An analytic model for the study of the impact of mirror segmentation on AO performance and application to a 30-m segmented telescope

Primary mirrors of the next generation of extremely large telescopes will be highly segmented. Since these telescopes will be equipped with adaptive optics (AO), it is very important to examine in details what are the consequences of different segmentation schemes on the delivered image quality after AO correction. We do so using our analytical AO simulation code PAOLA {Performance of Adaptive Optics for Large (or Little) Apertures}, upgraded to include AO correction of the primary mirror static aberrations. This study allows us to derive requirements on the geometry of the primary mirror, and the maximum acceptable segments positioning and figuring errors knowing that part of their amplitude will be corrected by the AO system offset. The first important issue is the influence of the segments size and gap width. These parameters have indeed a strong impact on the structure of the wings of the diffraction limited point spread function (PSF), but on the other hand, with the smooth AO residual halo superimposed onto it, the relative importance of the wing structures is decreased. To assess these effects, we consider the case of an AO system working in a near infrared classical mode on a 30-m segmented telescope and examine how the encircled energy radius and image contrast evolve with segment size and gap width. The second important issue is the effect of residual segment positioning & figuring errors after active optics and AO correction. Using appropriate metrics, we characterise the maximum acceptable segment positioning & figuring errors residuals in a classical AO mode, for one of the pupils studied in the first part of our work.