Guided waves in a photonic bandgap structure with a line defect

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 responsible 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 that is localized near the line defects that behave like guided modes. Moreover, these solutions can be parameterized by a frequency spectrum that is continuous and can cover parts of the original bandgap. The frequency of the guided modes depends on a wave number parameter and can be interpreted as a dispersion relation. We illustrate the main findings of the investigation in a numerical example.