Improved Analytical Theory of Ophthalmic Lens Design

A revisited form of the classic third-order ophthalmic lens design theory that provides a more precise and meaningful use of aspheric surfaces and a generalization of the standard oblique errors is presented. The classical third-order theory follows from the application of the Coddington equations to a ray trace through the lens and the expansion of the incidence angles and the surface sagittas appearing on them up to order two of the radial coordinate. In this work we show that the approximations for surface sagittas and angles can be decoupled, and the lens oblique powers predicted by the proposed theory provides a better fit to the numerical results obtained by exact raytracing and multi-parametric optimization than the classical third-order theory does. Modern ophthalmic lens design uses numerical optimization and exact ray tracing, but the methods presented in this paper provide a deeper understanding of the problem and its limitations. This knowledge and the more general merit functions that are also presented may help guide the numerical approaches.