Conservation of extremal ellipticity and analytical expressions for astigmatism and asymmetry in Gaussian beam transformations.

A Gaussian beam with initial astigmatism, waist, and M asymmetry propagates through a simple optical system consisting of a single thin, aberration-free lens. We derive simple analytical expressions for the asymmetry, astigmatism, and ellipticity of the beam after the lens, and show that the maximal ellipticity of the beam is a conserved quantity that is solely dependent on these input parameters. This appears to still hold for general astigmatic Gaussian beams.