Self-consistent theory of high-order harmonic generation by relativistic plasma mirror

We present a self-consistent semianalytical model of the relativistic plasma mirror, based on the exact computation of the laser-driven electron surface oscillations within the cold-fluid approximation. Valid for arbitrary solid densities, laser incidence angle, and a large set of laser intensities (${10}^{18}–{10}^{21}\text{W}/{\text{cm}}^2$), the model unravels different regimes of harmonic generation. In particular, it is found that efficient conversion of $p$-polarized laser pulses into high-order harmonics well above the plasma frequency requires either high laser intensities, low plasma densities, or incidence angles larger than a threshold value. This critical angle corresponds to a transition between a regime where the electron surface dynamics is mostly governed by the laser $J\times B$ force and a “cyclotron Brunel” regime, where electrons perform many cyclotron gyrations when moving into the vacuum. Under conditions relevant to current laser experiments, the latter regime gives rise to nonmonotonic variations of the harmonic yield with the laser field. Our predictions are supported by an extensive parametric study performed with highly resolved one-dimensional particle-in-cell simulations.

Figure 1

(a) and (b) EPB position, ${\lambda }_s\left(t\right)$, according to the theoretical model (solid lines) and the PIC simulations (dashed lines). (c) Reflected magnetic field in the vacuum, $b_{zr}(x+t)$ for the same parameters as in panel (a). (d) Filtered magnetic field map for $\nu >20$ (blue), electron density map (red), and EPB (green line) from PIC simulation ($a_0/\sqrt{n_0}=1, n_0=300, \theta ={65}^{\circ }$). The parameters are indicated.

Figure 2

(a) and (b) Odd harmonics spectra normalized to the reflected wave energy, as predicted by the model (solid line) and the PIC simulations (dashed line).

Figure 3

(a) Trajectory ${\lambda }_s$ and (b) reflected magnetic field $b_{zr}$ obtained from the fluid model. (c) Reflected light energy contained between $2\le \nu \le 100$ normalized to the full reflected light, against $a_0/\sqrt{n_0}$ and $\theta$, according to the model (c) and the simulation (d). Critical angle against $a_0/\sqrt{n_0}$ according to formula (5) (black solid line). The inset in panel (d) presents a more finely resolved grid between $0.1\le a_0/\sqrt{n_0}<1$ and ${30}^{\circ }\le \theta \le {75}^{\circ }$.

Figure 4

(a) and (b) Odd harmonics spectra normalized to the reflected wave energy, as predicted by the model (solid line) and the PIC simulations with $T_0=511\text{eV}$ (dashed line) and $T_0=5\text{eV}$ [dashed-dotted line, in panel (a) for $\theta =0^0$ and ${15}^{\circ }]$.