  A conventional free-electron laser is useful but large, driven by a beam with
many relativistic electrons. Although, recently, keV electron beams have been
used to excite broadband radiation from material chips, there remains a quest
for a chip-size free-electron laser capable of emitting coherent radiation.
Unfortunately, those keV emitters from electron microscopes or dielectric laser
accelerator usually deliver a small current with discrete moving electrons
separated by a distance of a few or tens of microns. To envisage a chip-size
free-electron laser as a powerful research tool, we study in this paper
achievable laser radiation from a single electron and an array of single
electrons atop a nano-grating dielectric waveguide. In our study, thanks to the
strong coupling between the electron and the guided wave in a structure with
distributed feedback, a single 50-keV electron generates 1.5-um laser-like
radiation at the Bragg resonance of a 31-um long silicon grating with a 400-nm
thickness and 310-nm period. When driven by a train of single electrons
repeating at 0.1 PHz, the nano-grating waveguide emits a strong laser radiation
at the second harmonic of the excitation frequency. A discrete spectrum of
Smith-Purcell radiation mediated by the waveguide modes is also predicted in
theory and observed from simulation in the vacuum space above the grating
waveguide. This study opens up the opportunity for applications requiring
combined advantages from compact high-brightness electron and photon sources.
