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)3301410.22363/2658-4670-2022-30-4-330-341ArticlesСтатьиResearch ArticleOn a dispersion curve of a waveguide filled with inhomogeneous substanceО дисперсионной кривой волновода, заполненного неоднородным веществомhttps://orcid.org/0000-0002-5691-7331KroytorOleg K.КройторО. К.

Senior lecturer of Department of Applied Probability and Informatics

kroytor-ok@rudn.ruhttps://orcid.org/0000-0001-6541-6603MalykhMikhail D.МалыхМ. Д.

Doctor of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics

malykh-md@rudn.ruPeoples’ Friendship University of Russia (RUDN University)Российский университет дружбы народовJoint Institute for Nuclear ResearchОбъединённый институт ядерных исследований26122022304VOL 30, NO4 (2022)ТОМ 30, №4 (2022)33034126122022Copyright ©; 2022, Kroytor O.K., Malykh M.D.Copyright ©; 2022, Кройтор О.К., Малых М.Д.2022Kroytor O.K., Malykh M.D.Кройтор О.К., Малых М.Д.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/33014

The paper discusses the relationship between the modes traveling along the axis of the waveguide and the standing modes of a cylindrical resonator, and shows how this relationship can be explored using the Sage computer algebra system. In this paper, we study this connection and, on its basis, describe a new method for constructing the dispersion curve of a waveguide with an optically inhomogeneous filling. The aim of our work was to find out what computer algebra systems can give when calculating the points of the waveguide dispersion curve. Our method for constructing the dispersion curve of a waveguide with optically inhomogeneous filling differs from those proposed earlier in that it reduces this problem to calculating the eigenvalues of a self-adjoint matrix, i.e., a well-studied problem. The use of a selfadjoint matrix eliminates the occurrence of artifacts associated with the appearance of a small imaginary addition to the eigenvalues. We have composed a program in the Sage computer algebra system that implements this method for a rectangular waveguide with rectangular inserts and tested it on SLE modes. The obtained results showed that the program successfully copes with the calculation of the points of the dispersion curve corresponding to the hybrid modes of the waveguide, and the points found fit the analytical curve with graphical accuracy even when with a small number of basis elements taken into account.

В статье рассматривается связь между модами, бегущими вдоль оси волновода, и стоячими модами цилиндрического резонатора. Показывается, как данная связь может быть исследована с помощью системы компьютерной алгебры Sage. В работе мы исследуем эту связь и на её основе описываем новый метод построения дисперсионной кривой волновода с оптически неоднородным заполнением. Целью нашей работы было выяснить, что могут дать системы компьютерной алгебры при вычислении (точек) дисперсионной кривой волновода. Метод построения дисперсионной кривой волновода с оптически неоднородным заполнением, предложенный нами, отличается от предложенных ранее тем, что сводит эту задачу к вычислению собственных значений самосопряжённой матрицы, то есть к задаче, хорошо изученной. Использование самосопряжённой матрицы исключает возникновение артефактов, связанных с появлением малой мнимой добавки у собственных значений. Мы составили программу в системе компьютерной алгебры Sage, в которой реализован этот метод для волновода прямоугольного сечения с прямоугольными вставками, и протестировали её на SLE-модах. Полученные результаты показали, что программа успешно справляется с вычислением точек дисперсионной кривой, отвечающих гибридным модам волновода, и найденные точки с графической точностью ложатся на аналитическую кривую даже при небольшом числе учитываемых базисных элементов.

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