Gaussian beams with optimal focal properties

Paraxial models of Gaussian beams can be used effectively in designing laser scanning systems for either reading or writing. As a first step, the Gaussian beam that maximizes the mean encircled energy fraction in a cylindrical region of space is considered here. The equations that result must be solved numerically in general, but asymptotic analysis leads in both the two- and three-dimensional geometries to a simple expression that accurately characterizes the optimal Gaussian for all symmetric configurations.