3-DIMENSIONAL THEORY OF THE SMALL-SIGNAL HIGH-GAIN FREE-ELECTRON LASER INCLUDING BETATRON OSCILLATIONS

We have developed a three-dimensional free-electron laser (FEL) theory in the small-signal high-gain regime based upon the Maxwell-Vlasov equations including the effects of the energy spread, the emittance, and the betatron oscillations of the electron beam. 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 electron distributions. The coupled Maxwell-Vlasov equations are then reduced to a matrix equation, from which a dispersion relation for the eigenvalues is derived. The growth rate for the fundamental mode can be obtained for any initial beam distribution including the hollow-beam, the water-bag, and the Gaussian distribution. Comparisons of our numerical solutions with simulation results and with other analytical approaches show good agreements except for the one-dimensional limit. We present a handy interpolating formula for the FEL gain of a Gaussian beam, as a function of the scaled parameters, that can be used for a quick estimate of the gain. The present theory can be applied to the beam-conditioning case by a few modifications.