Abstract

We study light transport in phosphor plates of white light-emitting diodes (LEDs). We measure the broadband diffuse transmission through phosphor plates of varying YAG:Ce3+ density. We distinguish the spectral ranges where absorption, scattering, and re-emission dominate. Using diffusion theory, we derive the transport and absorption mean free paths from first principles. We find that both transport and absorption mean free paths are on the order of the plate thickness. This means that phosphors in commercial LEDs operate well within an intriguing albedo range around 0.7. We discuss how salient parameters that can be derived from first principles control the optical properties of a white LED.

© 2014 Optical Society of America

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    [CrossRef]
  30. W. Meulebroeck, Y. Meuret, S. Heyvaert, H. Thienpont, “The experimental characterization of the absorption and scatter properties of photopolymers,” Proc. SPIE 8439, 84391Z (2012).
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2013

2012

W. Meulebroeck, Y. Meuret, S. Heyvaert, H. Thienpont, “The experimental characterization of the absorption and scatter properties of photopolymers,” Proc. SPIE 8439, 84391Z (2012).
[CrossRef]

2010

2009

C. Sommer, J. R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

2008

E. Alerstam, T. Svensson, S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

H. Bechtel, P. Schmidt, W. Busselt, B. S. Schreinemacher, “Lumiramic: a new phosphor technology for high performance solid state light sources,” Proc. SPIE 7058, 70580E (2008).
[CrossRef]

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Viña, D. S. Wiersma, C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

O. L. Muskens, A. Lagendijk, “Broadband enhanced backscattering spectroscopy of strongly scattering media,” Opt. Express 16, 1222–1231 (2008).
[CrossRef] [PubMed]

2007

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160–175 (2007).
[CrossRef]

1999

M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

J. Gómez Rivas, R. Sprik, C. M. Soukoulis, K. Busch, A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
[CrossRef]

1998

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

1996

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

1994

D. J. Durian, “Influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E 50, 857–866 (1994).
[CrossRef]

1992

N. Garcia, A. Z. Genack, A. A. Lisyansky, “Measurement of the transport mean free path of diffusing photons,” Phys. Rev. B 46, 14475–14479 (1992).
[CrossRef]

1991

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

1989

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

1988

M. B. van der Mark, M. P. van Albada, A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

1971

Akkermans, E.

E. Akkermans, G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).
[CrossRef]

Alerstam, E.

E. Alerstam, W. C. Y. Lo, T. D. Han, J. Rose, S. Andersson-Engels, L. Lilge, “Next-generation acceleration and code optimization for light transport in turbid media using GPUs,” Biomed. Opt. Express 1, 658–675 (2010).
[CrossRef]

E. Alerstam, T. Svensson, S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

Altube, A.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Viña, D. S. Wiersma, C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Andersson-Engels, S.

E. Alerstam, W. C. Y. Lo, T. D. Han, J. Rose, S. Andersson-Engels, L. Lilge, “Next-generation acceleration and code optimization for light transport in turbid media using GPUs,” Biomed. Opt. Express 1, 658–675 (2010).
[CrossRef]

E. Alerstam, T. Svensson, S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

Bechtel, H.

H. Bechtel, P. Schmidt, W. Busselt, B. S. Schreinemacher, “Lumiramic: a new phosphor technology for high performance solid state light sources,” Proc. SPIE 7058, 70580E (2008).
[CrossRef]

Bertolotti, J.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Viña, D. S. Wiersma, C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Blanco, Á.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Viña, D. S. Wiersma, C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Bohren, C. F.

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

Bret, B. P. J.

B. P. J. Bret, Multiple Light Scattering in Porous Gallium Phosphide, Ph.D. thesis (University of Twente, 2005).

Busch, K.

J. Gómez Rivas, R. Sprik, C. M. Soukoulis, K. Busch, A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
[CrossRef]

Busselt, W.

H. Bechtel, P. Schmidt, W. Busselt, B. S. Schreinemacher, “Lumiramic: a new phosphor technology for high performance solid state light sources,” Proc. SPIE 7058, 70580E (2008).
[CrossRef]

Craford, M. G.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160–175 (2007).
[CrossRef]

de Vries, P.

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

Durian, D. J.

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

D. J. Durian, “Influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E 50, 857–866 (1994).
[CrossRef]

Garcia, N.

N. Garcia, A. Z. Genack, A. A. Lisyansky, “Measurement of the transport mean free path of diffusing photons,” Phys. Rev. B 46, 14475–14479 (1992).
[CrossRef]

García, P. D.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Viña, D. S. Wiersma, C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Genack, A. Z.

N. Garcia, A. Z. Genack, A. A. Lisyansky, “Measurement of the transport mean free path of diffusing photons,” Phys. Rev. B 46, 14475–14479 (1992).
[CrossRef]

Gilray, C.

C. Gilray, I. Lewin, “Monte Carlo techniques for the design of illumination optics,” in Illuminating Engineering Society North America (IESNA) Annual Conference Technical Papers(July 1996), Paper no. 85, pp. 65–80.

Gómez Rivas, J.

J. Gómez Rivas, R. Sprik, C. M. Soukoulis, K. Busch, A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
[CrossRef]

Han, T. D.

Harbers, G.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160–175 (2007).
[CrossRef]

Hartmann, P.

C. Sommer, J. R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

Heyvaert, S.

W. Meulebroeck, Y. Meuret, S. Heyvaert, H. Thienpont, “The experimental characterization of the absorption and scatter properties of photopolymers,” Proc. SPIE 8439, 84391Z (2012).
[CrossRef]

Huffmann, D. R.

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

IJzerman, W. L.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vols. I and II.

Krames, M. R.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160–175 (2007).
[CrossRef]

Krenn, J. R.

C. Sommer, J. R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

Lagendijk, A.

W. L. Vos, T. W. Tukker, A. P. Mosk, A. Lagendijk, W. L. IJzerman, “Broadband mean free path of diffuse light in polydisperse ensembles of scatterers for white LED lighting,” Appl. Opt. 52, 2602–2609 (2013).
[CrossRef] [PubMed]

O. L. Muskens, A. Lagendijk, “Broadband enhanced backscattering spectroscopy of strongly scattering media,” Opt. Express 16, 1222–1231 (2008).
[CrossRef] [PubMed]

J. Gómez Rivas, R. Sprik, C. M. Soukoulis, K. Busch, A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
[CrossRef]

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

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

M. B. van der Mark, M. P. van Albada, A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

Lemieux, P.-A.

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

Lewin, I.

C. Gilray, I. Lewin, “Monte Carlo techniques for the design of illumination optics,” in Illuminating Engineering Society North America (IESNA) Annual Conference Technical Papers(July 1996), Paper no. 85, pp. 65–80.

Lilge, L.

Lisyansky, A. A.

N. Garcia, A. Z. Genack, A. A. Lisyansky, “Measurement of the transport mean free path of diffusing photons,” Phys. Rev. B 46, 14475–14479 (1992).
[CrossRef]

Liu, S.

Liu, Z.

Lo, W. C. Y.

López, C.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Viña, D. S. Wiersma, C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Luo, X.

Malacara, D.

D. Malacara, Color Vision and Colorimetry: Theory and Applications (SPIE, 2011).

Martín, M. D.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Viña, D. S. Wiersma, C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Meulebroeck, W.

W. Meulebroeck, Y. Meuret, S. Heyvaert, H. Thienpont, “The experimental characterization of the absorption and scatter properties of photopolymers,” Proc. SPIE 8439, 84391Z (2012).
[CrossRef]

Meuret, Y.

W. Meulebroeck, Y. Meuret, S. Heyvaert, H. Thienpont, “The experimental characterization of the absorption and scatter properties of photopolymers,” Proc. SPIE 8439, 84391Z (2012).
[CrossRef]

Montambaux, G.

E. Akkermans, G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).
[CrossRef]

Mosk, A. P.

Mueller, G. O.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160–175 (2007).
[CrossRef]

Mueller-Mach, R.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160–175 (2007).
[CrossRef]

Muskens, O. L.

Nagel, J. R.

Nieuwenhuizen, T. M.

M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

Pachler, P.

C. Sommer, J. R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

Pine, D. J.

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

Pinnow, D. A.

Rose, J.

Sapienza, R.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Viña, D. S. Wiersma, C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Scarpulla, M. A.

Schmidt, P.

H. Bechtel, P. Schmidt, W. Busselt, B. S. Schreinemacher, “Lumiramic: a new phosphor technology for high performance solid state light sources,” Proc. SPIE 7058, 70580E (2008).
[CrossRef]

Schreinemacher, B. S.

H. Bechtel, P. Schmidt, W. Busselt, B. S. Schreinemacher, “Lumiramic: a new phosphor technology for high performance solid state light sources,” Proc. SPIE 7058, 70580E (2008).
[CrossRef]

Schubert, E. F.

E. F. Schubert, Light Emitting Diodes (Cambridge University, 2006).
[CrossRef]

Schweighart, M.

C. Sommer, J. R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

Shchekin, O. B.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160–175 (2007).
[CrossRef]

Sommer, C.

C. Sommer, J. R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

Soukoulis, C. M.

J. Gómez Rivas, R. Sprik, C. M. Soukoulis, K. Busch, A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
[CrossRef]

Sprik, R.

J. Gómez Rivas, R. Sprik, C. M. Soukoulis, K. Busch, A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
[CrossRef]

Svensson, T.

E. Alerstam, T. Svensson, S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

Tasch, S.

C. Sommer, J. R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

Thienpont, H.

W. Meulebroeck, Y. Meuret, S. Heyvaert, H. Thienpont, “The experimental characterization of the absorption and scatter properties of photopolymers,” Proc. SPIE 8439, 84391Z (2012).
[CrossRef]

Tukker, T. W.

W. L. Vos, T. W. Tukker, A. P. Mosk, A. Lagendijk, W. L. IJzerman, “Broadband mean free path of diffuse light in polydisperse ensembles of scatterers for white LED lighting,” Appl. Opt. 52, 2602–2609 (2013).
[CrossRef] [PubMed]

T. W. Tukker, “Fluorescence modeling in remote and close LED illumination devices,” in SPIE International Optical Design Conference 2010, Jackson Hole, WY, U.S.A. (June 2010), Paper no. ITuE2.

van Albada, M. P.

M. B. van der Mark, M. P. van Albada, A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1957).

van der Mark, M. B.

M. B. van der Mark, M. P. van Albada, A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

M. B. van der Mark, Propagation of Light in Disordered Media: A Search for Anderson Localization, Ph.D. thesis (University of Twente, 1990).

van Rossum, M. C. W.

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Figures (9)

Fig. 1
Fig. 1

(a) Total relative intensity Trel(λ) versus wavelength for polymer plates with a phosphor density of 2 and 4 wt% YAG:Ce3+. (b) The spectrum of the reference intensity I0.

Fig. 2
Fig. 2

(a) Total relative intensity Trel(λ) versus wavelength for a phosphor density of 4 wt% YAG:Ce3+. The original data from Fig. 1 are indicated as green circles. The total transmission T (λ) obtained from a measurement with a high-pass filter is shown as red diamonds. λ2 indicates the wavelength cut-off of the filter. (b) The phosphor emission spectrum of YAG:Ce3+. λ1 is the wavelength where we estimate the onset of emission. (c) The total transmission T (λ) for a phosphor density of 2 and 4 wt% YAG:Ce3+. The spectral region from 490 to 540 nm is excluded due to the presence of emission and absorption simultaneously.

Fig. 3
Fig. 3

Transport mean free path tr versus wavelength in the range of weak absorption, extracted through Eq. (7) for different phosphor densities (symbols). By taking tr as a linear function of λ, we obtain by extrapolation tr(λ < λ2) from the measured data shown in Fig. 2 (lines).

Fig. 4
Fig. 4

Inverse of the transport mean free path tr versus phosphor density at λ = 460 nm. The line has a slopes of 0.45 ± 0.01 (mm wt%)−1. The reciprocal sample thickness 1/L is indicated by the dashed red line.

Fig. 5
Fig. 5

Inverse of the absorption mean free path abs versus YAG:Ce3+ density at λ = 460 nm. The line has a slope of 0.19 ± 0.01 (mm wt%)−1. The reciprocal sample thickness 1/L is indicated by the dashed red line.

Fig. 6
Fig. 6

Spectral distribution of the inverse absorption mean free path 1/abs for a phosphor plate containing 4 wt% YAG:Ce3+. The reciprocal sample thickness 1/L is indicated by the dashed red line.

Fig. 7
Fig. 7

The albedo a at a wavelength λ = 460 nm as a function of phosphor density.

Fig. 8
Fig. 8

Color coordinates of the phosphor plates with increasing phosphor density on the CIE 1960 u′, v′ chromaticity diagram (filled red circles). The plates were illuminated by a reference light source with u′i = 0.21, v′i = 0.30 (open red circle). The solid black line denotes the boundary of the u′, v′ color space. The dashed blue line denotes the Planckian locus. Combining phosphor plates of increasing density with the reference light source creates a color point sequence which intersects the Planckian locus at a color temperature of 5000 K.

Fig. 9
Fig. 9

Total transmission versus inverse absorption mean free path 1/abs for a polymer plate with 4 wt% YAG:Ce3+ density, at λ = 460 nm, within the absorption region of YAG:Ce3+. Calculated from Eq. (3) for a value of tr extracted from the total transmission measurements shown in Fig. 2. The red dashed line indicate the asymptotic value of the inverse absorption mean free path 1/abs for strong absorption.

Equations (7)

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T rel ( λ ) I tot ( λ ) I 0 ( λ )
T rel ( λ ) = T ( λ ) + T em ( λ ) = I ( λ ) I 0 ( λ ) + I em ( λ ) I 0 ( λ ) .
T = ( 1 R s ) 1 α z e sinh [ α ( z p + z e ) ] sinh ( α z e ) sinh [ α ( L + 2 z e ) ]
z e = 1 2 α ln ( 1 + α z 0 1 α z 0 ) .
z 0 = 2 3 t r ( 1 + R ¯ 1 R ¯ ) ,
T t r , α 0 = ( 1 R s ) 1 α z e α ( z p + z e ) ( α z e ) α ( L + 2 z e ) .
T t r 0 = ( 1 R s ) z p + z 0 L + 2 z 0

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