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# Can coherent Smith–Purcell radiation be used to determine the shape of an electron bunch?

Department of Physics, University of Essex, Colchester, UK CO4 3SQ

Nuclear Instruments and Methods in Physics Research Section A Accelerators Spectrometers Detectors and Associated Equipment (Impact Factor: 1.22). 05/2002; DOI: 10.1016/S0168-9002(02)00324-8 **ABSTRACT**

Recent measurements at Frascati of the Smith–Purcell radiation emitted from the interaction of a 2.3 MeV (total) electron beam with a 2.5 mm grating show clear evidence of coherent enhancement of the radiation, with power levels exceeding 100 mW at emission angles around 90° relative to the beam direction. The experimental results are in reasonable agreement with theoretical predictions and suggest that Smith–Purcell radiation may offer a simple way of determining the shape and duration of short (picosecond) electron bunches.

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**ABSTRACT:**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 approximately 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 approximately 18 MeV, 50 nC/m, 200 microm bunch traveling 0.6 mm above a ten-period echelle grating having a 2.-mm periodicity. - [Show abstract] [Hide abstract]

**ABSTRACT:**A simulation of the generation of Smith-Purcell (SP) radiation at microwave frequencies is performed using the two-dimensional particle-in-cell code MAGIC. The simulation supposes that a continuous, thin (but infinitely wide), monoenergetic electron beam passes over a diffraction grating, while a strong axial magnetic field constrains the electrons to essentially one-dimensional motion. The code computes the time-dependent electric and magnetic fields by solving the Maxwell equations using a finite element approach. We find that the passage of the beam excites an evanescent electromagnetic wave in the proximity of the grating, which in turn leads to bunching of the initially continuous electron beam. The frequency and wave number of the bunching are determined, and found to be close to those proposed by Brau and co-workers in recent work. This frequency is below the threshold for SP radiation. However, the bunching is sufficiently strong that higher harmonics are clearly visible in the beam current. These harmonic frequencies correspond to allowed SP radiation, and we see strong emission of such radiation at the appropriate angles in our simulation, again in agreement with Brau's predictions. We also find that at the ends of the grating, some of the evanescent wave is diffracted away from the surface, and radiation below the threshold occurs. ln addition, we observe a second evanescent wave at the same frequency, but with a different wave number. The existence of this wave is also predicted by the theory, although its presence in our simulation is unexpected. Numerical estimates of the growth of the evanescent wave are also in reasonable agreement with the predictions, although the precise form of the dependence of the gain on beam current remains hard to establish. - [Show abstract] [Hide abstract]

**ABSTRACT:**Smith-Purcell radiation (SPR), generated by an electron beam traveling above a grating, is characterized by a broad range of frequencies. The radiated wavelength depends on the angle of observation according to the SPR resonance relationship and the bandwidth is inversely proportional to the number of the grating grooves. A rigorous theoretical model of SPR from a three-dimensional bunch of relativistic electrons passing above a grating of finite length is presented by an electric-field integral equation method. The finite-length grating results are compared with the case of an infinitely long grating assumption in which periodic boundary conditions are rigorously applied and with a model based on the image-charge approximation. The SPR resonance relationship is the same in all three formalisms. Significant errors in the strength of the radiated energy are introduced by the two approximations. In particular, for gratings with less than ∼20 periods, the image-charge approximation and the infinitely long grating assumption result in an order of magnitude too high and too low radiated energy per groove, respectively, in the plane transverse to the grating groove lines. Numerical examples are calculated for an ∼18 MeV bunch traveling above different finite-length gratings with a period of 2.5 mm.