Radiation field of a uniformly moving charge in an anisotropic medium

Abstract Fourier integrals are set up for the field of a point charge moving uniformly in an arbitrary direction in a uniaxial medium anisotropic in ε only. The integrals break up into several parts two of which yield the ordinary and extraordinary cones with uniform azimuthal potential distribution. The remaining integrals neither contribute to the energy radiated nor affect the size and the shape of the cones, but merely distort the field within the cones. The integrals are evaluated exactly in the non-dispersive case and closed expressions for the potential are obtained. In the dispersive case, the radiation field is determined by using the asymptotic form of the Hankel functions occurring in the integrand. The resulting expressions exhibit the high azimuthal asymmetry characteristic of anisotropic fields. From the expressions derived for a pure dielectric the potential in a doubly anisotropic medium is obtained, without a fresh calculation, by making appropriate substitutions for the coordinates of the field point and the components of the dielectric tensor.