Spatial Properties of High-Order Harmonic Beams from Plasma Mirrors: A Ptychographic Study

Spatial properties of high-order harmonic beams produced by high-intensity laser-matter interactions carry rich information on the physics of the generation process, and their detailed understanding is essential for applications of these light beams. We present a thorough study of these properties in the case of harmonic generation from plasma mirrors, up to the relativistic interaction regime. In situ ptychographic measurements of the amplitude and phase spatial profiles of the different harmonic orders in the target plane are presented, as a function of the key interaction parameters. These measurements are used to validate analytical models of the harmonic spatial phase in different generation regimes, and to benchmark ultrahigh-order Maxwell solvers of particle-in-cell simulation codes.

Figure 1

Spatial properties of high-order harmonics and attosecond pulses. Panels (a) and (c) show results of 2D particle-in-cell simulations, respectively, in the CWE ($a_0=0.4$, $L={\lambda }_L/30$) and ROM ($a_0=3$, $L={\lambda }_L/10$) generation regimes, plotted here in a Lorentz frame [39] where the laser field is normally incident on the plasma ($\theta =45°$ in the laboratory frame). The gray scale map shows the plasma electron density at a given time in a laser optical cycle. The color map shows the wave fronts of a single attosecond pulse (harmonics 4 to 12), emitted during the same cycle but plotted at a slightly later time, after it has propagated in vacuum. Panels (b) and (d) show the absolute value of the wave front curvature $1/R$ (in units of ${\lambda }_L/w_L^2$) of the harmonic beam for these two generation mechanisms, predicted by analytical models [Eqs. (1) and (2)], as a function of $a_0$ and $L$. The black crosses indicate the physical conditions used in panels (a) and (c).

Figure 2

Typical ptychographic measurements of harmonic beams produced by plasma mirrors. Panels (a) and (b) show ptychographic data sets measured for the 12th harmonic, in the CWE ($a_0=0.2$, $L={\lambda }_L/35$) and ROM ($a_0=1.65$, $L=\lambda /8$) generation regimes. $d$ is the spatial period of the transient plasma grating used for the measurement. The amplitude (full lines) and phase (dashed lines) spatial profiles of the harmonic source retrieved from such measurements are shown in panels (c) and (d), for four harmonic orders in each case. The harmonic source size $w_n$ (red plot, in units of laser focal spot size $w_0$) and the wave front radius of curvature $|1/R|$ (orange plot, in units of ${\lambda }_L/w_0^2$) are, respectively, displayed in panels (e),(f) and (g),(h), for all harmonic orders observable in these scans. The source sizes obtained from 2D PIC simulations in the same interaction conditions as these experiments are shown in panels (e) and (f), in the cases of a second-order Yee solver (grey plot) and an ultrahigh-order ($\text{order}=128$ was used here) spectral solver (black plot) for Maxwell’s equations. The dashed lines in (g) and (h) correspond to the predictions of Eqs. (1) and (2).

Figure 3

Measured curvatures $|1/R|$ of the harmonic wave fronts in the CWE and ROM regimes, as a function of the laser amplitude $a_0$ and the density gradient scale length $L$. These points are the results of $\approx 11\times 100=1100$ laser shots, which provide $\approx 13000$ independent harmonic angular profiles (one for each harmonic order in each laser shot) used in the ptychographic analysis. The full lines show the predictions of the analytical models [Eq. (1) and Eq. (2)].