Theory of multidither adaptive optical systems operating with zonal control of deformable mirrors

A theory is developed for multidither adaptive optical systems that employ N discrete actuators driving a deformable control and dither mirror with the same shape of influence function. The major emphasis is on mirrors with a Gaussian influence function. It is shown, as a special case, that single-actuator displacements or errors of such systems yield error signals that can be described in closed form, via sine integral functions of the actuator displacement. Closed-form expressions for both the first- and second-harmonic content in the dither signal outputs are developed. Expressions for servo cross coupling (from adjacent actuator deformations) are developed and it is shown how the coupling is typically larger than the mechanical (actuator center) coupling. Effective modulation index comparisons are established between Gaussian and piston mirrors, for piston and Gaussian phasing error components. It is shown how a potential secondary maxima lock-up condition (the 2nπ problem) can occur when operating hill-climbing adaptive optical systems with deformable mirrors. In some cases a major loss in system Strehl ratio may result.