Abstract
Strongfield methods in solids enable new strategies for ultrafast nonlinear spectroscopy and provide alloptical insights into the electronic properties of condensed matter in reciprocal and real space. Additionally, solidstate media offers unprecedented possibilities to control highharmonic generation using modified targets or tailored excitation fields. Here we merge these important points and demonstrate circularlypolarized highharmonic generation with polarizationmatched excitation fields for spectroscopy of chiral electronic properties at surfaces. The sensitivity of our approach is demonstrated for structural helicity and terminationmediated ferromagnetic order at the surface of silicondioxide and magnesium oxide, respectively. Circularly polarized radiation emanating from a solid sample now allows to add basic symmetry properties as chirality to the arsenal of strongfield spectroscopy in solids. Together with its inherent temporal (femtosecond) resolution and nonresonant broadband spectrum, the polarization control of high harmonics from condensed matter can illuminate ultrafast and strong field dynamics of surfaces, buried layers or thin films.
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Introduction
Nonlinear spectroscopy has made a huge leap forward with the proof of highharmonic generation (HHG) in condensed matter^{1,2,3,4}. Recently, numerous excellent studies, involving band reconstructions^{5}, exciton analysis^{6}, momentum dependent phases^{7,8,9}, and valenceelectron mapping^{10} have shed new light on various solidstate phenomena. At the same time, the functionalization of solid targets through structural and chemical modifications^{11,12,13} or nanoconfined and tailored excitation fields^{13,14,15,16} has led to unprecedented possibilities for the controlled generation of high harmonics. The variation of the excitation field’s polarization, in particular, can provide access to crystal orientationdependent information^{17,18,19,20}. When the crystal symmetry is known, the study of phenomena in the sample can be conducted by combining linearly or circularly polarized excitation with a polarization analysis of the harmonic emission. Moreover, in such schemes, the generation of harmonic radiation with elliptical polarization is possible^{20,21,22}. The use of nontrivial tailored driving fields can significantly enrich solidstate HHG by providing a universal control of the highharmonic polarizations in arbitrary (symmetric) crystal structures, which is key to efficient generation of circularly polarized radiation and to symmetryresolved chiral spectroscopy.
In HHG from atoms and molecules, intricate field symmetries already proved their important role, including the probing of orbital angular momentum states with symmetries across the laser wavefront^{23,24,25} and symmetries at the level of local electric fields^{26,27,28}. Of great interest for crystalline solids are bichromatic light fields with controllable polarization, which can possess discrete rotational symmetries, and are known from circularlypolarized HHG in gases^{29,30}. In the particular case of a bicircular field comprising counterrotating fundamental wavelength and its second harmonic, the rotational symmetry is threefold, and the field’s angular momentum is imprinted on the generated harmonics. In solids, where the HHG is affected by the crystalline symmetry, such tailored excitation fields would allow for a symmetrysensitive probing of chiral surfaceband features, which is particularly relevant for the study of correlated electronic systems or magnetic properties.
Here, we demonstrate that bichromatic rotational symmetric driving fields^{29,30,31} probe rotational and chiral symmetries at the surfaces of bulkinsulating crystals via HHG. In particular, we match a threefold driving field with threefold, fourfold, and sixfold structures of specific crystal cuts of silicon dioxide (quartz) and magnesium oxide (MgO) and find a high sensitivity of circularlypolarized HHG in solids to structural helicity and surface magnetism. By evaluating the difference in the resonant, i.e., nearbandedge, absorption between left and rightcircular polarized harmonics we can measure chiralitymediated band shifts in the respective crystal systems. While chiral band shifts in quartz(0001) and at the polar MgO(111) oxygenterminated surface^{32} is expected due to the screwlike structural helicity (P321 space group)^{33} and the reconstructionmediated ferromagnetism at the polar surface^{34,35}, respectively, we also discover a chiral footprint also on the cubic nonpolar surface of MgO(100).
Results
HHG in solids with bicircular driving fields
In this study singlecrystalline quartz and MgO targets are excited under moderate vacuum conditions (< 10^{−6} mbar pressure) with bicircular twocolor laser pulses (50 fs pulses at 1 kHz repetition rate) comprising a circularlypolarized fundamental field at 800 nm wavelength and a counterrotating circularlypolarized second harmonic (400 nm wavelength). The field strengths we used (10.5 V nm^{−1} in quartz and 7.7 V nm^{−1} in MgO) are comparable to other studies^{4,17} that observed strongfield HHG in these materials. Figure 1a schematically illustrates the experimental setup, where highharmonic radiation is generated in reflection geometry from a crystal surface and collected with an extremeultraviolet spectrometer. The driving bicircular field exhibits a threefold helical polarization (see Supplementary Fig. S1 for details), which we utilized as a spectroscopic probe via circular HHG. Figure 1b depicts this principle, which involves rotational symmetry probing by changing the angle θ between the field and the crystal as well as chiral sensitivity through the circular polarization state of the generated harmonics. A typical spectrum spanning from the fifth harmonic (7.75 eV photon energy) to the tenth harmonic (15.5 eV) is shown in Fig. 1c for quartz(0001). The spectra for all crystals are shown in Supplementary Fig. S2. The nearperfect suppression of every third harmonic is the selection rule that corresponds to circular polarization of the unsuppressed harmonics^{29,30} (see Supplementary section selection rules). More precisely, the circular polarization of the harmonic orders \(q=3n+1\;\left(4,7,10,\ldots \right)\) exhibits the same sense of rotation as the fundamental driving field while the orders \(q=3n1\;\left(5,8,11,\ldots \right)\) have an opposite polarization helicity (see Fig. 1a–c). For a harmonic generation in isotropic targets, such as gases, the radiation symmetries are dominated by the threefold polarization shape of the driving bicircular field. Since threefold and sixfold rotation symmetries inherently conform to the threefold symmetry of the laser field, the same selection rules apply for the termination planes of quartz(0001) and MgO(111) (cf. spectrum in Supplementary Fig. S2). Conversely, crystals exhibiting other symmetries, e.g., the fourfold MgO(100), remove the constraints on the selection rules and lead to a less pronounced suppression of every third harmonic (see the spectrum in Supplementary Fig. S2 and supplementary information for details on the selection rules). On the other hand, the suppression of every third harmonic confirms the threefold symmetry of the driving field, which allows for the probing of the crystal axes by polarization rotations (see Fig. 1b left). The spectrogram in Fig. 1d shows allowed harmonic orders which peak every 120°, corresponding to the P3 pointgroup of the quartz crystal. We attribute the higher angular sensitivity, i.e., the stronger modulation of the signal, for increasing harmonic orders to the nonlinear intensity scaling of the generation process. More specifically, the angular resolution is determined by the spatial generation anisotropy of the harmonics (whether in the perturbative or nonperturbative regime) and the particular polarization contrast of the threefold excitation field (see Fig. S1a). While the influence of the underlying mechanism warrants further investigation, the observed phase differences between individual harmonics (cf. H8 and H10 in Fig. 1d) already suggest a nonperturbative generation of higher harmonics^{3,5,36} in quartz, which could be relevant for the study of diverse solidstate phenomena^{5,10,37}.
Inversionsymmetry probing
The lack of inversion symmetry of bicircular fields is particularly relevant for the probing of material inversions. Figure 2 analyzes the rotational dependence of harmonic generation driven by bicircular fields and inversionsymmetric linearpolarized fields for different crystals. The solid targets quartz(0001), MgO(111) and MgO(100) represent threefold, sixfold, and fourfold rotational symmetries, respectively. The shown polar plots of the eighth harmonic intensity (ninth harmonic in MgO for linear polarizations) are extracted from the spectrograms (see also Supplementary Figs. S2 and S3). In quartz, an inversionsymmetrybroken crystal, the bicircular laser field probes the threefold rotational symmetry and the crystal orientation accurately (Fig. 2a). Contrary, a polarization scan with a linearlypolarized field, which possesses a twofold symmetry, exhibits an ambiguous sixfold beating (Fig. 2d) corresponding to an inversionsymmetrized threefold rotation. For crystals with sixfold symmetry as MgO(111) (Figs. 2b, e) rotating the bicircular or linearly polarized field yields similar sixfold patterns. In the case of the fourfold MgO(100) (Fig. 2c, f), the harmonic emission from the linearlypolarized fundamental exhibits the expected, fourfold pattern, while for excitation with the bicircular field the signal loses the appearance of its fourfold symmetry. The convolution of the threefold field and the fourfold crystal should result in a 12fold signal^{38}, which is not resolvable. Since in MgO(111) the angulardependent emission peaks are narrow enough to resolve a 12fold symmetry, we conclude that the MgO(100) is more isotropic. It may be possible to refine the angular modulation of the harmonic intensity by tuning the parameters of the driving field, such as the amplitude ratio of fundamental and second harmonic, which has an influence on the symmetry modulation of the field, as discussed in “Methods”.
Structural and magnetic circular dichroism in solidstate HHG
Chiral electronic properties become apparent when comparing circular HHG for the left and right helicities of the bicircular field. We find chiral spectral signatures in quartz and MgO and attribute those to structural helicity and ferromagnetic order, respectively. The difference of spectrograms for left and right bicircular driving fields (see Supplementary Fig. S2) in quartz, shown in Fig. 3a, reveals a strong angledependent spectral shift of the seventh harmonic, as outlined in Fig. 3c (gray dots). A similar shift of the fifth harmonic is observed for both crystal cuts of MgO, which is exemplified in Fig. 3b, d for MgO(100). Importantly, regardless of the crystal termination planes in MgO (both 100 and 111, cf. Supplementary Fig. S4), the spectral shifts are independent of the field rotation angle. In both materials, the harmonic orders exhibiting a spectral shift are located near the bandgap energy and we refer to these harmonic orders as “onresonance” in the following.
Our observations can be explained by circular dichroism, i.e., a helicitydependent absorption of generated harmonic radiation. This absorption leads to red and blueshifted onresonance harmonics having left and rightcircular polarization, respectively (see red and blue dots in Fig. 3e, f). This effect is similar to dichroic absorption measurements with external sources, however here, the probe emanates from the sample itself and is sensitive to the crystal symmetry. To evaluate the chiral spectral response, we consider nearbandedge absorption of the opposite circularlypolarized harmonics, with different helicitydependent absorption channels in both materials. In quartz, the screwlike crystal arrangement possesses a structural chirality (see schematics for the 3_{1} and 3_{2} structure in Fig. 4a), where the two enantiomers exhibit different band structures^{33}, as qualitatively drawn in Fig. 4b. Hence, opposing helicities of emitted harmonics probe a different band structure in 3_{1} and 3_{2} (see purple arrows in the band diagram in Fig. 4b). On the other hand, in MgO—being an achiral crystal—the surface ferromagnetism dominates the helicityselective absorption. Spontaneous symmetrybreaking in the nonstoichiometric surface reconstructions^{32,35,39} lead to the formation of a ferromagnetic layer with a spinpolarized metallic surface^{39,40}. The oxygen termination of the polar MgO(111) may create a magnetic moment perpendicular at the surface^{41}, as schematically depicted in Fig. 4c. The band structure for the minority and majority spinpolarized charge carriers on the surface of MgO(111) is drawn in Fig. 4d. The annotated arrows mark the different absorption for left and rightcircular polarized harmonics (see purple arrows in Fig. 4d). In MgO(100), magnetic ordering may arise from vacancies and impurities or faceted surface reconstructions^{40,41}, while bulk ferromagnetism can be excluded at room temperature^{35}. The fifth harmonic is sensitive to the ferromagnetism at the MgO surface due to its absorption by excitons, allowing for a penetration depth of only \({\delta }_{{\rm{HHG}}}=15\) nm (ref. ^{42}). This attenuation length is comparable to the electronhole binding distance \({d}_{{\rm{ex}}}\) (about 6 nm in MgO (ref. ^{43})), which highlights the significance of these surfacelocalized excitons in our measured spectra.
In order to reproduce the spectral response in Fig. 3, we calculate the helicitydependent absorption of harmonics with left and rightcircular polarization by considering energeticallyshifted absorption coefficients \(k\left(E\pm \Delta E/2\right)\) for quartz^{44} and MgO (ref. ^{42}) The term \(\pm \Delta E/2\) represents the energy shift for opposite circular polarizations (see Supplementary Fig. S5). While in the calculation for quartz we consider the entire optical penetration depth (see Eq. (1) in “Methods”), in the case of MgO, we introduce an effective interaction length \(\delta\) as an additional parameter (see Eq. (2) in “Methods”) to model a surface sensitivity. We fit the absorption spectra shown in Fig. 3e, f to Eq. (1) and (2) by varying \(\Delta E\) (and \(\delta\), in the case of MgO), and present the final curve alongside the difference spectra in Fig. 3c, d (solid lines). In both materials, the absorption model yields significant spectral shifts for the onresonance harmonic orders while the other harmonic orders show only intensity differences, similar to what we observe in Fig. 3a, b. This is consistent with the derivative of the absorption coefficient being largest at the band edge, which justifies the extraction of the band splittings from the onresonance harmonic signals. The obtained band splitting values of \(\Delta E=192\) meV in quartz (maximum, at 215° field rotation), \(\Delta E=364\) meV in MgO(100), and \(\Delta E=395\) meV in MgO(111) (average over all field rotation angles), are in good agreement with the theoretically predicted changes of the band energy^{33,39,45} (cf. Fig. 4b, d). Furthermore, the obtained interaction length of \(\delta =47\) nm in MgO matches the expected exciton diameter, which suggests a coupling of the surfacemagnetic moment to exciton states. The intrinsically different origin of the circular dichroism in quartz and MgO is revealed by the remarkably different dependence of their band splitting on the fieldrotation angle, as shown in Fig. 5. In quartz (Fig. 5b), the band splitting is rotationally anisotropic with a threefold periodicity, indicating a crystalaxisdependent band structure^{33}. On the other hand, the isotropic band splitting in MgO (Fig. 5a) can be attributed to the ferromagnetic order on the surface^{35}. The accuracy in which the chiral excitations on the surface explain our observations suggests that further theoretical and experimental analysis using tailored strongfields under consideration of the energydependent emission phases (see Fig. 1d) could provide additional microscopic information on chiral systems, possibly even on the atomic level^{10}.
Discussion
In conclusion, we demonstrate a distinct symmetrydependent chiral sensitivity of high harmonics generated from bicircular laser fields near crystal surfaces. We show that the tailored symmetry and angular momentum of the driving field polarization can be utilized to generate circularlypolarized harmonic radiation in solids with arbitrary crystal structures and allow for symmetryresolved chiral spectroscopy of the generation medium. Specifically, the harmonic yield depends on the matching between the Lissajous polarization curve and the crystalline axes, which can be harnessed for the detection of inversion symmetry. The circular polarization of the generated harmonics enables probing of the crystalline chirality in quartz and ferromagnetism on the surface of MgO. Using harmonics at the band resonance, we extract the symmetryresolved energy shearing of helicitydependent (quartz) and spinpolarized (MgO) bands with a fitting model incorporating a polarizationdependent absorption coefficient. Compared to dichroic absorption spectroscopy with external light sources, the symmetrysensitive, local generation as well as the large bandwidth and wavelength tunability of the harmonics represent clear advantages. Furthermore, the relaxed requirements on the vacuum conditions and the sample preparation allow for a wider range of materials to be studied. This is particularly relevant for insulating materials for which chiral spectroscopy on thinfilm samples is possible but challenging with electronbased methods like spinresolved photoemission spectroscopy^{46,47} or spinpolarized scanning tunneling microscopy^{48,49,50} due to surface charge and surface contamination over time^{51}. Besides illustrating an alloptical scheme for surfacesensitive spectroscopy, our method also facilitates compact, gasfree extremeultraviolet light sources for experiments involving ultrahigh vacuum. Moreover, extending the spectroscopic capabilities of our approach, the inherent femtosecond pulse duration of the harmonics and the breaking of expected selection rules imply a powerful, singleshot probe for ultrafast dynamics. In addition to the detection of global properties such as orientation, chirality, or phase transitions, setting the HHG on a particular resonance, e.g., in semiconductors^{1,3,9,52} or insulators^{4,10}, will allow targeting other surfaces phenomena with the potential of direct or diffractive imaging^{11}. Thus, this work opens a path for detailed ultrafast surface spectroscopy and microscopy of dielectric and insulating materials.
Methods
Crystals
In our experiment we use commercially available (MTI crystals), singlecrystalline silicon dioxide (quartz), and magnesium oxide (MgO) targets with 10 mm × 10 mm lateral size and 300 µm thickness. The quartz crystal is oriented along the 0001direction perpendicular to the zplane (space group P3_{1}21 or P3_{2}21). Two additional MgO crystals are cut perpendicular to the [111] and [100]direction, respectively.
The laser system and twocolor bicircularlypolarized driving fields
The laser system used in this study (Coherent Elite amplifier with Vitesse seed oscillator) delivers milliJoulelevel, 50 femtoseconds (FWHM) pulses with 800 nm central wavelength (see the red spectrum in Supplementary Fig. S1b) at a 1 kHz repetition rate. We are able to control the excitation power with a variable attenuator and the incident polarization with a halfwave plate.
Bicircularly–polarized twocolor light fields are generated in an inline MAZELTOV (MAchZEhnderLess for Threefold Optical Virginia spiderwort) apparatus. Supplementary Fig. S1a depicts the setup, which consists of a 40 cm planoconvex lens and a BBO crystal for the generation of secondharmonic laser pulses with 400 nm wavelength (see the blue spectrum in Supplementary Fig. S1b) and a polarization perpendicular to the fundamental polarization direction. Two thick horizontally counterrotational tiltable calcite plates control the timing (overlap) of the two pulses and a dichroic quarterwave plate converts the linear polarized pulses to oppositely circularly polarized fields. This configuration allows for the generation of bicircularlypolarized twocolor laser pulses with a defined relative phase, which leads to a threefold symmetric field distribution with fixed orientation in space. Therefore, the quarterwave plate’s fast optical axis is set to ±45° relative to the linearly polarized fundamental where the sign determines the direction of rotation. A wave plate angle of 0° leaves the perpendicular linearpolarizations of the two pulses unchanged. An additional tiltable thin calcite plate mounted on a rotational stage is placed before the quarterwave plate, in order to finetune and scan the relative phase of the fundamental and the secondharmonic pulses. The phase delay between the fundamental and the secondharmonic pulses controls the relative rotational orientation of the threefold field distribution to the crystal axis (compare angle \(\theta\) in Fig. 1). More details on the MAZELTOV apparatus and the generation principle can be found elsewhere^{53}. Removing the MAZELTOV apparatus and the quarterwave plate allows for the use of linearlypolarized laser excitation at the fundamental wavelength, with variable polarization angle via the halfwave plate.
In the experiment, the relative power of the second harmonic field to its fundamental was 6% for the measurements on quartz and 1% in the case of MgO. Considering the wavelengthdependent focusing of the 800 nm and 400 nm fields, the field amplitude ratios are 1:2 and 1:5 in the quartz and MgO measurements, respectively, which determines the actual shape of the driving fields in both cases (see Fig. S1a). Supplementary Fig. S1c, d show the threefold driving Lissajous curves for a field amplitude ratio of 1:2 for right (c) and left (d) helical rotation. An altered amplitude ratio has no impact on the rotational symmetry of the field nor the helicity of emitted harmonic radiation. It does, however, influence the spatial isotropy, i.e., the degree of angular modulation, of the field, which has an impact on the angle resolution in polarization scans.
We calculated the driving field under consideration of nonoptimal quarterwave plate retardance (see Fig. 1b). Therefore, we quantify the specified frequencydependent retardance \(R\left(\omega \right)\) with two linear fits at 400 nm and 800 nm wavelength. In addition, the spectrum of the driving pulse was measured and fitted by two Gaussians. Under the assumption of a Fourier limited pulse, we determined the frequencydependent electric field \(F(\omega )\). The timedependent electric field can be calculated by a Fouriertransformation \(F\left(t\right){\mathfrak{=}}{\mathfrak{F}}{[F(\omega ){{\rm{e}}}^{{\rm{i}}R\left(\omega \right)}]}\) and is compared to the field generated with a perfect quarterwave plate. One optical cycle of the electric field \(F\left(t\right)\) and its difference \(\Delta F\) to an undistorted twocolor field is shown in Supplementary Fig. S1c, d. The total driving field as well as the difference exhibit a threefold cloverleaf shape. We find that a lefthelical field is accompanied by a righthelical residual component and vice versa (see Fig. S1c, d). As the residual field counter rotates a weak sixfold beating is expected when scanning the cloverleaf field orientation.
Vacuum system and extremeultraviolet spectrometer
We utilize a highvacuum system at a base pressure of 1e−6 mbar for the HHG and spectral detection of the emitted extremeultraviolet radiation. A lens with a 40 cm focal length steers the laser through an optical window into the vacuum chamber. The beam is then retroreflected of a dualband dielectric mirror (Eksma, 0524080i0) under nearly normal incidence (horizontal and vertical incidence angles <1°) and focused to a focal spot size of 120 µm in diameter (fullwidth at halfmaximum) on the target crystal surface. Due to the reflection geometry of the setup, the incident horizontal angle of the laser beam on the target is 5°, whereas the vertical incident angle is <1°. The generated highharmonic radiation is collected with a focusing flatfield spectrometer (McPherson, model 234, grating with 1200 grooves/mm) and recorded with a phosphorscreen microchannel plate detector using a CCD camera.
Highharmonic spectra and spectrograms
Highharmonic spectra from the three crystal targets are shown in Supplementary Fig. S2a–c. The intensity of highharmonics generated with the threefold bicircular driving field (red) is compared to the signal generated with a twocolor perpendicularlinearly polarized field (blue) to assess the suppression of individual harmonic orders. By measuring the spectra with bicircular excitation as a function of the threefold field rotation \({\rm{\theta }}\) we obtain the spectrograms, which are presented in Supplementary Fig. S2d–f. The spectrallyintegrated signal of the eighth harmonic is used to generate the polar plots in Fig. 2a–c.
Helicitydependent splitting of onresonance harmonic peaks
The difference spectrograms in Fig. 3a, b and Supplementary Fig. S4 are determined from spectrograms, similar to those shown in Supplementary Fig. S2d–f, which are recorded for both helicities of the fundamental. The fifth harmonic generated in both facets of MgO and the seventh harmonic generated in quartz shows a spectral shift that stems from a chiral response in both materials systems.
We model the effect of magnetic circular dichroism in MgO and structural circular dichroism in quartz via fitting of a helicitydependent absorption to the observed spectral shearing of the onresonance harmonics. Considering energy splitting ΔE in the absorption channels for the two polarization states of the harmonics (cf. Fig. 4b, d) the helicitydependent attenuation is described by energeticallyshifted literature values of the absorption coefficient k(E±ΔE/2) (Supplementary Fig. S5). Here, the signs determine the circular polarization of the highharmonic radiation. The respective fifth and seventh harmonics are close to the resonances in MgO and quartz, such that even small energetic shifts of k(E) lead to a pronounced change in the absorption of an initial harmonic intensity \({I}_{0}(E)\). The intensity of shifted harmonics with left and rightcircular polarization (\({I}_{{\rm{L}},{\rm{R}}}\left(E\right)\)) then result from the following two equations for the respective crystal systems:
where \({{I}}_{0}\) is the initially unshifted harmonic peak and λ is the wavelength of the harmonic. We used Eq. (1) and (2) to calculation the emitted harmonic intensity by estimating \({I}_{0}(E)\) as the mean of the recorded left and rightcircular polarized spectra. In the case of quartz, the spectrum was additionally fitted by a gaussian function to compensate for the saturated signal in the corresponding measurement. A helicitydependent bulk effect is expected in quartz harmonics generated from every depth x and weighted by their chiral absorption contribute to the emitted intensity. Contrary, in MgO we use an additional fit parameter \(\delta\) to describe the interaction length in a nearsurface region where the absorption is helicitydependent. We assume that all generated harmonic radiation passes this surface near the region, neglecting the generation in the layer itself. This is justified since the major portion of the detected radiation is generated beyond this surface region as can be approximated by the penetration depth \({\delta }_{{\rm{HHG}}}\).
In addition to a chiral material response, we also considered the energy splitting of excitonic states due to an optical stark effect from the residual field as a possible cause for shifted highharmonics. Since changing of the fundamental helicity lead to inversion of all helicities including the residual fields (see Fig. S1c, d), we can conclude that no symmetrybreaking field is present and no stark shift signal is expected in our case^{54,55}. Linear field components that result from the nearnormal incidence of the bicircular field on the crystal could lead to a symmetrybroken excitation. However, under consideration of the Fresnel coefficients, the calculated field values are many orders of magnitude too low for a significant optical stark effect.
Data availability
The data that support the finding of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors thank Hugo LourençoMartins for helpful comments on the manuscript. This work was funded with resources from the Gottfried Wilhelm Leibniz Prize. O.K. gratefully acknowledges funding from the European Union’s Horizon 2020 research and innovation program under the Marie SkłodowskaCurie grant agreement No. 752533. P.B.C. acknowledges funds from the United States Air Force Office of Scientific Research (AFOSR) under award numbers FA95501610109.
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M.S. conceived and designed the experiment with contributions from M.T. and O.K. M.S., T.H., and M.T. conducted the experiments with contributions from O.K., analyzed the data, and prepared the manuscript with contributions from O.K. P.B.C., A.S., and C.R. All authors discussed the results and interpretation.
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Heinrich, T., Taucer, M., Kfir, O. et al. Chiral highharmonic generation and spectroscopy on solid surfaces using polarizationtailored strong fields. Nat Commun 12, 3723 (2021). https://doi.org/10.1038/s41467021239999
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DOI: https://doi.org/10.1038/s41467021239999
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