Abstract

LED lighting has been a strongly growing field for the last decade. The outstanding features of LED, like compactness and low operating temperature take the control of light distributions to a new level. Key for this is the development of sophisticated optical elements that distribute the light as intended. The optics design method known as tailoring relies on the point source assumption. This assumption holds as long as the optical element is large compared to the LED chip. With chip sizes of 1 mm2 this is of no concern if each chip is endowed with its own optic. To increase the power of a luminaire, LED chips are arranged to form light engines that reach several cm in diameter. In order to save costs and space it is often desirable to use a single optical element for the light engine. At the same time the scale of the optics must not be increased in order to trivially keep the point source assumption valid. For such design tasks point source algorithms are of limited usefulness. New methods that take into account the extent of the light source have to be developed. We present two such extended source methods. The first method iteratively adapts the target light distribution that is fed into a points source method while the second method employs a full phase space description of the optical system.

© 2014 Optical Society of America

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References

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  1. S. J. Schruben, “Formulation of a Reflector-Design Problem for a Lighting Fixture,” J. Opt. Soc. Am. 62, 1498–1501 (1972).
    [CrossRef]
  2. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 590–595 (2002).
    [CrossRef]
  3. B. Parkyn and D. Pelka, “Free-form illumination lenses designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE 6338, 633808 (2006).
    [CrossRef]
  4. A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, Fresnel-type freeform optics,” Proc. SPIE 8485, 84850H (2012).
    [CrossRef]
  5. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20, 14477–14485 (2012).
    [CrossRef] [PubMed]
  6. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Optimization of single reflectors for extended sources,” Proc. of SPIE 7103, 71030I (2008).
    [CrossRef]
  7. P. Benítez, J.-C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” SPIE Proc. Optical Engineering 43, 1489–1502 (2004).
  8. A. Rabl and J. Gordon, “Reflector design for illumination with extended sources: the basic solutions,” Applied Optics 33, 6012–6021 (1994).
    [CrossRef] [PubMed]
  9. T. Glimm and V. Oliker, “Optical design of single reflector systems and the mongekantorovich mass transfer problem,” Journal of Mathematical Sciences 117, 4096–4108 (2003).
    [CrossRef]
  10. S. Seroka and S. Sertl, “Modeling of refractive freeform surfaces by a nonlinear PDE for the generation of a given target light distribution,” International Light Simulation Symposium (2012).
  11. R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation.” Optics letters 38, 229–231 (2013).
    [CrossRef]
  12. S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering dataII. Numerical solution,” Numer. Math. 79, 553–568 (1998).
    [CrossRef]
  13. F. R. Fournier, W. J. Cassarly, and J. P. Roland, “Designing freeform reflectors for extended sources,” Proc. of SPIE 7423, 743202 (2009).
  14. S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal Mass Transport for Registration and Warping,” Int. J. of Comput. Vision 60, 225–240 (2004).
    [CrossRef]
  15. A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21, 10563–10571 (2013).
    [CrossRef] [PubMed]
  16. J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE 6670, 66700A (2007).
    [CrossRef]
  17. R. Wester, A. Bruneton, A. Bäuerle, J. Stollenwerk, and P. Loosen, “Irradiance tailoring for extended sources using a point-source freeform design algorithm,” Proc. of SPIE, Optical Systems Design 8550, 85502S(2012).
    [CrossRef]

2013

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation.” Optics letters 38, 229–231 (2013).
[CrossRef]

A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21, 10563–10571 (2013).
[CrossRef] [PubMed]

2012

R. Wester, A. Bruneton, A. Bäuerle, J. Stollenwerk, and P. Loosen, “Irradiance tailoring for extended sources using a point-source freeform design algorithm,” Proc. of SPIE, Optical Systems Design 8550, 85502S(2012).
[CrossRef]

A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, Fresnel-type freeform optics,” Proc. SPIE 8485, 84850H (2012).
[CrossRef]

A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20, 14477–14485 (2012).
[CrossRef] [PubMed]

2009

F. R. Fournier, W. J. Cassarly, and J. P. Roland, “Designing freeform reflectors for extended sources,” Proc. of SPIE 7423, 743202 (2009).

2008

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Optimization of single reflectors for extended sources,” Proc. of SPIE 7103, 71030I (2008).
[CrossRef]

2007

J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE 6670, 66700A (2007).
[CrossRef]

2006

B. Parkyn and D. Pelka, “Free-form illumination lenses designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE 6338, 633808 (2006).
[CrossRef]

2004

P. Benítez, J.-C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” SPIE Proc. Optical Engineering 43, 1489–1502 (2004).

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal Mass Transport for Registration and Warping,” Int. J. of Comput. Vision 60, 225–240 (2004).
[CrossRef]

2003

T. Glimm and V. Oliker, “Optical design of single reflector systems and the mongekantorovich mass transfer problem,” Journal of Mathematical Sciences 117, 4096–4108 (2003).
[CrossRef]

2002

1998

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering dataII. Numerical solution,” Numer. Math. 79, 553–568 (1998).
[CrossRef]

1994

A. Rabl and J. Gordon, “Reflector design for illumination with extended sources: the basic solutions,” Applied Optics 33, 6012–6021 (1994).
[CrossRef] [PubMed]

1972

Angenent, S.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal Mass Transport for Registration and Warping,” Int. J. of Comput. Vision 60, 225–240 (2004).
[CrossRef]

Bäuerle, A.

A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21, 10563–10571 (2013).
[CrossRef] [PubMed]

R. Wester, A. Bruneton, A. Bäuerle, J. Stollenwerk, and P. Loosen, “Irradiance tailoring for extended sources using a point-source freeform design algorithm,” Proc. of SPIE, Optical Systems Design 8550, 85502S(2012).
[CrossRef]

A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, Fresnel-type freeform optics,” Proc. SPIE 8485, 84850H (2012).
[CrossRef]

A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20, 14477–14485 (2012).
[CrossRef] [PubMed]

Benítez, P.

P. Benítez, J.-C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” SPIE Proc. Optical Engineering 43, 1489–1502 (2004).

Blen, J.

P. Benítez, J.-C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” SPIE Proc. Optical Engineering 43, 1489–1502 (2004).

Bortz, J.

J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE 6670, 66700A (2007).
[CrossRef]

Bruneton, A.

A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21, 10563–10571 (2013).
[CrossRef] [PubMed]

R. Wester, A. Bruneton, A. Bäuerle, J. Stollenwerk, and P. Loosen, “Irradiance tailoring for extended sources using a point-source freeform design algorithm,” Proc. of SPIE, Optical Systems Design 8550, 85502S(2012).
[CrossRef]

A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20, 14477–14485 (2012).
[CrossRef] [PubMed]

A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, Fresnel-type freeform optics,” Proc. SPIE 8485, 84850H (2012).
[CrossRef]

Cassarly, W. J.

F. R. Fournier, W. J. Cassarly, and J. P. Roland, “Designing freeform reflectors for extended sources,” Proc. of SPIE 7423, 743202 (2009).

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Optimization of single reflectors for extended sources,” Proc. of SPIE 7103, 71030I (2008).
[CrossRef]

Chaves, J.

P. Benítez, J.-C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” SPIE Proc. Optical Engineering 43, 1489–1502 (2004).

Dross, O.

P. Benítez, J.-C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” SPIE Proc. Optical Engineering 43, 1489–1502 (2004).

Falicoff, W.

P. Benítez, J.-C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” SPIE Proc. Optical Engineering 43, 1489–1502 (2004).

Fournier, F. R.

F. R. Fournier, W. J. Cassarly, and J. P. Roland, “Designing freeform reflectors for extended sources,” Proc. of SPIE 7423, 743202 (2009).

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Optimization of single reflectors for extended sources,” Proc. of SPIE 7103, 71030I (2008).
[CrossRef]

Glimm, T.

T. Glimm and V. Oliker, “Optical design of single reflector systems and the mongekantorovich mass transfer problem,” Journal of Mathematical Sciences 117, 4096–4108 (2003).
[CrossRef]

Gordon, J.

A. Rabl and J. Gordon, “Reflector design for illumination with extended sources: the basic solutions,” Applied Optics 33, 6012–6021 (1994).
[CrossRef] [PubMed]

Haker, S.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal Mass Transport for Registration and Warping,” Int. J. of Comput. Vision 60, 225–240 (2004).
[CrossRef]

Hernández, M.

P. Benítez, J.-C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” SPIE Proc. Optical Engineering 43, 1489–1502 (2004).

Kochengin, S. A.

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering dataII. Numerical solution,” Numer. Math. 79, 553–568 (1998).
[CrossRef]

Li, H.

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation.” Optics letters 38, 229–231 (2013).
[CrossRef]

Liu, P.

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation.” Optics letters 38, 229–231 (2013).
[CrossRef]

Liu, X.

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation.” Optics letters 38, 229–231 (2013).
[CrossRef]

Loosen, P.

A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21, 10563–10571 (2013).
[CrossRef] [PubMed]

R. Wester, A. Bruneton, A. Bäuerle, J. Stollenwerk, and P. Loosen, “Irradiance tailoring for extended sources using a point-source freeform design algorithm,” Proc. of SPIE, Optical Systems Design 8550, 85502S(2012).
[CrossRef]

A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20, 14477–14485 (2012).
[CrossRef] [PubMed]

A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, Fresnel-type freeform optics,” Proc. SPIE 8485, 84850H (2012).
[CrossRef]

Miñano, J.-C.

P. Benítez, J.-C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” SPIE Proc. Optical Engineering 43, 1489–1502 (2004).

Mohedano, R.

P. Benítez, J.-C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” SPIE Proc. Optical Engineering 43, 1489–1502 (2004).

Muschaweck, J.

Oliker, V.

T. Glimm and V. Oliker, “Optical design of single reflector systems and the mongekantorovich mass transfer problem,” Journal of Mathematical Sciences 117, 4096–4108 (2003).
[CrossRef]

Oliker, V. I.

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering dataII. Numerical solution,” Numer. Math. 79, 553–568 (1998).
[CrossRef]

Parkyn, B.

B. Parkyn and D. Pelka, “Free-form illumination lenses designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE 6338, 633808 (2006).
[CrossRef]

Pelka, D.

B. Parkyn and D. Pelka, “Free-form illumination lenses designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE 6338, 633808 (2006).
[CrossRef]

Rabl, A.

A. Rabl and J. Gordon, “Reflector design for illumination with extended sources: the basic solutions,” Applied Optics 33, 6012–6021 (1994).
[CrossRef] [PubMed]

Ries, H.

Roland, J. P.

F. R. Fournier, W. J. Cassarly, and J. P. Roland, “Designing freeform reflectors for extended sources,” Proc. of SPIE 7423, 743202 (2009).

Rolland, J. P.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Optimization of single reflectors for extended sources,” Proc. of SPIE 7103, 71030I (2008).
[CrossRef]

Schruben, S. J.

Seroka, S.

S. Seroka and S. Sertl, “Modeling of refractive freeform surfaces by a nonlinear PDE for the generation of a given target light distribution,” International Light Simulation Symposium (2012).

Sertl, S.

S. Seroka and S. Sertl, “Modeling of refractive freeform surfaces by a nonlinear PDE for the generation of a given target light distribution,” International Light Simulation Symposium (2012).

Shatz, N.

J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE 6670, 66700A (2007).
[CrossRef]

Stollenwerk, J.

Tannenbaum, A.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal Mass Transport for Registration and Warping,” Int. J. of Comput. Vision 60, 225–240 (2004).
[CrossRef]

Traub, M.

A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, Fresnel-type freeform optics,” Proc. SPIE 8485, 84850H (2012).
[CrossRef]

Wester, R.

A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21, 10563–10571 (2013).
[CrossRef] [PubMed]

R. Wester, A. Bruneton, A. Bäuerle, J. Stollenwerk, and P. Loosen, “Irradiance tailoring for extended sources using a point-source freeform design algorithm,” Proc. of SPIE, Optical Systems Design 8550, 85502S(2012).
[CrossRef]

A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, Fresnel-type freeform optics,” Proc. SPIE 8485, 84850H (2012).
[CrossRef]

A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20, 14477–14485 (2012).
[CrossRef] [PubMed]

Wu, R.

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation.” Optics letters 38, 229–231 (2013).
[CrossRef]

Xu, L.

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation.” Optics letters 38, 229–231 (2013).
[CrossRef]

Zhang, Y.

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation.” Optics letters 38, 229–231 (2013).
[CrossRef]

Zheng, Z.

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation.” Optics letters 38, 229–231 (2013).
[CrossRef]

Zhu, L.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal Mass Transport for Registration and Warping,” Int. J. of Comput. Vision 60, 225–240 (2004).
[CrossRef]

Applied Optics

A. Rabl and J. Gordon, “Reflector design for illumination with extended sources: the basic solutions,” Applied Optics 33, 6012–6021 (1994).
[CrossRef] [PubMed]

Int. J. of Comput. Vision

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal Mass Transport for Registration and Warping,” Int. J. of Comput. Vision 60, 225–240 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Journal of Mathematical Sciences

T. Glimm and V. Oliker, “Optical design of single reflector systems and the mongekantorovich mass transfer problem,” Journal of Mathematical Sciences 117, 4096–4108 (2003).
[CrossRef]

Numer. Math.

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering dataII. Numerical solution,” Numer. Math. 79, 553–568 (1998).
[CrossRef]

Opt. Express

Optics letters

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation.” Optics letters 38, 229–231 (2013).
[CrossRef]

Proc. of SPIE

F. R. Fournier, W. J. Cassarly, and J. P. Roland, “Designing freeform reflectors for extended sources,” Proc. of SPIE 7423, 743202 (2009).

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Optimization of single reflectors for extended sources,” Proc. of SPIE 7103, 71030I (2008).
[CrossRef]

Proc. of SPIE, Optical Systems Design

R. Wester, A. Bruneton, A. Bäuerle, J. Stollenwerk, and P. Loosen, “Irradiance tailoring for extended sources using a point-source freeform design algorithm,” Proc. of SPIE, Optical Systems Design 8550, 85502S(2012).
[CrossRef]

Proc. SPIE

J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE 6670, 66700A (2007).
[CrossRef]

B. Parkyn and D. Pelka, “Free-form illumination lenses designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE 6338, 633808 (2006).
[CrossRef]

A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, Fresnel-type freeform optics,” Proc. SPIE 8485, 84850H (2012).
[CrossRef]

SPIE Proc. Optical Engineering

P. Benítez, J.-C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” SPIE Proc. Optical Engineering 43, 1489–1502 (2004).

Other

S. Seroka and S. Sertl, “Modeling of refractive freeform surfaces by a nonlinear PDE for the generation of a given target light distribution,” International Light Simulation Symposium (2012).

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Figures (5)

Fig. 1
Fig. 1

The rays emitted by a point source travel through the optical system to the target plane. The optical system is designed so that ray paths don’t cross.

Fig. 2
Fig. 2

Target irradiance distribution adaptation using the solution of a Monge-Ampère equation. The desired target distribution a) is fed into a point source method. The distribution b) is computed with the lens of step a) using a point source. Distribution c) shows the case with extended source. Now a mapping is computed via a solution of a Monge-Ampère equation using the method outlined in [14] to transform the distribution c) into distribution b). This mapping is then applied to the original distribution a) which results in distribution d). Distribution d) is now fed into the point source method which results in the distribution e) for the extended source case.

Fig. 3
Fig. 3

Example setup of a source, a spherical lens and a target plane. Rays are traced from the target plane back to the source plane. Rays that hit the source contribute to the integral Eq. (12). The squares on the left and right show the points in phase space of those rays that hit the source. The phase space on the source is densely occupied whereas on the target side only a small part of phase space is occupied. In order to avoid unnecessary computations the target side phase space has to be restricted to the relevant parts. This is achieved by first tracing forward from the source to the target. The curve designated by ”integration” is the result of integral Eq. (12). The dots represent results obtained by using a Monte-Carlo ray tracer.

Fig. 4
Fig. 4

The source has an extend of 6 mm, the lens is 14 mm wide, the target has a width of 50 mm and the distance from source to target is 25 mm. The output angle of the source is restricted to 46.5°.

Fig. 5
Fig. 5

Target irradiance distribution generated by the setup Fig. 4 computed using FRED (www.photonengr.com). The prescribed target distribution was uniform within ±25mm.

Tables (3)

Tables Icon

Table 1 Optical surface parametrizations.

Tables Icon

Table 2 Table of symbols used in this article.

Tables Icon

Table 3 Principal point source algorithms.

Equations (13)

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

n s L s ( u s , s s ) d Ω s cos ( θ s ) d A s = n t L t ( u t , s t ) d Ω t cos ( θ t ) d A t
n s d Ω s cos ( θ s ) d A s n t d Ω t cos ( θ t ) d A t = 1
L s ( u s , s s ) = L t ( u t , s t )
I s ( s x , s y ) d s x d s y = E t ( u x , u y ) d u x d u y
u x = u x ( s x , s x )
u y = u x ( s x , s y )
I s ( s x , s y ) d s x d s y = E t ( u x ( s x , s y ) , u y ( s x , s y ) ) det ( J ) d s x d s y
det ( J ) = ( u x s x u y s y u x s y u y s x )
( u x , u y ) = ϕ
I s ( s x , s y ) = E t ( ϕ , ϕ ) ( 2 ϕ s x 2 2 ϕ s y 2 ( 2 ϕ s x s y ) 2 )
Ω t L t ( u t , s t ) cos ( θ t ) d Ω t = Ω t L s ( u s , s s ) cos ( θ t ) d Ω t
E t ( u t ) = Ω t L s ( u s , s s ) cos ( θ t ) d Ω t
E t ( u t ) 1 2 i [ L s , i ( u s , s s ) cos θ t , i + L s , i + 1 ( u s , s s ) cos θ t , i + 1 ] Δ Ω t , i , i + 1

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