Tunnel Ionization in Tightly Focused Laser Fields at Intensity up to 3 x 10$^{\mathrm{23\thinspace }}$W/cm$^{\mathrm{2}}$*

Experimental tests of the Ammosov-Delone-Krainov (ADK) tunnel ionization model above 10$^{\mathrm{20}}$ W/cm$^{\mathrm{2}}$ should provide insight into relativistic effects in the tunneling process. We present simulations of the ion yields, ion dynamics, and electron dynamics in near-infrared laser fields with intensities ranging from 10$^{\mathrm{20\thinspace }}$W/cm$^{\mathrm{2}}$ to 3 x 10$^{\mathrm{23}}$ W/cm$^{\mathrm{2}}$ and how our results will influence the design of future experiments. We included the effects of the $f$/1 focal geometry required to reach 3 x 10$^{\mathrm{23}}$ W/cm$^{\mathrm{2\thinspace }}$in the near future, incorporating nonparaxial corrections to the laser fields up to fifth order in the diffraction angle. Simulations of the ion energy gained from the ion-laser interaction demonstrate the need to develop new ionization yield measurement techniques, as the Wiley-McLaren time-of-flight methods used previously have insufficient energy resolution when laser intensity exceeds 10$^{\mathrm{21}}$ W/cm$^{\mathrm{2}}$. When considering the ionization of Kr$^{\mathrm{35+}}$ at 3 x 10$^{\mathrm{23}}$ W/cm$^{\mathrm{2}}$, we find that the ponderomotive expulsion of ions from the laser focus will decrease the ionization yield. The highly charged krypton ions and their above-threshold ionization electrons can be accelerated to energies above 2 MeV/nucleon and 1.4 GeV, respectively.