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

In pulsed-beam free-electron devices, longitudinal space-charge fields result in collective effects leading to an expansion of short electron bunches along their trajectory. This effect restricts an application of intense ultrashort electron pulses in free-electron radiation sources. A careful theoretical treatment is required in order to achieve an accurate description of the self-fields and the resulted electron beam dynamics. In this paper, longitudinal space-charge fields are considered in the framework of a three-dimensional, space-frequency approach. The model is based on the expansion of the total electromagnetic field (including self-fields) in terms of transverse eigenmodes of the (cold) cavity, in which the field is excited and propagates. The electromagnetic field, originally obtained in the model as a solution of the wave equation, is shown to satisfy also Gauss's law. We applied the theory to derive an analytical expression for the longitudinal electric field of a pointlike charge, moving along a waveguide at a constant velocity. This enables consideration and study of the role played by different terms of the resulted expressions, such as components arising from forward and backward waves, propagating waves, and under cutoff frequencies, and so on. Possible simplifications in evaluation of longitudinal space-charge fields are discussed.