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Laser wakefield acceleration (LWFA) using high repetition rate mJ-class laser systems brings unique opportunities for a broad range of applications. In order to meet the conditions required for the electron acceleration with lasers operating at lower energies, one has to use high density plasmas and ultrashort pulses. In the case of a few-cycle pulse, the dispersion and the carrier envelope phase effects can no longer be neglected. In this work, the properties of the wake waves generated by ultrashort pulse lasers in near-critical density plasmas are investigated. The results obtained may lead to enhancement of the quality of LWFA electron beams using kHz laser systems.

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