Relativistic attosecond electron bunch emission from few-cycle laser irradiated nanoscale droplets

Attosecond electron bunches produced at the surface of nanometer-scale droplets illuminated by a two-cycle laser pulse are investigated for the purpose of determining their optimal emission characteristics. Significant departures from Mie theory are found for electron bunch emission from droplets whose radii satisfy the condition ${\delta }_r<R<10{\delta }_r$, where ${\delta }_r={\gamma }^{1/2}c/{\omega }_p$ is the plasma relativistic skin depth; an effect which can be accounted for by induced transparency. Scattering from such droplets is subject to a transitional regime which is neither accounted for by optical Mie theory valid for $R\gg \delta$, where $\delta$ is the usual plasma skin depth, nor with the Rayleigh regime ($R<\delta \ll \lambda$). Instead the angular emission of the bunches is to a good approximation described by the nonlinear ponderomotive scattering model. Subsequently, the bunches are subject to further deflection by the ponderomotive pressure of the copropagating laser field in vacuum, depending on the initial droplet parameters. Final emission angles are estimated, together with the energy spectrum of the bunches.

Figure 1

Laser electric field (bottom, in $\mathrm{V}/\mathrm{m}$), and electron number density (top, in ${\mathrm{m}}^{-3}$) plot for a 100 nm radius droplet hit by a laser pulse of intensity $I=10^{20}\mathrm{W}/{\mathrm{cm}}^2$ at times $\approx 3\mathrm{fs}$ (a), 8 fs (b), 16 fs (c). The cluster is centered at position (0,0). Angles are measured relative to the center of the most dense region of the bunch. The logarithmic density scale is normalized to the critical density $n_c=1.8\times 10^{21}{\mathrm{cm}}^{-3}$, so that the maximum value corresponds to $100n_c$. The maximum absolute value of the electric field is $3.5\times 10^{13}\mathrm{V}/\mathrm{m}$.

Figure 2

Logarithmic plot of the electron number density for a 100 times overdense droplet with $R=100\mathrm{nm}$ as it evolves in the field of a linearly polarized Gaussian laser pulse with $I=10^{20}\mathrm{W}/{\mathrm{cm}}^2$. The contoured regions represent the dense regions which can be identified as the proper nanobunches, at, respectively, $t=10\mathrm{fs}$ (left) and 15 fs (right) after the start of the laser-droplet interaction. Two main perspectives are represented: xy-plane view (up) and xz-plane (down).

Figure 3

Electron emission angles as a function of the laser intensities, for different droplets sizes, compared with available models. Color corresponds to different cluster radii, namely: $\text{red}=100\mathrm{nm}$, $\text{orange}=200\mathrm{nm}$, $\text{cyan}=300\mathrm{nm}$, $\text{green}=500\mathrm{nm}$, $\text{violet}=1\mathrm{micron}$. Solid lines represent simulation points, dashed horizontal lines Mie theory prediction. Black and blue line represent nonlinear ponderomotive scattering prediction and deviation formula of Eq. (3), respectively. The observed divergence with respect to central direction of the bunches is about 5 degrees.

Figure 4

Absolute value of the radial electric field for a 100 times overdense droplet with $R=100\mathrm{nm}$ (a), 200 nm (b), 500 nm (c), and 1 micron (d) in an incident plane wave of wavelength $\lambda =800\mathrm{nm}$ as a function of time and angle on the surface. The color scale indicates the enhancement factor of the incident electric field $E_0$, for a constant plane wave amplitude.

Figure 5

Number of particles $f(r)$ as a function of the radial distance $r$ from the center of the droplet, integrated over all angles for a droplet with radius $R=100\mathrm{nm}$ illuminated by a laser at intensity $I={10}^{20}\mathrm{W}/{\mathrm{cm}}^2$. The particles are probed at a time corresponding to Fig. 1.

Figure 6

Geometry of bunch deflection due to space charge.

Figure 7

Electric field energy density (in legend), superposed with the logarithmic plot of number electron density (in units of critical density $1.8\times 10^{27}{\mathrm{m}}^{-3}$) for a droplet with radius 100 nm (above) and 1 micron (below) respectively hit by a laser pulse of intensity $I=10^{20}\mathrm{W}/{\mathrm{cm}}^2$ at times $\approx 3\mathrm{fs}$ (a), 8 fs (b), 16 fs (c). Angles are measured considering the center of the most dense region of the bunch. In the logarithmic density scale, the maximum value corresponds to the critical density (see Fig. 1).

Figure 8

Angular energy distributions (above) and single bunch energy spectra (below) at $t\approx 20\mathrm{fs}$ after the start of the interaction, for a 100 nm radius droplet hit by a laser pulse of intensities $I=10^{19}\mathrm{W}/{\mathrm{cm}}^2$, $I=10^{20}\mathrm{W}/{\mathrm{cm}}^2$, and $I=10^{21}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 9

Electric transverse field (in $\mathrm{V}/\mathrm{m}$), and electron density (in ${\mathrm{m}}^{-3}$) plot for a 500 nm radius droplet hit by a laser pulse of intensity $I=10^{20}\mathrm{W}/{\mathrm{cm}}^2$ at times (a) $3\mathrm{fs}$, (b) 8 fs, (c) 16 fs, respectively. Angles are measured considering the center of the most dense region of the bunch. In the density scale, the maximum value corresponds to the critical density $1.8\times 10^{27}{\mathrm{m}}^{-3}$. The maximum absolute value of the electric field is $2.5\times 10^{13}\mathrm{V}/\mathrm{m}$.