Highly nonlinear ellipsoid bubble regime of the laser wake-field acceleration with high-intensity laser pulse is considered with analytical and numerical calculations. The important property of this regime is the production of the mono-energetic high-quality electron beam. We introduce a new twofold ellipsoid structure of the bubble (egg shape) by referring to some published two-dimensional (2D) and 3D simulations. In this paper, a new analytical formalism is introduced, in which dimensions of the front part of the ellipsoid bubble are related to the laser pulse and plasma parameters. These relationships are in agreement with 2D particle-in-cell code results in recent work (Benedetti et al., 2013).
