Mechanisms for the production of harmonics in free electron lasers

Harmonics in the radiation of a free electron laser are useful for extending the range of tuning, may originate in spontaneous or parametric processes, and can take part in stimulated emission or amplification. These mechanisms exhibit interesting analogies with those of nonlinear optics. Apart from the well known nonparametric gain analogous to stimulated Brillouin scattering, they include third-harmonic generation and other four-wave mixing parametric processes, with spatial harmonics of the electron density taking the place of the higher-order terms in the nonlinear dielectric susceptibility. We investigate these mechanisms using a one-dimensional model