Field theory on excitation of space-charge waves through a gridless modulating gap

A novel field-theoretic approach is presented to treat the system composed of a drift tube, a gridless modulating gap and an electron beam. It is mentioned that no space-charge wave can propagate just to the front of the plane of discontinuity of the electronic waveguide. The field component is expressed by a Fourier integral, and the amplitude of any of the excited space-charge waves is calculated from the residue at the corresponding pole, in a way similar to the one used for a passive waveguide. The analysis is applicable to the case of an arbitrary field distribution over the gap, not only the single gap but also the multiple gap. It is shown that the ratio of the amplitude of any excited space-charge wave to that of the lowest is independent of the field distribution over the gap. Finally, the beam loading is derived from the standpoint of the space-charge wave theory and is compared with this by the ballistic theory.