Improving the convergence of transverse ray aberration series

Abstract

Transverse ray aberration series are simply the Taylor series of functions which specify the configuration of a ray after passage through an optical system in terms of the initial configuration. These series are expansions about some specified base ray—the axis for a symmetric system. As is typical of Taylor series, the accuracy of aberration series (truncated at a particular order) deteriorates as the configuration of the initial ray moves away from the base ray. If the accuracy becomes inadequate, an obvious response is to compute the series to higher orders. This approach, however, relies on the assumption that the region of convergence of the series encloses the region of interest. In practice, using the conventional series for systems designed to image plane objects (possibly at infinity), this assumption is often invalid. The limited convergence of these series has hitherto restricted their use to the analysis of systems with (at most) moderate fields and apertures. The selection of an unconventional set of ray parameters as the arguments of the aberration series is shown to yield (i) convergence over the entire region of interest for more demanding systems (e.g., systems with half-fields of, say, 70°) and (ii) remarkable improvements in the accuracy of the truncated series.