The results of a theoretical consideration of positron acceleration in the plasma wakefield driven by asymmetric laser pulses are presented in this paper. It is found that the asymmetric laser pulse can significantly modify the phase portrait of the positron dynamics and enhance the maximum energy of the accelerated positrons. There exists an optimum ratio of the lengths of the rising and falling segments of the asymmetric laser pulse. There exists also an upper limit on plasma density to get the maximum wakefield potential.
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