Measurement and control of transverse photonic degrees of freedom via parity sorting and spin-orbit interaction

In this dissertation, several new methods for the measurement and control of transverse photonic degrees of freedom are developed. We demonstrate a mode sorter for two-dimensional (2-D) parity of transverse spatial states of light based on an out-of-plane Sagnac interferometer. The first experimental 2-D parity sorting measurements of Hermite-Gauss transverse spatial modes are presented. Due to the inherent phase stability of this type of interferometer, it provides a promising tool for the manipulation of higher order transverse spatial modes for the purposes of quantum information processing. We propose two such applications: the production of both spatial-mode entangled Bell states and heralded single photons, tailored to cover the entire Poincare sphere of first-order transverse modes.;In addition to the aforementioned transverse spatial manipulation based on free-space parity sorting, we introduce several more such techniques involving photons propagating in optical fibers. We show 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 novel 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 demonstrate how these phenomena can be used to reversibly transfer entanglement between the spin and OAM degrees of freedom of two-particle states.;In order to provide a deeper insight into the cause of the SOI for photons, we also investigate an analogous interaction for electrons in a cylindrical waveguide and find that each of the SOI effects mentioned above remain manifest for the electron case. We show 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 the particle has mass, charge, or magnetic moment.