Three-dimensional theory of the free-electron laser. I. Gain and evolution of optical modes

Abstract

A three-dimensional free-electron-laser theory in the small-signal regime is presented. In the first part, we derive an energy theorem which relates the gain and the loss of an optical beam of arbitrary shape, driven by an electron beam of arbitrary profile. We show how this theorem can be applied for gain calculation and optimum resonator design. In the second part, we introduce a generalized description of a Gaussian optical beam. This includes four real parameters: the on-axis intensity and phase, the beam waist, and its radius of curvature. These quantities are made to evolve self-consistently and are governed by a set of ordinary differential equations which are readily integrated numerically. The solution of these equations provides an easy and intuitive means for the understanding of transverse effects induced by the interaction with the electron beam in a way which is considerably simpler than the more accurate large-scale simulation techniques.