When two electromagnetic fields counterpropagate, they are modified due to mutual interaction via the polarised virtual electron-positron states of the vacuum. By studying how photon-photon scattering effects such as birefringence and four-wave mixing evolve as the fields pass through one another, we find a significant increase during overlap when both electromagnetic variants can be non-zero. The results have particular relevance for calculations based on a constant field background.

Electrodynamics becomes nonlinear and permits the self-interaction of fields when the quantised nature of vacuum states is taken into account. The effect on a plane probe pulse propagating through a stronger constant crossed background is calculated using numerical simulation and by analytically solving the corresponding wave equation. The electromagnetic shock resulting from vacuum high harmonic generation is investigated and a nonlinear shock parameter identified.

We report on experiments irradiating isolated plastic spheres with a peak laser intensity of 2-3×10^{20}Wcm^{-2}. With a laser focal spot size of 10 μm full width half maximum (FWHM) the sphere diameter was varied between 520 nm and 19.3 μm. Maximum proton energies of ∼25 MeV are achieved for targets matching the focal spot size of 10 μm in diameter or being slightly smaller. For smaller spheres the kinetic energy distributions of protons become nonmonotonic, indicating a change in the accelerating mechanism from ambipolar expansion towards a regime dominated by effects caused by Coulomb repulsion of ions. The energy conversion efficiency from laser energy to proton kinetic energy is optimized when the target diameter matches the laser focal spot size with efficiencies reaching the percent level. The change of proton acceleration efficiency with target size can be attributed to the reduced cross-sectional overlap of subfocus targets with the laser. Reported experimental observations are in line with 3D3V particle in cell simulations. They make use of well-defined targets and point out pathways for future applications and experiments.

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 fourand 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.

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