Introduction

Particle-in-cell (PIC) and Vlasov simulations both solve the Vlasov equation. The Vlasov equation (cf. Chapter 2) governs the evolution of the distribution function of charged particles (electrons, ions) in the six-dimensional phase space, consisting of three velocity (or momentum) dimensions and three position dimensions, plus time. It offers an accurate description of a plasma in the collisionless limit; that is, when the particles are affected by long-range electric and magnetic fields only, and when short-range fields from their nearest neighbors can be neglected.

PIC simulations resolve the distribution function statistically with macro-particles (or super-particles) and follows the solution over trajectories along which the distribution function is constant; the characteristics are given by the equations of motion for the charged particles. This is the Lagrangian description. Many PIC codes have been developed over the years; modern PIC codes include the plasma simulation code (PSC) originally developed by Hartmut Ruhl, the implicit iPIC3D code aimed at connecting kinetic and magnetohydrodynamic time scales, the EPOCH code, partially based on PSC, the VSIM/VORPAL code, the OSIRIS code, and QuickPIC. PIC simulations are very adaptive and efficient for many problems, such as high-energy beam–plasma and laser–plasma interactions. On the other hand, they also have limitations; the numerical noise and slow convergence with increasing number of particles are some issues. There is also the need to resolve the Debye length with particles to avoid artificial numerical heating.

A different strategy is followed in Vlasov simulations using a Eulerian description. Here, the distribution function is treated as a phase fluid resolved on a fixed numerical grid. Vlasov simulations do not have the statistical noise of PIC simulations; they can also more accurately resolve the high-velocity tail of the particle distribution functions. On the other hand, Vlasov simulations in higher dimensions are very memory demanding due to the need to resolve the six-dimensional phase space on a numerical grid. In some cases, the distribution function can also become oscillatory in phase space, leading to sharp gradients and a need to introduce numerical dissipation in velocity space whilst avoiding artificial numerical heating due to the broadening of the distribution in velocity space. Hence, the choice between PIC and Eulerian Vlasov simulations strongly depends on the physical problem at hand.

The laser, with its coherent, monochromatic, and well collimated character, has been a most remarkable discovery of the twentieth century. Along with semiconductors, its multifaceted applications have broadly touched and greatly improved our lives – it has made an indelible mark in the field of sensing, printing, barcode scanning, surgery, communications, and so on. It has also become a major tool for scientific research. For example, Thomson scattering and laser induced fluorescence are important tools for plasma diagnostics. Lasers have been used successfully for cooling of atoms and heating of plasmas.

The laser peak power has increased about a 1000 fold every decade since its invention. Starting from hundred watts in the 1960s, table top terawatt Ti: sapphire lasers became available in the 1990s following the discovery of the chirped pulse amplification (CPA) by Mourou and Strickland in 1985. These lasers do not only have high power but also very short pulses of a few femtoseconds, opening a new field of ultra-short pulse lasers and their interactions with matter, such as electron dynamics in molecules. In the past few years, we have seen worldwide efforts to build high power laser infrastructures. The Extreme Light Infrastructure (ELI) has been approved to construct three petawat laser facilities in Eastern Europe. Similar efforts are being made in Korea, Japan and China.

With the rise in laser power, there has been a phenomenal growth in the field of high power laser-plasma interaction with diverse applications, ranging from laser driven fusion and laser acceleration of charged particles to laser ablation of materials. The field has revealed a rich variety of fascinating new phenomena. Parametric coupling between lasers and plasma eigenmodes and quasi-modes gives rise to stimulated Raman, Brillouin, and Compton scattering, two-plasmon decay, and four-wave processes of filamentation, modulational, and oscillating two-stream instabilities of the laser. Nonlinear refraction gives rise to selffocusing and self-guiding of lasers over long distances in plasma and air, offsetting diffraction divergence. Laser interaction with rough metallic surfaces reveals surface-enhanced Raman scattering (SERS) where Raman scattered power from adsorbed molecules rises a million times due to surface plasmon resonance. Laser mode conversion to surface plasma waves (SPWs) on metallic surfaces has been shown to enhance the ablation yield and thin film deposition rates by orders of magnitude, making pulsed laser deposition a very attractive scheme.