Mathematical Principles in Photonic Crystals

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 conned 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-Schroedinger 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