Direct electron acceleration with tightly focused TM 0,1 beams: boundary conditions and non-paraxial corrections

Non-paraxial corrections for a scalar optical field that follows the Helmotz equation are extracted for the first time, to the best of our knowledge, in the angular spectrum representation by taking into account generic boundary conditions. Those integration constants are compared with closed-form solutions and approximate series expansions usually obtained by other authors. This method is particularized to the direct electron acceleration with a tightly focused TM0,1 laser beam to demonstrate that these constants have a strong effect on the final average energy and quality of the electron beam.