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High-harmonic generation (HHG) in bulk crystal is attributed to the sub-cycle electronic motion driven by an intense laser field2,3,4,5,6,7,8,9,10,11. There has been a growing interest in utilizing HHG to probe the electronic structure of solids8,9,11. Vampa et al. reconstructed the momentum-dependent bandgap of ZnO along the Γ–M direction using HHG from a two-colour driving field11. Luu et al. retrieved the energy dispersion of the lowest conduction band of SiO2 assuming that the harmonics are produced by the intraband currents8. The dependence of solid-state HHG on the coupling of multiple electronic bands has also been identified with the production of even harmonics in GaSe9 and the emergence of a second plateau in rare-gas solids12. These findings show the possibility of using solid-state HHG to probe the electronic band structures in solids, but the analyses are so far limited to one dimension. For a complete electronic structure, it is desirable to exploit the microscopic process to measure the periodic potential in three dimensions (real space). This is analogous to tomographic imaging of a molecule, where the three-dimensional spatial information (that is, orbital wavefunction) of the target molecule is extracted13,14,15. Those measurement techniques are based critically on the dependence of HHG efficiency on molecular alignment with respect to the laser field16.

In this letter, we demonstrate the strong sensitivity of HHG to the atomic-scale structure in the cubic wide-bandgap crystal MgO. First, using a linearly polarized field, we measure a highly anisotropic angular distribution in high-harmonic signal—despite the isotropic linear and weakly anisotropic nonlinear optical properties of the cubic crystal in the perturbative regime17. Second, we observe a strong ellipticity dependence of the HHG yield similar to the gas-phase HHG18 for small ellipticity when the major axis is aligned along the bonding direction. For other orientations and large ellipticity, we observe notable differences from the gas and other solids such as ZnO (ref. 2), including a significant signal for its circular polarization. To understand the origin of anomalous anisotropy we perform semi-classical calculations of real-space electron trajectories inside the MgO crystal for electrons that have tunnelled across the direct bandgap. We find that the harmonic signal is strongly enhanced or diminished when the electron wave packet is directed into or away from neighbouring atomic sites. For a perfect crystal, the dynamics are identical for all sites, leading to a simple real-space picture of HHG in solids as arising from coherent collisions of the delocalized charge carriers with the periodic potential.

In the experiment, we measure the crystal orientation-dependent high harmonics for a linear polarized pump by rotating the sample around the axis given by the laser propagation (〈001〉) (see Fig. 1a and Methods). This is similar to the tomographic imaging method in gas-phase molecules, where HHG in laser-aligned molecules is measured. Shown in Fig. 1b, the measured spectrum covers odd harmonics from 13th to 21st. A clear four-fold symmetry is seen on all harmonics, with a set of sharp maxima along θ = 0°, 90°, 180° and 270°, which are the Mg–O, 〈100〉 nearest-neighbour bonding directions. A second set of maxima can be seen along 45°, 135° and 225°, which are Mg–Mg (or O–O), 〈110〉 nearest-neighbour directions. Figure 1c shows the lineout for individual harmonics, which exhibits sharp features along the Mg–O bonding directions. The yield is higher and the angular distribution is sharper along the Mg–O bonding directions compared to the Mg–Mg (or O–O) directions. This effect becomes especially prominent for the near-cutoff harmonics. The angular width of the maxima along the Mg–O bonding direction is ∼10°, which approximates the angle subtended by the O atom to the Mg atom considering their atomic radii19.

Figure 1: Orientation-dependent high-harmonic generation in MgO crystal.
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a, A strong infrared laser beam with wavelength centred at 1.3 μm is focused onto a 200-μm-thick, 001-cut MgO crystal to generate extreme ultraviolet high-harmonic radiation. To study the orientation dependence of high-order harmonic emission, the crystal is rotated to various angles (θ) with respect to the fixed laser polarization. b, Measured harmonic spectrum for various orientations. A strong dependence is seen in all 4-fold bonding directions. The maxima along θ = 0°, 90°, 180° and 270° correspond to the cubic bonding direction. Secondary maxima along θ = 45°, 135° and 225° correspond to the O–O directions. c, Lineout (integrated intensity) corresponding to various harmonic orders from 13th to 21st. Measured yield on harmonics are scaled as shown in the legend.

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The strong anisotropy observed for the linear polarized pump with the crystal orientation cannot be explained just from the nonlinear optical susceptibility using perturbation theory. We note that Schubert et al. measured the angular distribution of below-threshold harmonics in GaSe, which followed the second-order optical susceptibility4. We also study the orientation dependence of the third harmonic, which corresponds to the lowest-order nonlinearity for a system with inversion symmetry. The orientation dependence of the third-harmonic generation17 is expected to be only slightly anisotropic based on the measured two independent components of the third-order susceptibility tensor20 (χ1111(3) and χ1122(3)). This is confirmed by our measurement at comparatively low field (∼0.25 V Å−1), shown in Fig. 2. However, when the field is increased to the ∼1 V Å−1 level, the orientation dependence deviates from the perturbative response seen at relatively low fields.

Figure 2: Orientation dependence of the third harmonic in MgO.
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The measurement procedure is similar to Fig. 1a. The green dots are for moderate field strength (∼0.25 V Å−1) and red dots (not to scale) are for the highest field strength (∼1 V Å−1). The solid lines are calculated results from the perturbative model considering third-order susceptibility tensors. At moderate field, the measurement reproduces the perturbative model, but at higher field additional maxima occur along 0°, 90°, 180° and 270°, which are the directions of the maxima in the high-order harmonics in Fig. 1b.

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We attribute the anomalous anisotropy of HHG to a nonperturbative, highly directional, field-induced nonlinear current. One potential contribution would be from the possible dependence of tunnelling rate on crystallographic directions if the reduced electron and hole mass were different in different directions21. In MgO, the effective mass of electron and hole are isotropic around the minimum bandgap at the Γ point22. Therefore, the strong anisotropy we observe cannot be explained just from tunnelling rates at the zone centre. In MgO, a large electronegativity gradient exists along the Mg–O, 〈100〉 bonding direction (Mg++/O−−); therefore, this could be a preferred crystallographic direction for field-induced charge transfer23,24. To analyse this possibility at the microscopic level, we consider semi-classical electron trajectory calculations in the crystal. In the gas phase, such analysis has proved useful in capturing basic recollision dynamics, including the anisotropy in molecular HHG13. We extend that analysis to the solid state by including the effects from the periodic potential that produce the non-parabolic energy dispersion and folding of the band structure. In particular, we seek to test whether high harmonics in solids could also be viewed from a real-space picture involving coherent collisions of Bloch electrons, with neighbouring atomic sites. In the limit that the wavefunction is delocalized in the solid state, we expect that identical semi-classical trajectories repeat at every unit cell. In MgO, we assume that valence to conduction band tunnelling originates at the zone centre from the O site, since the upmost valence band is primarily from the 2p orbital of O (ref. 25). Next, the laser field accelerates the electron depending on the laser phase and field strength. The electron is constrained to move in a plane that includes Γ–X (0°, 90°, 180° and 270°) and Γ–K directions (45°, 135°, 225° and 315°). In Fig. 3, we show the electron trajectories under laser fields polarized along θ = 0°, 12°, 45°, 78° and 90°. It is seen that along the cubic bonding directions (for example, 0° and 90°), the electron trajectories cross several Mg sites. This is the direction of large electronegativity differences connecting distinct atomic species and corresponding to the maxima in the HHG spectrum along with its sharp angular features. At the diagonal directions (for example, 45°), trajectories connect atomic sites of the same species, and therefore same electronegativity, resulting in secondary maxima in the harmonic signal. Finally, for angles such as 12° and 78°, the trajectories do not connect nearby atomic sites, compatible with the observed minima in the HHG signal. We note that the hole motion would be qualitatively similar to the electrons, but travelling in the opposite direction.

Figure 3: Semi-classical electron trajectories inside MgO crystal.
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Calculated semi-classical electron wave packet trajectories for linearly polarized laser fields at different crystal orientations. The origin represents the position of the electron wave packet about an arbitrary O atom immediately after tunnelling across the direct bandgap at zone centre and near the peak of the field. Magenta and blue lines are for 0° and 90°, where the electron trajectory connects the O–Mg–O sites. Red line is for 45°, where the electron trajectory connects O–O sites. Cyan and green lines are for 12° and 78°, respectively, where the electron does not effectively scatter from the atomic sites. These angles correspond to the observed minima in the angular distribution of the measured harmonic signal in Fig. 1b.

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The sensitivity of HHG to the atomic-scale structure of the crystal becomes more evident through the laser ellipticity dependence. Figure 4a, b shows the ellipticity dependence of the 19th harmonic when the major axis of the ellipse is kept along the Mg–O and O–O bonding direction, respectively. In the Mg–O case, the yield is maximum for linear polarization (ellipticity ɛ = 0) and decreases quickly, reaching the noise level at ɛ ∼ 0.4. This behaviour is very similar to the gas-phase harmonics in the strong-field limit18,26, albeit with a quicker decrease, potentially due to the confinement of the electron density around the atomic sites in the MgO crystal. A striking difference with respect to the solid-phase measurements occurs for ɛ > 0.4 as the signal begins to grow again, reaching a secondary maximum at ɛ = 1. This is compatible with our picture, based on real-space trajectories, which allows the generation of harmonics with an elliptical field through scattering at the nearest-neighbour site. In Fig. 4c, we show the semi-classical trajectories for ɛ = 0, 0.4 and 1. At ɛ = 0, the electron collides into Mg sites; for ɛ = 0.4, the electron remains collision-free in the unit cell; and for ɛ = 1, the electron collides into the O site. This is compatible with observed maxima, minima, and secondary maxima at ɛ = 0, 0.4, and 1, respectively, in Fig. 4a. In the case along the O–O bonding direction, the yield is maximum at ɛ = ±0.65, and becomes minimum at ɛ = ±1, compatible with the expectation from trajectory analysis, as shown in Fig. 4d: ɛ = ±0.65 trajectories connect to Mg sites, but ɛ = ±1 trajectories do not collide into atomic sites in the unit cell. We note that the semi-classical electron trajectory analysis does not predict the quantitative harmonic efficiency either for linear and elliptical polarizations.

Figure 4: Ellipticity dependence of HHG in MgO crystal.
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a, Measured 19th-order harmonic yield as a function of laser ellipticity when the major axis is fixed along the Mg–O bonding direction. Yield is maximum at linear polarization (ɛ = 0). It then decreases rapidly with increasing ellipticity, faster than the prediction from the perturbative model (shown in dotted blue line) and gas-phase HHG18. A stark difference to the gas phase arises when |ɛ| > 0.5, where yield increases until it reaches another maxima at circular polarization, |ɛ| = 1. b, Measurement results with the major axis along the O–O (or Mg–Mg) direction. HHG is not maximized with linear but elliptical polarization, at around |ɛ| ∼ 0.65. c, Semi-classical electron trajectories inside MgO crystal for various ellipticity values when the major axis is kept along the Mg–O direction. For linear polarization (red curve), the electron trajectory connects Mg sites efficiently. For |ɛ| = 0.4, the trajectory misses atomic sites until a propagation distance of about the unit cell. At |ɛ| = 1, the trajectory begins to connect the O site. Collision with the first neighbour (O) is expected to be more significant than with the second and so on. d, Semi-classical electron trajectories inside MgO crystal for various ellipticity values when the major axis is along the O–O direction. For linear polarization, electrons do not connect any Mg sites but only O sites. A fairly large ellipticity, |ɛ| = 0.65, is required for the trajectory to connect the Mg sites, as shown by the green and grey lines. Finally, at |ɛ| = 1, the trajectory does not connect any atomic sites within a few unit cells.

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Finally, to test this simple picture of HHG from coherent collisions in periodic solids, we perform a crystal rotation test while keeping the major axis of the polarization ellipse along the vertical direction. In this configuration, when the crystal is rotated to the left (−) by 15° along the laser direction, we find that the main peak is shifted towards the left elliptical polarization direction at about ɛ = −0.2 (see Fig. 5a). Similarly, when the crystal is rotated to the right (+) by 15°, the main peak is shifted towards the right elliptical polarization direction at about ɛ = 0.2 (see Fig. 5b). As shown in the corresponding electron trajectory analysis in Fig. 5c, d, in these rotated configurations, elliptical fields of appropriate handedness are necessary to steer the electron such that trajectories connect to the Mg atom.

Figure 5: Helicity dependence of HHG in a rotated MgO crystal.
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a, Ellipticity dependence with the major axis fixed in the vertical direction while the crystal is rotated by θ = −15°. This rotation maximizes the harmonic yield at ɛ ∼ −0.2. b, Ellipticity dependence when the crystal is rotated by θ = + 15°. This rotation maximizes the yield at ɛ = + 0.2. c,d, Semi-classical electron trajectories for angles θ = −15° and θ = + 15°, respectively. The electron trajectories show that it is much more likely for electron trajectories to connect the Mg sites at ɛ = ±0.2.

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To explain the observation of a harmonic signal for an elliptically polarized field—in particular the circularly polarized field—it is necessary to include the role of multiple atomic sites. This is an important difference from the generalized recollision model discussed recently by Vampa et al. 10,27 and from atomic HHG (see Supplementary Information). The ellipticity dependence in solids is also material-dependent; for example, it is relatively weak in ZnO (ref. 2), but fairly strong in rare-gas solids12. As these crystals exhibit different bonding character, the nature of the bonding in solids seems to play an important role in defining the solid-state strong-field response. In the atomic case, the ellipticity dependence is largely independent of the atom, as seen in Ar, Ne and Kr (ref. 17). In solids, electrons are never far from the atomic cores; thus, the usual strong-field approximation breaks down and the role of interatomic bonding and nearest neighbour becomes important. That is the origin of materials-dependent strong-field response.

We demonstrate strongly anisotropic high-harmonic generation in single-crystal MgO. For a linearly polarized field, the HHG signal is strongly enhanced along the Mg–O, 〈100〉 bonding direction. Harmonics also show a strong laser ellipticity dependence, including a non-negligible signal for circular polarization. A comparison to our simulated semi-classical, real-space electron trajectories show that harmonics are enhanced when the electron traverses across atomic sites inside the crystal. Generation of high harmonics even for the circularly polarized laser fields is consistent with scattering from neighbouring sites in the crystal. These results open up the possibility of measuring field-induced charge flow23,24,28,29 and imaging atomic orbitals in solids30, which we discuss in the Methods. In analogy to the conventional electron diffraction methods, solid-state HHG allows us to use electrons that are already present in the bulk matter to image the ångström-scale structure of the condensed matter. Since these measurements are in principle sensitive to almost instantaneous charge density, the implications are in probing optical field-induced multi-petahertz dynamics in solids.

Methods

Extreme ultraviolet (XUV) harmonics are produced by focusing a near-infrared laser beam into a 200-μm-thick, 001-cut MgO single crystal (010-cut on edge) at normal incidence, as shown schematically in Fig. 1a. The laser wavelength is 1,333 nm, and the pulse duration is ∼50 fs. The maximum peak intensity in the vacuum is ∼1013 W cm−2 (peak field ∼1 V Å−1), which is below the damage threshold for repetitive excitation at 1 kHz. We record the harmonic spectrum with an imaging spectrometer consisting of a flat-field variable groove density grating and micro channel plate. We measure odd-order harmonics from 13th to 21st, covering from ∼12 eV to ∼20 eV of photon energy in the XUV range. The lower photon energy is the detection limit, while the 21st harmonic is at the high-energy cutoff of the spectrum. The spectrum is not corrected for the sensitivity of the grating.

For the orientation dependence study, we fix the laser polarization and rotate the crystal so we can exclude the polarization sensitivity of the diffraction grating. For the ellipticity dependence study, we change the laser ellipticity with a combination of a quarter-wave plate and a half-wave wave plate such that the direction of the major axis is kept fixed. Similar to HHG, we study the orientation dependence of third-harmonic generation by fixing the laser polarization and rotating the crystal. We measure the third harmonic using a visible spectrometer without polarization analysis.

Here we discuss the possibility of using the results to image atomic orbitals in solids. In the interband picture6,27, the harmonic signal is proportional to the dipole moment30, q∫ unit cellUCB, k∗(r)r UVB, k(r)d3r, where q is the electronic charge, r is the position, and UCB, k∗ and UVB, k are the periodic part of the conduction band (CB) and valence band (VB) Bloch functions, respectively. Therefore, the VB Bloch function could be evaluated in principle with the knowledge of the CB Bloch function and vice versa. This points to the possibility of all-optical methods for measuring the charge density distributions in solids.

Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.