Abstract
Finite temperature effects of a Free-Electron -Laser amplifier are analysed. The formalism follows Bernstein-Hirshfield model by using the linearized relativistic Vlasov equation while keeping the exact pump potential except for its radial dependence. In addition to this, our formalism includes an arbitrary distribution function and a guide magnetic field. Calculations using a cold fluid with a guide magnetic field are performed. The resulting dispersion tensor is hermitian for real values of frequency and wave-vector and exhibits a new resonance frequency which explains the stability of the equilibrium orbits. The dispersion relation obtained from the kinetic model is solved numerically. The width of the distribution function causes a mismatch in the real parts of the wave-numbers of the coupled waves and consequently a decrease in the gain. The spectrum broadens and the central emission frequency decreases with temperature. A boundary-value problem is formulated and numerically solved in order to study how temperature affects the propagation of the modes and to calculate its relation to the coupling loss. Temperature effects become noticeable at a distance given by l(,(DELTA)) = (lamda)(,(omega))((mu)(,0)/(DELTA)), where (lamda)(,(omega)) is the wiggler wave-length, (mu)(,0) is the central momentum and (DELTA) is the momentum spread. In the amplifier experiment and for small pump parameters, only three modes propagate. When temperature increases there is a gradual transition from a three-mode into a two-mode situation. The coupling loss after an initial increase, decreases with temperature. This fact can alleviate the decrease in gain due to Landau damping of the plasma wave. The inclusion of the guide magnetic field does not alter the above description, except when operating close to magneto-resonance. For one type of stable orbits, there is a minimum momentum for the integrals in the dispersion tensor. This causes branch-cut contributions when inverting the Laplace transform using the residue theorem in addition to the usual poles. These contributions for zero order of the pump parameter represent two trains of waves with indices of refraction satisfying the dispersion relation of the cyclotron modes. These contributions become important when operating close to magnetoresonance.