Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)3301610.22363/2658-4670-2022-30-4-357-363Research ArticleProfile thickness synthesis of thin-film waveguide Luneburg lensLovetskiyKonstantin P.<p>Candidate of Physical and Mathematical Sciences, Associate Professor of Department of Applied Probability and Informatics</p>lovetskiy-kp@rudn.ruhttps://orcid.org/0000-0002-3645-1060SevastianovAnton L.<p>Candidate of Physical and Mathematical Sciences, Deputy Head of Department of Education digitalization</p>alsevastyanov@gmail.comhttps://orcid.org/0000-0002-0280-485XZorinAlexander V.<p>Doctor of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics</p>zorin-av@rudn.ruhttps://orcid.org/0000-0002-5721-4558Peoples’ Friendship University of Russia (RUDN University)Higher School of Economics (HSE University)2612202230435736326122022Copyright © 2022, Lovetskiy K.P., Sevastianov A.L., Zorin A.V.2022<p style="text-align: justify;">In the work the link between the focusing inhomogeneity of the effective refractive index of waveguide Luneburg lens and the irregularity of the waveguide layer thickness generating this inhomogeneity is demonstrated. For the dispersion relation of irregular thin-film waveguide in the model of adiabatic waveguide modes the problem of mathematical synthesis and computer-aided design of the waveguide layer thickness profile for the Luneburg thin-film generalized waveguide lens with a given focal length is being solved. The calculations are carried out in normalized (in a special way) coordinates to adapt the used relations to computer calculations. The obtained solution is compared with the same solution within the cross-section’s method. The performance of the algorithm implemented in Delphi, was demonstrated by plotting the dispersion curves and plotting a family of dispersion curves, demonstrating a critical convergence. As an additional result, the thickness profiles of additional (irregular in thickness) waveguide layer, forming a thin film generalized waveguide Luneburg lens were synthesized. This result generalizes Southwell’s results.</p>Luneburg waveguide lenseffective refractive index inhomogeneitysection methodwaveguide adiabatic mode modelволноводная линза Люнеберганеоднородность эффективного показателя преломленияметод сечениймодель адиабатических мод волновода[A. L. Sevastyanov, “Computer modeling of directed modes’ fields of thin-film generalized waveguide Luneburg lens. Candidate’s Dissertation in Physics and Mathematics [Komp’yuternoe modelirovanie polej napravlyaemyh mod tonkoplenochnoj obobshchennoj volnovodnoj linzy Lyuneberga],” in Russian, Ph.D. dissertation, Peoples’ Friendship University of Russia, Moscow, 2010.][L. A. Sevast’yanov and A. A. Egorov, “Theoretical analysis of the waveguide propagation of electromagnetic waves in dielectric smoothlyirregular integrated structures,” Optics and Spectroscopy, no. 105, pp. 576-584, 2008. DOI: 10.1134/S0030400X08100123.][A. A. Egorov and L. A. 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Southwell, “Index profiles for generalized Luneburg lenses and their use in planar optical waveguides,” Journal of the Optical Society of America, vol. 67, no. 8, pp. 1010-1014, 1977. DOI: 10.1364/JOSA.67.001010.][E. Fehlberg, “Low-order Classical Runge-Kutta formulas with step size control,” NASA Technical Report R-315, Tech. Rep., 1969.][A. L. Sevastyanov, “Structure of modes of smoothly irregular threedimensional integrated optical four-layer waveguide,” Physics of Particles and Nuclei Letters, vol. 8, no. 479, pp. 804-811, 2011. DOI: 10.1134/S1547477111050177.][A. A. Egorov et al., “Stable computer modeling of thin-film generalized waveguide Luneburg lens [Ustojchivoe komp’yuternoe modelirovanie tonkoplenochnoj obobshchennoj volnovodnoj linzy Lyuneberga],” Matem. modelirovaniye, vol. 26, no. 11, pp. 37-44, 2014, in Russian.]