Abstract

The general CIE color rendering index (CRI) of light is an important index to evaluate the quality of illumination. However, because of the complexity in measurement of the rendering ability under designated constraints, an approach for general mathematical formulation and global optimization of the rendering ability of light emitting diode (LED) light mixtures is difficult to develop. This study is mainly devoted to developing mathematical formulation and a numerical method for the CRI optimization. The method is developed based on the so-called complex method [Computer J. 8, 42 (1965) ; G. V. Reklaitis et al., Engineering Optimization—Methods and Applications (Wiley, 1983) ] with modifications. It is first applicable to 3-color light mixtures and then extended to a hierarchical and iterative structure for higher-order light mixtures. The optimization is studied under the constraints of bounded relative intensities of the light mixture, designated correlated color temperature (CCT), and the required approximate white of the light mixture. The problems of inconsistent constraints and solutions are addressed. The CRI is a complicated function of the relative intensities of the compound illuminators of the mixture. The proposed method requires taking no derivatives of the function and is very adequate for the optimization. This is demonstrated by simulation for RGBW LED light mixtures. The results show that global and unique convergence to the optimal within required tolerances for CRI and spatial dispersivity is always achieved.

© 2010 Optical Society of America

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References

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  1. Y. Ohno, “Optical metrology for LEDs and solid state lighting,” Proc. SPIE 6046, 604625 (2006).
    [CrossRef]
  2. A. Zukauskas, F. Ivanauskas, R. Vaicekauskas, M. S. Shur, and R. Gaska,“Optimization of multichip white solid-state lighting source with four or more LEDs,” Proc. SPIE 4445, 148–155 (2001).
    [CrossRef]
  3. H. Ries, I. Leike, and J. Muschaweck, “Optimized additive mixing of colored light-emitting diode sources,” Opt. Eng. (Bellingham) 43, 1531–1536 (2004).
    [CrossRef]
  4. T-h. Hsu, “Optimization on color characteristics for LED lighting system,” M.S. thesis (National Central University, Taiwan, 2005).
  5. X. Guo and K. W. Houser, “A review of colour rendering indices and their application to commercial light sources,” Light. Res. Technol. 36, 183–197 (2004).
    [CrossRef]
  6. CIE, Method of Measuring and Specifying Colour Rendering Properties of Light Sources, Publication 13.3 (Commission Internationale de l’Eclairage, Vienna, 1995). ISBN 978-3900734572.
  7. W. Davis and Y. Ohno, “Toward an improved color rendering metric,” Proc. SPIE 5941, 59411G (2005).
    [CrossRef]
  8. S. Boissard and M. Fontoynont, “Optimization of LED-based lighting blendings for object presentation,” Color Res. Appl. 34, 310–320 (2009).
    [CrossRef]
  9. R. Mirhosseini, M. F. Schubert, S. Chhajed, J. Cho, J. K. Kim, and E. F. Schubert, “Improved color rendering and luminous efficacy in phosphor-converted white light emitting diodes by use of dual-blue emitting active regions,” Opt. Express 17, 10806–10823 (2009).
    [CrossRef] [PubMed]
  10. S. P. Ying, C.-W. Tang, and B.-J. Huang, “Characterizing LEDs for mixture of colored LED light sources,” International Conference on Electronic Materials and Packaging, EMAP 2006 (IEEE, 2006), pp. 1–5.
    [CrossRef]
  11. H. Wu, N. Narendran, Y. Gu, and A. Bierman, “Improving the performance of mixed-color white LED systems by using scattered photon extraction technique,” Proc. SPIE 6669, 666905 (2007).
    [CrossRef]
  12. M. O. Holcomb, R. Mueller-Mach, G. O. Mueller, D. Collins, R. M. Fletcher, D. A. Steigerwald, S. Eberle, Y. K. Lim, P. S. Martin, and M. Krames, “The LED lightbulb: are we there yet? Progress and challenges for solid state illumination,” Conference on Lasers and Electro-Optics Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper CMQ2.
  13. A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska, and M. S. Shur, “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80, 234–237 (2002).
    [CrossRef]
  14. M. Dyble, N. Narendran, A. Bierman, and T. Klein, “Impact of dimming white LEDs: chromaticity shifts due to different dimming methods,” Proc. SPIE 5941, 291–299 (2005).
  15. X. Qu, S. C. Wong, and C. K. Tse, “Color control system for RGB LED light sources using junction temperature measurement,” The 33rd Annual Conference of the IEEE Industrial Electronics Society, IECON 2007, Taipei, Taiwan (IEEE, 2007).
    [CrossRef]
  16. M. J. Box, “A new method of constrained optimization and a comparison with other methods,” Comput. J. 8, 42–45 (1965).
  17. G. V. Reklaitis, R. Ravindran, and K. M. Ragsdell, Engineering Optimization-Methods and Applications (Wiley, 1983).

2009 (2)

2007 (2)

H. Wu, N. Narendran, Y. Gu, and A. Bierman, “Improving the performance of mixed-color white LED systems by using scattered photon extraction technique,” Proc. SPIE 6669, 666905 (2007).
[CrossRef]

X. Qu, S. C. Wong, and C. K. Tse, “Color control system for RGB LED light sources using junction temperature measurement,” The 33rd Annual Conference of the IEEE Industrial Electronics Society, IECON 2007, Taipei, Taiwan (IEEE, 2007).
[CrossRef]

2006 (2)

S. P. Ying, C.-W. Tang, and B.-J. Huang, “Characterizing LEDs for mixture of colored LED light sources,” International Conference on Electronic Materials and Packaging, EMAP 2006 (IEEE, 2006), pp. 1–5.
[CrossRef]

Y. Ohno, “Optical metrology for LEDs and solid state lighting,” Proc. SPIE 6046, 604625 (2006).
[CrossRef]

2005 (3)

T-h. Hsu, “Optimization on color characteristics for LED lighting system,” M.S. thesis (National Central University, Taiwan, 2005).

W. Davis and Y. Ohno, “Toward an improved color rendering metric,” Proc. SPIE 5941, 59411G (2005).
[CrossRef]

M. Dyble, N. Narendran, A. Bierman, and T. Klein, “Impact of dimming white LEDs: chromaticity shifts due to different dimming methods,” Proc. SPIE 5941, 291–299 (2005).

2004 (2)

H. Ries, I. Leike, and J. Muschaweck, “Optimized additive mixing of colored light-emitting diode sources,” Opt. Eng. (Bellingham) 43, 1531–1536 (2004).
[CrossRef]

X. Guo and K. W. Houser, “A review of colour rendering indices and their application to commercial light sources,” Light. Res. Technol. 36, 183–197 (2004).
[CrossRef]

2003 (1)

M. O. Holcomb, R. Mueller-Mach, G. O. Mueller, D. Collins, R. M. Fletcher, D. A. Steigerwald, S. Eberle, Y. K. Lim, P. S. Martin, and M. Krames, “The LED lightbulb: are we there yet? Progress and challenges for solid state illumination,” Conference on Lasers and Electro-Optics Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper CMQ2.

2002 (1)

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska, and M. S. Shur, “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80, 234–237 (2002).
[CrossRef]

2001 (1)

A. Zukauskas, F. Ivanauskas, R. Vaicekauskas, M. S. Shur, and R. Gaska,“Optimization of multichip white solid-state lighting source with four or more LEDs,” Proc. SPIE 4445, 148–155 (2001).
[CrossRef]

1983 (1)

G. V. Reklaitis, R. Ravindran, and K. M. Ragsdell, Engineering Optimization-Methods and Applications (Wiley, 1983).

1965 (1)

M. J. Box, “A new method of constrained optimization and a comparison with other methods,” Comput. J. 8, 42–45 (1965).

Bierman, A.

H. Wu, N. Narendran, Y. Gu, and A. Bierman, “Improving the performance of mixed-color white LED systems by using scattered photon extraction technique,” Proc. SPIE 6669, 666905 (2007).
[CrossRef]

M. Dyble, N. Narendran, A. Bierman, and T. Klein, “Impact of dimming white LEDs: chromaticity shifts due to different dimming methods,” Proc. SPIE 5941, 291–299 (2005).

Boissard, S.

S. Boissard and M. Fontoynont, “Optimization of LED-based lighting blendings for object presentation,” Color Res. Appl. 34, 310–320 (2009).
[CrossRef]

Box, M. J.

M. J. Box, “A new method of constrained optimization and a comparison with other methods,” Comput. J. 8, 42–45 (1965).

Chhajed, S.

Cho, J.

Collins, D.

M. O. Holcomb, R. Mueller-Mach, G. O. Mueller, D. Collins, R. M. Fletcher, D. A. Steigerwald, S. Eberle, Y. K. Lim, P. S. Martin, and M. Krames, “The LED lightbulb: are we there yet? Progress and challenges for solid state illumination,” Conference on Lasers and Electro-Optics Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper CMQ2.

Davis, W.

W. Davis and Y. Ohno, “Toward an improved color rendering metric,” Proc. SPIE 5941, 59411G (2005).
[CrossRef]

Dyble, M.

M. Dyble, N. Narendran, A. Bierman, and T. Klein, “Impact of dimming white LEDs: chromaticity shifts due to different dimming methods,” Proc. SPIE 5941, 291–299 (2005).

Eberle, S.

M. O. Holcomb, R. Mueller-Mach, G. O. Mueller, D. Collins, R. M. Fletcher, D. A. Steigerwald, S. Eberle, Y. K. Lim, P. S. Martin, and M. Krames, “The LED lightbulb: are we there yet? Progress and challenges for solid state illumination,” Conference on Lasers and Electro-Optics Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper CMQ2.

Fletcher, R. M.

M. O. Holcomb, R. Mueller-Mach, G. O. Mueller, D. Collins, R. M. Fletcher, D. A. Steigerwald, S. Eberle, Y. K. Lim, P. S. Martin, and M. Krames, “The LED lightbulb: are we there yet? Progress and challenges for solid state illumination,” Conference on Lasers and Electro-Optics Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper CMQ2.

Fontoynont, M.

S. Boissard and M. Fontoynont, “Optimization of LED-based lighting blendings for object presentation,” Color Res. Appl. 34, 310–320 (2009).
[CrossRef]

Gaska, R.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska, and M. S. Shur, “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80, 234–237 (2002).
[CrossRef]

A. Zukauskas, F. Ivanauskas, R. Vaicekauskas, M. S. Shur, and R. Gaska,“Optimization of multichip white solid-state lighting source with four or more LEDs,” Proc. SPIE 4445, 148–155 (2001).
[CrossRef]

Gu, Y.

H. Wu, N. Narendran, Y. Gu, and A. Bierman, “Improving the performance of mixed-color white LED systems by using scattered photon extraction technique,” Proc. SPIE 6669, 666905 (2007).
[CrossRef]

Guo, X.

X. Guo and K. W. Houser, “A review of colour rendering indices and their application to commercial light sources,” Light. Res. Technol. 36, 183–197 (2004).
[CrossRef]

Holcomb, M. O.

M. O. Holcomb, R. Mueller-Mach, G. O. Mueller, D. Collins, R. M. Fletcher, D. A. Steigerwald, S. Eberle, Y. K. Lim, P. S. Martin, and M. Krames, “The LED lightbulb: are we there yet? Progress and challenges for solid state illumination,” Conference on Lasers and Electro-Optics Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper CMQ2.

Houser, K. W.

X. Guo and K. W. Houser, “A review of colour rendering indices and their application to commercial light sources,” Light. Res. Technol. 36, 183–197 (2004).
[CrossRef]

Hsu, T-h.

T-h. Hsu, “Optimization on color characteristics for LED lighting system,” M.S. thesis (National Central University, Taiwan, 2005).

Huang, B.-J.

S. P. Ying, C.-W. Tang, and B.-J. Huang, “Characterizing LEDs for mixture of colored LED light sources,” International Conference on Electronic Materials and Packaging, EMAP 2006 (IEEE, 2006), pp. 1–5.
[CrossRef]

Ivanauskas, F.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska, and M. S. Shur, “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80, 234–237 (2002).
[CrossRef]

A. Zukauskas, F. Ivanauskas, R. Vaicekauskas, M. S. Shur, and R. Gaska,“Optimization of multichip white solid-state lighting source with four or more LEDs,” Proc. SPIE 4445, 148–155 (2001).
[CrossRef]

Kim, J. K.

Klein, T.

M. Dyble, N. Narendran, A. Bierman, and T. Klein, “Impact of dimming white LEDs: chromaticity shifts due to different dimming methods,” Proc. SPIE 5941, 291–299 (2005).

Krames, M.

M. O. Holcomb, R. Mueller-Mach, G. O. Mueller, D. Collins, R. M. Fletcher, D. A. Steigerwald, S. Eberle, Y. K. Lim, P. S. Martin, and M. Krames, “The LED lightbulb: are we there yet? Progress and challenges for solid state illumination,” Conference on Lasers and Electro-Optics Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper CMQ2.

Leike, I.

H. Ries, I. Leike, and J. Muschaweck, “Optimized additive mixing of colored light-emitting diode sources,” Opt. Eng. (Bellingham) 43, 1531–1536 (2004).
[CrossRef]

Lim, Y. K.

M. O. Holcomb, R. Mueller-Mach, G. O. Mueller, D. Collins, R. M. Fletcher, D. A. Steigerwald, S. Eberle, Y. K. Lim, P. S. Martin, and M. Krames, “The LED lightbulb: are we there yet? Progress and challenges for solid state illumination,” Conference on Lasers and Electro-Optics Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper CMQ2.

Martin, P. S.

M. O. Holcomb, R. Mueller-Mach, G. O. Mueller, D. Collins, R. M. Fletcher, D. A. Steigerwald, S. Eberle, Y. K. Lim, P. S. Martin, and M. Krames, “The LED lightbulb: are we there yet? Progress and challenges for solid state illumination,” Conference on Lasers and Electro-Optics Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper CMQ2.

Mirhosseini, R.

Mueller, G. O.

M. O. Holcomb, R. Mueller-Mach, G. O. Mueller, D. Collins, R. M. Fletcher, D. A. Steigerwald, S. Eberle, Y. K. Lim, P. S. Martin, and M. Krames, “The LED lightbulb: are we there yet? Progress and challenges for solid state illumination,” Conference on Lasers and Electro-Optics Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper CMQ2.

Mueller-Mach, R.

M. O. Holcomb, R. Mueller-Mach, G. O. Mueller, D. Collins, R. M. Fletcher, D. A. Steigerwald, S. Eberle, Y. K. Lim, P. S. Martin, and M. Krames, “The LED lightbulb: are we there yet? Progress and challenges for solid state illumination,” Conference on Lasers and Electro-Optics Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper CMQ2.

Muschaweck, J.

H. Ries, I. Leike, and J. Muschaweck, “Optimized additive mixing of colored light-emitting diode sources,” Opt. Eng. (Bellingham) 43, 1531–1536 (2004).
[CrossRef]

Narendran, N.

H. Wu, N. Narendran, Y. Gu, and A. Bierman, “Improving the performance of mixed-color white LED systems by using scattered photon extraction technique,” Proc. SPIE 6669, 666905 (2007).
[CrossRef]

M. Dyble, N. Narendran, A. Bierman, and T. Klein, “Impact of dimming white LEDs: chromaticity shifts due to different dimming methods,” Proc. SPIE 5941, 291–299 (2005).

Ohno, Y.

Y. Ohno, “Optical metrology for LEDs and solid state lighting,” Proc. SPIE 6046, 604625 (2006).
[CrossRef]

W. Davis and Y. Ohno, “Toward an improved color rendering metric,” Proc. SPIE 5941, 59411G (2005).
[CrossRef]

Qu, X.

X. Qu, S. C. Wong, and C. K. Tse, “Color control system for RGB LED light sources using junction temperature measurement,” The 33rd Annual Conference of the IEEE Industrial Electronics Society, IECON 2007, Taipei, Taiwan (IEEE, 2007).
[CrossRef]

Ragsdell, K. M.

G. V. Reklaitis, R. Ravindran, and K. M. Ragsdell, Engineering Optimization-Methods and Applications (Wiley, 1983).

Ravindran, R.

G. V. Reklaitis, R. Ravindran, and K. M. Ragsdell, Engineering Optimization-Methods and Applications (Wiley, 1983).

Reklaitis, G. V.

G. V. Reklaitis, R. Ravindran, and K. M. Ragsdell, Engineering Optimization-Methods and Applications (Wiley, 1983).

Ries, H.

H. Ries, I. Leike, and J. Muschaweck, “Optimized additive mixing of colored light-emitting diode sources,” Opt. Eng. (Bellingham) 43, 1531–1536 (2004).
[CrossRef]

Schubert, E. F.

Schubert, M. F.

Shur, M. S.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska, and M. S. Shur, “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80, 234–237 (2002).
[CrossRef]

A. Zukauskas, F. Ivanauskas, R. Vaicekauskas, M. S. Shur, and R. Gaska,“Optimization of multichip white solid-state lighting source with four or more LEDs,” Proc. SPIE 4445, 148–155 (2001).
[CrossRef]

Steigerwald, D. A.

M. O. Holcomb, R. Mueller-Mach, G. O. Mueller, D. Collins, R. M. Fletcher, D. A. Steigerwald, S. Eberle, Y. K. Lim, P. S. Martin, and M. Krames, “The LED lightbulb: are we there yet? Progress and challenges for solid state illumination,” Conference on Lasers and Electro-Optics Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper CMQ2.

Tang, C.-W.

S. P. Ying, C.-W. Tang, and B.-J. Huang, “Characterizing LEDs for mixture of colored LED light sources,” International Conference on Electronic Materials and Packaging, EMAP 2006 (IEEE, 2006), pp. 1–5.
[CrossRef]

Tse, C. K.

X. Qu, S. C. Wong, and C. K. Tse, “Color control system for RGB LED light sources using junction temperature measurement,” The 33rd Annual Conference of the IEEE Industrial Electronics Society, IECON 2007, Taipei, Taiwan (IEEE, 2007).
[CrossRef]

Vaicekauskas, R.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska, and M. S. Shur, “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80, 234–237 (2002).
[CrossRef]

A. Zukauskas, F. Ivanauskas, R. Vaicekauskas, M. S. Shur, and R. Gaska,“Optimization of multichip white solid-state lighting source with four or more LEDs,” Proc. SPIE 4445, 148–155 (2001).
[CrossRef]

Wong, S. C.

X. Qu, S. C. Wong, and C. K. Tse, “Color control system for RGB LED light sources using junction temperature measurement,” The 33rd Annual Conference of the IEEE Industrial Electronics Society, IECON 2007, Taipei, Taiwan (IEEE, 2007).
[CrossRef]

Wu, H.

H. Wu, N. Narendran, Y. Gu, and A. Bierman, “Improving the performance of mixed-color white LED systems by using scattered photon extraction technique,” Proc. SPIE 6669, 666905 (2007).
[CrossRef]

Ying, S. P.

S. P. Ying, C.-W. Tang, and B.-J. Huang, “Characterizing LEDs for mixture of colored LED light sources,” International Conference on Electronic Materials and Packaging, EMAP 2006 (IEEE, 2006), pp. 1–5.
[CrossRef]

Zukauskas, A.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska, and M. S. Shur, “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80, 234–237 (2002).
[CrossRef]

A. Zukauskas, F. Ivanauskas, R. Vaicekauskas, M. S. Shur, and R. Gaska,“Optimization of multichip white solid-state lighting source with four or more LEDs,” Proc. SPIE 4445, 148–155 (2001).
[CrossRef]

Appl. Phys. Lett. (1)

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska, and M. S. Shur, “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80, 234–237 (2002).
[CrossRef]

Color Res. Appl. (1)

S. Boissard and M. Fontoynont, “Optimization of LED-based lighting blendings for object presentation,” Color Res. Appl. 34, 310–320 (2009).
[CrossRef]

Comput. J. (1)

M. J. Box, “A new method of constrained optimization and a comparison with other methods,” Comput. J. 8, 42–45 (1965).

Light. Res. Technol. (1)

X. Guo and K. W. Houser, “A review of colour rendering indices and their application to commercial light sources,” Light. Res. Technol. 36, 183–197 (2004).
[CrossRef]

Opt. Eng. (Bellingham) (1)

H. Ries, I. Leike, and J. Muschaweck, “Optimized additive mixing of colored light-emitting diode sources,” Opt. Eng. (Bellingham) 43, 1531–1536 (2004).
[CrossRef]

Opt. Express (1)

Proc. SPIE (5)

Y. Ohno, “Optical metrology for LEDs and solid state lighting,” Proc. SPIE 6046, 604625 (2006).
[CrossRef]

A. Zukauskas, F. Ivanauskas, R. Vaicekauskas, M. S. Shur, and R. Gaska,“Optimization of multichip white solid-state lighting source with four or more LEDs,” Proc. SPIE 4445, 148–155 (2001).
[CrossRef]

W. Davis and Y. Ohno, “Toward an improved color rendering metric,” Proc. SPIE 5941, 59411G (2005).
[CrossRef]

M. Dyble, N. Narendran, A. Bierman, and T. Klein, “Impact of dimming white LEDs: chromaticity shifts due to different dimming methods,” Proc. SPIE 5941, 291–299 (2005).

H. Wu, N. Narendran, Y. Gu, and A. Bierman, “Improving the performance of mixed-color white LED systems by using scattered photon extraction technique,” Proc. SPIE 6669, 666905 (2007).
[CrossRef]

Other (6)

M. O. Holcomb, R. Mueller-Mach, G. O. Mueller, D. Collins, R. M. Fletcher, D. A. Steigerwald, S. Eberle, Y. K. Lim, P. S. Martin, and M. Krames, “The LED lightbulb: are we there yet? Progress and challenges for solid state illumination,” Conference on Lasers and Electro-Optics Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper CMQ2.

X. Qu, S. C. Wong, and C. K. Tse, “Color control system for RGB LED light sources using junction temperature measurement,” The 33rd Annual Conference of the IEEE Industrial Electronics Society, IECON 2007, Taipei, Taiwan (IEEE, 2007).
[CrossRef]

G. V. Reklaitis, R. Ravindran, and K. M. Ragsdell, Engineering Optimization-Methods and Applications (Wiley, 1983).

T-h. Hsu, “Optimization on color characteristics for LED lighting system,” M.S. thesis (National Central University, Taiwan, 2005).

S. P. Ying, C.-W. Tang, and B.-J. Huang, “Characterizing LEDs for mixture of colored LED light sources,” International Conference on Electronic Materials and Packaging, EMAP 2006 (IEEE, 2006), pp. 1–5.
[CrossRef]

CIE, Method of Measuring and Specifying Colour Rendering Properties of Light Sources, Publication 13.3 (Commission Internationale de l’Eclairage, Vienna, 1995). ISBN 978-3900734572.

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Figures (12)

Fig. 1
Fig. 1

Measured SPDs of the RGBW LED illuminators used for this study.

Fig. 2
Fig. 2

Mixture spaces of RGB and RGBW: p 1 p 4 : red, green, blue, and white LED chromaticities, respectively.

Fig. 3
Fig. 3

Two examples of inconsistent constraints: (a) 0.3 w 1 0.6 and CCT < 3000 K ; (b) w 4 > 0.5 and DC < 0.0054 . See text for additional details.

Fig. 4
Fig. 4

CRI distributions of 3-color light mixtures: (a) RGB; (b) RGW, GBW, and RBW.

Fig. 5
Fig. 5

CRI distributions of a RGBW light mixture at four values of w 4 : w 4 : ( a ) 0.1; (b) 0.2; (c) 0.3; (d) 0.4.

Fig. 6
Fig. 6

CRI maxima in the H 4 space of an RGBW mixture. See text for additional details.

Fig. 7
Fig. 7

Relative intensities and CRI during iterations for Example 1 without the CCT and DC constraints.

Fig. 8
Fig. 8

Fifty convergent chromaticities of Example 1 without the CCT and DC constraints.

Fig. 9
Fig. 9

Fifty convergent chromaticities of Example 1 with the CCT and DC constraints.

Fig. 10
Fig. 10

CRI maxima and chromaticities of RGBW light mixture versus w 4 .

Fig. 11
Fig. 11

Fifty convergent chromaticities of Example 2 with 4000 K CCT 6000 K .

Fig. 12
Fig. 12

Fifty convergent CRI maxima of Example 2 with 4000 K CCT 6000 K : (a) convergent to the global maximum; (b) convergent to the singularity.

Equations (47)

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

x = i = 1 n w i X i w i ( X i + Y i + Z i ) , y = i = 1 n w i Y i w i ( X i + Y i + Z i ) ,
i = 1 n w i = 1 ,
w i l w i w i u , i = 1 , 2 , , n ,
T 1 CCT T 2 ,
DC = ( u u r ) 2 + ( v v r ) 2 < 0.0054 ,
Max under the constraints of Eqs. ( 2 ) and ( 3 ) R a ( f , f ̃ ) ,
1 x 12 = 1 α 12 x 1 + α 12 x 2 , 1 y 12 = 1 β 12 y 1 + β 12 y 2 ,
α 12 = w 2 X 2 w 1 X 1 + w 2 X 2 , β 12 = w 2 Y 2 w 1 Y 1 + w 2 Y 2 .
1 x 123 = 1 α 123 x 12 + α 123 x 3 , 1 y 123 = 1 β 123 y 12 + β 123 y 3 ,
α 123 = w 3 X 3 w 1 X 1 + w 2 X 2 + w 3 X 3 , β 123 = w 3 Y 3 w 1 Y 1 + w 2 Y 2 + w 3 Y 3 .
w 1 = ( X 4 + Y 4 + Z 4 ) ( x 14 * x 4 ) ( X 4 + Y 4 + Z 4 ) ( x 14 * x 4 ) ( X 1 + Y 1 + Z 1 ) ( x 14 * x 1 ) = ( X 4 + Y 4 + Z 4 ) ( y 14 * y 4 ) ( X 4 + Y 4 + Z 4 ) ( y 14 * y 4 ) ( X 1 + Y 1 + Z 1 ) ( y 14 * y 1 ) ,
w 4 = 1 w 1 ,
w 1 = | X 4 x 1 x 3 ( x 134 * x 4 ) x 1 [ X 3 x 4 ( x 134 * x 3 ) X 4 x 3 ( x 134 * x 4 ) ] Y 4 y 1 y 3 ( y 134 * y 4 ) y 1 [ Y 3 y 4 ( y 134 * y 3 ) Y 4 y 3 ( y 134 * y 4 ) ] | | x 3 [ X 1 x 4 ( x 134 * x 1 ) X 4 x 1 ( x 134 * x 4 ) ] x 1 [ X 3 x 4 ( x 134 * x 3 ) X 4 x 3 ( x 134 * x 4 ) ] y 3 [ Y 1 y 4 ( y 134 * y 1 ) Y 4 y 1 ( y 134 * y 4 ) ] y 1 [ Y 3 y 4 ( y 134 * y 3 ) Y 4 y 3 ( y 134 * y 4 ) ] | ,
w 3 = | x 3 [ X 1 x 4 ( x 134 * x 1 ) X 4 x 1 ( x 134 * x 4 ) ] X 4 x 1 x 3 ( x 134 * x 4 ) y 3 [ Y 1 y 4 ( y 134 * y 1 ) Y 4 y 1 ( y 134 * y 4 ) ] Y 4 y 1 y 3 ( y 134 * y 4 ) | | x 3 [ X 1 x 4 ( x 134 * x 1 ) X 4 x 1 ( x 134 * x 4 ) ] x 1 [ X 3 x 4 ( x 134 * x 3 ) X 4 x 3 ( x 134 * x 4 ) ] y 3 [ Y 1 y 4 ( y 134 * y 1 ) Y 4 y 1 ( y 134 * y 4 ) ] y 1 [ Y 3 y 4 ( y 134 * y 3 ) Y 4 y 3 ( y 134 * y 4 ) ] | ,
w 4 = 1 w 1 w 3 ,
P = { P i | i = 1 , 2 , , N } , P i R ̃ n ,
w n = γ 1 ,
w i = ( 1 j = i + 1 n w j ) γ n i + 1 , i = n 1 , n 2 , , 2 ,
w 1 = 1 j = 2 n w j ,
P L * = P L + α ( P C P L ) ,
w i * = w i u if w i * > w i u , w i * = w i l if w i * < w i l , i = 1 , 2 , , n .
P L * = 0.5 ( P C + P L * ) .
R a ( P H ) R a ( P L ) R a ( P H ) ϵ R ,
P i P C ϵ C , i = 1 , 2 , , N ,
max P i P j > ϵ D , j = 1 , 2 , , i 1 ,
P L * = { P L + α ( P C P L ) , if P C P L > ϵ C P 2 L + α ( P 2 C P 2 L ) , otherwise } ,
P C P L * < ϵ C .
R a ( P L ) R a ( P L * ) R a ( P L ) < ϵ R .
W = { w i j | w i l w i j w i u , i = n , n 1 , , 4 , and j = 1 , 2 , , M n i + 1 } ,
w n j = γ 1 j , j = 1 , 2 , , M ,
w i j = ( 1 k = i + 1 n w k j ) γ ( n i + 1 ) j , i = n 1 , n 2 , , 4 and j = 1 , 2 , , M n i + 1 ,
W s = { w i j | j = [ s DIV ( M i 4 + 1 ) ] + 1 ; i = n , n 1 , , 4 } , s = 1 , 2 , , M n 3 .
W ̃ s = { w i s | i = 3 , 2 , 1 } , s = 1 , 2 , , M n 3 ,
P s = W s W ̃ s , s = 1 , 2 , , M n 3 .
P = { P s | s = k + M j 1 , k = 1 , 2 , , M } .
w i L * = w i L + α ( w i C w i L ) ,
w i L * = 0.5 ( w i C + w i L * ) .
w ¯ 1 = 0.268 , σ w 1 = 0.0004 , w ¯ 2 = 0.267 , σ w 2 = 0.0002 ,
w ¯ 3 = 0.465 , σ w 3 = 0.0006 , R ¯ a ( P h ) = 87.4 , σ CRI = 0.001 ,
w ¯ 1 = 0.449 , σ w 1 = 0.0004 , w ¯ 2 = 0.248 , σ w 2 = 0.0002 ,
w ¯ 3 = 0.303 , σ w 3 = 0.0005 , R ¯ a ( P h ) = 75.6 , σ CRI = 0.028 .
w ¯ 1 = 0.513 , σ w 1 = 0.0374 , w ¯ 2 = 0.031 , σ w 2 = 0.0188 , w ¯ 3 = 0.185 , σ w 3 = 0.0340 ,
w ¯ 4 = 0.271 , σ w 4 = 0.0154 , x ¯ = 0.364 , σ x = 0.0086 , y ¯ = 0.356 , σ y = 0.0028 ,
R ¯ a ( P h ) = 96.4 , σ CRI = 0.097 .
w ¯ 1 = 0.529 , σ w 1 = 0.0008 , w ¯ 2 = 0.023 , σ w 2 = 0.0009 , w ¯ 3 = 0.171 , σ w 3 = 0.0009 ,
w ¯ 4 = 0.277 , σ w 4 = 0.0011 , x ¯ = 0.368 , σ x = 0.0002 , y ¯ = 0.357 , σ y = 0.0001 ,
R ¯ a ( P h ) = 96.5 , σ CRI = 0.001 .

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