Light propagation in anisotropic crystals has long been described by the ellipsoid model of Fresnel. Its theoretical basis lies in Maxwell's equations, with the polarization density P included in the electric displacement

A consistent multipole theory is presented to describe light propagation in nonabsorbing non-magnetic crystals. Although valid for the 32 crystal classes, the theory is applied here to all except the five members of the triclinic and monoclinic systems. To account for the birefringence that has been observed in certain cubic crystals and also for the predicted Jones birefringence, the theory has to allow for electric octopoles and magnetic quadrupoles induced by the light wave. At the earlier stage of electric quadrupoles and magnetic dipoles, it is able to describe optical activity in crystals. An expression for this is derived which, when electric quadrupole contributions are omitted, yields the familiar Nye result. As a criterion for the correct inclusion in the theory of all relevant induced multipole moments, tensor expressions for observables are shown to be independent of the choice of origin. Finally, the concepts of O-ray and E-ray are found to break down beyond the electric dipole approximation and alternatives are proposed.