Fundamental limits to phase and amplitude estimation in the high-Strehl regime

Ground-based telescopes are susceptible to seeing, an atmospheric blur that reduces the resolving power of large observatories to only a few arcseconds. Compensating these effects is critical to realizing the potential of both existing and upcoming extremely large telescopes, a challenging task that requires precise wavefront control. Ultimately, this precision is limited by the wavefront sensor (WFS) design and its inherent capacity to accurately encode phase and amplitude aberrations. In this work, we employ statistical estimation theory to derive fundamental limits to phase and amplitude reconstruction, providing a closed-form expression for the minimum-achievable residual error. For circular apertures, we find that this bound can be saturated by an instrument we refer to as the piston-adapted WFS (PAWS). The PAWS uses a Zernike mode sorter built from spatially varying half-waveplates to isolate the Zernike piston mode, apply a controllable phase shift to it, and then reconstitute the pupil as a pair of irradiance patterns that are nearly linear in incident aberrations. For arbitrary apertures, one can use a single-mode converter to reshape the pupil's native piston mode into Zernike piston and apply the same procedure. We expect our results to improve the residual wavefront errors in future closed-loop adaptive optics systems, while simultaneously finding applications in free-space communication and microscopy.