Impact on a Seeded Harmonic Generation FEL of an Initial Chirp and Curvature in the Electron Bunch Energy Distribution

in a harmonic generation free electron laser (HG FEL), the electron bunch entering the undulator can have an initial energy curvature besides an initial energy chirp. Solving the Vlasov-Maxwell equations within the 1D model, we derive an expression for the Green function for the seeded HG FEL process for the case of the electron bunch having both an energy chirp and an energy curvature. We give an asymptotic closed form which is a good approximation in the exponential growth regime, and a series expression that allows the evaluation of the field envelope along the undu-lator in both lethargy and exponential growth regime. The latter is useful to study the HG FEL behavior in the short modulator, like that of the FERMI@Elettra project. The FEL radiation properties such as central frequency shift and frequency chirp are studied considering Gaussian laser seeds of different temporal duration with respect to that of the Green function. The energy chirp and curvature of the electron bunch result in a time dependent bunching factor for the FEL start-up process in the radiator of the HG FEL like the FERMI@Elettra. The coherence properties of the FEL are examined