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arxiv: 2604.21424 · v1 · submitted 2026-04-23 · ⚛️ physics.optics

Optical hopfions with arbitrary two winding numbers

Xinji Zeng , Jinwen Wang , Yun Chen , Guang Liu , Zhenyu Guo , Yongkun Zhou , Xin Yang , Chengyuan Wang
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Dong Wei Haixia Chen Yijie Shen Andrew Forbes Hong Gao
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Pith reviewed 2026-05-09 21:00 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords optical hopfionsLaguerre-Gaussian modestopological winding numberspolarization filamentstorus knotsfree space opticsthree-dimensional topology
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The pith

Optical fields can realize hopfions with arbitrary poloidal and toroidal winding numbers via Laguerre-Gaussian superpositions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to generate optical hopfions with any desired winding numbers by superposing Laguerre-Gaussian laser modes in free space. Hopfions are three-dimensional structures whose topology is set by two independent integers describing how field lines wind around each other. Earlier work achieved only simple cases with small winding numbers that could not be changed easily. The new construction reaches poloidal windings up to 5 and toroidal up to 3, and the structures are mapped out by tracking polarization directions along filaments. If the method works as described, it supplies a practical way to produce these topological objects on demand for use in light-based technologies and as models for similar phenomena elsewhere.

Core claim

The central discovery is that arbitrary-order hopfions can be realized as superpositions of Laguerre-Gaussian beams, where the beam parameters are chosen so that the resulting three-dimensional polarization texture has independently prescribed poloidal and toroidal winding numbers. This is achieved in free space and verified by direct observation of the linked filamentary structures.

What carries the argument

Tailored superpositions of Laguerre-Gaussian modes, which allow independent control over the poloidal and toroidal winding numbers in the resulting optical hopfion.

Load-bearing premise

That the designed superpositions of Laguerre-Gaussian modes produce the claimed three-dimensional topological textures with independently tunable winding numbers rather than only approximate or lower-order structures.

What would settle it

An experimental or simulated calculation of the Hopf invariant for one of the constructed fields that yields a value different from the product of the intended poloidal and toroidal winding numbers.

Figures

Figures reproduced from arXiv: 2604.21424 by Andrew Forbes, Chengyuan Wang, Dong Wei, Guang Liu, Haixia Chen, Hong Gao, Jinwen Wang, Xinji Zeng, Xin Yang, Yijie Shen, Yongkun Zhou, Yun Chen, Zhenyu Guo.

Figure 1
Figure 1. Figure 1: Schematic diagrams of the Hopf map, the structures of hopfions and torus knots. (a1) is the first-order Poincaré sphere with several polarization states marked by small spheres, the color map is based on the sphere’s azimuthal angle ψ and spatial angle χ. (a2) shows the nested torus structure corresponding to the Sz isosurface, with the torus at Sz = 0 displayed in (a3) and several torus knots drawn. (b1) … view at source ↗
Figure 2
Figure 2. Figure 2: Simulation and experimental results of optical hopfions with winding number (p,1). For (a), p = 1; (b), p = 3; (c), p = 4 and (d), p = 5. The middle column shows the polarization distributions in the xy plane at z = 0 and in the xz plane at y = 0; the upper left of each subplot is the torus knot corresponding to the respective order, and the lower left is a pair of orthogonal isopolar filaments with a fixe… view at source ↗
Figure 3
Figure 3. Figure 3: (a) and (b) are simulated and experimental results for (1,q) winding hopfions. The left side show the polarization distributions in the xy plane at z = 0 and in the xz plane at y = 0; the middle show individual isopolar filaments, with coordinates representing sampling points; the right side show all filaments for a fixed Sz , with the upper part being the simulation and the lower part the experimental res… view at source ↗
Figure 4
Figure 4. Figure 4: (a) shows the simulation and experimental results for the hopfion with winding number (p,q) = (3,2), and (b) shows the simulation results for (p,q) = (5,2). From left to right, they display the polarization distributions in the xy and xz planes, the mutually orthogonal isopolar filaments with S = (0.94,0,−0.34) and (−0.94,0,−0.34), and the eight isopolar fibers with Sz = −0.34. The impact of coefficient va… view at source ↗
Figure 5
Figure 5. Figure 5: Polarization distributions in the xy and xz planes for different α and β values. The middle shows three regions: hopfion, skyrmion tube, and topologically trivial. Points A, B, C, and D are given by (a), (b), (c), and (d), respectively. The reliability of the generated optical hopfion structure is highly sensitive to the relative weights α and β in the superposition. In this section, we systematically expl… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims to construct optical hopfions with independently tunable poloidal and toroidal winding numbers of arbitrary order using tailored superpositions of Laguerre-Gaussian modes in free space. Analytic field expressions are derived, the Hopf invariant is computed by direct 3D integration and shown to equal the product of the two winding numbers, and the resulting torus-knot structures are visualized both numerically and experimentally via polarization filaments up to orders (5,3).

Significance. If the construction and verification hold, the work supplies a systematic, experimentally realizable route to high-order hopfions in free-space optics, extending prior low-order demonstrations. The use of standard LG modes, explicit mode-selection rules, direct Hopf-invariant integration, and polarization-filament confirmation constitute clear strengths that could support applications in topological photonics and analogies to magnetic systems.

minor comments (3)
  1. The abstract states that the method achieves 'any order with any winding number,' yet demonstrations are limited to poloidal order 5 and toroidal order 3. A brief statement in the conclusions on the practical limits of the superposition approach (e.g., beam divergence or mode overlap at higher orders) would strengthen the claim of generality.
  2. [Experimental results] In the experimental section, the polarization-filament images are presented without explicit scale bars or reference to the beam waist; adding these would improve quantitative comparison with the numerical torus-knot geometry.
  3. [Theory] The notation for the two winding numbers (p, q) is introduced clearly, but the manuscript does not explicitly restate the Hopf invariant formula used for the numerical check; a one-line reminder of the integral definition would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The summary accurately reflects our construction of tunable optical hopfions via Laguerre-Gaussian superpositions, the direct computation of the Hopf invariant, and the experimental confirmation up to orders (5,3).

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper constructs hopfions via explicit superpositions of Laguerre-Gaussian modes whose indices are chosen to independently set the poloidal and toroidal winding numbers. The full analytic field expression is given, the Hopf invariant is obtained by direct volume integration (recovering the product of the two windings), and both numerical and experimental visualizations (polarization filaments) are supplied as confirmation. No step reduces to a fitted parameter renamed as prediction, a self-referential definition, or a load-bearing self-citation whose validity is assumed rather than independently verified. The derivation is self-contained against the stated external benchmarks of mode superposition and direct topological computation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the construction is described as a superposition of known Laguerre-Gaussian modes without additional postulated objects.

pith-pipeline@v0.9.0 · 5512 in / 1177 out tokens · 21574 ms · 2026-05-09T21:00:21.275027+00:00 · methodology

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Reference graph

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