# Magnetic Vortex Dynamics Probed by Time-Resolved Magnetic Helicoidal Dichroism

Mauro Fanciulli, Matteo Pancaldi, Anda-Elena Stanciu, Matthieu Guer, Emanuele Pedersoli, Dario de Angelis, Primož Rebernik Ribič, David Bresteau, Martin Luttmann, Pietro Carrara, et al.

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## To cite this version:

Mauro Fanciulli, Matteo Pancaldi, Anda-Elena Stanciu, Matthieu Guer, Emanuele Pedersoli, et al. Magnetic Vortex Dynamics Probed by Time-Resolved Magnetic Helicoidal Dichroism. Physical Review Letters, 2025, 134 (15), pp.156701. 10.1103/PhysRevLett.134.156701. hal-05127585

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Distributed under a Creative Commons Attribution 4.0 International License
Magnetic Vortex Dynamics Probed by Time-Resolved Magnetic Helicoidal Dichroism

Mauro Fanciulli,¹,²,³,*,† Matteo Pancaldi,⁴,⁵,* Anda-Elena Stanciu,⁶ Matthieu Guer,² Emanuele Pedersoli,⁴

Dario De Angelis,⁴ Primož Rebernik Ribič,⁴ David Bresteau,² Martin Luttmann,² Pietro Carrara,²,⁷

Arun Ravindran,⁸ Benedikt Rösner,⁹ Christian David,⁹ Carlo Spezzani,⁴ Michele Manfredda,⁴

Ricardo Sousa,⁶ Laurent Vila,⁶ Ioan Lucian Prejbeanu,⁶ Liliana D. Buda-Prejbeanu,⁶ Bernard Dieny,⁶

Giovanni De Ninno,⁴,⁸ Flavio Capotondi,⁴ Thierry Ruchon,² and Maurizio Sacchi,⁷,¹⁰,‡

¹CY Cergy Paris Université, CEA, LIDYL, 91191 Gif-sur-Yvette, France

²Université Paris-Saclay, CEA, LIDYL, 91191 Gif-sur-Yvette, France

³New Technologies Research Center, University of West Bohemia, 30100 Plzeň, Czech Republic

⁴Elettra-Sincrotrone Trieste S.C.p.A., 34149 Basovizza, Trieste, Italy

⁵Department of Molecular Sciences and Nanosystems, Ca' Foscari University of Venice, 30172 Venezia, Italy

⁶Université Grenoble Alpes, CNRS, CEA, Grenoble INP, IRIG-SPINTEC, 38000 Grenoble, France

⁷Sorbonne Université, CNRS, Institut des NanoSciences de Paris, INSP, 75005 Paris, France

⁸Laboratory of Quantum Optics, University of Nova Gorica, 5001 Nova Gorica, Slovenia

⁹PSI Center for Photon Science, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland

¹⁰Synchrotron SOLEIL, L'Orme des Merisiers, Saint-Aubin, B. P. 48, 91192 Gif-sur-Yvette, France

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(Received 25 November 2024; accepted 3 March 2025; published 15 April 2025)

The laser excitation of magnetization dynamics without applying external fields is a topic of high interest for its potential applications (e.g., magnetic memories, oscillators, and THz emitters). We explore the use of an ultrashort infrared laser pulse for triggering transient changes in a magnetic vortex, probing the induced dynamics by time-resolved magnetic helicoidal dichroism (MHD) in resonant extreme ultraviolet scattering. MHD describes the optical response of a magnetic sample upon sign reversal of either the light orbital angular momentum or the magnetization and was demonstrated to be sensitive to the sample spin texture. Here, we show that, in addition to the well-known ultrafast demagnetization and remagnetization laser-induced processes, the analysis of the MHD signal, supported by micromagnetic simulations, provides direct evidence of important transient reorganizations of the spin texture. In particular, we find that an ultrafast laser pulse of sufficient intensity can induce a surface transient magnetic texture where the vortex curling direction is reversed with respect to the bulk. This result provides insight into the preparation of metastable complex spin states in magnetic films by optical methods without applying an external field, which is of relevance for novel applications in data storage and manipulation.

DOI: 10.1103/PhysRevLett.134.156701

*Introduction*—The angular momentum of light beams entails both spin and orbital components. The spin angular momentum is associated with the polarization state of the light and finds many applications based on circular dichroism. The orbital angular momentum (OAM), associated with the wavefront shape, is a less frequently exploited property of electromagnetic radiation, which is drawing a steadily growing interest in both fundamental research [1–3] and applications [4–8]. The interest in producing Laguerre-

Gaussian beams carrying OAM and in understanding their potential for applications has covered spectral ranges from the infrared-visible to the shorter wavelengths of the extreme ultraviolet (XUV) [9–16], soft x-ray [17], and hard x-ray [18] radiation, as well as combination of different ranges [19–21], opening up a whole new and rapidly expanding field in x-ray spectroscopy [22,23]. OAM beams possess an intrinsically well-defined handedness that, as for circularly polarized beams carrying spin angular momentum, make them well suited to study chiral systems [24,25], as well as magnetic materials [5,26–28], when the beam size matches the dimension of the investigated object [29]. In particular, the concept of magnetic helicoidal dichroism (MHD) has been introduced recently for resonant XUV scattering [30] and its predictions have been experimentally confirmed [31] for a magnetic vortex configuration.

In this Letter, we report on the use of time-resolved MHD to address the dynamics of a magnetic vortex structure

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

<footnote>*These authors contributed equally to this work.</footnote>

<footnote>†Contact author: mauro.fanciulli@u-cergy.fr</footnote>

<footnote>‡Contact author: maurizio.sacchi@synchrotron-soleil.fr</footnote>
excited by a ≈60 fs infrared (IR) laser pulse. We use free electron laser (FEL) radiation and spiraling zone plates to generate XUV pulses with a well-defined topological charge (ℓ = ±1). We show that MHD can provide information not only on the well-known fast demagnetization-remagnetization process [32], but also on the transient local modification of the magnetic texture, offering new possibilities in the study of ultrafast magnetization dynamics and optically driven processes. In particular, OAM beams and the topological sensitivity of the MHD signal can be appealing to follow the spin texture dynamics in the early stage of processes involving all-optical switching or the formation of three dimensional skyrmionic structures [33]. This Letter is organized as follows: after introducing the experimental details of the time-resolved MHD technique, we present our experimental results and then discuss them in the context of micromagnetic simulations and of MHD calculations.

**MHD and experimental setup**—When a linearly $p$-polarized Laguerre-Gaussian beam is reflected by a magnetic structure of given symmetry, the magneto-optical Kerr effect signal [34] is modified by taking into account the topological charge $ℓ$, resulting in a scattered beam that contains a weighted sum of all $ℓ + n$ modes, $n$ being the index of the non-null coefficients of the azimuthal Fourier decomposition of the transverse magnetization [30]. This means that the intensity and phase profiles of the reflected beam depend on both the initial OAM topological charge and on the particular magnetization texture seen by the incoming beam. The reflectivity change observed when switching either the OAM sign or the magnetization sign at every point of the structure have been named MHD$ℓ$ and MHD$m$, respectively. The latter is easier to realize experimentally, since it does not involve any realignment of the sample or of the OAM beam, and it is less sensitive to optical aberrations [29]. By measuring the 2D intensity profile of the reflected beam and comparing it with analytical simulations [30], it is possible to retrieve the general topological properties of the magnetization structure probed by the OAM beam without requiring a fine spatial scan. This becomes particularly important and interesting for studying the dynamics of the formation of transient spin textures, when an external stimulus perturbs the magnetic order, inducing a depth-dependent ultrafast spin reorganization.

The experiment was performed at the DiProI beamline of the FERMI FEL, using conditions similar to those of the previous static MHD experiment [31], as sketched in Fig. 1. The samples were 80 nm thick permalloy (Py) dots of 15 µm lateral size deposited on Si. The exact shape (oval with a V-shaped indent) was optimized by micromagnetic simulations [35,36] in order to feature a remanent central magnetic vortex stabilized by two smaller vortices of opposite curling direction located at the top left and right corners of the indented zone (Fig. 1). The diameter of the central vortex is larger than 10 µm and its curling direction

<figure><img src="image_4.png" /><figcaption>FIG. 1. Sketch of the experimental setup. For each time delay between the IR (pump) and the XUV-OAM (probe) beams, the scattered XUV intensity is collected by a 2D detector. Magnetic pulses of opposite sign along *x* from an electromagnetic coil form, at remanence, magnetic vortices of opposite curling direction, allowing the measurement of MHD*m*. Top left: scanning electron microscope image of the sample (Py dot), with the results of micromagnetic simulations depicting the central magnetic vortex texture superimposed on it.</figcaption></figure>

can be switched by applying an external magnetic field pulse (~30 mT) along the *x* direction. The central vortex has also a core magnetized out of plane but its diameter is of the order of the exchange length (~5 nm) [36] and it is neglected in the following. As shown before [30,31], the case of a magnetic vortex structure has a simple cylindrical symmetry, simplifying the analysis of MHD signals, and the selectable and switchable curling direction facilitates simple and reproducible MHD*m* measurements at fixed *ℓ*. The *p*-polarized probe beam is an XUV 50 fs FEL pulse, tuned in energy to match the Fe 3*p* → 3*d* core resonance (52.8 eV) and focused to 4 µm (FWHM) on the center of the sample by one of the two spiraling zone plates [14,37], which imparts a topological charge of either +1 or −1. The sample excitation is provided by an almost collinear (2° off) pump IR pulse (780 nm wavelength, 60 fs, *p*-polarized) of variable intensity (1 to 2 µJ/pulse) and focused in a spot of 100 µm FWHM to illuminate homogeneously the whole sample area. Both pump and probe beams have a 50 Hz repetition rate, with a relative jitter lower than 10 fs [38]. We measured magnetic scattering in transverse magneto-optical Kerr effect geometry (i.e., most sensitive to the *m*<sub>y</sub> component in Fig. 1). The scattered intensity was collected by a two-dimensional detector (CCD) in a geometry close to the Brewster extinction condition (incidence at 48° from the surface normal). For each given delay value and pump fluence, the MHD*m* signal was measured by collecting, at remanence, two images of the reflected beam after magnetic pulses of opposite sign, i.e., for opposite curling direction of the remanent vortex structure.

**Experimental results**—Following the methodology introduced in [31] for the static analysis, here we studied the dynamics of the magnetic vortex by measuring the MHD*m* signal (defined as the difference divided by the sum of the
<figure><img src="image_5.png" /><figcaption>FIG. 2. (a) Sequence of MHD$m$ signals for $ℓ = -1$ and pump fluence $F_1$ as a function of the pump-probe delay (values in ps in black). (b) Corresponding ultrafast dynamics of the magnetic contrast (black circles), expressed as the average absolute value of the asymmetry ratio over an MHD$m$ image. On the same scale of delays, the corresponding values of the measured rotation of the MHD$m$ image (red squares, relative to negative delay signal as a reference) are also shown. (c) MHD$m$ signal at three selected time delays, shown on a normalized color scale for each image. (d) Corresponding azimuthal profiles of the integrated signal along the black dotted annulus shown in (c). The MHD$m$ rotation shown in (b) is quantified by measuring the zero-crossing azimuthal angle of each profile.</figcaption></figure>

reflectivity measured for opposite curling direction of the vortex) as a function of the pump-probe delay. We performed the experiment for $ℓ$ values of +1 and -1, and for three different pump fluences of $F_1 = 4.5$ mJ/cm$^2$, $F_2 = 6.3$ mJ/cm$^2$, and $F_3 = 8.1$ mJ/cm$^2$. Figure 2(a) shows the time series for the case $ℓ = -1$ and the lower fluence $F_1$. The magnetic signal, expressed as the average absolute value of the MHD$m$ over the entire image (the integral of the signal is expected to be zero with linearly polarized XUV), is plotted as a function of the delay in Fig. 2(b). We observe a strong reduction in the amplitude of the signal over less than a ps, followed by a slower recovery (time constant of ~10 ps), a typical behavior of Py demagnetized by a 780 nm laser pulse [32]. The images for three selected time delays are shown in Fig. 2(c), where the color scale of each image has been adjusted to the maximum absolute value. A careful analysis of the patterns highlights a change in the MHD$m$ signal shape at around 3.5 ps that can be associated with a spin texture reorganization in the very beginning of magnetic recovery. This change, which can be empirically described as a rotation of the MHD pattern, is better quantified by looking at the intensity profile along the azimuth, evaluated along the dotted annular path in Fig. 2(c), as reported in Fig. 2(d). The rotation angle is reported versus delay as a red trace in

<figure><img src="image_6.png" /><figcaption>FIG. 3. (a) MHD$m$ signals for different pump-probe delays for pump fluence $F_1$, as in Fig. 2(c), but for $ℓ = +1$. (b) Corresponding azimuthal line profiles as in Fig. 2(d). (c) Delay dependence of the MHD rotation for different OAM values and fluences. The high-fluence values (blue points) up to ≈5 ps are affected by high uncertainty due to the extremely low level of the signal.</figcaption></figure>

Fig. 2(b), together with the magnetic signal. The analysis shows that the rotation, close to the maximum of the demagnetization signal, is fairly large, up to ≈40° in the case of $F_1$, and its variation with delay closely follows the behavior of the magnetic signal amplitude. This suggests that the ultrafast demagnetization and remagnetization process is accompanied by a transient change of the spin texture, which is encoded in the MHD signal. It is worth noting that in our experimental geometry, we are mainly sensitive to the component of the magnetization normal to the scattering plane [31] ($m_y$ in Fig. 1), which means that from a symmetry point of view, MHD mainly provides information on the two domains of opposite $m_y$ values of a vortex configuration. Static numerical simulations [39] show that the observed rotation of the MHD pattern is indeed consistent with a rotation of such domains.

In order to corroborate this interpretation, Fig. 3(a) shows the results obtained by inverting the OAM sign to $ℓ = +1$, for the same pump fluence $F_1$ as in Fig. 2(c) (full dataset shown in [39]). In addition to the expected change of symmetry and sign of the static (i.e., at negative delays) MHD signal [31], we observe also a counterclockwise rotation of the MHD image at positive delays [Fig. 3(b)], i.e., opposite to the one observed for $ℓ = -1$. Figure 3(c) shows that the rotation angles consistently increase for increasing $F$ values. At the highest fluence (time sequence of MHD$m$ shown in [39]) one observes a saturation that lasts up to a delay of ≈20 ps, with a large rotation angle of the order of 180°, corresponding to a complete inversion of the MHD sign. This experimental observation indicates that the ultrafast demagnetization and remagnetization of the magnetic vortex is unexpectedly accompanied by a transient reversal of its curling direction, as will be further investigated in the following.

**Simulations and discussions**—For a pump pulse of $F_3$ fluence we observe a change of sign of the MHD$m$ images on a timescale of the order of 20 ps, as shown in Fig. 4(a) (selected data, see [39] for the full dataset). We strengthen the interpretation of this change as the transient reversal of $m_y$ and of the magnetic curling direction by means of
<figure><img src="image_7.png" /><figcaption>FIG. 4. (a) Experimental MHDm images for $F_3$ and $\ell = +1$. (b) Corresponding simulated MHD signal obtained from the calculated magnetic configurations shown in (c) for the closest time delay after alignment of time zero, considering a weighted average over the sample thickness to account for the XUV absorption length. The red-blue color scale is normalized by a scaling factor indicated in black for each image. (c) Micro-magnetic simulations of the time evolution of the $m_y$ component of the magnetization (pink-green color scale) as a function of time and depth $z$. Simulation parameters are detailed in [39]. Over each image, a quiver plot in black represents the corresponding in-plane magnetic texture $m_{xy}$. (d) Plots of the vortex toroidicity $\tau$ [39] as a function of time evaluated for each layer. Red circles highlight the transient perturbation of the magnetic vortex state, most visible at the topmost layer.</figcaption></figure>

micromagnetic simulations [36]. With a simplified model detailed in [39], we consider the sample to start from a static vortex state, and the effect of the pump pulse to be a sudden increase in temperature $T(z)$ [40], with $T$ decreasing through the different layers at depth $z$. As long as $T(z) > T_C$, with $T_C$ being the Py Curie temperature, the layers are assumed to be totally demagnetized, while they are in a randomly partially demagnetized state that retains part of the initial vortex configuration deeper in the sample when $T(z) < T_C$. Figure 4(c) shows the time evolution as a function of depth of the $m_y$ magnetization component in a green-pink color scale for our model system. Over each image, the in-plane magnetic texture $\vec{m}_{xy}$ is also shown as a quiver plot in black at a given radius $\tilde{r}$. We analyze the images in terms of the toroidal moment [41,42], and for a given $\tilde{r}$ and for each layer at depth $\tilde{z}$, Fig. 4(d) shows the vortex toroidicity $\tau$ as a function of time delay, evaluated as $\tau(t) = (1/2\pi) \int_0^{2\pi} [\vec{m}_{xy}(t)/m_{xy}(0)] \hat{u}_\theta d\theta$, with $\hat{u}_\theta$ being the unitary vector along the azimuthal angle coordinate $\theta$ [39]. The index $\tau$ quantifies the extent to which a vortex texture is formed, with $\tau = +1$ ($-1$) corresponding to a perfect vortex state with counterclockwise (clockwise) curling

direction. The calculations show that the remagnetization process as a function of time takes place upward from the bottom part to the surface of the sample, since the deeper layers in the bulk, less affected by the pump pulse, pin the initial curling direction of the vortex state. Notably, in the window 10–20 ps during which the system recovers the magnetization, a transient reversal of the curling direction occurs [e.g. the pink-green colors are inverted at the topmost layer and 13.5 ps in Fig. 4(c)]. It is interesting to observe that such perturbation “bounces” at the surface and propagates back into the depth (red circles in Fig. 4(d); see also [39]), providing some insight into the complex dynamical behavior of a magnetic vortex, possibly related to magnetoacoustic shock waves [43–45].

Starting from the micromagnetic simulations of Fig. 4(c), the corresponding MHDm signal has been calculated for each time step based on the previously developed analytical model [30,31]. The results, shown in Fig. 4(b), are obtained with a $z$-dependent weighted average of the reflected electric field for each magnetic texture at different layers in order to account for the XUV-OAM probe absorption length [39]. Large variations are observed in the sign and shape of the MHD signal during the remagnetization process, including a reversal of the sign. Although a quantitative comparison between experimental [Fig. 4(a)] and calculated [Fig. 4(b)] results is impractical given the simplified model and the low intensity of the experimental signal at the strongest demagnetization, qualitatively they compare fairly well. This corroborates the picture of a transient reversal of the magnetic vortex near the surface of the Py dot.

**Conclusions**—We have performed time-resolved MHD measurements of an optically excited Py magnetic vortex using XUV-OAM radiation tuned to the Fe $3p \rightarrow 3d$ resonance, with an overall sub-100 fs time resolution. We showed that, in addition to the expected fast demagnetization and remagnetization of the Py, the shape of the MHD signal undergoes transient changes that evidence a dynamic modification of the vortex structure. At high enough fluence, of the order of 8 mJ/cm$^2$, we observe a reversal of the signal sign over the 10–20 ps timescale. The results were interpreted by combining micromagnetic simulations, accounting for the optical pump penetration depth, with MHD calculations that considered the probing depth of the XUV-OAM beam, allowing us to trace their origin to the thickness-dependent laser-induced thermal demagnetization.

Our results have a twofold implication. On the one hand, they show the potential of the MHD scattering technique, introduced in [31] in the static regime and here extended to the ultrafast time domain, for the study of magnetization dynamics. Complementary to x-ray magnetic circular dichroism in terms of accessing the magnetization reflectivity matrix elements [30], MHD introduces new capabilities for investigating the magnetic textures thanks to the
additional degree of freedom provided by the OAM. On the other hand, our results reveal a complex behavior in the laser-excited ultrafast dynamics of a magnetic vortex, providing a new point of view on the way to optically prepare and characterize elaborate metastable magnetization states in thin films. This can possibly be extended to nanorings for nanomotor applications [46] and other nanoscale skyrmionic configurations, offering a novel way to manipulate and read high density magnetic topological information by all-optical methods, as long as the OAM beam focal spot can be shrunk to the size of the system under investigation.

**Acknowledgments**—This work was supported by the Agence Nationale de la Recherche (France), project HELIMAG, Grant No. ANR-21-CE30-0037, and project TORNADO, Grant No. ANR-23-EXLU-0004; the Programme ERC CZ, Project TWISTnSHINE, Grant No. LL2314; the Programme Johannes Amos Commenius, call of Excellent Research, Project “Quantum materials for applications in sustainable technologies (QM4ST),” Grant No. CZ.02.01.01/00/22_008/0004572; the Slovenian Research Agency (ARRS), Project “Structured light as a tool for triggering and probing new states of matter,” Grant No. J1-3012; the Indo-French Centre for the Promotion of Advanced Research (CEFIPRA), Grant No. 7104-3; the EU-H2020, project NFFA, Grant No. 654360; the Italian Ministry of University and Research, PRIN2020 funding program, Grant No. 2020PY8KTC; the European Cooperation in Science and Technology (COST) Action NEXT, Grant No. CA22148.

[1] G. Spektor, D. Kilbane, A. K. Mahro, B. Frank, S. Ristok, L. Gal, P. Kahl, D. Podbiel, S. Mathias, H. Giessen et al., Revealing the subfemtosecond dynamics of orbital angular momentum in nanoplasmonic vortices, Science **355**, 1187 (2017).

[2] Yijie Shen, Xuejiao Wang, Zhenwei Xie, Changjun Min, Xing Fu, Qiang Liu, Mali Gong, and Xiaocong Yuan, Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities, Light Sci. Appl. **8**, 90 (2019).

[3] V. V. Kotlyar, A. G. Nalimov, A. A. Kovalev, A. P. Porfirev, and S. S. Stafeev, Spin-orbit and orbit-spin conversion in the sharp focus of laser light: Theory and experiment, Phys. Rev. A **102**, 033502 (2020).

[4] Hiroyuki Fujita and Masahiro Sato, Ultrafast generation of skyrmionic defects with vortex beams: Printing laser profiles on magnets, Phys. Rev. B **95**, 054421 (2017).

[5] Florian Siegrist, Julia A. Gessner, Marcus Ossiander, Christian Denker, Yi-Ping Chang, Malte C. Schröder, Alexander Guggenmos, Yang Cui, Jakob Walowski, Ulrike Martens, J. K. Dewhurst, Ulf Kleineberg, Markus Münzenberg, Sangeeta Sharma, and Martin Schultze, Light-wave dynamic control of magnetism, Nature (London) **571**, 240 (2019).

[6] Xiaoqiang Zhang, Guanghao Rui, Yong Xu, Fan Zhang, Yinchang Du, Xiaoyang Lin, Anting Wang, and Weisheng Zhao, Fully controllable three-dimensional light-induced longitudinal magnetization using a single objective lens, Opt. Lett. **45**, 2395 (2020).

[7] Jean-Luc Bégin, Ashish Jain, Andrew Parks, Felix Hufnagel, Paul Corkum, Ebrahim Karimi, Thomas Brabec, and Ravi Bhardwaj, Nonlinear helical dichroism in chiral and achiral molecules, Nat. Photonics **17**, 82 (2023).

[8] Yijie Shen et al., Roadmap on spatiotemporal light fields, J. Opt. **25**, 093001 (2023).

[9] J Bahrdt, K. Holldack, P. Kuske, R. Müller, M. Scheer, and P. Schmid, First observation of photons carrying orbital angular momentum in undulator radiation, Phys. Rev. Lett. **111**, 034801 (2013).

[10] Carlos Hernández-García, Antonio Picón, Julio San Román, and Luis Plaja, Attosecond extreme ultraviolet vortices from high-order harmonic generation, Phys. Rev. Lett. **111**, 083602 (2013).

[11] Genevieve Gariepy, Jonathan Leach, Kyung Taec Kim, T. J. Hammond, E. Frumker, Robert W. Boyd, and P. B. Corkum, Creating high-harmonic beams with controlled orbital angular momentum, Phys. Rev. Lett. **113**, 153901 (2014).

[12] R Géneaux, A. Camper, T. Auguste, O. Gobert, J. Caillat, R. Taëb, and T. Ruchon, Synthesis and characterization of attosecond light vortices in the extreme ultraviolet, Nat. Commun. **7**, 12583 (2016).

[13] D Gauthier, P. Rebernik Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, Tunable orbital angular momentum in high-harmonic generation, Nat. Commun. **8**, 14971 (2017).

[14] Primož Rebernik Ribič, Benedikt Rösner, David Gauthier, Enrico Allaria, Florian Döring, Laura Foglia, Luca Giannessi, Nicola Mahne, Michele Manfredda, Claudio Masciovecchio, Riccardo Mincigrucci, Najmeh Mirian, Emilio Principi, Eléonore Roussel, Alberto Simoncig, Simone Spampinati, Christian David, and Giovanni De Ninno, Extreme-Ultraviolet Vortices from a Free-Electron Laser, Phys. Rev. X **7**, 031036 (2017).

[15] Fanqi Kong, Chunmei Zhang, Frédéric Bouchard, Zhengyan Li, Graham G. Brown, Dong Hyuk Ko, T. J. Hammond, Ladan Arissian, Robert W. Boyd, Ebrahim Karimi, and P. B. Corkum, Controlling the orbital angular momentum of high harmonic vortices, Nat. Commun. **8**, 14970 (2017).

[16] Laura Rego, Kevin M. Dorney, Nathan J. Brooks, Quynh L. Nguyen, Chen-Ting Liao, Julio San Román, David E. Couch, Allison Liu, Emilio Pisanty, Maciej Lewenstein, Luis Plaja, Henry C. Kapteyn, Margaret M. Murnane, and Carlos Hernández-García, Generation of extreme-ultraviolet beams with time-varying orbital angular momentum, Science **364**, eaaw9486 (2019).

[17] Justin S. Woods, Xiaoqian M. Chen, Rajesh V. Chopdekar, Barry Farmer, Claudio Mazzoli, Roland Koch, Anton S. Tremsin, Wen Hu, Andreas Scholl, Steve Kevan, Stuart Wilkins, Wai-Kwong Kwok, Lance E. De Long, Sujoy Roy, and J. Todd Hastings, Switchable x-ray orbital angular
momentum from an artificial spin ice, Phys. Rev. Lett. **126**,
117201 (2021).

[18] Andrew G. Peele, Keith A. Nugent, Adrian P. Mancuso, David Paterson, Ian McNulty, and Jason P. Hayes, X-ray phase vortices: theory and experiment, J. Opt. Soc. Am. A **21**, 1575 (2004).

[19] Sucharita Giri, Misha Ivanov, and Gopal Dixit, Signatures of the orbital angular momentum of an infrared light beam in the two-photon transition matrix element: A step toward attosecond chronoscopy of photoionization, Phys. Rev. A **101**, 033412 (2020).

[20] Giovanni De Ninno et al., Photoelectric effect with a twist, Nat. Photonics **14**, 554 (2020).

[21] Jonas Wätzel et al., Light-induced magnetization at the nanoscale, Phys. Rev. Lett. **128**, 157205 (2022).

[22] James B. Rosenzweig, Twisted light beyond the visible, Nat. Photonics **13**, 141 (2019).

[23] Dino Habibovič, Kathryn R. Hamilton, Ofer Neufeld, and Laura Rego, Emerging tailored light sources for studying chirality and symmetry, Nat. Rev. Phys. **6**, 663 (2024).

[24] Jérémy R. Rouxel, Benedikt Rösner, Dmitry Karpov, Camila Bacellar, Giulia F. Mancini, Francesco Zinna, Dominik Kinschel, Oliviero Cannelli, Malte Oppermann, Cris Svetina, Ana Diaz, Jérôme Lacour, Christian David, and Majed Chergui, Hard X-ray helical dichroism of disordered molecular media, Nat. Photonics **16**, 570 (2022).

[25] Jérémy R. Rouxel and Shaul Mukamel, Molecular chirality and its monitoring by ultrafast X-ray pulses, Chem. Rev. **122**, 16802 (2022).

[26] Thomas Bose and Jamal Berakdar, Nonlinear magneto-optical response to light carrying orbital angular momentum, J. Opt. **16**, 125201 (2014).

[27] Shirong Lin, Zhongquan Nie, Weichao Yan, Yao Liang, Han Lin, Qing Zhao, and Baohua Jia, All-optical vectorial control of multistate magnetization through anisotropy-mediated spin-orbit coupling, Nanophotonics **8**, 2177 (2019).

[28] Nicolò Maccaferri, Irina Zubritskaya, Ilya Razdolski, Ioan-Augustin Chioar, Vladimir Belotelov, Vassilios Kapaklis, Peter M. Oppeneer, and Alexandre Dmitriev, Nanoscale magnetophotonics, J. Appl. Phys. **127**, 080903 (2020).

[29] Matteo Pancaldi et al., High-resolution ptychographic imaging at a seeded free-electron laser source using OAM beams, Optica **11**, 403 (2024).

[30] Mauro Fanciulli, David Bresteau, Mekha Vimal, Martin Luttmann, Maurizio Sacchi, and Thierry Ruchon, Electromagnetic theory of helicoidal dichroism in reflection from magnetic structures, Phys. Rev. A **103**, 013501 (2021).

[31] Mauro Fanciulli et al., Observation of magnetic helicoidal dichroism with extreme ultraviolet light vortices, Phys. Rev. Lett. **128**, 077401 (2022).

[32] E Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Ultrafast spin dynamics in ferromagnetic nickel, Phys. Rev. Lett. **76**, 4250 (1996).

[33] Noah Kent, Neal Reynolds, David Raftrey, Ian T. G. Campbell, Selven Virasawmy, Scott Dhuey, Rajesh V. Chopdekar, Aurelio Hierro-Rodriguez, Andrea Sorrentino, Eva Pereiro, Salvador Ferrer, Frances Hellman, Paul

Sutcliffe, and Peter Fischer, Creation and observation of Hopfions in magnetic multilayer systems, Nat. Commun. **12**, 1562 (2021).

[34] H Ebert, Magneto-optical effects in transition metal systems, Rep. Prog. Phys. **59**, 1665 (1996).

[35] M. J. Donahue and D. G. Porter, OOMMF User's Guide, Version 1.0, Interagency Report NISTIR 6376, National Institute of Standards, Technology, Gaithersburg, MD, 1999.

[36] Arne Vansteenkiste, Jonathan Leliaert, Mykola Dvornik, Mathias Helsen, Felipe Garcia-Sanchez, and Bartel Van Waeyenberge, The design and verification of MuMax3, AIP Adv. **4**, 107133 (2014).

[37] Benedikt Rösner, Florian Döring, Primož R. Ribič, David Gauthier, Emiliano Principi, Claudio Masciovecchio, Marco Zangrando, Joan Vila-Comamala, Giovanni De Ninno, and Christian David, High resolution beam profiling of X-ray free electron laser radiation by polymer imprint development, Opt. Express **25**, 30686 (2017).

[38] Miltcho B. Danailov, Filippo Bencivenga, Flavio Capotondi, Francesco Casolari, Paolo Cinquegrana, Alexander Demidovich, Erika Giangrisostomi, Maya P. Kiskinova, Gabor Kurdi, Michele Manfredda, Claudio Masciovecchio, Riccardo Mincigrucci, Ivaylo P. Nikolov, Emanuele Pedersoli, Emiliano Principi, and Paolo Sigalotti, Towards jitter-free pump-probe measurements at seeded free electron laser facilities, Opt. Express **22**, 12869 (2014).

[39] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.134.156701 for additional experimental data and model calculations.

[40] Philippe Scheid, Quentin Remy, Sébastien Lebègue, Gregory Malinowski, and Stéphane Mangin, Light induced ultrafast magnetization dynamics in metallic compounds, J. Magn. Magn. Mater. **560**, 169596 (2022).

[41] Nicola A. Spaldin, Manfred Fiebig, and Maxim Mostovoy, The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect, J. Phys. Condens. Matter **20**, 434203 (2008).

[42] Satoru Hayami, The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect, Nanomaterials **21**, 1729 (2024).

[43] A. V. Scherbakov, A. S. Salasyuk, A. V. Akimov, X. Liu, M. Bombeck, C. Brüggemann, D. R. Yakovlev, V. F. Sapega, J. K. Furdyna, and M. Bayer, Coherent magnetization precession in ferromagnetic (Ga,Mn)As induced by picosecond acoustic pulses, Phys. Rev. Lett. **105**, 117204 (2010).

[44] Yunyoung Hwang, Jorge Puebla, Kouta Kondou, Carlos Gonzalez-Ballestero, Hironari Isshiki, Carlos Sánchez Muñoz, Liyang Liao, Fa Chen, Wei Luo, Sadamichi Maekawa, and Yoshichika Otani, Strongly coupled spin waves and surface acoustic waves at room temperature, Phys. Rev. Lett. **132**, 056704 (2024).

[45] Mingyu Hu, Ezio Iacocca, and Mark A. Hoefer, Spin-piston problem for a ferromagnetic thin film: Shock waves and solitons, Phys. Rev. B **105**, 104419 (2022).

[46] S. Prosandeev and L. Bellaiche, Electric field induced by dynamical change of dipolar configurations in ferromagnets, Phys. Rev. Lett. **102**, 097205 (2009).