Transverse Beam Evolution in a Plasma Wakefield Accelerator with Plasma Density Ramps

A plasma density ramp at the entrance and exit of a plasma-based accelerator can limit the chromatic emittance growth within the accelerator by controlling the focusing of the beam into and out of the plasma. It is challenging, however, to determine an effective plasma density profile. We present an analytic theory to describe the transverse evolution of an electron beam in a longitudinal plasma density profile. We classify the evolution into three distinct regimes: adiabatic, non-adiabatic, and perturbative. We show that analytic solutions can be calculated for beams with arbitrary chromatic content in the adiabatic and perturbative regimes, and for several cases in the non-adiabatic regime. Further, we show how the theory can be used to determine the effect the shape of the ramp has on the chromatic emittance growth in the ramp and the incoming beam parameters required for matching.