Description of continuous surface deformable mirrors by using orthogonal polynomials

By taking a piezoelectric Deformable Mirror (PDM) and a Micromachined Membrane Deformable Mirror (MMDM) as examples, a method to accurately characterize the surface of continuous surface deformable mirrors was discussed. The description characteristics of Zernike polynomial and Q polynomial methods using different fitting parameters were compared and the influence of higher polynomial terms on fitting accuracy was researched. By activating one or more actuators, the two groups of five surface data were obtained for PDM and MMDM respectively. Then, the number of sampling points and the number of polynomial orders were set as variables. The ten surface data were fitted using the least square method based on five types of sampling grids; either the Zernike polynomial and Q polynomial were adopted. The experimental results show that the edge-clustered sampling grid outperforms the four uniform grids in terms of fitting accuracy. The Q-polynomial fits produce smaller fitting residuals than Zernike fits when the polynomial order goes beyond 40. Moreover, the RMS (Root-Mean-Square) of fitting residuals stays in the 1×10^-5 mm order of magnitude when the polynomial order increases from 10 to 80, while the residuals away from the aperture edge decrease significantly. The above results can guide the choice of fitting parameters in deformable mirror applications, describe the surface characteristics of deformable mirrors and improve wavefront control accuracy.