Laser-driven ions have compelling properties and their potential use for medical applications has attracted a huge global interest. One of the major challenges of these applications is generating beams of the required energies. To date, there has been no systematic study of the effect of laser intensity on the generation of laser-driven ions from ultrathin foils during relativistic transparency. Here we present a scaling for ion energies with respect to the on-target laser intensity and in considering target thickness we find an optimum thickness closely related to the experimentally observed relativistic transparency. A steep linear scaling with the normalized laser amplitude a0 has been measured and verified with PIC simulations. In contrast to TNSA, this scaling is much steeper and has been measured for ions with Z > 1. Following our results, ion energies exceeding 100MeV/amu are already accessible with currently available laser systems enabling realization of numerous advanced applications.For more than a decade, intense short pulse lasers have been used to drive energetic ion beams [1][2][3][4][5][6][7]. These laser-driven ion beams have high particle numbers, energies up to several tens of MeV/amu [8] and a low transverse emittance [9] and a promise to be a competitive alternative to conventional radio frequency accelerators. The range of applications covers medicine with hadron cancer therapy [10], threat reduction (e.g. detection of fissile material) [11,12] or energy generation with concepts in ion fast ignition [13][14][15]. The major drawback so far is that ion energies are typically too low for any of these applications. For the hadron cancer therapy, protons of 250MeV or carbon C 6+ ions of 4-5 GeV are needed. In the regime of the Target Normal Sheath Acceleration (TNSA) [1-3] protons have been accelerated to 67MeV [16] with laser intensities exceeding 10 20 W/cm 2 . Heavier ions have merely reached several MeV/amu due to protons shielding the accelerating fields in this mechanism. Even with target cleaning, energies have not passed 10 MeV\amu [6]. Recent advances in laser intensities and contrast enabled exploration of new acceleration mechanisms such as the radiation pressure acceleration (RPA) [17][18][19][20][21] or the Break-Out Afterburner (BOA) [22][23][24][25] mechanism, which reach higher ion energies for both protons and heavier ions.Scaling laws for ion energies are a fundamental requirement for the design of future laser systems and the realization of advanced applications. So far, scaling laws have been discussed for the TNSA mechanism [5,7] and also for RPA dominated acceleration [18,27,41]. Here, we extend this research framework by presenting experimental data and theoretical analysis for scaling of ion energies in the relativistic transparent regime, where the BOA mechanism is operative. In a series of experiments we investigated how maximum ion energies scale for the BOA. Ion energies have been measured for intensities of 8 × 10 19 , 2 × 10 20 and 1.7 × 10 21 W/cm 2 at the Los Alamo...
