Longitudinal field components for laser beams in vacuum

The discovery of Lax, Louisell, and Knight (LLK) [Phys. Rev. 9, 378 (1974)] that electromagnetic beams in vacuum do have a longitudinal component can be proved experimentally from the polarization independence of the energy of electrons from the focus of a laser. For this purpose we had to develop the LLK paraxial approximation to a Maxwellian exact solution for a Gaussian beam. Inserting the exact solutions into the Maxwellian stress tensor expression of the nonlinear force for the electron acceleration demonstrates a polarization dependence if only the transversal optical components are used. Including the exact longitudinal fields results in the experimentally proven polarization independence.