Direct Laser Ion Acceleration and Above-Threshold Ionization at Intensities from $10^{21}$ W/cm$^{2}$ to $3 \times 10^{23}$ W/cm$^{2}$

Calculations on the dynamics of ions and electrons in near-infrared laser fields at intensities up to $3 \times 10^{23}$ W/cm$^2$ are presented. We explore the acceleration of ions in a laser focus by conservation of canonical momentum during ionization events and by the ponderomotive force in the f/1 focal geometry required to reach such intensity. At intensity exceeding 10$^{23}$ W/cm$^2$, highly charged ions are expelled from the laser focus before they can interact with the laser pulse at peak intensity, decreasing the predicted ionization yields of deeply-bound states. We consider the interaction of a tightly-focused, f/1 laser pulse with krypton at an intensity of $3 \times 10^{23}$ W/cm$^{2}$ and a pulse duration of 140 fs. We find that the ions and electrons are accelerated to energies in excess of 2 MeV/nucleon and 1.4 GeV, respectively. Ponderomotive expulsion of the parent ions decreases the total number of ultra-relativistic ATI electrons produced by tunneling ionization from the K-shell states of krypton but does not change their energy spectrum.