Accuracy Analysis for Shack–Hartmann Wave Front Sensing with Extended Sources

Abstract

Correlating the Shack–Hartmann wave front sensor (SH-WFS) with extended targets is widely used in solar adaptive optics systems. This paper aims to introduce a theoretical analysis that evaluates the accuracy of the SH-WFS on extended sources, with a specific focus on the implementation of the Normalized Cross-correlation (NCC) algorithm. To obtain an accurate error description, we utilized the calculation formula of the NCC algorithm to directly express the coordinates of the maximum value in the correlation function matrix. Furthermore, we determined the variance of the centroid position through the error transfer function, which quantifies the measurement error. In comparison with the previous findings of Michau et al., our result exhibits a coefficient disparity, specifically obtaining results 1.5 times higher than their work. The extensive solar granulation simulation and experimental results validate the theoretical error formulas. These error formulas can effectively estimate the accuracy of the SH-WFS, providing a theoretical foundation for the design of optical systems.