# Sub-laser-cycle control of relativistic plasma mirrors

L Chopineau, G Blaclard, A Denoeud, H Vincenti, F Quéré, S Haessler

► To cite this version:

L Chopineau, G Blaclard, A Denoeud, H Vincenti, F Quéré, et al.. Sub-laser-cycle control of relativistic plasma mirrors. Physical Review Research, 2022, 4 (1), pp.L012030. 10.1103/PhysRevResearch.4.L012030. hal-03767708

HAL Id: hal-03767708

https://hal.science/hal-03767708v1

Submitted on 2 Sep 2022

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Sub-laser-cycle control of relativistic plasma mirrors

L. Chopineau,¹,²,* G. Blaclard,¹,* A. Denoeud,¹,² H. Vincenti,¹ F. Quéré,¹ and S. Haessler³,†

¹LIDYL, CEA, CNRS, Université Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette, France

²CEA, DAM, DIF, F-91297 Arpajon, France

³Laboratoire d'Optique Appliquée, Institut Polytechnique de Paris,
ENSTA-Paris, Ecole Polytechnique, CNRS, 91120 Palaiseau, France

(Dated: January 19, 2022)

We present measurements of high-order harmonics and relativistic electrons emitted into the vacuum from a plasma mirror driven by temporally-shaped ultra-intense laser waveforms, produced by collinearly combining the main laser field with its second harmonic. We experimentally show how these observables are influenced by the phase delay between these two frequencies at the attosecond timescale, and relate these observations to the underlying physics through an advanced analysis of 1D/2D Particle-In-Cell simulations. These results demonstrate that sub-cycle shaping of the driving laser field provides fine control on the properties of the relativistic electron bunches responsible for harmonic and particle emission from plasma mirrors.

# I. INTRODUCTION

Plasma mirrors are overdense plasmas created at the surface of laser-ionized optically flat solid targets. They are versatile optical devices for the manipulation of ultra-intense femtosecond (fs) laser beams ($I_L \gtrsim 10^{16}$ W/cm$^2$). They are also considered as a promising path for the generation of intense attosecond light pulses through high-order harmonic generation (HHG) [1–5], as well as for laser-driven particle acceleration [6–9]. From a fundamental standpoint, plasma mirrors represent ideally simple testbeds for ultrahigh-intensity laser-plasma interaction physics because the dynamics is confined to a thin layer at the target surface, where the plasma particles are directly exposed to the ultraintense laser field, without any prior alteration of this field [10, 11].

Collective electronic motion at this plasma-vacuum interface on a sub-femtosecond timescale plays a key role in this physics. When a p-polarized laser is focused on a solid target at intensities $I_L \gtrsim 10^{18}$ W/cm$^2$, the laser-driven electron motion becomes relativistic and can be described by a push-pull process [12–14], repeating once per driving laser period. The incident laser field first pushes electrons into the plasma, piling up a dense electron bunch and creating a restoring internal plasma field. As the laser field changes sign, the combined plasma and laser fields accelerate the electron bunch to a relativistic velocity towards the vacuum. This can induce drastic temporal modulations to the reflected laser wave, which sensitively depend on the electron bunch properties and dynamics [15–20]: key physical parameters are the bunch charge, velocity, and spatial extent. In the spectral domain, these periodic temporal modulations result in HHG. We will refer to this process as *Relativistic Oscillating Mirror* (ROM) in the following.

While most of the electrons then get pushed back into

the plasma, a fraction of them are expelled into the vacuum at relativistic velocities. Their tight temporal locking with the laser field then lets them get further accelerated in the reflected laser field through *Vacuum Laser Acceleration* (VLA) [7]. This suggests a tight correlation of HHG and fast-electron emissions in the relativistic regime, which has indeed been observed experimentally [21, 22].

While the understanding of this relativistic laser-plasma interaction physics has greatly advanced over the last two decades, the means of its experimental control have remained rather limited. To date, the main control knob is the scale length $L_g$ of the plasma density gradient at plasma-vacuum interface, $n(x) \propto \exp[-x/L_g(\tau)]$, which can be varied by adjusting the delay $\tau$ between a weaker ionizing prepulse and the main ultra-intense driving pulse [10, 11]. On the relevant sub-femtosecond time scale, this is of course a static parameter. An adequately fast dynamic control could be achieved with a temporally tailored driving waveform. This concept has already had great success with strong-field dynamics driven on the single-atom level at much lower intensities $\sim 10^{14}$ W/cm$^2$ [23–27]. Its potential for relativistic plasma mirrors has recently been shown in numerical studies [28, 29] that predicted significant enhancements of the HHG efficiency, shortly after confirmed in a first experiment [30]. However, no other observable than the angle-integrated HHG spectrum has been studied, leaving mostly unexplored the detailed attosecond control afforded by the additional degree of freedom of the driving optical cycle shape.

In this Letter, we present experiments where high-order harmonics as well as relativistic electrons are generated through the interaction of a plasma mirror with ultra-intense temporally shaped optical cycles, generated by combining the fundamental laser field ($\omega_L$) with its second harmonic ($2\omega_L$). By adjusting their relative phase, we can dynamically control the collective plasma electron motion on the sub-femtosecond time scale. We simultaneously measure the angularly and spectrally resolved high-order harmonics generated around the spec-

<footnote>* These authors contributed equally</footnote>

<footnote>† stefan.haessler@ensta-paris.fr</footnote>
ular direction, as well as the high-energy electron beam emitted into the vacuum. Finally, we relate these observations to the underlying physics through an advanced analysis of 1D/2D Particle-In-Cell (PIC) simulations.

## II. EXPERIMENTAL SETUP

The experiments are carried out on the 100 TW-class Ti:sapphire laser UHI100 (LIDYL, CEA Saclay), delivering 25-fs pulses at central wavelength $\lambda_L = 800$ nm with an excellent temporal contrast $\gtrsim 10^{13}$ on a $\gtrsim 100$ ps timescale reached thanks to a double plasma mirror system [31]. In these conditions, the maximum energy on target is $\sim 0.5$ J, distributed in a 65-mm-diameter near-top-hat beam. A deformable mirror coupled to a wave-front sensor (Imagine Optic) corrects aberrations thus optimizing the focal spot on target.

The optical setup preparing the pulse sequence sent on target is shown on Fig. 1. The scale length $L_g$ of the plasma density gradient on the target surface, $n(x) \propto \exp[-x/L_g(\tau)]$ is controlled by a weak prepulse ($F \approx 10^3$ J/cm$^2$ on target) at an adjustable delay $\tau$ before the main pulse [10]. The value of $L_g$ was measured using spatial domain interferometry [11]. An aperture mask transmits the prepulse as well as a 33-mm-diameter top-hat main beam (limited by the available size of the calcite crystal mentioned below).

The two-color waveforms are generated by a combination of three transmissive optics [24, 30] (see Fig. 1): part of the fundamental light with frequency $\omega_L$ is frequency-doubled to $2\omega_L$ in an 800-µm thick KDP crystal (C₁; Gooch & Housego), which can be rotated away from the optimum phase-matching angle to reduce the conversion efficiency from its maximum of $\approx 25\%$ down to zero. Due to the top-hat beam profile we suppose a spatially uniform SHG efficiency. The temporal walkoff due to dispersion is partly compensated by shortening due to the nonlinearity of SHG [30] so that we suppose a 35-fs pulse duration for the $2\omega$-component.

The timing of the two color-components is controlled by a 1.5-mm thick calcite crystal (C₂; 35 mm×35 mm; Berhard Halle Nachfl.) with its optic axis oriented at 45° to the crystal front surface and perpedicular to the fundamental's polarization axis. As shown in Appendix 1, small rotations of the crystal about this axis lead to quasi-linear shifts of the relative group delay $\tau_g$ and phase delay $\tau_\phi$ of the two color components. Their temporal pulse envelopes are well overlapped for a 4.3° incidence angle on the calcite crystal. Scanning the calcite angle in small steps ($\sim 0.01°$) over a 0.13°-wide range around this position shifts the group delay in total by $\approx 1.7$ fs and the phase delay by $\tau_\phi = 1.34$ fs, the latter corresponding to $T' = 2\pi/2\omega_L$, i.e. one period of the second harmonic. Such a rotation therefore scans the full range of two-color optical-cycle shapes with excellent temporal stability and reproducibility while keeping the pulse envelopes well overlapped.

<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) Experimental setup with the prepulse (red) and main pulse (purple), with adjustable relative delay $\tau$ as set by the two-part mirror M / M<sub>P</sub>, and the three crystals C<sub>1</sub>, C<sub>2</sub>, WP for two-color waveform generation (see text). The inset shows waveforms obtained for $\tau_{\phi,h} = 0$ (solid gray line) and $\tau_{\phi,e} = -T'/4$ (dashed black line).</figcaption></figure>

Finally, a 273-µm thick quartz wave plate (WP; 6.5λ at 400 nm, 3λ at 800 nm; Eksma Optics) rotates the polarization direction of the second harmonic so as to be parallel to that of the fundamental. The total energy of the one- or two-color pulses on target is $\approx 125$ mJ.

Both the prepulse and the two-color main-pulse are unpolarized and focused by a $f = 200$ mm off-axis parabola onto a fused-silica target at 58° angle of incidence. The measured diameter (FWHM in intensity) of the optimized focal spot of the fundamental color component is 5.6 µm, whereas it is estimated to 3.3 µm for the second harmonic beam, leading to on-target peak intensities of $5 \times 10^{18}$ and $3.5 \times 10^{18}$ W/cm$^2$, respectively.

Two diagnostics for the plasma mirror emission have been implemented as displayed in Fig.1. First, the spatial profile of the electron beam, $S_e(\theta_x, \theta_y)$, was measured using an insertable LANEX screen, placed at 150 mm from the target and imaged by a CCD camera. This scintillating screen is protected with a 13 µm thick aluminium foil and thus detects only electrons with energies $\gtrsim 0.15$ MeV. The high-harmonic emission, $S_h(\omega, \theta_y)$, was characterized using an angularly-resolved XUV spectrometer with an angular acceptance of 200 mrad.

## III. EXPERIMENTAL RESULTS

### A. Single-color ($\omega_L$) driver

As a first step, we used a driving field of frequency $\omega_L$ only, obtained by rotating the KDP (C₁) crystal by 7° away from the optimum phase-matching angle, thus
<figure><img src="image_3.png" /></figure>

FIG. 2. Single-color ($\omega_L$) driver: Angular emission pattern of accelerated electrons (left) and angularly-resolved harmonic spectrum (right) obtained with the minimal gradient scale length $L_g \lesssim \lambda_L/50$ (a) and with a gradient scale length optimizing both ROM HHG and vacuum-laser-accelerated electron emission (b). The vertical black lines in the electron beam profiles mark the XUV spectrometer angular acceptance.

completely suppressing SHG while retaining the same $\approx 125$ mJ pulse energy on target. We measured the evolution of the experimental observables with the gradient scale length $L_g$, in otherwise equal interaction conditions to the subsequent two-color experiments. Letting the prepulse arrive after the main pulse minimizes $L_g$ to $\lesssim \lambda_L/50$, limited by the main pulse temporal contrast. As demonstrated earlier [10, 32], such a steep gradient strongly favors harmonic generation by the Coherent Wake Emission (CWE) process [33]. This mechanism leads to a sharp spectral cutoff at the maximum plasma frequency given by the target material density ($\sim 20\omega_L$ for silica), as well as rather high divergence [34] and marked spatial modulations of the emitted beam. A smoother density gradient ($L_g \approx \lambda/16$) provides optimal conditions for the ROM process [10, 21, 32], which at moderately relativistic driving intensities leads to lower-divergence, spatially smooth beams [10, 35, 36]. Fig. 2 shows this striking spatial difference in the measured HHG spectra. This evidences a transition from CWE to ROM harmonics with increasing $L_g$, although the spectra do not extend beyond the CWE spectral cutoff in our weakly relativistic regime ($a_0 \approx 1.5$).

For the high-energy electron emission, also shown in Fig. 2, we find a correlation of the ejected charge with the ROM efficiency, i.e. both are maximized for $L_g \sim \lambda_L/16$. The electron beam in the ROM-conditions was very similar to that measured in previous experiments [7], with a distinctive hole in the spatial beam profile centered on

<figure><img src="image_4.png" /></figure>

FIG. 3. Two-color ($\omega_L + 2\omega_L$) driver: Angular emission pattern of accelerated electrons (left) and angularly-resolved harmonic spectrum (right) obtained with a gradient scale length $L_g \lesssim \lambda_L/50$ and for phase delays maximizing either the electron emission $\tau_{\phi,e}$, (a), or the high-order harmonic emission $\tau_{\phi,h}$ (b). The vertical black lines in the electron beam profiles mark the XUV spectrometer angular acceptance.

the specular direction, resulting from the interaction between the ejected electrons and the reflected laser light during VLA. Both the ROM mechanism and electron acceleration benefit from the charge separation field produced at longer gradients. Their correlation for sinusoidal, i.e. single-color, driving waveforms has been reported before [21, 22] and interpreted in the framework of the push-pull model [13].

## B. Two-color ($\omega_L + 2\omega_L$) driver

Using the temporally-shaped two-color main beam, this clear correlation disappears: while the ejected electron charge varies very similarly with $L_g$ as with the $\omega_L$-only driver, the harmonic signal drops with increasing $L_g$ to be finally absent around $L_g \sim \lambda_L/16$ (see Appendix B). This is the case for all two-color phase delays $\tau_\phi$. The parameters optimizing the high-order harmonic generation are thus no longer the same as those for relativistic electron emission, which is a first indication that the two-color driver modifies the plasma mirror dynamics.

This is further corroborated by the measurement of the electron and harmonic signals for the shortest gradient ($L_g \lesssim \lambda_L/50$) as a function of the $\omega_L-2\omega_L$ phase delay $\tau_\phi$, i.e. of the optical cycle shape. The main experimental findings are shown in Fig. 3, by presenting the angular emission pattern of relativistic electrons $S_e(\theta_x, \theta_y)$ and the angularly-resolved harmonic spectrum $S_h(\omega, \theta_y)$ in
<figure><img src="image_5.png" /></figure>

FIG. 4. Experimental harmonic spectrum $\tilde{S}_h$ (a) and accelerated electron beam profile $\tilde{S}_e$ (b) in the incidence plane as a function of the phase delay $\tau_\phi$ (with an unknown experimental offset $\tau_0$). The total high-harmonic energy $W_h$ (blue circles) and electron charge $Q_e$ (red squares) obtained by integration over the vertical dimensions in the panels above are plotted in panel (c). The ejected electron charge and the harmonic energy similarly oscillate but with optima shifted by $\simeq 0.3T'$. The experimental parameters are the same as in Fig.3.

the cases where $\tau_\phi$ is optimized for either the high-energy electrons or the high-order harmonics. In our interaction conditions, we find clearly distinct optima for both observables. In particular, when the optical-cycle shape maximizes the electron charge (Fig.3a), a highly divergent harmonic beam is reminiscent of that obtained in the CWE regime, whereas this beam becomes much less divergent and brighter in the other case (Fig.3b). This suggests that these harmonics are now generated via the ROM process, which had not been observed with the $\omega_L$-single-color driver for such a short gradient (see Fig. 2). We conclude that on top of a less-bright CWE background, the ROM harmonic generation shows an on-off oscillation as function of the driving optical cycle shape. No significant change in the electron spatial distribution other than the total charge is noticed.

The details of the experimental phase-delay dependence of the ejected electron charge and the harmonic emission are presented in Fig.4. First, the emitted harmonic spectrum $\tilde{S}_h(\tau_\phi, \omega)$, shown in Fig.4a, represents the spectral intensity within the central 33-mrad wide divergence cone, i.e. preferentially the ROM signal. The total high-harmonic energy $W_h(\tau_\phi)$ is obtained by integration over the presented spectral range $(\omega/\omega_L \in [10, 20])$. A clear modulation with a period $T' \approx 1.35$ fs is observed (Fig.4c). The electron beam angular profile $\tilde{S}_e(\tau_\phi, \theta_x)$, obtained by integrating $S_e(\theta_x, \theta_y)$ over $\theta_y$, is shown in Fig.4b. Further integration over $\theta_x$ yields the

<figure><img src="image_6.png" /></figure>

FIG. 5. Ejected electron charge (red squares) and high-order harmonic energy (blue circles) as function of the phase delay $\tau_\phi$, obtained from 2D PIC simulations. Full blue and red lines show the normalized heuristic model predictions $S(\tau_\phi)$ and $P(\tau_\phi)$ for E-field steepness ($n=9$ here) and maximum pulling E-field, respectively. The periodic modulations of these electronic and harmonic signals are shifted by $\simeq T'/4$. The insets above show the driving fields for maximum electron (left) and high-harmonic (right) emission. A positive-valued electric field pulls electrons out of the plasma into vacuum.

emitted electron charge $Q_e(\tau_\phi)$ shown in Fig.4c. It oscillates with the same period as the harmonic signal, with lower contrast but nonetheless presenting clear extrema, which had never been observed before.

A striking finding is the dephasing of these periodic modulations: the optimum phase delays maximizing the ejected electron charge or ROM harmonic emission are shifted by $\approx 0.3T' = 0.4$ fs. Therefore, intense attosecond light pulses are not necessarily generated in correlation with high-charge electron beams, in contrast to experiments where control is solely achieved through the plasma density gradient scale length $L_g$. Temporally shaping the driving optical cycles thus adds a new dimension to the control parameter space, which allows optimizing properties of the ejected electron bunches whose effect on HHG outweighs that of their total charge.

As detailed in Appendix C, a ROM-HHG efficiency enhancement by a factor of $\approx 1.8$ is also measured. Compared to earlier simulations [28, 29] and experiments [30], this much more modest here certainly due to our weakly relativistic intensity.

# IV. NUMERICAL SIMULATIONS

## A. 2D PIC

In order to gain insight into the dynamically controlled sub-cycle plasma dynamics, we turn to Particle-In-Cell
simulations using the WARP+PXR code [37–43]. We consider a p-polarized incident laser pulse with electric field $E(t, \tau_\phi) = E_1(t) \sin(\omega_L t) + E_2(t+\tau_\phi) \sin[2\omega_L(t+\tau_\phi)]$, where $E_1(t)$ and $E_2(t)$ are 18-cycles long cosine-half-cycle envelopes ($\approx 30$ fs FWHM) with equal peak field strengths corresponding to a normalized vector potential of $a_0 = 1$ for an 800-nm wave. It impinges a solid target with gradient scale length $L_g = \lambda_L/50$ at $55^\circ$ angle of incidence. The main results of 2D PIC simulations are summarized in Fig.5, which shows the ejected electron charge (only electrons with energies $\gtrsim 0.1$ MeV), as well as the high-order harmonic energy (harmonics 7 to 13; a region just before the CWE spectral cutoff for the simulated plasma density, cf. Appendix D) as a function of the phase delay $\tau_\phi$. Similarly to the experiments (although with higher contrast), we retrieve clear oscillations of the harmonic and electron signals with period $T'$ and a relative shift by $\approx T'/4 = 0.35$ fs, in good agreement with the experimental results. Also shown in Fig. 5 are the incident driving waveforms that maximize the harmonic and electron emissions. In agreement with previous simulations done for similar as well as much higher laser intensities [28–30], we find that the harmonic emission is optimized for a maximized waveform steepness in the change from the push to the pull phase (from negative to positive E-field). The emitted electron charge is found to be maximized, in our weakly relativistic conditions, rather for a waveform with the strongest possible field in the pull-phase. This suggests a simple heuristic model: (i) a power-law dependence on the E-field steepness, $S(\tau_\phi) = (\max_t[\partial E(t, \tau_\phi)/\partial t|_{E(t)=0})]^n$, describes the harmonic signal modulation, and (ii) the maximum pulling field $P(\tau_\phi) = \max_t[E(t, \tau_\phi)]$, describes the electron charge modulation. Despite the extreme simplicity, these predictions are found to fit the signal modulations remarkably well, as shown in Fig. 5.

## B. 1D PIC

To gain a deeper physical insight, we have performed 1D PIC simulations which allow performing larger ensemble runs with better statistics. The simulated conditions are kept the same as in the 2D case discussed above. For each driving waveform characterized by $\tau_\phi$, we analyze the electron bunch properties during the central optical cycle. At every instant, only those electrons with velocity $> 0.1c$ towards the vacuum are selected to calculate their total charge, the bunch position as their spatial center of mass, and the bunch width as their spatial root mean square width. The time derivative of the bunch position gives the bunch velocity. We then select a time interval $\Delta t_p$ for the pull phase, starting at the instant of maximum bunch charge and ending at the instant when either the bunch velocity vanishes or when the bunch position has reached $z = 0.25\lambda_L$. The bunch properties plotted in Fig.6 as function of $\tau_\phi$ are the bunch charge and velocity, both averaged over $\Delta t_b$, and the bunch width at the

<figure><img src="image_7.png" /><figcaption>FIG. 6. Electron bunch properties during the pull phase as a function of the phase delay $\tau_\phi$, extracted from 1D PIC simulations: mean charge (a), mean velocity (b), and initial spatial width (c). Blue and red shadings mark the phase delay ranges maximizing harmonic and electron emissions, respectively, in the 2D simulations as shown in Fig.5. The insets show the temporal evolution of the plasma electron density (gray scale color map, in log scale) in each phase delay range, spatially resolved along the normal to the target surface. The emitted attosecond pulses are overlaid to this density map in purple. Note that the bunch charge (a) is not as that of the emitted VLA electron beam as displayed in Fig.5.</figcaption></figure>

beginning of $\Delta t_b$. Note that this bunch charge is not directly that of the emitted VLA electron beam: the bunch analyzed here has not yet left the plasma and, as clearly visible in the insets in Fig.6, part of it will turn back into the plasma during the subsequent pushing phase, in particular in our weakly relativistic conditions.

As in the 2D-simulations, the harmonic emission is maximized around $\tau_\phi = 0$ and thus for the fastest switch from pushing to pulling field. While this does not maximize the electron bunch charge, it creates both a well compressed and fast outgoing electron bunch, which are the key parameters for efficient ROM HHG [28, 29]. The electron velocity alone, as considered in [30], is not a sufficient criterion: while it is also boosted by the waveforms with enhanced pulling field, around $\tau_\phi = -T'/4$, this comes at the expense of a weakened push phase and thus a less compressed electron bunch. This is the situation we find to be optimal for electron emission in our conditions. As apparent in the left inset in Fig.6, the weaker field in the subsequent pushing phase lets a greater proportion of the bunch electrons fully escape the plasma with high velocity to be injected into the reflected laser field for VLA. Putting all emphasis on the pushing field
strength ($\tau_{\phi} = +T'/4$), leads to the narrowest bunch, but at the expense of a smaller velocity. Finally, around $\tau_{\phi} = +T'/2$, where the harmonic emission is minimized, pushing and pulling peak fields are approximately balanced but the switch between the two is so slow, that the compressed electron bunch breaks apart and is not effectively accelerated outward.

## V. CONCLUSIONS

In conclusion, we used ultra-intense temporally shaped optical cycles to experimentally realize the dynamic control on the sub-femtosecond time scale of collective plasma electron motion on plasma mirrors emitting high-order harmonics and relativistic electrons. The experimentally observed difference of the optimal two-color phase delays for harmonics and electrons in our conditions provides a strict gage for the comparison to PIC simulations. The fact that our simulations reproduce this difference thus significantly underpins the interpretations we draw from them, elucidating how the relative strength of pushing and pulling field half-cycles and the rapidity of the switch between them controls the formation and outward acceleration of the dense electron bunches that lead to HHG and seed VLA.

These results illustrate the new possibilities opened by the attosecond steering of collective plasma electron motion driven by ultra-intense multi-color waveforms. Our enhanced experimental characterization and fine control over extreme light-matter interactions together with their accurate modeling represent fundamental progress in high-field physics, of particular relevance, e.g. to the future quest of the Schwinger limit using PW-class lasers [44]. We expect that expanding this approach to other frequency combinations and polarization shaping will have a strong impact on the development of powerful plasma-based radiation and particle sources.

## ACKNOWLEDGMENTS

We thank Aurélien Houard of LOA for lending the KDP crystal. This work was funded by the European Research Council under Contract No. 694596 of the European Union Horizon 2020 Research and Innovation Programme, the EquipEx Attolab (ANR-11-EQPX-005-ATTOLAB), the LabEx PALM (ANR-10-LABX-0039-PALM) and the Région Île-de-France (SESAME-2012-ATTOLITE). An award of computer time was provided by the INCITE programme (project PlasmInSilico). This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under contract DE-AC02-06CH11357.

## APPENDIX A: TWO-COLOR TIMING CONTROL

Figure 7 shows the calculated relative group and phase delays between the 800-nm fundamental and the 400-nm second harmonic as a function of the incidence angle on the birefringent calcite crystal (C₂). At 4.3° incidence angle, the ≈ 90 fs group delay induced by the propagation through the KDP (C₁) and subsequent waveplate (WP) are compensated. Scanning the calcite angle over a 0.13°-wide range around this position shifts the group delay in total by ≈ 1.7 fs and the phase delay by $\tau_{\phi} = 1.34$ fs, the latter corresponding to one period of the second harmonic. In other words, the pulse envelopes' relative timing shifts together with a simultaneous drift of the pulses' CEPs. Since the rotations remain very small, the resulting change of the pulse-envelope overlap remains negligible (≈ 3.4 fs in the experimental scan over two modulation periods, vs. the pulse durations of ≈ 30 fs). We can therefore focus only on the scanned relative phase delay $\tau_{\phi}$ of the two carrier waves ($\omega$ and $2\omega$), which is the quantity that governs the shape of optical cycle under the combined pulse envelope.

<figure><img src="image_8.png" /><figcaption>FIG. 7. Calculated relative group (left) and phase delays (right) between the 800-nm fundamental and the 400-nm second harmonic as a function of the incidence angle on the birefringent calcite crystal (C₂).</figcaption></figure>

## APPENDIX B: GRADIENT DEPENDENCE WITH THE TWO-COLOR DRIVER

Figure 8 shows HHG spectra and electron beam profiles measured with $\omega_L + 2\omega_L$ two-color driving waveforms with the phase delays $\tau_{\phi}$ set so as to either optimize low-divergence ROM HHG or electron emission. For each driving waveform, the plasma gradient scale length $L_g$ was scanned. Figure 8 shows the data for the same two scale lengths as in Fig. 2, i.e. the shortest accessible value $L_g \le \lambda/50$ as well as $L_g \approx \lambda/16$, found optimal for both ROM HHG and electron emission with the $\omega_L$-single-color driver.

Here, with both two-color driving waveforms, we find that while the ejected electron charge barely changes when increasing $L_g$ up to ≈ $\lambda/16$, the HHG signal rapidly drops and has almost completely vanished for the scale length $L_g \approx \lambda/16$. Thus, the effect of the plasma density gradient scale length is different for single-color and two-color driving waveforms, and there is thus no longer a common set of optimal conditions for HHG and electron emissions in the two-color case.
<figure><img src="image_9.png" /></figure>

FIG. 8. Angular emission pattern of accelerated electrons (left) and angularly-resolved harmonic spectrum (right) obtained with the same two gradient scale lengths as in Fig.2 ($L_g \approx \lambda/50$ above and $L_g \approx \lambda/16$ below). The phase delay of the two-color driving waveforms is set such that it optimizes low-divergence ROM HHG emission (a) or fast electron emission (b). The vertical black lines in the electron beam profiles mark the XUV spectrometer angular acceptance.

## APPENDIX C: EFFICIENCY COMPARISON

Figure 9 summarizes a comparison of the HHG efficiency obtained with a $\omega_L$-single-color and $\omega_L + 2\omega_L$ two-color drivers. Clearly, the HHG radiant intensities obtained with the single- and two-color beams are rather similar and we do not observe an order-of-magnitude efficiency boost as predicted in earlier simulations [28, 29] and reported for the single previous experiment [30]. The reason may be the lower driving intensity in our case, as evident from the fact that the HHG spectra in ref. [30] extend beyond the CWE spectral cutoff at 30 eV (for

<figure><img src="image_10.png" /></figure>

FIG. 9. Angularly-resolved harmonic spectra obtained with minimal plasma gradient scale length $L_g \approx 0$ (a)-(c) and with $L_g \approx 0.06\lambda$ (d). Two-color ($\omega_L + 2\omega_L$) driver with a phase delay $\tau_\phi$ minimizing [(a), “2-color worst”] or maximizing [(b), “2-color best”] the low-divergence ROM HHG emission. Single-color ($\omega_L$) driver [(c), “1-color CWE”], [(d), “1-color ROM”]. The linear color scale is the same in all panels such that the signals are quantitatively comparable. The red boxes in (a)-(d) mark the angular-spectral integration range for the efficiency comparison in the text.

<figure><img src="image_11.png" /></figure>

FIG. 10. Electron beam spatial profiles $S(e)$ measured with minimal plasma gradient scale length $L_g \approx 0$ (a)-(c) and with $L_g \approx 0.06\lambda$ (d)-(f), and with a single-color ($\omega_L$) laser driver (a),(d), a two-color ($\omega_L + 2\omega_L$) driver with a phase delay $\tau_\phi$ maximizing (b),(e) or minimizing (c),(f) the electron signal.

SiO$_2$ targets) while ours do not.

Integrating the generated HHG emission spatiot spectrally (from 15.2 eV to 28.5 eV and over a 33-mrad wide cone around the spatial beam center, see the red boxes in Fig. 9), we find an HHG energy enhancement by a factor 1.8 for the “2-color best” case, i.e. the 2-color driver with optimal phase delay $\tau_\phi$ with the steepest initial plasma gradient (Fig. 9(b)), as compared to the “1-
color ROM" reference case, i.e. the 1-color driver (same on-target pulse energy of ≈ 125 mJ) with the optimized plasma gradient scale length $L_g ≈ 0.06λ$ (Fig. 9(d)). Relative to the same reference and in the same integration limits, the “1-color CWE” case yields a 1.3 times higher HHG energy (typical for the mildly relativistic regime near $a_0 = 1$, and the factor would increase further if a larger divergence cone were considered), while the “2-color worst” case reduces the HHG energy by a factor 0.6. We must however note that concluding from these observations on a useful ROM-HHG efficiency enhancement is made difficult by the presence of the high-divergence CWE background that remains present in the 2-color case.

Figure 10 summarizes a comparison of the detected electron beam profiles obtained with a $ω_L$-single-color and $ω_L + 2ω_L$ two-color drivers. Both for the steepest plasma gradient as well as for the scale length $L_g ≈ 0.06λ$, optimal for ROM HHG emission as well as vacuum laser electron acceleration, we find very similar accelerated electron charges for a single-color driver and for two-color drivers with phase delay $τ_{φ,e}$ maximizing the electron charge. For the phase delay $τ_{φ,e} + T'/2$, minimizing the electron charge, the charge drops to about 65% of the charge obtained with $τ_{φ,e}$.

## APPENDIX D: DETAILS ON PIC SIMULATIONS

The simulations presented in the paper were performed with the Particle-In-Cell code WARP+PXR that relies on a very accurate pseudo-spectral solver – PSATD (Pseudo-Spectral Analytic Time Domain) – where the fields are analytically integrated in Fourier space over a finite time step. The advantage of such a solver is that it largely mitigates most numerical effects; in particular it suppresses all effects related to numerical dispersion, present in all finite-difference based solvers. It then allows for a better convergence to the solution in term of computational resources [41].

Note that the two-color driving field $E(t, τ_φ) = E_1(t) \sin(ω_L t) + E_2(t + τ_φ) \sin[2ω_L(t + τ_φ)]$ used in the

simulations is a simplified expression that neglects dispersion by assuming equal group and phase delay shifts: both the envelope $E_2$ and the $2ω$-carrier wave are shifted by $τ_φ$. Since the effect of the small relative group delay shift should be negligible (cf. the discussion in Appendix A), the difference of 0.36 fs per $τ_φ$-period between experiment and simulations should also be negligible.

Note also that in the simulations we only scanned over one 1.34-fs period of $τ_φ$ and then duplicate the results to show two periods in Figs. 5 and 6 for better comparability with the experimental scan in Fig. 4 that was performed over two $τ_φ$-periods.

In table I, we summarize the numerical and physical parameters used for our 1D and 2D simulations. Note that the CWE spectral cutoff in the experiments is at harmonic 20 while it is at harmonic 14 in the simulations due to the lower assumed bulk plasma density. The harmonic spectral range in Fig. 5 is chosen to be comparable to the experiment such that the CWE and ROM mechanisms contribute significantly.

Note further that the 1D simulations were performed in a Lorentz boosted frame to emulate an angle of incidence $θ$ of the laser on target.

<table><caption>TABLE I. Numerical and physical parameters for simulations presented in the paper. Here, $θ$ is the angle of incidence, $Δx$ and $Δy$ the spatial steps, $Δt$ the temporal step, $c$ the celerity of light and $n$ is an index to tune to modify the delay between the two color components.</caption><thead><tr><th></th><th>1D</th><th>2D</th></tr></thead><tbody><tr><td>Maxwell solver</td><td>PSATD</td><td>PSATD</td></tr><tr><td>Δx (Δy)</td><td>λ/350</td><td>λ/160</td></tr><tr><td>cΔt</td><td>Δx</td><td>Δx</td></tr><tr><td>particles/cell</td><td>500</td><td>5 × 5</td></tr><tr><td>a<sub>0</sub> (ω<sub>L</sub>)</td><td>1</td><td>1</td></tr><tr><td>a<sub>0</sub> (2ω<sub>L</sub>)</td><td>0.5</td><td>0.5</td></tr><tr><td>laser duration</td><td>≈ 30 fs</td><td>≈ 30 fs</td></tr><tr><td>target density</td><td>220 n<sub>c</sub></td><td>220 n<sub>c</sub></td></tr><tr><td>L</td><td>λ/50</td><td>λ/50</td></tr><tr><td>θ</td><td>55°</td><td>55°</td></tr><tr><td>τ<sub>φ</sub></td><td>nT<sub>L</sub>/16</td><td>nT<sub>L</sub>/16</td></tr></tbody></table>

[1] G. D. Tsakiris, K. Eidmann, J. Meyer-ter Vehn, and F. Krausz, Route to intense single attosecond pulses, New Journal of Physics 8, 19 (2006).

[2] B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, et al., High harmonic generation in the relativistic limit, Nature Physics 2, 456 (2006).

[3] C. Thaury, F. Quéré, J.-P. Geindre, A. Levy, T. Cecotti, P. Monot, M. Bougeard, F. Réau, P. d'Oliveira, P. Audebert, et al., Plasma mirrors for ultrahigh-intensity optics, Nature Physics 3, 424 (2007).

[4] U. Teubner and P. Gibbon, High-order harmonics from laser-irradiated plasma surfaces, Reviews of Modern Physics 81, 445 (2009).

[5] L. Chopineau, A. Denoeud, A. Leblanc, E. Porat, P. Martin, H. Vincenti, and F. Quéré, Spatio-temporal characterization of attosecond pulses from plasma mirrors, Nature Physics 17, 968 (2021).

[6] A. Macchi, M. Borghesi, and M. Passoni, Ion acceleration by superintense laser-plasma interaction, Reviews of Modern Physics 85, 751 (2013).

[7] M. Thévenet, A. Leblanc, S. Kahaly, H. Vincenti,
A. Vernier, F. Quéré, and J. Faure, Vacuum laser acceleration of relativistic electrons using plasma mirror injectors, Nature Physics 12, 355 (2016).

[8] Y. Tian, J. Liu, W. Wang, C. Wang, A. Deng, C. Xia, W. Li, L. Cao, H. Lu, H. Zhang, Y. Xu, Y. Leng, R. Li, and Z. Xu, Electron Emission at Locked Phases from the Laser-Driven Surface Plasma Wave, Physical Review Letters 109, 115002 (2012).

[9] N. Zaïm, F. Böhle, M. Bocoum, A. Vernier, S. Haessler, X. Davoine, L. Videau, J. Faure, and R. Lopez-Martens, Few-cycle laser wakefield acceleration on solid targets with controlled plasma scale length, Physics of Plasmas 26, 033112 (2019).

[10] S. Kahaly, S. Monchocé, H. Vincenti, T. Dzelzainis, B. Dromey, M. Zepf, P. Martin, and F. Quéré, Direct observation of density-gradient effects in harmonic generation from plasma mirrors, Physical Review Letters 110, 175001 (2013).

[11] M. Bocoum, F. Böhle, A. Vernier, A. Jullien, J. Faure, and R. Lopez-Martens, Spatial-domain interferometer for measuring plasma mirror expansion, Optics Letters 40, 3009 (2015).

[12] A. A. Gonoskov, A. V. Korzhimanov, A. V. Kim, M. Marklund, and A. M. Sergeev, Ultrarelativistic nanoplasmonics as a route towards extreme-intensity attosecond pulses, Physical Review E 84, 046403 (2011).

[13] M. Thévenet, H. Vincenti, and J. Faure, On the physics of electron ejection from laser-irradiated overdense plasmas, Physics of Plasmas 23, 063119 (2016).

[14] A. Gonoskov, Theory of relativistic radiation reflection from plasmas, Physics of Plasmas 25, 013108 (2018).

[15] R. Lichters, M. t. Vehn, and A. Pukhov, Short-pulse laser harmonics from oscillating plasma surfaces driven at relativistic intensity, Physics of Plasmas 3, 3425 (1996).

[16] S. Gordienko, A. Pukhov, O. Shorokhov, and T. Baeva, Relativistic Doppler Effect: Universal Spectra and Zep-tosecond Pulses, Physical Review Letters 93, 115002 (2004).

[17] C. Thaury and F. Quéré, High-order harmonic and attosecond pulse generation on plasma mirrors: basic mechanisms, Journal of Physics B: Atomic, Molecular and Optical Physics 43, 213001 (2010).

[18] an der Brügge and A. Pukhov, Enhanced relativistic harmonics by electron nanobunching, Physics of Plasmas 17, 033110 (2010).

[19] J. M. Mikhailova, M. V. Fedorov, N. Karpowicz, P. Gibbon, V. T. Platonenko, A. M. Zheltikov, and F. Krausz, Isolated Attosecond Pulses from Laser-Driven Synchrotron Radiation, Physical Review Letters 109, 245005 (2012).

[20] M. R. Edwards and J. M. Mikhailova, The X-Ray Emission Effectiveness of Plasma Mirrors: Reexamining Power-Law Scaling for Relativistic High-Order Harmonic Generation, Scientific Reports 10, 1 (2020).

[21] L. Chopineau, A. Leblanc, G. Blaclard, A. Denoeud, M. Thévenet, J.-L. Vay, G. Bonnaud, P. Martin, H. Vincenti, and F. Quéré, Identification of coupling mechanisms between ultraintense laser light and dense plasmas, Physical Review X 9, 011050 (2019).

[22] S. Haessler, F. Böhle, M. Bocoum, M. Ouillé, J. Kaur, D. Levy, L. Daniault, A. Vernier, J. Faure, and R. Lopez-Martens, Relativistic high-harmonic generation and correlated electron acceleration from plasma mirrors at 1 khz repetition rate, arXiv:2005.01343 [physics] (2021).

[23] R. Bartels, S. Backus, E. Zeek, L. Misoguti, G. Vdovin, I. Christov, M. Murnane, and H. Kapteyn, Shaped-pulse optimization of coherent emission of high-harmonic soft x-rays, Nature 406, 164 (2000).

[24] N. Dudovich, O. Smirnova, J. Levesque, Y. Mairesse, M. Y. Ivanov, D. Villeneuve, and P. B. Corkum, Measuring and controlling the birth of attosecond xuv pulses, Nature Physics 2, 781 (2006).

[25] S. Haessler, T. Balčiunas, G. Fan, G. Andriukaitis, A. Pugžlys, A. Baltuška, T. Witting, R. Squibb, A. Zaïr, J.-W.-G. Tisch, J.-P. Marangos, and L.-E. Chipperfield, Optimization of quantum trajectories driven by strong-field waveforms, Physical Review X 4, 021028 (2014).

[26] C. Jin, G. Wang, H. Wei, A.-T. Le, and C. Lin, Waveforms for optimal sub-kev high-order harmonics with synthesized two-or three-colour laser fields, Nature Communications 5, 1 (2014).

[27] P. Wei, J. Miao, Z. Zeng, C. Li, X. Ge, R. Li, and Z. Xu, Selective enhancement of a single harmonic emission in a driving laser field with subcycle waveform control, Physical Review Letters 110, 233903 (2013).

[28] M. R. Edwards, V. T. Platonenko, and J. M. Mikhailova, Enhanced attosecond bursts of relativistic high-order harmonics driven by two-color fields, Optics Letters 39, 6823 (2014).

[29] M. R. Edwards and J. M. Mikhailova, Waveform-controlled relativistic high-order-harmonic generation, Physical Review Letters 117, 125001 (2016).

[30] M. Yeung, S. Rykovanov, J. Bierbach, L. Li, E. Eckner, S. Kuschel, A. Woldegeorgis, C. Rödel, A. Sävert, G. Paulus, et al., Experimental observation of attosecond control over relativistic electron bunches with two-colour fields, Nature Photonics 11, 32 (2017).

[31] A. Lévy, T. Ceccotti, P. D’Oliveira, F. Réau, M. Perdrix, F. Quéré, P. Monot, M. Bougeard, H. Lagadec, P. Martin, et al., Double plasma mirror for ultrahigh temporal contrast ultraintense laser pulses, Optics Letters 32, 310 (2007).

[32] C. Rödel, a. d. Brügge, J. Bierbach, M. Yeung, T. Hahn, B. Dromey, S. Herzer, S. Fuchs, A. G. Pour, E. Eckner, M. Behmke, M. Cerchez, O. Jäckel, D. Hemmers, T. Toncian, M. C. Kaluza, A. Belyanin, G. Pretzler, O. Willi, A. Pukhov, M. Zepf, and G. G. Paulus, Harmonic Generation from Relativistic Plasma Surfaces in Ultrasteep Plasma Density Gradients, Physical Review Letters 109, 125002 (2012).

[33] F. Quéré, C. Thaury, P. Monot, S. Dobosz, P. Martin, J.-P. Geindre, and P. Audebert, Coherent wake emission of high-order harmonics from overdense plasmas, Physical Review Letters 96, 125004 (2006).

[34] A. Leblanc, S. Monchocé, H. Vincenti, S. Kahaly, J.-L. Vay, and F. Quéré, Spatial properties of high-order harmonic beams from plasma mirrors: a ptychographic study, Physical Review Letters 119, 155001 (2017).

[35] B. Dromey, D. Adams, R. Hörlein, Y. Nomura, S. Rykov-anov, D. Carroll, P. Foster, S. Kar, K. Markey, P. McKenna, et al., Diffraction-limited performance and focusing of high harmonics from relativistic plasmas, Nature Physics 5, 146 (2009).

[36] H. Vincenti, S. Monchocé, S. Kahaly, G. Bonnaud, P. Martin, and F. Quéré, Optical properties of relativistic plasma mirrors, Nature Communications 5, 1 (2014).

[37] http://blast.lbl.gov/blast-codes-warp.

[38] https://picsar.net.
[39] J.-L. Vay, R. Lehe, H. Vincenti, B. Godfrey, I. Haber, and P. Lee, Recent advances in high-performance modeling of plasma-based acceleration using the full pic method, Nucl. Instrum. Methods Phys. Res. **829**, 353 (2016).

[40] H. Vincenti and J.-L. Vay, Detailed analysis of the effects of stencil spatial variations with arbitrary high-order finite-difference maxwell solver, Comput. Phys. Commun. **200**, 147 (2016).

[41] G. Blaclard, H. Vincenti, R. Lehe, and J.-L. Vay, Pseudospectral maxwell solvers for an accurate modeling of doppler harmonic generation on plasma mirrors with particle-in-cell codes, Physical Review E **96**, 033305 (2017).

[42] H. Vincenti, M. Lobet, R. Lehe, R. Sasanka, and J.-L. Vay, An efficient and portable simd algorithm for charge/current deposition in particle-in-cell codes, Comput. Phys. Commun. **210**, 145 (2017).

[43] H. Vincenti and J.-L. Vay, Ultrahigh-order maxwell solver with extreme scalability for electromagnetic pic simulations of plasmas, Comput. Phys. Commun. **228**, 22 (2018).

[44] L. Fedeli, A. Sainte-Marie, N. Zaïm, M. Thévenet, J.-L. Vay, A. Myers, F. Quéré, and H. Vincenti, Probing strong-field qed with doppler-boosted petawatt-class lasers, Physical Review Letters **127**, 114801 (2021).