Dynamics of optical vortices in 2D materials

, Optical vortices in planar geometries are a universal wave phenomenon, where electromagnetic waves possess topologically protected integer values of orbital angular momentum (OAM). The conservation of OAM governs their dynamics, including their rules of creation and annihilation. However, such dynamics remained so far beyond experimental reach. Here, we present a first observation of creation and annihilation of optical vortex pairs. The vortices conserve their combined OAM during pair creation/annihilation events and determine the field profile throughout their motion between these events. We utilize free electrons in an ultrafast transmission electron microscope to probe the vortices, which appear in the form of phonon polaritons in the 2D material hexagonal boron nitride. These results provide the first observation of optical vortices in any 2D material, which were predicted but never observed. Our findings promote future investigation of vortices in 2D materials and their use for chiral plasmonics, toward the control of selection rules in light-matter interactions and the creation of optical simulators of phase transitions in condensed matter physics.

Optical vortices are points in the light field with non-zero optical orbital angular momentum (OAM) [1]. In these points, the phase is undefined, creating a singularity with zero amplitude and integer values of OAM which equals the integrated phase of the field (over 2 ) in a closed contour around the vortex location [2,3]. Distinguished by dimensionality, two distinct families of optical vortices are known: vortex beams [4] and planar vortices [5]. Vortex beams showed a plethora of phenomena [6] from stimulated emission depletion (STED) microscopy [7] and optical tweezers [8], through nonlinear optical effects such as vortex solitons [9], to implications for quantum entanglement [10,11] and transfer of angular momentum from light to matter [12].
In planar systems, optical vortices are confined to the 2D-plane and evanescently decay in the out-of-plane direction, as famously shown with surface plasmon polaritons [5] and with other guided modes [13]. Their reduced dimensionality led to unique phenomena such as nanotweezers [14], optical skyrmions [15], and manipulation of selection rules in light-matter interactions [16,17]. Much of the research on 2D optical vortices is focused on controlling the vortex OAM and location [18][19][20]. The control is primarily based on engineering the boundary conditions and the laser excitation properties (e.g., polarization), even showing vortices with high OAM [21].
Delicate engineering of the interference of planar waves enabled sub-cycle ultrafast observations of the formation, dissipation, and rotation of individual vortices [21][22][23][24] and of topological plasmon vortices (and vortex arrays), as optical analogous to magnetic merons and skyrmions [23,24]. The latter works suggested to use active control of the excitation pulses to "enable the creation, manipulation and annihilation of plasmonic topological spin textures" [23].
Fundamental to the physics of vortices is the conservation of their topological OAM, implying that vortex pairs can be created or annihilated while maintaining a fixed overall OAM in each process [25]. These conservation laws were probed experimentally by sweeping over the frequency [26] or polarization [20] of time harmonic (monochromatic) fields. These approaches provide an indirect analogue of the temporal dynamics of vortices, showing that pairs of vortices can be created and annihilated, and change their location for slowly varying continuous wave fields [27].
The creation and annihilation of vortices are of particular interest in ultrafast optics: ultrafast optical phenomena require a wide bandwidth of frequency components and while each component separately can support vortices in a different location, their constructive interference is expected to destroy the vortices by eliminating their phase singularities. Thus, the conservation laws that exist in each individual frequency component might be broken for ultrafast dynamics.
To the best of our knowledge, no work so far accessed the full spatiotemporal dynamics of optical vortices, to show the in-plane motion of a vortex and its evolution from creation to annihilation.
In this work, we observe the spatiotemporal dynamics of 2D optical vortices, identifying events of vortex-pair creation and annihilation. The 2D optical vortices appear as points of zero field amplitude that are shown to move continuously inside the sample. All the features predicted by the theory of vortex temporal dynamics (  We use an ultrafast transmission electron microscope (UTEM) [29][30][31][32][33][34][35] to probe the spatiotemporal dynamics of the phonon polariton (PhP) vortices in boron-10 isotopically pure hexagonal boron nitride (h 10 BN). hBN is a widely studied 2D material which exhibits unique hyperbolic PhPs [36][37][38][39][40][41][42]. Besides PhPs in hBN, 2D materials span a much larger range of polaritons (including plasmon-polariton, phonon-polariton, exciton-polariton, and more [43,44]), which have dispersion relations that may be tuned via their thicknesses, surrounding environments, and material doping. Here, PhPs were chosen due to their slow group velocity that determined the vortex velocity and the relatively long lifetime of the PhPs in monoisotopic boron h 10 BN in room temperature [38,42]. The long lifetime was essential for showing other 2D optical phenomena in room temperature such as PhP cavity dynamics [39], wavepacket dynamics [41], and PhP lensing [40]. Our observation of PhP vortices in hBN is the first measurement of optical vortices in any 2D-polaritonic system. Fig. 2 shows how we extract the temporal dynamics of the vortices. We use a pump-probe technique, in which a single laser pulse is divided so that one part is converted to the mid infrared (IR) regime to pump the PhPs in the sample, while the other part is converted to the UV to excite the free electron probe in the UTEM (see details in SM section S1). Our probing approach is based on the technique called photon-induced near-field electron microscopy (PINEM) [29], which originally operated in the visible and near IR range [30][31][32][33][34][35]. The pulsed free electron interacts with the electric field along its trajectory, resulting in a widening of its energy spectrum. The image of the PhPs is produced when applying an energy filter which collects only the electrons that gained energy from their interaction with the PhPs. This technique is named energy-filtered transmission electron microscopy (EFTEM) [45][46][47]. The filtering creates a threshold for the minimum electric field that we can measure. Above this threshold, there is a quadratic connection between the integrated electric field along the electron trajectory and the number of counts in the image [30,31]. The dynamics of the field is probed when changing the time delay between the mid-IR pump pulse and the free-electron probe pulse. A similar approach to observe field spatiotemporal dynamics was first used in [41] to monitor the propagation of PhP wavepackets and extract their group velocities. We use time steps of 50 fs and find vortex dynamics for a duration of 4.5 ps, significantly longer than the mid IR pulse duration of 600 fs FWHM. The PhP spatiotemporal dynamics is shown in Fig. 2b and Movie S1. The PhP field is excited at the edges of the sample (coupling directly to the bulk is impossible due to momentum mismatch between the free-space photons and the PhP modes). From the edges, the PhP wavepackets propagate toward the sample center ( =0 to 0.4 ps) and interfere with each other   Since pairs of vortices can be created or annihilated, the total number of vortices is not conserved. The quantity that is conserved is the local OAM at each point inside the bulk. However, the total OAM of the field in the sample is not conserved due to a third mechanism of vortex creation and annihilation -along the edges, single vortices can be created or annihilated (Fig. 3 middle column). At the core of this mechanism is the effect of anomalous reflection of planar waves at the edges of the sample that adds a phase to the field [48,39].

Analysis of the measured spatiotemporal vortex dynamics
Our experimental observation of vortex creation and annihilation demonstrates the features predicted by the theory. Specifically, Fig. 4a shows selected timeframes to highlight vortex dynamics, in close agreement with the theory we showed in Fig. 1  The simulation field pattern in a specific timeframe, showing how a set of vortices can determine the qualitative shape of the field's pattern inside the sample. (e) An amplitude peak is formed when surrounded by a set of vortices with alternating orientations (pink). When the vortex orientations are not alternating, a saddle point is created, having a smaller amplitude compared to the peak (gray). (f) The number of vortices with an OAM of +1 (blue), -1 (red), their sum (purple), and difference (brown). The total OAM changes due to creation or annihilation of vortices on the edges while the total number of vortices includes pair creation or annihilation. The excitation pulse duration is shown in blue. The comparison with a full numerical COMSOL simulation is shown in Fig. S7. Fig. 4b-e show how we deduce the location and sign of the vortices. For each frame, we find the vortex locations by identifying the nodal points and nodal lines, appearing as areas with extremely low counts (Fig. 4b and Fig. S2). Then, we analyze the field pattern to determine the OAM sign of each vortex (Fig. 4c). We can verify from simulations (Fig. 4d-e) that each peak of the field is surrounded by a set of vortices of alternating signs (e.g., +-+-), forcing a uniformity in the phase within the peak. In contrast, when the vortices do not have alternating signs (e.g., ++--), the field that is surrounded by these vortices is not a peak but a saddle, having a smaller amplitude compared to a peak and a larger variation in the phase inside it. The saddle configuration can create a large area in the sample with very low counts (Fig. 4c bottom right). This area seems to have almost zero counts due to the minimal electric field that we can measure (~1 MV/m, SM section S1), but it is not strictly zero (see the average signal over time, Fig. S3). Finally, when comparing the estimated vortices in all timestamps, we recognize the continuous movement of the vortices and the events of vortex creation and annihilation.
The observation of vortex dynamics shows the prospects of our experimental platform as an optical simulator of hard-to-access spatiotemporal phenomena such as vortex dynamics and phase transitions in condensed matter systems. In these systems, vortex-antivortex correlations exist for a condensate in the low-temperature phase but disappear in the high-temperature phase after a Berezinskii-Kosterlitz-Thouless (BKT) phase transition [28]. In our measurement (Fig.   4f.) and simulations (Fig. S7), we identify the total number of vortices and the total OAM of the flake as a function of time. We observe that the laser excitation (cyan in Fig. 4f) acts as a source of order that decreases the total number of vortices when it is applied on the sample, analogous to reducing the effective temperature in a solid-state BKT-type system. The number of vortices then gradually increase again after the laser excitation is over. The degrees of freedom in controlling the optical excitations and the system's boundary conditions make planar optical platforms valuable simulators to promote our understanding of condensed-matter physics phenomena (e.g., [15,23]).

Discussion and outlook
Our experiment stands as an example for the universal nature of planar vortices. In most experiments that showed 2D optical vortices so far, the boundary conditions were designed for generating a specific vortex OAM at a specific location that remains constant in time (e.g., [21,22]). In our experiment, we used the natural edges of the sample to show that vortex spatiotemporal dynamics can appear in arbitrary samples with only a single requirement -that the sample is optically mesoscopic. We can divide all samples into three categories (see  ). Such dimensions support free-propagation dynamics (data taken from [41]), until the field disperses and decays. In all figures, the dashed lines denote the sample's boundaries. The inset at the bottom of each panel shows a hBN flake of the corresponding size with which the presented data was collected. The red bar denotes 5 m in all insets.
It is interesting to compare the features of PhP vortices to vortices in exciton-polariton condensates [50]. There, a strong interaction between matter and an optical cavity mode creates the condensate, which can form a superfluid [51]. Previous work in such systems show the creation and propagation of vortex pairs [52,53] but did not so far measure their spatiotemporal dynamics.
Although sharing certain polaritonic properties with PhPs, the behavior of exciton-polaritons arise from the Gross-Pitaevskii equation and requires extremely low temperatures, in contrast to the PhP vortices that arise from Maxwell's equations and were measured here at room temperature. Recent demonstrations of quantum simulations [54] and analogies of gravity [55] with exciton-polaritons raise intriguing possibilities for similar prospects with PhPs, especially once higher excitation intensities reach the regime where their nonlinear optical response cannot be neglected [56]. It is possible that nonlinear effects had already affected our measurements here (see Fig. S8 and SM section S4). Future investigations may deliberately introduce nonlinearity through atomic emitters or 2D quantum wells, exploiting their extremely strong light-matter interactions with 2D light [57], and specifically with vortices of 2D light [16,17].
Looking forward, desired vortex phenomena such as studying chiral plasmonics [58] with 2D polaritons could be created by engineering boundary conditions (for example using inverse design methods [59]). Moreover, if excited by non-classical light, a properly designed boundary condition could establish full entanglement between two vortex states of opposite charge (±1) at two different locations , : taking the form of |+1, ⟩|−1, ⟩ + |−1, ⟩|+1, ⟩. Together with the precise ultrafast coherent control and deep-subwavelength spatial resolution, 2D material platforms could be suitable for demonstrating and probing wide ranges of vortex phenomena in optics and condensed matter physics.