Analytical theory of pulsed relativistic electron beams entering a vacuum

An electron beam with a high-current density, passing from a conductor into a vacuum, generates strong electric fields which significantly influence further propagation of the electrons. An analytical solution for a one-dimensional pulsed electron beam with an arbitrary energy distribution is found which allows one to calculate electric field distributions, electron density distributions and potential distributions in the vacuum region. The analytical solution is fully relativistic and is applicable up to the time at which the interpenetration of electron paths occurs. It may be useful in research with fs lasers relating to generation of relativistic electron beams, ion beams, and the pumping of x-ray lasers.