The process of electron self-injection in the nonlinear bubble wake generated by a short and intense laser pulse propagating in an uniform underdense plasma is studied by means of fully self-consistent particle-incell simulations and test-particle simulations. We consider a wake generated by a non-evolving laser driver traveling with a prescribed velocity, which then sets the structure and the velocity of the wake, so the injection dynamics is decoupled from driver evolution but a realistic

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... cture for the wakefield is retained. We show that a threshold for self-injection into a non-evolving bubble wake exists, and we characterize the dependence of the self-injection threshold on laser intensity, wake velocity, and plasma temperature for a range of parameters of interest for current and future laser-plasma accelerators.