Classical harmonic oscillator approach of a helical-wiggler freeelectron laser with axial guide field

Electron beam dynamics in a helical-wiggler free-electron laser (PEL) with a uniform axial guide magnetic field are studied using a three-dimensional Hamiltonian approach. The basic feature of the analysis is the definition of a rotational variable, h, that plays the primordial role in lowering to the half the dimension of the quadratic Hamiltonian as a system of two uncoupled oscillators with definite frequencies and amplitudes. It is through applying this variable in the vicinity of a fixed point that the Heisenberg picture of the dynamics of the particles comes to light, leading thus to the association of the steady-stale ideal helical trajectories with arbitrary trajectories. The approach recognized the usual two constants of motion, one being the total energy while the other is the canonical axial angular momentum, Pz`. If the value of the latter is such that a fixed point exists, the Hamiltonian is expanded about the fixed point up to second order. The so-obtained oscillator characteristic frequencies allowed one to .study the different modes of propagation and to identify, and then avoid the problematic operating conditions of the FEL concerned. On the other hand, the amplitudes of the oscillations, which do depend on the frequencies, are fortunately found to be constants of motion and then controlled by the boundary conditions (initial conditions).