# Iterative wave-front reconstruction in the Fourier domain

## TL;DR

This paper introduces an iterative Fourier domain wave-front reconstruction method using a modified Gerchberg routine, improving accuracy and noise performance for adaptive optics systems in large telescopes.

## Contribution

It presents a novel iterative routine that adapts circular wave-front sensor data for Fourier analysis, compatible with existing reconstruction methods, enhancing performance.

## Key findings

- Increased Strehl ratio from 95.4% to 96.9% in K-band
- Reduced noise propagation in simulations
- Avoids high spatial frequency errors

## Abstract

The use of Fourier methods in wave-front reconstruction can significantly reduce the computation time for large telescopes with a high number of degrees of freedom. However, Fourier algorithms for discrete data require a rectangular data set which conform to specific boundary requirements, whereas wave-front sensor data is typically defined over a circular domain (the telescope pupil). Here we present an iterative Gerchberg routine modified for the purposes of discrete wave-front reconstruction which adapts the measurement data (wave-front sensor slopes) for Fourier analysis, fulfilling the requirements of the Fast Fourier Transform (FFT) and providing accurate reconstruction. The routine is used in the adaptation step only and can be coupled to any other Wiener-like or least-squares method. We compare simulations using this method with previous Fourier methods and show an increase in performance in terms of Strehl ratio and a reduction in noise propagation for a 40x40 SPHERE-like adaptive optics system. For closed loop operation with minimal iterations the Gerchberg method provides an improvement in Strehl, from 95.4% to 96.9% in K-band. This corresponds to ~40 nm improvement in rms, and avoids the high spatial frequency errors present in other methods, providing an increase in contrast towards the edge of the correctable band.