Knots and links, with their unique and robust topological structures, play a vital role across various disciplines, including physics, life sciences, and light–matter interactions. As a representative of topological configurations, structured light of knots and links provides new perspectives and degrees of freedom for advanced beam shaping and applications. Optical singularities—such as phase and polarization singularities—serve as essential building blocks for creating knot-like topological structures. Coherence singularities, a novel type of optical singularity, correspond to the zeros of statistical correlations in incoherent light fields. Their spatial distribution can be precisely controlled and is closely tied to the field's coherence properties. Coherence singularities have shown significant potential in applications such as secure optical communication and robust detection in turbulent environments. However, the construction of high-dimensional structured light fields—especially those with topological features—using coherence singularities remains largely unexplored.
In a new paper published in Light: Science & Applications, a team of scientists, led by Professor Chengliang Zhao from Soochow University, in collaboration with Professor Zhigang Chen from Nankai University and Professor Yangjian Cai from Shandong Normal University, introduced the concept of "incoherent links and knots"—three-dimensional topological knot structures formed by coherence singularities in incoherent light fields. Remarkably, despite the speckled and randomly fluctuating nature of the instantaneous electric field in incoherent light, those incoherent knots exhibit robust and stable topological configurations. This research, entitled "Topological links and knots of speckled light mediated by coherence singularities" was published in Light: Science & Applications.
As a branch of mathematics that investigates properties preserved under continuous deformations, the concept of topology has been widely applied in fields such as fluid dynamics, quantum field theory, and Bose–Einstein condensation. It has demonstrated unique advantages in ensuring the topological stability of geometric structures within physical systems, offering a solid foundation for high-fidelity communication and data storage. Among topological structures, links and knots—mathematically defined as closed curves embedded in three-dimensional space—are of particular significance. These intricate configurations, manifesting as interwoven loops or complex entanglements, are not only of profound mathematical interest but have also been experimentally observed in various physical systems, including knotted vortices in fluids and topological structures in liquid crystals.
In fully coherent fields, the concept of stable optical knots formed by phase singularities was first proposed by Berry and Dennis in 2001. Building on this theoretical foundation, Leach et al. experimentally realized such optical knots through the linear superposition of fully coherent Laguerre–Gaussian beams. Subsequent studies extended this framework by introducing singular lines in the polarization structure, leading to the construction of polarization knots based on polarization singularities. Throughout these developments, fully coherent light has been used as the primary source for both the design and experimental realization of optical knots. However, coherence—alongside phase and polarization—is a fundamental property of light. In fully coherent fields, phase singularities appear as points of zero intensity with undefined phase. In the polarization domain, polarization singularities occur at points where the polarization vector is undefined. Coherence singularities, by contrast, are embedded within the statistical properties of incoherent light fields, manifesting as points where the degree of coherence—relative to a reference point—vanishes.
Fig. 1 illustrates coherence singularities in an incoherent light field. While the instantaneous intensity and phase of a speckled field fluctuate randomly, the time-averaged intensity distribution of the incoherent field tends to become uniform due to statistical averaging. Using Young's double-slit interference as a diagnostic tool, one can evaluate the spatial degree of coherence with respect to a reference point—for example, the center of the field. Bright and high-contrast fringes indicate strong coherence, whereas the disappearance of fringes signifies a vanishing degree of coherence, identifying a coherence singularity. Despite their fundamental nature, optical singularities arising from coherence modulation, especially their corresponding three-dimensional topological field structures, remain largely unexplored.
The team of scientists proposed and experimentally realized a novel class of three-dimensional topological structures—incoherent links and knots—constructed from coherence singularities. Remarkably, despite the instantaneous electric field of the light being randomly distributed due to speckle, the topological structure of the incoherent knots remains stable, thanks to the stationary statistical nature of the field. In the experiments, the incoherent knot fields were generated by modulating statistically stationary dynamic speckle patterns using a computer-generated hologram. Their characterization was achieved through incoherent modal decomposition, which enabled the reconstruction of the incoherent knot topology from a set of orthogonal incoherent modes and their propagated counterparts at different planes. The results reveal a striking contrast between intensity and coherence-based structures: while the zero-intensity points in the instantaneous fields fluctuate significantly and vanish upon temporal averaging, coherence singularities exhibit remarkable robustness. Consequently, the incoherent knots constructed from coherence singularities maintain a stable topological configuration over time. This discovery not only broadens the scope of topological structured light research, but also opens a new avenue for applications in areas such as turbulence-resistant 3D structured light, incoherent light–matter interactions, and high-dimensional optical information encoding.
The results of the incoherent topological knots are illustrated in Fig. 2. In the experiment, a fully coherent beam is scattered by a rough surface, producing a speckle field. When the rough surface is subjected to vibration or rotation, a series of dynamic speckle patterns is generated. The incoherent superposition of these speckle fields forms an incoherent field. Once the condition of statistical stationarity is met, the amplitude of the resulting coherence structure approximates a Gaussian distribution. By applying a computer-generated hologram (Fig. 2b), specific phase information is introduced into the propagating field. In the three-dimensional space beyond the scattering surface, coherence singularities emerge at designated planes and locations, evolving into knot-like topological structures. As shown in Fig. 2a, there are no zero-intensity points in the field, and the dark regions are smoothly distributed. Instead, only the zeros of the degree of coherence persist stably, especially as revealed by the phase distribution of the mutual coherence function, which allows precise localization of the coherence singularities. As the spatial coherence of the field decreases, the outer envelope of the coherence structure—within which the coherence singularities reside—shrinks. However, the coherence singularities themselves remain stable, and the incoherent knot topological structure they form remains intact (Fig. 2e). This result further highlights the robustness of topological structures mediated by coherence singularities in incoherent light fields.
Knots in optical fields, as a unique class of three-dimensional topological structures, have attracted significant interest from both experimental and theoretical perspectives. The introduction of the concept of incoherent topological knots, along with their generation, evolution, and control, not only deepens our understanding of topological optical fields but also paves the way for advancing their applications in precision metrology, optical communication, and quantum information science. The team precisely controlled the distribution of coherence singularities by leveraging incoherent modal superposition, thereby constructing incoherent topological knots and links structures in light fields. Despite the strong fluctuations in the instantaneous field distribution, the incoherent topological structures, which are closely tied to the statistical properties of the field, remain remarkably stable. This offers a novel approach for realizing high-dimensional optical information encoding. Future research can further explore directions such as the implementation of incoherent topological information encoding, the transformation of topological structures, and their interactions with matter. Leveraging the robustness of incoherent topological structures under disordered and fluctuating environments, information encoding based on such structured light fields holds promise for enhancing the security and transmission efficiency of optical communication systems. Subsequent developments following this work are expected to play a significant role in the advancement of optical information processing and communication technologies.
