Small-Signal Gain for a Planar FEL with a Periodic Magnetic Field

We describe computation of the small-signal normal modes for a Compton regime free-electron laser (FEL) with an arbitrary periodic magnetic field. Our work shows how to extend the previously known sinusoidal results into the low-energy regime and also how to describe consequences of periodic magnetic fields at arbitrary e-beam energy. The formalism has an a posteriori requirement that the calculated gain or phase shift be small over a wiggler period. The concept of coupling coefficients applies to these generalized conditlons, and provides a unified description of this large class of cases. The formalism is based on the classical one-dimensional Vlasov-Maxwell equations, without an Abraham-Lorentz model of radiation reaction. These results are specialized to a number of important limiting cases: the cold-beam limit, the sinusoidal limit, and the highly relativistic sinusoidal field limit that we compare to known results. The results of this formalism show that coupling to harmonics can be enhanced far beyond that achievable by purely sinusoidal magnetic fields.