Foundations of second-order layout for asymmetric systems

Methods are described here that can be used to determine expressions for the second-order properties (with respect to a particular ray—the base ray) of any system composed of individual homogeneous, isotropic regions. Since asymmetric systems cannot be expected to form the best image when the object and image planes are normal to the associated base ray segments, the significance of object and image tilts, to second order, is also considered. With these results, it is possible to determine constraints on a system’s configuration that ensure a given set of second-order imaging properties. As an illustrative example, constraints on the configuration of a single interface—either refracting or reflecting—are determined when sharp, second-order imagery is re-quired. It is shown that the only nontrivial solutions are spherical refracting surfaces, and, for a given spherical surface, the basal object and image points must correspond to aplanatic points associated with the sphere.