For an X-ray free electron laser (FEL), a high-quality electron bunch with low emittance, high peak current and energy is needed. During the phases of acceleration, bunch compression and transportation, the electron beam is subject to radio frequency curvature and to wakefields
effects. Thus, the energy profile of the electron beam can present a parabolic profile, which has important electromagnetic effects on the FEL process. The quality of the electron beam is also degraded by the interaction with the low-gap undulator vacuum chamber. In our work we first analyze this interaction, deriving a formula to evaluate the longitudinal and the transversal wakefields for an elliptical cross section vacuum chamber, obtaining accurate results in the short range. Subsequently within the Vlasov-Maxwell one-dimensional model, we derive the Green functions necessary to evaluate the radiation envelope, having as initial conditions both an energy chirp and curvature on the electrons and eventually an initial bunching, which is useful to treat the harmonic generation FEL cascade configuration. This allows to study the impact of the elecron beam energy profile on the FEL performance. Using the derived Green functions we discuss FEL radiation properties such as bandwidth, frequency shift, frequency chirp and velocity of propagation. Finally, we propose a method to achieve ultra-short FEL pulses using a frequency chirp on the seed laser and a suitable electron energy profile.
