Ray-optical Poincaré sphere for structured Gaussian beams

A general family of scalar structured Gaussian beams naturally emerges from a consideration of families of rays. These ray families, with the property that their transverse profile is invariant upon propagation (except for a global rescaling), have two parameters, the first giving a position on an ellipse naturally represented by a point on a ray-family analog of the Poincaré sphere (familiar from polarization optics), and the other determining the position of a curve
traced out on this Poincaré sphere. This construction naturally accounts for thewell-known families of Gaussian beams, including Hermite-Gauss, Laguerre-Gauss and Generalized Hermite-Laguerre-Gauss beams, but is farmore general, opening the door for the design of a large variety of propagation-invariant beams. This ray-based description also provides a simple explanation for many aspects of these beams, such as “self-healing” and the Gouy and Pancharatnam-Berry phases. Further, through a conformal mapping between a projection of the Poincaré sphere and the physical space of the transverse plane of a Gaussian beam, the otherwise hidden geometric rules behind the beam’s intensity distribution are revealed. While the treatment is based on rays, a simple prescription is given for recovering exact solutions to the paraxial wave equation corresponding to these rays.