Experimental proposal of a mode sorter for vector vortex beams of arbitrary order

We propose an experimental scheme for sorting vector vortex beams using a system of three optical cavities. The method targets the separation of the four-vector vortex modes of a given order $m$ by exploiting their symmetry properties with respect to radial axes. The first cavity separates the modes into two pairs---one symmetric and one antisymmetric under horizontal flip---by transmitting one pair and reflecting the other. Each output is then directed into a second cavity, which further resolves the pairs into individual modes based on their parity. A key element between the cavities is a system introduced in this work, the profile rotator, which rigidly rotates the beam profile (including its polarization structure) using an appropriately calibrated K-mirror and a Faraday rotator. The theoretical framework is based on the analysis of mode symmetries and their interaction with cavity boundary conditions. This work lays the groundwork for experimental implementations of efficient mode sorting, with potential applications in high-dimensional quantum information processing, optical communications, and structured light manipulation.