Modification of transition radiation by three-dimensional topological insulators

We study how transition radiation is modified by the presence of a generic magnetoelectric medium with a special focus on topological insulators. To this end, we use the Green's function for the electromagnetic field in presence of a plane interface between two topological insulators with different topological parameters, permittivities and permeabilities. We employ the far-field approximation together with the steepest descent method to obtain approximate analytical expressions for the electromagnetic field. Through this method we find that the electric field is a superposition of spherical waves and lateral waves. Contributions of both kind can be attributed to a purely topological origin. After computing the angular distribution of the radiation, we find that in a region far from the interface the main contribution to the radiation comes from the spherical waves. We present typical radiation patterns for the topological insulator TlBiSe$_2$ and the magnetoelectric TbPO$_4$. In the ultra-relativistic case, the additional contributions from the magnetoelectric coupling appreciably enhance the global maximum of the angular distribution. We also present an analytic expression for the frequency distribution of the radiation for this case. We find that in the limit where the permittivities are equal there still exists transition radiation of the order of the square of the topological parameter with a pure topological origin.