Optical guiding in the separable beam limit

The nonlinear theory of optical guiding in an FEL amplifier is developed for the case in which the spatial dependence of the current source term in the wave equation can be separated into the product of a function of radius and a function of axial distance. Such a separation can be motivated if either the betatron wavelength is shorter than other lengths of interest (synchrotron wavelength, vacuum Rayleigh length) or if the radiation waist exceeds the beam radius. In this limit with the choice of a Gaussian profile for the electron beam density the wave equation can be solved exactly and the radiation field felt by the particles can be expressed as a one-dimensional convolution of the current source. With the given expression for the radiation field, the equations of motion can be solved in the trapped particle regime. Requiring consistency between the particle motion and the fields yields expressions describing nonlinear guided states. The adiabatic evolution of these guided states in the presence of a tapered wiggler is determined by conservation of the electrons' action and total (field and electron beam) energy. Using these relations the growth of the radiation waist as the beam is decelerated can be calculated. (orig.)