Discrete Chromatic Aberrations Arising from Photoinduced Electron

Article pubs.acs.org/JPCA

Discrete Chromatic Aberrations Arising from Photoinduced ElectronPhoton Interactions in Ultrafast Electron Microscopy Dayne A. Plemmons and David J. Flannigan* Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Avenue SE, Minneapolis, Minnesota 55455, United States S Supporting Information *

ABSTRACT: In femtosecond ultrafast electron microscopy (UEM) experiments, the initial excitation period is composed of spatiotemporal overlap of the temporally commensurate pump photon pulse and probe photoelectron packet. Generation of evanescent near-fields at the nanostructure specimens produces a dispersion relation that enables coupling of the photons (ℏω = 2.4 eV, for example) and freely propagating electrons (200 keV, for example) in the near-field. Typically, this manifests as discrete peaks occurring at integer multiples (n) of the photon energy in the low-loss/gain region of electron-energy spectra (i.e., at 200 keV ± nℏω eV). Here, we examine the UEM imaging resolution implications of the strong inelastic near-field interactions between the photons employed in optical excitation and the probe photoelectrons. We find that the additional photoinduced energy dispersion occurring when swift electrons pass through intense evanescent near-fields results in a discrete chromatic aberration that limits the spatial resolving power to several angstroms during the excitation period.

■

INTRODUCTION Recent advances in transmission electron microscopy (TEM) technology have resulted in the achievement of subangstrom spatial resolutions and energy resolutions reaching below 10 meV (80 cm−1).1,2 In combination with in situ holders and fast, sensitive direct-electron detectors, such capabilities enable studies of atomic-scale processes occurring on submillisecond time scales.3,4 Indeed, experiments enabled by such advances in instrumentation are numerous and include studies of nanocrystal nucleation and growth,5,6 catalytically driven reactions,7 and structural phase transformations.8,9 However, myriad atomic and nanoscale phenomena occur on time scales far shorter than currently accessible with fast detectors (e.g., plasmon excitation and propagation, electron−phonon coupling, nanoscale phonon dynamics, etc.). Importantly, such ultrafast processes originate from, and are strongly coupled to, femtosecond perturbation of atomic-scale electronic and lattice structure, and gaining access to this parameter space with TEM requires increases in temporal resolution of many orders of magnitude. Extension of TEM temporal resolution to the femtosecond regime is presently accomplished through application of a stroboscopic pump−probe methodology via a technique called ultrafast electron microscopy (UEM).10−12 This approach consists of interfacing an otherwise conventional microscope with a short-pulsed laser system such that optical-pump/ photoelectron-probe experiments can be conducted. In this way, real-space imaging with combined angstrom-femtosecond spatiotemporal resolutions is in principle possible, as © 2016 American Chemical Society

demonstrated via acquisition of lattice-fringe images with a pulsed photoelectron source at high laser-repetition rates.10,13,14 Ideally with UEM, one would have access to femtosecond time scales without compromising the resolution of the base-TEM instrument (e.g., 1.4 Å real-space line resolution for a 200 kV thermionic microscope). Accordingly, it is worth noting that real-space imaging is critical for visualizing dynamics that are otherwise inaccessible with methods that inherently operate via ensemble averaging and from which effects of atomic-scale heterogeneities are either inferred or ignored.12,15 Integrating the real-space, angstrom-scale imaging of conventional TEMs with the ultrafast temporal resolutions afforded by short-pulsed lasers represents a potential route to direct visualization of femtosecond dynamics without limits to local order, structural morphology, or chemical composition. Consequently, a thorough and quantitative understanding of the effects of photon-pump-pulse/photoelectron-probe-packet spatiotemporal overlap on image formation is critical for resolving initial ultrafast dynamics. In UEM, the narrow electron-energy distribution afforded by populating each packet with, on average, one photoelectron16 can be compromised during such spatiotemporal overlap at the specimen. Inelastic interaction of swift photoelectrons with intense, photogenerated transient evanescent near-fields results in significant population of discrete energies separated from the initial state Received: March 21, 2016 Revised: April 18, 2016 Published: April 25, 2016 3539

DOI: 10.1021/acs.jpca.6b02916 J. Phys. Chem. A 2016, 120, 3539−3546

Article

The Journal of Physical Chemistry A

Figure 1. Effects of discrete chromatic dispersion on a conventional point spread function (PSF). (a) Conical ray diagram depicting the spread of electrons having a discrete energy distribution on the Gaussian image plane. The green rays correspond to zero-loss (i.e., E0 = 200 keV) electrons, whereas the red and blue are those that have gained and lost one photon quantum of energy, respectively. The spread in each of the colored rays is meant to qualitatively represent the inherent energy distribution arising from the photoemission process (e.g., 1 eV, absent significant space-charge broadening). (b) Nonlinear envelope (dashed red line) of a typical UEM photon-induced near-field electron-energy distribution arising from a moderate interaction parameter, Ω. The x-axis corresponds to energy loss relative to E0; negative values indicate energy gains with respect to E0. (c) Annular chromatic point spread function (PSFc) at τ = 0 plotted as a function of r = x, y Å. (d) Static PSF as τ → ∞ [i.e., temporally far from electron-photon overlap, PSF∞(r)] encompassing the instrument spatial-resolution limit. (e) Convolution of PSF∞(r) and PSFc(τ=0,r) to produce a final PSF [PSFf (τ=0,r)] illustrating the effects of the UEM discrete chromatic dispersion at τ = 0. Note the factor of 2 increase in the range of the x and y axes relative to that in (d).

by integer multiples of the pump-photon energy.17−21 Here we show that the induced electron-energy envelope arising from absorption and emission of photons in the excited-specimen near-field becomes the dominant effect with respect to the spatial resolution of UEM during the initial excitation period. Moreover, we show that certain spatial frequencies become exaggerated due to the nonlinear nature of the near-field interaction.22,23 By applying a filter composed of discrete, transient chromatic aberrations resulting from near-field interactions to representative lattice-fringe images of ordered nanostructures, we demonstrate that the time-dependent point spread function produces significant blurring, the transient nature of which directly tracks the pump-pulse/probe-packet cross correlation.

where Cjn =

P(E) =

METHODS Electron-Energy Distributions. Population of nth-order inelastic scattering peaks was calculated via a solution to the time-dependent Schrödinger equation with application of a slowly varying pulsed-light approximation, following Park et al.22 At a fixed interaction parameter (Ω), population of an nthorder peak is numerically evaluated with the Riemann summation shown in eq 1.

⎡ −(E − nℏω)2 ⎤ ⎥ 2σE ⎣ ⎦

∑ P(n) exp⎢ n

(2)

Here, a fwhm of 1 eV (σE = 0.425 eV) has been used for the incident electron-energy distribution, and 519 nm (ℏω = 2.39 eV) has been used for the pump laser energy. The energy envelope is calculated by evaluating eq 1 on a continuousenergy interval and by generalizing factorials in the approximation of the Bessel function to the gamma function for nonintegral values. Calculation of Point Spread Functions and Transfer Functions. Projections of the electron-energy distribution in the Gaussian image plane are evaluated with the following TEM

⎡ ⎤ SnjkR τ 2 n n* 2 −1/2 ⎢ ⎥ C C (1 + S R ) exp − ∑∑ j k njk σ 2 ⎢⎣ 2(1 + SnjkR σ ) ⎥⎦ j=0 k=0 ∞

(1)

Here, Snjk = n + j + k, Rσ = σe/σp, and Rτ = τ/σp. Gaussian temporal profiles have been used for photon pulses and electron packets, with standard deviations of σp = 200 fs and σe = 400 fs, respectively, unless otherwise noted. The electronenergy distribution plots are generated from the scattering populations by convolution with the energy profiles of the incident electrons. For an initial electron-energy distribution of Gaussian shape, the resulting photon-induced energy distribution is described by eq 2.

■

P(n) =

⎛ Ω ⎞n + j ⎛ −Ω ⎞ j 1 ⎜ ⎟ ⎜ ⎟ (|n| + j) ! j! ⎝ 2 ⎠ ⎝ 2 ⎠

∞

3540


Article


images.32 Figure 1 summarizes the results of convoluting the UEM discrete chromatic dispersion with a conventional PSF. Owing to the velocity dependence of the Lorentz force, an energy distribution such as the one shown in Figure 1b will produce a radially symmetric projection in the Gaussian image plane consisting of concentric rings of varying intensity (ignoring conventional aberrations for the moment) (Figure 1c). Thus, the PSF describing the time-delay-dependent chromatic aberration, PSFc(τ,r), can be represented by spatial analogues to the energy-distribution parameters ΔE and ℏω (eqs 3−5). Here, τ is the delay time between the pump pulse and the probe packet, r is the position in the image plane, and ΔE is the electron-energy distribution, which is governed by the photoemission process at the electron gun (in the case of negligible space-charge effects).

operating parameters: a collection angle (β) of 4.5 mrad, a coefficient of chromatic aberration (Cc) of 1.2 mm, and an accelerating voltage (V) of 200 kV. The Fourier transforms of the point spread functions (PSF) are approximated with the discrete Cooley−Tukey numerical method.41 One-dimensional (1D) cross sections of the PSFs are sampled out to r = ±

Ccβ(240eV) V

with frequency FS =

V Ccβ(0.24eV)

to avoid low-

frequency sampling artifacts. The radial symmetry of the PSFs allows rotational sweeping of 1D cross sections of the FFT to produce two-dimensional transfer functions with increased numerical stability and accuracy. Image Simulations. The bright-field images of nanostructures shown in Figure 5 were acquired with a Tecnai Femto UEM (FEI Company) operated in thermionic mode at 200 kV. Images of Au nanocrystals were generated from a gold/ graphitized carbon calibration specimen (Ted Pella), whereas those of the carbon nanotube were obtained from a specimen prepared by drop casting from a dilute solution (Sigma-Aldrich; outer diameter 6−13 nm, length 2.5−20 μm) in ethanol. To simulate the effect of quantized energy dispersion, the experimental images were convoluted with the chromatic PSF using the imf ilter command with circular boundary options in Matlab. For accurate portrayal of blurring, the spatial sampling frequency of the experimental images, 93.6 pixels nm−1 for the Au nanocrystals and 76.4 pixels nm−1 for the carbon nanotube, is replicated in the calculation of the chromatic PSF. Fast Fourier transforms of the images are displayed on a logarithmic color scale mapped from counts in the 85th (black) to the 98th percentile (white).

δΔE = Cc

ΔE β V

(3)

δℏω = Cc

ℏω β V

(4)

PSFc(τ ,r ) =

⎡ −(r − nδ )2 ⎤ ℏω ⎥ 2δΔE 2 ⎣ ⎦

∑ P(τ ,nℏω) exp⎢ n

(5)

The width and separation between the concentric rings shown in Figure 1c are described by δΔE and δℏω, respectively (eqs 3 and 4, respectively), where β is the collection angle and V is the accelerating voltage. The amplitude of each ring is governed by the energy envelope describing the population of discrete states, P(τ,E) [Figure 1b, dotted red line]. Note that, as with the sideband peak amplitudes in energy space, the intensities of the rings comprising PSFc are spatially dependent owing to the variance of Ω at different points within the nearfield (Supporting Information).22,23 For a homogeneous section of the near-field, however, Ω is assumed to slowly vary over the few angstroms in which the PSF is nonzero. Evaluating populations at a fixed Ω then becomes a reasonable approximation. To examine the regime in which near-field induced chromatic effects are prevalent, an additional Gaussian PSF [PSF∞(r)], which encompasses the inherent instrumentresolution limit (i.e., far from temporal overlap), is introduced. For PSF∞(r), the radius of the disk in the image plane (rmin) is fixed at 1.5 Å. Convolution of PSF∞(r) and PSFc(r) returns the final time-delay-dependent UEM PSF [PSFf(r)], as described by eqs 6−8 and shown in Figure 1e.

■

RESULTS AND DISCUSSION Discrete Chromatic Dispersion and the Point-Spread Function. In UEM, nanoscale specimen dynamics are triggered in situ by excitation with a femtosecond laser pulse. Owing to the high peak intensities of ultrashort pulses (e.g., 100 GW cm−2), strong plasmonic excitations can be initiated in the specimen, which lead to scattering and near-field enhancement of the laser light.24 The resulting spatially confined evanescent waves have reduced phase velocity, and thus, energymomentum conservation requirements for coupling with freely propagating electrons can be satisfied.22,25−27 Consequently, photoelectrons in the probe packet can be optically driven by light in the near-field when spatiotemporally overlapped with pump photons at the specimen.21 Evidence of this effect is manifested in the low-loss/gain region of the electron-energy spectrum, where discrete states occurring at integer multiples of the pump-photon energy (±nℏω, where n is an integer, ℏ is the reduced Planck constant, and ω is the angular frequency) relative to the zero-loss peak are populated. That is, the incoming 200 keV electrons absorb and emit photons such that, when detected, they have energies of 200 keV ± nℏω eV. Typically, this effect is experimentally observed via a UEM variant of energy-filtered TEM dubbed photon-induced nearfield electron microscopy.17,19−21,28−31 To quantify the effects described above on UEM image formation, the point spread function (PSF) of the objective lens is determined using spherical- and chromatic-aberration coefficients (Cs and Cc, respectively) and operating parameters typical of a 200 kV TEM equipped with a LaB6 thermionic gun (see Methods section). The PSF for a particular imaging system describes its response to a point source and thus enables visualization of the effects of particular aberrations on the

δmin =

2rmin 2 2 ln(2)

(6)

⎡ −(r )2 ⎤ ⎥ PSF∞(r ) = exp⎢ ⎣ 2δmin 2 ⎦

(7)

PSFf (τ ,r ) = (PSF∞*PSFc)(τ ,r ) =

∫ PSF∞(ρ) PSFc(τ ,r−ρ) dρ

(8)

At maximum temporal overlap (i.e., τ = 0), a broad shoulder is observed in PSFf(r) that corresponds to the energy envelope resulting from the photon-induced near-field effect (Figure 1e). Note also that the area of the xy-plane has expanded by a factor of 4. Absent energy-filtering capabilities (which would 3541


Article


Figure 2. Transient chromatic transfer functions. (a) Two-dimensional transient chromatic transfer function at τ = 0 [Ĥ c(τ=0,s)]. Here, s = spatial frequency and τ = time delay. The horizontal red line marks the position from which the cross-section in panel (b) was extracted (solid blue). The scale bar corresponds to 1 Å−1. (b) Plot of the behavior of the normalized intensity dependence of Ĥ c [solid blue curve; extracted from (a)] as a function of spatial frequency, as dictated by the particular inverse energy envelope (P̂ , dashed red curve) around the distinct maxima occurring at ±n/δℏω. The τ → ∞ transfer-function envelope (Ĥ ∞, solid green curve) is included to illustrate that multiple P̂ sidebands fall within the Ĥ ∞ envelope. (c) Overall transfer function resulting from convolution of the transient component at τ = 0 with that inherent to the system (i.e., far from temporal overlap, τ → ∞) [Ĥ ∞Ĥ c(τ=0,s)]. Inclusion of Ĥ c produces depletion of discrete intensities within the τ → ∞ transfer-function envelope. The scale bar corresponds to 1 Å−1.

Figure 3. Temporal dependence of frequency depletions. (a) Variation of Ĥ c as a function of spatial frequency and time delay. The transient chromatic transfer function evolves from a Gaussian shape with width 1/δΔE to nonmonotonic behavior around τ = 0. (b) Time traces of Ĥ ∞Ĥ c(τ) at select spatial frequencies (labeled above each curve in Å−1). The cross-correlation of the electron packet and the pump-laser pulse temporal profiles [Pe ⊗ Ip(τ); dotted pink line] is shown for reference. (c) Contour plot of the fwhm of Ĥ ∞Ĥ c(τ) at s = 0.28 Å−1 with Ω = 10 for relevant electronpacket and laser-pulse parameters. Linear coupling occurs in the upper regime (i.e.,

Δe Δp

> 2.5), and Δt ≅ Δe 2 + Δp2 . As

inflection point is observed, which is indicative of nonlinear coupling and results in Δt >

introduce additional challenges associated with signal acquisition times), PSFc(r) and, thus, PSFf(r) will significantly impact UEM image quality during the initial excitation period, as discussed below. It is worth noting that the value of rmin does not necessarily correspond to the same focus conditions considered in the calculations of PSFc(r); the overall plane of least confusion does not generally occur in the Gaussian image plane but rather at the Scherzer defocus.33 Indeed, these calculations represent a best-case scenario for chromatic aberrations, as minor focus deviations from the image plane will result in increased radii of the concentric rings that comprise PSFc(r). Discrete Chromatic Dispersion in Reciprocal Space. It is also useful to quantify spatial resolution in Fourier space, where nonvanishing intensity at spatial frequencies furthest from zero correspond to the smallest resolvable features in realspace images. In reciprocal space, transfer functions that represent the imaging-system response are multiplicatively applied to the intensity spectrum of the object (eqs 9 and 10).32

Δe Δp

decreases, however, an

Δe 2 + Δp2 .

Iobserved(τ ,r ) = (Iobject*PSFf )(τ ,r ) =

̂ (τ ,s) = Iobserved

∫ Iobject(ρ) PSFf (τ ,r−ρ) dρ

(9)

∫ Iobserved(τ ,r) exp(2π irs) dr

̂ Ĥ∞Ĥc(τ ,s) = Iobject

(10)

̂ Here, Iobserved and Iobserved are the image intensity in real and ̂ reciprocal space, respectively; Iobject and Iobject are the object intensity in real and reciprocal space, respectively; Ĥ c and Ĥ ∞ are the Fourier transforms of PSFc and PSF∞, respectively; and s is the spatial frequency. For any arbitrary object [Iobject(r)], ̂ (s)] are modulated by features in the intensity spectrum [Iobject the envelope resulting from the product of the transfer functions upon formation of the observed image and intensity ̂ spectrum [Iobserved(τ,r) and Iobserved (τ,s), respectively]. Thus, resolution limits set by separate phenomena can be evaluated by examining the behavior of the appropriate transfer function. 3542


Article


Figure 4. Image resolution for varying interaction parameters (i.e., near-field strengths). (a) Evolution of the inverse energy envelope (spatial frequency) as a function of interaction parameter. Widths of P̂ (τ=0) rapidly become narrower than the envelope width of the static lens transfer function (inset; dashed green) with increasing interaction parameter, indicating the transient chromatic dispersion governs spatial resolution. (b) Temporal evolution of the effective resolution reff(τ) for varying interaction parameters (representative values labeled above each curve). For modest near-field strengths, the effective resolution is limited to several angstroms near τ = 0 and endures for a majority of the subpicosecond temporal regime until converging toward the static resolution (rmin, green line at reff = 1.5 Å).

Figure 5. Simulated UEM chromatic blurring in nanostructures. (a, c) Conventional TEM lattice-fringe images of Au nanocrystals (a) and a fewwalled carbon nanotube (c). Intensity peaks in the FFTs (insets) illustrate the symmetry and the spacing of the observed fringes. (b, d) Simulated images at τ = 0 (i.e., with Ĥ c applied) illustrating the distinct photoinduced blurring that is predicted to occur during femtosecond UEM. The 2.3 Å (111) atomic planes of the Au nanocrystals are almost entirely unresolvable, whereas the (002) graphitic planes in the carbon nanotube are faintly visible, as the corresponding peak in the FFT falls within a side peak of the chromatic transfer envelope (inset, d). Scale bars: images = 5 nm; FFTs = 5 Å−1.

Figure 2 summarizes the effects of convolving Ĥ c with Ĥ ∞ and illustrates the behavior as a function of time delay. As can be seen in Figure 2a, Ĥ c exhibits distinct maxima at spatialfrequency values of ±n/δℏω, with amplitude dictated by the width of the zero-loss peak. When this is convolved with Ĥ ∞, spatial frequencies falling between those that share intensity with Ĥ c are significantly depleted. A one-dimensional crosssection of Ĥ c (Figure 2b, solid blue; from red horizontal line in

Figure 2a) resembles the Fourier transform of the population envelope P̂ near the distinct maxima. This indicates that the energy envelope governs the behavior of Ĥ c at relevant spatial frequencies (i.e., those frequencies falling within nonvanishing values of Ĥ ∞). Indeed, Figure 2b shows distinct depletion at frequencies within the envelope of Ĥ ∞, confirming that spatial resolution near time-zero is limited by the increased chromatic dispersion. 3543


Article

The Journal of Physical Chemistry A Transient Evolution of Spatial Frequencies. Plotting the cross-section of Ĥ c for time delays where pulse overlap occurs (Figure 3a) illustrates the temporal behavior of the effect; Ĥ c is observed to gradually shift from its static form with width 1/δΔE to its transient form with shape dictated by P̂ approaching zero time delay. Because photoinduced chromatic effects are entirely dependent on spatiotemporal overlap of the pump laser pulse and photoelectron packet, significant intensity depletion occurs over time scales roughly corresponding to the quadrature of their temporal durations (full-width half-maxima, fwhm;

To visually illustrate the effect of photoinduced chromatic dispersion on spatial resolution, PSFc(τ=0) was applied to experimental TEM images (Figure 5). Lattice-fringe images of Au nanocrystals and a few-walled carbon nanotube were acquired in conventional TEM mode to represent the static resolution limit (i.e., images containing Ĥ ∞). The chromatic filter Ĥ c was applied in postprocessing to exemplify the effect of laser-pulse/electron-packet overlap for a given time delay and near-field strength. Figure 5 compares the images and FFTs of the nanostructures both far from (τ → ±∞) and at maximum overlap (τ = 0). The τ → ±∞ image of Au nanocrystals (Figure 5a) shows distinct fringes corresponding to the (111) planes (d = 2.3 Å). Because this planar spacing falls well outside the main envelope of Ĥ c, these fringes are almost entirely unresolved in the resulting τ = 0 image (Figure 5b). For larger spacings, such as the (002) graphitic planes of a few-walled carbon nanotube (d = 3.4 Å; Figure 5c), distinct blurring is still observed in the τ = 0 image (Figure 5d), but because this feature falls within the secondary envelope, or side-peak of Ĥ c (Figure 5d inset), these planes are still faintly discernible. Although some information beyond the effective resolution defined previously (reff) is obtained, the observations described above have greater consequences for subpicosecond atomicscale imaging. Because high-resolution TEM relies on phase contrast,32 transient modulations of the chromatic contrasttransfer function will appear as deviations in focus conditions in the images and thus convolute the interpretation of structure for a given time delay. It is worth noting that a spatially homogeneous near-field over the selected area in the x,y-plane has been considered here. For a more general case (e.g., the near-fields imaged by Piazza et al.),19 a projection of the x,y spatial features of the field in the form of blurring is expected, such that distinct chromatic distortions occurring in both strong and weak regions are only marginally affected. Nevertheless, these images embody the loss of most information pertaining to atomic planes as a result of the discrete, transient UEM chromatic aberration. The photoinduced chromatic effects in UEM imaging discussed here represent another challenge in resolving concurrently the structure of materials in space and their ultrafast modulations in time, particularly for specimens which exhibit strong near-fields upon photoexcitation. Indeed, maintaining beam coherency and reducing temporal spread of the probing electron packet throughout the photogeneration, propagation, and imaging processes has been a theme in improving femtosecond electron-imaging capabilities.16,34 Notably, new iterations of hardware and methodology are being implemented, which seek to achieve brighter, more coherent, and shorter electron packets through increased accelerating voltages (e.g., MeV) and radio frequency (RF) temporal-compression techniques.35−39 Though these schemes combat complications arising during photogeneration and propagation, the resulting electron probes are still subject to effects arising at the specimen and subsequent imaging system.40 Indeed, operating at MeV accelerating voltages generally enhances electron-photon coupling and the commensurate interaction strength. This, however, should in principle reduce the relative energy spread, which is encompassed by the PSF parameter δΔE. Likewise, the condensed temporal envelope of RF-compressed electron packets are subject to strong coupling with evanescent fields (i.e., the nonlinear regime shown in Figure 3c) but are nonetheless a route to improved instrument-response times.

Δe 2 + Δp2 , where Δe = duration of electron packet and Δp = duration of photon pulse). Thus, for properties typical of femtosecond UEM (e.g., Δp ≅ 100−1000 fs and Δe ≅ 500− 1500 fs), chromatic blurring will appear for 100s to ∼1000 fs around the initial excitation period (depending upon the precise durations). As depicted in Figure 3c, contours of the temporal duration of intensity depletion (Δt, fwhm) at fixed spatial frequency and fluence confirm the above-stated behavior follows for the majority of applicable femtosecond UEM parameters. Devia-

tions from linear coupling (where Δt ≅ Δe 2 + Δp2 ; see Figure S1 in Supporting Information) can be attributed to the increased portion of the optical pulsewith intense fields and associated with a condensed time envelopecapable of inducing several absorption and emission events.23 Note that typical experimental cases, in which Δe > Δp, have been highlighted here. For the inverse case (i.e., Δe < Δp), a condensed photoelectron packet samples only strong coupling regions of the optical pulse leading to highly nonlinear population envelopes.21 Accordingly, the chromatic transfer function exhibits an increase in the quantity and intensity of side bands around the main peak. Nevertheless, these calculations suggest the transient chromatic aberration will be present for a large fraction of the subpicosecond regime. Effect of Discrete Chromatic Dispersion on Spatial Resolution. The transfer-function approach can be extended to determine the amount of chromatic blurring that is expected to occur, and thus the effect on overall spatial resolution, as a function of photoinduced near-field strength. Figure 4a portrays the reciprocal envelope P̂ for appropriate values of the interaction parameter Ω at maximum pulse overlap. For increasing strength, the width of the main peak becomes exceedingly narrow, limiting the resolvable spatial frequencies. It is clear that even for modest interaction parameters, the envelope width is on the order of the width of Ĥ ∞ (Figure 4a, green dashed line), indicating that a large spatial-resolution regime is impacted. The extent of this impact is also dependent on the pulse overlap occurring at different time delays. Consequently, the effective resolution (reff) at each time point is estimated by adding in quadrature the reciprocal widths of the main peak of P̂ (ΔP̂, fwhm) and Ĥ ∞ such that reff = 1 2

(2rmin)2 + 1/(Δ P ̂)2 . The temporally varying resolution for various interaction parameters is poorest at zero time delay, with minimum resolvable features limited to several Å for stronger fields (Figure 4b). Note also that the resolution does not recover until hundreds of femtosecond after τ = 0. Despite maintaining subnanometer resolution, these results suggest subpicosecond UEM imaging of few-angstrom lattice fringes may require phase retrieval with a known chromatic-dispersion filter. 3544


Article


■

(3) Ramachandramoorthy, R.; Bernal, R.; Espinosa, H. D. Pushing the Envelope of In Situ Transmission Electron Microscopy. ACS Nano 2015, 9, 4675−4685. (4) Zheng, H.; Meng, Y. S.; Zhu, Y. Frontiers of In Situ Electron Microscopy. MRS Bull. 2015, 40, 12−18. (5) Yuk, J. M.; Park, J.; Ercius, P.; Kim, K.; Hellebusch, D. J.; Crommie, M. F.; Lee, J. Y.; Zettl, A.; Alivisatos, A. P. High-Resolution EM of Colloidal Nanocrystal Growth Using Graphene Liquid Cells. Science 2012, 336, 61−64. (6) Liao, H.-G.; Zherebetskyy, D.; Xin, H.; Czarnik, C.; Ercius, P.; Elmlund, H.; Pan, M.; Wang, L.-W.; Zheng, H. Facet Development during Platinum Nanocube Growth. Science 2014, 345, 916−919. (7) McDowell, M. T.; Lu, Z.; Koski, K. J.; Yu, J. H.; Zheng, G.; Cui, Y. In Situ Observation of Divergent Phase Transformations in Individual Sulfide Nanocrystals. Nano Lett. 2015, 15, 1264−1271. (8) Lin, Y.-C.; Dumcenco, D. O.; Huang, Y.-S.; Suenaga, K. Atomic Mechanism of the Semiconducting-to-Metallic Phase Transition in Single-Layered MoS2. Nat. Nanotechnol. 2014, 9, 391−396. (9) Baldi, A.; Narayan, T. C.; Koh, A. L.; Dionne, J. A. In situ Detection of Hydrogen-Induced Phase Transitions of Individual Palladium Nanocrystals. Nat. Mater. 2014, 13, 1143−1148. (10) Zewail, A. H. Four-Dimensional Electron Microscopy. Science 2010, 328, 187−193. (11) Piazza, L.; Masiel, D. J.; LaGrange, T.; Reed, B. W.; Barwick, B.; Carbone, F. Design and implementation of a fs-resolved transmission electron microscope based on thermionic gun technology. Chem. Phys. 2013, 423, 79−84. (12) Plemmons, D. A.; Suri, P. K.; Flannigan, D. J. Probing Structural and Electronic Dynamics with Ultrafast Electron Microscopy. Chem. Mater. 2015, 27, 3178−3192. (13) Park, H. S.; Baskin, J. S.; Kwon, O.-H.; Zewail, A. H. AtomicScale Imaging in Real and Energy Space Developed in Ultrafast Electron Microscopy. Nano Lett. 2007, 7, 2545−2551. (14) Barwick, B.; Park, H. S.; Kwon, O.-H.; Baskin, J. S.; Zewail, A. H. 4D Imaging of Transient Structures and Morphologies in Ultrafast Electron Microscopy. Science 2008, 322, 1227−1231. (15) Cremons, D. R.; Plemmons, D. A.; Flannigan, D. J. Femtosecond Electron Imaging of Defect-Modulated Phonon Dynamics. Nat. Commun. 2016, 7, 11230. (16) Lobastov, V. A.; Srinivasan, R.; Zewail, A. H. Four-Dimensional Ultrafast Electron Microscopy. Proc. Natl. Acad. Sci. U. S. A. 2005, 102, 7069−7073. (17) Barwick, B.; Flannigan, D. J.; Zewail, A. H. Photon-Induced Near-Field Electron Microscopy. Nature 2009, 462, 902−906. (18) Howie, A. Photon-Assisted Electron Energy Loss Spectroscopy and Ultrafast Imaging. Microsc. Microanal. 2009, 15, 314−322. (19) Piazza, L.; Lummen, T. T.; Quinonez, E.; Murooka, Y.; Reed, B. W.; Barwick, B.; Carbone, F. Simultaneous Observation of the Quantization and the Interference Pattern of a Plasmonic Near-Field. Nat. Commun. 2015, 6, 6407. (20) Barwick, B.; Zewail, A. H. Photonics and Plasmonics in 4D Ultrafast Electron Microscopy. ACS Photonics 2015, 2, 1391−1402. (21) Feist, A.; Echternkamp, K. E.; Schauss, J.; Yalunin, S. V.; Schäfer, S.; Ropers, C. Quantum Coherent Optical Phase Modulation in an Ultrafast Transmission Electron Microscope. Nature 2015, 521, 200− 203. (22) Park, S. T.; Lin, M.; Zewail, A. H. Photon-Induced Near-Field Electron Microscopy (PINEM): Theoretical and Experimental. New J. Phys. 2010, 12, 123028. (23) Plemmons, D. A.; Park, S. T.; Zewail, A. H.; Flannigan, D. J. Characterization of Fast Photoelectron Packets in Weak and Strong Laser Fields in Ultrafast Electron Microscopy. Ultramicroscopy 2014, 146, 97−102. (24) Maier, S. A.; Atwater, H. A. Plasmonics: Localization and Guiding of Electromagnetic Energy in Metal/Dielectric Structures. J. Appl. Phys. 2005, 98, 011101. (25) Weingartshofer, A.; Holmes, J. K.; Caudle, G.; Clarke, E. M.; Krüger, H. Direct Observation of Multiphoton Processes in LaserInduced Free-Free Transitions. Phys. Rev. Lett. 1977, 39, 269−270.

CONCLUSIONS In conclusion, it has been shown here how the additional energy dispersion present when photoelectron packets interact with evanescent near-fields in the vicinity of the specimen results in an additional resolution-limiting chromatic aberration in UEM imaging. Though the magnitude and the duration are a function of the excitation pulse properties, it is expected that this phenomenon will be especially apparent at magnifications that enable subnanometer resolutions. Consequently, robust imaging of atomic planes and their modulations on the subpicosecond time scale (i.e., during the initial photoexcitation period) may require precise knowledge of the particular transient chromatic-dispersion function such that phase retrieval of lattice fringes may be performed. Looking forward, continued identification of such fundamental physical processesabsent in conventional TEMare expected not only to enable combined angstrom-femtosecond real-space imaging but also to lead to additional discoveries and development of new experimental approaches in UEM.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.6b02916. Quantitative description of the dimensionless interaction parameter and its effect on electron−photon coupling and the low-loss/gain energy profile is provided, as is a discussion of the effects of the parameter within both the weak- and strong-interaction regimes. Also provided is a figure depicting the transient behavior of the discrete chromatic aberrations within the linear-coupling regime. (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*D. J. Flannigan. E-mail: fl[email protected]. Office: +1 (612) 625-3867. Fax: +1 (612) 626-7246. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported primarily by the National Science Foundation through the University of Minnesota MRSEC under Award Number DMR-1420013, in part by a 3M Nontenured Faculty Award under Award Number 13673369, and in part by the Arnold and Mabel Beckman Foundation through a Beckman Young Investigator Award. Acknowledgement is made to the Donors of the American Chemical Society Petroleum Research Fund for partial support of this research under Award Number 53116-DNI7. D.A.P. acknowledges support in the form of a Doctoral Dissertation Fellowship from the University of Minnesota Graduate College.

■

REFERENCES

(1) Krivanek, O. L.; Lovejoy, T. C.; Dellby, N.; Aoki, T.; Carpenter, R. W.; Rez, P.; Soignard, E.; Zhu, J.; Batson, P. E.; Lagos, M. J.; et al. Vibrational Spectroscopy in the Electron Microscope. Nature 2014, 514, 209−212. (2) Batson, P. E.; Dellby, N.; Krivanek, O. L. Sub-Angstrom Resolution Using Aberration Corrected Electron Optics. Nature 2002, 418, 617−620. 3545


Article

The Journal of Physical Chemistry A (26) Miaja-Avila, L.; Lei, C.; Aeschlimann, M.; Gland, J. L.; Murnane, M. M.; Kapteyn, H. C.; Saathoff, G. Laser-Assisted Photoelectric Effect from Surfaces. Phys. Rev. Lett. 2006, 97, 113604. (27) García de Abajo, F. J.; Asenjo-Garcia, A.; Kociak, M. Multiphoton Absorption and Emission by Interaction of Swift Electrons with Evanescent Light Fields. Nano Lett. 2010, 10, 1859− 1863. (28) Yurtsever, A.; van der Veen, R. M.; Zewail, A. H. Subparticle Ultrafast Spectrum Imaging in 4D Electron Microscopy. Science 2012, 335, 59−64. (29) Yurtsever, A.; Zewail, A. H. Direct Visualization of Near-Fields in Nanoplasmonics and Nanophotonics. Nano Lett. 2012, 12, 3334− 3338. (30) Park, S. T.; Zewail, A. H. Relativistic Effects in Photon-Induced Near Field Electron Microscopy. J. Phys. Chem. A 2012, 116, 11128− 11133. (31) Liu, H. H.; Baskin, J. S.; Zewail, A. H. Infrared PINEM Developed by Diffraction in 4D UEM. Proc. Natl. Acad. Sci. U. S. A. 2016, 113, 2041−2046. (32) Ludwig, R.; Helmut, K. Transmission Electron Microscopy: Physics of Image Formation; Springer: New York, 2008. (33) Hawkes, P. W. Electron Optics and Electron Microscopy; Taylor & Francis: New York, 1972. (34) Kieft, E.; Schliep, K. B.; Suri, P. K.; Flannigan, D. J. Effects of Thermionic-Gun Parameters on Operating Modes in Ultrafast Electron Microscopy. Struct. Dyn. 2015, 2, 051101. (35) Chatelain, R. P.; Morrison, V. R.; Godbout, C.; Siwick, B. J. Ultrafast Electron Diffraction with Radio-Frequency Compressed Electron Pulses. Appl. Phys. Lett. 2012, 101, 081901. (36) Gao, M.; Jean-Ruel, H.; Cooney, R. R.; Stampe, J.; de Jong, M.; Harb, M.; Sciaini, G.; Moriena, G.; Miller, R. J. D. Full Characterization of RF Compressed Femtosecond Electron Pulses using Ponderomotive Scattering. Opt. Express 2012, 20, 12048−12058. (37) Li, R. K.; Musumeci, P. Single-Shot MeV Transmission Electron Microscopy with Picosecond Temporal Resolution. Phys. Rev. Appl. 2014, 2, 024003. (38) Weathersby, S. P.; Brown, G.; Centurion, M.; Chase, T. F.; Coffee, R.; Corbett, J.; Eichner, J. P.; Frisch, J. C.; Fry, A. R.; Gühr, M.; et al. Mega-Electron-Volt Ultrafast Electron Diffraction at SLAC National Accelerator Laboratory. Rev. Sci. Instrum. 2015, 86, 073702. (39) Gliserin, A.; Walbran, M.; Krausz, F.; Baum, P. Sub-PhononPeriod Compression of Electron Pulses for Atomic Diffraction. Nat. Commun. 2015, 6, 8723. (40) Chatelain, R. P.; Morrison, V.; Godbout, C.; van der Geer, B.; de Loos, M.; Siwick, B. J. Space-Charge Effects in Ultrafast Electron Diffraction Patterns from Single Crystals. Ultramicroscopy 2012, 116, 86−94. (41) Cooley, J. W.; Tukey, J. W. An Algorithm for the Machine Calculation of Complex Fourier Series. Math. Comput. 1965, 19, 297− 301.

3546


Discrete Chromatic Aberrations Arising from Photoinduced Electron

Recommend Documents