## 3 Particle in Cell Simulations of High Intensity Laser Pulses with Snow Whiskers
Few \"hole\"-like spots are seen in the protons density distribution. Charge separation between electrons and ions is maintained, continuing the protons acceleration, while the un-neutralized heavier oxygen ions remain behind and add a pushing field. At further longer times, the charge separation diminishes, the acceleration stops and the protons move at constant velocity. Figure 4 displays the structure of the electric field at early time, 235 fs after the laser hit the whisker. At the bottom left corner the bright yellow region represents the electric field of the laser surrounded by a cloud of electrons. The field associated with the electrons is seen as an external weak region, due to the first part of electrons expelled out of the whisker and an inner spherical shell. Inside this shell, there is a double ellipsoidal shell associated with the protons front, outer shell and, inner shell with size similar to the whisker before the interaction, associated with the oxygen ions, that almost did not move by that time. To emphasize the role of the special geometry and density gradient for the enhancement of laser-snow whisker interaction, Figure 6 shows four whisker shapes: Figure 6A is the whisker considered in the figures above; Figure 6B has ellipsoidal shape, to capture the effect of mass limited curved target, and constant solid density; Figure 6C is similar to A, but longer, to mimic interaction of the laser with plasma gradient, but planar surface; and Figure 6D has solid density and planar shape, similar to targets where the protons acceleration is driven by the TNSA mechanism. Figure 4: Spatial dependence of the electric field in units of a\({}_{0}\) = 3.4, at 235 fs after the start of interaction of a laser pulse at intensity \(2.5\times 10^{19}\,\mathrm{W/cm^{2}}\) at 45\({}^{\circ}\) irradiation angle. Figure 5: Spatial density distribution of electrons (**left**); protons (**middle**); oxygen ions (**right**) at times a-212 fs, b-424 fs and c-850 fs. The laser hit the whisker at 141 fs. Color map scale for density is logarithmic normalized to critical density. Vertical (x) and horizontal (z) coordinates are given in microns. The laser-snow whiskker enhancement of the interaction, resulting from improved laser absorption, field enhancement near the tip and significantly larger charge separation for the case A, induces proton  
Figure 6: Various whisker models considered in the simulations, with the following properties: (**A**) ellipsoidal shape and density gradient; (**B**) ellipsoidal shape and constant density; (**C**) density gradient and approximately planar shape; (**D**) constant density and planar shape. X, Y axes are in \(\upmu\)m. Color map scale is logarithmic normalized related to the critical density. Figure 7: Simulated proton energy spectra for the cases defined in Figure 6.  
acceleration to much higher energies. The spectrum of the accelerated protons, calculated for the same laser intensity and irradiation angle, is shown in Figure 7, for the different whisker shapes shown in Figure 6.  
The effect of the initial plasma whisker density gradient on the charge separation is further seen in Figure 8, comparing the electrons and protons distributions for cases A and B. At time about 320 fs after the start of the interaction, the electrons spatial distribution (Figure 8, left) is almost similar to the protons spatial distribution (Figure 8, right), resulting in much weaker charge separation.