On the spectrum of waveguides in planar photonic bandgap structures

TL;DR

This paper investigates how introducing linear defects into two-dimensional photonic crystals can create guided electromagnetic modes within band gaps, revealing that even small perturbations can generate such modes and that only finitely many eigenvalues appear for each quasi-momentum.

Contribution

It proves that arbitrarily small perturbations can induce guided modes in photonic crystal waveguides and characterizes the finite number of eigenvalues generated for each quasi-momentum.

Findings

Small perturbations can create guided modes inside band gaps.

For each fixed quasi-momentum, only finitely many eigenvalues are generated.

Guided mode spectrum can be induced by arbitrarily small defects.

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. 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.