Geometric scaling of laser-driven proton focusing from hemispherical foils

We systematically characterize the focusing behavior of laser-driven proton beams from hemispherical targets of various diameters using mesh radiography. The proton focal location is inferred to be near the geometrical center for the smallest tested hemisphere ($\Psi=D_{hemi}/D_{Laser}=6.1$). However, larger hemispheres ($\Psi=14.6$) degrade the focusing behavior and behave more like flat foils with focal location significantly inside the hemisphere. We also infer a tight virtual focus of $9\pm3~\mu$m through a mesh transition analysis.