Attosecond streaking metrology with isolated nanotargets

The development of attosecond metrology has enabled time-resolved studies on atoms, molecules, and (nanostructured) solids. Despite a wealth of theoretical work, attosecond experiments on isolated nanotargets, such as nanoparticles, clusters, and droplets have been lacking. Only recently, attosecond streaking metrology could be extended to isolated silica nanospheres, enabling real-time measurements of the inelastic scattering time in dielectric materials. Here, we revisit these experiments and describe the single-shot analysis of velocity-map images, which permits to evaluate the recorded number of electrons. Modeling of the recorded electron histograms allows deriving the irradiated nanoparticle statistics. Theoretically, we analyze the influence of the nanoparticle size on the field-induced delay, which is one of the terms contributing to the measured streaking delay. The obtained new insight into attosecond streaking experiments on nanoparticles is expected to guide wider implementation of the approach on other types of nanoparticles, clusters, and droplets.


Introduction
Recent advances in laser technology and ultrashort pulse generation have permitted to record electron dynamics on timescales reaching into the attosecond domain [1,2]. Since the advent of attosecond physics [3,4], the field has come very far and several metrologies for the real-time observation of electron dynamics have been developed and demonstrated (for an overview see e.g. [5][6][7][8]). Among the most commonly used techniques is attosecond streaking spectroscopy [9][10][11].
Attosecond streaking spectroscopy has also been suggested as a suitable metrology for measuring collective electron dynamics in nanostructures [24][25][26][27][28][29][30]. The spatial homogeneity of the driving laser field is one of the key aspects in attosecond streaking of atoms [2,13,15]. Nanosystems, in contrast, exhibit spatially inhomogeneous nearfields, which strongly vary in amplitude and phase from the exciting laser field. The local fields can be enhanced by orders of magnitude and their spatial properties generally depend on material properties, shape, and environment [31,32].
The application of attosecond streaking spectroscopy to nanosystems was introduced by Stockman et al [24] for metallic nanostructures, where localized or propagating surface plasmons can be excited, exhibiting such large local oscillator strength that they can dominate the acceleration of released electrons. In these cases, the energy of the external, exciting laser field can be localized to nanometer scales, below the diffraction limit of the exciting light [33]. Since the first theoretical proposal in 2007 [24], much progress has been made in measuring the near-field driven photoemission from metallic nanotips with streaking spectroscopy on femtosecond [34,35] and attosecond timescales [36,37], and in advancing attosecond photoemission electron microscopy [38][39][40]. Despite this progress and a wealth of theoretical studies [25,[27][28][29][30], the translation of attosecond streaking spectroscopy to isolated nanoparticles has posed a formidable challenge. Only very recently, the first experiment has been successfully implemented [41].
In the proof-of-principle experiment [41], the delayed photoemission from dielectric nanoparticles was measured and served as a stopwatch for electron scattering. It is noteworthy that such measurements could not be implemented on stationary dielectric solids owing to the accumulative charging from the XUV-driven electron emission. In the experiments on nanoparticles, a continuous stream of aerosolized nanospheres circumvented this problem, enabling the chronoscopy of inelastic scattering in dielectrics [41].
Here, we detail the conditions and technological advances that have enabled the implementation of attosecond streaking spectroscopy on isolated nanoparticles. We extend the analysis of the data shown in [41] to obtain single-shot electron statistics, which in turn provides information on the laser-irradiated particle statistics within the laser interaction. Furthermore, we extend our modeling of the field-induced delay contribution to the measured streaking delay as a function of nanoparticle size. The new information paves the way towards related experiments on other material classes and other nanoscopic targets such as clusters and droplets.

Attosecond setup
The attosecond setup is especially designed to achieve sufficiently high photon flux to perform time-resolved spectroscopy of low-density targets such as isolated nanoparticles. The setup is driven by NIR, sub 4 fs, 2.5 mJ pulses obtained by hollow-core fiber compression of 25 fs, 6 mJ, 780 nm pulses from a commercial Ti:sapphire laser system (Femtopower V Pro CEP, Spectra-Physics). To achieve maximum coupling efficiency, the hollow-core fiber compressor is operated in pressure-gradient configuration. The residual single-shot carrier-envelope phase (CEP) fluctuation of the driving pulses is ∼200 mrad (rms) [42]. A schematic of the attosecond setup is shown is figure 1. A portion of the NIR beam (70%) is focused by a 1 m radius-of-curvature spherical mirror into a static gas cell filled with a noble gas to produce XUV radiation by high-order harmonic generation. A continuous emission spectrum corresponding to isolated attosecond pulses is achieved with the polarization gating technique [43]. A pair of fused silica wedges is used to properly adjust dispersion and CEP of the NIR driving pulses. The fundamental radiation and the energy region of the spectrum below 16 eV are filtered out by a 100 nm thick aluminum filter. The remaining portion of the NIR beam (30%) is properly delayed with attosecond resolution by using a piezo-stage and then collinearly recombined with the XUV beam by using a drilled mirror with a 3 mm diameter central hole in an interferometric configuration. The interferometer is actively stabilized using a frequency stabilized He-Ne laser.
In a typical time-resolved measurement, the pulse delay (Δt) therefore measures the delay of the NIR pulse with respect to the XUV pulse. An additional pair of fused silica wedges is used to properly adjust dispersion and CEP of the NIR probe pulses. A gold-coated toroidal mirror ( f=90 cm) is used to focus both the XUV and NIR pulses into a beam of nanoparticles and the resulting electron photoemission is detected with a single-shot velocity-map imaging (VMI) spectrometer (see section 2.2) [44,45]. The toroidal mirror is operated in Rowland configuration with unity magnification and provides an almost aberration-free image of the XUV source, with a negligible temporal smearing of the attosecond pulses. The spectral characterization of the XUV radiation is simultaneously achieved by using a high-resolution flat-field soft x-ray spectrometer consisting of a second gold-coated toroidal mirror, followed by a grating, a microchannel plate (MCP), a phosphor screen, and a charge-coupled device camera [46]. The angle of incidence of the toroidal mirror is 86°and two gratings (Hitachi 001-0639 and 001-0640) are used to cover the energy range 12-100 eV. As can be seen from figure 2, completely tunable XUV emission covering the energy region between 16 and 45 eV can be achieved by using different generation gases (xenon, krypton or argon). The XUV photon flux has been measured on target by using a National Institute of Standard and Technology calibrated open photo-diode. The current from the photodiode, proportional to the number of incidence photons, is read by a Keithley amperemeter. When generating XUV pulses in xenon, their energy in the case of continuous spectra is ∼4 nJ, measured after the 100 nm thick aluminium filter. Such an energy corresponds to ∼8×10 8 photons/pulse (or ∼8×10 11 photons s −1 for a repetition rate of 1 kHz of the laser system).
To temporally characterize the XUV and NIR pulses, the attosecond beamline is also equipped with a time-of-flight spectrometer to perform attosecond streaking in argon gas. For reconstruction of the laser fields, the frequency-resolvedoptical-gating for complete-reconstruction-of-attosecondbursts (FROG-CRAB) technique is used [47]. Measured and reconstructed streaking traces are presented in figures 3(a) and (b), respectively. The retrieved temporal intensity profile and phase of the XUV pulses are shown in figure 3(c). A pulse duration of 250 as is typically achieved using the described setup.

Single-shot VMI
The setup for single-shot attosecond VMI with nanoparticles is schematically shown in figure 4. The delay-controlled XUV and NIR pulses propagate collinearly in positive direction along the x-axis, and are polarized along the y-axis. The nanoparticle stream is injected into the interaction region by an aerodynamic lens along the y-axis. The homogeneous static electric fields between the electrodes accelerate photoelectrons towards the MCP/phosphor screen detector with an active diameter of 80 mm (F2226-24PX, Hamamatsu). The supply voltages of the detector were gated by a fast highvoltage switch with a gate width of 300 ns (HTS-41-03 GSM, Behlke) to reduce background contributions. The VMI on the phosphor screen were recorded by a high-speed digital complementary metal-oxide-semiconductor (CMOS) camera. The camera was operated at the laser repetition rate (1 kHz) with a resolution of 800×600 pixels, and the exposure time was 0.1 ms. For each image, the camera software (Marathon Pro, GS Vitec) applied a flat-field correction, and up to 1024 pixels with a brightness above a defined threshold were recorded (see [48] for details). This approach significantly reduced the amount of data and enabled storage of single-shot images at the full repetition rate. The camera and the fast high voltage switch are both synchronized to the laser with suitably delayed TTL-trigger signals.

Aerosol generation
In the studies described here, the aerosol generation system consisted of an evaporator, silica drying stage, impactor, pressure equalizer (figure 5), and the aerodynamic lens with subsequent differential pumping. The aerosol is evaporated from a nanoparticle suspension by a commercial aerosol generator (model 3076, TSI). The carrier gas is neon, and the evaporation pressure is 1.75 bar. The nanoparticle density in the aerosol is determined by the nanoparticle concentration in the suspension and the evaporation gas pressure. The aerosol is sent through the diffusion dryer (model 3062, TSI) to remove residual solvent from the aerosol. The impactor consists of a sharp 90°turn, and blocks nanoparticle clusters. A pressure equalizer (HEPA Capsule Filter 1602051, TSI) is installed in front of the aerodynamic lens system to level the input pressure to one atmosphere. An aerodynamic lens system typically consists of a flow control orifice, a series of focusing lenses and an acceleration Continuous XUV spectra generated in xenon (red curve), krypton (green curve) and argon (blue curve). The spectra have been acquired by using a high-resolution flat-field soft x-ray spectrometer. Full tunability between 16 and 45 eV is achieved. nozzle. We used the design reported in [49,50], which has also been successfully employed in photoemission studies from isolated nanoparticles in intense few-cycle fields [51][52][53][54]. The initial orifice determines the gas flow through the lens system and reduces pressure from atmosphere to the value required for aerodynamic focusing. In the current setup an orifice of 130 μm is installed. The gas flow is converged and diverged through the focusing lenses which include five apertures (schematically shown in figure 4). With given particle size and gas flow properties, the nanoparticles can efficiently be separated from the carrier gas due to their inertia [55,56]. The nanoparticles are focused into a tight particle beam after the acceleration nozzle, while the carrier gas is injected effusively, and therefore efficiently removed after some distance from the last aperture of the aerodynamic lens. Three stages of differential pumping system behind the lens system maintain the pressure in the experimental chamber below 2×10 −7 mbar with operating nanoparticle source.
The nanoparticle density in the interaction region is limited by multiple factors: (i) the nanoparticle density in the dispersion, (ii) the aerosol generation step where evaporation conditions have to be met to avoid cluster formation, (iii) the aerosol transport system including liquid reflow, drying stage(s), and pressure equalization, (iv) impactor(s) to reduce the amount of clusters in the beam, (v) restrictions of the aerodynamic lens system, including aperture sizes and nanoparticle-size dependent throughput, and finally (vi) the distance from the last aperture to the interaction region. The use of nanoparticle dispersions with small size distributions (below 10%) makes it unnecessary to use differential mobility analysis, increasing the nanoparticle density in the interaction region. In experiments on SiO 2 , the 50 nm diameter particles (c) Reconstructed temporal intensity profile of the XUV pulse. The retrieved pulse duration was 250±20 as full-width-at-half-maximum with a residual parabolic phase indicating the presence of small second-order dispersion. The second-order dispersion value was determined in situ in the main experiments from residual gas data. A beam of isolated nanoparticles is illuminated with few-cycle NIR and XUV pulses along the y-axis. The inset shows the transmission electron micrograph of the SiO 2 nanoparticles with a diameter of 50 nm. The XUV and NIR pulses propagate in positive direction along the x-axis, and are polarized along the y-axis. The electron emission is projected with a static electric field onto the MCP/phosphor screen assembly of the VMI spectrometer and each shot is detected with a fast CMOS-camera outside of the vacuum chamber (not shown).
were prepared by wet chemistry methods based on the Stöber procedure and a seeded growth process [57]. The inset of figure 4 shows the transmission electron micrograph of the SiO 2 nanoparticles with a diameter of 50 nm. The polydispersity of the particles was about 8%.

Single-shot data discrimination
A distinct advantage of the single shot detection is the ability to record attosecond photoemission from aerosolized nanotargets and residual gas at the very same experimental conditions. The gas thereby provides a temporal ruler, which permits relative attosecond streaking delay measurements, even under conditions, where the resulting electron momenta overlap. The single-shot data discrimination is related to recent coincidence experiments, where the attosecond streaking from two different gases could be simultaneously measured and distinguished in the data analysis [14]. Without such approaches, either the emission bands have to be spectrally well separated (see e.g. [13,58]), or measurements have to be taken consecutively, which is, however, difficult due to typically limited long-term interferometer stability. The latter would pose a severe problem for measurements on dilute targets, where acquisition times are long (typically in the range of 45-60 min at 1 kHz for one streaking measurement on nanoparticles).
Due to the magnification setting of the imaging lens, a single electron hit on the MCP/phosphor assembly can illuminate several pixels on the CMOS chip. For sufficiently low density of hits on the detector per laser shot, a centroiding algorithm can be used to uncover the number of electrons recorded in single-shot images [45]. Examples for post-processed single-shot images of the 2D electron momenta (x and y components), as obtained from the residual gas only and for SiO 2 nanoparticles, are shown in figures 6(a) and (b), respectively. Note that this analysis differs from the analysis shown in [41], where the centroiding method was not applied.
The laser pulses propagated along the p x axis, and were polarized along the p y axis. The number of electrons emitted per laser shot from nanoparticles (figure 6(b)) was typically much larger than from the residual gas ( figure 6(a)). For the residual gas frames, single ionization by the XUV is expected to dominate, and the detected electrons therefore originate predominantly from different gas atoms or molecules present in the interaction region. For nanoparticles hit by the XUV, typically more electrons are emitted and show an asymmetry along the laser propagation direction, as depicted in figure 6(b). The asymmetry manifests as more electrons appearing on the left side of the image, corresponding to the incident side along the laser propagation axis. This effect is  known as shadowing [29,50,59], and originates from the asymmetric absorption of the attosecond XUV light in the nanoparticle.
In figure 7(b), we plot the histogram of detected electrons per shot for nanoparticles (red line), compared to just residual gas (blue line). Both measurements were performed under identical conditions (laser intensity and experimental chamber pressure (<2×10 −7 mbar)), by using the evaporator with just ethanol and with a nanoparticle-ethanol dispersion. Without nanoparticles, the number of electrons per shot has a sharp upper cutoff N 50.
cutoff Gas = For the nanoparticle dispersion, the maximum number of electrons per shot goes up to N 270, cutoff SiO 2 = and 15% of the shots recorded more than 50 electrons, which clearly arise from emission from SiO 2 particles.
For the typically employed low nanoparticle densitities, on average, less than one nanoparticle is present in the central, high density region of the laser focus. Therefore, even when using the nanoparticle dispersion, the frames contain a signal from residual injection gas and possibly one or a few nanoparticles. In order to quantify the effects of the nanoparticle density, nanoparticle beam parameters and the focal laser intensity profile on the single-shot electron number distribution, we introduce a simplified hit statistics model that is based on the schematic setup shown in figure 7(a). The number density of SiO 2 nanoparticles in the beam is modeled by a Gaussian distribution the beam width at position x on the optical axis, =´. We now assume that every photon falling into the geometrical cross section σ geo = πR 2 of a nanoparticle with radius R = 25 nm is absorbed and generates a photoelectron. This assumption is well justified considering the corresponding attenuation length for the XUV radiation. Considering that each photoelectron is detected with probability η, we can introduce a reference fluence that specifies the fluence needed to generate on average one detected electron per nanoparticle. As a result, the absolute fluence F can conveniently be expressed as a relative fluence f F F single = / that specifies the on average expected measurable electron number per nanoparticle.
In the next step, a simplified rate equation model is employed to describe the probability p q for measuring q electrons at a given relative fluence. Departing from the initial condition p q = δ q,0 for a vanishing fluence, the corresponding probability distribution p q ( f ) follows from integrating the coupled rate equations for measuring a specific electron number q from a nanoparticle. Note that the distribution P q so far reflects only the result for a single nanoparticle and is normalized automatically to P 1 The probability to hit a selected specific nanoparticle in the control volume is P P q q hit 1 å = = ¥ . Based on this individual hit probability and using a known number of nanoparticles N V in the control volume, we can introduce the multihit probability to simultaneously hit i nanoparticles as The resulting multihit probabilities for the given parameters and for i = 0...3 are displayed in the inset of figure 7(d). Note that for the case of i = 0, only the residual gas signal is considered. Starting from the corresponding electron number distribution for residual gas P q (0) , which can be taken from the experiment, the electron number distribution for a given ) from this simulation and the final result is compared to the experiment in figure 7(b).
The simulated electron number distribution shows similar features to the experimental data, i.e. the pronounced residual gas peak and a plateau-like feature up to high electron numbers. However, the slope of the distribution in the plateau region is steeper in the experiment. Scaling the calculated results with an exponential damping function (dashed curve in figure 7(d)) can reproduce the experimental data. While the origin of such scaling effects in the experiment is unclear, we like to point out that the model has neglected contributions, such as shot to shot fluctuations of the XUV pulse energy, quenched electron emission resulting from Coulomb blockade effects or saturation in the electron detection.
In order to select the frames that contain photoemitted electrons from SiO 2 nanoparticles, we calculate the average momentum component in the propagation direction p p to quantify the asymmetry in the electron emission. Here, N is the number of electrons per shot, and p x i , is the x-component of the momentum of the i-th detected electron as illustrated in figure 6(a). The single frame distributions as a function of p x á ñ and the number of electrons per frame are presented in figure 8(a) for a measurement with nanoparticles injected in the interaction region. The black solid line indicates the weighted average of p x á ñ of the frames with different numbers of electrons. The frames below the blue line (36 electrons) contain less electrons symmetrically distributed, and are assigned to residual-gas-only frames. This can be verified by an independent residual-gas-only measurement with similar laser conditions, as shown in figure 8(b). The residual-gas-only frames contain less than 30 electrons and the average momentum distribution is symmetric with respect to the laser propagation direction.
Due to the limited number of electrons in each frame, the average projected momentum, p x á ñ of the residual-gas-only frames shows a relatively wide distribution. A significant number of frames above the red line (70 electrons) contain larger amount of electrons and an asymmetric distribution due to the photoemission from nanoparticles (see also figure 6(b)). The frames with hit numbers in between the blue and red lines barely contain electrons from nanoparticles and are disregarded. The discussed features of the nanoparticle frames, i.e. the combination of the number of electrons and an asymmetric distribution, provide an efficient discrimination method for the post-selection of residual-gas-only and nanoparticle frames. The shot-averaged images following this approach are shown in figures 8(c) and (d). The residual gas momentum distribution is symmetric with respect to the laser propagation direction ( figure 8(d)), while the nanoparticle emission clearly has an asymmetric distribution due to the XUV shadowing, see figure 8(c).

Analysis of nanoparticle streaking spectrograms
By measuring momentum images for different pulse delays of the NIR with respect to the XUV pulse, streaking spectrograms can be obtained. In the measurements on 50 nm SiO 2 particles, the pulse delay step size was set to 150 as, and 40 000 laser shots were recorded at each delay step. The spectrograms depicted in figure 9 were derived from angular integration of the projected momentum distributions over an angular range of 25   around the laser polarization direction (see figure 8(c) and (d)). We note here that the VMI images were not inverted and the spectrogram was directly obtained from converting the projected radial momenta to energies. This needs to be taken into account when comparing the data to theoretical simulations. Figure 9 shows streaking spectrograms for (a) residual gas and (b) nanoparticles, obtained from the discriminated single-shot data. Both spectrograms look very similar and exhibit the characteristic oscillations with respect to the pulse delays. Two contour lines for a selected asymptotic energy (25 eV) are shown as solid cyan lines.
To determine relative streaking delays between the nanoparticle and residual-gas data, we fit contour lines in the streaking spectrograms. Furthermore, each contour line was filtered to remove high frequency noise (circles in figures 10(a) and (b)) by employing a low-pass filtering. The filtered contour lines were then fit with a few-cycle waveform of the following form: where E is the asymptotic energy of the considered contour line, A is the streaking amplitude, w is the carrier angular frequency, t d is the streaking delay, t 0 is the center of the Gaussian envelope, and t is the pulse width. The solid lines in figures 10(a) and (b) show the fitting results corresponding to the filtered contour lines in figures 9(a) and (b), respectively. Both curves provide a very good representation of the data and reveal a relative time shift between the curves from the residual gas and SiO 2 nanoparticle. Meanwhile, an effective electric field amplitude of the streaking NIR laser pulse can be reconstructed from the fitted curve for the residual gas. The maximum electric field is E 2.6 10 V m , =´corresponding to an instantaneous intensity of 9.0 10 W cm . 11

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The energy-dependent temporal streaking delays t gas d for the residual gas and t SiO 2 d for SiO 2 nanoparticles are shown with blue and red dots in figure 10(c). Both of the curves exhibit a characteristic tilt originating from the chirp of the XUV pulses, which can be fully taken into account in theoretical simulations (for details, see [41]).
In the single-shot attosecond streaking experiments, the simultaneous measurement of the residual gas and nanoparticles within the same experiment offers retrieving the energy-dependent relative streaking delay t t t , rel SiO gas which reveals a photoemission time shift between the gas reference and nanoparticles.
In the data discrimination, we accounted for the fact that the recorded events in nanoparticle frames also contain contributions from the residual gas. In order to calculate the time delay of the photoelectrons from the nanoparticles, it is crucial to uncover the ratio between these two contributions. For this purpose, we compared the averaged single-frame spectra from figures 8(d) and (c), which correspond to the residual gas and nanoparticle frames, respectively. Figure 11(a) shows the resulting spectrum from residual gas (blue line) and the spectrum obtained from the nanoparticle frames (red line). The black line shows the nanoparticle single frame spectrum after subtracting the gas contribution, reflecting the pure nanoparticle signal. The nanoparticle spectrum has a higher intensity and higher spectral cutoff than the residual gas data. The ratio between the nanoparticle and residual gas signal as function of energy is shown in figure 11(b). The contribution of photoelectrons from the SiO 2 nanoparticles increases above 20 eV, and dominates the signal beyond the cutoff energy of the residual gas. Limited by too strong an overlap between the residual gas and nanoparticle signals below 20 eV and the signal-to-noise ratio of the streaking spectra near the cutoff energies (see figure 9), the relative streaking delay between SiO 2 and the reference gas can be extracted in the window between 20 and 30 eV. We note that this particular energy window was accessible with the XUV pulses generated in Kr. Using other generation gases, the window can be tuned, see figure 2.

Discussion
To unravel the physics resulting in the delayed photoemission from the dielectric nanoparticles, semi-classical trajectory simulations based on the mean-field Mie Monte-Carlo (M 3 C) model were employed [53,54,60,61]. In these simulations, the propagation of the XUV and NIR pulses was evaluated using a spectral decomposition of the incident fields and calculating the spatial modes for the individual spectral components using the Mie-solutions for a sphere including dispersion. Photoelectron trajectories were launched in the sphere via Monte-Carlo sampling of the local instantaneous spectral photoionization rate (calculated from the local XUV near-field) and integrated in the NIR near-field using classical equations of motion. For trajectories inside the sphere, elastic electron-atom and inelastic electron-electron collisions were included as instantaneous, isotropic scattering events using energy-dependent mean-free paths and sampled with Monte-Carlo methods (for details see [41]). As ionization due to the XUV field was weak in the investigated scenario, charge interaction effects could be neglected.
From M 3 C simulations for the experimental parameters, we calculated streaking spectrograms and extracted energydependent streaking delays using the same projections and analysis as for the experimental data. The excellent agreement of simulations and experiment (compare curves and dots in figure 10(c)) motivated a systematic analysis of the contributions to the streaking delays. We found that in our scenario the streaking delay for the SiO 2 nanoparticles t t t t t  (experimentally unknown) absolute offset of the XUV/NIR delay axis as well as the CEP of the NIR pulse ( t offset d ). Second, an energy-dependent delay generated by the chirp of the attosecond pulse ( t chirp d ). Third, a delay induced by the retardation and spatial inhomogeneity of the local near-fields ( t , fields d details below) and fourth, a delay induced by the electron collisions inside the medium ( t coll d ). Since the first two contributions also define the delay of the reference measurement ( t t t is fully determined by the fieldand collision-induced contributions. The field contributions are relevant for nanoscopic targets such as the nanoparticles discussed here, but do not play a role in the photoemission from atoms. Note that additional contributions due to Wigner delays [62] and Coulomb-laser coupling [63][64][65] are assumed to be comparable for nanoparticles and gas as we evaluate similar electron energies and the longrange interactions are Coulombic in both cases. These contributions thus cancel in the relative delay. The remaining short-range effects are assumed negligible away from resonances [66].
In order to extract information on the electron scattering within the material the field-induced contribution to the relative streaking delay must be either negligible or known. In the following we describe how this contribution can be estimated for the example of 50 nm silica spheres. We split the field-induced delay t t t field retard. inhom.
into the parts corresponding to retardation of the XUV and NIR fields ( t retard. d ) and the part reflecting the effect of the spatial inhomogeneities of the NIR field ( t inhom. d ), i.e. its decay with increasing distance from the nanosphere.
The retardation effect can be estimated by comparing the evolution of the local near-fields at the sphere surface (in the z=0 plane, see figure 12(a)) with respective vacuum solutions. The XUV field enters the dynamics only via photoionization, which is determined by the pulse envelope. Therefore, we evaluate the group delay t group,XUV = t t ,   where the full waveform enters the dynamics, we estimate the retardation effect via the phase delay t t phase,NIR phase,NIR SiO 2 = t phase,NIR vac (red curve in figure 12(c)). The overall retardation induced streaking delay then reads t t retard. group,XUV d = t , phase,NIR where the XUV/NIR parts contribute with positive/ negative sign as the incident NIR-field needs to be delayed/ advanced to compensate for the respective retardation effect. We found that the retardation induced delay (black curve in figure 12(d)) is smaller than 40 as over the whole surface of the sphere. However, due to the shadowing of the XUV radiation, the dominant part of the detected photoelectrons is emitted from the front side of the sphere (see figure 10(b)). In particular, at the effective birth angle 101 x q »  á ñ (see dashed vertical line in figures 12(b)-(d)), the retardation induced delay is particularly small (<15 as).
To demonstrate that the delay due to field retardation directly enters the streaking delay, we employed the following simplified trajectory model. Trajectories start at the peak of the local XUV field at the surface with initial momenta pointing away from the sphere and are propagated in the NIR surface field, i.e. without including its decay into free space. In this case, the propagation is not influenced by collisions inside or by field inhomogeneities outside of the particle. Therefore, the resulting streaking delay includes only the retardation effect and resembles the delay extracted directly from the fields (compare black dots to black line in figure 12(d)). The additional impact of the NIR inhomogeneity can be extracted from the simplified trajectory model when sampling the full NIR field along the trajectories. In that case, the streaking delay includes the retardation and the inhomogeneity effects (cyan dots in figure 12(d)).
For a systematic comparison of the different fieldinduced contributions to the streaking delay, we calculated angle-averaged streaking delays in dependence of the nanosphere diameter, see figure 12(f). The mean birth angles for the investigated sphere diameters, obtained from M 3 C simulations, are shown in figure 12(e). The oscillation of the field retardation induced streaking delay as function of sphere diameter (black curve and dots in figure 12(f)) is attributed to the successive excitation of higher order modes of the NIR near-field. The offset of the streaking delay of around 10 as in the limit of small spheres is a pure dispersion induced feature of the XUV near field and not a propagation effect. The additional effect of the NIR inhomogeneity (compare green to black dots in figure 12(f)) is negligible for large spheres but strongly modifies the field induced streaking delay for small spheres. In particular, for d=50 nm the inhomogeneity effect cancels the retardation effect t 2.5 as d < ( ) such that field-induced delays are negligible ( t t fields coll d d  ), making the relative streaking delay to a direct measure of the collisional delay ( t t rel coll d d » ). To study the physics behind the collisional delay, we systematically analyzed the impact of different elastic and inelastic scattering times (average times between two scattering events, corresponding to their respective mean-free paths) on the calculated streaking delays, see figure 13(a)). We found that for a broad range of dielectric materials (sampled by scanning the relative permittivity at the NIR wavelength) the collisional streaking delay is almost entirely determined by the inelastic scattering time. Our analysis revealed that the additional effect of the elastic scattering time strongly depends on the materials permittivity. While for low permittivities ( 5 r e < ) larger elastic scattering times result in larger streaking delays the opposite trend is found for high permittivities ( 5 r e > ). Thus, the effect of the elastic scattering time cancels in the intermediate permittivity region of typical dielectric materials. This result shows that experimentally accessible relative streaking delays serve as direct clock for inelastic scattering times in SiO 2 . The approach for clocking inelastic scattering times should be transferable to other dielectrics.
Due to experimental and conceptual difficulties in measuring electron inelastic scattering times in solids directly (e.g. by the overlayer method [67]), obtaining information on inelastic electron scattering in solids relied to a large extend on model-based fits of optical transmission and absorption data [68] or simulations, limiting the accuracy of the predictions for electron energies below 50 eV. Figure 13(b)) shows a comparison of the inelastic scattering times obtained via attosecond streaking spectroscopy on SiO 2 nanoparticles in the energy window between 20 and 30 eV to literature values [69][70][71][72]. Note that best agreement is found with the work of Tanuma et al who state that their data in the range of 10-40 eV is unreliable and only serves as a trend [69]. With inelastic scattering times obtained via attosecond streaking on nanoparticles the predictive capability of theoretical models for scattering in dielectrics can now be thoroughly tested.

Conclusions
In conclusion, we have shown that attosecond streaking metrology is a powerful tool for real-time photoemission measurements on isolated nanotargets, and can reveal new insight into e.g. electron scattering. In cases where experiments on solids suffer from accumulative charging, nanoparticle beams provide a solution as a refreshable target. Most importantly, the collective and/or nonlinear dynamics in nanoparticles, such as e.g. localized plasmons [25,[27][28][29][30] and plasma waves [73], rapid inner and outer ionization [74], and electron density fluctuations created by, for example, shock waves [75] are interesting phenomena that await their exploration with sub-femtosecond precision. We expect the outlined approaches for the implementation of attosecond streaking in nanoparticles to contribute to further advances in these directions. acknowledge the computing time provided by the North-German super-computing center HLRN (project ID mvp00011).