Phase-matched frequency conversion in ionizing atomic gases

We solve the one-dimensional wave equation for high-order mixing of the type ${\omega }_h=q{\omega }_0-l{\omega }_c(q\gg l>=1)$ using a Keldysh-type model to describe the high-field atomic nonlinear susceptibility. When ${\omega }_c$ is selected to give static phase matching, we show that the conversion efficiency of the mixed field can be increased by about an order of magnitude compared to non-phase-matched high-order harmonic generation but that fundamental limitations are introduced by axial electron-density gradients.