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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  1. A. Zukauskas, M. S. Schur, 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, W. L. Wolfe, Eds (McGraw-Hill, 2001).
  6. M. Strojnik, G. Paez, "Radiometry," in Handbook of Optical Engineering, D. Malacara, B. Thompson, Eds. (2001).
  7. Y. Uchida, 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, 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, I. E. Ashdown, eds., Proc. SPIE 6337, 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, 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, E. Fred Schubert; Eds., Proc. SPIE 5739, p. 33-41 (2005).
    [CrossRef]
  11. H. Ries, I, Leike, 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, 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, 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, R. Ivanov, "Uniform illumination of distant targets using a spherical LED array," Opt. Eng. 46 (3), (In press2007).
    [CrossRef]
  16. Y. Uchida, 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, J. Koshel, eds., Proc. SPIE 6342, 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, Y. S. Park, eds., Proc. SPIE 4776, 51-60 (2002).
    [CrossRef]
  20. J. Hernández-Andrés, R. L. Lee, Jr., 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, 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, I. E. Ashdown, eds., Proc. SPIE 5941, 359-365 (2005).

2007

I. Moreno, J. Muñoz, R. Ivanov, "Uniform illumination of distant targets using a spherical LED array," Opt. Eng. 46 (3), (In press2007).
[CrossRef]

2006

2005

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

Y. Uchida, 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]

H. Y. Chou, T. H. Hsu, 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, E. Fred Schubert; Eds., Proc. SPIE 5739, p. 33-41 (2005).
[CrossRef]

2004

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

2002

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R Gaska, M. S. Shur, "Optimization of white polychromatic semiconductor lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

1999

1992

C. S. McCamy, "Correlated color temperature as an explicit function of chromaticity coordinates," Color Res. Appl. 17, 142-144 (1992).
[CrossRef]

Avendaño-Alejo, M.

Chou, H. Y.

H. Y. Chou, T. H. Hsu, 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, E. Fred Schubert; Eds., Proc. SPIE 5739, p. 33-41 (2005).
[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, M. S. Shur, "Optimization of white polychromatic semiconductor lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Hernández-Andrés, J.

Hsu, T. H.

H. Y. Chou, T. H. Hsu, 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, E. Fred Schubert; Eds., Proc. SPIE 5739, p. 33-41 (2005).
[CrossRef]

Ivanauskas, F.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R Gaska, M. S. Shur, "Optimization of white polychromatic semiconductor lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Ivanov, R.

I. Moreno, J. Muñoz, R. Ivanov, "Uniform illumination of distant targets using a spherical LED array," Opt. Eng. 46 (3), (In press2007).
[CrossRef]

Lee, R. L.

McCamy, C. S.

C. S. McCamy, "Correlated color temperature as an explicit function of chromaticity coordinates," Color Res. Appl. 17, 142-144 (1992).
[CrossRef]

Moreno, I.

I. Moreno, J. Muñoz, R. Ivanov, "Uniform illumination of distant targets using a spherical LED array," Opt. Eng. 46 (3), (In press2007).
[CrossRef]

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

Muñoz, J.

I. Moreno, J. Muñoz, R. Ivanov, "Uniform illumination of distant targets using a spherical LED array," Opt. Eng. 46 (3), (In press2007).
[CrossRef]

Ohno, Y.

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

Ries, H.

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

Romero, J.

Shur, M. S.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R Gaska, M. S. Shur, "Optimization of white polychromatic semiconductor lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Taguchi, T.

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

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

Tzonchev, R. I.

Uchida, Y.

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

Y. Uchida, 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, M. S. Shur, "Optimization of white polychromatic semiconductor lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Yang, T. H.

H. Y. Chou, T. H. Hsu, 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, E. Fred Schubert; Eds., Proc. SPIE 5739, p. 33-41 (2005).
[CrossRef]

Zukauskas, A.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R Gaska, M. S. Shur, "Optimization of white polychromatic semiconductor lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R Gaska, M. S. Shur, "Optimization of white polychromatic semiconductor lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Color Res. Appl.

C. S. McCamy, "Correlated color temperature as an explicit function of chromaticity coordinates," Color Res. Appl. 17, 142-144 (1992).
[CrossRef]

Light. Res. Technol.

J. M. Gaines, "Modelling of multichip LED packages for illumination," Light. Res. Technol. 38, 153-165 (2006).
[CrossRef]

Opt. Eng.

I. Moreno, J. Muñoz, R. Ivanov, "Uniform illumination of distant targets using a spherical LED array," Opt. Eng. 46 (3), (In press2007).
[CrossRef]

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

Y. Uchida, 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]

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

Proc. SPIE

H. Y. Chou, T. H. Hsu, 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, E. Fred Schubert; Eds., Proc. SPIE 5739, p. 33-41 (2005).
[CrossRef]

Other

Y. Gu, N. Narendran, T. Dong, 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, I. E. Ashdown, eds., Proc. SPIE 6337, 63370J:1-7 (2006).
[CrossRef]

A. Zukauskas, M. S. Schur, 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, W. L. Wolfe, Eds (McGraw-Hill, 2001).

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

I. Moreno, "Spatial distribution of LED radiation," in The International Optical Design Conference, G. Gregory, J. Howard, J. Koshel, eds., Proc. SPIE 6342, 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, Y. S. Park, eds., Proc. SPIE 4776, 51-60 (2002).
[CrossRef]

I. Moreno, 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, I. E. Ashdown, eds., Proc. SPIE 5941, 359-365 (2005).

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

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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.

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.

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 uv’ 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).

Fig. 4.
Fig. 4.

Simulations with batwing-type LEDs for the arrays analyzed in Fig. 3. (a) CIE uv’ 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).

Fig. 5.
Fig. 5.

(a) Circular ring array of LEDs with 3 rings. (b) CIE uv’ 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.

Fig. 6.
Fig. 6.

(a) Rectangular RGB array with 36 side emitting LEDs. (b) CIE uv’ 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.

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 uv’ 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).

Equations (21)

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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 ) , Y i = k 400 nm 700 nm E i ( λ ) y ¯ ( λ ) , Z i = k 400 nm 700 nm E i ( λ ) z ¯ ( λ ) ,
X i = kE peak i 400 nm 700 nm U i ( λ ) x ¯ ( λ ) = kE peak i X ̂ i , Y i = kE peak i 400 nm 700 nm U i ( λ ) y ¯ ( λ ) = kE peak i Y ̂ i ,
Z i = kE peak i 400 nm 700 nm U i ( λ ) z ¯ ( λ ) = 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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