Abstract

We investigated, both analytically and numerically, the irradiance formation of an asymmetrically located Lambertian light source in hollow straight light pipes with square and circular shapes. The uniform irradiance distribution in a square light pipe and hot-spot localization in a circular light pipe were examined and determined semianalytically. Typical factors of influence, such as light-pipe length, width, and source size, were identified with extensive simulation. When the ratio of light-pipe length and width was less than 0.5, the deviation from uniformity could be more than 20%. But once the source size was large enough (approximately half of the incident port), such that the Lambertian characteristics of the source dominated the irradiance distribution, the uniformity deviation was reduced. Furthermore, a quantity of root-mean-square circular differences was defined in order to identify the shape deformation of the light pipe; it was found that the peak value of the hot spot decreased exponentially with the deformation scale. The influence of nonperfect reflectivity of the pipe wall on irradiance formation was also examined for a square light pipe; when the reflectivity is larger than 90%, the difference in uniformity is less than 10% and uniform irradiance remains, provided that the ratio of light-pipe length and width is larger than 1; even the source is located asymmetrically.

© 2006 Optical Society of America

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References

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  1. K. K. Li, "Illumination engine for a projection display using a tapered light pipe," U.S. patent 6,739,726 (25 May 2004).
  2. N. Takahashi and S. Umemoto, "Liquid crystal display apparatus having light pipe with reflective polarizer," U.S. patent 6,778,235 (17 August 2004).
  3. J. Lee and J. E. Greivenkamp, "Modeling of automotive interior illumination systems," Opt. Eng. (Bellingham) 43, 1537-1544 (2004).
    [CrossRef]
  4. H. Delattre, "Motor vehicle headlight with light pipe," U.S. patent 6,547,428 (15 April 2003).
  5. J. F. V. Derlofske and T. A. Hough, "Analytical model of flux propagation in light-pipe systems," Opt. Eng. (Bellingham) 43, 1503-1510 (2004).
    [CrossRef]
  6. A. Gupta, J. Lee, and R. J. Koshel, "Design of efficient lightpipes for illumination by an analytical approach," Appl. Opt. 40, 3640-3648 (2001).
    [CrossRef]
  7. D. G. Hawthorn and T. Timusk, "Transmittance of skew rays through metal light pipes," Appl. Opt. 38, 2787-2794 (1999).
    [CrossRef]
  8. E. Fu, "Transmission of submillimeter waves through metal light pipes," J. Opt. Soc. Am. B 13, 702-705 (1996).
    [CrossRef]
  9. F. Zhao, N. Narendran, and J. Van Derlofske, "Optical elements for mixing colored LEDs to create white light," in Solid State Lighting II, I.T.Ferguson, N.Narendran, S.P.Den Baars, and Y.-S.Park, eds., Proc. SPIE 4776, 206-214 (2003).
  10. More information on the TracePro software can be found at http://www.lambdares.com.

2004 (2)

J. Lee and J. E. Greivenkamp, "Modeling of automotive interior illumination systems," Opt. Eng. (Bellingham) 43, 1537-1544 (2004).
[CrossRef]

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

2001 (1)

1999 (1)

1996 (1)

Delattre, H.

H. Delattre, "Motor vehicle headlight with light pipe," U.S. patent 6,547,428 (15 April 2003).

Derlofske, J. F. V.

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

Fu, E.

Greivenkamp, J. E.

J. Lee and J. E. Greivenkamp, "Modeling of automotive interior illumination systems," Opt. Eng. (Bellingham) 43, 1537-1544 (2004).
[CrossRef]

Gupta, A.

Hawthorn, D. G.

Hough, T. A.

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

Koshel, R. J.

Lee, J.

J. Lee and J. E. Greivenkamp, "Modeling of automotive interior illumination systems," Opt. Eng. (Bellingham) 43, 1537-1544 (2004).
[CrossRef]

A. Gupta, J. Lee, and R. J. Koshel, "Design of efficient lightpipes for illumination by an analytical approach," Appl. Opt. 40, 3640-3648 (2001).
[CrossRef]

Li, K. K.

K. K. Li, "Illumination engine for a projection display using a tapered light pipe," U.S. patent 6,739,726 (25 May 2004).

Narendran, N.

F. Zhao, N. Narendran, and J. Van Derlofske, "Optical elements for mixing colored LEDs to create white light," in Solid State Lighting II, I.T.Ferguson, N.Narendran, S.P.Den Baars, and Y.-S.Park, eds., Proc. SPIE 4776, 206-214 (2003).

Takahashi, N.

N. Takahashi and S. Umemoto, "Liquid crystal display apparatus having light pipe with reflective polarizer," U.S. patent 6,778,235 (17 August 2004).

Timusk, T.

Umemoto, S.

N. Takahashi and S. Umemoto, "Liquid crystal display apparatus having light pipe with reflective polarizer," U.S. patent 6,778,235 (17 August 2004).

Van Derlofske, J.

F. Zhao, N. Narendran, and J. Van Derlofske, "Optical elements for mixing colored LEDs to create white light," in Solid State Lighting II, I.T.Ferguson, N.Narendran, S.P.Den Baars, and Y.-S.Park, eds., Proc. SPIE 4776, 206-214 (2003).

Zhao, F.

F. Zhao, N. Narendran, and J. Van Derlofske, "Optical elements for mixing colored LEDs to create white light," in Solid State Lighting II, I.T.Ferguson, N.Narendran, S.P.Den Baars, and Y.-S.Park, eds., Proc. SPIE 4776, 206-214 (2003).

Appl. Opt. (2)

J. Opt. Soc. Am. B (1)

Opt. Eng. (Bellingham) (2)

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

J. Lee and J. E. Greivenkamp, "Modeling of automotive interior illumination systems," Opt. Eng. (Bellingham) 43, 1537-1544 (2004).
[CrossRef]

Other (5)

H. Delattre, "Motor vehicle headlight with light pipe," U.S. patent 6,547,428 (15 April 2003).

K. K. Li, "Illumination engine for a projection display using a tapered light pipe," U.S. patent 6,739,726 (25 May 2004).

N. Takahashi and S. Umemoto, "Liquid crystal display apparatus having light pipe with reflective polarizer," U.S. patent 6,778,235 (17 August 2004).

F. Zhao, N. Narendran, and J. Van Derlofske, "Optical elements for mixing colored LEDs to create white light," in Solid State Lighting II, I.T.Ferguson, N.Narendran, S.P.Den Baars, and Y.-S.Park, eds., Proc. SPIE 4776, 206-214 (2003).

More information on the TracePro software can be found at http://www.lambdares.com.

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Figures (10)

Fig. 1
Fig. 1

(a) Ray transported in a straight light pipe, (b) ray transported in a folding light pipe, (c) functional graph of Eq. (1), (d) irradiance distribution of a one-dimensional source at the light-pipe output.

Fig. 2
Fig. 2

(a) Ray transported in a straight light pipe when source position has a shift distance d, (b) ray transported in a straight light pipe when the incident angle is negative, (c) irradiance distribution of two-dimensional half-angle and full-angle sources at light-pipe output when the source position has a shift distance d, (d) simulation results of a one-dimensional source at a light-pipe output for half-angle and full-angle Lambertian sources.

Fig. 3
Fig. 3

Schematic diagram of a two-dimensional Lambertian skew ray propagated in a three-dimensional light pipe: (a) square case, (b) circular case.

Fig. 4
Fig. 4

Schematic diagram of pixel density for a circular light pipe.

Fig. 5
Fig. 5

Irradiance at different locations h for a two-dimensional light pipe with length scale L D : (a) analytical result, (b) and (c) simulation results.

Fig. 6
Fig. 6

(a) and (b) Irradiance distribution of L D = 5 light pipes with sources in different incident positions. (a) Source at input center, (b) source has a horizontal shift, (c) cross-section plots in either the x or y directions of (a).

Fig. 7
Fig. 7

Simulation result of a circular light pipe: (a) hot-spot irradiance distribution of a circular light pipe, (b) cross-section profile.

Fig. 8
Fig. 8

(a) Uniformity deviation versus light-pipe scale L D for a square light pipe; (b) and (c) are the irradiance distributions with different source size S = 0.01 and 0.5, respectively, where L D = 0.5 .

Fig. 9
Fig. 9

(a) Uniformity deviation versus reflectivity with different source sizes, where L D = 0.5 , P = ( 1 , 1 ) ; (b) uniformity deviation versus L D where S = 0.01 and 0.2 with two different reflectivities R = 99 % and 90% are shown, where P = ( 1 , 1 ) ; (c) uniformity versus L D where S = 0.01 and 0.2 with three different source positions P = ( 1 , 1 ) , (0.5,1), and (0.5,0.5), where reflectivity R is 90%.

Fig. 10
Fig. 10

Simulation result for a modified-shape light pipe. (a) Schematic diagram of a modified-shape light pipe; (b) influence of the deformation scale on the hot-spot peak value; (c)–(f) distribution of light at the output port when α is (c) 10°, (d) 20°, (e) 30°, (f) 40°.

Equations (10)

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h ( θ ) = k + ( 1 ) m ( L tan θ m D ) ,
E ( h 0 ) = i = 1 cos θ i .
h ( θ , d ) = k + ( 1 ) m ( L tan θ + d m D ) ,
h ( θ , d ) = k + ( 1 ) m ( L tan θ d m D ) ,
h ( θ , d ) = D h ( θ , D d ) ,
E ( h ) full angle = E ( h ) + θ + E ( h ) θ ,
P x ( θ , ϕ ) , P y ( θ , ϕ ) = [ h ( θ ) sin ϕ , h ( θ ) cos ϕ ] .
E ( R ) = total flux of one circle total area of one circle = ( F N d ) N s π R 2 π ( R Δ D ) 2 = F D Δ D 2 π Δ ϕ 2 π R Δ D π Δ D 2 = ( F D Δ ϕ 1 R ) ( 1 Δ D 2 R ) 1 ( F D Δ ϕ 1 R ) [ 1 + Δ D 2 R ( Δ D 2 R ) 2 + ] .
δ = [ 1 n i = 1 n E i E ¯ ] E ¯ ,
Δ = 1 2 π 0 2 π ( D m ( θ ) R s ) 2 d θ ,

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