Super-resolution wavefront reconstruction

Abstract

Context. Cutting-edge, ground-based astronomical instruments are fed by adaptive optics (AO) systems that are aimed at providing high performance down to the visible wavelength domain on 10 m class telescopes and in the near infrared for the first generation instruments of Extremely Large Telescopes (ELTs). Both applications lead to a large ratio between the telescope diameter, D, and the coherence length or Fried parameter, r0, that is D/r0. As the parameter that defines the required number of degrees of freedom of the AO system, D/r0 drives the requirement to reconstruct the incoming wavefront with ever-higher spatial resolution. In this context, super-resolution (SR) appears as a potential game changer. Indeed, SR promises to dramatically expand the range of spatial frequencies that can be reconstructed from a set of lower resolution measurements of the wavefront. Aims. As a technique that seeks to upscale the resolution of a set of measured signals, SR retrieves higher-frequency signal content by combining multiple lower resolution sampled data sets. It is well known both in the temporal and spatial domains and widely used in imaging to reduce aliasing and enhance the resolution of coarsely sampled images. This study applies the SR technique to the bidimensional wavefront reconstruction. In particular, we show how SR is intrinsically suited for tomographic multi-wavefront sensor (WFS) AO systems, revealing many of its advantages with minimal design effort. Methods. We provide a direct space and Fourier optics description of the wavefront sensing operation and we demonstrate how SR can be exploited through signal reconstruction, especially within the framework of periodic non-uniform sampling. We investigate both meta-uniform and non-uniform sampling schemes and we show that under some conditions, both sampling schemes enable a perfect reconstruction of band-limited signals. We also provide a SR bi-dimensional model for a Shack-Hartmann (SH) WFS, along with an analysis of the characteristics of the sensitivity function. We validated the SR concept with numerical simulations of representative multi-WFS SH AO systems. Finally, we explored the extension of the method to pyramid WFSs. Results. Our results show that combining several WFS samples in a SR framework grants access to a greater number of modes than the native one offered by a single WFS (despite the fixed sub-aperture size across samples). We show that the wavefront reconstruction achieved with four WFSs can be equivalent to a single WFS providing a sampling resolution that is twice greater (linear across the telescope aperture). We also show that the associated noise propagation is not degraded under SR. Finally, we show that the concept can be extended to the signal produced by single pyramid WFS, with its four re-imaged pupils serving as multiple non-redundant samples. Conclusions. We find that SR applied to wavefront sensing and reconstruction (WFR) offers a new parameter space to explore, as it decouples the size of the sub-aperture from the desired wavefront sampling resolution. By shifting away from outdated assumptions, new and more flexible, better-performing AO designs have now become possible.