Velocity Modulation of a Relativistic Electron Beam

Velocity modulation of a one-dimensional electron beam is treated relativistically, starting from the collisionless Boltzmann equation and using first-order perturbation theory. A general expression for the perturbation current density j1 is applied first to a mono-energetic unperturbed beam. Compared to the non-relativistic case, the magnitude of j1 is reduced by the factor (1−u2/c2)3/4, where u is the unperturbed beam velocity and c the velocity of light. The distance between the maxima of j1 is increased by the inverse of the same factor. Relativistic effects decrease the efficiency of modulation η, which is defined as the energy carried by the electric field generated by the perturbations, divided by the energy flux in the unperturbed beam. For an unperturbed beam with a rectangular velocity distribution of narrow width w and a fixed mean velocity um, the results are independent of w to first order. Expressions for j1 and η are also found for an unperturbed beam with a rectangular momentum distribution of arbitrary width. When this width is small and increasing as um is fixed, j1 increases in second order if the beam is highly relativistic or if its plasma frequency is larger than the frequency of modulation, while it decreases otherwise. Under either condition η decreases in second order.