Abstract

An analytical method of researching the light distribution properties on the output end of a hollow n-sided polygonal light pipe and a light source with a Gaussian distribution is developed. The mirror transformation matrices and a special algorithm of removing void virtual images are created to acquire the location and direction vector of each effective virtual image on the entrance plane. The analytical method is demonstrated by Monte Carlo ray tracing. At the same time, four typical cases are discussed. The analytical results indicate that the uniformity of light distribution varies with the structural and optical parameters of the hollow n-sided polygonal light pipe and light source with a Gaussian distribution. The analytical approach will be useful to design and choose the hollow n-sided polygonal light pipe, especially for high-power laser beam homogenization techniques.

© 2013 Optical Society of America

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  1. B. Kohler, A. Noeske, T. Kindervater, A. Wessollek, T. Brand, and J. Biesenbach, “11 kW direct diode laser system with homogenized 55×20  mm2 top-hat intensity distribution,” Proc. SPIE 6456, 64560O (2007).
    [CrossRef]
  2. F. Zhang, C. Wang, R. Geng, Z. Tong, T. Ning, and S. Jan, “Anamorphic beam concentrator for linear laser-diode bar,” Opt. Express 15, 17038–17043 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. Y.-K. Cheng and J.-L. Chern, “Irradiance formations in hollow straight light pipes with square and circular shapes,” J. Opt. Soc. Am. A 23, 427–434 (2006).
    [CrossRef]
  7. I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Proc. SPIE 743074300M (2004).
    [CrossRef]
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    [CrossRef]
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  15. I. Moreno, “Output irradiance of tapered lightpipes,” J. Opt. Soc. Am. A 27, 1985–1993 (2010).
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  16. Y. Li, “Beam shaping by superposition of fundamental mode Gaussian beams,” Proc. SPIE 5525, 128–137 (2004).
    [CrossRef]

2010 (3)

2008 (2)

2007 (3)

F. Zhang, C. Wang, R. Geng, Z. Tong, T. Ning, and S. Jan, “Anamorphic beam concentrator for linear laser-diode bar,” Opt. Express 15, 17038–17043 (2007).
[CrossRef]

B. Kohler, A. Noeske, T. Kindervater, A. Wessollek, T. Brand, and J. Biesenbach, “11 kW direct diode laser system with homogenized 55×20  mm2 top-hat intensity distribution,” Proc. SPIE 6456, 64560O (2007).
[CrossRef]

Y. Qu, J. R. Howell, and O. A. Ezekoye, “Monte Carlo modeling of a light-pipe radiation thermometer,” IEEE Trans. Semicond. Manuf. 20, 39–50 (2007).
[CrossRef]

2006 (1)

2004 (3)

Y. Li, “Beam shaping by superposition of fundamental mode Gaussian beams,” Proc. SPIE 5525, 128–137 (2004).
[CrossRef]

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

I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Proc. SPIE 743074300M (2004).
[CrossRef]

2003 (1)

2002 (1)

2001 (1)

1963 (1)

Abdou-Ahmed, M.

Akiyama, D.

Bartnicki, E.

Berkowitz-Mattuck, J. B.

Biesenbach, J.

B. Kohler, A. Noeske, T. Kindervater, A. Wessollek, T. Brand, and J. Biesenbach, “11 kW direct diode laser system with homogenized 55×20  mm2 top-hat intensity distribution,” Proc. SPIE 6456, 64560O (2007).
[CrossRef]

Bourdet, G. L.

Brand, T.

B. Kohler, A. Noeske, T. Kindervater, A. Wessollek, T. Brand, and J. Biesenbach, “11 kW direct diode laser system with homogenized 55×20  mm2 top-hat intensity distribution,” Proc. SPIE 6456, 64560O (2007).
[CrossRef]

Cassarly, W. J.

Chen, M. M.

Cheng, C.-M.

Cheng, Y.-K.

Chern, J.-L.

Ezekoye, O. A.

Y. Qu, J. R. Howell, and O. A. Ezekoye, “Monte Carlo modeling of a light-pipe radiation thermometer,” IEEE Trans. Semicond. Manuf. 20, 39–50 (2007).
[CrossRef]

Forbes, A.

I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Proc. SPIE 743074300M (2004).
[CrossRef]

Fournier, F.

Geng, R.

Glaser, P. E.

Gouveia, H.

Graf, T.

Gupta, A.

Hough, T. A.

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

Howell, J. R.

Y. Qu, J. R. Howell, and O. A. Ezekoye, “Monte Carlo modeling of a light-pipe radiation thermometer,” IEEE Trans. Semicond. Manuf. 20, 39–50 (2007).
[CrossRef]

Jan, S.

Kindervater, T.

B. Kohler, A. Noeske, T. Kindervater, A. Wessollek, T. Brand, and J. Biesenbach, “11 kW direct diode laser system with homogenized 55×20  mm2 top-hat intensity distribution,” Proc. SPIE 6456, 64560O (2007).
[CrossRef]

Kohler, B.

B. Kohler, A. Noeske, T. Kindervater, A. Wessollek, T. Brand, and J. Biesenbach, “11 kW direct diode laser system with homogenized 55×20  mm2 top-hat intensity distribution,” Proc. SPIE 6456, 64560O (2007).
[CrossRef]

Koshel, R. J.

Kreske, K.

Lee, J.

Li, Y.

Y. Li, “Beam shaping by superposition of fundamental mode Gaussian beams,” Proc. SPIE 5525, 128–137 (2004).
[CrossRef]

Liang, D.

Litvin, I. A.

I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Proc. SPIE 743074300M (2004).
[CrossRef]

Matsuura, Y.

Miyagi, M.

Morais, P. J.

Moreno, I.

Ning, T.

Noeske, A.

B. Kohler, A. Noeske, T. Kindervater, A. Wessollek, T. Brand, and J. Biesenbach, “11 kW direct diode laser system with homogenized 55×20  mm2 top-hat intensity distribution,” Proc. SPIE 6456, 64560O (2007).
[CrossRef]

Pereira, R.

Qu, Y.

Y. Qu, J. R. Howell, and O. A. Ezekoye, “Monte Carlo modeling of a light-pipe radiation thermometer,” IEEE Trans. Semicond. Manuf. 20, 39–50 (2007).
[CrossRef]

Rolland, J. P.

Tong, Z.

Van Derlofske, J. F.

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

Voss, A.

Wang, C.

Weichelt, B.

Wessollek, A.

B. Kohler, A. Noeske, T. Kindervater, A. Wessollek, T. Brand, and J. Biesenbach, “11 kW direct diode laser system with homogenized 55×20  mm2 top-hat intensity distribution,” Proc. SPIE 6456, 64560O (2007).
[CrossRef]

Zhang, F.

Appl. Opt. (7)

IEEE Trans. Semicond. Manuf. (1)

Y. Qu, J. R. Howell, and O. A. Ezekoye, “Monte Carlo modeling of a light-pipe radiation thermometer,” IEEE Trans. Semicond. Manuf. 20, 39–50 (2007).
[CrossRef]

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

Opt. Eng. (1)

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

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (3)

B. Kohler, A. Noeske, T. Kindervater, A. Wessollek, T. Brand, and J. Biesenbach, “11 kW direct diode laser system with homogenized 55×20  mm2 top-hat intensity distribution,” Proc. SPIE 6456, 64560O (2007).
[CrossRef]

I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Proc. SPIE 743074300M (2004).
[CrossRef]

Y. Li, “Beam shaping by superposition of fundamental mode Gaussian beams,” Proc. SPIE 5525, 128–137 (2004).
[CrossRef]

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Figures (18)

Fig. 1.
Fig. 1.

Principle of operation of a light pipe and virtual images on the entrance of the light pipe mentioned by Cheng and Chern [14].

Fig. 2.
Fig. 2.

Schematic of a regular octagonal light pipe and a light source in Cartesian coordinates.

Fig. 3.
Fig. 3.

First layer image process and light distribution on the output end of a regular octagonal light pipe: (a) first layer image process and (b) superposition of irradiances on the output end.

Fig. 4.
Fig. 4.

Effective and void virtual images imaged from the first side of an octagonal light pipe for the two layer image process.

Fig. 5.
Fig. 5.

Arithmetic for removing the second kind of void virtual images in an octagonal light pipe: (a) first step, (b) second step, (c) third step, and (d) fourth step.

Fig. 6.
Fig. 6.

Effective virtual images and virtual octagon images on the entrance plane of a light pipe with different image layers: (a) 1 layer image, (b) 2 layer image, and (c) 3 layer image.

Fig. 7.
Fig. 7.

Effective virtual images and virtual polygon images at the entrance plane of the light pipe with four layer images for different numbers of sides: (a) triangle, (b) square, (c) pentagon, (d) hexagon, (e) heptagon, (f) octagon, (g) enneagon, (h) decagon, and (i) hendecagon.

Fig. 8.
Fig. 8.

Light distribution pattern of a square light pipe with length L=10mm (light source is located on the center of the input end and the transmission direction is parallel to the Z axis). (a) Light distribution generated with the mathematic model, (b) light distribution generated with Monte Carlo ray tracing but with only 106 rays.

Fig. 9.
Fig. 9.

Light distribution pattern of a square light pipe with length L=10mm [light source is located on the corner of the input end (x0=0.15, x0=0.15) and the transmission direction is parallel to the Z axis]. (a) Light distribution generated with the mathematic model, (b) light distribution generated with Monte Carlo ray tracing but with only 106 rays.

Fig. 10.
Fig. 10.

Light distribution pattern of a square light pipe with length L=10mm (light source is located on the center of the input end and the vector of the transmission direction is [0.3, 0, 1]). (a) Light distribution generated with the mathematic model, (b) light distribution generated with Monte Carlo ray tracing but with only 106 rays.

Fig. 11.
Fig. 11.

Light distribution pattern of a regular triangular light pipe with length (a) L=5mm, (b) L=20mm, and (c) L=50mm (light source is located on the center of the input end and the transmission direction is parallel to the Z axis).

Fig. 12.
Fig. 12.

Light distribution pattern of a pentagonal light pipe with length (a) L=5mm, (b) L=20mm, and (c) L=50mm (light source is located on the center of the input end and the transmission direction is parallel to the Z axis).

Fig. 13.
Fig. 13.

Standard deviation of the light distribution versus the length of different polygonal light pipes (light source is located on the center of the input end and the transmission direction is parallel to the Z axis).

Fig. 14.
Fig. 14.

Light distribution pattern on the output end of a square light pipe with a light source located on the corner of the input end (x0=0.15, y0=0.15), and the transmission direction is parallel to the Z axis for different lengths (a) L=5mm, (b) L=20mm, and (c) L=100mm.

Fig. 15.
Fig. 15.

Light distribution pattern on the output end of a square light pipe with a light source located on the center of the input end, and the vector of the transmission direction is [0.3, 0, 1] for different lengths (a) L=5mm, (b) L=20mm, and (c) L=100mm.

Fig. 16.
Fig. 16.

Light distribution pattern on the output end of a square light pipe with a light source located on the corner of the input end (x0=0.15, y0=0.15), and the vector of the transmission direction is [0, 0.3, 1] for different lengths (a) L=5mm, (b) L=20mm, and (c) L=100mm.

Fig. 17.
Fig. 17.

Light distribution pattern on the output end of an octagonal light pipe with a light source with a 10° divergence angle for different lengths (a) L=5mm, (b) L=20mm, and (c) L=50mm.

Fig. 18.
Fig. 18.

Light distribution pattern on the output end of an octagonal light pipe with a light source with a 30° divergence angle for different lengths (a) L=5mm, (b) L=20mm, and (c) L=50mm.

Tables (1)

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Table 1. Basic Parameters of this Mathematic Model

Equations (16)

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{xk=R·cos[2πN(k1)]yk=R·sin[2πN(k1)],
Aix+Biy+Ci=0.
{Ai=yi+1yiBi=xixi+1Ci=xi+1yixiyi+1(i=1,2,N1){AN=y1yNBN=xNx1CN=x1yNxNy1(i=N),
V(x,y,z)=cw(z)exp(r2w2(z))exp{i[k(z+r22R)artctg(zf)]},
{r2=x2+y2k=2πf,f=πw02λ,w(z)=w01+(zf)2R=R(z)=z+f2z,
V(x,y,z)=cw(z)exp(r2w2(z))exp{i[k(z+r22R)arctg(zf)]},
{r2=x2+y2k=2πf,f=πw02λ,w(z)=w01+(zf)2R=R(z)=z+f2z
x=x·cosγ·cosβy·sinγ·cosβ+z·sinβ+cosβ·(txcosγtysinγ)+tzsinβ=cosβ·[(x+tx)·cosγ(y+ty)·sinγ]+(z+tz)·sinβ,y=x·sinγ+y·cosγ+txsinγ+tycosγ=(x+tx)·sinγ+(y+ty)·cosγ,z=x·sinβ·cosγ+y·sinβ·sinγ+z·cosβsinβ·(txcosγtysinγ)+tzcosβ=sinβ·[(x+tx)·cosγ+(y+ty)·sinγ]+(z+tz)cosβ,sinβ=a2+b2a2+b2+c2;cosβ=ca2+b2+c2;sinγ=ba2+b2cosγ=aa2+b2;tx=x0,ty=y0,tz=0.
V(x,y,L)Total=VSource+VImages,
Pj+1,i=Pj,i×IMgTrMLj,iPj,iobjPj,iLout,
IMgTrMLj,i=[cos(2αi)sin(2αi)0sin(2αi)cos(2αi)0CiAi[cos(2αi)1]CiAisin(2αi)1](Ai0),IMgTrMLj,i=[10001002CiBi1](Ai=0),
JD=A1×xj+1,1+B1×yj+1,1+C1{>0=0<0.
Vertexj,i=Vertexj,i×IMgTrMLj,iVertexj,iobjVertexj,iLout,
n⃗p=[pAipBipCi],
LVj+1,i=LVj,i×IMgTrMPj,i,
IMgTrMPj,i=[12pAj,i22pAj,i·pBj,i2pAj,i·pCj,i2pAj,i·pBj,i12pBj,i22pBj,i·pCj,i2pAj,i·pCj,i2pBj,i·pCj,i12pCj,i2],

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