The free-electron laser is described by the self-consistent solution of the coupled system of the Lorentz force equation for the electrons and the Maxwell equation for the laser field. We show that, near saturation and at both low- and high-gain regime, the evolution of each longitudinal mode in the optical cavity can be written as an integral equation. We present numerical solutions of the integral equations for a free-electron-laser oscillator.
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Gallardo et al. (1987) studied this question.
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