We present an analysis of the beam dynamics in a Smith-Purcell free-electron laser (FEL). In this system, an electron beam interacts resonantly with a copropagating surface electromagnetic mode near the grating surface. The surface mode arises as a singularity in the frequency dependence of the reflection matrix. Since the surface mode is confined very close to the grating surface, the interaction is significant only if the electrons are moving very close to the grating surface. The group velocity of the surface mode resonantly interacting with a low-energy electron beam is in the direction opposite to the electron beam. The Smith-Purcell FEL is therefore a backward wave oscillator in which, if the beam current exceeds a certain threshold known as start current, the optical intensity grows to saturation even if no mirrors are employed for feedback. We derive the coupled Maxwell-Lorentz equations for describing the interaction between the surface mode and the electron beam, starting from the slowly varying approximation and the singularity in the reflection matrix. In the linear regime, we derive an analytic expression for the start current and calculate the growth rate of optical power in time. The analysis is extended to the nonlinear regime by performing a one-dimensional time-dependent numerical simulation. Results of our numerical calculation compare well with the analytic calculation in the linear regime and show saturation behavior in the nonlinear regime. We find that a significant amount of power grows in the surface mode due to this interaction. Several ways to outcouple this power to freely propagating modes are discussed.

A Smith-Purcell device can operate as a backward-wave oscillator for intense, narrow-bandwidth, continuous wave radiation at terahertz wavelengths. We determine the requirements on electron beam current and emittance for the system to oscillate based on a three-dimensional extension of our previous two-dimensional analysis. It is found that specially designed electron beams are required with a current that exceeds a certain threshold value and a flat transverse profile that allows the beam to travel very close to the grating surface. Two methods for producing electron beams with the required characteristics are discussed.

Operation of the Smith-Purcell backward wave oscillator requires a flat electron beam. Without the electron beam focusing, the requirement leads to a very stringent criterion on vertical emittance. In this paper, we discuss a way to relax the criterion by introducing an external focusing.

Reflection matrix elements of a grating play an important role in the study of Smith-Purcell (SP) free-electron lasers (FELs). Especially, the matrix element R 00 , which couples the incident co-propagating evanescent wave to the reflected co-propagating evanescent wave, is important for evaluation of the gain of an SP-FEL system [1]. We present a calculation of R 00 for rectangular grating and study its frequency dependence for a given phase velocity of incident wave. For the numerical calculation, we use the modal expansion method and extend it to include waves having slowly varying amplitude. The singularity of R 00 is studied in some detail and we find that it is possible to get a simple formula for the location of the singularity if we choose the eigenmodes of the groove as a basis set as done by Andrews et al. [2].

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