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We introduce a first full analytical bubble and blow-out model for a radiallyinhomogeneous plasma in a quasi-static approximation. For both cases wecalculate the accelerating and the focusing fields. In our model we also assumea thin electron layer that surrounds the wake field and calculate the fieldconfiguration within. Our theory holds for arbitrary radial density profilesand reduces to known models in the limit of a homogeneous plasma. From aprevious study of hollow plasma channels with smooth boundaries forlaser-driven electron acceleration in the bubble regime we know thatpancake-like laser pulses lead to highest electron energies [Pukhov et al, PRL113, 245003 (2014)]. As it was shown, the bubble fields can be adjusted tobalance the laser depletion and dephasing lengths by varying the plasma densityprofile inside a deep channel. Now we show why the radial fields in the vacuumpart of a channel become defocussing.

A non-linear theory for the bubble regime of plasma wake fields in tailored plasma channels