## 3 Particle in Cell Simulations of High Intensity Laser Pulses with Snow Whiskers
To explore the influence of the plasma density, density gradients and surface curvature we simulated the laser-whisker interaction for different values of whisker aspect ratios and density gradients. For comparison, we considered planar shape and constant solid density whiskers as well. To obtain the scaling law of the protons energy with the laser intensity, our multi-parameters simulations considered different laser intensities in the range 2.5 \(\times\) 10\({}^{17}\) W/cm\({}^{2}\) to 2.5 \(\times\) 10\({}^{21}\) W/cm\({}^{2}\), corresponding to normalized vector potential \(a_{0}\) = \(\frac{e\cdot E}{m\cdot c\cdot\omega}\) in the range 0.34 to 34. A laser pulse of 88 fs total duration with linear rise time and fall time of 32 fs each, wavelength \(\lambda\) of 0.8 \(\upmu\)m and spot size of 4 \(\upmu\)m was used in the simulations. An example of the particles distribution at different times after the laser-whisker interaction is shown in Figure 4. The laser pulse at intensity 2.5 \(\times\) 10\({}^{19}\) W/cm\({}^{2}\) (a\({}_{0}\) = 3.4) propagates perpendicular to the major axis of an ellipsoid whisker of the type shown in Figure 5a with density gradient described before. The beam has Gaussian spatial distribution and its center is few microns above the middle of the major axis of the whisker (the center of the simulation box). The simulation region is 100 \(\upmu\)m \(\times\) 60 \(\upmu\)m. Spatial density distributions of the electrons (left), protons (middle) and oxygen ions (right) are shown in Figure 5 at three different times, 212 fs, 424 fs and 850 fs. The laser pulse enters the simulation box from the left and hits the whisker at time \(t\) = 141 fs. As seen in the upper row of Figure 5, the electrons  
Figure 3: Spatial distribution of the electron plasma of the whisker before the start of the interaction with the main pulse. The upper half of the whisker is displayed. The density of the plasma is logarithmic scaled by the critical density. One can see a central dense core, surrounded by gradually lower density region. The laser electric field is schematically shown and not to scale (Adapted with permission from [16], published by American Physical Society, 2013). are accelerated by the laser's pondermotive potential. Part of the electrons escape out of the initial plasma cloud, generating a spherical low density cloud with accumulation near the tip of the whisker, which induces charge separation and generates an electrostatic field. We notice that the density distribution is not fully symmetrical relatively to the minor axis of the whisker, as the laser hits the whisker above its center of symmetry and is reflected from curved surfaces. It is seen from the electrons density distribution that there is not clear distinction between the back and the front surfaces as there are mixed over the whisker's tip, unlike in the TNSA mechanism. The middle row of Figure 5 displays the particles density distribution at time \(t=424\) fs. It is seen that a significant part of the electrons have left the initial whisker cloud. The charge separation is higher and the protons are accelerated by the electrostatic field. The asymmetry relatively to the whisker's minor axis is observed in the protons density distribution as well. The charge separation between the hot electron cloud and the positively charged whisker generates also an accelerating field in the lateral direction, and the total amplitude of the field is enhanced. At this time the heavier oxygen ions are starting to react to the electrostatic field and their density distribution is disturbed. The bottom row of Figure 5 displays the density distribution at time \(t=850\) fs. The electrons and protons distributions have highly expanded. The density distribution of the electrons near the two tips looks similar, their accumulation is reduced and decreases gradually over all directions. A sharp front is observed in the density distribution of the protons and the oxygen ions. Their upper side density distribution is wider.