Harmonic emission from cluster nanoplasmas subject to intense short laser pulses

Harmonic emission from cluster nanoplasmas subject to short intense infrared laser pulses is studied. In a previous publication [M. Kundu et al., Phys. Rev. A 76, 033201 (2007)] we reported particle-in-cell simulation results showing resonant enhancements of low-order harmonics when the Mie plasma frequency of the ionizing and expanding cluster resonates with the respective harmonic frequency. Simultaneously we found that high-order harmonics were barely present in the spectrum, even at high intensities. The current paper is focused on the analytical modeling of the process. We show that dynamical stochasticity owing to nonlinear resonance inhibits the emission of high-order harmonics.

Figure 1

(Color online) Time-frequency diagrams for an ${\text{Ar}}_{17256}$ cluster at laser intensities $2.5\times {10}^{15}\text{W}/{\text{cm}}^2$ (a), $7.5\times {10}^{17}\text{W}/{\text{cm}}^2$ (b). Panel (c) shows the TF diagram for an ${\text{Ar}}_{92096}$ cluster at the intensity $7.5\times {10}^{17}\text{W}/{\text{cm}}^2$. The scaled time-dependent Mie frequency ${\omega }_{\text{Mie}}\left(t\right)/{\omega }_l$ is included in all plots (solid lines).

Figure 2

Harmonic spectra (normalized to the signal at the fundamental frequency) calculated within the RSM for a cluster of radius $R_0=6.2\text{nm}$, $a=R_0/R_e=2$, and $\text{max}\left[{\omega }_{\text{eff}}\right]\approx 3.2{\omega }_l$ (a),(d), $\text{max}\left[{\omega }_{\text{eff}}\right]\approx 4.4{\omega }_l$ (b),(e), and $\text{max}\left[{\omega }_{\text{eff}}\right]\approx 5.4{\omega }_l$ (c),(f). The laser intensities are given in the panels in units of ${10}^{16}\text{W}/{\text{cm}}^2$.

Figure 3

Dynamical characteristics of the potential (11): the scaled eigenfrequency $\Omega \left(ϵ\right)/\Omega \left(0\right)$ (solid line) and the nonlinearity parameter (14) (dashed line) vs the scaled energy $ϵ/U_0$. Positions of the first- and third-order resonances are shown by dotted lines. The value of the parameter $\beta$ at the resonant energies is $\beta \left({ϵ}_1\right)\approx 1.2$ and $\beta \left({ϵ}_3\right)\approx 4.5$, respectively.

Figure 4

The phase space of the Hamiltonian (19). A separatrix (thick solid lines) subdivides the phase space into domains of finite (2) and infinite (1, 3) motion. Two typical trajectories are shown by dashed lines. Dotted lines indicate the positions of the thresholds $\pm P_{\text{max}}$ determined in Eq. (20).

Figure 5

Upper panel: electron coordinate vs dimensionless time, $x\left(\phi ={\omega }_{l}t\right)$, calculated from the numerical solution of Eq. (10) for $U_0=5.0$, ${\omega }_l=1.0$, and $R_0=1.0$, assuming the initial conditions $x\left(-40\right)=0.0$, $0\le \alpha \le \pi /2$ (black solid and dashed curves), and $\alpha =\pi$ (gray curves). Other parameters are ${ϵ}_0=0.2U_0$, $\alpha =0.0$, ${\mathcal{E}}_0=0.5$ (a), ${ϵ}_0=0.3U_0$, $\alpha =1.0$, ${\mathcal{E}}_0=0.5$ (b), and ${ϵ}_0=0.2U_0$, ${\mathcal{E}}_0=0.8$ (c) with $\alpha =0.0$ for a solid curve and $\alpha =0.05$ for a dashed curve. The electric field $\mathcal{E}\left({\omega }_{l}t\right)$ is shown in the inset. Lower panel: scaled electron energy vs dimensionless time for the parameters of the upper panel, (a)–(c). Thin horizontal lines show the position of the first-order $\left({ϵ}_1\approx 0.48U_0\right)$ and the third-order $\left({ϵ}_3\approx 0.77U_0\right)$ resonances. Dotted lines show positions of the time-dependent separatrices (21) for the first resonance. A thick horizontal line shows the continuum threshold $ϵ=U_0$.

Figure 6

Same as the lower panel of Fig. 5b but for half the frequency, ${\omega }_l^{\prime }=0.5$, and the duration two times longer, so that the dimensionless time $\phi ={\omega }_l^{\prime }t$ varies within the same limits.

Figure 7

TF diagrams corresponding to three [the black curve of Fig. 5a and both curves of Fig. 5b] electron trajectories.