(43-5) 01 * << * >> * Russian * English * Content * All Issues

Formulation of the inverse problem of calculating the optical surface for an illuminating beam with a plane wavefront as the Monge–Kantorovich problem

L.L. Doskolovich1,2, A.A. Mingazov1, D.A. Bykov1,2, E.A. Bezus1,2

IPSI RAS – Branch of the FSRC “Crystallography and Photonics” RAS,  
Molodogvardeyskaya 151, 443001, Samara, Russia,
Samara National Research University, Moskovskoye Shosse 34,
443086, Samara, Russia

 PDF, 807 kB

DOI: 10.18287/2412-6179-2019-43-5-705-713

Pages: 705-713.

Full text of article: Russian language.

A problem of calculating a refractive surface that forms a required irradiance distribution in the far field in the case of a plane illuminating beam is considered. We show that this problem can be formulated as a mass transportation problem. The specific form of the cost function for this problem is obtained. It is shown that with a certain choice of coordinates, the cost function becomes quadratic. The resulting mass transportation problem also describes a problem of calculating a mirror, which can be considered as a special case of the problem of calculating a refractive surface.

geometrical optics, optical design, nonimaging optics, illumination design, Monge-Kantorovich problem, mass transportation problem.

Doskolovich LL, Mingazov AA, Bykov DA, Bezus EA. Formulation of the inverse problem of calculating the optical surface for an illuminating beam with a plane wavefront as the Monge-Kantorovich problem. Computer Optics 2019; 43(5): 705-713. DOI: 10.18287/2412-6179-2019-43-5-705-713.

This work was supported by the Russian Foundation for Basic Research (RFBR) under grants ## 18-07-00982, 18-29-03067, 18-07-00514 (formulation of the problem of calculating refractive or reflective optical surface as an optimal mass transportation problem) and the RF Ministry of Science and Higher Education within the State assignment to FSRC “Crystallography and Photonics” RAS under agreement 007-ГЗ/Ч3363/26 (formulation of the weak solution of the problem).


  1. Wu R, Feng Z, Zheng Z, Liang R, Benítez P, Miñano JC. Design of freeform illumination optics. Laser Photon Rev 2018; 12(7): 1700310. DOI: 10.1002/lpor.201700310.
  2. Wu R, Liu P, Zhang Y, Zheng Z, Li H, Liu X. A mathematical model of the single freeform surface design for collimated beam shaping. Opt Express 2013; 21(18): 20974-20989. DOI: 10.1364/OE.21.020974.
  3. Wu R, Xu L, Liu P, Zhang Y, Zheng Z, Li H, Xiu X. Freeform illumination design: a nonlinear boundary problem for the elliptic Monge–Ampère equation. Opt Lett 2013; 38(2): 229-231. DOI: 10.1364/OL.38.000229.
  4. Wu R, Zhang Y, Sulman MM, Zheng Z, Benítez P, Miñano JC. Initial design with L2 Monge–Kantorovich theory for the Monge–Ampère equation method in freeform surface illumination design. Opt Express 2014; 22(13): 16161-16177. DOI: 10.1364/OE.22.016161.
  5. Ma Y, Zhang H, Su Z, He Y, Xu L, Lui X, Li H. Hybrid method of free-form lens design for arbitrary illumination target. Appl Opt 2015; 54(14): 4503-4508. DOI: 10.1364/AO.54.004503.
  6. Mao X, Xu S, Hu X, Xie Y. Design of a smooth freeform illumination system for a point light source based on polar-type optimal transport mapping. Appl Opt 2017; 56(22): 6324-6331. DOI: 10.1364/AO.56.006324.
  7. Wu R, Chang S, Zheng Z, Zhao L, Liu X. Formulating the design of two freeform lens surfaces for point-like light sources. Opt Lett 2018; 43(7): 1619-1622. DOI: 10.1364/OL.43.001619.
  8. Glimm T, Oliker V. Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem. J Math Sci 2003; 117(3): 4096–108. DOI: 10.1023/A:1024856201493.
  9. Wang XJ. On the design of a reflector antenna II. Calc Var Partial Dif 2004; 20(3): 329-341. DOI: 10.1007/s00526-003-0239-4.
  10. Gutiérrez CE. Refraction problems in geometric optics. In Book: Gutiérrez CE, Lanconelli E, eds. Fully nonlinear PDEs in real and complex geometry and optics. Springer; 2014: 95-150. DOI: 10.1007/978-3-319-00942-1_3.
  11. Gutiérrez CE, Huang Q. The refractor problem in reshaping light beams. Arch Ration Mech Anal 2009; 193(2): 423-443. DOI: 10.1007/s00205-008-0165-x.
  12. Rubinstein J, Wolansky G. Intensity control with a free-form lens. J Opt Soc Am A 2007; 24(2); 463-469. DOI: 10.1364/JOSAA.24.000463.
  13. Oliker V. Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport. Arch Ration Mech Anal 2011; 201(3): 1013-1045. DOI: 10.1007/s00205-011-0419-x.
  14. Oliker V, Doskolovich LL, Bykov DA. Beam shaping with a plano-freeform lens pair. Opt Express 2018; 26(15): 19406-19419. DOI: 10.1364/OE.26.019406.
  15. Doskolovich LL, Bykov DA, Andreev ES, Bezus EA, Oliker V. Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems. Opt Express 2018; 26(19): 24602-24613. DOI: 10.1364/OE.26.024602.
  16. Doskolovich LL, Mingazov AA, Bykov DA, Andreev ES, Bezus EA. Variational approach to calculation of light field eikonal function for illuminating a prescribed region. Opt Express 2017; 25(22): 26378-26392. DOI: 10.1364/OE.25.026378.
  17. Bykov DA, Doskolovich LL, Mingazov AA, Bezus EA, Kazanskiy NL. Linear assignment problem in the design of freeform refractive optical elements generating prescribed irradiance distributions. Opt Express 2018; 26(21): 27812-27825. DOI: 10.1364/OE.26.027812.
  18. Mingazov AA, Bykov DA, Doskolovich LL, Kazanskiy NL. Variational interpretation of the eikonal calculation problem from the condition of generating a prescribed irradiance distribution [In Russian]. Computer Optics 2018; 42(4): 568-573. DOI: 10.18287/2412-6179-2018-42-4-568-573.
  19. Sulman MM, Williams JF, Russell RD. An efficient approach for the numerical solution of the Monge–Ampère equation. Appl Numer Math 2011; 61(3): 298-307. DOI: 10.1016/j.apnum.2010.10.006.
  20. Doskolovich LL, Dmitriev AY, Moiseev MA, Kazanskiy NL. Analytical design of refractive optical elements generating one-parameter intensity distributions. J Opt Soc Am A 2014; 31(11): 2538-2544. DOI: 10.1364/JOSAA.31.002538.
  21. Eisenhart LP. A treatise on the differential geometry of curves and surfaces. Schwarz Press; 2008.


© 2009, IPSI RAS
Россия, 443001, Самара, ул. Молодогвардейская, 151; электронная почта: ko@smr.ru ; тел: +7 (846) 242-41-24 (ответственный секретарь), +7 (846) 332-56-22 (технический редактор), факс: +7 (846) 332-56-20