On the spectrum of waveguides in planar photonic bandgap structures

Abstract

We study a Helmholtz-type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a two-dimensional periodic medium; the defect is infinitely extended and aligned with one of the coordinate axes. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. In the first part of the paper, we prove that guided mode spectrum can be created by arbitrarily ‘small’ perturbations. Secondly, we show that, after performing a Floquet decomposition in the axial direction of the waveguide, for any fixed value of the quasi-momentum k x , the perturbation generates at most finitely many new eigenvalues inside the gap.