Short impulse of transition radiation excited by a relativistic electron bunch

The theoretical investigations and simulation are performed for the emission of a relativistic electron bunch during the injection into a semi-infinite vacuum/dielectric waveguide. The exact solution of the problem is obtained for the first time. Contrary to the previously used saddle-point technique, we applied a number of successive conform transformations of integration area in order to carry out the inverse Fourier transformation. The power and the frequency spectrum of high intensity pulse of ultra-wideband transition radiation, excited by a finite size electron bunch in a vacuum waveguide, are calculated numerically. It is shown that in a dielectric waveguide the short pulse of Cerenkov wake field drifts behind the relativistic bunch with the group velocity.