Collective Instability of a Free Electron Laser including Space Charge and Harmonics

Recently there has been interest in the collective instability regime of the free electron laser (FEL) [ 1,2]. There is also some work on the collective regime appearing in the Russian literature [3,4]. Bonifacio, Narducci and Pellegrini [5] (referred to hereafter as BNP) have discussed the collective instability regime of an FEL studying the growth of the radiation field from noise and the saturation level for the case of a cold electron beam, ignoring the effects of space charge and higher harmonics of the fundamental operating frequency, o = 2ck,y2/(1 + iK2). There has also been some analytic work done on the effects of energy spread on the collective instability [6] _ We shall extend BNP’s work to include the effects of higher harmonics, space charge and an energy spread in the electron beam. A planar undulator is assumed throughout the discussion. We shall use the notation of BNP: z is the direction of propagation of the electron beam; x and y are the transverse coordinates; the electron beam is characterized by its energy, y, in units of the rest energy, nz,, c2, and n, the electron number density which are combined to give the relativistic plasma frequency fip = (4rmee2/my0) 3 ‘j2 ; the undulator magnetic field strength, Bo, and period ho can be combined to yield the undulator parameter, K = eBoXo/2nmc 2 ; the undulator gives