Inverse problem approach in Extreme Adaptive Optics: analytical model of the fitting error and lowering of the aliasing

We present the results obtained with an end-to-end simulator of an Extreme Adaptive Optics (XAO) system control loop. It is used to predict its on-sky performances and to optimise the AO loop algorithms. It was first used to validate a novel analytical model of the fitting error, a limit due to the Deformable Mirror (DM) shape. Standard analytical models assume a sharp correction under the DM cutoff frequency, disregarding the transition between the AO corrected and turbulence dominated domains. Our model account for the influence function shape in this smooth transition. Then, it is well-known that Shack-Hartmann wavefront sensors (SH-WFS) have a limited spatial bandwidth, the high frequencies of the wavefront being seen as low frequencies. We show that this aliasing error can be partially compensated (both in terms of Strehl ratio and contrast) by adding priors on the turbulence statistics in the framework of an inverse problem approach. This represents an alternative to the standard additional optical filter used in XAO systems. In parallel to this numerical work, a bench was aligned to experimentally test the AO system and these new algorithms comprising a DM192 ALPAO deformable mirror and a 15x15 SH-WFS. We present the predicted performances of the AO loop based on end-to-end simulations.