On using the coherent far IR radiation produced by a charged-particle bunch to determine its shape: I Analysis

Because a short bunch of relativistic charged particles produces characteristic far infrared radiation when appropriately perturbed, the resulting spectrum can be related to the bunch form factor to provide information on the longitudinal shape. An important question which we address here regards the accuracy of the shape determined from such a spectroscopic measurement. Once the frequency dependence of the intensity of the emitted radiation has been obtained, there are two analysis methods which have been used to produce the longitudinal shape. Both make use of extrapolation into frequency regions where data is not available. One approach relies on the assumption that the bunch is symmetric so that a cosine Fourier transform can be used to find the shape. In the second approach, which we have proposed, a Kramers-Kronig relation is applied to the spectral form-factor data to find the minimal phase and then the asymmetric bunch shape is determined from the complete Fourier transform. By studying a variety of possible symmetric bunch shapes and extrapolations we have been able to identify the source of possible errors inherent in this phase determination process. For all reasonable shaped bunches and extrapolations we find that the actual phase is well represented by the minimal phase obtained from the Kramers-Kronig analysis. A straightforward extension illustrates how spectral measurements at different angles with respect to the beam trajectory may be used to define the 3-D bunch shape.