The dynamics of charge carriers (electrons, ions, holes) are central to numerous scientific and technological areas, including chemistry, solid-state physics, and the development of high-brightness electron sources. These dynamics often occur on ultrafast timescales, necessitating advanced visualization tools with high spatial and temporal resolution (Å-µm and fs-ns, respectively). Ultrafast electron microscopy (UEM) has emerged as a powerful technique in this domain, allowing direct visualization of photo-induced processes in nanoscale systems. UEM involves exciting a material with a short laser pulse followed by a short electron pulse for probing the subsequent dynamics through imaging, diffraction, or spectroscopy within a transmission electron microscope (TEM). This research utilizes UEM to observe the ultrafast evolution of a 3D photoemitted electron gas under a static magnetic field. Confined electron gases exhibit fascinating properties, such as high electron mobilities and quantum Hall effects, crucial for quantum science and technology. Two-dimensional (2D) electron gases, under external magnetic fields, have been studied using THz spectroscopy to investigate cyclotron resonances. This work extends this research by using UEM to visualize cyclotron dynamics in 3D electron gases, which provides a foundation for future studies involving spatially resolved imaging of carrier density variations in 2D electron gases (e.g., GaAs/AlGaAs or graphene) with fs-ps temporal resolution. This capability also has significant implications for plasma physics and the development of advanced electron sources.

Previous studies using ultrafast electron diffraction and microscopy have observed deflection effects due to transient electric fields from photocreated electron plumes. However, this research presents a detailed investigation of space-charge dynamics under a magnetic field and the resulting image changes. The literature establishes the importance of understanding confined electron gases and their properties (high mobilities, quantum Hall effects, etc.) in quantum science and technology. THz and microwave spectroscopies have been used to study these systems, but UEM offers the advantage of spatial resolution alongside temporal resolution. The use of UEM to study electron gases is also relevant to plasma physics and the development of high-brightness electron sources for X-ray facilities.

The experiments were conducted using a modified environmental TEM operating at 300 keV, coupled with a high-repetition-rate fs laser system. Laser pump pulses (~200 fs, 528 nm, ~30 mJ/cm²) were directed onto a tilted copper grid sample, with probe electron pulses (<1 ps) generated by a UV laser beam impinging on a LaB6 photocathode. The pump and probe pulses were synchronized, and real-space movies of the charge density dynamics were recorded. Photoelectrons were emitted through two-photon absorption. The sample consisted of a copper grid with hexagonal holes significantly smaller than the laser footprint. The experiments were performed at varying objective lens currents (OLC) and intermediate lens currents (ILC) to control the imaging conditions. Region-of-interest (ROI) analysis and Fast Fourier Transform (FFT) were used to analyze the data. An analytical model was developed to describe the electron gas evolution, including the time-dependent focal length of the electron-gas lens. This model was then refined by numerical N-body simulations which included Coulomb interactions, a static magnetic field, the copper grid (as an electrostatic boundary condition), and positive image charges. These simulations allowed for simulating the ultrafast electron microscopy images directly.

UEM movies revealed localized, periodic barrel distortions of the grid images, recurring at intervals of ~165 ps for over 20 cycles (~4 ns). This periodic distortion was attributed to a transient lensing effect caused by cyclotron oscillations of the photoemitted electron gas within the magnetic field of the objective lens. ROI analysis and FFT confirmed a fundamental frequency of 6.05 GHz. The lensing effects were only observable under out-of-focus image conditions. By adjusting the ILC, the lensing effect could be tuned from magnifying (barrel distortion) to demagnifying (pincushion distortion). At intermediate ILC, an astigmatic focal point was observed, attributed to the tilted sample surface and the elliptical laser footprint. The analytical model described the transverse radius of the electron gas and derived a formula for the time-dependent focal length, revealing a periodic dependence related to cyclotron oscillations. Fitting the ROI difference intensity trace to this model provided estimates of the velocity spread and number of electrons in the cloud. Numerical N-body simulations, incorporating Coulomb interactions and image charges, provided a holistic picture of the electron dynamics, demonstrating a rapid oblate-to-prolate shape transformation of the 3D electron gas (aspect ratio change of ~10^4) within 100-200 ps. Simulations matched experimental findings regarding the depletion of intensity, bright ring formation, and magnification at cyclotron resonance peaks. The simulations underscored the importance of including the copper grid and image charges to accurately model the early-time dynamics (<50 ps).

The findings directly address the research question by demonstrating the visualization of cyclotron dynamics and associated transient lensing effects in a 3D photoemitted electron gas using UEM. The observed periodic distortions and their dependence on lens currents confirm the transient lensing effect created by the oscillating electrons. The analytical model provides a quantitative framework, albeit with limitations, for describing the electron gas dynamics. The N-body simulations significantly improve the accuracy of the model, particularly for the early time dynamics, by incorporating the effects of Coulomb interactions, image charges, and electron absorption by the grid. The oblate-to-prolate shape transformation is a key finding highlighting the interplay between Coulomb explosion and magnetic confinement. This study is a significant advance in ultrafast electron microscopy, demonstrating the potential for spatiotemporal visualization of charge carrier dynamics in complex systems.

This study successfully visualized the picosecond-resolved cyclotron dynamics and transient lensing of a 3D hot electron gas using UEM. The oblate-to-prolate shape transformation and the resulting lensing effect were characterized experimentally and through analytical and numerical modeling. This work paves the way for future studies of photoemitted charged-particle beams, spatiotemporal visualization of cyclotron dynamics in 2D electron gases, and a deeper understanding of electron/plasma dynamics. Further research could explore different laser wavelengths and fluences, incorporate electrical bias for extended control, and adapt techniques for low-temperature investigations of 2D electron gas materials.

The analytical model simplifies certain aspects of the electron gas dynamics, neglecting Coulomb interactions and image charge effects in the initial phases (<50 ps). While N-body simulations address these aspects, limitations remain in precisely modeling the photoemission process and the exact characteristics of the TEM lens system. The temporal resolution is currently estimated to be ~500 fs, which could affect the resolution of extremely rapid processes. The assumption of a uniform static magnetic field simplifies the real experimental conditions where the field is non-uniform. The study focuses on a specific material (copper) and laser parameters; broader applicability needs further investigation.