FEL dispersion equation with account for relativistic space-charge effects

We give a qualitative derivation of a free-electron laser (FEL) dispersion equation, which takes into account not only non-relativistic (potential) but also relativistic (rotational) space-charge effects. The derivation is based on a generalisation of a dispersion equation for a Compton FEL obtained by us earlier, a certain weakly-relativistic quasistatic approximation of the space-charge field (the Darwin model), and uses an analogy between FEL and a travel wave tube (TWT). An analysis of the obtained linear growth rates is presented.