Coherence toroidal vortices and statistic-veiled correlation topologies

Abstract: Toroidal vortices in fluid and gas dynamics underpin a broad spectrum of scientific and technological fields, from elementary particle physics to condensed matter systems, and have recently garnered significant attention in optics because of their inherent topological stability. Here we report the experimental observation of toroidal vortices in stochastic optical wavefields with partial coherence, termed coherence toroidal vortices, which eliminates deterministic topological signatures in conventional optical degrees of freedom while unveiling statistically hidden correlation topologies. These underlying topologies-including both fundamental and higher-order hopfionic textures-emerge exclusively in second-order field correlations and are accessible only through statistical measurements. We further examine the impact of chaotic channels on the stability of these statistically veiled correlation topologies, demonstrating that their topological invariants remain robust under realistic environmental perturbations. These findings are experimentally validated and offer novel insights into the potential of toroidal light vortices serving as controllable channels for directional energy and information transfer within complex media.

• The paper introduces the concept of coherence toroidal vortices in partially coherent optical fields, showing that second-order correlations encode topological invariants.
• It employs coherent-mode decomposition, phase-only SLM encoding, and Mach-Zehnder interferometry to reveal 3D toroidal vortex cores and quantized hopfionic textures.
• Results indicate that these statistically veiled topologies remain robust under turbulence, offering promising disorder-immune encoding for optical communications.

Summary of "Coherence toroidal vortices and statistic-veiled correlation topologies" (2604.21269)

Introduction and Conceptual Landscape

This study presents the first experimental realization of toroidal vortex structures in optical fields with partial coherence, introducing the concept of "coherence toroidal vortices." Unlike prior work limited to deterministic, fully coherent optical fields, this work elucidates the persistence of topological invariants at the level of second-order field correlations within stochastic wavefields. The research uncovers that topological features, such as hopfionic textures and associated coherence singularities, are not manifest in single-shot instantaneous fields but are uniquely accessible via statistical correlation measurements.

Theoretical Framework

The construction of toroidal topological states in optics has classically relied on paraxial beam propagation via Laguerre-Gaussian (LG) modal superpositions. In deterministic fields, toroidal vortices are revealed through the phase structure of the field itself. This work establishes that, under partial coherence (modeled by random complex field modulations with finite transverse coherence length), the deterministic toroidal topology is lost at the first-order field amplitude but is transferred into second-order correlations. Mathematically, if $E(\mathbf{r},z)$ is the stochastic field, the correlation function $$G(\mathbf{r}_1,\mathbf{r}_2,z) = \langle E(\mathbf{r}_1,z)E^*(\mathbf{r}_2,z) \rangle$$ encodes stationary vortex structures—coherence singularities—that parallel deterministic toroidal vortices but are statistically veiled.

Experimental Realization

The study's methodology integrates coherent-mode decomposition with phase-only spatial light modulator (SLM) encoding, Mach-Zehnder interferometry, and digital propagation (angular spectrum method). Stochastic fields are generated by encoding ensembles of phase holograms; their second-order correlation functions are empirically reconstructed by averaging over thousands of speckle realizations. These correlation functions unveil 3D toroidal vortex cores and quantized crossing numbers in the phase structure, confirming the existence of hopfions embedded within the coherence domain rather than realized in the instantaneous field.

The experimental data robustly demonstrate:

• The formation of fundamental ($\ell=0$) and higher-order ($\ell \geq 1$) coherence toroidal vortices, where $\ell$ is the OAM charge.
• The direct observation of statistical-veiled hopfion topologies, as evidenced by linked equiphase lines on the toroidal surface with quantized crossing numbers $N = 2\ell$.

Topological Robustness Under Perturbed Channels

A focal aspect is the robustness of correlation-encoded topologies under perturbations characteristic of practical systems. Using a dynamically heated turbulence chamber, the experiments interrogate the stability of coherence toroidal vortices and their hopfionic topologies across varying turbulence strengths. The results show that, while the deterministic field's phase structure is degraded, the topological invariants of the correlation function (specifically, the crossing number $N$) remain unchanged. This demonstrates that the statistical encoding confers intrinsic resilience to environmental disorder, a property not shared by conventional deterministic topologies. This invariance suggests possible advantages for information transmission and manipulation within turbulent or scattering media, relevant for optical communications and robust topological photonic devices.

Implications and Prospects

On a theoretical level, this work defines a new regime where 3D topological features are statistically veiled in the correlation structure, expanding the landscape of singular optics to stochastic wavefields. The results challenge the prevailing notion that only coherent fields can support meaningful topological structures, showing that higher-order statistical measures can host and reveal rich topologies otherwise inaccessible.

From an applied standpoint, the demonstrated robustness of correlation-based topologies under turbulence and noise suggests a pathway toward disorder-immune topological encoding. This is pertinent for classical and quantum communication protocols in complex or variable environments, where deterministic topological states are vulnerable to degradation. The observed encoding of hopfionic invariants in the second-order coherence could also spur further exploration into hybrid OAM-coherence-based information channels and topologically protected photonic functionalities.

Conclusion

This work establishes the existence and experimental accessibility of coherence toroidal vortices and their associated hopfionic topologies in stochastic optical fields. The primary findings reveal that topological structures, previously thought exclusive to deterministic coherent optical fields, persist as statistically veiled invariants in the second-order correlation function, offering both theoretical insight and practical robustness against environmental disorder. These results position statistical correlation topology as a new paradigm for structuring and manipulating light in complex systems, opening prospects for stable, disorder-immune topological photonics and novel coherence-based information processing schemes.