  In the collision of a high-energy proton beam and a strong laser field,
merging of the laser photons can occur due to the polarization of vacuum. The
probability of photon merging is calculated by accounting exactly for the laser
field and presents a highly non-perturbative dependence on the laser intensity
and frequency. It is shown that the non-perturbative vacuum-polarization
effects can be experimentally measured by combining the next-generation of
table-top petawatt lasers with presently available proton accelerators.
