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Support quadric method in non-imaging optics problems that can be reformulated as a mass transfer problem
A.A. Mingazov 1, L.L. Doskolovich 1,2, D.A. Bykov 1,2, E.V. Byzov 1

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

 PDF, 1118 kB

DOI: 10.18287/2412-6179-CO-1055

Pages: 353-365.

Full text of article: Russian language.

Abstract:
The article deals with problems of generating desired illumination patterns, formulated in a special way. More precisely, we consider problems that can be reformulated as a Monge–Kantorovich mass transfer problem with some cost function. For all problems of this type, we uniformly formulate the support quadric method and show that it coincides with the gradient method for finding the maximum of a certain concave function.

Keywords:
non-imaging optics, geometric optics, inverse problem, Monge–Kantorovich problem, support quadric method.

Citation:
Mingazov AA, Doskolovich LL, Bykov DA, Byzov EV. Support quadric method in non-imaging optics problems that can be reformulated as a mass transfer problem. Computer Optics 2022; 46(3): 353-365. DOI: 10.18287/2412-6179-CO-1055.

Acknowledgements:
The work was funded by the Ministry of Science and Higher Education of the Russian Federation under a government project of  the FSRC “Crystallography and Photonics” RAS (numerical implementation of the calculation algorithm) and the Russian Science Foundation under grant #18-19-00326 (proof of the coincidence with the gradient method for the corresponding functional).

References:
  1. Chang S, Wu R, Li A, Zheng Z. Design beam shapers with double freeform surfaces to form a desired wavefront with prescribed illumination pattern by solving a Monge–Ampère type equation. J Opt 2016; 18(12): 125602. DOI: 10.1088/2040-8978/18/12/125602.
  2. Feng Z, Froese BD, Huang C-Y, Ma D, Liang R. Creating unconventional geometric beams with large depth of field using double freeform-surface optics. Appl Opt 2015; 54(20): 6277-6281. DOI: 10.1364/AO.54.006277.
  3. Feng Z, Froese BD, Liang R, Cheng D, Wang Y. Simplified freeform optics design for complicated laser beam shaping. Appl Opt 2017; 56(33): 9308-9314. DOI: 10.1364/AO.56.009308.
  4. Bösel C, Worku NG, Gross H. Ray-mapping approach in double freeform surface design for collimated beam shaping beyond the paraxial approximation. Appl Opt 2017; 56(13): 3679-3688. DOI: 10.1364/AO.56.003679.
  5. Bösel C, Gross, H. Double freeform illumination design for prescribed wavefronts and irradiances. J Opt Soc Am A 2018; 35(2): 236-243. DOI: 10.1364/JOSAA.35.000236.
  6. Mao X, Li J, Wang F, Gao R, Li X, Xie Y. Fast design method of smooth freeform lens with an arbitrary aperture for collimated beam shaping. OSA Technical Digest 2019; JT5A.2. Washington: DC United States; 2019. ISBN: 978-1-943580-60-6.
  7. Wei S, Zhu Z, Fan Z, Yan Y, Ma D. Double freeform surfaces design for beam shaping with non-planar wavefront using an integrable ray mapping method. Opt Express 2019; 27(19): 26757-26771. DOI: 10.1364/OE.27.026757.
  8. Kochengin SA, Oliker VI. Computational algorithms for constructing reflectors. Comput Vis Sci 2003; 6: 15-21. DOI: 10.1007/s00791-003-0103-2.
  9. Doskolovich LL, Moiseev MA, Kazanskiy NL. On using a supporting quadric method to design diffractive optical elements. Computer Optics 2015; 39(3): 339-346. DOI: 10.18287/0134-2452-2015-39-3-339-346.
  10. Andreeva KV, Moiseev MA, Kravchenko SV, Doskolovich LL. Design of optical elements with TIR freeform surface. Computer Optics 2016; 40(4): 467-474. DOI: 10.18287/2412-6179-2016-40-4-467-474.
  11. Mingazov AA, Bykov DA, Bezus EA, Doskolovich LL. On the use of the supporting quadric method in the problem of designing double freeform surfaces for collimated beam shaping. Opt Express 2020; 28(15): 22642-22657. DOI: 10.1364/OE.398990.
  12. Glimm T, Oliker VI. Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem. J Math Sci 2003; 117: 4096-4108. DOI: 10.1023/A:1024856201493.
  13. Wang X-J. On design of a reflector antenna II. Calc Variations Partial Differ Equ 2004; 20: 329-341. DOI: 10.1007/s00526-003-0239-4.
  14. Doskolovich LL, Bykov DA, Mingazov AA, Bezus EA. Optimal mass transportation and linear assignment problems in the design of freeform refractive optical elements generating far-field irradiance distributions. Opt Express 2019; 27(9): 13083-13097. DOI: 10.1364/OE.27.013083.
  15. Yadav NK. Monge-Ampere problems with non-quadratic cost function: application to freeform optics. Eindhoven: Technische Universiteit Eindhoven; 2018.
  16. Bykov DA, Doskolovich LL, Mingazov AA, Bezus EA. Optimal mass transportation problem in the design of freeform optical elements generating far-field irradiance distributions for plane incident beam. Appl Opt 2019; 58(33): 9131-9140. DOI: 10.1364/AO.58.009131.
  17. Glimm T, Oliker VI. Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat’s principle. Indiana Univ Math J 2004; 53(5): 1255-1277. DOI: 10.1512/iumj.2004.53.2455.
  18. 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.
  19. Oliker V, Rubinstein J, Wolansky G. Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light. Adv Appl Math 2015; 62: 160-183. DOI: 10.1016/j.aam.2014.09.009.
  20. Glimm T. A rigorous analysis using optimal transport theory for a two-reflector design problem with a point source. Inverse Probl 2010; 26(4): 045001. DOI: 10.1088/0266-5611/26/4/045001.
  21. Mingazov AA, Doskolovich LL, Bykov DA, Kazanskiy NL. The two reflector design problem for forming a flat wavefront from a point source as an optimal mass transfer problem. Computer Optics 2019; 43(6): 968-975. DOI: 10.18287/2412-6179-2019-43-6-968-975.
  22. 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.
  23. 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.
  24. Makarov B, Podkorytov A. Real analysis: Measures, integrals and applications. London: Springer-Verlag; 2013.
  25. Merigot Q. A multiscale approach to optimal transport. Comput Graph Forum 2011; 30(5): 1584-1592. DOI: 10.1111/j.1467-8659.2011.02032.x.
  26. TracePro – software for design and analysis of illumination and optical systems. Source: <https://www.lambdares.com/tracepro/>.

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