Mode analysis of a class of spatiotemporal photonic crystals

A detailed analysis is presented of the modes of a class of spatiotemporal photonic crystal. The structure analyzed is a perfect dielectric with periodic variation of {\epsilon} in a single spatial direction, as well as periodic variation in time. Analytic solutions are presented for {\epsilon} being separated in both variables, that is {\epsilon}(z,t)={\epsilon}0{\epsilon}r(z){\epsilon}t(t), and dispersion diagrams are presented.