Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia2470510.22363/2658-4670-2020-28-3-252-273Research ArticleAsymptotic method for constructing a model of adiabatic guided modes of smoothly irregular integrated optical waveguidesSevastianovAnton L.<p>Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics</p>sevastianov-al@rudn.ruPeoples’ Friendship University of Russia (RUDN University)1512202028325227328092020Copyright © 2020, Sevastianov A.L.2020<p>The paper considers a class of smoothly irregular integrated optical multilayer waveguides, whose properties determine the characteristic features of guided propagation of monochromatic polarized light. An asymptotic approach to the description of such electromagnetic radiation is proposed, in which the solutions of Maxwells equations are expressed in terms of the solutions of a system of four ordinary differential equations and two algebraic equations for six components of the electromagnetic field in the zero approximation. The gradient of the phase front of the adiabatic guided mode satisfies the eikonal equation with respect to the effective refractive index of the waveguide for the given mode.The multilayer structure of waveguides allows one more stage of reducing the model to a homogeneous system of linear algebraic equations, the nontrivial solvability condition of which specifies the relationship between the gradient of the radiation phase front and the gradients of interfaces between thin homogeneous layers.In the final part of the work, eigenvalue and eigenvector problems (differential and algebraic), describing adiabatic guided modes are formulated. The formulation of the problem of describing the single-mode propagation of adiabatic guided modes is also given, emphasizing the adiabatic nature of the described approximate solution of Maxwells equations.</p>smoothly irregular integrated optical multilayer waveguideseigenvalue and eigenvector problemsingle-mode propagation of adiabatic guided modesплавно нерегулярные интегрально-оптические многослойные волноводызадачи на собственные значения и собственные векторыодномодовый режим распространения адиабатических волноводных мод<p>Introduction Fundamental results in the theory of regular waveguides were obtained for closed (metallic) waveguides by A. N. Tikhonov and A. A. Samarskii [1], and for open (dielectric) waveguides by A. G. Sveshnikov [2] and V. V. Shevchenko [3]. Among the irregular waveguides, one can distinguish transversely irregular and longitudinally irregular waveguides. For transversely irregular waveguides, the equations and the corresponding solutions allow the separation of variables [4]. Sevastianov A. L., 2020 This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/ Here the incomplete Galerkin method developed by A. G. Sveshnikov [2], [5], [6] received the greatest recognition. For closed longitudinally irregular waveguides, B. Z. Katsenelenbaum de- veloped the method of cross sections [7], which was generalized for open longitudinally irregular waveguides by V. V. Shevchenko [8]. These models do not describe depolarization and hybridization of guided modes in irregular sec- tions of waveguides. A. A. Egorov, L. A. Sevastyanov and A. L. Sevastyanov developed the foundations of the theory of smoothly irregular 3D dielectric and, in particular, integrated optical waveguides [9], [10], which was success- fully applied to a number of three-dimensional integrated optical waveguides and smoothly irregular 3D waveguide devices based on them [11]-[13]. The mathematical basis of the model of adiabatic guided modes (AGMs) is the as- ymptotic method and the method of coupled modes. The asymptotic method for solving a boundary value problem for a system of differential equations with respect to a small parameter</p>[A. A. Samarskii and A. N. Tikhonov, “Representation of the field in a waveguide as the sum of the TE and TM fields [O predstavlenii polya v volnovode v vide summy polej TE i TM],” Zhurnal tekhnicheskoy fiziki, vol. 18, no. 7, pp. 959-970, 1948, in Russian.][A. G. 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