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

TL;DR

This paper investigates high order harmonic generation in vacuum caused by intense electromagnetic fields using the Heisenberg-Euler formalism, deriving expressions for different beam configurations and relating to previous collision studies.

Contribution

It provides new analytical expressions for HOH generation in vacuum for various electromagnetic configurations, including realistic Gaussian beams, within a perturbative quantum electrodynamics framework.

Findings

Optimal geometry identified for maximum HOH at given power

Explicit formulas derived for $4 ext{-} ext{dipole}$ and crossing Gaussian beams

Results connect with earlier plane wave collision studies

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 $4\pi$-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.