Ultrafast Relativistic Electron Nanoprobes

One of the frontiers in electron scattering is to couple ultrafast temporal resolution with highly localized probes to investigate the role of microstructure on material properties. Here, taking advantage of the high average brightness of our electron source, we demonstrate the generation of ultrafast relativistic electron beams with picometer-scale emittance and their ability to probe nanoscale features on materials with complex microstructures. The electron beam is tightly focused at the sample plane by a custom in-vacuum lens system, and its evolution around the waist is accurately reconstructed. We then use the focused beam to characterize a Ti-6 wt% Al polycrystalline sample by correlating the diffraction and imaging modality, showcasing the capability to locate grain boundaries and map adjacent crystallographic domains with sub-micron precision. This work provides a paradigm for ultrafast electron instrumentation, enabling characterization techniques such as relativistic ultrafast electron nano-diffraction and ultrafast scanning transmission electron microscopy. Electrons have been used to map the structural properties of materials since the discovery of the particle-wave duality, while recent advances in ultrafast electron sources enabled time-resolved electron scattering techniques to probe atomic-scale structural dynamics with femtosecond temporal accuracy. The authors demonstrate ultrafast nano-diffraction with relativistic beams as well as scanning transmission electron microscopy enabling them to probe the micro-texture in complex heterogeneous materials.

S ince the discovery of the particle-wave duality 1 , electrons have been extensively used to probe matter at atomic scales. Owing to their very short (sub-Å) wavelength and large scattering cross section compared to X-rays, electron diffraction and imaging are today well established techniques for structure determination. More recently, the advent of ultrafast lasers (subpicosecond) sparked the development of intense ultrashort electron sources which, in turn, paved the way to a new generation of time-resolved electron scattering techniques such as ultrafast electron diffraction and microscopy (UED/M) [2][3][4] . These are now capable of probing atomic-scale structural dynamics with femtosecond-scale temporal accuracy.
Recent developments in this field include the introduction of methods and technology common in particle accelerator science. Radio frequency (RF) based electron sources have been successfully used for generating few-femtosecond electron probe beams 5,6 and for gathering information about ultrafast structural changes in solids and gases 7,8 . Here, electrons are generated and rapidly accelerated to relativistic energies by using high accelerating gradients, increasing the maximum achievable electron current density 9,10 and minimizing the temporal broadening caused by Coulomb repulsion and initial energy bandwidth, which are the main challenges for low-energy electron sources. Furthermore, owing to the relativistic electron speed, the fluctuations of arrival time at the sample caused by shot-to-shot energy jitters are rapidly dumped with increasing electron energy 11 . In terms of signal quality, shorter electron wavelengths imply a larger radius of the Ewald sphere, which in turn results in larger signal in the high-order diffraction peaks 12 , enhancing the sensitivity to lattice dynamics.
Notwithstanding this significant progress, ultrafast electronbased instrumentation is still far from reaching spatial resolution similar to what can be achieved in static electron microscopes. At low energies, setups using tip-based photoemission guns in standard transmission electron microscope columns can take advantage of very small source areas and direct current (DC) accelerating fields [13][14][15] . On the other hand, all pump-probe studies with MeV electrons up to date have so far been limited to systems with long-range order in the tens of micrometers or more 16,17 , as a consequence of the limited transverse coherence length and average transverse beam brightness B n,av of the source. In particular the latter quantity, defined as the number density of electrons in transverse phase space (i.e., per unit solid angle and unit area, also called 4D emittance), sets a limit to how many electrons in a beam can be strongly focused before their intrinsic divergence overwhelms any deflections due to scattering from the sample. For a pulsed electron source, the most direct way to improve the average transverse brightness is by increasing the repetition rate, which, for very high gradient RF guns, is typically limited to few hundred Hz.
In the present study, we use relativistic ultrafast electron pulses to map structural variations in microstructured materials, demonstrating a relativistic UED probe with nanoscale spatial resolution. The setup benefits from a unique electron beam source 18 capable of generating a B n,av more than 2 orders of magnitude beyond previous setups. First, we demonstrate lateral focusing of the electron beam down to the nanometer scale. The electron beam is opportunely collimated and then injected into the experimental chamber for final focusing using custom invacuum permanent-magnet lenses. The sub-nm emittance beam coupled with strong focusing required the development of a novel beam characterization technique, based on knife-edge scanning measurements complemented by a detailed data analysis. We obtained a full reconstruction of the beam transverse phase space evolution near the waist yielding transverse spot sizes smaller than 100 nm and normalized emittance in the sub-nanometer range. We then show the potential for such beams in ultrafast nano-diffraction (relativistic nano-UED) and scanning transmission electron microscopy (relativistic USTEM) by recording high quality diffraction patterns and mapping grain orientation with sub-micrometer resolution, pinpointing a grain boundary with sub-spot size resolution. In this context, we first characterize the ultrafast point-projection microscopy mode of the instrument (relativistic UPPM), obtaining a spatial resolution consistent with the knife-edge waist size measurements [19][20][21] , and then elucidate the critical role of this modality in the context of the nano-UED experiments. UPPM in fact, is shown to provide key information which can be used for correlating the ultrafast structural dynamics data from nano-UED with static information retrievable with other techniques such as conventional transmission electron microscopy (TEM).

Results
Lateral squeezing of ultrafast electron beams. Nanoscience applications of ultrafast relativistic electron pulses pose stringent requirements on the electron beam properties. In the case of diffraction, resolution can be described using the resolving power R ¼ R hkl =ΔR hkl 22 , where R hkl is the distance of a specific diffraction point/ring with Miller indices (hkl) from the zero-order beam while ΔR hkl represents the minimum distance at which a second point/ring can be discriminated. The resolving power in diffraction is ultimately limited by the angular spread of the beam, σ u′ , which is inversely linked to the spot radius σ u at the waist by the beam normalized emittance (ie. ε n;u ¼ βγσ u σ u′ ), with γ and β being respectively the relativistic Lorentz factor and the ratio between the speed of the electrons and the velocity of light. Variables u and u' refer to lateral position and angle coordinates respectively. To achieve a given resolving power at a particular spot size, the emittance requirement is ε n;u ¼ λ c σ u 2d hkl R , where λ c is the electron Compton wavelength and d hkl is the inter-atomic separation distance. For example, to achieve R ¼ 10 with a nanoscale beam (σ u ¼ 500 nm) in a sample with an inter-atomic distance d hkl ¼ 2 Å, the required normalized emittance is ε n;u ¼ 300 pm. Such a value is more than one order of magnitude beyond the smallest emittance experimentally measured at present date in relativistic ultrafast electron beamlines 23 .
This work utilizes the High Repetition-rate Electron Scattering (HiRES) beamline, a recently developed UED/M instrument that employs a unique RF-based electron source to achieve ultrashort, low-emittance electron beams. The details and performance of HiRES are described elsewhere 11 .
In Fig.1a, b, we show a cartoon of the apparatus, summarizing the beamline elements and functions relevant for this work. Short bursts of electrons are generated via photoemission at a 1 MHz rate, and instantaneously accelerated to relativistic energy of 735 keV via rapidly oscillating electromagnetic fields with 20 MV m −1 amplitude 18 . They then travel through the 4.5 mlong electron transport line, which performs spatial filtering, energy collimation and longitudinal compression before reaching the sample. In particular, an RF-based compression cavity (RF buncher) imparts a negative energy-time correlation to the electron distribution which results in beam temporal compression through vacuum dispersion. Figure 1a includes a schematic view of the electron pulse length evolution along the beamline. The optimal electric field amplitude depends on initial beam energy modulation and pulse length 11 , and it is carefully tuned so the pulse is shortest at the sample position (260 fs RMS in Fig. 1c).
The transverse electron beam properties are shaped at several points along the beamline. First, a stream of electron beam pulses with an average current of 60 nA is generated using a laser pulse with 300 fs full width at half maximum (FWHM) transversely focused to a 50 μm root-mean-square (RMS) spot on the photocathode. The combination of the first solenoid and a collimating aperture with a fixed diameter of 500 μm (A1 in Fig. 1a) downstream from the source selects the particles with low transverse momentum, thereby filtering the transverse phase space. The transverse normalized emittance of the resulting 320 pA beam is about 3 nm, measured by reconstructing the transverse phase space at the aperture position via TEM grid shadowgraph analysis 24 . While such beam parameters could be very interesting for a large range of UED experiments, lower emittance values are required to reach sub-500 nm spot size at the sample maintaining a transverse coherence sufficient to revolve typical unit cells in solid state samples, as discussed above. Therefore the electron beam is spatially filtered again by a second aperture with variable diameter from 1 mm down to 10 μm just upstream of the experimental chamber (A2 in Fig. 1a) to reach sub-nanometer emittance values. The electron optics downstream of A1 (not shown) are tuned to modulate the transverse aspect ratio of the beam at A2 and, consequently, partition the four-dimensional emittance to create round or flat beam waists. Typical current values after the second collimator are in the range of 100-200 fA.
Focusing relativistic electron beams to nanoscale spots requires strong confining magnetic fields. In this experimental work, we explore the use of an in-vacuum lens assembly composed by 3 permanent magnet-based focusing elements 25,26 as a compact alternative to large solenoid lenses.
We designed and fabricated permanent magnetic quadrupole (PMQ) lenses (Fig. 2a, b) with focusing gradients in excess of Here the position of the lens is such that the waist is produced upstream the sample plane, producing a shadowgraph of the specimen. The green Gaussian waveform represents a qualitative behaviour of the beam temporal evolution. b The same schematic with the setup operating in diffraction mode, with coincident electron beam focus and sample planes. c Electron beam dynamics simulations showing the behaviour of the electron beam emittance and pulse length throughout the beamline. The two apertures A1 and A2 decrease the electron beam emittance by about 1 order of magnitude each. At the same time, a negative energy-time correlation is imprinted on the electron beam by the radio-frequency buncher, which causes the beam to compress in the subsequent vacuum drift, reaching a minimum value at the sample plane 100 T m −1 ( Fig. 2c and inset) using Neodymium-based permanent magnets (Nd 2 Fe 14 B, remanence B r ¼ 1:25 T). We then arranged them in a triplet configuration to achieve an overall focal length of f e ¼ 2:5 cm in both planes. Figure 2a shows a single quadrupole element of the focusing assembly, with four 3 mm-thick permanent magnets held together by a round aluminum holder with a bore aperture diameter of 4 mm. The relative longitudinal distance between midpoints of the 3 elements was optimized using particle tracking simulations 27 . Figure 2d reports an example of simulation where a round input beam with an 50 μm RMS size and 600 pm normalized emittance is focused down to 400 nm RMS about 2.5 cm downstream the exit of the third focusing element, for relative distances between the quadrupoles of 5 and 6.5 mm, respectively. Precise transverse alignment and field measurements of the PMQ lenses are required to achieve the target focal length and beam size. The three quadrupole magnets were aligned using the pulse-wire technique to a common axis with a tolerance of < 15 μm (<3 mm transverse kick at the detector) and their gradient was measured with an error better than <1% of the specification (<5 mm error in longitudinal position of focal plane), within the acceptance tolerances set by a parametric simulation study of focal length variation and transverse dipolar kicks. Figure 2b shows the final in-vacuum mechanical configuration for the experiment. The flexure structure embracing each quadrupolar element was designed to allow such a precise transverse alignment of the magnet while minimizing longitudinal footprint. To allow compensation for small errors in focal length, an in-vacuum linear piezo-actuator was added to the system to adjust the longitudinal position of the third element during the experiment (labeled as PMQ-3). In addition, the entire  The outer square flexure is specifically designed to minimize the footprint, and provide at the same time high precision control on transverse alignment of the magnetic element. b Three-dimensional view of the sample area. The sample holder can be moved horizontally and vertically with 100 nm precision. The permanent magnetic quadrupole (PMQ) focusing system can be moved longitudinally and horizontally with better than 10 nm precision. Additionally, the longitudinal position of the last quadrupole element can be adjusted with sub-μm precision. c Transverse and longitudinal alignment of magnetic elements. The plot shows the magnetic field magnitude moving along the focusing system. With no offset (x = 0) the peak measured field was below 1.5 mT, corresponding to a transverse misalignment error of less than 14 μm. The inset shows the measured focusing strength of one of the elements. d Beam dynamics simulation of a collimated electron beam entering the focusing system using general particle tracer (GPT). Such simulations were performed using the actual measured gradient profiles, with the distances between the elements optimized to minimize the overall focal length of the system triplet assembly can be translated longitudinally over 2 cm, and horizontally in and out of the beam path, while the sample holder can be moved horizontally and vertically.
Characterization of electron beam evolution near the waist. Measurement and control of relativistic electron beams with nanometer resolution is an active field of research. Recently, measurement of sub-micron electron beams has been reported using the beam-loss monitor signal generated by interaction of the electron beam with a nano-fabricated wire 28 . In our setup, we use a similar approach -a knife-edge target is inserted gradually into the beam along the horizontal and vertical direction -but we record the full beam image at the detector for each step (Fig. 3a,  b). This allowed for a detailed analysis yielding the full phase space reconstruction, uncovering important correlations between the horizontal and vertical plane which would not be seen otherwise. The PMQ lens was moved along the direction of electron propagation, acquiring data at different longitudinal locations (Fig. 4a, b).
The knife-edge target used in the measurements is shown in Fig. 3b. Focused ion beam (FIB) was used to mill 10 × 10 μm square holes from 30 nm SiN windows with 75 nm of gold deposited via thermal evaporation. Scanning electron microscopy (SEM) images of the square edges reveal a roughness of about 10 nm, together with a 50 nm-wide area with rapidly varying gold thickness We perform a 10-parameter global fit on this dataset to reconstruct the coupled four-dimensional particle distribution in the canonical phase space ðx; p x ; y; p y Þ 29 , the related 4 × 4 second order beam matrix and its RMS volume ε n4D ¼ 0:0144 ± 0:0065 (nm rad) 2 . The evolution of the eigenvalues of the beam matrix in the xy plane allow the determination of the position and size of the beam waists (Fig. 4c), together with its rotation angle. The beam size minima were found to be 363 nm and 609 nm at the specific longitudinal planes shown by the transverse sections in Fig. 4d.
Asymmetric emittance and spot sizes can be achieved by changing the beam aspect ratio at the second aperture (A2 in Fig.1c) to take advantage of the dependence of the apertured beam transverse emittance on the angular divergence distribution before the aperture plane. By re-tuning the upstream quadrupoles to control this divergence, we were then able to generate beams with spot size aspect ratio up to 10 and minimum dimension at the focal point of 91 nm (Fig. 5a, b), with a total RMS volume of ε n4D ¼ 0:0086 ± 0:004 (nm rad) 2 . Such electron probes could be particularly useful in situations where high resolution is only needed in one dimension.
An electron beam rotation in the transverse plane is observed and reported in Fig. 5c. This is a result of an initial non-vanishing correlation between transverse planes, which can be due to various factors, including an asymmetric initial electron beam distribution at the cathode coupled with rotation by upstream solenoid lenses and different focal lengths of the final PMQ lens in the two transverse directions. Once measured, and characterized, differences in PMQ focal lengths can be eliminated by adjusting the longitudinal position of the last PMQ quadrupole and by adding a skew-quadrupole compensation optics. It is worth noting that, after the subtraction of 90 degrees originated by the crossing of the horizontal axis by the beam and the consequent exchange of the minor and major axis of the beam ellipse (Fig. 5a), the beam angle in space at the two waists position is off by about 3 degrees. Such small residual angle could be given by a small angular momentum introduced for example by a rolling alignment error of the PMQ quadrupoles.
Nanoscale femtosecond relativistic point-projection microscopy. When the beam focus position is set upstream the sample plane, the instrument operates in imaging mode, performing ultrafast point-projection microscopy (UPPM). The focal plane is positioned upstream of the sample so the resulting image at the detector represents a magnified mass-contrast shadowgraph of the specimen. To study the resolution of our system in imaging mode, we extended the concept of Ronchi ruling to electron  30,31 . We fabricated horizontal and vertical three-bar rulings with width and spacing ranging from 1.1 μm to 300 nm (Fig. 6a). Such rulings provide targets of known spatial frequency composition which can be used to determine the image contrast as a function of frequency, known as the contrast transfer function (CTF). The material is 50 nm of AuPd alloy sputtered onto a 30 nm SiN membrane, and the gaps were milled through with a focused Ga ion beam. With the electron beam focused in the configuration depicted in Fig. 4, we formed an image of the target at the detector by accumulating ultrafast point-projection images at 1 MHz repetition rate for 1 s to obtain Fig. 6b. The target longitudinal position was chosen to minimize shear and stretching distortions in the final image (z = 0 in Fig. 4c). From this image, we determine the contrast transfer function of the instrument, shown in Fig. 6c. We extracted contrast values from the three largest rulings at their fundamental frequency f 0 and f 0 2 (see "Methods" section for detailed procedure), and we fit a Gaussian CTF. We verified this CTF by applying it to model gratings with the SEM-measured ruling dimensions to generate the profiles shown in the inset of Fig. 6c. These profiles reproduce the shapes of the overlaid measured profiles. From this fit, we find the spatial frequency resolution at 5% contrast to be 0.725 ± 0.012 μm −1 in X and 0.960 ± 0.017 μm −1 in Y.
The UPPM resolution is set by the angular divergence of the electron probe, which in turn is defined by the beam size at its waist. Therefore we expect the point spread function (PSF) of the system to be closely related to the beam waist size (see "Methods" section). For the round beam case of Fig. 4, we computed the PSF from our fit CTF and found its standard deviation to be 406 ± 7 and 539 ± nm, respectively in the horizontal and vertical planes, in fair agreement with the minimum beam sizes measurements.
Ultrafast relativistic scanning electron nano-diffraction. When focused at the sample plane, the low-emittance electron beam produces high quality diffraction patterns providing structure and orientation information at the nanoscale. We used the beam described in Fig. 5 to map grain orientation and boundaries in a hexagonal close-packed Ti-6 wt.% Al (Ti-6Al in the following) polycrystal. The sample was thinned by jet polishing to create a hole surrounded by ultrathin regions. Electron backscatter diffraction (EBSD) in an SEM was used to create a reference map of the grain orientations near the hole (Fig. 7a). Using the pointprojection microscopy mode described above, we located and imaged the largest protrusion into the hole to determine the sample in-plane orientation (Fig. 7b). We then focused the beam onto the protrusion and obtained a diffraction pattern indicating [0001] in that grain is nearly normal to the sample, matching the orientation determined using EBSD.
We then demonstrated the ability of the ultrafast beam to locate a grain boundary with sub-diameter precision, by scanning the sample stage along one axis and acquiring a diffraction pattern at each 250 nm step. The orientation fraction, the fraction of electrons that pass through a grain, is f j ¼ ðI j =I j Þ P k ðI k =I k Þ ; I j is the total intensity of a chosen set of diffraction spots due to grain j and I j is I j when the probe is entirely within grain j. We compute I j by fitting symmetric 2D Gaussian functions to the selected peaks (circled in the figure) and summing their  Finally, we formed an ultrafast scanning transmission electron microscopy (USTEM) image by mapping the crystal orientation throughout a continuous bend in the sample over a 20 ×14 μm area in 2 μm steps (Fig. 7d). We identified four on-zone orientations along this bend in the diffraction patterns, made possible by the nanoscale beam size. We approximate the orientation at each location to be an average of these four zone axes weighted by their orientation fraction as defined above. The colors in the USTEM orientation map correspond to those used in the EBSD map. The USTEM map shows a gradual bend matching the sample bending found around the hole in the EBSD map of the same region. This example demonstrates potential for spatio-temporal mapping with nanoscale spatial resolution.

Discussion
We have presented experimental results demonstrating ultrafast relativistic electron scattering at the nanoscale. Electron beams as small as 90 nm have been measured, nearly two orders of magnitude smaller than previous attempts, with four-dimensional emittance values in the range of 0.01 (nm rad) 2 resulting in unprecedented degree of lateral coherence. A relevant figure of merit in such regard is the relative coherence length 32 , defined as where λ e is the de Broglie wavelength, σ * is the beam transverse size at the focus and σ θ its angular spread. From the width of the diffraction spots in Fig. 7 (250 μm RMS) we obtain σ θ ¼ 0:42 mrad. Combining this information with the beam size at focus reported in Fig.4 provides a relative coherence length of 0.72 and 0.43% for the two planes respectively, two orders of magnitude better than previously published results 16,23 . We can also define an effective normalized emittance along the two directions of minimum spot size multiplying the waist size in Fig.4 with the calculated σ θ at the detector. Such quantity, although much larger than the real diagonal 2D emittance, is considered more relevant for applications in USTEM and nano-UED in the case of beam with coupled transverse planes, as the width of the diffraction spots and the beam size at the detector ultimately determine respectively the reciprocal space and real space resolution. In the case of the beam in Fig.4 we found such values to be respectively 370 and 620 pm for the two planes.
Ultrafast electron-based instrumentation is not expected to have the same spatial resolution of static electron microscopes any time soon. On the other hand, ultrafast electron probes with sizes in the 100 nm range as demonstrated in this work are smaller than the typical grain size in most materials, and can therefore be used to probe local dynamics as a function of orientation or proximity to grain boundaries. As the USTEM measurement demonstrates, this technique can be applied to both sharp boundaries and graded regions, supporting spatio-temporal mapping of complex microstructures with heterogeneous orientation, composition, and phase. Relativistic nano-UED provides a new mean for accessing local structural information in real time, with femtosecond-nanometer resolution. The enormous scientific potential of the technique includes the study of real-time energy transfer in materials through all possible degrees of freedom, nanoscale thermal transport 33,34 , ultrafast dynamics of individual nanowires, nanoparticles or in low-dimensional functional nanomaterials, such as the coherent interlayer phonon excitations in epitaxial transition metal dichalcogenide (TMDC) micro-crystallites, the ultrafast manipulation of mirror domain walls in charge density wave (CDW) materials 35 , and the Moiré pattern dynamics in unconventional superconducting magic-angle graphene superlattices 36 .
We also note how there are a plethora of techniques and probes that rely on information from the microscopic and submicroscopic world including atomic force microscopy, neutron, X-ray and electron diffraction, transmission electron microscopy and all of their variants. Often the information from one of these probes needs to be integrated and complemented with the information from the other or even better a study with one probe can be used to inform another one. This correlative microscopy can only take place if image references can be used to crosscorrelate the observed position on the sample. For this reason, for example, it is essential that an instrument capable of nanodiffraction also is equipped with an imaging modality that can be effectively used in this correlation task such as the pointprojection microscopy demonstrated here. One possible direction taking advantage of this correlative electron microscopy would be to couple high-spatial resolution maps performed at TEMs with the high temporal resolution of nano-UED setup. In this situation, the region of interest for time-resolved studies would first be selected from the TEM images, based on a specific spatial feature, and then moved to UED setups where the local dynamics can be studied. In this work, we have demonstrated how this can be enabled by UED instrumentation with both diffraction and projection microscopy capabilities, each with sub-micrometer resolution.
Lastly, these results find application also beyond the field of novel ultrafast electron scattering instrumentation in the development of ultrahigh brightness sources for injection in laserdriven micrometer aperture dielectric structures 37 .

Methods
UPPM contrast transfer function determination. We define the contrast transfer function (CTF) to be the % of initial contrast observed for an infinite sinusoidal feature as a function of its spatial frequency f. Also called the modulation transfer function, this CTF can be used to compute the expected image of any feature from its spatial frequency composition 38 . The three-bar rulings fabricated here are finite and thus composed of many spatial frequencies; it is necessary to extract the modulation at particular spatial frequencies to determine the CTF. We base our approach on an existing procedure for three-bar gratings 39 .
To prepare a ruling image for analysis, we first select a 5 pixel wide region spanning the ruling and average along its width to obtain a line profile. We then obtain a similar profile from a nearby background region, smooth with a 50 pixel Gaussian filter, and subtract the background from the ruling profile. Such a large kernel is used to avoid modifying the spatial frequency composition of the line profile in the grating frequencies of interest when subtracting the background. We then take the Fourier transform of the line profile to obtain its spatial frequency composition. We also take the Fourier transform of computer-generated model grating profiles having the lateral dimensions determined for the actual gratings by SEM. The modulation at a particular frequency is the ratio of the Fourier component at that frequency in the measured grating relative to that in the model grating, where both components are normalized by the component at f = 0.
To compare the image to the model, the image magnification must be calibrated. We take advantage of a property of three-bar gratings: the Fourier component at f 0 3 is zero. The spatial frequency at which this feature appears is not modified by applying the CTF, so we can directly determine the scale of the image from the location of the first zero point in the Fourier transform of the measured profile. We find the magnification is 145x in both directions.
For each three-bar ruling, the modulation can be most reliably determined at the fundamental frequency, f 0 , since that is the strongest frequency component. This f 0 is given by the reciprocal of the grating period, which is the sum of the bar width and spacing. It is also possible to determine the modulation at other frequencies; Three-bar gratings have an f 0 2 component that is about 1/3 the strength of the f 0 component. We have computed the modulation at both f 0 and f 0 2 for the three largest rulings as shown in Fig. 6c, outlining a CTF.
We estimated the standard error in modulation measurements by simulated measurements. We consider an error dominated by random noise. We first modeled noiseless grating profiles using step functions with the lateral dimensions of the actual gratings modified by a Gaussian CTF. We then computed the noise level from a flat region of the image and added random noise at this level to the model profiles. 5000 simulated measurements were generated for each ruling by sampling the model profile with the experimental detector resolution and then adding random noise. We computed the modulation at f 0 and f 0 2 from all the measurements and determined the standard error represented by the error bars in Fig. 6c. The standard error is lowest in measurements using f 0 because it is the strongest frequency component.
Relationship between UPPM resolution and beam size. The point spread function (PSF) in point-projection microscopy is influenced by angular spread of the imaging beam (Fig. 8). For a Gaussian beam phase space, the PSF in sample coordinates is Gaussian with σ u;psf ¼ Lσ u′;zs M u , where σ u′;z is the local angle distribution, z s is the sample location, u refers to x or y coordinate, L is the distance between sample and detector, and M u is the magnification in u.
Assuming for simplicity no correlation between transverse planes, we can derive a direct relation between the PSF and the minimum beam size under the conditions depicted in Fig. 8. We first note that the magnification M u ¼ σ u;z det σ u;zs , with σ u;z det the beam size at the detector plane and σ u;z s the beam size at the sample plane. Also, the beam geometric emittance is constant along line (assuming no acceleration and negligible space charge forces), i.e., ε g ¼ σ u;z s σ u′;z s ¼ σ u;z 0 σ u′;z 0 , with z 0 being the beam waist position. Using these definitions and recognizing that σ u;z det ¼ Lσ u′;z 0 , we find: Diffraction pattern analysis. The high-resolution orientation map (Fig. 7a) was acquired using scanning electron backscatter diffraction (EBSD) on an FEI Strata 235 dual beam FIB/SEM. Orientations at each position were automatically determined from the acquired EBSD patterns by the vendor's built-in pattern fitting routine. Any pattern fits with a confidence index below a threshold of 0.7 were set to black in the map: these positions are seen to correspond either to the central hole or occasionally to grain boundaries. A 3 × 3 pixel median filter was applied to the final RGB colormap to remove erroneously labeled single pixels within the large, homogeneous grains. To identify the zone axis for each ultrafast electron diffraction pattern (DP) shown in Fig. 7b-d, we first computed a table of reciprocal lattice basis vector pairs and corresponding zone axis based on hexagonal close-packed (HCP) Ti. We then computed the basis vector lengths and angle from each measured DP and determined the best match in the table. All patterns were confirmed by generating the expected locations of diffraction spots accounting for HCP selection rules and ensuring the measured DP satisfied these rules.
For the nano-UED line scan (Fig. 7c), we used the single diffraction pattern shown from each pure grain to identify the zone axes. We only used the circled peaks to compute orientation fraction. The total diffraction signal from the grain was determined by fitting circular 2D Gaussian functions to the peaks and summing their volumes. For each zone axis, peak fitting was performed sequentially along the scan line starting from the pure grain. For the first pattern, the initial guess peak width, position, and background were set manually. For each subsequent pattern, the best-fit parameters from the previous pattern were used as the next initial guess.
For the USTEM orientation map (Fig. 7d), the grain orientation is continually rotating. We first identified the four displayed single diffraction patterns corresponding to discrete zone axes along the bend. We then estimated intermediate orientations by computing the orientation fraction of the identified discrete zone axes at each position and considering the orientation to be a linear combination of the discrete constituent zone axes. This allows us to reproduce the bending feature from the EBSD map (Fig. 7a). In this case, the initial guess for peak parameters for a given orientation at each position used the best-fit values from the corresponding representative single diffraction pattern.  Fig. 8 Schematic of point-projection microscopy. This illustrates the quantities and relationships used to derive the direct relation between point spread function, σ u;psf , and beam size at the waist, σ u;z 0 . The beam propagates along the longitudinal z axis, with the u axis representing the x or y lateral position coordinate and u′ representing the x or y angular coordinate. σ u′;z 0 and σ u′;z s are the angular spread at the beam waist and the sample plane respectively. σ u;z s and σ u;z det are the beam size at the sample plane and the detector plane respectively. L is the sample-detector distance, and M is the magnification of the image at the detector

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability
A detailed description of the analysis performed can be found online 29 . The source code is available from the corresponding author upon reasonable request.