  Direct laser acceleration of electrons in ion channels is investigated in a
general case when the laser phase velocity is greater than (or equal to) the
speed of light. Using the similarity of the equations of motion for
ultra-relativistic electrons, we develop a universal scaling theory that gives
the maximum possible energy that can be attained by an electron for given laser
and plasma parameters. The theory predicts appearance of forbidden zones in the
phase space of the particle, which manifests itself as an energy gain
threshold. We apply the developed theory to find the conditions needed for an
energy enhancement via a resonant interaction between the third harmonic of
betatron oscillations and the laser wave. The theory is also used to analyze
electron dynamics in a circularly polarized laser.
