Numerical modelling of a thin deformable mirror for laser beam control

Abstract

For intra-cavity laser beam control, a small, low-cost deformable mirror is required. This mirror can be used to correct for time- dependent phase aberrations to the laser beam, such as those caused by thermal expansion of materials. A piezoelectric unimorph design is suitable for this application. The proposed unimorph consists of a copper disc with mirror finish, bonded to a piezoelectric disc. The deformations that the mirror is required to perform are routinely (at least in optical applications) described using Zernike polynomials, which are a complete set of orthogonal functions defined on a unit disc. The challenge is to design a device that can represent selected polynomials as accurately as possible with a specified amplitude. To assist in the design process, numerical modelling is required to predict the deformation shapes that can be achieved by a unimorph mirror with a particular electrode pattern. In this paper a previously proposed axisymmetric Rayleigh-Ritz formulation, is extended to account for non-axisymmetric voltage distributions, and therefore non-axisymmetric displacements. The Rayleigh-Ritz model, which uses the Zernike polynomials directly to describe the displacements, produced a small model (stiffness matrix dimension equal to the number of polynomials used) that predicts the deformations of the piezoelectric mirror with remarkable accuracy. The results using this Rayleigh-Ritz formulation are compared to results from a traditional finite element analysis using a commercial finite element package. Both numerical models were applied to model a prototype deformable mirror and produced good agreement with experimental results.