Ray-wave structured vortex beams have attracted increasing attention due to their unique spatial geometric coupling to control complex orbital angular momentum (OAM). Still, current models were constrained by circular symmetry with limited modulation freedom. Herein, we propose a generalized class of ray-wave light fields called Mathieu geometric modes (MGMs) fulfilling the form of a stationary coherent state but based on a set of helical Mathieu modes (HMMs), in which geometrically tunable elliptical accelerating vortices are obtained by tuning their eccentricity-related parameters. MGMs also possess intriguing properties of coordinate transformation, self-healing, and multilayer tunable angular acceleration upon propagation. MGMs have higher degrees of freedom to control spatial accelerating vortices, paving the way for higher-dimensional optical tweezers and complex particle manipulation.
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