Covariant formulation of the transition radiation energy spectrum of an electron beam at a normal angle of incidence onto a round metallic screen

In the formal expression of the transition radiation (TR) energy spectrum of an N electron bunch, the electron spatial coordinates leave the mark of the temporal causality and covariance constraints. In the case of normal incidence, the distribution of the N electron longitudinal coordinates determining indeed the temporal sequence of the N electron collisions onto the metallic screen rules the emission phases of the N single electron radiation field amplitudes from the metallic surface according to a causality constraint. The role of the N electron transverse coordinates goes beyond the determination of the relative phase distribution of the N electron field amplitudes at the observation point as a function of the transverse displacement of the N electrons with respect to the beam axis. The distribution of the transverse coordinates of the N electrons being a relativistic invariant under a Lorentz transformation with respect to the direction of motion of the beam is expected to leave a covariant mark on the radiation field amplitudes of the N single electrons and, consequently, on both the temporal coherent and incoherent parts of the TR energy spectrum. Compared to the ideal case of a single electron hitting an infinite metallic screen, the covariance of the TR energy spectrum is expected to manifest itself, with the decrease of the beam transverse size, as an asymptotic increase and broadening, respectively, of the radiation spectral intensity and of the corresponding angular distribution. In the case of a round metallic screen with an arbitrary radius, the formal expression of the TR energy spectrum will be derived and numerical results will be presented.