Modal bases of coaxial electromagnetic step index fibers
We consider the eigenvalue problem to find the modes of an electromagnetic coaxial step index fiber. More specific, we consider a closed (meaning PEC boundary conditions) cylindrical waveguide with circular cross section $\Gamma$, wave propagation modeled by the time-harmonic Maxwell's equations with frequency $\omega$, the permeability $\mu$ and the permittivity $\epsilon$ being scalar, uniformly positive, piece-wise constant and depending only on the radial variable of the cross section. We prove that if the deviation from the homogeneous case is small, i.e., $\delta_{\epsilon,\mu}:=\|\epsilon-\epsilon_0\|_{L^\infty}+\|\mu-\mu_0\|_{L^\infty}\ll1$, then the tangential electric (magnetic) fields of the modes form a Riesz basis in $\mathbf{H}_{0}(\operatorname{curl}_{\Gamma};\Gamma)$ ($\mathbf{H}(\operatorname{curl}_{\Gamma};\Gamma)$). For a constant permeability (permittivity) the Riesz basis property for the tangential electric (magnetic) fields holds also in the natural trace space $\mathbf{H}_{0}^{-1/2}(\operatorname{curl}_{\Gamma};\Gamma)$ ($\mathbf{H}^{-1/2}(\operatorname{curl}_{\Gamma};\Gamma)$). These results hold also for complex frequencies $\omega$. In addition, if $\omega\in\mathbb{R}$, then for small enough $\delta_{\epsilon,\mu}$ all wavenumbers are located on the axes and there exist no backward modes. Key tools in the analysis are a particular reformulation of the eigenvalue problem, the perturbation theory for selfadjoint operators under a local subordinate condition and uniform properties of Bessel functions.