Thick beam free electron laser

A 3-D theory is presented for a free electron laser that employs an electron beam of a thickness comparable to both the wiggler wavelength and the waveguide radius. The time-independent and the linearized time-dependent cold fluid and Maxwell equations are expanded in a small parameter, which is the ratio of the perpendicular to parallel electron momentum. The stability problem is reduced to a nonlinear eigenvalue problem of a fourth-order system of linear ordinary differential equations. A perturbation method is justified and used to solve these equations. A dispersion relation is derived which results from the solvability condition for the first-order equations in the perturbation. The orders of magnitude of the beam density and wave frequency, for which the growth rate of the instability scales as in the strong-pump regime of the 1-D analysis, are determined. An equation, which the beam energy radial profile has to satisfy, is also derived