Table 1. Important parameters (first column), in addition to gas density profile, that affect ion acceleration in underdense plasma and the major sub-processes controlled by them (second column).

Parameters	Controlled sub-processes	Our case	Reference10
τpi/τ	Ion dynamics within laser pulse duration	∈[26.74,245.2] » 1	∈[0.95, 5.6] ~ 1
cτ/λp	Electron acceleration mechanisms28 and the subsequent electron energy distributions32,33	∈[0.3, 2.3] ~ 1(cτ ~ 10.5 μm; λp ∈ [4.5, 42] μm)	∈[10.9,64.3] » 1(cτ ~ 180 μm; λp ∈ [2.8, 16.5] μm)
PL/Pc	Self-focusing26,29 and the degree of electron evacuation from the laser drilled plasma channel10,18	∈[0.5, 50]	∈[60, 2080]
dμ/λp	Wake field curvature28,32,41 determining the radial field in the laser wake affecting electron and ion dynamics13,16,18	∈[0.5, 4.4]	∈[0.6, 3.6]

The operating parameter space in our case (third column) is compared with a previous experiment¹⁰ for which ion energy scaling is available (last column).