A theoretical model for laser-accelerated ion focusing from spherically curved targets using a Gaussian self-similar solution is presented. This model describes the evolution of the focal location and focal radius with final ion energy and the initial radial boundary of the ions, the latter being a function of both target geometry and the ion acceleration radius for a given ion energy. The theory is supported by particle-in-cell simulations of a variety of target shapes with varying radii of curvature and target opening angles, as well as variations in the injected electron beam radius and energy spectrum. The theory and simulations suggest that the focal location varies linearly with the radius of curvature, with the square root of the ratio of ion energy to effective electron temperature, and monotonically increases with the energy-dependent ratio of the initial ion radial boundary to the radius of curvature. Considering ponderomotive acceleration, this suggests that the focal length should scale inversely with intensity, (ILλL2)−1/4, suggesting a 10× increase in intensity will reduce the focal distance by ̃1.8× for a given ion energy.
Higginson et al. (2026) studied this question.