  The advent of Petawatt-class laser systems allows generating electro-magnetic
fields of unprecedented strength in a controlled environment, driving
increasingly more efforts to probe yet unobserved processes through their
interaction with the quantum vacuum. Still, the lowest intensity scale
governing these effects lies orders of magnitude beyond foreseen capabilities,
so that such endeavor is expected to remain extremely challenging. In recent
years, however, plasma mirrors have emerged as a promising bridge across this
gap, by enabling the conversion of intense infrared laser pulses into
coherently focused Doppler harmonic beams lying in the X-UV range. In this
work, we present quantitative predictions on the quantum vacuum signatures
produced when such beams are focused to intensities between $10^{24}$ and
$10^{28}\ \mathrm{W.cm}^{-2}$ . These signatures, which notably include
photon-photon scattering and electron-positron pair creation, are obtained
using state-of-the-art massively parallel numerical tools. In view of
identifying experimentally favorable configurations, we also consider the
coupling of the focused harmonic beam with an auxiliary optical beam, and
provide comparison with other established schemes. Our results show that a
single coherently focused harmonic beam can produce as much scattered photons
as two infrared pulses in head-on collision, and confirm that the coupling of
the harmonic beam to an auxiliary beam gives rise to significant levels of
inelastic scattering, and hence holds the potential to strongly improve the
attainable signal to noise ratios in experiments.
