Simultaneous ultrabroadband quasi-phase-matching for high-order harmonic generation

We propose a simple quasi-phase-matching (QPM) scheme in high-order harmonic generation (HHG) based on the approximately linear dependence of the phase mismatch on harmonic order. With this scheme, essentially all the harmonic orders are phase matched simultaneously, with harmonic order $q$ experiencing a $q\mathrm{th}$-order QPM. We validate this proposal by simulations using a semiclassical numerical model of HHG.

Figure 1

HHG with simultaneous QPM of all harmonic orders using modulation of the polarization vector by the function $S_1\left(z\right)$ (see text). (a) Evolution of the intensity $I_q$ of various harmonic orders, (b) the same evolution in logarithmic scale, (c) the HHG spectrum at the end of the interaction without (black) and with (colored line) the application of $S_1\left(z\right)$ modulation at the fundamental phase mismatch frequency, (d) relative enhancement of each harmonic order (blue circles) in a logarithmic scale, where the solid red line shows the theoretical prediction according to Eq. (2) and the vertical dash-dotted line indicates the cutoff location, and (e) the ${\text{Ar}}^{+}$ transmission [33] for a 5-mm propagation.

Figure 2

Same as Fig. 1, but the QPM modulation is of the form of $S_2\left(z\right)$ (see text) set to phase match all harmonic orders with a preference to harmonic order $q\prime =35$. In this case, $l_0=\pi /\mathrm{\Delta }{k}_0=857\mu \text{m}$, and $l_{ph}=\pi /q\prime \mathrm{\Delta }{k}_0=24.5\mu \text{m}$ is the coherence length of the 35th harmonic. The thick black arrows denote the values related to harmonic order $q\prime =35$.