Acceleration of electron beams in finite-amplitude wake waves

The energy spectra of the electrons accelerated in a nonlinear finite-ampli tude wake field are studied analytically and numerically by means of three- dimensional simulations. It is shown that the deformation of the wave front leads to a substantial extension of the region in which the focusing and a ccelerating phases of a nonlinear wake wave overlap; as a result, the maxim um final energy of the accelerated electrons increases. The bunching of the electrons accelerated in a nonlinear wake wave near the point of maximum e nergy in energy space is studied. Simple approximating formulas are derived that make it possible to determine the most important (highest energy) par t of the energy spectrum of the accelerated electrons from the shape of the potential profile.