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Extremely Large Telescopes have overwhelmingly opted for the Pyramidwavefront sensor (PyWFS) over the more widely used Shack-Hartmann WaveFrontSensor (SHWFS) to perform their Single Conjugate Adaptive Optics (SCAO) mode.The PyWFS, a sensor based on Fourier filtering, has proven to be highlysuccessful in many astronomy applications. However, it exhibits non-linearitybehaviors that lead to a reduction of its sensitivity when working withnon-zero residual wavefronts. This so-called Optical Gains (OG) effect,degrades the close loop performance of SCAO systems and prevents accuratecorrection of Non-Common Path Aberrations (NCPA). In this paper, we aim atcomputing the OG using a fast and agile strategy in order to control the PyWFSmeasurements in adaptive optics closed loop systems. Using a novel theoreticaldescription of the PyFWS, which is based on a convolutional model, we are ableto analytically predict the behavior of the PyWFS in closed-loop operation.This model enables us to explore the impact of residual wavefront error onparticular aspects such as sensitivity and associated OG. The proposed methodrelies on the knowledge of the residual wavefront statistics and enablesautomatic estimation of the current OG. End-to-End numerical simulations areused to validate our predictions and test the relevance of our approach. Wedemonstrate, using on non-invasive strategy, that our method provides anaccurate estimation of the OG. The model itself only requires AO telemetry datato derive statistical information on atmospheric turbulence. Furthermore, weshow that by only using an estimation of the current Fried parameter r_0 andthe basic system-level characteristics, OGs can be estimated with an accuracyof less than 10%. Finally, we highlight the importance of OG estimation in thecase of NCPA compensation. The proposed method is applied to the PyWFS.

Pyramid wavefront sensor optical gains compensation using a convolutional model