We first simulate the interaction of two focused counter-propagating pump pulses and we calculate the vaccum refractive index induced by the interference of the two pump pulses, using formula derived from Euler-Heisenberg Lagrangian, given in our article.

We then calculate the corresponding integrated phase induced by the vacuum index gradient, assuming a probe pulse propagating in phase and parallel to the pump pulse and transversally shifted (at focus) by *w _{0 }*/2 = 5 μm. Result is presented in Figure 1. This phase corresponds to an average refraction angle of the order of 0.2 picoradian.

The Sagnac interferometer is then simulated assuming an extinction factor *E*=10^{-5}, corresponding to the extinction measured with our current interferometer prototype. The extinction factor is defined as the ratio of the intensity at the dark output to the intensity at the entrance of the interferometer. The dimensions of the Sagnac interferometer are 20×10 cm2. The PSD photodiode in the dark output is located 10 cm away from the beam splitter, or 20 cm from the focus. The diameter of the beam on the PSD is then 4 mm (FWHM).

The transverse displacement *δx* of the intensity profile on the PSD photodiode, calculated from the simulation, is *δx* = 0.03 nm.

As validated by the simulations, the shift *δx* varies as the inverse of *w _{0}^{3}* and as the inverse of the extinction factor

*Figure 1: integrated phase induced by the interference of the two counter-propagating pump pulses, delivered by LASERIX. A schematic view of the refracted plane waves of the probe pulse is superposed (white lines). The probe pulse is shifted transversally to introduce a transversally asymmetric phase.*

*Figure 2: Transversal shift on the PSD photodiode as a function of the *

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