## 5 Discussion  
The dynamics of the PCDS in ultra-intense laser interactions with thin foils, ranging in thickness from tens to hundreds of nanometres, has been investigated experimentally and numerically. The maximum PCDS velocity has been determined experimentally via measurements of the relativistic Doppler shift of second harmonic light produced at the laser focus. Coupled 2D PIC simulations are used to explore the spatial and temporal evolution of the PCDS for each target thickness investigated. Comparison of the experimental and numerical results with corresponding analytical models of the HB and LS modes of RPA reveal the target thickness ranges for which each mode dominates. The numerical results show that a hybrid model, starting with HB and transitioning to LS (when the compressed electron layer reaches the rear surface of the target) is required for thicknesses between the extreme cases. Measurements of the transmitted laser light reveal that at the thinnest targets considered, transparency reduces the radiation pressure, resulting in a lower maximum PCDS velocity.  
These results confirm the picture emerging of ion acceleration driven by radiation pressure in thin foil targets. They demonstrate that the PCDS velocity, and thus the maximum ion energy, driven by HB is comparably smaller than that produced by LS in thinner foils. The maximum ion energy driven by LS can be expected to increase with decreasing foil thickness, until the thickness at which the target expands to become relativistically transparent early in the interaction with the laser pulse. The overall maximum ion energy achievable via RPA for a given set of laser parameters is thus effectively set by the condition for the onset of transparency. In principle, compared to linear polarization, the use  
Figure 6: Comparison of experiment and simulation results, and HB and LS model calculations, for the maximum PCDS velocity (on a logarithmic scale) as a function of target thickness. Three distinct regimes: HB-dominant; LS-dominant; and RSIT-onset; are identified, as delineated by the background shading.  
of circularly polarized light reduces heating of the plasma electrons, and thus the threshold foil thickness for the onset of RSIT is reduced, increasing the PCDS velocity and ion energy. However, we note that the strong target deformation induced by the variation in radiation pressure across the laser spatial-intensity profile (as observed in Figure 3) results in significant target heating despite the application of circular polarization. Circumventing this issue to achieve higher RPA-ion energies may require engineering of the laser focus intensity distribution or the target profile to reduce the heating and thus extend the target thickness range for which LS dominates.  
We thank the Gemini team at the Central Laser Facility of the Rutherford Appleton Laboratory for their support during the experiment. We acknowledge the use of the ARCHIE-WeST and ARCHER high performance computers. This work is supported by EPSRC (grants: EP/R006202/1, EP/M018091/1, EP/K022415/1, EP/P007082/1 and EP/L000237/1), STFC (grant number ST/K502340/1) and the European Union Horizon 2020 research and innovation program (grant agreement No 654148 Laserlab-Europe). EPOCH was developed under EPSRC grant EP/G054940/1. Data associated with research published in this paper can be accessed at: [[http://dx.doi.org/10.15129/c5490074-6](http://dx.doi.org/10.15129/c5490074-6) faa-4487-a209-b6a75c2 bf3 bd]([http://dx.doi.org/10.15129/c5490074-6](http://dx.doi.org/10.15129/c5490074-6) faa-4487-a209-b6a75c2 bf3 bd)  
[PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON] performed the experiment, with guidance from [PERSON] and [PERSON], and under the supervision of [PERSON]; [PERSON] analysed the data; [PERSON], [PERSON] and [PERSON] performed the simulations, modelling and analysed the data; [PERSON] and [PERSON] wrote the paper.  
The authors declare no conflict of interest.