Accuracy doubling in the determination of final ray configurations

It is proposed that the extremal property of a trajectory be exploited to accelerate the analysis of systems governed by an extremum principle (e.g., Fermat’s principle). Given any procedure that is able to determine trajectories through such a system to a specific accuracy, a prescription is derived that allows the endpoints of these trajectories to be determined to twice the original accuracy. That is, if a path is originally found to n decimal places, this prescription yields the final configuration (at some fixed plane, say) to ˜2n decimal places. It is emphasized that, in fact, a differential path is required; that is, both the approximate trajectory and its derivatives with respect to initial conditions are needed. Such accuracy doubling is valuable only when paths are difficult to determine (e.g., in the analysis of gradient index systems or when the path is required in the form of an aberration series). As both a first demonstration and a proof of principle, it is shown that, for systems of homogeneous media, this method can reduce the number of operations needed in the computation of high-order Lagrangian aberration series by a factor of 2 or 3.