Adaptive optics (AO) systems are essential for compensating atmospheric and optical path aberrations in applications such as high-resolution astronomical imaging, retinal diagnostics, and free-space optical communication (FSOC). By using wavefront sensor (WFS) measurements to control deformable mirrors (DMs) in real-time, AO systems enable substantial improvements in image quality and optical performance.
In this work, we investigate the recently introduced iQuad WFS, which employs a four-quadrant phase mask in the focal plane. We develop a rigorous mathematical framework for the iQuad sensor, beginning with the derivation of nonlinear forward models. Using Fréchet-derivative linearizations, our analysis reveals a close connection between the linearized iQuad operator and the 2d finite Hilbert transform. It is also shown that the linear iQuad operator is self-adjoint - an unusual and advantageous property for model-based wavefront reconstruction algorithms. We further introduce the double iQuad, a dual-path configuration with mutually rotated sensors that mitigates poorly observed phase modes while also yielding an analytically simpler formulation.
Numerical simulations for telescope-scale AO systems illustrate the performance of the proposed wavefront sensing strategies.