Nature of lossy Bloch states in polaritonic photonic crystals

We examine the effects of absorption losses in photonic crystal structures composed of polar materials which exhibit transverse phonon-polariton excitations. In order to explore the Bloch states of such a system, we study the two subspaces of the complete set of complex $(k,\omega )$ states consisting of either real frequency, accessible through a frequency-domain method, or real-wave vector, which we determine using a frequency-dependent time-domain method. We describe analytically the conditions under which the imaginary frequency component of a real-wave-vector state is related to the imaginary-wave-vector component of a real-frequency state through a factor of the group velocity, and we present a one-dimensional lossy crystal as an example that satisfies these constraints. We also discover that the real-frequency states of a two-dimensional crystal bear little resemblance to the class of real-wave-vector states, due to interplay between the prohibitively large spatial decay of the states near the edge of the Brillouin zone and the existence of metalliclike states localized to the surrounding ambient dielectric region with much lower levels of loss. We then put these results in the context of possible experiments, including reflection of a plane-wave from a slab structure, and discuss the viability for observing the node switching and flux expulsion phenomena previously discovered in lossless crystals.