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In this work, we introduce a new ellipsoid model to describe bubble acceleration of electrons and discuss the required conditions of forming it. We have found that the electron trajectory is strongly related to background electron energy and cavity potential ratio. In the ellipsoid cavity regime, the quality of the electron beam is improved in contrast to other methods, such as that using periodic plasma wakefield, spherical cavity regime, and plasma channel guided acceleration. The trajectory of the electron motion can be described as hyperbola, parabola, or ellipsoid path. It is influenced by the position and energy of the electrons and the electrostatic potential of the cavity. In the experimental part of this work, a 20 TW power and 30 fs laser pulse was focused on a pulsed He gas jet. We have focused the laser pulse in the best matched point above the nozzle gas to obtain a stable ellipsoid bubble. The finding of the optimum points will be described in analytical details.
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