Emittance preservation through density ramp matching sections in a plasma wakefield accelerator

In plasma wakefield acceleration, the witness beam's emittance needs to be preserved when it propagates through a plasma stage. The plasma includes density ramps at both the entrance and the exit. Using the Wentzel-Kramers-Brillouin solution of a single particle's motion, analytical expressions for the evolution of the beam emittance and the Twiss parameters in an arbitrary adiabatic plasma profile are provided neglecting the acceleration of the beam inside the plasma. It is shown that the beam emittance can be preserved under the matching condition even when the beam has an initial energy spread. It is also shown that the emittance growth for an unmatched beam is minimized when it is focused to the same vacuum plane for a matched beam. The emittance evolution from 3D QuickPIC simulation results agree well with the theoretical results. In the some of the proposed experiments on nearly completed FACET II facility, the matching condition may not be perfectly satisfied and the wake may not be perfectly symmetric. It is shown that for a given set of beam parameters that are consistent with FACET II capabilities, even when the assumptions of the theory are not satisfied, the emittance growth can still be minimized by choosing the optimal focal plane. Last, another issue that may cause emittance growth in realistic plasmas is also examined. When using a lithium plasma source in FACET II experiments a helium buffer gas is used. The plasma is formed from field ionization which can lead to a nonlinear focusing force when there are nonuniform helium ions due to its high ionization potential. For an initial beam emittance of $20\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$, the helium ionization is found to be small and the witness beam's emittance can be preserved.