Research ArticleOPTICS

Table-top extreme ultraviolet second harmonic generation

See allHide authors and affiliations

Science Advances  19 May 2021:
Vol. 7, no. 21, eabe2265
DOI: 10.1126/sciadv.abe2265


The lack of available table-top extreme ultraviolet (XUV) sources with high enough fluxes and coherence properties has limited the availability of nonlinear XUV and x-ray spectroscopies to free-electron lasers (FELs). Here, we demonstrate second harmonic generation (SHG) on a table-top XUV source by observing SHG near the Ti M2,3 edge with a high-harmonic seeded soft x-ray laser. Furthermore, this experiment represents the first SHG experiment in the XUV. First-principles electronic structure calculations suggest the surface specificity and separate the observed signal into its resonant and nonresonant contributions. The realization of XUV-SHG on a table-top source opens up more accessible opportunities for the study of element-specific dynamics in multicomponent systems where surface, interfacial, and bulk-phase asymmetries play a driving role.


Nonlinear interactions between light and matter not only form the basis for the generation of light at wavelengths spanning the tetrahertz to x-ray regimes but also enable spectroscopies that yield unique insight into fundamental material properties (13). When describing the nonlinear light-matter interaction, a material’s polarization response to incident light of frequency ω can be approximated as a power series in increasing powers of the electric field E(ω), where only noncentrosymmetric materials, interfaces, and surfaces permit nonvanishing even-order terms. For this reason, second harmonic generation (SHG) and sum-frequency generation (SFG) are inherently sensitive to broken inversion symmetry (4, 5). Up to a constant factor, the frequency-dependent effective second-order nonlinear-susceptibility χ(2) can be extracted experimentally via the relationshipI(2ω)χ(2)(2ω;ω+ω)2 I(ω)2(1)where I(ω) is the intensity at frequency ω. While optical nonlinear spectroscopies have been highly insightful probes of interfacial chemistry (69) and broken symmetries in solid-state systems (10), optical light can couple efficiently to multiple excitation pathways, rendering spectra difficult to interpret in multi-element samples (11, 12). The desire to gain core-level specificity in nonlinear spectroscopies is thus motivated by the study of complex systems with wide-ranging applications including all-solid-state batteries with multiple buried interfaces, ferroelectric materials that, by definition, have bulk-phase structural asymmetries, and low-dimensional heterostructures, to name a few examples.

Popular element-specific probing methods include photoelectron and linear x-ray spectroscopies, which are sensitive to either surface or bulk properties, respectively, but not both, prompting recent efforts to apply nonlinear spectroscopies to the x-ray regime (13). The first promising experiments at free-electron lasers (FELs) have demonstrated soft x-ray (SXR) SHG as a viable technique to study noncentrosymmetric materials (1416), surfaces (17), and buried organic-inorganic interfaces (18) when the energy of the fundamental or its SHG counterpart matches an allowed electronic transition. With FEL sources, the inherent intensity jitter lends itself naturally to calculating the nonlinear response via Eq. 1. The next frontier in x-ray SHG lies in the development of compact table-top sources capable of delivering high enough x-ray photon fluxes for nonlinear techniques. Given the moderate input intensities (~1012 W/cm2) required for FEL-SHG that are in line with numerical estimations of intensities required for two-photon absorption (19), it is not unreasonable to think that high-power table-top HHG (high harmonic generation)-based sources could be used in a similar scheme (17, 2023).

Here, we report the first extreme ultraviolet second harmonic generation (XUV-SHG) experiment above the Ti M edge (32.6 eV), which also represents the first table-top demonstration of SHG at photon energies beyond the UV regime. A comparison of our observations with density functional perturbation theory (DFPT) and real-time time-dependent density functional theory (RT-TDDFT) calculations suggests resonant contributions to SHG from Ti-3p to Ti-3d transitions at the Ti surface. This proof-of-concept experiment demonstrating XUV-SHG aims to present on viable parameter ranges that will enable XUV-SHG at similar high-flux table-top sources.


In the experimental setup depicted in Fig. 1A, a 37.8-eV HHG-SXRL (soft x-ray laser) (20, 24) with an average input energy of 111 ± 23 nJ, pulse duration of 1.73 ± 0.13 ps, and Gaussian-like beam profile (Fig. 1A, inset) was tightly focused using a Au-coated ellipsoidal mirror onto a Ti foil (hexagonal centrosymmetric, 6/mmm point group) placed at the rear focal plane of the ellipsoid with an on-target spot size of 4.5 ± 1.5 μm and average intensity of 4.1 ± 1.9 × 1011 W/cm2 (Fig. 1B). Given the 50-nm thickness of the Ti foil and the estimated 13-nm attenuation depth of 37.8-eV photons in Ti (25), it is likely that the SHG signal was predominately generated on the front face of the foil, and SHG emission from the rear surface can be disregarded. These estimations are supported by the fact that measurements were also performed with 100-nm foils, for which no statistically significant SHG could be found, and by complementary DFPT simulations discussed below. Given that the SHG emission is confined to the Ti surface layer, the SHG emission is always phase-matched. Spectra recorded without the sample present lacked SHG signal between the 0th and ± 1st diffraction order, thus confirming that SHG emission was not a result of second harmonic contamination in incoming SXRL beam (fig. S2). This is an advantage of plasma-based sources as opposed to undulator based ones, since harmonic contributions to the incident beam are expected in the latter but not the former (26).

Fig. 1 Experimental setup and energy diagram for SXRL-SHG at the Ti M2,3 edge.

(A) From left, the setup includes the incoming 37.8-eV HHG-SXRL pulse (red beam, far-field beam profile in inset) that is focused by an ellipsoidal mirror onto the Ti foil surface, where SHG is generated (blue beam) and propagates collinearly with the fundamental. The divergent field is refocused with a toroidal mirror onto a grating that disperses the SHG and fundamental beams simultaneously onto a CCD camera. (B) Schematic diagram of the origin of the SHG emission. Inversion symmetry is broken on the front Ti surface, allowing for SHG. The fundamental and SHG beams exit from the rear foil surface. (C) Ti orbital-resolved density of states (DOS) (40). The SHG emission results from on-resonant excitation of 3p core states (−32.6 eV) to an empty intermediate state of 3d character (+5.2 eV) and subsequently to a virtual state (+43 eV).

The on-target fluence of 0.7 ± 0.3 J/cm2 surpassed the single-shot damage threshold as evidenced by consistent sample damage, which required moving the sample to an unexposed spot for each laser shot. Since the Ti conduction band (CB) consists primarily of 3d states, the 37.8-eV linear absorption can be attributed in part to a resonant dipole-allowed inter-shell transition from 3p core states to empty 3d CB states with SHG emerging from a subsequent transition to a virtual state 43 eV above the Fermi energy EF (Fig. 1C).

The outgoing photons were refocused using a Au toroidal mirror and dispersed with a transmission grating (1000 lines/mm) onto a deep-cooled charge-coupled device (CCD; Princeton-MTE), enabling a simultaneous analysis of the fundamental and SHG peaks. The pulse energy was calibrated with respect to the counts on the CCD by measuring the shot-to-shot fluctuations of the SXRL and correlating the statistics with those observed on the CCD. A characteristic spectrum featuring the fundamental and SHG peaks is shown in Fig. 2A. Measurements of the seeded SXRL linewidth using a high-resolution spectrometer confirmed a linewidth at full width at half maximum of 2.6 meV, which is poorly resolved but expected since the spectrometer was optimized to cover more than one octave at low resolution to be sensitive for SHG. The broadened linewidth of the second harmonic is due to aberrations of the imaging spectrometer but otherwise expected to be the same order of magnitude as that of the fundamental. The background-corrected spectra were analyzed by integrating the peaks observed on the CCD and aggregating single-shot spectra according to the intensity of the fundamental. The slope of a plot relating the on-target energies of the fundamental and SHG peaks allows retrieval of χ(2) via Eq. 1 (Fig. 2B). To understand the SHG process in detail, first-principles DFPT (27) and RT-TDDFT calculations were performed using the exciting all-electron full-potential computer package using the formalism of Sharma and Ambrosch-Draxl (28) that uses linearized augmented planewave and local orbital methods, the results of which will be discussed below (29, 30).

Fig. 2 Representative spectrum of fundamental and second harmonic peaks where the second harmonic response scales quadratically with that of the fundamental.

(A) Representative spectrum of the fundamental and SHG signals (black). The linewidth is not resolved in this spectrum because of the spectrometer being optimized for high sensitivity exhibiting poor spectral resolution. The true linewidth of the SXRL was separately measured (∆E ≈ 2.6 meV) with a high-resolution spectrometer (red line). The linewidth of the SHG signal is assumed to be of the same order of magnitude as that of the fundamental. (B) The nonlinear energy dependence of the SHG signal with respect to the fundamental. A linear equation (red line) is fit to experimental data plotted as |E(ω)|2 versus E(2ω) (black dots with 1 SD error bars, determined by the SD in photon counts on the CCD resulting from averaging shots at the same fundamental pulse energy together).


The quadratic modulo intensity dependence of the observed emission peak at 75.6 eV with respect to the energy of the fundamental (37.8 eV) was confirmed by a coefficient of determination (R2) value of 0.87 (Fig. 2B) when plotting the incident pulse energies, |E(ω)|2 versus E(2ω). These results are in agreement with RT-TDDFT calculations performed using a development version of the SIESTA (31) electronic structure code. Here, a Ti slab was driven under an XUV monochromatic pulse to investigate its interaction with varying laser field intensity. The nonlinear second harmonic response was confirmed by fitting the current density, J(2ω), to a quadratic equation in the driving electric field strength (fig. S8). Hence, we ascribe the observed feature at 75.6 eV to a second-order nonlinear process involving frequency-doubling of the incoming SXRL beam. Using the mean yield of the SHG, fundamental, and input parameters of the SXRL, the SHG conversion efficiency was estimated to be η = 0.023 ± 0.005.

Given the monochromaticity of the SXRL beam, it was not possible to measure the SHG spectrum across the Ti M2,3 edge. However, linear absorption measurements performed with a table-top HHG source (Fig. 3) suggest that the efficiency of SHG at 37.8 eV could have a contribution from a resonantly-enhanced excitation process. From the linear absorption measurements, it can be seen that 37.8 eV lies well within the primary absorption feature due to the dipole-allowed 3p to 3d transition.

Fig. 3 The linear absorption spectrum from a broadband HHG source of the Ti foil.

The experimentally measured onset of the Ti M2,3 edge can clearly be seen (gray) and is confirmed by a calculated absorption spectrum (red). The incident energy used in the SHG experiment of 37.8 eV (black dashed line) lies well within the resonantly enhanced region of the absorption spectrum. a.u., arbitrary units.

Previous FEL experiments estimated a threshold power for nonlinear effects in the hard x-ray regime exceeding 1016 W/cm2 (14). In the SXR regime, efficient SHG at incident intensities of ≈1012 W/cm2 was observed (17, 32). Our results demonstrate efficient SHG with smaller intensities (≈1011 W/cm2). The relatively lower intensities required here for SHG as opposed to those in the SXR regime could be due to the fewer number of K-shell electronic states and the weaker coupling of dipole moments between K-shell and valence electrons (17, 18). Although beyond the scope of this first experimental work in the XUV, it appears worthwhile to explore how the nonlinear conversion efficiencies scale with the energies of core states in future theoretical work.

As for the surface specificity of this technique, DFPT calculations performed within the dipole approximation of the nonlinear susceptibility across the Ti M2,3 edge find a bulk response more than nine orders of magnitude smaller than that originating from the surface (fig. S9). These results agree with symmetry considerations that predict SHG exclusively from the surface given that the vacuum-sample interface is noncentrosymmetric, while the bulk is not. This disparity in contributions points to the possibility that measuring relative differences in XUV-SHG emission after altering the surface chemistry of a centrosymmetric sample would be an element-specific probe of a changing surface environment.

In conclusion, we have shown the first successful demonstration of XUV-SHG with surface specificity indicated by first-principles calculations and performed nonlinear spectroscopy on a table-top SXRL system. In contrast with the energies required for SHG at FEL sources, it was possible that a table-top SXRL with nanojoule input energies and tight focusing could generate second harmonic radiation at 75.6 eV. In addition, the large number of transition metal elements with M edges in the XUV window potentially makes XUV-SHG a feasible technique for investigating the intrinsic properties of transition metal molecular complexes, heterojunctions, interfaces, and noncentrosymmetric materials with elemental specificity. The findings presented here hint at opportunities to perform time-resolved femtosecond or even attosecond nonlinear XUV spectroscopy experiments using, for example, the high-energy, wavelength-tunable XUV-HHG–based sources at the Extreme Light Infrastructure (33), up-scalable sources based on relativistic surface harmonics (34), or even broadband attosecond HHG-based table-top sources (3537). With a small focus, peak intensities using the latter have reached 1014 W/cm2 and have demonstrated two-photon absorption in atoms and molecules (38). Our first demonstration of nonlinear XUV spectroscopy on the table-top holds great promise to expand the nonlinear suite to tunable XUV-SFG to record broadband surface and interface spectra and reduce the temporal resolution by using few-femtosecond compressed optical gate pulses.


The goal of the experiment was to measure SHG from a Ti surface at an SXRL. The linear XUV absorption spectrum was first measured with a broadband table-top HHG source to confirm the location of the Ti M2,3 edge. A sub-4-fs, broadband NIR (near infrared) pulse centered at 730-nm was focused into an Ar gas jet, producing high harmonics. The linear absorption spectrum was collected by measuring the broadband XUV transmission through vacuum (Tvac) and through a 50-nm-thick Ti foil (TTi). Sets of these two measurements were repeated 127 times to determine the average absorbance A = −log(TTi/Tvac).

The SHG measurements were subsequently conducted at the Laboratoire d’Optique Appliquée using the Salle Jaune Ti:Sapphire laser system (24), which delivers three independently compressed multiterawatt femtosecond pulses at a repetition rate of 10 Hz. The SXRL works by coupling a resonant HHG pulse into an SXR amplifier. The amplifier is a plasma of Kr8+ ions emitting at the 3d94d(J = 0) ➔ 3d94p(J = 1) transition at 32.8-nm (37.8-eV) generated by optical field ionization of a high-density Kr gas jet by an ultrashort infrared pump pulse (1.5 J, 30-fs, focused at 3 × 1018W/cm2). As a result of the high electron density of the plasma (up to 1020 e/cm3), the pump pulse cannot propagate in the plasma. A waveguide was therefore implemented beforehand by focusing a 0.7-J, 0.6-ns infrared pulse in the gas jet using an axicon lens (39). The HHG was seeded with a 15-mJ, 30-fs infrared pulse focused in an Ar-filled gas cell and coupled into the Kr8+ amplifier using a grazing incidence toroidal mirror. The 25th harmonic of the HHG spectrum was tuned to the lasing transition by chirping the driver pulse, and the injection delay was set to 1.2 ps to match the SXRL gain peak position.

The SXRL was focused with an ellipsoidal mirror of a focal length of 33 cm (3-μm diffraction limit) onto the sample. An upper limit on the focal spot of 6-μm was estimated by the distance it required to move the position of the incident beam in the sample plane between laser shots because of the sample damage. The irradiated spot size was thus estimated be 4.5 ± 1.5 μm. To ensure that the sample plane was aligned with the focal position, the z position (optical axis) of the sample was varied until a threshold intensity strong enough to burn holes into the foil was reached. After hitting the sample, the fundamental and SHG beams were focused with a toroidal mirror (f = 33 cm) onto a cooled CCD camera (2048 × 2048 pixels of which a 1024 × 1024 quadrant was used because of the low pixel quality in the middle of the camera, pixel pitch of 13.5 μm, −50°C). A total of 394 spectra were recorded, 248 of which had full resolution of the CCD chip and 146 of which with a hardware binned camera of 2 × 2 pixels.


Supplementary material for this article is available at

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.


Acknowledgments: We thank D. Attwood, C. Schwartz, C. Uzundal, and F. Tuitje for fruitful discussions. Funding: The research leading to these results has received funding from the European Community’s Horizon 2020 research and innovation program under grant agreement no. 654148 (Laserlab Europe). M.Z. acknowledges support by the Max Planck Society (Max Planck Research Group) and the Federal Ministry of Education and Research (BMBF) under “Make our Planet Great Again - German Research Initiative” (grant no. 57427209 “QUESTforENERGY”) implemented by DAAD. This work is supported by Investissements d’Avenir Labex PALM (ANR-10-LABX-0039-PALM). J.O. is supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (2019R1A6A3A03032979). H.-T.C. acknowledges support from the U.S. Air Force Office of Scientific Research nos. FA9550-19-1-0314 and FA9550-20-1-0334 and the W. M. Keck Foundation award no. 046300-002. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under contract no. DE-AC02-05CH11231. This work also used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by NSF grant no. ACI-1548562. C.S. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG; German Research Foundation) under Germany’s Excellence Strategy–EXC 2051 (no. 390713860), “Balance of the Microverse.” M.Z. acknowledges funding by the W. M. Keck Foundation (no. 993922), funding from the UC Office of the President within the Multicampus Research Programs and Initiatives (no. M21PL3263), and funding from Laboratory Directed Research and Development Program at Berkeley Lab (no. 107573). Author contributions: T.H., L.H., A.K., J.G., F.T., J.-P.G., S.S., and M.Z. performed the experiments. A.K., J.G., F.T., J.-P.G., and S.S. built, optimized, and operated the SXRL. A.K. performed detailed characterization of the SXRL and focus geometry. T.H., A.K., E.B., and M.Z. analyzed the data. S.J., C.D.P., and T.A.P. performed the simulations. J.O. and H.-T.C. performed linear absorption measurements. M.Z. conceived the experiment. C.S. and M.Z. supervised the project. T.H., E.B., and M.Z. wrote the manuscript with input from all authors. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are available on the Dryad repository under DOI Additional data related to this paper may be requested from the authors.

Stay Connected to Science Advances

Navigate This Article