Modelling Chromatic Emittance Growth in Staged Plasma Wakefield Acceleration to 1 TeV using Nonlinear Transfer Matrices

A framework for integrating transfer matrices with particle-in-cell simulations is developed for TeV staging of plasma wakefield accelerators. Using nonlinear transfer matrices in terms up to ninth order in normalized energy spread $\sqrt{\langle\delta\gamma^2\rangle}$ and deriving a compact expression for the chromatic emittance growth in terms of the nonlinear matrix, plasma wakefield accelerating stages simulated using the three-dimensional particle-in-cell framework OSIRIS 4.0 were combined to model acceleration of an electron beam from 10 GeV to 1 TeV in 85 plasma stages of meter scale-length with long density ramps and connected by simple focusing lenses. In this calculation, we find that for relative energy spreads below $10^{-3}$ and normalized emittance below mm-mrad, the chromatic emittance growth can be minimal. The technique developed here may be useful for plasma collider design, and could be expanded to encompass non-linear wake structures and include other degrees of freedom such as lepton spin.