Techniques to generate intense isolated attosecond pulses from relativistic plasma mirrors

Doppler harmonic generation of a high-power laser on a relativistic plasma mirror is a promising path to produce bright attosecond light bursts. However, a major challenge has been to find a way to generate isolated attosecond pulses, better suited to timed-resolved experiments, rather than trains of pulses. A promising technique is the attosecond lighthouse effect, which consists in imprinting different propagation directions to successive attosecond pulses of the train, and then spatially filtering one pulse in the far field. However, in the relativistic regime, plasma mirrors get curved by the radiation pressure of the incident laser and thus focus the generated harmonic beams. This increases the harmonic beam divergence and makes it difficult to separate the attosecond pulses angularly. In this article, we propose two novel techniques readily applicable in experiments to significantly reduce the divergence of Doppler harmonics, and achieve the generation of isolated attosecond pulses from the lighthouse effect without requiring few-cycle laser pulses. Their validity is demonstrated using state-of-the-art simulations, which show that isolated attosecond pulses with $10\text{TW}$ peak power in the XUV range can be generated with PW-class lasers. These techniques can equally be applied to other generation mechanisms to alleviate the constraints on the duration on the laser pulses needed to generate isolated attosecond pulses.

Figure 1

Principle of the attosecond lighthouse effect: separation criterion. The incident laser beam is shown in red and the multiple attosecond beamlets generated through the highly nonlinear interaction are shown in purple. The wavefront rotation (WFR) of the laser field at focus is sketched in the upper inset.

Figure 2

PM denting by laser radiation pressure. This figure shows simulation results of a 2D particle-in-cell (PIC) simulation. In this simulation, the laser impinges on the PM at a ${45}^{\circ }$ angle of incidence. The PM has an exponential density profile of gradient scale length $L={\lambda }_0/15$. The simulation results are displayed in a special Lorenz boosted frame [43] where the laser is at normal incidence on the PM, thereby simplifying the visualization of simulation results. The gray colorscale represents a snapshot of the PM electron density. The colorscale represents the generated Doppler harmonic field (zoom on a single attosecond pulse within the generated train of pulses), where we filtered harmonic orders 8 to 22.

Figure 3

2D PIC simulations of attosecond pulse emission by a relativistic PM. Simulation results are displayed in the same boosted frame as in Fig. 2, where the laser is normally incident on the PM. Panels (a) and (b): Schematic representation of driving laser wavefronts. Panels (c) and (d): The colorscale represents the electromagnetic field of one attosecond pulse of the train, when the PM surface is placed at the best focus of the laser beam, and at a distance $\delta z=0.6Z_r$ after this best focus ($Z_r$ Rayleigh length of the laser beam), respectively. The gray colorscale represents the electron density of the PM.

Figure 4

Optimal defocusing distance as a function of the gradient density scale length $L$ at the PM surface. The red crosses are obtained from 2D PIC simulations, and the black dashed line corresponds to the prediction of Eq. (20) for a laser amplitude $a_0=30$ (without WFR), an angle of incidence $\theta ={45}^{\circ }$, a laser duration ${\tau }_0=16\text{fs}$, and a PFT parameter $\xi ={\xi }_0$.

Figure 5

(a) Evolution of ${\eta }_{\xi }$ as a function of $\xi$. The black curve represents ${\eta }_{\xi }$ as given by Eq. (22) for harmonic order $n=20$. Red circles correspond to ${\eta }_{\xi }$ obtained from 2D PIC simulations for harmonic order $n=20$. In both cases, we assumed an angle of incidence ${\theta }_0={55}^{\circ }$, a laser duration ${\tau }_0=16\text{fs}$ (without WFR), a PM gradient scale length $L=\frac{{\lambda }_0}{20}$, and a normalized laser amplitude $a_0=30$ (at best focus, without WFR). (b) Evolution of the actual laser amplitude at the target surface obtained using Eq. (21), with WFR and optimal defocusing.

Figure 6

Angular profiles of Doppler harmonic beams (orders 20 to 30) obtained from 2D PIC simulations for two different pulse-front tilt values: (a) $\xi ={\xi }_0$ and (b) $\xi =1.5{\xi }_0$. All other laser and plasma parameters are kept constant otherwise: ${\theta }_0={55}^{\circ }$, $L=\frac{{\lambda }_0}{20}$, a laser duration ${\tau }_0=16\text{fs}$ (without WFR, at best focus), and $a_0=30$ (at best focus, without WFR). On each panel, the red dots highlight the direction of emission of the successive attosecond pulses of the train.

Figure 7

Illustration of (a) PM denting for a spatially flat laser beam intensity profile and (b) of the Doppler harmonic beam generated by such a flat PM.

Figure 8

(a) Intensity of a laser beam with SC (but without any spectral shaping) in the spatio-spectral domain $(x,\omega )$. (b) Spectrally integrated spatial profile of this beam along the $x$ direction. (c, d) Same quantities, now with the application of the spectral shaping of Fig. 9.

Figure 9

Effect of spectral shaping on the laser spectrum. The red curve is the laser spectrum without any spectral shaping. Using a programmable acousto-optic filter, the gain modulation function shown in black is applied to the whole beam, leading to the spectrum plotted in blue. This beam is then shaped spatiotemporally (application of PFT) and focused to induce the attosecond lighthouse effect on target.

Figure 10

Angular profile of the Doppler harmonic beam (between harmonic orders 15–20) obtained for each simulation case of Table 1. Panels (a) to (c) correspond to simulation cases 1 to 3 (see text and Table 1), respectively. On panels (b) and (c), the red dots highlight the direction of emission of the successive attosecond pulses of the train.

Figure 11

Angular profiles of Doppler harmonics obtained from 3D PIC simulations for (a) harmonic orders $\ge 16$ and (b) harmonic orders $\ge 26$. The side graphs represent lineouts (integrated along ${\theta }_y$) of the harmonic angular profiles as a function of ${\theta }_x$ (direction of WFR). On each panel inset, the red dots highlight the angles of emission of the successive attosecond pulses of the train.

Figure 12

Temporal profile of the attosecond pulse train in the 3D simulation. The blue curve represents the temporal evolution of the reflected field amplitude (where we filtered harmonic orders $\ge 26$). The red curve corresponds to the reflected laser field (fundamental frequency only).

Figure 13

(a, c): Spatiotemporal profile of central attosecond pulses along the ${\theta }_x$ axis, for harmonic orders (a) $\ge 16$ and (b) $\ge 26$. (b, d) Intensity profile of the attosecond pulse obtained after the 6 mrad slit filter [sketched in panels (a) and (c)]. After the slit, we obtain a central attosecond pulse with a duration of 182 as and 200 as (FWHM), respectively and an energy of 1.5 and $0.18\text{mJ}$ leading to a peak power of $7\mathrm{TW}$ for the first harmonic range (harmonic orders $>16$) and of $0.7\text{TW}$ for the second one (harmonic orders $>26$).