  Driving laser wakefield acceleration with extremely short, near single-cycle
laser pulses is crucial to the realisation of an electron source that can
operate at kHz-repetition rate while relying on modest laser energy. It is also
interesting from a fundamental point of view, as the ponderomotive
approximation is no longer valid for such short pulses. Through
particle-in-cell simulations, we show how the plasma response becomes
asymmetric in the plane of laser polarization, and dependent on the
carrier-envelope phase (CEP) of the laser pulse. For the case of
self-injection, this in turn strongly affects the initial conditions of
injected electrons, causing collective betatron oscillations of the electron
beam. As a result, the electron beam pointing, electron energy spectrum and the
direction of emitted betatron radiation become CEP-dependent. For injection in
a density gradient the effect on beam pointing is reduced and the electron
energy spectrum is CEP-independent, as electron injection is mostly
longitudinal and mainly determined by the density gradient. Our results
highlight the importance of controlling the CEP in this regime for producing
stable and reproducible relativistic electron beams and identify how CEP
effects may be observed in experiments. In the future, CEP control may become
an additional tool to control the energy spectrum or pointing of the
accelerated electron beam.
