Variational approach to eikonal function computation
Doskolovich L.L., Mingazov A.A., Bykov D.A., Andreev E.S.

 

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

Abstract:
The problem of calculating the eikonal function from the condition of focusing into a prescribed region is formulated as a variational problem and as a Monge-Kantorovich mass transportation problem. It is found that the cost function in the Monge-Kantorovich problem corresponds to the distance between a point of the original region (in which the eikonal function is defined) and a point of the focal region. The formalism proposed in this work makes it possible to reduce the calculation of the eikonal function to a linear programming problem. In this case, the calculation of the “ray mapping” corresponding to the eikonal function is reduced to the solution of a linear assignment problem. The proposed variational approaches are illustrated by examples of calculation of optical elements for focusing a circular beam into a rectangular region.

Keywords:
geometrical optics, eikonal function, variational problem, Monge-Kantorovich mass transportation problem.

Citation:
Doskolovich LL, Mingazov AA, Bykov DA, Andreev ES. Variational approach to eikonal function computation. Computer Optics 2018; 42(4): 557-567. DOI: 10.18287/2412-6179-2018-42-4-557-567.

References:

  1. Wu R, Xu L, Liu P, Zhang Y, Zheng Z, Li H, Liu H. Freeform illumination design: a nonlinear boundary problem for the elliptic Monge–Ampére equation. Opt Lett 2013; 38(2): 229-231. DOI: 10.1364/OL.38.000229.
  2. Wu R, Benítez P, Yaqin Z, Mi>ano JC. Influence of the characteristics of a light source and target on the Monge–Ampére equation method in freeform optics design. Opt Lett 2014; 39(3): 634-637. DOI: 10.1364/OL.39.000634.
  3. 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.
  4. Bösel C, Gross H. Ray mapping approach for the efficient design of continuous freeform surfaces. Opt Express 2016; 24(13): 14271-14282. DOI: 10.1364/OE.24.014271.
  5. Oliker VI. Mathematical aspects of design of beam shaping surfaces in geometrical optics. In Book: Oliker VI, Kirkilionis M, Krömker S, Rannacher R, Tomi F, eds. Trends in nonlinear analysis. Chap 4. Berlin, Heidelberg: Springer; 2003: 197-224. DOI: 10.1007/978-3-662-05281-5_4.
  6. Kochengin SA, Oliker VI. Computational algorithms for constructing reflectors. Computing and Visualization in Science 2003; 6(1): 15-21. DOI: 10.1007/s00791-003-0103-2.
  7. Oliker V. Controlling light with freeform multifocal lens designed with supporting quadric method (SQM). Opt Express 2017; 25(4): A58-A72. DOI: 10.1364/OE.25.000A58.
  8. Oliker V, Rubinstein J, Wolansky G. Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light. Advances in Applied Mathematics 2015; 62: 160-183. DOI: 10.1016/j.aam.2014.09.009.
  9. Fournier FR, Cassarly WJ, Rolland JP. Fast freeform reflector generation using source-target maps. Opt Express 2010; 18(5): 5295-5304. DOI: 10.1364/OE.18.005295.
  10. Michaelis D, Schreiber P, Bäuer A. Cartesian oval representation of freeform optics in illumination systems. Opt Lett 2011; 36(6): 918-920. DOI: 10.1364/OL.36.000918.
  11. Glimm T, Oliker V. Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem. J Math Sci 2003; 117(3): 4096-4108. DOI: 10.1023/A:1024856201493.
  12. Doskolovich LL, Moiseev MA, Bezus EA, Oliker V. On the use of the supporting quadric method in the problem of the light field eikonal calculation. Opt Express 2015; 23(15): 19605-19617. DOI: 10.1364/OE.23.019605.
  13. Soifer V, Kotlyar V, Doskolovich L. Iterative methods for diffractive optical elements computation. London: Taylor & Francis Ltd; 1997. ISBN: 0-7484-0634-4.
  14. Doskolovich LL, Dmitriev AYu, Kharitonov SI. Analytic design of optical elements generating a line focus. Opt Eng 2013; 52(13): 091707. DOI: 10.1117/1.OE.52.9.091707.
  15. Wang XJ. On the design of a reflector antenna II. Calculus of Variations and Partial Differential Equations 2004; 20(3): 329-341. DOI: 10.1007/s00526-003-0239-4.
  16. Tardos E. A strongly polynomial algorithm to solve combinatorial linear programs. Operation Research 1986; 34(2): 250-256. DOI: 10.1287/opre.34.2.250.
  17. Canavesi C, Cassarly WJ, Rolland JP. Direct calculation algorithm for two-dimensional reflector design. Opt Lett 2012; 37(18): 3852-3854. DOI: 10.1364/OL.37.003852.
  18. Canavesi C, Cassarly WJ, Rolland JP. Observations on the linear programming formulation of the single reflector design problem. Opt Express 2012; 20(4): 4050-4055. DOI: 10.1364/OE.20.004050.
  19. Munkres J. Algorithms for the assignment and transportation problems. Journal of the Society for Industrial and Applied Mathematics 1957; 5(1): 32-38. DOI: 10.1137/0105003.
  20. Trace Pro. Source: <https://www.lambdares.com/tracepro/>.
  21. Rhinoceros®: design, model, present, analyze, realize... Source: <http://www.rhino3d.com>.
  22. MathWorks. File exchange. Munkres assignment algorithm. Source: <https://www.mathworks.com/matlabcentral/fileexchange/20328-munkres-assignment-algorithm>.

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