C. Van Der Mee, P. Pintus and S. Seatzu

Mathematical principles in photonic crystals

dedicated to the memory of Giulio Di Cola

Pages 99-137
Received: 11 February 2008
Mathematics Subject Classification (2000): Primary 78A46 - 78A60; Secondary 34L25.
Keywords: Helmholtz-Schrödinger equation, periodic media with impurity, photonic crystals, inverse scattering.

Research supported in part by the Italian Ministery of University and Research (MIUR) under PRIN grant no. 2006017542-003 and INdAM-GNCS.

Abstract In this article we introduce a mathematical model to describe light propagation in a mono-dimensional photonic crystal under the hypothesis of a linear, stationary, isotropic and lossless medium. We study the typical band structure and spectral properties. In addition, we analyse a crystal with an impurity confined to a bounded region and study the change in its spectrum as a result of introducing the impurity. The asymptotic expressions for the solution of the Helmholtz-Schrödinger model equation with impurity are analysed to derive the scattering matrix. We introduce the period map matrix and derive it from the scattering matrix. We pay particular attention to a photonic crystal with a piecewise constant index of refraction and recover it from the scattering matrix in a few important special cases.