Strongly driven molecules: probing the tunneling phase in the over-the-barrier regime and prevalence of different double ionization pathways

Using a three-dimensional quasiclassical technique we explore molecular double ionization by a linearly polarized, infrared (800~nm) 27~fs laser pulse. For intensities ranging from the tunneling to the over-the-barrier regime, we identify the double ionization pathways in a unified way as a function of total electron energy. For the tunneling regime, we discuss the differences in the interplay of double ionization (DI) pathways between strongly driven He and strongly driven $N_{2}$. For intermediate intensities in the over-the-barrier regime, we find that both the correlated momenta and the double ionization probability distribution as a function of total energy probe the tunneling phase of the re-colliding electron. This allows for a direct verification of the re-colliding electron tunneling at a large phase of the laser field in the over-the-barrier regime in contrast to a small tunneling phase in the tunneling regime.