Highly efficient coherent XUV generation via high-order frequency mixing

Considering the combined effect of the phase-matching and of the blue shift of the driving field during its propagation in a rapidly ionising medium, we show that the latter can be the dominant mechanism limiting the growth of the XUV efficiency with increasing laser intensity. We introduce a new spatial scale, the blue-shift length, which sets the upper bound for the quadratic intensity growth of the generated harmonics. Moreover, we show that this seemingly fundamental restriction can be overcome by using an additional weak mid-IR field. For specific combinations of frequencies of the generating fields, the corresponding high-order frequency-mixing (HFM) process does not suffer from the blue shift of the drivers and the corresponding phase mismatch, and thus its efficiency grows quadratically with propagation distance.