Scaling laws for laser-driven ion acceleration from nanometer-scale ultrathin foils

Laser-driven ion acceleration has attracted global interest for its potential towards the development of a new generation of compact, low-cost accelerators. Remarkable advances have been seen in recent years with a substantial proton energy increase in experiments, when nanometer-scale ultrathin foil targets and high-contrast intense lasers are applied. However, the exact acceleration dynamics and particularly the ion energy scaling laws in this novel regime are complex and still unclear. Here, we derive a scaling law for the attainable maximum ion energy from such laser-irradiated nanometer-scale foils based on analytical theory and multidimensional particle-in-cell simulations, and further show that this scaling law can be used to accurately describe experimental data over a large range of laser and target parameters on different facilities. This provides crucial references for parameter design and experimentation of the future laser devices towards various potential applications.