Thermal effects on the linear gain in free-electron lasers

In this paper the effect of an axial energy spread on the linearized gain in free-electron lasers is considered for configurations which employ both helical and planar wiggler fields. The analysis includes collective effects and is valid for either the Raman or high-gain Compton regimes. A thermal function is obtained which applied to both the helical and planar wiggler configurations at the fundamental, and which is generalized to treat the thermal effect on the harmonics for a planar wiggler. It is assumed that the displacement of the electron beam from the axis of symmetry for a helical wiggler, or the plane of symmetry for a planar wiggler, is much less than the wiggler period, and an idealized one-dimensional model is considered. The electron-beam model used to describe the axial energy spread is based upon the assumption of a monoenergetic beam which exhibits a pitch angle spread. This is described in the analysis by the inclusion of nonvanishing components of the canonical momenta in the single-particle trajectories of the electrons, and the specific distribution used is that of a Gaussian spread in the canonical momenta. The linearized Vlasov-Maxwell equations are then used to derive the dispersion equations, including collective Raman effects, for both the helical and planar wigglers. The analysis treats the interaction at the fundamental resonance frequency in the case of the helical wiggler, and a general thermal function is derived which describes the effect of the axial energy spread. The planar wiggler configuration admits interactions at odd harmonics as well as the fundamental, and a general dispersion equation is derived which includes the thermal effect at each harmonic as well as the fundamental. In addition, the nonvanishing canonical momenta results in an oscillation in the axial velocity at the wiggler period which gives rise to emission at all harmonics