Classical Propagation of Light in Spatio-Temporal Periodic Media

We analyze the propagation of electromagnetic waves in media where the dielectric constants undergo rapid temporal periodic modulation. Both spatially homogeneous and periodic media are studied. Fast periodic temporal modulation of the dielectric constant of a homogeneous medium leads to existence of photonic band-gap like phenomena. In the presence of both spatial and tem- poral periodicity the electromagnetic spectrum is described in a four-dimensional cube, defining an effective Brillouin zone. In the case of incommensurability between space and time periodicities, completely dispersed point spectra exist.