An electron injected sideways into the focal region of a tightly-focused laser beam encounters strong field variations in both space and time. Thus, momentum may be imparted to it by the field in the nature of a small number of violent impulses. Results from single-particle calculations are presented which demonstrate electron acceleration to GeV energies in petawatt beams focused down to around ten-micron spot size. We conclude that a captured electron moves effectively in phase with the field and absorbs energy from it.
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