Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia3095310.22363/2658-4670-2022-30-2-149-159Research ArticleInvestigation of adiabatic waveguide modes model for smoothly irregular integrated optical waveguidesSevastyanovAnton L.<p>PhD in Physical and Mathematical Sciences, Deputy head of department: Department of Digitalization of Education</p>alsevastyanov@gmail.comhttps://orcid.org/0000-0002-0280-485XHigher School of Economics0305202230214915903052022Copyright © 2022, Sevastyanov A.L.2022<p style="text-align: justify;">The model of adiabatic waveguide modes (AWMs) in a smoothly irregular integrated optical waveguide is studied. The model explicitly takes into account the dependence on the rapidly varying transverse coordinate and on the slowly varying horizontal coordinates. Equations are formulated for the strengths of the AWM fields in the approximations of zero and first order of smallness. The contributions of the first order of smallness introduce depolarization and complex values characteristic of leaky modes into the expressions of the AWM electromagnetic fields. A stable method is proposed for calculating the vertical distribution of the electromagnetic field of guided modes in regular multilayer waveguides, including those with a variable number of layers. A stable method for solving a nonlinear equation in partial derivatives of the first order (dispersion equation) for the thickness profile of a smoothly irregular integrated optical waveguide in models of adiabatic waveguide modes of zero and first orders of smallness is described. Stable regularized methods for calculating the AWM field strengths depending on vertical and horizontal coordinates are described. Within the framework of the listed matrix models, the same methods and algorithms for the approximate solution of problems arising in these models are used. Verification of approximate solutions of models of adiabatic waveguide modes of the first and zero orders is proposed; we compare them with the results obtained by other authors in the study of more crude models.</p>smoothly irregular thin-film dielectric waveguidesadiabatic waveguide modesregularized methods for calculating field strengthsмодели квантовых измеренийвозмущение дискретного спектракомплексные собственные значенияпучки операторов1. Introduction The adiabatic waveguide propagation of optical radiation was previously described in optical fibers using the method of cross sections in the papers by B.Z. Katsenelenbaum [1], V.V. Shevchenko [2], M.V. Fedoruk [3], and in integrated optical waveguides using the method of adiabatic waveguide modes - in the papers by A.A. Egorov, L.A. Sevastyanov and their coauthors [4]-[6]. In the papers by A.L. Sevastyanov [7], [8], the model of adiabatic waveguide modes was substantiated. It should be noted that in the last decade there has been an interest in the adiabatic waveguide propagation of electromagnetic radiation for the study of coherent quantum effects in atomic, molecular or condensed matter systems. These effects are difficult to investigate because of dephasing effects or fast temporal dynamics. Optical Bloch oscillations [9], quantum-mechanical analogy of dynamic mode stabilization and radiation loss suppression [10], quantum enhancement and suppression of tunneling in directional optical couplers [11], [12], as well as Landau-Zener tunneling in coupled waveguides [13] can serve as optical models of coherent quantum effects. An interesting example is the three-level system with stimulated Raman adiabatic passage (STIRAP), which vividly illustrates counterintuitive quantum effects [14]- [19]. 2. 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