Abstract

This study exhibited the correlated color temperature (CCT)- and color-rendering index (CRI)-tuning behavior of light emission from white light-emitting diodes (WLEDs) using three-dimensional non-close-packed (3D NCP) colloidal photonic crystals (CPhCs). The CCT of approximately 5300 K (characteristic of cold WLEDs) of white light propagated through the NCP CPhCs dropped to 3000 K (characteristic of warm WLEDs) because of the photonic stop-bands based on the photonic band structures of NCP CPhCs. This study successfully developed a novel technique that introduces lower-cost CCT- and CRI-tuning cold WLEDs with a CRI of over 90 that of warm WLEDs by using 3D NCP CPhCs.

© 2013 OSA

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2012 (3)

J. Kwak, W. K. Bae, D. Lee, I. Park, J. Lim, M. Park, H. Cho, H. Woo, Y. Yoon, K. Char, S. Lee, C. Lee, “Bright and efficient full-color colloidal quantum dot light-emitting diodes using an inverted device structure,” Nano Lett. 12(5), 2362–2366 (2012).
[CrossRef] [PubMed]

Y. Zhang, L. Wu, M. Ji, B. Wang, Y. Kong, J. Xu, “Structure and photoluminescence properties of KSr4(BO3)3:Eu3+ red-emitting phosphor,” Opt. Mater. Express 2(1), 92–102 (2012).
[CrossRef]

J. H. Oh, S. J. Yang, Y. G. Sung, Y. R. Do, “Excellent color rendering indexes of multi-package white LEDs,” Opt. Express 20(18), 20276–20285 (2012).
[CrossRef] [PubMed]

2010 (1)

2009 (2)

C. F. Lai, J. Y. Chi, H. C. Kuo, H. H. Yen, C. E. Lee, C. H. Chao, H. T. Hsueh, W. Y. Yeh, “Far-field of GaN film-transferred green light-emitting diodes with two-dimensional photonic crystals,” Opt. Express 17(11), 8795–8804 (2009).
[CrossRef] [PubMed]

C. F. Lai, J. Y. Chi, H. C. Kuo, H. H. Yen, C. E. Lee, C. H. Chao, W. Y. Yeh, T. C. Lu, “Far-Field and Near-Field Distribution of GaN-Based Photonic Crystal LEDs With Guided Mode Extraction,” IEEE J. Sel. Top. Quantum Electron. 15(4), 1234–1241 (2009).
[CrossRef]

2008 (1)

S. G. Romanov, M. Bardosova, I. M. Povey, M. E. Pemble, C. M. S. Torres, “Optical photonic crystals fabricated from colloidal systems,” Appl. Phys. Lett. 92, 191106 (2008).
[CrossRef]

2007 (1)

S. Nizamoglu, T. Ozel, E. Sari, H. V. Demir, “White light generation using CdSe/ZnS core–shell nanocrystals hybridized with InGaN/GaN light emitting diodes,” Nanotechnology 18(6), 065709 (2007).
[CrossRef]

2006 (1)

J. K. Park, K. J. Choi, J. H. Yeon, S. J. Lee, C. K. Kim, “Embodiment of the warm white-light-emitting diodes by using a Ba2+ codoped Sr3SiO5:Eu phosphor,” Appl. Phys. Lett. 88(4), 043511 (2006).
[CrossRef]

2005 (2)

E. Pavarini, L. C. Andreani, C. Soci, M. Galli, F. Marabelli, D. Comoretto, “Band structure and optical properties of opal photonic crystals,” Phys. Rev. B 72(4), 045102 (2005).
[CrossRef]

E. F. Schubert, J. K. Kim, “Solid-state light sources getting smart,” Science 308(5726), 1274–1278 (2005).
[CrossRef] [PubMed]

2003 (1)

R. Fenollosa, F. Meseguer, “Non-close-packed artificial opals,” Adv. Mater. 15(15), 1282–1285 (2003).
[CrossRef]

2002 (2)

F. López-Tejeira, T. Ochiai, K. Sakoda, J. Sánchez-Dehesa, “Symmetry characterization of eigenstates in opal-based photonic crystals,” Phys. Rev. B 65(19), 195110 (2002).
[CrossRef]

D. A. Steigerwald, J. C. Bhat, D. Collins, R. M. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8(2), 310–320 (2002).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

Andreani, L. C.

E. Pavarini, L. C. Andreani, C. Soci, M. Galli, F. Marabelli, D. Comoretto, “Band structure and optical properties of opal photonic crystals,” Phys. Rev. B 72(4), 045102 (2005).
[CrossRef]

Bae, W. K.

J. Kwak, W. K. Bae, D. Lee, I. Park, J. Lim, M. Park, H. Cho, H. Woo, Y. Yoon, K. Char, S. Lee, C. Lee, “Bright and efficient full-color colloidal quantum dot light-emitting diodes using an inverted device structure,” Nano Lett. 12(5), 2362–2366 (2012).
[CrossRef] [PubMed]

Bardosova, M.

S. G. Romanov, M. Bardosova, I. M. Povey, M. E. Pemble, C. M. S. Torres, “Optical photonic crystals fabricated from colloidal systems,” Appl. Phys. Lett. 92, 191106 (2008).
[CrossRef]

Bhat, J. C.

D. A. Steigerwald, J. C. Bhat, D. Collins, R. M. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8(2), 310–320 (2002).
[CrossRef]

Busch, K.

Chao, C. H.

C. F. Lai, J. Y. Chi, H. C. Kuo, H. H. Yen, C. E. Lee, C. H. Chao, H. T. Hsueh, W. Y. Yeh, “Far-field of GaN film-transferred green light-emitting diodes with two-dimensional photonic crystals,” Opt. Express 17(11), 8795–8804 (2009).
[CrossRef] [PubMed]

C. F. Lai, J. Y. Chi, H. C. Kuo, H. H. Yen, C. E. Lee, C. H. Chao, W. Y. Yeh, T. C. Lu, “Far-Field and Near-Field Distribution of GaN-Based Photonic Crystal LEDs With Guided Mode Extraction,” IEEE J. Sel. Top. Quantum Electron. 15(4), 1234–1241 (2009).
[CrossRef]

Char, K.

J. Kwak, W. K. Bae, D. Lee, I. Park, J. Lim, M. Park, H. Cho, H. Woo, Y. Yoon, K. Char, S. Lee, C. Lee, “Bright and efficient full-color colloidal quantum dot light-emitting diodes using an inverted device structure,” Nano Lett. 12(5), 2362–2366 (2012).
[CrossRef] [PubMed]

Chi, J. Y.

C. F. Lai, J. Y. Chi, H. C. Kuo, H. H. Yen, C. E. Lee, C. H. Chao, W. Y. Yeh, T. C. Lu, “Far-Field and Near-Field Distribution of GaN-Based Photonic Crystal LEDs With Guided Mode Extraction,” IEEE J. Sel. Top. Quantum Electron. 15(4), 1234–1241 (2009).
[CrossRef]

C. F. Lai, J. Y. Chi, H. C. Kuo, H. H. Yen, C. E. Lee, C. H. Chao, H. T. Hsueh, W. Y. Yeh, “Far-field of GaN film-transferred green light-emitting diodes with two-dimensional photonic crystals,” Opt. Express 17(11), 8795–8804 (2009).
[CrossRef] [PubMed]

Cho, H.

J. Kwak, W. K. Bae, D. Lee, I. Park, J. Lim, M. Park, H. Cho, H. Woo, Y. Yoon, K. Char, S. Lee, C. Lee, “Bright and efficient full-color colloidal quantum dot light-emitting diodes using an inverted device structure,” Nano Lett. 12(5), 2362–2366 (2012).
[CrossRef] [PubMed]

Choi, K. J.

J. K. Park, K. J. Choi, J. H. Yeon, S. J. Lee, C. K. Kim, “Embodiment of the warm white-light-emitting diodes by using a Ba2+ codoped Sr3SiO5:Eu phosphor,” Appl. Phys. Lett. 88(4), 043511 (2006).
[CrossRef]

Collins, D.

D. A. Steigerwald, J. C. Bhat, D. Collins, R. M. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8(2), 310–320 (2002).
[CrossRef]

Comoretto, D.

E. Pavarini, L. C. Andreani, C. Soci, M. Galli, F. Marabelli, D. Comoretto, “Band structure and optical properties of opal photonic crystals,” Phys. Rev. B 72(4), 045102 (2005).
[CrossRef]

Demir, H. V.

S. Nizamoglu, T. Ozel, E. Sari, H. V. Demir, “White light generation using CdSe/ZnS core–shell nanocrystals hybridized with InGaN/GaN light emitting diodes,” Nanotechnology 18(6), 065709 (2007).
[CrossRef]

Do, Y. R.

Doosje, M.

Fenollosa, R.

R. Fenollosa, F. Meseguer, “Non-close-packed artificial opals,” Adv. Mater. 15(15), 1282–1285 (2003).
[CrossRef]

Fletcher, R. M.

D. A. Steigerwald, J. C. Bhat, D. Collins, R. M. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8(2), 310–320 (2002).
[CrossRef]

Galli, M.

E. Pavarini, L. C. Andreani, C. Soci, M. Galli, F. Marabelli, D. Comoretto, “Band structure and optical properties of opal photonic crystals,” Phys. Rev. B 72(4), 045102 (2005).
[CrossRef]

Hoenders, B. J.

Holcomb, M. O.

D. A. Steigerwald, J. C. Bhat, D. Collins, R. M. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8(2), 310–320 (2002).
[CrossRef]

Hsueh, H. T.

Ji, M.

Joannopoulos, J. D.

John, S.

Johnson, S. G.

Kim, C. K.

J. K. Park, K. J. Choi, J. H. Yeon, S. J. Lee, C. K. Kim, “Embodiment of the warm white-light-emitting diodes by using a Ba2+ codoped Sr3SiO5:Eu phosphor,” Appl. Phys. Lett. 88(4), 043511 (2006).
[CrossRef]

Kim, J. K.

E. F. Schubert, J. K. Kim, “Solid-state light sources getting smart,” Science 308(5726), 1274–1278 (2005).
[CrossRef] [PubMed]

Knoester, J.

Kong, Y.

Kuo, H. C.

C. F. Lai, J. Y. Chi, H. C. Kuo, H. H. Yen, C. E. Lee, C. H. Chao, H. T. Hsueh, W. Y. Yeh, “Far-field of GaN film-transferred green light-emitting diodes with two-dimensional photonic crystals,” Opt. Express 17(11), 8795–8804 (2009).
[CrossRef] [PubMed]

C. F. Lai, J. Y. Chi, H. C. Kuo, H. H. Yen, C. E. Lee, C. H. Chao, W. Y. Yeh, T. C. Lu, “Far-Field and Near-Field Distribution of GaN-Based Photonic Crystal LEDs With Guided Mode Extraction,” IEEE J. Sel. Top. Quantum Electron. 15(4), 1234–1241 (2009).
[CrossRef]

Kwak, J.

J. Kwak, W. K. Bae, D. Lee, I. Park, J. Lim, M. Park, H. Cho, H. Woo, Y. Yoon, K. Char, S. Lee, C. Lee, “Bright and efficient full-color colloidal quantum dot light-emitting diodes using an inverted device structure,” Nano Lett. 12(5), 2362–2366 (2012).
[CrossRef] [PubMed]

Lai, C. F.

C. F. Lai, J. Y. Chi, H. C. Kuo, H. H. Yen, C. E. Lee, C. H. Chao, W. Y. Yeh, T. C. Lu, “Far-Field and Near-Field Distribution of GaN-Based Photonic Crystal LEDs With Guided Mode Extraction,” IEEE J. Sel. Top. Quantum Electron. 15(4), 1234–1241 (2009).
[CrossRef]

C. F. Lai, J. Y. Chi, H. C. Kuo, H. H. Yen, C. E. Lee, C. H. Chao, H. T. Hsueh, W. Y. Yeh, “Far-field of GaN film-transferred green light-emitting diodes with two-dimensional photonic crystals,” Opt. Express 17(11), 8795–8804 (2009).
[CrossRef] [PubMed]

Lee, C.

J. Kwak, W. K. Bae, D. Lee, I. Park, J. Lim, M. Park, H. Cho, H. Woo, Y. Yoon, K. Char, S. Lee, C. Lee, “Bright and efficient full-color colloidal quantum dot light-emitting diodes using an inverted device structure,” Nano Lett. 12(5), 2362–2366 (2012).
[CrossRef] [PubMed]

Lee, C. E.

C. F. Lai, J. Y. Chi, H. C. Kuo, H. H. Yen, C. E. Lee, C. H. Chao, H. T. Hsueh, W. Y. Yeh, “Far-field of GaN film-transferred green light-emitting diodes with two-dimensional photonic crystals,” Opt. Express 17(11), 8795–8804 (2009).
[CrossRef] [PubMed]

C. F. Lai, J. Y. Chi, H. C. Kuo, H. H. Yen, C. E. Lee, C. H. Chao, W. Y. Yeh, T. C. Lu, “Far-Field and Near-Field Distribution of GaN-Based Photonic Crystal LEDs With Guided Mode Extraction,” IEEE J. Sel. Top. Quantum Electron. 15(4), 1234–1241 (2009).
[CrossRef]

Lee, D.

J. Kwak, W. K. Bae, D. Lee, I. Park, J. Lim, M. Park, H. Cho, H. Woo, Y. Yoon, K. Char, S. Lee, C. Lee, “Bright and efficient full-color colloidal quantum dot light-emitting diodes using an inverted device structure,” Nano Lett. 12(5), 2362–2366 (2012).
[CrossRef] [PubMed]

Lee, S.

J. Kwak, W. K. Bae, D. Lee, I. Park, J. Lim, M. Park, H. Cho, H. Woo, Y. Yoon, K. Char, S. Lee, C. Lee, “Bright and efficient full-color colloidal quantum dot light-emitting diodes using an inverted device structure,” Nano Lett. 12(5), 2362–2366 (2012).
[CrossRef] [PubMed]

Lee, S. J.

J. K. Park, K. J. Choi, J. H. Yeon, S. J. Lee, C. K. Kim, “Embodiment of the warm white-light-emitting diodes by using a Ba2+ codoped Sr3SiO5:Eu phosphor,” Appl. Phys. Lett. 88(4), 043511 (2006).
[CrossRef]

Lim, J.

J. Kwak, W. K. Bae, D. Lee, I. Park, J. Lim, M. Park, H. Cho, H. Woo, Y. Yoon, K. Char, S. Lee, C. Lee, “Bright and efficient full-color colloidal quantum dot light-emitting diodes using an inverted device structure,” Nano Lett. 12(5), 2362–2366 (2012).
[CrossRef] [PubMed]

López-Tejeira, F.

F. López-Tejeira, T. Ochiai, K. Sakoda, J. Sánchez-Dehesa, “Symmetry characterization of eigenstates in opal-based photonic crystals,” Phys. Rev. B 65(19), 195110 (2002).
[CrossRef]

Lu, T. C.

C. F. Lai, J. Y. Chi, H. C. Kuo, H. H. Yen, C. E. Lee, C. H. Chao, W. Y. Yeh, T. C. Lu, “Far-Field and Near-Field Distribution of GaN-Based Photonic Crystal LEDs With Guided Mode Extraction,” IEEE J. Sel. Top. Quantum Electron. 15(4), 1234–1241 (2009).
[CrossRef]

Ludowise, M. J.

D. A. Steigerwald, J. C. Bhat, D. Collins, R. M. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8(2), 310–320 (2002).
[CrossRef]

Marabelli, F.

E. Pavarini, L. C. Andreani, C. Soci, M. Galli, F. Marabelli, D. Comoretto, “Band structure and optical properties of opal photonic crystals,” Phys. Rev. B 72(4), 045102 (2005).
[CrossRef]

Martin, P. S.

D. A. Steigerwald, J. C. Bhat, D. Collins, R. M. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8(2), 310–320 (2002).
[CrossRef]

Meseguer, F.

R. Fenollosa, F. Meseguer, “Non-close-packed artificial opals,” Adv. Mater. 15(15), 1282–1285 (2003).
[CrossRef]

Nizamoglu, S.

S. Nizamoglu, T. Ozel, E. Sari, H. V. Demir, “White light generation using CdSe/ZnS core–shell nanocrystals hybridized with InGaN/GaN light emitting diodes,” Nanotechnology 18(6), 065709 (2007).
[CrossRef]

Ochiai, T.

F. López-Tejeira, T. Ochiai, K. Sakoda, J. Sánchez-Dehesa, “Symmetry characterization of eigenstates in opal-based photonic crystals,” Phys. Rev. B 65(19), 195110 (2002).
[CrossRef]

Oh, J. H.

Ozel, T.

S. Nizamoglu, T. Ozel, E. Sari, H. V. Demir, “White light generation using CdSe/ZnS core–shell nanocrystals hybridized with InGaN/GaN light emitting diodes,” Nanotechnology 18(6), 065709 (2007).
[CrossRef]

Park, I.

J. Kwak, W. K. Bae, D. Lee, I. Park, J. Lim, M. Park, H. Cho, H. Woo, Y. Yoon, K. Char, S. Lee, C. Lee, “Bright and efficient full-color colloidal quantum dot light-emitting diodes using an inverted device structure,” Nano Lett. 12(5), 2362–2366 (2012).
[CrossRef] [PubMed]

Park, J. K.

J. K. Park, K. J. Choi, J. H. Yeon, S. J. Lee, C. K. Kim, “Embodiment of the warm white-light-emitting diodes by using a Ba2+ codoped Sr3SiO5:Eu phosphor,” Appl. Phys. Lett. 88(4), 043511 (2006).
[CrossRef]

Park, M.

J. Kwak, W. K. Bae, D. Lee, I. Park, J. Lim, M. Park, H. Cho, H. Woo, Y. Yoon, K. Char, S. Lee, C. Lee, “Bright and efficient full-color colloidal quantum dot light-emitting diodes using an inverted device structure,” Nano Lett. 12(5), 2362–2366 (2012).
[CrossRef] [PubMed]

Pavarini, E.

E. Pavarini, L. C. Andreani, C. Soci, M. Galli, F. Marabelli, D. Comoretto, “Band structure and optical properties of opal photonic crystals,” Phys. Rev. B 72(4), 045102 (2005).
[CrossRef]

Pemble, M. E.

S. G. Romanov, M. Bardosova, I. M. Povey, M. E. Pemble, C. M. S. Torres, “Optical photonic crystals fabricated from colloidal systems,” Appl. Phys. Lett. 92, 191106 (2008).
[CrossRef]

Povey, I. M.

S. G. Romanov, M. Bardosova, I. M. Povey, M. E. Pemble, C. M. S. Torres, “Optical photonic crystals fabricated from colloidal systems,” Appl. Phys. Lett. 92, 191106 (2008).
[CrossRef]

Romanov, S. G.

S. G. Romanov, M. Bardosova, I. M. Povey, M. E. Pemble, C. M. S. Torres, “Optical photonic crystals fabricated from colloidal systems,” Appl. Phys. Lett. 92, 191106 (2008).
[CrossRef]

Rudaz, S. L.

D. A. Steigerwald, J. C. Bhat, D. Collins, R. M. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8(2), 310–320 (2002).
[CrossRef]

Sakoda, K.

F. López-Tejeira, T. Ochiai, K. Sakoda, J. Sánchez-Dehesa, “Symmetry characterization of eigenstates in opal-based photonic crystals,” Phys. Rev. B 65(19), 195110 (2002).
[CrossRef]

Sánchez-Dehesa, J.

F. López-Tejeira, T. Ochiai, K. Sakoda, J. Sánchez-Dehesa, “Symmetry characterization of eigenstates in opal-based photonic crystals,” Phys. Rev. B 65(19), 195110 (2002).
[CrossRef]

Sari, E.

S. Nizamoglu, T. Ozel, E. Sari, H. V. Demir, “White light generation using CdSe/ZnS core–shell nanocrystals hybridized with InGaN/GaN light emitting diodes,” Nanotechnology 18(6), 065709 (2007).
[CrossRef]

Schubert, E. F.

E. F. Schubert, J. K. Kim, “Solid-state light sources getting smart,” Science 308(5726), 1274–1278 (2005).
[CrossRef] [PubMed]

Shi, F. G.

Soci, C.

E. Pavarini, L. C. Andreani, C. Soci, M. Galli, F. Marabelli, D. Comoretto, “Band structure and optical properties of opal photonic crystals,” Phys. Rev. B 72(4), 045102 (2005).
[CrossRef]

Steigerwald, D. A.

D. A. Steigerwald, J. C. Bhat, D. Collins, R. M. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8(2), 310–320 (2002).
[CrossRef]

Sung, Y. G.

Torres, C. M. S.

S. G. Romanov, M. Bardosova, I. M. Povey, M. E. Pemble, C. M. S. Torres, “Optical photonic crystals fabricated from colloidal systems,” Appl. Phys. Lett. 92, 191106 (2008).
[CrossRef]

Tran, N. T.

Wang, B.

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Supplementary Material (3)

» Media 1: MOV (3694 KB)     
» Media 2: AVI (3608 KB)     
» Media 3: MOV (3694 KB)     

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic diagram of the 3D NCP CPhCs deposited onto the 5630 WLED package. The inset shows the OM image of c-WLEDs with 3D NCP CPhCs. (b) The fcc Brillouin zone with symmetry points. The z-axis is along the [111] direction.

Fig. 2
Fig. 2

FESEM images of the 3D NCP CPhCs prepared using PS nanospheres with D of (a) 225 nm, and (b) 250 nm. (c) The FESEM surface images of the NCP CPhCs showing that the nanospheres stack with a well-organized (111) plane and fcc structures. The first Brillouin zone with the symmetry points indicated. Inset: cross-section of the NCP CPhCs into domains along the (110) plane. (d) Schematic showing the NCP CPhC of the fcc structure viewed along the (100) plane.

Fig. 3
Fig. 3

(a) Current-dependent luminous flux and luminous efficiency of the four types of WLEDs. (b) Luminescence spectra of the four types of WLEDs. The WLEDs have an input power of approximately 0.384 W.

Fig. 4
Fig. 4

(a) and (c) Unpolarized angular-resolved transmission measurements of the c-WLEDs with NCP CPhC of a = 349 nm (Media 1) and 385 nm. (b) and (d) The photonic dispersion curves determined from the data shown in (a) and (c) and compared with the photonic band structures. Inset: The photonic band structure of the infinite NCP CPhCs of the fcc structure calculated using the PWE method. With respect to mirror symmetry, the continuous blue lines represent asymmetric bands, and the dashed red lines represent symmetric bands.

Tables (1)

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Table 1 The optical characteristics of the four types of WLEDs.

Equations (4)

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λ R ( D )=2 d 111 n eff 2 sin 2 θ 111
d 111 = a 3 = 2 3 ( D+x )
n eff 2 = n PS 2 f PS + n air 2 ( 1 f PS )
f PS = 4 V sphere V cubic = 4 2 3 π [ D 2( D+x ) ] 3

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