Scaling of the longitudinal electric fields and transformer ratio in a non-linear Plasma Wakefield Accelerator

In a world where finances are playing an increasing role in deciding which scientific questions of interest will be addressed, large particle accelerators are in danger of becoming instruments of the past. New methods must be examined to make them smaller and cheaper. We present, in detail, here one such method, known as the Plasma Wakefield Accelerator or PWFA, using the large electric fields plasmas can support to accelerate charge.;We derive the expected longitudinal field as Ez ∼ mcwpe IpIcr , and show that this is consistent with experiment, where Ez ∼ I0.623+/-0.007p . We discuss the effect of erosion of the bunch head, showing that this will have limited effect on the wake magnitude measurement, but will reduce the achievable transformer ratio in the system. We show that, despite erosion, the unloaded transformer ratio must be greater than unity for our system, and that this is consistent with the measurement. We culminate with a measurement of the unloaded transformer ratio for two different plasma lengths, 10 and 85 cm, showing transformer ratios consistently greater than unity and weakly varying with Ip, the first time this has ever been achieved.