Photonic crystal waveguides as discrete systems

Photonic crystals (PCs) are regarded as a promising and variable material base for a future photonic integration. Here we present a much simpler mathematical description, which often allows for analytical solutions. We start from the Bloch-modes of the unperturbed PCW. Using a kind of reciprocity theorem we obtain a set of evolution equations to describe the spatial-temporal evolution of the amplitudes of these Bloch-modes in the presence of arbitrary perturbations. Another important result of the reciprocity theorem is a new orthogonality relation for Bloch-modes, which allows to derive analytical expressions to quantify the coupling between PCWs and conventional waveguides. Reflection coefficient between two consecutive PCWs determined by means of the theory of a strongly coupled discrete system is presented.