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SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHOTON.2013.288

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Supplementary Information for

„Carrier-envelope phase effects on the strong-field photoemission of electrons from metallic nanostructures“

Björn Piglosiewicz1,2, Slawa Schmidt1,2, Doo Jae Park1,2, Jan Vogelsang1,2, Petra Groß1,2,

Cristian Manzoni3, Paolo Farinello3, Giulio Cerullo3, and Christoph Lienau1,2*

1 Institut für Physik, Carl von Ossietzky Universität, 26129 Oldenburg, Germany 2 Center of Interface Science, Carl von Ossietzky Universität, 26129 Oldenburg, Germany

3 IFN-CNR, Dipartimento di Fisica, Politecnico di Milano, 20133 Milano, Italy

*Corresponding author: [email protected]

1. Ultrashort NIR pulse generation with passive CEP stabilization

In our experiments, few-cycle pulses at a wavelength around 1.3 to 1.9 m and with a stable and con-trollable carrier-envelope phase (CEP) are used for the generation and acceleration of electron pulses from sharp metal tips. Such pulses are generated by difference frequency generation (DFG) of pulses from two non-collinear optical parametric amplifier (NOPA) stages, which are both pumped by the same femtosecond laser and seeded by the same white light. This concept ensures that the output of the two NOPAs displays the same shot-to-shot CEP variation, which cancels out during the DFG pro-cess1,2.

The experimental setup of the NOPA-DFG system is based on the one presented in Ref. 1 and is shown schematically in Fig. S1. Pulses from an amplified Titanium:Sapphire (Ti:Sa) regenerative amplifier (Newport, Spitfire Pro) with an energy of 0.5 mJ, 120-fs duration, 1-kHz repetition rate, and with a center wavelength of 800 nm are split into three parts to form the pump sources for the white light (WL) generation and, after frequency doubling, for the two NOPA stages. WL is generated in a 2-mm thick sapphire plate. A pulse energy of 1.5 J from the Ti:Sa is sufficient to sustain a single, stable filament. The WL is split into two parts using a reflective neutral density grey filter (NDF), and each part is overlapped with a pump pulse in a nonlinear crystal. The first NOPA stage (NOPA#1) is pumped by 60 J of second harmonic (SH) generated in a 1-mm thick, type-I -barium borate (BBO) crystal ( = 29°). The NOPA#1 stage consists of a 1-mm thick, type-I BBO crystal cut at = 32°, to sustain broad-bandwidth amplification in a spectral range from 540 nm to 650 nm. The pulses from NOPA#1 are compressed to nearly transform-limited 7-fs duration by chirped mirrors. The second NOPA stage (NOPA#2) is pumped by 80 J of SH generated in a 1-mm thick, type-I BBO crystal ( = 29°). A spectral range from 870 nm to 890 nm is selected for amplification in a 1-mm thick, type-I-cut BBO crystal ( = 29°) by tuning the phase-matching angle. Finally, the output pulses of both NOPA stages are collinearly aligned and are temporally overlapped in a 0.3-mm thick, type-II cut BBO crystal ( = 32°, = 30°) to generate the difference frequency (DF), which results in pulses as short as 14 fs, tunable between 1.3 and 1.9 m, and with a pulse energy of up to 220 nJ.

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SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHOTON.2013.288

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Fig. S1: Schematic of the experimental setup displaying the phases of the pulses at relevant posi-tions. Constant phase delays due to propagation have been omitted. The inset depicts energy level schemes of the involved frequency conversion processes and their frequency and phase preserva-tion laws.

In order to explain the passively achieved CEP stability of the generated DF pulses, we follow the phases of the generated pulses through the relevant frequency conversion stages. Denoting the CEP of the fundamental Ti:Sa pulses with F, this phase is not constant but varies over time, as the oscillator is not CEP stabilized. WL generation in solid state media is dominated by intra-pulse four-wave mixing (FWM) and self-phase modulation (SPM), which can be regarded as degenerate FWM processes3. These processes are expected to be phase preserving4,5, and the phase of the driving pulse is thus passed on to the WL, offset only by –/2 due to the frequency conversion process6 and a constant phase delay cWL acquired during propagation, yielding

WL = F – /2 + cWL. (1)

The generated WL serves as the seed in the two NOPA stages, each of which is pumped by the SH of the Ti:Sa pulses with SH = 2F. The NOPA process can be described as a DFG process, where the pump pulses are down-converted to generate light at the WL seed frequency WL and at the idler fre-quency I = SH - WL (compare inset in Fig. S.1). In this process, the phase of the amplified WL pulses is fixed (again, except for a constant phase delay due to propagation) to

NOPA#1,2 = WL + c1,2, (2)

while the idler phase is coupled to the phase SH of the frequency-doubled pump pulses by the relation I = SH - WL – /2 + cI. In the final DFG stage, the NIR pulse for tip illumination is generated as DF = NOPA#1 - NOPA#2, and the according phase relationship is

DF = NOPA#1 - NOPA#2 – /2 + cDFG = c1 – c2 – /2 + cDFG. (3)

From Eq. 3 one can see immediately that any initial temporal fluctuations of the laser CEP cancel out during the DFG process. The CEP is governed only by the phase delay that is acquired during the propagation of the individual pulses. Neglecting noise that may occur during the frequency conversion




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stages, e.g., by intensity-to-phase coupling during WL generation, the CEP of our NOPA-DFG system is inherently stable, with the stability practically being limited only by drift and fluctuations of the optical beam path lengths.

2. CEP stability measurement by spectral interferometry in an f-to-2f interferometer

Fig. S2: Passive CEP stability and CEP control. a, Spectra (vertical) recorded over a time span of 10 min, leaving the CEP unchanged. The spectral fringes show a high stability. b, CEP extracted from a, giving a noise of 66 mrad (root-mean-square) over the interval of 10 min. c, spectral inter-ference fringes recorded over a time span of 60 s. The wedges for CEP control were inserted dur-ing the time span between 10 and 60 s. d, The CEP extracted from c shows the corresponding line-ar increase and a total shift over 8.8.

In order to show the inherent CEP stability of the NOPA-DFG system and to determine the residual CEP fluctuations, the CEP was measured using nonlinear spectral interferometry in a conventional f-to-2f interferometer7,8. Spectral interferometry was performed in the wavelength region between 820 and 980 nm by overlapping two pulses derived from the NOPA-DFG output pulses with a center wavelength of 1.65 m. One of the two pulses was generated by extending the bandwidth of the DFG pulses to shorter wavelengths through white-light generation, which is a CEP-preserving effect: WL = DF – /2. The second pulse was generated by frequency-doubling the DF pulses, thereby also dou-bling the CEP phase: SH = 2DF – /2. By overlapping the two pulses with a fixed temporal delay , an interference pattern emerges in the spectrum, which is sensitive to the phase difference of the two pulses, i.e., to the CEP of the NOPA-DFG system, DF:

)cos()()(2)()()( DFSHWLSHWL IIIII (4)

Experimentally, a 6-mm thick yttrium aluminum garnet (YAG) crystal is used to generate the WL continuum while the SH is generated in a 100-m thick type-I BBO crystal ( = 20.8°). YAG was chosen as the nonlinear material because it supports a WL filament already at relatively low pump power and because it has been demonstrated to generate a very flat SC extending well into the visible when pumped at MIR wavelengths9. WL and SH pulses are directed into a spectrometer (Acton Spec-traPro 2500i equipped with a Princeton Instruments PIXIS: 100BR camera). For determination of the passive CEP stability of the NOPA-DFG system output, the spectrum in the overlap region was ob-served over a time span of 10 min. During this time, a sequence of 600 individual spectra was record-ed (Fig. S2a), each being an average over 1000 laser shots. The stability of the fringe pattern is excel-lent, and is disturbed mainly by intensity fluctuations of the NOPA-DFG system rather than by phase




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fluctuations. We have extracted the CEP variation from the spectral interference pattern (Fig. S2b) and have found fluctuations as small as 66 mrad (root-mean-square) during the 10-min-interval.

With such a good passive stability, the CEP could easily be controlled by simply inserting dispersive material (fused silica) via DC-step motor-driven wedges. We used a step-size of 0.1 mm, increasing the inserted wedge thickness by 7 m per step, and integrated the spectra over 500 ms for every step. The spectra shown in Fig. S2c (also Fig. 1d in the manuscript) were recorded over a time span of 60 s. After 10 s the wedges were inserted by one step per second, such that the glass thickness increased by 350 m during 50 s. The CEP extracted from the spectral interference fringes at a wavelength of 875 nm (Fig. S2d) displays a linear shift over 8.8, corresponding to a shift of 0.18/step or of 2 per 79.3 m of inserted material. Based on the dispersion relation of fused silica10, a glass thickness of 74 µm glass is changing by 2 at a wavelength of 1.75 m, which compares well with the experimen-tally obtained value of 79.3 µm. The stability and the high contrast of the interference pattern shown in Fig. S2 clearly demonstrate the good shot-to-shot stability of the CEP stabilization mechanism in the DFG process of the NOPA.

3. Strong-field photoemission from sharp metallic tips

In atomic and molecular physics, the Keldysh parameter plays a dominant role for the electron dy-namics in the presence of ultrashort and strong laser fields. A value of unity marks the transition be-tween multiphoton ionization and weak-field above threshold ionization ( 1 ) and various strong-field phenomena ( 1 ) such as high harmonic generation and attosecond pulse generation. During recent years, these strong-field phenomena have been of utmost interest as they are at the heart of the entire field of attoscience.

In metallic nanostructures, the electron dynamics induced by ultrashort pulse photoemission are more involved. It is now known that the dynamics and the resulting angle-resolved photoemission spectra are largely governed by the two different and in principle independently tunable parameters, the Keldysh parameter , and the spatial adiabaticity parameter /F ql l , where lF is the near-field de-

cay length and lq is the electron quiver amplitude11-14.

Distinctly different electron dynamics are observed when tuning and/or from values above unity to below unity. Four different regimes can be distinguished, as schematically shown in Fig. S3.

1. For 1 and 1 , multiphoton ionization (MPI) and above threshold ionization (ATI) pre-vail and exponentially decaying photoemission spectra, possibly superimposed by different above threshold emission peaks are observed15,16.

2. For 1 and 1 , the quiver motion of the photoemitted electrons within the spatially es-sentially homogeneous optical field becomes more important. This gives rise to recollisions with the nanostructure, resulting in pronounced plateaus in the photoelectron spectra with a high-energy cutoff of ~10 pU (cf. Fig. S4). Here pU is the ponderomotive energy (see Sec. 5)

3. For 1 and 1 , the ponderomotive energy is weak in comparison to the work function. MPI and ATI lead to electron emission from the metallic nanostructure. Spatial field gradients in the near field of the nanostructure exist but the photoemitted electrons are only weakly ac-celerated along the field lines. Therefore, the effect of the laser field on the electron spectra is weak and spectra similar to those in case (1) are expected. This regime has not been studied in much detail and is likely to be seen when using long-wavelength (THz) radiation for photoe-mission.




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4. For 1 and 1 , strong-field tunnel ionization is thought to dominate the emission pro-cess. The photoemitted electrons are strongly accelerated along the field lines of the optical driving field, locally enhanced due to the presence of the metallic nanostructure. This “sub-cycle” regime has been discovered only recently11-14 and is unique to solid state nanostruc-tures, which give rise to pronounced local field enhancement and, correspondingly, strong field gradients. In this sub-cycle regime, the electron dynamics are fundamentally different from those known from strong-field experiments on atomic and molecular systems. Recolli-sions, a key feature in atomic and molecular systems, are suppressed. Instead, strong-field ac-celeration along the electric fields lines in the vicinity of the nanostructure dominates and characteristic plateau-like photoelectron spectra are observed.

Earlier experiments15 studying CEP effects on electron emission from a single tungsten tip have been performed in regime (1), for values of ~100 and ~ 3 14. In this regime, above threshold ionization prevails. The observed CEP effects are ascribed to changes of the interference pattern created by elec-trons emitted in subsequent cycles, while the effect of the electric field amplitude on the kinetic energy spectra is weak.

In our experiments, we study, for the first time, CEP effects in the strong-field, sub-cycle regime. We observe clear variations in the energetic width of the plateau-like kinetic energy spectra that are char-acteristic for the sub-cycle regime. Our analysis indicates that the CEP-induced variations mainly re-flect the effect of the electric field amplitude on the electron acceleration, showing that we can use the electric field of the laser to directly control the motion of electrons emitted by metal nanostructures.

Fig. S3. Schematic illustration of the effect of the Keldysh parameter and the spa-tial adiabaticity parameter on electron emission from sharp metallic nanostruc-tures. The horizontal and vertical lines denote 1 and 1 , respectively. For

1 and 1 (lower left), multiphoton ionization (MPI) and above threshold ion-ization (ATI) prevail. For 1 and 1 (upper left), the quiver motion of the elec-trons and recollisons become more important. For 1 and 1 (lower right), MPI and ATI are followed eby weak-field acceleration along field line gradients. For 1 and 1 , tunnel-ionization and strong-field acceleration of sub-cycle electrons result in plateau-like energy spectra.




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4. Simulation of photoelectron spectra

The simulation of CEP dependent electron kinetic energy spectra is based on the calculation of elec-tron trajectories within an extended semiclassical 3-dimensional Simpleman model11,15,17. Calculations are performed for both single-electron and multi-electron emission. The numerical implementation of the model follows three steps: First, the oscillating electric field ( , )E r t near the tip apex is derived.

Using the electric field strength directly at the tip apex, a finite number pN of electrons are generated

at randomly chosen positions in space and moments in time based on the Fowler-Nordheim photoe-mission model. The generation probability 0( , )Bp r t for photoemitted electrons is calculated as a func-

tion of the position on the tip 0r

and the birth time /B Bt (with the emission phase B and the laser frequency ). In the final step, the classical trajectories for each of the generated electrons are simulated under the action of the force field , ( , ) ,CF r t eE r t F r t . Here, e is the elementary

charge and ,CF r t denotes the sum of the repulsive Coulomb forces exerted by all other electrons and the attractive forces between the electron and image charges in the metallic tip. All forces are evaluated classically. For every electron, a mirror charge 0' /q e r R ( 0r : tip radius, R : distance of

the electron from the tip center) is placed at a distance 20' /R r R from the center of the spherically

shaped tip end. Possible effects of the Coulomb interaction on the photoemission yield have been ne-glected.

When calculating ( , )E r t , a quasi-static approximation is applied and the radiating far-field compo-

nent is neglected. Under this condition, the electric field ( , )E r t is described as follows:

2 20( , ) ( ) exp( 2ln2 / )cos( )CEPE r t E r t t . (5)

0 ( )E r is a 3-dimensional static electric field derived from existing analytic expressions18, assuming a

hyperbolic tip shape with a cone opening angle of = 30° and a variable tip radius 0r . The field is

obtained by applying the numerical derivative yx, to the electrostatic potential V 18. The parameter

is the experimentally measured pulse duration and CEP is the carrier-envelope phase which varies

in our simulation from 0 to 2 with 0.2resolution. The electrostatic potential V outside the tip is obtained by solving the Laplace equation in a prolate spheroidal coordinate system ( , ) in the vicin-ity of a hyperbolic tip:

( ) ( )V P P , (6)

where ( )P and ( )P are Legendre functions of fractional order . This fractional order is ob-tained from the boundary condition.

The electron generation probability 0( , )Bp r t is calculated by using a Fowler-Nordheim equation in quasi-static approximation:

23 3/2

00 0 2

0

( , ) 4 2( , ) ( , ) exp16 3 ( , )

BB B

B

e E r t mp r t E r te E r t

. (7)




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Here, 0( , )BE r t is the amplitude of the oscillating electric field at emission site 0r at birth time Bt .

The work function of the gold is taken as =5.5 eV 19, is the reduced Planck constant, m the elec-

tron rest mass, and 0( , )BE r t is the Heaviside step function introduced to restrict electron emission

to the negative phase of the electric field, when the electric force drives the electron away from the tip surface11.

To simulate the electron motion under the action of the (combined) electric field, the 3-dimensional classical equations of motion ( , ) ,i i c imr t eE r t F r t , 1,..., pi N , are integrated numerically for

all of the pN electrons generated by the laser pulse. The integration is performed by applying a 4th

order Runge-Kutta method with a temporal step size of <30 as (~1/180 cycle of the incident laser field) and an integration time of 30 fs which is sufficiently long to obtain the terminal electron veloci-ty.

From the terminal electron velocity , , , ,, ,i x i y i z ir r r r at the end of the simulation

run, both the terminal kinetic energy ,kin iE and the emission angles ,x i and ,y i are calculated for

each of the pN electrons in the non-relativistic limit as:

2, ,

12kin i iE m r , (8)

,1/ ,

/ ,tan z i

x y ix y i

r

r

. (9)

Using these relations, angle-resolved kinetic energy spectra ( , )kin xP E are calculated by averaging

over a sufficient number of simulations and over all emission angles ,| ( ) | 13y ir , as limited by the

detector geometry in our experiment.

Angle-integrated spectra for a given CEP are then obtained by numerically integrating ( , )kin xP E along the angle axis

( ) ( , )kin kin x xP E P E d . (10)

5. Field enhancement and near-field decay length

In this section, we briefly describe how we have estimated crucial parameters of our experiment, spe-cifically the local field enhancement factor at the tip apex and the near-field decay length from the data shown in Fig. 3 of the manuscript. These parameters serve as input parameters of the Simpleman simulations which are shown in Fig. 4 of the manuscript.

5.1 Field enhancement factor

The measured spectra show a distinctly plateau-like shape (Fig. 3(b)), where the broad width of the spectra reflects the strong-field acceleration of photoemitted electrons within the steep near-field gra-




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dient at the tip apex. The plateau-like shape is a clear indication of strong-field emission and the emer-gence of sub-cycle electrons. This means that the described experiment must take place in a certain regime which is classified by 1 and 1 . Here is the Keldysh parameter, which characterizes the transition from photon-driven (multi-photon ionization, 1 ) to field-driven (tunnelling, 1 )

photoemission.11,20 The adiabaticity parameter /F ql l is the ratio between near field decay length

Fl and the quiver amplitude ql . For 1 , the quiver motion of the electrons in the oscillating laser

field is suppressed and the electrons can escape the near field in a time shorter than one half-cycle of the field oscillation11,17.

The width of the plateau-like spectra gives a direct measure of the maximum kinetic energy acquired by the electrons when traversing the near field. The width of the spectra, cU , is directly proportional

to the ponderomotive Energy pU :

c pU U (11)

with the ponderomotive energy depending on the local electric field strength at the tip apex, F, and the carrier frequency of the applied laser field ,

2 2

24pe

e FUm

. (12)

In the sub-cycle regime, which is of interest here, recollisions of the photoemitted electrons with the tip are strongly suppressed11,13,14. The width cU therefore measures the maximum kinetic energy acquired by the non-recolliding electrons. If spatial field gradients are weak, the proportionality factor takes the usual value of 2 . For 1 , this value is reduced and we estimate 1.2( 0.2, 0.3) for the data in Fig. 312,14. From the measurement in Fig. 3 we find the width of the kinetic energy spec-tra, i.e., the difference between the high- and the low-energy cutoff, to be 10.5cU eV. With

1.2( 0.2, 0.3) we obtain 8.8( 2.1, 1.8)pU eV from Eq. 11 and estimate a local field strength

F from Eq. 12 of 16 2, 1F V/nm.

We can determine the peak electric field strength 0E applied to the tip during the experiment quite accurately from the external pulse parameters pulse energy, duration and focus diameter21 and yield

0 1.7VE nm . Comparison yields the field enhancement factor 0

Ff E to be 9 1f . This pro-

vides a rather direct measure of the field enhancement factor 9 1f in the strong-field, sub-cycle

regime. Since 1/f , the deduced enhancement factor is not very sensitive to the exact value of .

We thus conclude that the field enhancement factor 9f in the strong-field regime ( 10F V/nm) is large and close to the values deduced from analytical or numerical near field calculations, at least in the regime that we study in our work ( 18F V/nm). Also, the nonlinearity of the electron yield measured in the MPI regime (Fig. 2a), 2

0(( ) )nN fE , with 7.5 1n is close to what we expect from the work function of 5.0 to 5.5 eV and the photon energy of 0.75 eV without the need to introduce any intensity dependence of f .




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5.2 Field decay length

In earlier work it was found11,13 that the strong-field, sub-cycle regime is reached for 1F

q

ll

, i.e., if

the field decay length Fl becomes shorter than the quiver length ql . Hence, the quiver length can

serve as an upper limit for an estimation of the decay length. The quiver length is given by

2qe

e Flm

(13)

and, with 16VF nm , this amounts to 2.2ql nm .

For tips with smaller field enhancement factor, such as, e.g., tungsten tips, one would need to apply an even larger external laser field to reach the same value of the quiver length. This makes it more diffi-cult to reach the strong-field sub-cycle regime before tip destruction sets in. Also for tips with larger radii (Fig. S4), with a correspondingly larger near-field decay length, it becomes more difficult to reach the sub-cycle regime at these rather short near-infrared wavelengths.

In principle, this rather short near-field decay length may slightly be enlarged due to effect of the small static electric field (~ 1V/nm at the tip surface) arising from the small DC bias voltage (~ 50 V) ap-plied to our tips. Both experimentally and theoretically, we find that the bias field has a rather weak influence on the spectral shape of the kinetic energy spectra as long as the maximum field strength at the tip surface is so weak that DC tunnel-emission is negligible (< 5 V/nm, see e.g. Ref. 22). Such a DC bias field is included in our model simulations and we find that we can model the plateau-like energy spectra seen in Fig. 3a best if we assume a short near-field decay length of slightly shorter than 2 nm or, correspondingly, a tip radius 0r of 2 nm.

5.3. Effect of the tip radius on the electron emission spectra

To independently test the effect of the tip radius on electron spectra, we have performed a series of new control experiments on gold tips with a similar taper geometry, but with an increased radius of curvature of 15 – 20 nm, greatly increasing the decay length of the near-field at the taper apex. Repre-sentative kinetic energy spectra recorded for a laser pulse energy of 0.4 nJ, corresponding to an inci-dent field 0 2E V/nm, are shown in Fig. S4. In this regime, the energy dependence of the photoemis-sion spectra is fundamentally different from that observed for the sharp tips. We see a narrow peak at low energies and broad and weak plateau extending beyond 80 eV. The spectral shape is very reminis-cent of that seen in HHG spectra in atomic systems. The spectral shape is essentially quantitatively reproduced using the same Simpleman model that has also been used to explain the data in the main manuscript. The only parameter that has been adapted was the tip radius, which was increased to 15 nm. Increasing the tip radius increases the -parameter (here: 6 ) and brings the system from the sub-cycle into the quiver regime. Consequently, rescattering becomes important and the rescattered electrons gain much higher kinetic energy (up to 10 pU ) than the directly emitted quiver electrons (up to 2 pU ). This spectrum is qualitatively different from spectra recorded with sharper tips and for simi-lar local field strength (Fig. S4, inset, data taken from Ref. 12). We believe that the observation of this qualitative change of the spectral shape of the photoemission spectra by varying the tip shape (while keeping the local field strength essentially constant) provides strong experimental support for the as-sumption that the plateau-like kinetic energy spectra in Fig. 3 are indeed a unique signature of the sub-cycle regime.




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Fig. S4. a, Photoemission spectrum recorded for a gold tip with a radius of curvature of 15-20 nm. The tip is illuminated with short laser pulses centered at 1.65 µm. The pulse duration is 30 fs and the pulse energy about 400 nJ. Inset: Spectrum recorded for a sharp tip for simi-lar laser parameters. b, Simpleman simulation of the spectrum in a. Note the important con-tribution of rescattered electrons marked in red.

6. Influence of the CEP on the low- and high-energy cutoff

The simulated photoelectron spectra (Fig. 4 of the main manuscript) display qualitatively the same influence of the CEP as our experimental observations. When choosing a sufficiently short near-field decay length (~2 nm), the simulations account well for the plateau-like spectra and reveal the domi-nant contribution of the sub-cycle electrons, which explains the variation of the high-energy cutoff. In addition, the simulations show a weak out-of-phase modulation of the low-energy cutoff, similar to that seen in experiment.

The behavior of the high- and low-energy cutoff can be understood from a calculation displayed in Fig. S5, which shows the generation probability (red curves) and the terminal kinetic energy (black curves) as a function of the birth time of the electrons for 0CEP (Fig. S5a) and for CEP (Fig. S5b). The fastest electrons for 0CEP are those that are accelerated by the main laser field cycle. That can be either sub-cycle electrons generated at the beginning of the main cycle (between -1 and 0 fs), or electrons generated during the previous negative half-cycle (at -5 fs), which then recollide with the tip during the following positive half-cycle and obtain the main acceleration in the near field gradient during the complete main cycle. These electrons are potentially even faster than the sub-cycle electrons, however, there are only few of them due to the much lower generation probability.




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Fig. S5. Calculated terminal kinetic energy (black curves) and generation probability (red curves) as a function of the birth time of the electrons, a, for 0CEP and b, for CEP

For 0CEP , the low-energy cutoff is also governed by electrons emitted during the main negative half-cycle. Electrons emitted at about 0.5 fs are accelerated weakly, but traverse the near field far enough such that their acceleration during the following half cycles is negligible. The contribution of electrons emitted during the other cycles is too weak to significantly alter the low-energy cutoff.

For CEP (Fig. S5b), the electric field strength during the two center negative half-cycles is equal but much less than that of the main cycle for 0CEP . Consequently the maximum energy gained by sub-cycle acceleration is reduced in comparison with 0CEP and the high energy cutoff is lower. The low energy cutoff is governed by electrons emitted during the first cycle (around -2.5 fs). These electrons gain most of their energy via acceleration during the subsequent half cycle. Therefore, their kinetic energy is slightly higher than that of the slow electrons for 0CEP . Electrons emitted during the second cycle (at around 3 fs) can in principle have lower kinetic energy but the generation proba-bility for these slow electrons is low, as seen in Fig. S5b.

7. Effect of electron-electron interactions on the CEP-dependent photoelectron spectra

The section briefly discusses the possible influence of space-charge effects on strong-field photoemis-sion23 from sharp metallic tips. So far, these effects have not yet been analyzed in great detail, neither experimentally nor theoretically. Herink et al.11 conclude that space charge effects are insignificant since kinetic energy distributions measured at different wavelengths and constant emitted charge show a strong increase in cutoff energy with wavelength. This confirms that the dynamics are governed not by the total emitted charge, but rather by the properties of the driving field. We observe very similar dependencies in our experiments (see e.g., Fig. 2 in Ref. 13).

In order to further support those conclusions, we have analyzed the potential effect of Coulomb inter-actions on strong-field photoemission in the sub-cycle regime by means of extended multi-electron Simpleman simulations, using the model outlined in Section 4. In these calculations, we adopt a tip radius 0 2r nm and a pulse duration of 16 fs in order to best match the experimental conditions in




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Fig. 3. The simulations shown in Fig. S6 are performed for pN 1, 3, 10 and 30. Experimentally, we

measure maximum count rates of 1-3 electrons per pulse from these sharp tips, as compared to 20-50 electrons seen for less sharp tips with radii of curvature of 15-20 nm.

Fig. S6. Simulation of electron emission performed for a gold tip with 2-nm apex radius, illuminated with 18-fs laser pulses centered at 1.65 µm. The maximum field amplitude at the tip apex is 16 V/nm. The panels show angular resolved kinetic en-ergy spectra, i.e., the electron yield color-coded as a function of kinetic energy and the terminal emission angle, for an emission of 1, 3, 10, and 30 electrons per pulse.

With increasing number of electrons, two main effects are observed:

(i) For small numbers of electrons, the overall shape of the plateau-like spectrum does not change very much, but there is a slight increase in the high-energy cutoff due to electron-electron repulsion. For larger pN , the spectrum splits into two peaks: a fast-electron-peak

from electrons being pushed forward by a cloud of slower electrons and a slow-electron-peak from electrons which are decelerated by the cloud of faster electrons. Also both the low-energy side and the high energy side of the spectrum are increasingly broadened with increase in pN .

(ii) There is a clear increase in the angular distribution of the photoelectron spectra when in-creasing pN . The electrons are no longer accelerated along the field lines, the characteris-

tic of the sub-cycle regime, but their trajectories are getting randomized by electron-electron collisions.

Both of the space-charge effects are not (or at most weakly) observed in our data. In the experiment, we find rather sharp and steep edges of the photoemission spectra both on the low energy side (edge sharpness (10%-90%) less than 1-1.5 eV) and on the high energy side (edge sharpness (10%-90%) less than 2 eV). Such sharp edges of the plateau-like spectra are even seen when increasing the local field




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beyond 20 V/nm, enhancing the width of the plateau region to more than 15 eV (see e.g., Ref. 12). Al-so, we have recently demonstrated first angle-resolved strong-field photoemission spectra recorded in the sub-cycle regime13. These measurements explicitly demonstrate strong-field induced steering of the sub-cycle electrons along the field lines, as opposed to the diverging emission of quiver electrons. We therefore conclude that our experimental results reflect and are dominated by the effects of the local near-field at the tip apex on the electron motion. Electron interaction and space charge broaden-ing effects are weak. Even though they may slightly alter the shape of the photoemission spectra and in particular their angular distribution, their overall effect is much smaller than that of the strong nano-localized driving field – even for these very sharp tips studied in our work.

Fig. S7. Effect of electron-electron interactions on the CEP-dependent photoemission spectra. The parameters are the same as in Fig. S6, the number of photoemitted elec-trons per pulse is 1, 3, 10 and 30, respectively.

These conclusions are largely confirmed by studying the role of electron-electron interactions on the CEP dependence of the photoemission spectra. We find that for sharp tips and sufficiently short puls-es, a variation of carrier envelope phase of the laser pulse has a quite pronounced influence on the shape of the photoemission spectra when a sufficient number of electrons are emitted per pulse (Figs. S7 to S9). Essentially, for 0CEP , most electrons are emitted during one cycle and space charge broadening results in a large increase of the high-energy cutoff. The plateau-like spectrum seen in case of single electron emission is transformed into a double-peak structure with a clearly pronounced high-energy peak (s. Fig. S8 for kinetic energy spectra recorded for a tip with a radius of 2 nm). Since recol-lision effects are weak, the electron repulsion among slow and fast electrons tends to enhance the ki-netic energy of the fast electrons, resulting in the second high-energy peak.




14

Fig. S8. Photoemission spectra for a tip with a radius of curvature of 2 nm at a CEP phase of 0CEP . The parameters are the same as in Fig. S6.

For CEP , the electron cloud is launched in two subsequent cycles. Hence, the first bunch of elec-trons can already leave the near field within the first half-cycle before the second cloud of electrons is generated. This effectively reduces the electron-electron repulsion among electrons emitted in differ-ent cycles and thus results in a narrowing of the emission spectra in comparison to those observed for

0CEP (s. Fig. S9).

Fig. S9. Photoemission spectrum simulated for same parameters as in Fig. S8 at CEP .

These simulations predict that, for a sufficiently large number of electrons emitted per cycle, the ef-fects of electron-electron repulsion on the kinetic energy spectra depend sensitively on the CEP. The simulations also indicate that, in this regime, a CEP variation substantially alters the spectral shape of the kinetic energy spectra. In stark contrast, we observe in Fig. 3 plateau-like kinetic energy spectra with a shape that is largely independent on the CEP. This confirms our conclusion that the CEP effects seen in Fig. 3 are induced by the CEP-variation of the local near-field at the tip apex.




15

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