| Summary: | We present an analytic theory for the small-signal operation of a grating-based free-electron laser that includes the effects of transverse diffraction on the evanescent wave. In this device, the electron beam interacts with an evanescent wave of the grating that bunches the beam and creates superradiant Smith-Purcell radiation. We find that the evanescent wave is guided by the electron beam, giving an optical-mode width that depends on the gain. We consider the cases of very wide and very narrow electron beams. For a wide electron beam, the cubic dispersion relation previously found for slow-wave structures is recovered. When the electron beam is narrow, so that gain guiding is important, a fifth-order dispersion relation is found instead. Diffraction in a system where the group velocity is very different (sometimes negative) from the phase velocity leads to unexpected results. The Brillouin zone subdivides into four regions; only two physically allowed (gain-guided) roots are obtained in the regions near the center of the Brillouin zone, but three are found in the regions away from the center. In the left half of the Brillouin zone, corresponding to high electron energy, the device operates on a convective instability, as an amplifier. In the right half of the Brillouin zone, where the group velocity is negative, the device operates on an absolute instability, as an oscillator. In the region where only two guided modes exist, oscillator operation will be more difficult.
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