Necklace-structured high-harmonic generation for low-divergence, soft x-ray harmonic combs with tunable line spacing

Abstract

The extreme nonlinear optical process of high-harmonic generation (HHG) makes it possible to map the properties of a laser beam onto a radiating electron wave function and, in turn, onto the emitted x-ray light. Bright HHG beams typically emerge from a longitudinal phased distribution of atomic-scale quantum antennae. Here, we form a transverse necklace-shaped phased array of linearly polarized HHG emitters, where orbital angular momentum conservation allows us to tune the line spacing and divergence properties of extreme ultraviolet and soft x-ray high-harmonic combs. The on-axis HHG emission has extremely low divergence, well below that obtained when using Gaussian driving beams, which further decreases with harmonic order. This work provides a new degree of freedom for the design of harmonic combs-particularly in the soft x-ray regime, where very limited options are available. Such harmonic beams can enable more sensitive probes of the fastest correlated charge and spin dynamics in molecules, nanoparticles, and materials.

Fig. 1.. Tunable, low-divergence high-harmonic combs via necklace-driven HHG.

(A) Two linearly polarized vortex beams carrying OAM of ℓ₁ = 2 and ℓ₂ = −3 are overlapped to create a necklace-structured intensity and phase profile and focused into He/Ar gas to drive HHG. The intensity lobes at the necklace focal plane represent a phased array of EUV/SXR emitters, which interfere on-axis to form a comb of harmonics with a spacing dependent on the OAM of the driving fields. The harmonic intensity profile shows a strong emission on-axis (detailed in the figure for the 25th harmonic). (B) Time-frequency analysis of the simulated on-axis harmonic comb for Gaussian- and necklace-driven (ℓ₁ = 2, ℓ₂ = −3) HHG in He at 800 nm. The dashed color lines indicate the corresponding phase from (A) for each emission event. (C) HHG spectrum for the case of necklace- and Gaussian-driven HHG simulated in (B). The harmonic spacing, ∆ω = 10ω₀ (15.5 eV), in the necklace-driven on-axis HHG spectrum is a result of OAM conservation and is tunable by varying the OAM content of the driving laser (top). The on-axis harmonics are emitted with a divergence, which is substantially reduced compared to that of Gaussian-driven HHG and decreases with increasing harmonic order (bottom).

Fig. 2.. Harmonic frequency combs with tunable line spacing controllable through the drivers’ OAM content.

(A) Representation of the line spacing allowed by the selection rules for different values of |ℓ₁| and |ℓ₂|. The color scale represents the line spacing, being 6ω₀ (blue), 10ω₀ (green),14ω₀ (yellow), and 18ω₀ (red). (B and C) Simulation results of the high-harmonic spectra obtained in He for (B) 800-nm and (C) 2-μm wavelength drivers, respectively, for the driver’s OAM combinations: ℓ₁ = 1, ℓ₂ = −2 (blue); ℓ₁ = 2, ℓ₂ = −3 (green); ℓ₁ = 3, ℓ₂ = −4 (yellow); and ℓ₁ = 4, ℓ₂ = −5 (red). The line spacing corresponds to that predicted in (A). The driving beam waists of the different OAM modes are chosen to overlap at the radius ($30/\sqrt{2}\mathrm{\mu m}$) of maximum intensity (6.9 × 10¹⁴ W/cm² for 800 nm, and 5 × 10¹⁴ W/cm² for 2 μm) at the focal plane. The laser pulses are modeled with a trapezoidal envelope with 26.7 fs of constant amplitude.

Fig. 3.. Experimental and theoretical high-harmonic combs in Ar gas using a pair of 790-nm OAM driving lasers with opposite parity (ℓ₁ = 1 and ℓ₂ = −2).

The intensity spatial structure for the 15th (H15) to 21st (H21) harmonics is shown for (A) theory and (B) experiment. On-axis emission is allowed for H15 and H21 as a result of OAM selection rules and transverse phase-matching conditions. Thus, the on-axis component is transmitted for H15 and H21, while H17 and H19 are blocked [insets in (A) and (B)]. (C) Simulated (top) and experimental (bottom) HHG spectra of necklace-driven on-axis emission. By measuring the spectrum along a line in the dispersion plane that intersects the optical axis for all orders, we find that the line spacing of the transmitted harmonics, ∆ω = 6ω₀ (9.5 eV), is consistent with that predicted by OAM conservation laws. Small HHG signals experimentally observed at other harmonic orders are due to leak-through of the components carrying higher topological charges and could be further suppressed by using a smaller aperture. The difference in the ratio H15/H21 is due to slightly different cutoff energies between simulation and experiment.

Fig. 4.. Low divergence of the necklace-driven harmonic frequency combs.

(A) Simulation results of the spatial intensity profile (top) and OAM content (bottom) of the 27th harmonic generated in He for ℓ₁ = 1, ℓ₂ = −2 (left) and ℓ₁ = 4, ℓ₂ = −5 (right) driving fields. The OAM spectra are obtained from the azimuthal Fourier transform at each divergence angle. (B) Simulation results of the full width at half maximum (FWHM) divergence of the high-order harmonics for different OAM driving combinations (color dots)—with the gas jet placed at the focus position—and for a Gaussian driving beam where the gas jet is placed at different positions relative to the focus: from 2 mm before the focus (z = −2 mm) to 1 mm after the focus (z = 1 mm). The orange dashed line indicates the estimation given by Eq. 5. The top insets show the divergence profile of two sample harmonics (27th and 45th) for different OAM driving combinations (showing the ℓ_q = 0 contribution in solid and the total one in dashed line) and for a Gaussian beam where the gas jet is placed at the focus position. Simulation parameters correspond to those of Fig. 2B.

Fig. 5.. Ar-driven harmonic divergences.

(A) Theoretical (Th) and experimental (Exp) comparison of the intensity spatial profiles for the 790-nm Gaussian- and necklace-driven (ℓ₁ = 1 and ℓ₂ = −2) in Ar gas. The white dashed circles indicate the on-axis emission of the 15th (H15) and 21st (H21) harmonics for both theory and experiment. We applied an angular integration radially to these profiles to extract the divergences. (B) Angularly integrated radial profiles for H15 and H21 for the on-axis emission, necklace-driven case compared to the equivalent Gaussian. The vertical blue line and double-headed arrow indicate half of the FWHM of the dual-vortex, necklace-driven profile. The intensities of the theoretically predicted on-axis divergence are rescaled for H15 and H21, respectively, to match the intensities of the experimental profiles. (C) Theoretical (Eq. 5) and measured FWHM for necklace-driven on-axis emission profile indicates a decrease in the divergence with increasing harmonic order. This is in contrast to Gaussian-driven HHG, where divergence increases at higher harmonic orders.