Beyond the ponderomotive limit: direct laser acceleration of   relativistic electrons in sub-critical plasmas
  We examine a regime in which a linearly-polarized laser pulse with relativistic intensity irradiates a sub-critical plasma for much longer than the characteristic electron response time. A steady-state channel is formed in the plasma in this case with quasi-static transverse and longitudinal electric fields. These relatively weak fields significantly alter the electron dynamics. The longitudinal electric field reduces the longitudinal dephasing between the electron and the wave, leading to an enhancement of the electron energy gain from the pulse. The energy gain in this regime is ultimately limited by the superluminosity of the wave fronts induced by the plasma in the channel. The transverse electric field alters the oscillations of the transverse electron velocity, allowing it to remain anti-parallel to laser electric field and leading to a significant energy gain. The energy enhancement is accompanied by development of significant oscillations perpendicular to the plane of the driven motion, making trajectories of energetic electrons three-dimensional. Proper electron injection into the laser beam can further boost the electron energy gain. 
A. V. Arefiev and V. N. Khudik and A. P. L. Robinson and G. Shvets and   L. Willingale and M. Schollmeier
physics.plasm-ph