Self-calibration of adaptive optics systems. Application to the THEMIS solar telescope
The interaction matrix models the effects of deformable mirror (DM) commands as perceived by the wavefront sensor (WFS) and is the cornerstone of the Adaptive Optics (AO) control. Difficulties to calibrate this matrix arise due to the number of degrees of freedom, the non-linearity, and temporal evolution of the AO system. An affine approximation of the unknown mapping between the DM commands and the WFS measurements is considered. Optimal estimators of the parameters of this affine model are obtained by fitting, in the weighted least squares sense, WFS data acquired with random probe commands sent to the DM. Several calibration methods are being considered depending on whether the model is directly fit to the WFS data or to the differences between successive WFS data. We derive closed-form expressions of the estimators of the components of the model. The proposed calibration methods can be applied under different conditions: before observing, on an internal source, or on-sky in open- or closed-loop. By introducing forgetting factors to reduce the weight of data as they age, we show that the model can be learned continuously using simple recurrence rules. The low computational complexity of these rules makes them suitable for real-time at the same frequency as the AO loop. We derive simple expressions for the mean squared errors (MSE) of the proposed estimators. We apply the proposed calibration methods to real telemetry data from the AO system of the THEMIS solar telescope. Our results show that the simple differences method is the method of choice: not only does it produce estimators with the least MSE, but it is also very simple to implement compared to the push-pull method which is widely used in AO systems.