Coupling coefficients of photonic crystal waveguides

Starting from Maxwell's equations we derive a new orthogonality relation and a recirpocity theorem for photonic crystal waveguide modes. The orthogonality relation implies an integration over a plain nonparallel to the waveguide direction. This is in contrast to the common orthogonality relations where an integration over the whole photonic crystal is needed. By means of this new orthogonaly relation coupling coefficients between different photonic crystal waveguides as well as photonic crystal waveguides and conventional wavguides are deduced. The considerations are valid for 2D as well as 3D geometries. In the case of monomode waveguides simple quasi-analytical approximations for the reflection and transmission coefficients are obtained, which generalize the well known formulas for coupling between conventional waveguides. The reciprocity theorem can be used for the efficient simulation or even analytical description of photonic crystal waveguides in the presence of arbitrary perturbations. Depending on the kind of perturbation and the number of modes a set of strongly coupled discrete equations for the field amplitudes in the photonic crystal unit cells results. In the frame of this paper we study two problems by means of this theory. First we investigate the transission - reflection - problem of two consecutive photonic crystal waveguides analytically. Furthermore we study the influence of the dispersion of the dielectric function of the photonic crystal material on the band structure. Especially a polariton lead to strong deformation or even a splitting of the band structure close to an optical resonance.