Study of nonlinear theory refinements for small plasma bubbles

Beam-driven plasma wakefield accelerators (PWFAs) have demonstrated exceptionally high acceleration gradients, reaching GV/m in the nonlinear blowout regime. While the well-established nonlinear theory provides accurate approximations for bubble structure, wake potential, and the longitudinal electric field near the bubble center, it is primarily suited for large plasma bubbles. However, this approximation becomes less accurate for small-radius bubbles (k_pR_b<4, where k_p is the plasma wavenumber and R_b is the maximum plasma bubble radius). This proceeding presents a modified model tailored for small-radius plasma bubbles, informed by simulation studies. The proposed corrections are compared with Particle-In-Cell (PIC) simulations, focusing on bubbles with k_pR_b<4. The refined model is further applied to calculate witness beam profiles that enabling complete beam loading in small bubbles.