  We explore a regime of laser-driven plasma acceleration of electrons where
the radial envelope of the laser-pulse incident at the plasma entrance is
strongly mismatched to the nonlinear plasma electron response excited by it.
This regime has been experimentally studied with the gemini laser using f/40
focusing optics in August 2015 and f/20 in 2008. The physical mechanisms and
the scaling laws of electron acceleration achievable in a laser-plasma
accelerator have been studied in the radially matched laser regime and thus are
not accurate in the strongly mismatched regime explored here. In this work, we
show that a novel adjusted-a0 model applicable over a specific range of
densities where the laser enters the state of a strong optical shock, describes
the mismatched regime. Beside several novel aspects of laser-plasma interaction
dynamics relating to an elongating bubble shape and the corresponding
self-injection mechanism, importantly we find that in this strongly mismatched
regime when the laser pulse transforms into an optical shock it is possible to
achieve beam-energies that significantly exceed the incident intensity matched
regime scaling laws.
