Swings and Roundabouts: Operators and Rays for structured Gaussian beams
Mark Dennis (University of Bristol)
There is a deep analogy between the Poincaré sphere describing polarization, and the scalar Gaussian modes of laser cavities. Mathematically, the Poincaré sphere describes the orbits of a classical orbits of a classical two-dimensional harmonic oscillator, in terms of the formalism of Hamiltonian mechanics. Many properties of Hermite-, Laguerre- and Ince-Gaussian beam families can be understood with these as eigenfunctions of certain operators, such as the angular momentum operator, corresponding to Hamiltonian constants of the motion (themselves related to Stokes parameters). This correspondence can be approached semiclassically, where any self-similar Gaussian beam family can be constructed from a hyperboloidal family of straight rays. This ray-based description also provides a simple explanation for many aspects of structured Gaussian beams, such as “self-healing” and the Gouy and Pancharatnam–Berry phases. In particular the analysis of the Ince-Gaussian mode family reveals surprising Kepler-orbit like geometry in the representation of the ray families, and connects to a well-studied model in atomic physics.
Please contact M.Gradhand@bristol.ac.uk for further information.