Stability of Optical Vortex Knots and Flower Beams in Turbulence

Abstract

Optical vortex knots offer a promising avenue for robust optical information transfer, thanks to their unique topological properties and the potential to encode complex data in a structured, multidimensional manner. This capability is especially valuable in modern telecommunications, sensing, imaging, and related fields that demand high-capacity, interference-resistant transmission methods. From a theoretical standpoint, the inherent topology of knots can protect data from external perturbations, positioning them as attractive candidates for next-generation optical communication systems. However, we found that when propagating through turbulent media, vortex knots face significant stability challenges. Turbulence triggers reconnection events and phase distortions that undermine traditional topological invariants, such as the number of crossings, and broaden the spectral signatures of mode decompositions. This degradation not only reduces the effectiveness of spectral encoding methods but also complicates the accurate measurement and practical deployment of optical knots in free-space communication, sensing, and imaging applications.These challenges call for a detailed investigation into the behavior of singular structures such as optical vortex knots in turbulence, as well as the development of strategies to counteract these effects. Enhancing the stability of optical knots requires advanced techniques that go beyond traditional spectral encoding. For example, we show that one might exploit the inherent topological information – i.e. the discrete knot‐invariants (crossing number, linking number, Alexander polynomials, etc.) that remain unchanged under any smooth deformation – in a novel way to achieve robust data encoding even in adverse turbulent environments. This work presents a unified framework that combines a comprehensive theoretical approach, advanced numerical modeling, and detailed experimental validation. Atmospheric turbulence is modeled using both Kolmogorov and von Kármán power spectra, implemented via phase screen techniques within a split-step propagation framework that accurately reproduces realistic conditions. Experimentally, a Mach-Zehnder interferometer paired with a controlled turbulence chamber enables single-shot complex field measurements, ensuring that our simulations accurately represent laboratory observations. Isolated optical vortex knots are generated by a precisely engineered superposition of Laguerre-Gaussian beams, whose amplitudes and phases are derived from Milnor polynomial representations obtained through braided constructions using stereographic projections. Building on this method, we introduce a rigorous theoretical framework for sculpting three-dimensional, topological particle-like objects, such as optical knots or links, with precise control over their spatial features. This approach enables the creation of intricate knot geometries with high accuracy, providing a versatile platform for studying vortex dynamics and harnessing the properties of optical singular structures. In addition, we propose a novel iterative optimization algorithm, comprising both intensity optimization and singularity position optimization, that maximizes the separation between phase singularity lines and enhance intensity contrast. This method dramatically improves the stability of optical knot structures in turbulent environments, as evidenced by significantly higher recovery rates observed in both simulations and experiments under challenging conditions. Furthermore, we introduce a deep learning framework employing fully connected neural networks and three-dimensional convolutional neural networks to assess the geometric robustness of singularity line shapes. Our combined analysis reveals that the spatial trajectories of vortex singularities are inherently more resilient to turbulence than traditional topological invariants or spectral contents. This insight not only deepens our understanding of vortex dynamics but also paves the way for a new class of structured optical fields. Our findings in controlling optical singularity line shapes enable us to manipulate and tighten knots with unprecedented freedom, as well as to create entirely novel structures. Termed 'flower beams,' these newly developed beams exploit the untapped potential of shape stability in singularity lines. They exhibit broad scalability, exceptional controllability, and markedly higher stability than classical optical knots, even under strong turbulence, offering a scalable and robust alternative for free-space optical communication, sensing, and imaging. In summary, our integrated approach demonstrates that combining advanced simulation techniques, optimization algorithms, and deep learning can substantially enhance the stability of optical vortex knots. By shifting the focus toward shape-based encoding – exemplified by Flower beams – we open new avenues for resilient information transfer in turbulent environments.