Discrete and Continuous Models and Applied Computational ScienceDiscrete 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)2470510.22363/2658-4670-2020-28-3-252-273ArticlesСтатьиResearch ArticleAsymptotic method for constructing a model of adiabatic guided modes of smoothly irregular integrated optical waveguidesАсимптотический метод построения модели адиабатических волноводных мод плавно-нерегулярных интегрально-оптических волноводовSevastianovAnton L.СевастьяновА. Л.

Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics

sevastianov-al@rudn.ruPeoples’ Friendship University of Russia (RUDN University)Российский университет дружбы народов15122020283VOL 28, NO3 (2020)ТОМ 28, №3 (2020)25227328092020Copyright ©; 2020, Sevastianov A.L.Copyright ©; 2020, Севастьянов А.Л.2020Sevastianov A.L.Севастьянов А.Л.http://creativecommons.org/licenses/by/4.0https://journals.rudn.ru/miph/article/view/24705

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 Maxwell’s 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 Maxwell’s equations.

В работе рассмотрен класс плавно нерегулярных интегрально-оптических многослойных волноводов, свойства которых определяют характерные черты волноводного распространения в них монохроматического поляризованного света. Предложен асимптотический подход к описанию данного вида электромагнитного излучения, в результате которого решения системы уравнений Максвелла редуцируется к такому виду, который выражается через решения системы четырёх обыкновенных дифференциальных уравнений и двух алгебраических уравнений для шести компонент электромагнитного поля в нулевом приближении. Градиент фазового фронта адиабатической волноводной моды удовлетворяет уравнению эйконала относительно эффективного показателя преломления волновода относительно данной моды.Многослойная структура волноводов позволяет произвести ещё один этап редукции системы уравнений модели к однородной системе линейных алгебраических уравнений, условие нетривиальной разрешимости которой задаёт связь градиента фазового фронта излучения с градиентами поверхностей раздела между тонкими однородными слоями.В завершающей части работы сформулированы задачи (дифференциальная и алгебраическая) на собственные значения и собственные векторы для описания адиабатических волноводных мод. Приведена также формулировка задачи описания одномодового режима распространения адиабатических волноводных мод, подчёркивающая адиабатический характер описываемого приближенного решения уравнений Максвелла.

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