Pyramid wavefront sensor optical gains compensation using a convolutional model

Context. Extremely Large Telescopes have chosen the Pyramid wavefront sensor (PyWFS) over the more widely used Shack-Hartmann WaveFront Sensor (SHWFS) to perform their Single Conjugate Adaptive Optics (SCAO) mode. The PyWFS is a Fourier-filtering based sensor, which has proven to be strongly efficient for astronomical purposes. However, it shows non-linearity behaviors that lead to a reduction of its sensitivity when working around a non-null phase. This effect, called Optical Gains (OG), degrades the performance of the closed loop and prevents accurate correction of Non-Common-Path Aberrations (NCPA). Aims. We aim at computing these so-called OG with a fast and agile technique in order to control the PyWFS measurements for adaptive optics closed-loop systems. Methods. Thanks to a new theoretical description of the PyFWS, which uses a convolutional model to describe the sensor, we analytically predict the behavior of the PyWFS in closed-loop operation. This model allows us to explore the impact of residual phases on the properties of the PyFWS measurements in terms of sensitivity, and associated OG. The proposed method relies on the knowledge of the residual phase statistics and allows to automatically estimate the current OG. End-to-End numerical simulations are used to validate our predictions and test the relevance of our approach. Results. We show that an accurate estimation of the OGs is possible by only using the AO telemetry data to derive statistical information of the turbulence. The method is then fully non-invasive. We further show that by only having an estimation of the current Fried parameter r 0 and the basic system characteristics, OGs can be estimated within 10% accuracy. The proposed method applies to Pyramid WFS, but also to any Fourier-Filtering WFS suffering from OG variations.