Spatiotemporal imaging of 2D polariton wave packet dynamics using free electrons

Imaging polariton dynamics Two-dimensional (2D) materials can confine light to volumes much shorter than the wavelength, and, together, the long propagation lengths make them attractive materials for developing nanophotonic platforms. Characterizing the spatiotemporal control of 2D polariton wave packets has been hindered for the same reasons that make their potential applications exciting: They have extremely small wavelengths and are strongly confined inside the material. Kurman et al. developed a new pump-probe technique based on electron emission that provides access to the spatiotemporal dynamics of 2D polaritons. The nanometric spatial resolution and femtosecond temporal resolution will be useful for probing the excitation dynamics of these materials. Science, abg9015, this issue p. 1181 A pump-probe electron emission technique is used image the dynamics of polaritons in 2D materials. Coherent optical excitations in two-dimensional (2D) materials, 2D polaritons, can generate a plethora of optical phenomena that arise from the extraordinary dispersion relations that do not exist in regular materials. Probing of the dynamical phenomena of 2D polaritons requires simultaneous spatial and temporal imaging capabilities and could reveal unknown coherent optical phenomena in 2D materials. Here, we present a spatiotemporal measurement of 2D wave packet dynamics, from its formation to its decay, using an ultrafast transmission electron microscope driven by femtosecond midinfrared pulses. The ability to coherently excite phonon-polariton wave packets and probe their evolution in a nondestructive manner reveals intriguing dispersion-dependent dynamics that includes splitting of multibranch wave packets and, unexpectedly, wave packet deceleration and acceleration. Having access to the full spatiotemporal dynamics of 2D wave packets can be used to illuminate puzzles in topological polaritons and discover exotic nonlinear optical phenomena in 2D materials.

There is motivation to use the distinctive properties of 2D polaritons and integrate them into ultrafast optical technologies that rely on the spatiotemporal control of light. Conventional areas of ultrafast optics achieve such control using pulse-shaping (12) and dispersion engineering, which are instrumental, for example, in photonic waveguides (13). The attainment of similar control with 2D polaritons could promote the integration of 2D polaritonic materials in mature areas of science and technology and contribute to their fundamental understanding. However, the spatiotemporal control of 2D polariton wave packets has remained out of reach for exactly the same reasons that make their potential applications exciting: They have extremely small wavelengths and are strongly confined inside the material. New capabilities are necessary for accessing the spatiotemporal dynamics of 2D polaritons and their wave packets with nanometric spatial resolution and femtosecond temporal resolution.
In this regard, it is particularly interesting to consider wave packets in materials that exhibit hyperbolic dispersion (14,15): Polaritons in hyperbolic materials show rich physical behavior, ranging from negative refraction (16) and subdiffraction imaging (14) to effective Hawking radiation. Although hyperbolic dispersion was originally observed in metamaterials, phonon-polariton (PhP) excitations in 2D materials also exhibit hyperbolic dispersion (8,17); the phononic resonance creates a dispersion relation that contains multiple branches (see Fig. 1B), which were shown to be tunable by the 2D material geometry, thickness, and surrounding environment (4,5,17), reaching high confinement with relatively low losses, even at room temperature (6).
The peculiar dynamics of hyperbolic wave packets can be seen by looking at the group velocity: The derivative of the dispersion, @w/@k (where w is the radial frequency and k is the wave number), approaches zero in a hyperbolic medium, because the medium supports excitations with (in principle) arbitrarily large momenta. In practicality, the group velocity of PhPs is well defined, given an excitation of a finite bandwidth exciting a specific branch. However, in 2D hyperbolic materials, any coherent excitation is expected to simultaneously stimulate multiple PhP branches, each having different dynamics. Moreover, even a small variation in the excitation's bandwidth or frequency results in a large change in the polariton group velocity and the entire wave packet propagation dynamics. These prospects are especially intriguing in 2D hyperbolic materials such as thin flakes of hexagonal boron nitride (hBN), where the group velocity of polaritons was shown be as low as c/500 (with c being the speed of light in vacuum) over a relatively wide bandwidth (18). To explore these prospects of hyperbolic wave packets and reveal their physics, we need access to the field comprising the wave packet during its evolution inside the 2D material.

Coherent wave packet dynamics
Our approach to measuring the spatiotemporal dynamics of hyperbolic PhP wave packets inside isotopically pure hBN ( 11 B) flakes (6) is made possible by exploiting the interaction between free electrons and PhP wave packets. The temporal dynamics is obtained using a pump-probe technique with a mid-infrared (IR) laser pump and a free-electron probe in an ultrafast transmission electron microscope (UTEM) (19-28) (Fig. 1A). The pulsed free electron penetrates the sample and consequently changes its energy spectrum according to the integrated electric field along its path (insets of Fig. 1C). Through energy filtering of the postinteraction electron, we can reconstruct the image of the PhP wave packet. The use of laser-driven energy-filtered transmission electron microscopy (EFTEM) was first demonstrated in photoninduced near-field electron microscopy (PINEM). Our approach takes PINEM to the mid-IR range and combines it with 2D materials, providing a test of dispersion models of 2D materials in the time domain.
We recorded the PhP wave packet creation and propagation [ Fig. 1D and movies S1 and S2 (29)], revealing rare physical behaviors, such as multibranch wave packet splitting, acceleration, and deceleration. The measured acceleration and deceleration dynamics are especially surprising because wave packets conventionally have a fixed group velocity. In certain cases, as in the thin hyperbolic hBN flakes, the group velocity is expected to be very slow-indeed, we observed group velocities from c/45 to c/850and still, each wave packet is expected to propagate with a fixed group velocity. In contrast to this expectation, we show that the dispersive nature of PhPs (i.e., their momenta change rapidly in frequency) facilitates acceleration and deceleration dynamics. This result serves as a key example for the rich physical phenomena that can be found when probing the spatiotemporal dynamics of 2D polaritons in a nondestructive manner when combining the femtosecond temporal and nanometer spatial resolution of the UTEM.

Imaging 2D polariton excitations
Among the various experimental techniques used in the field of 2D polaritons, scanning nearfield optical microscopy (SNOM) and its variants have had the most impact so far on the direct near-field imaging of 2D polaritons (6,17,30). Recent advances in time-resolved SNOM also allowed the polariton's properties as group velocity to be extracted from the interference of scattered polaritons with different time delays (18,(31)(32)(33). However, this interferometric technique cannot image the wave packet dynamics, because it does not include the spectral phase, i.e., the phase difference between photons of different wavelengths. Other important experimental approaches, such as photoemission electron microscopy (34), are also used for near-field imaging in plasmonics; these approaches, thus far, have not accessed the mid-and far-IR regions. We discuss the different experimental approaches in the supplementary text (29). Importantly, techniques based on TEM stand out from all of the above because the electron penetrates through the sample and also becomes sensitive to the buried field rather than only to the field on the surface (23), an advantage for probing the highly confined 2D polaritons.
Our approach for the observation of PhP wave packet dynamics adds to the toolbox of electron-beam spectroscopy and microscopy (35). Of particular importance for our approach are the advances in the imaging of polaritons using electron energy-loss spectroscopy (EELS) (36), which is able to measure IR excitations in vibrational electron spectroscopy (37). EELS enabled the measurement of purely vibrational modes (phonons) in bulk media and on surfaces (38) and the extraction of their dispersion relations using electron imaging and diffraction (39). In recent years, the improvements in the energy resolution of EELS (37) enabled the extraction of the PhP dispersion in extremely thin samples (40). Such experiments translated methodologies in electron microscopy, once applied to plasmons in the visible range [e.g., (41)], to phonons in the mid-IR range.
In all these EELS experiments, one obtains static, time-independent information on the polaritonic modes and other excitations of the sample, all of which are triggered by the free electron. By contrast, PINEM-based techniques such as ours use the electron only as a timedependent probe (and not as a trigger of the excitation); thus, PINEM allows the extraction of time-dependent information on the temporal dynamics of the polaritons that are excited by a separate laser pulse.
Wave packet creation and propagation Figure 1D and movie S1 present an example of the measurements made of the wave packet during its creation and propagation inside the flake. The video is created by repeating the measurement for a range of time delays between the laser pump and the electron probe. Such measurements of wave packet dynamics rely on stimulated free electron-PhP interactions. For the wave packet, the electron is a nondestructive probe: The interaction alters the wave packet Kurman (29) in a negligible manner and hence does not interfere with the wave packet evolution dynamics. Thus, the wave packet propagates across the sample uninterrupted, starting from a single edge [chosen by optimizing the laser coupling (29)]. At each time delay, the wave packet profile is reconstructed from the electron energy distribution: At points where the PhP's out-of-plane electric field is stronger, there is a larger probability for the probing electron to gain energy and pass an energy filter. The energy filter is chosen to maximize the signal (see fig. S4); it creates a threshold for the detectable PhP field and thus reduces the signal-to-noise ratio. Consequently, as we describe in the theory below, the connection between the number of counts and the electric field is nonlinear.
To model the free electron-PhP interaction, we find it essential to generalize the singlefrequency theory of conventional PINEM (20,21) that was used to describe most PINEM experiments to date. The need to go beyond the successful PINEM theory lies in the finite bandwidth of the excitation laser (necessary for creating the pulsed PhP wave packet) that excites the highly dispersive hBN PhPs. To capture the spectral bandwidth, we describe the free electron-PhP coupling through a generalized coupling function g(x, y, w). This coupling function g quantifies the strength of the interaction for each in-plane coordinate (x, y) and each frequency w. According to this continuous-PINEM theory (42) where e and v are the electron charge and velocity, respectively, and ħ is the reduced Planck's constant. The integral is performed along the electron propagation direction z, on the z component of the electric field phasor E z w ð Þ ¼ ∫E z t ð Þe iwt dt , which includes all the PhP modes due to their transverse magnetic (TM) polarization.
The PhP wave packet is imprinted on the electron as a time-dependent phase modulation: This phase modulation multiplies the initial electron wave function with a time delay (t d ) relative to the PhP wave packet excitation; t d is shifted for recording a video of the dynamics. Our theory predicts the measured electron energy spectra as the Fourier transform (time→ energy) of the resulting electron wave function. Equation 2 shows how larger g values imply stronger modulation in the phase of the freeelectron wave function, equivalent to the electron gaining and losing more energy (29). The energy required for a detectable signal is related to the incoherent energy width of the preinteraction electron (also called zero-loss peak). Because the width (0.9 eV in our system; right inset of Fig. 1C) is larger than the energy of a single PhP quanta, the postinteraction EELS spectrum (left inset of Fig. 1C) does not have discrete peaks as in PINEM experiments in the visible or near-IR range (26). Nevertheless, the change in the electron's energy is sufficient for probing the PhP wave packet: The electron image in the x-y plane is filtered by energy for different time delays t d to extract the PhP spatiotemporal dynamics.
In the analysis of the measured PhP wave packet dynamics, we first extract the field profile along the direction of propagation ( Fig. 2A). When averaging the signal along the sample's excited edge, we reduce the signal-to-noise ratio [described in (29)]. We find that a combination of a Gaussian and an exponential decay due to the edge effect provide a good fit to the measured wave packet, which can also capture the slowly moving higher branches quite accurately. This analysis reveals the formation of the wave packet during the arrival of the excitation pulse. Figure 2B shows an intriguing phenomenon: The wave packet appears to remain stuck at the edge for a certain time duration and does not immediately propagate away from the boundary. Instead, the wave packet's width and amplitude gradually increase. Thus, the gradually forming wave packet increases in amplitude while remaining near the edge. Only toward the end of the excitation pulse (when its tail arrives) does the wave packet start to move away from the edge more quickly, exhibiting phenomena of acceleration and deceleration that vary between samples and excitation wavelengths. Once the excitation pulse has ended, we can extract the stable group velocities.
The wave packet properties during its formation, propagation, and gradual decay are summarized for three different samples and a range of excitation pulses (Fig. 3, A to C). As expected, the group velocities become smaller as the sample thickness decreases. The lowest measured group velocity (light blue in Fig. 3A)   lower than the speed of light in vacuum. In a thicker sample, the fastest recorded group velocity is 6.7 mm/ps, which is 45 times lower than the speed of light in vacuum but still sufficiently low for our free-electron probing technique to record the dynamics. An additional measurement includes the propagation over a duration of more than 2.5 ps in a 7.5-nmthick sample and propagation lengths over distances of more than 12 mm for a 55-nm-thick sample, crossing the entire length of the sample. These propagation distances and durations are a merit of the isotopically pure hBN ( 11 B), which encounters smaller losses than normal hBN flakes (6). The data show a first demonstration of a change in the group velocities of PhP wave packets during their propagation, for which we use the terminology wave packet acceler-ation and deceleration. We observed this effect in all sample thicknesses (Fig. 3, A to C). For example, in the 7.5-nm sample, the group velocity decreased by a factor of 5 (Fig. 3A). This effect is surprising because the PhP dispersion is expected to cause the slower modes to decay faster, which would result in acceleration rather than deceleration.
We find that this effect reveals a more general S-shaped trajectory for the PhP wave packets: During the pulse excitation (shaded backgrounds in Fig. 3, A to C and E), the wave packet first remains near the edge of the sample (extraordinarily slow group velocity), then experiences fast propagation, and finally decelerates to reach its final steady propagation velocity. The first two regimes are related to the excitation pulse duration and chirp, which pumps the PhP with different spectral components at different times. The measured phenomena of wave packet deceleration and acceleration are consistent with the continuous-PINEM analysis of a set of finite-difference time-domain (FDTD) simulations, as shown in Fig. 3D, fig. S5, and movies S3 and S4. The simulations show that when the pulse is longer, and/or when the chirp is stronger, the change in velocity becomes more dominant, in agreement with our measurements.

Analysis of wave packet dynamics
Our experiment identifies intriguing features of the PhP wave packets during the wave packet formation and free propagation. We integrate over the entire observable signal at each time delay to analyze the competition between the laser pumping to the dispersion and decay. We identify two features in the PhP wave packet dynamics: The first is the gradual buildup of the wave packet when the pump overcomes the PhP wave packet dispersion and intrinsic ohmic losses (shown as the blue-shaded background in Fig. 3E). We identify cases for which the integrated signal continues to grow even when the wave packet peak is already 10 mm from the edge and is propagating at a stable group velocity (green curves in Fig. 3, C and E). The second feature is that the measured wave packets disperse and decay at different rates (Fig. 3, E and F), quantified by a dispersion time t (Fig. 3E). The dispersion causes wave packet broadening that reduces the field amplitude and thus also reduces the detected electron signal. The detected electron signal also varies with frequency, because different PhP modes have different confinements [see (29) for a quantitative analysis of the signal dependence on PhP confinement]. As a result, the decay in the integrated signal is caused mostly by the wave packet dispersion (subpicosecond time scale) and not by intrinsic ohmic loss (few-picosecond time scale). As expected from theory, shorter dispersion times (i.e., larger dispersion) occur for slower group velocities. The efficient PINEM interaction enables the measurement of the propagation of multi-branch wave packets that split over time into distinguishable Gaussian-like wave packets of different group velocities ( Fig. 4 and movie S2). The multibranch wave packets are created because the excitation at each wavelength can couple into more than a single branch in the dispersion relations (Fig. 1B), a phenomenon which was also observed in plasmonic structures (34). Figure 4A shows the propagation of a multibranch PhP wave packet that splits into two single-branch wave packets, propagating as a double Gaussian (Fig. 4B). From the location of the double Gaussian peaks (shown in Fig. 4C), we extract two different group velocities for the first-and second-branch wave packets (6 and 1.1 mm/ps, respectively). We enhanced the observation of this effect by changing the electron energy slit to reduce the threshold electric field. As confirmed by the FDTD simulations (Fig. 4, D to G), at short times, the two wave packets completely overlap. Then, at longer times, the wave packets gradually split from one another. Interestingly, as also confirmed by the FDTD simulation, the PINEM-type measurement shows a clear spatial separation between the wave packets while their fields still appear to overlap. It is the electron's sensitivity to the field inside the hBN that enables one to distinguish the individual profiles of the two partially overlapping wave packets.
We found that the PINEM interaction remains efficient for polariton wave packets at higher-order branches. This feature is seen in the experimental data: The first-and secondorder PhP branches provide a signal of a similar magnitude (Fig. 4B). Our numerical simulations confirmed this result: Despite the second branch having an overall smaller energy (owing to less efficient coupling), the PINEM signal is of comparable strength. This result arises from the nature of the interaction, whereby the electron integrates along its trajectory and thus includes the field inside the hBN, which is larger because the field is more confined. Higher-branch modes have larger confinement and thus higher field amplitudes, and yet they decay faster along the z direction; thus, the integrated signal remains comparable. Most near-field imaging techniques probe the surface of the material and are thus essentially less efficient when the goal is to image wave packets in higher-order branches, because they decay more rapidly along the z direction. (The inefficient light out-coupling of the higher branches reduces the signal further.) By using the penetration of free electrons, PINEM-type techniques such Kurman  as ours bypass these limits, becoming especially advantageous for imaging higher-branch polaritonic modes and highly confined polaritons in general.

Concluding remarks and outlook
The measured phenomena are fully captured by the linear optical response of hyperbolic media, as verified by our simulations. Finding the nonlinear optical response of hBN is still a topic of ongoing investigation (43,44) and should require stronger fields compared with the values in our experiments [electric field of 1.5 MV/m (29)]. Future investigation of optical nonlinearities in 2D materials holds potential for the observation of spatiotemporal soliton wave packets of 2D polaritons and previously unobserved forms of 2D shockwaves and light bullets.
Our work builds on key advances in the areas of PINEM, ultrafast electron microscopy, and ultrafast electron diffraction (25). Whereas PINEM techniques have examined the temporal dynamics of optical excitations by interference patterns of plasmon polaritons (26) and decay of photonic cavity modes (27,28), our experiments resolve the dynamics of the entire optical wave packet due to the slower group velocities and shorter wave packet extents of PhPs. The slow group velocity causes the wave packet envelope to remain almost static within the duration of the electron interaction (the wave packet moves only a fraction of a micron during the electron's duration; 0.13-ps standard deviation); thus, we can record the wave packet dynamics. In this manner, our experiment is operated at frequencies that are in between the optical range and the range of mechanical vibrations (typically at gigahertz frequencies), where the dynamics of phonons was observed because of their slow oscillation relative to the duration of the electron probe (22,24). The PhPs in our experiment differ from these mechanical vibrations by being coherent optical excitations, which are in fact hybrids of the mechanical vibrations (optical phonons) and photons in the IR region. Consequently, the imaging of polaritons in 2D materials uses advantages from both plasmon imaging (26) and phonon imaging (22,24).