Main

Measuring the energy of electrons or photons emitted from a sample under laser irradiation leads to electron or photon spectra. They contain tell-tale features such as multiphoton orders1,2, the plateau of high-harmonic generation and rescattering physics2,3,4,5,6, the featureless cut-off indicating the generation of individual attosecond pulses7 or the low-energy structure8,9, evidencing soft electron recollision mechanisms10,11,12. Discovering and understanding these qualitatively new spectral features facilitated new dimensions of insights, giving birth to strong field and attosecond science3 and often providing new control knobs to the motion of electron wave packets on suboptical-cycle timescales13,14,15.

Most of these spectral features have been measured not only with atoms and molecules in the gas phase but also at nanoparticles16,17 or nanometre sharp metal structures13,14,15,18,19,20,21,22,23,24,25,26. In contrast to atoms or molecules, nanometric objects like spheres or needle tips generate an optical near field when illuminated with light. This near field typically leads to a substantial field enhancement at the surface of the structure and, from there, decays away into the vacuum. Hence, a dramatic optical field inhomogeneity arises, making the optical near field a temporally and spatially sharply varying field. However, it is this near field in which electrons emitted from the nanostructure move. Thus, it has a pivotal influence on electron motion, which leads to, for example, the quenching of the electron’s quiver motion19,27,28. Resting on these localized and enhanced optical near fields, metal nanostructures are today the basis for initial attosecond-fast on-chip devices. They enable suboptical-cycle field sampling and light-driven current generation29,30,31,32,33,34,35 as well as ultrafast tunnelling microscopes36,37.

The intense optical near field leads to a ponderomotive potential and the fast spatial decay to a steep ponderomotive potential gradient, exerting a force on a photoemitted electron. We will show that a photoemitted electron experiences an additional drift velocity from the quickly (spatially) varying ponderomotive potential that continuously decreases the later the electron is born into the laser pulse. Especially electrons born early into the optical field always gain energy, even if they are born into the field at the crest of the optical field cycle. This is in contrast to most cases known from atomic physics in which these electrons would not show any drift momentum—here they do, resulting in a new minimum energy curve (MEC). Together with the temporal localization of the electron’s time of birth in the tunnelling regime and the few-cycle nature of the driving field, near-field-induced low-energy stripes (NILES) result in the direct part of the electron spectrum, as we will show below in detail. Interestingly, this goes far beyond the quenching of the quiver motion19, which also shows up, for example, for long driving pulses. NILES do not.

NILES

We trigger electrons from a sharp tungsten needle tip with 10-nm radius of curvature illuminated with 11.5-fs-long (2.2 cycles) laser pulses at 1,570 nm (Fig. 1a and Methods). The near field’s peak intensity of 2.0 × 1013 W cm−2 is chosen such that we are well in the strong-field regime (Keldysh γ ≈ 0.7)18. This is substantiated by the clearly visible hallmarks of recollision physics (cut-offs of direct and recollision electrons and a recollision plateau) in the measured electron energy spectra, shown for two selected carrier-envelope phases (CEPs; Fig. 1b). The energy of 30 eV resulting from the static bias voltage applied to the tip is subtracted.

Fig. 1: NILES.
Fig. 1: NILES.The alternative text for this image may have been generated using AI.
Full size image

a, Electrons are photoemitted by a two-cycle laser pulse and propagate in the steeply decaying ponderomotive potential resulting from the optical near field (shown in red in the zoomed-in view) at the apex of a sharp tungsten tip. With only about two emission events per laser pulse, this leads to NILES (see the main text). b, Measured electron energy spectra for two selected CEPs (as indicated) shown on a linear scale (orange, left axis) and a logarithmic scale (grey, right axis). c, CEP-resolved spectra. The high-energy electrons show the well-known CEP-dependent energy modulation in the plateau (~10–40 eV) and the cut-off (~40–50 eV). In the low-energy region between –2 eV and 5 eV, new spectral stripes appear prominently: NILES. d, Zoomed-in view of the low-energy region of direct electrons showing NILES on a linear yield scale. The large modulation depth of the integrated yield with CEP, reaching up to 33%, is noteworthy (Methods). Lineouts of the low-energy region are shown in Extended Data Fig. 2. The energy of 30 eV resulting from the static voltage of –30 V applied to the tip is subtracted from the energy axis in c and d.

Source data

Intriguingly, the CEP-resolved spectra in Fig. 1c reveal a pronounced CEP dependence over the entire spectrum, not only in the plateau and cut-off region but also for the direct electrons, that is, below 5 eV (shown on a linear scale in Fig. 1d). We observe stripes that extend over a CEP of ~1.4 × (2π), that is, over more than one period. This low-energy region showing NILES contains 51% of all the emitted electrons (range from –1 eV to 4 eV). When only one single peak is visible (CEP = 1.65 × (2π)), the energy width is ~2.2 eV (full-width at half-maximum (FWHM); Extended Data Fig. 2). The integrated yield in an energy window of ±0.4 eV around the maximum shows a modulation depth of 33% (Methods), that is, the total current modulation is about an order of magnitude higher than what is known from cut-off modulation-dominated works13.

The root cause for NILES is most straightforwardly understood when focusing on a comparison between electrons moving in a quickly decaying ponderomotive potential, like in front of a sharp nanostructure, and in a spatially homogeneous ponderomotive potential, such as when the electrons are emitted from individual atoms or molecules, or from a metallic structure with a large radius of curvature. Figure 2a shows the ponderomotive potential UP(x, t) of an intense, short and spatially homogeneous optical field as a function of position and time in one spatial dimension, that is, constant in space and with a Gaussian envelope in time (Methods). Because we are in the optical tunnelling emission regime, electrons are preferentially emitted at the crests of the negative half-cycles of the laser field. Hence, we first concentrate on trajectories of these electrons.

Fig. 2: Origin of NILES.
Fig. 2: Origin of NILES.The alternative text for this image may have been generated using AI.
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a,c, Position- and time-dependent ponderomotive potential UP(x, t) of a homogeneous optical field profile (a) and an optical near field with a decay length of ζ = 10 nm as in the experiment (c). The projections (blue curves) indicate the respective spatial enhancement profiles and temporal intensity envelopes. The spatial gradient of this potential dUP(x, t)/dx leads to a force acting on the emitted electrons (surface colour). The red curves show trajectories of electrons liberated at instants of vanishing vector potential in the dominant field half-cycles and resulting from the ponderomotive (cycle-averaged) force acting on the electrons. These trajectories are called drift trajectories, resulting in drift momentum. The grey curves, by contrast, show the corresponding trajectories obtained when the full field is taken into account. Importantly, they result in the same final momentum—the drift momentum. b,d, Projections of the trajectories in position and time, corresponding to a (b) and c (d), for homogeneous and inhomogeneous optical near fields. The lowest-possible drift momentum and the respective minimum final energy is realized for electrons liberated at vector potentials A(t) = 0 and becomes zero for a vanishing ponderomotive force, whereas additional ponderomotive acceleration results in non-vanishing minimum energies (compare the red dots in b and d). The blue curve connecting the red dots represents the minimum energy determined from full trajectories when varying the CEP. e, Zoomed-in view of d: for increasing CEP, the minimum energies shift continuously (as indicated by the arrows). This results in two almost-identical MECs for the full trajectories and drift trajectories (compare the blue and dashed red curves). Clearly, the drift trajectories already contain the full physics. Importantly, the separated stripes of NILES result from the fact that the red dots are sparse on this curve and, hence, display a large energy difference exceeding ~1.2 eV. The orange curve shows the analytical upper estimate for the MEC, as detailed in the main text.

Source data

For these electrons, we can describe the action of the laser field twofold. (1) If we consider the (cycle-resolved) optical field, the quiver motion of the electron eventually ceases and no net momentum is acquired because the vector potential \(A(t)=-\mathop{\int}\nolimits_{-\infty }^{t}E({t}^{{\prime} }){\rm{d}}{t}^{{\prime} }\) vanishes at the field crests. (2) In the cycle-averaged intensity picture, the electron is born into a ponderomotive potential UP(x, t) that is space independent in the case of a homogeneous field. As its spatial gradient is ∇UP(x, t) = 0, no force can act on the electron. The trajectories of both pictures are shown in Fig. 2a,b, one including the quiver motion (full trajectories, case (1), grey curves) and the other only including the drift motion in the ponderomotive potential picture (drift trajectories, case (2), red curves). Importantly, both pictures yield the same result: an electron emitted at the field crest exhibits no final kinetic energy. This repeats for every laser cycle (Fig. 2b, red dots).

The situation at the sharp metal tip (Fig. 2c,d) is entirely different because of the strong spatial decay of the optical near field. The field in front of the tip can be described by the position-dependent field enhancement factor \(\xi (x)=1+({\xi }_{0}-1)\times \exp (-x/\zeta )\), with the field enhancement factor ξ0 at the surface of the tip, the distance from the tip apex x and the decay length ζ, equalling the tip radius38,39. The optical near field acting on the emitted electron is, thus, ENF(x, t) = ξ(x) × Ein(t), with incident laser field Ein(t) (Methods). Consequently, the ponderomotive potential (\(\propto {\langle {E}_{{\rm{NF}}}^{2}(x,t)\rangle }_{t}\)) exhibits a large spatial gradient that photoemitted electrons can roll down (Fig. 2c). This leads to non-zero drift momenta despite the trajectories being launched at vanishing vector potentials40 and causes birth-time-dependent final energies (Fig. 2d, red dots). Increasing the CEP and thus decreasing the birth times leads to a continuous shift in these dots (Fig. 2e, arrows). This forms what we call the MEC (Fig. 2d,e, blue curves) because electrons emitted before or after the field crests gain more energy due to the non-vanishing vector potential. Most importantly, the MEC matches the behaviour extracted from drift trajectories almost exactly (compare the red dashed to blue curves in Fig. 2e), substantiating the dominance of the ponderomotive acceleration over higher-order effects of the quiver motion for NILES to appear.

In the following, we provide an intuitive picture behind the MEC, which reflects a temporal integration of the spatiotemporal ponderomotive force along the numerically propagated trajectories. An analytical upper limit can be obtained when considering only the temporal evolution of the ponderomotive force sampled at the tip surface. For electrons starting at rest at the crests of the optical field cycles, this upper limit for the MEC reads (Methods)

$${{\mathcal{E}}}_{\min .{\rm{final}}}({t}_{{\rm{b}}})=\frac{2}{m}{\left({U}_{{\rm{P}}}^{\;{\rm{inc}}}{\xi }_{0}{\xi }_{0}^{{\prime} }\right)}^{2}{{\mathcal{F}}}^{2}({t}_{{\rm{b}}}).$$
(1)

Here the electron energies are pivotally determined by the laser parameters (ponderomotive potential \({U}_{{\rm{P}}}^{{\rm{inc}}}\) of the incident field) as well as the magnitude ξ0 and gradient \({\xi }_{0}^{{\prime} }(={\rm{d}}\xi (x)/{\rm{d}}x{| }_{x = 0})\) of the near-field enhancement at the tip surface. The birth-time-dependent shape of the MEC is governed by the remaining normalized pulse fluence \({\mathcal{F}}({t}_{\rm{b}})=\frac{1}{{I}_{0}}\mathop{\int}\nolimits_{{t}_{\rm{b}}}^{\infty }I(t){\rm{d}}t\) associated with the intensity envelope I(t) and experienced by the electron after its birth. Importantly, for a Gaussian I(t), \({\mathcal{F}}({t}_{\rm{b}})\) decreases continuously with tb in the shape of a complementary error function. The resulting analytical MEC (Fig. 2e, orange curve) provides an upper limit as the ponderomotive force is the maximum at the surface. MECs extracted from the trajectories show the same shape but with a smaller magnitude, which is because electrons leave the vicinity of the tip quickly for the experimental near-field parameters. Importantly, the analytical upper bound represents a decent estimate as it predicts the order of magnitude of the minimal energies and the curve’s shape and, most importantly, provides a direct link between the continuous increase in the minimal electron energy for decreasing (earlier) birth times and the pulse intensity envelope.

We emphasize that despite the conceptual equivalence of the considered ponderomotive acceleration to that reported for atoms41,42, NILES is a specific feature of combined subwavelength field localization and few-cycle driving. For few-cycle pulses, only around two tunnelling emission events arise during the interaction of the laser pulse with the tip, spaced in time by the period of the optical field. Because the duration of the few-cycle laser pulses is extremely short, the kinetic energy gained by the electrons (equation (1)) varies substantially between the emission events. The appearance of NILES, thus, results from the sparsity of the emission events within a few-cycle laser pulse together with the ponderomotive force. Intriguingly, when varying the CEP, the emission events continuously map out the MEC, resulting in the spectrally isolated stripe-like features, which enable subcycle resolution—the key novelty of NILES. We add in passing that although this effect is fully general, it is easier to observe with mid-infrared driving wavelengths compared with visible or near-infrared driving because of the \({U}_{{\mathrm{p}}}^{2}\) scaling of the MEC (equation (1) and Methods).

To inspect the parameter dependencies of NILES, we simulated the CEP-resolved spectra for three different near-field profiles (Fig. 3): one spatially homogeneous near field (ζ→∞; Fig. 3a), one slowly decaying near field (ζ = 21 nm; Fig. 3b) and one fast decaying near field (ζ = 10 nm; Fig. 3c). Clearly, NILES shows up for the two inhomogeneous field profiles (Fig. 3b,c), whereas NILES is absent if there is no spatial gradient of the near-field-generated ponderomotive potential (Fig. 3a). As indicated by the four highlighted CEPs (circles 1–4), the continuous shifting of one relevant cycle can be observed over ~1.5 CEP periods, that is, even beyond one full period, similar to the experiment (Fig. 1d).

Fig. 3: Decay length dependence of NILES.
Fig. 3: Decay length dependence of NILES.The alternative text for this image may have been generated using AI.
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a–c, Semiclassical simulations of electron spectra for three near-field decay lengths, as indicated above each panel and corresponding to a planar surface (a), a medium-sharp tip (b) and a sharp tip (c), as used in the experiment. The near-field peak intensity was fixed at 2 × 1013 W cm−2. Clearly, NILES appears only for inhomogeneous near-field profiles (b and c) and extend up to 2 eV for the experimental parameters (c). The slope of NILES sampled in the central region increases from 0.76 eV/(2π) in b to 1.5 eV/(2π) in c. We note that the slope scales in indirect proportion to the pulse duration, resulting in vanishing NILES features for long pulses (Methods). The circles labelled 1–4 indicate different emission times within a selected half-cycle. d, Tracking of the emission time by modulating the CEP. With respect to the homogeneous field case (energy E = 0, black line), emission before the centre of the pulse (1 and 2) leads to an energy upshift, whereas a later emission (4) results in an energy downshift (indicated by the coloured arrows). This behaviour is given mainly by the MEC (Fig. 2e); the shift to energies below E = 0 is due to the fact that we now included the d.c. bias voltage applied to the tip (Methods). The labels 1–4 relate directly to the corresponding one in c. We stress the excellent overall agreement of the numerical results in c with the experimental NILES data shown in Fig. 1. We have applied a Gaussian blurring filter of 0.3 eV (FWHM) to the semiclassical simulation results to account for the finite energy resolution of the detector in the experiment. As shown in Fig. 1, the energy resulting from the tip bias voltage is subtracted from the energy axis (Methods).

Source data

Because of the strongly nonlinear nature of the electron emission as a function of the field strength, only the central cycles of a laser pulse contribute substantially to the emission (Extended Data Fig. 3). For this reason, for any given CEP, no more than two NILES are visible in the experimental spectra and no more than three in the simulation (for an optical pulse duration of τ = 11.5 fs at 1,570 nm), corresponding to the two–three mainly emitting optical cycles for the respective CEP.

The structure of NILES strongly depends on the pulse duration τ, which determines the number of visible stripes. The slope of these stripes in the central region scales approximately with 1/τ, which can be straightforwardly shown from equation (1). Further crucial parameters for the MEC are the optical near-field decay length ζ and the peak optical field strength (Extended Data Fig. 4 and Methods). Although the optical near-field decay is the root cause for NILES, strong static electric fields often present at metal needle tips can further influence the precise shape (Methods).

Extracting electron momentum widths

So far, we discussed NILES based on classical trajectories. In an extended semiclassical model, we can include quantum features by, for example, assigning a finite momentum width43 to the electron trajectories to account for the finite birth time window within the field cycle14,15. To this end, each trajectory contributes to the final spectrum via a Gaussian momentum distribution with a width σp centred at the final momentum of the classical trajectory. We determine the optimal value of this width by fitting the simulated spectra to the experiment (Methods). In particular, this process allows us to match the sharpness of the measured NILES features. The resulting width of σp = 4.8 × 10−26 kg m s−1 = 0.024 a.u. is considerably smaller than that predicted by conventional models for atomic above-threshold ionization spectra σp,theory ≈ 0.4–0.5 a.u. (refs. 44,45). Hence, NILES calls for a description beyond conventional broadening concepts and may enable investigating quantum diffusion effects and momentum broadening at low energies in a so-far unexplored scenario, representing an exciting outlook of forthcoming NILES-based experiments.

Individual attosecond electron burst

As shown by the simulation results in Fig. 2c,d, NILES leads to an emission time-sensitive energy shift in the direct electrons within the laser pulse. This shift allows us to select electrons from a specific cycle just by filtering the associated energies, as shown in Fig. 4a,b (red band). For this energy band, ranging from –0.7 to 0.3 eV, other dominant emission cycles are energetically well separated. This selection of energy (Fig. 4c), thus, translates to emission time filtering. Figure 4d shows the emission times of the filtered electrons (red) versus all electrons (blue). Clearly, from the multicycle laser pulses, we isolate an individual 430-as-long electron emission burst in the simulation. This burst contains a substantial 17% of the emitted electrons. Energy filtering is readily available in electron microscopes, facilitating to single out an individual attosecond electron emission burst. To harness this temporal confinement, available technologies for electron dispersion compensation have to be used46,47,48,49. We note that the exact width of the emission window strongly depends on the experimental conditions, filtering and intrinsic energy width of the electron emitter. Last (Methods), we note that we expect this work to result in integrated or even on-chip CEP measurement and stabilization devices because of the large CEP-dependent current.

Fig. 4: Isolation of individual attosecond electron bursts through NILES filtering.
Fig. 4: Isolation of individual attosecond electron bursts through NILES filtering.The alternative text for this image may have been generated using AI.
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a, Energy of the emitted electrons as a function of the emission time for parameters similar to those in Fig. 3c, obtained from the simulation model. The colour code indicates the emission yield, obtained from the Yudin–Ivanov rate50. We note that the values above 5 eV were multiplied by a factor of 10 for better visibility. b, Zoomed-in view of a, showing that the cycle-dependent energy shift through NILES allows us to single out electron emission from one cycle when filtering an energy band from –0.3 to 0.7 eV (red lines). c, Resulting electron energy spectra. The red area again shows the filtered electrons. d, Emission time of the energy-filtered electrons (red) and all electrons (blue). With these filter settings, 17% of the emitted electrons are confined within a window of 430 as. Six per cent of the filtered electrons are in another cycle. The simulations include the bias voltage applied to the tip, but no spectral blurring filter as in the simulation data shown in Fig. 3.

Source data

Methods

Experiment

We use an amplified erbium fibre oscillator system as the laser source with a 170-fs pulse duration operating at a repetition rate of 80 MHz. We reduce the repetition rate with a pulse picker down to 100 kHz (Extended Data Fig. 1). We broaden the spectrum by a highly nonlinear normal dispersion fibre51. After compression with fused silica, we obtain a pulse duration of 11.5 fs measured by frequency-resolved optical gating (FROG). The laser pulses are passively CEP stable, and we vary the CEP by two motorized glass wedges in the beam path.

These laser pulses are then focused by an off-axis parabolic mirror (f = 15 mm) to a focal spot size of 3 μm (1/e2-intensity radius) onto a tungsten needle tip. The tip is placed inside an ultrahigh-vacuum chamber at a pressure of 8 × 10−10 hPa. For the measurement shown in Fig. 1, we use an electrochemically etched tungsten tip post-processed by focused-ion-beam milling, leading to even cleaner electron spectra. The NILES feature also shows up in just-etched tungsten tips (Extended Data Fig. 10), which is recorded using a non-focused-ion-beamed tip. To remove the remnants of oxides at the very tip apex, we used in situ cleaning by field evaporation before the experiments in all cases52.

The laser-triggered electrons are accelerated from the negatively biased tip (Utip = –30 V) towards a delay-line detector. By the time of flight and position of electrons at the detector, we calculate the energy of each individual electron and record the electron energy spectra. Typically, we choose the laser power such that the electron count rate lies around 0.1 to 0.5 electrons per pulse on average. In this way, most of the electron events are single electrons and potential Coulomb effects are suppressed53.

For the measurement shown in Fig. 1, we used a step size of 0.26 rad in CEP and recorded 105 electrons per phase step over a range of ~1.6 × 2π at a near-field peak intensity of 2.1 × 1013 W cm−2. We determine this intensity from the measured cut-off position of 42 eV.

Depending on the exact count rate and the CEP range, we can record a full map like that in Fig. 1 on the few-minute timescale. We note, however, that the electron emission yield and the spectra are typically very stable on the hour timescale. Here we profit from the low repetition rate of our laser system compared with typical oscillators operating around 80 MHz, which leads to fewer heating effects and consequently longer lifetimes of the metal needle tips.

We match the CEP axes (offset) of the experiment and theory based on a two-dimensional convolution of the low-energy region, exactly where the NILES feature shows up, similar to schemes that use the cut-off position or the photoionization yield13,54,55,56,57,58. The resulting phase error of this scheme can be as low as 120 mrad.

Analytical form of the MEC

We consider the ponderomotive energy at the tip surface (that is, in the maximally enhanced field and defined for peak field strength) as

$${U}_{{\rm{P}}}={\xi }_{0}^{2}{U}_{{\rm{P}}}^{\;{\rm{inc}}}={\xi }_{0}^{2}\frac{{e}^{2}{E}_{0}^{2}}{4m{\omega }^{2}}={\xi }_{0}^{2}\frac{{e}^{2}{I}_{0}}{2c{\epsilon }_{0}m{\omega }^{2}},$$

where \({U}_{{\rm{P}}}^{{\rm{inc}}}\) is the ponderomotive energy of the incident laser pulse without field enhancement.

Considering the temporal intensity envelope and the spatial near-field profile, the dynamical (spatiotemporal) ponderomotive energy reads

$${U}_{{\rm{P}}}(x,t)=\xi {(x)}^{2}\frac{{e}^{2}I(t)}{2c{\epsilon }_{0}m{\omega }^{2}}.$$

This results in the (also spatiotemporal) ponderomotive force

$${F}_{{\rm{P}}}(x,t)=-\frac{\,\text{d}}{\text{d}\,x}{U}_{{\rm{P}}}(x,t)=-\frac{{e}^{2}I(t)}{2c{\epsilon }_{0}m{\omega }^{2}}\times 2\xi (x){\xi }^{{\prime} }(x).$$

Solving this equation yields the drift trajectories shown in Fig. 2, which include the full near-field decay but neglect the oscillations of the optical field because we average over one optical cycle. Note that electrons start one quiver length away from the surface in the drift picture.

As an upper estimate, we now only evaluate the ponderomotive force at the surface, where ξ(x = 0) = ξ0 and \({\xi }^{{\prime} }(x=0)={\xi }_{0}^{{\prime} }\), and find the temporal ponderomotive force at the surface as

$${F}_{{\rm{P}}}^{\;{\rm{surf}}}(t)=-\frac{{e}^{2}I(t)}{2c{\epsilon }_{0}m{\omega }^{2}}\times 2{\xi }_{0}{\xi }_{0}^{{\prime} }.$$

The final momentum of an electron starting at rest at birth time tb follows from the integration of the classical equation of motion and yields

$${p}_{{\rm{final}}}({t}_{\mathrm{b}})=-2{U}_{{\rm{P}}}^{\;{\rm{inc}}}{\xi }_{0}{\xi }_{0}^{{\prime} }{\mathcal{F}}({t}_{\mathrm{b}}),$$

where \({\mathcal{F}}({t}_{\mathrm{b}})=\frac{1}{{I}_{0}}\mathop{\int}\nolimits_{{t}_{b}}^{\infty }I(t){\rm{d}}t\) is the remaining normalized pulse fluence the electron may experience after its birth into the optical field.

The drift energy of the electrons then reads

$${E}_{{\rm{final}}}({t}_{\mathrm{b}})=\frac{2}{m}{\left({U}_{{\rm{P}}}^{\;{\rm{inc}}}{\xi }_{0}{\xi }_{0}^{{\prime} }\right)}^{2}{{\mathcal{F}}}^{2}({t}_{\mathrm{b}}).$$

Trajectory simulations

One common way to simulate the electron energies of electrons triggered from metal needle tips is based on the three-step model4,5,43. It is based on a three-dimensional semiclassical point-particle trajectory analysis and allows us to simulate the electron energies for different laser intensities, as a function of CEP, and for spatially inhomogeneous optical fields. The initial conditions of the electrons in the three spatial dimensions are given by the projection of a two-dimensional Gaussian distribution on a hemisphere to model the tip. The starting times and the yield of the electrons are based on a closed-form quantum-mechanical model, often called the Yudin–Ivanov rate50,59. Originally, this model was developed for the electron emission from atoms. The metal–vacuum interface at the tips breaks the symmetry as compared with an atom. We assume, therefore, that electron emission takes place only in negative half-cycles as we are in the subcycle tunnelling regime (Fig. 4d). To avoid Coulomb repulsion effects, we artificially set the total rate to one electron per pulse in these simulations. After the emission, the electrons are propagated classically by numerically integrating the equations of motion based on a standard fourth-order Runge–Kutta scheme. Here we take into account the static field applied to the tip as well as the oscillating field. We model the static field using a spherical capacitor with the outer sphere set to infinity60. We choose Utip = –10 V as the bias voltage for all our simulations, if not specified otherwise. This voltage leads to a static field of 1 V nm−1 at the apex of the tip. For the oscillating near field, we choose a pulse with a Gaussian envelope representing the laser pulse given by

$$\begin{array}{rcl}E(r,t)&=&\exp \left(-\frac{2\ln (2){t}^{2}}{{\tau }^{2}}\right)\cos \left(\omega t+{\varPhi }_{{\rm{CEP}}}\right)\\ &&\times \left[1+({\xi }_{0}-1)\exp \left(-\frac{r}{\zeta }\right)\right].\end{array}$$

Here the pulse duration τ is given as the intensity FWHM. We chose a near-field shape with a decay constant of ζ = 10 nm, unless stated otherwise. For the simulations shown in Fig. 3a–c, the field enhancement factors chosen for the simulation are ξ0 = 1 (Fig. 3a), ξ0 = 7 (Fig. 3b) and ξ0 = 15 (Fig. 3c). We chose these values to account for the fact that the field enhancement factor increases for decreasing tip radius.

Extended Data Fig. 3a shows the CEP-resolved energy spectrum as obtained from the simulation. We colour code the emission distributions from the different optical cycles, where each colour corresponds to a specific cycle and the rate is given by the opacity of each colour. Extended Data Fig. 3b–d depicts which optical cycle corresponds to which colour for three different CEP values. This illustration clarifies that the number of NILES corresponds to the number of emitting cycles. Furthermore, we find that the lowest-energy part from –1 to 1 eV is dominated by the purple emission burst for a CEP of ~1.7 × 2π to 2.3 × 2π, for example. This is the basis to obtain a single attosecond emission burst (Fig. 4).

Parameter dependencies of the MEC

Figure 2e shows how NILES arises for one set of parameters, when the near field is inhomogeneous. Here the MEC (solid lines), representing the origin of NILES, is influenced by all parameters that change the optical field the electron experiences as it performs a quiver motion.

Extended Data Fig. 4a–c shows the dependence of the MEC on the most relevant parameters: peak optical near-field intensity, near-field decay length ζ and laser pulse duration τ. We discuss these parameters in the following. We use ζ = 10 nm, τ = 11.5 fs, a near-field intensity of 2.3 × 1013 W cm−2, a central wavelength of 1,570 nm and a tip voltage of –10 V, if not stated otherwise.

To quantify the effect of these parameters on NILES, we further calculate the maximum amplitude for each MEC, given by the difference in the maximum energy and minimum energy. We refer to this amplitude as the MEC amplitude and show these amplitudes in Extended Data Fig. 4d–f.

We find that the MEC amplitude increases for increasing intensity (Extended Data Fig. 4a). Extended Data Fig. 4a shows that it does so in a linear fashion. This increase reflects that the electrons are driven further away from the surface in the optical near field with increasing intensity; thus, they experience a larger variation in the optical near field, leading to an enhanced NILES effect. We note that this linear scaling is similar to the scaling of the low-energy peak for one fixed emission time, investigated in ref. 28. We further note that for our set of parameters, NILES is not detectable in the multiphoton region, as the amplitude would be too small to be detected for γ > 2.

When we sweep ζ from 150 nm (dark blue) towards smaller decay lengths (Extended Data Fig. 4b), we first observe an increase in the MEC amplitude, down to a value of ζ = 13.3 nm (green curve). When ζ is further decreased, however, the amplitude reduces again. Physically, this reduction is a consequence of an increasingly strong quenching of the quiver motion for smaller decay lengths, that is, the amplitude of the oscillatory motion decreases.

Extended Data Fig. 4c shows the MEC for different pulse durations τ. Although the amplitude of the MEC only slightly increases with a decreasing pulse duration (Extended Data Fig. 4f), the width of the MEC clearly broadens for longer pulses. This broadening along the time axis is to be expected, as the MEC is mainly affected by the time that the electron needs to escape the optical near field, as well as the optical pulse duration. In the limit of an infinitely long pulse, that is, the continuous-wave case, the MEC would be constant above zero for our case, because without the temporal envelope, all electrons born with a time difference of one optical period traverse the same near field. In the other case of constant pulse duration and increasing ζ, the NILES amplitude vanishes to zero (Extended Data Fig. 4b).

Influence of static electric fields

NILES is an effect entirely based on the quickly decaying optical near field. However, the shape and amplitude of NILES are influenced by the static bias field at the tip apex occurring when biasing the tip with a voltage Utip. This voltage results in an additional contribution to the final electron energies of Ed.c. = e × Utip, which we subtract when displaying the energies in our plots. To show this influence in more detail, we simulate the final energies of electron trajectories for four different cases: with and without a static field for a homogeneous field (ζ = ∞, ξ0 = 1) and a decaying near field (ζ = 10 nm, ξ0 = 10). The low-energy part of these simulations (Extended Data Fig. 7) reveals that only in the presence of a static field, energies below zero (that is, the final kinetic energies below Ed.c.) can be reached.

This result may be surprising when one thinks about an electron emitted without a laser field that surfs down only the static potential, leading to an energy of Ed.c. (in our case, 10 eV). It is the interplay of the laser-induced near field and the static field that enables emission with final energies below the d.c. voltage. However, this effect can only be observed clearly in the case of a strong inhomogeneity of the static field, given by the small dimensions of the tip.

Analogous to the MEC shown in Fig. 2e, the result including a static potential like in the experiment is shown in Extended Data Fig. 9, illustrating its effect on the NILES shape.

Extraction of momentum width

For the extraction of momentum width, we simulated the spectra similar to that detailed above with minor modifications to the model. First, we weighted the trajectories using the Wentzel–Kramers–Brillouin tunnelling rate. Second, we include the experimental energy resolution of the detector in the simulations. The resulting low-energy spectral features (Extended Data Fig. 5a) qualitatively capture the experimental NILES signatures (Extended Data Fig. 5c), but do not agree quantitatively. For a quantitative reconstruction, we further modify our simulations and perform an optimization (Nelder–Mead simplex method) for three additional parameters: first, the width σp of a Gaussian momentum distribution that we associate with each trajectory after propagation; second, the exponent n of a now considered exponential ionization rate R = En(x = 0, t), where E(x = 0, t) is the locally enhanced electric field at the surface; and third, a random CEP shift φCE. In this way, we optimized for the best overlap between the simulated and measured low-energy region of the spectra, including the NILES features. Out of 100 initial conditions for the optimization the best agreement (compare Extended Data Fig. 5b,c) was found for σp = 4.8 × 10−26 kg m s−1 = 0.024 a.u., n = 4.24 and φCE = 0.05π. Especially including the momentum width paves the way to examine inter- and intracycle interference effects to a large extent with extended semiclassical simulations43.

CEP-dependent current

A CEP-dependent current is of the highest interest for direct CEP locking of lasers without the need of nonlinear interferometers. Although CEP-dependent currents have been demonstrated, so far they are too small for direct CEP locking13,22,30,32. In the following, we illustrate how NILES can be used to realize such a device, based on the direct electrons and, thus, with a much larger fraction of electrons contributing to the CEP-dependent current.

Extended Data Fig. 6a shows again the results of the measurement similar to Fig. 1b, from which we select two energy bands for the rescattered (blue) and direct (red) electrons. The chosen energy band for the rescattered electrons is 40–60 eV and 0.4–1.2 eV for the direct electrons. Extended Data Fig. 6b,c shows the integrated yield of the electrons of these bands, which both show a sine modulation as a function of the CEP. Although the modulation depth for the direct electrons is 33% and, thus, smaller than 46% for the rescattered electrons, the total yield of the direct electrons is 17 times higher than rescattered electrons. Hence, in total, NILES yields an approximately nine times better signal-to-noise ratio (SNR) than the rescattered electrons.

To stabilize a laser oscillator, an SNR of 30 dB at a bandwidth of 100 kHz is required61,62. Assuming this bandwidth and a repetition of 80 MHz, we obtain an SNR of 8 dB for NILES, too low for stabilization by a factor of over 100 in current. However, using a nanotip array, the current can be directly enhanced by orders of magnitude63. We foresee that this scheme can even be miniaturized to fit into the package of a small standard photodiode. There are already systems that measure the CEP based on the electron emission in a planar tip or bow-tie antenna configurations32,64,65. Especially for more than two-cycle laser pulses the integrated CEP-modulated current of such devices is small, leading to a small SNR. By using NILES and energy filtering, the sensitivity of such schemes can be considerably improved. Choosing an even larger energy window for the low-energy electrons (Extended Data Fig. 6), for example, the CEP-modulated current decreases, which is exactly what happens when measuring the integrated CEP current. Further, it was shown that for certain peak intensities, the CEP sensitivity of the integrated current can drop by over an order of magnitude, the so-called vanishing points32. This drop results from equally strongly emitting cycles within the laser pulse for different CEPs. Because of the energy shift of the different cycles, NILES can energetically separate different cycles and help to avoid such CEP-insensitive points.

Effect of near field on rescattered electrons

Figure 3 shows the influence of different optical near fields on NILES. Extended Data Fig. 8 shows the same simulations but on a logarithmic scale and a larger energy range, including the plateau electrons.

For the homogeneous case (Extended Data Fig. 8a), the direct electrons in the low-energy range do not show any CEP dependence, as expected. Yet, the tell-tale feature of strong field physics, the plateau, is shown, like the two others (Extended Data Fig. 8b,c). Since the actual waveform determines the maximum achievable energies, we observe a modulation of the highest energies as a function of the CEP, leading to the well-known arches in the cut-off region54. The electron energy spectrum in the homogeneous case reaches up to the expected ten times the ponderomotive energy UP of the laser field, that is, roughly 42 eV (ref. 6).

This cut-off energy cannot be reached for the inhomogeneous near-field profiles (Extended Data Fig. 8b,c), because of a quenched quiver motion19. This energy reduction is even better visualized by observing the final energies from one optical cycle (Extended Data Fig. 9a), where the maximum energies only reach 8.3UP for the chosen parameters. In addition, the reduced near field leads to a delayed starting time of rescattering (Extended Data Fig. 9a, stars).

The energy reduction demonstrates by how much NILES is different: although the optical near field affects only the energy of the rescattered electron, its presence changes the spectral structure of direct electrons completely.

Further, the simulations depicted in Extended Data Fig. 8a–c show that the near-field decay changes the CEP \({\phi }_{\max }\) at which the electrons are born that reach the highest energies (indicated by coloured arrows).

To determine \({\phi }_{\max }\) experimentally, we recorded further spectra with a CEP step size of 0.26 rad over a range of more than three periods with 5 × 105 electrons per phase step and rate of 0.4 electrons per pulse (Extended Data Fig. 10a). We determine the cut-off position for each CEP by fitting two linear functions to the plateau and cut-off region13,38.

We fit a sine to the determined cut-offs (Extended Data Fig. 10a) and find an experimental value of \({\phi }_{\max ,\exp }=1.8\pm 0.5\) rad and a cut-off position of 40.7 eV with a modulation depth of 5.4 ± 0.2 eV (twice the amplitude of the sine fit). The resulting near-field intensity in this measurement is then 2.0 × 1013 W cm−2.

Extended Data Fig. 10b shows the simulated phase \({\phi }_{\max }\) as a function of ζ. We find that \({\phi }_{\max }\) monotonically decreases for increasing ζ. We observe that \({\phi }_{\max }(\zeta )\) is matched by a power law, where the offset is given by the homogeneous field case, with a phase of ϕmax(ζ→∞) = 0.9 rad. The measured phase of 1.8 ± 0.5 rad matches the simulated data within the error bars (black dot).

Because the experimentally determined phase \({\phi }_{\max ,\exp }\) is 0.9 rad larger than the simulated case with a homogeneous optical field (ζ = ∞), we see that the CEP at which the highest electron energies are created cannot be considered constant in the presence of an optical near field. This result shows that schemes measuring the CEP by the cut-off electrons clearly have to take decaying near fields into account. If not, the CEP can be grossly misinterpreted.

Moreover, the evaluated modulation depth of the cut-off of 5.4 ± 0.2 eV serves as an intrinsic measure for the near-field pulse duration present at the tip’s apex. The modulation depth is in good approximation given by the difference in the ponderomotive energies associated with the maximum field at CEP = 0 and CEP = π, for one fixed pulse envelope. Extended Data Fig. 10c shows the modulation depth as a function of the cut-off energy, that is, ten times the ponderomotive energy, and the pulse duration. On the basis of this map, the experimental pulse duration is 11.1 ± 0.4 fs, in good agreement with the FROG-based measurement of 11.5 fs.

Similar maps can be generated using the semiclassical simulation including both optical near field and static field. The map produced by such a simulation gives a pulse duration of 11.5 ± 0.4 fs, perfectly matching the result from FROG. The better agreement is due to the fact that quenching effects are not taken into account when we only consider the difference in ponderomotive energies at CEP = 0 and CEP = π. However, we emphasize that using this difference provides experimenters with a simple tool to measure the pulse duration within the typically required accuracy in the experiment, without the need for simulations.

Impact of coherent phenomena on NILES

So far, all simulations were based on a semiclassical treatment in which only the emission process was modelled quantum mechanically. To rigorously include quantum interference and quantum diffusion, we now simulate the energy spectra based on a numerical solution of the one-dimension time-dependent Schrödinger equation (TDSE)15,39. We obtain the wavefunction of the electron by integrating the TDSE numerically by a Crank–Nicolson method (details and code are provided in ref. 15). As the ground state, we assume an electron that is bound in a box potential with a half-sided linear decaying potential, mimicking the static potential present at the tip. The potential is chosen such that the electron has an energy of 20 eV without any laser field after traversing the full potential. The width of the box is chosen to match the work function of tungsten, which we assume as 6 eV (ref. 15). The assumption of a single bound state may seem surprising because metals have, in general, many available states that are occupied according to the Fermi–Dirac distribution. However, because of the high nonlinearity of the electron emission, one energy level will be dominant and all other lower-lying states are strongly suppressed66.

Extended Data Fig. 10d shows the result for an incident intensity of 1.8 ×1013 W cm−2, ζ = 10 nm and ξ0 = 5, similar to the experiment shown in Extended Data Fig. 10a. The main features of the simulated spectrum are very close to the spectra simulated with the semiclassical model. However, the TDSE is fully coherent; hence, we find multiphoton photoemission (MPP) peaks spaced by the photon energy, effectively visible as fine stripes over the entire spectrum. NILES also shows up in the low-energy region, smoothly transitioning into the spacing of the MPP peaks above 2 eV (Extended Data Fig. 10e). It is interesting to note that the whole low-energy region (–1 to 5 eV) experiences a CEP-dependent energy shift similar to the shape of NILES, even above 2 eV in which we classically observe no NILES feature. The fact that the semiclassical and quantum simulation agree except for the appearance of MPP peaks shows that the dynamics of the emitted electron wave packets is dominated by the centre-of-mass motion in the optical near field (Ehrenfest dynamics).

In the experiment, we have so far not observed MPP peaks together with NILES, which can be for two reasons. For longer pulse durations without CEP variation, we have measured such MPP peaks already in the same setup67. The broader bandwidth of our few-cycle laser pulses in the present case broadens the width of MPP peaks68. Further, the coherence time at such high intensities is possibly reduced so heavily that already two emission events in two consecutive cycles cannot interfere anymore. Investigating details of the electron coherence will be the subject of future work.