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The problem of oscillator evolution and mode competition in free electron lasers is studied. Relativistic quantum field theory is used to calculate electron wave functions, the angular distribution of spontaneous emission, and the transition rates for stimulated emission and absorption in each mode. The photon rate equation for the weak-field regime is presented. This rate equation is applied to oscillator evolution with a conventional undulator, a two-stage optical klystron, and a tapered undulator. The effects of noise are briefly discussed.
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Bosco et al. (1983) studied this question.
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