Geometric phases are ubiquitous in physics; they act as memories of the transformation of a physical system. In optics, the most prominent examples are the Pancharatnam-Berry phase and the spin-redirection phase. Recent technological advances in phase and polarization structuring have led to the discovery of additional geometric phases of light. The underlying mechanism for all of these is provided by fiber bundle theory. This Colloquium reviews how fiber bundle theory not only sheds light on the origin of geometric phases of light but also lays the foundations for the exploration of high-dimensional state spaces, with implications for topological photonics and quantum communications.

• Received 10 November 2021

DOI:https://doi.org/10.1103/RevModPhys.94.031001