Analysis of Smith-Purcell free-electron lasers

Abstract

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.

Small signal gain of a Smith-Purcell free electron laser

“…The electron beam is not assumed to be infinitely thin as it was considered in Refs. [12,16] and it does not occupy the entire space above the grating as was assumed in Ref. [13].…”

“…For the thin electron beam k 0 = 0 1, which for magnetized electron beam ( 1=2) corresponds to the setup of Ref. [16], from Eq. (56) we get…”

“…In [16], the authors have revised the work [12] with proper evaluation of singularities of reflection matrix for rectangular grating. The authors considered an infinitely thin electron beam extending analysis also to the nonlinear regime by performing one-dimensional numerical simulations.…”

Enhancement of coherent Smith-Purcell radiation at terahertz frequency by optimized grating, prebunched beams, and open cavity

“…When the beam current exceeds a certain threshold value, called the starting current I s [25,26,45,46,49], the net gain of the signal can build up in time the BWO oscillation in the structure, and the beam gets bunched by the enhanced evanescent wave. For a given injection beam current density, we run the simulation and check whether the electron beam is bunched after 3 ns (which is about 100 times of the transit time for one electron to travel across the length of the whole grating).…”

“…It was achieved due to the appearance of periodic electron bunching, induced by strong nonlinear interaction of the electron beam with an oscillating-evanescent wave (or grating surface mode) that is excited on the grating structure. The physics of coherent SPR has been studied extensively [21][22][23][24][25][26]. Many simulation works have been performed to investigate coherent SPR from continuously injected electron beam [27][28][29][30].…”