Linearly polarized laser beam with generalized boundary condition and non-paraxial corrections.

Linearly polarized Gaussian beams, under the slowly varying envelope approximation, tightly focused by a perfect parabola modeled with the integral formalism of Ignatovsky are found to be well approximated with a generalized Lax series expansion beyond the paraxial approximation. This allows obtaining simple analytic formulas of the electromagnetic field in both the direct and momentum spaces. It significantly reduces computing time, especially when dealing with the problem of simulating direct laser acceleration. The series expansion formulation depends on integration constants that are linked to boundary conditions. They are found to depend significantly on the region of space over which the integral formulation is fit. Consequently, the net acceleration of electrons initially at rest is extremely sensitive to the chosen set of initial parameters due to the extreme focusing investigated here. This suggests avoiding too tight focusing schemes in order to obtain reliable predictions when the process of interest is sensitive mainly to the field and not the intensity.