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The interaction of a multi-Petawatt, pancake-shaped laser pulse with anunmagnetized plasma is studied analytically and numerically in the regime offully relativistic electron jitter velocities and in the context of the laserwakefield acceleration scheme. The study is applied to the specificationsavailable at present time, or planned for the near future, of the Ti:SaFrascati Laser for Acceleration and Multidisciplinary Experiments (FLAME) inFrascati. Novel nonlinear equation is derived by a three-timescale description,with an intermediate timescale associated with the nonlinear phase of the laserwave. They describe on an equal footing both the strong and moderate laserintensity regimes, pertinent to the core and the edges of the pulse. These havefundamentally different dispersive properties since, in the core, the electronsare almost completely expelled by a very strong ponderomotive force and theelectromagnetic wave packet is imbedded in a vacuum channel and has (almost)linear properties, while at the pulse edges the laser amplitude is smaller andthe wave is dispersive. The nonlinear phase provides a transition to anondispersive electromagnetic wave at large intensities and the saturation ofthe previously known nonlocal cubic nonlinearity, without the violation of theimposed scaling laws. The temporal evolution of the laser pulse is studied bythe numerical solution of the model equations in a two-dimensional geometry,with the spot diameter presently used in the self-injection test experiment(SITE) with FLAME. The most stable initial pulse length is found to be around 1$\mu$m, which is several times shorter than presently available. A stretchingof the laser pulse is observed, followed by the development of a vacuum channeland a very large electrostatic wake potential, as well as the bending of thelaser wave front.

Ultrarelativistic regime in the propagation of an ultrastrong, femtosecond laser pulse in plasmas