Ionization injection is attractive as a controllable injection scheme for generating high quality electron beams using plasma-based wakefield acceleration. Due to the phase dependent tunneling ionization rate and the trapping dynamics within a nonlinear wake, the discrete injection of electrons within the wake is nonlinearly mapped to discrete final phase space structure of the beam at the location where the electrons are trapped. This phenomenon is theoretically analyzed and examined by three-dimensional particle-in-cell simulations which show that three dimensional effects limit the wave number of the modulation to between $> 2k_0$ and about $5k_0$, where $k_0$ is the wavenumber of the injection laser. Such a nano-scale bunched beam can be diagnosed through coherent transition radiation upon its exit from the plasma and may find use in generating high-power ultraviolet radiation upon passage through a resonant undulator.
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