Longitudinal bunch profile diagnostics in the 50fs range using coherent Smith-Purcell radiation

We have considered the possibility of using coherent Smith-Purcell radiation for the single-shot determination of the longitudinal profile of 50 fs (FWHM) long electron bunches. This length is typical for the bunches currently produced by Laser Wakefield Acceleration and is at the limit of what is achievable by alternative techniques, such as Electro-Optic sampling. It is concluded that there are no obstacles, either theoretical or experimental, in the implementation of this technique. A set of three gratings, with periods of 15, 85 and 500 micrometres, will produce detectable energy in the wavelength region 10-1000 micrometres, which should be adequate for the reconstruction of the bunch shape by the Kramers-Kronig technique. For bunch charges of 109 electrons, or more, the radiated energy can be detected by room temperature pyroelectric detectors. The limits of possible extension of the technique to even shorter lengths are also considered.