A Unified Theory of Electron Beam Interaction with Slow Wave Structures, with Application to Cut-off Conditions†

Abstract

An analysis is given of slow wave structures coupled to electron beams, which is based on their resonant properties rather than their traveling wave properties. Thus the analysis is particularly appropriate for narrow band slow wave structures. The fields of a general symmetrical cavity chain are expanded in a set of solenoidal and irrotational short-circuit modes, defined within electric shorting planes on the coupling surfaces. The amplitudes of the modes are evaluated in terms of the beam current and tangential electric field on the coupling surfaces. The latter field is expanded in a set of open-circuit modes which amplitudes are related back to those of the shortcircuit modes with the aid of a Variational Principle. Alternately, the fields of a cavity are expanded in a set of open-circuit modes, driven by the beam current and by tangential magnetic field on the coupling surfaces. This latter field is expanded in a set of short-circuit modes, which amplitudes are related to the open-circuit mode amplitudes with the aid of the Variational Principle. The accuracy of the procedure is verified for a narrow passband by proving the relation power flow equals stored energy times group velocity. It is possible to prove the equivalencemore » of the dispersion relations for a cold structure obtained from the short- and open-circuit mode expanstons. Small signal transmission line equations are obtatned for a longitudinally confined beam, including relativistic effects, under the action of cavity solenoidal electric field and space charge irrotational field. Analysis is applied to a study of the beam-circuit interaction when the slow space charge beam mode is nearly synchronous with the cold circuit near the latter's cut-off. This is the situation which often leads to oscillations in high-power structures, interfering with their operation at a lower frequency. On a small-signal basis, four waves exist, two of constant amplitude and a pair of growing and decaying waves. The cut-off oscillations, which exist in a relatively wide voltage range such that the slow beam mode is in synchronism with the cold circuit near the cut-off, are attributed to the presence of the growing wave of forward circuit power and negative beam power. As the voltage is lowered (in a structure of forward group velocity from zero to pi phase shift), the negative power of the growing wave changes to positive power at about the condition marking the boundary of the cut-off oscillations. As the voltage is lowered further, two constant amplitude waves of negative beam power appear, enabling backward wave oscillations to exist within a narrow voltage range. This distinction between cut-off oscillations and backward wave oscillations should be made in the general class of slow wave structures.« less