Modal description of longitudinal space-charge fields in pulse-driven free-electron devices

Longitudinal space-charge fields taking place in free-electron devices are considered in the framework of a three-dimensional, space-frequency approach. The model is based on the expansion of the total electromagnetic field (radiation and space-charge waves) in terms of transverse eigen-modes of the (cold) cavity, in which the field is excited and propagates. Using the model, an analytical expression for longitudinal electric field is derived for a pointlike charge, moving along a waveguide with a constant velocity. This enables comparison of different components of the resulted longitudinal electric field, including contributions of forward and backward going waves, of near and under cut-off frequencies, and others.