Three dimensional radiation fields in free electron lasers using Lienard-Wiechert fields

In a free electron laser a relativistic electron beam is bunched under the action of the ponderomotive potential and is forced to radiate in close phase with the input wave. Until recently, most theories of the FEL have dealt solely with electron beams of infinite transverse dimension radiating only one-dimensional E.M. waves (plane waves). Although these theories describe accurately the dynamics of the electrons during the FEL interaction process, neither the three dimensional nature of the radiated fields nor its non-monochromatic features can be properly studied by them. As a result of this, very important practical issues such as the gain per gaussian-spherical optical mode in a free electron laser have not been well addressed, except through a one dimensional field model in which a filling factor describes crudely the coupling of the FEL induced field to the input field.