Simulation of input electron noise in the free-electron laser

Numerical integration of the free-electron laser equations usually starts with a small arbitrary value of input field, to allow the FEL instability to show itself. This procedure fails to reproduce the start-up of the real laser and the time or distance to saturation will depend on the arbitrary initial value. The procedure is obviously valid only for initial field-values larger than the random fluctuations in the real system; even if the level of these fluctuations is used as an initial value, the time to saturation for start-up from noise will be incorrectly predicted, since the initial evolution of the electron bunching and field is an incoherent process with linear growth, which is superseded only gradually by a coherent process with exponential growth. The purpose of the present communication is to explain how noise in the electron beam should be simulated on a computer. The FEL equations consist of a Maxwell wave equation for the radiation field envelope a coupled with a number of Lorentz equations for the ponderomotive phases 0j and energies of the electron. In the Compton limit, for example, using the normalization of ref. [ 1 ] and the coordinate z to represent the electron-beam local time, the equations are