Extension of the convergence of multivariate aberration series

The region of convergence of the Lagrangian aberration series of symmetric optical systems designed to image plane objects (possibly at infinity) is investigated. Using the conventional variables, the region of convergence of these series is often found to be limited by complex singularities that, unlike real singularities, do not correspond to physically significant events (such as a physical ray grazing a surface). These complex singularities have restricted the use of aberration series to the analysis of systems with at most moderate fields and apertures. It is shown that, by choosing a new set of variables, it is possible to move these singularities and obtain convergence over the entire region of interest for more demanding systems (e.g., systems with half-fields of, say, 70°). As a consequence of this extended convergence, the accuracy of the truncated aberration series is greatly improved. Some numerical examples are given to illustrate graphically the superiority of the aberration series in the new variables.