Field propagation-induced directionality of carrier-envelope phase-controlled photoemission from nanospheres

Near-fields of non-resonantly laser-excited nanostructures enable strong localization of ultrashort light fields and have opened novel routes to fundamentally modify and control electronic strong-field processes. Harnessing spatiotemporally tunable near-fields for the steering of sub-cycle electron dynamics may enable ultrafast optoelectronic devices and unprecedented control in the generation of attosecond electron and photon pulses. Here we utilize unsupported sub-wavelength dielectric nanospheres to generate near-fields with adjustable structure and study the resulting strong-field dynamics via photoelectron imaging. We demonstrate field propagation-induced tunability of the emission direction of fast recollision electrons up to a regime, where nonlinear charge interaction effects become dominant in the acceleration process. Our analysis supports that the timing of the recollision process remains controllable with attosecond resolution by the carrier-envelope phase, indicating the possibility to expand near-field-mediated control far into the realm of high-field phenomena.

N anostructures enable the concentration of laser light in highly localized near-fields with dimensions far below the incident wavelength 1 . Utilizing optical near-fields for the control of electron motion in nanostructures with attosecond resolution is a major prospect and challenge in ultrafast light-wave driven nanoelectronics 2 . Enhanced strong-field photoemission in near-fields has been demonstrated for metal nanotips [3][4][5][6][7] , dielectric nanospheres 8,9 and surface-assembled nanoantennas 10 . In analogy to atomic above-threshold ionization, electron backscattering dominates the high-energy electron emission if the electron quiver amplitude is small compared with the near-field extension into free space. Many-particle charge interaction effects can increase the electron cutoff energy far beyond the values expected from the linear field enhancement, as demonstrated in previous experiments on small nanospheres 8,9 . The coherent nature of the near-field driven acceleration has been revealed by the fact that photoelectron spectra depend on the laser's electric field waveform, controlled by the carrier-envelope phase (CEP) 4,5,8,9 .
Key prerequisite for near-field-mediated tailoring of the underlying electronic strong-field dynamics is knowledge about its dependence on and feedback to the spatiotemporal near-field evolution. A detailed exploration of near-field control up to high field intensities is offered by studying unsupported reproducible nanosystems in the gas phase. Theoretical studies of the electron acceleration from droplets support that propagation-induced near-field effects have a strong impact on the electron dynamics, including the directionality of the emission, and can be utilized for the generation of attosecond electron bunches up to relativistic intensities 11,12 . Control of photoelectron angular distributions has also been demonstrated for atoms 13 and chiral molecules 14 via polarization-shaped femtosecond laser pulses. In the latter cases, however, the directionality results from selection rules and the chirality of the electronic initial or/and continuum states, respectively, and can be described in dipole approximation. The physics is therefore fundamentally different from control of electronic motion via field propagation-induced near-fields as considered in the present work.
Here we employ isolated 50-550 nm SiO 2 nanospheres (Fig. 1a) and demonstrate the size-dependent effect of field propagation-induced near-field deformation on the directionality of the strong-field photoemission. We observe systematic directional tunability of the electron emission with respect to the propagation direction via the sphere size and find evidence for the persistence of robust attosecond control of the dominant surface backscattering process via the CEP. Our combined experimental and theoretical analysis shows that dynamical many-particle charge interaction results in substantial quenching of the electron emission and becomes dominant for the electron acceleration for the largest investigated sphere sizes. A systematic trajectory analysis based on semiclassical transport simulations enables for a clear discrimination of the impact of near-field enhancement, vectorial field properties and self-consistent collective electron dynamics on the strong-field photoemission process.

Results
Propagation-induced near field deformation. Exposing nanospheres to few-cycle pulses with known CEP allows the generation of well-defined near-fields, whose linear response structure is accurately described by the Mie solution 15 . For a given refractive index, the latter depends mainly on the dimensionless Mie size parameter r ¼ pd/l (where d is the sphere diameter and l is the wavelength) and resembles Rayleigh's quasi-static dipole solution 16 for small spheres (ro o1). Propagation-induced near-field deformation arises and becomes significant for r\1 because of substantial excitation of higher order multipole modes 15 , resulting in a gradual shift of the region of maximal field enhancement in propagation direction (Fig. 1b). Increasing the size parameter to the range rc1 leads first to nanojet-type focusing employed in superlenses 17,18 followed by the regime of geometric optics 19 . Such systematic modifications can be achieved by changing the excitation wavelength or the sphere size. We varied the sphere size to realize scale parameters between rE0.2 and 2.4, ensuring peak field enhancement to occur at the surface; the employed widebandgap dielectric material ensures minimal pulse broadening such that the few-cycle character of the near-field is preserved. Nevertheless, field propagation induces a nontrivial elliptic local field (see Fig. 1d,e and the Methods).
Size-and carrier-envelope phase-dependent directionality. We measured the angle-resolved photoemission from SiO 2 nanospheres (dE50 À 550 nm, cf. Fig. 1a) via velocity-map imaging (VMI) using a setup similar to Zherebtsov et al. 9 . The photoelectron dynamics was controlled by the CEP, j CE , of 4 fs few-cycle laser fields at 720 nm central wavelength (see Methods). Near-field induced symmetry breaking of the photoemission for r\1 is revealed by the asymmetry of CEP-averaged photoelectron momentum projections with respect to the laser propagation direction (left-to-right in Fig. 1f). Below the 10 U p backscattering cutoff (see shaded areas in Fig. 1f,g), the momentum maps may contain spurious photoemission signal from residual background gas. The CEP-dependent signal from nanospheres shows that the electron emission can be effectively switched upwards or downwards while the left-right asymmetry remains (Fig. 1g). From the modulus of the projected momentum up to which phase-dependent VMI signal is observed we determined the global cutoff momentump c (solid circles) to define the cutoff energy E c ¼p 2 c =2m, where m is the electron mass. This cutoff is attributed to surface backscattering, see Fig. 1c.
Most importantly, we observed a size-dependent directionality of the phase-controlled photoemission. Maps of the high-energy electron yield show strong phase dynamics and signal concentration in a narrow angular range for the upward and downward direction for all investigated sphere diameters, see examples in Fig. 2a,b. From each map we extracted the critical final emission angle, y crit , and critical CEP, j crit CE , to characterize the directionality and CEP dependence (see circular symbols). Comparison of results for 95 and 550 nm spheres reveals similar phase dynamics but a significant shift of the critical emission directions from nearly 90°to almost 45°.
Simple man's model. To explore the impact of the linear near-field we describe the strong-field photoemission dynamics classically 20 with the simple man's model (SMM). Electron trajectories launched at rest at the nanosphere surface are integrated under the local field obtained from Mie's solution and assuming elastic specular reflection at the surface. The resulting cutoff momenta as function of the asymptotic emission angle, p c (y), indicate maximal energies for single recollision trajectories (n ¼ 1) and substantially lower energies for direct (n ¼ 0) and higher order (n41) recollision electrons (Fig. 3). The critical birth angle of most energetic electrons (n ¼ 1, small circle) coincides with the angle of peak radial near-field enhancement and deviates only weakly from the critical final angle (large circle), underlining the dominance of the radial field in the recollision-based process. Comparison with experiment shows that the SMM reasonably explains the critical emission angles but substantially underestimates the observed global cutoff momenta (blue filled circle).
Self-consistent simulation model. For a realistic description we employed quasi-classical Mean-field Mie Monte-Carlo (M 3 C) simulations (see Methods) and compared the results with the measured data (Fig. 2). Both the CEP-dependent switching of the photoemission and the size-dependent critical emission angles are well reproduced. M 3 C energy spectra (Fig. 4a) show that single recollision electrons are dominant over direct and higher order recollisions for all energies and the cutoff energy by far exceeds the SMM prediction. For classifying M 3 C trajectories, we counted each electron re-entry as one recollision event even if several microscopic collisions are involved. Both the quenching of direct electrons (because of the local trapping potential 8 ) and the enhanced acceleration reflect the importance of charge interaction, which is analysed in more detail below. The resulting global M 3 C cutoff momentum (red filled circle in Fig. 3) is close to the experimental result regarding both magnitude and direction, indicating that all major effects contributing to the electron acceleration have been captured by the model.
Attosecond recollision dynamics. To characterize the attosecond dynamics, we analysed CEP-dependent M 3 C data for emission into the upper half space. Energy spectra for single recollision electrons (Fig. 4b) visualize the cutoff energy modulation underlying the switching effect. By timing analysis (Fig. 4e) we find that, irrespective of CEP, essentially all fast electrons are born by tunnelling within a narrow time interval (o300 as) in a single The incident laser field E y ðtÞ ¼ E 0 f ðtÞ cosðot þ j CE Þ with 4 fs (intensity full-width at half-maximum) Gaussian envelope f(t) at centre wavelength l ¼ 720 nm propagates along the x axis. (c) Schematic illustration of the electron recollision process. (d) Vectorial representation of the field evolution in the x À y plane normalized to incident peak amplitude E 0 , sampled at the point with peak radial near-field enhancement (y ¼ 61.0°, CEP as indicated).
Coloured arrows indicate the local reference frame for radial (red) and tangential (blue) fields to illustrate the evolution of the field ellipticity. (e) Evolution of radial and tangential electric field components E r (blue) and E t (red). (f) CEP-averaged VMI electron momentum projection (momenta in atomic units) measured for d ¼ 400 nm at 2.7 Â 10 13 Wcm À 2 . (g) Phase-resolved VMI images (CEP as indicated) after subtraction of the CEP-averaged spectra. Solid circles in (f) and (g) indicate the extracted cutoff momentum and dashed circles the uncertainty estimated from the deviation of the results for the upper and lower half of the momentum distribution. The shaded circular areas in (f, g) indicate the momentum range that corresponds to electron energies below the 10 U p classical rescattering cutoff and may contain residual signal from background gas.
dominant half-cycle. The nonlinear near-field contribution (see short-term evolution in Fig. 4d) reveals strong dynamics and has a significant effect on the local radial field (Fig. 4c). In particular, the resulting trapping field quenches tunnel ionization, as discussed in more detail later. However, although the ionization dynamics and energetics are substantially changed in the M 3 C treatment, the excursion length ( Fig. 4d) and average timing ( Fig. 4e) of recollision trajectories remain similar to the SMM prediction, substantiating the applicability of the recollision picture. Altogether, the half-cycle selectivity, the smooth systematic shift of birth and recollision times with CEP, and the resulting pronounced high-energy signal modulation give evidence for robust attosecond control.
Systematic comparison of experiment and theory. The sizedependent impact of propagation-induced near-field deformation on the electron emission is analysed in Fig. 5. First, the evolution of measured critical angles (Fig. 5a) agrees well with both the SMM and M 3 C simulations, substantiating a systematically tunable directionality; the remaining offset between experiment and theory is attributed to a systematic error (for example, inhomogeneous VMI detector response). Second, the measured weakly size-dependent critical phase (Fig. 5b) shows a notable offset from the SMM result but is well described by M 3 C theory, supporting persistence of robust attosecond control and a proper description of the electron dynamics up to strongly deformed near-fields. Third, the measured cutoff energies are reasonably captured by the full M 3 C simulations but exceed the predictions  Predicted electron cutoff momenta as function of emission angle, p c (y f ), for n ¼ 0, 1, 2 recollisions at optimal CEP for upward emission; p 0 ¼ eE 0 =mo is the free space quiver momentum. Small/large circular symbols denote the critical birth/emission angles associated with respective highest cutoff momenta (see inset for corresponding radial trajectories); blue/red filled circles show critical emission angles and cutoff momenta from experiment and M 3 C simulations; shaded rings are guides to the eye.
of SMM and M 3 C theory with charge interaction switched off by up to a factor of two (Fig. 5c).
Selective analysis of acceleration mechanisms. Finally, we disentangle and quantify the different many-particle contributions to the acceleration process using a selective energy gain analysis, see Methods for technical details. The resulting size-dependent analysis is depicted in Fig. 5c. The relative cutoff energy enhancement because of the local trapping potential is only weakly size-dependent and results mainly from the radial electron motion (compare black and blue lines). This behaviour supports that the trapping potential is only determined by the local electron density (surface charge density) and thus insensitive to the system size. The tangential field effect is notable only for large spheres (dashed versus solid blue line) but remains small even if tangential and radial field amplitudes become comparably strong, for example at d ¼ 500 nm (see Methods). This supports that the tangential field is neither crucial for the emission angles nor for the acceleration process. The relative enhancement due to the space-charge repulsion increases strongly with size and becomes dominant for large spheres. This trend can be explained with stronger Coulomb repulsion due to an increasing number of electrons in the escaping bunch, being an effect that is sensitive to the full (non-local) electron distribution. The time-resolved analysis of the energy gain contributions in full M 3 C simulations shows the following dynamics, see examples in Fig. 5d,e. Compared with the SMM results (solid black curves), the enhancement of the gain associated with the radial Mie field (dotted blue curves) develops shortly after the moment of birth within the recollision process. The fact that no substantial change of the relative energy is found in later stages of the pulse supports that the impact of the trapping potential unfolds close to the surface. The gains calculated from the full Mie field (dashed blue curves) show that the tangential field effect also develops on short time-scales, that is, during the pulse. Finally, the energy gains with the full M 3 C field (red curves) reveal that the additional acceleration due to Coulomb explosion develops on a substantially longer timescale as it results solely from electron repulsion within the emitted bunch.
Mean-field-induced quenching of the electron emission. In the investigated range of laser intensities and particle sizes we find a strong impact of the Coulomb field on the electron emission. The yield predicted by simulations without the Coulomb interaction scales roughly exponentially with sphere diameter and laser intensity, reflecting the highly nonlinear tunnelling rate. Inclusion of Coulomb effects decreases the yield by up to two orders of magnitude (red versus black dashed curves in the insets of Fig. 6a,b). The reduction is a direct consequence of the trapping field, which quenches tunnel ionization at the surface and limits the number of electrons that can escape with a given initial kinetic energy. This pivotal influence of the Coulomb field on the actual ionization dynamics also explains why the M 3 C results are only weakly affected from changes of the (certainly approximate) tunnelling rate (for example, by slight variations of the effective ionization potential) as soon as a substantial trapping field fields (blue lines) show the enhancement due to the local trapping potential. The difference between gains for radial and full Mie fields (dotted versus dashed blue line) quantifies the tangential field contribution. Space-charge repulsion is included when using the full self-consistent near-field for the energy gain integration, recovering the cutoff energy of the full M 3 C runs. For details see Methods. Note that because of low overall count rate for d ¼ 550 nm particles (low aerodynamic lens transmission) we might underestimate the cutoff energy for this size. The slope of the critical SMM phase in (b) mainly reflects the small trivial CEP shift, that is, the propagation induced CEP offset of the effective radial field relative to the incoming pulse. Horizontal error bars in (a-c) indicate the standard deviation of the particle diameter. Vertical error bars in (a) and (b) are composed by the standard deviations of upper and lower detector sides combined with systematic errors estimated to be 4°for y crit and 0.1 p for j crit CE , respectively. The error for E c in (c) is dominated by the uncertainty associated with laser intensity which is approximated to be ± 15%. Uncertainties (t5%) in the refractive index and work function because of surface contamination modify the simulation results by less than the experimental error bars. (d,e) Time evolution of the energy gains for two selected sphere sizes (as indicated); note the different scaling of the time axes before and after the black vertical lines. The pulse peak (in free space and at x ¼ 0) is set to t ¼ 15 fs. specific to nanoparticles and free from background gas contributions, we counted only electrons with momenta beyond the thresholdp c = ffiffi ffi 2 p , which includes only signal well above the gas signal cutoff (see Fig. 1g). Further, focal averaging is circumvented by assigning most intense single-shot VMI images to the peak laser intensity. Because of the low particle density in the beam, such images reflect the emission from a single nanoparticle. Considering the estimated experimental electron detection efficiency of (30±20)%, the resulting measured nearcutoff electron yield as function of particle size and laser intensity is compatible with the corresponding M 3 C predictions with mean-field, see symbols and solid red lines in Fig. 6a,b. The remaining discrepancies for small particles and at low intensities are attributed to residual background signal. The agreement strongly supports the M 3 C prediction that the number of emitted electrons evolves nearly linearly with size and intensity in the presence of substantial Coulomb interaction.

Discussion
The present results suggest that the size-dependent directionality of the strong-field photoemission from nanospheres directly relates to the field propagation-induced near-field deformation. The analysis reveals the dominance of the radial field driven recollision dynamics up to large sizes with substantial near-field ellipticity and gives evidence for the persistence of attosecond control of the electron dynamics. As the near-field deformation could in principle be manipulated directly via the excitation wavelength, our findings indicate feasibility of near-field-induced photoemission with optically tailored directionality with respect to the beam propagation axis.
Furthermore, we identify the transition from local-fielddominated dynamics 8,9 to a regime where charge interaction effects due to the non-local structure of the emitted electron bunches become equally important or even dominant for the energetics of the electron acceleration. These results are of general relevance for strong-field electron dynamics in nanosystems (nanoparticles, -jets, -solids and -tips) and indicate the extension of near-field-mediated waveform control into the extremely nonlinear regime 21,22 . We anticipate that this enables steering of attosecond electron bunch emission from droplets 11,12 via phase control and near-field enhancement of surface high-harmonic generation 23 from nanostructured targets. Eventually, correlating the near-field driven electron dynamics with ion spectra 24 and imaging the resulting particle damage via single-particle X-ray scattering promises unprecedented insights into the poorly understood processes of sub-wavelength laser machining in dielectrics 25 .

Methods
Sample preparation. Silica nanoparticles with diameters between 50 and 550 nm and a narrow size distribution were prepared by wet chemistry approaches. All chemicals (ethanol (Berkel AHK ultrapure, 100%), tetraethoxysilane (TES, Fluka, purum, 98%), ammonia solution (Merck, p.a., 28-30%)) were used as received without further purification. The reaction flasks were cleaned by hydrofluoric acid (8 vol.%) and ultrapure water before use. First, small seed nanoparticles were prepared by the Stöber method 26 . In a typical seed preparation procedure 35.41 g of TES and 43 ml of ammonia solution were added to 1,000 ml of ethanol and stirred for 12 h. After cleaning the sample by centrifugation and redispersion in ethanol, a further shell was grown on the silica nanoparticles by the seeded growth methods 27,28 until the desired particle size between 50 and 550 nm was reached. In every step the dispersions were diluted to silica volume fractions of 0.5% and the ammonia and water concentrations were kept at 0.69 M NH 3 and 1.56 M H 2 O, respectively. TES (30 ml (0.134 mol)) was added for each step. All samples have been stored in ultrapure ethanol or an ethanol ultrapure water mixture (80:20) after cleaning. Characterization by transmission electron microscopy and scanning electron microscopy yielded a polydispersity of B15% for small particles around 50 nm and decreases substantially to 3% for the large particles under study. The surface of silica nanoparticles prepared by the Stöber method are typically covered by silanols, that is, Si-OH groups 28 . Depending on the pH value of the surrounding environment the silanols can be protonated or deprotonated. At pH 7 the surface of silica nanoparticles is negatively charged and slightly hydrophilic 29 .
Experimental approach. The few-cycle laser pulses were generated by spectral broadening of a Ti:Sa amplifier output (25 fs, 790 nm) using a Ne filled hollow-core fibre and chirped mirror compression. A fraction of the laser beam was split off and sent to a stereo atomic above-threshold ionization phasemeter 30 for single-shot CEP measurement. The remaining laser beam was intersected with the nanoparticle beam in the focus of the VMI apparatus 9,31 . The jet of isolated nanospheres was generated from a dispersion of SiO 2 particles in ethanol by where counted in the experiment. The horizontal error bar indicates the estimated ± 15% uncertainty of the intensity; vertical error bars for the yield are defined as in (a). We checked via the M 3 C data that the differences between results using projected and full momenta are insignificant for this analysis. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms8944 ARTICLE aerodynamic lensing 32 . This technique ensured that the sample was refreshed between consecutive laser shots. The hit rate of nanoparticles was well below unity, ensuring single-particle conditions. Emitted electrons were projected onto a microchannel plate/phosphor screen assembly and recorded by a camera capable of acquiring single images at the laser repetition rate of 1 kHz. The VMI data was synchronized with the phase information from the phasemeter. In addition, laser shots containing no nanoparticle signal could be split off during analysis based on the number of electron events on the detector. No (Abel) inversion of the obtained momentum distributions was applied as the requirement of axial symmetry around the laser polarization axis is not met for larger nanoparticles. The background electron signal (mostly N 2 ) of a nanoparticle scan served for laser parameter calibration (peak intensity, CEP offset) by comparison with separate scans in Xe.
Linear few-cycle near-fields of nanospheres. For the time-domain description of the near-fields at the SiO 2 nanospheres we employ Mie's continuous wave solution of the medium Maxwell equations combined with a spectral field decomposition. In linear response and spectral complex field representation, the spatiotemporal electric field evolution can be expressed as with E 0 the field peak amplitude, f(o) the normalized amplitude spectrum, Eðr; oÞ the spatial mode structure as function of angular frequency, o, and the CEP j CE . For a bandwidth-limited few-cycle laser pulse (polarized along y axis, propagating along x axis) with Gaussian temporal field envelope, the spectrum and mode structure read and E inc ðr;oÞ ¼ e y e ikx ; ð3Þ with Do ¼ ffiffiffiffiffiffiffiffiffiffi 2 ln 2 p =t fwhm , where t fwhm is the full-width at half-maximum of the temporal intensity envelope. The incident plane wave modes E inc ðr;oÞ provide the boundary conditions for the Helmholtz equation of a dielectric sphere and lead to the well-known Mie solutions where the summation over l runs over the multipole orders of the expansion into spherical vector wave functions 33 and d and e(o) describe the diameter and relative permittivity of the sphere. We evaluate the integral in equation (1) numerically by a discrete sum over 25 spectral components and by including multipole order up to l ¼ 5 in the Mie modes of equation (4). We checked that the dispersion of SiO 2 is sufficiently small to justify the use of a fixed relative permittivity e ¼ 2.12 sampled at the angular frequency o 0 corresponding to the laser central wavelength 34 .
Near-field structure. To characterize the size-dependent near-field we consider 4 fs pulses at 720 nm central wavelength and analyse the radial and tangential fields E r ðtÞ and E t ðtÞ in the z ¼ 0 plane at (outside) the nanosphere surface for y40, see white arc in Fig. 7a, where the field component along the z axis vanishes for symmetry reasons. In the relevant size range, highest field enhancement occurs on this arc under the characteristic angleỹðdÞ, see Fig. 7b. With increasing sphere size and under this characteristic angle, the radial and tangential peak amplitude show a modest and rapid increase, respectively (Fig. 7c). The resulting local fields become increasingly elliptic with size and exhibit a tilt of the major field axis with respect to the local surface normal (Fig. 7d). The pulse duration (intensity envelope) grows by o4% in the investigated size range and can thus be considered as essentially constant (Fig. 7e). Finally, the propagation effect introduces a small shift of the effective CEP of the radial field.
Simulation model. In the quasi-classical M 3 C model, electron trajectories are generated via Monte-Carlo sampling of the surface tunnel ionization. In each time step, a tunnelling path is determined for randomly chosen surface atoms using an effective ionization potential of 9 eV to describe the band gap of SiO 2 and following the local field direction. The ionization probability is determined from Ammosov-Delone-Krainov atomic tunnel ionization rates 35 using the field gradient averaged over the tunnelling path. Trajectories are launched at the classical tunnel exit and integrated classically in the local electric fields. The latter contain the timedependent Mie solution and the instantaneous, self-consistent mean-field. The mean-field describes the Coulomb interaction with free charges (electrons and residual ions) in the presence of the dielectric sphere via high-order multipole expansion up to multipole order l ¼ 10. In the mean-field solver, the sphere polarization is described by the same relative permittivity as in the Mie solver. Elastic electron-atom collisions are described by isotropic scattering events using an energy-dependent mean free path derived from quantum mechanical scattering cross-sections for the atomic potentials of Si and O atoms. Inelastic collisions are modelled with Lotz's electron impact ionization cross-sections 36 . Electron spectra for escaped electrons are calculated from trajectories with positive single-particle energy and positive normal velocity. A typical single high-resolution simulation run contains 2.5 Â 10 6 trajectories and takes B1 week on a single CPU core. The systematic scans over CEP and particle size performed in the current study required several tens of CPU years on the HLRN supercomputer, taking maximum advantage of massive parallel computation techniques. For the investigated laser intensities (It4 Â 10 13 W cm À 2 ) adiabatic metallization 37 , breakdown effects 38 and the nonlinear optical response of the dielectric sphere material 21 can be neglected.
Energy gain analysis. The contributions from the local trapping potential and space charge repulsion to the acceleration process are mediated by the self-consistent mean-field potential in the M 3 C simulations. To separate and quantify these effects we employ a selective energy gain analysis. Therefore, we consider the kinetic energy gain DEðtÞ ¼ R t tb _ rðt 0 Þ Á E½rðt 0 Þdt 0 resulting from different field contributions E. Considering the results of the SMM, evaluation of this integral for a given electron trajectory (and using only the Mie fields) is equal to the instantaneous kinetic energy. Likewise, integration over the full M 3 C trajectories and using the full near-field (Mie and dynamic mean-field) converges to the final asymptotic kinetic energy. For the analysis we select trajectories with final energy equal to the cutoff energy. To separate the effect of the local trapping potential, we use the full M 3 C trajectories but include only the Mie field in the energy gain integration. This includes the modification of the trajectories due to the presence of the mean-field, but excludes the acceleration mediated by its dynamical evolution, for example, due to Coulomb explosion of the escaping electron bunch.