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 ($\mathrm{\Psi }=D_{hemi}\mathrm{/}{D}_{Laser}=6.1$). However, larger hemispheres ($\mathrm{\Psi }=14.6$) 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 $9\pm 3\mu$m through a mesh transition analysis.