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We describe the radiation properties of an X-ray, free electron laser (FEL) oscillator starting with its start-up from noise through saturation. We decompose the initially chaotic undulator radiation into the longitudinal modes of the resonator whose properties are largely determined by the transverse gain profile and the bandwidth of the Bragg mirror. Because the radiation is initially comprised of several modes whose growth rates are comparable, we show that only after many oscillator passes is the output pulse dominantly characterized by the lowest-order, gaussian mode. We also comment on the transverse structure of the radiation, and discuss how the FEL gain can modify the free space resonator modes. Understanding the full longitudinal and transverse structure during the initial amplification will be critical in assessing the tolerances on electron beam, undulator, and optical cavity required for robust operation.
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