My research centres around the use of symmetry techniques in optics. Many of the symmetry methods familiar from undergraduate mechanics like continuous symmetries and conserved quantities have powerful generalisations to the idea of group actions on different kinds of manifolds and become applicable to many physical theories, including optics. I use this geometry to investigate the relationship between ray and wave optics, primarily using the tools of geometric quantisation and semiclassical physics. Optical physics provided the initial systems and physical motivation for my work. The physical pictures of the quantisation of such systems is performed under the auspices of the correspondence between classical wave optics and 2+1D quantum mechanics described by the paraxial approximation.
More explicitly I am interested in applications of geometrical methods to the quantisation of pathological systems, such as those with noncompact Bohr-Sommerfeld leaves and also what the extra geometry such methods furnish (like Lagrangian foliations) can help us say about caustic singularities in phase space. At the moment I am trying to construct quantisations of harmonic potentials in 2D and use them to interpret various nice properties that emerge from the caustics of the semiclassical geometry.
The tools I spend the most time playing around with and some of their standard reference texts are
Optics
Born & Wolf - Principles of Optics
Gbur - Singular Optics
Geometrical (Hamiltonian) Mechanics
Abraham & Marsden - Foundations of Mechanics
Marsden & Ratiu - Introduction to Mechanics and Symmetry
Geometric Quantization
Woodhouse - Geometric Quantization