Abstract

The behaviors of the light propagation in the phosphor play a vital role in determining the optical performance of the phosphor-converted light-emitting diodes (pc-LEDs). In this paper, we presented a general model based on the radiative transfer equation integrated with fluorescence (FRTE) to describe the overall light propagating properties in the phosphor layer in terms of light absorption, strong forward scattering, and fluorescence. The model was established by accounting for general boundary conditions including the LED Lambertian incidence, the diffuse reflection at the substrate/reflector, and the Fresnel reflection at the phosphor-air interface. The spectral element method (SEM) was extended to numerically solve FRTE. The radiant intensity at any location and direction of blue and yellow light was iteratively calculated, in which case the angular properties could be further evaluated. The model was validated by comparing the light extraction efficiency (LEE) and angular correlated color temperature (CCT) calculated by the presented model with the experimental results. Good agreements were achieved between model predictions and measurements with the corresponding maximum deviation of 4.9% and 3.7% for LEE and CCT, respectively. We also conducted a comparison between our model and the previous Kubelka-Munk (KM) theory. It has been revealed that the KM theory may overestimate the phosphor heating due to lacking of the blue light scattering effect.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

Y. P. Ma, W. Lan, B. Xie, R. Hu, and X. B. Luo, “An optical-thermal model for laser-excited remote phosphor with thermal quenching,” Int. J. Heat Mass Transf. 116, 694–702 (2018).
[Crossref]

2017 (2)

Y. Peng, R. X. Li, H. Zhen Chen, H. Li, and M. X. Chen, “Facile preparation of patterned phosphor-in-glass with excellent luminous properties through screen-printing for high-power white light-emitting diodes,” J. Alloys Compd. 693, 279–284 (2017).
[Crossref]

Y. P. Ma, R. Hu, X. J. Yu, W. C. Shu, and X. B. Luo, “A modified bidirectional thermal resistance model for junction and phosphor temperature estimation in phosphor-converted light-emitting diodes,” Int. J. Heat Mass Transf. 106, 1–6 (2017).
[Crossref]

2016 (3)

A. Correia, P. Hanselaer, and Y. Meuret, “An efficient optothermal simulation framework for optimization of high-luminance white light sources,” IEEE Photonics J. 8(4), 1–15 (2016).
[Crossref]

X. B. Luo, R. Hu, S. Liu, and K. Wang, “Heat and fluid flow in high-power LED packaging and applications,” Pror. Energy Combust. Sci. 56, 1–32 (2016).
[Crossref]

J. Sun, H. L. Yi, and H. P. Tan, “Local RBF meshless scheme for coupled radiative and conductive heat transfer,” Numer. Heat Transf. A 69(12), 1390–1404 (2016).
[Crossref]

2015 (1)

2014 (4)

S. W. Jeon, J. H. Noh, K. H. Kim, W. H. Kim, C. Yun, S. B. Song, and J. P. Kim, “Improvement of phosphor modeling based on the absorption of Stokes shifted light by a phosphor,” Opt. Express 22(S5Suppl 5), A1237–A1242 (2014).
[Crossref] [PubMed]

A. Lenef, J. Kelso, M. Tchoul, O. Mehl, J. Sorg, and Y. Zheng, “Laser-activated remote phosphor conversion with ceramic phosphors,” Proc. SPIE 9190, 919000C (2014).

A. Lenef, J. F. Kelso, and A. Piquette, “Light extraction from luminescent light sources and application to monolithic ceramic phosphors,” Opt. Lett. 39(10), 3058–3061 (2014).
[Crossref] [PubMed]

X. B. Luo and R. Hu, “Calculation of the phosphor heat generation in phosphor-converted light-emitting diodes,” Int. J. Heat Mass Transf. 75, 213–217 (2014).
[Crossref]

2013 (4)

R. Hu, B. Cao, Y. Zou, Y. M. Zhu, S. Liu, and X. B. Luo, “Modeling the light extraction efficiency of bi-layer phosphor in white LEDs,” IEEE Photonics Technol. Lett. 25(12), 1141–1144 (2013).
[Crossref]

R. Hu, Y. M. Wang, Y. Zou, X. Chen, S. Liu, and X. B. Luo, “Study on phosphor sedimentation effect in white LED packages by modeling multi-layer phosphors with the modified Kubelka-Munk theory,” J. Appl. Phys. 113(6), 063108 (2013).
[Crossref]

R. Hu, H. Zheng, J. Y. Hu, and X. B. Luo, “Comprehensive study on the transmitted and reflected light through the phosphor layer in light-emitting diode packages,” J. Disp. Technol. 9(6), 447–452 (2013).
[Crossref]

A. Lenef, J. Kelso, Y. Zheng, and M. Tchoul, “Radiance limits of ceramic phosphors under high excitation fluxes,” Proc. SPIE 8841, 884107 (2013).

2012 (2)

2011 (2)

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]

T. J. Tarvainen, V. Kolehmainen, S. R. Arridge, and J. P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport–diffusion model,” J. Quant. Spectrosc. 112(16), 2600–2608 (2011).
[Crossref]

2010 (4)

2008 (1)

2007 (2)

J. M. Zhao and L. H. Liu, “Spectral element approach for coupled radiative and conductive heat transfer in semitransparent medium,” J. Heat Transfer 129(10), 1417–1424 (2007).
[Crossref]

J. M. Zhao and L. H. Liu, “Discontinuous spectral element method for solving radiative heat transfer in multidimensional semitransparent media,” J. Quant. Spectrosc. 107(1), 1–16 (2007).
[Crossref]

2006 (3)

J. M. Zhao and L. H. Liu, “Least-squares spectral element method for radiative heat transfer in semitransparent media,” Numer. Heat Transf. B 50(5), 473–489 (2006).
[Crossref]

E. F. Schubert, J. K. Kim, H. Luo, and J. Q. Xi, “Solid state lighting–a benevolent technology,” Rep. Prog. Phys. 69(12), 3069–3099 (2006).
[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]

2004 (1)

L. H. Liu, “Finite element simulation of radiative heat transfer in absorbing and scattering media,” J. Thermophys. Heat Transfer 18(4), 555–557 (2004).
[Crossref]

2003 (2)

J. C. Chai, “One-dimensional transient radiation heat transfer modeling using a finite-volume method,” Numer. Heat Transf. B 44(2), 187–208 (2003).
[Crossref]

T. Shakespeare and J. Shakespeare, “A fluorescent extension to the Kubelka–Munk model,” Color Res. Appl. 28(1), 4–14 (2003).
[Crossref]

1997 (2)

W. E. Vargas and G. A. Niklasson, “Applicability conditions of the Kubelka-Munk theory,” Appl. Opt. 36(22), 5580–5586 (1997).
[Crossref] [PubMed]

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. van 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]

1995 (1)

W. A. Fiveland and J. P. Jessee, “Comparison of discrete ordinates formulations for radiative heat transfer in multidimensional geometries,” J. Thermophys. Heat Transf. 9(1), 47–54 (1995).
[Crossref]

1994 (1)

1992 (1)

S. McCamy, “Correlated color temperature as an explicit function of chromaticity coordinates,” Color Res. Appl. 17(2), 142–144 (1992).
[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)

W. A. Fiveland, “Three-dimensional radiative heat-transfer solutions by the discrete-ordinates method,” J. Thermophys. Heat Transf. 2(4), 309–316 (1988).
[Crossref]

1968 (1)

J. R. Howell, “Application of Monte Carlo to heat transfer problems,” Adv. Heat Transf. 5, 1–54 (1968).

1941 (1)

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

Aalders, M.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. van 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]

Arridge, S. R.

T. J. Tarvainen, V. Kolehmainen, S. R. Arridge, and J. P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport–diffusion model,” J. Quant. Spectrosc. 112(16), 2600–2608 (2011).
[Crossref]

Beek, J. F.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. van 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]

Blokland, P.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. van 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]

Burch, C. L.

Cao, B.

R. Hu, B. Cao, Y. Zou, Y. M. Zhu, S. Liu, and X. B. Luo, “Modeling the light extraction efficiency of bi-layer phosphor in white LEDs,” IEEE Photonics Technol. Lett. 25(12), 1141–1144 (2013).
[Crossref]

Chai, J. C.

J. C. Chai, “One-dimensional transient radiation heat transfer modeling using a finite-volume method,” Numer. Heat Transf. B 44(2), 187–208 (2003).
[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, M. X.

Y. Peng, R. X. Li, H. Zhen Chen, H. Li, and M. X. Chen, “Facile preparation of patterned phosphor-in-glass with excellent luminous properties through screen-printing for high-power white light-emitting diodes,” J. Alloys Compd. 693, 279–284 (2017).
[Crossref]

Chen, X.

R. Hu, Y. M. Wang, Y. Zou, X. Chen, S. Liu, and X. B. Luo, “Study on phosphor sedimentation effect in white LED packages by modeling multi-layer phosphors with the modified Kubelka-Munk theory,” J. Appl. Phys. 113(6), 063108 (2013).
[Crossref]

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]

Correia, A.

A. Correia, P. Hanselaer, and Y. Meuret, “An efficient optothermal simulation framework for optimization of high-luminance white light sources,” IEEE Photonics J. 8(4), 1–15 (2016).
[Crossref]

Deconinck, G.

Durinck, G.

Fiveland, W. A.

W. A. Fiveland and J. P. Jessee, “Comparison of discrete ordinates formulations for radiative heat transfer in multidimensional geometries,” J. Thermophys. Heat Transf. 9(1), 47–54 (1995).
[Crossref]

W. A. Fiveland, “Three-dimensional radiative heat-transfer solutions by the discrete-ordinates method,” J. Thermophys. Heat Transf. 2(4), 309–316 (1988).
[Crossref]

Greenstein, J. L.

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

Hanselaer, P.

A. Correia, P. Hanselaer, and Y. Meuret, “An efficient optothermal simulation framework for optimization of high-luminance white light sources,” IEEE Photonics J. 8(4), 1–15 (2016).
[Crossref]

S. Leyre, G. Durinck, B. Van Giel, W. Saeys, J. Hofkens, G. Deconinck, and P. Hanselaer, “Extended adding-doubling method for fluorescent applications,” Opt. Express 20(16), 17856–17872 (2012).
[Crossref] [PubMed]

Henyey, L. G.

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

Hofkens, J.

Howell, J. R.

J. R. Howell, “Application of Monte Carlo to heat transfer problems,” Adv. Heat Transf. 5, 1–54 (1968).

Hu, J. Y.

R. Hu, H. Zheng, J. Y. Hu, and X. B. Luo, “Comprehensive study on the transmitted and reflected light through the phosphor layer in light-emitting diode packages,” J. Disp. Technol. 9(6), 447–452 (2013).
[Crossref]

Hu, R.

Y. P. Ma, W. Lan, B. Xie, R. Hu, and X. B. Luo, “An optical-thermal model for laser-excited remote phosphor with thermal quenching,” Int. J. Heat Mass Transf. 116, 694–702 (2018).
[Crossref]

Y. P. Ma, R. Hu, X. J. Yu, W. C. Shu, and X. B. Luo, “A modified bidirectional thermal resistance model for junction and phosphor temperature estimation in phosphor-converted light-emitting diodes,” Int. J. Heat Mass Transf. 106, 1–6 (2017).
[Crossref]

X. B. Luo, R. Hu, S. Liu, and K. Wang, “Heat and fluid flow in high-power LED packaging and applications,” Pror. Energy Combust. Sci. 56, 1–32 (2016).
[Crossref]

X. B. Luo and R. Hu, “Calculation of the phosphor heat generation in phosphor-converted light-emitting diodes,” Int. J. Heat Mass Transf. 75, 213–217 (2014).
[Crossref]

R. Hu, B. Cao, Y. Zou, Y. M. Zhu, S. Liu, and X. B. Luo, “Modeling the light extraction efficiency of bi-layer phosphor in white LEDs,” IEEE Photonics Technol. Lett. 25(12), 1141–1144 (2013).
[Crossref]

R. Hu, Y. M. Wang, Y. Zou, X. Chen, S. Liu, and X. B. Luo, “Study on phosphor sedimentation effect in white LED packages by modeling multi-layer phosphors with the modified Kubelka-Munk theory,” J. Appl. Phys. 113(6), 063108 (2013).
[Crossref]

R. Hu, H. Zheng, J. Y. Hu, and X. B. Luo, “Comprehensive study on the transmitted and reflected light through the phosphor layer in light-emitting diode packages,” J. Disp. Technol. 9(6), 447–452 (2013).
[Crossref]

R. Hu and X. B. Luo, “A model for calculating the bidirectional scattering properties of phosphor layer in white light-emitting diodes,” J. Lightwave Technol. 30(21), 3376–3380 (2012).
[Crossref]

Hung, C. H.

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]

Jeon, S. W.

Jessee, J. P.

W. A. Fiveland and J. P. Jessee, “Comparison of discrete ordinates formulations for radiative heat transfer in multidimensional geometries,” J. Thermophys. Heat Transf. 9(1), 47–54 (1995).
[Crossref]

Joo, J. Y.

Kaipio, J. P.

T. J. Tarvainen, V. Kolehmainen, S. R. Arridge, and J. P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport–diffusion model,” J. Quant. Spectrosc. 112(16), 2600–2608 (2011).
[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]

Kelso, J.

A. Lenef, J. Kelso, M. Tchoul, O. Mehl, J. Sorg, and Y. Zheng, “Laser-activated remote phosphor conversion with ceramic phosphors,” Proc. SPIE 9190, 919000C (2014).

A. Lenef, J. Kelso, Y. Zheng, and M. Tchoul, “Radiance limits of ceramic phosphors under high excitation fluxes,” Proc. SPIE 8841, 884107 (2013).

Kelso, J. F.

Kim, J. K.

E. F. Schubert, J. K. Kim, H. Luo, and J. Q. Xi, “Solid state lighting–a benevolent technology,” Rep. Prog. Phys. 69(12), 3069–3099 (2006).
[Crossref]

Kim, J. P.

Kim, K. H.

Kim, W. H.

Kolehmainen, V.

T. J. Tarvainen, V. Kolehmainen, S. R. Arridge, and J. P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport–diffusion model,” J. Quant. Spectrosc. 112(16), 2600–2608 (2011).
[Crossref]

Lan, W.

Y. P. Ma, W. Lan, B. Xie, R. Hu, and X. B. Luo, “An optical-thermal model for laser-excited remote phosphor with thermal quenching,” Int. J. Heat Mass Transf. 116, 694–702 (2018).
[Crossref]

Lee, D. H.

Lee, S. K.

Lenef, A.

A. Lenef, J. F. Kelso, and A. Piquette, “Light extraction from luminescent light sources and application to monolithic ceramic phosphors,” Opt. Lett. 39(10), 3058–3061 (2014).
[Crossref] [PubMed]

A. Lenef, J. Kelso, M. Tchoul, O. Mehl, J. Sorg, and Y. Zheng, “Laser-activated remote phosphor conversion with ceramic phosphors,” Proc. SPIE 9190, 919000C (2014).

A. Lenef, J. Kelso, Y. Zheng, and M. Tchoul, “Radiance limits of ceramic phosphors under high excitation fluxes,” Proc. SPIE 8841, 884107 (2013).

Leyre, S.

Li, H.

Y. Peng, R. X. Li, H. Zhen Chen, H. Li, and M. X. Chen, “Facile preparation of patterned phosphor-in-glass with excellent luminous properties through screen-printing for high-power white light-emitting diodes,” J. Alloys Compd. 693, 279–284 (2017).
[Crossref]

Li, R. X.

Y. Peng, R. X. Li, H. Zhen Chen, H. Li, and M. X. Chen, “Facile preparation of patterned phosphor-in-glass with excellent luminous properties through screen-printing for high-power white light-emitting diodes,” J. Alloys Compd. 693, 279–284 (2017).
[Crossref]

Liu, L. H.

J. M. Zhao and L. H. Liu, “Discontinuous spectral element method for solving radiative heat transfer in multidimensional semitransparent media,” J. Quant. Spectrosc. 107(1), 1–16 (2007).
[Crossref]

J. M. Zhao and L. H. Liu, “Spectral element approach for coupled radiative and conductive heat transfer in semitransparent medium,” J. Heat Transfer 129(10), 1417–1424 (2007).
[Crossref]

J. M. Zhao and L. H. Liu, “Least-squares spectral element method for radiative heat transfer in semitransparent media,” Numer. Heat Transf. B 50(5), 473–489 (2006).
[Crossref]

L. H. Liu, “Finite element simulation of radiative heat transfer in absorbing and scattering media,” J. Thermophys. Heat Transfer 18(4), 555–557 (2004).
[Crossref]

Liu, S.

X. B. Luo, R. Hu, S. Liu, and K. Wang, “Heat and fluid flow in high-power LED packaging and applications,” Pror. Energy Combust. Sci. 56, 1–32 (2016).
[Crossref]

R. Hu, B. Cao, Y. Zou, Y. M. Zhu, S. Liu, and X. B. Luo, “Modeling the light extraction efficiency of bi-layer phosphor in white LEDs,” IEEE Photonics Technol. Lett. 25(12), 1141–1144 (2013).
[Crossref]

R. Hu, Y. M. Wang, Y. Zou, X. Chen, S. Liu, and X. B. Luo, “Study on phosphor sedimentation effect in white LED packages by modeling multi-layer phosphors with the modified Kubelka-Munk theory,” J. Appl. Phys. 113(6), 063108 (2013).
[Crossref]

Z. Liu, S. Liu, K. Wang, and X. Luo, “Measurement and numerical studies of optical properties of YAG:Ce phosphor for white light-emitting diode packaging,” Appl. Opt. 49(2), 247–257 (2010).
[Crossref] [PubMed]

Z. Liu, K. Wang, X. Luo, and S. Liu, “Precise optical modeling of blue light-emitting diodes by Monte Carlo ray-tracing,” Opt. Express 18(9), 9398–9412 (2010).
[Crossref] [PubMed]

Liu, Z.

Luo, H.

E. F. Schubert, J. K. Kim, H. Luo, and J. Q. Xi, “Solid state lighting–a benevolent technology,” Rep. Prog. Phys. 69(12), 3069–3099 (2006).
[Crossref]

Luo, X.

Luo, X. B.

Y. P. Ma, W. Lan, B. Xie, R. Hu, and X. B. Luo, “An optical-thermal model for laser-excited remote phosphor with thermal quenching,” Int. J. Heat Mass Transf. 116, 694–702 (2018).
[Crossref]

Y. P. Ma, R. Hu, X. J. Yu, W. C. Shu, and X. B. Luo, “A modified bidirectional thermal resistance model for junction and phosphor temperature estimation in phosphor-converted light-emitting diodes,” Int. J. Heat Mass Transf. 106, 1–6 (2017).
[Crossref]

X. B. Luo, R. Hu, S. Liu, and K. Wang, “Heat and fluid flow in high-power LED packaging and applications,” Pror. Energy Combust. Sci. 56, 1–32 (2016).
[Crossref]

X. B. Luo and R. Hu, “Calculation of the phosphor heat generation in phosphor-converted light-emitting diodes,” Int. J. Heat Mass Transf. 75, 213–217 (2014).
[Crossref]

R. Hu, B. Cao, Y. Zou, Y. M. Zhu, S. Liu, and X. B. Luo, “Modeling the light extraction efficiency of bi-layer phosphor in white LEDs,” IEEE Photonics Technol. Lett. 25(12), 1141–1144 (2013).
[Crossref]

R. Hu, Y. M. Wang, Y. Zou, X. Chen, S. Liu, and X. B. Luo, “Study on phosphor sedimentation effect in white LED packages by modeling multi-layer phosphors with the modified Kubelka-Munk theory,” J. Appl. Phys. 113(6), 063108 (2013).
[Crossref]

R. Hu, H. Zheng, J. Y. Hu, and X. B. Luo, “Comprehensive study on the transmitted and reflected light through the phosphor layer in light-emitting diode packages,” J. Disp. Technol. 9(6), 447–452 (2013).
[Crossref]

R. Hu and X. B. Luo, “A model for calculating the bidirectional scattering properties of phosphor layer in white light-emitting diodes,” J. Lightwave Technol. 30(21), 3376–3380 (2012).
[Crossref]

Ma, Y. P.

Y. P. Ma, W. Lan, B. Xie, R. Hu, and X. B. Luo, “An optical-thermal model for laser-excited remote phosphor with thermal quenching,” Int. J. Heat Mass Transf. 116, 694–702 (2018).
[Crossref]

Y. P. Ma, R. Hu, X. J. Yu, W. C. Shu, and X. B. Luo, “A modified bidirectional thermal resistance model for junction and phosphor temperature estimation in phosphor-converted light-emitting diodes,” Int. J. Heat Mass Transf. 106, 1–6 (2017).
[Crossref]

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S. McCamy, “Correlated color temperature as an explicit function of chromaticity coordinates,” Color Res. Appl. 17(2), 142–144 (1992).
[Crossref]

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A. Lenef, J. Kelso, M. Tchoul, O. Mehl, J. Sorg, and Y. Zheng, “Laser-activated remote phosphor conversion with ceramic phosphors,” Proc. SPIE 9190, 919000C (2014).

Meuret, Y.

A. Correia, P. Hanselaer, and Y. Meuret, “An efficient optothermal simulation framework for optimization of high-luminance white light sources,” IEEE Photonics J. 8(4), 1–15 (2016).
[Crossref]

Nicolaï, B. M.

Niklasson, G. A.

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Peng, Y.

Y. Peng, R. X. Li, H. Zhen Chen, H. Li, and M. X. Chen, “Facile preparation of patterned phosphor-in-glass with excellent luminous properties through screen-printing for high-power white light-emitting diodes,” J. Alloys Compd. 693, 279–284 (2017).
[Crossref]

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J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. van 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]

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Piquette, A.

Posthumus, P.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. van 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]

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E. F. Schubert, J. K. Kim, H. Luo, and J. Q. Xi, “Solid state lighting–a benevolent technology,” Rep. Prog. Phys. 69(12), 3069–3099 (2006).
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Shakespeare, J.

T. Shakespeare and J. Shakespeare, “A fluorescent extension to the Kubelka–Munk model,” Color Res. Appl. 28(1), 4–14 (2003).
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Shakespeare, T.

T. Shakespeare and J. Shakespeare, “A fluorescent extension to the Kubelka–Munk model,” Color Res. Appl. 28(1), 4–14 (2003).
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Y. P. Ma, R. Hu, X. J. Yu, W. C. Shu, and X. B. Luo, “A modified bidirectional thermal resistance model for junction and phosphor temperature estimation in phosphor-converted light-emitting diodes,” Int. J. Heat Mass Transf. 106, 1–6 (2017).
[Crossref]

Song, S. B.

Sorg, J.

A. Lenef, J. Kelso, M. Tchoul, O. Mehl, J. Sorg, and Y. Zheng, “Laser-activated remote phosphor conversion with ceramic phosphors,” Proc. SPIE 9190, 919000C (2014).

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. van 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]

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J. Sun, H. L. Yi, and H. P. Tan, “Local RBF meshless scheme for coupled radiative and conductive heat transfer,” Numer. Heat Transf. A 69(12), 1390–1404 (2016).
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Tan, H. P.

J. Sun, H. L. Yi, and H. P. Tan, “Local RBF meshless scheme for coupled radiative and conductive heat transfer,” Numer. Heat Transf. A 69(12), 1390–1404 (2016).
[Crossref]

Tarvainen, T. J.

T. J. Tarvainen, V. Kolehmainen, S. R. Arridge, and J. P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport–diffusion model,” J. Quant. Spectrosc. 112(16), 2600–2608 (2011).
[Crossref]

Tchoul, M.

A. Lenef, J. Kelso, M. Tchoul, O. Mehl, J. Sorg, and Y. Zheng, “Laser-activated remote phosphor conversion with ceramic phosphors,” Proc. SPIE 9190, 919000C (2014).

A. Lenef, J. Kelso, Y. Zheng, and M. Tchoul, “Radiance limits of ceramic phosphors under high excitation fluxes,” Proc. SPIE 8841, 884107 (2013).

Thennadil, S. N.

Tien, C. H.

van Gemert, M. J.

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

R. Hu, Y. M. Wang, Y. Zou, X. Chen, S. Liu, and X. B. Luo, “Study on phosphor sedimentation effect in white LED packages by modeling multi-layer phosphors with the modified Kubelka-Munk theory,” J. Appl. Phys. 113(6), 063108 (2013).
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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).
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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]

Xi, J. Q.

E. F. Schubert, J. K. Kim, H. Luo, and J. Q. Xi, “Solid state lighting–a benevolent technology,” Rep. Prog. Phys. 69(12), 3069–3099 (2006).
[Crossref]

Xie, B.

Y. P. Ma, W. Lan, B. Xie, R. Hu, and X. B. Luo, “An optical-thermal model for laser-excited remote phosphor with thermal quenching,” Int. J. Heat Mass Transf. 116, 694–702 (2018).
[Crossref]

Yi, H. L.

J. Sun, H. L. Yi, and H. P. Tan, “Local RBF meshless scheme for coupled radiative and conductive heat transfer,” Numer. Heat Transf. A 69(12), 1390–1404 (2016).
[Crossref]

Yu, X. J.

Y. P. Ma, R. Hu, X. J. Yu, W. C. Shu, and X. B. Luo, “A modified bidirectional thermal resistance model for junction and phosphor temperature estimation in phosphor-converted light-emitting diodes,” Int. J. Heat Mass Transf. 106, 1–6 (2017).
[Crossref]

Yudovsky, D.

Yun, C.

Zhao, J. M.

J. M. Zhao and L. H. Liu, “Spectral element approach for coupled radiative and conductive heat transfer in semitransparent medium,” J. Heat Transfer 129(10), 1417–1424 (2007).
[Crossref]

J. M. Zhao and L. H. Liu, “Discontinuous spectral element method for solving radiative heat transfer in multidimensional semitransparent media,” J. Quant. Spectrosc. 107(1), 1–16 (2007).
[Crossref]

J. M. Zhao and L. H. Liu, “Least-squares spectral element method for radiative heat transfer in semitransparent media,” Numer. Heat Transf. B 50(5), 473–489 (2006).
[Crossref]

Zhen Chen, H.

Y. Peng, R. X. Li, H. Zhen Chen, H. Li, and M. X. Chen, “Facile preparation of patterned phosphor-in-glass with excellent luminous properties through screen-printing for high-power white light-emitting diodes,” J. Alloys Compd. 693, 279–284 (2017).
[Crossref]

Zheng, H.

R. Hu, H. Zheng, J. Y. Hu, and X. B. Luo, “Comprehensive study on the transmitted and reflected light through the phosphor layer in light-emitting diode packages,” J. Disp. Technol. 9(6), 447–452 (2013).
[Crossref]

Zheng, Y.

A. Lenef, J. Kelso, M. Tchoul, O. Mehl, J. Sorg, and Y. Zheng, “Laser-activated remote phosphor conversion with ceramic phosphors,” Proc. SPIE 9190, 919000C (2014).

A. Lenef, J. Kelso, Y. Zheng, and M. Tchoul, “Radiance limits of ceramic phosphors under high excitation fluxes,” Proc. SPIE 8841, 884107 (2013).

Zhu, Y. M.

R. Hu, B. Cao, Y. Zou, Y. M. Zhu, S. Liu, and X. B. Luo, “Modeling the light extraction efficiency of bi-layer phosphor in white LEDs,” IEEE Photonics Technol. Lett. 25(12), 1141–1144 (2013).
[Crossref]

Zou, Y.

R. Hu, Y. M. Wang, Y. Zou, X. Chen, S. Liu, and X. B. Luo, “Study on phosphor sedimentation effect in white LED packages by modeling multi-layer phosphors with the modified Kubelka-Munk theory,” J. Appl. Phys. 113(6), 063108 (2013).
[Crossref]

R. Hu, B. Cao, Y. Zou, Y. M. Zhu, S. Liu, and X. B. Luo, “Modeling the light extraction efficiency of bi-layer phosphor in white LEDs,” IEEE Photonics Technol. Lett. 25(12), 1141–1144 (2013).
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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).
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IEEE Photonics J. (1)

A. Correia, P. Hanselaer, and Y. Meuret, “An efficient optothermal simulation framework for optimization of high-luminance white light sources,” IEEE Photonics J. 8(4), 1–15 (2016).
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R. Hu, B. Cao, Y. Zou, Y. M. Zhu, S. Liu, and X. B. Luo, “Modeling the light extraction efficiency of bi-layer phosphor in white LEDs,” IEEE Photonics Technol. Lett. 25(12), 1141–1144 (2013).
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Y. P. Ma, W. Lan, B. Xie, R. Hu, and X. B. Luo, “An optical-thermal model for laser-excited remote phosphor with thermal quenching,” Int. J. Heat Mass Transf. 116, 694–702 (2018).
[Crossref]

Y. P. Ma, R. Hu, X. J. Yu, W. C. Shu, and X. B. Luo, “A modified bidirectional thermal resistance model for junction and phosphor temperature estimation in phosphor-converted light-emitting diodes,” Int. J. Heat Mass Transf. 106, 1–6 (2017).
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X. B. Luo and R. Hu, “Calculation of the phosphor heat generation in phosphor-converted light-emitting diodes,” Int. J. Heat Mass Transf. 75, 213–217 (2014).
[Crossref]

J. Alloys Compd. (1)

Y. Peng, R. X. Li, H. Zhen Chen, H. Li, and M. X. Chen, “Facile preparation of patterned phosphor-in-glass with excellent luminous properties through screen-printing for high-power white light-emitting diodes,” J. Alloys Compd. 693, 279–284 (2017).
[Crossref]

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R. Hu, Y. M. Wang, Y. Zou, X. Chen, S. Liu, and X. B. Luo, “Study on phosphor sedimentation effect in white LED packages by modeling multi-layer phosphors with the modified Kubelka-Munk theory,” J. Appl. Phys. 113(6), 063108 (2013).
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R. Hu, H. Zheng, J. Y. Hu, and X. B. Luo, “Comprehensive study on the transmitted and reflected light through the phosphor layer in light-emitting diode packages,” J. Disp. Technol. 9(6), 447–452 (2013).
[Crossref]

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J. M. Zhao and L. H. Liu, “Spectral element approach for coupled radiative and conductive heat transfer in semitransparent medium,” J. Heat Transfer 129(10), 1417–1424 (2007).
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J. Quant. Spectrosc. (2)

J. M. Zhao and L. H. Liu, “Discontinuous spectral element method for solving radiative heat transfer in multidimensional semitransparent media,” J. Quant. Spectrosc. 107(1), 1–16 (2007).
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T. J. Tarvainen, V. Kolehmainen, S. R. Arridge, and J. P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport–diffusion model,” J. Quant. Spectrosc. 112(16), 2600–2608 (2011).
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J. M. Zhao and L. H. Liu, “Least-squares spectral element method for radiative heat transfer in semitransparent media,” Numer. Heat Transf. B 50(5), 473–489 (2006).
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J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42(11), 2255–2261 (1997).
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Proc. SPIE (2)

A. Lenef, J. Kelso, M. Tchoul, O. Mehl, J. Sorg, and Y. Zheng, “Laser-activated remote phosphor conversion with ceramic phosphors,” Proc. SPIE 9190, 919000C (2014).

A. Lenef, J. Kelso, Y. Zheng, and M. Tchoul, “Radiance limits of ceramic phosphors under high excitation fluxes,” Proc. SPIE 8841, 884107 (2013).

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X. B. Luo, R. Hu, S. Liu, and K. Wang, “Heat and fluid flow in high-power LED packaging and applications,” Pror. Energy Combust. Sci. 56, 1–32 (2016).
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E. F. Schubert, J. K. Kim, H. Luo, and J. Q. Xi, “Solid state lighting–a benevolent technology,” Rep. Prog. Phys. 69(12), 3069–3099 (2006).
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Figures (8)

Fig. 1
Fig. 1 Left: the schematic of one-dimensional phosphor model and boundary conditions. Conditions 1, 2 and 3 denote the incident LED Lambertian intensity, diffuse reflection at the substrate/reflector surface, and the Fresnel reflection at the phosphor-air interface, respectively. Right: the schematic of anisotropic scattered blue and yellow light and isotropic converted yellow light.
Fig. 2
Fig. 2 The flowchart of solving the fluorescent radiative transfer equations using the spectral element method.
Fig. 3
Fig. 3 Schematic of the experimental setup for angular CCT distribution and the inset shows the remote pc-LED with planar phosphor plate.
Fig. 4
Fig. 4 (a) Calculated absorption coefficient and (b) scattering coefficient for both blue and yellow lights versus phosphor concentration using Mie–Lorenz theory.
Fig. 5
Fig. 5 Upper: The effect of (a) Nsol, (b) Nθ, and (c) Nφ on the computing time. Bottom: The effect of (d) Nsol, (e) Nθ, and (f) Nφ on the radiant flux at z = 0 and z = d for both blue and yellow lights. The reference values for those three numbers are Nsol = 20, Nθ = 20, and Nφ = 20, respectively.
Fig. 6
Fig. 6 Angular intensity distribution under varying invasion depth for (a) blue and (b) yellow light, and (c) angular specular reflectivity distribution at the phosphor-air interface, (d) the output angular intensity of blue and yellow light. The phosphor thickness and concentration used in this case are 0.6 mm and 0.15 g/cm3, respectively.
Fig. 7
Fig. 7 (a) The calculated radiant flux for blue and yellow lights and (b) the normalized phosphor heat generation density versus the normalized invasion depth.
Fig. 8
Fig. 8 (a) Comparison of light extraction efficiency versus phosphor concentration between the experiment and model under thickness of 0.60 mm and 1.0 mm; (b) comparison of the normalized phosphor heating power between KM and our model; comparison of angular CCT distribution between experiment and model under thickness of 0.6 mm and concentration of (c) 0.10 g/cm3 and (d) 0.15 g/ cm3.

Equations (18)

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

Ω I B ( R,Ω )=( κ a B + κ s B ) I B ( R,Ω )+ κ s B 4π 4π I B ( R,Ω ) p( Ω, Ω ' )d Ω '
Ω I Y ( R,Ω )=( κ a Y + κ s Y ) I Y ( R,Ω )+ κ s Y 4π 4π I Y ( R,Ω ) p( Ω, Ω ' )d Ω ' + η con 4π κ a B 4π I B ( R,Ω ) dΩ
I B,LED ( 0,Ω )= P in cosθ/π ,Ω n w >0
I i,sub ( 0,Ω )= γ i π Ω ' n w <0 I sub ( 0, Ω ' )| Ω ' n w |d Ω ' ,Ω n w >0,i=B,Y
I i,phair ( d,Ω )=R( Ω ' n ) I i,phair ( d, Ω ' ),Ωn>0,i=B,Y
R( cos θ ' )={ 1 2 [ sin 2 ( θ ' θ t ) sin 2 ( θ ' + θ t ) + tan 2 ( θ ' θ t ) tan 2 ( θ ' + θ t ) ]for θ ' θ c 1for θ ' > θ c
I B ( 0,Ω )= I B,LED ( 0,θ )+ I B,sub ( 0,Ω ), I B ( d,Ω )= I B,phair ( d,Ω )
I Y ( 0,Ω )= I Y,sub ( 0,Ω ), I Y ( d,Ω )= I Y,phair ( d,Ω )
μ m d I B m dz =( κ a B + κ s B ) I B m + κ s B 4π m'=1 M I B m' p m',m ω m'
μ m d I Y m dz =( κ a Y + κ s Y ) I Y m + κ s Y 4π m'=1 M I Y m' p m',m ω m' + η con 4π κ a B m=1 M I B m ω m
I g ( z )= j=1 N sol I j ξ j ( z )
Φ i ( z )= 4π I i ( z,Ω )dΩ = m=1 M I i ( z,Ω ) ω m ,i=B,Y.
I i ( d, θ t ,φ )=[ 1R( cosθ ) ] I i ( d,θ,φ ),θ< θ c ,i=B,Y.
P out,i = m=1 M ' I i ( d, θ t m ) ϖ m ,i=B,Y
LEE= P out / P in = ( P out,B + P out,Y )/ P in .
Q ph = P in ( P out,B + P out,Y ).
q( z )= κ a B ( 1 η con ) Φ B ( z )+ κ a Y Φ Y ( z ).
p( cosΘ )= 1 g 2 ( 1+ g 2 2gcosΘ ) 3/2

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