The Review covers recent progress in laser-matter interaction at intensities above 10¹⁸ W cm⁻². At these intensities electrons swing in the laser pulse with relativistic energies. The laser electric field is already much stronger than the atomic fields, and any material is instantaneously ionized, creating plasma. The physics of relativistic laser-plasma is highly non-linear and kinetic. The best numerical tools applicable here are particle-in-cell (PIC) codes, which provide the most fundamental plasma model as an ensemble of charged particles. The three-dimensional (3D) PIC code Virtual Laser-Plasma Laboratory runs on a massively parallel computer tracking trajectories of up to 10⁹ particles simultaneously. This allows one to simulate real laser-plasma experiments for the first time.

When the relativistically intense laser pulses propagate through plasma, a bunch of new physical effects appears. The laser pulses are subject to relativistic self-channelling and filamentation. The gigabar ponderomotive pressure of the laser pulse drives strong currents of plasma electrons in the laser propagation direction; these currents reach the Alfvén limit and generate 100 MG quasistatic magnetic fields. These magnetic fields, in turn, lead to the mutual filament attraction and super-channel formation. The electrons in the channels are accelerated up to gigaelectronvolt energies and the ions gain multi-MeV energies. We discuss different mechanisms of particle acceleration and compare numerical simulations with experimental data.

One of the very important applications of the relativistically strong laser beams is the fast ignition (FI) concept for the inertial fusion energy (IFE). Petawatt-class lasers may provide enough energy to isochorically ignite a pre-compressed target consisting of thermonuclear fuel. The FI approach would ease dramatically the constraints on the implosion symmetry and improve the energy gain. However, there is a set of problems to solve before the FI will work. The laser pulse cannot reach the dense core of the target directly. The laser energy must be converted into fast particles first and then transported through the overdense plasma region. The energy spectra of the laser-generated particle beams, their emittance and transport problems are discussed here.

The laser–particle interaction at relativistic intensities is highly non-linear and higher laser harmonics are generated. In plasma, the high-harmonic generation is a collective effect—it appears to be quite effective when an intense laser pulse is reflected from the overdense plasma layer. The plasma boundary is then driven by the laser ponderomotive force and works as a relativistically oscillating mirror.

Another interesting application is the amplification of short-pulse laser in plasma by a counter-propagating pump pulse. 3D PIC simulations suggest that multi-terawatt pulses of sub-10 fs duration can be generated this way.