Polarization-state-resolved high-harmonic spectroscopy of solids
Attosecond metrology 1 sensitive to sub-optical-cycle electronic and structural dynamics is opening up new avenues for ultrafast spectroscopy of condensed matter. Using intense lightwaves to precisely control the extremely fast carrier dynamics in crystals holds great promise for next-generation electronics and devices operating at petahertz bandwidth 2; 3. The carrier dynamics can produce high-order harmonics of the driving light field extending up into the extreme-ultraviolet region 4; 5; 6. Here, we introduce polarization-state-resolved high-harmonic spectroscopy of solids, which provides deeper insights into both electronic and structural dynamics occuring within a single cycle of light. Performing high-harmonic generation measurements from silicon and quartz samples, we demonstrate that the polarization states of high-order harmonics emitted from solids 4 are not only determined by crystal symmetries, but can be dynamically controlled, as a consequence of the intertwined interband and intraband electronic dynamics 7; 40; 41, responsible for the harmonic generation. We exploit this symmetry-dynamics duality to efficiently generate circularly polarized harmonics from elliptically polarized driver pulses. Our experimental results are supported by ab-initio simulations 40; 41, providing clear evidence for the microscopic origin of the phenomenon. This spectroscopy technique might find important applications in future studies of novel quantum materials 10 such as strongly correlated materials. Compact sources of bright circularly polarized harmonics in the extreme-ultraviolet regime will advance our tools for the spectroscopy of chiral systems, magnetic materials, and 2D materials with valley selectivity.
The study of lightwave-driven electronic dynamics occurring on sub-optical-cycle time scales in condensed matter and nanosystems is a fascinating frontier of attosecond science originally developed in atoms and molecules 1; 11. Adapting attosecond metrology techniques to observe and control the fastest electronic dynamics in the plethora of known solids and novel quantum materials 10 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; 3 or strong-field optoelectronics 12; 13.
The nonlinear process of high-order harmonic generation (HHG) in gases is one of the cornerstones of attosecond science and is well understood by the semiclassical three-step model 1. In solids, nonperturbative HHG up to 25th harmonic order without irreversible damage was first reported in . This work triggered extensive research activities aimed at unraveling the microscopic interband and intraband dynamics underlying HHG from crystals (for a comprehensive overview, see ), thereby extending attoscience techniques to solids. The prevailing strong-field 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 films 5; 6, or investigated HHG from 2D materials such as graphene 14, 2D transition-metal dichalcogenides 14; 15, and monolayer hexagonal boron nitride 16.
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 many-body electronic dynamics and real crystal structure is using ab-initio time-dependent density-functional theory (TDDFT) simulations 40; 41. 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 41. 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 40.
The symmetry properties of the light-matter 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 propeller-shaped bichromatic waveforms produces circular (see remark on terminology 17) harmonics 46. In molecules, both the point group and the driving field determine the symmetries of the coupled light-matter system. Consequently, depending on the molecular symmetries and the molecule’s orientation with respect to the light propagation direction, elliptical high-order harmonics can be produced by linear 19; 20 or elliptical driver pulses 21. 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 22; 23; 24. In crystals, several recent works studied the high-harmonic response on driver pulse ellipticity , which can strongly differ qualitatively from the atomic and molecular cases. Whereas earlier work 25 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 41; 26; 27 and reported circular HHG from a single-color driver field 26; 27, which is symmetry forbidden in atoms.
Here, we present a combination of HHG experiments and first-principles 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 40; 41. Most importantly, we demonstrate for the first time the strong-field control of the harmonics’ polarization states. Our findings indicate that polarization-state-resolved high-harmonic spectroscopy of solids provides deeper insights into both electronic and structural dynamics as well as symmetries on sub-cycle time scales.
In our experiments, we irradiated free-standing, 2-m-thin, (100)-cut silicon samples with 120 fs, 2.1 m (0.59 eV) pulses with tunable ellipticity and peak intensities up to 0.7 TW cm in vacuum (see Methods section and Supplementary Fig. 1). At this intensity, the harmonics are generated nonperturbatively (see Supplementary Fig. 3) up to harmonic order 19 (HH19) in the XUV regime for our experimental conditions, as shown by our TDDFT simulations (see Supplementary Fig. 5). Only harmonics up to HH9 are detected by the spectrometer used in our experiments. We also irradiated 50-m-thin, z-cut quartz with an estimated intensity of 40 TW cm in vacuum.
Fig. 1 shows the measured high-harmonic response in Si of HH5, HH7 and HH9 as function of driver ellipticity and sample orientation ; panels a-c display normalized harmonic intensities, d-f harmonic ellipticities , where and correspond to the intensities at the minor and major axis of the polarization ellipse. In all panels, , refer to the major axis of the driving field ellipse along the directions  (X) and  (K) 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 , and the harmonic yields peak for different sample rotations. HH5 exhibits Gaussian-shaped, atomic-like ellipticity profiles for all sample rotations. The distribution is symmetric around (see white dotted center-of-mass curve) for all sample rotations. The intensity distribution of HH7 (Fig. 1b) shows intriguing, non-atomic-like features with maximum yield at non-zero ellipticity for certain sample rotations, similar to experiments on MgO.25 HH9 (Fig. 1c) exhibits the most pronounced deviations from a Gaussian-like ellipticity profile, with non-monotonic non-atomic-like profiles for wide ranges of sample rotation. Its yield is strongly asymmetric with respect to for all sample orientations (different from mirror planes), and displays strong non-sinusoidal oscillations of the center-of-mass 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 low-energetic electrons, that mostly remain within the parabolic region of the bands, leading to an atomic-like behavior. For above-bandgap harmonics, the joint density of states (JDOS) (see Supplementary Fig. 6), i.e., the density of optical transitions at a given energy, determines the relative weight of interband compared to intraband mechanisms 40. 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 40. Interestingly, these harmonics are more efficiently generated with different helicities, as can be seen from the different signs of the center-of-mass curves for certain sample rotations (see Supplementary Fig. 7). This clearly indicates different generation mechanisms of HH7 and HH9, as predicted in Ref. . For HH9, for which interband transitions are strongly suppressed by the low JDOS at this energy, higher-energetic electrons explore larger non-parabolic regions in the bands, which results in pronounced non-atomic-like ellipticity profiles.
Fig. 1d-f reports the measured harmonics’ polarization states as function of driving ellipticity and sample rotation. Whereas linear drivers yield almost linear harmonics, we observe astonishing deviations of the harmonic ellipticities from the driver ellipticity . Consistent with our TDDFT predictions 41 and selection rules in , for circular driver pulses, , all harmonics become circular . 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 and 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 BZ depending on and . The measured harmonics’ polarizations contain the complete information on the - and -components of the harmonics’ amplitudes and their relative phases.
Fig. 2 summarizes our findings on circular harmonics from circular drivers. In both silicon (Fig. 2a) and -quartz (Fig. 2b), all harmonic intensities remain constant while rotating a polarizer by , 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  point group 28 of -quartz. The selection rules also manifest themselves in the helicities of the circular harmonics. In accordance with group-theoretical considerations 28 and TDDFT simulations 41, the odd harmonics from Si have alternating helicities as Si has point group . This was confirmed with a tunable quarter-wave plate behind the sample, which converts circular to linear polarization, with the polarization angle depending on the helicity (see Fig. 2d). The trigonal crystal structure of -quartz results in different selection rules, leading to alternating helicities of HH4 and HH5 in Fig. 2e. As shown in Section VIII of the Supplementary Information, we extracted the Stokes polarization parameters of the harmonics from these measurements and estimated a value of the degree of polarization (DOP) of for all harmonics, similar to reported values for the generation of circular harmonics from atomic and molecular gases 46; 47. Moreover, we find in Si that the harmonic ellipticities are all close to 1, independent of sample rotation (see Fig. 2f). Minor deviations from are likely due to small deviations from perfectly circular driver pulses.
Fig. 3a shows two polarizer scans under excitation conditions, for which HH9 and HH7 are circular for elliptical driver polarization. The measured harmonic ellipticities are 0.93 in both cases. We also found similarly high ellipticities for HH5 (not shown). Fig. 3b shows the intensity-dependence of the harmonic ellipticities for and . 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, that sensitively depend on the nonperturbative dynamics of the system, can not be explained by symmetry arguments only, but clearly indicates strong-field control of the harmonic ellipticities through the lightwave-driven electron dynamics. This might find applications, e.g., in polarization-controlled high-harmonic sources.
The total harmonic intensities for exemplary cases of circular harmonics for different and are compared in Fig. 3c. As discussed above, for Si, the harmonic yield tends to decrease (apart from non-monotonic exceptions) with increasing . Therefore, the generation of circular harmonics using elliptical driver pulses () is expected to be significantly more efficient than for circular ones (), as indeed observed in Fig. 3c. For HH5 and HH7, the circular harmonics generated for and 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 to .
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.
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 and , we computed ab initio the high-harmonic 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 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 ab-initio 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.
In conclusion, after the first works on circular HHG from solids 41; 26; 27, 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 high-harmonic generation from silicon and quartz accompanied by ab-initio 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 point-group 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, magnetic materials, and 2D materials with valley selectivity 15. Circular isolated attosecond pulses from solids also seem in reach employing appropriate gating techniques. Finally, polarization-state-resolved high-harmonic spectroscopy offers the unique advantage of sensitivity to both electronic and structural dynamics on sub-cycle time scales, thus opening up new avenues for the spectroscopy of quantum materials on extreme time scales 10; 30.
Experimental high-harmonic generation setup
Supplementary Fig. 1 shows the experimental setup used for high-order harmonic generation (HHG) from crystalline solids. Passively carrier-envelope phase (CEP)-stabilized 31, 120-fs pulses at 2.1 m (0.59 eV photon energy) are generated in a Ti:sapphire-pumped white-light-seeded optical parametric amplifier (OPA) 32; 33. These 2.1-m driver pulses pass through a wire-grid polarizer, a quarter-wave plate (QWP) and a half-wave plate (HWP), which allow setting the driver ellipticity while keeping the major axis of the polarization ellipse constant (see Supplementary Fig. 2). The pulses are focused onto the sample with a 25-cm CaF lens, resulting in a focus diameter of m. After cm of propagation, an iris is used to spatially suppress the otherwise very strong third harmonic. A curved UV-enhanced Al mirror is used to direct the output light to an Ocean Optics UV-VIS HR4000 spectrometer with a slit width of 10 m. 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 , measuring a spectrum every . To detect the helicity of the circular harmonics, a tunable zero-order QWP (from Alphalas) is placed between sample and polarizer. For post-processing the polarizer scans, the harmonic intensities are fitted with a sin-square curve offset from zero, the ellipticity calculated as and the major-axis rotation as . The driving-intensity scan in Supplementary Fig. 3 is performed employing reflective neutral-density filters.
Ab-initio TDDFT simulations of high-harmonic generation in solids
Within the framework of time-dependent density functional theory (TDDFT), the evolution of the wavefunctions and the evaluation of the time-dependent current are computed by propagating the Kohn-Sham equations
where is a Bloch state, a band index, a point in the first Brillouin zone (BZ), and 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 exchange-correlation potential, that 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 exchange-correlation potential using the adiabatic approximation. We employ norm-conserving pseudo-potentials. We emphasize that within TB09 MGGA, the experimental band gap of common semiconductors and insulators is well reproduced 37, which is an important improvement over the local-density approximation (LDA) used in [40,41], permitting direct comparison between experiment and theory. As shown in Ref. , dynamical correlations do not affect the HHG spectra of Si. The excitonic effects in Si mainly come from the long-range part of the exchange-correlation potential 38, i.e., a renormalization of the Hartree term (which does not play any role in HHG from Si 40), 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 band gap, or in strongly correlated materials 30; 39.
All calculations for bulk Si are performed using the primitive cell of bulk Si, using a real-space spacing of 0.484 atomic units. We consider a laser pulse of 50-fs FWHM duration with a sin-square envelope and a carrier wavelength of 2.08 m, corresponding to 0.60 eV carrier photon energy. We employ an optimized 363636 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 of for computing the intensity in matter. We use the experimental lattice constant leading to a MGGA band gap (direct) of silicon of 3.09 eV. In all our calculations, we assume a CEP of .
We compute the total electronic current from the time-evolved wavefunctions, the HHG spectrum is then directly given by
where denotes the Fourier transform.
Supplementary Fig. 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 UV-VIS HR4000). Our TDDFT calculations predict that harmonics up to HH19 in the XUV spectral region are generated for our experimental conditions.
The data that support the findings of this study are available from the corresponding authors on reasonable request, and will be deposited on the NoMaD repository (https://nomad-repository.eu/).
The OCTOPUS code is available from http://www.octopus-code.org.
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We acknowledge support by the excellence cluster ’The Hamburg Centre of Ultrafast Imaging-Structure, Dynamics and Control of Matter at the Atomic Scale’, the priority program QUTIF (SPP1840 SOLSTICE) of the Deutsche Forschungsgemeinschaft, and financial support from the European Research Council (ERC-2015-AdG-694097), Grupos Consolidados (IT578-13), and European Unions 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 X-ray diffraction characterization of samples.
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 IR-OPA driver source. N.K., Y.Y., G.D.S. and O.D.M. conceived the setup and performed the HHG experiments. 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.
Appendix A Experimental high-harmonic generation setup
Supplementary Fig. 1 shows the experimental setup used for high-order harmonic generation (HHG) from crystalline solids. For a detailed discussion of this setup, see Methods section.
Appendix B Precision of ellipticity and major-axis angle calibration
When measuring the harmonic response maps as function of driver-pulse ellipticity and sample rotation , we require that the major axis of the driving field polarization ellipse remains constant ideally over the entire range of values. Furthermore, also the precise value of is important. As can be seen from the calibration data shown in Supplementary Fig. 2, the combination of a quarter-wave plate (QWP) and a half-wave plate (HWP) allows us to vary to ellipticity within [0, 0.98], while the major axis of the polarization ellipse remains constant to within and for .
Appendix C Intensity scaling of the harmonics in silicon
Perturbatively, the -harmonic intensity, , scales as . Here, is the driving pulse intensity. Supplementary Fig. 3 shows the intensities of HH5-HH9 in Si when varying the driving intensity. For intensities above TW cm, all harmonics clearly deviate from the perturbative power law. All our experiments and simulations are performed above this intensity in the nonperturbative regime.
Appendix D Laue X-ray diffraction of silicon samples
In our experiments, we mainly use free-standing 2-m-thin (100)-cut Si samples (by Norcada). To verify the monocrystallinity and to determine and align the exact crystal orientation, in-house Laue X-ray diffraction is performed employing a Mo anode X-ray tube on these samples. Supplementary Fig. 4 shows a typical diffraction pattern.
Appendix E Comparison of TDDFT computed and experimentally measured HHG spectra
Supplementary Fig. 5 shows a comparison of a computed HHG spectrum to a corresponding experimental spectrum. Note that in our experiments we only detect harmonics up to HH9 due to the spectrometer used (Ocean Optics UV-VIS HR4000). Our TDDFT calculations, described in the Methods section, predict that harmonics up to HH19 in the XUV spectral region are generated for our experimental conditions.
Appendix F Joint density of states (JDOS)
The generation of harmonic photons by the interband mechanism corresponds to the recombination of an excited electron in a conduction band with a hole in a valence band. Therefore, the possibility of emitting a photon by the interband mechanism at a given energy is directly linked to the density of optical transitions available at this photon energy 40. This quantity is also known as the joint density of states (JDOS). Due to the nonperturbative nature of the process of high-harmonic generation, carriers intially created by interband transitions near the point in Si, start to explore a region of the BZ around the minimal band gap. In order to estimate the region of the BZ seen by the driven electrons, one can resort to the so-called “acceleration theorem”, which gives an upper limit to this region. In fact, the scattering between electrons, as well as the coupling to other bands (interband transitions) reduces the exploration of the electron wavepacket 41. The JDOS is given by
where the indices v,c denote valence and conduction bands, denotes a sum over the explored region of the BZ determined by the acceleration theorem in the case of independent electrons, i.e., .
In Supplementary Fig. 6, we show the evolution of the JDOS with the ellipticity of the driving field and sample rotation. We found that for all sample rotations and ellipticities, the JDOS at the energy of HH7 is much higher than at the energy of HH9, indicating that interband transitions are stronger for HH7 compared to HH9.
Appendix G Calculation of the center of mass
The center of mass (COM) curves displayed in Fig. 1a-c are calculated using
where is the yield of the respective harmonic order, and , are again driver ellipticity and sample rotation. The centers of mass are calculated in an interval to emphasize the asymmetric response in the most intense region of the ellipticity profiles. Supplementary Fig. 7 shows the COM curves of HH5-HH9 in Si as shown in Fig. 1a-c in the main text. Note that is required by mirror symmetry along the symmetry axes X and K and that this is well reproduced in our data.
Appendix H Stokes polarization parameters of harmonics from silicon
and denote the electric field components along the - and -directions. The degree of polarization (DOP) is defined as
for completely polarized light, and for completely unpolarized light.
If denotes the transmitted intensity after a polarizer with angle and an introduced phase shift between and , then the Stokes parameters can be determined from
Hence, , and can be measured with a polarizer only. For measuring , an additional QWP is required to introduce . The normalized Stokes parameters (, , ) can be measured throughout the IR to XUV regions 44; 45.
We measured the Stokes parameters of the harmonics from Si in two experiments: for circular driver pulses versus sample rotation (see Supplementary Fig. 8), and for fixed sample orientation along as function of driver ellipticity (Supplementary Fig. 9). Note that the measurement of is known to be prone to experimental errors 43 due to the insertion of a QWP and comparison with another independent measurement without the QWP. Our tunable QWP (from Alphalas) itself might absorb or reflect light and might also not introduce a precise phase shift for the different harmonics. We believe that this is the reason for the scatter of values in Supplementary Fig. 8. However, as stated in the main text and as can be seen from Supplementary Fig. 2, experimentally a circular polarization was only realized to , therefore small variations of might also be caused by a minor deviation from perfectly circular driver polarization.
The different signs of of the individual harmonics observed in Supplementary Fig. 8 are another proof of circular harmonics with alternating helicities. and are near zero, as one would expect for circular harmonics. The DOP values, that are comparable to reported values for the generation of circular harmonics from atomic and molecular gases 46; 47, show that the harmonics are highly polarized.
Supplementary Fig. 9 shows the evolution of the Stokes parameters of HH7 and HH9 when varying the driver-pulse ellipticity , while keeping the major-axis orientation fixed. In Supplementary Fig. 9, also the measured Stokes parameters of the driving field are displayed. In order to also estimate the experimental uncertainty, we repeated the same measurements twice. One can see clear deviations from the driver in the polarization-response of the various harmonics in all Stokes parameters . For and , both measurements yield almost the same values. For and DOP, the insertion of the QWP introduces some uncertainty that, however, does not change the qualitative behavior.
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