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A theoretical analysis is made of the transformation of the dispersionrelation of waves in artificial crystals under the influence of loss, includingthe case of photonic and phononic crystals. Considering a general dispersionrelation in implicit form, an analytic procedure is derived to obtain thetransformed dispersion relation. It is shown that the dispersion relation isgenerally shifted in the complex $(k,\omega)$ plane, with $k$ the wavenumberand $\omega$ the angular frequency. The value of the shift is obtainedexplicitly as a function of the perturbation of material constants accountingfor loss. The method is shown to predict correctly the transformation of thecomplex band structure $k(\omega)$. Several models of the dispersion relationnear a symmetry point of the Brillouin zone are analyzed. A lower bound for thegroup velocity, related to the local shape of the band around symmetry points,is derived for each case.

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