Time- and frequency-domain models for Smith-Purcell radiation from a two-dimensional charge moving above a finite length grating

Smith-Purcell radiation (SPR), formed by an electron beam traveling above a grating, is a very promising source of coherent radiation from the THz to the optical regime. We present two theoretical calculations of the SPR from a two-dimensional bunch of relativistic electrons passing above a grating of finite length. The first calculation uses the finite-difference time-domain approach with the total-field/scattered-field procedure for fields incident on the grating. This calculation allows good physical insight into the radiation process and also allows arbitrary geometries to be treated. The second calculation uses an electric-field integral equation method. Good agreement is obtained between these two calculations. The results of these theoretical calculations are then compared with a theoretical formalism based on an infinite-length grating. The latter formalism allows periodic boundary conditions to be rigorously applied. For gratings with less than ∼50 periods, a significant error in the strength of the radiated field is introduced by the infinite-grating approximation. It is shown that this error disappears asymptotically as the number of periods increases. The Wood-Rayleigh anomalies, predicted in the infinite-grating approximation, were not seen in our finite-grating calculations. The SPR resonance condition is the same in all three formalisms. Numerical examples are presented for an ∼18 MeV, 50 nC∕m, 200 μm bunch traveling 0.6 mm above a ten-period echelle grating having a 2.1-mm periodicity.