Estimating M4 Petal Modes in SCAO: Beyond Atmospheric Turbulence

In the context of the Single Conjugate Adaptive Optics (SCAO) modules such as the one of ArmazoNes high

Dispersion Echelle Spectrograph (ANDES) instrument for the ESO Extremely Large Telescope (ELT), the

presence of differential piston errors, so-called petal modes, between the six segments of the M4 deformable

mirror represents a significant challenge. Unlike atmospheric turbulence, which produces smooth and continuous

wavefronts, petal-like aberrations introduce sharp phase discontinuities that cannot be effectively captured by

standard phase reconstruction techniques. A naive approach based on averaging the phase over each segment

leads to incorrect estimation, as it primarily reflects atmospheric components, such as low-order modes like tilt

and trefoil, which can significantly alter the segment-averaged phase without involving any actual inter-segment

discontinuity. In this work, we explore and compare three alternative methods aimed at isolating and estimating

the petal mode content from reconstructed wavefront phases. The first is a minimum mean square error (MMSE)

estimator specifically designed to minimize cross-contamination from atmospheric modes. The second employs

a first-derivative-based filter that is blind to edge discontinuities across segment boundaries. The third is a

data-driven method based on machine learning, using a neural network trained to recognize petal patterns under

various conditions. We present the problem framework and the specifics of each approach, along with simulation

results that compare their accuracy, robustness, and sensitivity under realistic observing conditions. This work

provides insight into possible strategies for petal mode control in SCAO systems, with direct impact on the

stability and performance of ELT adaptive optics.