Quasi-phase-matching of high-order harmonics using a modulated atomic density

We present a numerical study of high-order harmonic generation in a gas medium with spatial modulation of the density. We show that, by adjusting the modulation period, the contributions of the short and long quantum paths to harmonic emission can be selectively quasi-phase-matched on axis. In particular, quasi-phase-matching of the short path contribution dramatically improves the spatial and spectral properties of the harmonic beam, increasing peak brightness by a factor close to 40 for the 45th harmonic of a Ti:sapphire laser at $17.8\mathrm{nm}$. Finally, simultaneous or selective quasi-phase-matching of the short and long paths makes it possible to control their relative weight.

Figure 1

(a) Number of photons in H45 generated in neon as a function of the modulation period $\beta$ for $\alpha =1$, $P_0=10\text{Torr}$, $I_0=6\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$; (b) evolution of the $M^2$ factor; (c) $\tau \times \Delta \nu$ product; (d) brightness normalized to $\alpha =0$ case. In (a), the solid (dotted) line corresponds to the case without (with) ionization, and the dot-dashed line to $P_0=60\text{Torr}$ with ionization. The dashed lines emphasize quasi-phase-matching for the long and the short quantum paths, respectively.

Figure 2

H45 intensity increase along propagation for $\alpha =0$ (dotted curve), and for $\alpha =1$ and $\beta =390\mu \mathrm{m}$ (solid curve).

Figure 3

(a) Near-field profiles of H45 calculated for $\alpha =0$ (dotted curve), and for $\alpha =1$ and $\beta =390\mu \mathrm{m}$ (solid curve). (b) Corresponding far-field profiles.

Figure 4

(a) Power spectra of H45 calculated for $\alpha =0$ (dotted curve), and for $\alpha =1$ and $\beta =390\mu \mathrm{m}$ (solid curve). (b) Corresponding temporal profiles.