On the Problem of Electromagnetic Waves Propagating along a Nonlinear Inhomogeneous Cylindrical Waveguide

Abstract

Electromagnetic TE wave propagation in an inhomogeneous nonlinear cylindrical waveguide is considered. The permittivity inside the waveguide is described by the Kerr law. Inhomogeneity of the waveguide is modeled by a nonconstant term in the Kerr law. Physical problem is reduced to a nonlinear eigenvalue problem for ordinary differential equations. Existence of propagating waves is proved with the help of fixed point theorem and contracting mapping method. For numerical solution, an iteration method is suggested and its convergence is proved. Existence of eigenvalues of the problem (propagation constants) is proved and their localization is found. Conditions of k waves existence are found.

Nonlinear Double-Layer Bragg Waveguide: Analytical and Numerical Approaches to Investigate Waveguiding Problem

“…Here, we are going to demonstrate this numerical approach for special waveguiding problem. As Bragg (or multilayered) waveguides have application (see, e.g., [14,15]), we demonstrate here that the approach from [13] can be easily extended to be applied to more complicated problems. In order to justify the analytical approach given here, we will widely use the results of the paper [13].…”

“…Let us formulate the transmission eigenvalue problem (problem ) to which the problem of surface waves propagating in a cylindrical waveguide has been reduced. The goal is to find quantities such that, for given 1 ̸ = 0 (or 2 ̸ = 0), there is a nonzero function ( ; ) that is defined by formulas (12) and (13) for < 1 and > 2 , respectively, and solves equation (11) for 1 < < 2 ; moreover, the function ( ; ) thus defined satisfies conditions (8) and transmission conditions (14).…”

“…For big numbers it is true that the eigenvalue = ( 2 ) (see [22]). Using the same technique as in [13], we obtain an integral representation for the solution ( ) of (11) for ∈ [ 1 , 2 ]:…”

“…., with 0 = 0. The iterative procedure together with the convergence theorem is given in [13]. These iterated solutions can be applied to determine approximated eigenvalues of problem .…”

“…In the paper [13], we considered integral equation approach to derive dispersion equations in a nonlinear waveguiding problem. This approach can be used for numerical implementation.…”