Abstract

The efficient design of continuous freeform surfaces, which maps a given light source to an arbitrary target illumination pattern, remains a challenging problem and is considered here for collimated input beams. A common approach are ray-mapping methods, where first a ray mapping between the source and the irradiance distribution on the target plane is calculated and in a subsequent step the surface is constructed. The challenging aspect of this approach is to find an integrable mapping ensuring a continuous surface. Based on the law of reflection/refraction and an integrability condition, we derive a general condition for the surface and ray mapping for a collimated input beam. It is shown that in a small-angle approximation a proper mapping can be calculated via optimal mass transport - a mathematical framework for the calculation of a mapping between two positive density functions. We show that the surface can be constructed by solving a linear advection Eq. with appropriate boundary conditions. The results imply that the optimal mass transport mapping is approximately integrable over a wide range of distances between the freeform and the target plane and offer an efficient way to construct the surface by solving standard integrals. The efficiency is demonstrated by applying it to two challenging design examples, which shows the ability of the presented approach to handle target illumination patterns with steep irradiance gradients and numerous gray levels.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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  1. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002).
    [Crossref]
  2. R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illuminaton design: a nonlinear boundary problem for the elliptic Monge-Ampère equation,” Opt. Lett. 38(2), 229–231 (2013).
    [Crossref] [PubMed]
  3. R. Wu, K. Li, P. Liu, Z. Zheng, H. Li, and X. Liu, “Conceptual design of dedicated road lighting for city park and housing estate,” Appl. Opt. 52(21), 5272–5278 (2013).
    [Crossref] [PubMed]
  4. R. Wu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “A mathematical model of the single freeform surface design for collimated beam shaping,” Opt. Express 21(18), 20974–20989 (2013).
    [Crossref] [PubMed]
  5. Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with Monge-Ampère equation method,” Opt. Comm. 331, 297–305 (2014).
    [Crossref]
  6. R. Wu, Y. Zhang, M. M. Sulman, Z. Zheng, P. Benítez, and J. C. Miñano, “Initial design with L2 Monge-Kantorovich theory for the Monge–Ampère equation method in freeform surface illumination design,” Opt. Express 22(13), 16161–16177 (2014).
    [Crossref] [PubMed]
  7. C. R. Prins, J. H. M. ten Thije Boonkkamp, J. Van Roosmalen, W. L. I. Jzerman, and T. W. Tukker, “A Monge-Ampère-Solver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
    [Crossref]
  8. K. Brix, Y. Hafizogullari, and A. Platen, “Designing illumination lenses and mirrors by the numerical solution of Monge–Ampère equations,” J. Opt. Soc. Am. A 32(11), 2227–2236 (2015).
    [Crossref]
  9. V. I. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics,” in Trends in Nonlinear Analysis, M. Kirkilionis, S. Kromker, R. Rannacher, and F. Tomi, eds. (Springer-Verlag, 2003), pp. 193–222.
    [Crossref]
  10. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express 18(5), 5295–5304 (2010).
    [Crossref] [PubMed]
  11. C. Canavesi, W. J. Cassarly, and J. P. Rolland, “Target flux estimation by calculating intersections between neighboring conic reflector patches,” Opt. Lett. 38(23), 5012–5015 (2013).
    [Crossref] [PubMed]
  12. D. Ma, Z. Feng, and R. Liang, “Tailoring freeform illumination optics in a double-pole coordinate system,” Appl. Opt. 54(9), 2395–2399 (2015).
    [Crossref] [PubMed]
  13. D. Michaelis, D. Schreiber, and A. Brauer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. 36(6), 918–920 (2011).
    [Crossref] [PubMed]
  14. J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24(2), 463–469 (2007).
    [Crossref]
  15. V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy 3(1), 035599 (2013).
    [Crossref]
  16. V. Oliker, “Differential equations for design of a freeform single lens with prescribed irradiance properties,” Opt. Eng. 53(3), 031302 (2014).
    [Crossref]
  17. V. Oliker and B. Cherkasskiy, “Controlling light with freeform optics: recent progress in computational methods for optical design of freeform lenses with prescribed irradiance properties,” Proc. SPIE 9191, 919105 (2014).
    [Crossref]
  18. A. Bauerle, A. Bruneton, P. Loosen, and R. Wester, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20(13), 14477–14485 (2012).
    [Crossref] [PubMed]
  19. A. Bruneton, A. Bauerle, P. Loosen, and R. Wester, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21(9), 10563–10571 (2013).
    [Crossref] [PubMed]
  20. Z. Feng, L. Huang, G. Jin, and M. Gong, “Beam shaping system design using double freeform optical surfaces,” Opt. Express 21(12), 14728–14735 (2013).
    [Crossref] [PubMed]
  21. Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express 21(23), 28693–28701 (2013).
    [Crossref]
  22. J.C. Miñano, P. Benítez, W. Lin, J. Infante, F. Muñoz, and A. Santamaría, “An application of the SMS method for imaging designs,” Opt. Express 17(26), 24036–24044 (2009).
    [Crossref]
  23. A. Hofmann, J. Unterhinninghofen, H. Ries, and S. Kaiser, “Double tailoring of freeform surfaces for off-axis aplanatic systems,” Proc. SPIE 8550, 855014 (2012).
    [Crossref]
  24. F. Duerr, P. Benítez, J.C. Miñano, Y. Meuret, and H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express 205, 5576–5585 (2012).
  25. V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Anal. 201(3), 1013–1045 (2011).
    [Crossref]
  26. J.-D. Benamou, Y. Brenier, and K. Guittet, “The Monge-Kantorovich mass transfer and its Compuational Fluid Mechanics formulation,” Int. J. Numer. Meth. Fluids 40(1–2), 21–30 (2002).
    [Crossref]
  27. S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comp. Vis. 60(3), 225–240 (2004).
    [Crossref]
  28. M. M. Sulman, J. F. Williams, and R. D. Russel, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
    [Crossref]
  29. D. Kuzmin, A Guide to Numerical Methods for Transport Equations (University Erlangen-Nuremberg, 2010).

2015 (2)

2014 (5)

R. Wu, Y. Zhang, M. M. Sulman, Z. Zheng, P. Benítez, and J. C. Miñano, “Initial design with L2 Monge-Kantorovich theory for the Monge–Ampère equation method in freeform surface illumination design,” Opt. Express 22(13), 16161–16177 (2014).
[Crossref] [PubMed]

V. Oliker, “Differential equations for design of a freeform single lens with prescribed irradiance properties,” Opt. Eng. 53(3), 031302 (2014).
[Crossref]

V. Oliker and B. Cherkasskiy, “Controlling light with freeform optics: recent progress in computational methods for optical design of freeform lenses with prescribed irradiance properties,” Proc. SPIE 9191, 919105 (2014).
[Crossref]

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with Monge-Ampère equation method,” Opt. Comm. 331, 297–305 (2014).
[Crossref]

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. Van Roosmalen, W. L. I. Jzerman, and T. W. Tukker, “A Monge-Ampère-Solver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

2013 (8)

V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy 3(1), 035599 (2013).
[Crossref]

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illuminaton design: a nonlinear boundary problem for the elliptic Monge-Ampère equation,” Opt. Lett. 38(2), 229–231 (2013).
[Crossref] [PubMed]

A. Bruneton, A. Bauerle, P. Loosen, and R. Wester, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21(9), 10563–10571 (2013).
[Crossref] [PubMed]

Z. Feng, L. Huang, G. Jin, and M. Gong, “Beam shaping system design using double freeform optical surfaces,” Opt. Express 21(12), 14728–14735 (2013).
[Crossref] [PubMed]

R. Wu, K. Li, P. Liu, Z. Zheng, H. Li, and X. Liu, “Conceptual design of dedicated road lighting for city park and housing estate,” Appl. Opt. 52(21), 5272–5278 (2013).
[Crossref] [PubMed]

R. Wu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “A mathematical model of the single freeform surface design for collimated beam shaping,” Opt. Express 21(18), 20974–20989 (2013).
[Crossref] [PubMed]

Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express 21(23), 28693–28701 (2013).
[Crossref]

C. Canavesi, W. J. Cassarly, and J. P. Rolland, “Target flux estimation by calculating intersections between neighboring conic reflector patches,” Opt. Lett. 38(23), 5012–5015 (2013).
[Crossref] [PubMed]

2012 (3)

2011 (3)

D. Michaelis, D. Schreiber, and A. Brauer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. 36(6), 918–920 (2011).
[Crossref] [PubMed]

M. M. Sulman, J. F. Williams, and R. D. Russel, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Anal. 201(3), 1013–1045 (2011).
[Crossref]

2010 (1)

2009 (1)

2007 (1)

2004 (1)

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comp. Vis. 60(3), 225–240 (2004).
[Crossref]

2002 (2)

J.-D. Benamou, Y. Brenier, and K. Guittet, “The Monge-Kantorovich mass transfer and its Compuational Fluid Mechanics formulation,” Int. J. Numer. Meth. Fluids 40(1–2), 21–30 (2002).
[Crossref]

H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002).
[Crossref]

Angenent, S.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comp. Vis. 60(3), 225–240 (2004).
[Crossref]

Bauerle, A.

Benamou, J.-D.

J.-D. Benamou, Y. Brenier, and K. Guittet, “The Monge-Kantorovich mass transfer and its Compuational Fluid Mechanics formulation,” Int. J. Numer. Meth. Fluids 40(1–2), 21–30 (2002).
[Crossref]

Benítez, P.

Brauer, A.

Brenier, Y.

J.-D. Benamou, Y. Brenier, and K. Guittet, “The Monge-Kantorovich mass transfer and its Compuational Fluid Mechanics formulation,” Int. J. Numer. Meth. Fluids 40(1–2), 21–30 (2002).
[Crossref]

Brix, K.

Bruneton, A.

Canavesi, C.

Cassarly, W. J.

Cherkasskiy, B.

V. Oliker and B. Cherkasskiy, “Controlling light with freeform optics: recent progress in computational methods for optical design of freeform lenses with prescribed irradiance properties,” Proc. SPIE 9191, 919105 (2014).
[Crossref]

Duerr, F.

Feng, Z.

Fournier, F. R.

Gong, M.

Guittet, K.

J.-D. Benamou, Y. Brenier, and K. Guittet, “The Monge-Kantorovich mass transfer and its Compuational Fluid Mechanics formulation,” Int. J. Numer. Meth. Fluids 40(1–2), 21–30 (2002).
[Crossref]

Hafizogullari, Y.

Haker, S.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comp. Vis. 60(3), 225–240 (2004).
[Crossref]

Hofmann, A.

A. Hofmann, J. Unterhinninghofen, H. Ries, and S. Kaiser, “Double tailoring of freeform surfaces for off-axis aplanatic systems,” Proc. SPIE 8550, 855014 (2012).
[Crossref]

Huang, L.

Infante, J.

Jin, G.

Jzerman, W. L. I.

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. Van Roosmalen, W. L. I. Jzerman, and T. W. Tukker, “A Monge-Ampère-Solver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

Kaiser, S.

A. Hofmann, J. Unterhinninghofen, H. Ries, and S. Kaiser, “Double tailoring of freeform surfaces for off-axis aplanatic systems,” Proc. SPIE 8550, 855014 (2012).
[Crossref]

Kuzmin, D.

D. Kuzmin, A Guide to Numerical Methods for Transport Equations (University Erlangen-Nuremberg, 2010).

Li, H.

Li, K.

Liang, R.

Lin, W.

Liu, P.

Liu, X.

Loosen, P.

Ma, D.

Meuret, Y.

Michaelis, D.

Miñano, J. C.

Miñano, J.C.

Muñoz, F.

Muschaweck, J.

Oliker, V.

V. Oliker, “Differential equations for design of a freeform single lens with prescribed irradiance properties,” Opt. Eng. 53(3), 031302 (2014).
[Crossref]

V. Oliker and B. Cherkasskiy, “Controlling light with freeform optics: recent progress in computational methods for optical design of freeform lenses with prescribed irradiance properties,” Proc. SPIE 9191, 919105 (2014).
[Crossref]

V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy 3(1), 035599 (2013).
[Crossref]

V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Anal. 201(3), 1013–1045 (2011).
[Crossref]

Oliker, V. I.

V. I. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics,” in Trends in Nonlinear Analysis, M. Kirkilionis, S. Kromker, R. Rannacher, and F. Tomi, eds. (Springer-Verlag, 2003), pp. 193–222.
[Crossref]

Platen, A.

Prins, C. R.

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. Van Roosmalen, W. L. I. Jzerman, and T. W. Tukker, “A Monge-Ampère-Solver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

Ries, H.

A. Hofmann, J. Unterhinninghofen, H. Ries, and S. Kaiser, “Double tailoring of freeform surfaces for off-axis aplanatic systems,” Proc. SPIE 8550, 855014 (2012).
[Crossref]

H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002).
[Crossref]

Rolland, J. P.

Rubinstein, J.

V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy 3(1), 035599 (2013).
[Crossref]

J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24(2), 463–469 (2007).
[Crossref]

Russel, R. D.

M. M. Sulman, J. F. Williams, and R. D. Russel, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

Santamaría, A.

Schreiber, D.

Sulman, M. M.

Tannenbaum, A.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comp. Vis. 60(3), 225–240 (2004).
[Crossref]

ten Thije Boonkkamp, J. H. M.

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. Van Roosmalen, W. L. I. Jzerman, and T. W. Tukker, “A Monge-Ampère-Solver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

Thienpont, H.

Tukker, T. W.

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. Van Roosmalen, W. L. I. Jzerman, and T. W. Tukker, “A Monge-Ampère-Solver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

Unterhinninghofen, J.

A. Hofmann, J. Unterhinninghofen, H. Ries, and S. Kaiser, “Double tailoring of freeform surfaces for off-axis aplanatic systems,” Proc. SPIE 8550, 855014 (2012).
[Crossref]

Van Roosmalen, J.

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. Van Roosmalen, W. L. I. Jzerman, and T. W. Tukker, “A Monge-Ampère-Solver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

Wester, R.

Williams, J. F.

M. M. Sulman, J. F. Williams, and R. D. Russel, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

Wolansky, G.

V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy 3(1), 035599 (2013).
[Crossref]

J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24(2), 463–469 (2007).
[Crossref]

Wu, R.

Xu, L.

Zhang, Y.

Zheng, Z.

Zhu, L.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comp. Vis. 60(3), 225–240 (2004).
[Crossref]

Appl. Numer. Math. (1)

M. M. Sulman, J. F. Williams, and R. D. Russel, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

Appl. Opt. (2)

Arch. Ration. Mech. Anal. (1)

V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Anal. 201(3), 1013–1045 (2011).
[Crossref]

Int. J. Comp. Vis. (1)

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comp. Vis. 60(3), 225–240 (2004).
[Crossref]

Int. J. Numer. Meth. Fluids (1)

J.-D. Benamou, Y. Brenier, and K. Guittet, “The Monge-Kantorovich mass transfer and its Compuational Fluid Mechanics formulation,” Int. J. Numer. Meth. Fluids 40(1–2), 21–30 (2002).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Photonics Energy (1)

V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy 3(1), 035599 (2013).
[Crossref]

Opt. Comm. (1)

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with Monge-Ampère equation method,” Opt. Comm. 331, 297–305 (2014).
[Crossref]

Opt. Eng. (1)

V. Oliker, “Differential equations for design of a freeform single lens with prescribed irradiance properties,” Opt. Eng. 53(3), 031302 (2014).
[Crossref]

Opt. Express (9)

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express 18(5), 5295–5304 (2010).
[Crossref] [PubMed]

R. Wu, Y. Zhang, M. M. Sulman, Z. Zheng, P. Benítez, and J. C. Miñano, “Initial design with L2 Monge-Kantorovich theory for the Monge–Ampère equation method in freeform surface illumination design,” Opt. Express 22(13), 16161–16177 (2014).
[Crossref] [PubMed]

R. Wu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “A mathematical model of the single freeform surface design for collimated beam shaping,” Opt. Express 21(18), 20974–20989 (2013).
[Crossref] [PubMed]

F. Duerr, P. Benítez, J.C. Miñano, Y. Meuret, and H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express 205, 5576–5585 (2012).

A. Bauerle, A. Bruneton, P. Loosen, and R. Wester, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20(13), 14477–14485 (2012).
[Crossref] [PubMed]

A. Bruneton, A. Bauerle, P. Loosen, and R. Wester, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21(9), 10563–10571 (2013).
[Crossref] [PubMed]

Z. Feng, L. Huang, G. Jin, and M. Gong, “Beam shaping system design using double freeform optical surfaces,” Opt. Express 21(12), 14728–14735 (2013).
[Crossref] [PubMed]

Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express 21(23), 28693–28701 (2013).
[Crossref]

J.C. Miñano, P. Benítez, W. Lin, J. Infante, F. Muñoz, and A. Santamaría, “An application of the SMS method for imaging designs,” Opt. Express 17(26), 24036–24044 (2009).
[Crossref]

Opt. Lett. (3)

Proc. SPIE (2)

V. Oliker and B. Cherkasskiy, “Controlling light with freeform optics: recent progress in computational methods for optical design of freeform lenses with prescribed irradiance properties,” Proc. SPIE 9191, 919105 (2014).
[Crossref]

A. Hofmann, J. Unterhinninghofen, H. Ries, and S. Kaiser, “Double tailoring of freeform surfaces for off-axis aplanatic systems,” Proc. SPIE 8550, 855014 (2012).
[Crossref]

SIAM J. Sci. Comput. (1)

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. Van Roosmalen, W. L. I. Jzerman, and T. W. Tukker, “A Monge-Ampère-Solver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

Other (2)

V. I. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics,” in Trends in Nonlinear Analysis, M. Kirkilionis, S. Kromker, R. Rannacher, and F. Tomi, eds. (Springer-Verlag, 2003), pp. 193–222.
[Crossref]

D. Kuzmin, A Guide to Numerical Methods for Transport Equations (University Erlangen-Nuremberg, 2010).

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Figures (4)

Fig. 1
Fig. 1 a) For the source and target irradiance distributions IS(x,y) and IT (x,y) the OMT ray mapping u(x,y) is calculated. The mapping defines a vector field s3 between the source plane z = 0 and the target plane z = zT. ΩS and ΩT are the source and target irradiance distribution boundaries, respectively. b) In the second process step the freeform surface z(x,y) is constructed in a way that it is continuous and redirects the incoming collimated beam, described by the vector field s1, according to the given OMT map. Because of energy conservation, the boundary of the freeform corresponds to the boundary shape ΩS of IS(x,y).
Fig. 2
Fig. 2 The boundary ΩS is parameterized by s. At each point of the boundary the local coordinate system is spanned by the tangential vector t ^ and normal vector r ^ to the boundary as well as the unit vector ez. Since the boundary values z(s) of the freeform surface only determine the tangential deflection of the rays hitting the boundary, the projection of the law of refraction (4) on the t ^ ( s ) e z plane can be used for the calculation of the boundary values.
Fig. 3
Fig. 3 The figure shows one possible way of solving the linear advection Eq. (12) by the simple integration of Eq. (16) along straight lines. First the value z0 = z(x0,y0) is fixed and used for the integration along the green line. The values of z(x,y) on the green line serve as starting values for the line-by-line integration along the red lines in the orthogonal direction. Then the blue lines are integrated by using the values of z(x,y) on the last red line.
Fig. 4
Fig. 4 a), b): given target irradiance distribution. In both cases, the incoming collimated beam was chosen to have a uniform irradiance distribution. c), d): calculated freeform lenses with side length of one arb. unit. The integration constant of Eq. (16) at the upper left side (x,y) = (−0.5,−0.5) was z0 = 0 arb. unit and the source-target distance zT = 5 arb. unit. The imported numerical data was interpolated automatically by ZEMAX. e), f): irradiance distributions from a ZEMAX raytracing with 3 · 108 rays. g), h): absolute difference between the given target irradiance distributions and·the irradiance distributions from the raytracing with ZEMAX. Especially in the regions of steep gradients the quality of the simulation results is reduced. The main reasons for that are the limited precision of the OMT algorithm, the linear interpolation of the lens data by ZEMAX and the finite distance between the lens and the target plane according to the condition (11).

Equations (25)

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2 I S ( x ) d x = 2 I T ( x ) d x
d e t ( u ( x ) ) I T ( u ( x ) ) = I S ( x )
d ( I S , I T ) 2 = inf u M | u ( x ) x | 2 I S ( x ) d x ,
n = n 1 s ^ 1 n 2 s ^ 2 ,
n ( × n ) = 0 .
s 1 = ( 0 0 z ( x , y ) ) , s 3 = ( u x ( x , y ) x u y ( x , y ) y z T ) , s 2 = s 3 s 1 ,
s 2 ( × s 1 ) = n 1 { s 2 × [ ( s 2 ) s 2 ] } z n s 2 + s 2 ( × s 3 ) ,
v z ( x , y ) = n 1 v [ ( v ) v ] n s 2 ( z T z ( x , y ) ) v ,
( z z ( x , y ) ) = ! n ( n ) z ( x z ( x , y ) y z ( x , y ) 1 ) = ! ( n 2 | s 2 | ( n ) z v y n 2 | s 2 | ( n ) z v x 1 ) .
n 1 v [ ( v ) v ] n s 2 ( z T z ( x , y ) ) v ! 0 .
n s 2 n 1 v [ ( v ) v ]
v z ( x , y ) = ( v z ( x , y ) ) = 0 , v = ( u ( x , y ) Id )
t = d d s ( x ( s ) y ( s ) 0 ) , r = d d s ( y ( s ) x ( s ) 0 ) , e z = ( 0 0 1 ) .
n ( n ) z = ! ( l z ( l ) 1 ) , l ( s ) : = 0 s ( d x d t ) 2 + ( d y d t ) 2 d t ,
s z ( s ) = s 2 t ( z T z ( s ) ) n 1 n 2 ( s 2 t ^ ) 2 + ( z T z ( s ) ) 2
s z ( s ) ~ z T s 2 t ( z T z 0 ) ( 1 n 1 n 2 ) = v x s y v y s x ( z T z 0 ) ( 1 n 1 n 2 ) ,
n ( × s 2 | s 2 | ) = n ( 1 | s 2 | × s 2 s 2 × 1 | s 2 | ) = 0
1 | s 2 | = 1 2 | s 2 | 3 ( s 2 s 2 ) = 1 | s 2 | 3 [ s 2 × ( × s 2 ) + ( s 2 ) s 2 ]
s 2 × 1 | s 2 | = 1 | s 2 | 3 { s 2 × [ s 2 × ( × s 2 ) ] = [ s 2 ( × s 2 ) ] s 2 | s 2 | 2 ( × s 2 ) + s 2 × [ ( s 2 ) s 2 ] }
n { [ s 2 ( × s 2 ) ] s 2 + s 2 × [ ( s 2 ) s 2 ] } = 0 .
( n s 2 ) [ s 2 ( × s 2 ) ] + n 1 { s 2 × [ ( s 2 ) s 2 ] } z = 0
s 2 ( × s 1 ) = n 1 { s 2 × [ ( s 2 ) s 2 ] } z n s 2 + s 2 ( × s 3 ) .
s 2 ( × s 1 . ) = ( ( s 2 ) y ( s 2 ) x ) ( x z ( x , y ) y z ( x , y ) ) = v z ( x , y )
{ s 2 × [ ( s 2 ) s 2 ] } z = ( s 2 ) x [ ( s 2 ) s 2 ] y ( s 2 ) y [ ( s 2 ) s 2 ] x = ( ( s 2 ) y ( s 2 ) x ) [ ( s 2 ) ( ( s 2 ) x ( s 2 ) y ) ] = v [ ( v ) v ]
s 2 ( × s 3 ) = ( z T z ( x , y ) ) [ x ( s 2 ) y y ( s 2 ) x ] = ( z T z ( x , y ) ) v .

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