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We present a model for a Smith-Purcell free-electron laser, including nonlinear effects, that can be solved in closed form. The model is shown to obey the pendulum equation. The average electron efficiency calculated by means of this model is compared with that obtained from an exact numerical calculation and from a simpler "phase-space" model for a wide range of physical parameters.
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Crowne et al. (1981) studied this question.
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