Band structure of absorptive photonic crystals

The band structure of absorptive dielectric photonic crystals is investigat ed. Provided the frequency-dependent electric permeability epsilon(x, omega ) satisfies certain analyticity requirements as a function of frequency, we show that no bandgaps exist in frequency regions where absorption takes pl ace, i.e. where epsilon(x, omega) has a non-zero imaginary part. In this ca se real eigenvalues of the Helmholtz operator in the Bloch-decomposed forma lism are absent. Using a suitable analytic continuation procedure, we find that the former change into resonances, i.e. complex numbers depending on k , the wavevector from the first Brillouin zone, thus leading to complex ban ds in the lower half plane. This is confirmed numerically for a simple, one -dimensional example.