Advances in Mathematical Physics

Volume 2015, Article ID 614976, 11 pages

http://dx.doi.org/10.1155/2015/614976

## Guided Electromagnetic Waves Propagating in a Two-Layer Cylindrical Dielectric Waveguide with Inhomogeneous Nonlinear Permittivity

Department of Mathematics and Supercomputing, Penza State University, Krasnaya Street 40, Penza 440026, Russia

Received 12 November 2014; Accepted 22 January 2015

Academic Editor: Ivan Avramidi

Copyright © 2015 E. Yu. Smol’kin and D. V. Valovik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper focuses on the problem of monochromatic electromagnetic TM wave propagation in a two-layer circular cylindrical dielectric waveguide. The space outside the
waveguide is filled with isotropic medium having constant permittivity. The inner core of
the waveguide is filled with isotropic medium having constant permittivity; the cladding
of the core is filled with isotropic inhomogeneous nonlinear permittivity (the nonlinear
term is expressed by Kerr law). Existence of guided modes which depend harmonically
on *z* (the waveguide axis coincides with *z*-axis) is proved and their localization is found.
Numerical results including different type of nonlinearities are presented. A comparison
with the linear case is given. The existence of a new propagation regime is predicted.

#### 1. Introduction

The paper studies the problem of monochromatic electromagnetic TM (transverse-magnetic) wave propagation in a two-layered circle cylindrical dielectric waveguide with nonlinear permittivity inside one of its layers. Here we talk only about intensity-dependent permittivity. We do not consider multiple harmonic generation or other nonlinear effects that in rigorous statement involve time-dependent Maxwell’s equations. The nonlinear permittivity is described by the Kerr law. Kerr law is one of the most important dependencies in nonlinear optics; see, for example, [1–3], and for newest experimental observation see [4–6].

The physical problem is reduced to a nonlinear transmission eigenvalue problem for a system of nonlinear ordinary differential equations. Eigenvalues of the problem correspond to propagation constants (PCs) of the waveguide. The full set of PCs of a waveguide is one of the most important characteristics of the waveguide; this characteristic is used for waveguide’s designing. One of the main methods to study the problem is the small parameter method. Since the Kerr law is characterised by a small constant factor in front of the nonlinear term (coefficient of the nonlinearity), then this approach is justified. Numerical results are based on a numerical method that does not depend on the smallness of the parameter [7]. As is known the Kerr law is described by an unbounded function; in order to demonstrate difference between unbounded and bounded nonlinear permittivities we presented numerical results for both cases; we also gave a comparison between linear and nonlinear cases. Numerical results given here demonstrate not only those eigenvalues that are predicted by the main theorem of this work but new eigenvalues that correspond to a new nonlinear propagation regime.

Currently there has been significant progress in studying of polarised electromagnetic TE and TM wave propagation in waveguide structures (for a planar waveguide, see [8–11]; for a circular cylindrical waveguide, see [1, 8, 12–15]; for a circular two-layered cylindrical waveguide, see [7, 16]) filled with nonlinear dielectric permittivities. In particular, theorems on existence and localization of eigenvalues in some of these problems have been proved.

Most of these papers are devoted to studying of polarized waves in waveguides filled with a homogeneous nonlinear medium. From aforementioned studies only [7, 13, 15, 16] focus on inhomogeneous nonlinear permittivity.

Multilayered cylindrical linear homogeneous waveguide was studied in [17, 18], and one of the practical applications for nonlinear two-layered waveguides is shown in [19].

Results obtained in this paper together with the results given in [16] give an opportunity to consider a very intriguing phenomenon of coupled electromagnetic TE-TM wave propagation in a two-layered cylindrical waveguide. This problem will be treated in a separate paper. For different types of phenomena of coupled wave propagation see [20–22].

#### 2. Governing Equations

Consider three-dimensional space with cylindrical coordinate system . In this space a two-layer circular cylindrical waveguide is placed; the waveguide axis coincides with . (The waveguide is unlimited in direction.) The waveguide is filled with isotropic inhomogeneous nonlinear medium. The space outside is filled with isotropic medium characterised by constant permittivity. Throughout the paper we assume that everywhere , where is the permeability of free space. Geometry of the problem is shown in Figure 1.