Limits of linear plasma wakefield theory for electron or positron beams

Abstract

The validity and usefulness of linear wakefield theory for electron and positron bunches is investigated. Starting from the well-known Green’s function for a cold-fluid plasma, engineering formulas for the maximum accelerating field for azimuthally symmetric bi-Gaussian beams of the form nb=nbe−r2∕2σr2e−z2∕2σz2 are derived. It is also found that for fixed beam parameters the optimum wake is obtained for kpσz=21∕2, for kpσr⩽1. The validity and usefulness of linear-fluid theory is studied using fully nonlinear particle-in-cell simulations. It is found that linear theory can be useful beyond the nominal range of validity for narrow bunches. The limits of usefulness differ significantly between electron and positron bunches. For electron bunches, scaling laws are found for three limits for optimal plasma density (kpσz=21∕2), characterized by the normalized spot size kpσr and the normalized charge per unit length of the beam, Λ≡(nb∕np)kp2σr2. These are ε≡eE∕mcωp=1.3(nb∕np) for kpσr>1 and nb∕np<1, ε=1.3Λln(1∕kpσr), for (Λ∕10)1∕2<kpσr<1 and Λ<1, and ε=1.3Λln([10∕Λ]1∕2), for kpσr<(Λ∕10)1∕2 and Λ<1. Linear theory breaks down for nb∕np≅10. On the other hand, for positron drivers linear-fluid theory breaks down for nb∕np⩾1 independent of spot size.