Precision-controlled ultrafast electron microscope platforms. A case study: Multiple-order coherent phonon dynamics in 1T-TaSe2 probed at 50 fs–10 fm scales

TL;DR

A new ultrafast electron microscope system is tested, achieving high temporal resolution to study atomic vibrations in a material.

Contribution

A new RF controller system enables high temporal resolution and sensitivity in ultrafast electron microscopy.

Findings

Temporal resolution of 50 fs and detection sensitivity better than 1% were achieved.

Multi-terahertz coherent phonon dynamics in 1T-TaSe2 were probed effectively.

High-precision phase-space manipulation enhances detection of low-contrast signals.

Abstract

We report on the first detailed beam tests attesting the fundamental principle behind the development of high-current-efficiency ultrafast electron microscope systems where a radio frequency (RF) cavity is incorporated as a condenser lens in the beam delivery system. To allow for the experiment to be carried out with a sufficient resolution to probe the performance at the emittance floor, a new cascade loop RF controller system is developed to reduce the RF noise floor. Temporal resolution at 50 fs in full-width-at-half-maximum and detection sensitivity better than 1% are demonstrated on exfoliated 1T-TaSe2 system under a moderate repetition rate. To benchmark the performance, multi-terahertz edge-mode coherent phonon excitation is employed as the standard candle. The high temporal resolution and the significant visibility to very low dynamical contrast in diffraction signals via high-precision phase-space manipulation give strong support to the working principle for the new high-brightness femtosecond electron microscope systems.

Figures

The ultrafast electron microscope system with RF optics. (a) Schematic of ultrafast microscope utilizing RF cavities as electron lenses in a TEM column to temporally and spectrally focus the photo-electron pulses generated at photocathode. The fs laser system provides timing to pump pulse initiating material transformation as well as for seeding the RF signals sent to the RF cavity that delivers the short electron probe pulses. A phase-locked and control loop serves to ensure laser and RF are synchronized to each other. (b) The phase-space manipulation to deliver time or energy compressed electron pulses. The key stages involve the phase-space chirp resetting following space-charge-led expansion, (i)–(iii), and condenser lens aperture slicing to adjust aspect ratios of the phase space structure while maintaining high core brightness. (c) The mechanism of aperture slicing of high-intensity electron pulses that manifest in the laminar flow. The inter-dimensional effect is shown where removal of non-mixing particle streams, exemplified with trajectories 1 and 2 in the peripheral areas, with a condenser aperture causes tightening in both transverse and longitudinal phase space. Adapted with permission from C.-Y. Ruan, Structural Dynamics with X-Ray and Electron Scattering (The Royal Society of Chemistry, 2023).19 Copyright 2023 RSC Publishing.

Spatiotemporal resolutions from RF-optics-enabled UEM platform. (a) The ultrafast diffraction experiment carried out with Ne of 106 electrons on layered CeTe3 thin film. The lower panel shows the change in the integrated intensity of CDW satellites peaks I(q,t), which is normalized by the ground state intensity determined at t < 0, i.e., ml = Il(q, t)/Il(q, t < 0), which inform the corresponding structure factors. The change in correlation length is plotted via the shaded area, and its value can be tracked via the inserted scale bar. The correlation length ξ(t), which sets a limit on the beam coherent length Xc, is determined from the peak width and is given along with ml(q, t) via the shaded area. (b) The ultrafast imaging experiment carried out with Ne of 5 × 105 on layered 1T-TaS2 thin film. The spatial (ΔR) and temporal (Δt) resolution can be deduced from the intensity contrast and the dynamics of the bend fringe oscillations. (c) The schematic intercorrelates the UED and UEM performance on the complementarity between different measurements. The scheme as detailed in the main text unites Xc, dose, Δt, ΔR over the bunch transverse (εx) and longitudinal (εz) emittances. These results recorded prior to implementing the higher-level cascade RF noise suppression scheme have limited performance set by the stochastic noise floor in the RF focusing optics; see discussion in Sec. II B. Panel (a) is reproduced with permission from Zhou et al., Nat. Commun. 12(1), 566 (2021).36 Copyright 2021 Authors, licensed under a CC BY 4.0 License. Panel (b) is reproduced with permission from Sun et al., C. R. Phys. 22(S2), 15–73 (2021).37 Copyright 2021 Authors, licensed under a CC BY 4.0 License.

UEM and UED performance from tuning the condenser system. The left panels give the performance in the coherence length and electron dose under the tuning of the beam convergence half angle, a control parameter set by the magnetic condenser current and aperture size. The right panels give the performance in the imaging and time resolution under the tuning of the RF condenser. The model considers the beam energy of 60 keV with Ne = 105. The UED and UEM categories are classified based on the tuning ranges. For UEM, the convergence half angle from 0.6 to 3.5 mrad is varied, subject to the demagnification of the beam by the condensers as a virtual source at the front focal distance of the objective prefield. The angle is subject to the value of εx, the condenser settings, and the prefield focal distance (1.4 mm). For UED, this angle is 0.1 to 1.1 mrad. Alternatively, in the tuning of the RF condenser, the longitudinal focusing Δt is set for UEM from 250 to 750 fs (FWHM) at the low emittance, but nominally we increase this figure by the square root of εz to simulate spatial resolution. For UED, the setting for Δt is smaller, starting from 25 to 75 fs at low εz. The microscope resolution function from the derivation of ΔR is calculated using the lens aberration coefficients CS = 1 mm and CC = 1.9 mm. The shaded areas represent the spans estimated from the experimental results operating the RF beamline under UED36 (in red) and UEM30,31 (in blue) modalities.

The impacts from the RF instabilities on the temporal and spatial resolutions. The left panel gives the temporal resolution for UEM and UED modalities modeled for the nominal values of εx = 10 nm and εz = 1 nm for Ne = 105. The results are calculated for the beam energy varied from 40 to 120 and 200 keV. The right panel gives the corresponding changes in the imaging resolution. The different RF optical settings for UED and UEM reflect different phase-space aspect ratio prioritized for either imaging or ultrafast diffraction. For imaging, the energy spread is 0.3 eV to allow for sub-10 nm imaging resolution, while for UED it increases to 250 eV to give a better temporal resolution set by the emittance floor.

Circuit schematics for the two-level PID system for RF phase control and noise suppression. fREF is the reference frequency selected from the frequency comb of the laser, used as the reference frequency for the RF system. fDRV is the driving frequency sent to the RF cavity that compresses the electron beam. fREF is the frequency of signals at the pickup port.

RF noise characterizations before and after applying PID feedback controls. The main panel shows the integrated RMS noise measured from PID-1 and PID-0 loops separately, and the performance with active PID correct and without PID correct is compared. The inserts show the corresponding noise power spectrum plotted with the same color code. Dotted area represents the shared responsibility region for the two PID system.

Interactions between the two PID control loops in the 10 Hz to 1 kHz range.

Noise characterization under different PID scenarios. (a) The noise time sequences obtained by PID-0 phase detector with increasing levels of PID control; see the color-coded figure legend for the different PID settings. A pronounced source of noises is given at the level of PID-1 feedback control, where the uncompensated noises from PID-1 subsystem overflow into the master loop in telegraph noise-like steps at a relatively low frequency, ≈0.06 Hz; see the inset. (b) The phase noise histograms obtained from corresponding time sequences in panel a. (c) The accumulated RMS noises derived from integrating the noise power spectrum obtained from the time sequences. The left sub-panel shows the results with integration starting at the high frequency, whereas the right sub-panel gives the results with integration starting at the low frequency.

RF phase stabilities tracked by phase detector at the UEM station.

Coherent phonon dynamics probed by ultrafast diffraction modality. (a) The diffraction pattern from the exfoliated 1T-TaSe2 sample. The dashed lines outline the corresponding triangular lattice and CDW supercells periodicities. (b) The intensity modulations registering the coherent phonon dynamics and the incoherent channels including the thermal phonon signatures and the density-wave state evolutions; see text for details. The coherent multiple-order nonlinear phonon excitation involving zone-edge (K) acoustic phonon in periodic high and low intensity modulation is visible in the low fluence data set. (c) The modal structure of the K-edge phonon schematically reproduced from the previous study.54 (d) The baseline-subtracted transients from the main and superlattice structure factors indicating phase-locking. (e) The Fourier analyses conducted to retrieve the coherent phonon frequencies.

Full text

INTRODUCTION

I.

Transmission electron microscope (TEM) is a powerful materials characterization tool that offers nanoscale information on a wide range of materials and devices, which has been crucial for materials research.1 The main advantage with electron-based technology in comparison with x rays as a structural probe lies in its flexible use of field-based electron optics for multimodal operation, uniting crystallography, imaging, and spectroscopy with relative ease. However, TEM systems traditionally come with a very limited time resolution. Earlier efforts in upgrading the TEM with ultrashort time resolution encountered daunting roadblocks presented by the space-charge effect, leading to a significant trade-off between the resolution and the sensitivity.2 Recently, a paradigm for obtaining high spatiotemporal resolution in ultrafast electron microscope (UEM) systems based on lightly triggered photometers with few or less than one electron per pulse at high repetition rates (typically 1–100 MHz)3 is pioneered by Zewail et al.3,4 This approach not only elegantly preserves the key features traditionally offered by continuous wave (CW) TEM systems, but also provides a baseline performance at sub-ps and sub-nm levels. Since the initial works, various UEM systems employing such one-electron-at-a-time paradigm have significantly impacted many areas of research, including nanophotonics,5–9 plasmonics,10,11 and micro-analysis.12–15

Meanwhile, boosting the current efficacy by operating at a higher bunch charge is necessary for a broad range of material research that typically does not have a short recovery time.1,2 In pursuing such high current efficiency systems,16–19 significant progress has recently been made through constructive dialogs1,2,20 between ultrafast electron diffraction (UED), traditional electron microscopy and the accelerator physics communities. Especially notable are the progresses made in the ultrafast electron diffraction (UED) sector by incorporating radio frequency (RF) cavity compression to improve temporal resolution in keV UED beamlines,21–25 and the advances in the MeV-UED systems with relativistic RF photoinjectors planned for the FEL injector at SLAC,26 BNL,27 DESY,28 and Shanghai Jiaotong University29 among many others. A parallel development has been carried out at MSU, where the RF compression element is used to condense the beam longitudinally in a high throughput arrangement, in order to realize ultrafast electron microscopes, targeting multimodalities.30,31

Here, we present the first detailed beam tests designed to benchmark the RF-beamline performance at the emittance floor, attesting the principles behind developing high current efficiency ultrafast electron microscope systems. The article is organized into principle (Sec. II) and technical development sections (Sec. III). In the principle section, arguments derived from recent numerical studies are given for the laminar space-charge flow concept and the high-brightness beam delivery based on bunch phase-space manipulation. Derivations for the spatial and temporal resolutions from the RF-microscope are introduced along with model simulations to set the performance baselines both in imaging and diffraction modalities. In the technical development section, we consider the practical implementation in realizing the performance goals set by the beam phase-space areas, or so-called emittances. We introduced a key advance in stable control through a new cascade RF feedback-control mechanism, which effectively identifies and suppresses the phase noises to a level that, when transcribed into the beam incoherence envelope, is no more than the contribution expected from the bunch emittance floor. This implementation offers a means to test whether the performance is already at fundamental stochastic limits, which is set by the longitudinal and transverse emittances. At bunch particle level of 10^5^, the expected stochastic limits are under the scales of 50 fs and 10 nm in ultrafast diffraction and imaging modalities. The beam tests are carried out by operating the microscope in the ultrafast electron crystallography modality on exfoliated 1T-TaSe_2_ thin film. With improved resolution and brightness, we target a more subtle scattering regime, involving multiple-order edge-mode phonons coupling to the intervalley scattering, which requires a gentler excitation and is much less explored by a structural probe. We successfully record, at 5 kHz pump–probe repetition rate, the edge-mode coherent phonon excitation at 50 fs full-width-at-half-maximum (FWHM) resolution, 5 nm in coherence length, and sub-picometer detection sensitivity achieved via diffraction contrast. These levels of performance are highly commensurate with the theoretical predictions and thus validate the key concepts of implementing the high-current efficiency ultrafast microscope systems.

RF-ENABLED UEM/UED METHODOLOGY

II.

Here, we demonstrate, by judiciously employing a beam phase-space manipulation protocol, one can approach a high current efficiency for running the UEM experiments without suffering major loss in spatial and temporal resolutions. Conventionally, the beam delivery in a transmission electron microscope builds on small area emitter where the electrons are extracted to form continuous single-particle stream. The narrow emitters with subsequent condenser demagnetization provide continuous wave TEM the capability to deliver low-emittance and high-coherence beam. However, in the consideration of photoemission with short electron bunches that drives the ultrafast microscope, the probe phase space is necessarily extended by the particle number in the bunch (N_e_) and the beam brightness depends heavily on the packing density. Normally, in a stochastic source, where the individual particle trajectories may be considered as independent, the associated phase-space area (bunch emittance) would increase linearly with particle number. However, due to strong Coulombic interparticle interactions, the beam formation can develop fluid-like behavior. The best delivered beam is one where the bunch emittance grows sub-linearly with increasing N_e_, leading to an increase in beam brightness—a regime which can be achieved via facilitating and maintaining laminar flow from the source to the specimen target. The condition to promote laminar space-charge flow under the flat cathode geometry has been investigated numerically based on the multi-level fast multiple method (ML-FMM).32,33 Here, we will focus on the subsequent beam delivery and the validation of the beamline performance leveraging the high brightness laminar flow. Specifically, phase-space manipulation is required to reverse the velocity chirp effect derived from particle streams in order to form coherent illumination from space-charge-dominated beam. This, in turn, reduces the electron footprint but not the emittance size.

Emittance-limited beam delivery

A.

The phase-space manipulation is carried out through the joint application of magnetic (transverse) condenser and the new RF-cavity-based longitudinal condenser, and appropriate phase-space slicing to locally boost the brightness. To illustrate the concept, the RF-enabled TEM optical beamline is schematically shown in Fig. 1(a), where the high-intensity electron bunch is generated at a Pierce-gun photocathode via front-illumination by 266 nm drive laser pulses up to the virtual cathode limit. The flat cathode geometry, and its proximity-coupling to a source condenser lens, facilitate laminar flow emission just below the virtual cathode effect.33,34 The beam is fed into the condenser system of the UEM column, where an RF cavity is specifically inserted between magnetic lenses and serves as a new optical element to condense the bunch profile longitudinally. Meanwhile, the magnetic condenser and objective lenses demagnify the beam transversely. With two types of condensers acting together, a pancake beam is typically formed through different lens settings. Just prior to illumination, a variable-size aperture array is employed to slice the emittance (and particle number) as required.

The ultrafast electron microscope system with RF optics. (a) Schematic of ultrafast microscope utilizing RF cavities as electron lenses in a TEM column to temporally and spectrally focus the photo-electron pulses generated at photocathode. The fs laser system provides timing to pump pulse initiating material transformation as well as for seeding the RF signals sent to the RF cavity that delivers the short electron probe pulses. A phase-locked and control loop serves to ensure laser and RF are synchronized to each other. (b) The phase-space manipulation to deliver time or energy compressed electron pulses. The key stages involve the phase-space chirp resetting following space-charge-led expansion, (i)–(iii), and condenser lens aperture slicing to adjust aspect ratios of the phase space structure while maintaining high core brightness. (c) The mechanism of aperture slicing of high-intensity electron pulses that manifest in the laminar flow. The inter-dimensional effect is shown where removal of non-mixing particle streams, exemplified with trajectories 1 and 2 in the peripheral areas, with a condenser aperture causes tightening in both transverse and longitudinal phase space. Adapted with permission from C.-Y. Ruan, Structural Dynamics with X-Ray and Electron Scattering (The Royal Society of Chemistry, 2023).19 Copyright 2023 RSC Publishing.

An important design principle for running the new UEM system, as discussed below, is for delivering the bunch in proper aspect ratio that best projects the particle stream into the prioritized resolution window. This operates on two extrema scenarios of focusing as illustrated in Fig. 1(b), based on the longitudinal phase-space structure carried out by the RF optics—a similar picture equally applies to the bunch transverse focusing carried out by the magnetic condenser lenses. By running the RF-cavity field to reverse the longitudinal velocity chirp, the RF condenser refocuses the pulse to a smaller footprint along with positional or momentum direction to create a focused or temporally coherent pulse at the specimen plane. Similar action from the magnetic condenser sets the illumination conditions as either spatially more focused (for imaging) or coherent (for diffraction). However, the applications of RF and magnetic condensers are not entirely independent since the longitudinally tightened beam, tends to expand transversely due to internal space-charge forces.

The subtle cross-dimensional effect is a key element in considering the beam delivery in the new UEM system. We may contrast the different operations in terms of how the space-charge effect unfolds in a drastically different fashion in the high-density beam system from those operating with fewer particles. In the latter case, the presence of excessive independent particles generally accelerates the stochastic growth in emittance via pairwise scattering causing degradation in performance. Meanwhile, the recent detailed fast multiple method simulation presented a different picture when propelling the photoemission into the high-density laminar flow regime.33,35 The basic physics behind these builds on the strong inter-particle correlation mediated through Coulombic interaction, which tends to develop bi-directional flows: the thermal particles with higher kinetic energy flow to the exterior whereas the laminar flow develops from condensing the low-entropy particles at the core region. On the basis of this bi-directional flow concept, a high-brightness regime for operating the UEM beamline can be conceived. This strategy, of applying an appropriately sized condenser aperture near the specimen, targets the high-brightness core of the particle streams, as conceptually drawn in Fig. 1(c). At this stage, the bi-directional flow has fully developed and the slicing along the transverse axis simultaneously removes longitudinally divergent particle streams in the circumference region more than the core. This laminar-flow high-brightness strategy thus could result in a local boost in brightness for the beam delivered to the specimen. Such phase-space manipulation of high-intensity electron bunches becomes the foundation for operating the new type of RF-enabled UEM system with a high current efficiency.

The bunch emittances estimated by the recent beam dynamics simulations under the laminar flow scenarios could be examined via methods targeting multimodalities. Based on the theoretical principles, one can cross-correlate multiple figures of merit, including the coherence length (X_c_), the dose, the imaging resolution (ΔR), and the temporal resolutions (ΔR), relevant to the UED/UEM experiments based on the emittances of the bunch and the aberration coefficients of the electron optics. In another word, using the complementarily obtained performance metrics as the feedback, one could derive the bunch emittance; see Fig. 2(c) and discussion below. Such phase-space-centered beam optics arguments set a bound on the performance based on the stochastic emittance strongly influenced by the bunch charge. Following the scaling law established previously,33 it is predicted that, concerning the temporal resolution, sub-100 fs-level (in full-width half maximum, FWHM) RF focusing is achievable with N_e_ up to 10^6^ (also shown in Ref. 38) and accordingly, by slicing N_e_ down to 10^5^, from the reduced emittances19,24 sub-30 fs-level temporal resolution and ≈1 nm-level spatial resolution could be obtained with the systems optimized for diffraction and imaging modalities. This level of performance has yet been fully demonstrated in the RF-optics-enabled UEM systems being implemented.30,37 The best recorded joint spatial-temporal resolution at MSU UEM systems is thus far beyond 100 fs–10 nm (as shown in Fig. 2a and Fig. 2b) even though the independently verified emittance figures support a higher level of performance. In the following, we will examine how the bunch stochastic emittance and the instabilities in the RF optics control couple to the beam illumination incoherence envelope, so we could better evaluate the impacts from these factors on the performance. Such evaluations help distinguish the influence by the fundamental limits and other sources derived from the laboratory practice, such as a mismatched optical setting or instabilities from other parts of the beamlines. Addressing the respective technical issues shall further allow the system performance to reach the stochastic floor set by the fundamental physics.

Spatiotemporal resolutions from RF-optics-enabled UEM platform. (a) The ultrafast diffraction experiment carried out with Ne of 106 electrons on layered CeTe3 thin film. The lower panel shows the change in the integrated intensity of CDW satellites peaks I(q,t), which is normalized by the ground state intensity determined at t < 0, i.e., ml = Il(q, t)/Il(q, t < 0), which inform the corresponding structure factors. The change in correlation length is plotted via the shaded area, and its value can be tracked via the inserted scale bar. The correlation length ξ(t), which sets a limit on the beam coherent length Xc, is determined from the peak width and is given along with ml(q, t) via the shaded area. (b) The ultrafast imaging experiment carried out with Ne of 5 × 105 on layered 1T-TaS2 thin film. The spatial (ΔR) and temporal (Δt) resolution can be deduced from the intensity contrast and the dynamics of the bend fringe oscillations. (c) The schematic intercorrelates the UED and UEM performance on the complementarity between different measurements. The scheme as detailed in the main text unites Xc, dose, Δt, ΔR over the bunch transverse (εx) and longitudinal (εz) emittances. These results recorded prior to implementing the higher-level cascade RF noise suppression scheme have limited performance set by the stochastic noise floor in the RF focusing optics; see discussion in Sec. II B. Panel (a) is reproduced with permission from Zhou et al., Nat. Commun. 12(1), 566 (2021).36 Copyright 2021 Authors, licensed under a CC BY 4.0 License. Panel (b) is reproduced with permission from Sun et al., C. R. Phys. 22(S2), 15–73 (2021).37 Copyright 2021 Authors, licensed under a CC BY 4.0 License.

Electron focusing in the UEM system

B.

To understand the TEM optical manipulation with magnetic and RF condenser lenses, we resort to the coupling between the phase space, considered as a stochastically filled area or emittance, to the optical transfer function modeled in the context of objective aberration function.19 To illustrate the theoretical principles, we first ignore the contributions from a noisy RF source. We trace the effects from a finite-sized electron bunch phase space to the resolution function of the TEM. In this phase-space-based picture, the beam incoherent illumination causes information loss and degrades the resolution which can be modeled through the spatial and temporal incoherence envelopes:39 $[eqn]$ and $[eqn]$ , expressed in the Fourier spatial frequency $[eqn]$ . As part of the (objective) lens transfer function,40 these incoherence envelopes restrict the information at high $[eqn]$ , given by

$$[eqn]$$

where $[eqn]$ gives the phase factor of the distorted wavefront. The envelope functions are given by

$$[eqn]$$

where $[eqn]$ is the electron de Broglie wavelength. The direct impact from the incoherence phase-space envelope is seen in the coupling between exponent $[eqn]$ , the root mean square (RMS) of the beam convergence angle, and the spherical aberration coefficient $[eqn]$ , which results in an expanded cone of incoherent illumination. Namely, with a large transverse emittance size ε_x_, which propagates into a large transverse extremum size through focusability [Fig. 1(b)], it is expected to limit the spatial resolution (ΔR) or the coherence length (X_c_) from the bunch illumination. Meanwhile, taking temporal incoherence envelope into consideration $[eqn]$ is defined by

$$[eqn]$$

with $[eqn]$ and $[eqn]$ the RMS deviation of the beam energy ( $[eqn]$ ) and objective current ( $[eqn]$ ) coupling to the lens chromatic aberration coefficient $[eqn]$ . The expanded $[eqn]$ limits the spatial resolution due to the cross-dimensional effect; namely, beam temporal incoherence causes an imprecision in spatial focusing. It is easy to see the instability of the RF optics will couple to the lens optics through an increase in $[eqn]$ . Hence, an imprecise RF lens not only causes a loss in temporal resolution, but also impacts the spatial resolution directly in an UEM. The resolution loss is given by the complementarity (Fourier-pair) relation between the resolution function R(r) and the transfer function t(k),19

$$[eqn]$$

In Fig. 3, we present the results modeled for the UEM and UED modalities where the optical tunings of the transverse and longitudinal phase-space structures are carried out through the magnetic and RF condensers, measured in terms of the convergence half angle and the compressed pulse width Δt. For evaluating the effect on the focusing property, the results are given for different transverse and longitudinal emittances.

UEM and UED performance from tuning the condenser system. The left panels give the performance in the coherence length and electron dose under the tuning of the beam convergence half angle, a control parameter set by the magnetic condenser current and aperture size. The right panels give the performance in the imaging and time resolution under the tuning of the RF condenser. The model considers the beam energy of 60 keV with Ne = 105. The UED and UEM categories are classified based on the tuning ranges. For UEM, the convergence half angle from 0.6 to 3.5 mrad is varied, subject to the demagnification of the beam by the condensers as a virtual source at the front focal distance of the objective prefield. The angle is subject to the value of εx, the condenser settings, and the prefield focal distance (1.4 mm). For UED, this angle is 0.1 to 1.1 mrad. Alternatively, in the tuning of the RF condenser, the longitudinal focusing Δt is set for UEM from 250 to 750 fs (FWHM) at the low emittance, but nominally we increase this figure by the square root of εz to simulate spatial resolution. For UED, the setting for Δt is smaller, starting from 25 to 75 fs at low εz. The microscope resolution function from the derivation of ΔR is calculated using the lens aberration coefficients CS = 1 mm and CC = 1.9 mm. The shaded areas represent the spans estimated from the experimental results operating the RF beamline under UED36 (in red) and UEM30,31 (in blue) modalities.

Here, the categorization of modality is based on the starting values of the tuning parameters. As the tuning parameters are varied, the performance in ultrafast diffraction and imaging may be prioritized; however, regimes where UED and UEM experiments are carried out at similar optical settings can be approached, namely, a joint multimodal scheme under the same beam illumination condition may be developed, but with some compromises in resolution from each channel. For operating in the imaging modality, the primary goal is set to optimize the spatial resolution ΔR. In this case, a larger convergence half-angle is set to promote the dose over the coherence length and a high temporal coherence is preferred over the temporal resolution to reach sub-10 nm spatial resolution. In the case of ultrafast diffraction, the system is prioritized for higher X_c_ and short pulse width. Consequently, the dose and direct-space resolution may suffer. The divergent trends from these simulations over the emittance provide the guidance, based off the delivery of performance in UED and UEM modalities, to set bounds on the size of emittance. The shaded areas represent the spans estimated from the experimental results operating the RF beamline under UED36 (in red) and UEM30,31 (in blue) modalities; e.g., see Fig. 3. While the assignments give only the order-magnitude precision on the beam emittance, it is instructive to point out the assigned characteristic emittance is consistent with the source emittance given by the multi-level fast multipole method simulation32,33 optimized for the laminar flow regime.

We now trace the effects on the resolution from a noisy RF source feeding the RF-cavity longitudinal lens. To understand this effect on the performance, we relate the RF phase instability $[eqn]$ to an increase in temporal incoherence envelope given by the beam velocity spread,24

$$[eqn]$$

where γ and β are the relativistic factors, and $[eqn]$ is the RF focusing parameter set by the cavity RF resonance field $[eqn]$ at

$$[eqn]$$

where $[eqn]$ , d, and $[eqn]$ are the bunch velocity, the gap size, and electric field amplitude of the cavity. Specifically, the values of $[eqn]$ to reach the two extrema shown in Fig. 1(b) are set by the focal distance $[eqn]$ , with $[eqn]$ and $[eqn]$ at bunch zero-crossing [ $[eqn]$ upon arrival at RF gap, see Fig. 1(b)], where a0 is the bunch phase-space chirp, respectively, for spectral and temporal compression. The corresponding arrival time jitter influencing the deliverable temporal resolution is given by

$$[eqn]$$

Based on the nominal emittance deduced for N_e_ = 10^5^ (Fig. 3), and the RF lens settings where f0 = 1.013 GHz, d = 2.1 cm, we calculate the incoherence envelops following Eqs. (5)–(7) at the respective extrema for the UED and UEM experiments. The additional temporal incoherence from the RF cavity $[eqn]$ is convoluted into the overall incoherence width by $[eqn]$ , from which we determine the spatial resolution. The results calculated for different beam energies are presented in Fig. 4 where we find the impact on the temporal resolution is largest in the ultrafast diffraction. We notice based on the simulation it is possible to compress the bunch down to 22 fs (FWHM) which only weakly depends on the beam energy. This weak energy dependence can be seen in the time-to-phase ratio, $[eqn]$ . With $[eqn]$  ≈ 0.4 m in our setups, $[eqn]$ of 4 to 4.5 is estimated for beam energy from 40 to 100 keV. Meanwhile, the impact on the ultrafast imaging lies in the spatial resolution; see right panel of Fig. 4. Irrespectively, the high sensitivity to the RF noise at the fs-nm scale measurements is quite noticeable, which highlight the importance for further improving the phase stability of the RF system. For testing the emittance floor based on $[eqn]$ given in both scenarios, an RMS noise close to the level of 0.005° would be required.

The impacts from the RF instabilities on the temporal and spatial resolutions. The left panel gives the temporal resolution for UEM and UED modalities modeled for the nominal values of εx = 10 nm and εz = 1 nm for Ne = 105. The results are calculated for the beam energy varied from 40 to 120 and 200 keV. The right panel gives the corresponding changes in the imaging resolution. The different RF optical settings for UED and UEM reflect different phase-space aspect ratio prioritized for either imaging or ultrafast diffraction. For imaging, the energy spread is 0.3 eV to allow for sub-10 nm imaging resolution, while for UED it increases to 250 eV to give a better temporal resolution set by the emittance floor.

NEW DEVELOPMENT IN THE PHASE PRECISION CONTROL

III.

While the core optical technologies combining magnetic and RF condense lenses promise to deliver a bright beam with sub-100 fs temporal resolution (e.g., the simulation presented in Fig. 4 promises 22 fs FWHM at the specimen plane), translating such short bunch delivery to the performance requires a high stability and precision on the control system. For the UEM systems, the synchronization between the laser pump and the RF system seeding the RF cavity is achieved through a phase-locked circuit. Given schematically in Fig. 1(a), the circuit consists of a fast-rectifying photodiode that converts the light pulse signal from the laser oscillator into a frequency comb up to multi-GHz. The resonance frequency f0 of the RF-cavity optics is locked on to the frequency comb at an integer harmonic at 1.013 GHz. As the signals running from the laser frontend to the UEM cavity port is over an extensive distance, the success of the RF phased-locked circuit depends on identifying the spurious noise sources and their removal, which often involves proportional–integral–derivative (PID) control loops over the appropriate timescales of the experiments.

Several effective strategies based on feedback-control have been reported in the literature for running the RF systems used in the keV-scale UED41,42 and UEC/UEM24,36 experiments. Nominally, these systems have achieved a temporal resolution from 80 fs for the relativistic beams29,43,44 to close to 100 fs for sub-relativistic systems,21–25 but the actual performance likely deviates from system to system due to varying beamline designs and the specific types of noise sources present in these locations.

RF noise suppression with the cascade PID control system

A.

Here, we will focus on the development of a new PID scheme for the active noise reduction to reach the level of 0.005° RMS to provide sufficient precision to deliver the bunch focusing down to the emittance floor. The new scheme has a two-level cascade design as schematically drawn in Fig. 5.

Circuit schematics for the two-level PID system for RF phase control and noise suppression. fREF is the reference frequency selected from the frequency comb of the laser, used as the reference frequency for the RF system. fDRV is the driving frequency sent to the RF cavity that compresses the electron beam. fREF is the frequency of signals at the pickup port.

Specially, a higher frequency PID-1 inner loop with a digital phase-locked loop (PLL) is proximity-coupled to the UEM column. It deals with the local phase noise $[eqn]$ measured between the signal fed into the low-level RF (LLRF) controller and the signal at the cavity pickup port. This high-speed LLRF controller is designed and fabricated by the RF group at Facility for Rare Isotope Beams (FRIB) to support the new 644 MHz superconducting cavity for future FRIB400 upgrade and 1.013 GHz room-temperature UEM cavity.45 An important feature of this LLRF controller is the System-on-Chip (SoC) field programmable gate array (FPGA) design, fast data converters that allow direct sampling at 1.013 GHz and PLL chips with good phase noise performance up to 100 kHz that generates clocks for data converters and FPGA.46 Thus, the LLRF controller can easily suppress any local phase noise within the cavity bandwidth ≈0.1 MHz.

Meanwhile, PID-0 is the master feedback-loop overseeing the entire length of the RF circuit. It compensates the phase difference, $[eqn]$ , measured between the laser frontend and the RF pickup port; the output for the phase compensation is fed into the PID-1 subsystem. The PID-0 feedback-control employs a 4 MHz digitizer with averaging over 1k samples to reach sufficient phase detection precision to run the PID. The sub-millisecond detection time allows the system to detect noise spectrum beyond 1 kHz, yet the system operates at 10 Hz to give the overall phase stability it needs for longer term operation.

The nested feedback loops detect and compensate for the active noise sources over their own detectable spectral ranges. Importantly, the cascade loop design could synergistically reduce the noise floor as well as ensuring the longer-term stability through the existence of a shared responsibility spectrum region for joint optimization. This spectral region is allocated in the most active frequency domain from 10 Hz to 1 kHz. To inspect this effect, we first carry out the noise spectrum analyses at different levels of feedback controls by the two PID sub-systems. The results are given in Fig. 6, first shown with the scenario of open loops. In this free-running scenario, the intrinsic noise spectrum given by the PID-1 sub-loop carries an accumulated RMS noise that rises to 0.013° from 1 to 10^6^ Hz (in green). Meanwhile, the phase noise detected in the master loop rises to 0.039° RMS from 10^−6^ to 10 Hz (in pink). The corresponding noise power spectra are presented in the inset panels below, which show temporal correlations with the characteristics of a power-law distribution extended over several decades.47 We then investigate the synergistic effect from running the two PID control loops simultaneously with the set parameters used in PID-0 optimized for the betterment of overall performance viewed from $[eqn]$ , which necessarily encompasses the increased noise from the active PID-1 sub-loop for reducing the high frequency noises. At this stage of implementation, only the PID-0 parameters are tuned.

RF noise characterizations before and after applying PID feedback controls. The main panel shows the integrated RMS noise measured from PID-1 and PID-0 loops separately, and the performance with active PID correct and without PID correct is compared. The inserts show the corresponding noise power spectrum plotted with the same color code. Dotted area represents the shared responsibility region for the two PID system.

By activating both PID loops, the noise levels drop significantly. For the PID-1 subsystem, the integrated noise reaches the 0.0055° noise floor. In the PID-0 master loop, noises are added in the range above 10 kHz, reaching an overall integrated RMS noise of 0.0089°. By comparing the corresponding noise power spectrum, shown in the sub-panels below, it is evident that the improvements are achieved mainly by suppressing the diverging spectral power at lower frequencies; respectively, the feedback action defines a corner frequency f_c_ of 0.5 Hz and 3 kHz for the effective noise suppression in the two loops. By setting the goal for reducing the integrated noise, the noise below f_c_ is markedly reduced; however, a slight increase in noise power above f_c_ may be observed as the result of active intervention.

To study the interaction between the two control loops, we examine the time traces over different levels of PID controls. The blue curve in Fig. 7 shows the noise power spectrum from independently running the high-frequency feedback loop, where one can see the noises spectrum levels off at around 1 kHz. However, by turning on the PID-0 feedback control with the goal of reducing the overall integrated noise, the noise spectrum seen at frequency below 1 kHz drops markedly. We view this synergistic effect also from the perspective of PID-0 loop. The results are given at different levels of feedback controls, first presented in the time traces in Fig. 8(a) over the span of 2000 s. Here, as the reference the top trace (in red) shows the free-running scenario. The trace (in pink) below gives the results from activating just the PID-1 feedback control. The noise level is reduced but not significantly. This surprising result is understood by closer examination of the noise fluctuations (inset panel) where a visible telegraph-like noise with switching behavior is shown. The switching frequency of this noise is below 0.1 Hz, thus undetected at PID-1 level. Locally within each plateau, the noise level is significantly lower than the free running case. From the histogram analysis, these noise features give bi-modal distribution and carry a significant part of the integrated noise; see Fig. 8(b). We also examine the case with activating just PID-0; see Fig. 8(a) in green. In this case, the accumulated RMS noise reaches a level close to 0.01°, better than the previous scenarios; see Fig. 8(c). However, the local noise is higher than the case with just turning on the PID-1 feedback control albeit no signature of the telegraph noise can be traced here. We thus can conclude the presence of the telegraph noise to be a spillover effect from active intervention in the higher frequencies—one that turning on PID-0 feedback control can help remediate. Indeed, by activating the two feedback loops simultaneously, both the local and integrated global noise are reduced.

Interactions between the two PID control loops in the 10 Hz to 1 kHz range.

Noise characterization under different PID scenarios. (a) The noise time sequences obtained by PID-0 phase detector with increasing levels of PID control; see the color-coded figure legend for the different PID settings. A pronounced source of noises is given at the level of PID-1 feedback control, where the uncompensated noises from PID-1 subsystem overflow into the master loop in telegraph noise-like steps at a relatively low frequency, ≈0.06 Hz; see the inset. (b) The phase noise histograms obtained from corresponding time sequences in panel a. (c) The accumulated RMS noises derived from integrating the noise power spectrum obtained from the time sequences. The left sub-panel shows the results with integration starting at the high frequency, whereas the right sub-panel gives the results with integration starting at the low frequency.

We further consider the ramification of the noise suppression protocols on actual experimental implementation where ultrafast diffraction or imaging carried the information frame-by-frame over the acquisition time of seconds. From this perspective, we examine the RF phase independently monitored at the UEM column while running the experiments shown in Fig. 9. By comparing the results between just activating PID-1 and joint feedback control (PID-0 + PID-1), one can judge the main improvement made in integrated RMS phase noise comes from reducing the active period of the telegraph noise, and upon averaging their contribution is minimized to a small pedestal region in the histogram representing the resolution function in the temporal response. Clearly, based on the Gaussian sigma value extracted from the histogram, the precision of the RF system has reached within 0.01° RMS. At this low level of jitters, it is essential to test the sampling error; currently to be sensitive to noise from the higher frequency region for joint feedback control the phase detector integration time is at 60 μs—which clearly is able to pickup the sharp rise of the telegraph noises (and hence correcting the effect in the feedback loop), but the short integration time may introduce a detector noise higher than the noise floor of the device. For this purpose, we deliberately raise the acquisition window to 350 μs; see Fig. 9 colored in green. At this acquisition time, the detector is still sensitive to kHz noises and can faithfully detect the residual noise spikes but long enough to reduce the sampling noises. The noise distribution as given in the corresponding histogram gives a much-reduced Gaussian width of 0.0053°. Given we are confident that the noise spectrum above 1 kHz is effectively suppressed by the PID-1 feedback control (Fig. 6), we believe the measurement here does reflect the noise floor from the joint PID control loop.

RF phase stabilities tracked by phase detector at the UEM station.

With real-time recording of the phase noise, one may adopt a strategy of rejecting the effects from spiky features by setting the acquisition primarily over the low noise regions. Keeping the acquisition time well below the timescale of the noise spikes allows such discrimination. Meanwhile, any residual noise distributed over the acquisition time can be effectively compensated through data processing using the recorded phase as the time stamp to shift the timing, following Eq. (7). Judicious applications of these strategies could reduce the noise floor for the ultrafast measurements even further.

Performance validation: Multiple-order coherent phonon dynamics in 1T-TaSe2

B.

The reduction of the RF system phase instabilities offers a platform to investigate the ultrafast structure dynamics down to the emittance floor; for particle numbers as few as 10^5^, one expects a new limit in the temporal resolution well below 100 fs to be investigated by the sensitive RF-optics-based UEM systems. Here, we explore this fundamental limit of probing with quasi-2D layered quantum materials. The coherent ultrafast lattice responses well into the multi-THz regime have been reported and may serve as an independent ultrafast clock to calibrate our system. Especially, in the transition metal dichalcogenide (TMD) system, the Brillouin Zone edges host massive edge phonon modes with a high-density of states. Due to the reduced dimensionality, these edge modes strongly interact with valley electrons48,49 and are responsible for establishing and detecting lattice chirality at the valley points50,51 in mono- or few layer TMD. The relevant edge-phonon-assisted valley electronic processes could be investigated via ultrafast pump–probe scheme to synchronize the events, particularly at a low laser flux, to preserve the signatures of quantum scattering. The synchronously launched coherent phonons under this low-flux mechanism differ significantly from the ones driven by the collective effects, carried out at higher laser fluence. In such scenarios, the more massive electronic excitations directly impact the band structure and impose a swift deformation force that prompts a larger scale lattice movement in response. Investigating such impulsively driven nonequilibrium dynamics within a typically very short time window presents a niche vista point to investigate the fundamental coupling hierarchy.52,53

A particularly interesting case involving zone-edge coherent phonon generation was reported in a recent ultrafast optical measurement for 1T-MoSe_2_ where a non-sinusoidal signature of coherent phonon generation at ≈ 4.7 THz was isolated as a leading candidate responsible for valley depolarization in this system.54 Guided by the density-functional theory calculation, the authors were able to trace the signature differential transmittance signals to the longitudinal acoustic (LA) edge mode in a multiple-order response that directly couples to the intervalley scattering from the K to K′ valley corners.54

We selected a different TMD system 1T-TaSe_2_, which hosts similar in-layer zone-edge modes as MoSe_2_. However, the formation of the commensurate charge-density wave impacts the electronic structure in this system.55,56 In this case, its Fermi surface of the 2D Brillouin zone is gapped out with density wave modulation, and the electronic dispersion is only along the z axis. The Γ-point now is closer the Fermi level and the K and K′ valley points are upshifted by ≈1.5 eV from the Γ-point.57 This allows us to directly launch the K-edge mode via an indirect optical transition from the Γ-point to the K-valley with near-infrared photons. The experiments were conducted employing the generation-I UEM system at MSU. The 1T-TaSe_2_ specimen was prepared through tape-exfoliation to ≈40 nm thickness over ≈25 μm in lateral size and pumped with 30 fs near-IR (800 nm) laser pulses at 45° incidence under the vacuum. We performed the pump–probe UED experiments with bunch charge limited to 10^5^ to confine the emittance in the nm scale to approach the required sub-100 fs temporal resolution. The scattering data integration was carried out at 5 kHz repetition rate over an 8-s integration time to gather more than 10^7^ electrons in the relevant Bragg scattering. The overall sensitivity thus is expected to reach sub-1% over the intensity modulation to resolve the LA phonon dynamics expected at smaller than picometer scale.54 We note, in this feasibility study, that our work focused on illustrating the previously revealed edge-mode coherent phonon signatures well below the CDW phase transition threshold. The coupled ultrafast structural and carrier dynamics represents another interesting area that was recently investigated by others;58–60 such dynamics from the perspective of fs coherent scattering and imaging will be reported elsewhere.

The key results are given in Fig. 10, obtained by employing the ultrafast crystallography modality.36 Figure 10(a) gives the six-cycle-averaged diffraction patterns, where the TaSe_2_ triangular lattice and CDW superlattice formation are identified by the sharp scattering features resulting in corresponding structure factors $[eqn]$ and $[eqn]$ located by the wave-vectors Ghk and Qj in reciprocal space (s). For investigating the dynamics, we obtain differential scattering intensities integrated over the $[eqn]$ and $[eqn]$ , given in Fig. 10(b). The changes are normalized to the ground-state levels obtained at delays before the pump–probe zero-of-time. With this self-normalizing scheme, indeed multi-THz signatures of coherent phonon dynamics modulating the integrated intensity of the two structure factors can be disentangled well below 1% level.

Coherent phonon dynamics probed by ultrafast diffraction modality. (a) The diffraction pattern from the exfoliated 1T-TaSe2 sample. The dashed lines outline the corresponding triangular lattice and CDW supercells periodicities. (b) The intensity modulations registering the coherent phonon dynamics and the incoherent channels including the thermal phonon signatures and the density-wave state evolutions; see text for details. The coherent multiple-order nonlinear phonon excitation involving zone-edge (K) acoustic phonon in periodic high and low intensity modulation is visible in the low fluence data set. (c) The modal structure of the K-edge phonon schematically reproduced from the previous study.54 (d) The baseline-subtracted transients from the main and superlattice structure factors indicating phase-locking. (e) The Fourier analyses conducted to retrieve the coherent phonon frequencies.

We notice that at the low fluence level the data shows the peculiar non-sinusoidal oscillation, in alternating high and low intensity modulations. These transient signals indeed bear a remarkable resemblance to the transient optical signal pattern retrieved from MoSe_2_, identified as the K-edge anti-symmetric multiple-order acoustic phonon. The modal structure is schematically reproduced in Fig. 10(c).54 In the case of MoSe_2_, the non-sinusoidal beating pattern consists of a fundamental wave at 4.65 THz and the higher-order contributions, predominantly from the harmonic at ≈9.71 THz.54 Here, this fundamental frequency is downshifted, which is expected due to heavier Ta ions in the lattice.57 Nonetheless, the characteristic multiple-order excitation signature is well preserved in layered 1T-TaSe_2_.

To better decipher the coherent signatures and the related dynamics resulted from the perturbative changes in the broken-symmetry order as well as coupling to the lattice phonon bath, we resort to properly assigning the different evolution channels to different co-factors in the structure factors. The structure factor associated with the broken-symmetry CDW order, $[eqn]$ , is given in the basis of the unmodified lattice form factor $[eqn]$ ,

$$[eqn]$$

where $[eqn]$ denotes the position of the undistorted lattice.37 The dynamical evolution shown in lattice and superlattice peaks are given by lattice displacement field $[eqn]$ , depicted by two different types of lattice waves: the lattice phonons ( $[eqn]$ ) or collective field modes of the broken-symmetry order ( $[eqn]$ ) with momentum wavevector $[eqn]$ and $[eqn]$ given by $[eqn]$ and $[eqn]$ , referencing to the central Bragg reflection $[eqn]$ .

In this independent mode picture, we first understand the carrier wave effects, unrelated to the broken-symmetry collective order changes. Such effects underscoring the coherent $[eqn]$ (t) dynamics are given by the common co-factors in $[eqn]$ and $[eqn]$ . Indeed, the appearance of the phase locking in the observed $[eqn]$ and $[eqn]$ reflects this. To better evaluate the lattice displacement and frequency responses, we first subtract the non-oscillating transient (in dashed line) to examine the coherent modes. The key oscillating signatures can be isolated as a superposition of two LA mode frequencies at 3.1 and 6.2 THz from direct Fourier analysis as well as model reconstruction given in Fig. 10(e) in gray curve. We could determine the LA mode amplitude based on the intensity modulation.37 The ≈0.5% amplitude change, as observed in Fig. 10(b), gives the coherent phonon amplitude at the level of 5 × 10^−4 ^Å. Meanwhile, incoherent coupling to lattice phonon bath is presented in the $[eqn]$ **-**channel given as the Debye–Waller factor (DWF) from incoherent summing. The relevant dynamics is given as the “thermal” RMS amplitude $[eqn]$ that contributes as an exponential decay given by momentum dependence $[eqn]$ . This direct coupling to the phonon bath could be retrieved from the sub-ps drop in the baseline levels both in $[eqn]$ and $[eqn]$ . Accordingly, an RMS incoherent amplitude $[eqn]$ ≈ 2 × 10^−4 ^Å is deduced from the decays at different momenta.

Now, we turn our attention to the corresponding evolution of the density wave state. We expect even in the gentle excitation regime, the direct promotion of carries condensed within the CDW collective to the excited orbitals shall weaken the strength of CDW. From the symmetry-breaking perspective, this effect, involving $[eqn]$ , manifests in the counteracting movements from $[eqn]$ (a reduction) and $[eqn]$ (an increase);61 see the complementary trends of baselines. The $[eqn]$ intensity change reflects the order parameter suppression, by 3 × 10^−3 ^Å, to occur on the same sub-100 fs timescales. The rapid, collective response is an indication that an alternate displacive effect is already at play, even at a low excitation level. This is checked by doubling the excitation fluence. As a result, the nonlinear quantum scattering signatures are indeed nearly washed out and, in its place, a broader range of phonon excitation is visible. We note that responses dominated by the displacive mechanism have been very recently investigated by ultrafast angle-resolved photoemission spectroscopy.58 Upon applying much higher IR pulse excitation above the gap energy (0.6 eV), a transient nonequilibrium valence-band electronic manifold is shown to promptly settle in, with a transient electronic temperature up to 0.4 eV. Under this deformation stress, a reduction of CDW energy gap, by an amount of up to ≈22%, occurs in ≈0.3 ps, followed by a short recovery to ground state on just 1 ps timescale. Meanwhile, a separate UED investigation focused on the phase transition and excited the system with a significantly higher intensity, which is in a different regime and find charge transport across the stacking direction, establishing a 3D response associated with commensurate-to-incommensurate transition.60

Basing off the transient dynamics, the key subject of evaluating the RF-optics-incorporated beamline performance is discussed here. First, we determine the dose to be 3 e/μm^2^ and a coherence length of 5 nm based off the “atomic grating” approach19,24 adopted previously in carrying out the electron bunch characterization. These give a transverse emittance ε_x_ ≈ 5.7 nm, which largely is aligned with the previous results.19 The increased electron dose leads to a better resolution in discerning the extraordinary low level of structure dynamics. To evaluate the corresponding resolution from the signal floor in the diffraction contrast, we take the nominal noise level $[eqn]$ ≈ 0.1% over the intensity of $[eqn]$ , providing the distinction of the multiple-order oscillations. The amplitude scale corresponds to this minimum distinguishing scale can be estimated by the Overhauser's formula for CDW structure factor61 $[eqn]$ . Since the corresponding CDW amplitude a ≈ 0.2 Å in TaSe_2_, the noise floor gives the minimum detectable amplitude at the level of 10 fm. For the temporal resolution, it may be evident the RF compression has reached the sub-100 fs resolution judging from the initial response time in the UED signatures. As it depends on the knowledge of the physical initial decay, a more independent evaluation could be based on the visibility for discerning the well-established features at higher frequencies. Following the convolution principle, a bound for the response time is given using $[eqn]$ in RMS, where the specific $[eqn]$ is determined from carrying out convolution of the data with a Gaussian sigma $[eqn]$ , until the high harmonic transient signatures become indistinguishable. Through this method, we give a temporal resolution of ≈50 fs (FWHM) based on the minimum level convolution at ≈30 fs RMS to smear out the high harmonic signatures, as given in the results plotted as the dashed line in Fig. 10(e).

SUMMARY

IV.

We have presented justification for the high-current efficiency systems and set bounds for the performance in the RF-optics-enabled UEM systems being implemented at MSU. It may seem reasonable to expect an even more stable RF circuit seeding the RF optics can be conceived with better tuning of the PID control, particularly within the PID-1 system, by effectively removing the telegraph noises in the current implementation. As the system performance shall continue to improve, perhaps, a more important understanding gathered here is the correlation we have established between reducing the RF circuit instability and the performance. This confirms the notion that the RF-optics-enabled UEM system can be operated in regimes near the low emittance floor offered by the laminar space-charge flow. Such realization has ramifications in the future implementation of multimodality UEM based on the phase-space manipulation protocol as outlined here. In the case of studying coherent phonons, the successful implementation of the time-domain spectroscopy in conjunction with diffraction and imaging modalities shall complement optical studies. In particular, the direct structure-based probe can set the scale of change in the lattice degree of freedom and potentially, the symmetry of the excited modal structures, e.g., see Refs. 54 and 62, which has been a central topic in understanding the degree in which the momentum-dependent quantum scattering process may constitute viable means for coherent control of electronic processes. Continuously mapping the active coherent modes and their transformation upon increasing laser fluence, en route to phase transition, would likely provide deeper insight into symmetry-breaking mechanisms in layered quantum materials.