Nonperturbative Twist in the Generation of Extreme-Ultraviolet Vortex Beams

High-order harmonic generation (HHG) has been recently proven to produce extreme-ultraviolet (XUV) vortices from the nonlinear conversion of infrared twisted beams. Previous works have demonstrated a linear scaling law of the vortex charge with the harmonic order. We demonstrate that this simple law hides an unexpectedly rich scenario for the buildup of orbital angular momentum (OAM) due to the nonperturbative behavior of HHG. The complexity of these twisted XUV beams appears only when HHG is driven by nonpure vortex modes, where the XUV OAM content is dramatically increased. We explore the underlying mechanisms for this diversity and derive a general conservation rule for the nonperturbative OAM buildup. The simple scaling found in previous works corresponds to the collapse of this scenario for the particular case of pure (single-mode) OAM driving fields.

Figure 1

(a) Schematic view of HHG driven by a combination of two vortex beams. An intense IR OAM pulse is focused into a gas jet with transverse coordinates ($\rho$, $\varphi$), where each atom emits harmonic radiation that, upon propagation, results in the far-field emission of XUV vortices with some divergence and azimuth ($\beta$, $\phi$). (b) The IR driving beam is composed by a combination of ${\mathrm{LG}}_{1,0}$ and ${\mathrm{LG}}_{2,0}$, with parameters chosen so the amplitude and radii at the ring maximum are the same.

Figure 2

Nonperturbative OAM generation in the 21st harmonic vortex beam. The upper row represents the far-field spatial intensity distribution (logarithmic scale), and the lower row shows the OAM spectrum (logarithmic scale). The results were obtained through (a) quantum SFA simulations and (b)–(f) different versions of the TSM, where we have considered (b) perturbative ($q$th power, no intrinsic phase), (c) nonperturbative $p$th power (no intrinsic phase), and full nonperturbative (intrinsic phase included) with (d) short, (e) long, and (f) short + long trajectory contributions. The excellent agreement between (a) and (f) supports the use of the TSM to understand the nonperturbative OAM contributions in the quantum calculation.

Figure 3

$\beta$-integrated OAM spectrum (logarithmic scale) of the 21st harmonic vortex, obtained with the TSM using (a) the perturbative $q$th power [dark blue, Fig. 2] and the nonperturbative $p$th power scaling [light blue, Fig. 2]; (b),(c) nonperturbative channels $n=10$ and $n=11$, respectively, including the intrinsic phase of short (purple) and long (green) trajectories; and (d) all nonperturbative channel contributions [corresponding to Fig. 2]. The arrows indicate the maximum OAM interval given by the conservation rules for the short (purple) and long (green) trajectory contributions.

Figure 4

Quantum SFA simulations of the 21st-harmonic vortex structure driven by (a) a pure $\ell =1$ mode, and nonpure OAM beams, with combination of (b) 95% ${\ell }_1=1$ and 5% ${\ell }_1=2$, and (c) 90% ${\ell }_1=1$ and 10% ${\ell }_1=2$. The beam waists are the same as in Fig. 1b. The top row represents the harmonic intensity profiles, which are found to be similar. In the bottom row, the OAM spectra of 21st-harmonic are shown, presenting a clear broadening around the single-vortex value ${\ell }_{21}=21$ from (a) to (c).