3-dimensional semi-classical theory of free-electron laser

The three-dimensional Klein-Gordon equation is solved by means of perturbation theory. The zeroth-order wave functions of the electrons are characterized by two discriminate numbers of m, n and one continuous p. The small signal gain obtained from the first order perturbation agrees with that obtained from one-dimension theory, but the filling factor F is physically more reasonable. The results show that there is a 'matching' problem between the electron states and the laser modes.