Madey’s gain-spread theorem for the free-electron laser and the theory of stochastic processes

Publisher Website
Google Scholar
Abstract
We elucidate the correspondence between Madey’s gain-spread theorem for the free-electron laser, and a similar theorem obtained for the Brownian motion of a stochastically driven oscillator. By using suitable changes of variables, these two different theorems can be shown to be special cases of a more general result valid for a mechanical system described by a Hamiltonian of the form H = H0(p,t)+λH1(q,p,t). For such a system, when terms up to second order in λ are kept, under certain specified conditions it follows that 〈Δp〉 = (1/2) (∂/∂pi) 〈(Δp)2〉, where Δp is the change in p in time Δt, and the average is over the initial value of the coordinate qi, for fixed time and initial momentum pi. In the case of the free-electron laser, the canonical momentum p must be chosen as the total energy E of the electron, while for the driven oscillator, the necessary choice is the action variable J corresponding to the unperturbed periodic motion.