Classical theory of the free-electron laser in a moving frame

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Abstract
This paper presents a fully classical theory of the free-electron laser. The theory is formulated in a moving frame where the pseudoradiation field due to static wiggler magnet is in resonance with the laser field imposed by an optical cavity. The basic amplification process is then one of elastic scattering. In this frame both fields are treated classically, as is the electron motion. When the laser and wiggler fields are taken to be constant during the interaction period, it is shown that the laser operation can be described by the classical pendulum equation. This is used to evaluate the unsaturated gain and momentum distribution. For the saturated situation numerical results are given, which have been obtained by the use of the Jacobi elliptic functions. The new features found are singularities in the momentum distribution and their evolution with intensity, saturation enhancement of the gain and the possibilities of bistability and hysteresis entailed by it, and the possibility of discussing and interpreting the various features in a unified way. Finally, a detailed discussion of the approximations and their validity is given.