The radio-frequency current distribution in Brillouin flow

It has been shown that in an electron beam with Brillouin focusing, two pairs of space-charge waves are possible. One pair has the peculiar property of having no RF charge density in the volume of the beam, with most of the current being caused by ripples of the boundary. This paper shows that, in the case of modulation by the gridless gap of a klystron, it is only these space-charge waves which are excited in the electron beam. This result has also been verified elsewhere by experiment. In addition to the detailed calculation, a simple proof is given which demonstrates why one gets the particular behavior predicted by the detailed theory, namely, a modulated beam with no RF charge density in the volume. This effect arises from the fact that the modulation is produced by an electric field with zero divergence, and therefore the RF velocity produced also has zero divergence. Zero divergence of the velocity is the condition for an incompressible fluid; i.e., constant density. In the case of modulation by a grid, the electric field does not have zero divergence and this kind of behavior does not occur.