Counter-acting image quality degradation caused by phase aberrations through physical correction requires the phase field to be known. As imaging hardware captures real-valued intensity, defined as the wave amplitude squared, obtaining this lost phase information is known as the phase retrieval problem and is a non-linear and non-convex optimisation problem.

Literature treats this problem from two points of view: reconstruction using indirect phase sampling, and reconstruction using Fourier amplitude sampling. The former employs wavefront sensors such as the Shack-Hartmann sensor, which encodes phase gradient information in the form of the displacement of imaged spots. These methods, while fast, discard information such as interference and are limited to low-order reconstruction. The latter, also called wavefront sensorless phase retrieval, uses the full point-spread function (PSF) to obtain high-accuracy reconstruction at the cost of computational speed, number of intensity images required, or limited aberration magnitude. These two points of view have remained largely separated, but can be made compatible through a modelling technique called Shack-Hartmann diversity. This thesis explores the merging of wavefront sensorless methods with Shack-Hartmann intensity patterns to leverage more information from a single captured image frame.

Firstly, a low-order modal reconstruction technique is presented applied to phase gradient fields obtained from Fourier demodulation of a Shack-Hartmann intensity pattern. A method of minimising the amount of redundant data-points used for reconstruction is illustrated through the removal of Fourier-interpolated data to speed up performance.

Secondly, a novel extension of the Fourier demodulation technique to hexagonal Shack-Hartmann arrays is presented, allowing phase gradient extraction and modal reconstruction of hexagonal array intensity images using Fourier demodulation.

Thirdly, Shack-Hartmann diversity is used to extend intensity-based modal phase retrieval using Taylor approximation of the intensity function to Shack-Hartmann intensity patterns. This bridges the gap between wavefront sensorless methods and Shack-Hartmann intensity patterns.

Lastly, a novel hybrid method is presented for high-accuracy phase reconstruction based on applying the above intensity-based method to a single Shack-Hartmann intensity pattern with low-order pre-conditioning obtained from Fourier demodulation. The method is demonstrated to obtain highly accurate reconstruction on low-order aberrations, and better reconstruction accuracy on small-magnitude high-order aberrations with dominating large-magnitude low-order terms than traditional methods. Potential use cases are discussed, such as open-loop turbulence reconstruction for use in turbulence modelling.