Quantum description of free electron laser radiation and nonlinear amplitude equations

Abstract

A relativistic quantum mechanical model to describe the quantum free electron laser dynamics has been developed. Neglecting the spin of electrons in the impacting beam, this model is based on the Klein–Gordon equation coupled to the Poisson equation for the space-charge potential and the wave equation for the transverse components of the radiation field. Furthermore, a system of coupled nonlinear envelope equations for the slowly varying amplitudes of the electron beam distribution and the radiation field has been derived. The fundamental system of basic equations has been cast into a suitable hydrodynamic formulation. In the framework of the hydrodynamic representation, a new dispersion relation has been derived and analyzed in both the quantum and the quasi-classical regimes, where the space-charge oscillations of the electron beam are taken into account.