What is the exact mechanism ruling the predicted nonlinear optical process in vacuum at super high peak field? What exactly are the experimental proofs?
I think you are talking about ripping apart the vacuum? This is known as the Schwinger limit (after Julian Schwinger 's paper 1951 http://dx.doi.org/10.1103%2FPhysRev.82.664 ) This is essentially quantum electrodynamics, and such an experiment would test the fundamental validity of the theory. Plans are underway to build a laser capable of exceeding this limit, which is 1.3E18 V/m electric field, this field is equivalent to the rest mass energy of the electron over a Compton wavelength, or equivalently 5E29 W/cm^2 laser intensity, but effects should start around 1E27 W/cm^2, and vacuum birefringence becomes measurable at 1E22 W/cm^2 (inelastic photon scattering has been measured at SLAC by C. Bamber et al. PRD 60 092204 (1999)) The laser is part of the European ELI consortium, ( http://www.extreme-light-infrastructure.eu/ ), which seeks to build 4 large laser facilities in Europe for different physics reasons. The facility to test QED will be the ELI-Ultra High Field laser to be determined this year. QED theory predicts (unlike classical Maxwell equations) that photons will interact elastically with each other (photon-photon scattering), which creates virtual electron-positron pairs (Feynman diagrams describe this). ELI will still likely have to use tricks to reach the Schwinger limit like attosecond pulse generation, but can reach the 1E22 W/cm^2 turn on. This is explored in section B.VI.1 of the ELI science document. ( http://www.extreme-light-infrastructure.eu/pictures/ELI-scientific-case-id17.pdf ) These experiments could resolve the issue of dark matter and dark energy, or lead to new physics beyond the standard model by measuring the expected yield of particles and anti-particles from the focal spot particle showers.
References:
S.S. Bulanov et al PRL 105 220407 (2010)http://prl.aps.org/pdf/PRL/v105/i22/e220407
ELI Document : http://www.extreme-light-infrastructure.eu/pictures/ELI-scientific-case-id17.pdf
J. Schwinger PR 82 664 (1951) http://dx.doi.org/10.1103%2FPhysRev.82.664
C. Bamber et al., "Studies of nonlinear QED in collisions of 46.6 GeV electrons with intense laser pulses", Phys. Rev. D, 60 (1999) 092204. doi:10.1103/PhysRevD.60.092004
Marco, Kirk Flippo provided a pretty comprehensive answer as far as the state experimental tools necessary to produce NLO effects in vacuum. Let me add a few things. To make it a bit more clear -- when we talk about NLO in vacuum, a commonly expected effect would be photon-photon scattering, and the process that would facilitate it is the electron-positron pair formation. The critical scale of the required field is provided by Kirk ; it would be great to have a laser to go to that limit and beyond, but we are still way below it (at this point -- by about 8 orders of magnitude). There were a few proposals to see some effects (e.g. high harmonics generation) using avalable (or soon available) lasers. One of the proposed approaches was to use a (quasi)dc magnetic field and look for expected 2-nd harmonics by using filed-gradient induced effect. Here is a pointer to our paper on the subject (PRA, 62, 043805, 2000):