Wavefront reconstruction for ELT-sized telescopes with pyramid wavefront sensors

The new generation of ground-based telescopes relies on real-time adaptive optics systems to compensate for atmospheric perturbations arising during the imaging process. Pyramid wavefront sensors are planned to be part of many instruments currently under development for ELT-sized telescopes. The high number of correcting elements to be controlled in real-time and the segmented pupils of the ELTs lead to unprecedented challenges posed to the control algorithms. Based on various (approximate) models, several algorithms were developed in the last decades for linear and non-linear wavefront correction from pyramid sensor data. Among those, we emphasize interaction-matrix-based approaches, Fourier domain methods, iterative algorithms, and algorithms based on the inversion of the Finite Hilbert transform. We briefly present the core ideas of the algorithms and provide the necessary theoretical background like, e.g., the Fourier domain filters, or the direct inversion formulas. We give a detailed comparison of the presented methods with respect to underlying pyramid sensor models, the computational complexities, and reconstruction qualities. The performance of our algorithms is demonstrated in the context of an XAO system on the EPICS instrument and a SCAO system on the METIS instrument on the ELT. In the simulations, realistic features as the ELT spiders and the hexagonal M4 geometry are partially taken into account.