  Staging laser wake-field acceleration is considered as a necessary technique
for developing full-optical jitter-free electron accelerators. Splitting of the
acceleration length into several technical parts with their lengths smaller
than the dephasing length and with independent laser drivers allows generation
of stable, reproducible acceleration fields. Temporal and spatial coupling of
pre-accelerated electron bunches for their injection in the acceleration phase
of a successive laser pulse wake field is the key part of the staging
laser-driven acceleration. Here, characterization of the coupling is performed
with dense, stable, a narrow energy band <3% and energy selectable electron
beams with charges ~1.6 pC and energy ~10 MeV generated from a laser plasma
cathode. Cumulative focusing of electron bunches in a low density pre-plasma,
exhibiting the Budker- Bennett effect, is shown to result in the efficient
injection of electrons even with a long distance between the injector and the
booster in the laser pulse wake. Measured characteristics of electron beams
modified by the booster wake field agree well with those obtained by
multidimensional particle-in-cell simulations.
