Spectra of electromagnetic excitations in periodic dielectric structures with space and temporal dispersion

We have studied the influence of space and time dispersion on the frequency dependence of the wave vectors of electromagnetic waves propagating in periodic dielectric structures. Two types of structures are considered: media with weak periodic modulation of the permittivity and photonic crystals composed of periodically arranged identical resonant dielectric particles embedded into nontransparent host medium. It is shown that in these systems, different types of excitations exist, for example, a peculiar kind of polaritons arises in the photonic crystals due to the interaction of the electromagnetic field, eigenoscillations of the dielectric medium, and the Debye resonance. Additional transparency bands and gaps appear in the frequency spectrum of photons due to the dispersion, and their widths depend on the wave vector. The widths of the transparency zones and of the band gaps have been calculated as functions of the frequency and of the parameters of the media. The interaction of different types of waves brings into existence a discontinuity of the frequency at some surfaces in k space. If such a surface is not a plane, the classical Bragg condition does not hold.