Numerical experiment of phase-matched ion-based high-order harmonic generation in the water-window x-ray spectral region via electromagnetic envelope equation

We present a phase-matching scheme for efficient high-order harmonic generation in the water-window x-ray spectral region using a 405-nm driving pulse. A high-intensity pulse (∼10¹⁶ W/cm²) is used to produce He¹⁺ ions as the target medium, increasing the cutoff photon energy to the water-window x-ray spectral region. By adjusting the driving pulse divergence, the positive dipole phase variation balances the negative plasma dispersion and geometrical phase shift, achieving phase matching. Using the electromagnetic envelope equation coupled with the Keldysh ionization model, numerical experiments identify the optimal conditions. Results show that the relative conversion efficiencies of the 95^th harmonic (4.26 nm) and the 169^th harmonic (2.4 nm) reach 66% and 79% of the perfect phase-matching conditions, respectively.