 
Summary: Guided Waves in a Photonic Bandgap Structure
with a Line Defect
Habib Ammari Fadil Santosa y
Abstract
Numerical simulations have shown that when a line of defects is
introduced into a photonic bandgap structure, waves can be guided
along the line. It has been conjectured that the mechanism respon
sible for the guidance phenomenon is the introduction of a spectrum
in the bandgap by the defect. The purpose of this work is to give
a mathematical framework for understanding this phenomenon. We
show that there exist solutions of the scalar wave equation which is
localized near the line defects which behave like guided modes. More
over, these solutions can be parameterized by a frequency spectrum
which is continuous, and can cover parts of the original bandgap. The
frequency of the guided modes depend on a wave number parameter
and can be interpreted as a dispersion relation. We illustrate the main
ndings of the investigation in a numerical example.
1 Introduction
The purpose of this work is the study of wave propagation in an innite
periodic structure with a line defect. The medium under consideration is
