An Eulerian Vlasov code is applied for the numerical solution of the one-dimensional Vlasov–Poisson system of equations for electrons, and with ions forming an immobile background. We study the non-linear evolution of the bump-on-tail instability in the case when the system length L is greater than the wavelength λ of the unstable mode, with a beam density of 10% of the total density, nb = 0.1. We follow the growth and the saturation of an initially unstable wave perturbation, and the formation of a traveling Bernstein–Greene–Kruskal (BGK) mode, which evolves out of the instability. This first stage is followed by sidebands growing from round-off errors which develop and disrupt the BGK equilibrium. In the excited spectrum, mode coupling is mediated by the oscillating resonant particles and results in the electric energy of the system flowing to the longest wavelengths (inverse cascade), and reaching in the asymptotic state a steady state with constant amplitude oscillation modulated by the persistent oscillation of the trapped particles. Coherent phase-space electron holes are formed, which are localized phase-space regions of reduced density on trapped electron orbits, where the electron density is lower than the surrounding plasma electron density. The distribution function evolves to a shape with stationary inflection points of zero slope, at the phase velocities of the excited waves. The longest wavelengths show oscillations at frequencies below the plasma frequency, with phase velocities higher than that of the injected beam, which can accelerate electrons to energies in excess of the initial beam energy. The present work makes a connection between the formation of electron holes, the existence of inflection points of zero slopes in the electron distribution function at the phase velocities of the dominant waves, and at frequencies below the plasma frequency. A fine resolution grid is used in the Eulerian Vlasov code in the phase space and time to allow an accurate calculation of the time history of the system and of the dynamic and oscillation of the trapped particles in the low-density regions of the phase space, and of those particles at the separatrix regions of the vortex structures which evolve periodically between trapping and untrapping states and which can only be accurately studied using a fine-resolution phase-space grid.

We use an Eulerian Vlasov code, which solves the one-dimensional relativistic Vlasov–Maxwell equations for both electrons and ions, to follow in details the evolution of the distribution functions and the mechanism of the formation and evolution of double layers during ion acceleration driven by a high-intensity circularly polarized short laser pulse (12 ω−1 where ω is the laser angular frequency) normally incident on a thin dense foil. We compare three cases with a high-density deuterium plasma target of total thickness 1.767 cω−1 and constant n/n cr = 100, where n cr is the critical density, and where the laser intensity is varied from a situation where the target is opaque to the laser pulse (normalized vector potential or quiver momentum a 0 = 80), to a situation where, above a critical laser intensity, a very small fraction of the laser pulse is transmitted through the target (a 0 = 90), and finally to a situation where a more important fraction is transmitted through the target (a 0 = 100). The dynamics of ion and electron acceleration are quite different in the three cases, and are followed in detail by the Eulerian Vlasov code, which allows an accurate representation of the distribution function. In the intermediate case, the Vlasov code has revealed a remarkably well-developed spiral structure in the phase space of the electron distribution function, which is associated with large sawtooth modulations in the electron density profiles.

We apply an Eulerian Vlasov code to study the amplification of an ultra-short seed pulse via stimulated Raman and Brillouin backscattering of energy from a long pump pulse, assumed at constant amplitude, in a plasma embedded in an external magnetic field. Detailed analysis of the spectra developed during the amplification process are presented, together with the evolution showing the pump depletion, accompanied by the counter-propagating seed-pulse amplification, compression and increased steepness of the waveform. In addition to the problem of the amplification of ultra-short seed pulses, there is an obvious academic interest in the study of problems of amplification of electromagnetic waves observed in many situations in laboratory plasmas and in the magnetosphere and other geophysical situations, such as in the environments of planets, where important variations in the presence and strength of magnetic fields are observed. The numerical code solves a one-dimensional relativistic Vlasov–Maxwell set of equations for a plasma in a magnetic field for both electrons and ions. We also apply the code to the problem of wakefield acceleration. The absence of noise in the Eulerian Vlasov code allows one to follow the evolution of the system with an accurate representation of the phase-space structures of the distribution functions.

We use an Eulerian Vlasov code to study the efficient ion acceleration in dense targets by the ponderomotive force of a high-intensity circularly polarized laser beam, normally incident on a dense plasma. The code solves the one-dimensional relativistic Vlasov–Maxwell equations for both electrons and ions. We follow in details the mechanism of formation and evolution of a double-layer structure, where electrons are pushed steadily in the forward direction by the ponderomotive force of the laser beam, trapping an ion population, while an induced space charge electric field pulls ions behind them, forming a double-layer structure supported by the strong ponderomotive pressure of the intense laser beam. We consider the case of a high-density deuterium plasma with n/ncr = 100, where ncr is the critical density. Three cases are studied, by varying the width of the dense target and the intensity of the laser beam (with the normalized vector potential or quiver momentum a0 = 50 and a0 = 100), to follow the physical processes involved in the ion acceleration and the final formation of a neutral plasma jet ejected from the back of the target. We follow the transition from a situation where the laser pulse radiation pressure is acting on the double layer in the target, to a situation where below a given thickness a fraction of the laser energy is transmitted through the target. The absence of noise in the Eulerian Vlasov code allows us to follow accurately the evolution of the phase-space structures of the distribution functions.

We study the problem of the radiation pressure acceleration of ions and the formation of plasma jets, driven by a high-intensity circularly polarized laser beam normally incident on thin plasma targets. We use an Eulerian Vlasov code to solve the one-dimensional relativistic Vlasov-Maxwell equations for both electrons and ions. We consider the case of a high density plasma with n/ncr = 100, where ncr is the critical density. Three cases are studied with different target thicknesses, to investigate the physical processes involved when decreasing the target thickness from several electron skin depths down to the order of one skin depth. The results show a more important acceleration of the ions when the thickness is decreased. Although we observe in all cases a neutral plasma jet ejected from the back of the target, the evolution of the system which leads to the formation of this neutral plasma jet is different in the three cases considered. In each case, this evolution will be studied in details. Also, a leak or ejection of electrons from the back of the target is observed in the thinnest case treated (thickness of the order of the skin depth), before the formation of the neutral plasma jet, a regime called leaky light sail radiation pressure acceleration. The absence of noise in the Eulerian Vlasov code allows an accurate representation of the phase-space structures of the distribution functions.

We present a numerical study of the acceleration of ions in the interaction of a high intensity circularly polarized laser beam normally incident on an overdense plasma target, and the subsequent formation of neutral plasma ejected toward the rear side of the target. We compare the results obtained from two different numerical codes. We use an Eulerian Vlasov code for the numerical solution of the one-dimensional relativistic Vlasov-Maxwell set of equations, for both electrons and ions, and a particle-in-cell code applied to the same problem. We consider the case when the laser free space wavelength λ0 is greater than the scale length of the jump in the plasma density at the target plasma edge Ledge (λ0 ≫ Ledge), and the ratio of the plasma density to the critical density is such that n/ncr ≫ 1. The ponderomotive pressure due to the incident high-intensity laser radiation pushes the electrons at the target plasma surface, producing a sharp density gradient at the plasma surface, which gives rise to a charge separation. The resulting electric field accelerates the ions that reach a free streaming expansion phase, where they are neutralized by the electrons. A neutral plasma jet is thus ejected toward the rear side of the target. Two cases are studied: In the first case, the laser intensity rises to a maximum and then remains constant, and in the second case, the laser intensity is a Gaussian-shaped pulse. The results show substantial differences in the phase-space structure of the ions and the electrons between these two cases. There is good agreement between the quantitative macroscopic results obtained by the two codes, and good qualitative agreement between the results showing the kinetic details of the phase-space structures. The low noise level of the Eulerian Vlasov code allows a more detailed representation of the phase-space structures associated with this system, especially in the low density regions of the phase-space where ions are accelerated.