Abstract
Attosecond metrology sensitive to subopticalcycle electronic and structural dynamics is opening up new avenues for ultrafast spectroscopy of condensed matter. Using intense lightwaves to precisely control the fast carrier dynamics in crystals holds great promise for nextgeneration petahertz electronics and devices. The carrier dynamics can produce highorder harmonics of the driving field extending up into the extremeultraviolet region. Here, we introduce polarizationstateresolved highharmonic spectroscopy of solids, which provides deeper insights into both electronic and structural subcycle dynamics. Performing highharmonic generation measurements from silicon and quartz, we demonstrate that the polarization states of the harmonics are not only determined by crystal symmetries, but can be dynamically controlled, as a consequence of the intertwined interband and intraband electronic dynamics. We exploit this symmetrydynamics duality to efficiently generate coherent circularly polarized harmonics from elliptically polarized pulses. Our experimental results are supported by abinitio simulations, providing evidence for the microscopic origin of the phenomenon.
Introduction
The study of lightwavedriven electronic dynamics occurring on subopticalcycle time scales in condensed matter and nanosystems is a fascinating frontier of attosecond science originally developed in atoms and molecules^{1}. Adapting attosecond metrology techniques^{2} to observe and control the fastest electronic dynamics in the plethora of known solids and novel quantum materials^{3} is very promising for studying correlated electronic dynamics (e.g., excitonic effects, screening) on atomic length and time scales, thereby potentially impacting future technologies such as emerging petahertz electronic signal processing^{2,4} or strongfield optoelectronics^{5,6}.
The nonlinear process of highorder harmonic generation (HHG) in gases is one of the cornerstones of attosecond science and is well understood by the semiclassical threestep model^{1}. In solids, nonperturbative HHG up to 25th harmonic order without irreversible damage was first reported in ref. ^{7}. This work triggered extensive research activities aimed at unraveling the microscopic interband and intraband dynamics underlying HHG from crystals (for a comprehensive overview, see ref. ^{8}), thereby extending attoscience techniques to solids. The prevailing strongfield dynamics were successfully identified in specific cases, even if a global picture has not yet emerged. Other works demonstrated isolated attosecond extreme ultraviolet (XUV) pulses emitted from thin SiO_{2} films^{9}, or investigated HHG from twodimensional (2D) materials such as graphene^{10}, 2D transitionmetal dichalcogenides^{10,11}, and monolayer hexagonal boron nitride^{12}.
Elucidating the complex microscopic electronic dynamics producing HHG without making a priori severe assumptions poses a challenge for theory. Indeed, the theory must capture at the same time the transitions between discrete electronic bands, and the ultrafast motion of electrons within the bands; two mechanisms usually decoupled in the description of either optical properties or transport in semiconductors and insulators. An effective way to account for the full interacting manybody electronic dynamics and real crystal structure is using abinitio timedependent density functional theory (TDDFT) simulations^{13,14,15}. Some of us recently used this theoretical framework to reveal how the microscopic mechanisms governing HHG in solids depend on the ellipticity of the driving field and the underlying band structure^{15}. That work predicted that different harmonics react differently to the driver ellipticity, as they can either originate mainly from intraband contributions or from coupled interband and intraband dynamics^{14}.
The symmetry properties of the lightmatter interaction Hamiltonian distinguishes HHG in crystals from atoms and molecules, with major ramifications for the selection rules of different harmonics and their polarization states. HHG from atoms driven by propellershaped bichromatic waveforms produces circular harmonics^{16} (for convenience, we often use the terminology linear/circular/elliptical instead of linearly/circularly/elliptically polarized to refer to the polarization state). In molecules, both the point group and the driving field determine the symmetries of the coupled lightmatter system. Consequently, depending on the molecular symmetries and the molecule’s orientation with respect to the light polarization direction, elliptical highorder harmonics can be produced by linear^{17} or elliptical driver pulses^{18}. For bichromatic bicircular driver fields, circular harmonics with alternating helicities can be obtained, provided that the molecule’s symmetries are compatible with that of the driving field^{19}. In crystals, several recent works studied the highharmonic response on driver pulse ellipticity ε, which can strongly differ qualitatively from the atomic and molecular cases. Whereas earlier work^{20} looked exclusively at the harmonic yield, later research also investigated the polarization states and selection rules of the higher harmonics from various solids of different crystal symmetries^{15,21,22} and reported circular HHG from a singlecolor driver field^{21,22}, which is symmetry forbidden in atoms.
Here, we present a combination of HHG experiments and firstprinciples TDDFT simulations for silicon and quartz, demonstrating that a complete understanding of the harmonics’ polarization states requires, beside knowledge of the crystal’s symmetries, a microscopic understanding of the underlying complex, coupled interband and intraband dynamics^{14,15}. Most importantly, we demonstrate strongfield control of the harmonics’ polarization states. Our findings indicate that polarizationstateresolved highharmonic spectroscopy of solids provides deeper insights into both electronic and structural dynamics as well as symmetries on subcycle time scales. This spectroscopy technique therefore might find important applications in future studies of novel quantum materials^{3} such as strongly correlated materials^{23,24}, topological insulators^{25}, and magnetic materials^{26}. Moreover, compact sources of bright circularly polarized harmonics in the XUV regime might advance our tools for the spectroscopy of chiral systems^{18} and 2D materials with valley selectivity^{11}.
Results
Highharmonic generation experiments
In our experiments, we irradiated freestanding, 2μmthin, (100)cut silicon samples with 120fs, 2.1μm (0.59 eV) pulses with tunable ellipticity ε and peak intensities up to 0.7 TW cm^{−2} in vacuum (see Methods section and Supplementary Figure 1). At this intensity, the harmonics are generated nonperturbatively (see Supplementary Figure 3) up to harmonic order 19 (HH19) in the XUV regime for our experimental conditions, as shown by our TDDFT simulations (see Supplementary Figure 5). Only harmonics up to HH9 are detected by the spectrometer used in our experiments. We also irradiated 50μmthin, zcut quartz with an estimated intensity of 40 TW cm^{−2} in vacuum.
Figure 1 shows the measured highharmonic response in Si of HH5, HH7, and HH9 as a function of driver ellipticity ε and sample orientation θ; panels a–c display normalized harmonic intensities, panels d–f harmonic ellipticities \(\varepsilon _{{\mathrm{HH}}} = \sqrt {I_{{\mathrm{min}}}/I_{{\mathrm{max}}}}\), where I_{min} and I_{max} correspond to the intensities at the minor and major axes of the polarization ellipse. In all panels, θ = 0°, 45° refer to the major axis of the driving field ellipse along the directions [100] (ΓX) and [110] (ΓK) in real (reciprocal) space. The crystal symmetries are recovered in all maps shown in Fig. 1.
All harmonics respond in a distinctly different way to the driver pulse ellipticity ε, and the harmonic yields peak for different sample rotations. HH5 exhibits monotonically decreasing yield versus ellipticity profiles for all sample rotations, resembling the Gaussianshaped profile in the atomic case^{27}. We therefore call such profiles atomiclike. The distribution is symmetric around ε = 0 (see white dotted centerofmass curve) for all sample rotations. The intensity distribution of HH7 (Fig. 1b) shows intriguing, nonatomiclike features with maximum yield at nonzero ellipticity for certain sample rotations, similar to experiments on MgO^{20}. HH9 (Fig. 1c) exhibits the most pronounced deviations from a Gaussianlike ellipticity profile, with nonmonotonic nonatomiclike profiles for wide ranges of sample rotation. Its yield is strongly asymmetric with respect to ε = 0 for all sample orientations (different from mirror planes), and displays strong nonsinusoidal oscillations of the centerofmass curve.
The overall behavior can be understood by inspecting the Si band structure: HH5 (2.95 eV) is below the direct Si bandgap of 3.1 eV. This harmonic thus originates purely from intraband dynamics of lowenergetic electrons, which mostly remain within the parabolic region of the bands, leading to an atomiclike behavior. For abovebandgap harmonics, the joint density of states (JDOS) (see Supplementary Figure 6), i.e., the density of optical transitions at a given energy, determines the relative weight of interband compared to intraband mechanisms^{14}. Around 5.3 eV (HH9), the JDOS is significantly lower than for 4.1 eV (HH7). Therefore, while coupled inter and intraband dynamics lead to the emission of HH7, HH9 is mostly produced by intraband effects^{14}. Interestingly, these harmonics are more efficiently generated with different helicities, as can be seen from the different signs of the centerofmass curves for certain sample rotations (see Supplementary Figure 7). This clearly indicates different generation mechanisms of HH7 and HH9, as predicted in ref. ^{15}. For HH9, for which interband transitions are strongly suppressed by the low JDOS at this energy, higherenergetic electrons explore larger nonparabolic regions in the bands, which results in pronounced nonatomiclike ellipticity profiles.
Figure 1d–f reports the measured harmonics’ polarization states as a function of driving ellipticity and sample rotation. Whereas linear drivers yield almost linear harmonics, we observe astonishing deviations of the harmonic ellipticities ε_{HH} from the driver ellipticity ε. Consistent with our TDDFT predictions^{15} and selection rules in ref. ^{28}, for circular driver pulses, ε ≈ 1, all harmonics become circular ε_{HH} ≈ 1. Most importantly, for all observed harmonics, circular harmonics can be generated from elliptical driving polarizations, as elaborated on below. These islands of high ellipticity sensitively depend on ε and θ in the cases of HH5 and HH9; however, for HH7 this sensitivity is less pronounced. This observation is again consistent with a strong dependence of the microscopic mechanisms on the polarization state of the driving field, as the electrons explore different regions of the Brillouin zone (BZ) depending on ε and θ. The measured harmonics’ polarizations contain the complete information on the x and ycomponents of the harmonics’ amplitudes and their relative phases.
Figure 2 summarizes our findings on circular harmonics from circular drivers. In both silicon (Fig. 2a) and αquartz (Fig. 2b), all harmonic intensities remain constant while rotating a polarizer by 360°, thus confirming circular harmonic polarization. In Fig. 2c, we observe a strong intensity suppression of HH3 going from linear to circular driver, as expected from the selection rules for the D_{3}[32] group^{28} of αquartz. The selection rules also manifest themselves in the helicities of the circular harmonics. In accordance with grouptheoretical considerations^{28} and TDDFT simulations^{15}, the odd harmonics from Si have alternating helicities as Si has point group O_{h} [m3m] and is fourfold symmetric for our [001]cut sample. This was confirmed with a tunable quarterwave plate (QWP) behind the sample, which converts circular to linear polarization, with the polarization angle δ depending on the helicity (see Fig. 2d). The trigonal crystal structure of αquartz results in different selection rules, leading to alternating helicities of HH4 and HH5 in Fig. 2e. As shown in Supplementary Note 9, we extracted the Stokes polarization parameters of the harmonics from these measurements and estimated a value of the degree of polarization of 0.8 ± 0.2 for all harmonics, similar to reported values for the generation of circular harmonics from atomic and molecular gases^{29,30}. Moreover, we find in Si that the harmonic ellipticities ε_{HH} are all close to 1, independent of sample rotation θ (see Fig. 2f). This isotropic behavior supports that driver pulses are almost perfectly circularly polarized.
Figure 3a shows two polarizer scans under excitation conditions, for which HH9 and HH7 are circular for elliptical driver polarization. The measured harmonic ellipticities ε_{HH} are ~0.93 in both cases. We also found similarly high ellipticities for HH5 (see Supplementary Figure 8). Figure 3b shows the intensity dependence of the harmonic ellipticities for ε = 0.4 and θ = ΓX + 10°. By varying the intensity of the driving field, we achieve a high degree of control over the harmonics’ polarization states. This key result has two important consequences: First, it shows that the relative importance of interband and intraband mechanisms is not a material property only, but strongly depends on excitation conditions, thus offering a broader perspective on the controversial debate about the dominant mechanism responsible for HHG in solids. Second, the observation of circular harmonics for elliptical driver polarization, which sensitively depend on the nonperturbative dynamics of the system, can not be explained by symmetry arguments only, but clearly indicates strongfield control of the harmonic ellipticities ε_{HH} through the lightwavedriven electron dynamics. This might find applications, e.g., in polarizationcontrolled highharmonic sources.
The total harmonic intensities for exemplary cases of circular harmonics for different ε and θ are compared in Fig. 3c. As discussed above, for Si, the harmonic yield tends to decrease (apart from nonmonotonic exceptions) with increasing ε. Therefore, the generation of circular harmonics using elliptical driver pulses (ε < 1) is expected to be significantly more efficient than for circular ones (ε = 1), as indeed observed in Fig. 3c. For HH5 and HH7, the circular harmonics generated for ε = 0.3–0.4 and θ = ΓX + 5° are 10× brighter than for circular driver pulses. In the case of HH9, circular harmonics were even produced with 40% efficiency compared to maximum yield obtained for linear polarization, which corresponds to an 18× yield enhancement going from ε = 1 to ε = 0.2. We have also experimentally confirmed the harmonics’ temporal and spatial coherence from Si (Supplementary Figs. 13 and 14), for both linear and circular harmonics. We found the coherence time to be independent of the harmonics’ polarization state. Circular harmonics exhibit spatial coherence.
Three scenarios are in principle possible to explain the observation of circular harmonics from elliptical driver pulses shown in Figs. 3 and 4: First, the harmonic emission occurs directly with this polarization state. Second, the harmonics are emitted with elliptical polarization and subsequently changed during propagation. Third, the driving pulse’s polarization is altered during propagation due to induced birefringence. Moreover, the presence of the surface and a possible oxide layer might affect the polarization of the harmonics.
Abinitio TDDFT simulations
To address this question, we performed extensive microscopic TDDFT simulations (see Methods section), which at this point do neither account for propagation nor surface effects, computing only the nonlinear microscopic response of the crystal to the incident electric field. For varying ε and θ = ΓX, we computed ab initio the highharmonic response from Si and compared it to our measurements. The results shown in Fig. 4a–c display a remarkable agreement between experimental data and TDDFT calculations. This is true for harmonic yield, harmonic ellipticity as well as the rotation of the harmonics’ major axes. We find minor deviations between calculations and experiments, mostly for HH7 and HH9, which can be expected by the increasing role of light propagation effects for photon energies above the bandgap. However, even in the presence of a surface and propagation effects in experiment, the calculations yield circular harmonics from elliptical drivers exactly for the conditions in which they are observed experimentally. This is shown for HH7 in Fig. 4d. Therefore our abinitio simulations confirm unambiguously that the measured polarization states of the harmonics have a microscopic origin in the coupled inter and intraband dynamics, and is not due to macroscopic propagation effects or induced birefringence. From the comparison between experiments and simulations, it seems that the surface does not play a major role in determining the polarization states of the emitted harmonics.
Discussion
In conclusion, after the first works on circular HHG from solids^{15,21,22}, we aimed at advancing our understanding and to demonstrate that a high degree of control over the polarization states of HHG from solids can be achieved. We found that both crystal symmetry and the nonperturbative coupled interband and intraband dynamics underlying harmonic emission play decisive roles in the polarization states of the emitted harmonics. We have elucidated this duality between symmetry and dynamics in experiments on highharmonic generation from silicon and quartz accompanied by abinitio TDDFT simulations. Our investigation has revealed that both the yields and polarization states of the higher harmonics sensitively respond differently to driver pulse ellipticity, sample rotation, and intensity. In a broader perspective, our results demonstrate that the relative importance of intraband and interband mechanisms is not only determined by the driving wavelength and the material itself, but can be dynamically controlled by the laser intensity.
Circular harmonics can be produced for both circular and elliptical driver polarizations: For circular driver pulses, the circular harmonics have alternating helicities, consistent with the selection rules derived from the crystallographic pointgroup symmetry^{28}. For elliptical driver pulses, circular harmonics were generated for the first time to our knowledge, with up to 40% efficiency compared to linear driver pulses in Si, corresponding to an 18× enhancement compared to circular harmonics from circular drivers. Compact sources of bright circular harmonics from solids extending into the XUV regime might open up appealing new applications in the spectroscopy of chiral systems^{18} and 2D materials with valley selectivity^{11}. Circular isolated attosecond pulses from solids also seem in reach employing appropriate gating techniques. Finally, polarizationstateresolved highharmonic spectroscopy offers the unique advantage of sensitivity to both electronic and structural dynamics on subcycle time scales, thus opening up new avenues for the spectroscopy of quantum materials on extreme time scales^{3,23,24,25,26}.
Methods
Experimental highharmonic generation setup
Supplementary Figure 1 shows the experimental setup used for HHG from crystalline solids. Passively carrierenvelope phase (CEP)stabilized^{31}, 120fs pulses at 2.1 μm (0.59 eV photon energy) are generated in a Ti:sapphirepumped whitelightseeded optical parametric amplifier (OPA^{32,33}. These 2.1μm driver pulses pass through a wiregrid polarizer, a QWP and a halfwave plate, which allow setting the driver ellipticity while keeping the major axis of the polarization ellipse constant (see Supplementary Figure 2). The pulses are focused onto the sample with a 25cm CaF_{2} lens, resulting in a 1/e^{2} focus diameter of 2w_{0} = 95 μm. After 50 cm of propagation, an iris is used to spatially suppress the otherwise very strong third harmonic. A curved UVenhanced Al mirror is used to direct the output light to an Ocean Optics UVVIS HR4000 spectrometer with a slit width of 10 μm. To determine the ellipticities and major axes of the generated harmonics, a Rochon polarizer is placed between sample and iris and rotated in total by 360°, measuring a spectrum every 18°. To detect the helicity of the circular harmonics, a tunable zeroorder QWP (from Alphalas) is placed between sample and polarizer. For postprocessing the polarizer scans, the harmonic intensities are fitted with a sinsquare curve offset from zero, the ellipticity calculated as \(\varepsilon _{{\mathrm{HH}}} = \sqrt {I_{{\mathrm{min}}}/I_{{\mathrm{max}}}}\) and the majoraxis rotation as ϕ_{HH} = arctan(I_{y}/I_{x}). The drivingintensity scan in Supplementary Figure 3 is performed employing reflective neutraldensity filters.
Abinitio TDDFT simulations of highharmonic generation in solids
Within the framework of TDDFT, the evolution of the wavefunctions and the evaluation of the timedependent current are computed by propagating the Kohn–Sham equations
where ψ_{n,k} is a Bloch state, n a band index, k a point in the first Brillouin zone (BZ), and H_{KS} is the Kohn–Sham Hamiltonian given by
The different terms correspond to the kinetic energy, the ionic potential, the Hartree potential, that describes the classical electron–electron interaction, and exchangecorrelation potential, which contains all the correlations and nontrivial interactions between the electrons. The latter needs to be approximated in practice^{34}.
We perform the calculations using the Octopus code^{35}, employing the TB09^{36} meta generalized gradient approximation (MGGA) functional to approximate the exchangecorrelation potential using the adiabatic approximation. To ensure the stability of our timepropagation, we solved the timedependent Kohn–Sham equations selfconsistently at every time step using the enforced timereversal symmetry propagator^{37}. The cvalue entering in the TB09 functional is recomputed at each time step using the gaugeinvariant kinetic energy density. We employ normconserving pseudopotentials. We emphasize that within TB09 MGGA, the experimental bandgap of common semiconductors and insulators is well reproduced^{38}, which is an important improvement over the localdensity approximation (LDA) used in refs. ^{14,15}, permitting direct comparison between experiment and theory. As shown in ref. ^{14} for adiabatic LDA (ALDA), localfield effects and dynamical correlations (at the level of the ALDA functional) do not seem to affect the HHG spectra of Si. The excitonic effects in Si mainly come from the longrange part of the exchangecorrelation potential^{39}, i.e., a renormalization of the Hartree term (which does not play any role in HHG from Si^{14}); therefore, excitonic effects are not expected to modify the HHG spectra of materials such as Si. We note that this is not necessarily true for all materials, in particular materials with strongly localized excitons, for which bound states will form in the bandgap, or in strongly correlated materials^{23,24}.
All calculations for bulk Si are performed using the primitive cell of bulk Si, using a realspace spacing of 0.484 atomic units, corresponding to 15 points along each primitive axis. We consider a laser pulse of 50fs FWHM duration with a sinsquare envelope and a carrier wavelength λ of 2.08 μm, corresponding to 0.60 eV carrier photon energy. We employ an optimized 36 × 36 × 36 grid shifted four times to sample the BZ, and we use the intensity corresponding to the experimental intensity, using the value for the optical index n of ~3.4 for computing the intensity in matter. The four shifts of the kpoint grid are (in reduced coordinates) (0.5, 0.5, 0.5), (0.5, 0.0, 0.0), (0.0, 0.5, 0.0), (0.0, 0.0, 0.5). We use the experimental lattice constant a leading to a MGGA bandgap (direct) of silicon of 3.09 eV. In all our calculations, we assume a CEP of ϕ = 0.
We compute the total electronic current j(r, t) from the timeevolved wavefunctions, the HHG spectrum is then directly given by
where FT denotes the Fourier transform.
Supplementary Figure 5 shows a comparison of a computed HHG spectrum to a corresponding experimental spectrum. Note that, as mentioned above, in our experiments we only detect harmonics up to HH9 due to the spectrometer used (Ocean Optics UVVIS HR4000). Our TDDFT calculations predict that harmonics up to HH19 in the XUV spectral region are generated for our experimental conditions.
Code availability
The OCTOPUS code is available from http://www.octopuscode.org.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request, and will be deposited on the NoMaD repository (https://doi.org/10.17172/NOMAD/2019.03.041).
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Acknowledgements
We acknowledge support by the cluster of excellence “Advanced Imaging of Matter” (EXC 2056—project ID 390715994) and the priority program QUTIF (SPP1840 SOLSTICE) of the Deutsche Forschungsgemeinschaft, as well as financial support from the European Research Council (ERC2015AdG694097), Grupos Consolidados (IT57813), and European Union’s H2020 program under GA no. 676580 (NOMAD). N.T.D., A.R., and O.D.M. thank M. Altarelli for very fruitful discussion. We thank M. Spiwek for help with the Laue Xray diffraction characterization of samples.
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N.T.D., A.R., F.X.K., and O.D.M. conceived, designed, and coordinated the project. G.M.R. and R.E.M. implemented the IROPA driver source. N.K., Y.Y., G.D.S., and O.D.M. conceived the setup and performed the HHG experiments. F.S. and N.K. performed the 2DSI characterization. N.T.D. carried out the code implementation and numerical calculations. N.K., N.T.D., and O.D.M. analyzed and interpreted the experimental and theoretical results. N.K., N.T.D., A.R., F.X.K., and O.D.M. participated in the discussion of the results and contributed to the manuscript with revisions by all.
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Klemke, N., TancogneDejean, N., Rossi, G.M. et al. Polarizationstateresolved highharmonic spectroscopy of solids. Nat Commun 10, 1319 (2019). https://doi.org/10.1038/s41467019093281
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