Nonlinear Propagation of Leaky TE-Polarized Electromagnetic Waves in a Metamaterial Goubau Line

. Propagation of leaky TE-polarized electromagnetic waves in the Goubau line (a perfectly conducting cylinder covered by a concentric dielectric layer) ﬁlled with nonlinear metamaterial medium is studied. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function of an auxiliary boundary value problem on an interval. The existence of propagating nonlinear leaky TE waves for the chosen nonlinearity (Kerr law) is proved using the method of contraction. For the numerical solution, a method based on solving an auxiliary Cauchy problem (a version of the shooting method) is proposed. New propagation regimes are discovered.


Introduction
For many years, the theory of the polarized electromagnetic wave propagation in planar and cylindrical waveguides in the linear case has been developing [2,26,28]. This theory is important because, firstly, such problems describe real physical processes that are important both for applied and theoretical sciences and, secondly, from the mathematical point of view, this theory is an affluent source of sophisticated and interesting mathematical problems.
The problem of the wave propagation in open waveguides constitutes an important class of vector electromagnetic problems. A perfectly conducting cylinder covered by a concentric dielectric layer, the Goubau line (GL), is the simplest type of such guiding structures. A mathematical investigation of symmetric surface modes in a GL is performed in [4,5,6,27]. It may be important both from the theoretical standpoint and in view of further applications of GLs, to extend these results for much broader families of GLs with nonlinear metamaterial media. This will be the first necessary step to analyze more complicated open metal-dielectric waveguides with such fillings. The occurrence and analysis of leaky waves together with their various applications in microwave engineering have been a subject of numerous studies since the early 1950s (see [8,9,10]).
In 1967, Russian physicist V.G. Veselago predicted an extraordinary electromagnetic wave phenomenon which is related to materials with a simultaneously negative permittivity and negative permeability [29]. He hypothetically created a lossless meta-material and showed the extraordinary properties of this material which is not found in nature, in particular, negative group velocity, negative refraction, the reversal of the Doppler effect and Cherenkov radiation.
The present study focuses on the analysis of leaky electromagnetic wave propagation in a Goubau Line filled with a nonlinear metamaterial medium. Metamaterial is an artificial material with negative permittivity and negative permeability (see [12,29,30]). The nonlinearity is expressed by the Kerr law [3,7,13]. The main task which we resolve is to elaborate mathematically correct problem statements for nonlinear differential equations that enable one to introduce and investigate eigenmodes in an open nonlinear guiding medium, calculate propagation constants, and reveal fundamental properties of eigenmodes. The propagation constants will be determined as roots of the commonly used dispersion equations.
This framework has not been addressed in the literature, to the best of our knowledge. A background study concerning nonlinear guided waves in media with Kerr and Kerr-like nonlinearities can be found in [14,15,20,24]. In papers [16,17,18,19,21,22] numerical study in layered nonlinear dielectric and metal-dielectric waveguides is performed. Numerical results are obtained with the help of a version of the shooting method (see e.g. [16,23,25]).

Statement of the problem
Let us consider three-dimensional space R 3 with a cylindrical coordinate system Oρϕz. The space is filled with an isotropic and nonmagnetic medium (it is supposed that everywhere µ = µ 0 , where µ 0 is the permeability of free space) having the constant permittivity ε 0 = const where ε 0 is the permittivity of free space, without sources. A perfectly conducting cylinder covered by a dielectric layer with a cross-section Σ := {(ρ, ϕ) : r 0 ρ r, 0 ϕ < 2π} and a generating line parallel to the axis Oz is placed in R 3 .
The cross section of GL consists of two concentric circles of radii r 0 and r (see Figure 1): r 0 is the radius of the internal (perfectly conducting) cylinder and r 0 − r is the thickness of the external (dielectric) cylindrical shell. Complex amplitudes E, H of the electromagnetic field satisfy Maxwell's equations [11] rot H = −iω εε 0 E, have continuous tangential field components on the media interface ρ = r; and obey the radiation condition at infinity: the electromagnetic field increase as ρ → ∞ in the region ρ > r. We assume that permittivity and permeability in the entire space have the form εε 0 and µµ 0 , where and ; e k is the orthonormal vector in the k direction; (·, ·) is the Euclidean scalar product; and ε, µ and α are real positive constants.

TE-polarized leaky waves
Consider TE-polarized waves in the harmonic mode (see [11]), where E, H are complex amplitudes, and γ is an unknown (spectral) parameter.
Let k 2 0 = ω 2 µ 0 ε 0 . Substituting components (3.1) into (2.1) and using the notation u(ρ) := E ϕ (ρ) we obtain where ε and µ are defined by formula (2.2). Let us denote by β 2 := γ 2 − k 2 0 a new spectral parameter. In the domain ρ > r equation (3.2) takes the form In the domain r 0 ρ r where α := αω 2 µε 0 , χ 2 := k 2 0 (εµ − 1) > 0. In the general case the solution of equation (3.3) for ρ > r has the form u = CI 1 (βρ) + CK 1 (βρ), ρ > r, (3.5) where I 1 and K 1 are the modified Bessel function (Infield and Macdonald function) and C, C are arbitrary constants. We will consider leaky waves. More precisely, the choice of CI 1 (βρ) at ρ > r determines the leaky waves (see [26]) increasing at infinity because I 1 (βρ) → ∞ as ρ → ∞ (see [1]). In accordance with the condition at infinity solution (3.5) for leaky waves takes the form Continuity conditions for functions u and u result from the transmission conditions for the tangential field components (E ϕ and H z ) and have the form Formulate the transmission eigenvalue problem (problem P ) to which the problem of leaky waves propagating in a metamaterial rod is reduced. Find quantities β ∈ R + such that, for a given C = 0, there is a nonzero function u(ρ; β) defined by formula (3.6) for ρ > r which solves equation (3.4) for r 0 < ρ < r and satisfies transmission and boundary conditions (3.7) and (3.8), respectivelly.

Nonlinear integral equation
To show the existence of eigenvalues of problem P we will study solvability of the equation Construct the Green functions G for the boundary value problems: It can be proved [14] that the Green functions have the forms Using the same technique as in [20] we obtain an integral representation of solution u(s) to equation (3.4) for s ∈ [r 0 , r]: It can be shown (see [24]) that the following assertions are valid, which will be used below. least one solution such that u x * .
Using Propositions 1 and 2, it can be shown (see [24]) that the following theorems hold.
The left-hand side of (5.1) is bounded, positive, and continuous for positive β [1]. It is known [1] that the right-hand side of (5.1) has simple poles β is the nth positive root of the equation n = 1, 2, . . . . It follows from the properties of the Bessel functions that the right-hand side of (5.1) is a decreasing function.
We denote by β (n) 1 the nth positive root of the equation In addition, assume that δ i are chosen so that the inequalities g β In other words, we choose δ i so that to move away from the poles of the Green function β (n) 1 but preserve solutions of the DE g (β) = 0 for the linear medium. Thus, we can choose a sufficiently small value of α such that the nonlinear DE Cg(β) − αF (β) = 0 will have a solution which lies close to the "linear" solution.
It should be pointed out that F (β) is bounded and the product αF (β) can be made sufficiently small by choosing a suitable α 0 . Then, it can be affirmed that Cg(β) − αF (β) = 0 has a solution if α 0 is small enough.
Consider the DE Φ(β) = 0. Function g(β) is continuous and changes its sign when β varies from β to β (n) 1 + δ i , we can always assure, by choosing a suitable α, that the equation Φ(β) = 0 has at least n roots β i such that β i ∈ B i , i = 1, n.
The main result of the present work is the following theorem.
Theorem 3. Let B i , i = 1, n be a segment defined by formula (5.2). Then there is a number α 0 > 0 such that, for any α α 0 , problem P has at least n eigenvalues β i such that β i ∈ B i .

, then
Cg β which completes the proof.
Theorem 3 implies that, under the above-formulated conditions, there exist leaky TE-polarized wave propagating without attenuation in a metamaterial Goubau Line filled with an isotropic Kerr-nonlinear medium.

Numerical results
Numerical results are obtained with the help of a version of the shooting method (see e.g. [16,23]). In the figures below, normalized eigenvalues β/k 0 are shown calculated with respect to frequency ω, as well as eigenfunctions u(ρ). In the figures for eigenfunctions the colour of a curve duplicates the colour of the marked eigenvalue on the dispersion curves. The larger an eigenvalue, the higher is the maximum of the corresponding eigenfunction. The vertical grey dashed line corresponds to the dielectric-free space boundaries.
If we compare blue (linear case) and red (nonlinear case) dispersion curves, we see that there are eigenvalues that are close to those occurring in the linear    case; the latter can be determined with the help of perturbation theory and existence of these very eigenvalues is rigorously proved in this paper.

Conclusions
We have developed a method of analysis of the leaky TE wave propagation in a Goubau Line with a nonlinear metamaterial filling. The existence of these waves has been proved by reducing the problem to a nonlinear integral equation and applying the method of contraction. An analytical method proposed in this study allows one to prove the existence of eigenvalues of the nonlinear problem that are close to eigenvalues of the linear problem and are actually perturbations of the latter. Numerical solution is performed using a version of the shooting method developed specifically for this class of problems. The proposed technique and mathematical method can be developed and applied to rigorous validation of the existence of other types of waves and analysis of metamaterial dielectric waveguides having different cross sections.