We investigate the focal plane wavefront sensing technique, known as Phase Diversity, at the scientific focal plane of a segmented mirror telescope with an adaptive optics (AO) system. We specifically consider an optical system imaging a point source in the context of (i) an artificial source within the telescope structure and (ii) from AO-corrected images of a bright star. From our simulations, we reliably disentangle segmented telescope phasing errors from non-common path aberrations (NCPA) for both a theoretical source and on-sky, AO-corrected images where we have simulated the Keck/NIRC2 system. This quantification from on-sky images is appealing, as its sensitive to the cumulative wavefront perturbations of the entire optical train; disentanglement of phasing errors and NCPA is therefore critical, where any potential correction to the primary mirror from an estimate must contain minimal NCPA contributions. Our estimates require a one-minute sequence of short-exposure, AO-corrected images; by exploiting a slight modification to the AO-loop, we find that 75 defocused images produces reliable estimates. We demonstrate a correction from our estimates to the primary and deformable mirror results in a wavefront error reduction of up to 67% and 65% for phasing errors and NCPA, respectively. If the segment phasing errors on the Keck primary are on the order of ~130 nm RMS, we show we can improve the H-band Strehl ratio by up to 10% by using our algorithm. We conclude our technique works well to estimate NCPA alone from on-sky images, suggesting it is a promising method for any AO-system.
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