we take the injected beam to have σ$_{x}$ ≃ 77nm( n$_{b}$/n$_{0}$ ≃ 28), corresponding to the average β encountered in the plasma. These simulations were carried out using the 2D electro- magnetic PIC code OOPIC Pro [9], with the predictions for the longitudinal plasma wake E$_{z}$ ( r, ξ ) shown in Fig. 1. This scenario produces unprecedented large fields, in excess of 1.3 TV/m. Based on such accelerating gradients, one may envision a key component of a TeV-class collider existing on a table-top. Further, this field amplitude is remarkably large even by internal atomic physics standards. Indeed, the collective field of such beams can readily ionize matter, allowing both plasma formation, and a unique probe of atomic physics in an extreme field limit. The size of E$_{z}$ is not surprising, however, as the transverse vacuum E -field at beam edge for σ$_{x}$ = 77 nm can be estimated as E$_{r,max}$ ≃ eN$_{b}$/ 2(4 π ) 1 / $^{2}$ϵ$_{0}$ σ$_{x}$σ$_{z}$ = 1 . 5TV/m, or approximately that of the induced longitudinal plasma wakefield field. With this scenario, we should restrict our interaction to ∼ 1mm, giving a maximum incurred energy loss of < 1 GeV; in this case chromatic effects in transport do not cause beam loss in the experimentally sensitive region downstream of the LCLS undulator. 

We now examine the focusing needed to yield sufficiently small σ$_{x}$$_{(}$$_{y}$$_{)}$ that produces con- ditions for plasma creation and efficient wake formation. Further, to implement this exper- iment in the spatially-limited region downstream of the LCLS undulator, we should make the focusing system compact (1.2 m in length) and easily implemented. As such, we have designed an ultra-high field gradient, permanent magnet quadrupole (PMQ) based focusing system similar to that developed for previous UCLA/LLNL experiments on inverse Comp- ton scattering [10]. This short focal length system, which is based on quadrupole gradients of over 700 T/m, has the further advantage of evading performance restrictions due to chro- matic aberrations. It satisfies space restrictions, and is simple and light-weight, giving the possibility of beamline insertion with minimum downtime. A collimating focusing system downstream of the spot waist, having reflection symmetry with the final focus system can be employed to capture and transport the beam exiting the plasma region. The final fo- cus system design parameters are: beam energy 14.3 GeV, rms relative momentum spread σ$_{δp/p}$ = 3 × 10 − $^{5}$, initial β -function of 400 m, and ϵ$_{n}$ = 4 × 10 − $^{7}$m-rad. With the PMQs arranged in a triplet configuration of 12.5, 25, and 25 cm magnet lengths, ELEGANT sim- ulations indicate achievable beam sizes of σ$_{x}$ ( y ) = 130nm. This value is well suited for injection into the plasma, with a final β = 1 . 2mm, a bit longer than the proposed plasma length. 

