Ions can be effectively accelerated during the interaction of an ultra-intense ultra-short laser pulse irradiating a thin solid target via the so-called target normal sheath acceleration (TNSA) mechanism. One of the pivotal questions at this stage of the research is how to predict the properties of the accelerated ions, both from a fundamental point of view and in the light of foreseen applications. In this context, it is desirable to have a simple but reliable description to be used to extrapolate current results to future regimes, which will be made available in the near future, thanks to developments in laser technology. In this paper, the possible approaches for an analytical description of TNSA are discussed, and a theoretical TNSA model is developed. This model is then used to investigate the maximum ion energy as a function of laser parameters. Detailed comparisons with available experimental data and scaling laws are presented. In particular, the relative role played by both the laser pulse energy and irradiance in determining the ion features is investigated.
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