  A general family of 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 cycling of
the rays and a global rescaling), have two parameters, the first giving a
position on an ellipse naturally represented by a point on the Poincar\'e
sphere (familiar from polarization optics), and the other determining the
position of a curve traced out on this Poincar\'e sphere. This construction
naturally accounts for the familiar families of Gaussian beams, including
Hermite-Gauss, Laguerre-Gauss and Generalized Hermite-Laguerre-Gauss beams, but
is far more general. The conformal mapping between a projection of the
Poincar\'e sphere and the physical space of the transverse plane of a Gaussian
beam naturally involves caustics. In addition to providing new insight into the
physics of propagating Gaussian beams, the ray-based approach allows effective
approximation of the propagating amplitude without explicit diffraction
calculations.
