Abstract

We consider the problem of calculating the eikonal function defined on a certain curved surface from the condition of generating a prescribed irradiance distribution on a target surface. We show that the calculation of the “ray mapping” corresponding to the eikonal function is reduced to the solution of a linear assignment problem (LAP). We propose an iterative algorithm for calculating a refractive optical surface from the condition of generating a prescribed near-field irradiance distribution in a non-paraxial case. The algorithm is based on sequential calculation of eikonal functions defined on curved surfaces using the LAP-based approach. The proposed algorithm is applied to the calculation of refractive optical elements generating uniform irradiance distributions in a rectangular region and in a region in the form of the letters “IPSI” in the case of a circular incident beam. The presented ray-tracing simulations of the designed optical elements demonstrate high efficiency of the proposed iterative algorithm.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, and J. C. Miñano, “Design of freeform illumination optics,” Laser & Photonics Rev. 12(7), 1700310 (2018).
    [Crossref]
  2. 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]
  3. R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Xiu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge–Ampère equation,” Opt. Lett. 38(2), 229–231 (2013).
    [Crossref] [PubMed]
  4. 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]
  5. Y. Ma, H. Zhang, Z. Su, Y. He, L. Xu, X. Lui, and H. Li, “Hybrid method of free-form lens design for arbitrary illumination target,” Appl. Opt. 54(14), 4503–4508 (2015).
    [Crossref] [PubMed]
  6. X. Mao, S. Xu, X. Hu, and Y. Xie, “Design of a smooth freeform illumination system for a point light source based on polar-type optimal transport mapping,” Appl. Opt. 56(22), 6324–6331 (2017).
    [Crossref] [PubMed]
  7. R. Wu, S. Chang, Z. Zheng, L. Zhao, and X. Liu, “Formulating the design of two freeform lens surfaces for point-like light sources,” Opt. Lett. 43(7), 1619–1622 (2018).
    [Crossref] [PubMed]
  8. C. R. Prins, J. H. M. ten Thije Boonkkamp, J. van Roosmalen, W. L. IJzerman, and T. W. Tukker, “A Monge–Ampère-solver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
    [Crossref]
  9. C. Bösel and H. Gross, “Ray mapping approach for the efficient design of continuous freeform surfaces,” Opt. Express 24(13), 14271–14282 (2016).
    [Crossref] [PubMed]
  10. C. Bösel and H. Gross, “Single freeform surface design for prescribed input wavefront and target irradiance,” J. Opt. Soc. Am. A 34(9), 1490–1499 (2017).
    [Crossref]
  11. C. Bösel and H. Gross, “Double freeform illumination design for prescribed wavefronts and irradiances,” J. Opt. Soc. Am. A 35(2), 236–243 (2018).
    [Crossref]
  12. C. R. Prins, R. Beltman, J. H. M. ten Thije Boonkkamp, W. L. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the Monge–Ampère equation,” SIAM J. Sci. Comput. 37(6), B937–B961 (2015).
    [Crossref]
  13. 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]
  14. K. Brix, Y. Hafizogullari, and A. Platen, “Solving the Monge–Ampère equations for the inverse reflector problem,” Math. Model. Methods Appl. Sci. 25(6), 803–837 (2015).
    [Crossref]
  15. M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of the Monge– Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
    [Crossref]
  16. L. L. Doskolovich, A. A. Mingazov, D. A. Bykov, E. S. Andreev, and E. A. Bezus, “Variational approach to calculation of light field eikonal function for illuminating a prescribed region,” Opt. Express 25(22), 26378–26392 (2017).
    [Crossref] [PubMed]
  17. T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
    [Crossref]
  18. X.-J. Wang, “On the design of a reflector antenna II,” Calc. Var. 20(3), 329–341 (2004).
    [Crossref]
  19. C. E. Gutiérrez, “Refraction problems in geometric optics,” Fully Nonlinear PDEs in Real and Complex Geometry and Optics, Vol. 2087 of the Series Lecture Notes in Mathematics (Springer, 2014), pp. 95–150.
    [Crossref]
  20. C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
    [Crossref]
  21. J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24(2), 463–469 (2007).
    [Crossref]
  22. V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. for Ration. Mech. Analysis 201(3), 1013–1045 (2011).
    [Crossref]
  23. L. L. Doskolovich, D. A. Bykov, E. S. Andreev, E. A. Bezus, and V. Oliker, “Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems,” Opt. Express 26(19), 24602–24613 (2018).
    [Crossref]
  24. J. Munkres, “Algorithms for the assignment and transportation problems,” Journal of the Society for Industrial and Applied Mathematics 5(1), 32–38 (1957).
    [Crossref]
  25. D. P. Bertsekas, “The auction algorithm: A distributed relaxation method for the assignment problem,” Ann. Oper. Res. 14(1), 105–123 (1988).
    [Crossref]
  26. C. E. Gutiérrez and Q. Huang, “The near field refractor,” Ann. Inst. Henri Poincaré C Non Linear Anal. 31(4), 655–684 (2014).
    [Crossref]
  27. S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13(2), 363–373 (1997).
    [Crossref]
  28. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
    [Crossref]
  29. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
    [Crossref]
  30. L. C. Evans, “Partial differential equations and Monge–Kantorovich mass transfer,” in Current Developments in Mathematics, R. Bott, A. Jaffe, D. Jerison, G. Lusztig, I. Singer, and S.-T. Yau, eds. (International Press of Boston, 1999).
  31. A. A. Mingazov, D. A. Bykov, L. L. Doskolovich, and N. L. Kazanskiy, “Variational interpretation of the eikonal calculation problem from the condition of generating a prescribed irradiance distribution,” Comput. Opt. 42(4), 567–573 (2018).
  32. X. Mao, H. Li, Y. Han, and Y. Luo, “Polar-grids based source-target mapping construction method for designing freeform illumination system for a lighting target with arbitrary shape,” Opt. Express 23(4), 4313–4328 (2015).
    [Crossref] [PubMed]
  33. Y. Ding, X. Liu, Z.-R. Zheng, and P.-F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
    [Crossref] [PubMed]
  34. L. L. Doskolovich, N. L. Kazansky, S. I. Kharitonov, and V. A. Soifer, “A method of designing diffractive optical elements focusing into plane areas,” J. Mod. Opt. 43(7), 1423–1433 (1996).
    [Crossref]
  35. M. Balzer, T. Schölmer, and O. Deussen, “Capacity-constrained point distributions: a variant of Lloyd’s method,” ACM Trans. Graph. 28(3), 86 (2009).
    [Crossref]
  36. F. de Goes, K. Breeden, V. Ostromoukhov, and M. Desbrun, “Blue noise through optimal transport,” ACM Trans. Graph. 31(6), 171 (2012).
    [Crossref]
  37. J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley, 1997).
  38. V. A. Soifer, V. V. Kotlyar, and L. L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, 1997).
  39. Implementation of Bertsekas’ auction algorithm. http://www.mathworks.com/matlabcentral/fileexchange/48448 .
  40. A. M. Oberman and Y. Ruan, “An efficient linear programming method for optimal transportation,” https://arxiv.org/abs/1509.03668 .
  41. B. Schmitzer and C. Schnörr, “A hierarchical approach to optimal transport,” in Scale Space and Variational Methods in Computer Vision, A. Kuijper, K. Bredies, T. Pock, and H. Bischof, eds. (Springer, 2013), pp. 452–464.
    [Crossref]

2018 (5)

2017 (3)

2016 (1)

2015 (5)

2014 (3)

C. E. Gutiérrez and Q. Huang, “The near field refractor,” Ann. Inst. Henri Poincaré C Non Linear Anal. 31(4), 655–684 (2014).
[Crossref]

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]

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

2013 (2)

2012 (1)

F. de Goes, K. Breeden, V. Ostromoukhov, and M. Desbrun, “Blue noise through optimal transport,” ACM Trans. Graph. 31(6), 171 (2012).
[Crossref]

2011 (2)

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

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

2009 (2)

C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[Crossref]

M. Balzer, T. Schölmer, and O. Deussen, “Capacity-constrained point distributions: a variant of Lloyd’s method,” ACM Trans. Graph. 28(3), 86 (2009).
[Crossref]

2008 (1)

2007 (1)

2004 (1)

X.-J. Wang, “On the design of a reflector antenna II,” Calc. Var. 20(3), 329–341 (2004).
[Crossref]

2003 (1)

T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
[Crossref]

1997 (1)

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13(2), 363–373 (1997).
[Crossref]

1996 (1)

L. L. Doskolovich, N. L. Kazansky, S. I. Kharitonov, and V. A. Soifer, “A method of designing diffractive optical elements focusing into plane areas,” J. Mod. Opt. 43(7), 1423–1433 (1996).
[Crossref]

1988 (1)

D. P. Bertsekas, “The auction algorithm: A distributed relaxation method for the assignment problem,” Ann. Oper. Res. 14(1), 105–123 (1988).
[Crossref]

1957 (1)

J. Munkres, “Algorithms for the assignment and transportation problems,” Journal of the Society for Industrial and Applied Mathematics 5(1), 32–38 (1957).
[Crossref]

Andreev, E. S.

Balzer, M.

M. Balzer, T. Schölmer, and O. Deussen, “Capacity-constrained point distributions: a variant of Lloyd’s method,” ACM Trans. Graph. 28(3), 86 (2009).
[Crossref]

Beltman, R.

C. R. Prins, R. Beltman, J. H. M. ten Thije Boonkkamp, W. L. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the Monge–Ampère equation,” SIAM J. Sci. Comput. 37(6), B937–B961 (2015).
[Crossref]

Benítez, P.

Bertsekas, D. P.

D. P. Bertsekas, “The auction algorithm: A distributed relaxation method for the assignment problem,” Ann. Oper. Res. 14(1), 105–123 (1988).
[Crossref]

Bezus, E. A.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[Crossref]

Bösel, C.

Breeden, K.

F. de Goes, K. Breeden, V. Ostromoukhov, and M. Desbrun, “Blue noise through optimal transport,” ACM Trans. Graph. 31(6), 171 (2012).
[Crossref]

Brix, K.

K. Brix, Y. Hafizogullari, and A. Platen, “Solving the Monge–Ampère equations for the inverse reflector problem,” Math. Model. Methods Appl. Sci. 25(6), 803–837 (2015).
[Crossref]

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]

Bykov, D. A.

Chang, S.

de Goes, F.

F. de Goes, K. Breeden, V. Ostromoukhov, and M. Desbrun, “Blue noise through optimal transport,” ACM Trans. Graph. 31(6), 171 (2012).
[Crossref]

Desbrun, M.

F. de Goes, K. Breeden, V. Ostromoukhov, and M. Desbrun, “Blue noise through optimal transport,” ACM Trans. Graph. 31(6), 171 (2012).
[Crossref]

Deussen, O.

M. Balzer, T. Schölmer, and O. Deussen, “Capacity-constrained point distributions: a variant of Lloyd’s method,” ACM Trans. Graph. 28(3), 86 (2009).
[Crossref]

Ding, Y.

Doskolovich, L. L.

A. A. Mingazov, D. A. Bykov, L. L. Doskolovich, and N. L. Kazanskiy, “Variational interpretation of the eikonal calculation problem from the condition of generating a prescribed irradiance distribution,” Comput. Opt. 42(4), 567–573 (2018).

L. L. Doskolovich, D. A. Bykov, E. S. Andreev, E. A. Bezus, and V. Oliker, “Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems,” Opt. Express 26(19), 24602–24613 (2018).
[Crossref]

L. L. Doskolovich, A. A. Mingazov, D. A. Bykov, E. S. Andreev, and E. A. Bezus, “Variational approach to calculation of light field eikonal function for illuminating a prescribed region,” Opt. Express 25(22), 26378–26392 (2017).
[Crossref] [PubMed]

L. L. Doskolovich, N. L. Kazansky, S. I. Kharitonov, and V. A. Soifer, “A method of designing diffractive optical elements focusing into plane areas,” J. Mod. Opt. 43(7), 1423–1433 (1996).
[Crossref]

V. A. Soifer, V. V. Kotlyar, and L. L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, 1997).

Evans, L. C.

L. C. Evans, “Partial differential equations and Monge–Kantorovich mass transfer,” in Current Developments in Mathematics, R. Bott, A. Jaffe, D. Jerison, G. Lusztig, I. Singer, and S.-T. Yau, eds. (International Press of Boston, 1999).

Feng, Z.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, and J. C. Miñano, “Design of freeform illumination optics,” Laser & Photonics Rev. 12(7), 1700310 (2018).
[Crossref]

Glimm, T.

T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
[Crossref]

Gross, H.

Gu, P.-F.

Gutiérrez, C. E.

C. E. Gutiérrez and Q. Huang, “The near field refractor,” Ann. Inst. Henri Poincaré C Non Linear Anal. 31(4), 655–684 (2014).
[Crossref]

C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[Crossref]

C. E. Gutiérrez, “Refraction problems in geometric optics,” Fully Nonlinear PDEs in Real and Complex Geometry and Optics, Vol. 2087 of the Series Lecture Notes in Mathematics (Springer, 2014), pp. 95–150.
[Crossref]

Hafizogullari, Y.

K. Brix, Y. Hafizogullari, and A. Platen, “Solving the Monge–Ampère equations for the inverse reflector problem,” Math. Model. Methods Appl. Sci. 25(6), 803–837 (2015).
[Crossref]

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]

Han, Y.

He, Y.

Hu, X.

Huang, Q.

C. E. Gutiérrez and Q. Huang, “The near field refractor,” Ann. Inst. Henri Poincaré C Non Linear Anal. 31(4), 655–684 (2014).
[Crossref]

C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[Crossref]

IJzerman, W. L.

C. R. Prins, R. Beltman, J. H. M. ten Thije Boonkkamp, W. L. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the Monge–Ampère equation,” SIAM J. Sci. Comput. 37(6), B937–B961 (2015).
[Crossref]

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

Kazanskiy, N. L.

A. A. Mingazov, D. A. Bykov, L. L. Doskolovich, and N. L. Kazanskiy, “Variational interpretation of the eikonal calculation problem from the condition of generating a prescribed irradiance distribution,” Comput. Opt. 42(4), 567–573 (2018).

Kazansky, N. L.

L. L. Doskolovich, N. L. Kazansky, S. I. Kharitonov, and V. A. Soifer, “A method of designing diffractive optical elements focusing into plane areas,” J. Mod. Opt. 43(7), 1423–1433 (1996).
[Crossref]

Kharitonov, S. I.

L. L. Doskolovich, N. L. Kazansky, S. I. Kharitonov, and V. A. Soifer, “A method of designing diffractive optical elements focusing into plane areas,” J. Mod. Opt. 43(7), 1423–1433 (1996).
[Crossref]

Kochengin, S. A.

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13(2), 363–373 (1997).
[Crossref]

Kotlyar, V. V.

V. A. Soifer, V. V. Kotlyar, and L. L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, 1997).

Kravtsov, Yu. A.

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
[Crossref]

Li, H.

Liang, R.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, and J. C. Miñano, “Design of freeform illumination optics,” Laser & Photonics Rev. 12(7), 1700310 (2018).
[Crossref]

Liu, P.

Liu, X.

Lui, X.

Luo, Y.

Ma, Y.

Mao, X.

Miñano, J. C.

Mingazov, A. A.

A. A. Mingazov, D. A. Bykov, L. L. Doskolovich, and N. L. Kazanskiy, “Variational interpretation of the eikonal calculation problem from the condition of generating a prescribed irradiance distribution,” Comput. Opt. 42(4), 567–573 (2018).

L. L. Doskolovich, A. A. Mingazov, D. A. Bykov, E. S. Andreev, and E. A. Bezus, “Variational approach to calculation of light field eikonal function for illuminating a prescribed region,” Opt. Express 25(22), 26378–26392 (2017).
[Crossref] [PubMed]

Munkres, J.

J. Munkres, “Algorithms for the assignment and transportation problems,” Journal of the Society for Industrial and Applied Mathematics 5(1), 32–38 (1957).
[Crossref]

Oliker, V.

L. L. Doskolovich, D. A. Bykov, E. S. Andreev, E. A. Bezus, and V. Oliker, “Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems,” Opt. Express 26(19), 24602–24613 (2018).
[Crossref]

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

T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
[Crossref]

Oliker, V. I.

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13(2), 363–373 (1997).
[Crossref]

Orlov, Yu. I.

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
[Crossref]

Ostromoukhov, V.

F. de Goes, K. Breeden, V. Ostromoukhov, and M. Desbrun, “Blue noise through optimal transport,” ACM Trans. Graph. 31(6), 171 (2012).
[Crossref]

Platen, A.

K. Brix, Y. Hafizogullari, and A. Platen, “Solving the Monge–Ampère equations for the inverse reflector problem,” Math. Model. Methods Appl. Sci. 25(6), 803–837 (2015).
[Crossref]

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]

Prins, C. R.

C. R. Prins, R. Beltman, J. H. M. ten Thije Boonkkamp, W. L. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the Monge–Ampère equation,” SIAM J. Sci. Comput. 37(6), B937–B961 (2015).
[Crossref]

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

Rubinstein, J.

Russell, R. D.

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

Schmitzer, B.

B. Schmitzer and C. Schnörr, “A hierarchical approach to optimal transport,” in Scale Space and Variational Methods in Computer Vision, A. Kuijper, K. Bredies, T. Pock, and H. Bischof, eds. (Springer, 2013), pp. 452–464.
[Crossref]

Schnörr, C.

B. Schmitzer and C. Schnörr, “A hierarchical approach to optimal transport,” in Scale Space and Variational Methods in Computer Vision, A. Kuijper, K. Bredies, T. Pock, and H. Bischof, eds. (Springer, 2013), pp. 452–464.
[Crossref]

Schölmer, T.

M. Balzer, T. Schölmer, and O. Deussen, “Capacity-constrained point distributions: a variant of Lloyd’s method,” ACM Trans. Graph. 28(3), 86 (2009).
[Crossref]

Soifer, V. A.

L. L. Doskolovich, N. L. Kazansky, S. I. Kharitonov, and V. A. Soifer, “A method of designing diffractive optical elements focusing into plane areas,” J. Mod. Opt. 43(7), 1423–1433 (1996).
[Crossref]

V. A. Soifer, V. V. Kotlyar, and L. L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, 1997).

Su, Z.

Sulman, M. M.

ten Thije Boonkkamp, J. H. M.

C. R. Prins, R. Beltman, J. H. M. ten Thije Boonkkamp, W. L. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the Monge–Ampère equation,” SIAM J. Sci. Comput. 37(6), B937–B961 (2015).
[Crossref]

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

Tukker, T. W.

C. R. Prins, R. Beltman, J. H. M. ten Thije Boonkkamp, W. L. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the Monge–Ampère equation,” SIAM J. Sci. Comput. 37(6), B937–B961 (2015).
[Crossref]

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

Turunen, J.

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley, 1997).

van Roosmalen, J.

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

Wang, X.-J.

X.-J. Wang, “On the design of a reflector antenna II,” Calc. Var. 20(3), 329–341 (2004).
[Crossref]

Williams, J. F.

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

Wolansky, G.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[Crossref]

Wu, R.

Wyrowski, F.

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley, 1997).

Xie, Y.

Xiu, X.

Xu, L.

Xu, S.

Zhang, H.

Zhang, Y.

Zhao, L.

Zheng, Z.

Zheng, Z.-R.

ACM Trans. Graph. (2)

M. Balzer, T. Schölmer, and O. Deussen, “Capacity-constrained point distributions: a variant of Lloyd’s method,” ACM Trans. Graph. 28(3), 86 (2009).
[Crossref]

F. de Goes, K. Breeden, V. Ostromoukhov, and M. Desbrun, “Blue noise through optimal transport,” ACM Trans. Graph. 31(6), 171 (2012).
[Crossref]

Ann. Inst. Henri Poincaré C Non Linear Anal. (1)

C. E. Gutiérrez and Q. Huang, “The near field refractor,” Ann. Inst. Henri Poincaré C Non Linear Anal. 31(4), 655–684 (2014).
[Crossref]

Ann. Oper. Res. (1)

D. P. Bertsekas, “The auction algorithm: A distributed relaxation method for the assignment problem,” Ann. Oper. Res. 14(1), 105–123 (1988).
[Crossref]

Appl. Numer. Math. (1)

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

Appl. Opt. (2)

Arch. for Ration. Mech. Analysis (1)

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

Arch. Ration. Mech. Anal. (1)

C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[Crossref]

Calc. Var. (1)

X.-J. Wang, “On the design of a reflector antenna II,” Calc. Var. 20(3), 329–341 (2004).
[Crossref]

Comput. Opt. (1)

A. A. Mingazov, D. A. Bykov, L. L. Doskolovich, and N. L. Kazanskiy, “Variational interpretation of the eikonal calculation problem from the condition of generating a prescribed irradiance distribution,” Comput. Opt. 42(4), 567–573 (2018).

Inverse Probl. (1)

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13(2), 363–373 (1997).
[Crossref]

J. Math. Sci. (1)

T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
[Crossref]

J. Mod. Opt. (1)

L. L. Doskolovich, N. L. Kazansky, S. I. Kharitonov, and V. A. Soifer, “A method of designing diffractive optical elements focusing into plane areas,” J. Mod. Opt. 43(7), 1423–1433 (1996).
[Crossref]

J. Opt. Soc. Am. A (4)

Journal of the Society for Industrial and Applied Mathematics (1)

J. Munkres, “Algorithms for the assignment and transportation problems,” Journal of the Society for Industrial and Applied Mathematics 5(1), 32–38 (1957).
[Crossref]

Laser & Photonics Rev. (1)

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, and J. C. Miñano, “Design of freeform illumination optics,” Laser & Photonics Rev. 12(7), 1700310 (2018).
[Crossref]

Math. Model. Methods Appl. Sci. (1)

K. Brix, Y. Hafizogullari, and A. Platen, “Solving the Monge–Ampère equations for the inverse reflector problem,” Math. Model. Methods Appl. Sci. 25(6), 803–837 (2015).
[Crossref]

Opt. Express (7)

L. L. Doskolovich, A. A. Mingazov, D. A. Bykov, E. S. Andreev, and E. A. Bezus, “Variational approach to calculation of light field eikonal function for illuminating a prescribed region,” Opt. Express 25(22), 26378–26392 (2017).
[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]

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]

C. Bösel and H. Gross, “Ray mapping approach for the efficient design of continuous freeform surfaces,” Opt. Express 24(13), 14271–14282 (2016).
[Crossref] [PubMed]

X. Mao, H. Li, Y. Han, and Y. Luo, “Polar-grids based source-target mapping construction method for designing freeform illumination system for a lighting target with arbitrary shape,” Opt. Express 23(4), 4313–4328 (2015).
[Crossref] [PubMed]

Y. Ding, X. Liu, Z.-R. Zheng, and P.-F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
[Crossref] [PubMed]

L. L. Doskolovich, D. A. Bykov, E. S. Andreev, E. A. Bezus, and V. Oliker, “Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems,” Opt. Express 26(19), 24602–24613 (2018).
[Crossref]

Opt. Lett. (2)

SIAM J. Sci. Comput. (2)

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

C. R. Prins, R. Beltman, J. H. M. ten Thije Boonkkamp, W. L. IJzerman, and T. W. Tukker, “A least-squares method for optimal transport using the Monge–Ampère equation,” SIAM J. Sci. Comput. 37(6), B937–B961 (2015).
[Crossref]

Other (9)

C. E. Gutiérrez, “Refraction problems in geometric optics,” Fully Nonlinear PDEs in Real and Complex Geometry and Optics, Vol. 2087 of the Series Lecture Notes in Mathematics (Springer, 2014), pp. 95–150.
[Crossref]

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
[Crossref]

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[Crossref]

L. C. Evans, “Partial differential equations and Monge–Kantorovich mass transfer,” in Current Developments in Mathematics, R. Bott, A. Jaffe, D. Jerison, G. Lusztig, I. Singer, and S.-T. Yau, eds. (International Press of Boston, 1999).

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley, 1997).

V. A. Soifer, V. V. Kotlyar, and L. L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, 1997).

Implementation of Bertsekas’ auction algorithm. http://www.mathworks.com/matlabcentral/fileexchange/48448 .

A. M. Oberman and Y. Ruan, “An efficient linear programming method for optimal transportation,” https://arxiv.org/abs/1509.03668 .

B. Schmitzer and C. Schnörr, “A hierarchical approach to optimal transport,” in Scale Space and Variational Methods in Computer Vision, A. Kuijper, K. Bredies, T. Pock, and H. Bischof, eds. (Springer, 2013), pp. 452–464.
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Geometry of the problem.
Fig. 2
Fig. 2 Schematic representation of two iterations of the algorithm for calculating the working surface of the optical element (CF stands for the cost function).
Fig. 3
Fig. 3 Rectangular mesh on a circle (a) and its image in the target rectangle (b) calculated at the first iteration of the algorithm. The inset shows the magnified fragment along with the discrete mapping obtained by solving the LAP of Eq. (11). Only the images of the points lying on the square mesh (a) are shown in the inset.
Fig. 4
Fig. 4 Irradiance distributions generated by the designed optical elements at the iterations 1–4 (a). Optical element calculated at the fifth iteration (b) and the corresponding irradiance distribution (c).
Fig. 5
Fig. 5 Irradiance distributions generated by the optical elements calculated at the iterations 1–4 (a). Optical element calculated at the fifth iteration (b), and the corresponding irradiance distribution (c). The curves on the element surface show the sharp bends caused by the discontinuous character of the mapping P(u).
Fig. 6
Fig. 6 Gaussian curvature of the surface of the resulting optical element shown in Fig. 5(b).

Tables (1)

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Table 1 Algorithm

Equations (16)

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{ Φ ( u ) u 1 = p 1 ( u ) + p 3 ( u ) g ( u ) u 1 , Φ ( u ) u 2 = p 2 ( u ) + p 3 ( u ) g ( u ) u 2 , p 3 ( u ) = 1 [ p 1 ( u ) ] 2 [ p 2 ( u ) ] 2 .
p ( u ) = ( P ( u ) u , f ( P ( u ) ) g ( u ) ) ρ ( u , P ( u ) ) .
Φ ( u ) = P ( u ) u ρ ( u , P ( u ) ) + g ( u ) f ( P ( u ) ) g ( u ) ρ ( u , P ( u ) ) ,
Ψ ( v ) = v P 1 ( v ) ρ ( P 1 ( v ) , v ) + f ( v ) f ( v ) g ( P 1 ( v ) ) ρ ( P 1 ( v ) , v ) ,
P Φ 1 ( ω ) E G ( u ) d G = ω E F ( v ) d F ,
P Φ 1 ( ω ) E 0 ( u ) d u = P Φ 1 ( ω ) E ( P Φ ( u ) ) J P Φ ( u ) d u ,
J P Φ ( u ) = | P Φ , 1 u 1 P Φ , 2 u 2 P Φ , 1 u 2 P Φ , 2 u 1 | .
E 0 ( u ) = E ( P Φ ( u ) ) J P Φ ( u ) .
ρ ( u , v ) = | v u | 2 + [ f ( v ) g ( u ) ] 2
𝒞 ( P ) = Ω ρ ( u , P ( u ) ) E 0 ( u ) d u .
( P , λ ) = Ω { ρ ( u , P ( u ) ) E 0 ( u ) + λ ( u ) [ E ( P ( u ) ) J P ( u ) E 0 ( u ) ] } d u ,
( M ) i , j = ρ ( u i , v j ) , i , j = 1 , , N .
𝒞 d ( j 1 , , j N ) = i ρ ( u i , v j i ) min ,
z el ( u ) = 1 n 0 1 [ Φ ( u ) g ( u ) ] ,
Φ ( u ) = m , n c m n B m ( u 1 ) D n ( u 2 ) ,
Φ ( u ) = max v Θ [ Ψ ( v ) ρ ( u , v ) ] .

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