Beam loading in the bubble regime in plasmas with hollow channels
  Based on the already existing analytical theory of the strongly-nonlinear wakefield (which is called "bubble") in transversely inhomogeneous plasmas, we study particular behavior of non-loaded (empty) bubbles and bubbles with accelerated bunches. We obtain an analytical expression for the shape of a non-loaded bubble in a general case and verify it with particle-in-cell (PIC) simulations. We derive a method of calculation of the acceleration efficiency for arbitrary accelerated bunches. The influence of flat-top electron bunches on the shape of a bubble is studied. It is also shown that it is possible to achieve acceleration in a homogeneous longitudinal electric field by the adjustment of the longitudinal density profile of the accelerated electron bunch. The predictions of the model are verified by 3D PIC simulations and are in a good agreement with them. 
A. A. Golovanov, I. Yu. Kostyukov, J. Thomas, A. Pukhov
physics.plasm-ph