Three-dimensional particle-in-cell simulations are used to demonstrate a lens-like focusing effect of ions accelerated from parabolically shaped laser pulses interacting with planar hydrogen foils. An ultrashort, multipetawatt shaped laser pulse with a peak intensity of 1.7× 1022 W/cm2 accelerates 35% of the target within the focal spot area into a high energy, high density, focused ion beam through radiation-pressure acceleration. At single-petawatt laser powers, with a peak intensity of 6×1020 W/cm2, a pulse with a parabolic pulse front generated from the phase of a stepped concentric echelon is shown to collimate ions accelerated from a planar target. The high-power pulse demonstrates the capability of next-generation laser systems for creating high-energy density focused ions, while the lower power pulse highlights the potential of current petawatt-class lasers for producing collimated and dense ions compared to standard Gaussian pulses. The requirements for creating parabolically shaped laser pulses at the focus and their effects on ion acceleration are explored.
Charbonnet et al. (2026) studied this question.