Geometric spectral properties of electromagnetic waveguides

Source abstract

Abstract We consider a homogeneous, isotropic electromagnetic waveguide with simply connected cross‐section embedded in a perfect conductor. We investigate the influence of geometric deformations, namely bending and twisting, on the real spectrum of the associated self‐adjoint Maxwell operator. We first establish sufficient conditions on the asymptotic behavior of the curvature and twist to ensure preservation of the essential spectrum of a reference configuration given by a waveguide with zero curvature and constant twist. The spectrum of the reference operator is purely essential and symmetric with respect to zero. It exhibits a spectral gap around the origin and contains zero, an isolated eigenvalue of infinite multiplicity. To obtain this result, our approach relies on a Birman–Schwinger‐type principle, which may be of independent interest and applicable in other contexts. We then focus on purely bent waveguides and derive sufficient conditions (including one depending on the geometry of the cross‐section) under which bending induces the emergence of discrete spectrum in the spectral gap. Such eigenvalues, associated with trapped modes, are isolated and of finite multiplicity. We study their localization. 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, particularly for rectangular cross‐sections.