Collisionless energy absorption in the short-pulse intense laser-cluster interaction

In a previous paper [Phys. Rev. Lett. 96, 123401 (2006)] we have shown by means of three-dimensional particle-in-cell simulations and a simple rigid-sphere model that nonlinear resonance absorption is the dominant collisionless absorption mechanism in the intense, short-pulse laser cluster interaction. In this paper we present a more detailed account of the matter. In particular we show that the absorption efficiency is almost independent of the laser polarization. In the rigid-sphere model, the absorbed energy increases by many orders of magnitude at a certain threshold laser intensity. The particle-in-cell results display maximum fractional absorption around the same intensity. We calculate the threshold intensity and show that it is underestimated by the common overbarrier ionization estimate.

Figure 1

Laser energy absorption vs driver amplitude in the rigid sphere-model for ${({\omega }_{\text{Mie}}∕{\omega }_{\mathrm{l}})}^2=10∕3$, an $n=8$ cycle laser field $E_{\mathrm{l}}\left(\tau \right)=E_0{\sin }^2(\tau ∕2n)\cos \left(\tau \right)$, and cluster radius $R=3.2\mathrm{nm}$. Within a narrow field strength interval (here $\simeq 0.5-1$) the absorbed energy per particle (solid line) increases by many orders of magnitude. The dashed line represents the absorbed energy (7) by a purely harmonic oscillator driven by the same laser field.

Figure 2

(Color online) Typical behavior of ${({\omega }_{\mathrm{eff}}\left(\tau \right)∕{\omega }_{\mathrm{l}})}^2$ (red, upper solid line) vs laser cycles above the threshold driver strength for ${({\omega }_{\text{Mie}}∕{\omega }_{\mathrm{l}})}^2=40∕3$. Here we take $E_0∕R{\omega }_{\mathrm{l}}^2\simeq 7.48$ corresponding to a laser intensity $\simeq 2.5\times {10}^{16}\mathrm{W}∕{\mathrm{cm}}^2$, an $n=8\text{cycle}{\sin }^2$ pulse $E_{\mathrm{l}}\left(\tau \right)=E_0{\sin }^2(\tau ∕2n)\cos \left(\tau \right)$ of wavelength $\lambda =1056\mathrm{nm}$, and a cluster radius $R=3.2\mathrm{nm}$. Excursion $x∕R$ (green, middle solid line) and energy of the electron sphere $E_{\mathrm{tot}}∕R^2{\omega }_{\mathrm{l}}^2$ (blue, lower dashed line) are included in the plot. Outer ionization (i.e., $E_{\mathrm{tot}}∕R^2{\omega }_{\mathrm{l}}^2⩾0$) and occurrence of NLR ${[{\omega }_{\mathrm{eff}}\left(\tau \right)∕{\omega }_{\mathrm{l}}]}^2=1$ always coincide (dashed vertical line).

Figure 3

(Color online) Effective frequency ${\omega }_{\mathrm{eff}}∕{\omega }_{\mathrm{l}}$ vs the excursion amplitude $\widehat{r}$ of the electronic sphere for different cluster charge densities $\rho ∕{\rho }_{\mathrm{c}}=10–40$. The solid lines are computed from the numerical solution of Eq. (3). The dashed lines are the corresponding analytical approximations from Eq. (11) for $k$ up to 6, and the dash-dotted lines are from (12). For the charge density $\rho ∕{\rho }_{\mathrm{c}}=40$ one expects NLR (horizontal dashed line) to occur at the excursion $\widehat{r}\simeq 2.88$.

Figure 4

(Color online) Typical behavior of ${\mathbf{(}{\omega }_{\mathrm{eff}}\left(\tau \right)∕{\omega }_{\mathrm{l}}\mathbf{)}}^2$ (green, top left solid line) vs laser cycles above the threshold intensity $\simeq 3\times {10}^{16}\mathrm{W}∕{\mathrm{cm}}^2$ for circular polarization, ${({\omega }_{\text{Mie}}∕{\omega }_{\mathrm{l}})}^2=40∕3$, and cluster radius $R=3.2\mathrm{nm}$. Here, $E_0∕R{\omega }_{\mathrm{l}}^2\simeq 8.2$, corresponding to a laser intensity $\simeq 5\times {10}^{16}\mathrm{W}∕{\mathrm{cm}}^2$, $n=8\text{cycle}{\sin }^2$ pulse with components $E_{\mathrm{l}}^x\left(\tau \right)=E_0{\sin }^2(\tau ∕2n)\cos \left(\tau \right)∕\sqrt{2}$, $E_{\mathrm{l}}^y\left(\tau \right)=E_0{\sin }^2(\tau ∕2n)\sin \left(\tau \right)∕\sqrt{2}$, and wavelength $\lambda =1056\mathrm{nm}$. Excursion $r$ (blue, middle left solid line) and energy $E_{\mathrm{tot}}∕R^2{\omega }_{\mathrm{l}}^2$ (red, dashed line) of the electron sphere are also plotted. The energy $E_{\mathrm{tot}}∕R^2{\omega }_{\mathrm{l}}^2$ is scaled down by a factor of 10 to display within the excursion and the frequency range. Outer ionization (i.e., $E_{\mathrm{tot}}∕R^2{\omega }_{\mathrm{l}}^2⩾0$) and the occurrence of NLR ${[{\omega }_{\mathrm{eff}}\left(\tau \right)∕{\omega }_{\mathrm{l}}]}^2=1$ always coincide (dashed vertical).

Figure 5

(Color online) NLR threshold intensities in the RSM vs the cluster charge density $\rho ∕{\rho }_{\mathrm{c}}$ for linearly (LP) and circularly (CP) polarized laser fields. Results from the full numerical solution of Eq. (3) with LP (gray, thick solid), Eq. (13) CP (thin solid), the vanishing barrier approximation (VBA-LP, line with bullets), Eq. (18), and the VBA corrected for CP (upper thick dashed) are shown (see text for a discussion). The over-the-barrier approximation (OBA) (19) (lower thin dashed) underestimates the exact threshold intensities.

Figure 6

(Color online) Kinetic energy (KE) and total energy (TE) absorbed per electron in units of $U_{\mathrm{p}}$ vs laser intensity for charge density 40 times the critical density and a ${\mathrm{Xe}}_{1611}$ cluster ($N=5498$ electrons, mean ionic charge state $\simeq 3.4$, and a cluster radius $R=3.2\mathrm{cm}$). Solid lines are PIC results; dashed lines are the RSM results for a $n=8\text{cycle}$ pulse $E_{\mathrm{l}}\left(t\right)=E_0{\sin }^2({\omega }_{\mathrm{l}}t∕2n)\cos \left({\omega }_{\mathrm{l}}t\right)$ of wavelength $\lambda =1056\mathrm{nm}$.

Figure 7

(Color online) Energy and self-consistent potential $\Phi$ from the PIC simulation vs the excursion $x∕R$ in the laser polarization direction. Yellow (solid gray) lines represent cuts of the potential $\Phi$ at time ${\omega }_{\mathrm{l}}t∕2\pi =4.81$ for different $y$ and $z=0$. The circles (red) represent the total energy $E_{\mathrm{tot},i}$ of individual PIC electrons located within the simulation box at that time. The total energy (TE) of a PIC electron (black, bold) that is outer ionized when the excursion $x∕R\simeq 2.88$ meets the condition ${\omega }_{\mathrm{eff}}={\omega }_{\mathrm{l}}$ at the same time (see Fig. 3 for the charge density $\rho ∕{\rho }_{\mathrm{c}}=40$) is shown, arrows indicating the time evolution (starting from $t=0$). The RSM potential is included (dashed line). The peak laser intensity is $I_0=2.5\times {10}^{16}\mathrm{W}∕{\mathrm{cm}}^2$. Other parameters as in Fig. 6.

Figure 8

(Color online) Effective frequency squared ${({\omega }_{\mathrm{eff},i}∕{\omega }_{\mathrm{l}})}^2$, excursion $r_i∕R$, and the total energy $E_{\mathrm{tot},i}={\stackrel{\mathrm{̇}}{\mathit{r}}}_i^2\left(t\right)∕2-\Phi ({\mathit{r}}_i,t)$ for a PIC electron vs time in laser cycles. The total energy becomes positive only when the NLR is crossed (indicated by the vertical dashed line). This result resembles the RSM result in Fig. 2 when NLR is met. The charge density is $\rho ∕{\rho }_{\mathrm{c}}=40$, the peak laser intensity is $I_0=2.5\times {10}^{16}\mathrm{W}∕{\mathrm{cm}}^2$. Other parameters as in Fig. 6.

Figure 9

Snapshots of PIC electrons in the frequency vs energy plane at times (a) $t=1.5$, (b) $t=2.0$, (c) $t=2.5$, (d) $t=3.0$, (e) $t=3.5$, (f) $t=4.0$, (g) $t=4.5$, and (h) $t=5.0$ laser cycles for LP, laser intensity $2.5\times {10}^{16}\mathrm{W}{\mathrm{cm}}^{-2}$, and ${({\omega }_{\text{Mie}}∕{\omega }_{\mathrm{l}})}^2=40∕3$. Other parameters as in Fig. 8. Gray scaling of the symbols is used to indicate the radial positions (in units of $R$). Electrons become free upon crossing the NLR; i.e., $({\omega }_{\mathrm{eff}}^2∕{\omega }_{\mathrm{l}}^2,E_{\mathrm{tot}}∕U_{\mathrm{p}})=(1,0)$.

Figure 10

(Color online) Effective frequency squared (${\omega }_{\mathrm{eff},i}∕{\omega }_{\mathrm{l}}{)}^2$, excursion $r_i∕R$, and the total energy $E_{\mathrm{tot},i}={\stackrel{\mathrm{̇}}{\mathit{r}}}_i^2\left(t\right)∕2-\Phi ({\mathit{r}}_i,t)$ for a PIC electron in a CP field vs time in laser cycles. The total energy becomes positive only when the NLR is crossed (indicated by vertical, dashed line). This result resembles the RSM result in Fig. 4. The charge density is $\rho ∕{\rho }_{\mathrm{c}}=40$; the peak laser intensity is $I_0=2.5\times {10}^{16}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 11

Snapshots of PIC electrons in the frequency vs energy plane for CP at times (a) $t=1.5$, (b) $t=2.0$, (c) $t=2.5$, (d) $t=3.0$, (e) $t=3.5$, (f) $t=4.0$, (g) $t=4.5$, and (h) $t=5.0$ laser cycles for a laser intensity $2.5\times {10}^{16}\mathrm{W}{\mathrm{cm}}^{-2}$ and ${({\omega }_{\text{Mie}}∕{\omega }_{\mathrm{l}})}^2=40∕3$. Other parameters as in Fig. 6. Gray scaling of the symbols is used to indicate the radial positions (in units of $R$). Electrons become free upon crossing the NLR—i.e., $({\omega }_{\mathrm{eff}}^2∕{\omega }_{\mathrm{l}}^2,E_{\mathrm{tot}}∕U_{\mathrm{p}})=(1,0)$.

Figure 12

(Color online) Total absorbed energy per electron in units of $R^2{\omega }_{\mathrm{l}}^2$ vs laser intensity for charge densities (a) $\rho ∕{\rho }_{\mathrm{c}}=3$ (linear resonance), (b) $\rho ∕{\rho }_{\mathrm{c}}=5$, (c) $\rho ∕{\rho }_{\mathrm{c}}=20$, and (d) $\rho ∕{\rho }_{\mathrm{c}}=40$. The absorbed energies for LP (red, lighter gray) and CP (blue, darker gray) using PIC (dashed) and the RSM (solid) are shown. Plots (e)–(f) show the same results scaled by the ponderomotive energy.

Figure 13

PIC results for the outer ionization degree $n_{\mathrm{out}}∕N$ (where $n_{\mathrm{out}}$ is the number of electrons with $r>2R$) after the laser pulse for linear (solid grey) and circular polarization (dashed black) for different charge densities $\rho ∕{\rho }_{\mathrm{c}}=3$ (a), 5 (b), 20 (c), and 40 (d).