Knotted optical fields: From phase singularities to coherence singularities

Knotted optical fields represent a distinctive class of topological structured light, which can be constructed through closed, self-entangled field lines embedded in real three-dimensional space. Originating from phase singularities and optical vortices, the customized knots and links provide a powerful framework for realizing nontrivial topology in wave fields. This Review surveys recent advances in knotted optical fields, spanning coherent and incoherent regimes. We first outline the fundamental concepts of phase and coherence singularities, emphasizing their physical origins, mathematical definitions, detection, and applications. We then review the generation, robustness, measurement, and optimization of coherent knotted optical fields, from paraxial beams to tightly focused configurations. Moving beyond fully coherent beams, we highlight the emerging paradigm of incoherent knotted optical fields, where topology is embedded in a second-order statistical quantity: complex degree of coherence. By shifting the topological carrier from phase singularities to coherence singularities, incoherent topology offers enhanced robustness against environmental perturbations and introduces coherence as an additional controllable degree of freedom. Finally, we discuss future opportunities for coherence-enabled topology in multidimensional light control and practical applications.