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)2518210.22363/2658-4670-2020-28-4-361-377Research ArticleSingle-mode propagation of adiabatic guided modes in smoothly irregular integral 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)1512202028436137709122020Copyright © 2020, Sevastianov A.L.2020<p>This paper investigates the waveguide propagation of polarized electromagnetic radiation in a thin-film integral optical waveguide. To describe this propagation, the adiabatic approximation of solutions of Maxwells equations is used. The construction of a reduced model for adiabatic waveguide modes that retains all the properties of the corresponding approximate solutions of the Maxwell system of equations was carried out by the author in a previous publication in DCM ACS, 2020, No 3. In this work, for a special case when the geometry of the waveguide and the electromagnetic field are invariant in the transverse direction. In this case, there are separate nontrivial TEand TM-polarized solutions of this reduced model. The paper describes the parametrically dependent on longitudinal coordinates solutions of problems for eigenvalues and eigenfunctions - adiabatic waveguide TE and TM polarizations. In this work, we present a statement of the problem of finding solutions to the model of adiabatic waveguide modes that describe the stationary propagation of electromagnetic radiation. The paper presents solutions for the single-mode propagation of TE and TM polarized adiabatic waveguide waves.</p>waveguide propagation of polarized lightintegral optical waveguideadiabatic approximationeigenvalues and eigenfunctionsKantorovich methodsingle-mode regimeволноводное распространение поляризованного светаинтегральный оптический волноводадиабатическое приближениесобственные значения и собственные функцииметод Канторовичаодномодовый режим<p>1. Introduction In works [1]-[5] a cycle of studies of the propagation of polarized light in integrated-optical smoothly irregular thin-film waveguides was carried out within the framework of the model of adiabatic waveguide waves. They showed the advantages of the model and its advantages over other models in the description of open dielectric waveguides [6]-[8]. At the same time, until recently, the question of substantiating this model remained open. In work [9] the substantiation of the model was carried out, which is a reduction of a more complex in use general model based on Maxwells equations. In the present work, within the framework of the model of adiabatic waveguide waves, the problem of stationary propagation of polarized light in a smoothly irregular Sevastianov A. L., 2020 This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/ integral-optical waveguide is posed, an auxiliary problem for eigenvalues and eigenfunctions (adiabatic waveguide modes) is formulated and solved. The solution of the stationary problem by the generalized Kantorovich method is proposed, its solution is obtained in the single-mode propagation mode. 2. Basic concepts and notation Waveguide propagation of monochromatic polarized electromagnetic radiation in integrated optical waveguides is described by Maxwells equations. The electromagnetic field is described using complex amplitudes. The material environment is considered, consisting of dielectric subdomains that fill the entire three-dimensional space. The latter means that the dielectric constants of the subdomains are different and real, and the magnetic permeability is everywhere equal to the magnetic permeability of the vacuum. It follows from the foregoing that in the absence of external currents and charges, the induced currents and charges are equal to zero. In the absence of external charges and currents, the scalar Maxwell equations follow from the vector ones, and the boundary conditions for the normal components follow from the boundary conditions for the tangential components. The constitutive equations of connection in the case under consideration are assumed to be linear. Thus, the electromagnetic field in a space filled with dielectrics in the Gaussian system of units is described by the equations [10]: rotE = - 1</p>[L. A. 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