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One-dimensional analysis of the spiking behavior in a free-electron laser.

We derive here a one-dimensional generalized dispersion relation for a free-electron laser, describing, at the same time, the steady-state instability, the superradiant instability, and the sideband instability. Intermediate forms of instability are discussed, where the sideband instability and the superradiant instability are seen to occur together. Numerical simulations have been carried out in order to test the analytical results and investigate the nonlinear evolution of the perturbation of different equilibrium states. We show that both the superradiant instability and the sideband instability can lead to the growth of superradiant spikes