Solving the inverse problem of focusing laser radiation in a plane region using geometrical optics
S.I. Kharitonov, L. L. Doskolovich, N.L. Kazanskiy


Image Processing Systems Institute оf RAS, – Branch of the FSRC “Crystallography and Photonics” RAS, Samara, Russia,
Samara National Research University, Samara, Russia

Full text of article: Russian language.

We proposed a method for calculating the eikonal function of a light field from the condition of generating a desired intensity distribution in a predetermined region of the focal plane. To improve the efficiency of solving the problem of focusing we developed a modification of a method of rectangle matching, which enables designing diffractive optical elements (DOE) intended to focus into a plane region. The novelty of our approach consists in a method of reconstructing the eikonal function from the known ray map of points on the DOE onto points in the focal plane. We presented the results of the DOE-aided focusing into a circle, a rectangle, a rhombus, and a superellipse. The simulation results showed the high quality of focusing, thus corroborating the efficiency of the proposed method. We expect to use the diffractive optical elements (DOE) for lighting the area of interest from low-flying drones used for night-time Earth remote sensing.

geometrical optics, the eikonal function, focusing, method of matched rectangles, ray mapping.

Kharitonov SI, Doskolovich LL, Kazanskiy NL. Solving the inverse problem of focusing laser radiation in a plane region using geometrical optics. Computer Optics 2016; 40(4): 439-450. DOI: 10.18287/2412-6179-2016-40-4-439-450.


  1. Bryngdahl O, Lee WH. Laser beam scanning using computer generated holograms. Appl Opt 1976; 15(1): 183-194.
  2. Bryngdahl O. Geometrical transforms in optics. J  Opt Soc Am 1974; 64(8): 1092-1099.
  3. Golub MA, Karpeev SV, Prokhorov AM, Sisakyan IN, Soifer VA. Focusing light into a specified volume by computer synthesized holograms. Soviet Technical Physics Letters 1981; 7(10): 264-266.
  4. Golub MA, Degtyarova VP, Klimov AN, Popov VV, Prokhorov AM, Sisakyan IN, Soifer VA. Machine synthesis of focusing elements for CO2-laser [in Russian]. Soviet Technical Physics Letters 1982; 8(13): 449-451.
  5. Danilov VA, Popov VV, Prokhorov AM, Sagatelyan DM, Sisakyan IN, Soifer VA. Synthesis of optical elements, that create focal free-form line [in Russian]. Soviet Technical Physics Letters 1982; 8(13): 810-815.
  6. Goncharsky AV, Danilov VA, Popov VV, Prokhorov AM, Sissakian IN, Soifer VA, Stepanov VV. Solution of the inverse problem of focusing of laser radiation into an arbitrary curve [in Russian]. Doklady AN SSSR 1983; 273(3): 605-608.
  7. Danilov VA, Kinber BE, Shishlov AV. Theory of coherent focusers [In Russian]. Computer Optics 1989; 1(1): 29-37.
  8. Golub MA, Sisakyan IN, Soifer VA. Infra-red radiation focusators. Optics and Lasers in Engineering 1991; 15(5): 297-309. DOI: 10.1016/0143-8166(91)90017-N.
  9. Doskolovich LL, Dmitriev AYu, Kharitonov SI. Analytic design of optical elements generating a line focus. Optical Engineering 2013; 52(9): 091707. DOI: 10.1117/1.OE.52.9.091707.
  10. Hilton PA, Lloyd D, Tyrer JR. Use of a diffractive optic for high power laser cutting. Journal of Laser Applications 2016; 28: 012014. DOI: 10.2351/1.4938279.
  11. Abul'khanov SR. Technological applications of focusing optical elements. Proc SPIE 2014; 9533: 95330B. DOI: 10.1117/12.2180564.
  12. Doskolovich LL, Kazanskiy NL, Kharitonov SI, Usplenjev GV. Focusators for laser-branding. Optics and Lasers in Engineering 1991; 15(5): 311-322. DOI: 10.1016/0143-8166(91)90018-O.
  13. Grewell D, Benatar A. Diffractive optics as beam-shaping elements for plastics laser welding. Optical Engineering 2007; 46(11): 118001. DOI: 10.1117/1.2802588.
  14. Aslanov ER, Doskolovich LL, Moiseev MA, Bezus EA, Kazanskiy NL. Design of an optical element forming an axial line segment for efficient LED lighting systems. Optics Express 2013; 21(23): 28651-28656. DOI: 10.1364/OE.21.028651.
  15. Popov SB. The intellectual lighting for optical information-measuring systems. Proc SPIE 2014; 9533: 95330P. DOI: 10.1117/12.2181168.
  16. Kazanskiy NL, Skidanov RV. Binary beam splitter. Appl Opt 2012; 51(14): 2672-2677. DOI: 10.1364/AO.51.002672.
  17. Doskolovich LL, Moiseev MA, Byzov EV, Kravchenko SV. Computation of light field eikonal to focus into a set of points. Computer Optics 2014; 38(3): 443-448.
  18. Kazanskiy NL, Murzin SP, Osetrov YL, Tregub VI. Synthesis of nanoporous structures in metallic materials under laser action. Optics and Lasers in Engineering 2011; 49(11): 1264-1267. DOI: 10.1016/j.optlaseng.2011.07.001.
  19. Murzin SP. Formation of nanoporous structures in metallic materials by pulse-periodic laser treatment. Optics and Laser Technology 2015: 72: 48-52. DOI: 10.1016/j.optlastec.2015.03.022.
  20. Murzin SP. Local laser annealing for aluminium alloy parts. Lasers in Engineering 2016; 33(1-3): 67-76.
  21. Doskolovich LL, Bezus EA, Moiseev MA, Bykov DA, Kazanskiy NL. Analytical source-target mapping method for the design of freeform mirrors generating prescribed 2D intensity distributions. Optics Express 2016; 24(10): 10962-10971. DOI: 10.1364/OE.24.010962.
  22. Rodgers KJ, Glen HG, Salz JJ, Maguen E, Berry MJ. Improved method of laser thermal keratoplasty to overcome presbyopia. Proc SPIE 2011; 7885: 78850N. DOI: 10.1117/12.878330.
  23. Sobol EN, Baum OI, Omelchenko AI, Vorobyeva NN, Zakharkina OL, Soshnikova YM, Yuzhakov AV, Shcherbakov EM, Shekhter AB, Guller AE, Bolshunov AV, Baskov AV, Svistushkin VM. Promising laser technology for modification of structure and reconstruction of tissues in otolaryngology, orthopedics and ophthalmology. In book: Panchenko VYa, Lebedev VF, eds. Modern laser-information technology [In Russian]. Moscow: “Interkontakt Nauka” Publisher; 2015: 567-614.
  24. Veldkamp WB, Kastner CJ. Beam profile shaping for laser radars that use detector arrays. Appl Opt 1982; 21: 345-356.
  25. Veldkamp WB. Laser beam profile shaping with interlaced binary gratings. Appl Opt 1982; 21: 3209-3212.
  26. Han CY, Ishii Y, Murata K. Reshaping collimated laser beams with Gaussian profile to uniform profiles. Appl Opt 1983; 22(22): 3644-3647.
  27. Roberts NC. Beam shaping by holographic filters. Appl Opt 1989; 28(1): 31-32.
  28. Eismann MT, Tai AM, Cederquist JN. Iterative design of holographic beamformer. Appl Opt 1989; 28(13): 2541-2650.
  29. Golub MA, Doskolovich LL, Kazanskiy NL, Sisakyan IN, Soifer VA, Kharitonov SI. Computational experiment with the focusator of Gaussian beam into a rectangle with constant intensity [In Russian]. Computer Optics 1990; 7: 42-49.
  30. Aleksoff CC, Ellis KK, Neagle BD. Holographic conversion of a Gaussian beam to a near-field uniform beam. Optical Engineering 1991; 30(5): 537-543.
  31. Sweatt WC. Transforming a circular laser beam into a square or trapezoid-almost. Optical Engineering 1992; 31(2): 245-250.
  32. Roberts NC. Multilevel computer-generated holograms with separable phase functions for beam shaping. Appl Opt 1992; 31(17): 3198-3200.
  33. Golub MA, Doskolovich LL, Kazanskiy NL, Soifer VA, Kharitonov SI. Study of focusators in a rectangle by the method of computational experiment [In Russian]. Computer Optics 1992; 10-11: 110-122.
  34. Doskolovich LL, Kazanskiy NL. The study of holographic optical elements, the focusing in a two-dimensional rectangular region [In Russian]. Computer Optics 1992; 12: 14-17.
  35. Cordingley J. Application of a binary diffractive optic for beam shaping in semiconductor processing by lasers. Appl Opt 1993; 32(14): 2538-2542.
  36. Doskolovich LL, Golub MA, Kazanskiy NL, Kharitonov SI, Soifer VA. Diffraction investigation of focusators into plane area. Proc SPIE 1993; 1983(2): 656-657.
  37. Stevenson RM, Norman MJ, Bett TH, Pepler DA, Danson CN, Ross IN. Binary-phase zone plate arrays for the generation of uniform focal profiles. Opt Lett 1994; 19(6): 363-365.
  38. Kazanskiy NL, Kotlyar VV, Soifer VA. Computer-aided design of diffractive optical elements. Optical Engineering 1994; 33(10): 3156-3166. DOI: 10.1117/12.178898.
  39. Doskolovich LL, Kazanskij NL, Sojfer VA. Calculation of binary diffraction optical elements for focusing into a given two-dimensional region [in Russian]. Avtometriya 1995; (5): 42-50.
  40. Duparré M, Golub MA, Lüdge B, Pavelyev VS, Soifer VA, Uspleniev GV, Volotovskii SG. Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2 laser beams. Applied Optics 1995; 34(14): 2489-2497. DOI: 10.1364/AO.34.002489.
  41. Khonina SN, Kotlyar VV, Skidanov RV, Soifer VA. Levelling the focal spot intensity of the focused Gaussian beam. Journal of Modern Optics 2000; 47(5): 883-904.
  42. Doskolovich LL, Khonina SN, Kotlyar VV, Nikolsky IV, Soifer VA, Uspleniev GV. Focusators into a ring. Optical and Quantum Electronics 1993; 25(11): 801-814.
  43. Greisukh GI, Bobrov ST, Stepanov SA. Optics of diffractive and gradient-index elements and systems. Bellingham: SPIE Press; 1997.
  44. Komenda O, Skeren M. Design of rotationally symmetric diffractive beam shapers using IFTA. Frontiers in Optics 2006: JSuA48. DOI: 10.1364/FIO.2006.JSuA48.
  45. Umhofer U, Jäger E, Bischoff C. Refractive and diffractive laser beam shaping optics. Laser Technik Journal 2011; 8: 24-27. DOI: 10.1002/latj.201190020.
  46. Yao PH, Chen CH, Chen CH. Low speckle laser illuminated projection system with a vibrating diffractive beam shaper. Optics Express 2012; 20(15): 16552-16566. DOI: 10.1364/OE.20.016552.
  47. Hsu KH, Lin HY. Trade-off between diffraction efficiency and uniformity for design of binary diffractive laser beam shaper. Optical Review 2013; 20(4): 296-302. DOI: 10.1007/s10043-013-0054-x.
  48. Doskolovich LL, Kazansky NL, Kharitonov SI, Soifer VA. A method of designing diffractive optical elements focusing into plane areas. Journal of Modern Optics 1996; 43(7): 1423-1433. DOI: 10.1080/09500349608232815.
  49. Koshlyakov NS, Gleaner EB, Smirnov MM. Partial differential equations of mathematical physics [In Russian]. Moscow: “Vyshaya Shkola” Publisher; 1970. 712 p.

© 2009, IPSI RAS
Institution of Russian Academy of Sciences, Image Processing Systems Institute of RAS, Russia, 443001, Samara, Molodogvardeyskaya Street 151; e-mail:; Phones: +7 (846 2) 332-56-22, Fax: +7 (846 2) 332-56-20