State feedback control with quadratic output for wavefront correction in adaptive optics

We propose an efficient aberration correction algorithm for systems with a linear state equation and a quadratic measurement based only on recorded images. We make a step further from the approaches that linearize the point spread function (PSF) and use a second order Taylor series expansion of the nonlinear PSF around zero aberration, assuming that the residual phase aberration is small. The Taylor expansion is performed for two arbitrary (large) diversities, in order to optimize the phase retrieval. We consider both the static and the dynamic cases, when the wavefront over the pupil does not change and when it varies in time, respectively. In the static case, the estimate is obtained either by solving a quadratic program or by using an extended Kalman filter (EKF) recursively. In the dynamic case, we use the EKF to estimate the state and a quadratic controller obtained from maximizing the intensity in the images to reduce the wavefront aberration.