# Measurement and Control of Transverse Photonic Degrees of Freedom via Parity Sorting and Spin-Orbit Interaction

We discuss new methods for the measurement and control of transverse photonic degrees of freedom. We demonstrate an interferometric mode sorter for two-dimensional (2-D) parity of transverse spatial states of light, present the first experimental 2-D parity sorting measurements of Hermite-Gauss transverse spatial modes, and discuss applications to quantum information processing. We also introduce techniques involving photons propagating in optical fibers, demonstrating theoretically that when a photon propagates in a cylindrically symmetric waveguide, its spin angular momentum and its orbital angular momentum (OAM) interact. This spin-orbit interaction (SOI) leads to the prediction of several rotational effects: the spatial or time evolution of the photonic polarization vector is controlled by its OAM quantum number or, conversely, its spatial wave function is controlled by its spin. We propose an experiment to characterize these phenomena, and demonstrate how they can be used to reversibly transfer entanglement between the spin and OAM degrees of freedom of two-particle states. Finally, we explore an analogous spin-orbit interaction experienced by electrons, showing that the SOI dynamics are quantitatively described by a single expression applying to both electrons and photons, and explain their common origin in terms of a universal geometric phase associated with the interplay between either particle's spin and OAM. This implies that these SOI-based effects occur for any particle with spin, and thereby exist independently of whether or not theparticle has mass, charge, or magnetic moment.