to the laser intensity $I_{\text{cr}}=4.6\times 10^{29}\;\text{W/cm$^2$}$ [@Di_Piazza_2012]. Relativistic quantum effects also allow for the nonlinear interaction of a photon with a laser field, as in the case of electron-positron pair photo-production (nonlinear Breit-Wheeler pair production (NBWPP)) [@Ritus_1985; @Heinzl_2010; @Titov_2012; @Krajewska_2013]. This process, as well as any QED process occurring in the collision of a photon with a strong laser field[^1], is essentially controlled by the two Lorentz- and gauge-invariant parameters $\xi=E_0/E_{\text{rel}}$ and $\varkappa=[(k_0k)/m\omega_0]E_0/E_{\text{cr}}$. Here, $(k_0k)=\omega_0\omega-\bm{k}_0\cdot\bm{k}$, with $k_0^{\mu}=(\omega_0,\bm{k}_0)$ and $k^{\mu}=(\omega,\bm{k})$ being the four-momentum of the laser photons and of the incoming photon, respectively. It is worth observing that in the so-called “ultra-relativistic” limit $\xi\to \infty$, the net number of laser photons absorbed in NBWPP is very large and of the order of $\xi^3$ [@Ritus_1985]. Since presently available optical lasers allow for values of $\xi$ of the order of $10^2$ [@Yanovsky_2008], unprecedented degrees of nonlinearity of the order of one million are in principle achievable.

Refractive QED processes in a strong laser field involve only initial and final photons, and the background field [@Dittrich_b_2000]. Such processes of genuinely quantum nature are a unique tool for testing the predictions of strong-field QED on the nonlinear evolution of the electromagnetic field in vacuum. Vacuum 