General framework for incoherent topological structured light and optical information encoding
Topology provides a powerful language for describing global invariants in physical systems, yet optical topology has been explored predominantly with fully coherent light. Recent studies have shown that incoherent light can host topological structures mediated by coherence singularities; however, a general framework for their construction and control has been lacking. Here, we introduce an incoherent Milnor polynomial, which establishes a theoretical framework for real-space incoherent topological structured light, in which topology and statistical coherence emerge as independent and jointly addressable degrees of freedom. This framework overcomes a fundamental limitation of coherent topological structured light, enabling arbitrary intensity engineering without altering the underlying topological configuration. Experimentally, we realize incoherent Hopf-linked and trefoil-knotted coherence singularities with programmable statistical coherence. We further demonstrate a robust optical information-encoding scheme inspired by Rubik's-cube-like rotations, where statistical coherence determines far-field intensity patterns associated with the cube's initial states, and topological structures govern controlled rotations acting as encryption keys. Our results advance incoherent topological structured light from a physical curiosity to a programmable photonic platform, opening new avenues for optical information encoding, statistical photonics, and coherence-engineered functionalities beyond coherent optical topology.