Classical multimode theory of the free electron laser

A calculation of the small signal gain of a free-electron laser (FEL) operating in a multimode regime is carried out. The calculation is performed in the framework of the single-article classical theory of Bambini and Renieri (1978). One has multimode operation of the FEL whenever the stimulating field can be expressed in terms of a sum of longitudinal modes of the optical cavity. The Weiszacker-Williams approximation is used and the wiggler field is treated as a pseudo radiation field. Invariants are obtained from the nonrelativistic Hamiltonian. The regime in which spontaneous emission can be neglected in comparison with the stimulated one is considered.