A plasma wave and accelerated electrons from a self-modulated laser wakefield accelerator were studied. A 400 fs, 1.053 {dollar}mu{dollar}m laser pulse was focused into an under-dense plasma {dollar}(3times 10sp{lcub}19{rcub}{dollar} cm{dollar}sp{lcub}-3{rcub}){dollar} where it self-modulated and formed a plasma wave with an accelerating gradient of 1 GeV/cm. The plasma wave was temporally and spatially characterized with a variety of diagnostics. Approximately 1 nC of MeV energy, accelerated electrons were observed in a directed beam. When the laser power exceeds the critical power for relativistic self-focusing by a factor of ten, the laser was found to self-guide and extend beyond the natural limit of diffraction. This was found to increase the energy of the accelerated electrons and reduce their emittance. This indicates that acceleration of electrons beyond the conventional diffraction limit was achieved. Ionization was found to play an important role in the evolution of the plasma wave and laser focusing.
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