Numerical ray-tracing methods for gradient index media

Abstract

The accuracies of several numerical ray-tracing methods for graded-index optical media are investigated. Tracing is done by numerically solving the ray equation for the classical Luneburg lens. For such an index distribution, an analytic solution exists, with which the numerical methods are compared. Five previously published, numerical ray-tracing algorithms, along with two new methods, based on the multistep predictor and predictor–corrector formulas, are studied, and their relative accuracies are compared by using three criteria. Additional comparison is made by tracing rays through different sections of the lens, giving an indication of their dependence on the position of the ray. We conclude that the point-by-point numerical methods that use physical-path, rather than optical-path, increments are more accurate. It is shown that when the focussing properties of the lens are of chief concern, Sharma's Runge–Kutta method and our new, more efficient, multistep methods have comparable accuracies. When the optical-path length of the ray is important, Montagnino's Taylor-series method and our new predictor method are best.