Abstract

We developed an analytical method of illuminance formation for mixed-color LEDs in a rectangular light pipe in order to derive American National Standards Institute (ANSI) light uniformity, ANSI color uniformity, and color difference of light output using photometry, nonimaging, and colorimetry. The analytical results indicate that the distributions of illuminance and color difference vary with different geometric structures of light pipes and the location of the light sources. It was found that both the ANSI light and the ANSI color uniformity on the exit plane of the light pipe are reduced exponentially with the increase in length of the light pipe. It is evident that a length scale LA greater than unity assures that the mixed-color LED sources on the entrance plane are uniformly illuminated with acceptable uniform brightness and color on the exit plane of the rectangular light pipe, where L is the length of the light pipe, and A is a constant, which is a geometric parameter for the scale unit of the light pipe's input face. Furthermore, the ANSI light uniformity can be minimized, and the ANSI color uniformity can be maximized under the condition of multilight-source locations P=Q=±A/4, where P and Q are the coordinates along the long axis and the short axis, respectively, with one being the entrance plane of the light pipe. We can conclude that the optimum form factor of the light pipe is a square shaped cross section, with the length scale LA being equal to unity and with multilight sources located individually on positions of A∕4 in order to achieve very uniform illuminations with the highest light efficiency and compact package for the optical system with mixed-color LEDs, where L is the length of the light pipe.

© 2008 Optical Society of America

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References

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  1. D. G. Pelka and K. Patel, "Invited paper: an overview of LED applications for general illumination," Proc. SPIE 5186, 15-26 (2003).
    [CrossRef]
  2. W. Cassarly, "Nonimaging optics: concentration and illumination," in OSA Handbook of Optics, 2nd ed. (McGraw-Hill, 2001) Vol. III, pp. 2.28-2.32.
  3. C. M. Chang and H. P. D. Shieh, "Design of illumination and projection optics for projectors with single digital micromirror device," Appl. Opt. 39, 3202-3208 (2000).
    [CrossRef]
  4. W. N. Partlo, P. J. Tompkins, P. G. Dewa, and P. F. Michaloski, "Depth of focus and resolution enhancement of i-line and deep-UV lithography using annular illumination," Proc. SPIE 1927, 137-157 (1993).
    [CrossRef]
  5. H. H. Hopkins, "Physics of the fiberoptics endoscope," in Endoscopy, G.Berci, ed. (Appleton-Century-Croft, 1976), pp. 27-63.
  6. T. Hough, J. F. V. Derlofske, and L. Hillman, "Management of light in thick optical waveguides for illumination: an application of radiometric principles," SAE Tech. Pap. Ser. 940512 (1994).
  7. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000), pp. 279-281.
  8. Y. K. Cheng and J. L. Chern, "Irradiance formation in hollow straight light pipes with square and circular shapes," J. Opt. Soc. Am. A 23, 427-434 (2006).
    [CrossRef]
  9. J. F. V. Derlofske and T. A. Hough, "Analytical model of flux propagation in light pipe system," Opt. Eng. 43, 1503-1510 (2004).
    [CrossRef]
  10. F. Zhao, N. Narendram, and J. F. V. Derlofske, "Optical elements for mixing colored LEDs to create white light," Proc. SPIE 4776, 206-214 (2002).
    [CrossRef]
  11. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).
  12. G. Wyszecki and W. S. Stiles, Color Science--Concept and Methods, Quantitative Data and Formulae (Wiley, 2000), Chap. 3, pp. 143-167.
    [PubMed]
  13. American National Standard for Audiovisual Systems--Electronic Projection--Fixed Resolution Projectors, ANSI/NAPM IT7.228-1997 (American National Standards Institute, 1997).
  14. C. M. Cheng and J. L. Chern, "Optical transfer functions for specific-shaped apertures generated by illumination with a rectangular light pipe," J. Opt. Soc. Am. A 23, 3123-3132 (2006).
    [CrossRef]
  15. See http://www.osram-os.com/ostar-projection/technical.php for technical information.
  16. Mathematica version 4, Wolfram Research, Inc., 100 Trade Center Drive, Champaign, Ill. 61820-7237, USA.

2006 (2)

2004 (1)

J. F. V. Derlofske and T. A. Hough, "Analytical model of flux propagation in light pipe system," Opt. Eng. 43, 1503-1510 (2004).
[CrossRef]

2003 (1)

D. G. Pelka and K. Patel, "Invited paper: an overview of LED applications for general illumination," Proc. SPIE 5186, 15-26 (2003).
[CrossRef]

2002 (1)

F. Zhao, N. Narendram, and J. F. V. Derlofske, "Optical elements for mixing colored LEDs to create white light," Proc. SPIE 4776, 206-214 (2002).
[CrossRef]

2000 (1)

1994 (1)

T. Hough, J. F. V. Derlofske, and L. Hillman, "Management of light in thick optical waveguides for illumination: an application of radiometric principles," SAE Tech. Pap. Ser. 940512 (1994).

1993 (1)

W. N. Partlo, P. J. Tompkins, P. G. Dewa, and P. F. Michaloski, "Depth of focus and resolution enhancement of i-line and deep-UV lithography using annular illumination," Proc. SPIE 1927, 137-157 (1993).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

J. F. V. Derlofske and T. A. Hough, "Analytical model of flux propagation in light pipe system," Opt. Eng. 43, 1503-1510 (2004).
[CrossRef]

Proc. SPIE (3)

F. Zhao, N. Narendram, and J. F. V. Derlofske, "Optical elements for mixing colored LEDs to create white light," Proc. SPIE 4776, 206-214 (2002).
[CrossRef]

W. N. Partlo, P. J. Tompkins, P. G. Dewa, and P. F. Michaloski, "Depth of focus and resolution enhancement of i-line and deep-UV lithography using annular illumination," Proc. SPIE 1927, 137-157 (1993).
[CrossRef]

D. G. Pelka and K. Patel, "Invited paper: an overview of LED applications for general illumination," Proc. SPIE 5186, 15-26 (2003).
[CrossRef]

SAE Tech. Pap. Ser. (1)

T. Hough, J. F. V. Derlofske, and L. Hillman, "Management of light in thick optical waveguides for illumination: an application of radiometric principles," SAE Tech. Pap. Ser. 940512 (1994).

Other (8)

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000), pp. 279-281.

H. H. Hopkins, "Physics of the fiberoptics endoscope," in Endoscopy, G.Berci, ed. (Appleton-Century-Croft, 1976), pp. 27-63.

R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).

G. Wyszecki and W. S. Stiles, Color Science--Concept and Methods, Quantitative Data and Formulae (Wiley, 2000), Chap. 3, pp. 143-167.
[PubMed]

American National Standard for Audiovisual Systems--Electronic Projection--Fixed Resolution Projectors, ANSI/NAPM IT7.228-1997 (American National Standards Institute, 1997).

W. Cassarly, "Nonimaging optics: concentration and illumination," in OSA Handbook of Optics, 2nd ed. (McGraw-Hill, 2001) Vol. III, pp. 2.28-2.32.

See http://www.osram-os.com/ostar-projection/technical.php for technical information.

Mathematica version 4, Wolfram Research, Inc., 100 Trade Center Drive, Champaign, Ill. 61820-7237, USA.

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Figures (14)

Fig. 1
Fig. 1

Schematic and dimension of the light pipe and the single LED light source.

Fig. 2
Fig. 2

(a) Principle of the operation of a light pipe. (b) Virtual image at the entrance of the light pipe. The light pipe is made with parallel reflective sides with a rectangular cross section. The multiple reflections of the light source through the pipe can produce a spatial checkerboard-array-shaped light distribution.

Fig. 3
Fig. 3

(a) Schematic and dimension of the light pipe. (b) Locations of red, green, and blue LED light sources, i.e., mixed-color LEDs, on the entrance of the light pipe.

Fig. 4
Fig. 4

Illustration of an LED light source radiating into the exit plane of the light pipe for the different virtual light spots on the entrance plane of the light pipe.

Fig. 5
Fig. 5

Illustration of a Lambertian light source radiating into the exit plane of the light pipe for the different virtual light spots on the entrance plane of the light pipe.

Fig. 6
Fig. 6

Measurement locations at the center of nine equal rectangles of a 100% exit plane of a light pipe. The four corner points 10, 11, 12, and 13 are located at 10% of the distance from the corner itself to the center of the measurement location 5 [13].

Fig. 7
Fig. 7

Relative spectral concentrations of the radiant powers S R ( λ ) , S G ( λ ) , and S B ( λ ) . For OSTAR—Projection (Type name: LE ATB A2A) as our LED light sources.

Fig. 8
Fig. 8

Distributions and contours of illuminance for single LEDs under the condition of P = Q = 0 and a = b = A with (a) and (b) L = 0.1 A , (c) and (d) L = 0.5 A , (e) and (f) L = 1.0 A . Also, the variation of ANSI light uniformity versus the length L of a light pipe with (g) the linear chart and (h) exponential chart.

Fig. 9
Fig. 9

Distributions and contours of illuminance for mixed-color LEDs in the condition of P = Q = A / 4 and a = b = A with (a) and (b) L = 0.1 A , (c) and (d) L = 0.5 A , (e) and (f) L = 2.0 A . Also, the variation of ANSI light uniformity versus the length L of a light pipe with (g) the linear chart and (h) exponential chart.

Fig. 10
Fig. 10

Distributions and contours of the color difference for mixed-color LEDs in the condition of P = Q = A / 4 and a = b = A with (a) and (b) L = 0.1 A , (c) and (d) L = 0.5 A , (e) and (f) L = 1.0 A , (g) and (h) L = 2.0 A . Also, (i) the variation of ANSI color uniformity versus the length L of a light pipe with the linear chart and exponential chart.

Fig. 11
Fig. 11

Distributions and contours of illuminance for mixed-color LEDs in the condition of P = Q = A / 4 and a = L = A with (a) and (b) b = 1.5 A , (c) and (d) b = 2.0 A , (e) and (f) b = 3.0 A . Also, the variation of ANSI light uniformity versus the height b of a light pipe with (g) the linear chart and (h) exponential chart.

Fig. 12
Fig. 12

Distributions and contours of the color difference for mixed-color LEDs in the condition of P = Q = A / 4 and a = L = A and (a) and (b) b = 1.5 A , (c) and (d) b = 2.0 A , (e) and (f) b = 3.0 A . Also, (g) the variation of ANSI color uniformity versus the height b of a light pipe with the linear chart and exponential chart.

Fig. 13
Fig. 13

Distributions and contours of illuminance for mixed-color LEDs in the condition of a = b = L = A with (a) and (b) P = Q = A / 8 , (c) and (d) P = Q = A / 4 , (e) and (f) P = Q = 3 A / 8 A , (g) and (h) P = Q = 1 A . Also, (i) the variation of ANSI light uniformity versus the locations P and Q of a light pipe with the linear chart and exponential chart.

Fig. 14
Fig. 14

Distributions and contours of the color difference for mixed-color LEDs in the condition of a = b = L = A with (a) and (b) P = Q = A / 8 , (c) and (d) P = Q = A / 4 , (e) and (f) P = Q = 3 A / 8 , (g) and (h) P = Q = 1 A . Also, (i) the variation of ANSI color uniformity versus the locations P and Q of a light pipe with the linear chart and exponential chart.

Equations (33)

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J Ω = J ( θ ) = J 0   cos   θ ,   90 ° θ 90 ° ,
cos   θ = L L 2 + ( x 2 + y 2 ) = L R .
H ( x , y ) = J ( θ ) cos 3 θ L 2 = J 0 × L 2 ( L 2 + x 2 + y 2 ) 2 .
H 00 ( x , y ) = J 0 × L 2 ( L 2 + x 2 + y 2 ) 2 ,   a 2 x a 2 , b 2 y b 2 .
H 10 ( x , y ) = J 0 × ρ L 2 ( L 2 + x 2 + y 2 ) 2 ,   a 2 x 3 a 2 , b 2 y b 2 ,
H 20 ( x , y ) = J 0 × ρ 2 L 2 [ L 2 + x 2 + y 2 ] 2 ,   3 a 2 x 5 a 2 , b 2 y b 2 ,
H 10 ( x , y ) = J 0 × ρ L 2 [ L 2 + ( x + a ) 2 + y 2 ] 2 , a 2 x a 2 , b 2 y b 2 ,
H 20 ( x , y ) = J 0 × ρ 2 L 2 [ L 2 + ( x + 2 a ) 2 + y 2 ] 2 ,
a 2 x a 2 , b 2 y b 2 .
H ± 1 ± 1 ( x , y ) = J 0 × ρ ρ L 2 [ L 2 + ( x ± a ) 2 + ( y ± b ) 2 ] 2 ,   a 2 x a 2 , b 2 y b 2 ,
H ± 2 ± 2 ( x , y ) = J 0 × ρ | ± 2 | ρ | ± 2 | ρ L 2 [ L 2 + ( x ± 2 a ) 2 + ( y ± 2 b ) 2 ] 2 ,
a 2 x a 2 , b 2 y b 2 .
H 0 ( x , y ) = j = i = J 0 ρ | i | + | j | L 2 { L 2 + [ ( 1 ) i x + i a ] 2 + [ ( 1 ) j y + j b ] 2 } 2 ,   a 2 x a 2 , b 2 y b 2 ,
H G 1 ( x , y ) = H 0 ( x P , y Q ) ,
H G 2 ( x , y ) = H 0 ( x + P , y + Q ) ,
H R ( x , y ) = H 0 ( x P , y + Q ) ,
H B ( x , y ) = H 0 ( x + P , y Q ) ,
H T ( x , y ) = H G 1 ( x , y ) + H G 1 ( x , y ) + H R ( x , y ) + H B ( x , y ) .
U r + = ( Maximum [ H T ( x l , y l ) ] l = 10,11,12,13 Average [ H T ( x l , y l ) ] l = 1,2,     ,9 1 ) × 100 % ,
U r = ( Minimum [ H T ( x l , y l ) ] l = 10,11,12,13 Average [ H T ( x l , y l ) ] l = 1,2,     ,9 1 ) × 100 % ,
X m ( x , y ) = k H m ( x , y ) × λ β m ( λ ) S m ( λ ) x ¯ ( λ ) Δ λ ,
Y m ( x , y ) = k H m ( x , y ) × λ β m ( λ ) S m ( λ ) y ¯ ( λ ) Δ λ ,   m = R , G 1 , G 2 , B ,
Z m ( x , y ) = k H m ( x , y ) × λ β m ( λ ) S m ( λ ) z ¯ ( λ ) Δ λ ,
X w ( x , y ) = X R ( x , y ) + X G 1 ( x , y ) + X G 2 ( x , y ) + X B ( x , y ) ,
Y w ( x , y ) = Y R ( x , y ) + Y G 1 ( x , y ) + Y G 2 ( x , y ) + Y B ( x , y ) ,
Z w ( x , y ) = Z R ( x , y ) + Z G 1 ( x , y ) + Z G 2 ( x , y ) + Z B ( x , y ) ,
u ( x , y ) = 4 X w X w + 15 Y w + 3 Z w ,
v ( x , y ) = 9 X w X w + 15 Y w + 3 Z w .
Δ u v = [ ( u 1 u 0 ) 2 + ( v 1 v 0 ) 2 ] 1 / 2 ,
Δ E a b * ( x , y ) = 200 [ ( Δ L * ) 2 + ( Δ a * ) 2 + ( Δ b * ) 2 ] 1 / 2 ,
L * = 116 ( Y w Y n ) 1 / 3 16 ,
a * = 500 [ ( X w X n ) 1 / 3 ( Y w Y n ) 1 / 3 ] ,
b * = 200 [ ( Y w Y n ) 1 / 3 ( Z w Z n ) 1 / 3 ] 1 / 2 ,

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