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 $\ensuremath{\sim}20$ 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 $\ensuremath{\sim}18\text{ }\text{ }\mathrm{MeV}$ bunch traveling above different finite-length gratings with a period of 2.5 mm.
Highlights
Smith-Purcell radiation (SPR) [1] is typically formed by a charge passing above a periodic grating
A very good agreement was obtained, on a relative scale, between the measured radiation patterns and the calculations using the three-dimensional electricfield integral equation (EFIE) model described in this paper
The objectives of this paper are to (a) extend the twodimensional EFIE model in [9] to the general case of a three-dimensional bunch moving above a finite-length grating, (b) demonstrate consistency with van den Berg’s model [4] for the special case of an infinitely long periodic grating assumption, and (c) compare the radiated energy per groove calculated by the finite-length grating, van den Berg, and the image-charge models
Summary
Smith-Purcell radiation (SPR) [1] is typically formed by a charge passing above a periodic grating. A very good agreement was obtained, on a relative scale (peak energy normalized to unity), between the measured radiation patterns and the calculations using the three-dimensional electricfield integral equation (EFIE) model described in this paper This comparison allowed to evaluate bunch lengths of 1:0 0:1 ps and 0:6 0:1 ps at different accelerator operating parameters. The objectives of this paper are to (a) extend the twodimensional EFIE model in [9] to the general case of a three-dimensional bunch moving above a finite-length grating, (b) demonstrate consistency with van den Berg’s model [4] for the special case of an infinitely long periodic grating assumption, and (c) compare the radiated energy per groove calculated by the finite-length grating, van den Berg, and the image-charge models. A preliminary work on the three-dimensional EFIE model described in this paper is reported in [24]
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