Strong-field coherent control of isolated attosecond pulse generation

Attosecond science promises to reveal the most fundamental electronic dynamics occurring in matter and it can develop further by meeting two linked technological goals related to high-order harmonic sources: improved spectral tunability (allowing selectivity in addressing electronic transitions) and higher photon flux (permitting to measure low cross-section processes). New developments come through parametric waveform synthesis, which provides control over the shape of field transients, enabling the creation of highly-tunable isolated attosecond pulses via high-harmonic generation. Here we demonstrate that the first goal is fulfilled since central energy, spectral bandwidth/shape and temporal duration of isolated attosecond pulses can be controlled by shaping the laser waveform via two key parameters: the relative-phase between two halves of the multi-octave spanning spectrum, and the overall carrier-envelope phase. These results not only promise to expand the experimental possibilities in attosecond science, but also demonstrate coherent strong-field control of free-electron trajectories using tailored optical waveforms.


S1 Optical waveform characterization via 2DSI
Here we report on the synthesized waveform characterization based on the two-dimensional spectral shearing interferometry (2DSI) technique 1 . We performed 2DSI on the two individual pulses (NIR and IR) that compose the waveforms. The 2DSI setup allows one to measure spectralintensity and spectral-phase of the ultrabroadband pulses of each channel, leaving the corresponding carrier-envelope phases (CEPs), and the relative phase (RP), unknown. Since CEPs of both NIR and IR pulses are linked to that of the common seed, they can be adjusted by controlling the CEP of the seed, in the seeding front end. The pulse-to-pulse CEP variation of the overall synthesized waveform is measured via f -2f interferometry at the synthesis point and stabilized by a feedback loop. The absolute CEP value of the synthesized waveform at the interaction point can be inferred from the observation of CEP-dependent HHG spectra or directly measured via attosecond streaking.
In order to reassemble the synthesized waveforms from the individual pulses we need to determine the CEP as well as the RP (or delay) between the two pulses. The latter can be obtained by broadening the IR pulse spectrum and observe the interference with the NIR pulse. The RP and the "relative delay" among the two pulses are effectively synonyms, since one can convert from one to the other by multiplying by the speed of light in vacuum and dividing by the wavelength at which the RP beating occurs (RP = relative delay× c/λ RP ). We prefer the term RP over "relative delay" since what we observe via the phase meter 2 is indeed the RP extracted from the spectral interference. To stabilize and control the RP, one pulse is delayed with respect to the other via piezo-actuated delay-lines. A more detailed description of the multi-phase meter that simultane- Figure S1: Pulse temporal characterization with 2DSI. Spectro-temporal characterization of the NIR a, and IR b, pulses obtained via 2DSI.
ously detects RP and CEP of the synthesized waveforms can be found in G. M. Rossi et. al. 2 , while additional information on the phase stabilization and control system can be found in section S6 of this supplementary information. The complete set of waveforms, and the corresponding intensity profiles, that is possible to synthesize can be calculated from the individual pulses in Fig. S1 by numerically varying the CEP and the RP. The RP between both channels is imprinted by adding a linear term on the spectral phase of the IR channel. A zero-order phase is simultaneously added to both NIR and IR channels to mimic the CEP variation. As shown in Fig. S2, these two parameters allow access to a wide variety of intensity profiles and waveforms.  Figure S2: Numerical relative-phase scan. Evolution of the intensity profile of the numerically synthesized waveform during a RP scan for a, CEP=0, b, CEP=π/4, c, CEP=π/2. d-f, Corresponding evolutions of the synthesized field for RP values between -7 rad (bottom) and 7 rad (top) (each step 0.7 rad or ≈ 380 as). Delay = RP × λ RP /c, λ RP = 990 nm.

S2 HHG spectral analysis
The HHG spectrum was recorded with a McPherson 251MX extreme ultraviolet (XUV) grating spectrometer equipped with an Andor Newton 940 CCD camera. There are two variable-linespacing (VLS) gratings (300 groove/mm and a 1200 groove/mm) for resolving different spectral ranges (≈ 10 -60 eV and ≈ 60 -250 eV). The absorption edges of various metallic filters (Al, Be and Zr) were used for photon energy calibration and thereby for the corresponding Jacobiancorrection. When measuring broadband HHG emission, the low-energy side of the first-diffraction order (FDO) of the spectrum can overlap with the second (SDO) diffraction order of the highenergy side, resulting in a modified spectral shape. Fig. S3 shows the procedure of removing the SDO contribution when using the 300 groove/mm grating. We start with a narrowband spectrum to easily atttribute spectral features to the FDO and the SDO. For example, as shown in Fig. S3, the two main peaks around 42 eV and 21 eV are the same spectral feature and manifest as the FDO and the SDO of the grating. The SDO-to-FDO signal amplitude ratio is inferred by comparing the SDO and FDO peaks, and assuming the ratio is constant over the entire spectral range. A replica of the spectrum shifted to half the photon energy is then multiplied by the measured SDOto-FDO amplitude ratio (= 0.2). Finally, by subtracting this photon energy-shifted and attenuated replica from the original spectrum, the SDO contribution to the relevant portion of the spectrum is removed. Any residual contribution is filtered out with a Tukey-window (see Fig. S3b). A similar correction was not required for data acquired with the 1200 groove/mm grating since there were no observable higher order diffraction signals. a b Figure S3: Second diffraction order (SDO) removal on 300 groove/mm grating. SDO removal of a, narrowband and b, broadband spectra. a, A narrowband XUV input spectrum (shaded area) showing both its first diffraction (FDO) and SDO, which serves to compute the SDO-to-FDO amplitude ratio. The background due to SDO, (red dashed line) is the product between the SDO factor (0.2) and a replica of input narrowband (a) or broadband (b) spectra at half the energy.
Its subtraction from the input signal yields the SDO-corrected spectra (blue solid line). A Tukey window (black dotted line in b) is applied to remove residual contributions to the spectra.
The effect of the CCD quantum efficiency (QE) and grating diffraction efficiency on the shape of the Jacobian-corrected spectra is shown in Fig. S4. For the 300 groove/mm grating, the product of the CCD-QE and the grating efficiency curve is close to constant over the relevant spectral range.
Consequently, the shape of the calibrated spectrum is almost identical to the uncalibrated one (see Fig. S4a (i-ii)) and the corresponding spectra presented in the main text (Fig. 2) are only Jacobiancorrected. Meanwhile, for the 1200 groove/mm grating cases (see Fig. S4b (i-ii)), both the CCD and the grating responses are taken into account in Fig. 2 b, e, f and i of the main text.

S3 HHG photon flux and pulse energy
Measurement of the photon flux and pulse energy based solely on the XUV CCD detector can often carry several errors. The precise spectral response of each reflective or transmissive elements (e.g., metallic filters, gratings, etc.) needs to be known. For instance, due to oxidation, manufacturing imperfection and/or contamination these can differ to what is reported in published databases [3][4][5] .
Careful alignment and beam pointing stability through apertures coupling the beam into the spectrometer also has to be assured, owing to the fact that small deviations in the HHG beam pointing can lead to truncation of the beam and thereby to a considerably different energy throughput. Furthermore, the latter is of utmost importance, since the divergence of the XUV/soft X-ray beam depends on the precise HHG generation conditions (i.e., driving synthesized waveform, gas pressure, gas-cell position, etc.) and can result in a clipped beam reaching the CCD. Thus, the lack of inspection on these details can easily lead to a significant underestimation of photon number and pulse energy values. A calibrated XUV photodiode (XUV-PD) can give a more accurate measurement of these values, provided that the spectral shape of the pulse is known. As in our experiments we acquired hundreds of spectra, alternating between the CCD and the XUV-PD for each spectrum is impossible.
Therefore, we measured the number of photons of a reference HHG spectrum employing both a photodiode as well as a XUV spectrometer. The ratio between the two measurements was used as a calibration factor to estimate the photon flux associated with the attosecond pulses based in their spectra, shown in Fig. 4 of the main article. The procedure used to derive the calibration factor is described in the following paragraphs. For estimating the number of photons after the gold toroidal mirror via the CCD measurement, we first compute the number of incident photons on the CCD from the CCD signal S CCD (see Fig.   S5b) as where σ is the sensitivity in electrons per CCD count, η QE is the CCD quantum efficiency (see Fig.   S4) and n e-h (ω) =hω/3.65 is the number of electron-hole pairs freed per incident photon with energyhω 5, 6 . Next, we calculate the number of photons at the location of the XUV-PD, i.e., after the gold toroidal mirror (without considering any transmission drop caused by the holey mirror and input slit) by further taking into account the grating efficiency η g and filter transmission t f , i.e., It is necessary to consider the transmission of the grating spectrometer and of the filter in front of the spectrometer since different gratings and different filters were used. Fig. S5c displays the n CCD (blue curve) and n CCD (red curve) as function of the photon energy. Accordingly, the total number of photons per pulse is determined as which outputs a value of N CCD = 475 photons per pulse. As both the photodiode and CCD measurement need to yield the same value, we force N CCD = N PD , which results in a calibration factor given by α 0 := N PD /N CCD ≈ 118. Finally, the spectrally-resolved number of photons on target is n target = α 0 · n CCD and, thus, the total number of photons per pulse is evaluated as Equivalently, the pulse energy on target is  8 3.33×10 4 --500.00 Ref. 9 --5.00×10 5 -Ref. 10 1.00×10 7 300.00 --  The Jacobian correction factor was then applied to calibrate the spectral intensity. The Shirley background, resulting from low-kinetic energy electrons (below ≈ 10 eV), was also subtracted from photoelectron spectra. Lastly, with the purpose of minimizing inaccuracies in the streaking trace reconstruction, the intrinsic high-frequency measurement noise along the delay axis was eliminated by filtering out frequencies above the second-harmonic of the highest frequency in the synthesized spectrum, that is ≈ 950 THz (see Fig. S1a and Fig. S6).
After applying the procedure described above, the data-sets are ready to be reconstructed. To this end, we used the Volkov-Transform Generalized Projections Algorithm (VTGPA) 11 . The algorithm includes the complex-valued and energy-dependent dipole matrix elements of neon (p-electron orbital; I p ≈ 21.7 eV). The initial guess of the XUV-pulse was represented by a Gaussian envelope, with its central energy being inferred from the input spectrogram. The vector potential of the streaking field was represented as the product of an envelope function and a carrier wave, A(t) = A N (t) cos α(t). The envelope function A N (t) was described as a cubic-spline interpolation of N points while the carrier wave as a polynomial expansion with k coefficients, namely, α(t) = α 0 + α 1 t + α 2 t 2 + ... + α k t k . We found that to reconstruct the envelope function of the synthesized streaking fields up to N = 60 and k = 3 points were necessary (see Fig. S7). The time sampling step size in the reconstruction was set to be 4-5 times shorter than the main XUV oscillation period.  and due to their spatial divergence 19 , thus they were excluded during the Lewenstein integration and only the contribution from short trajectories was considered. The electric field E L (t) of the input driving waveforms was obtained from the temporal superposition of the measured NIR and IR channel electric fields, with the RP and the CEP being numerically varied as described in Sec.S1.
The peak intensity was set as 2.5×10 14 W/cm 2 when the RP and the CEP are both set to 0 rad. The IR channel intensity was chosen to be about four times the NIR channel intensity. Here, we did not use the retrieved IR waveforms from attosecond streaking measurements as simulation inputs.
The reason is that the waveform replica used in the streaking experiment is not identical to the one  In the attosecond streaking measurements shown in the main text, a 100-nm thick aluminum foil was employed to block the HHG driving field. We explored the effects of the aluminum foil on the attosecond bursts by introducing its absorption and dispersion 22,23 (see Fig. S11a) in the same way as in the gas target case. Fig. S11 b shows the influence of a 100-nm thick aluminum foil on a XUV spectrum reaching energies slightly above 80 eV and with a FTL pulse duration of 90 as (see Fig. S11c). This narrower spectrum (compare with Fig. S10) was obtained by setting the RP and the CEP to 5.7 rad (3 fs) and 1.57 rad, respectively. The absorption and dispersion due to 300 mbar, 2 mm-thick column of argon were also included in the simulation. Within this spectral range, the response of the aluminum foil only introduces a negligible difference (≈ 2 as) to the pulse duration. The absorption and refraction of a 2 mm-thick argon gas target at 300 mbar was also included when the aluminum foil was applied.

S6 Data Acquisition Procedure for CEP/RP-scans
The synthesis parameters (CEP, RP) are measured and actively stabilized by an FPGA-based system which generates feedback signals controlling multiple delay lines in the parametric waveform synthesizer. On top, a graphical user interface on the control PC is providing more complex measurement sequences via scripts commanding the FPGA-system. The main detector consists of a home-made dual-phase meter measuring two spectral beat signals (f -f and f -2f ) derived from the beam at the secondary beam-combiner output (main beam directed to experiment) 24 . The two beat signals overlap spectrally but are set to be at different beat-frequencies allowing separation of the two signals. To this end, the measured single-shot spectra are Fourier transformed and the two corresponding phases are extracted and unwrapped. One of these two phases represents the RP and the other one the sum of CEP and RP. These phase values are fed to a PI-feedback system giving distributed feedback to piezo-driven actuators in our CEP-stable front-end (affecting the CEP) and in one of the synthesizer channels (affecting the RP). Those fast actuators allow to stabilize and control the phases in a limited range within less than 2 laser shots of latency (2 ms). The CEP can be modulated by up to 2 cycles (4π), which is sufficient to perform a full CEP scan due to its 2π-periodicity. For the RP this 2-cycle range is too limited, for that reason a long-range delay line additionally allows to displace the RP by hundreds of cycles (stage-travel: 25 mm). A long-range RP-scan (see Fig. S12) is performed by modulating the locked CE-phase over a 1.75 cycle range with an asymmetric saw-tooth function.
After two of those CEP-scan cycles, the RP-set-point is incremented step-wise by π/4. This scan- HHG-spectrogram (i) is recorded while the PWS control system performs phase-scans. The CEP is constantly modulated (ii, red) while the RP is incremented step-wise after two CEP-modulation cycles (ii, black). The measured phases (CEP in red, CEP+RP in blue) are actively stabilized by fast PZT-actuators (iii, red/blue). The RP-setpoint is scanned over a range wider than the RP-actuator allows. This actuator is brought back to its centre position by correcting with long delay-line (b iii, black).
ning procedure of the 2D synthesis parameter space is similar to the raster scan of a cathode ray tube display. During such a scan, the control computer observes the position of the fast RP-piezo actuator and, if displaced further than 1/4 of the dynamic range, brings it back to the centre position by moving the long-range delay line. Meanwhile the HH-emission from these synthesized waveforms is continuously recorded with our XUV spectrometer. The linear ramp of the CEP modulation extends over 90% of its period and during the short return time a LED driven by our feedback system flashes into the spectrometer, imprinting a time grid and absolute time markers directly onto the HHG-spectral trace.
In post-processing, this allows us to synchronize the synthesis parameters corresponding to every laser shot with a certain recorded HHG-spectrum. Therefore, the merged data-set contains our synthesis parameters (CEP and RP) and the corresponding response of the HHG. The measured CE-phase here can be wrapped due to its 2π-periodicity. A resorting algorithm then sorts every recorded spectrum onto a bin of a CEP-RP matrix. While the bin-size for the RP is set by the discrete step-size in the sequence controller, the CEP is continuously scanned and put on a grid of π/8 wide bins, optimizing both the number of spectra in each bin, as well as only grouping HHG-responses for very similar synthesized waveforms. The resorted data matrix is checked for consistency by comparing the spectra sorted in each bin for similarity of their shape. While a certain overall XUV-intensity fluctuation is observed (and expected), the spectral shape shows a very good reproducibility. Each bin contains 4-8 spectra and the mean spectrum is calculated.
These resorted spectra undergo calibration of the spectral characteristics of our spectrometer (see Section S2) and yields the basis for the data presented. A GUI allows to interactively browse the 4-dimensional data-set and to create constant-CEP and constant-RP cuts through the data (see Fig.   S13). Figure S13: GUI for data resorting and extraction. a (i-iv), raw HHG-spectra (i), corresponding phase and gating functions (ii), Spectra-density in each phase bin (iii) and all spectra sorted in one phase-bin (iv). b,c shows a cut through constant CEP and constant RP (corresponding to red markers in a).