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The similarity theory is derived for ultra-relativistic laser-plasmainteractions. First, it is shown that the most fundamental S-similarity isvalid for both under- and overdense plasmas. Then, the particular case oftenious plasma is considered in great detail. It is shown that the electrondynamics in this case has two characteristic scales. The fast scale correspondsto relaxation to some attractor solution. The slow dynamics describes anadiabatic evolution of this attractor. This leads to a remarkable wave breakingexclusion rule in the 3D geometry. A similarity theory for the slow dynamicsallows obtaining simple ``engineering'' scalings for the maximum electronenergies, the number of accelerated electrons, the electron beam density, andfor the acceleration distance. These scalings are aimed at design of ahigh-energy laser-plasma accelerator generating electron beams with superiorproperties

Scalings for ultrarelativistic laser plasmas and quasimonoenergetic electrons