Nonlinear optical guiding in the separable beam limit

The nonlinear theory of optical guiding in a free electron laser amplifier is developed for the case in which the axial and radial dependence of the fluctuating current in the electron beam can be separated. 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 such cases, assuming a Gaussian profile for the electron beam density enables the wave equation to be solved exactly and the radiation field experienced by the particles to be expressed as a one-dimensional convolution of the current source. Analytical and numerical calculations of the effect of diffraction on the gain and nonlinear saturation of a free electron laser amplifier are presented. The constraints imposed by diffraction on efficiency enhancement by tapering are examined.