Analytical Optimization of Tetrahedral Mirror Corners as Geometrical Reference for High-Precision Nanopositioning Systems

To meet rising requirements of integrated circuit fabrication for positioning accuracy, resolution, repeatability, and efficiency, nanopositioning and nanomeasuring machines must achieve tighter tolerances for their critical components and improved dynamic performance of moving elements. The mirror stage, as an interferometrically tracked geometric reference for stage motion, plays a central role in this context. Since its volume directly affects the system’s dynamic behavior, minimizing mirror size while maintaining measurement capability is crucial. This article proposes an analytical approach to optimize the design of a tetrahedral mirror stage dependent on an arbitrary measuring volume. Therefore, we conducted a comparative analysis of three geometries—the regular tetrahedron, the cube corner, and a general tetrahedron—with the objective of reducing mirror volume. The results show that depending on the given measurement volume, the optimized general tetrahedron enables significant reductions in mirror volume compared to both the regular tetrahedron and the cube corner for single-beam as well as and multi-beam interferometer configurations.