Abstract
We present a three-dimensional formulation of the cw free-electron laser amplifier in the small-signal domain. The formalism is used to calculate expressions for the power gain in both small-gain and large-gain situations, and differences from the corresponding formulas of the one-dimensional theory are studied in detail for an input optical beam of Gaussian cross section. In the small-gain case, the maximum of the gain profile is shifted from that of the one-dimensional theory, in accord with previous results. In the large-gain case, we show that the primary effect of diffraction of the fields is conveniently expressed in terms of radially dependent shifts in the resonance frequency and the overall phase. This leads to the result that an optical beam, as it amplifies in the electron-wiggler system, undergoes a smaller diffractive spreading than what it would have in free space.
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