On a problem of the synthesis of nano-optical protection elements





nano-optical elements, flat computer optics, electron beam lithography, inverse problems, computer generated hologram, protection against counterfeiting, pattern recognition


This paper deals with the mathematical modeling and the synthesis of nano-optical elements for the forming of 2D images used for automated identification. The inverse problem of synthesis is considered in Fresnel’s scalar wave approximation. The problem is reduced to the solution of a nonlinear Fredholm operator equation of the first kind. Some efficient numerical iterative methods are developed to solve the inverse problem. When illuminated by a coherent light, the designed nano-optical elements produce 2D images used for the automated identification. The proposed identification procedure is invariant with respect to rotations and shifts of images. The electron beam lithography was used to produce samples of nano-optical elements for the case of 650 nm wavelength. The microfabrication technology allows forming a microrelief with 20 nm accuracy. Such nanooptical elements are resistant against the damage of microreliefs and can be used for the verification of banknotes, documents, etc.

Author Biographies

A.A. Goncharsky

S.R. Durlevich


  1. H. Sauer, P. Chavel, and G. Erdei, “Diffractive Optical Elements in Hybrid Lenses: Modeling and Design by Zone Decomposition,” Appl. Opt. 38 (31), 6482-6486 (1999).
  2. F. Belloni and S. Monneret, “Quadrant Kinoform: An Approach to Multiplane Dynamic Three-Dimensional Holographic Trapping,” Appl. Opt. 46 (21), 4587-4593 (2007).
  3. Q. Tan, Y. Yan, G. Jin, and D. Xu, “Fine Design of Diffractive Optical Element for Beam Transform,” Proc. SPIE 4443, 184-188 (2001).
    doi 10.1117/12.446750
  4. X. Duan, G. Zhou, Y. Huang, et al., “Theoretical Analysis and Design Guideline for Focusing Subwavelength Gratings,” Opt. Express 23 (3), 2639-2646 (2015).
  5. P. Karvinen, D. Grolimund, M. Willimann, et al., “Kinoform Diffractive Lenses for Efficient Nano-Focusing of Hard X-rays,” Opt. Express. 22 (14), 16676-16685 (2014).
  6. P. Macko and M. P. Whelan, “Fabrication of Holographic Diffractive Optical Elements for Enhancing Light Collection from Fluorescence-Based Biochips,” Opt. Lett. 33 (22), 2614-2616 (2008).
  7. J.-H. Jang, C. K. Ullal, M. Maldovan, et al., “3D Micro- and Nanostructures via Interference Lithography,” Adv. Funct. Mater. 17 (16), 3027-3041 (2007).
  8. N. J. Jenness, R. T. Hill, A. Hucknall, et al., “A Versatile Diffractive Maskless Lithography for Single-Shot and Serial Microfabrication,” Opt. Express 18 (11), 11754-11762 (2010).
  9. A. V. Goncharsky and A. A. Goncharsky, Computer Optics and Computer Holography (Mosk. Gos. Univ., Moscow, 2004) [in Russian].
  10. A. A. Goncharsky and S. R. Durlevich, “A Problem of Synthesis of Nano-Optical Elements for the Formation of Dynamic Images,” Vychisl. Metody Programm. 14, 343-347 (2013).
  11. A. A. Goncharsky, “On the Problem of Synthesis of Nano-Optical Elements,” Vychisl. Metody Programm. 9, 405-408 (2008).
  12. A. V. Goncharsky, I. V. Kochikov, and A. N. Matvienko, Reconstructive Processing and Analysis of Images in the Problems of Computational Diagnostics (Mosk. Gos. Univ., Moscow, 1993) [in Russian].
  13. G. A. Boutry, “Augustin Fresnel: His Time, Life and Work, 1788-1827,” in Science Progress (Murray, London, 1949), Vol. 36, pp. 587-604.
  14. A. N. Tikhonov, “Solution of Incorrectly Formulated Problems and the Regularization Method,” Dokl. Akad. Nauk SSSR 151 (3), 501-504 (1963) [Sov. Math. Dokl. 5 (4), 1035-1038 (1963)].
  15. A. N. Tikhonov, “On the Problems with Approximately Specified Information,” in Ill-Posed Problems in the Natural Sciences (Mir, Moscow, 1987), pp. 13-20.
  16. M. M. Lavrentyev, Some Improperly Posed Problems of Mathematical Physics (Sib. Otdel. Akad. Nauk SSSR, Novosibirsk, 1962; Heidelberg, Springer, 1967).
  17. V. K. Ivanov, “The Approximate Solution of Operator Equations of the First Kind,” Zh. Vychisl. Mat. Mat. Fiz. 6 (6), 1089-1094 (1966) [USSR Comput. Math. Math. Phys. 6 (6), 197-205 (1966)].
  18. A. Bakushinsky and A. Goncharsky, Ill-Posed Problems: Theory and Applications (Kluwer, Dordrecht, 1994).
  19. A. A. Goncharsky and D. V. Tunitsky, “The Inverse Problem of Synthesis of Optical Elements for Laser Radiation,” Vychisl. Metody Programm. 7, 138-162 (2006).
  20. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 105-155 (1969).
  21. A. V. Kuzmenko, “Weighting Iterative Fourier Transform Algorithm of the Kinoform Synthesis,” Opt. Lett. 33 (10), 1147-1149 (2008).
  22. S. Yang and T. Shimomura, “Iterative Multikinoform Method for Improving the Reconstruction of Kinoform,” Opt. Rev. 4 (6), 660-665 (1997).
  23. S. Nozaki, Y.-W. Chen, and Z. Nakao, “A New Approach Based on Simulated Annealing to Kinoform Optimization,” Proc. SPIE 3956, 160-166 (2000).
    doi 10.1117/12.379990
  24. A. V. Goncharsky, V. V. Popov, and V. V. Stepanov, Introduction to Computer Optics (Mosk. Gos. Univ., Moscow, 1991) [in Russian].
  25. D. Sarid, Scanning Force Microscopy (Oxford Univ. Press, New York, 1991).



How to Cite

Гончарский А., Дурлевич С. On a Problem of the Synthesis of Nano-Optical Protection Elements // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2015. 16. 290-297. doi 10.26089/NumMet.v16r228



Section 1. Numerical methods and applications

Most read articles by the same author(s)