The charge separation between electrons and ions that exists within an electron plasma density wave can create large electric fields. In 1979 Tajima and Dawson first recognized that the longitudinal component of the field of a so-called “relativistic” wave (one propagating with a phase velocity close to c), could be used to accelerate charged particles to high energies in a short distance [1]. The accelerating gradient of such a plasma wave, Eo, can be approximated—assuming a total separation of electrons and ions in such a wave with wavelength λp = 2πc/ωp—as