(a) and (b): Normalized plasma electron density (a) and return current density (b) from a quasi-static PIC simulation. (c): Azimuthal magnetic field . (d): Normalized radial profiles of (dashed line) and return current (solid line) for fixed ξ. The driving beam (not shown) propagates from right to left.
(a): Plasma electron density and (b) return current density calculated from the analytic model. The following parameters were used: , , , . The shape of the plasma bubble is given in the Appendix.
Trajectories of plasma electrons with (solid blue and red lines) skirting the outer edge of a non-evolving plasma bubble with different thicknesses of the return current layer. Solid black line: bubble's radius . Solid blue line: (artificially over-shielded magnetic field), solid red line: (realistic shielding). Remaining parameters: , , and with . The electron-bubble interaction length is shorter in the over-shielded case than in the realistic one where the electron's trajectory follows along the edge of the bubble (see insets).
Maximum energy gain for a non-evolving bubble as a function of the normalized bubble size rm ≡ kp r bubble for an idealized model of the bubble (no electric or magnetic fields outside the bubble: blue line) and the more realistic model with finite return width of the current layer (red solid line) described in this paper. Ideal bubble: ; realistic bubble: . In both models, the bubble is assumed to be moving with .
Electron trapping by a bubble growing due to a density downramp. (a) Variation of the background plasma density along the propagation distance z. The downramp here is very gradual, i.e., . (b) Electron relativistic factor (blue) and moving-frame Hamiltonian (red). The electron becomes trapped once . (c) Initial (solid black) and final (dashed black) size of the bubble and trajectory of trapped electron (red). Here, , and . The impact radius of the trapped electron is .
Final moving-frame Hamiltonian of electrons versus impact radius from (a) the model and (b) WAKE. Here, , and . Electrons with are trapped.
Comparison of bubble expansion rates required for trapping in the simplified spherical bubble model (blue) and the more realistic model described in this paper (red). While the idealized spherical bubble model overestimates the expansion rate required for trapping, the new model agrees closely with PIC simulation (black markers).
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