Design of Focusing Catadioptric Systems using Differential Geometry

Abstract

In recent years catadioptric systems, consisting of lenses and mirrors, have gained increasing popularity for the task of environmental perception. However, focusing of such systems is a common problem as it is often not considered during the design process of the optical system. This paper presents a novel approach to address focus in the design of optics with rotational symmetry. The approach does not only adress the construction of catadioptric systems but can also be used to calculate conventional optics. The approach is based on the calculation of the first order approximation of meridional and sagittal focus using differential geometry. Additional conditions like a single-viewpoint can be considered as well. The derived equations are combined to a set of ordinary differential equations that is used to calculate the shape of the optical system via numerical integration. The design concept has been verified by multi-chromatic ray tracing simulations.