THREE-DIMENSIONAL THEORY OF THE CERENKOV FREE-ELECTRON LASER

We present an analytical theory for the operation of a Cerenkov free-electron laser which includes diffraction of the optical mode in the direction transverse to the elec-tron beam. Because the width of the optical mode depends on gain, the usual cubic dispersion relation is replaced by a 5/2-power dispersion relation, which allows two roots. These roots both have positive real parts, indicating that they are slow waves. For a narrow electron beam, the op-tical mode is much wider than the beam, thus reducing the gain by an order of magnitude from that predicted by the two-dimensional theory. In the limit of a wide electron beam, the two-dimensional theory is recovered