Madey's theorems for free-electron devices, spontaneous emission, and applications

Abstract

Two theorems due to Madey occupy a central position in free-electron laser physics: one relates the gain to the derivative of the spontaneous emission line shape and the other one relates it to the derivative of the electron energy spread in stimulated emission. We use quantum mechanical perturbation thoery of first order in the radiation field to give a general derivation of the theorems based on(a) the hermiticity of the electronfield interaction,(b) the applicability of lowest order perturbation theory, and(c) the assumption that the emitted photon have a sufficiently low energy. Assumption(b) restricts the validity of the theorems to the small-signal weak-field regime,(c) to the small recoil regime where the gain is classical. We use scalar quantum electrodynamics in the Furry picture in order to keep effects which are nonlinear in the undulator field, e.g. higher harmonic emission. We consider a fairly general one-dimensional (i.e. not having transverse variations) monochromatic undulator field (magnetic or optical undulator, linear or circular polarization, possible presence of a diffractive medium). An appendix considers nonmonochromatic fields. We derive explicit results for the linearly polarized and the helical undulator allowing for an arbitrary orientation of the undulator axis, the electron beam and the emitted radiation with respect to each other. In particular, we discuss the case of Gaussian modes where the applicability of the first theorem has been questioned. It turns out that the theorem is applicable provided that spontaneous emission into the Gaussian mode in question is considered (more generally, into whatever mode is of interest for the gain).