Estimating differential pistons for the Extremely Large Telescope using focal plane imaging and a residual network

TL;DR

This paper tackles the challenge of measuring differential piston (petaling) in the ELT's M4 by estimating the piston difference between petals from images of a 2×2 Shack-Hartmann WFS. A ResNet-based neural network is trained on extensive simulated data to map sub-PSF intensity patterns to differential piston values, with careful handling of the 2π ambiguity and a focus on realistic observing conditions, including independent and correlated turbulence, polychromatism, and detector noise. The results show that temporal averaging significantly improves RMSE, polychromatism degrades performance by less than 5%, and detector noise is not the limiting factor; the method remains robust under training/inference mismatches and can generalize to other focal-plane PSFs like LIFT. Practically, the approach provides a viable, data-driven avenue to measure and potentially correct petaling in ELT-scale AO systems, with implications for achieving near-diffraction-limited imaging and guiding future instrument designs.

Abstract

As the Extremely Large Telescope (ELT) approaches operational status, optimising its imaging performance is critical. A differential piston, arising from either the adaptive optics (AO) control loop, thermomechanical effects, or other sources, significantly degrades the image quality and is detrimental to the telescope's overall performance. In a numerical simulation set-up, we propose a method for estimating the differential piston between the petals of the ELT's M4 mirror using images from a 2x2 Shack-Hartmann wavefront sensor (SH-WFS), commonly used in the ELT's tomographic AO mode. We aim to identify the limitations of this approach by evaluating its sensitivity to various observing conditions and sources of noise. Using a deep learning model based on a ResNet architecture, we trained a neural network (NN) on simulated datasets to estimate the differential piston. We assessed the robustness of the method under various conditions, including variations in Strehl ratio, polychromaticity, and detector noise. The performance was quantified using the root mean square error (RMSE) of the estimated differential piston aberration. This method demonstrates the ability to extract differential piston information from 2x2 SH-WFS images. Temporal averaging of frames makes the differential piston signal emerge from the turbulence-induced speckle field and leads to a significant improvement in the RMSE calculation. As expected, better seeing conditions result in improved accuracy. Polychromaticity only degrades the performance by less than 5% compared to the monochromatic case. In a realistic scenario, detector noise is not a limiting factor, as the primary limitation rather arises from the need for sufficient speckle averaging. The network was also shown to be applicable to input images other than the 2x2 SH-WFS data.

Figures (16)

• Figure 1: (a) Initial pupil configuration. The petals are numbered using the ESO convention. (b) Pupil configuration for a pupil rotation angle $\theta = 15\degree$. (c) and (d) Respective resulting sub-PSFs as measured by the $2\times2$ SH-WFS with atmospheric turbulence residuals.
• Figure 2: Outliers and range definition. Top: Residual differential piston as a function of the true differential piston for single frame SP10 configuration. The blue points represents the inliers, while the red points are the outliers. Bottom: Histogram of the outliers distribution.
• Figure 3: Residual blocks used in the ResNet. Left: A-block type, with the skip layer being a $1\times1$ convolution. Right: B-block type, with the skip layer being the identity.
• Figure 4: ResNet architecture. The number of filters is shown to the right of each convolutional layer.
• Figure 5: Influence of the number of averaged frames on the differential piston RMSE. Differential piston RMSE versus number of averaged images. Results are presented for the three turbulence configurations SP10, SP25, and SP50 listed in Table \ref{['tab:datasets']} in a monochromatic uncorrelated regime.