  Enhancement of the trapping and optimization of the beam quality are two key
issues of Laser Wake Field Acceleration (LWFA). The influence of stochastic
acceleration on the trapping of electrons is compared to the one of cold
injection. It is shown that when considering a high intensity wave perturbed by
a low intensity counter-propagating wave, in the non-linear blowout regime, the
influence of the colliding pulses polarizations (either parallel linear or
positive circular) on the beam quality seems weak when the electron density is
below $\sim 10^{-3}$ critical density. The effect of a homogenous constant
magnetic field $B_0$, parallel to the direction of propagation of the pump
pulse, is studied in the blowout regime. Transverse currents are generated at
the rim of the bubble, which results in the amplification of the $B_0$ field at
the rear of the bubble. Without $B_0$ field the beam periodically explodes and
re-confines, this phenomenon is suppressed when $B_0$ reaches some threshold,
which is a function of the laser pulses parameters (intensity, waist,
duration). Therefore the dynamics of the beam is modified, its maximum energy
is slightly boosted and above all transverse emittance reduced. Moreover the
low energy tail, observed in the non magnetized case, can be completely
suppressed leading to very sharp mono-energetic beam when $B_0$ is applied. If
the available $B_0$ field is limited then one has to fine-tune the
spatio-temporal shape and intensity of the colliding pulse in order to get an
acute control on the beam quality.
