The coherent build-up of harmonics through QPM based on structured gas density distribution was theoretically proposed by Auguste et al.4949. T. Auguste, B. Carré, and P. Salières, “Quasi-phase-matching of high-order harmonics using a modulated atomic density,” Phys. Rev. A 76, 011802 (2007). https://doi.org/10.1103/physreva.76.011802 The experimental demonstration of this effect has been reported by Seres et al.5151. I. Hadas and A. Bahabad, “Periodic density modulation for quasi-phase-matching of optical frequency conversion is inefficient under shallow focusing and constant ambient pressure,” Opt. Lett. 41, 4000–4003 (2016). https://doi.org/10.1364/ol.41.004000 In their work, the modulation of the gas density was obtained by adding two gas sources in a focused laser beam. However, in a tight-focusing geometry, scaling this scheme to a larger number of gas sources is challenging. Furthermore, based on the theoretical work by Hadas and Bahabad,5151. I. Hadas and A. Bahabad, “Periodic density modulation for quasi-phase-matching of optical frequency conversion is inefficient under shallow focusing and constant ambient pressure,” Opt. Lett. 41, 4000–4003 (2016). https://doi.org/10.1364/ol.41.004000 on HHG in a shallow focusing regime, the nonlinear optical conversion by QPM in a periodically modulated gas medium may be strongly inefficient. This result suggests investigating the possibility to overcome the theoretical limitations of a periodic geometry by the use of a nonperiodic gas density modulation.

In this framework, we developed microfluidic devices equipped with a more sophisticated gas-delivery module, composed of an array of identical De Laval micronozzles, whose supersonic outflow improves the spatial focusing of the gas sources at the inlets of the main channel. These micronozzles are directly interfaced with the hollow waveguide, such that local gas density peaks can be realized along the beam path. The tailoring of the gas density in this kind of device can be engineered by changing the shape, the number, and the position of the micronozzles.

A. HHG with integrated arrays of gas jets

The micronozzles are characterized by a convergent–divergent profile with diameters of 220, 60, and 90 µm in the input, throat, and output, respectively. The convergent zone has a length of 55 µm, and the divergent one of 75 µm. We realized two different multi-jet microfluidic sources, the first composed of four evenly distributed gas jets (relative distance L = 1.2 mm), and the second composed of three gas jets accommodated at different relative distances (L1 = 1.9 mm and L2 = 1.4 mm). In front of the nozzle outlets, we arranged an exhaust rectangular opening to allow free gas expansion and to reduce gas stagnation between two adjacent gas jets. To preserve the optical structure of the hollow waveguide, the width of the opening was set at 90 µm, smaller than the diameter of the waveguide. A scheme of the two layouts is reported in Fig. 5(a).In these devices, we obtained focused and confined gas jets, providing the desired gas density peaks along the waveguide axis. In both configurations (with three and four nozzles), we arranged the gas jets at a distance much larger than their transversal size to obtain a high gas density contrast between the jets and the background. The geometry was preliminarily optimized by running computational fluid dynamics (CFD) simulations. The Navier–Stokes–Fourier (NSF)5252. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, 1987). equations for a viscous and compressible fluid were used to study the steady gas flow through the device. NSF equations were solved numerically on the commercial Comsol Multiphysics™ CFD platform.5353. COMSOL Multiphysics, version 5.4, www.comsol.com, COMSOL AB, Stockholm, Sweden A laminar flow regime was assumed due to the relatively low value of the reference Reynolds number. No-slip boundary conditions were applied to walls due to the small value of the Knudsen number5454. C. Cercignani, Rarefied Gas Dynamics (Cambridge University Press, 2000). in the whole flow domain. Details about the adopted grid structure are given in the supplementary material (Sec. S1). Under these conditions, a moderately supersonic gas flow inside the waveguide is obtained at the output of the nozzles.Figure 3(b) reports a sectional view of the numerical gas density distributions computed at a backing pressure in the reservoir of 1 bar. The gas distribution is composed of highly dense and confined gas jets interposed to the low-density background, with a high peak-to-background contrast ratio: the density contrast is >7 in the middle of the waveguide, while it is >20 on the two extremities of the waveguide. Under these conditions, we can minimize the contribution to HHG from the gas background and can study the dependence of the harmonic emission on the gas density modulation by directly comparing the harmonics spectra in different gas-jets configurations.

The use of two different devices with three and four nozzles required changing the device from one measurement to the other and aligning them independently. To allow for a reliable comparison of HHG in the two configurations, we constantly checked the laser parameters (energy, spectrum, pulse compression) and the coupling efficiency (output power over input power) to be stable from one measurement to the other. The beam coupling inside the devices is monitored by an auxiliary beamline equipped with a mirror mounted on a motorized translational stage that can be inserted into the beamline to send the output mode onto a beam profiler. Moreover, we used to make a precise adjustment of the beam-to-device alignment through the remotely controlled five-axis motion system, by direct optimization of the harmonic signal.

Figure 4 shows the experimental high-order harmonic spectra generated in the two devices filled with helium at a backing pressure of 1 bar. A clear difference between the two spectra can be observed throughout the whole spectral range. In fact, the yield is not monotonically related to the number of sources, but it exhibits a counterintuitive re-shaping. In the three-jet aperiodic configuration, we observe a higher generation yield at the cutoff. In fact, in the periodic four-nozzles arrangement, harmonic components up to 140 eV are efficiently generated, while, in the aperiodic three-jets arrangement, the spectrum approaches 200 eV, with an exponential decrease of the harmonic intensity above the cutoff, at 190 eV.

A phenomenological picture of the mechanism behind the yield enhancement of selected portions of the XUV spectrum induced by phase matching can be drawn in terms of the coherence length of the harmonic components.

Since the gas jets are very confined in space along the waveguide axis, with a longitudinal length of ∼100 µm, they can be considered as point sources of harmonics emission. During the propagation from one nozzle to the next one, the harmonics radiation undergoes dephasing from the fundamental field because of the dispersion produced by the residual gas background inside the waveguide. This dephasing is quantified by the phase-mismatch parameter Δkq. The coherence length of the q-th harmonic component is related to Δkq according to the following definition, Lc,q = 2π/Δkq. In a simple picture, point sources that are exactly placed at a distance of n · Lc,q are perfectly matched in phase for the q-th harmonic component, and a constructive sum of their emission occurs. A deviation from ideal conditions might result in significant suppression of the spectral intensity of specific spectral regions.

A one-dimensional propagation model intended to give a description of the harmonic field propagation along the waveguide is reported in Sec. .

B. A simple-man 1D propagation model of HHG in a multi-jet

To investigate the growth of the harmonic components at higher energies in a tailored gas density, we calculate the phase-mismatch Δkq between the fundamental and harmonic fields for each harmonic order q. We used a one-dimensional approximation, in which we assumed the fundamental coupled to the EH11 mode of the hollow waveguide. The pulse is modeled in the temporal domain as a Gaussian envelope with a Full width at Half Maximum (FWHM) duration and intensity of 25 fs and 9.5 · 1014 W/cm2, respectively. Due to the short propagation distance, we neglected the radiative losses through the hollow waveguide boundaries. The ionization fraction η was estimated by the Yudin and Ivanov model,5555. G. L. Yudin and M. Y. Ivanov, “Nonadiabatic tunnel ionization: Looking inside a laser cycle,” Phys. Rev. A 64, 013409 (2001). https://doi.org/10.1103/physreva.64.013409 leading to ηp = 5.9% at the pulse peak in helium.In the model, we assumed the fundamental pulse to propagate in a medium with a constant ion fraction ηp within the full length of the waveguide. The phase mismatch for the q-th harmonic is written as1515. C. G. Durfee III, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187 (1999). https://doi.org/10.1103/physrevlett.83.2187Δkqx=−qu112λ04πa2+k0qnoρ−nqρ1−ηp−ρρatmηpωp2ω021−1q2,where ρatm, a, u11, k0, ω0, and ωp represent the gas density (in standard conditions), the waveguide radius, the first zero of the Bessel function J0, the fundamental wavevector, the fundamental frequency, and the plasma frequencies for a fully ionized gas at 1 atm in standard conditions, respectively. n(ρ) and nq(ρ) are the refractive indices calculated as a function of the local gas density ρ(x). x is the laser propagation direction.The harmonic field is computed by solving numerically the growth equation,5151. I. Hadas and A. Bahabad, “Periodic density modulation for quasi-phase-matching of optical frequency conversion is inefficient under shallow focusing and constant ambient pressure,” Opt. Lett. 41, 4000–4003 (2016). https://doi.org/10.1364/ol.41.004000dEqdx=−αqxEq+b(x)ei∫−x0xΔkq(x′)dx′,where αq is the medium absorption as a function of the gas density distribution and b(x) = iqω02ρ(x)dq(x) (1 − ηp)/2ε0c0. The second term is the polarization field of the q-th order. To isolate the role of the phase mismatch in the development of the harmonic field, the nonlinear dipole moment dq(x) is assumed to be constant along x for each harmonic order.

To take into account the three-dimensional gas density distribution inside the waveguide, the 1D model was applied along the waveguide axis on a bundle of lines parallel to the waveguide axis and regularly distributed inside the waveguide volume.

Specifically, for sampling the gas density ρ(x,y,z), we divided the 3D waveguide volume into a discrete number of n sub-volumes, with n = 44, that form a Cartesian grid. Each sub-volume has a squared base with ∆y · ∆z = (R/4)2 (∆y = ∆z = R/4, R waveguide radius) and a length of 8 mm (waveguide length). For each point along x, we extrapolated the average density value within the sub-volumes and obtained a set of n gas density profiles that accounts for the non-uniform gas distribution along different axial lines with fixed (yn,zn)-coordinates inside the waveguide, ρn(x). We used the ρn(x) profiles to calculate the harmonic field along x in each n-th domain, according to the above-mentioned dEq/dx equation. The phase-mismatch parameter Δkq(x) is density-dependent, thus providing different phase field conditions depending on n. To account for the non-uniform electric field intensity on the (y,z) planes, we calculated the driving field amplitude En and the corresponding nonlinear dipole moment dq,n for each n-th volume by assuming a Bessel-like radial profile for the field mode inside the waveguide (EH11). By applying the corresponding n-th input conditions on gas density and driving field amplitude, we obtained a set of harmonic spectra and summed them to obtain the final HHG spectrum.

Figure 5(a) shows the numerical HHG spectra calculated from the single-atom nonlinear dipole5656. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117 (1994). https://doi.org/10.1103/physreva.49.2117 by applying to each harmonic component an amplitude modulation according to the above-described phase-matching model for the two different nozzle distributions, with three (red curve) and four (black curve) jets.The model reproduced well the enhancement of the cutoff yield in the three-nozzle configuration compared to the four-nozzle case. The nonlinear field growth along the propagation direction as a function of the harmonic order as predicted by this simple-man model for harmonics 95 (147 eV) and 109 (169 eV) is shown in Figs. 5(c) and 5(d), respectively.

The fields have a characteristic step-like behavior in correspondence to the gas peaks, overlapped with periodic amplitude modulation. The modulation is due to the residual gas background inside the waveguide, and the periodicity is related to the effective coherence length of the harmonic as they propagate in the residual gas. The gas density peaks produce a change in both the harmonic amplitude. In particular, the phase accumulated by the q-th harmonic during the propagation in the gas background may induce destructive or constructive interference with the newly generated field inside the gas jet. This results in a local increase or a decrease in the amplitude of the up-converted field, depending on the gas jet position. The four-nozzle scheme allows a favorable field growth in the range of 140–150 eV, while it provides a less favorable quasi-phase-matching for harmonics between 160 and 200 eV. Due to the different inter-nozzle distances, phase-matching is achieved on a broader spectral region in the three-nozzle device. If the growth of a harmonic component between two jets is hindered because of an unfavorable jet distance, phase-matching can be recovered by detuning the distance of the third jet. Thus, the components at the cutoff are not suppressed and a higher conversion yield is measured in this spectral region. We estimated that in both devices, the role of absorption in the gas-jet configurations is negligible, leading to an attenuation of less than 5% of the harmonic field.