to produce long optical paths, we find that media containing a vanishingly small amount of scatterers can still provide high path length enhancements due to the very long trajectories sustained by total internal reflection at the slab interfaces. The path length distributions are then used to calculate the light absorption efficiency of media with varying absorption coefficients. We find that maximum absorption enhancement is obtained at an optimal scattering strength, in-between the single-scattering and the diffusive (strong multiple-scattering) regimes. This study can guide experimentalists towards more efficient and potentially low-cost solutions in photovoltaic technologies.

© 2015 Optical Society of America

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References

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  2. S. B. Mallick, N. P. Sergeant, M. Agrawal, J. Lee, and P. Peumans, “Coherent light trapping in thin-film photovoltaics,” MRS Bull. 36, 453–460 (2011).
    [Crossref]
  3. M. A. Green and S. Pillai, “Harnessing plasmonics for solar cells,” Nat. Photonics 6, 130–132 (2012).
    [Crossref]
  4. F. Priolo, T. Gregorkiewicz, M. Galli, and T. F. Krauss, “Silicon nanostructures for photonics and photovoltaics,” Nat. Nanotechnol. 9, 19–32 (2014).
    [Crossref] [PubMed]
  5. A. Goetzberger and W. Greube, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
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  8. E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Devices 29, 300–305 (1982).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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2014 (3)

F. Priolo, T. Gregorkiewicz, M. Galli, and T. F. Krauss, “Silicon nanostructures for photonics and photovoltaics,” Nat. Nanotechnol. 9, 19–32 (2014).
[Crossref] [PubMed]

F. E. Gálvez, P. R. F. Barnes, J. Halme, and H. Míguez, “Dye sensitized solar cells as optically random photovoltaic media,” Energy Environ. Sci. 7, 689–697 (2014).
[Crossref]

R. Pierrat, P. Ambichl, S. Gigan, A. Haber, R. Carminati, and S. Rotter, “Invariance property of wave scattering through disordered media,” Proc. Natl. Acad. Sci. U. S. A. 111, 17765–17770 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (3)

2011 (2)

T. Svensson, E. Adolfsson, M. Lewander, C. T. Xu, and S. Svanberg, “Disordered, strongly scattering porous materials as miniature multipass gas cells,” Phys. Rev. Lett. 107, 143901 (2011).
[Crossref] [PubMed]

S. B. Mallick, N. P. Sergeant, M. Agrawal, J. Lee, and P. Peumans, “Coherent light trapping in thin-film photovoltaics,” MRS Bull. 36, 453–460 (2011).
[Crossref]

2010 (1)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

2009 (1)

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62, 24–29 (2009).
[Crossref]

2008 (1)

O. L. Muskens, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and A. Lagendijk, “Design of light scattering in nanowire materials for photovoltaic applications,” Nano Lett. 8, 2638–2642 (2008).
[Crossref] [PubMed]

2004 (1)

R. Elaloufi, R. Carminati, and J. J. Greffet, “Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach,” J. Opt. Soc. Am A 21, 1430–1437 (2004).
[Crossref]

2003 (1)

M. Grätzel, “Dye-sensitized solar cells,” J. Photochem. Photobiol. C,  4, 145–153 (2003).
[Crossref]

2002 (1)

M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Prog. Photovoltaics 10, 235–241 (2002).
[Crossref]

2000 (1)

A. Usami, “Theoretical simulations of optical confinement in dye-sensitized nanocrystalline solar cells,” Sol. Energ. Mat. Sol. Cells 64, 73–83 (2000).
[Crossref]

1999 (1)

G. Rotherberger, P. Conte, and M. Grätzel, “A contribution to the optical design of dye-sensitized nanocrystalline solar cells,” Sol. Energ. Mat. Sol. Cells 58, 321–336 (1999).
[Crossref]

1997 (1)

D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation,” I. Theory,” Appl. Opt. 36, 4587–4599 (1997).

1995 (2)

C. Heine and R. H. Morf, “Submicrometer gratings for solar energy applications,” Appl. Opt. 34, 2476–2482 (1995).
[Crossref] [PubMed]

L. Wang, S. L. Jacques, and L. Zheng, “MCML – Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Bio. 47, 131–146 (1995).
[Crossref]

1992 (1)

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[Crossref] [PubMed]

1991 (1)

J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A. 44, 3948–3959 (1991).
[Crossref] [PubMed]

1989 (1)

A. Lagendijk, R. Vreeker, and P. de Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[Crossref]

1982 (2)

E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72, 899–907 (1982).
[Crossref]

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Devices 29, 300–305 (1982).
[Crossref]

1977 (1)

A. Goetzberger and W. Greube, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[Crossref]

1976 (1)

1974 (1)

D. Redfield, “Multiple-pass thin-film silicon solar cell,” Appl. Phys. Lett. 25, 647–648 (1974).
[Crossref]

Adolfsson, E.

T. Svensson, E. Adolfsson, M. Lewander, C. T. Xu, and S. Svanberg, “Disordered, strongly scattering porous materials as miniature multipass gas cells,” Phys. Rev. Lett. 107, 143901 (2011).
[Crossref] [PubMed]

Agrawal, M.

S. B. Mallick, N. P. Sergeant, M. Agrawal, J. Lee, and P. Peumans, “Coherent light trapping in thin-film photovoltaics,” MRS Bull. 36, 453–460 (2011).
[Crossref]

Alerstam, E.

Algra, R. E.

O. L. Muskens, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and A. Lagendijk, “Design of light scattering in nanowire materials for photovoltaic applications,” Nano Lett. 8, 2638–2642 (2008).
[Crossref] [PubMed]

Ambichl, P.

R. Pierrat, P. Ambichl, S. Gigan, A. Haber, R. Carminati, and S. Rotter, “Invariance property of wave scattering through disordered media,” Proc. Natl. Acad. Sci. U. S. A. 111, 17765–17770 (2014).
[Crossref] [PubMed]

Andreani, L. C.

Arridge, S. R.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[Crossref] [PubMed]

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

Bakkers, E. P. A. M.

O. L. Muskens, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and A. Lagendijk, “Design of light scattering in nanowire materials for photovoltaic applications,” Nano Lett. 8, 2638–2642 (2008).
[Crossref] [PubMed]

Barnes, P. R. F.

F. E. Gálvez, P. R. F. Barnes, J. Halme, and H. Míguez, “Dye sensitized solar cells as optically random photovoltaic media,” Energy Environ. Sci. 7, 689–697 (2014).
[Crossref]

Burresi, M.

Carminati, R.

R. Pierrat, P. Ambichl, S. Gigan, A. Haber, R. Carminati, and S. Rotter, “Invariance property of wave scattering through disordered media,” Proc. Natl. Acad. Sci. U. S. A. 111, 17765–17770 (2014).
[Crossref] [PubMed]

R. Elaloufi, R. Carminati, and J. J. Greffet, “Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach,” J. Opt. Soc. Am A 21, 1430–1437 (2004).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover Publications, 1960).

Cody, G. D.

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Devices 29, 300–305 (1982).
[Crossref]

Conte, P.

G. Rotherberger, P. Conte, and M. Grätzel, “A contribution to the optical design of dye-sensitized nanocrystalline solar cells,” Sol. Energ. Mat. Sol. Cells 58, 321–336 (1999).
[Crossref]

Contini, D.

D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation,” I. Theory,” Appl. Opt. 36, 4587–4599 (1997).

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[Crossref] [PubMed]

de Vries, P.

A. Lagendijk, R. Vreeker, and P. de Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[Crossref]

Delpy, D. T.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[Crossref] [PubMed]

Elaloufi, R.

R. Elaloufi, R. Carminati, and J. J. Greffet, “Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach,” J. Opt. Soc. Am A 21, 1430–1437 (2004).
[Crossref]

Fusco, R.

Galli, M.

F. Priolo, T. Gregorkiewicz, M. Galli, and T. F. Krauss, “Silicon nanostructures for photonics and photovoltaics,” Nat. Nanotechnol. 9, 19–32 (2014).
[Crossref] [PubMed]

Gálvez, F. E.

F. E. Gálvez, P. R. F. Barnes, J. Halme, and H. Míguez, “Dye sensitized solar cells as optically random photovoltaic media,” Energy Environ. Sci. 7, 689–697 (2014).
[Crossref]

Gigan, S.

R. Pierrat, P. Ambichl, S. Gigan, A. Haber, R. Carminati, and S. Rotter, “Invariance property of wave scattering through disordered media,” Proc. Natl. Acad. Sci. U. S. A. 111, 17765–17770 (2014).
[Crossref] [PubMed]

Goetzberger, A.

A. Goetzberger and W. Greube, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[Crossref]

Grätzel, M.

M. Grätzel, “Dye-sensitized solar cells,” J. Photochem. Photobiol. C,  4, 145–153 (2003).
[Crossref]

G. Rotherberger, P. Conte, and M. Grätzel, “A contribution to the optical design of dye-sensitized nanocrystalline solar cells,” Sol. Energ. Mat. Sol. Cells 58, 321–336 (1999).
[Crossref]

Green, M. A.

M. A. Green and S. Pillai, “Harnessing plasmonics for solar cells,” Nat. Photonics 6, 130–132 (2012).
[Crossref]

M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Prog. Photovoltaics 10, 235–241 (2002).
[Crossref]

Greffet, J. J.

R. Elaloufi, R. Carminati, and J. J. Greffet, “Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach,” J. Opt. Soc. Am A 21, 1430–1437 (2004).
[Crossref]

Gregorkiewicz, T.

F. Priolo, T. Gregorkiewicz, M. Galli, and T. F. Krauss, “Silicon nanostructures for photonics and photovoltaics,” Nat. Nanotechnol. 9, 19–32 (2014).
[Crossref] [PubMed]

Greube, W.

A. Goetzberger and W. Greube, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[Crossref]

Haber, A.

R. Pierrat, P. Ambichl, S. Gigan, A. Haber, R. Carminati, and S. Rotter, “Invariance property of wave scattering through disordered media,” Proc. Natl. Acad. Sci. U. S. A. 111, 17765–17770 (2014).
[Crossref] [PubMed]

Halme, J.

F. E. Gálvez, P. R. F. Barnes, J. Halme, and H. Míguez, “Dye sensitized solar cells as optically random photovoltaic media,” Energy Environ. Sci. 7, 689–697 (2014).
[Crossref]

Heine, C.

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML – Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Bio. 47, 131–146 (1995).
[Crossref]

Kowalczewski, P.

Krauss, T. F.

F. Priolo, T. Gregorkiewicz, M. Galli, and T. F. Krauss, “Silicon nanostructures for photonics and photovoltaics,” Nat. Nanotechnol. 9, 19–32 (2014).
[Crossref] [PubMed]

Lagendijk, A.

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62, 24–29 (2009).
[Crossref]

O. L. Muskens, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and A. Lagendijk, “Design of light scattering in nanowire materials for photovoltaic applications,” Nano Lett. 8, 2638–2642 (2008).
[Crossref] [PubMed]

A. Lagendijk, R. Vreeker, and P. de Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[Crossref]

Lambe, J.

Lee, J.

S. B. Mallick, N. P. Sergeant, M. Agrawal, J. Lee, and P. Peumans, “Coherent light trapping in thin-film photovoltaics,” MRS Bull. 36, 453–460 (2011).
[Crossref]

Lewander, M.

T. Svensson, E. Adolfsson, M. Lewander, C. T. Xu, and S. Svanberg, “Disordered, strongly scattering porous materials as miniature multipass gas cells,” Phys. Rev. Lett. 107, 143901 (2011).
[Crossref] [PubMed]

Liscidini, M.

Malfanti, I.

Mallick, S. B.

S. B. Mallick, N. P. Sergeant, M. Agrawal, J. Lee, and P. Peumans, “Coherent light trapping in thin-film photovoltaics,” MRS Bull. 36, 453–460 (2011).
[Crossref]

Martelli, F.

D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation,” I. Theory,” Appl. Opt. 36, 4587–4599 (1997).

Míguez, H.

F. E. Gálvez, P. R. F. Barnes, J. Halme, and H. Míguez, “Dye sensitized solar cells as optically random photovoltaic media,” Energy Environ. Sci. 7, 689–697 (2014).
[Crossref]

Morf, R. H.

Mupparapu, R.

Muskens, O. L.

O. L. Muskens, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and A. Lagendijk, “Design of light scattering in nanowire materials for photovoltaic applications,” Nano Lett. 8, 2638–2642 (2008).
[Crossref] [PubMed]

Peumans, P.

S. B. Mallick, N. P. Sergeant, M. Agrawal, J. Lee, and P. Peumans, “Coherent light trapping in thin-film photovoltaics,” MRS Bull. 36, 453–460 (2011).
[Crossref]

Pierrat, R.

R. Pierrat, P. Ambichl, S. Gigan, A. Haber, R. Carminati, and S. Rotter, “Invariance property of wave scattering through disordered media,” Proc. Natl. Acad. Sci. U. S. A. 111, 17765–17770 (2014).
[Crossref] [PubMed]

Pillai, S.

M. A. Green and S. Pillai, “Harnessing plasmonics for solar cells,” Nat. Photonics 6, 130–132 (2012).
[Crossref]

Pine, D. J.

J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A. 44, 3948–3959 (1991).
[Crossref] [PubMed]

Polman, A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

Priolo, F.

F. Priolo, T. Gregorkiewicz, M. Galli, and T. F. Krauss, “Silicon nanostructures for photonics and photovoltaics,” Nat. Nanotechnol. 9, 19–32 (2014).
[Crossref] [PubMed]

Redfield, D.

D. Redfield, “Multiple-pass thin-film silicon solar cell,” Appl. Phys. Lett. 25, 647–648 (1974).
[Crossref]

Rivas, J. G.

O. L. Muskens, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and A. Lagendijk, “Design of light scattering in nanowire materials for photovoltaic applications,” Nano Lett. 8, 2638–2642 (2008).
[Crossref] [PubMed]

Rotherberger, G.

G. Rotherberger, P. Conte, and M. Grätzel, “A contribution to the optical design of dye-sensitized nanocrystalline solar cells,” Sol. Energ. Mat. Sol. Cells 58, 321–336 (1999).
[Crossref]

Rotter, S.

R. Pierrat, P. Ambichl, S. Gigan, A. Haber, R. Carminati, and S. Rotter, “Invariance property of wave scattering through disordered media,” Proc. Natl. Acad. Sci. U. S. A. 111, 17765–17770 (2014).
[Crossref] [PubMed]

Savo, R.

Scudo, P.

Sergeant, N. P.

S. B. Mallick, N. P. Sergeant, M. Agrawal, J. Lee, and P. Peumans, “Coherent light trapping in thin-film photovoltaics,” MRS Bull. 36, 453–460 (2011).
[Crossref]

Sprafke, A. N.

A. N. Sprafke and R. B. Wehrspohn, “Current concepts for optical path enhancement in solar cells,” in Photon Management in Solar Cells, R. B. Wehrspohn, U. Rau, and A. Gombert, eds. (Wiley-VCH, 2015), pp. 1–20.

Svanberg, S.

T. Svensson, E. Adolfsson, M. Lewander, C. T. Xu, and S. Svanberg, “Disordered, strongly scattering porous materials as miniature multipass gas cells,” Phys. Rev. Lett. 107, 143901 (2011).
[Crossref] [PubMed]

Svensson, T.

Usami, A.

A. Usami, “Theoretical simulations of optical confinement in dye-sensitized nanocrystalline solar cells,” Sol. Energ. Mat. Sol. Cells 64, 73–83 (2000).
[Crossref]

van Tiggelen, B.

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62, 24–29 (2009).
[Crossref]

Vignolini, S.

Vreeker, R.

A. Lagendijk, R. Vreeker, and P. de Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[Crossref]

Vynck, K.

Wang, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML – Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Bio. 47, 131–146 (1995).
[Crossref]

Weber, W. H.

Wehrspohn, R. B.

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Figures (4)

Fig. 1
Fig. 1 Illustration of possible approaches to enhance light absorption in dielectric slabs. (a) Random roughness is introduced on the surfaces of the slab, leading to an efficient spreading of light in the medium. At broad angles, light follows long trajectories. (b) Volume scattering is exploited to create a broad distribution of path lengths in the medium, consisting of trajectories that may be smaller or much larger than the sample thickness.
Fig. 2
Fig. 2 Path length distributions P(l/L) (normalized to the sample thickness) in media with different optical thicknesses. (a) Strong multiple scattering in optically thick media (here, L/ℓs = 100) leads to a broad and smooth distribution of path lengths. Some trajectories are shorter than the sample thickness, and others are much longer, see the inset. (b) In optically thin media (here, L/ℓs = 0.0125), most of the light leaves the slab without being scattered, but the small portion of scattered light can perform very long trajectories thanks to total internal reflection, see the inset.
Fig. 3
Fig. 3 Path length enhancement ⟨l/L determined from MC simulations (black dots) versus the optical thickness L/ℓs, ranging from optically thin to optically thick media (left to right). The gray colored region indicates region where diffusion theory applies, the red and blue lines indicate the path length enhancement expected in the limit of optically thick (L/ℓs ≫ 1) and optically thin (L/ℓs ≪ 1) media. MC simulations are in very good agreement with theoretical predictions, Eqs. (11)(20).
Fig. 4
Fig. 4 (a) Effect of volume scattering on light absorption in dielectric slabs for three different absorption strengths L/ℓi = 1,0.1,0.01 (black, orange and blue dots, respectively). The gray dots indicate the upper bound for light absorption enhancement, according to Eq. (24). The red dashed line corresponds to media containing no scatterers (enhancement equals 1 by definition). The absorption enhancement strongly depends on the optical thickness of the medium, exhibiting an optimal regime where the enhancement is maximum. (b) Total transmission (red points) and reflection (black points) through disordered media versus their optical thickness. Transmission and reflection are comparable in an intermediate regime between single scattering and diffusion (strong multiple scattering). It is in this scattering range that light absorption enhancement is found to be maximum.

Tables (1)

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Table 1 Total absorption in a bare slab and in slabs with varying optical thicknesses (L/ℓs), chosen from the optical scattering regime, and absorption strengths (L/ℓi).

Equations (24)

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u ( r , t ) t = D 2 u ( r , t ) ,
T ( t ) = 2 π D ( L + 2 z e ) 2 n = 1 n sin [ n π t + z e L + 2 z e ] cos [ n π L + z e L + 2 z e ] exp [ n 2 π 2 D t ( L + 2 z e ) 2 ] ,
R ( t ) = 2 π D ( L + 2 z e ) 2 n = 1 n sin [ n π t + z e L + 2 z e ] cos [ n π z e L + 2 z e ] exp [ n 2 π 2 D t ( L + 2 z e ) 2 ] .
T t o t = 0 T ( t ) d t = 2 π n = 1 1 n sin [ n π t + z e L + 2 z e ] cos [ n π L + z e L + 2 z e ] ,
R t o t = 0 R ( t ) d t = 2 π n = 1 1 n sin [ n π t + z e L + 2 z e ] cos [ n π z e L + 2 z e ] ,
t T = 0 t T ( t ) d t T t o t = 2 ( L + 2 z e ) 2 π 3 D T t o t n = 1 1 n 3 sin [ n π t + z e L + 2 z e ] cos [ n π L + z e L + 2 z e ] ,
t R = 0 t R ( t ) d t R t o t = 2 ( L + 2 z e ) 2 π 3 D R t o t n = 1 1 n 3 sin [ n π t + z e L + 2 z e ] cos [ n π z e L + 2 z e ] .
t = t T T t o t + t R R t o t
= 2 ( L + 2 z e ) 2 π 3 D n = 1 1 n 3 sin [ n π t + z e L + 2 z e ] ( cos [ n π z e L + 2 z e ] cos [ n π L + z e L + 2 z e ] ) .
t 2 L π 2 D n = 1 1 n 2 ( t + z e ) ( 1 ( 1 ) n ) = L ( t + z e ) 2 D .
l = ( 3 2 + 1 + r i 1 r i ) L .
l = N i , b i , b + ( 1 N i , b ) i , s + ( 1 N i , b ) P o u t o u t + ( 1 N i , b ) ( 1 P o u t ) i n + ( 1 N i , b ) ( 1 P o u t ) P o u t o u t + ( 1 N i , b ) ( 1 P o u t ) 2 i n + + ( 1 N i , b ) ( 1 P o u t ) n 1 P o u t o u t + ( 1 N i , b ) ( 1 P o u t ) n i n .
l = N i , b i , b + ( 1 N i , b ) i , s + ( 1 N i , b ) o u t + ( 1 N i , b ) 1 P o u t P o u t i n .
I ˜ s ( l ) = exp [ l / s ] .
I ˜ o ( l ) = exp [ l ( 1 r ( θ ) ) cos ( θ ) L ] .
N i , b = I ˜ s ( l ) I ˜ o ( l ) d l I ˜ o ( l ) d l = s ( 1 r 0 ) L + s ( 1 r 0 ) ,
i , b = l I ˜ s ( l ) I ˜ o ( l ) d l I ˜ s ( l ) I ˜ o ( l ) d l = L s L + s ( 1 r 0 ) .
θ c = arcsin [ n 1 n 2 ] ,
P o u t = 2 4 π 0 2 π d ϕ 0 θ c sin θ d θ = 1 1 n 1 2 n 2 2 .
l = ( n 1 + n 2 ) 2 4 n 1 n 2 ( 1 1 n 1 2 n 2 2 ) L .
A = ( 1 r 0 ) [ 1 P ( l ) exp ( 1 i ) d l ] .
P b ( l ) = ( 1 r 0 ) [ δ ( l L ) + r 0 δ ( l 2 L ) + + r 0 n δ ( l ( n + 1 ) L ) ] ,
Γ = 1 P ( l ) exp ( l / i ) d l [ 1 exp ( L / i ) ] / [ 1 r 0 exp ( L / i ) ] .
Γ = l L ( 1 r 0 ) ,

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