Zernike–Galerkin method: efficient computational tool for elastically deformable optics

Abstract

We present the Zernike-Galerkin method, a tool for the discretization of partial differential equations (PDEs) on thin membranes in polar coordinates. The use of a truncated Zernike series as ansatz yields a semianalytical compact and parametric solution of the PDE. We demonstrate its use for the solution of the Poisson equation in polar coordinates, which is the equation of governing a thin strained membrane's deformation, or the flow of heat, both of which are important influences for the deformable membrane design. The obtained solution is directly expressed in terms of the components of the wavefront error, which highly facilitates the formulation of design questions. The method is computational highly efficient due to the sparsity and recursivity of the ansatz, is applicable to other PDEs, and can be efficiently combined with geometric optical and optimization methods. Its application to model a pressure-driven adaptive lens membrane is demonstrated.