High-order harmonic generation in a strongly overdriven regime

High-harmonic generation (HHG) normally requires a careful adjustment of the driving laser intensity (typically $10^{14} - 10^{15}$ W/cm$^2$) and gas medium parameters to enable good phase matching conditions. In contrast with conventional wisdom, we present experimental results indicating phase-matched HHG in all rare gases, using high-density driver around $10^{16}$ W/cm$^2$. The are corroborated by theoretical simulations, which indicate that ionization-induced self-phase modulation plasma defocusing self-regulate level is appropriate for matching. A ten-fold broadening NIR spectrum observed, continuous spectra from $18-140$ eV spite 50-fs-long pulses. presented scheme represents simple versatile concept XUV soft X-ray continua, ideally suited transient absorption reflection spectroscopy.