Time dependent formulation of the energy loss by an accelerated intense electron beam just emitted by the cathode of RF-FEL photoinjector

The energy loss by an accelerated electron bunch of a conical shape propagating in the laser-driven RF-photoinjector is expressed in terms of an expansion of the vector and scalar potentials into a series of eigenfunctions of the empty unit “pill-box” cavity. A versatile and simple analytical formula which can be easily applied to a bunch of any shape is obtained.