We use an Eulerian Vlasov code, which solves the one-dimensional relativistic Vlasov–Maxwell equations for both electrons and ions, to follow in details the evolution of the distribution functions and the mechanism of the formation and evolution of double layers during ion acceleration driven by a high-intensity circularly polarized short laser pulse (12 ω−1 where ω is the laser angular frequency) normally incident on a thin dense foil. We compare three cases with a high-density deuterium plasma target of total thickness 1.767 cω−1 and constant n/n
cr = 100, where n
cr is the critical density, and where the laser intensity is varied from a situation where the target is opaque to the laser pulse (normalized vector potential or quiver momentum a
0 = 80), to a situation where, above a critical laser intensity, a very small fraction of the laser pulse is transmitted through the target (a
0 = 90), and finally to a situation where a more important fraction is transmitted through the target (a
0 = 100). The dynamics of ion and electron acceleration are quite different in the three cases, and are followed in detail by the Eulerian Vlasov code, which allows an accurate representation of the distribution function. In the intermediate case, the Vlasov code has revealed a remarkably well-developed spiral structure in the phase space of the electron distribution function, which is associated with large sawtooth modulations in the electron density profiles.