Tip–tilt error for extremely large segmented telescopes: detailed theoretical point-spread-function analysis and numerical simulation results

We present an analysis of point-spread functions for segmented mirrors affected by random tip-tilt errors on each segment. In addition to Strehl ratio evaluation, this analysis considers key characteristics such as the intensity and the location of speckles and secondary peaks and the relative energy distribution between these features. We develop a method to describe the shape of a nonaveraged point-spread function and deduce the final expressions for ensemble-averaged characteristics. Based on Keck-type hexagonal segmentation geometry, our study is extended to an arbitrary number of segments, and we describe qualitatively the transition from the case of a mirror with few segments to that of a mirror with several hundred segments--prototype of the next generation of Extremely Large Telescopes.