Three-dimensional theory of the Cerenkov free-electron laser

We describe an analytical theory for the gain of a Cerenkov free-electron laser including diffraction of the optical beam in the direction transverse to the electron beam, parallel to the surface of the dielectric. Since the width of the optical beam depends on the gain, the usual cubic dispersion relation of two-dimensional slow-wave structures is replaced by a 5 ∕ 2 -power dispersion relation, but three of the five roots are extraneous. The results show that for a narrow electron beam, the optical beam is much wider than the electron beam. This reduces the gain by an order of magnitude. Moreover, in the three-dimensional theory the allowed roots of the dispersion relation have positive real parts, so they correspond to slow waves; when transverse diffraction is included, fast waves are forbidden.