Abstract

In this paper a fast, yet accurate method to estimate the spectral and angular distribution of the scattered radiation of a fluorescent material is described. The proposed method is an extension of the adding-doubling algorithm for non-fluorescent samples. The method is validated by comparing the spectral and angular transmittance and reflectance characteristics obtained with the extended algorithm with the results obtained using Monte Carlo simulations. The agreement using both methods is within 2%. However, the adding-doubling method achieves a reduction of the calculation time by a factor of 400. Due to the short calculation time, the extended adding-doubling method is very useful when fluorescent layers have to be optimized in an iterative process.

© 2012 OSA

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2011 (3)

E. Klampaftis and B. S. Richards, “Improvement in multi-crystalline silicon solar cell efficiency via addition of luminescent material to EVA encapsulation layer,” Prog. Photovolt. Res. Appl.19(3), 345–351 (2011).
[CrossRef]

Y. Shuai, N. T. Tran, and F. G. Shi, “Nonmonotonic phosphor size dependence of luminous efficacy for typical white LED emitters,” IEEE Photon. Technol. Lett.23(9), 552–554 (2011).
[CrossRef]

J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography-computational efficiency,” Med. Phys.38(10), 5788–5798 (2011).
[CrossRef] [PubMed]

2010 (3)

D. Bera, S. Maslov, L. Qian, J. S. Yoo, and P. H. Holloway, “Optimization of the yellow phosphor concentration and layer thickness for down-conversion of blue to white light,” J. Disp. Technol.6(12), 645–651 (2010).
[CrossRef]

I. Seo, J. Jung, B. J. Oh, and K. Whang, “Improvement of luminance and luminous efficacy of mercury-free, flat fluorescent lamp by optimizing phosphor profile,” IEEE Trans. Plasma Sci.38(5), 1097–1100 (2010).
[CrossRef]

D. Yudovsky and L. Pilon, “Modeling the local excitation fluence rate and fluorescence emission in absorbing and strongly scattering multilayered media,” Appl. Opt.49(31), 6072–6084 (2010).
[CrossRef]

2009 (2)

J. Y. Chi, J.-S. Chen, C.-Y. Liu, C.-W. Chu, and K.-H. Chiang, “Phosphor converted LEDs with omni-directional-reflector coating,” Opt. Express17(26), 23530–23535 (2009).
[CrossRef] [PubMed]

J. P. You, N. T. Tran, Y. Lin, Y. He, and F. G. Shi, “Phosphor-concentration-dependent characteristics of white LEDs in different current regulation modes,” J. Electron. Mater.38(6), 761–766 (2009).
[CrossRef]

2008 (3)

2007 (4)

W.-T. Chien, C.-C. Sun, and I. Moreno, “Precise optical model of multi-chip white LEDs,” Opt. Express15(12), 7572–7577 (2007).
[CrossRef] [PubMed]

J. H. Park and J. H. Ko, “Optimization of the emitting structure of flat fluorescent lamps for LCD backlight applications,” J. Opt. Soc. Korea11(3), 118–123 (2007).
[CrossRef]

P. Chung, H. Chung, and P. H. Holloway, “Phosphor coatings to enhance Si photovoltaic cell performance,” J. Vac. Sci. Technol. A25(1), 61–66 (2007).
[CrossRef]

Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf.105(2), 243–263 (2007).
[CrossRef]

2006 (1)

D.-Y. Kang, E. Wu, and D.-M. Wang, “Modeling white light-emitting diodes with phosphor layers,” Appl. Phys. Lett.89(23), 231102 (2006).
[CrossRef]

2005 (2)

W. G. J. H. M. van Sark, “Enhancement of solar cell performance by employing planar spectral converters,” Appl. Phys. Lett.87(15), 151117 (2005).
[CrossRef]

C. C. Chang, R.-L. Chern, C. C. Chang, C.-C. Chu, J. Y. Chi, J.-C. Su, I.-M. Chan, and J.-F. T. Wang, “Monte Carlo simulation of optical properties of phosphor-screened ultraviolet light in a white light-emitting device,” Jpn. J. Appl. Phys.44(8), 6056–6061 (2005).
[CrossRef]

1997 (2)

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med.21(2), 166–178 (1997).
[CrossRef] [PubMed]

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol.42(11), 2255–2261 (1997).
[CrossRef] [PubMed]

1993 (2)

1991 (1)

A. Rosema, W. Verhoef, J. Schroote, and J. F. H. Snel, “Simulating fluorescence light-canopy interaction in support of laser-induced fluorescence measurements,” Remote Sens. Environ.37(2), 117–130 (1991).
[CrossRef]

1990 (1)

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review on the optical properties of biological tissues,” IEEE J. Quantum Electron.26(12), 2166–2185 (1990).
[CrossRef]

1988 (1)

P. J. Flatau and G. L. Stephens, “On the fundamental solution of the radiative transfer equation,” J. Geophys. Res.93(D9), 11037–11050 (1988).
[CrossRef]

1976 (1)

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transf.16(8), 637–658 (1976).
[CrossRef]

1973 (1)

1965 (1)

R. G. Young and E. G. F. Arnott, “The effect of phosphor coating weight on the lumen output of luorescent lamps,” J. Electrochem. Soc.112(10), 982–984 (1965).
[CrossRef]

1941 (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J.93, 70–83 (1941).
[CrossRef]

1860 (1)

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. Lond.11(0), 545–556 (1860).
[CrossRef]

Aalders, M.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol.42(11), 2255–2261 (1997).
[CrossRef] [PubMed]

Arnott, E. G. F.

R. G. Young and E. G. F. Arnott, “The effect of phosphor coating weight on the lumen output of luorescent lamps,” J. Electrochem. Soc.112(10), 982–984 (1965).
[CrossRef]

Baum, B. A.

Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf.105(2), 243–263 (2007).
[CrossRef]

Beek, J. F.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol.42(11), 2255–2261 (1997).
[CrossRef] [PubMed]

Bera, D.

D. Bera, S. Maslov, L. Qian, J. S. Yoo, and P. H. Holloway, “Optimization of the yellow phosphor concentration and layer thickness for down-conversion of blue to white light,” J. Disp. Technol.6(12), 645–651 (2010).
[CrossRef]

Blokland, P.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol.42(11), 2255–2261 (1997).
[CrossRef] [PubMed]

Catchings, F. E.

Chan, E.

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med.21(2), 166–178 (1997).
[CrossRef] [PubMed]

Chan, I.-M.

C. C. Chang, R.-L. Chern, C. C. Chang, C.-C. Chu, J. Y. Chi, J.-C. Su, I.-M. Chan, and J.-F. T. Wang, “Monte Carlo simulation of optical properties of phosphor-screened ultraviolet light in a white light-emitting device,” Jpn. J. Appl. Phys.44(8), 6056–6061 (2005).
[CrossRef]

Chang, C. C.

C. C. Chang, R.-L. Chern, C. C. Chang, C.-C. Chu, J. Y. Chi, J.-C. Su, I.-M. Chan, and J.-F. T. Wang, “Monte Carlo simulation of optical properties of phosphor-screened ultraviolet light in a white light-emitting device,” Jpn. J. Appl. Phys.44(8), 6056–6061 (2005).
[CrossRef]

C. C. Chang, R.-L. Chern, C. C. Chang, C.-C. Chu, J. Y. Chi, J.-C. Su, I.-M. Chan, and J.-F. T. Wang, “Monte Carlo simulation of optical properties of phosphor-screened ultraviolet light in a white light-emitting device,” Jpn. J. Appl. Phys.44(8), 6056–6061 (2005).
[CrossRef]

Chen, J.

J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography-computational efficiency,” Med. Phys.38(10), 5788–5798 (2011).
[CrossRef] [PubMed]

Chen, J.-S.

Cheong, W.-F.

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review on the optical properties of biological tissues,” IEEE J. Quantum Electron.26(12), 2166–2185 (1990).
[CrossRef]

Chern, R.-L.

C. C. Chang, R.-L. Chern, C. C. Chang, C.-C. Chu, J. Y. Chi, J.-C. Su, I.-M. Chan, and J.-F. T. Wang, “Monte Carlo simulation of optical properties of phosphor-screened ultraviolet light in a white light-emitting device,” Jpn. J. Appl. Phys.44(8), 6056–6061 (2005).
[CrossRef]

Chi, J. Y.

J. Y. Chi, J.-S. Chen, C.-Y. Liu, C.-W. Chu, and K.-H. Chiang, “Phosphor converted LEDs with omni-directional-reflector coating,” Opt. Express17(26), 23530–23535 (2009).
[CrossRef] [PubMed]

C. C. Chang, R.-L. Chern, C. C. Chang, C.-C. Chu, J. Y. Chi, J.-C. Su, I.-M. Chan, and J.-F. T. Wang, “Monte Carlo simulation of optical properties of phosphor-screened ultraviolet light in a white light-emitting device,” Jpn. J. Appl. Phys.44(8), 6056–6061 (2005).
[CrossRef]

Chiang, K.-H.

Chien, W.-T.

Chu, C.-C.

C. C. Chang, R.-L. Chern, C. C. Chang, C.-C. Chu, J. Y. Chi, J.-C. Su, I.-M. Chan, and J.-F. T. Wang, “Monte Carlo simulation of optical properties of phosphor-screened ultraviolet light in a white light-emitting device,” Jpn. J. Appl. Phys.44(8), 6056–6061 (2005).
[CrossRef]

Chu, C.-W.

Chung, H.

P. Chung, H. Chung, and P. H. Holloway, “Phosphor coatings to enhance Si photovoltaic cell performance,” J. Vac. Sci. Technol. A25(1), 61–66 (2007).
[CrossRef]

Chung, P.

P. Chung, H. Chung, and P. H. Holloway, “Phosphor coatings to enhance Si photovoltaic cell performance,” J. Vac. Sci. Technol. A25(1), 61–66 (2007).
[CrossRef]

Criswell, G.

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med.21(2), 166–178 (1997).
[CrossRef] [PubMed]

Feld, M. S.

Flatau, P. J.

P. J. Flatau and G. L. Stephens, “On the fundamental solution of the radiative transfer equation,” J. Geophys. Res.93(D9), 11037–11050 (1988).
[CrossRef]

Gardner, C.

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med.21(2), 166–178 (1997).
[CrossRef] [PubMed]

Gemert, M. J. C.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol.42(11), 2255–2261 (1997).
[CrossRef] [PubMed]

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J.93, 70–83 (1941).
[CrossRef]

Greenwald, T.

Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf.105(2), 243–263 (2007).
[CrossRef]

He, Y.

J. P. You, N. T. Tran, Y. Lin, Y. He, and F. G. Shi, “Phosphor-concentration-dependent characteristics of white LEDs in different current regulation modes,” J. Electron. Mater.38(6), 761–766 (2009).
[CrossRef]

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J.93, 70–83 (1941).
[CrossRef]

Holloway, P. H.

D. Bera, S. Maslov, L. Qian, J. S. Yoo, and P. H. Holloway, “Optimization of the yellow phosphor concentration and layer thickness for down-conversion of blue to white light,” J. Disp. Technol.6(12), 645–651 (2010).
[CrossRef]

P. Chung, H. Chung, and P. H. Holloway, “Phosphor coatings to enhance Si photovoltaic cell performance,” J. Vac. Sci. Technol. A25(1), 61–66 (2007).
[CrossRef]

Hu, Y.

Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf.105(2), 243–263 (2007).
[CrossRef]

Huang, H.-L.

Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf.105(2), 243–263 (2007).
[CrossRef]

Intes, X.

J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography-computational efficiency,” Med. Phys.38(10), 5788–5798 (2011).
[CrossRef] [PubMed]

Jung, J.

I. Seo, J. Jung, B. J. Oh, and K. Whang, “Improvement of luminance and luminous efficacy of mercury-free, flat fluorescent lamp by optimizing phosphor profile,” IEEE Trans. Plasma Sci.38(5), 1097–1100 (2010).
[CrossRef]

Kang, D.-Y.

D.-Y. Kang, E. Wu, and D.-M. Wang, “Modeling white light-emitting diodes with phosphor layers,” Appl. Phys. Lett.89(23), 231102 (2006).
[CrossRef]

Kattawar, G.

Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf.105(2), 243–263 (2007).
[CrossRef]

Kattawar, G. W.

Klampaftis, E.

E. Klampaftis and B. S. Richards, “Improvement in multi-crystalline silicon solar cell efficiency via addition of luminescent material to EVA encapsulation layer,” Prog. Photovolt. Res. Appl.19(3), 345–351 (2011).
[CrossRef]

Ko, J. H.

Li, J.

Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf.105(2), 243–263 (2007).
[CrossRef]

Liebert, A.

Lin, Y.

J. P. You, N. T. Tran, Y. Lin, Y. He, and F. G. Shi, “Phosphor-concentration-dependent characteristics of white LEDs in different current regulation modes,” J. Electron. Mater.38(6), 761–766 (2009).
[CrossRef]

Liu, C.-Y.

Macdonald, R.

Maslov, S.

D. Bera, S. Maslov, L. Qian, J. S. Yoo, and P. H. Holloway, “Optimization of the yellow phosphor concentration and layer thickness for down-conversion of blue to white light,” J. Disp. Technol.6(12), 645–651 (2010).
[CrossRef]

Moreno, I.

Nicolaï, B. M.

Oh, B. J.

I. Seo, J. Jung, B. J. Oh, and K. Whang, “Improvement of luminance and luminous efficacy of mercury-free, flat fluorescent lamp by optimizing phosphor profile,” IEEE Trans. Plasma Sci.38(5), 1097–1100 (2010).
[CrossRef]

Park, J. H.

Pfefer, J.

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med.21(2), 166–178 (1997).
[CrossRef] [PubMed]

Pickering, J. W.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol.42(11), 2255–2261 (1997).
[CrossRef] [PubMed]

Pilon, L.

Plass, G. N.

Posthumus, P.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol.42(11), 2255–2261 (1997).
[CrossRef] [PubMed]

Prahl, S. A.

S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid mediaby using the adding-doubling method,” Appl. Opt.32(4), 559–568 (1993).
[CrossRef] [PubMed]

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review on the optical properties of biological tissues,” IEEE J. Quantum Electron.26(12), 2166–2185 (1990).
[CrossRef]

Qian, L.

D. Bera, S. Maslov, L. Qian, J. S. Yoo, and P. H. Holloway, “Optimization of the yellow phosphor concentration and layer thickness for down-conversion of blue to white light,” J. Disp. Technol.6(12), 645–651 (2010).
[CrossRef]

Ramon, H.

Rava, R. P.

Richards, B. S.

E. Klampaftis and B. S. Richards, “Improvement in multi-crystalline silicon solar cell efficiency via addition of luminescent material to EVA encapsulation layer,” Prog. Photovolt. Res. Appl.19(3), 345–351 (2011).
[CrossRef]

Richards-Kortum, R.

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med.21(2), 166–178 (1997).
[CrossRef] [PubMed]

Rosema, A.

A. Rosema, W. Verhoef, J. Schroote, and J. F. H. Snel, “Simulating fluorescence light-canopy interaction in support of laser-induced fluorescence measurements,” Remote Sens. Environ.37(2), 117–130 (1991).
[CrossRef]

Saeys, W.

Schroote, J.

A. Rosema, W. Verhoef, J. Schroote, and J. F. H. Snel, “Simulating fluorescence light-canopy interaction in support of laser-induced fluorescence measurements,” Remote Sens. Environ.37(2), 117–130 (1991).
[CrossRef]

Seo, I.

I. Seo, J. Jung, B. J. Oh, and K. Whang, “Improvement of luminance and luminous efficacy of mercury-free, flat fluorescent lamp by optimizing phosphor profile,” IEEE Trans. Plasma Sci.38(5), 1097–1100 (2010).
[CrossRef]

Shi, F. G.

Y. Shuai, N. T. Tran, and F. G. Shi, “Nonmonotonic phosphor size dependence of luminous efficacy for typical white LED emitters,” IEEE Photon. Technol. Lett.23(9), 552–554 (2011).
[CrossRef]

J. P. You, N. T. Tran, Y. Lin, Y. He, and F. G. Shi, “Phosphor-concentration-dependent characteristics of white LEDs in different current regulation modes,” J. Electron. Mater.38(6), 761–766 (2009).
[CrossRef]

Shuai, Y.

Y. Shuai, N. T. Tran, and F. G. Shi, “Nonmonotonic phosphor size dependence of luminous efficacy for typical white LED emitters,” IEEE Photon. Technol. Lett.23(9), 552–554 (2011).
[CrossRef]

Snel, J. F. H.

A. Rosema, W. Verhoef, J. Schroote, and J. F. H. Snel, “Simulating fluorescence light-canopy interaction in support of laser-induced fluorescence measurements,” Remote Sens. Environ.37(2), 117–130 (1991).
[CrossRef]

Stephens, G. L.

P. J. Flatau and G. L. Stephens, “On the fundamental solution of the radiative transfer equation,” J. Geophys. Res.93(D9), 11037–11050 (1988).
[CrossRef]

Sterenborg, H. J. C. M.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol.42(11), 2255–2261 (1997).
[CrossRef] [PubMed]

Stokes, G. G.

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. Lond.11(0), 545–556 (1860).
[CrossRef]

Su, J.-C.

C. C. Chang, R.-L. Chern, C. C. Chang, C.-C. Chu, J. Y. Chi, J.-C. Su, I.-M. Chan, and J.-F. T. Wang, “Monte Carlo simulation of optical properties of phosphor-screened ultraviolet light in a white light-emitting device,” Jpn. J. Appl. Phys.44(8), 6056–6061 (2005).
[CrossRef]

Sun, C.-C.

Thennadil, S. N.

Tran, N. T.

Y. Shuai, N. T. Tran, and F. G. Shi, “Nonmonotonic phosphor size dependence of luminous efficacy for typical white LED emitters,” IEEE Photon. Technol. Lett.23(9), 552–554 (2011).
[CrossRef]

J. P. You, N. T. Tran, Y. Lin, Y. He, and F. G. Shi, “Phosphor-concentration-dependent characteristics of white LEDs in different current regulation modes,” J. Electron. Mater.38(6), 761–766 (2009).
[CrossRef]

van Gemert, M. J. C.

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

W. G. J. H. M. van Sark, “Enhancement of solar cell performance by employing planar spectral converters,” Appl. Phys. Lett.87(15), 151117 (2005).
[CrossRef]

Velazco-Roa, M. A.

Verhoef, W.

A. Rosema, W. Verhoef, J. Schroote, and J. F. H. Snel, “Simulating fluorescence light-canopy interaction in support of laser-induced fluorescence measurements,” Remote Sens. Environ.37(2), 117–130 (1991).
[CrossRef]

Wabnitz, H.

Wang, D.-M.

D.-Y. Kang, E. Wu, and D.-M. Wang, “Modeling white light-emitting diodes with phosphor layers,” Appl. Phys. Lett.89(23), 231102 (2006).
[CrossRef]

Wang, J.-F. T.

C. C. Chang, R.-L. Chern, C. C. Chang, C.-C. Chu, J. Y. Chi, J.-C. Su, I.-M. Chan, and J.-F. T. Wang, “Monte Carlo simulation of optical properties of phosphor-screened ultraviolet light in a white light-emitting device,” Jpn. J. Appl. Phys.44(8), 6056–6061 (2005).
[CrossRef]

Warren, S.

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med.21(2), 166–178 (1997).
[CrossRef] [PubMed]

Welch, A. J.

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med.21(2), 166–178 (1997).
[CrossRef] [PubMed]

S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid mediaby using the adding-doubling method,” Appl. Opt.32(4), 559–568 (1993).
[CrossRef] [PubMed]

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review on the optical properties of biological tissues,” IEEE J. Quantum Electron.26(12), 2166–2185 (1990).
[CrossRef]

Whang, K.

I. Seo, J. Jung, B. J. Oh, and K. Whang, “Improvement of luminance and luminous efficacy of mercury-free, flat fluorescent lamp by optimizing phosphor profile,” IEEE Trans. Plasma Sci.38(5), 1097–1100 (2010).
[CrossRef]

Wiscombe, W. J.

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transf.16(8), 637–658 (1976).
[CrossRef]

Wu, E.

D.-Y. Kang, E. Wu, and D.-M. Wang, “Modeling white light-emitting diodes with phosphor layers,” Appl. Phys. Lett.89(23), 231102 (2006).
[CrossRef]

Wu, J.

Yang, P.

Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf.105(2), 243–263 (2007).
[CrossRef]

Yoo, J. S.

D. Bera, S. Maslov, L. Qian, J. S. Yoo, and P. H. Holloway, “Optimization of the yellow phosphor concentration and layer thickness for down-conversion of blue to white light,” J. Disp. Technol.6(12), 645–651 (2010).
[CrossRef]

You, J. P.

J. P. You, N. T. Tran, Y. Lin, Y. He, and F. G. Shi, “Phosphor-concentration-dependent characteristics of white LEDs in different current regulation modes,” J. Electron. Mater.38(6), 761–766 (2009).
[CrossRef]

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R. G. Young and E. G. F. Arnott, “The effect of phosphor coating weight on the lumen output of luorescent lamps,” J. Electrochem. Soc.112(10), 982–984 (1965).
[CrossRef]

Yudovsky, D.

Zhang, Z.

Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf.105(2), 243–263 (2007).
[CrossRef]

Zhou, D. K.

Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf.105(2), 243–263 (2007).
[CrossRef]

Zolek, N.

Appl. Opt. (5)

Appl. Phys. Lett. (2)

W. G. J. H. M. van Sark, “Enhancement of solar cell performance by employing planar spectral converters,” Appl. Phys. Lett.87(15), 151117 (2005).
[CrossRef]

D.-Y. Kang, E. Wu, and D.-M. Wang, “Modeling white light-emitting diodes with phosphor layers,” Appl. Phys. Lett.89(23), 231102 (2006).
[CrossRef]

Astrophys. J. (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J.93, 70–83 (1941).
[CrossRef]

IEEE J. Quantum Electron. (1)

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review on the optical properties of biological tissues,” IEEE J. Quantum Electron.26(12), 2166–2185 (1990).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

Y. Shuai, N. T. Tran, and F. G. Shi, “Nonmonotonic phosphor size dependence of luminous efficacy for typical white LED emitters,” IEEE Photon. Technol. Lett.23(9), 552–554 (2011).
[CrossRef]

IEEE Trans. Plasma Sci. (1)

I. Seo, J. Jung, B. J. Oh, and K. Whang, “Improvement of luminance and luminous efficacy of mercury-free, flat fluorescent lamp by optimizing phosphor profile,” IEEE Trans. Plasma Sci.38(5), 1097–1100 (2010).
[CrossRef]

J. Disp. Technol. (1)

D. Bera, S. Maslov, L. Qian, J. S. Yoo, and P. H. Holloway, “Optimization of the yellow phosphor concentration and layer thickness for down-conversion of blue to white light,” J. Disp. Technol.6(12), 645–651 (2010).
[CrossRef]

J. Electrochem. Soc. (1)

R. G. Young and E. G. F. Arnott, “The effect of phosphor coating weight on the lumen output of luorescent lamps,” J. Electrochem. Soc.112(10), 982–984 (1965).
[CrossRef]

J. Electron. Mater. (1)

J. P. You, N. T. Tran, Y. Lin, Y. He, and F. G. Shi, “Phosphor-concentration-dependent characteristics of white LEDs in different current regulation modes,” J. Electron. Mater.38(6), 761–766 (2009).
[CrossRef]

J. Geophys. Res. (1)

P. J. Flatau and G. L. Stephens, “On the fundamental solution of the radiative transfer equation,” J. Geophys. Res.93(D9), 11037–11050 (1988).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Korea (1)

J. Quant. Spectrosc. Radiat. Transf. (2)

Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf.105(2), 243–263 (2007).
[CrossRef]

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transf.16(8), 637–658 (1976).
[CrossRef]

J. Vac. Sci. Technol. A (1)

P. Chung, H. Chung, and P. H. Holloway, “Phosphor coatings to enhance Si photovoltaic cell performance,” J. Vac. Sci. Technol. A25(1), 61–66 (2007).
[CrossRef]

Jpn. J. Appl. Phys. (1)

C. C. Chang, R.-L. Chern, C. C. Chang, C.-C. Chu, J. Y. Chi, J.-C. Su, I.-M. Chan, and J.-F. T. Wang, “Monte Carlo simulation of optical properties of phosphor-screened ultraviolet light in a white light-emitting device,” Jpn. J. Appl. Phys.44(8), 6056–6061 (2005).
[CrossRef]

Lasers Surg. Med. (1)

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med.21(2), 166–178 (1997).
[CrossRef] [PubMed]

Med. Phys. (1)

J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography-computational efficiency,” Med. Phys.38(10), 5788–5798 (2011).
[CrossRef] [PubMed]

Opt. Express (3)

Phys. Med. Biol. (1)

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol.42(11), 2255–2261 (1997).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. (1)

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. Lond.11(0), 545–556 (1860).
[CrossRef]

Prog. Photovolt. Res. Appl. (1)

E. Klampaftis and B. S. Richards, “Improvement in multi-crystalline silicon solar cell efficiency via addition of luminescent material to EVA encapsulation layer,” Prog. Photovolt. Res. Appl.19(3), 345–351 (2011).
[CrossRef]

Remote Sens. Environ. (1)

A. Rosema, W. Verhoef, J. Schroote, and J. F. H. Snel, “Simulating fluorescence light-canopy interaction in support of laser-induced fluorescence measurements,” Remote Sens. Environ.37(2), 117–130 (1991).
[CrossRef]

Other (2)

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

H. G. Völz, Industrial color testing, Fundamentals and Techniques (Wiley-VCH, 2001)

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Figures (7)

Fig. 1
Fig. 1

Subdivision of the flux into n conical segments

Fig. 2
Fig. 2

Schematic representation of the vectors and matrices for a single non-fluorescent slab

Fig. 3
Fig. 3

Schematic representation of the contribution of all excitation wavelength intervals to one emission wavelength interval

Fig. 4
Fig. 4

Schematic representation of the vector-matrix equation for a fluorescent slab for one excitation wavelength ΔλXi (solid lines) and one emission wavelength ΔλMj (dashed lines). The equations represented are Eq. (2) for the excitation wavelength interval and Eq. (8) for the emission wavelength interval.

Fig. 5
Fig. 5

Schematic representation of the contribution of one excitation wavelength interval to all emission wavelength intervals, the contribution to each emission wavelength region is denoted with the factor w(ΔλMj).

Fig. 6
Fig. 6

Scattered intensities for θi = 0° obtained through Monte Carlo simulations (marks) and calculated with the adding-doubling method (lines).

Fig. 7
Fig. 7

Total reflection and transmission obtained with Monte Carlo simulations (marks) and adding-doubling calculations (lines)

Equations (68)

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

[ I ( θ 1 ) I ( θ n ) ]=[ R( θ 1 , θ 1 ) R( θ 1 , θ n ) R( θ n , θ 1 ) R( θ n , θ n ) ]x[ I + ( θ 1 ) I + ( θ n ) ]
I 1 + = R 10 I 1 + T 01 I 0 +
I 0 = R 01 I 0 + + T 10 I 1
I 2 + = R 21 I 2 + T 12 I 1 +
I 1 = R 12 I 1 + + T 21 I 2
R 20 = R 21 + T 12 (E R 10 R 12 ) 1 R 10 T 21
T 02 = T 12 (E R 10 R 12 ) 1 T 01
I 1 + = R 10 I 1 + T 01 I 0 + + i=1 N R 10 c (Δ λ i X ,Δ λ j M ) I 1 (Δ λ i X )+ T 01 c (Δ λ i X ,Δ λ j M ) I 0 + (Δ λ i X )
I 0 = R 01 I 0 + + T 10 I 1 + i=1 N R 01 c (Δ λ i X ,Δ λ j M ) I 0 + (Δ λ i X )+ T 10 c (Δ λ i X ,Δ λ j M ) I 1 (Δ λ i X )
R 20 = R 21 + T 12 (E R 10 R 12 ) 1 R 10 T 21
T 02 = T 12 (E R 10 R 12 ) 1 T 01
R 20 c ( Δ λ i X ,Δ λ j M )= T 12 (E R 10 R 12 ) 1 [ R 10 ( R 12 c ( Δ λ i X ,Δ λ j M ){ [ E R 10 c ( Δ λ i X ,Δ λ j M ) R 12 c ( Δ λ i X ,Δ λ j M ) ] 1 R 10 c ( Δ λ i X ,Δ λ j M ) T 21 c ( Δ λ i X ,Δ λ j M ) } + T 21 c ( Δ λ i X ,Δ λ j M ) ) + R 10 c ( Δ λ i X ,Δ λ j M ) [ E R 12 c ( Δ λ i X ,Δ λ j M ) R 10 c ( Δ λ i X ,Δ λ j M ) ] 1 T 21 ( Δ λ i X ) ] + R 21 c ( Δ λ i X ,Δ λ j M )+ T 12 c ( Δ λ i X ,Δ λ j M ) [ E R 10 ( Δ λ i X ) R 12 ( Δ λ i X ) ] 1 R 10 ( Δ λ i X ) T 21 ( Δ λ i X )
T 02 c ( Δ λ i X ,Δ λ j M )= T 12 (E R 10 R 12 ) 1 { R 10 R 12 c ( Δ λ i X ,Δ λ j M ) [ E R 10 ( Δ λ i X ) R 12 ( Δ λ i X ) ] 1 T 01 ( Δ λ i X ) + R 10 c ( Δ λ i X ,Δ λ j M ) [ E R 12 ( Δ λ i X ) R 10 ( Δ λ i X ) ] 1 R 12 ( Δ λ i X ) T 01 ( Δ λ i X ) + T 01 c ( Δ λ i X ,Δ λ j M ) } + T 12 c ( Δ λ i X ,Δ λ j M ) [ E R 10 ( Δ λ i X ) R 12 ( Δ λ i X ) ] 1 T 01 ( Δ λ i X )
ν ϑI(ν) ϑz =( μ a + μ s )I(ν)+ μ s 1 1 P(ν,ν')I(ν')dν'
P HG (ϒ)= 1 2 1 g 2 ( 1+ g 2 2gϒ ) 3 2
( E+α ) I 1 + =( Eα ) I 0 + +β( I 0 + I 1 )
( E+α ) I 0 =( Eα ) I 1 +β( I 0 + + I 1 + )
α= M 1 ( E μ s Δd 2 p ++ c )
β= M 1 μ s Δd 2 p + c
M(i,j)= ν i δ ij
c(i,j)= c i δ ij
p ij +± =P( ν i ,± ν j )
R 10 =2Γβ ( E+α ) 1
T 01 =2ΓE
Γ= [ E+α-β ( E+α ) -1 β ] -1
ν ϑI(ν) ϑz =( μ a + μ s )I(ν)+ μ s 1 1 P(ν,ν')I(ν')dν' +w(Δ λ j M ) i=1 N { μ a (Δ λ i X )QE(Δ λ i X ) λ j M λ i X [ 1 1 1 2 I(ν', Δ λ i X )dν' ]Δ λ i X }
j=1 L w(Δ λ j M )=1
( E+α ) I 1 + =( Eα ) I 0 + +β( I 0 + I 1 ) + i=1 N γ i [ I 0 + (Δ λ i X )+ I 1 + (Δ λ i X )+ I 0 (Δ λ i X )+ I 1 (Δ λ i X ) ]
( E+α ) I 0 =( Eα ) I 1 +β( I 0 + + I 1 + ) + i=1 N γ i [ I 0 + (Δ λ i X )+ I 1 + (Δ λ i X )+ I 0 (Δ λ i X )+ I 1 (Δ λ i X ) ]
γ i =w(Δ λ j M ) μ a (Δ λ i X )QE(Δ λ i X ) λ M λ Xi Δd 4 p x c
p x (i,j)=1
R 10 =2Γβ ( E+α ) 1
T 01 =2ΓE
R 10 c (Δ λ i X ,Δ λ j M )= T 01 c (Δ λ i X ,Δ λ j M )= [ Γβ ( E+α ) -1 +E ] γ i [ E+T(Δ λ i X )+R(Δ λ i X ) ]
I 1 + = R 10 I 1 + T 01 I 0 + + i=1 N R 10 c (Δ λ i X ,Δ λ j M ) I 1 (Δ λ i X )+ T 01 c (Δ λ i X ,Δ λ j M ) I 0 + (Δ λ i X )
I 0 = R 01 I 0 + + T 10 I 1 + i=1 N R 01 c (Δ λ i X ,Δ λ j M ) I 0 + (Δ λ i X )+ T 10 c (Δ λ i X ,Δ λ j M ) I 1 (Δ λ i X )
I 2 + = R 21 I 2 + T 12 I 1 + + i=1 N R 21 c (Δ λ i X ,Δ λ j M ) I 2 (Δ λ i X )+ T 12 c (Δ λ i X ,Δ λ j M ) I 1 + (Δ λ i X )
I 1 = R 12 I 1 + + T 21 I 2 + i=1 N R 12 c (Δ λ i X ,Δ λ j M ) I 1 + (Δ λ i X )+ T 21 c (Δ λ i X ,Δ λ j M ) I 2 (Δ λ i X )
I 1 + = ( E R 10 R 12 ) 1 { R 10 [ T 21 I 2 + i=1 N R 12 c (Δ λ i X ,Δ λ j M ) I 1 + (Δ λ i X )+ T 21 c (Δ λ i X ,Δ λ j M ) I 2 (Δ λ i X ) ] + T 01 I 0 + + i=1 N R 10 c (Δ λ i X ,Δ λ j M ) I 1 (Δ λ i X )+ T 01 c (Δ λ i X ,Δ λ j M ) I 0 + (Δ λ i X ) }
I 1 (Δ λ i X )= [ E R 12 (Δ λ i X ) R 10 (Δ λ i X ) ] 1 [ T 21 (Δ λ i X ) I 2 (Δ λ i X )+ R 12 (Δ λ i X ) T 01 (Δ λ i X ) I 0 + (Δ λ i X ) ]
I 1 + (Δ λ i X )= [ E R 10 (Δ λ i X ) R 12 (Δ λ i X ) ] 1 [ T 01 (Δ λ i X ) I 0 + (Δ λ i X )+ R 10 (Δ λ i X ) T 21 (Δ λ i X ) I 2 (Δ λ i X ) ]
I 2 + = R 21 I 2 + T 12 ( E R 10 R 12 ) 1 [ R 10 ( T 21 I 2 + i=1 N { R 12 c (Δ λ i X ,Δ λ j M ) [ E R 10 (Δ λ i X ) R 12 (Δ λ i X ) ] 1 [ T 01 (Δ λ i X ) I 0 + (Δ λ i X )+ R 10 (Δ λ i X ) T 21 (Δ λ i X ) I 2 (Δ λ i X ) ] + T 21 c (Δ λ i X ,Δ λ j M ) I 2 (Δ λ i X ) } ) + T 01 I 0 + + i=1 N { R 10 c (Δ λ i X ,Δ λ j M ) [ E R 12 (Δ λ i X ) R 10 (Δ λ i X ) ] 1 [ T 21 (Δ λ i X ) I 2 (Δ λ i X )+ R 12 (Δ λ i X ) T 01 (Δ λ i X ) I 0 + (Δ λ i X ) ] + T 01 c (Δ λ i X ,Δ λ j M ) I 0 + (Δ λ i X ) } ] +{ i=1 N R 21 c (Δ λ i X ,Δ λ j M ) I 2 (Δ λ i X )+ T 12 c (Δ λ i X ,Δ λ j M ) [ E R 10 (Δ λ i X ) R 12 (Δ λ i X ) ] 1 [ T 01 (Δ λ i X ) I 0 + (Δ λ i X )+ R 10 (Δ λ i X ) T 21 (Δ λ i X ) I 2 (Δ λ i X ) ] }
I 2 + =[ R 21 + T 12 ( E R 10 R 12 ) 1 R 10 T 21 ] I 2 +[ T 12 ( E R 10 R 12 ) 1 T 01 ] I 0 + + i=1 N [ T 12 ( E R 10 R 12 ) 1 ( R 10 { R 12 c (Δ λ i X ,Δ λ j M ) [ E R 10 (Δ λ i X ) R 12 (Δ λ i X ) ] 1 R 10 (Δ λ i X ) T 21 (Δ λ i X )+ T 21 c (Δ λ i X ,Δ λ j M ) } + R 10 c (Δ λ i X ,Δ λ j M ) [ E R 12 (Δ λ i X ) R 10 (Δ λ i X ) ] 1 T 21 (Δ λ i X ) ) + R 21 c (Δ λ i X ,Δ λ j M )+ T 12 c (Δ λ i X ,Δ λ j M ) [ E R 10 (Δ λ i X ) R 12 (Δ λ i X ) ] 1 R 10 (Δ λ i X ) T 21 (Δ λ i X ) ] I 2 (Δ λ i X ) + i=1 N ( T 12 ( E R 10 R 12 ) 1 { R 10 R 12 c (Δ λ i X ,Δ λ j M ) [ E R 10 (Δ λ i X ) R 12 (Δ λ i X ) ] 1 T 01 (Δ λ i X ) + R 10 c (Δ λ i X ,Δ λ j M ) [ E R 12 (Δ λ i X ) R 10 (Δ λ i X ) ] 1 R 12 (Δ λ i X ) T 01 (Δ λ i X ) + T 01 c (Δ λ i X ,Δ λ j M ) } + T 12 c (Δ λ i X ,Δ λ j M ) [ E R 10 (Δ λ i X ) R 12 (Δ λ i X ) ] 1 T 01 (Δ λ i X ) ) I 0 + (Δ λ i X )
R 20 = R 21 + T 12 ( E R 10 R 12 ) 1 R 10 T 21
T 02 = T 12 ( E R 10 R 12 ) 1 T 01
R 20 c (Δ λ i X ,Δ λ j M )= T 12 ( E R 10 R 12 ) 1 ( R 10 { R 12 c (Δ λ i X ,Δ λ j M ) [ E R 10 (Δ λ i X ) R 12 (Δ λ i X ) ] 1 R 10 (Δ λ i X ) T 21 (Δ λ i X )+ T 21 c (Δ λ i X ,Δ λ j M ) } + R 10 c (Δ λ i X ,Δ λ j M ) [ E R 12 (Δ λ i X ) R 10 (Δ λ i X ) ] 1 T 21 (Δ λ i X ) ) + R 21 c (Δ λ i X ,Δ λ j M )+ T 12 c (Δ λ i X ,Δ λ j M ) [ E R 10 (Δ λ i X ) R 12 (Δ λ i X ) ] 1 R 10 (Δ λ i X ) T 21 (Δ λ i X )
T 02 c (Δ λ i X ,Δ λ j M )= T 12 ( E R 10 R 12 ) 1 { R 10 R 12 c (Δ λ i X ,Δ λ j M ) [ E R 10 (Δ λ i X ) R 12 (Δ λ i X ) ] 1 T 01 (Δ λ i X ) + R 10 c (Δ λ i X ,Δ λ j M ) [ E R 12 (Δ λ i X ) R 10 (Δ λ i X ) ] 1 R 12 (Δ λ i X ) T 01 (Δ λ i X ) + T 01 c (Δ λ i X ,Δ λ j M ) } + T 12 c (Δ λ i X ,Δ λ j M ) [ E R 10 (Δ λ i X ) R 12 (Δ λ i X ) ] 1 T 01 (Δ λ i X )
ν ϑI(ν) ϑz =( μ a + μ s )I(ν)+ μ s 1 1 P(ν,ν')I(ν')dν' +w(Δ λ j M ) i=1 N { μ a (Δ λ i X )QE(Δ λ i X ) λ j M λ i X [ 1 1 1 2 I(ν', Δ λ i X )dν' ]Δ λ i X }
± ν i ϑI(± ν i ) ϑz =( μ a + μ s )I(± ν i )+ μ s j=1 M p ij + c j I( ν j )+ p ij ++ c j I(± ν j ) +w(Δ λ j M ) i=1 N { μ a (Δ λ i X )QE(Δ λ i X ) λ j M λ i X [ j=1 M 1 2 c j I( ± ν j ,Δ λ i X ) + 1 2 c j I( ν j ,Δ λ i X ) ]Δ λ i X }
p ij +± =P( ν i ,± ν j )
±M ϑ I ± ϑz =( μ a + μ s ) I ± + μ s ( p ++ c I ± + p + c I ) +w(Δ λ j M ) i=1 N { μ a (Δ λ i X )QE(Δ λ i X ) λ j M λ i X 1 2 ( p x c I ± (Δ λ i X )+ p x c I (Δ λ i X ) )Δ λ i X }
M(i,j)= ν i δ ij
c(i,j)= c i δ ij
p x (i,j)=1
±M( I 1 ± I 0 ± )=( μ a + μ s )Δd I 1 2 ± + μ s Δd( p ++ c I 1 2 ± + p + c I 1 2 ) +w(Δ λ j M ) i=1 N { μ a (Δ λ i X )QE(Δ λ i X ) λ j M λ i X 1 2 Δd ( p x c I 1 2 ± (Δ λ i X ) + p x c I 1 2 (Δ λ i X ) )Δ λ i X }
I 1 2 ± = 1 Δz Z0 Z1 I ± dz
±M( I 1 ± I 0 ± )=( μ a + μ s )Δd 1 2 ( I 0 ± + I 1 ± ) + μ s Δd 1 2 [ p ++ c( I 0 ± + I 1 ± )+ p + c( I 0 + I 1 ) ] +w(Δ λ j M ) i=1 N ( μ a (Δ λ i X )QE(Δ λ i X ) λ j M λ i X 1 4 Δd { p x c[ I 0 ± (Δ λ i X )+ I 1 ± (Δ λ i X ) ] + p x c[ I 0 (Δ λ i X )+ I 1 (Δ λ i X ) ] }Δ λ i X )
( E+α ) I 1 + =( Eα ) I 0 + +β( I 0 + I 1 ) + i=1 N γ i [ I 0 + (Δ λ i X )+ I 1 + (Δ λ i X )+ I 0 (Δ λ i X )+ I 1 (Δ λ i X ) ]
( E+α ) I 0 =( Eα ) I 1 +β( I 0 + + I 1 + ) + i=1 N γ i [ I 0 + (Δ λ i X )+ I 1 + (Δ λ i X )+ I 0 (Δ λ i X )+ I 1 (Δ λ i X ) ]
α= M 1 ( E μ s Δd 2 p ++ c )
β= M 1 μ s Δd 2 p + c
γ i =w(Δ λ j M ) μ a (Δ λ i X )QE(Δ λ i X ) λ j M λ i X Δd 4 p x c
( E+α ) I 1 + =( Eα ) I 0 + +β{ ( E+α ) 1 ( Eα ) I 1 + ( E+α ) 1 β I 0 + + I 1 + + ( E+α ) 1 i=1 N γ i [ I 0 + (Δ λ i X )+ I 1 + (Δ λ i X ) + I 0 (Δ λ i X )+ I 1 (Δ λ i X ) ] + I 1 } + i=1 N γ i [ I 0 + (Δ λ i X )+ I 1 + (Δ λ i X )+ I 0 (Δ λ i X )+ I 1 (Δ λ i X ) ]
I 1 + =( 2ΓE ) I 0 + +2Γβ ( E+α ) 1 I 1 + i=1 N { [ Γβ ( E+α ) -1 +E ] γ i [ E+T(Δ λ i X )+R(Δ λ i X ) ][ I 0 + (Δ λ i X )+ I 1 + (Δ λ i X ) ] }
Γ= [ E+α-β ( E+α ) -1 β ] -1
R 10 =2Γβ ( E+α ) 1
T 01 =2ΓE
R 10 c (Δ λ i X ,Δ λ j M )= T 01 c (Δ λ i X ,Δ λ j M )= [ Γβ ( E+α ) -1 +E ] γ i [ E+T(Δ λ i X )+R(Δ λ i X ) ]

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