Modal bases of coaxial electromagnetic step index fibers

TL;DR

This paper proves that the electromagnetic modes of a coaxial step index fiber form a Riesz basis under small deviations from homogeneous materials, ensuring stable mode representations and no backward modes for real frequencies.

Contribution

It establishes the Riesz basis property of fiber modes for small material deviations and extends results to complex frequencies, improving understanding of mode stability and spectral properties.

Findings

Modes form a Riesz basis under small deviations

No backward modes for real frequencies with small deviations

Results hold for complex frequencies

Abstract

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.