Modelling and optimization of a deformable mirror for laser beam control

Abstract

An intra-cavity adaptive mirror is required to compensate for time-dependent phase aberrations to the laser beam, such as those caused by thermal lensing. A piezoelectric unimorph design can provide a small, low-cost deformable mirror for this application. The unimorph consists of a metallic disc, with a mirror finish, bonded to a piezoelectric disc. In adaptive optics the deformations that the mirror is required to perform are described by the Zernike polynomials, which are a complete set of orthogonal functions. The challenge is to design a device that can represent selected polynomials as accurately as possible with specified amplitude. Numerical modelling is required to predict the deformation shapes that can be achieved by a unimorph mirror with a particular electrode pattern. The results from a Rayleigh-Ritz model and a finite element model employing elements including rotational degrees of freedom were compared to results from a conventional finite element model. The Rayleigh-Ritz model, which used 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. While this method requires some effort to implement and is not very flexible, it does provide insight into the operation of the deformable mirror and can be used to optimize the design in an elegant manner. The finite element model including rotational degrees of freedom is more efficient than the conventional finite element model but retains the flexibility of this model. This method was applied to model a prototype deformable mirror and produced good agreement with experimental results.