Role of beam quality in free‐electron lasers

A kinetic theory of the scaling of free‐electron lasers is presented. The results are based on a variational solution of an eigenvalue equation that is derived from the Vlasov–Maxwell equations and includes the effects of optical guiding, betatron oscillations and emittance, energy spread, and self‐fields. The dependence of growth rate and filling factor on betatron oscillations and emittance is displayed in terms of scaled variables. A scaled thermal velocity, S, is defined as a measure of axial beam quality and shown to provide a quantitative distinction between the cold and the kinetic regimes of operation. The possibility of enhancing the growth rate by compressing the beam longitudinally is explored and discussed on the basis of the variation of S. Finally, we discuss a surprising result. It is shown that self‐fields, by opposing the betatron oscillations due to wiggler gradients, tend to reduce the spread in the axial velocity of the electrons. That is, self‐fields effectively cool the beam.