Flipping the helicity of X-rays from an undulator at unprecedented speed

X-ray circular dichroism (XMCD), one of the main tools to study magnetism, benefits enormously from the capability of a fast alterable helicity of circularly polarized X-ray photons. Here we present a method for boosting the alternating frequency between right- and left-handed photons to the MHz regime, more than three orders of magnitude faster than state-of-the-art technologies. The method is based on a twin elliptical undulator installed in an electron storage ring being operated in a novel mode where the electron optics is tuned close to a resonance with electrons captured in transverse resonance island buckets. Propagating through the twin undulator, electrons from different islands emit photons of the same wavelength but of opposite helicity. These two helicity components can be alternated as fast as 2 ns. In a proof-of-principle experiment at BESSY II, we demonstrate XMCD at the L2,3 absorption edges of Ni with an 800 ns helicity flip. Rapidly switching the helicity of polarised x-rays is desirable for studying magnetic dynamics via circular dichroism spectroscopy, but has not yet been realised from synchrotron radiation sources. Here, switching between ultrafast emission of monoenergetic, circularly polarized x-rays with opposite helicity is shown within a few nanoseconds.

O ver the past decades, X-ray circular dichroism (XMCD) 1-3 , the difference in absorption of the left and right circularly polarized X-rays in magnetic material, has become an indispensable tool to study element-selective static and dynamic magnetism on magnetic recording materials 4,5 , quantum materials 6,7 , clusters 8 , as well as molecular materials relevant for spintronics and catalysis 9 . To measure XMCD, absorption spectra across atomic resonances are recorded with different relative orientation of X-ray helicity and sample magnetization. Sum rules relate integrals over the XMCD spectrum, i.e., the difference between these spectra, to the ordered spin and orbital magnetic moment 10 . At synchrotron light sources, insertion devices producing circularly polarized X-rays 11 have been successfully employed, combined with proper tools at the experiment to manipulate the magnetization of the sample. In most XMCD experiments, the time needed to change the X-ray helicity or the magnetization is longer than the time needed to change the photon energy and to acquire the absorption signal at a certain energy/polarization setting. As the slow processes determine the time needed for the experiment, typically two energy-dependent spectra are recorded subsequently with a helicity or magnetization change in between. Although this scheme is acceptable when sample and all other experimental parameters are stable over the whole time of the experiment, studies from dynamical processes and experiments addressing small effects such as in molecular spin systems on surfaces, spatially resolved magnetic domains, and quasi-particles in solids 12 significantly benefit from a scheme where very fast helicity alternation in the MHz regime allows for a quasisimultaneous acquisition of both X-ray helicities such that the XMCD spectrum can be recorded in a single energy sweep. This experimental scheme does not only reduce the total acquisition time per spectrum (allowing, e.g., to study dynamical processes) but also improves the signal-to-noise ratio considerably, because the X-ray pointing stability in modern light sources improves by four to five orders of magnitude 13 between 0.1 Hz and MHz acquisition frequency, which promises that previously inaccessible effects in magnetism become measurable.
Although practically all XMCD experiments are carried out in the soft X-ray range, where excitations into the 3d states of transition metals and 4f states of lanthanides occur, a helicity alternation at a rate of 2 kHz has so far only been demonstrated in the hard X-ray regime 14 . A diamond crystal-based quarter wave plate behind a planar undulator has been oscillated around the Bragg angle utilizing a galvanometer scanner. Efficient phase retarders are not available for soft X-rays and are also difficult to operate during energy scans. In this energy range, the polarization must be flipped within the undulator.
Today, all accelerator-based light sources with variable polarization are realized either as permanent magnet undulators (PMUs) or as room temperature electromagnetic undulators (EMUs), or as a combination of both technologies 15 . Only recently, a superconducting undulator prototype with variable polarization, SCAPE (Superconducting Arbitrarily Polarizing Emitters), was built at the Advanced Photon Source (APS) 16 . The alternation frequency of PMUs is limited to a few Hz 17 , because magnet arrays must be accelerated in the presence of strong magnetic forces. EMUs are limited in frequency by the coil selfinductance and the eddy currents induced into the vacuum chamber, where the eddy currents can be compensated with dedicated coils 18 . The helicity alternation rate of a SCAPE is restricted by the quench behavior of the wire. Double undulator systems that generate both helicities simultaneously can further enhance the alternation frequency. The photon beams of the two undulators can be separated statically, and the selection of the specific helicity is accomplished with a chopper in the beamline close to the focal point. A static separation can be realized as an angle orbit bump 19,20 or as an orbit displacement 21 . Alternatively, a dynamic local electron orbit bump in combination with an aperture in the front end chooses the light from one or the other undulator, where both beams are on-axis. A dynamic orbit displacement has been realized with a five kicker bump 22,23 . In other concepts, the two undulator beams of opposite helicity are alternatively modified to be blocked efficiently with the monochromator bandwidth. This can be accomplished with two methods: (i) phase shifters in one or the other module split the spectrum with zero intensity on resonance via the introduction of phase jumps of π 24 ; (ii) additional coils on top of the poles detune the undulator harmonic slightly. The latter design is part of the APS upgrade project utilizing two SCAPE undulators (Y. Ivanyushenkov, personal communication, 2019). For further details on various undulator technologies, we refer to ref. 20 . All coilbased designs are limited to a few 10 Hz and existing chopper systems limit the alternation frequency to about 100 Hz. Recently, MHz choppers 25 have been used in soft X-ray beamlines to extract single camshaft bunches from the fill pattern. However, they need a small intermediate focus (not standard at soft X-ray beamlines), a gap in the storage ring filling pattern, which is not foreseen in most low-emittance machines of the next generation and-in the specific case mentioned above-a special double slit geometry to switch between two separated beams.
In contrast, the method described in this study makes use of both a bunch-by-bunch or periodic turn-by-turn electron orbit variation and a separation in the time domain. Inherently, our new method offers helicity alternation rates in the MHz regime without any detrimental effect on the electron beam stability.

Results
General layouts. The basic idea of the scheme reported here uses the recently implemented option to store electron bunches on two different stable orbits in the so-called transverse resonance island buckets (TRIBs; see Methods) [26][27][28] and to use an undulator to generate different helicities from the different orbits. In an ideal scenario, one would use a special cloverleaf-like magnetic structure (Fig. 1a) where electron bunches that pass on two canted orbits emit photon beams of opposite helicity. This scheme represents a particularly clean case, because the light from both orbits is emitted from the same source point in the center of the undulator and is focused automatically on the same spot of the sample. Intersecting beams in the center of the straight section is a matter of the electron optics that supports transversely separated beams.
However, a cloverleaf-like magnet design as in Fig. 1a does not exist yet and its design admittedly has to overcome considerably difficulties as strong field gradients are required to switch from the right to left circular field within a short lateral distance. Diffraction-limited light sources with round beams, small divergence, and small magnetic gaps will help in this aspect. Nevertheless, the maximum field is reduced compared with a conventional Advanced Planar Polarized Light Emitter-II (APPLE II) design and, furthermore, the intersection between the le-and right-handed array must be sufficiently long here to permit a clean separation of the electron beams and a proper shielding of the fields. Thinking about the feasibility with existing undulator designs, one might horizontally displace only one module of a twin APPLE undulator similar to that in Fig. 1b. Here, electron beams traverse the axes of separated helical magnet structures emitting light of opposite helicity if the shift parameters of the two APPLE II-type undulators 10 are set to an opposite sign. Re-aligning one single undulator is quickly done, if the module is mounted on a remotely controllable motorized carriage. It is important to mention that the two sources are longitudinally shifted and the beamline must accept and focus two strongly separated beams that would need two separated premirror optics or require even two monochromators.
Pilot experiment at BESSY II. For our proof-of-principle experiment, we use a scheme that does not even require the displacement of an undulator. It is displayed in Fig. 1c (with fill patterns in Fig. 1d) and described in the following: if an on-axis electron beam passes two helical undulators with identical gap setting and opposite shift, the harmonics for both helicities appear at the same energy. Beams passing both devices with a lateral offset from the undulator axis will still produce identical harmonics in both undulators, but at a blue-shifted energy as simulated 29 in Fig. 2a and measured in Fig. 2b, because the electron beam propagates in the fringe fields (see Supplementary Note 1 and Supplementary Fig. 1). Figure 2 reveals that the blue shift is indeed considerable and amounts to ca. 20 eV at the Ni L 3 edge at only 3 mm horizontal offset (blue curve) at preserved degree of circular polarization of 60% (see also Supplementary Note 2 and Supplementary Fig. 2) By tuning the gap, we can compensate for this energy shift and move the harmonics back to the original energy of the on-axis beam. Our approach is based on a spectral separation of the two radiation cores using this energy shift.
Detuning only the gap of, e.g., the downstream device such that the off-axis beam emits the same photon energy as the on-axis beam in the upstream device (black line in Fig. 2a), the experiment sees in its narrow spectral window of ΔE = 200 meV width at E Ph = 852 eV, determined by the monochromator, only light from the on-axis beam in the upstream device and from the off-axis beam in the downstream undulator. The harmonics from the other beams (off-axis in the upstream device and onaxis in the downstream one) are both shifted in energy out of the monochromator acceptance. Setting both APPLE II devices to opposite helicity with appropriate phasing 11 , an on-axis beam in the upstream device will only emit one helicity and an offaxis beam in the downstream device the counter helicity. The experimental proof of this scheme is the subject of the present paper.
In all three schemes presented in Fig. 1, the electron bunches pass the two undulators on different trajectories, in the best case on a bunch-by-bunch basis resulting in 500 MHz helicity alternation rate or at least turn-by-turn, i.e., 1.25 MHz at BESSY II. In contrast to dynamic orbit bumps 22,23 or phase shifters 24 by fast kickers, where the residual closed-orbit distortion sets a limit to the helicity alternation rate, we use a novel entirely static twoorbit operation mode based on TRIBs [26][27][28] . In addition to the main orbit, which is defined by the center of all quadrupole magnets, a second stable island orbit is established at a horizontal  third-order resonance winding around the main orbit in the horizontal plane and closing after three revolutions, as depicted in Fig. 3a and their temporal x-ray emission in Fig. 3b. This way, three additional island buckets, i.e., potential wells, are generated where electrons can be stored. Both orbits have different tunes and can simultaneously be populated with electron bunches as in Fig. 3c. Moreover, by a resonant tune excitation, electron bunches can be pushed from one orbit to the other one without losing charge. The hard X-ray source image from a dipole in Fig. 3d shows the situation when all electron bunches are stored on the island orbit. It is noteworthy that the vertical distance of the three source points observed in a dipole practically vanishes in the straight section (by 2% coupling, see Supplementary Note 3 and Supplementary Figs. 3 and 4).
There are two modes to populate the island orbit with electron bunches: (i) individual bunches from the main orbit (black) are divided in equal parts onto the three island buckets (1-blue, 2-red, 3-green), resulting in three almost equally charged bunches passing a beamline within the same time window. A beamline that only accepts one single island spot will see each turn a signal reduced by 1/3, but from different bunches (i.e., 1,2,3,1,2,3 etc.). (ii) Alternatively, the entire charge of the main orbit bunch (black) is pushed into only one island bucket (e.g., the blue one from Fig. 3). Then, a beamline accepting only one island spot will see a full signal only every third turn. Such Single Island Population (SIP) has already been used at the Metrology Light Source to increase the revolution time for time-of-flight electron spectroscopy experiments 30 .
Combining the twin undulator setup (Fig. 1c) with the TRIBs and electrons stored in the islands, there are two different scenarios possible as depicted in Fig. 3: (i) selected bunches, e.g., successive ones, are populated alternatingly in the core orbit and in the islands at about the same bunch charge. If one accepts one of the permanently populated islands and the core beam in the beamline, then both would emit alternately opposite helicity with 500 MHz flip rate (2 ns). (ii) SIP mentioned above with all bunches (the complete fill pattern) in the ring leads to a closed orbit of this macro-pulse only after three turns as in Fig. 3 and, as indicated by blue and red radiation cones, a turn-by-turn flip of the helicity after 800 ns between the first and second turn (see fill patterns in Fig. 1d) but with 800 ns break if the green third turn is not accepted like in the present case. As a detection with <2 ns time resolution was not available in our current setup to address scenario (i), we used a nearly complete filling of the BESSY II ring (50 mA, 350 of 400 possible bunches) and demonstrate the proofof-principle of method (ii) as follows.
Turn-by-turn detection of the helicity change. To detect the Xray helicity, we use the XMCD effect in a Permalloy (Ni 80 Fe 20 ) sample that we magnetized along the beam direction using an electromagnet. As described in the Methods section, the X-ray signal transmitted through the sample is detected with a fast avalanche photo diode (APD); the signal from the first and second revolution is temporally sorted using a digital oscilloscope. As there is no beamline capable of focusing on-axis and off-axis beam simultaneously on the same spot, we use the first mirror M 1 (Supplementary Note 4 with optical layout in Supplementary  Fig. 5) of the beamline to horizontally steer either one or the other beam on the sample. Figure 4a, b show the signals (and Fig. 4c, d the corresponding magnetic asymmetry) transmitted through the sample as a function of the angular position of the M 1 -mirror in the undulator arrangement including gap detuning (by 1.1 mm) and opposite shifts (phases) of ±24 mm described above. As the mirror M1 (at 17 m) images the source on a scale of ca. 1 : 1, the horizontal offset of the electron beam from the first and second revolution is directly measured here as Δx = 3.7 mm (distance of the 2 peaks in Fig. 4). By alternating the out-of-plane magnetization (+,−) of the sample with the photon energy tuned to the Ni L 3 resonance, it is possible to estimate the X-ray helicity emitted by the two source points. As shown in Fig. 4c, the magnetic contrast between the two subsequent turns (i.e., the two source points) is obviously reversed. The sign of the magnetic contrast is reversed in the right graph of Fig. 4d upon flipping the   Fig. 3 Principle of bunch-by bunch or turn-by-turn helicity flip of elliptically polarized X-rays from an undulator in a storage ring using transverse island buckets from a third-order resonance. a Sketch of the electron orbit that closes after three turns passing the magnetic rows of a helical undulator at displaced position. b Thus, a number of photons N ph of different helicity is emitted in time from a number N e− of electrons from the first (blue) and second turn (red). X-rays from the third turn (green in c) are blocked and not accepted by the beamline. c Hard X-ray pinhole camera image of the four source points in a dipole observed when the core beam and the islands are populated by the same number of electrons and d when only the island orbit is populated.
shift parameters of downstream and upstream device according to the layout Fig. 1c.
The same information in time domain is shown in Fig. 5. Here, the traces show the raw signals emitted from the SIP with a special filling of a long macro-pulse and with a single bunch arriving 100 ns before the macro-pulse (Supplementary Note 5 and Supplementary Figs. 6 and 7). The corresponding maximum magnetic contrast A = (I + − I − )/(I + + I − ) shows that bunches from the first and second revolution emit here the opposite helicity. Using a benchmark XMCD measurement of the same sample from a bending magnet beamline PM3 that yielded A = 7.4% corresponding to a circular polarization |S 3 /S 0 | = 75% (see Supplementary Note 6 and Supplementary Fig. 8), we find here that the bunch train in Fig. 5  The small intensity differences in a become evident in the normalized difference signal (magnetic contrast), which indicates the helicity flip between first (blue) and first turn (red). There is no dichroic signal at the third turn since island 3 is not accepted by the beamline. It is noteworthy that a single bunch filled 100 ns before the macro bunch likewise shows a turn-by-turn helicity flip of its X-ray emission, useful for pump-probe applications 4,5 . According to the simulations in Fig. 2, Fig. 5 we expect from the individual undulators |S 3 /S 0 | = 60% but measure less than half of this value. The reason for this is the small horizontal separation, which causes transverse and spectral overlap effects of the radiation cones, as proven by simulations using the code named WAVE 29 with particle tracking along the magnetic field gradient (see Methods). In our setting, we could not further increase the horizontal distance of the two source points because of the limited horizontal acceptance of the existing beamline. There are, however, two straightforward technical solutions to improve |S 3 /S 0 |: (i) increase of the horizontal acceptance of the beamline or (ii) a double slit aperture in the front end to accept only the individual undulator cones, to clip red shifted fringes. In a dedicated setup at a diffraction-limited light source, the overlap reductions will be naturally alleviated by a much lower emittance and a point-like source.
To demonstrate a usual XMCD measurement at the selected settings, the mean values of the signals from the first and second gate windows of Fig. 5 were now displayed in Fig. 6 during an energy scan (combined undulator-monochromator ride with magnetic field alternation). The XMCD difference-over-sum signal clearly shows that the helicity is reversed between the two turns. It is noteworthy that at an optimized beamline the two spectra for one magnetization would be recorded quasi-simultaneously with only an 800 ns time delay of the acquisition gate windows and both would have the same incident photon flux. The absolute averaged |S 3 /S 0 | at the Ni L 3 resonance energy is <25% here, because the mean value of the macro-pulse from Fig. 5 shows some background effects of the APD (slow fall time) and, likewise, owing to a finite population purity (see Methods/TRIBs). Similar to the limited acceptance mentioned above, this technical issue can also be solved with a dedicated beamline and at faster detection.

Discussion
To conclude, we provided the first demonstration of a turn-by-turn flip of the X-ray helicity from an electron storage ring, which is the highest alternation frequency from an undulator in a storage ring ever reported. The experiment further demonstrated that a stable operation of a low-emittance light source under TRIBs settings combined with state-of-the-art helical undulator technology at diffraction-limited light sources, currently coming to life worldwide, might open new insights into the physics of magnetic materials at ultimate signal-to-noise and time resolution.

Methods
Beamline and circular dichroism. To probe the helicity of X-rays from the undulator, the UE56-2 PGM2 beamline 31 was used with the third harmonic from the twin APPLE II undulator UE56-2 at a shift parameter of ±24 mm in the range of 700-900 eV at a resolution of ΔE = 200 meV (1200 l/mm grating) addressing XMCD in transmission at the Nickel L 3 and L 2 absorption edges (Ni, 2p 3/2 → 3d, 2p 1/2 → 3d), respectively. For spectral separation, the downstream undulator was set to a gap of 26.024 mm and the upstream on to 27.124 mm tuning the third harmonic of both devices to the Ni L 3 transition 2p 3/2 → 3d, i.e, E Ph = 852 eV photon energy. For energy scans across L 3 and L 2 , both gap motors were driven in parallel with the spectral window of ΔE = 200 meV of the monochromator. This beamline is a plane grating monochromator 31 (see Supplementary Fig. 5) specially designed for large horizontal angular acceptance of ≈1 mrad as determined by the first mirror and ≈ 4 mm horizontal acceptance (given by the refocusing mirror) with respect to the source displacement. It is important to note that the first toroidal mirror M1 produces horizontally a ≈1 : 1 image of the source point(s) on the detector.
A dedicated XMCD endstation 3 allowing for magnetizing the sample with up to B = ±0.5 T is used. The detection employs a silicon APD (SAR 3000 from Laser Components) together with a 4 GHz oscilloscope (LeCroy) and a trigger hardware fed by the 500 MHz RF master clock frequency, which divides the bunch clock of 1.25 MHz (revolution frequency) by a further factor of 3 to detect average standing waveforms of the APD signal from three subsequent revolutions. As specimen, a magnetron-sputtered thin film of 15 nm-thick Ni 80 Fe 20 (Permalloy) on a 0.5 µm freestanding Aluminum foil of 4 mm diameter was used in normal incidence at out-of-plane magnetizationM by the external axial magnetic field along the pointing vectorS of the X-rays. The sample was benchmarked previously at the PM3 beamline at BESSY II, a plane grating monochromator on a dipole magnet optimized for XMCD experiments 32 . To detect the helicity and estimate the polarization degree, we use the fact that for a thin film, the magnetic contrast A = (I + − I − )/(I + + I − ) at the resonances is proportional to |S 3 /S 0 | , where I + and I − are the transmitted intensities forM parallel or antiparallel toS, respectively. A maximum XMCD signal from that sample of A = 7.4% at the Ni L 3 edge was found here corresponding to |S 3 /S 0 | = 75% degree of circular polarization (see Supplementary Note 6).
Accelerator settings and transverse island buckets. The main orbit in storage rings is defined by the center of all quadrupole magnets, which focus off-center particles in the transverse planes leading to betatron oscillations in the x and y plane at fixed frequencies. The number of these oscillations is called tune and typically chosen not to be in resonance with the revolution of the storage ring; since then, magnetic errors and/or misalignments would sum up turn-by-turn and drive the beam to large amplitudes until it is lost at an aperture. At BESSY II, the tunes are chosen to be (Q x , Q y ) = (17.848, 6.724) for the standard user optics. For the TRIBs optics 26,27 , the storage ring is operated in the present case close to the third-order integer resonance Q xRsonance = 17.666 and, by a special setting of the nonlinear sextupole magnets, three additional so-called resonant island buckets appear in the horizontal phase space in addition to the main bucket (main orbit), representing a second stable "island" orbit closing after three revolutions as depicted in Fig. 3 (in more detail in Supplementary Fig. 4). Both beams have different tunes and for this experiment Q xCore = 17.649, Q xIsland = 17.684, and separated by 33 Q xResonance = 17 2 / 3 . The tune deviation defines the separation of both orbits, which results in displacements of −4.4 mm, 4.5 mm, and 1.0 mm of the three folded island orbit (in the relevant straight section) as shown in the Supplementary Fig. 3. There is a minimum tune separation causing both beams to merge, because then the orbit/bucket separation becomes smaller than the equilibrium beam emittance. Likewise, there is a maximum tune separation, where electrons from the island orbit get lost far off-axis. Displacements far beyond 10 mm are possible at BESSY II. The equilibrium beam emittance for the main orbit in the TRIBs setting is nearly the same as for the standard user setting ε x = 7 nmrad, whereas the emittance of the TRIBs orbit is slightly increased at a small separation to ε x = 8 nmrad and can further rise by up to ε x = 12 nmrad at large separation 33 .
As both orbits have different tunes, a bunch-by-bunch resolved resonant tune excitation, in our case generated with the Bunch-by-Bunch Feedback System 34,35 , can be used to populate and depopulate the orbits with electron bunches. By exciting each turn resonantly at the tune of a bunch of the main orbit, the particles a b Fig. 6 A typical X-ray circular dichroism (XMCD) measurement using the gated detector signal from first and second turn for both magnetization directions of the sample. a XMCD spectra normalized to the 800 eV value and shifted for clarity across the L 3 and L 2 , 2p→3d resonances of Nickel in the Permalloy sample and the corresponding magnetic contrast (asymmetry) b from the first (blue) and second turn (red). Here, the signals (mean values) from the corresponding gate windows set to the first and second turn are acquired for the two different magnetization directions (+) and (−), parallel and antiparallel orientation of the pointing vector of X-rays and out-of-plane sample magnetization vector, respectively. are driven to higher transverse oscillation amplitudes leaving the main bucket getting equally captured in the three island buckets. Exciting the bunches on the island orbit, only two of three turns will push them back to the main orbit leading to a SIP 26,33 . For the proof-of-principle experiment described here, the ring was filled with 350 of 400 buckets at an injected current of I = 50 mA in decay mode with a lifetime of~15 h. A good SIP of about 85% purity was reached with 10% and 5% residual charge captured in the other two island buckets (see Fig. 5). Tuning the separation of the TRIBs optics, the displacement and angle of the electron beam in the straight section can be changed such that undulator light from the selected islands points into the beamline. In the present case, a slight reduction to 3.7 mm distance as measured from Fig. 4 between the first and second turn with respect to the standard setting of 5.4 mm (see Supplementary Fig. 3), −4.4 mm and +1.0 mm) was necessary to accept X-rays from two turns assisted by a horizontal local orbit bump of x = 1 mm and x' = −0.14 mrad.
Numerical simulation of undulator radiation from TRIBs. The numerical calculation of the spectral flux S 0 and polarization degree expressed as Stokes parameter ratio |S 3 /S 0 | have been carried out with the code WAVE 28 . The magnetic field of the undulators was calculated for a model of permanent magnet blocks, as it exists in reality at the two UE56 undulators (30 periods) including fringe fields. The field of each block is evaluated via the current sheet method and the remanence value of the magnets was slightly adjusted here to be consistent with the measured photon energy of the third on-axis harmonic of the UE56 at the Ni L 3 absorption edge of E Ph = 852 eV. The beam emittance of ε x = 8 nmrad at an energy spread of δγ/γ = 7 × 10 −4 at γ = 1.72 GeV electron energy of the TRIBs electron optics in BESSY II have been taken into account by tracking 10 5 electrons generated randomly from the appropriate phase-space distribution. To speed up calculations for each electron, a single transverse photon observation point has been randomized within the experimentally used aperture of horizontal ±15 mm and vertical of ±1 mm at 11 m from the source point (aperture plane). The vertical elliptical mode at shift 24 mm, the large angular acceptance (of red shifted radiation) together with a larger horizontal source size in the high β-sections, then lead to a theoretical degree of circular polarization of |S 3 /S 0 | ≈ 60% at the maximum of the third harmonic at E Ph = 852 eV (see Supplementary Fig. 2).

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.