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Optimized laser ion acceleration at the relativistic critical density surface

In the effort of achieving high-energetic ion beams from the interaction of ultrashort laser pulses with a plasma, volumetric acceleration mechanisms beyond Target Normal Sheath Acceleration have gained attention. A relativisticly intense laser can turn a near critical density plasma slowly transparent, facilitating a synchronized acceleration of ions at the moving relativistic critical density front. While simulations promise extremely high ion energies in in this regime, the challenge resides in the realization of a synchronized movement of the ultra-relativistic laser pulse ($a_0\gtrsim 30$) driven reflective relativistic electron front and the fastest ions, which imposes a narrow parameter range on the laser and plasma parameters. We present an analytic model for the relevant processes, confirmed by a broad parameter simulation study in 1D- and 3D-geometry. By tayloring the pulse length plasma density profile at the front side, we can optimize the proton acceleration performance and extend the regions in parameter space of efficient ion acceleration at the relativistic relativistic density surface.