Vacuum high-harmonic generation and electromagnetic shock

Abstract

When one takes into account the presence of virtual charged states in the quantum vacuum, a nonlinear self-interaction can arise in the propagation of electromagnetic fields. This self-interaction is often referred to as ‘real photon–photon scattering’. When the centre-of-mass energy of colliding photons is much lower than the rest energy of an electron–positron pair, this quantum effect can be included in the classical field equations of motion as a vacuum current and charge density using the Heisenberg–Euler Lagrangian. Using analytical and numerical methods for subcritical fields, the intrinsic solution to Maxwell’s equations has been found for counterpropagating probe and pump plane waves in the presence of vacuum four- and six-wave mixing. In the corresponding all-order solution for the scattered probe, a route to vacuum high-harmonic generation is identified in which a long phase length can compensate for the weakness of interacting fields. The resulting shocks in the probe carrier wave and envelope are studied for different parameter regimes and polarisation set-ups. In this special issue, we study two additional set-ups: that of a slowly varying single-cycle background to highlight the effect of an oscillating background on the probe harmonic spectrum, and that of a few-cycle probe to highlight the smoothing of the harmonic peaks produced by a wider spectrum of probe photons. We also correct sign errors in an earlier publication.