Adaptive optics simulation with mechanically motivated basis functions

For high accuracy simulation of Adaptive Optics (AO), multi-conjugate AO (MCAO), and ground layer AO (GLAO) analytic models have proven to be of significant importance. Usually, these models employ a finite set of Zernike basis functions that allow replacing point-by-point computation of phase maps by algebraic manipulation of basis function coefficients. For closed loop simulation of AO systems it is essential to consider the spatial and temporal dynamics of deformable mirrors and wavefront sensors. In this case, simulations with Zernike basis functions have several drawbacks. First of all, they become computationally intractable when high order and high frequency behavior is analyzed. Additionally, the spatial dynamics of deformable mirrors cannot be approximated well by Zernike functions when mechanical constraints are considered. In this paper, a set of orthogonal basis functions formed by spatial eigenmodes of deformable mirrors is proposed for simulation of large scale AO systems. It is shown that an analytic approximation of deformable mirror bending modes can be derived by solving a partial differential equation (PDE) and an inclusion of appropriate boundary conditions. Three sets of basis functions from different boundary conditions are studied in detail: the cases of a clamped edge, free edge, and flexible support of a circular mirror plate. The basis functions are compared to the Zernike functions and their mathematical properties are discussed.