On a dispersion curve of a waveguide filled with inhomogeneous substance

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Abstract

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.

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1. Introduction In classical electrodynamics, there are two related spectral problems - the problem of the normal modes of the waveguide and the problem of eigenmodes of the resonator. Recall their formulations. Let
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About the authors

Oleg K. Kroytor

Peoples’ Friendship University of Russia (RUDN University)

Email: kroytor-ok@rudn.ru
ORCID iD: 0000-0002-5691-7331

Senior lecturer of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Mikhail D. Malykh

Peoples’ Friendship University of Russia (RUDN University); Joint Institute for Nuclear Research

Author for correspondence.
Email: malykh-md@rudn.ru
ORCID iD: 0000-0001-6541-6603

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

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation

References

  1. M. M. Karliner, Microwave electrodynamics: Lecture course [Elektrodinamika SVCH: Kurs lektsiy]. Novosibirsk: Novosibirsk state iniversity, 2006, In Russian.
  2. G. Duvaut and J. L. Lions, Les inéquations en mécanique et en physique. Paris: Dunod, 1972.
  3. I. E. Mogilevsky and A. G. Sveshnikov, Mathematical problems of diffraction theory [Matematicheskiye zadachi teorii difraktsii]. Moscow: MSU, 2010, In Russian.
  4. W. C. Chew, Lectures on theory of microwave and optical waveguides. 2012. doi: 10.48550/arXiv.2107.09672.
  5. A. N. Bogolyubov and T. V. Edakina, “Application of variationdifference methods for calculation of dielectric waveguides [Primeneniye variatsionno-raznostnykh metodov dlya rascheta dielektricheskikh volnovodov],” Bulletin of Moscow University. Series 3: Physics. Astronomy, vol. 32, no. 2, pp. 6-14, 1991, In Russian.
  6. A. N. Bogolyubov and T. V. Edakina, “Calculation of dielectric waveguides with complex cross-section shape by variation-difference method [Raschet dielektricheskikh volnovodov so slozhnoy formoy poperechnogo secheniya variatsionno-raznostnym metodom],” Bulletin of Moscow University. Series 3: Physics. Astronomy, vol. 34, no. 3, pp. 72-74, 1992, In Russian.
  7. P. Deuflhard, F. Schmidt, T. Friese, and L. Zschiedrich, “Adaptive Multigrid Methods for the Vectorial Maxwell Eigenvalue Problem for Optical Waveguide Design,” in Mathematics - Key Technology for the Future, W. Jäger and H. J. Krebs, Eds., Berlin-Heidelberg: Springer, 2011, pp. 279-292. doi: 10.1007/978-3-642-55753-8_23.
  8. F. Schmidt, S. Burger, J. Pomplun, and L. Zschiedrich, “Advanced FEM analysis of optical waveguides: algorithms and applications,” Proc. SPIE, vol. 6896, 2008. doi: 10.1117/12.765720.
  9. E. Lezar and D. B. Davidson, “Electromagnetic waveguide analysis,” in Automated solution of differential equations by the finite element method, The FEniCS Project, 2011, pp. 629-643.
  10. A. N. Bogolyubov, A. L. Delitsyn, and A. G. Sveshnikov, “On the completeness of the set of eigen- and associated functions of a waveguide [O polnote sistemy sobstvennykh i prisoyedinennykh funktsiy volnovoda],” Computational Mathematics and Mathematical Physics, vol. 38, no. 11, pp. 1815-1823, 1998, In Russian.
  11. A. N. Bogolyubov, A. L. Delitsyn, and A. G. Sveshnikov, “On the problem of excitation of a waveguide filled with an inhomogeneous medium [O zadache vozbuzhdeniya volnovoda s neodnorodnym zapolneniyem],” Computational Mathematics and Mathematical Physics, vol. 39, no. 11, pp. 1794-1813, 1999, In Russian.
  12. A. L. Delitsyn, “An approach to the completeness of normal waves in a waveguide with magnetodielectric filling,” Differential Equations, vol. 36, no. 5, pp. 695-700, 2000. doi: 10.1007/BF02754228.
  13. A. L. Delitsyn, “On the completeness of the system of eigenvectors of electromagnetic waveguides,” Computational Mathematics and Mathematical Physics, vol. 51, no. 10, pp. 1771-1776, 2011. doi: 10.1134/S0965542511100058.
  14. N. A. Novoselova, S. B. Raevsky, and A. A. Titarenko, “Calculation of propagation characteristics of symmetrical waves of round waveguide with radial non-uniform dielectric filling [Raschet kharakteristik rasprostraneniya simmetrichnykh voln kruglogo volnovoda s radial’noneodnorodnym dielektricheskim zapolneniyem],” Proceedings of Nizhny Novgorod State Technical University named after R.E. Alekseev, vol. 2, no. 81, pp. 30-38, 2010, In Russian.
  15. A. L. Delitsyn and S. I. Kruglov, “Mixed finite elements used to analyze the real and complex modes of cylindrical waveguides,” Moscow University Physics Bulletin, vol. 66, no. 6, pp. 546-551, 2011. doi: 10.3103/S0027134911060063.
  16. A. A. Tiutiunnik, D. V. Divakov, M. D. Malykh, and L. A. Sevast’yanov, “Symbolic-Numeric Implementation of the Four Potential Method for Calculating Normal Modes: An Example of Square Electromagnetic Waveguide with Rectangular Insert,” Lecture notes in computer science, vol. 11661, pp. 412-429, 2019. doi: 10.1007/978-3-030-26831-2_27.
  17. D. V. Divakov, M. D. Malykh, L. A. Sevast’yanov, and A. A. Tiutiunnik, “On the Calculation of Electromagnetic Fields in Closed Waveguides with Inhomogeneous Filling,” Lecture notes in computer science, vol. 11189, pp. 458-465, 2019. doi: 10.1007/978-3-030-10692-8_52.
  18. M. D. Malykh and L. A. Sevast’yanov, “On the representation of electromagnetic fields in discontinuously filled closed waveguides by means of continuous potentials,” Computational Mathematics and Mathematical Physics, vol. 59, no. 2, pp. 330-342, 2019. doi: 10.1134/S0965542519020118.
  19. O. K. Kroytor, M. D. Malykh, and L. A. Sevast’yanov, “On normal modes of a waveguide,” Computational Mathematics and Mathematical Physics, vol. 62, no. 3, pp. 393-410, 2022. doi: 10.1134/S0965542522030083.

Copyright (c) 2022 Kroytor O.K., Malykh M.D.

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