Optical properties of crystals with spatial dispersion – Josephson plasma resonance in layered superconductors

– We derive the transmission coefficient, T (ω), for grazing incidence of crystals with spatial dispersion accounting for the excitation of multiple modes with different wave vectors k for a given frequency ω. The generalization of the Fresnel formulas contains the refraction indices of these modes as determined by the dielectric function ǫ(ω,k). Near frequencies ωe, where the group velocity vanishes, T (ω) depends also on an additional parameter determined by the crystal microstructure. The transmission T is significantly suppressed, if one of the excited modes is decaying into the crystal. We derive these features microscopically for the Josephson plasma resonance in layered superconductors. Usually propagation of light in crystals is sensitive to average properties described by the dielectric function, but not to the specific atomic structure of the crystal, because the wave length of light is much larger than the interatomic distance. However, in crystals with spatial dispersion the atomic structure may affect optical properties at special extremal frequencies, ωe, where the group velocity, vg = ∂ω(k)/∂k, of an eigenmode with dispersion ω(k) vanishes. Near ωe the effective wave length, λg = vg/ω, becomes comparable with the interatomic distance, and then optical properties such as the reflectivity may feel the crystal microstructure. Pekar [1] and Ginzburg [2], already realized that several eigenmodes with different wave vectors, k, at given ω may be excited by the incident light inside a crystal with spatial dispersion. Then the Maxwell boundary conditions (continuity of the components of the electric, E, and of the magnetic field, H, parallel to the surface), are not sufficient to find the relative amplitudes of these modes and hence the reflection coefficient. To overcome this problem within the macroscopic approach Pekar and Ginzburg introduced so called additional boundary conditions (ABC), which are supposed to be related somehow to the crystal microstructure near the surface. In this phenomenological approach the choice of these ABC is only heuristic and may be controversial, see ref. [3] and Comments to this paper. It is only a microscopic model which can determine the solutions inside the crystal unambiguously and demonstrate explicitly how the reflectivity depends on the crystal microstructure. In this paper we present for the first time general macroscopic expressions for the transmission coefficient T (ω) in uniaxial crystals in the vicinity of ωe to show how the Fresnel formula