A formula is derived for the small-signal gain of a Smith-Purcell free-electron laser. The theory describes the electron beam as a moving plasma dielectric, and assumes that the electron beam interacts with an evanescent mode traveling along the surface of a periodic waveguide with a rectangular profile. The phase velocity of the evanescent wave is synchronous with the electron velocity, but the group velocity is actually negative. The electron beam amplifies the evanescent wave, which does not itself radiate. According to this picture, the radiation observed emanating from the grating is Smith-Purcell radiation enhanced by the bunching of the electrons due to the interaction with the evanescent mode. There will also be radiation from the part of the evanescent mode that is outcoupled from the ends of the grating. This radiation appears at a lower frequency than the Smith-Purcell radiation. The new results explain both the gain and the radiation observed in the experiments of Urata and Walsh, and the cube-root current dependence of the gain inferred by Bakhtyari, Walsh, and Brownell.

Smith-Purcell (SP) radiation is emitted when an electron passes close to the surface of a metallic grating. The radiation becomes coherent (fluence proportional to the square of the number of electrons) when the electrons are in bunches whose dimensions are smaller than the wavelength of the radiation. This has been observed in experiments in which the electrons are prebunched by an rf linac. The enhancement of the spectral intensity is accompanied by large changes in the angular and spectral distribution of the radiation when the electrons appear in periodic bunches. This is called superradiance. Recently, superradiant SP radiation has been observed from a so-called Smith-Purcell free-electron laser (SP-FEL) in which the electrons are bunched by the lasing process. As in other slow-wave structures, the electron beam in a SP-FEL interacts with an evanescent wave for which the phase velocity matches the electron velocity and amplifies it. The frequency of this wave lies below the range of SP radiation and the wave is not radiated except from the ends of the grating. However, the bunching of the electrons by the interaction with the evanescent wave enhances the ordinary Smith-Purcell radiation and changes the angular and spectral distribution due to superradiant effects. In this article, we introduce a new method for computing the SP radiation in three dimensions, including the effects of finite grating length and superradiance due to periodic electron bunching at an arbitrary frequency. We show that the SP radiation develops spectrally and angularly narrow peaks at the harmonics of the bunching frequency. In rf linacs, where the bunches are widely spaced, several closely spaced harmonics lie under the spectral envelope of the emission from a single electron. In a SP-FEL the harmonics are widely spaced and the SP radiation appears in narrow cones at the SP angles corresponding to the harmonics of the bunching frequency. Finally, we calculate the angular spectral fluence radiated by an electron passing over a lamellar grating of finite length, examine its coherent enhancement in SP-FELs and rf linacs, and compare the results with numerical simulations and available experimental data.

We present an analytical theory for the operation of a Cerenkov free-electron laser which includes diffraction of the optical mode in the direction transverse to the electron beam. Because the width of the optical mode depends on gain, the usual cubic dispersion relation is replaced by a 5/2-power dispersion relation, which allows two roots. These roots both have positive real parts, indicating that they are slow waves. For a narrow electron beam, the optical mode is much wider than the beam, thus reducing the gain by an order of magnitude from that predicted by the two-dimensional theory. In the limit of a wide electron beam, the two-dimensional theory is recovered.

We present experimental observations of evanescent waves in a Smith-Purcell free-electron laser (FEL). These waves, predicted by both theory and simulations, have wavelengths longer than the Smith-Purcell radiation, group velocity antiparallel to the electron beam, and for sufficiently high current, provide feedback to bunch the electron beam. This feedback is the basis of oscillator operation of the SmithPurcell FEL. The wavelengths observed agree with theoretical predictions, and strong radiation from the upstream end of the grating confirms the negative group velocity. Radiation observed at the second harmonic may indicate electron bunching by the evanescent wave.

We present an analytic theory for the exponential-gain (growth) regime of a Smith-Purcell free-electron laser amplifier (oscillator), which includes the effects of transverse diffraction in the optical beam. The optical mode is guided by the electron beam, having a mode width that depends upon the gain length. For the case of a wide electron beam, the dispersion relation converges with that of the 2-D theory. When the electron beam is narrow, the conventional cubic-dispersion relation is replaced by a five-halves dispersion. The dispersive properties of the grating divide device operation into four distinct regions, two amplifier and two oscillator. The number and location of physically allowed roots changes depending on operating region. Additionally, in the narrow-beam case, new challenges arise in satisfying the boundary conditions required for operation as an oscillator