Critical Laser Intensity of Phase-Matched High-Order Harmonic Generation in Noble Gases

The efficient generation of high-order harmonic radiation has been a challenging task since the early days of strong-field physics. An essential requirement to achieve efficient high-order harmonic generation inside a gas medium is the phase matching of the high-order harmonic radiation and the incident laser pulse. The dominant contribution to the wave-vector mismatch Δk is associated with the ionization probability of the medium. In this work, we derive two analytical formulas to calculate the critical intensity of a general linearly polarized laser pulse that obey the phase-matching condition Δk=0. The analytic formulas are valid in the tunneling regime (ADK model) and the regime of the tunnel and multi-photon ionization (PPT model), respectively. We compare our results to numerical computations and discuss the scaling of the critical intensity depending on the pulse duration and the wavelength of a realistic incident laser pulse. The analytical approach demonstrated in this work is highly accurate and can compete with the existing numerical computational methods by an error of less than 1% and a decrease in the computation time of approximately 4 to 6 orders of magnitude. This enables complex theoretical studies of the efficiency scaling in HHG or to consider the effects of ground state depletion efficiently.