Acceleration of electron beams in finite-amplitude wake waves

The energy spectra of the electrons accelerated in a nonlinear finite-amplitude 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 accelerating phases of a nonlinear wake wave overlap; as a result, the maximum final energy of the accelerated electrons increases. The bunching of the electrons accelerated in a nonlinear wake wave near the point of maximum energy in energy space is studied. Simple approximating formulas are derived that make it possible to determine the most important (highest energy) part of the energy spectrum of the accelerated electrons from the shape of the potential profile.