Analytic inverse solutions for Risley prisms in four different configurations for positing and tracking systems

The conventional ray deviation formula and analytical inverse solution for Risley prisms are traditionally derived by the first-order paraxial method, which is simplified and does not provide sufficient accuracy. When a light ray enters the wedge prism, the component that is perpendicular to the base side is deviated by the wedge prism and the component parallel to the base side is uninfluenced. In this study, the problem is presented and analyzed from the scalar form of Snell's law, and 2D vector algebra is used to solve the problem. The rotation and refracting angles of a single wedge prism are obtained by analyzing the propagation through the prism of incident rays from different directions, and ray deviation formulas are derived. Four configurations are described with ray deviation formulas and analytical inverse solutions deduced from the corresponding formulas for a single wedge prism. In the final section, two examples are considered for the four configurations. The numeric results demonstrate the accuracy of the inverse solution provided by the proposed method. (C) 2018 Optical Society of America.