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In a recent paper, we have shown that the QED nonlinear corrections imply aphase correction to the linear evolution of crossing electromagnetic waves invacuum. Here, we provide a more complete analysis, including a full numericalsolution of the QED nonlinear wave equations for short-distance propagation ina symmetric configuration. The excellent agreement of such a solution with theresult that we obtain using our perturbatively-motivated Variational Approachis then used to justify an analytical approximation that can be applied in amore general case. This allows us to find the most promising configuration forthe search of photon-photon scattering in optics experiments. In particular, weshow that our previous requirement of phase coherence between the two crossingbeams can be released. We then propose a very simple experiment that can beperformed at future exawatt laser facilities, such as ELI, by bombarding a lowpower laser beam with the exawatt bump.

Detecting photon-photon scattering in vacuum at exawatt lasers