Length-explicit stability analysis of Helmholtz problems in leaky circular waveguides

TL;DR

This paper provides a length-explicit stability analysis for Helmholtz problems in leaky circular waveguides, incorporating heterogeneity and modal eigenvalue analysis to improve understanding of wave propagation in optical fibers.

Contribution

It introduces a stability estimate explicitly dependent on waveguide length, using modal analysis and perturbation theory to handle heterogeneity and nonselfadjoint operators.

Findings

Stability estimate explicitly depends on waveguide length

Modes form a Riesz basis in L^2 and H^1 spaces

Analysis includes heterogeneous waveguide sections

Abstract

Motivated by the study and simulation of long, coiled optical fibers we consider in this article a simplified model that is prevalent in the engineering community. Mathematically, the problem is specified as follows: Time-harmonic wave propagation is modeled by the Helmholtz equation; the waveguide is a bounded circular section with a transparent boundary condition on one end; the dissipation of energy is modeled by an impedance boundary condition on the outer hull of the waveguide. We show a stability estimate that is explicit in terms of the angular length of the waveguide. The analysis is based on a separation of variables ansatz and the study of the related (nonselfadjoint) modal eigenvalue problem. The key property there is to show that the modes form a Riesz basis in both $L^2$ and $H^1$ spaces. To this end we apply perturbation theory for selfadjoint operators and the concept of local subordination of perturbations [B. Mityagin and P. Siegl, JAM 139 (2019)]. Since the possibility of nontrivial Jordan chains cannot be ruled out, our whole methodology is conducted accordingly. In addition, in contrast to previous works, we include a bounded but heterogeneous part of the waveguide into our considered setting.