Topological Polarisation States

Polarisation states are described by spin expectation values, known as Stokes parameters, whose trajectories in a rotationally symmetric system form a sphere named after Poincaré. Here, we show that the trajectories of broken rotational symmetric systems can exhibit distinct topological structures in polarisation states. We use a phase-shifter to form a polarisation circle (${\mathbb{S}}^1$), which interferes with the original input due to the phase change of the output state upon the rotation. By rotating the circle using a rotator, the trajectories become a polarisation torus (${\mathbb{S}}^1\times {\mathbb{S}}^1$), which was experimentally confirmed in a simple set-up using passive optical components together with Mach-Zehnder interferometers. We also discuss about realisations of other topological features, such as Möbius strip, Hopf-links, and topological Dirac bosons with a bulk-edge correspondence.