# Gap Localization of TE-Modes by arbitrarily weak defects - multiband case

## TL;DR

This paper analyzes how arbitrarily weak defects in photonic crystal waveguides create localized TE-modes, providing a precise characterization of eigenvalues using Floquet-Bloch theory in complex media.

## Contribution

It introduces a novel analysis allowing discontinuities in coefficients and characterizes defect-induced eigenvalues in multiband photonic crystals.

## Key findings

- Eigenvalues created by weak defects are precisely characterized.
- Discontinuities in coefficients are incorporated into the model.
- The analysis applies to multiband photonic crystal waveguides.

## Abstract

This paper considers the propagation of TE-modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coefficients of the operator, which is required to model many photonic crystals. Using the Floquet-Bloch theory in negative order Sobolev spaces, we characterize the precise number of eigenvalues created by the line defect in terms of the band functions of the original periodic background medium for arbitrarily weak defects.