Synchrotron and Smith-Purcell radiations from a charge rotating around a cylindrical grating

We study the radiation for a charge rotating around a diffraction grating on a cylindrical surface with metallic strips parallel to the cylinder axis. The effect of the grating on the radiation intensity is approximated by the surface currents induced on the strips by the field of the rotating charge. The expressions are derived for the vector potential and for the angular density of the radiation intensity on a given harmonic. We show that the behavior of the radiation intensity on large harmonics can be essentially different from that for a charge rotating in the vacuum. For the geometry of diffraction grating the radiation intensity on higher harmonics does not vanish for small angles with respect to the cylinder axis. For given characteristics of the charge, by the choice of the parameters of the diffraction grating, one can have highly directional radiation near the normal to the plane of the charge rotation. With decreasing energy, the relative contribution of the synchrotron radiation decreases and the Smith-Purcell part is dominant.