Mode growth and competition in the x-ray free-electron laser oscillator start-up from noise

We describe the radiation properties of an x-ray free-electron laser (FEL) oscillator, beginning with its start-up from noise through saturation. We first decompose the initially chaotic undulator radiation into the growing longitudinal modes of the composite system consisting of the electron beam, the undulator, and the Bragg mirror resonator cavity. Because the radiation initially comprises several modes whose growth rates are comparable, we find that only after many oscillator passes is the output pulse dominantly characterized by the lowest-order Gaussian mode. We verify our analytic results with a novel, reduced one-dimensional FEL code (derived in the text), and with two-dimensional FEL simulations. Understanding the full longitudinal structure during the initial amplification will be critical in assessing the tolerances on the electron beam, undulator, and optical cavity required for robust operation.