What is the confocal parameter?

A novel derivation is presented of the Gaussian beam as a limit of the solution to the full wave equation. Usually, the functional form of the Gaussian beam is found by a two-step process. First, the Green's function of the paraxial wave equation is identified. Then, since the paraxial wave equation is invariant under translation, the z-axis variable is replaced by z+jb. It is shown that when starting with a solution of the full three-dimensional Helmholtz equation in spherical coordinates, performing the transformation z→z+jb corresponds physically to causing the phase fronts of the solution to become ellipsoids. The separation of the foci of the ellipsoids is 2b, where b is the confocal parameter of the beam. In the paraxial limit the ellipsoidal solution becomes a Gaussian beam. Adopting this approach to Gaussian beams allows a simple, geometrical interpretation of the optical resonator stability criterion