Modeling of deformable MEMS mirrors in modern telescopes

Ground based modern telescopes with AO (Adaptive Optics) are at par with the space borne telescopes in providing unprecedented quality images. Recently, deformable mirrors using MEMS (Micro-Electro-Mechanical Systems) have become a popular choice for adaptive mirrors due to various advantages. The continuous facesheet of the MEMS mirror can be modeled with the help of theory developed for thin plates [1], [2] and [3]. In this paper we discuss the modeling techniques using energy principles and variational methods [4] and [5]. For modeling and simulations we will follow the specifications of a commercially available 144 actuator continuous facesheet deformable MEMS mirror by Boston Micromachines Corporation [6]. The dynamics of this mirror is very fast and hence is neglected when compared to the rate of corrections to be applied and it is assumed that the boundaries of the mirror are simply supported. Thus our problem simplifies to that of a simply supported thin plates static under equilibrium condition. The MEMS mirror equation under the influence of point load matrix is obtained using superposition principle and Navier solution method is used for solving the deformation matrix for a given force matrix. In the case of an AO system, first the atmospheric wavefront is measured which then gives the desired shape of the mirror. Hence the deformation matrix is known and it is required to derive the force matrix, which essentially means solving the inverse problem. If the measured wavefront has noise which is normally the case, or the transformation matrix is rank deficient, the inverse problem becomes ill-posed [7] and [8].