Three-dimensional free electron laser dispersion relation including betatron oscillations

We have developed a 3-D FEL theory based upon the Maxwell-Vlasov equations including the effects of the energy spread and emittance of the electron beam, and of betatron oscillations. The radiation field is expressed in terms of the Green's function of the inhomogeneous wave equation and the distribution function of the electron beam. The distribution function is expanded in terms of a set of orthogonal functions determined by the unperturbed particle distribution. The coupled Maxwell-Vlasov equations are then reduced to a matrix equation, from which a dispersion relation for the eigenvalues is derived. In the limit of small betatron oscillation frequency, the present dispersion relation reduces to the well-known cubic equation of the one-dimensional theory in the limit of large beam size, and it gives the correct gain in the limit of small beam size. Comparisons of our numerical results with other approaches show good agreement. We present a handy empirical formula for the FEL gain of a 3-D Gaussian beam, as a function of the scaled parameters, that can be used for a quick estimate of the grain. 5 refs., 2 figs