The generation of very high quality electron bunches (high brightness and low energy spread) from a plasma-based accelerator in the three-dimensional blowout regime using self-injection in tailored plasma density profiles is analyzed theoretically and with particle-in-cell simulations. The underlying physical mechanism that leads to the generation of high quality electrons is uncovered by tracking the trajectories of the electrons as they cross the sheath and are trapped by the wake. Details on how the intensity of the driver and the density scale-length of the plasma control the ultimate beam quality are described. Three-dimensional particle-in-cell simulations indicate that this concept has the potential to produce beams with peak brightnesses between $10^{20}$ and $10^{21}$ $mathrm{A}/mathrm{m}^2/mathrm{rad}^2$and with absolute projected energy spreads of $sim 0.3~mathrm{MeV}$ using existing lasers or electron beams to drive nonlinear wakefields.
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