Enhanced multi-colour gating for the generation of high-power isolated attosecond pulses

Isolated attosecond pulses (IAP) generated by high-order harmonic generation are valuable tools that enable dynamics to be studied on the attosecond time scale. The applicability of these IAP would be widened drastically by increasing their energy. Here we analyze the potential of using multi-colour driving pulses for temporally gating the attosecond pulse generation process. We devise how this approach can enable the generation of IAP with the available high-energy kHz-repetition-rate Ytterbium-based laser amplifiers (delivering 180-fs, 1030-nm pulses). We show theoretically that this requires a three-colour field composed of the fundamental and its second harmonic as well as a lower-frequency auxiliary component. We present pulse characterization measurements of such auxiliary pulses generated directly by white-light seeded OPA with the required significantly shorter pulse duration than that of the fundamental. This, combined with our recent experimental results on three-colour waveform synthesis, proves that the theoretically considered multi-colour drivers for IAP generation can be realized with existing high-power laser technology. The high-energy driver pulses, combined with the strongly enhanced single-atom-level conversion efficiency we observe in our calculations, thus make multi-colour drivers prime candidates for the development of unprecedented high-energy IAP sources in the near future.

during which the efficient XUV generation remains possible (through a combination of efficient tunnel ionization launching electron trajectories, transient phase-matching with a tolerable free-electron density, and depletion of the ground state) 10,11 Finally, a sufficiently fast wavefront rotation of the driver pulses leads to the individual attosecond pulses of an APT to be emitted at varying angles, such that an IAP may be selected by a spatial mask in the far field 12,13 .
All these techniques require extremely short  2-cycle driver pulse durations. Due to gain-narrowing, these cannot be achieved directly from laser amplifiers, so an additional post-compression stage after the amplifier is required, where the laser spectrum broadens by self-phase modulation during the propagation through a gas-filled hollow-core fiber and is subsequently compressed with chirped mirrors 14,15 This typically limits the driver pulse energy to a few mJ 16 due to the onset of ionisation. Even with the best optimization effort, the attainable IAP energy is thus limited to the nJ-level simply through the limited driver pulse energy.
While very advanced OPCPA schemes with careful phase control do permit the generation of a sub-3-cycle pulses with very high pulse energy 17 or average power 18 , this complex technology remains rather inefficient.
The resulting low power/pulse energy of current IAP sources hampers a much wider applicability of current attosecond science methods to lower sample densities (like molecules, clusters or nanoparticles that have to be evaporated or ablated to be brought into the gas phase, or excited species that only exist as a small fraction of the target medium) or to processes with low cross-sections. It also makes it very hard to reach a non-linear interaction regime for the XUV attosecond pulses 19,20 .
The gating technique known as multi-colour gating has lead to a breakthrough by enabling IAP generation with ≈10-cycle drivers. It is based on the coherent mixing of two or more laser pulses with different carrier wavelengths (and usually parallel linear polarization directions). Already in 2006, it was demonstrated that the broken symmetry of a two-colour field composed of a fundamental with its second harmonic doubles the spacing of attosecond pulses from one half-cycle to one full laser cycle (2.7 fs in the case of an 800-nm fundamental), which correspondingly allows generating IAP with twice as long (i.e. ≈4-cycle) driver pulses 21,22 . The usable driver duration was further increased to ≈10-cycles by either adding polarization shaping to this scheme to perform "double optical gating" 23 , or by detuning the second harmonic 24 , which further increases the periodicity of the two-colour field.
While such "incommensurate" two-colour combinations turned out to be barely realizable by optical harmonic generation with its very limited tunability, optical parametric amplification (OPA), creating a frequency down-converted additional colour, lends itself ideally to this task. The price to pay is the fact that the phase-delay between the fundamental and the auxiliary colour component is not passively stable, as in optical harmonic generation. Using a white-light-seeded OPA 25 together with an actively CEP-locked fundamental (pump) laser does however yield a stable phase delay between the pump and the signal/ idler wave and thus stable multi-colour waveforms 26 . Without CEP-locking, the phase delay fluctuates and the gating efficiency thus varies shot-to-shot 27,28 Conditions can however be found, notably including sufficiently short pulse envelopes, which yield an IAP for a very wide CEP-range 20,29 thus relaxing the CEP-locking requirement, albeit for the price of strong shot-to-shot fluctuation of the IAP intensity. The latter method yielded record-high µJ-level energy per IAP, constituting a 100-times enhancement compared to previous achievements 11 resulting mainly from the fact that the longer driver pulses can be taken directly from laser amplifiers or OPA. Note that multi-colour gating is much more energy-efficient than polarization-or ionization-based gating techniques. The latter "waste" a lot of field-energy in parts of the pulse where HHG is shut off by elliptical polarization or excessive ionization, whereas the multi-colour beating automatically concentrates the available field energy in a few very strong field oscillations which then efficiently drive the desired HHG process. There are even indications that undesired ionization might be reduced 29 , which would allow reaching higher saturation intensities and thus higher HHG efficiency and cutoff.
Here, we are going to expand on the temporal gating aspect of our recent work on optimizing HHG at the single-atom-level through shaped multi-colour driver waveforms 26 . We derive a clear understanding from a simple time-domain picture and then demonstrate, using single-atom-level numerical simulations with the Lewenstein model of HHG 30 , how the multi-colour gating technique combined with (rough) CEP-control so that stable multi-colour waveforms are realized enables the generation of IAP based on 180-fs long pulses typically supplied by Yb-based laser amplifier technology. The extremely efficient temporal gating is accompanied by strongly enhanced IAP intensities compared to a single-colour driver, achieved through advantageous shaping of the central driving waveform cycle as experimentally demonstrated in our recent work 26 . While the single-atom-level calculations we discuss here cannot quantitatively simulate an experimental HHG source, recent experimental work demonstrated that the qualitative predictions of such a simplified theoretical treatment are indeed realized in the lab 29,26 . Our results thus suggest that high-energy/high-flux sources of isolated attosecond pulses based on Ytterbium-laser technology without a need for pulse-post-compression are within immediate reach. Finally, we discuss the experimental implementation.

Theoretical analysis
Temporal gating by two-colour "few-cycle-pulse trains". A formal dynamical symmetry analysis 31 shows that incommensurate two-colour combintations, i.e. of fields with frequencies ω 0 and ω ηω = 1 0 with η . < < 0 5 1, leads in the spectral domain to a densification of high-harmonic peaks-independently of the colour-components' phase delay. The derived spectral selection rules have been experimentally verified 32 and the potential for temporal gating of HHG has been underlined.
While this potential has already been examined in a number of (parametric) numerical studies published during the last years [33][34][35] we will here, in contrast, base our reasoning on an intuitive time-domain picture, inspired by a brief mention in Ref. 31. We find this picture to give a very accessible understanding of the temporal gating effect and its dependence of the colour components' phase delay, which enables us to devise and understand very easily the optimal colour combinations for generating an IAP over a broad CEP-range.
Coherently combining two laser waves with equal strength and frequencies ω 0 and ω 1 leads to a beating which can be understood by considering i.e. the factorization into a slowly oscillating "envelope" term and a faster oscillating "carrier" term. We are particularly interested in frequency ratios η . < < . 0 5 0 9 since these produce waveforms that can be considered as a train of few-cycle pulses with carrier frequency ω ω ( + )/2 i.e. twice the number of carrier-cycles per few-cycle sub-pulse. For integer N , the few cycle pulses repeat with constant CEP when N is odd (i.e. the waveform is strictly π ω ω /( − ) 2 0 1 -periodic, see Fig. 1b), or with a π CEP flip between successive pulses when N is even (see Fig. 1c). For non-integer N , there is a corresponding fraction-of-π CEP slip between successive few-cycle pulses (see Fig. 1a). For targeting a particular ratio N , the relation will be useful. Since we consider the ω 1 -auxiliary-component to be generated from a white-light-seeded OPA pumped by the ω 0 -fundamental-component, we set ϕ ϕ ϕ = = 0 1 CEP so that the phase delay between the two colour components, τ The "envelope" term in equation (1) is then always maximal at the pulse center, = t 0, and the "carrier" term has a CEP of ϕ CEP . This means that the phase delay variation caused by a drift of ϕ CEP can also be thought of as a CEP-drift of the few-cycle sub-pulses. Figure 1b shows an illustrative example of two-colour pulses composed of 1030 nm and 1545 nm, which corresponds to the particular case of odd = N 5. We obtain a train of near-single-cycle sub-pulses with 1236 nm carrier wavelength (corresponding to a 4.1-fs oscillation period) spaced by a 10.3-fs period. Due to the odd N , the sub-pulses all have the same CEP, ϕ CEP . Even with the rather long 60-fs pulse envelopes, the large sub-pulse spacing leads to a significant variation of the peak field strength from one sub-pulse to the next, which underlines the potential for efficient temporal gating.
Detuning the auxiliary wavelength has a number of coupled consequences. Bringing ω 1 closer to the fundamental ω 0 (see Fig. 1a) increases the spacing of the sub-pulses (e.g. to 13 fs for a 1400-nm auxiliary wavelength), as well as their effective duration, while decreasing their carrier wavelength (e.g. to 1187 nm for a 1400-nm auxiliary wavelength)-the sub-pulses thus contain more cycles. Bringing ω 1 farther away from ω 0 has the opposite effect (see Fig. 1c). For temporal gating this means that there is an optimum auxiliary wavelength to be found where (i) the sub-pulses are neither too close, meaning that adjacent ones would contribute to the near-cutoff photon energy range, thus forming an APT, nor (ii) where the sub-pulses contain too many cycles of the carrier wave, meaning that within the central sub-pulse, more than one attosecond pulses in the near-cutoff energy range would be generated. Point (i) can be mitigated by an ω 1 -value correponding to a non-integer N , so that the adjacent sub-pulses experience a CEP-shift. If the central sub-pulse has an optimal CEP for efficiently generating high photon energies, the adjacent ones would then be sub-optimal and their contribution suppressed (cp. Fig. 1a).
This picture is complementary to the one proposed in 36 where a "phase accordance" parameter of the two colour components is defined, which takes small values in gating windows corresponding to our sub-pulses. The authors of 36 , however, did not study the experimentally very important CEP-dependence of the gating and generation efficiency.

Numerical simulations.
Our numerical simulations treat HHG on the single-atom level with the non-adiabatic quantum path analysis of the Lewenstein model for HHG 30,37 (see Methods-section). For quantitative predictions on the performance of an experimental source, one has to also treat the propagation of the driving and generated XUV fields in the macroscopic ionizing HHG medium, as done, e.g. by Jin et al. 38,39 . This additional layer adds a large number of additional parameters that need to be optimized, which is beyond the scope of the present work. Instead, we focus here on the single-atom level dynamics, from which we select the short-trajectory contribution to the HHG emission (see Methods-section), as this is typically the one macroscopically phase-matched in an experimental attosecond pulse source.
For our numerical simulations, we start with 60-fs FWHM-intensity envelopes for all colour components before exploring even longer durations. We consider a total peak intensity (i.e. the sum of the peak intensities of the colour components) of = × − I 1 10 Wcm tot 14 2 . This means that, unless stated otherwise, we always compare driver pulses of the same total pulse energy. While we consider the ionization potential of argon atoms, = . I 15 76 p eV, note that hydrogenic dipole matrix elements d k [ ] are used so as to avoid an influence of the particular atomic structure on our conclusions. The driving fields are always linearly polarized and all colour components have parallel polarization directions. The attosecond emission is calculated from the HHG spectra after application of a 15-eV wide spectral filter selecting the region of highest generated photon energies near the cutoff.
As a reference, we show in Fig. 2 the HHG spectra generated by single-colour drivers with varying carrier wavelengths, as well as the temporal profile of the attosecond emission. Unsurprisingly, even for the longest wavelength of 1800 nm, the many-cycle pulse duration of the driving fields leads to the emission forming an APT. The well-known increase of the attainable photon energy cutoff accompanied by a rapid drop of conversion efficiency 40,41 is also clearly apparent.
Two-colour gating. We consider two-colour fields of the form with the intensity-FWHM duration τ = 60 fs. We consider a fundamental frequency ω 0 corresponding to the emission wavelength of Yb-based lasers of 1030 nm. Figure 3 shows for = R 1 the generated high-harmonic spectra and attosecond emission as a function of ϕ CEP , i.e. of the phase delay between the two colour components, for a range of auxiliary wavelengths, λ 1 , from 1200 nm to 1800 nm. As the carrier-wavelength of the two-colour waveform increases (with increasing auxiliary wavelength), the achievable HHG cutoff photon energy increases accompanied by a drop in conversion efficiency, as expected from the well-known wavelength scaling of HHG 40,41 As for temporal gating, the trends expected from the discussion of equation (1) are clearly observed: (i) For the shorter auxiliary wavelengths, the central sub-pulse becomes so long that it generates, for most CEP values, two attosecond pulses separated by a half-period of its carrier wave. As the auxiliary wavelength gets longer, the transition to a single-cycle central sub-pulse is observed, effectively limiting the attosecond pulse generation to a single half-cycle. (ii) For the shorter auxiliary wavelengths, the long spacing of the sub-pulses effectively suppresses the attosecond emission by the sub-pulses adjacent to the central one. (iii) For the auxiliary wavelengths corresponding to (close-to) integer N , 1545 nm and 1700 nm, the sub-pulses all have the same CEP and the attosecond emission is correspondingly strongest at the same CEP-values for the central sub-pulse and its adjacent ones. In contrast, the CEP-shift between subsequent sub-pulses for non-integer N is beneficial for temporal gating since it shifts the maxima of the attosecond emission of the adjacent sub-pulses to different CEP values as that of the central sub-pulse (e.g. compare Fig. 3d,e,f).
We now select optimal CEP-values from the scans shown in Figure 3 as follows: if there are CEP-values for which the contrast ratio between the intensity I max of the strongest attosecond pulse and that of the second-strongest one is ρ > 10, then only these are considered, and the CEP with the highest "fitness" equal to ρ I max 4 is selected. This ensures that we enforce a good contrast ρ > 10 while giving preference to the highest attosecond peak intensity. If no CEP-value provides ρ > 10, we relax the intensity preference and selected the CEP value with the highest fitness ρ I max . The optimal CEP-values so selected are marked by red dashed lines in Fig. 3; lineouts of the HHG spectra and attosecond emission for these CEPs are shown in the Supplementary information. With the exception of λ =1700 1 nm (due to an unfavorable combination of small driving sub-pulse spacing and near-integer N ), all considered auxiliary wavelengths permit, within a limited CEP-range, to generate an IAP. Here we define IAP as a single attosecond pulse which dominates over potential pre-or post-pulses by more than an order of magnitude, i.e. ρ > 10. The highest attosecond pulse intensities are found for λ = 1300 1 nm and 1400 nm, of which the latter will be the practically most advantageous because it offers a broader CEP-range (∆ϕ = . 1 7 CEP rad) within which an IAP is retained, i.e. ρ > 10 (cp. Fig. 3b,c). The obtained attosecond pulse intensities are two orders of magnitude higher than in the APT with similar photon energy obtained with single-colour drivers (to be generated by OPA) with the same total pulse energy and peak intensity (cp. Fig. 2). This underlines the extremely useful temporal gating action of well-designed two-colour drivers: we do not force the isolation of a very weak attosecond pulse by picking an ever so narrow spectral slice in the very cutoff region, but we obtain, from ⪆10-eV-wide spectra, IAP orders of magnitude more intense than the APT generated with a single-colour driver. Furthermore, reaching higher photon energies, like the ≈100 eV in the examples presented here, with "efficient" HHG gas media offering higher recombination cross sections, like argon, generally requires longer driving wavelengths that need to be generated by OPA. It is thus certain that a two-colour field composed of the OPA signal and part of the pump laser emission can be generated with higher total pulse energy. Not only does the two-colour field reach the same high HHG cutoff with higher efficiency, but it will contain more driving energy to begin with, further increasing its advantage in terms of attainable attosecond pulse energy and/or intensity.
For a practical implementation, it will be interesting to know how much the relative weight of the auxiliary colour component can be decreased while still keeping the temporal gating effect intact. While the factorization in equation (1) is valid only for equal amplitudes of the beating colour-components, we find that the conclusions drawn from it remain valid guides also for a much weaker auxiliary component. Having repeated the same numerical calculations as shown in Fig. 3 for = / R 1 2 (results shown in the Supplementary Information), we find that, except for the lowest λ = 1200 1 nm, which no longer leads to an IAP, the results are very similar to those obtained for = R 1 (attosecond pulse intensities remain the same within ±10%, the CEP range for IAP generation for λ = 1400 1 nm remains ∆ϕ = . 1 7 CEP rad), if one adapts the position of the spectral filter for the attosecond pulses to the small ≈5 eV decrease of the achieved cutoff photon energies. Upon further lowering the peak intensity ratio to = / R 1 9 (results shown in the Supplementary Information), we find the temporal gating to become significantly less effective, i.e. the CEP-range for which the central driving sub-pulse generates a single attosecond pulse only is now significantly reduced and the emission from the adjacent sub-pulses is less strongly suppressed. Nonetheless, with the spectral filter positions adapted to the further lowered cutoff photon energies, we can still find optimal CEP-values that allow generating an IAP for the auxiliary wavelengths λ = 1300 1 nm, 1400 nm, 1600 nm and 1800 nm, with IAP peak intensities reduced by a factor ⪅3 compared to the = R 1 and = / R 1 2-cases. The auxiliary wavelength λ = 1400 1 nm is still identified as the best choice, uniting highest attosecond pulse intensity and suppression of satellites. The requirement on CEP-stability (∆ϕ = 1 CEP rad) is now more stringent than for the cases of higher relative intensity of the auxiliary colour component, but still well achievable in real-world experimental setups 26 . It will thus be possible to relax the required OPA output energy and retain the majority of the driving energy in the fundamental laser pulse.
Having shown the effectiveness of two-colour temporal gating for 60-fs pulses and having identified the non-integer-N auxiliary wavelengths λ = 1300 1 nm, 1400 nm and 1600 nm as most promising choices, we can now explore even longer pulse durations approaching realistically achievable values for Ytterbium-based amplifiers 15, 42-46 . Since we have found the temporal gating to be very robust against variations of the relative strength of the auxiliary colour component, we can very well consider different pulse envelope durations for the two colour components. This is experimentally relevant since the OPA output can be significantly shorter than the pump pulses (see section "Experimental feasibility" below). Comparing calculations for 180-fs fundamental and 60-fs auxiliary pulses (with superposed envelope maxima) with = R 1 so as to maximise the gating efficiency (results shown in the Supplementary  Information), to the case where both colour components have 60-fs envelopes (Fig. 3) , we find, unsurprisingly, that the action of the central sub-pulse is barely modified while the suppression of the attosecond emission from the adjacent sub-pulses is less efficient. It is however still sufficient for the generation of IAP, albeit within narrowed CEP-ranges: nm. Since a 1300-nm auxiliary wavelength is not easily generated in an OPA pumped by a 1030-nm laser (see section "Experimental feasibility" below), and the CEP-ranges found for the longer auxiliary wavelengths are incompatible with real-world CEP-jitter, further enhancement of temporal gating will be required for IAP generation with experimentally realistic driver parameters.
Three-colour gating and enhancement. Adding a third colour component-the second harmonic of the fundamental-will provide this required enhancement. We extend equation (3) to i.e. we assume for simplicity that the second harmonic has the same pulse envelope, ( ) f t 0 , as the fundamental. The second harmonic has a phase delay ϕ ω /( ) 2 2 0 with respect to the fundamental wave. Since it is independent of the CEP, it can be considered perfectly stable even in realistic experimental conditions 47 .
The enhanced gating with this third colour component is illustrated in Fig. 4 for different auxiliary wavelengths around the value of 1400 nm, which we had found to provide a good compromise between wide inter-pulse spacing of the few-cycle-pulse train and short few-cycle duration. On the scale of the individual few-cycle pulses, the ω 2 0 -component functions as a "detuned second harmonic" of the carrier wave 24 , effectively "de-activating" all but a single half-cycle of the carrier wave for HHG. This doubles the periodicity of the CEP-dependence of HHG from π (as in Fig. 3) to π 2 (see Fig. 5). On the time-scale of the "envelope-term" of the few-cycle-pulse train, the effect depends on N . For integer N (Fig. 4a,c), the ω 2 0 -component has the same phase at every few-cycle-pulse maximum. Therefore, for odd N , its phase relative to the carrier-wave of the few-cycle-pulse train is also the same at each few-cycle-pulse maximum, whereas it changes by π from one few-cycle-pulse maximum to the next for even N . This means that if for a given phase ϕ 2 , for which the ω 2 0 -component enhances HHG in the central few-cycle pulse, it will have a quenching effect in the adjacent few-cycle-pulses if the auxiliary wavelength is chosen such that N is even. This is obviously beneficial for temporal gating since it effectively results in the generation of an IAP only every other few-cycle pulse in the train.
In the following calculations, we will thus consider three-colour drivers with the even-N auxiliary wavelengths λ = 1325 1 nm ( = N 8) or λ = 1440 1 nm ( = N 6). We set a 180-fs intensity-FWHM duration for the fundamental and its second harmonic, while for the auxiliary colour component, we set a 100-fs duration. The peak intensity of the second harmonic pulse is chosen to be 10% of that of the (4) . Figure 5 shows results of these calculations for an intensity ratio = / R 1 2 and a phase ϕ π = .
0 1 2 , selected because it yields the highest "fitness" of the IAP as defined earlier (i.e. best compromise between attosecond pulse intensity and suppression of satellites). Comparing the CEP-dependences with those for the two-colour drivers (Fig. 3c, note the different CEP-ranges), one clearly confirms the gating of attosecond pulse generation to a single carrier half-cycle per few-cycle driver sub-pulse. We also find, as expected, that the CEP range for efficient attosecond pulse generation by the central few-cycle sub-pulse is neatly separated from that where the two adjacent sub-pulses generate efficiently. This leaves a very wide CEP-range for IAP generation of ∆ϕ = 2 CEP rad for both auxiliary wavelengths. Within this CEP range, the pulse energy of the IAP varies by less than a factor 2 (see right panels in Fig. 5). A CEP-jitter of 1 rad, which is currently experimentally feasible 26 , would reduce the IAP-energy fluctuations to 10%. The IAP duration is 320 as in both cases, without considering possible compensation of their intrinsic GDD ("atto-chirp") 48,49 . The same calculations for an intensity ratio = R 1 give the same results, only with ≈5 eV higher HHG cutoff energies. The enhanced three-colour temporal gating thus enables the generation of IAP even with 100-fs auxiliary pulses.
We also note a further increase by a factor ≈4 of the obtained attosecond pulse intensities as compared to the two-colour results (Fig. 3b,c) at the same total peak intensity. This more than compensates for the higher total driver pulse energy of the considered longer pulses, i.e. the effective driver-to-IAP conversion efficiency is enhanced. As compared to the single-colour drivers (cp. Fig. 2), our calculations predict an enhancement of about three orders of magnitude. It is however important to again note that serious predictions about experimental conversion efficiencies would need to involve propagation calculations 38,39 , beyond the scope of this work.
The choice of an auxiliary wavelength corresponding to an even-valued N is indeed of key importance. Simulations for λ = 1300 1 nm ( = .  Analysis of quantum paths. The inner workings of the driver waveforms for optimal temporal gating and efficiency enhancement are well understood by studying the calculated quantum paths. Figure 6 shows the waveform and computed quantum paths corresponding to the results shown in Fig. 5b, i.e. for the even-N auxiliary λ = 1440 1 nm with ϕ π = .
0 1 2 . The photon energies >90 eV selected for the IAP are generated most efficiently only in a single event during the central driving sub-pulse. This event is launched by the strongest field crest (marked by the ionization instants), which is followed by a strong and broad field-crest efficiently accelerating the returning electron. The sub-pulse is short enough so that the sub-sequent field crest does only lead to much lower recollision energy and thus photon energy. The nearest adjacent sub-pulses have a different, sub-optimal cycle-shape leading to reduced ionization and subsequent electron-acceleration by the driving field. The next adjacent sub-pulses then repeat the cycle-shape of the central one, but the pulse envelopes reduce the peak field strengths so that the >90-eV photon energies are only reached with exponentially dropping trajectory contributions in the cutoff region.
The central sub-pulse has a very similar cycle shape as that described and realized in our recent experimental work 26 (where the optimal cycle shape was found to be given by the phases ϕ π = .
0 2 2 ), which demonstrated that such three-colour waveforms can indeed greatly enhance the HHG efficiency by implementing the optimisation-principles laid out by the "perfect wave" for HHG 50 . While in that work, we have used an odd-N auxiliary wavelength in order to create perfectly repeating cycle-shapes whose enhancing effect was then studied, here we have targeted the strongest possible temporal gating. The fact that both approaches lead to very similar cycle shapes underlines that not only is the generation of IAP possible with surprisingly long driving laser pulses, but it is at the same time achieved with enhanced efficiency. Note that CEP-jitter which is tolerable in terms of temporal gating does of course lead to deviations from the enhancing cycle shape discussed here. The resulting IAP yield fluctuation is quantified in Fig. 5. Experimental feasibility. The theory calculations described in this paper were performed for conditions converging to a realistic implementation using currently available laser technology. Ytterbium active  [10,20] fs (red, blue and cyan line, respectively). The grey shaded area marks the 2-rad-wide CEP-range within which an IAP is generated. laser materials are a particularly promising trend in high-average-power ultrafast laser development due to low quantum defect and the availability of diode stacks for pumping, which allows extracting unprecedentedly high average power from a laser amplifier 51,52 , and is compatible with scaling towards Joule-level pulse energies 45,46 . The femtosecond pulse duration allows simple implementation of white-light seeded OPA, as in our recent experimental work 26 .
As an illustration of such a setup and the OPA tuning range, we show in Fig. 7 spectral and temporal characterization measurements of the signal output from a white-light seeded KTA-based OPA pumped by a 180-fs Yb:CaF 2 regenerative amplifier. The OPA is built in a two-stage configuration, where the first 4 mm thick crystal is used as a pre-amplifier for the white-light seed generated in a 2-mm thick YAG plate 53 . The pre-amplified seed (5 µJ) is then subsequently amplified in the second OPA stage with a 2 mm thick crystal. The central wavelength can be tuned within the range of 1400-2000 nm simply by changing the phase matching angle of the KTA crystals. The tuning range on the short wavelength side is limited by the absorption of the idler wave in the KTA crystal above 4000 nm and the related thermal effects. The bandwidth of the signal pulse stays almost the same throughout the tuning range and is limited by the phase-matching bandwidth of the OPA crystal. A notable exception is the ≈1400-nm central wavelength where the bandwidth becomes significantly broader (see Fig. 7a) due to matched group velocities of the signal and idler waves 54 .
The pulses from the OPA are much shorter than the pump laser pulses due to the broad amplification bandwidth and broadband seed pulse derived via white-light generation, as well as the nonlinearity of the parametric gain. In particular for the amplified signal pulse with 1440 nm central wavelength, for which a frequency-resolved optical gating (FROG) measurement is shown in Fig. 7, we find a 70-fs FWHM duration right after the OPA. This pulse has some residual positive chirp due to self-phase modulation during white-light generation, which can in principle be compensated using a ≈15-mm BK7 glass block. However, this further increases the third-order phase with the effect that, although the compressed pulse duration (45 fs) is comparable to the transform limit (40 fs), this simple setup comes at the price of relatively poor pulse quality. The high third order phase (apparent as a quadratic group delay in Fig. 7c) delays the spectral components in the center of the spectrum and only the spectral wings are synchronized. For best compression more complicated pulse shaping schemes can be used, such as specially designed chirped mirrors or an acousto-optic programmable dispersive filter 55 .
We have thus generated an auxiliary pulse that is much shorter than the 100-fs pulse considered theoretically in Fig. 5. The experimentally realistic situation thus turns out to be quite a bit more comfortable than the extreme case which, according to our single-atom simulation, still supports robust generation of IAP. It should thus be possible to select a significantly broader XUV-continuum, or to obtain an IAP in a broader CEP-range. This is confirmed by simulations made for a three-colour driver including a 50-fs, 1440-nm auxiliary pulse, shown in Fig. 8, where we find an IAP to be generated within a very broad CEP-range of ∆ϕ π = CEP rad. Figure 6. Waveform (zoom-in around pulse peak (a) or full pulse (b)) and driven quantum paths for IAP generation (a). The quantum paths, shown for the short-trajectory branches only, are represented by the realparts of their ionization and recollision instants. These form matching pairs of curves, connected by arrows for the first of the shown driver sub-pulses. The colour of the data points shows on logarithmic scale the contribution of the corresponding quantum path to the generated HHG intensity. The blue-shaded energyrange marks the spectral filter used for the selecting an IAP in Fig. 5b Our recent experimental work 26 has shown that the three-colour drivers we have considered here can in principle be generated with available multi-mJ kHz-repetition-rate Yb-based laser-amplifiers 15,42-44 which can be actively CEP-locked 26,56,57 Currently available Yb-based laser technology should thus be able to robustly generate IAP with very high power driver pulses without any need for pulse post-compression. We identify two technical challenges that will be worth tackling for further improvements: (i) The (long-term) stability of the CEP-locking. While our calculations predict that large jitter around the optimal ϕ CEP is tolerable, the better the experimental stability, the more stable the generated IAP characteristics will be. For a reliable source of IAP, the robust CEP-locking of the Yb-driver-laser needs to be achieved over long time periods of ∼hours. While this has not yet been demonstrated, we do not foresee any principle obstacles. It should be noted that the width of the CEP range for generating an IAP can of course always be increased by shifting the spectral filter towards higher photon energies-though usually at the price of a decreasing IAP peak intensity. In our study, we have avoided to adapt the filter position at each modification of the parameters in order to retain a reasonable comparability of our results.
(ii) The lower end of the tunability range of a KTA-based OPA pumped by a 1030-nm laser is at λ ≈ 1400 1 nm. The case of λ = 1440 1 nm, considered in Fig. 5b and 8, is thus experimentally feasible, while the case of λ = 1325 1 nm considered in Fig. 5a would require a more complex OPA scheme should be used. For example, a BBO-based OPA pumped by the frequency-doubled 1030-nm fundamental would allow generating a 1325-nm idler wave with very wide bandwidth. The cost is of course a reduced overall OPA efficiency due to the frequency-doubling efficiency typically being below 60%. A shorter auxiliary wavelength would however be advantageous since it increases the temporal spacing of the few-cycle sub-pulses and thus improves the suppression of satellite attosecond pulses (the presence of the second harmonic prevents the generation of two attosecond pulses per driver sub-pulse). This would increase the XUV-bandwidth available for the IAP, and thus enable both stronger and shorter IAP (compression of the intrinsic attochirp provided).
Both mentioned challenges, OPA-tunability and CEP-locking, are essentially removed by a setup such as that demonstrated in 58 , where two colour-components with relatively close frequencies are each generated in separate OPA, and derived from a common passively CEP-locked seed. While this arrangement promises to provide great flexibility and more robust CEP-locking, it does of course require a great abundance of fundamental pump laser power and does not make use of the directly available laser output. Since it lifts the requirement of active CEP-locking of the fundamental laser, it can however be implemented with essentially every very-high-power laser system.