Abstract

We discuss the behavior of the free-electron laser in a quantum formulation. This is carried out under the condition of weak saturation, when the intensity is not too large. Then, a multiple-time-scale perturbation expansion is possible, which is shown to give good agreement with exact numerical calculations for asymptotically long times. We calculate the first two moments of the electron momentum distribution, which give the gain of the laser and the spread of the electron momentum. Already the zeroth approximation gives a nontrivial and analytically simple expression. The expansion is carried out to first order. In addition to the asymptotic agreement for long times, this approximation is shown to coincide exactly with the straightforward perturbation expansion for short times. The solution is presented and discussed.

© 1988 Optical Society of America

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