Perturbation Expansion for High-Gain Free-Electron Laser Saturation

We develop a perturbation expansion for the solution of the nonlinear one-dimensional free-electron laser equations. For a monochromatic wave, the radiation field is expanded in a Taylor series having a finite radius of convergence. Analytic continuation using Pade' approximates yields accurate results well into the saturation regime. We also formulate the perturbation expansion for finite bandwidth, self-amplified spontaneous-emission (SASE), and determine the lowest order correction to the well-known linear theory. Motivated by an approximation to the expansion coefficients, we introduce a simplified model for the SASE radiation field, and use it to discuss SASE statistics in the saturation regime, before the onset of the sideband instability