Design of optical systems using derivatives of rays: Derivatives of variable vector of spherical boundary surfaces with respect to system variable vector

A computational scheme based on differential geometry was proposed in a previous study [Appl. Opt. 52, 4151 (2013)] for determining the first- and second-order derivative matrices of a skew ray reflected/ refracted at a flat boundary surface. The present study extends this methodology to the case of a skew ray reflected/refracted at a spherical boundary surface. The validity of the proposed approach is demonstrated using two retro-reflectors for illustration purposes. The results show that the proposed method provides an effective means of determining the search direction required to minimize the merit function during the optimization stage of the optical system design process.