  In the study of laser-driven electron acceleration, it has become customary
to work within the framework of paraxial wave optics. Using an exact solution
to the Helmholtz equation as well as its paraxial counterpart, we perform
numerical simulations of electron acceleration with a high-power TM01 beam. For
beam waist sizes at which the paraxial approximation was previously recognized
valid, we highlight significant differences in the angular divergence and
energy distribution of the electron bunches produced by the exact and the
paraxial solutions. Our results demonstrate that extra care has to be taken
when working under the paraxial approximation in the context of electron
acceleration with radially polarized laser beams.
