Generation of High Order Harmonics in Vacuum for Various Configurations of Interacting Electromagnetic Field

arXiv:2508.09214v1 Announce Type: new
Abstract: High order harmonic (HOH) generation by interacting extremely intense electromagnetic waves in the quantum vacuum is investigated within the framework of the Heisenberg-Euler formalism. We consider here the process in the lowest order of a perturbation theory relative to the electromagnetic (EM) beam intensity, giving contribution to the HOH generation. The main expressions are obtained for a general geometry, whyle polarizations of different sub-beams forming the EM beam focus are almost the same. Nevertheless, explicit expressions for the HOH generation are derived for the $4pi$-dipole in-coming waves and for the two crossing Gaussian beams. The former geometry of the EM beam is optimal at a given EM wave power, whereas the latter one is more realistic from the experimental point of view. We consider also a relationship of our present general results with the results, obtained earlier for the HOH generation during of collision of two plane electromagnetic waves.