Three-dimensional analysis of an FEL amplifier operating with a continuous e beam

The authors discuss a three-dimensional theory of an FEL operating with a continuous e beam. The theory is based on a generalized method of solution of the FEL Helmholtz equation in the high-gain small-signal regime. The method consists basically of expanding the optical field in terms of Hermite-Gaussian modes, taken at the entrance to the undulator. The coefficients of that expansion are expressed in the form of a superposition of plane waves with complex frequencies. As a result, one gets a set of recursive relations, partly reproducing those obtained within the context of the single-mode evolution and partly displaying the effect (revealed by a coupling among the modes) of both the diffraction and the interaction with the e-beam