Abstract
Computational problems of electrodynamics require an approximate solution of the system of Maxwell’s vector equations for regions with different geometries. The main methods for solving problems with the Maxwell equations are either finite difference methods or methods based on the Galerkin and Kantorovich expansions, or the finite element method. Each of the classes of methods is characterised by a wide range of permissible objects, but in each of the methods, the solution contains a large number of quantities known only in numerical form.
We have chosen a different approach, in which to describe the waveguide propagation of electromagnetic radiation we propose using the model of adiabatic waveguide modes. This model allows reducing Maxwell equations to a system of ordinary differential equations, which allows analysis of its solutions at the symbolic level.
A fundamental system of solutions of the system is constructed in symbolic form. A numerical method for computing the guided modes of a planar three-layer open waveguide is formulated and implemented using a vector model of the adiabatic waveguide modes. Phase constants calculated in the framework of the model of adiabatic waveguide modes were verified by comparison with those calculated in the framework of the scalar model.
The publication has been prepared with the support of the “RUDN University Program 5-100” and funded by RFBR according to the research projects No. 18-07-00567 and 19-01-00645.
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References
Babich, V.M., Buldyrev, V.S.: Asymptotic Methods in Short-Wave Diffraction Problems. Nauka, Moscow (1972). [English translation: Springer Series on Wave Phenomena 4. Springer, Berlin Heidelberg New York 1991]
Kantorovich, L.V., Krylov, V.I.: Approximate Methods of Higher Analysis. Wiley, New York (1964)
Fletcher, C.A.J.: Computational Galerkin Methods. Springer, Heidelberg (1984). https://doi.org/10.1007/978-3-642-85949-6
Gusev, A.A., et al.: Symbolic-numerical algorithms for solving the parametric self-adjoint 2D elliptic boundary-value problem using high-accuracy finite element method. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2017. LNCS, vol. 10490, pp. 151–166. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66320-3_12
Sevastyanov, L.A., Sevastyanov, A.L., Tyutyunnik, A.A.: Analytical calculations in maple to implement the method of adiabatic modes for modelling smoothly irregular integrated optical waveguide structures. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2014. LNCS, vol. 8660, pp. 419–431. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10515-4_30
Bathe, K.J.: Finite Element Procedures in Engineering Analysis. Prentice Hall, Englewood Cliffs (1982)
Ciarlet, P.: The Finite Element Method for Elliptic Problems. North Holland Publishing Company, Amsterdam (1978)
Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)
Bogolyubov, A.N., Mukhartova, Yu.V., Gao, J., Bogolyubov, N.A.: Mathematical modeling of plane chiral waveguide using mixed finite elements. In: Progress in Electromagnetics Research Symposium, pp. 1216–1219 (2012)
Bogolyubov, A.N., Mukhartova, Y.V., Gao, T.: Calculation of a parallel-plate waveguide with a chiral insert by the mixed finite element method. Math. Models Comput. Simul. 5(5), 416–428 (2013)
Mukhartova, Y.V., Mongush, O.O., Bogolyubov, A.N.: Application of the finite-element method for solving a spectral problem in a waveguide with piecewise constant bi-isotropic filling. J. Commun. Technol. Electronics 62(1), 1–13 (2017)
Adams, M.J.: An Introduction to Optical Waveguides. Wiley, New York (1981)
Marcuse, D.: Light Transmission Optics. Van Nostrand, New York (1974)
Tamir, T.: Guided-Wave Optoelectronics. Springer, Berlin (1990). https://doi.org/10.1007/978-3-642-97074-0
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)
Hartman, P.: Ordinary Differential Equations, Classics in Applied Mathematics, 38. Society for Industrial and Applied Mathematics, Philadelphia (2002). [1964]
Johnson, W.: A Treatise on Ordinary and Partial Differential Equations. Wiley, New York (1913). In University of Michigan Historical Math Collection
Polyanin, A.D., Zaitsev, V.F., Moussiaux, A.: Handbook of First Order Partial Differential Equations. Taylor & Francis, London (2002)
Zwillinger, D.: Handbook of Differential Equations, 3rd edn. Academic Press, Boston (1997)
Mathematics-based software and services for education, engineering, and research. https://www.maplesoft.com/
Lovetskiy, K.P., Gevorkyan, M.N., Kulyabov, D.S., Sevastyanov, A.L., Sevastyanov, L.A.: Waveguide modes of a planar optical waveguide. Math. Model. Geom. 3(01), 43–63 (2015)
Ayryan, E.A., Egorov, A.A., Michuk, E.N., Sevastyanov, A.L., Sevastianov, L.A., Stavtsev, A.B.: Representations of Guided Modes of Integrated-Optical Multilayer Thin-Film Waveguides. E11–2011-31, LIT preprints (2011)
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Divakov, D.V., Sevastianov, A.L. (2019). The Implementation of the Symbolic-Numerical Method for Finding the Adiabatic Waveguide Modes of Integrated Optical Waveguides in CAS Maple. In: England, M., Koepf, W., Sadykov, T., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2019. Lecture Notes in Computer Science(), vol 11661. Springer, Cham. https://doi.org/10.1007/978-3-030-26831-2_8
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