Green function for a seeded energy chirped free electron laser

In a free electron laser (FEL), the electron bunch at the undulator entrance can have a correlated energy spread. In this paper we consider the case of electrons having a linear energy chirp and derive an expression for the FEL Green function for the seeded (i.e., the FEL startup is due to a seed laser) process. The Green function is found within the one dimensional Vlasov-Maxwell model and allows to evaluate the envelope of the radiated electromagnetic field along the undulator. In previous works the formula was derived using a saddle point asymptotic approximation, and is suitable in the exponential growth regime. Instead the new proposed formula works in the lethargy as well as in the exponential growth regime. Comparisons between the exact formula and the approximated one are given.