Adiabatic Vlasov theory of ultrastrong femtosecond laser pulse propagation in plasma. The scaling of ultrarelativistic quasi-stationary states: Spikes, peakons, and bubbles

The interaction of an ultrashort (femtosecond), pancake-shaped laser pulse with underdense unmagnetized plasma is studied analytically and numerically in a regime with ultrarelativistic electron jitter velocities. The adiabatic evolution of the quasistationary electron distribution function is resolved by following particles along their nonlinear trajectories in phase space. It is estimated that under typical laser-plasma accelerator conditions, the quasistationary distribution is established after the pulse has penetrated ? 1 mm into the plasma. The main particle effect is found to be the beam loading associated with the electrons reflected from the maxima of the effective potential, a fraction of which can also be entrapped in the time dependent potential minima. Owing to the relativistic mass increase, their participation in the plasma dynamics is reduced. The number density of reflected particles is exponentially small for nonrelativistic electron temperatures, but at the wavebreak, it can become comparable to the density of the background plasma if the temperature of electrons exceeds several percent of their rest energy m 0 c 2. Numerical calculations in the presence of beam loading and in the strong intensity regime, based on the three-time scale approximation[Jovanovi? et al., Phys. Plasmas 22, 043110.1 (2015)], reveal the creation of a bubble in the electron density, along with the steepening and the breaking of the nonlinear Langmuir wake that occurs simultaneously with the creation of a sharp spike in the distribution function and a peak in the electron density, located at the position of the wavebreak.