Optics, Quantisation and the Superintegrable Zernike System

The idea of quantisation underpins much of the structure of modern physics. Roughly speaking, quantisation builds a new mathematical structure out of an old one, in such a way that the symmetries of the old theory are respected. Quantisation process are the foundational constructions in quantum mechanics and quantum field theory, have found applications in abstract algebraic geometry and, in this project, we will argue for their use in describing purely classical physics. In particular, in describing the transition between ray optics and wave optics occurring in laser cavities. Such a model includes much of the structure of quantum mechanics (via the paraxial approximation), while retaining crucial differences that make is useful to study. We hope the ray-optical quantisation of systems exhibiting typically pathological features (notably the presence of unbounded solutions, curvature and nonlinearites) will lead to useful insight into the theory of quantisation, its interpretation and contribute to the current flourishing of geometry in optics.