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In 1979 Tajima and Dawson, Tajima & Dawson (1979), on the basis of theoretical work and simulations, have shown that relativistic plasma waves are suitable for the development of compact accelerators. In the proposed schemes, electrons were injected externally and were accelerated through the very high electric field (GV/m) sustained by relativistic plasma waves driven by lasers, in the laser beatwave regime or in the laser wakefield regime. Several experiments have been performed following their idea and injected electrons at the few MeV level have indeed been accelerated by electric fields in the GV/m range in a plasma medium using the beat wave and laser wakefield schemes. With the development of more powerful lasers, much higher electric fields were achieved, from few GV/m to more than 1 TV/m, Malka et al. (2002).