Impact of an initial energy chirp and an initial energy curvature on a seeded free electron laser: the Green's function

In a free electron laser (FEL), the electron bunch energy profile at the undulator entrance can have temporal structures. In this paper, we derive the FEL Green's function for the case of the electron bunch having both energy chirp and energy curvature by solving the coupled Vlasov-Maxwell equations. We give an integral representation as well as an analytic expression for the Green's function. The analytical expression is compared with direct numerical results. Evolution of the Green's function temporal duration and the frequency bandwidth are studied.