  Direct laser acceleration of electrons is an important energy deposition
mechanism for laser-irradiated plasmas that is particularly effective at
relativistic laser intensities in the presence of quasi-static laser-driven
plasma electric and magnetic fields. These radial electric and azimuthal
magnetic fields provide transverse electron confinement by inducing betatron
oscillations of forward-moving electrons undergoing laser acceleration.
Electrons are said to experience a betatron resonance when the frequency of
betatron oscillations matches the average frequency of the laser field
oscillations at the electron position. In this paper, we show that the
modulation of the laser frequency caused by the betatron oscillation can be
another important mechanism for net energy gain that is qualitatively different
from the betatron resonance. Specifically, we show that the frequency
modulation experienced by the electron can lead to net energy gain in the
regime where the laser field performs three oscillations per betatron
oscillation. There is no net energy gain in this regime without the modulation
because the energy gain is fully compensated by the energy loss. The modulation
slows down the laser oscillation near transverse stopping points, increasing
the time interval during which the electron gains energy and making it possible
to achieve net energy gain.
