## 3 Particle in Cell Simulations of High Intensity Laser Pulses with Snow Whiskers  
Numerical 2D PIC code simulations of the main laser pulse with the pre-formed plasma generated by the prepulse explain the role of the plasma density gradient and were found in agreement with the experimental scaling law shown in Figure 1. The increased proton energy may result from localized enhancement of the laser field intensity near the tip of the microstructured whisker. In addition, in the mass-limited whisker target mass-limited like phenomenon, _i.e._, the absence of a high density cold electron cloud in the vicinity of the whisker cannot compensate for the expelled electrons. The heated electrons remain in the vicinity of the positively charged whisker, and strong accelerating electrostatic fields are produced, pulling the protons out. The Coulomb explosion of the positively charged whisker, add longtime acceleration to the protons. A one dimensional fluid model, demonstrating the field enhancement by an order of magnitude by the local plasma density near the whisker tip, was presented in our previous work [15]. Aimed to optimize the laser-snow interaction and enhance proton acceleration, we conducted a multi-variables numerical study of laser-snow structured target interaction, using the 3D particle in cell (PIC) code TURBOWAVE [27, 28]. We considered intense laser pulse interaction with snow whiskers with different sizes, aspect ratios from 1 to 100, planar and ellipsoid shapes, density from step-like solid density to under-dense plasma with smooth Gaussian density gradients. Electrons, protons and oxygen ions acceleration was examined for laser intensity in the range 2.5\(\times\) 10\({}^{17}\) W/cm\({}^{2}\) to 2.5\(\times\) 10\({}^{21}\) W/cm\({}^{2}\) for different angles between the laser propagation and the whisker axis of symmetry. Our simulations show that maximum proton acceleration may occur due to combination of optimal conditions of the whisker's density gradient, surface curvature, aspect ratio and illumination conditions. The laser spot size in our experiments was about 4-5 \(\upmu\)m, therefore the laser spot may have interacted with one or sequentially with few whiskers, based on the SEM characterization of the snow targets [20, 21]. Here, for simplicity, we consider interaction with a single snow whisker. Following the prepulse illumination, the whisker is partially vaporized and ionized hence a non-uniform plasma cloud is formed. To study the influence of plasma density gradients on proton acceleration we model the non-uniform plasma cloud composed of protons, triply ionized oxygen ions and electrons, as an ellipsoid. We assumed that the snow whisker consists of a solid core surrounded by gradually decreasing density plasma, such that the critical density region is located at a distance larger by about a factor of 10 than the width of the solid core. An ellipsoid shape density enables as well study of expected laser-snow interaction enhancement with curved surfaces, in particular near the tip of the whisker [15], extending previous studies that considered only spherical and cylindrical targets [29]. Figure 3 shows the electron density distribution and the laser beam before the interaction for one of the cases considered in our simulations. In this example, the high density part of the whisker, of 100\(\cdot n_{\mathrm{cr}}\), where \(n_{\mathrm{cr}}\) = 1.71 \(\times\) 10\({}^{21}\) cm\({}^{-3}\) is the critical density at laser wavelength 0.8 \(\upmu\)m, is an ellipsoid of the order of 0.1-0.2 \(\upmu\)m width (minor axis) and 1-2 \(\upmu\)m length (major axis), and the critical density contour is an ellipsoid of the order of 1-2 \(\upmu\)m width (minor axis) and 10 \(\upmu\)m length (major axis). Our simulations assume that the polarization vector of the laser electric field and the major axis of the whisker are in the same plane.