A TWO-DIMENSIONAL THEORY OF PULSE PROPAGATION IN THE FEL OSCILLATOR

Abstract
We present an axial symmetric analysis of the free electron laser (FEL) oscillator in the low gain regime. The electron beam consists of short pulses where the axial pulse shape is arbitrary and the transverse profile is Gaussian. The radius of the electron beam is taken to be much smaller than the radius of the radiation beam. We will consider the case where the resonator is designed to operate in the Gaussian TEM_oo mode. The portion of the stimulated radiation of interest is a superposition of the Gaussian mode, i.e., the vector potential of the radiation pulse can be written as [MATH] (1) where G_oo (r,k,z) is the normalized complex amplitude associated with the TEM_oo mode and ω_o is the resonant laser frequency. The equation governing a_oo can be summarized in a rather compact form, [MATH] (2) where F(k,t,[MATH]) is the self-consistent complex filling factor, α is a constant, h (ζ_o) is the arbitrary axial electron profile, ζ_o is the axial position of the electron relative to the center of the Nth electron pulse at t=t_N, t_N is the time the Nth electron pulse entered the wiggler, and [MATH]( ζ_o, k, t) is the phase of the electron. The particle dynamics of the electrons enter the calculation through the phase [MATH]. Taking a constant wiggler as an example, the phase equation can be written as [MATH] (3) where [MATH] is the axial electron velocity, [MATH] is the axial position of the electron at time t, the "~" over the variables stand for Langrangian variables which are functions of (ζo, t), and c_k are constants. Equations (2) and (3) form a complete set of nonlinear self-consistent equations which govern the dynamics of the radiation pulses in the oscillator.