A solid state theoretical approach to the optical properties of photonic crystals

We outline a theoretical framework that allows qualitative as well as quantitative analysis of the optical properties of Photonic Crystals (PCs) and which is derived from solid state theoretical concepts. Starting from photonic bandstructure computations which allow us to obtain dispersion relations and associated Bloch functions, we show how related physical quantities such as densities of states and group velocities can be calculated. In addition, defect structures embedded in PCs can be efficiently treated with the help of photonic Wannier functions that are derived from photonic Bloch functions by means of a lattice Fourier transform. Nonlinear PCs may be investigated by an appropriate multi-scale analysis utilizing Bloch functions as carrier waves together with an adaptation of k · p-perturbation theory. This leads to a natural generalization of the slowly varying envelope approximation to the case of nonlinear wave propagation in PCs.