Calculation of zonal power and astigmatism of a freeform gradient index lens with freeform surfaces

Freeform gradient index (F-GRIN) lenses have been recently shown to enable compact optical design. However, aberration theory is only fully developed for rotationally symmetric distributions with a well-defined optical axis. The F-GRIN has no well-defined optical axis, and rays are continuously perturbed along their trajectory. Optical performance can be understood without abstracting optical function to numerical evaluation. The present work derives freeform power and astigmatism along an axis through a zone of an F-GRIN lens with freeform surfaces. Zonal power and astigmatism can be assessed without tracing any rays, capturing mixed contributions of the F-GRIN and freeform surface. Theory is compared with a commercial design software numerical raytrace evaluation. The comparison shows that the raytrace-free (RTF) calculation represents all raytrace contributions within a margin of error. In one example, it is demonstrated that linear terms of index and surface alone in an F-GRIN corrector can correct the astigmatism of a tilted spherical mirror. Accounting for the induced effects of the spherical mirror, RTF calculation provides the amount of astigmatism correction of the optimized F-GRIN corrector.