Configuring an electrostatic membrane mirror by least-squares fitting with analytically derived influence functions

Abstract

Electrostatic membrane mirrors are potentially useful as aberration generators. To realize this capability, it is necessary to know the influence functions (each giving mirror response to a single electrode) and then to use a fitting procedure to determine the optimal voltage settings for reproducing a desired surface shape. We show that an approximate analytical solution of Poisson’s equation exists that can generate the influence functions for a circular electrostatic mirror. We also show, by demonstration, that it is computationally feasible to calculate these influence functions and to use them with a fitting procedure to fit the surface of a 109-electrode mirror to a desired shape. Our methods allow one to test the theoretical performance of the 109-electrode mirror; we find that good fits are obtainable for Zernike polynomials of up to degree 6.