  Our goal is to study optical signatures of quantum vacuum nonlinearities in
strong macroscopic electromagnetic fields provided by high-intensity laser
beams. The vacuum emission scheme is perfectly suited for this task as it
naturally distinguishes between incident laser beams, described as classical
electromagnetic fields driving the effect, and emitted signal photons encoding
the signature of quantum vacuum nonlinearity. Using the Heisenberg-Euler
effective action, our approach allows for a reliable study of photonic
signatures of QED vacuum nonlinearity in the parameter regimes accessible by
all-optical high-intensity laser experiments. To this end, we employ an
efficient, flexible numerical algorithm, which allows for a detailed study of
the signal photons emerging in the collision of focused paraxial high-intensity
laser pulses. Due to the high accuracy of our numerical solutions we predict
the total number of signal photons, but also have full access to the signal
photons' characteristics, including their spectrum, propagation directions and
polarizations. We discuss setups offering an excellent background-to-noise
ratio, thus providing an important step towards the experimental verification
of quantum vacuum nonlinearities.
