  We introduce a first full analytical bubble and blow-out model for a radially
inhomogeneous plasma in a quasi-static approximation. For both cases we
calculate the accelerating and the focusing fields. In our model we also assume
a thin electron layer that surrounds the wake field and calculate the field
configuration within. Our theory holds for arbitrary radial density profiles
and reduces to known models in the limit of a homogeneous plasma. From a
previous study of hollow plasma channels with smooth boundaries for
laser-driven electron acceleration in the bubble regime we know that
pancake-like laser pulses lead to highest electron energies [Pukhov et al, PRL
113, 245003 (2014)]. As it was shown, the bubble fields can be adjusted to
balance the laser depletion and dephasing lengths by varying the plasma density
profile inside a deep channel. Now we show why the radial fields in the vacuum
part of a channel become defocussing.
