Extension of the bright high-harmonic photon energy range via nonadiabatic critical phase matching

The concept of critical ionization fraction has been essential for high-harmonic generation, because it dictates the maximum driving laser intensity while preserving the phase matching of harmonics. In this work, we reveal a second, nonadiabatic critical ionization fraction, which substantially extends the phase-matched harmonic energy, arising because of the strong reshaping of the intense laser field in a gas plasma. We validate this understanding through a systematic comparison between experiment and theory for a wide range of laser conditions. In particular, the properties of the high-harmonic spectrum versus the laser intensity undergoes three distinctive scenarios: (i) coincidence with the single-atom cutoff, (ii) strong spectral extension, and (iii) spectral energy saturation. We present an analytical model that predicts the spectral extension and reveals the increasing importance of the nonadiabatic effects for mid-infrared lasers. These findings are important for the development of high-brightness soft x-ray sources for applications in spectroscopy and imaging.