## 1 Introduction  
Significant progress has been made over the past decade in elucidating the physics of ultra-intense laser pulse interactions with solid targets. This has been motivated by fundamental investigations of plasma generation in extreme conditions and the development of compact laser-driven particle and radiation sources [1; 2; 3]. The case of thin foil (\(\sim\)0.2-20 \(\upmu\)m) targets has received particular attention due to their use in the development of ion acceleration based on sheath field formation [4]. In addition, ultra-thin (\(\sim\)1-200 nm) foils give rise to promising new ion acceleration mechanisms utilising the radiation pressure of the intense laser light, collectively referred to as radiation pressure acceleration (RPA) [5; 6]. Foils that are sufficiently thin enough can undergo relativistic self-induced transparency (RSIT) [7; 8; 9] during the interaction, which can be detrimental to RPA, but can enhance TNSA [10; 11]. It can also result in phenomena such as streaming instabilities [12; 13], relativistic electron jets [14; 15; 16] and diffraction [17; 18; 19], which can be applied to enhance or manipulate the beams of fast electrons and ions produced. In the RPA scheme, electrons at the critical density surface (\(n_{c}=m_{e}\epsilon_{0}\omega_{L}^{2}/e^{2}\); where \(m_{e}\) is the electron mass, \(\epsilon_{0}\) is the vacuum permittivity, \(\omega_{L}\) is the laser angular frequency and \(-e\) is the electron charge), are driven forward into the overdense plasma by the laser pressure, producing a charge separation layer and creating strong electrostatic fields that in turn accelerate the ions. If the target thickness is much greater than the plasma skin depth (\(c/\omega_{p}\), where \(c\) is the speed of light in vacuum and \(\omega_{p}\) is the plasma frequency), then this mode of RPA is known as hole-boring (HB) [20; 21; 22]. The recession velocity of the plasma critical density surface (PCDS) in this mechanism is recognized conventionally as the hole-boring velocity. This velocity is well described analytically in 1D (spatial) interactions, for relativistic laser intensities, as [21; 22]:  
\[\frac{v_{hb}}{c}=\frac{B}{1+B} \tag{1}\]  
where  
\[B=a_{0}\sqrt{\frac{Zm_{e}}{m_{a}n_{e}/n_{c}}} \tag{2}\]  
and where \(a_{0}=\frac{eE}{m_{e}c\omega_{L}}\) is the normalized vector potential, \(n_{e}\) is the electron density, \(Z\) is the ionization state, \(m_{a}=m_{i}+Zm_{e}=Am_{p}+Zm_{e}\approx Am_{p}\), \(A\) the mass number and \(m_{p}\) is the proton mass. The mode of RPA for which the laser pressure is able to push all of the plasma electrons from across the target thickness (in the region of the most intense part of the laser focus) is known as light sail (LS) [23; 24]. This RPA mode is expected to dominate over HB for the case of target thickness on the order of the plasma skin depth (i.e., nanometres for solid density targets). An important parameter for LS is the pulse energy per unit surface, or pulse fluence, \(F\):  
\[F=\int_{0}^{t}I(t^{\prime})dt^{\prime} \tag{3}\]  
The light sail velocity is defined as [23]:  
\[\frac{v_{ls}}{c}=\frac{[1+\epsilon(t)]^{2}-1}{[1+\epsilon(t)]^{2}+1} \tag{4}\]  
where  
\[\epsilon(t)=\frac{2F(t)}{\rho lc^{2}} \tag{5}\]  
and where the target density \(\rho=m_{a}n_{i}\approx Am_{p}n_{i}=\frac{\phi}{Z}m_{p}n_{e}\), for the case of perfect reflection. It follows that \(v_{hb}\) and \(v_{ls}\), and thus the ion kinetic energy, scales favourably with laser intensity. In principle, in the LS scenario, quasi-monoenergetic ion beams with energies in the hundreds-of-MeV to few-GeV per nucleon range should be possible when driving this mechanism with multi-petawatt lasers presently under development [24; 25; 26; 27]. The onset of RSIT, however, reduces the radiation pressure and thus the ion energies achievable. A precise characterization of the onset of RSIT and the resulting changes to the PCDS dynamics is thus essential to gauge the viability of RPA as an efficient acceleration mechanism for a given laser intensity and target thickness.