Effect of loss on the dispersion relation of photonic and phononic crystals

A theoretical analysis is made of the transformation of the dispersion relation of waves in artificial crystals under the influence of loss, including the case of photonic and phononic crystals. Considering a general dispersion relation in implicit form, an analytic procedure is derived to obtain the transformed dispersion relation. It is shown that the dispersion relation is generally shifted in the complex (k,omega) plane, with k the wave number and omega the angular frequency. The value of the shift is obtained explicitly as a function of the perturbation of material constants accounting for loss. The method is shown to predict correctly the transformation of the complex band structure k(omega). Several models of the dispersion relation near a symmetry point of the Brillouin zone are analyzed. A lower bound for the group velocity, related to the local shape of the band around symmetry points, is derived for each case.