Linearized Zernike-based phasing for segmented mirrors: application to the ELT

Achieving diffraction-limited performance with the Extremely Large Telescope (ELT) requires phasing the 798 segments of its primary mirror with tens of nanometer precision. This task is carried out by the phasing and diagnostic station (PDS), which is composed of multiple sensors, including two wavefront sensors dedicated to segment phasing: SHAPS, a modified Shack-Hartmann sensor operated in the visible (R band), and ZEUS, a Zernike phase sensor working in the near-infrared (J, H, and K bands). ZEUS complements SHAPS by validating phasing performance at longer wavelengths and by providing a fallback solution when SHAPS becomes unreliable, for example, in the presence of pupil misalignments. Unlike Shack-Hartmann-type sensors, ZEUS is tolerant to pupil motion on the detector. In this work, we develop a calibration technique and data-reduction algorithms that extract signals varying linearly with segment-edge phase discontinuities. This key property enables a linear phasing approach based on interaction and command matrices, avoiding signal fitting and iterative reconstruction schemes. We validate the method through numerical simulations of the ELT and through experimental measurements obtained on the MELT test bench. The results demonstrate accurate retrieval of edge phase jumps and segment piston values. This work establishes Zernike-based phasing as a linear and robust solution for the phasing of highly segmented mirrors.