Neural network algorithm for under-sampled wavefront reconstruction: mathematical analysis and implementation.

The Shack-Hartmann wavefront sensor (SHWFS) is critical in adaptive optics (AO) for measuring wavefronts via centroid shifts in sub-apertures. Under extreme conditions like strong turbulence or long-distance transmission, wavefront information degrades significantly, leading to undersampled slope data and severely reduced reconstruction accuracy. Conventional algorithms struggle in these scenarios, and existing neural network approaches are not sufficiently advanced. To address this challenge, we propose a mathematically interpretable neural network-based wavefront reconstruction algorithm designed to mitigate the impact of slope loss. Experimental results demonstrate that our algorithm achieves what is believed to be unprecedented fidelity in full-aperture aberration reconstruction with up to 70% wavefront undersampling, representing a precision improvement of approximately 89.3% compared to modal methods. Moreover, the algorithm can be fully trained using simulation data alone, eliminating the need for real data acquisition and significantly enhancing practical applicability.