  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.
