Conformal mapping is an important mathematical tool that can be used to solve various physical and engineering problems in many fields, including electrostatics, fluid mechanics, classical mechanics, and transformation optics. It is an accurate and convenient way to solve problems involving two terminals. However, when faced with problems involving three or more terminals, which are more common in practical applications, existing conformal mapping methods apply assumptions or approximations. A general exact method does not exist for a structure with an arbitrary number of terminals. This study presents a conformal mapping method for multiple terminals. Through an accurate analysis of boundary conditions, additional terminals or boundaries are folded into the inner part of a mapped region. The method is applied to several typical situations, and the calculation process is described for two examples of an electrostatic actuator with three electrodes and of a light beam splitter with three ports. Compared with previously reported results, the solutions for the two examples based on our method are more precise and general. The proposed method is helpful in promoting the application of conformal mapping in analysis of practical problems.

A pseudo-three-layer electrostatic repulsive out-of-plane actuator is proposed. It combines the advantages of two-layer and three-layer repulsive actuators, i.e., fabrication requirements and fill factor. A theoretical model for the proposed actuator is developed and solved through the numerical calculation of Schwarz-Christoffel mapping. Theoretical and simulated results show that the pseudo-three-layer actuator offers higher performance than the two-layer and three-layer actuators with regard to the two most important characteristics of actuators, namely, driving force and theoretical stroke. Given that the pseudo-three-layer actuator structure is compatible with both the parallel-plate actuators and these two types of repulsive actuators, a 19-element two-layer repulsive actuated deformable mirror is operated in pseudo-three-layer electrical connection mode. Theoretical and experimental results demonstrate that the pseudo-three-layer mode produces a larger displacement of 0–4.5 μm for a dc driving voltage of 0–100 V, when compared with that in two-layer mode.