Numerical analysis of an electrostatically formed membrane mirror

Abstract

Abstract The paper presents a numerical algorithm developed to determine the deformed shape of a membrane mirror controlled by electrostatic forces. Deformable mirrors are key components that are used in combination with wavefront sensors and real-time control systems in adaptive optics. The electrostatic membrane mirror concept implies using a thin conductive reflective membrane stretched over a solid flat frame and deformed electrostatically by applying control voltages to electrostatic actuators positioned behind the membrane. The proposed algorithm implies solving a coupled structural-electrostatic problem by using finite element and boundary element methods. Small deflections of a membrane are described by Poisson’s equation. The electric charge distribution over the membrane surface having a prescribed potential is governed by a Fredholm integral equation of the first kind. The coupled problem is solved iteratively, and a criterion for terminating iterations when searching for a steady-state solution is presented. The distinctive feature of this approach is that it allows us to take into account electrical edge effects typical for conducting thin-walled structures of very small thickness. Illustrative examples are provided to show the applicability and validity of the proposed method as well as its advantage over some existing techniques.