Abstract

We describe a fully-analytical, simple yet sufficiently accurate method to compute the color pattern of the light emitted from multicolor light-emitting diode (LED) assemblies. Spatial distributions for both color variation and correlated color temperature (CCT) as a function of typical parameters of influence, such as LED spectrum, spatial distribution of LED radiation, target distance, LED-to-LED spacing, and number of LEDs, are shown. To illustrate the method, we simulate and analyze the color patterns of linear, ring, and square RGB (red, green, and blue) arrays for Lambertian-type, batwing, and side emitting LEDs. Our theory may be useful to choose the optimal value for both the array configuration and the array-diffuser distance for lighting systems with color mixing devices.

©2007 Optical Society of America

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References

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  • Publication
  1. A. Zukauskas, M. S. Schur, and R. Gaska, Introduction to Solid State Lighting (Wiley-Interscience, New York, 2002)
  2. I. Ashdown, “Solid-state lighting design requires a system-level approach,” SPIE Newsroom (2006). http://newsroom.spie.org/x2235.xml?highlight=x531
  3. D. Malacara, Color Vision and Colorimetry, SPIE Press, Bellingham WA, USA, 2002.
  4. G. Wyszecki and W. S. Styles , Color Science: Concepts and Methods, Quantitative Data and Formulae (2nd ed.), (Wiley, New York , 1982).
  5. Y. Ohno, “Radiometry and Photometry Review for Vision Optics,” in Handbook of Optics (Vol. III, 2nd ed.), M. Bass, J. M. Enoch, E. W. Van Stryland, and W. L. Wolfe, Eds (McGraw-Hill, 2001).
  6. M. Strojnik and G. Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara and B. Thompson, Eds. (2001).
  7. Y. Uchida and T. Taguchi, “Lighting theory and luminous characteristics of white light-emitting diodes,” Opt. Eng. 44,124003–1 (2005).
    [Crossref]
  8. Y. Gu, N. Narendran, T. Dong, and H. Wu, “Spectral and luminous efficacy change of high-power LEDs under different dimming methods,” in Sixth International Conference on Solid State Lighting, I. T. Ferguson, N. Narendran, T. Taguchi, and I. E. Ashdown, eds., Proc. SPIE6337,63370J:1–7 (2006).
    [Crossref]
  9. Y. Ohno, “Spectral design considerations for white LED color rendering,” Opt. Eng. 44,111302-1 (2005).
    [Crossref]
  10. H. Y. Chou, T. H. Hsu, and T. H. Yang, “Effective method for improving illuminating properties of white-light LEDs,” in Light-Emitting Diodes: Research, Manufacturing, and Applications IX; Steve A. Stockman, H. Walter Yao, and E. Fred Schubert; Eds., Proc. SPIE5739, p.33–41 (2005).
    [Crossref]
  11. H. Ries, I, Leike, and J. Muschaweck, “Optimized additive mixing of colored light-emitting diode sources,” Opt. Eng. 43,1531–1356 (2004).
    [Crossref]
  12. A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R Gaska, and M. S. Shur , “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80,234–236 (2002).
    [Crossref]
  13. E. F. Schubert, Light-emitting diodes (Cambridge University Press, Cambridge, 2003).
  14. I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45,2265–2272 (2006).
    [Crossref] [PubMed]
  15. I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical LED array,” Opt. Eng. 46 (3), (In press 2007).
    [Crossref]
  16. Y. Uchida and T. Taguchi, “Lighting theory and luminous characteristics of white light-emitting diodes,” Opt. Eng. 44,124003–1 (2005).
    [Crossref]
  17. I. Moreno, “Spatial distribution of LED radiation,” in The International Optical Design Conference, G. Gregory, J. Howard, and J. Koshel, eds., Proc. SPIE6342,634216:1–7 (2006).
  18. I. Moreno, “Simple functions for intensity and irradiance distribution from LEDs,” in preparation.
  19. I. Ashdown, “Chromaticity and color temperature for architectural lighting,” in Solid State Lighting II, I. T. Ferguson, N. Narendran, S. P. DenBaars, and Y. S. Park, eds., Proc. SPIE4776,51–60 (2002).
    [Crossref]
  20. J. Hernández-Andrés, R. L. Jr. Lee, and J. Romero, “Calculating correlated color temperatures across the entire gamut of daylight and skylight chromaticities,” Appl. Opt. 38,5703–5709 (1999).
    [Crossref]
  21. C. S. McCamy, “Correlated color temperature as an explicit function of chromaticity coordinates,” Color Res. Appl. 17,142–144 (1992).
    [Crossref]
  22. J. M. Gaines, “Modelling of multichip LED packages for illumination,” Light. Res. Technol. 38,153–165 (2006).
    [Crossref]
  23. I. Moreno and L.M. Molinar, “Color uniformity of the light distribution from several cluster configurations of multicolor LEDs.” in Fifth International Conference on Solid State Lighting, I. T. Ferguson, J. C. Carrano, T. Taguchi, and I. E. Ashdown, eds., Proc. SPIE5941,359–365 (2005).

2007 (1)

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical LED array,” Opt. Eng. 46 (3), (In press 2007).
[Crossref]

2006 (2)

I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45,2265–2272 (2006).
[Crossref] [PubMed]

J. M. Gaines, “Modelling of multichip LED packages for illumination,” Light. Res. Technol. 38,153–165 (2006).
[Crossref]

2005 (3)

Y. Uchida and T. Taguchi, “Lighting theory and luminous characteristics of white light-emitting diodes,” Opt. Eng. 44,124003–1 (2005).
[Crossref]

Y. Uchida and T. Taguchi, “Lighting theory and luminous characteristics of white light-emitting diodes,” Opt. Eng. 44,124003–1 (2005).
[Crossref]

Y. Ohno, “Spectral design considerations for white LED color rendering,” Opt. Eng. 44,111302-1 (2005).
[Crossref]

2004 (1)

H. Ries, I, Leike, and J. Muschaweck, “Optimized additive mixing of colored light-emitting diode sources,” Opt. Eng. 43,1531–1356 (2004).
[Crossref]

2002 (1)

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R Gaska, and M. S. Shur , “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80,234–236 (2002).
[Crossref]

1999 (1)

J. Hernández-Andrés, R. L. Jr. Lee, and J. Romero, “Calculating correlated color temperatures across the entire gamut of daylight and skylight chromaticities,” Appl. Opt. 38,5703–5709 (1999).
[Crossref]

1992 (1)

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

Ashdown, I.

I. Ashdown, “Solid-state lighting design requires a system-level approach,” SPIE Newsroom (2006). http://newsroom.spie.org/x2235.xml?highlight=x531

I. Ashdown, “Chromaticity and color temperature for architectural lighting,” in Solid State Lighting II, I. T. Ferguson, N. Narendran, S. P. DenBaars, and Y. S. Park, eds., Proc. SPIE4776,51–60 (2002).
[Crossref]

Avendaño-Alejo, M.

I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45,2265–2272 (2006).
[Crossref] [PubMed]

Chou, H. Y.

H. Y. Chou, T. H. Hsu, and T. H. Yang, “Effective method for improving illuminating properties of white-light LEDs,” in Light-Emitting Diodes: Research, Manufacturing, and Applications IX; Steve A. Stockman, H. Walter Yao, and E. Fred Schubert; Eds., Proc. SPIE5739, p.33–41 (2005).
[Crossref]

Dong, T.

Y. Gu, N. Narendran, T. Dong, and H. Wu, “Spectral and luminous efficacy change of high-power LEDs under different dimming methods,” in Sixth International Conference on Solid State Lighting, I. T. Ferguson, N. Narendran, T. Taguchi, and I. E. Ashdown, eds., Proc. SPIE6337,63370J:1–7 (2006).
[Crossref]

Gaines, J. M.

J. M. Gaines, “Modelling of multichip LED packages for illumination,” Light. Res. Technol. 38,153–165 (2006).
[Crossref]

Gaska, R

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R Gaska, and M. S. Shur , “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80,234–236 (2002).
[Crossref]

Gaska, R.

A. Zukauskas, M. S. Schur, and R. Gaska, Introduction to Solid State Lighting (Wiley-Interscience, New York, 2002)

Gu, Y.

Y. Gu, N. Narendran, T. Dong, and H. Wu, “Spectral and luminous efficacy change of high-power LEDs under different dimming methods,” in Sixth International Conference on Solid State Lighting, I. T. Ferguson, N. Narendran, T. Taguchi, and I. E. Ashdown, eds., Proc. SPIE6337,63370J:1–7 (2006).
[Crossref]

Hernández-Andrés, J.

J. Hernández-Andrés, R. L. Jr. Lee, and J. Romero, “Calculating correlated color temperatures across the entire gamut of daylight and skylight chromaticities,” Appl. Opt. 38,5703–5709 (1999).
[Crossref]

Hsu, T. H.

H. Y. Chou, T. H. Hsu, and T. H. Yang, “Effective method for improving illuminating properties of white-light LEDs,” in Light-Emitting Diodes: Research, Manufacturing, and Applications IX; Steve A. Stockman, H. Walter Yao, and E. Fred Schubert; Eds., Proc. SPIE5739, p.33–41 (2005).
[Crossref]

Ivanauskas, F.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R Gaska, and M. S. Shur , “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80,234–236 (2002).
[Crossref]

Ivanov, R.

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical LED array,” Opt. Eng. 46 (3), (In press 2007).
[Crossref]

Lee, R. L. Jr.

J. Hernández-Andrés, R. L. Jr. Lee, and J. Romero, “Calculating correlated color temperatures across the entire gamut of daylight and skylight chromaticities,” Appl. Opt. 38,5703–5709 (1999).
[Crossref]

Leike, I,

H. Ries, I, Leike, and J. Muschaweck, “Optimized additive mixing of colored light-emitting diode sources,” Opt. Eng. 43,1531–1356 (2004).
[Crossref]

Malacara, D.

D. Malacara, Color Vision and Colorimetry, SPIE Press, Bellingham WA, USA, 2002.

McCamy, C. S.

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

Molinar, L.M.

I. Moreno and L.M. Molinar, “Color uniformity of the light distribution from several cluster configurations of multicolor LEDs.” in Fifth International Conference on Solid State Lighting, I. T. Ferguson, J. C. Carrano, T. Taguchi, and I. E. Ashdown, eds., Proc. SPIE5941,359–365 (2005).

Moreno, I.

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical LED array,” Opt. Eng. 46 (3), (In press 2007).
[Crossref]

I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45,2265–2272 (2006).
[Crossref] [PubMed]

I. Moreno, “Spatial distribution of LED radiation,” in The International Optical Design Conference, G. Gregory, J. Howard, and J. Koshel, eds., Proc. SPIE6342,634216:1–7 (2006).

I. Moreno, “Simple functions for intensity and irradiance distribution from LEDs,” in preparation.

I. Moreno and L.M. Molinar, “Color uniformity of the light distribution from several cluster configurations of multicolor LEDs.” in Fifth International Conference on Solid State Lighting, I. T. Ferguson, J. C. Carrano, T. Taguchi, and I. E. Ashdown, eds., Proc. SPIE5941,359–365 (2005).

Muñoz, J.

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical LED array,” Opt. Eng. 46 (3), (In press 2007).
[Crossref]

Muschaweck, J.

H. Ries, I, Leike, and J. Muschaweck, “Optimized additive mixing of colored light-emitting diode sources,” Opt. Eng. 43,1531–1356 (2004).
[Crossref]

Narendran, N.

Y. Gu, N. Narendran, T. Dong, and H. Wu, “Spectral and luminous efficacy change of high-power LEDs under different dimming methods,” in Sixth International Conference on Solid State Lighting, I. T. Ferguson, N. Narendran, T. Taguchi, and I. E. Ashdown, eds., Proc. SPIE6337,63370J:1–7 (2006).
[Crossref]

Ohno, Y.

Y. Ohno, “Spectral design considerations for white LED color rendering,” Opt. Eng. 44,111302-1 (2005).
[Crossref]

Y. Ohno, “Radiometry and Photometry Review for Vision Optics,” in Handbook of Optics (Vol. III, 2nd ed.), M. Bass, J. M. Enoch, E. W. Van Stryland, and W. L. Wolfe, Eds (McGraw-Hill, 2001).

Paez, G.

M. Strojnik and G. Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara and B. Thompson, Eds. (2001).

Ries, H.

H. Ries, I, Leike, and J. Muschaweck, “Optimized additive mixing of colored light-emitting diode sources,” Opt. Eng. 43,1531–1356 (2004).
[Crossref]

Romero, J.

J. Hernández-Andrés, R. L. Jr. Lee, and J. Romero, “Calculating correlated color temperatures across the entire gamut of daylight and skylight chromaticities,” Appl. Opt. 38,5703–5709 (1999).
[Crossref]

Schubert, E. F.

E. F. Schubert, Light-emitting diodes (Cambridge University Press, Cambridge, 2003).

Schur, M. S.

A. Zukauskas, M. S. Schur, and R. Gaska, Introduction to Solid State Lighting (Wiley-Interscience, New York, 2002)

Shur, M. S.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R Gaska, and M. S. Shur , “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80,234–236 (2002).
[Crossref]

Strojnik, M.

M. Strojnik and G. Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara and B. Thompson, Eds. (2001).

Styles, W. S.

G. Wyszecki and W. S. Styles , Color Science: Concepts and Methods, Quantitative Data and Formulae (2nd ed.), (Wiley, New York , 1982).

Taguchi, T.

Y. Uchida and T. Taguchi, “Lighting theory and luminous characteristics of white light-emitting diodes,” Opt. Eng. 44,124003–1 (2005).
[Crossref]

Y. Uchida and T. Taguchi, “Lighting theory and luminous characteristics of white light-emitting diodes,” Opt. Eng. 44,124003–1 (2005).
[Crossref]

Tzonchev, R. I.

I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45,2265–2272 (2006).
[Crossref] [PubMed]

Uchida, Y.

Y. Uchida and T. Taguchi, “Lighting theory and luminous characteristics of white light-emitting diodes,” Opt. Eng. 44,124003–1 (2005).
[Crossref]

Y. Uchida and T. Taguchi, “Lighting theory and luminous characteristics of white light-emitting diodes,” Opt. Eng. 44,124003–1 (2005).
[Crossref]

Vaicekauskas, R.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R Gaska, and M. S. Shur , “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80,234–236 (2002).
[Crossref]

Wu, H.

Y. Gu, N. Narendran, T. Dong, and H. Wu, “Spectral and luminous efficacy change of high-power LEDs under different dimming methods,” in Sixth International Conference on Solid State Lighting, I. T. Ferguson, N. Narendran, T. Taguchi, and I. E. Ashdown, eds., Proc. SPIE6337,63370J:1–7 (2006).
[Crossref]

Wyszecki, G.

G. Wyszecki and W. S. Styles , Color Science: Concepts and Methods, Quantitative Data and Formulae (2nd ed.), (Wiley, New York , 1982).

Yang, T. H.

H. Y. Chou, T. H. Hsu, and T. H. Yang, “Effective method for improving illuminating properties of white-light LEDs,” in Light-Emitting Diodes: Research, Manufacturing, and Applications IX; Steve A. Stockman, H. Walter Yao, and E. Fred Schubert; Eds., Proc. SPIE5739, p.33–41 (2005).
[Crossref]

Zukauskas, A.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R Gaska, and M. S. Shur , “Optimization of white polychromatic semiconductor lamps,” Appl. Phys. Lett. 80,234–236 (2002).
[Crossref]

A. Zukauskas, M. S. Schur, and R. Gaska, Introduction to Solid State Lighting (Wiley-Interscience, New York, 2002)

Appl. Opt (2)

I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45,2265–2272 (2006).
[Crossref] [PubMed]

J. Hernández-Andrés, R. L. Jr. Lee, and J. Romero, “Calculating correlated color temperatures across the entire gamut of daylight and skylight chromaticities,” Appl. Opt. 38,5703–5709 (1999).
[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. 80,234–236 (2002).
[Crossref]

Color Res. Appl (1)

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

Light. Res. Technol (1)

J. M. Gaines, “Modelling of multichip LED packages for illumination,” Light. Res. Technol. 38,153–165 (2006).
[Crossref]

Opt. Eng (4)

I. Moreno, J. Muñoz, and R. Ivanov, “Uniform illumination of distant targets using a spherical LED array,” Opt. Eng. 46 (3), (In press 2007).
[Crossref]

Y. Uchida and T. Taguchi, “Lighting theory and luminous characteristics of white light-emitting diodes,” Opt. Eng. 44,124003–1 (2005).
[Crossref]

Y. Uchida and T. Taguchi, “Lighting theory and luminous characteristics of white light-emitting diodes,” Opt. Eng. 44,124003–1 (2005).
[Crossref]

Y. Ohno, “Spectral design considerations for white LED color rendering,” Opt. Eng. 44,111302-1 (2005).
[Crossref]

Opt. Eng. (1)

H. Ries, I, Leike, and J. Muschaweck, “Optimized additive mixing of colored light-emitting diode sources,” Opt. Eng. 43,1531–1356 (2004).
[Crossref]

Other (13)

E. F. Schubert, Light-emitting diodes (Cambridge University Press, Cambridge, 2003).

H. Y. Chou, T. H. Hsu, and T. H. Yang, “Effective method for improving illuminating properties of white-light LEDs,” in Light-Emitting Diodes: Research, Manufacturing, and Applications IX; Steve A. Stockman, H. Walter Yao, and E. Fred Schubert; Eds., Proc. SPIE5739, p.33–41 (2005).
[Crossref]

Y. Gu, N. Narendran, T. Dong, and H. Wu, “Spectral and luminous efficacy change of high-power LEDs under different dimming methods,” in Sixth International Conference on Solid State Lighting, I. T. Ferguson, N. Narendran, T. Taguchi, and I. E. Ashdown, eds., Proc. SPIE6337,63370J:1–7 (2006).
[Crossref]

A. Zukauskas, M. S. Schur, and R. Gaska, Introduction to Solid State Lighting (Wiley-Interscience, New York, 2002)

I. Ashdown, “Solid-state lighting design requires a system-level approach,” SPIE Newsroom (2006). http://newsroom.spie.org/x2235.xml?highlight=x531

D. Malacara, Color Vision and Colorimetry, SPIE Press, Bellingham WA, USA, 2002.

G. Wyszecki and W. S. Styles , Color Science: Concepts and Methods, Quantitative Data and Formulae (2nd ed.), (Wiley, New York , 1982).

Y. Ohno, “Radiometry and Photometry Review for Vision Optics,” in Handbook of Optics (Vol. III, 2nd ed.), M. Bass, J. M. Enoch, E. W. Van Stryland, and W. L. Wolfe, Eds (McGraw-Hill, 2001).

M. Strojnik and G. Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara and B. Thompson, Eds. (2001).

I. Moreno, “Spatial distribution of LED radiation,” in The International Optical Design Conference, G. Gregory, J. Howard, and J. Koshel, eds., Proc. SPIE6342,634216:1–7 (2006).

I. Moreno, “Simple functions for intensity and irradiance distribution from LEDs,” in preparation.

I. Ashdown, “Chromaticity and color temperature for architectural lighting,” in Solid State Lighting II, I. T. Ferguson, N. Narendran, S. P. DenBaars, and Y. S. Park, eds., Proc. SPIE4776,51–60 (2002).
[Crossref]

I. Moreno and L.M. Molinar, “Color uniformity of the light distribution from several cluster configurations of multicolor LEDs.” in Fifth International Conference on Solid State Lighting, I. T. Ferguson, J. C. Carrano, T. Taguchi, and I. E. Ashdown, eds., Proc. SPIE5941,359–365 (2005).

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Figures (7)

Fig. 1
Fig. 1 Multicolor LED assembly that produces a color distribution over a flat surface located in front.

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Fig. 2.
Fig. 2. Examples of simulated LED intensity distribution obtained from analytical irradiance functions, Eqs. (10)-(12). —Lambertian type emitter with θ ½ =5°, ┄ Luxon® batwing emitter, and ···· Luxon® side emitter.

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Fig. 3.
Fig. 3. (a) Schematic of the 3 configurations used for simulating a RGB linear array. Parameter d is the spacing between LEDs. (b-d) Simulations with Lambertian-type LEDs. (b) Spatial distribution of CCT(x,0,z) for z=2d, 6d, and 10d. (c) CIE u’ v’ 1976 coordinates along the x direction at y=0 for -x 90<x<x 90. (d) The calculated color uniformity Δuv(z)RMS in function of the target distance. Inset: color distribution Δuv(x,0,d) and Δuv(x,0,10d).

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Fig. 4.
Fig. 4. Simulations with batwing-type LEDs for the arrays analyzed in Fig. 3. (a) CIE u’v’ 1976 coordinates along the x direction at y=0 for -x 90<x<x 90. (b) The calculated color uniformity Δuv(z)RMS in function of the target distance. Inset: color distribution Δuv(x,0,d) and Δuv(x,0,10d).

Download Full Size | PPT Slide | PDF

Fig. 5.
Fig. 5. (a) Circular ring array of LEDs with 3 rings. (b) CIE u’v’ 1976 coordinates along the radial direction (<r 90) for 3 RGB configurations. (c) The calculated color uniformity Δuv(z)RMS in function of the target distance. Inset: Spatial distribution of CCT(x,0,z) for z=3ρ.

Download Full Size | PPT Slide | PDF

Fig. 6.
Fig. 6. (a) Rectangular RGB array with 36 side emitting LEDs. (b) CIE u’v’ 1976 coordinates along the x direction at y=0 for -x 90<x<x 90. (c) The calculated color distribution, Δuv(x,y,0.2d) and Δuv(x,y,2d), on an illuminated surface within an area of 2x 90 × 2y 90.

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Fig. 7.
Fig. 7. (a) Experimental setup to measure the color distribution of the light emitted from a linear RGB array. (b) Shows the measured intensity distribution (normalized) of red, green, and blue LEDs. (c) Measured and calculated CIE u’v’ 1976 coordinates along the x direction at y=0 for -x 90<x<x 90 (x 90 is calculated for theory and it is measured for experiment). (d) The color uniformity Δuv(z)RMS in function of the target distance. Inset: Spatial distribution of CCT(x,0,z).

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Equations (21)

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

x T = Σ i = 1 S X i Σ i = 1 S ( X i + Y i + Z i ) , y T = Σ i = 1 S Y i Σ i = 1 S ( X i + Y i + Z i ) , z T = 1 − x T − y T .
X i = k ∫ 400 nm 700 nm E i ( λ ) x ¯ ( λ ) dλ , Y i = k ∫ 400 nm 700 nm E i ( λ ) y ¯ ( λ ) dλ , Z i = k ∫ 400 nm 700 nm E i ( λ ) z ¯ ( λ ) dλ ,
X i = kE peak − i ∫ 400 nm 700 nm U i ( λ ) x ¯ ( λ ) dλ = kE peak − i X ̂ i , Y i = kE peak − i ∫ 400 nm 700 nm U i ( λ ) y ¯ ( λ ) dλ = kE peak − i Y ̂ i ,
Z i = kE peak − i ∫ 400 nm 700 nm U i ( λ ) z ¯ ( λ ) dλ = kE peak − i Z ̂ i .
U i ( λ ) = 1 3 ⌊ g i ( λ ) + 2 g i 5 ( λ ) ⌋ ,
x T = Σ i = 1 S e i 1 X ̂ i Σ i = 1 S e i 1 ( X ̂ i + Y ̂ i + Z ̂ i ) , y T = Σ i = 1 S e i 1 Y ̂ i Σ i = 1 S e i 1 ( X ̂ i + Y ̂ i + Z ̂ i ) , z T = 1 − x T − y T .
Σ i = 1 S e i 1 [ ( 1 − x T − 1 ) X ̂ i + Y ̂ i + Z ̂ i ] = 0 , Σ i = 1 S e i 1 [ X ̂ i + ( 1 − y T − 1 ) Y ̂ i + Z ̂ i ] = 0 , Σ i = 1 S e i 1 [ X ̂ i + Y ̂ i + ( 1 − z T − 1 ) Z ̂ i ] = 0 .
e G − R = x T ( Y ̂ R Z ̂ B − Y ̂ B Z ̂ R ) + y T ( X ̂ B Z ̂ R − X ̂ R Z ̂ B ) + z T ( X ̂ R Y ̂ B − X ̂ B Y ̂ R ) x T ( Y ̂ B Z ̂ G − Y ̂ G Z ̂ B ) + y T ( X ̂ G Z ̂ B − X ̂ B Z ̂ G ) + z T ( X ̂ B Y ̂ G − X ̂ G Y ̂ B ) ,
e B − R = x T ( Y ̂ G Z ̂ R − Y ̂ R Z ̂ G ) + y T ( X ̂ R Z ̂ G − X ̂ G Z ̂ R ) + z T ( X ̂ G Y ̂ R − X ̂ R Y ̂ G ) x T ( Y ̂ B Z ̂ G − Y ̂ G Z ̂ B ) + y T ( X ̂ G Z ̂ B − X ̂ B Z ̂ G ) + z T ( X ̂ B Y ̂ G − X ̂ G Y ̂ B ) .
ε G , B − R ( x , y , z ) = e G , B − R [ Σ i N G , B E G , B ( x , y , z ; x i , y i ) Σ i N G , B E G , B ( 0,0 , z 0 ; x i , y i ) ] [ Σ i N R E R ( x , y , z ; x i , y i ) Σ i N R E R ( 0,0 , z 0 ; x i , y i ) ] − 1 ,
E ( x , y , z ; x i , y i ) = z m + 1 [ ( x − x i ) 2 + ( y − y i ) 2 + z 2 ] m + 3 2 .
E ( x , y , z ; x i , y i ) = Cz 3 [ ( x − x i ) 2 + ( y − y i ) 2 + z 2 ] 5 2 + z [ z cos α + ( x − x i ) 2 + ( y − y i ) 2 sin α ] n [ ( x − x i ) 2 + ( y − y i ) 2 + z 2 ] n + 3 2 .
E ( x , y , z ; x i , y i ) = Σ i = 1 5 C j z [ z cos α + ( x − x i ) 2 + ( y − y i ) 2 sin α ] Mi [ ( x − x i ) 2 + ( y − y i ) 2 + z 2 ] Mi + 3 2 .
x d ( x , y , z ) = X ̂ R + X ̂ G ε GR ( x , y , z ) + X ̂ B ε BR ( x , y , z ) ( X R ̂ + Y R ̂ + Z R ̂ ) + ( X G ̂ + Y G ̂ + Z G ̂ ) ε GR ( x , y , z ) + ( X B ̂ + Y B ̂ + Z B ̂ ) ε BR ( x , y , z ) ,
y d ( x , y , z ) = Y ̂ R + Y ̂ G ε GR ( x , y , z ) + Y ̂ B ε BR ( x , y , z ) ( X R ̂ + Y R ̂ + Z R ̂ ) + ( X G ̂ + Y G ̂ + Z G ̂ ) ε GR ( x , y , z ) + ( X B ̂ + Y B ̂ + Z B ̂ ) ε BR ( x , y , z ) ,
CCT ( x , y , z ) = A 0 + Σ i = 1 3 A i exp ⁡ [ − n ( x , y , z ) / t i ] ,
CCT ( x , y , z ) = a n 3 ( x , y , z ) + b n 2 ( x , y , z ) + c n ( x , y , z ) + d ,
n ( x , y , z ) = x d ( x , y , z ) − x e y d ( x , y , z ) − y e .
Δ uv ( x , y , z ) = [ u ´ ( x , y , z ) − u ´ ( 0,0 , z ) ] 2 + [ v ´ ( x , y , z ) − v ´ ( 0,0 , z ) ] 2 ,
u ´ ( x , y , z ) = 4 x d ( x , y , z ) − 2 x d ( x , y , z ) + 12 y d ( x , y , z ) + 3 , v ´ ( x , y , z ) = 9 y d ( x , y , z ) − 2 x d ( x , y , z ) + 12 y d ( x , y , z ) + 3 ,
Δ uv ( z ) RMS = 1 M Σ x Σ y Δ uv ( x , y , z ) 2 ,

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