The Effects of Dimensionality on Maximum Proton Energy in 1D, 2D, and 3D Particle-in-Cell Simulations

Particle-in-Cell simulations are often used to model laser-plasma interactions. Due to the high cost of running 3D simulations, 1D and 2D simulations are often run instead. It is well-known that the particle energies seen in 1D and 2D simulations of laser-ion acceleration differ from those seen in 3D. We build upon existing work by considering a wider range of laser intensities ranging from 10¹⁷ W/cm² to 10²¹ W/cm². The simulations run in 1D and 2D consistently overestimate the maximum proton energy compared to 3D simulations, but not by a constant factor. We applied known time-dependent models for 1D, 2D, and 3D simulations for maximum proton energy and found that the 2D and 3D models matched the simulation data well. These models can allow researchers to save computational resources by running shorter simulations and extrapolating to later times. Although the 1D, 2D, and 3D simulations have similar qualitative features, more work needs to be done to develop a model that would account for the quantitative differences between their maximum particle energies.