http://mmg.tversu.ru ISSN 2311-1275 M M G Mathematical Modelling and Geometry Volume 5, No 1, p. 1 – 20 (2017) Guided Modes of a Planar Gradient Waveguide M. N. Gevorkyan 1,a , D. S. Kulyabov 1,2,b , K. P. Lovetskiy 1,c N. E. Nikolaev 4,d A. L. Sevastianov 1,e and L. A. Sevastianov 1,3,f 1 Department of Applied Probability and Informatics, RUDN University, Miklukho-Maklaya str. 6, 117198 Moscow, Russia 2 Laboratory of Information Technologies, Joint Institute for Nuclear Research, Joliot-Curie 6, 141980 Dubna, Moscow region, Russia 3 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Joliot-Curie 6, 141980 Dubna, Moscow region, Russia 4 Department of Applied Physics RUDN University, Miklukho-Maklaya str. 6, 117198 Moscow, Russia e-mail: a mngevorkyan@sci.pfu.edu.ru, b dharma@sci.pfu.edu.ru, c lovetskiy@gmail.com, d nnikolaev@sci.pfu.edu.ru, e alsevastyanov@gmail.com, f sevast@sci.pfu.edu.ru Received 4 February 2017, in ﬁnal form 22 February. Published 24 February 2017. Abstract. The mathematical model of light propagation in a planar gradient op- tical waveguide consists of the Maxwell’s equations supplemented by the matter equations and boundary conditions. In the coordinates adapted to the waveguide geometry, the Maxwell’s equations are separated into two independent sets for the TE and TM polarizations. Each of the systems can be transformed to a second order ordinary diﬀerential equation. The boundary conditions for Maxwell’s equa- tions are reduced to two pairs of boundary conditions for the obtained equations. Thus, the problem of describing a complete set of modes in a regular planar waveg- uide is formulated in terms of an eigenvalue problem. For each polarization there are three types of waveguide modes: guided modes, substrate radiation modes, and cover radiation modes. In this work we implement the numerical-analytical calculation of all types of waveguide modes. Keywords: waveguide propagation of electromagnetic radiation, equations of waveguide modes of regular waveguide, complete set of modes of a planar gra- dient waveguide MSC numbers: 65Fxx, 65Hxx, 65L10, 65L15, 78A40, 78Mxx The work was partially supported by RFBF grants No 15-07-08795, No 16-07-00556. The work was supported by the Ministry of Education of the Russian Federation (the Agreement No 02.a03.21.0008 of 24.06.2016). c  The author(s) 2017. Published by Tver State University, Tver, Russia