The optical M\"{o}bius strip cavity: Tailoring geometric phases and far fields

The M"{o}bius strip, a long sheet of paper whose ends are glued together after a ${180}^{\circ }$ twist, has remarkable geometric and topological properties. Here, we consider dielectric M"{o}bius strips of finite width and investigate the interplay between geometric properties and resonant light propagation. We show how the polarization dynamics of the electromagnetic wave depends on the topological properties, and demonstrate how the geometric phase can be manipulated between $0$ and $\pi$ through the system geometry. The loss of the M"{o}bius character in thick cavities and for small twist segment lengths allows one to manipulate the polarization dynamics and the far-field emission, and opens the venue for applications.