### Experimental

#### Attosecond EUV streaking

For attosecond EUV streaking measurements (Prolonged Information Fig. 1a), the sub-cycle transients are centered onto the neon gasoline jet to generate EUV pulses by high-harmonic technology. The collinearly propagating EUV and optical pulses are spatially separated by a Zr filter, which additionally acts as an EUV high-pass spectral filter (>70 eV), enabling the isolation of a single attosecond pulse. The EUV and optical pulses are mirrored off a dual-mirror meeting, which consists of a Mo/Si inside mirror (centred at round 85 eV) and a nickel outer mirror, respectively. Inside and outer mirrors might be delayed with nanometric decision (Prolonged Information Fig. 1a). EUV and optical pulses are centered onto a second Ne gasoline jet. Photoelectron spectra recorded as a operate of the delay between the inside and outer mirrors permit the composition of attosecond streaking spectrograms, which permit the detailed characterization of the attosecond EUV pulse and, notably for these experiments, the sector waveform of the optical pulse. Particulars in regards to the related methods might be present in refs. ^{35,41,48}.

#### HAS

HAS measurements are carried out on the identical setup by way of (1) automated removing of the Ne gasoline used to generate excessive harmonics and the Zr filter, (2) automated alternative of the inside mirror within the dual-mirror module by a Ni-coated one of many identical focal lengths (Prolonged Information Fig. 1b) and (3) the streaking gasoline nozzle is changed by an electrically grounded tungsten nanotip. The above setup modifications are executed in a fraction of a minute and warrant equivalent circumstances for all related measurements. Within the HAS configuration of the setup in Prolonged Information Fig. 1b, Ni-coated inside and outer mirrors spatially divide the optical pulse into pump (inside mirror beam) and gate (outer mirror) pulses. A delay between the pulses mirrored off the inside and outer mirrors, respectively, is launched by a precision transitional stage (see inset in Prolonged Information Fig. 1b). The HAS measurements in addition to the intensity-scaling measurements of the electron spectrum and yields have been carried out with driving-field intensities beneath a crucial depth, at which the irreversible optical harm of the nanotips happens. This higher restrict of depth was experimentally recognized by observing sudden and irreversible drops of electron cutoff vitality in intensity-scaling measurements.

#### Measurement of absolutely the electron yield within the optical area emission

For the measurement of the entire electron counts per pulse generated in our setup, a skinny electrode (measurement roughly 5 mm × 5 mm) is launched about 2 mm above the nanotip (Prolonged Information Fig. 2). This configuration permits detection of launched electrons over a stable angle *Ω* > π steradians. The induced voltage on the skinny plate is measured by a lock-in amplifier on the reference frequency of the repetition charge of the driving laser (about 3 KHz). The digital present is evaluated by dividing the induced voltage by the system impedance (10 MΩ). The entire electron rely per pulse is in flip obtained by dividing the present by the repetition charge of the laser and the electron cost.

#### Pattern robustness and measurement stability research

Tungsten nanotips have been uncovered to intensities of as much as *I* ≈ 45 TW cm^{−2} with out noticing any harm. This was verified by performing the depth–cutoff vitality research proven in Fig. 1b, of each rising in addition to reducing depth, and recorded equivalent curves. When the crucial depth reached *I* ≈ 45 TW cm^{−2}, the tip is broken and the cutoff vitality irreversibly drops to a lot decrease values with out the chance for restoration except a brand new tip is put in.

To discover short-term and long-term stability of our system, we recorded electron spectra as a operate of time underneath equivalent circumstances and for intensities usually increased than these used within the HAS measurements. Information proven in Prolonged Information Fig. 3 recommend a wonderful stability of cutoff vitality and electron yield, implying the structural robustness of the nanotip on the timescale of typical measurements (a couple of minutes) in addition to over a number of hours.

### One-dimensional, semiclassical simulations of the optical area emission

The time-dependent ionization chance from a tungsten nanotip was approximated by the Fowler–Nordheim system as^{5,17,49,50}:

$$p(E(t))=Ntheta (-E(t))E(t)proper^{2},exp ,left[-frac{4sqrt{2m}{phi }^{3/2}}right],$$

(3)

by which *E*(*t*) is the electric-field waveform of the driving pulses, *ϕ* is the work operate of the steel and *m*, *ħ* and *e* are electron mass, diminished Planck’s fixed and electron cost, respectively. We calculated electron trajectories utilizing the classical equations of movement within the single-electron restrict as^{7,8,11,17,36}:

$$mfrac{{{rm{d}}}^{2}{z}_{i}}{{rm{d}}{t}^{2}}=-e{f}_{0}E(t)$$

(4)

Right here *i* is the index of every particular person trajectory and *f*_{0} is the field-enhancement issue. On the finish of the heart beat, an electron spectrum is evaluated by a spectral binning of the energies of all trajectories weighted by the ionization charge on the cases of their births.

For the experiments described right here, we simulated electron spectra from tungsten (*ϕ* = 4.53 eV) versus peak area depth of the driving pulse (Prolonged Information Fig. 4). The driving area (crimson curve in Prolonged Information Fig. 4a) utilized in our simulations was measured by an EUV attosecond streaking setup^{35,48} built-in in our equipment. A field-enhancement issue of *f*_{0} = 3.46 utilized in these simulations was derived experimentally as described in the primary textual content. The quiver size of electrons in our depth vary (the longest size of about 1.7 nm on the highest depth of about 41 TW cm^{−2}) is significantly shorter than the decay size of the close to area (about 30 nm). Therefore the launched electrons expertise an almost homogeneous close to area and due to this fact quenching results^{17} owing to the near-field decay might be uncared for.

In settlement with the info of Fig. 1c, the simulated electron spectra exhibit two well-discernible vitality cutoffs (crimson and blue dashed traces in Prolonged Information Fig. 4b) related to the backscattered (purple line in Prolonged Information Fig. 4a) and the direct (inexperienced line in Prolonged Information Fig. 4a) electrons. The slopes of excessive (({s}_{{rm{W}},{rm{excessive}}}^{({rm{th}})}={{rm{d}}E}_{{rm{c}}}/{{rm{d}}U}_{{rm{p}}}=130.3)) and low (({s}_{{rm{W}},{rm{low}}}^{({rm{th}})}=26.1)) cutoff energies agree effectively with these in our measurements (Fig. 1c). The speculation exhibits additional emission cutoffs at energies decrease than that of the direct electrons. As a result of these are comparatively weak, they don’t go away any direct signatures within the photoelectron spectra. But such contributions change into seen in HAS spectrograms, as mentioned in Fig. 3.

### FDTD simulations of the sector enhancement

To theoretically estimate the near-field enhancement within the neighborhood of the tungsten nanotip, we numerically solved Maxwell’s equations by way of three-dimensional finite-difference time-domain (FDTD) simulations. The nanotip was modelled as proven in Prolonged Information Fig. 5a as a sphere with radius of 35 nm that easily transitions to a cone with a gap angle (single facet) of 15° and contemplating optical properties for tungsten^{51}. The simulations predict a peak field-enhancement issue of about 3.8 near the floor on the tip apex. For comparability, the spatial distribution of the enhancement at a respective tungsten nanosphere (that’s, excluding the cone) is proven in Prolonged Information Fig. 5b, with a barely decrease peak enhancement issue of roughly 2.7.

### Three-dimensional, semiclassical trajectory simulation together with cost interplay

To examine whether or not cost interplay considerably impacts the electron emission dynamics for the thought of parameters, we carried out semiclassical trajectory simulations utilizing the Mie–imply area–Monte Carlo (M^{3}C) mannequin^{52}. The latter has been used extensively for the research of strong-field ionization in dielectric nanospheres^{53,54,55} and has just lately additionally been adopted for the outline of metallic nanotips^{56}. In short, we mimic the apex of the nanotip as one-half of a sphere with corresponding radius. The close to area is evaluated because the mixed linear polarization area owing to the incident pulse (evaluated by way of the Mie resolution of Maxwell’s equations) and an additional nonlinear contribution owing to cost interplay handled as a imply area in electrostatic approximation (evaluated by high-order multipole growth). The latter contains Coulomb interactions among the many emitted electrons in addition to their picture fees (that’s, an additional sphere polarization attributable to the free electrons). Photoelectron trajectories are generated by Monte Carlo sampling of ionization occasions on the sphere floor, at which we consider tunnelling possibilities inside WKB approximation by integration by means of the barrier offered by the native close to area. Trajectories are propagated within the close to area by integration of classical equations of movement and accounting for electron–atom collisions by means of respective scattering cross-sections for electrons transferring inside the materials. To imitate the marginally increased peak enhancement of the linear response close to area at a tungsten tip (≈3.8) as in contrast with a sphere (≈2.7), see Prolonged Information Fig. 5, we rescaled the incident laser depth by an element of 1.4. The carried out M^{3}C simulations predict about thrice fewer emitted electrons than the experiments, which we attribute to contributions of gradual electrons originating from the shank of the nanotip. That is substantiated by comparability of the entire electron yields predicted for the nanotip and the half-sphere, obtained by means of integration of the native ionization charges over the respective floor areas and the heart beat length. Nonetheless the cost densities on the pole of the sphere and the tip apex are comparable, enabling to examine the impression of cost interactions inside the simplified simulation mannequin. Prolonged Information Fig. 6a compares multielectron spectra simulated with (stable curves) and with out (dashed curves) accounting for cost interactions among the many particular person electrons and for 4 consultant settings of the height intensities (and corresponding electron yields) of the driving area, as indicated within the legend. The presence of cost interplay is primarily manifested by a noticeable decline of the yield of direct electrons (<50 eV), which—in accordance with earlier works^{52,57}—might be attributed to a partial trapping of those electrons within the neighborhood of the tip. Quasi-static electrical fields generated by the trapping of low-energy electrons in flip have an effect on the dynamics of the recolliding electrons and provides rise to a rise of the terminal electron-energy cutoff^{52,57}. For the best depth and corresponding electron yield, this shift is roughly 8% in vitality.

Though clear manifestations of such results can’t be discerned in our experiments, it’s helpful to grasp potential implications of cost interactions on the characterization of attosecond electron pulses utilizing HAS. To this finish, we prolonged our evaluation to the time area for pulses of depth (about 31 TW cm^{−2}) and electron yield (about 600 electrons per pulse), mimicking the experimental circumstances in our HAS measurements.

Prolonged Information Fig. 6b exhibits the recollision vitality distribution of the electron ensemble versus launch instances with out (high) and with (backside) inclusion of cost interactions. The instantaneous launch vitality is evaluated by taking the primary momentum of the time-resolved spectra (black dashed and stable curves) and the temporal phases (blue dashed and stable curves) by the temporal integration of the instantaneous vitality. Their comparability, proven in Prolonged Information Fig. 6b, suggests marginal variations and, thus, corresponding negligible results on the temporal traits of the electron pulse on the time of recollision.

When the time-domain evaluation is prolonged to the terminal energies of the recolliding electrons versus launch time, the house–cost interactions are manifested by a uniform upshift of terminal energies by about 8% however go away the temporal part of the terminal electron wave packet unaffected (Prolonged Information Fig. 6c).

To analyze how this vitality shift might doubtlessly have an effect on the retrieval of the electron pulse on the occasion of recollision, we utilized the simulated part results on our experimental knowledge (see the part ‘HAS reconstruction methodology’) and evaluated as soon as once more the spectral and temporal properties of the recolliding attosecond electron pulse. Key observations related to house–cost interactions embody a weak, uniform shift of the central vitality of the recolliding electron by a number of electronvolts (Prolonged Information Fig. 6d) and a refined change within the temporal profile of the electron pulse (Prolonged Information Fig. 6e), leading to an roughly 4-as elongation of its length on the full width at half most, which is inside the error of the experimental reconstruction (about 5 as).

### Mathematical formulation of HAS

The important thing goal of HAS is to retrieve the temporal construction of an attosecond electron pulse wave packet *ψ*_{r}(*t*) in the mean time of its recollision on its mother or father floor. As this wave operate shouldn’t be straight accessible, it’s essential to hyperlink it to different portions which are both straight measured within the experiments (such because the terminal spectral depth (I(p)={left|{widetilde{psi }}_{{rm{t}}}(p)proper|}^{2}) at a detector) or might be reconstructed from the experimental knowledge.

#### Description of strong-field electron emission

Contemplating an electron launched from and pushed again to a floor by a powerful pump area *E*_{p}(*t*), its recolliding wave packet *ψ*_{r}(*t*) might be linked to its terminal spectral amplitude ({widetilde{psi }}_{{rm{t}}}(p)) on the finish of interplay with the driving pulse. The time-dependent recollision wave packet *ψ*_{r}(*t*) is expressed by way of its Fourier illustration ({widetilde{psi }}_{{rm{r}}}(p)={int }_{-infty }^{infty }{psi }_{{rm{r}}}({t}_{{rm{r}}})exp [{rm{i}}frac{{p}^{2}}{2}{t}_{{rm{r}}}]{rm{d}}{t}_{{rm{r}}}) and, following the recollision, the spectral amplitude is remodeled to the terminal type^{38,39,40,58} (in atomic models):

$${widetilde{psi }}_{{rm{t}}}(p)propto {rm{i}}{int }_{-infty }^{infty ,}{psi }_{{rm{r}}}({t}_{{rm{r}}})exp ,left[{rm{i}}frac{{p}^{2}}{2}{t}_{{rm{r}}}right],exp [-{rm{i}}S(p,infty ,{t}_{{rm{r}}}{rm{;}}{A}_{{rm{p}}}(t))]{rm{d}}{t}_{{rm{r}}}$$

(5)

Right here *S* is the Volkov part imparted to the electron wave packet solely by the vector potential *A*_{p}(*t*) of the pump pulse after recollision at an occasion *t*_{r}, at which the overall type of the Volkov part accrued from a time occasion *t*_{1} to a later occasion *t*_{2} by an electrical area with vector potential *A*(*t*) is expressed as^{59}:

$$S(p,{t}_{2},,{t}_{1}{rm{;}}A(t))={int }_{{t}_{1}}^{{t}_{2}}left[frac{1}{2}right.{left[p+A(t)right]}^{2}-left.frac{1}{2}{p}^{2}proper]{rm{d}}t$$

(6)

Word that equation (6) excludes free-space propagation, that’s, it vanishes within the absence of the sector, and equation (5) displays the momentum-dependent wave operate on the floor, together with phases accrued solely by the pump area.

Earlier semiclassical theories of strong-field emission^{39,40,58} in atoms have recommended that the recolliding wave packet *ψ*_{r}(*t*) might be expressed by integration over ionization occasion *t*′ earlier than recollision at time *t*_{r} and over canonical momenta *p*′ by way of the ionization amplitude, dictated by the dipole transition *E*_{p}(*t*′)*d*(*p*′ *+* *A*_{p}(*t*)), the scattering amplitude usually described as *g*(*p*′ *+* *A*_{p}(*t*_{r})) and the Volkov part that the electron accumulates from *t*′ to *t*_{r} as:

$${psi }_{{rm{r}}}({t}_{{rm{r}}})={int }_{-infty }^{{t}_{{rm{r}}}}int g({p}^{{prime} }+{A}_{{rm{p}}}({t}_{{rm{r}}})){E}_{{rm{p}}}({t}^{{prime} })d({p}^{{prime} }+{A}_{{rm{p}}}({t}^{{prime} }))instances exp ,[-{rm{i}}S({p}^{{prime} },{t}_{{rm{r}}},{t}^{{prime} }{rm{;}}{A}_{{rm{p}}}(t))-{rm{i}}frac{{p}^{2}}{2}({t}_{{rm{r}}}-{t}^{{prime} })+{rm{i}}phi {t}^{{prime} }]{rm{d}}{p}^{{prime} }{rm{d}}{t}^{{prime} }$$

(7)

Right here *d* and *g* are the dipole and scattering matrix factor, respectively, as outlined in refs. ^{39,40,58}, and e^{iϕt} displays the additional phases acquired in the course of the time evolution of the certain state earlier than ionization. In our experimental setting of HAS by which the Keldysh parameter of *γ* ≈ 0.38 suggests the tunnelling regime^{60}, these three processes (ionization, propagation and backscattering) might be independently handled with out non-adiabatic corrections on the Volkov part, *S* (refs. ^{61,62}).

#### Description of electron wave packets underneath addition of a weak gate area

Equation (5) implies that entry to *ψ*_{r}(*t*) is feasible if ({widetilde{psi }}_{{rm{t}}}(p)) and *A*_{p}(*t*) are recognized. Due to this fact our aim is to explain how these portions might be accessed utilizing a phase-gating strategy of the optical area emission by a weak duplicate of the driving pulse (gate pulse). Now we examine the results of including the gate pulse on the dynamics of the electron described in equations (5)–(7). We outline the gate pulse by a area *E*_{g}(*t* + *τ*) and its vector potential *A*_{g}(*t* + *τ*), by which *τ* is the delay between the pump and gate pulses as described above. By changing the pump fields and its vector potentials by the superposition of pump and gate pulses in equations (5)–(7) as *E*_{p}(*t*) → *E*_{p}(*t*) + *E*_{g}(*t* + *τ*) and *A*_{p}(*t*) → *A*_{p}(*t*) + *A*_{g}(*t* + *τ*), the terminal spectral amplitude perturbed by the gate might be rewritten as:

$${widetilde{psi }}_{{rm{t}}}(p,tau )propto {rm{i}}{int }_{-infty }^{infty }{psi }_{{rm{r}}}^{({rm{g}})}({t}_{{rm{r}}},tau )exp left[{rm{i}}frac{{p}^{2}}{2}{t}_{{rm{r}}}right],exp left[-{rm{i}}Sleft(p,infty ,{t}_{{rm{r}}};{A}_{{rm{p}}}(t)+{A}_{{rm{g}}}(t+tau )right.right]{rm{d}}{t}_{{rm{r}}}$$

(8)

by which ({psi }_{{rm{r}}}^{({rm{g}})}({t}_{{rm{r}}},tau )) denotes the recolliding electron wave packet perturbed by the additional gate pulse as marked by the superscript (g) to be distinguished from the gate-free counterpart *ψ*_{r}(*t*) (see equation (7)). As a result of the gate-free amount *ψ*_{r}(*t*) is of curiosity on this dialogue, the topic on this part is the best way to categorical ({widetilde{psi }}_{{rm{t}}}(p,tau )) by way of *ψ*_{r}(*t*) with part phrases launched by the gate.

First we examine the perturbed recolliding electron wave packet ({psi }_{{rm{r}}}^{({rm{g}})}({t}_{{rm{r}}},tau )) in equation (8) and the best way to hyperlink it with the gate-free electron wave packet *ψ*_{r}(*t*). If the gate area is sufficiently weak, that’s, (eta ={left|{A}_{{rm{g}}}(t)proper|}^{2}/{left|{A}_{{rm{p}}}(t)proper|}^{2}ll 1), the dipole transition and scattering amplitudes might be thought of invariant, that’s, (left[{E}_{{rm{p}}}(t)+{E}_{{rm{g}}}(t+tau )right]d(p+{A}_{{rm{p}}}(t),+)({A}_{{rm{g}}}(t+tau ))approx {E}_{{rm{p}}}(t)g(p+{A}_{{rm{p}}}(t))) and (g(p+{A}_{{rm{p}}}(t)+{A}_{{rm{g}}}(t+tau ))approx g(p+{A}_{{rm{p}}}(t))) within the expression of the recolliding electron wave packet (equation (7)). In such case, the gate solely modifies the part imparted on the wave packet between ionization and recollision:

$${psi }_{{rm{r}}}^{({rm{g}})}({t}_{{rm{r}}},tau )approx {int }_{-infty }^{{t}_{{rm{r}}}}{rm{d}}{t}^{{prime} }int {rm{d}}{p}^{{prime} }g({p}^{{prime} }+{A}_{{rm{p}}}({t}_{{rm{r}}})){E}_{{rm{p}}}({t}^{{prime} })d({p}^{{prime} }+{A}_{{rm{p}}}({t}^{{prime} }))instances exp ,left[-{rm{i}}S({p}^{{prime} },{t}_{{rm{r}}},{t}^{{prime} }{rm{;}}{A}_{{rm{p}}}(t)+{A}_{{rm{g}}}(t+tau ))-{rm{i}}frac{{p}^{2}}{2}({t}_{{rm{r}}}-{t}^{{prime} })+{rm{i}}phi {t}^{{prime} }right]$$

(9)

In view of those concerns, the variation of the part (see equation (6)) might be expressed as:

$$start{array}{l}S(p,{t}_{2},{t}_{1};{A}_{p}(t)+{A}_{g}(t+tau ))={int }_{{t}_{1}}^{{t}_{2}}[frac{1}{2}{[p+{A}_{p}(t)+{A}_{g}(t+tau )]}^{2}-frac{1}{2}{p}^{2}]{rm{d}}t approx mathop{underbrace{{int }_{{t}_{1}}^{{t}_{2}}[frac{1}{2}{[p+{A}_{p}(t)]}^{2}-frac{1}{2}{p}^{2}]{rm{d}}t}}limits_{=S(p,,{t}_{2},{t}_{1};{A}_{p}(t))}+mathop{underbrace{{int }_{{t}_{1}}^{{t}_{2}}[p+{A}_{p}(t)]{A}_{g}(t+tau ){rm{d}}t}}limits_{equiv Delta S(p+{A}_{p}(t),{t}_{2},{t}_{1};{A}_{g}(t+tau ))}finish{array}$$

(10)

Right here the sq. time period of *A*_{g}(*t* + *τ*) is ignored, as its contribution is negligible in contrast with the opposite phrases. Equation (10) implies that the gate area introduces an additional part of (Delta S(p+{A}_{{rm{p}}}(t),{t}_{2},{t}_{1},;{A}_{{rm{g}}}(t+tau ))) to the gate-free case. Because of this, equation (9) might be rewritten as:

$${psi }_{{rm{r}}}^{({rm{g}})}({t}_{{rm{r}}},tau ),approx {int }_{-infty }^{{t}_{{rm{r}}}}{rm{d}}{t}^{{prime} }int {rm{d}}{p}^{{prime} }g({p}^{{prime} }+{A}_{{rm{p}}}({t}_{{rm{r}}})){E}_{{rm{p}}}({t}^{{prime} })d({p}^{{prime} }+{A}_{{rm{p}}}({t}^{{prime} }))instances exp left[-{rm{i}}S({p}^{{prime} },{t}_{{rm{r}}},{t}^{{prime} }{rm{;}}{A}_{{rm{p}}}(t))-{rm{i}}frac{{p}^{2}}{2}({t}_{{rm{r}}}-{t}^{{prime} })+{rm{i}}phi {t}^{{prime} }right]exp left[-{rm{i}}Delta S({p}^{{prime} }+{A}_{{rm{p}}}(t),{t}_{{rm{r}}},{t}^{{prime} }{rm{;}}{A}_{{rm{g}}}(t+tau ))right]$$

(11)

Word that if the final part time period e^{−iΔS} was lacking, the expression can be equivalent to equation (7) and, thus, the gate-free recollision wave operate *ψ*_{r}(*t*_{r}). Therefore it will clearly be handy to take away the e^{−iΔS} time period from the integrals, as this could allow to specific the recolliding electron wave packet ({psi }_{{rm{r}}}^{({rm{g}})}({t}_{{rm{r}}},tau )) in equation (8) by means of the unperturbed counterpart *ψ*_{r}(*t*_{r}) and an additional part. To proceed additional with this concept, we think about two approximations.

First, following the well-known saddle-point approximation, the dominant contribution within the integration over the canonical momenta *p*′ is offered by the kinetic momentum *p*′ + *A*_{p}(*t*) that equals the kinetic momentum *p*_{r} of the recolliding electron on the floor. Therefore the kinetic momentum time period in Δ*S* could also be approximated as (Delta S({p}^{{prime} }{+A}_{{rm{p}}}(t),{t}_{{rm{r}}},{t}^{{prime} }{rm{;}}{A}_{{rm{g}}}(t+tau ))approx Delta S({p}_{{rm{r}}},{t}_{{rm{r}}},{t}^{{prime} }{rm{;}}{A}_{{rm{g}}}(t+tau ))).

Second, as a result of the exponent time period of the additional part e^{−iΔS} is oscillating slowly in contrast with e^{−iS} within the time integration over *t*′ in equation (11), the additional part Δ*S* might be approximated by a time common (overline{Delta S}) inside a time window Δ*t*:

$$overline{Delta S}=frac{1}{int }_{{t}_{{rm{r}}}-Delta t}^{{t}_{{rm{r}}}}Delta S({p}_{{rm{r}}},{t}_{{rm{r}}},{t}^{{prime} },{A}_{{rm{g}}}(t+tau )){rm{d}}{t}^{{prime} }$$

(12)

As our fundamental aim is to reconstruct attosecond electron wave packets that contribute to the spectral cutoff, we select Δ*t* because the time interval between ionization and recollision of the classical backscattering trajectory that ends in the best ultimate kinetic vitality. To guage the averaged part (overline{Delta S}) and thereby simplify the analytical type of equation (12), Δ*S* might be expressed as:

$$start{array}{c}Delta S({p}_{{rm{r}}},{t}_{{rm{r}}},{t}^{{prime} },{A}_{{rm{g}}}(t+tau ))={int }_{{t}^{{prime} }}^{{t}_{{rm{r}}}}{p}_{{rm{r}}}{A}_{{rm{g}}}(t+tau ){rm{d}}t=-{int }_{{t}_{{rm{r}}}}^{{rm{infty }}}{p}_{{rm{r}}}{A}_{{rm{g}}}(t+tau ){rm{d}}t ,,,,,+{int }_{{t}^{{prime} }}^{{rm{infty }}}{p}_{{rm{r}}}{A}_{{rm{g}}}(t+tau ){rm{d}}have a tendency{array}$$

(13)

By inserting equation (13) into equation (12), the efficient (averaged) part variation (overline{Delta S},) can now be evaluated:

$$start{array}{l}overline{Delta S}=-{int }_{{t}_{r}}^{infty }{p}_{r}{A}_{g}(t+tau ){rm{d}}t+{int }_{{t}_{r}}^{infty }{p}_{r}mathop{underbrace{left[frac{1}{Delta t}{int }_{-Delta t}^{0}{A}_{g}(t+t{prime} +tau ),{rm{d}}t{prime} right]}}limits_{equiv {bar{A}}_{g}^{(b)}(t+tau )}{rm{d}}t ,=-{int }_{{t}_{r}}^{infty }{p}_{r}[{A}_{g}(t+tau )-{bar{A}}_{g}^{(b)}(t+tau )],{rm{d}}t ,=-Delta {rm{S}}({p}_{r},,infty ,{{rm{t}}}_{{rm{r}}};{A}_{g}(t+tau )-{bar{A}}_{g}^{(b)}(t+tau ))finish{array}$$

(14)

by which ({bar{A}}_{{rm{g}}}^{({rm{b}})}(t)) is outlined as:

$${bar{A}}_{{rm{g}}}^{({rm{b}})}(t)=frac{1}{Delta t}{int }_{-Delta t}^{0}{A}_{{rm{g}}}(t+{t}^{{prime} }){rm{d}}{t}^{{prime} }$$

(15)

Utilizing the above-described approximations now permits to tug the additional part time period out of the integrations in equation (11) and, contemplating the signal flip (overline{Delta S}to -overline{Delta S}) on the backscattering occasion, the perturbed recolliding electron wave packet might be expressed by means of the gate-free wave packet and the additional averaged part time period ({psi }_{{rm{r}}}^{({rm{g}})}({t}_{{rm{r}}},tau )approx {psi }_{{rm{r}}}({t}_{{rm{r}}}){{rm{e}}}^{{rm{i}}overline{Delta S}}) in equation (8).

We now transfer on to debate how the electron wave packet might be described on the finish of the interplay within the presence of the pump and gate fields (equation (8)). Taking the outcomes of equations (11)–(14) and restoring *p*_{r} with the kinetic momentum *p* + *A*_{p}(*t*), equation (8) might be rewritten as:

$${widetilde{psi }}_{{rm{t}}}(p,tau )propto {rm{i}}{int }_{-infty }^{infty }{psi }_{{rm{r}}}({t}_{{rm{r}}})exp left[{rm{i}}frac{{p}^{2}}{2}{t}_{r}right]exp left[-{rm{i}}S(p,infty ,{t}_{{rm{r}}}{rm{;}}{A}_{{rm{p}}}(t)+{A}_{{rm{g}}}(t+tau ))right]instances exp [-{rm{i}}Delta S(p+{A}_{{rm{p}}}(t),infty ,{t}_{{rm{r}}}{rm{;}}{A}_{{rm{g}}}left(t+tau right)-{bar{A}}_{{rm{g}}}^{({rm{b}})}(t+tau ))]{rm{d}}{t}_{{rm{r}}}$$

(16)

As a result of the mixing ranges for *S* and Δ*S* are equivalent (from *t*_{r} to ∞), the 2 phases might be merged right into a single equation (*S*′ = *S* + Δ*S*),

$$start{array}{c}S{prime} (p,infty ,{t}_{r},tau )={int }_{{t}_{r}}^{infty }[frac{1}{2}{[p+{A}_{p}(t)+{A}_{g}(t+tau )]}^{2}-frac{1}{2}{p}^{2}]{rm{d}}t ,,,+,{int }_{{t}_{r}}^{infty }(p+{A}_{p}(t))({A}_{g}(t+tau )-{bar{A}}_{g}^{(b)}(t+tau )){rm{d}}t ,,,approx ,{int }_{{t}_{r}}^{infty }[frac{1}{2}{[p+{A}_{p}(t)+mathop{underbrace{2{A}_{g}(t+tau )-{bar{A}}_{g}^{(b)}(t+tau )}}limits_{equiv {A}_{HAS}(t+tau )}]}^{2} ,,,-,frac{1}{2}{p}^{2}]{rm{d}}t=S(p,infty ,{t}_{r},;{A}_{p}(t)+{A}_{HAS}(t+tau ))finish{array}$$

(17)

by which *A*_{HAS}(*t*) is hereafter known as the efficient HAS vector potential and reads:

$${A}_{{rm{HAS}}}(t)=2{A}_{{rm{g}}}(t)-{bar{A}}_{{rm{g}}}^{({rm{b}})}(t)$$

(18)

This expression of the efficient HAS vector potential is appropriate with the classical momentum accumulation in the course of the tour from the ionization to the detection, (Delta p=-,{rm{e}}left[2A({t}_{{rm{r}}})-A({t}_{{rm{r}}}-Delta t)right]), underneath the rescattering situation, ({int }_{{t}_{{rm{r}}}-Delta t}^{{t}_{{rm{r}}}}{A}_{{rm{g}}}(t){rm{d}}t=Delta t{A}_{{rm{g}}}({t}_{{rm{r}}}-Delta t))(refs. ^{11,36,63}). Utilizing equation (17), the terminal electron amplitude ({widetilde{psi }}_{{rm{t}}}(p,tau )) might be expressed as:

$$start{array}{c}{widetilde{psi }}_{{rm{t}}}(p,tau )propto {rm{i}}{int }_{-infty }^{infty }{psi }_{{rm{r}}}({t}_{{rm{r}}}),exp ,left[{rm{i}}frac{{p}^{2}}{2}{t}_{{rm{r}}}right],exp [-{rm{i}}S(p,infty ,{t}_{{rm{r}}}{rm{;}}{A}_{{rm{p}}}(t) ,,+{A}_{{rm{HAS}}}(t+tau ))]{rm{d}}{t}_{{rm{r}}}finish{array}$$

(19)

Equation (19) implies that the gate additionally contributes to the terminal momentum of the electron wave packet by *A*_{HAS}(*t*_{r} + *τ*), which will depend on the time delay *τ*. As described in equation (5), the momentum contribution *A*_{p}(*t*) of the pump area is already integrated within the gate-free terminal electron spectral amplitude ({widetilde{psi }}_{{rm{t}}}(p)), whose depth is straight accessible in experiments. Due to this fact it’s handy for the evaluation of HAS knowledge to specific equation (19) with the terminal type of ({widetilde{psi }}_{{rm{t}}}(p).) We decompose the part in equation (19) by way of (S(p,{t}_{2},{t}_{1},;{A}_{{rm{p}}}(t)+{A}_{{rm{g}}}(t+tau ))approx S(p,{t}_{2},{t}_{1},;{A}_{{rm{p}}}(t))+Delta S(p+{A}_{{rm{p}}}(t),{t}_{2},{t}_{1},;{A}_{{rm{HAS}}}(t+tau ))) and rewrite equation (19) as:

$${widetilde{psi }}_{{rm{t}}}(p,tau )propto {rm{i}}{int }_{-infty }^{infty }{psi }_{{rm{r}}}({t}_{{rm{r}}})exp left[{rm{i}}frac{{p}^{2}}{2}{t}_{{rm{r}}}right]exp left[-{rm{i}}S(p,infty ,{t}_{{rm{r}}}{rm{;}}{A}_{{rm{p}}}(t))right]instances exp left[-{rm{i}}Delta S(p+{A}_{{rm{p}}}(t),infty ,{t}_{{rm{r}}}{rm{;}}{A}_{{rm{HAS}}}(t+tau ))right]{rm{d}}{t}_{{rm{r}}}$$

(20)

In analogy to equation (11), if the e^{−iΔS} time period vanishes, the above equation is equivalent to equation (5), which hyperlinks the recolliding electron wave packet *ψ*_{r}(*t*) to the terminal spectral amplitude ({widetilde{psi }}_{{rm{t}}}(p)). Right here, just like the Fourier illustration of the recollision wave packet, we outline the Fourier pair of the terminal electron wave packet as:

$$start{array}{c}{widetilde{psi }}_{{rm{t}}}(p)equiv ,{int }_{-infty }^{infty }{rm{d}}t,{psi }_{{rm{t}}}(t),exp left[ifrac{{p}^{2}}{2}tright], ,{psi }_{{rm{t}}}(t)equiv ,{int }_{-infty }^{infty }p{rm{d}}p,{widetilde{psi }}_{{rm{t}}}(p),exp left[-ifrac{{p}^{2}}{2}tright]finish{array}$$

(21)

Word that the terminal electron wave packet *ψ*_{t}(*t*) is an auxiliary electron wave packet that incorporates time–construction data of the recolliding electron wave packet *ψ*_{r}(*t*) on the recollision floor with the momenta translated by the Volkov propagation with the exponent (exp left[-{rm{i}}S(p,infty ,{t}_{{rm{r}}}{rm{;}}{A}_{{rm{p}}}(t))right]) (see equations (1) and (5)), however with out the part from house propagation to the detection. Utilizing the terminal electron wave packet *ψ*_{t}(*t*) (equation (21)), the terminal electron spectral amplitude (equation (20)) might be additional simplified as:

$${widetilde{psi }}_{{rm{t}}}(p,tau )propto {rm{i}}{int }_{-infty }^{infty }{psi }_{{rm{t}}}({t}_{{rm{r}}})exp left[{rm{i}}frac{{p}^{2}}{2}{t}_{{rm{r}}}right]exp left[-{rm{i}}S(p,infty ,{t}_{{rm{r}}}{rm{;}}{A}_{{rm{HAS}}}(t+tau ))right]{rm{d}}{t}_{{rm{r}}}$$

(22)

underneath the situation that the variation of the vector potential is weak in the course of the time window of the recollision. The HAS spectrogram equation then reads:

$$I(p,tau )={left|{widetilde{psi }}_{{rm{t}}}(p,tau )proper|}^{2}propto {left|{int }_{-infty }^{infty }{psi }_{{rm{t}}}({t}_{{rm{r}}})exp left[{rm{i}}frac{{p}^{2}}{2}{t}_{{rm{r}}}right]exp left[-{rm{i}}S(p,infty ,{t}_{{rm{r}}}{rm{;}}{A}_{{rm{HAS}}}(t+tau ))right]{rm{d}}{t}_{{rm{r}}}proper|}^{2}$$

(23)

Equation (23) describes a spectrogram whose reconstruction permits entry to the ultimate electron wave packet *ψ*_{t}(*t*) and, correspondingly, ({widetilde{psi }}_{{rm{t}}}(p)) in addition to *A*_{HAS}(*t*).

#### The efficient HAS vector potential *A*

_{HAS}(*t*)

An express relationship between the incident gate vector potential *A*_{g}(*t*) and the efficient HAS vector potential *A*_{HAS}(*t*) might be finest understood within the Fourier area. Utilizing the Fourier growth, ({A}_{{rm{g}}}(t)={int }_{-{rm{infty }}}^{{rm{infty }}}{rm{d}}omega {widetilde{A}}_{{rm{g}}}(omega ){{rm{e}}}^{{rm{i}}omega t}), the efficient HAS vector potential might be expressed as,

$${A}_{{rm{HAS}}}(t)=2{int }_{-{rm{infty }}}^{{rm{infty }}}{widetilde{A}}_{{rm{g}}}(omega ){{rm{e}}}^{iomega t}{rm{d}}omega -frac{1}{Delta t}{int }_{-Delta t}^{0}{int }_{-{rm{infty }}}^{{rm{infty }}}{widetilde{A}}_{{rm{g}}}left(omega proper){e}^{iomega (t+{t}^{{prime} })}{rm{d}}omega {rm{d}}t{prime} ={int }_{-{rm{infty }}}^{{rm{infty }}}{widetilde{A}}_{{rm{g}}}left(omega proper)left[2-frac{i}{omega Delta t}({e}^{-iomega Delta t}-1)right]{e}^{iomega t}{rm{d}}omega ={int }_{-{rm{infty }}}^{{rm{infty }}}{widetilde{A}}_{{rm{g}}}left(omega proper)widetilde{g}(omega ){e}^{iomega t}{rm{d}}omega $$

(24)

by which the newly launched multiplier (widetilde{g}(omega )) is outlined as:

$$widetilde{g}(omega )=left[2-frac{{rm{i}}}{omega Delta t}({{rm{e}}}^{-{rm{i}}omega Delta t}-1)right]$$

(25)

As proven in equation (24), the Fourier parts of the efficient HAS vector potential ({widetilde{A}}_{{rm{HAS}}}(omega )) is said to these of the incident gate vector potential ({widetilde{A}}_{{rm{g}}}(omega )) by multiplication of (widetilde{g}(omega ))

$${widetilde{A}}_{{rm{HAS}}}(omega )={widetilde{A}}_{{rm{g}}}(omega )widetilde{g}(omega )$$

(26)

The gate multiplier (widetilde{g}(omega )) is unbiased from ({widetilde{A}}_{{rm{g}}}(omega )). This enables the potential of the entire characterization of *A*_{g}(*t*) from *A*_{HAS}(*t*) imprinted in a HAS spectrogram.

To higher visualize the idea of *A*_{HAS}(*t*) and to confirm the validity of the assumptions used within the above derivation, a semiclassical simulation of a HAS spectrogram was carried out utilizing single-cycle pulses. The photoelectron spectrum cutoff vitality variation evaluated by the HAS spectrogram is in contrast with the efficient HAS vector potential *A*_{HAS}(*t*) calculated utilizing equation (24) (Prolonged Information Fig. 7). The multiplier (widetilde{g}(omega )) will depend on the tour time Δ*t* between ionization and the backscattering occasion of the highest-energy electron. On the premise of the well-established recollision mannequin, 0.685 instances the central tour interval^{64,65} (central interval of (E(omega )/left.{omega }^{2}proper))), which corresponds to about 0.85*T*_{L}, was used for Δ*t* to guage (widetilde{g}(omega )). Right here *T*_{L} is the centroid interval of the laser pulse. Prolonged Information Fig. 7b–d exhibits that the cutoff vitality variation in a HAS spectrogram intently follows *A*_{HAS}(*t*) (black curve), as calculated by the unmodified vector potential of the incident pulse (dashed crimson curve), whatever the carrier-envelope part.

### Retrieval of the vector potential *A*

_{g}(*t*) from a HAS spectrogram

The above dialogue means that, by tracing the variation of the cutoff vitality in a HAS spectrogram, we are able to acquire *A*_{HAS}(*t*) (crimson curves in Prolonged Information Fig. 8a,b). Due to this fact entry to the Fourier parts of the efficient HAS vector potential permits the characterization of the vector potential of the incident gate: ({widetilde{A}}_{{rm{g}}}(omega )={widetilde{g}}^{-1}(omega ){widetilde{A}}_{{rm{HAS}}}(omega )) (Prolonged Information Fig. 8c,d). The retrieved incident vector potential *A*_{g}(*t*) is proven in blue in Prolonged Information Fig. 8b.

#### Identification of absolutely the zero delay in a HAS spectrogram

Identification of the zero delay between pump and gate pulses in a HAS spectrogram might be obtained with varied strategies. Right here we opted for a technique that permits absolutely the delay to be derived straight from the HAS spectrogram. Regardless that the distinction between the intensities of pump and gate pulses is greater than two orders of magnitude, discernible modulations (roughly 5–10%) of the spectral amplitude of the spectrogram stay. In a HAS spectrogram, the entire photoelectron yield variation might be evaluated by spectral integration at every delay level (Prolonged Information Fig. 8e). Absolutely the zero-delay level might be discovered because the delay level at which the yield is maximally diversified (vertical dashed line in Prolonged Information Fig. 8e).

#### Benchmarking HAS by means of EUV attosecond streaking

EUV attosecond streaking gives entry to the detailed area waveform of a pulse^{33,34,35}. As a result of this system of area characterization is built-in in our experimental setup, it permits us to benchmark HAS as a field-characterization technique.

Prolonged Information Fig. 9a,b exhibits the HAS and EUV attosecond streaking measurements, respectively. The vector potential waveform of the incident gate pulse retrieved from the cutoff evaluation in HAS (crimson curve in Prolonged Information Fig. 9c) and that from EUV attosecond streaking (blue curve in Prolonged Information Fig. 9c) present wonderful settlement, as verified by the diploma of similarity of about 0.95 (ref. ^{43}), and help the notion that the gate pulse certainly acts as a part gate.

### The gate pulse as a part gate

The compact description of HAS as a spectrogram implied by equations (2) and (23) assumes that the weak duplicate of the pump area acts as an almost pure part gate on the electrons launched by the pump. In different phrases, it will possibly modify the momentum of electrons launched by the pump area however doesn’t enormously affect the method of electron ionization. But, except the ionization nonlinearities are effectively understood (as an illustration, in atoms), a theoretical estimate of the required ratio between pump and gate pulses for attaining a sufficiently pure part gate requires experimental validation.

To determine protected limits inside which the above situation is met, we carried out HAS measurements underneath completely different gate strengths and in contrast the vector potential waveforms extracted from HAS to these characterised by EUV attosecond streaking. As proven in Prolonged Information Fig. 10a,b, the vector potential waveforms from two methods obtain finest settlement at low gate/pump depth ratio (*η* < 10^{−2}). At increased depth ratios, we observe a progressively rising disagreement between the reconstructed waveforms with the 2 strategies (Prolonged Information Fig. 10c,d), implying that the gate pulse not serves as a weak perturbation. These measurements recommend that, for the studied system, HAS measurements require a gate pulse whose depth is about 10^{−2} decrease than the pump depth.

### HAS reconstruction methodology

On the first stage of the reconstruction of the HAS spectrogram, the terminal electron wave packet *ψ*_{t}(*t*) is retrieved, as its spectral depth ({left|{widetilde{psi }}_{{rm{t}}}(p)proper|}^{2}) might be straight obtained by a gate-free photoelectron spectrum. Due to this fact the reconstruction downside is diminished to retrieval of the spectral part.

As motivated in the primary textual content, we remoted a spectral space of curiosity (AOI) from 80 to 230 eV (Prolonged Information Fig. 11a,b). With this area of curiosity, the terminal wave packet might be expressed as:

$${psi }_{{rm{t}}}^{({rm{AOI}})}(t)={int }_{-{rm{infty }}}^{{rm{infty }}}left|{widetilde{psi }}_{{rm{t}}}^{({rm{AOI}})}(omega )proper|{{rm{e}}}^{-{rm{i}}{varphi }left({rm{omega }}proper)}{{rm{e}}}^{{rm{i}}omega t}{rm{d}}omega $$

(27)

Right here *φ*(*ω*) is the spectral part of the electron wave packet modelled as a polynomial collection as much as the sixth order,

$$varphi (omega )=mathop{sum }limits_{n}^{N=6}{D}_{n}{(omega -{omega }_{{rm{c}}})}^{n}$$

(28)

by which *D*_{n} and *ω*_{c} are the *n*th order dispersion and central frequency, respectively. The reconstruction relies on a least-squares algorithm written in MATLAB, which goals on the whole minimization of the distinction among the many experimental (Fig. 4a and Prolonged Information Fig. 11a) and reconstructed spectrogram (Fig. 4b and Prolonged Information Fig. 11b). To additional enhance the constancy of the reconstruction, we additionally concurrently match the differential map *D*(*E*, *τ*) of a HAS spectrogram *I*(*E*, *τ*), which is outlined as:

$$D(E,tau )=int frac{partial I(E,tau )}{partial tau }{rm{d}}tau $$

(29)

The differential map is beneficial as a result of it will possibly get rid of the unmodulated depth alongside the delay axis and permits the retrieval algorithm to reconstruct high-quality particulars of the experimental hint (Prolonged Information Fig. 11c–e). As an preliminary guess for the part, zero part was used. The retrieval of the terminal electron wave packet is proven in Prolonged Information Fig. 11f,g.

In a subsequent stage of the reconstruction, the recolliding electron pulse, which is the important thing amount on this work, is evaluated by the inverse Volkov propagation of the retrieved terminal electron wave packet as:

$${psi }_{{rm{r}}}^{({rm{AOI}})}(t)={int }_{-infty }^{infty }{rm{d}}p,{widetilde{psi }}_{{rm{t}}}^{({rm{AOI}})}(p)exp ,left[-{rm{i}}frac{{p}^{2}}{2}tright],exp left[{rm{i}}S(p,infty ,t{rm{;}}{A}_{{rm{p}}}(t))right]$$

(30)

The Volkov foundation is reconstructed by exactly measuring the pump-field waveform and its timing with respect to the emission. The retrieved recolliding electron pulse is proven in Fig. 4.