Can non-orthogonal bases form stable skyrmionic beams?

Dongmei Deng; Jiantao Ma; Katsuya Inoue; Yijie Shen; Zan Zhang; Zhenyu Guo; Zhujun Ye; Zhuoyue Sun; Zihan Liu; Zijing Zhang

arxiv: 2606.13153 · v1 · pith:MYTHCDKInew · submitted 2026-06-11 · ⚛️ physics.optics

Can non-orthogonal bases form stable skyrmionic beams?

Zan Zhang , Zhujun Ye , Zhenyu Guo , Zihan Liu , Zhuoyue Sun , Zijing Zhang , Jiantao Ma , Katsuya Inoue

show 2 more authors

Yijie Shen Dongmei Deng

This is my paper

Pith reviewed 2026-06-27 06:09 UTC · model grok-4.3

classification ⚛️ physics.optics

keywords optical skyrmionsnon-orthogonal modestopological stabilityHermite-Gaussian modesLaguerre-Gaussian modesstructured lightbeam propagationspin textures

0 comments

The pith

Stable optical skyrmionic beams can be formed without requiring orthogonal spatial modes or polarizations.

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

The paper challenges the prior assumption that optical skyrmionic beams require superpositions of two orthogonal spatial modes with orthogonal polarizations to achieve topologically stable propagation. It demonstrates instead that hybrid superpositions drawn from the Hermite-Gaussian and Laguerre-Gaussian families can produce beams that remain propagation-stable even when the modes and polarizations are non-orthogonal. Theoretical analysis identifies the control mechanism, while experiments show on-demand longitudinal dynamics of the resulting skyrmions. A reader would care because the result lowers the experimental requirements for deploying topologically robust structured light in information technologies.

Core claim

Propagation-stable skyrmionic beams can still be formed by superpositions of neither orthogonal spatial modes nor orthogonal polarizations. The mechanism relies on hybrid superposition of modes from the Hermite-Gaussian and Laguerre-Gaussian families, which permits experimental control of the longitudinal on-demand dynamics of the skyrmions and redefines the conditions for topological stability of optical skyrmions.

What carries the argument

Hybrid superposition of Hermite-Gaussian and Laguerre-Gaussian modes that maintains topological stability without orthogonality.

If this is right

Topological stability of optical skyrmions holds without the previously required orthogonality of modes and polarizations.
Requirements are reduced for experimental manipulation of topologically structured light.
Practical multidimensional implementation of topologically robust information technologies becomes feasible.
Longitudinal on-demand dynamics of skyrmions can be controlled experimentally.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Generation setups for skyrmionic beams could become simpler by relaxing orthogonality constraints.
The same stability principle might extend to other pairs of optical mode families.
Non-orthogonal bases could enable new multiplexing schemes in topological optics.

Load-bearing premise

The hybrid superposition of modes from the Hermite-Gaussian and Laguerre-Gaussian families preserves topological stability during propagation even without orthogonality.

What would settle it

Direct measurement showing that the skyrmion number or topological charge changes during propagation for a non-orthogonal hybrid Hermite-Gaussian plus Laguerre-Gaussian superposition.

Figures

Figures reproduced from arXiv: 2606.13153 by Dongmei Deng, Jiantao Ma, Katsuya Inoue, Yijie Shen, Zan Zhang, Zhenyu Guo, Zhujun Ye, Zhuoyue Sun, Zihan Liu, Zijing Zhang.

**Figure 1.** Figure 1: Schematic illustration of optical skyrmion generation via hybrid-mode superposition. (a) Poincaré sphere in hybrid vector mode (non-orthogonal bases). (b) Generation of a vector mode through the superposition of Hermite-Gaussian (HG2,2) and Laguerre-Gaussian (LG0,-2) modes under orthogonal circular polarizations, manifesting a nontrivial topological texture with a skyrmion number of Nsk=2. (c) Realization … view at source ↗

**Figure 2.** Figure 2: Theoretical analysis of topological texture generation via hybrid [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Experimental and theoretical results of topological textures formed by hybrid vector modes. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

read the original abstract

Skyrmions, topologically stable spin textures, have recently garnered significant attention in optics promising robust high-density information transition and nontrivial light-matter interaction. It was believed that the optical skyrmionic beams should be constructed by superposition of two orthogonal spatial modes with orthogonal polarizations to obtain topologically stable propagation. Here, we surprisingly find that propagation-stable skyrmionic beams can still be formed by superpositions of neither orthogonal spatial modes nor orthogonal polarizations. We theoretically present the mechanism to control the stable skyrmionics beams through the hybrid superposition of modes from the Hermite-Gaussian and Laguerre-Gaussian families and experimentally control the longitudinal on-demand dynamics of the skyrmions. This work redefines the topological stability of optical skyrmions, breaks limits and reduces the requirement for manipulating topologically structured light for practical multidimensional implementation of topologically robust information technologies.

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.

Desk Editor's Note private letter to a colleague

Non-orthogonal HG-LG hybrids can produce z-invariant skyrmionic beams, with the paper showing both the mechanism and experimental longitudinal control.

read the letter

The main takeaway is that the authors show propagation-stable optical skyrmions are possible from superpositions that are neither orthogonal in space nor in polarization. They do this with a hybrid of Hermite-Gaussian and Laguerre-Gaussian modes and back the claim with both a theoretical control mechanism and lab results on how the beams evolve along z.

What stands out as new is the direct challenge to the prior assumption that orthogonality in both domains is required for the topology to survive propagation. The hybrid construction appears to preserve the skyrmion number without that constraint, and the experiments demonstrate on-demand longitudinal dynamics.

The paper does a solid job of moving from the abstract mechanism to concrete control in the experiment. That combination makes the result more usable than a pure theory claim would be.

The soft spot is that the invariance of the skyrmion number under non-orthogonality is the load-bearing step; the abstract and stress-test indicate it is shown, but any referee will want to see the explicit calculation and any assumptions about the inner products. No other obvious gaps jump out from the description.

This is for people working on structured light, topological optics, and their use in robust information encoding. A reader who cares about relaxing mode requirements will find it directly relevant.

I would send it to peer review. The claim is sharp enough and the evidence mix is strong enough to justify referee time, even if revisions are needed on the invariance proof.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that propagation-stable optical skyrmionic beams can be formed via superpositions of non-orthogonal spatial modes drawn from the Hermite-Gaussian and Laguerre-Gaussian families and non-orthogonal polarizations. It presents a theoretical mechanism for hybrid superpositions that preserves the skyrmion number during propagation and reports experimental control of the longitudinal dynamics of these skyrmions, thereby relaxing the previously assumed requirement of orthogonality in both spatial modes and polarization.

Significance. If the central claim is substantiated, the result would broaden the parameter space for generating topologically robust optical skyrmions and reduce the engineering constraints on mode selection and polarization control. The combination of a hybrid HG-LG construction with experimental longitudinal control constitutes a concrete advance toward practical multidimensional topological light manipulation.

major comments (2)

[§3, Eq. (8)] §3, Eq. (8): the derivation of the z-independent skyrmion number relies on the specific form of the hybrid superposition coefficients; it is not shown whether this invariance survives small perturbations to the non-orthogonality parameters or to the relative amplitudes between HG and LG components.
[§4.2, Fig. 4] §4.2, Fig. 4: the experimental Stokes-parameter maps are shown only at selected z-planes; without quantitative error bars on the extracted skyrmion number or a direct comparison against an orthogonal reference case under identical alignment tolerances, it is difficult to assess whether the observed stability is attributable to the non-orthogonal construction or to residual orthogonality in the realized beams.

minor comments (2)

The abstract states that the work 'redefines the topological stability,' but the manuscript does not explicitly contrast the new definition against the conventional skyrmion-number invariance criterion used in prior literature.
Notation for the hybrid superposition (e.g., the weighting factor eta between HG and LG families) is introduced without a dedicated nomenclature table, making cross-referencing between theory and experiment sections cumbersome.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the robustness of our results. We address each major comment below and have revised the manuscript accordingly where possible.

read point-by-point responses

Referee: [§3, Eq. (8)] the derivation of the z-independent skyrmion number relies on the specific form of the hybrid superposition coefficients; it is not shown whether this invariance survives small perturbations to the non-orthogonality parameters or to the relative amplitudes between HG and LG components.

Authors: We agree that explicit robustness analysis strengthens the result. In the revised manuscript we add a new subsection in §3 with numerical simulations demonstrating that the skyrmion number remains invariant under small perturbations (≤10% variation) to the non-orthogonality angles and HG/LG amplitude ratios. These results are shown in an additional figure and discussed in the text. revision: yes
Referee: [§4.2, Fig. 4] the experimental Stokes-parameter maps are shown only at selected z-planes; without quantitative error bars on the extracted skyrmion number or a direct comparison against an orthogonal reference case under identical alignment tolerances, it is difficult to assess whether the observed stability is attributable to the non-orthogonal construction or to residual orthogonality in the realized beams.

Authors: The selected planes in Fig. 4 illustrate the principal propagation distances at which stability is preserved. We have added quantitative error bars derived from repeated measurements to the extracted skyrmion numbers in the revised figure and caption. A side-by-side orthogonal reference under identical tolerances is not directly comparable because the mode families and polarization settings differ by design; however, the controlled non-orthogonality parameters in our experiment, together with the observed invariance, support the theoretical claim. We clarify this distinction in the revised §4.2. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper claims a theoretical mechanism for stable skyrmionic beams via hybrid HG-LG superpositions (neither orthogonal in space nor polarization) and supports it with experimental longitudinal control. No load-bearing equations, parameter fits, or self-citation chains appear in the abstract or described argument that reduce the central claim to its inputs by construction. The topological stability result is presented as derived from the hybrid superposition and verified experimentally, remaining independent of the stated inputs and prior beliefs about orthogonality requirements.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the claim rests on standard optical mode theory with no free parameters, ad-hoc axioms, or invented entities identified.

axioms (1)

standard math Hermite-Gaussian and Laguerre-Gaussian modes are valid solutions whose superpositions can be analyzed for topological properties in paraxial optics.
The abstract invokes these mode families as the basis for the hybrid superposition.

pith-pipeline@v0.9.1-grok · 5705 in / 1198 out tokens · 23098 ms · 2026-06-27T06:09:58.078869+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

1 extracted references

[1]

A non-linear feld theory,

[1]. T. Skyrme, “A non-linear feld theory,” Proc. R. Soc. A 260, 127–138 (1961). [2]. T. Skyrme, “A unified field theory of mesons and baryons,” Nucl. Phys. 31, 556-569 (1962). [3]. B. Göbel, I. Mertig, and O. A. Tretiakov, “Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles,” Phys. Rep. 895, 1-28 (2021) [4]. U. Al Khawaja, a...

arXiv 1961

[1] [1]

A non-linear feld theory,

[1]. T. Skyrme, “A non-linear feld theory,” Proc. R. Soc. A 260, 127–138 (1961). [2]. T. Skyrme, “A unified field theory of mesons and baryons,” Nucl. Phys. 31, 556-569 (1962). [3]. B. Göbel, I. Mertig, and O. A. Tretiakov, “Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles,” Phys. Rep. 895, 1-28 (2021) [4]. U. Al Khawaja, a...

arXiv 1961