A novel design and selection scheme for surface-parallel actuators for ultrathin, lightweight mirrors is presented. The actuation system consists of electrodes printed on a continuous layer of piezoelectric material bonded to an optical-quality substrate. The electrodes provide almost full coverage of the piezoelectric layer, in order to maximize the amount of active material that is available for actuation, and their shape is optimized to maximize the correctability and stroke of the mirror for a chosen number of independent actuators and for a dominant imperfection mode. The starting point for the design of the electrodes is the observation that the correction of a figure error that has at least two planes of mirror symmetry is optimally done with twin actuators that have the same optimized shape but are rotated through a suitable angle. Additional sets of optimized twin actuators are defined by considering the intersection between the twin actuators, and hence an arbitrarily fine actuation pattern can be generated. It is shown that this approach leads to actuator systems with better performance than simple, geometrically based actuators. Several actuator patterns to correct third-order astigmatism aberrations are presented, and an experimental demonstration of a 41-actuator mirror is also presented
