Spatial dispersion in CaF$_2$ caused by the vicinity of an excitonic bound state

The microscopic mechanism beyond the optical anisotropy of an ionic crystal which occurs for short wavelengths is investigated. The electron-hole, two particle propagator and its analytical behaviour close to the band edge of the one particle continuum plays a major role for the mechanism of this optical anisotropy. Especially for an ionic crystal the two particle bound state, the exciton, is of special importance. In this way we argue that the so called ``intrinsic birefringence'' in CaF$_2$ is neither intrinsic to the material nor it is birefringence. Instead it is spatial dispersion caused by the vicinity of a dispersive optical absorption given by the excitonic bound state. We propose a model which connects the bound state dispersion with the band structure and a model potential for a screened coulomb interaction. Based on these considerations we predict a wavelength dependence of the dielectric function approaching close to the bound state level $\epsilon \sim (\lambda - \lambda_0)^{-1}$, where $\lambda_0$ is the wavelength of the excitonic bound state level.