Symbolic-Numeric Computation of the Eigenvalues and Eigenfunctions of the Leaky Modes in a Regular Homogeneous Open Waveguide

Dmitriy Divakov,Anastasiia Tiutiunnik,Anton Sevastianov,H Akca,R Luca Tudorache,L Abdullah

doi:10.1051/matecconf/201818601009

Dmitriy Divakov, Anastasiia Tiutiunnik + Show 4 more

Open Access

https://doi.org/10.1051/matecconf/201818601009

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Abstract

In this paper the algorithm of finding eigenvalues and eigenfunctions for the leaky modes in a three-layer planar dielectric waveguide is considered. The problem on the eigenmodes of open three-layer waveguides is formulated as the Sturm-Liouville problem with the corresponding boundary and asymptotic conditions. In the case of guided and radiation modes of open waveguides, the Sturm-Liouville problem is formulated for self-adjoint second-order operators on the axis and the corresponding eigenvalues are real quantities for dielectric media. The search for eigenvalues and eigenfunctions corresponding to the leaky modes involves a number of difficulties: the boundary conditions for the leaky modes are not self-adjoint, so that the eigenvalues can turn out to be complex quantities. The problem of finding eigenvalues and eigenfunctions will be associated with finding the complex roots of the nonlinear dispersion equation. In the present paper, an original scheme based on the method of finding the minimum of a function of several variables is used to find the eigenvalues. The paper describes the algorithm for searching for eigenvalues, the algorithm uses both symbolic transformations and numerical calculations. On the basis of the developed algorithm, the dispersion relation for the weakly flowing mode of a three-layer open waveguide was calculated in the Maple computer algebra system.

Highlights

The paper presents the formulation of the problem of numerical calculation of the leaky modes for planar three-layer waveguides
The propagation of waveguide radiation in open waveguides is described by Maxwell's equations, boundary conditions at media interface, constitutive equations and asymptotic conditions [1,2,3]
Taking into account the structure of the three-layer waveguide, the solution of the equation for the transverse part can be written in an analytical form in each of the regions, and for regions x h and x 0 it is necessary to choose a solution that satisfies the asymptotic conditions

Summary

Introduction

The paper presents the formulation of the problem of numerical calculation of the leaky modes for planar three-layer waveguides. The propagation of waveguide radiation in open waveguides is described by Maxwell's equations, boundary conditions at media interface, constitutive equations and asymptotic conditions [1,2,3]. The asymptotic conditions for the leaky modes in planar waveguide structure have the following form: Ey x ik0. Taking into account the structure of the three-layer waveguide, the solution of the equation for the transverse part can be written in an analytical form in each of the regions, and for regions x h and x 0 it is necessary to choose a solution that satisfies the asymptotic conditions. Time harmonic fields, under the assumption of the field invariance along the direction Oy , the Maxwell's equations are reduced to a pair of the Helmholtz equations: TE-mode and TM-mode; and two pairs of additional relations [1,2,3,4,5,6]:

Solution algorithm

Conclusion