Application of derivative matrices of skew rays to direct and inverse problems of Risley and tilting orthogonal double-prism systems

Risley systems and tilting orthogonal double-prism systems are optical systems consisting of two prisms in series. The analysis of such systems typically involves both direct and inverse problems. Problems of the former type can be easily solved using ray-tracing equations. However, inverse problems are comparatively more difficult if the search direction is not properly given prior to iteration. In the present study, the two optical systems are modeled using the homogeneous coordinate notation. A method is then proposed for solving the direct and inverse problems of both optical systems using a ray-tracing approach and the first- and second-order derivative matrices of the skew rays. In addition, four optimization methods based on the two derivative matrices are proposed for determining the search direction in the inverse problem. Eight illustrative examples are given. It is shown that the proposed method can not only determine the scan patterns and sensitivity coefficients in the direct problem, but also determine the search direction in the inverse problem.