Maxwell-consistent, symmetry- and energy-preserving solutions for ultrashort-laser-pulse propagation beyond the paraxial approximation

We analytically and numerically investigate the propagation of ultrashort tightly focused laser pulses in vacuum, with particular emphasis on Hermite-Gaussian and Laguerre-Gaussian modes. We revisit the Lax series approach for forward-propagating linearly polarized laser pulses, to obtain Maxwell-consistent and symmetry-preserving analytical solutions for the propagation of all field components beyond the paraxial approximation in four-dimensional geometry (space and time). We demonstrate that our solution conserves the energy, which is set by the paraxial-level term of the series. The full solution of the wave equation towards which our series converges is calculated in the Fourier space. Three-dimensional numerical simulations of ultrashort tightly focused pulses validate our analytical development