# On the spectrum of waveguides in planar photonic bandgap structures

Brown, Brian Malcolm, Hoang, Vu, Plum, Michael, Wood, Ian (2015) On the spectrum of waveguides in planar photonic bandgap structures. Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, . ISSN 1364-5021. E-ISSN 1471-2946. (doi:10.1098/rspa.2014.0673) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided)

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. The perturbation is expected to introduce guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. In the first part of the paper, we prove that, somewhat unexpectedly, 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.