Geometric spectral properties of electromagnetic waveguides

TL;DR

This paper investigates how bending and twisting affect the spectral properties of electromagnetic waveguides, providing conditions under which the essential spectrum is preserved or discrete eigenvalues emerge, with analytical and numerical insights.

Contribution

It offers new criteria for the preservation of the essential spectrum and the creation of discrete eigenvalues due to geometric deformations, using a Birman-Schwinger-type approach.

Findings

Conditions for preserving the essential spectrum under bending and twisting.

Criteria for the emergence of discrete eigenvalues within spectral gaps.

Numerical validation for waveguides with rectangular cross-sections.

Abstract

Consider a reference homogeneous and isotropic electromagnetic waveguide with a simply connected cross-section embedded in a perfect conductor. In this setting, when the waveguide is straight, the spectrum of the associated self-adjoint Maxwell operator with a constant twist (which may be zero) lies on the real line and is symmetric with respect to zero and exhibits a spectral gap around the origin. Moreover, the spectrum is purely essential, and contains 0 which is an eigenvalue of infinite multiplicity. In this work, we present new results on the effects of geometric deformations, specifically bending and twisting, on the spectrum of the Maxwell operator. More precisely, we provide, on the one hand, sufficient conditions on the asymptotic behavior of curvature and twist that ensure the preservation of the essential spectrum of the reference waveguide. Our approach relies on a Birman-Schwinger-type principle, which may be of independent interest and applicable in other contexts. On the other hand, we give sufficient conditions (involving in particular the geometrical shape of the cross-section of the waveguide) so that the geometrical deformation creates discrete spectrum (namely isolated eigenvalues of finite multiplicity) within the gap of the essential spectrum. In addition, we give some results on the localization of these discrete eigenvalues. The sufficient condition involving the cross-section is then studied both analytically and numerically. Finally, we examine its stability under shape deformations of the cross-section, focusing in particular on the case of a waveguide with a rectangular cross-section.