Focal-shape effects on the efficiency of the tunnel-ionization probe for extreme laser intensities

We examine the effect of laser focusing on the potential of a recently discussed scheme [M.F. Ciappina {\em et al}, Phys. Rev. A 99, 043405 (2019); Las. Phys. Lett. 17, 025301 (2020)] for in situ determination of ultra-high intensities of electromagnetic radiation delivered by multi-petawatt laser facilities. Using two model intensity distributions in the focus of a laser beam, we show how the resulting yields of highly charged ions generated in the process of multiple sequential tunneling of electrons from atoms, depend on the shape of these distributions. Our findings lead to the conclusion that an accurate extraction of the peak laser intensity can be made either in the near-threshold regime when the production of the highest charge state happens only in a small part of the laser focus close to the point where the intensity is maximal, or through the determination of the points where the ion yields of close charges become equal. We show that, for realistic parameters of the gas target, the number of ions generated in the central part of the focus in the threshold regime should be sufficient for a reliable measurement with highly sensitive time-of-flight detectors. Although positions of the intersection points generally depend on the focal shape, they can be used to localize the peak intensity value in a certain interval. Additionally to this analysis, we discuss the method in comparison to other recently proposed approaches for direct measurement of extreme laser intensities.