Focusing regions evolution under linear transformations in the boundary condition

The properties of optical surfaces generated by diffraction are studied analyzing the changes in the curvature function of the boundary condition. The study leads to establish a classification for the optical fields as elliptical, hyperbolical or parabolic. The elliptical regions are separated from hyperbolic regions by a region of parabolic type, which corresponds to optical focusing regions. The main result of the analysis leads us to describe and to control bifurcations and vortices effects allowing to geometrize and to control the topological properties of the optical field. For this feature is important to control the geometry of the parabolic region, obtained by controlling the curvature function in the boundary condition. The model is implemented experimentally applying a linear transformation in transmittances of zone plate type. The experimental results corroborate the theoretical predictions.