A mathematical framework for nonlinear wavefront reconstruction in adaptive optics systems with Fourier-type wavefront sensing

Abstract

Abstract Advanced adaptive optics (AO) instruments have applications in ophthalmic imaging, free-space optical communications and the future generation of extremely large telescopes. These AO systems are designed to perform real-time corrections of dynamic wavefront aberrations. The corrections can be performed by converting wavefront measurements into deformable mirror (DM) actuator commands. The role of the DM is to mitigate aberrations by restoring a planar wavefront. Optimal DM actuator commands therefore require precise phase measurements across the entire wavefront. Reconstructing a wavefront from wavefront sensor (WFS) data is an inverse problem that depends on the type of WFS implemented. Nonlinear Fourier-type WFSs are included in the design of many current and upcoming AO systems. Conventionally, these sensors perform AO control based on simplifications and linearisations of the underlying models. However, in nonlinear regimes, approximation errors critically degrade image quality. This study looks at overcoming nonlinear wavefront sensing regimes by introducing a nonlinear, iterative algorithm for Fourier-type wavefront reconstruction. The algorithm used is well-known in the field of inverse problems. The underlying mathematical theory for modelling Fourier-type WFSs is provided, along with how these models can be used to perform nonlinear wavefront reconstruction. A significant advantage of the analysis presented is its generalised applicability to any Fourier-type sensor. The only input required is the mathematical expression for the optical element transfer function. The generalised and full mathematical model of Fourier-type WFSs is introduced in a Sobolev space setting. Necessary inputs are derived for the nonlinear iterative algorithms, such as Fréchet derivatives and adjoints. The generalised theory is then expanded to solve the inverse problem of wavefront reconstruction for all Fourier-type WFSs. Moreover, the study concentrates on the pyramid WFS (PWFS)—one of the most well-known Fourier-type WFSs—and shows a Hilbert transform representation of the amplitude of the incoming light on its detector. The developed theory is demonstrated using a simulated PWFS to measure an example wavefront.