Transient lensing from a photoemitted electron gas imaged by ultrafast electron microscopy

Understanding and controlling ultrafast charge carrier dynamics is of fundamental importance in diverse fields of (quantum) science and technology. Here, we create a three-dimensional hot electron gas through two-photon photoemission from a copper surface in vacuum. We employ an ultrafast electron microscope to record movies of the subsequent electron dynamics on the picosecond-nanosecond time scale. After a prompt Coulomb explosion, the subsequent dynamics is characterized by a rapid oblate-to-prolate shape transformation of the electron gas, and periodic and long-lived electron cyclotron oscillations inside the magnetic field of the objective lens. In this regime, the collective behavior of the oscillating electrons causes a transient, mean-field lensing effect and pronounced distortions in the images. We derive an analytical expression for the time-dependent focal length of the electron-gas lens, and perform numerical electron dynamics and probe image simulations to determine the role of Coulomb self-fields and image charges. This work inspires the visualization of cyclotron dynamics inside two-dimensional electron-gas materials and enables the elucidation of electron/plasma dynamics and properties that could benefit the development of high-brightness electron and X-ray sources.


INTRODUCTION
Understanding the non-equilibrium dynamics of charge carriers (electrons/ions/holes) is of uttermost importance in a vast range of fundamental and technological fields, including chemistry, solid-state physics, plasma physics, and high-brightness electron sources. Carrier motion often unfolds on ultrafast time scales and requires tools that can directly visualize the dynamics with appropriate spatial and temporal resolutions, i.e. Ångstroms-micrometers (Å-μm) and femtoseconds-nanoseconds (fs-ns), respectively. In this regard, ultrafast electron microscopy (UEM) has recently emerged as a powerful technique for the study of ultrafast photoinduced processes in nanoscale systems [1][2][3][4][5][6][7][8][9][10][11][12][13][14] . The material is excited by a short fs-ns laser pulse, which is followed by a similarly short electron pulse that probes the ensuing dynamics by means of imaging, diffraction, or spectroscopy inside a transmission electron microscope (TEM).
Here, we use UEM to visualize the ultrafast evolution of a hot three-dimensional (3D) photoemitted electron gas under a static magnetic field in real time and real space. Confined electron gases 15 can exhibit intriguing properties such exceptionally high electron mobilities 16 , quantum Hall effects 17,18 , Shubnikov-de Haas oscillations 19 , anomalous de Haas-van Alphen effects 20 , and superradiant damping 21 . Understanding and controlling these phenomena is of fundamental importance in diverse fields of quantum science and technology 22,23 . For example, two-dimensional (2D) electron gases at semiconducting heterointerfaces or in 2D materials, that are subject to an external magnetic field, have been studied by frequency-and time-domain THz spectroscopies 17,21,[24][25][26] . An electron gas in a uniform magnetic field executes circular Larmor orbits in a plane perpendicular to the magnetic field. Transitions between the eigenstates (Landau levels) of electron gases confined by a magnetic field are called cyclotron resonances, whose frequencies, line widths, and decays have been used to determine band structures, effective masses, carrier densities, mobilities and scattering times in semiconducting materials 21,25,[27][28][29][30] . Quantum effects arising from Landau levels are dominant when the mean thermal energy of the gas is smaller than the energy level separation, which means experiments are often performed at low temperatures and under strong magnetic fields.
The proof-of-principle UEM experiments on 3D electron gases in uniform magnetic fields presented in this work pave the way for the direct visualization of cyclotron oscillations inside materials, in particular 2D electron-gas systems such as GaAs/AlGaAs 18,25,31 or graphene 26,32,33 . In contrast to frequency-or time-domain THz/microwave spectroscopic investigations, performing such experiments inside an ultrafast electron microscope enables spatially resolving photoexcited electron density variations, similar to previous scanning probe microscopy experiments 31,34-36 but with fs-picosecond (ps) temporal resolution. Furthermore, the capability to image and temporally resolve photoemitted carriers is highly relevant in the plasma physics community [37][38][39][40] , and for the development and characterization of high-brightness electron sources for fourth-generation X-ray facilities or ultrafast electron diffraction and microscopy setups [41][42][43][44][45][46][47][48][49] . The analytic model we develop here allows rough approximation of the number of electrons in the photoemitted gas, which is directly correlated with the electron lens magnification, as well as their velocity spread.
Systematic variations of the laser fluence and wavelength, and adding a bias to the sample, will enable obtaining valuable insight into the electron emission process and the subsequent processes that affect electron beam properties such as emittance.

RESULTS
Direct imaging of electron cyclotron oscillations on the picosecond time scale. We performed our experiments using a modified environmental TEM operating at 300 keV (Fig. 1a), which is interfaced with a high repetition-rate, fs laser system (see Methods section for more details). Laser pump pulses (~200 fs, 528 nm, ~33 mJ/cm 2 ) are guided onto the sample using a mirror/lens system that inserts into the energy-dispersive spectroscopy (EDS) port of the TEM. Short probe electron pulses (<1 ps) are generated by impinging a UV laser beam onto a LaB6 photocathode. The laser pump and electron probe pulses are precisely synchronized in time at a repetition rate of 490 kHz and their relative delay is adjusted by means of an optical delay stage. In this way, we record realspace movies of the charge density dynamics after laser excitation with an integration time of 1 with an objective lens current of 0.7 A. The time delays correspond to the first few local maxima and minima in the ROI difference intensity trace in Fig. 3. The difference images were generated by subtracting an averaged image before time zero (Δt = 0). A typical region-of-interest (ROI) circle that is used to make plots of the intensity changes due to lensing is indicated in the last frame. with an electron cyclotron period of * = 2,-. /! ⁄ , where ' is the velocity vector of the electrons, me is their mass, and e is their charge (Fig. 1b). The (non-relativistic) radius of gyration for each electron is given by 1 = 2 3 -. /! ⁄ , with 2 3 the transverse (4, 6) velocity component in the direction perpendicular to the magnetic field (7). Each electron therefore circulates with a different radius, depending on its initial velocity, but all electrons reconvene to their initial positions in the 4, 6-plane after a full cyclotron period T. Slightly before this point in time, the collective width of the electrons reaches a minimum and the transverse electric mean-field maximizes resulting in a pronounced transient lensing effect that is observed in the probe image.
While the magnetic field confines the electron gas in the transverse direction, the longitudinal dynamics is affected by the z-velocity profile and the boundary conditions at the surface of the copper grid. We show later that this anisotropic confinement causes a rapid oblate-to-prolate shape transformation of the 3D electron gas and a large concurrent increase of the lensing strength on the time scale of ~100 ps. Fig. 4a shows ROI difference intensity traces recorded at various objective lens currents (OLC). Using the cyclotron period formula above and a FFT analysis of the ROI traces, we constructed a plot of the OLC versus cyclotron period and magnetic field (Fig.   4b). A linear relationship between OLC and magnetic field is obtained, which matches data from the TEM manufacturer (see SI1). The ROI intensity trace for OLC = 0 A (no magnetic field) merely shows the first peak, as expected.
We note that deflection effects due to transient electric fields from photocreated electron plumes have been observed previously in ultrafast electron diffraction and microscopy setups 37,38,[53][54][55][56][57][58][59] . However, we report for the first time a detailed study of the space-charge dynamics  Dependence on the imaging conditions and electron-gas astigmatism. The transient lensing effects are only visible in the images if the projection lens system of the TEM is set to out-of-focus image conditions. This is a consequence of the geometry. Because the laser impinges on the grid from above, and because the fill fraction of the grid is quite high (66%), nearly all the photoemitted electrons will be above the grid. The grid itself will act as an electrostatic boundary condition preventing the space-charge electric fields from penetrating significantly below. Thus, the lensing effect of the electron cloud will deflect electrons radially before they strike the grid but will have essentially no effect on the grid pattern itself or on the focusing action produced by the TEM lenses below the grid. In a focused real-space image, the post-sample lenses map the (4, 6) spatial positions of electrons as they emerge from the back of the sample linearly onto the camera, suppressing information about the angles of the electron trajectories. Thus, an in-focus image should just produce an image of the grid, as we observe. To detect the space-charge lensing effect, we defocus the imaging system so that the resulting image is a linear combination of the spatial and angular coordinates of the electrons emerging from the back of the sample. Because the OLC is an extremely important parameter for the space-charge dynamics, we instead adjust the current of the first intermediate lens (IL), i.e. the first lens after the objective lens (see Fig. 1a). We recorded lensing movies (see movies S2-S4) for a range of IL excitation strengths. In this way, we are able to tune the electron-gas lensing effect from a magnifying, barrel image distortion for low IL excitations (Fig. 5a), to a demagnifying, pincushion image distortion for high IL excitations ( Fig. 5c). For intermediate IL strength (Fig. 5b) we can image the post-sample crossover of the objective lens onto the detector. Thus, the defocused images reveal the shift in the post-sample crossover caused by the lensing effect of the electron gas, manifesting as a magnification or demagnification of the affected region relative to the rest of the grid.
Since the electron cloud acts like a diverging lens, the probe electrons that pass through the cloud are focused by the objective lens after/below the probe electrons that pass further away and are not affected by the electron gas. If the IL current (ILC) is set to a value such that a plane between the post-sample crossover of the unaffected probe electrons and the post-sample crossover of the lens-affected probe electrons is imaged onto the detector (Fig. 6), the ROI difference where S = /! 2-. ⁄ is the cyclotron angular frequency, E V is the velocity spread in the transverse direction, and E U is the minimum transverse radius of the electron gas. Eq. (1) shows that at times F = 2>, S ⁄ (> = 0,1,2, …) the transverse radius reaches its smallest value E U , and the electron number density concurrently maximizes. These periodic electron density peaks are responsible for the transient lensing effects in the probe images (e.g. Fig. 2).
Using Maxwell-Gauss law, we derive an expression for the radial electric field ℇ U (F) associated with the cylindrical Gaussian charge density (see SI2.2) and find ℇ U ∝ 1 E 3 close to the center of the electron cloud the field is linear with radius r which imparts the lensing effect on the probe electrons. Assuming that the duration of the interaction between the relativistic probe electrons and the electron gas is short compared to the evolution time scale of the gas, and also using the thin-lens approximation, we derive the focal length of the electron gas as (see SI2.3) where i j is the vacuum permittivity, γ = 1.6 is the relativistic Lorentz factor, 2 G = 2. The resulting fit is shown in Fig. 7, together with the corresponding electron-gas focal length. Interestingly, the focal length magnitude varies by about a factor of ten, ranging from ~0.5 m at the lensing maxima to ~4-5 m in between cyclotron resonances. From the fit we are able to determine the angular frequency S = 37.97 ± 0.01 GHz, and the transverse velocity spread E V = 4.91 ± 0.01 • 10 w m/s (the standard deviations on the fit parameters do not reflect the inaccuracies of the model itself), the latter of which largely determines the width of the resonance peaks. The decay constant τ is on the order of several ns, but could not be determined with high accuracy due to the limited fitting window. We note that the quantities E V and E U are not expected to vary with time because the magnetic field does not do work and we are treating the self-interaction as negligible resulting in no appreciable electric field. Furthermore, the excellent agreement between the model and the data indicates that space-charge effects play a negligible role in this time regime (>100 ps). On the other hand, the amplitude of the ROI intensity change at the first peak predicted by theory is much larger than the experimentally measured amplitude. In fact, the ratio of the first in the experiment. The electromagnetic forces arise from self-Coulomb fields, the external magnetic field, as well as positive image charges due to the existence of the copper grid that acts as a planar conductor held at zero potential. We assume that electrons that hit the grid will be absorbed and hence omitted from the rest of the calculation. Snapshots taken from a simulation with ! = 0.22 T at three time delays are superimposed in Fig. 8a (full movie S5). The frame at 16 ps after photoexcitation shows a flat electron distribution close to the copper grid, that has already significantly expanded due to Coulomb explosion of the gas during the first few ps. The distribution in the intermediate 83 ps frame, which corresponds to the first minimum in the electron cloud density, is homogeneously spread out over tens of μm in all (4, 6, 7) directions. Finally, for the frame at 165 ps, which corresponds to the first cyclotron resonance peak, the electron gas regains its narrow transverse size, but it is severely elongated along the z-axis. Quantum phenomena such as Landau energy level quantization are therefore not expected to be observed experimentally.
The rapid oblate-to-prolate shape transformation of the electron gas, evidenced by the numerical simulations, has profound influence on its transient lensing strength. Indeed, the first peak in the ROI difference intensity traces is consistently lower in amplitude than the second peak.
Qualitatively, we can attribute this to two things: first, the oblate shape of the electron gas at early times leads to a small transverse electric field component, and therefore a reduced impulse on the probing electrons. Second, the transverse electric field component is further reduced by the positive image charges, effectively creating a parallel-plate capacitor at early times. When the electron gas adopts a prolate shape elongated along the z-axis, the effect of the image charges is largely reduced since the electrostatic dipole force along z scales with ~1~N ⁄ where ~ is the distance between the two charges.
In order to confirm this interpretation on a more quantitative basis, we simulated the UEM lensing movies by sending a regular grid of relativistic probe electrons through each frame of the N-body simulation (see Methods section for details). Here, we neglect Coulomb interactions between probe electrons, as well as any perturbations of the electron gas by the probe electrons.
Representative snapshots of these probe simulations are shown in Fig. 9a (full movie S6), which can be compared to the experimental movie frames in Fig. 2b. All features are reproduced well, including the depletion of the probe intensity in the center, a bright ring around the depletion area in the first peak at ΔF = 4 ps (12 ps in the experiment), as well as a profound magnification of the grid images at the cyclotron resonance peaks (ΔF = 165 ps, 330 ps). The lensing is much stronger at the cyclotron resonance peaks, when the electron gas adopts a prolate shape, than at the first peak, when it has an oblate shape. Corresponding ROI difference intensity traces, with and without the copper grid included, are shown in Fig. 9b, together with an experimental ROI trace taken at low IL currents. The agreement is satisfactory, in particular the ratio of the first and second peak amplitudes is reproduced very well, as well as the shape of the resonance peaks. The discrepancy in width of the peaks is assigned to differences in the electron velocity spread and the number of electrons, which are difficult to get right without explicitly including the photoemission process itself. We emphasize that the first-to-second peak amplitude ratio is only simulated well if the copper grid is included in the simulation. This shows that the image charges, as well as the absorption of electrons by the grid during the Coulomb explosion, play a significant role in the dynamics <50 ps.

CONCLUSIONS
Using a newly developed ultrafast electron microscope, we observed the ps-resolved cyclotron dynamics and lensing of a 3D hot electron gas created by photoemission from a copper target with intense fs laser pulses. Within 100-200 ps after photoexcitation, the gas undergoes an oblate-toprolate shape transformation with a change in aspect ratio of a factor of 10 z , and subsequent transverse expansions and contractions due to the gyration of individual electrons around the static magnetic field axis in the microscope. The cigar-shaped electron cloud acts as a diverging lens to the probe electrons, with focal lengths ranging from ~0.5-5 m during one cyclotron oscillation. We show that the observed lensing is dominated by a cooperative mean-field effect, as opposed to particle-particle scattering of individual probe and cloud electrons. Specifically, the granular nature of the electron distribution can effectively be ignored and instead can be replaced by the mean-field it creates (at least at the velocities we are considering here). Our current analytical treatment allows us to estimate the velocity spread and number of electrons in the gas, but it excludes the influence of Coulomb interactions inside the cloud, positive image charges, as well as the absorption of electrons by the grid. We performed numerical N-body simulations to take these effects into account, which proves to be crucial to understand and simulate the early dynamics before 50 ps. An analytical treatment including Coulomb interactions and image charges, and a more quantitative description of the TEM lensing system, will be part of future work.
These experiments inspire a plethora of future studies in at least three distinct fields. First, they present a unique way to directly visualize and characterize photoemitted charged-particle beams, which is of importance in the fields of high-brightness electrons sources for ultrafast microscopy and fourth-generation X-ray facilities, and plasma physics. Future experiments will focus on systematically investigating the dependence on laser wavelength (tuning the regime from two-photon, to one-photon and three-photon emission), and laser fluence. Furthermore, using an electrical TEM holder, one could apply a bias to the sample which enables studies below the virtual cathode limit 43,64 . Second, our work paves the way for the study of charge carrier cyclotron dynamics inside photoexcited materials using UEM. Such experiments would need to be performed at low temperatures, and need materials with large carrier diffusion lengths, such as InSb, InAs, GaAs/AlGaAs, or transition metal oxides 16,25 . Intense photoexcitation can create electron-hole plasmas, in which the electron and holes gyrate with different frequencies, direction, and spatial extent. Furthermore, the implementation of quantum point contacts using a custom MEMS-based TEM holder, could enable the spatiotemporal visualization of coherent flow and magnetic focusing of charge carriers in 2D electron-gas materials 31,[34][35][36] . In this context, it is important to note that in conventional electron diffraction experiments in solids, the discrete structure of the particles in the target is extracted by taking a Fourier transform of the scattering data 65 . The local electric field in these systems is dominated by the local charge density, because the system is close to charge neutral. In contrast, in systems that are not charge neutral, such as charged particle bunches in free space, the long-range Coulomb force leads to local electric fields that are strongly influenced by all of the particles in the bunch. In this case the probe electron deflections are dominated by cooperative mean-field space-charge effects, and the scattering due to local charge inhomogeneities typical of scattering from solids is a second order effect. In considering cooperative lensing effects in electron systems confined in the solid state, such as at heterointerfaces or at surfaces, several factors arise; including scattering from the atomic structure of the hosting solid, and screening of the Coulomb interactions due to charge polarization in the solid. Though this modifies the lensing effects, especially on long length scales, qualitatively similar cooperative lensing effects may still be expected in cases where high density interfacial electron gases can be generated.

Ultrafast electron microscopy setup, experimental conditions, and data treatment.
We employ a custom-modified environmental Hitachi H9500 TEM operating at 300 keV, interfaced with a high-repetition rate fs laser system (Light Conversion PHAROS with ORPHEUS-F OPA) that allows excitation of the sample with wavelengths between 260 and 2600 nm and variable repetition rates up to 1 MHz. In the experiments reported here, the sample is excited using 528 nm, ~200 fs laser pulses, with fluences of ~30 mJ/cm 2 . Short probe electron pulses are generated using the photoelectric effect by impinging 256 nm, ~200 fs UV laser pulses onto a graphite guard-ring LaB6 photocathode with a diameter of 50 μm (Kimball Physics). Laser pump and electron probe pulses impinge the sample with a repetition rate of 490 kHz and their relative delay is controlled using an optical delay line (Aerotech). The data acquisition software is provided by IDES Inc.
Typical integration times per image were 1 s, corresponding to 4.9 • 10 w pump/probe shots. We note that the exact temporal resolution of the setup is not known yet. However, we excite the photocathode with a low pulse energy of 16 nJ, which puts us into a regime where tens-hundreds of electrons are emitted at the photocathode, and only a few electrons will reach the sample. This so-called "single-electron" mode has previously been shown to yield instrumental response functions (IRF) that are almost entirely limited by the pump and probe laser pulse durations 66  Images are normalized to their total integrated intensity in order to compensate for slight variations in the probe electron intensity. A median filter of 5 × 5 pixels is applied to the images to mitigate random noise (the detector has 2000 × 2000 pixels). Difference images were generated by subtracting an averaged pre-time zero image from all subsequent frames. We also subtracted a frame recorded without probe electrons, but with pump laser beam in order to remove pump laser scatter that reaches the detector. Circular ROI radii were chosen with the goal of simultaneously optimizing the visibility and signal-to-noise ratio.
Analytical model. Our analytical derivations are based on the use of a Gaussian model for the charge distribution. For a non-interacting system it is a straightforward proof that the evolutions of the statistics of an ensemble of particles are independent of the spatial distribution of the particles; therefore, to most closely resemble the experimental conditions, we treat the spatial distribution as Gaussian in our analysis. The time-dependent density of the electron gas is described by where E 3 (F) is defined in Eq. (1) (see SI2 for derivation). The N-body simulation results indicate that E G (F) becomes significantly larger than E 3 (F) within 150 ps, which means at each time the charge distribution can be approximated by an infinitely long charged cylinder, whose electric field can be obtained from Maxwell-Gauss law as (see SI2 for derivation) i.e. the transverse electric field is linear in the radial coordinate r. A linear electric field imparts a lensing effect on the probe electrons. Further derivations that lead to Eq. (2) are provided in the SI Section 2.
Numerical simulations. We start our N-body simulations with a very oblate (2 × 2 × 0.01 μm 3 ) 3D Gaussian slab containing 10 z electrons. The oblate electron slab is placed at a distance of 30 nm away from the copper surface before starting the simulations. The photoemission process itself is not included in this approach, but the rapid photoemission of the electrons renders the longitudinal dimension of the bunch very small, resulting in a pancake-like bunch after the photoemission process is complete. We assume such a bunch for the initial conditions of our simulations.
The initial velocity distribution is a Gaussian with a mean of (0, 0, 6 • 10 w m/s) in the (4, 6, 7) directions and an isotropic spread with a standard deviation of 6 • 10 w m/s in each axis. These values were chosen to approximately match the velocity spread obtained from the experimental data, as well as to avoid that all electrons are absorbed by the grid. Since the copper grid is grounded, the potential at its surface is zero. Therefore, there is an electric dipole field formed between the photoemitted electron gas and its positive image charge, which is mostly aligned parallel to the propagation direction of the probe electrons. Image charges are approximated by calculating the dipole field for each electron and its positive counter-charge located at the opposite site of the copper grid surface, i.e. the electron coordinates are mirrored at the sample plane to find the coordinates of the image charges. Finally, the copper grid is approximated as a plane surface neglecting the effects of the holes. The measured cyclotron frequencies (GHz) correspond to centimeter wavelengths, which is much larger than the hole size of the grid (~4 μm), which justifies the use of a homogeneous slab instead of a grid in the simulations. An electron that hits the conductive copper plate is absorbed and excluded from the simulation. The tilt of the grid is neglected. The Lorentz force $ = &y + &' × ) is calculated for each electron in the gas, and the equation of motion is solved using the finite difference method. A time step of 80 fs was chosen such that reducing its value further would not change the results considerably. It is assumed that the electrons move much slower than the speed of light and hence it is not necessary to use retarded electric fields or account for losses due to electromagnetic radiation.
The effect of the photoemitted electron gas on the probe electrons is simulated by placing 11025 electrons equally spaced on a square grid with 14 μm sides, centered on the optical axis, and starting their motion at 3E G (F) above the copper grid. The kinetic energy of the probe electrons is 300 keV, corresponding to a speed of 2 G = 2.3 • 10 m m/s (or 0.77c). The detector was placed at 1 m below the electron gas, and a ROI circle radius of 7 μm was used to plot the difference intensity traces. Except for the electron gas lens, no other lenses inside the TEM are considered.
The comparison between experimental and simulated data is therefore only qualitative. We neglect Coulomb interactions between probe electrons, as well as any perturbations of the electron gas by the probe electrons.

SUPPLEMENTARY INFORMATION
Movie S1: Difference-image movie belonging to Fig. 2a Movie S2: Full-image movie belonging to Fig. 2b Movie S3: Full-image movie belonging to Fig. 4b Movie S4: Full-image movie belonging to Fig. 4c Movie S5: N-body 3D numerical simulation movie

SI1 Magnetic field data from the manufacturer
In Fig. S1a we compare the magnetic field that is determined in this study using the cyclotron resonance frequency and the Larmor formula ! = 2$% & '( ⁄ , with the magnetic field for three different objective lens currents (OLC) obtained from the TEM manufacturer (courtesy Hitachi High-Tech). The data is fitted with linear functions that pass through (0,0). The slopes differ by ~5%, which is within the tolerance of magnetic field difference between the lens models and microscopes. Fig. S1b shows the magnetic field as a function of z-coordinate obtained from the manufacturer. The field is uniform for a region ±1 mm away from the eucentric height, where the sample is placed, at z = 0.
with y > and y ? being the standard deviations of the zero-centered distribution in 2 and 7, respectively, which is determined by the laser pulse profile. Therefore, the charge density will be determined by a(2, 7, 9, /) = z z a > V ,? V (2, 7, 9, /) (S12) If y Z O ⁄ > y > , y ? , the transverse profile of the electron cloud will be mostly determined by the electron velocity distribution rather than the laser profile. Therefore, for the sake of simplicity, we assume that y > = y ? = y ë and for \^= 2^+ 7^, we find  (S16) For \ < y ñ , the first term of the Taylor expansion of Eq. (S16) is The electron probe equation of motion in the transverse direction is where ¢ ë is the transverse momentum of the probe electron, / 3 is the time the probe electron arrives at the sample plane, ℇ ë is the transverse component of the electric field and £ ú is the time that the probe electron traverses the sample area and can be approximated by where y @ (/) is the std of the electron cloud in the z direction and = @ the probe electron velocity.
Solving Eq. (S18) gives where ¢ ë3 is the initial momentum of the electrons and indeed represents the divergence of the probe electron beam determined by its brightness (at the TEM condenser stage). For an almost parallel beam, we set ¢ ë3 = 0. Solving for transverse displacement, we have where, as is shown in Fig. (S3), Δ\ is the time-dependent transverse distance of the probe electrons from the optical axis, and \ is the initial transverse distance. Inserting the electric field from Eq.

(4) Derivation of relationship between ©™´¨≠ and electron-gas focal length:
Suppose we replace all the lenses after the cloud (sample plane) by a single lens whose focal length is x ßAE and we call it the equivalent lens (EL). In the thin lens approximation, the total focal length of the lensing system (equivalent lens and electron-gas lens) is where { is the distance between the cloud and the EL. We assume { (cm) is negligible in comparison to other dimensions (m). Fig. S4 shows the simplified imaging system where probe electrons with the initial radius \ ∂ pass through the lens and hit the detector.
where \ ø is the radius of the probe beam on the detector and ¡ is the distance between the lensing system and the detector. The density of the probe electrons on the detector is where b ú is the number of probe electrons. The detected intensity in the ROI is where we have assumed x ß® (/) ≫ x ßAE − { (i.e. the electron-gas lens is much weaker than the TEM projection lenses). Therefore, if we excite the intermediate lens strongly, while the effect of the cloud lensing is still observable, the detected signal in the region of interest is approximately inversely proportional to the focal length of the cloud.
We note that this description of the lens system is highly simplified. It serves as a phenomenological model that gives us the right scaling in the focal length and the number of electrons (see below). Future work will focus on making this treatment more quantitative by taking into account the divergence of the incoming electron beam, and the distances between and excitation of all TEM lenses. In addition, geometric effects, such as the rather large dimension of the electron cloud along the z-direction, also need to be taken into account in order to reach quantitative agreement.

(5) Estimation of the number of electrons in the cloud:
From Eq.'s (S16) and (S17), the cloud radial electric field maximizes when y ñ (/) is minimum at A maximum magnification of ~2 is derived. The center of the grid holes are found by fitting Gaussians.

SI3 Details of the fit of the ROI difference intensity trace
Eq. (S30) suggest that we can fit the inverse of the focal length function to the ROI signal in the focused regime with high excitation of the IL (current 1.1 A). Accordingly, we fit the function other and therefore they cannot be determined independently. We therefore fit the ratio … = y Z y ë ⁄ , and set y ë = 12/√2 μm, which is obtained from the experimental laser spot size of ~29 μm FWHM or σ = 12 μm (average of major and minor axes of elliptical footprint), and considering that the electrons are emitted through a two-photon process that scales quadratically with photon intensity. Using a non-linear least-square fitting procedure (built in the Matlab curve-fitting toolbox) for time delays 100-900 ps (i.e. passed the Coulomb explosion regime) we then obtain: Ÿ = 1.05 ± 0.01 m -1 , O = 37.97 ± 0.01 GHz, y Z = 4.91 ± 0.01 • 10 ◊ m/s, / 3 = 7.8 ± 0.1 ps, and £ = 8.2 ± 0.2 ns.

SI4 Simulated absorption of electrons by the copper grid
In Fig. S6 we plot the number of electrons in the electron cloud as a function of time. The simulation starts with 10 Ä electrons whose center is placed at a distance of 30 nm away from the copper grid. During the first few ps, the electron gas undergoes a Coulomb explosion due to the large density and electron-electron repulsion. This leads to a large fraction of electrons being absorbed by the grid. As expected, more electrons are absorbed when image charges are included in the simulation. At later times, the electron absorption rate decreases until the fraction of electrons left in the simulation levels off at ~50%.

SI6 Simulations of ROI intensity traces
In Fig. S8 we show a set of simulated ROI difference intensity traces extracted from N-body probe simulations (see main text for details) for three cases: (1) without a grid (no absorption of electrons, no image charges); (2) with an absorbing grid, but without image charges; (3) with grid and image charges. It is clear that the majority of the amplitude reduction of the first peak is coming from the image-charge effect, which includes the increased absorption of electrons due to the dipole field between electrons and image charges (see Fig. S6). Fig. S8b shows a zoom into the first tens of ps after photoexcitation. It is seen that the rise time of the ROI intensity depletion signal is prolonged in the case of image charges, even though the creation process of the electron cloud was not explicitly included in the simulation. The latter could prolong the rise time even further.  Before After