Eikonal Approximation and the Conical Effect in Smith-Purcell Radiation

The diffraction and transition radiation under normal incidence of a particle to the semi-infinite dielectric plate had been considered previously using the eikonal approximation in the transition radiation theory. This approach is valid for the high radiation frequencies domain (where the dielectric permittivity of the plate material is close to unit), including X-ray domain. In the present report this method is applied to the case of oblique incidence of the particle on the plate (but parallel to the plate's edge) as well as to the periodic set of such plates (Smith-Purcell radiation). The oblique incidence of the particle (both in the case of the single plate and in the case of the periodic grating of such plates) leads to the so called conical effect in the radiation angular distribution observed recently. We outline that the origin of this effect lies in the superluminal motion along the plate's edge of the disturbance produced by the incident particle's field in the plates. The analogous effects arise in various physical situations. Also we compare our results to ones obtained using different technique.