  We study the direct acceleration of a free electron in infinite vacuum along
the axis of a pulsed radially-polarized laser beam. We find that net energy
transfer from laser pulse to electron is maximized with the tightest focusing.
We show that the net energy gain of an electron initially moving at a
relativistic velocity may exceed more than half the theoretical limit of energy
transfer, which is not possible with an initially stationary electron in the
parameter space studied. We determine and analyze the power scaling of maximum
energy gain, extending our study to include a relatively unexplored regime of
low powers and revealing that substantial acceleration is already possible
without the use of petawatt peak-power laser technology.
