Incoherent emission of an electron beam in a free electron laser rectangular waveguide

A calculation is presented of the incoherent radiation emitted by an electron beam of finite radius passing through a linearly polarized magnetic undulator in a rectangular waveguide with perfect conducting walls. This analysis is based on the Fourier component of the Lienard--Wiechert fields. The metallic boundary conditions are exactly incorporated in the formalism by means of the ''image currents'' method. In the Fresnel approximation, we show that the double infinite sum over ''images'' is transformed into one over ''vertical modes,'' consisting of trigonometric functions. The discrete frequency spectrum and angular distribution of the radiation are studied for several energy spread and electron beam radius cases. The predicted radiation patterns provide useful diagnostics to determine the electron beam properties (trajectories, energy and angular spread, betatron oscillation). The amount of relative power emitted on each mode is a gaussian function exp(-an2)