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Optical Skyrmions in Waveguides

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Within a conducting waveguide, a skyrmion's topological charge survives propagation whenever the transverse electric field stays nonzero, and adding a TE1n or TM1n stabilizing mode can guarantee that condition.

desk verdict Real idea and useful numerics, but the central theorem is under-specified: nonvanishing at z=0 does not imply nonvanishing along the whole waveguide, so the topological protection claim needs a uniform-in-z condition or a dominance bound. read the letter →

arxiv 2505.06735 v1 pith:LHHGIFTM submitted 2025-05-10 physics.optics

An Aloysius Wang , Yifei Ma , Yuxi Cai , Ji Qin , Bowei Dong , Chao He This is my paper · ORCID
classification physics.optics PACS 42.79.Gn42.25.Ja
keywords opticalskyrmionstopologicalprotectionconductingwaveguidesmodaldispersionskyrmionnumbertopologicallystabilizingmodesgeneralizedpolarizationtopology
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes conditions under which the skyrmion number of an optical polarization field survives propagation inside a conducting waveguide. It argues that if the transverse electric field is nowhere zero, the skyrmion number integral is constant along the guide even when modal dispersion reshapes the field, because propagation is then a continuous deformation of a compactifiable map. From this it derives a practical classification: TE1n and TM1n modes can act as topologically stabilizing modes that, when added to a superposition, remove field zeros and make the topological charge robust to small coefficient changes. The paper also shows that when the ordinary skyrmion number is not preserved, a generalized skyrmion number tied to components carved out by boundary and singularity curves can remain protected. If correct, this gives multi-mode waveguides a route to carry topologically protected, high-dimensional information.

What carries the argument

The central object is the skyrmion number integral, the degree of the map from the waveguide cross-section to the Poincaré sphere, computed from the Stokes parameters; its quantization rests on a canonical compactification of the boundary polarization field fixed by the conducting walls. The load-bearing mechanism is the observation that a transverse electric field that is everywhere nonzero makes the polarization map smooth and defined on the whole cross-section, so propagation along the waveguide becomes a homotopy that cannot change the integer. The named tool is the topologically stabilizing mode: a solution, in practice TE1n or TM1n, that is nonzero everywhere across the cross-section and can be added to any superposition to eliminate zeros, thereby enforcing both propagation stability and robustness to coefficient variations. For fields with persistent singularities, the paper invokes the generalized skyrmion number, which assigns separate integer charges to each connected component of the Poincaré sphere carved out by the images of physical boundaries and singularity boundaries, and is preserved as long as that component structure does not split.

What would settle it

Take a cylindrical conducting waveguide, launch a TE11 plus TE21 superposition in a parameter region predicted to be stable, and measure the Stokes-resolved transverse field at several propagation distances; finding the skyrmion number integral changing to a different integer while the transverse electric field remains everywhere nonzero would refute the central claim. Alternatively, an explicit numerical search could look for a coefficient path between two regions of different skyrmion number that avoids any transverse zero, which the theory says is impossible.

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Extended reading notes

Core claim

Within a hollow conducting waveguide, the fixed polarization state on the metal boundary lets the interior polarization field be canonically extended to a compactified sphere, so the interior skyrmion number integral is an integer homotopy invariant. The paper's central claim is that this invariant is preserved during propagation whenever the transverse electric field has no zeros: field zeros are the only way the domain gets punctured, and a smooth nonzero field makes propagation a homotopy. Therefore, for a superposition of TE and TM modes, any coefficient set whose transverse field is everywhere nonzero yields a constant skyrmion number along the guide, and nearby coefficient sets with the same property yield the same integer. Because all m not equal to 1 TE and TM modes vanish at the waveguide center, only TE1n and TM1n modes can serve as stabilizing additions that remove zeros; adding one to an unstable state recovers both propagation stability and parameter robustness. The paper also demonstrates numerically stable skyrmion numbers of 1, -1, and -2 in TE11/TE21 superpositions, and shows that when the ordinary skyrmion number varies, a generalized skyrmion number assigned to each connected component on the Poincaré sphere can remain constant or lose only the charge of a disappearing component.

Load-bearing premise

The argument rests on the premise that the skyrmion number integral inside the waveguide is quantized through a canonical extension of the fixed boundary polarization to a compactified sphere; if that quantization fails for the waveguide's boundary curve, the conclusion that nonzero fields cannot change their skyrmion number does not follow.

Editorial extensions

If this is right

  • Multi-mode conducting waveguides can carry topologically protected polarization patterns despite modal dispersion, so information encoded in skyrmion number need not be disrupted by mode beating.
  • Coupling precision can be relaxed: any perturbation of the mode coefficients that keeps the transverse field nonzero leaves the transported skyrmion number unchanged.
  • Superposing a topologically stabilizing mode can stabilize otherwise singular modes and create stable skyrmion numbers, such as -2, that neither mode alone can produce.
  • When the ordinary skyrmion number fails, the generalized skyrmion number can still protect several integer charges simultaneously, raising the information density per field.
  • Weakly stable single modes remain protected in propagation but not against coefficient changes; only genuinely stabilizing superpositions give both kinds of robustness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the same reasoning carries to dielectric waveguides, where the boundary polarization is not fixed, the generalized skyrmion number should provide protection over finite propagation distances as long as the boundary-curve topology does not change; an experimental check would be to launch a known TE/TM superposition and measure Stokes-resolved cross-sections at successive propagation distances.
  • The phase diagrams suggest a testable bifurcation structure: the skyrmion number can only change when a transverse zero crosses the domain, so one could predict critical coefficient surfaces where the integer jumps and verify that no jump occurs without a zero.
  • The TE1n/TM1n stabilizing condition may extend to other confining geometries, such as rectangular or ridge waveguides, wherever a mode is nonzero everywhere; engineering such modes could become a design rule for topologically robust integrated photonics.
  • An implicit consequence is that the topological charge carried by a disappearing component in the generalized setting is genuinely lost, so redundancy or error-correction schemes would be needed if multiple charges are used for data encoding.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript claims to establish the first theory of topological protection of optical skyrmions in conducting waveguides. It argues that when the transverse electric field of a waveguide mode superposition is everywhere nonzero, the skyrmion number integral is quantized and remains constant during propagation because propagation is a homotopy of the polarization field. It introduces a classification of solutions into topologically stabilizing, weakly stable, and unstable modes, identifies TE1n and TM1n as candidate stabilizing modes, and supports the claims with numerical phase diagrams for TE11, TE21, and their superpositions, including propagation along z. It further invokes a generalized skyrmion number from an unpublished preprint to show that a multi-valued topological charge can be preserved even when the usual skyrmion number fails, in two numerical examples.

Significance. If correct, this is a useful conceptual step toward using polarization topology in multimode waveguides, and the proposed classification of modes could guide experimental designs. The numerical simulations are a strength: the paper provides quantitative phase diagrams, tests robustness to coefficient perturbations, and demonstrates a concrete mechanism by which a stabilizing mode changes the topology of a superposition. The generalized-skyrmion examples are also valuable as a demonstration. However, the central theorem's hypothesis is under-specified with respect to the propagation coordinate, and the quantization argument rests on an unpublished framework; these issues prevent the paper from fully establishing the advertised claims.

major comments (4)
  1. [Section 1, Eq. (1) and the three numbered statements] The condition that 'E0 is everywhere non-zero' is not tied to the propagation coordinate z. Equation (1) is written without explicit z-dependence, but the actual fields in Eq. (3) contain factors exp(ik_n z) and exp(iκ_n z), so the relative phases between modes change with z. Non-vanishing at a single cross-section does not imply non-vanishing for all z; destructive interference between modes with different propagation constants can create a zero at a later z. The homotopy argument requires E0(r,θ,z) ≠ 0 for every z in the propagation interval, or a dominance threshold such as |c| > C that guarantees this uniformly. The paper states no such threshold in the abstract or in the propagation claims associated with Fig. 4. This is load-bearing because a zero punctures the domain and can change the skyrmion number.
  2. [Section 1, first paragraph] The quantization of the skyrmion number integral is asserted via a 'canonical extension' of the boundary polarization field to a compactifiable S^2-valued function, citing refs. [23] and [30]. For a conducting waveguide whose boundary curve is the equator traced twice, this extension is not constructed and the homotopy invariance is not proved. Since the subsequent statement that a nonzero field cannot change its skyrmion number under propagation depends on this quantization, the paper should either provide a self-contained derivation for the waveguide geometry or state precisely which theorem from [30] applies and what assumptions it requires.
  3. [Section 1, numbered statement 3 and the paragraph on TE1n/TM1n] The claim that adding a sufficiently large multiple of a nonzero mode to any solution yields a field that is everywhere nonzero is stated without proof. It is plausible if the stabilizing mode has a positive lower bound on its magnitude over the compact cross-section, but this lower bound and the threshold on |c| are not given. Also, the identification of TE1n and TM1n as 'topologically stabilizing' is based only on non-vanishing at r = 0; Fig. 1 shows that TE11 itself can develop zeros for |B11| ≈ 1, so the classification needs a quantitative criterion rather than the m ≠ 1 observation.
  4. [Section 2 and Fig. 5, generalized skyrmion numbers] The preservation of the generalized skyrmion numbers in Fig. 5 is asserted on the basis of visual inspection of the boundary curves and the framework of ref. [30]. No definition of the generalized skyrmion number is given in this paper, and no numerical value is computed from an integral; the reported values (0, −2, −4) appear to be inferred from the component structure rather than from a calculation. The reader cannot verify the claim without consulting an unpublished preprint. A self-contained definition and a computation recipe are needed.
minor comments (5)
  1. [Fig. 2 caption] The caption writes 'ImB11' in panel b, but the panel varies B21; this should be 'ImB21'. The word 'skrymion' in the caption should be 'skyrmion'.
  2. [Fig. 3 caption] The caption says 'As in Figure 4, there is a critical region between the two stable modes'; the reference appears to be to Fig. 1 rather than Fig. 4, and should be corrected.
  3. [Fig. 4 caption] The sentence 'The figure is organized in a similar way to (a)' is unclear because panel (a) is a phase diagram while panels (b)-(d) are propagation plots; the intended comparison should be stated explicitly.
  4. [Fig. 5 caption] The phrase 'which cases the usual skyrmion number to fluctuate' should read 'which causes the usual skyrmion number to fluctuate'.
  5. [Main text, Eq. (3)] The coefficients A_mn, B_mn, C_mn, D_mn and the mode functions E_TE and E_TM are not fully defined until Methods; a brief definition or a forward reference at the point of Eq. (3) would improve readability.

Circularity Check

2 steps flagged · score 4.0 of 10

The paper's central topological quantization step is imported from same-author prior work, but the conventional-skyrmion numerics are independent simulations, making the circularity partial.

  1. self citation load bearing [Section 1 (Main), first two paragraphs]
    "As explained in [23], the topological nature of a traditional skyrmion arises from constraints on its boundary values... the geometry of the waveguide entirely determines the polarization state at the air-conductor interface, and this restriction is sufficient in recovering non-trivial topologies [30]."

    The paper's central premise, that the boundary-fixed polarization state permits canonical compactification and hence quantized skyrmion numbers, is not derived for the conducting waveguide. Instead, it is assigned to the authors' own refs [23] and [30], the latter an unpublished same-author preprint. Every subsequent claim of topological protection during propagation, and especially the generalized skyrmion numbers used to 'recover' protection, inherits this imported framework. The manuscript provides no independent proof or external verification of the compactification step, so the main theoretical backbone reduces to a self-citation.

  2. self citation load bearing [Section 2 (Discussion), paragraph on generalized skyrmions]
    "The framework introduced above offers a comprehensive solution to skyrmion transport within a conducting waveguide and can be extended to overcome changing boundary conditions and polarization singularities by adopting the generalized skyrmion number introduced in [30]."

    The generalized-skyrmion demonstration in Fig. 5 is not a prediction derived from the waveguide equations in this paper; it is the [30] construction applied to TE/TM superpositions. The asserted preservation 'as long as the number of connected components carved out by the boundary remains unchanged' is exactly the preservation theorem of [30]. Since [30] is a same-author preprint and its generalized skyrmion numbers are adopted rather than re-derived, the figure's conclusion reproduces the cited framework rather than independently establishing it.

full rationale

The conventional-skyrmion numerical simulations (phase diagrams and propagation curves) are genuine computations with no fitted parameters: skyrmion numbers are evaluated from the Stokes fields of TE/TM superpositions, and the observed stability of integer values in the expected regions is not manufactured from the input coefficients. Those simulations provide independent content and support the paper's claim that nonzero transverse fields protect the ordinary skyrmion number. The circularity concern is limited to the theoretical framing: the boundary-compactification quantization theorem is taken from the authors' earlier work [23] and, for the generalized skyrmion extension, from the unpublished preprint [30]. The latter is load-bearing for the final claim that a generalized skyrmion number recovers topological protection when the usual number fails; that claim is essentially an application of [30]'s own theorem rather than a self-contained derivation in this manuscript. I do not count the possible z-uniformity gap in the nonzero-field hypothesis as circularity; it is a correctness risk, not a reduction of the conclusion to the input. Overall, the central numerical content is independent, but the foundational topology is substantially self-cited, so the paper is partially but not wholly circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the conducting-boundary compactification picture (self-cited prior work), the homotopy argument for nonzero fields, and the generalized skyrmion framework from an unpublished preprint. No free parameters are fitted to data; the numerical coefficients are illustrative inputs. No new physical entities are introduced.

assumptions (4)
  • domain assumption The transverse polarization state at the boundary of a perfectly conducting waveguide is fixed and identical on every cross-section, enabling compactification.
    Invoked in Section 1 to justify quantization of the skyrmion number integral. The boundary condition E_theta=0 fixes the Stokes vector on the equator traced twice.
  • domain assumption Propagation along the waveguide is a homotopy of the transverse Stokes field as long as the transverse electric field is everywhere non-zero.
    Used to conclude topological protection from the absence of singularities; the z-evolution of the field is treated as a continuous deformation.
  • ad hoc to paper The generalized skyrmion number framework of ref [30] assigns stable integer topological charges to fields that cannot be compactified.
    The final section's claims about generalized skyrmion numbers depend entirely on this self-cited, at-the-time unpublished preprint.
  • standard math Standard Bessel-function solutions of the Helmholtz equation in a cylindrical conducting waveguide (Methods 1).
    Basis of all mode superpositions; standard separation of variables for TE and TM modes.

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Cite this review

Pith. "Pith review of Optical Skyrmions in Waveguides." pith.science (2026). https://pith.science/paper/LHHGIFTM

@misc{pith2026250506735,
  author       = {Pith},
  title        = {Pith review of: Optical Skyrmions in Waveguides},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LHHGIFTM}},
  note         = {Machine review of arXiv:2505.06735}
}
read the original abstract

Optical skyrmions are topologically non-trivial polarization fields which have recently attracted attention due to their potential use in high density data applications such as optical communications, photonic computing and more. An important hurdle in utilizing optical skyrmions for such applications is establishing conditions under which their topological structure remains preserved during propagation: while results of this type already exist for paraxial beams in free-space propagation, the critical case relevant to modern applications involves propagation in confined media, such as waveguide systems. In this paper, we demonstrate for the first time that, within a conducting waveguide, the preservation of the skyrmion number during propagation is determined by the presence of so-called topologically stabilizing modes. If such a mode is present, not only will topological protection hold despite the transverse polarization profile changing due to modal dispersion, but there is also a degree of robustness to variations in the coefficients of TE and TM modes present. Lastly, we demonstrate how a generalized skyrmion number can recover topological protection in situations where the usual skyrmion number is not preserved. Our methods open new avenues for robust high-dimensional information manipulation in waveguiding systems.

Figures

Figures reproduced from arXiv: 2505.06735 by the authors.

Figure 1
Figure 1. Phase diagram of TE11 mode. a, (Left) Skyrmion number at different values of B11. (Right) Absolute difference between computed skyrmion number and its nearest integer. The skyrmion number is stable with the value of 1 in the region |B11| ≲ 1 and stable with a value of −1 in the region |B11| ≳ 1. Between the two stable regions, there is a critical region where the behaviour of the skyrmion number is unstable, with th… view at source ↗
Figure 2
Figure 2. Phase diagram of TE21 mode. a, (Left) Skyrmion number at different values of B21. (Right) Absolute difference between computed skyrmion number and its nearest integer. The phase diagram exhibits no stable regions, but is exactly 0 along the circle given by |B21| = 1. b, Skyrmion number along the Im B11 = 0 line. Note that the skyrmion number is exactly 1 when B21 = 0, and approaches -1 when |B21| approaches infinity… view at source ↗
Figure 3
Figure 3. Phase diagram of a superposition of TE11 and TE21 modes. a, (Left) Skyrmion number at different values of B21. (Right) Absolute difference between computed skyrmion number and its nearest integer. The skyrmion number is stable with the value of 1 in the inner circular region and stable with a value of −2 in the outer annulus. Between the two stable regions, there is a critical region where the behaviour of the skyrm… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Skyrmion number of dispersive modes along a waveguide. a, Chosen stable and unstable modes marked on the relevant phase diagram. b, Skyrmion number and stokes field of an unstable superposition of TE11 and TE21 modes (A11 = 1, B21 = 0.7 − 1.5i) against distance along t…
Figure 5
Figure 5. Figure 5: Generalized skyrmion number in propagation. a, The skyrmion number and generalized skyrmion number of a superposition of TE21 and TE22 modes with coefficients B21 = 1, A22 = 0.5 and B22 = 0.3 at different z. The stokes fields within the waveguide at different z drawn u…

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Reference graph

Works this paper leans on

31 extracted references · 28 canonical work pages

  1. [23]

    A. A. Wang et al. Topological protection of optical skyrmions through complex media.Light: Science & Applications, 13:314, 2024

    work page 2024

  2. [30]

    A. A. Wang et al. Generalized skyrmions. Preprint athttps://arxiv.org/abs/2409.17390, 2024

    arXiv 2024

  3. [1]

    T. H. R. Skyrme. A non-linear field theory.Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 260(1300):127–138, 1961

    work page 1961

  4. [2]

    A. Fert, N. Reyren, and V. Cros. Magnetic skyrmions: Advances in physics and potential applica- tions.Nature Reviews Materials, 2(7):17031, 2017

    work page 2017

  5. [3]

    Nagaosa and Y

    N. Nagaosa and Y. Tokura. Topological properties and dynamics of magnetic skyrmions.Nature Nanotechnology, 8(12):899–911, 2013

    work page 2013

  6. [4]

    Al Khawaja and H

    U. Al Khawaja and H. Stoof. Skyrmions in a ferromagnetic Bose–Einstein condensate.Nature, 411(6840):918–920, 2001

    work page 2001

  7. [5]

    Foster et al

    D. Foster et al. Two-dimensional skyrmion bags in liquid crystals and ferromagnets.Nature Physics, 15(7):655–659, 2019

    work page 2019

  8. [6]

    D. A. Smirnova, F. Nori, and K. Y. Bliokh. Water-wave vortices and skyrmions.Physical Review Letters, 132(5):054003, 2024

    work page 2024

Show all 31 references
  1. [7]

    Tsesses et al

    S. Tsesses et al. Optical skyrmion lattice in evanescent electromagnetic fields.Science, 361(6406):993—-996, 2018

    2018

  2. [8]

    C. He, Y. Shen, and A. Forbes. Towards higher-dimensional structured light.Light: Science & Applications, 11(1):205, 2022

    2022

  3. [9]

    Y. Shen, E. C. Mart´ ınez, and C. Rosales-Guzm´ an. Generation of optical skyrmions with tunable topological textures.ACS Photonics, 9(1):296–303, 2022. 12

    2022

  4. [10]

    Bai et al

    C. Bai et al. Dynamic tailoring of an optical skyrmion lattice in surface plasmon polaritons.Optics Express, 28(7):10320, 2020

    2020

  5. [11]

    Du et al

    L. Du et al. Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum.Nature Physics, 15(7):650–654, 2019

    2019

  6. [12]

    Lei et al

    X. Lei et al. Photonic Spin Lattices: Symmetry Constraints for Skyrmion and Meron Topologies. Physical Review Letters, 127(23):237403, 2021

    2021

  7. [13]

    P. Shi, L. Du, and X. Yuan. Spin photonics: from transverse spin to photonic skyrmions.Nanopho- tonics, 10(16):3927–3943, 2021

    2021

  8. [14]

    Gao et al

    S. Gao et al. Paraxial skyrmionic beams.Physical Review A, 102(5):053513, 2020

    2020

  9. [15]

    A. A. Wang et al. Unlocking new dimensions in photonic computing using optical skyrmions. Preprint athttps://arxiv.org/abs/2407.16311, 2024

    2024

  10. [16]

    H. A. Teng et al. Physical conversion and superposition of optical skyrmion topologies.Photonics Research, 11(12):2042–2053, 2023

    2023

  11. [17]

    W. B. Lin et al. Microcavity-based generation of full Poincar´ e beams with arbitrary skyrmion numbers.Physical Review Research, 3(2):023055, 2021

    2021

  12. [18]

    P. Shi, L. P. Du, and X. C. Yuan. Strong spin–orbit interaction of photonic skyrmions at the general optical interface.Nanophotonics, 9(15):4619–4628, 2020

    2020

  13. [19]

    He et al

    C. He et al. A universal optical modulator for synthetic topologically tuneable structured matter. Preprint athttps://arxiv.org/abs/2311.18148, 2023

    2023 arXiv

  14. [20]

    Shen et al

    Y. Shen et al. Optical skyrmions and other topological quasiparticles of light.Nature Photonics, 18(1):15–25, 2023

    2023

  15. [21]

    Ornelas et al

    P. Ornelas et al. Non-local skyrmions as topologically resilient quantum entangled states of light. Nature Photonics, 18(3):258–266, 2024

    2024

  16. [22]

    C. X. Liu et al. Disorder-induced topological state transition in the optical skyrmion family.Physical Review Letters, 129(26):267401, 2022

    2022

  17. [24]

    Meng et al

    Y. Meng et al. Optical meta-waveguides for integrated photonics and beyond.Light: Science & Applications, 10:235, 2021

    2021

  18. [25]

    D. M. Graham et al. Laser written waveguide photonic quantum circuits.Opt. Express, 17(15):12546–12554, Jul 2009

    2009

  19. [26]

    C. L. Li, D. J. Liu, and D. X. Dai. Multimode silicon photonics.Nanophotonics, 8(2):227–247, 2019

    2019

  20. [27]

    He et al

    T. He et al. Optical skyrmions from metafibers with subwavelength features.Nature Communica- tions, 15(10141), 2024

    2024

  21. [28]

    He et al

    C. He et al. Complex vectorial optics through gradient index lens cascades.Nature Communications, 10(1):4264, 2019

    2019

  22. [29]

    Shen et al

    Y. Shen et al. Topologically controlled multiskyrmions in photonic gradient-index lenses.Physical Review Applied, 21(2):024025, 2024

    2024

  23. [31]

    L. C. Evans.Partial Differential Equations.2nd edn. Berkeley: American Mathematical Society, 2010. 13

    2010

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