We investigate the extension of self-injecting laser wakefield experiments to
the regime that will be accessible with the next generation of petawatt class
ultra-short pulse laser systems. Using linear scalings, current experimental
trends and numerical simulations we determine the optimal laser and target
parameters, i.e. focusing geometry, plasma density and target length, that are
required to increase the electron beam energy (to > 1 GeV) without the use of
external guiding structures.Comment: 15 pages, 8 figure