  A synoptic view on the long-established theory of light propagation in
crystalline dielectrics is presented, providing a new exact solution for the
microscopic local electromagnetic field thus disclosing the role of the
divergence-free (transversal) and curl-free (longitudinal) parts of the
electromagnetic field inside a material as a function of the density of
polarizable atoms. Our results enable fast and efficient calculation of the
photonic bandstructure and also the (non-local) dielectric tensor, solely with
the crystalline symmetry and atom-individual polarizabilities as input.
