Optical hopfions with arbitrary two winding numbers

Abstract: Hopfions, as three-dimensional topologically nontrivial structures described by poloidal and toroidal winding numbers, hold promise as robust information carriers in spintronics, functional materials, and optical communications. Although they have been experimentally realized in various physical systems, such realizations have been restricted to low orders, with the winding numbers lacking tunability. Here, using optical fields as our platform, we outline how to make tunable hopfions in any order with any winding number. We use tailored superpositions of Laguerre-Gaussian modes in free-space as our construction, achieving effective control for arbitrary-order poloidal and toroidal winding numbers, which we demonstrate up to orders 5 and 3, respectively, for a new state-of-the-art. The resulting torus-knot structures are visualized experimentally via polarization filaments, confirming the designed topological textures. Our work reports an exotic optical topologies observed in free space, provides a systematic route hopfions of any order, with implications for topological photonics, optical communications, and analogies in magnetic and condensed-matter systems.

• The paper introduces a novel method for synthesizing optical hopfions with arbitrary (p,q) winding numbers via controlled superpositions of Laguerre-Gaussian modes.
• It employs precise polarization modulation and Gouy phase engineering to map three-dimensional torus knot topologies onto optical fields.
• Numerical Fourier integration and polarization-resolved tomography confirm the topological linking and scalability of the Hopf index in complex configurations.

Optical Hopfions with Arbitrary Two Winding Numbers: A Technical Review

Introduction and Motivation

Hopfions are three-dimensional topological solitons characterized by linked field lines whose linking number is defined by two winding numbers: the poloidal and toroidal numbers $(p,q)$. While their stability and topological protection have spurred interest across disciplines (magnetism, spintronics, ferroelectrics, liquid crystals, cosmology), practical realizations—especially in magnetic systems—have been limited to low-order (mainly $(1,1)$) hopfions, with no routes for independent and arbitrary control of both $p$ and $q$ in a general physical platform. Given the flexibility of optical fields constrained only by Maxwell's equations and the possibility of encoding topological textures in spatial and polarization degrees of freedom, this work addresses an outstanding challenge: the synthesis and experimental observation of optical hopfions in free-space with fully programmable winding numbers, both poloidal and toroidal, thereby expanding the accessible family of three-dimensional topological solitons and topological charge carriers.

Mathematical Foundations and Topological Properties

The construction exploits the polarization degree of freedom in vector optical beams. The polarization texture is described by the Stokes vector $\mathbf{S} = (S_x, S_y, S_z)$, which maps a physical three-dimensional spatial domain to the surface of the Poincaré sphere. The defining analytical forms for $(p,q)$-order hopfion polarization textures are

$$S_z = P \frac{1-\cosh^{2p}{(\eta)}\tanh^{2q}{(\eta)}}{1+\cosh^{2p}{(\eta)}\tanh^{2q}{(\eta)}},\quad S_x + iS_y = \sqrt{1 - S_z^2}\exp[i(p\theta + q\phi)]$$

with $P = \pm 1$ the polarity and $(\eta, \theta, \phi)$ the toroidal coordinates mapped to cylindrical coordinates. The field lines of constant polarization (isopolarization lines) form torus knots, their topological invariant given by the Hopf index $N_H$, which for ideal hopfions satisfies the integer relation $(1,1)$0.

Figure 1: Schematic diagrams of the Hopf map, hopfion energy surfaces, and torus knots for representative $(1,1)$1 orders, as well as Stokes vector distributions and filament structures for selected cases.

Optical Field Construction and Experimental Realization

The experimental approach leverages the superposition of Laguerre-Gaussian (LG) modes, for which the spatial amplitude and phase properties are well-defined and compatible with spatial light modulation and digital propagation techniques. The beam is synthesized in right- and left-circular polarization (RCP/LCP) components using amplitude and phase modulation via a spatial light modulator (SLM). The key design principles for the superposition are:

• For $(1,1)$2-order hopfions, adjacent orbital angular momentum (OAM) values are chosen for RCP and LCP components to ensure modal overlap even at high order, solving a previously identified challenge where increasing LCP OAM suppresses overlap.
• For $(1,1)$3-order hopfions, independent control of toroidal winding is accomplished not via OAM but via precise Gouy phase engineering, realized through superpositions of LG modes with radial and azimuthal indices tailored to achieve $(1,1)$4 windings of the polarization filaments around the propagation ($(1,1)$5) axis.

Explicit forms for the electric field are derived for both cases, with generalization to arbitrary $(1,1)$6 by combining strategies.

Numerical and Experimental Validation

The polarization textures and isopolarization filament topologies are fully reconstructed via polarization-resolved tomography. Key numerical evaluations include:

• Polarization field distributions (planar and volumetric)
• 3D filament trajectories showing topological linking
• Calculation of Hopf index $p$3 via a numerical Fourier-integral method, with measured $p$4 matching theoretical expectations up to the limitations of finite-volume sampling

For instance, for $p$5 the measured Hopf index approaches unity; for higher-order $p$6, measured indices scale accordingly (e.g., $p$7 for $p$8, $p$9 for $q$0).

Robustness and tunability are evaluated via variation of modal superposition weights $q$1; parameter regimes leading to skyrmion tubes (non-closed filaments) and topologically trivial fields are mapped.

Synthesis of Arbitrary $q$2 Hopfions

Generalization to arbitrary $q$3-order hopfions is realized by combining the OAM superposition strategy for $q$4 and the Gouy phase engineering for $q$5. The resulting polarization field supports torus knots and Hopf maps with programmable winding. The coprimality condition between $q$6 and $q$7 is necessary for forming singly connected filament structures; otherwise, the configuration decomposes into multiply connected filament families with degenerate winding.

Implications and Future Directions

This study establishes a general and flexible photonic method to generate and observe hopfions with arbitrary winding numbers, thus enabling systematic exploration of higher-order and exotic three-dimensional topological textures in light. The methodology is not limited to optics; its transferability to, e.g., chiral magnetization fields or nematic directors in soft matter, via analogous modal superpositions, is explicit. This connects structured light with broader topological material science and information encoding paradigms.

Potential applications include:

• High-dimensional topological encoding for optical communications (each $\mathbf{S} = (S_x, S_y, S_z)$0 as a robust logic state)
• Implementation of robust information carriers in next-generation 3D spintronic or photonic devices
• Systematic study of topological protection, transformation, and stability in complex and disordered media

Work remains to be done in extending modal superposition strategies to other platforms, optimizing parameter regimes for maximum topological charge, and exploring dynamical and quantum correlations of hopfionic states in structured fields.

Conclusion

This work provides a complete route to the design, synthesis, and experimental validation of optical hopfions with arbitrary poloidal and toroidal winding numbers $\mathbf{S} = (S_x, S_y, S_z)$1, demonstrating precise control of topological invariants in three-dimensional optical fields. The approach is verified by both numerical calculation and experimental polarization-resolved tomography, with direct visualization of torus knot topology in the polarization field. The implications for topological photonics and cross-disciplinary analogies are significant, opening new directions for high-capacity encoding, robust topological information carriers, and the study of complex topological excitations in physical systems.

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