Transverse mode dynamics in a free-electron laser

We derive the most general equations of motion for the electrons and the electromagnetic field in a free-electron laser including the effects of diffraction and pulse propagation. The field evolution is expressed in terms of the amplitudes and phases of a complete set of transverse modes. The analytic solution is given in the small-signal regime, where the theory is shown to be in excellent agreement with a recent experiment at Orsay.