Gaussian mode families from systems of rays

Gaussian mode families, including Laguerre–Gaussian, Hermite–Gaussian and generalized Hermite–Laguerre–Gaussian beams, can be described via a geometric optics construction. Ray families crossing the focal plane are represented as one-parameter families of ellipses, parametrized by curves on an analog Poincaré sphere for rays. We derive the optical path length weighting the rays, and find it to be related to the Pancharatnam–Berry connection on the Poincaré sphere. Dressing the rays with Gaussian beams, the approximation returns the Laguerre-, Hermite- and generalized Hermite–Laguerre–Gaussian beams exactly. The approach strengthens the connection between structured light and Hamiltonian optics, opening the possibility to new structured Gaussian beams.