Explicit representations of all refractive optical interfaces without spherical aberration.

The following explicit model, valid for high aperture refraction with homogenous and isotropic materials, encompasses all explicit solutions of the first-order nonlinear differential equation representing the perfect image-forming process of any axial object point into its axial image point. Solutions include well-known cases, such as flats, spheres, prolate ellipsoids, prolate hyperboloids, and other sections of nondegenerate Cartesian ovals of revolution, now classified according to the recurrent explicit solution introduced herein. We also present some series expansions, given in cylindrical coordinates z(r), for more efficient computation. Explicit solutions allow accurate and expedite thickness calculation as compared to the regular series, parametric, or implicit solutions commonly used. The results of this study are useful in the design of centered optical systems that are perfectly aligned.