The interaction of free electrons with intense laser beams in vacuum was studied using 3D test particle simulation instead of analytically solving the relativistic Newton-Lorentz equation of motions. We found a group of solutions for the equation, which reveal very interesting and unusual characteristics different from any previously reported. The fundamental characteristics of those trajectories are that an electron can be captured into the high-intensity region, rather than expelled from it and that the captured electron can be accelerated to GeV energy with an acceleration gradient of 1–50 GeV/cm. These solutions emerges only when the laser intensity is a0\\gtrsim100, where a0≡eE0/meωc is a measure of the laser intensity. The accelerated GeV electron bunch is a macropulse composed of multiple micropulses, which is analogous to the structure of bunches produced by conventional linacs. The paraxial approximation equations for the Gaussian laser beam used in the simulation are highly accurate and the contribution of the high-order correction is almost negligible when the laser beam width is w0\\geqslant60.
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