Figure 1

A comparison of the real and imaginary polarizability for our model (a),(b) and the hydrocode of Milchberg (c),(d). In each case, a cluster of initial radius $30\mathrm{nm}$ is irradiated with an $800\mathrm{nm}$, $100\mathrm{fs}$ FWHM laser pulse for three different peak intensities $5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ (dashed), $8\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ (dotted), and $1\times {10}^{15}\mathrm{W}∕{\mathrm{cm}}^2$ (solid, dark). The pulse profile is shown in (a) as a thin solid line.Reuse & Permissions

Figure 2

Contour plot of (a) real and (b) imaginary part of polarizability. The ellipse in the center of each plot marks FWHM points.Reuse & Permissions

Figure 3

The quantities $H_r\left(\xi =0\mathrm{fs}\right)$ (solid) and $H_i\left(\xi =0\mathrm{fs}\right)$ (dashed with cross markers) are plotted at $z=0.006\mathrm{cm}$. Here the cluster density set to $n_c=3\times {10}^{11}{\mathrm{cm}}^{-3}$ and the spot size of the pulse was varied from $10\mu \mathrm{m}$ to $200\mu \mathrm{m}$. All other parameters were initialized to the conditions in Table . The intersection of $H_r$ with the $1∕k_0{R}_0^2$ curve (dashed) gives the equilibrium values of $R$ for the chosen cluster density.Reuse & Permissions

Figure 4

Cluster density required for equilibrium of a pulse as a function of the pulse spot size for three different initial energies $1.93\mathrm{mJ}$ (dash-dot), $9.65\mathrm{mJ}$ (dashed), and $19.3\mathrm{mJ}$ (solid). All other parameters were initialized to the conditions in Table . Note that for a pulse with given initial energy the same cluster density can occur for two values of $R$.Reuse & Permissions

Figure 5

Stability expression ${\left(2H_r+R\partial H_{r}l\partial R\right)}_{R=R0}$ vs spot size $R$ for a laser beam for three different initial pulse energies $1.93\mathrm{mJ}$ (dash-dot), $9.65\mathrm{mJ}$ (dashed), and $1.93\mathrm{mJ}$ (solid) All other parameters initialized to the conditions in Table . The range of $R$ over which the plotted quantity is positive is stable. For example, in the $1.93\mathrm{mJ}$ case, the region of stability corresponds to $14\mu \mathrm{m}<R<82\mu \mathrm{m}$ and $R<8\mu \mathrm{m}$.Reuse & Permissions

Figure 6

Power within the pulse (initial parameters of Table ) at $z=0\mathrm{cm}$, $0.3\mathrm{cm}$, $0.6\mathrm{cm}$, $0.9\mathrm{cm}$, and $1.2\mathrm{cm}$. The front of the pulse, traveling through unionized clusters, does not get absorbed and hence propagates unattenuated while the trail end (where the imaginary part of polarizability is high) is strongly attenuated.Reuse & Permissions

Figure 7

Self-guiding of the laser pulse in cluster medium. The figure shows the spot size at the center of the pulse $\left(\xi =0\mathrm{fs}\right)$ (dashed with square marker) for propagation through $2.0\mathrm{cm}$ of clustered gas for the initial conditions of Table . The result for propagation through vacuum (solid with circle markers) is plotted for comparison. The rms (root mean squared) spot size is also shown (dashed with diamond markers). The center of the pulse remains focused for roughly $1.5\mathrm{cm}$. The energy within the pulse (solid with cross markers) is plotted in $\mathrm{mJ}$ (right axis). We note that about 83% of the pulse energy is absorbed by the clusters in $1.5\mathrm{cm}$ of propagation, after which the rate of absorption levels off.Reuse & Permissions

Figure 8

rms spot sizes as a function of propagation distance for pulses with six different initial peak intensities: $5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, $8\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, $1\times {10}^{15}\mathrm{W}∕{\mathrm{cm}}^2$, $2\times {10}^{15}\mathrm{W}∕{\mathrm{cm}}^2$, $5\times {10}^{15}\mathrm{W}∕{\mathrm{cm}}^2$, and $1\times {10}^{16}\mathrm{W}∕{\mathrm{cm}}^2$. Here cluster density was set to $n_c=3\times {10}^{11}{\mathrm{cm}}^{-3}$, other initial conditions being those in Table . We note that the guiding effect is strongest around peak intensity of $2\times {10}^{15}\mathrm{W}∕{\mathrm{cm}}^2$. The evolution of spot size for a pulse propagating in vacuum is plotted for comparison.Reuse & Permissions

Figure 9

Variation of energy within the pulse with propagation for pulses of different initial peak intensities: $5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, $8\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, $1\times {10}^{15}\mathrm{W}∕{\mathrm{cm}}^2$, $2\times {10}^{15}\mathrm{W}∕{\mathrm{cm}}^2$, $5\times {10}^{15}\mathrm{W}∕{\mathrm{cm}}^2$, and $1\times {10}^{16}\mathrm{W}∕{\mathrm{cm}}^2$. Here cluster density was set to $n_c=3\times {10}^{11}{\mathrm{cm}}^{-3}$, other initial conditions being those in Tab1e . Pulses with higher initial energy have a higher rate of energy absorption initially. As the pulse energy gets depleted the rate of absorption falls.Reuse & Permissions

Figure 10

(a) Phase, (b) chirp developed, (c) spot size of the pulse, (d) radially averaged frequency shift, and (e) power weighted radial averaged frequency shift, at $z=0.3\mathrm{cm}$, $z=0.72\mathrm{cm}$, and $0.9\mathrm{cm}$ (for initial conditions of Table ). The rise in phase at $\xi =-80\mathrm{fs}$ is due to ionization and appears in (b) and (d) as a redshift. The $z=0.72\mathrm{cm}$ curve shows a kink in phase at around $\xi =86\mathrm{fs}$. This is due to the sharp focusing of the pulse as seen in (c) and provides further redshift as seen in (b) and (d). The frequency shift weighted with power gives the effect of the propagation on the pulse spectrum.Reuse & Permissions

Figure 11

Variation with propagation distance of spot size (a), curvature (b), and phase (c) of the pulse at three different locations within the pulse ( $\xi =0\mathrm{fs}$, $86\mathrm{fs}$, $101\mathrm{fs}$). The moving focus seen in (a) causes a sharp fall in phase (c).Reuse & Permissions

Figure 12

Power spectrum of the pulse (in logarithmic scale) for $z=0\mathrm{cm}$, $0.3\mathrm{cm}$, and $0.9\mathrm{cm}$. We see the spectrum broadening with propagation and significant redshifting of the initial spectrum.Reuse & Permissions