Abstract
Context. The deployment of meter-scale (hitherto pre-focal) adaptive deformable mirrors finds some prominent examples in the leading ground-based visible to near-infrared facilities (e.g. the Very Large Telescope (VLT), the Large Binocular Telescope (LBT), or the Magellan Telescope) and is being adopted by several others (e.g. the Multiple Mirror Telescope (MMT) or Subaru). Furthermore, two out of the three giant segmented-mirror telescopes now under design will feature them. In all these cases, the proprietary technology is based on voice-coils and is limited in force, stroke, and velocity. Aims. Because of the nature of their purpose, that is, adaptive wave-front correction, any kind of optimality relies on the control of a subset of principal wave-front components or eigenmodes, for short, a basis of functions in a mathematical sense. Here we provide algorithmic procedures for generating such eigenbases, also called Karhunen–Loève (KL) modes, that integrate force limitations in their definitions whilst maintaining standard orthonormality, statistical independence, and deformable mirror span. Methods. The double-diagonalisation method was revisited to build KL modes ranked by the force applied on the actuators. Results. We analysed this new KL basis for von Kármán turbulence statistics and present the fitting error and the distribution of positions and forces. We further illustrate their use in the case of the quaternary mirror control for the European Extremely Large Telescope, and we include the outer actuator minioning and force policy constraints. © The Authors 2024.

Abstract
Capturing high-resolution imagery of the Earth's surface often calls for a telescope of considerable size, even from low Earth orbits (LEOs). A large aperture often requires large and expensive platforms. For instance, achieving a resolution of 1 m at visible wavelengths from LEO typically requires an aperture diameter of at least 30 cm. Additionally, ensuring high revisit times often prompts the use of multiple satellites. In light of these challenges, a small, segmented, deployable CubeSat telescope was recently proposed creating the additional need of phasing the telescope's mirrors. Phasing methods on compact platforms are constrained by the limited volume and power available, excluding solutions that rely on dedicated hardware or demand substantial computational resources. Neural networks (NNs) are known for their computationally efficient inference and reduced onboard requirements. Therefore, we developed a NN-based method to measure co-phasing errors inherent to a deployable telescope. The proposed technique demonstrates its ability to detect phasing errors at the targeted performance level [typically a wavefront error (WFE) below 15 nm RMS for a visible imager operating at the diffraction limit] using a point source. The robustness of the NN method is verified in presence of high-order aberrations or noise and the results are compared against existing state-of-the-art techniques. The developed NN model ensures its feasibility and provides arealistic pathway towards achieving diffraction-limited images. (c) 2024 Optica Publishing Group