Quantum coherence in electron beams

Following the experimental results on the interaction of electron beams with solids and molecules, the properties of the electron beam after the interaction with discrete changes of the energy have been studied; such studies are important for beam modulation or for free electron lasers. The quantum mechanical description of the electron beam results in a coherence property which is basic. ally different from optical beams. The earlier discussion of coherence of electron beams was correct only in first order to be analogous to the optical beam (correlation coherence). The quantum coherence of the electron beam includes the complete temporal expansion of a wave packet-though well-known-had not yet been evaluated for the problems of coherence. The essential properties found for electron beams are the different components depending on amplitudes with and without modulation to interpret the preceding experiments of beam-matter interaction. This is a second order extension of the analogy of Morpurgo of the quantum modulation with the Forrester effect. The importance of quantum modulated electron beams for a free electron laser are of importance in the case of interaction as in demodulation, inverse Cerenkov radiation, or transition radiation. For the free electron laser amplifier without wiggler field at transverse electron beam injection, the momentum balance is derived confirming the switching-on and -off process for the time averaged nonlinear-force process (contrary to the Kibble process). For a not-time averaged phase process the Kibble process can be used for a transversal free electron laser.