Learning Superoscillatory Point Spread Functions

Abstract

Superoscillatory (SO) functions offer a promising pathway to sub-diffraction optical imaging that is far-field, single-shot, and label-free, requiring only a passive optical element. However, the design of SO point spread functions (PSFs) remains an open problem—existing design techniques are guided by PSF-space heuristics rather than by the quality of the resulting image, and current SO PSF parameterizations are not directly amenable to gradient-based optimization. Specifically, previous work creates SO PSFs using basis functions that are either not physically-realizable in optics, or do not reliably yield SO behavior when optimized. As such, prior work has relied on limited SO PSF designs to explore their potential for sub-diffraction imaging. In this work, we introduce a first-principles approach to designing SO PSFs computationally and the first learning-based method for optimizing their imaging performance. Our framework provides direct control over where superoscillations appear in the spatial domain while preserving an explicit analytic mapping to the Fourier-domain pupil. This enables the SO design space to be systematically characterized for the first time: SO constraints are enforced at every iteration, guaranteeing a valid SO solution, and each design corresponds to a physically realizable optical element. We also present an end-to-end optimization pipeline and evaluate our approach in simulation on grayscale objects and dense structured patterns at varying spatial frequencies. Our results show that our framework produces SO PSFs that resolve features at 3× below the diffraction limit on standard targets (Ronchi grating) and significantly outperform existing methods on widely used image quality metrics. Finally, we characterize the noise regimes in which SO PSF engineering outperforms diffraction-limited imaging. By providing a computational roadmap to SO design and optimization, we hope to encourage further exploration by the computational imaging community of superoscillations as a tool for passive sub-diffraction imaging.