Wake field of electron beam accelerated in a RF-gun of free electron laser “ELSA”

Wake field effects driven by a coasting relativistic charged particle beam have been studied for various cavity geometries. In the particular case of a cylindrical “pill-box” cavity, an analytical expression of the (E, B)(x, t) map has been obtained as a development on the complete base cavity normal modes. We extend this method to the case of an accelerated beam, which leaves the downstream face of the cavity with a thermal velocity, and becomes relativistic in a few cm. This situation is very different from the classical wake of an ultrarelativistic beam for two reasons: (a) in the case of an ultrarelativistic beam, the field directly generated by beam particles in their wake can be neglected, and the so-called wake field is the electromagnetic linear reponse of the cavity to the exciting signal which is the beam. For a transrelativistic beam, the direct field must be taken into account and added to cavity reponse, which is no longer linear, except for low-intensity beam; (b) causality prevents any beamʹs field influence at a distance from the emissive cathode greater than ct.