A generalized formulation of free-electron lasers in the low-gain regime including transverse velocity spread and wiggler incoherence

It is shown that in the small-signal, low-gain regime, an analytic expression for the gain can be derived which is valid for a general electron distribution function and general longitudinal variation of the transverse magnetic field pump. This expression is used to evaluate the effects on the gain curve due to transverse momentum spread in the beam as well as inaccuracies (incoherence) in the magnetic pump phase and amplitude. The restricting criteria for the neglect of these effects are derived.