Optical fractional topological spatial differentiation

Optical spatial differentiations hold the capability of ultrahigh speed and low power consumption, providing an efficient solution for edge detection, feature extraction, and pattern recognition. However, existing optical spatial differentiators predominantly focus on integer-order operations, and fractional spatial differentiation remains challenging. In this work, we demonstrate fractional topological spatial differentiation through complex amplitude modulation in Fourier space. We derive and reveal the transfer function for fractional spatial differentiation, characterized by fractional power amplitude and fractional spiral phase. This distinctive feature is validated theoretically and experimentally in both spatial and frequency domains, and the underlying mechanism driving fractional differential imaging is elucidated. We apply fractional spatial differentiation to image texture and edge detection, resulting in enhanced texture features and dynamic edge evolution. This proof-of-principle study introduces a paradigm for optical fractional spatial differentiators, with potential applications in artificial intelligence, machine learning, and biomedical imaging.