Generation and control of Doppler harmonics approaching $10^{22}\textrm{W/cm}^2$ on plasma mirrors

In this letter, we report Doppler harmonic generation with a relativistic plasma mirror at unprecedented intensities $>10^{21} ~\textrm{W/cm}^2$ using a PetaWatt-class laser. We show that beyond a few $10^{21} ~\textrm{W/cm}^2$ a precise control of the laser contrast at the sub-picosecond time scale becomes essential to drive the efficient generation of high-order harmonics. Such control is paramount for deploying plasma mirrors in high-field applications at PetaWatt-class laser facilities, including, for instance, their use as intensity boosters in the pursuit of the strong-field regime of quantum electrodynamics.