Linking extended vector wave fields with momentum space topology

2); (2) present adress: University of W\"urzburg; (3) Delft University of Technology; 4); (4) present address; 5; (5) University of Stuttgart; 6); (6) University of Melbourne; 7) ((1) University of Duisburg-Essen

arxiv: 2604.26610 · v1 · submitted 2026-04-29 · ⚛️ physics.optics

Linking extended vector wave fields with momentum space topology

A. Neuhaus (1) , P. Gessler (1) , P. Dreher (1 , 2) , D. Janoschka (1) , A. R\"odl (1) , M. Manten (1) , Th. Bauer (3

show 27 more authors

4) M. Azhar (1) B. Frank (5) T. J. Davis (1 5 6) H. Giessen (5) K. Everschor-Sitte (1) F. Meyer zu Heringdorf (1 7) ((1) University of Duisburg-Essen Duisburg Germany (2) present adress: University of W\"urzburg W\"urzburg (3) Delft University of Technology Delft Netherlands (4) present address QphoX (5) University of Stuttgart Stuttgart (6) University of Melbourne Melbourne Australia (7) ICAN University of Duisburg-Essen Germany)

This is my paper

Pith reviewed 2026-05-07 12:42 UTC · model grok-4.3

classification ⚛️ physics.optics

keywords vector wave fieldsmomentum spacelinking numberBerry phaseHelmholtz decompositiontopological invariantssurface wavesaperiodic waves

0 comments

The pith

A linking number in momentum space serves as a topological invariant for aperiodic Helmholtz-decomposable vector wave fields.

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

Vector wave fields that lack position-space periodicity but remain Helmholtz-decomposable still possess only their intrinsic wave periodicity. The paper establishes that these fields carry a global topological invariant in momentum space, identified as the linking number and interpreted as a Berry phase. This invariant stays constant under continuous deformations that preserve the intrinsic periodicity. Experiments with electromagnetic and hydrodynamic surface waves demonstrate its robustness and reveal discrete jumps between distinct topological sectors. The result supplies a unified classification scheme for vector wave fields based on a single momentum-space quantity.

Core claim

For infinite vector waves that are aperiodic yet Helmholtz-decomposable, possessing only the wave's intrinsic periodicity, a topological invariant exists in momentum space. This invariant is the linking number, which represents a Berry phase. It captures the topology of the vector wave fields across both continuous and discrete momentum spaces, remains robust against deformations, and exhibits discrete transitions between distinct topological sectors as confirmed by electromagnetic and hydrodynamic surface-wave experiments.

What carries the argument

The linking number of extended vector wave fields computed in momentum space, functioning as a Berry phase that remains invariant under continuous deformations preserving intrinsic periodicity.

If this is right

Aperiodic vector wave fields can be classified globally by a single momentum-space topological number.
The invariant persists across electromagnetic and hydrodynamic realizations and survives continuous deformations that keep intrinsic periodicity.
Discrete jumps between topological sectors become observable when the wave field is deformed in a controlled way.
The same linking number supplies a common topological descriptor for both continuous and discrete momentum-space representations.

Where Pith is reading between the lines

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

The framework may apply to other vector fields obeying similar decomposition rules, such as acoustic or elastic waves in inhomogeneous media.
Momentum-space linking numbers could connect to existing classifications of polarization singularities or skyrmionic structures in wave optics.
Numerical extraction of the linking number from measured field data offers a practical diagnostic for topological transitions in laboratory wave systems.

Load-bearing premise

The wave fields are Helmholtz-decomposable and the linking number remains unchanged under continuous deformations that preserve the wave's intrinsic periodicity.

What would settle it

A continuous deformation of an aperiodic Helmholtz-decomposable vector wave field that alters the computed linking number without producing a discrete transition, or a mismatch between the linking number and the Berry phase calculated directly from the momentum-space field configuration.

read the original abstract

Topology describes properties of physical systems that remain constant under continuous deformations. For infinite vector waves, global topological invariants in position space are typically associated with periodic patterns. We demonstrate that even for aperiodic Helmholtz-decomposable wave fields, possessing only the wave's intrinsic periodicity, a topological invariant can be found in momentum space. This invariant, the linking number, represents a Berry phase. By utilizing electromagnetic and hydrodynamic surface waves, we confirm the robustness of the linking number against deformations, and experimentally observe discrete transitions between distinct topological sectors. The linking number captures the topology of vector wave fields across both continuous and discrete momentum spaces. Our work introduces a unified topological framework for vector wave fields, enabling their classification via a global invariant.

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

The paper defines a momentum-space linking number as a Berry-phase invariant for aperiodic Helmholtz-decomposable vector waves and backs it with EM and hydrodynamic experiments.

read the letter

The main point is that they extend the linking number to aperiodic vector wave fields that only have the wave's own wavelength as periodicity, place it in momentum space, and identify it with a Berry phase. They show it stays fixed under continuous deformations and demonstrate discrete jumps between sectors in both electromagnetic and hydrodynamic surface-wave experiments. That shift from position-space periodic patterns to momentum-space invariants for non-periodic fields is the concrete step forward. The experiments add value by showing the quantity is measurable and robust in two different physical systems. The analytic and numerical parts appear to support the construction without obvious internal contradictions. The main soft spot is that the abstract leaves the exact mapping from the vector field to the linking number somewhat implicit, so it is not yet clear how restrictive the Helmholtz-decomposability condition turns out to be in real setups or how cleanly the Berry-phase identification avoids double-counting with existing phase definitions. The claim of a unified framework is reasonable given the scope, but it will need the full derivations to stand up. This is worth sending to peer review for readers working on topological classification of waves in optics or fluids; the experimental confirmation gives it enough grounding to deserve referee time even if revisions are needed on the generality.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that aperiodic Helmholtz-decomposable vector wave fields possessing only intrinsic (wavelength) periodicity admit a topological invariant in momentum space, identified as the linking number and equivalent to a Berry phase. This invariant is shown to be robust under continuous deformations that preserve the intrinsic periodicity. The claim is supported by analytic construction, numerical examples, and experiments on electromagnetic and hydrodynamic surface waves that demonstrate invariance and discrete transitions between topological sectors. The linking number is presented as a global classifier for vector wave fields in both continuous and discrete momentum spaces.

Significance. If the central construction holds, the result supplies a unified topological framework that extends invariants beyond periodic position-space patterns to aperiodic vector fields via momentum space. The experimental confirmation across two distinct physical systems and the reported observation of sector transitions constitute concrete strengths that directly test the deformation invariance.

minor comments (3)

[Abstract] The abstract states experimental confirmation but does not specify the quantitative metric used to extract the linking number from measured fields; a brief parenthetical reference to the relevant figure or equation would improve clarity.
Figure captions should explicitly label the momentum-space trajectories or surfaces on which the linking number is computed to allow readers to verify the reported discrete transitions without consulting the main text.
Notation for the Helmholtz decomposition and the intrinsic periodicity condition should be introduced once in a dedicated paragraph rather than piecemeal across sections to aid readers unfamiliar with the vector-wave setup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We appreciate the recognition that the momentum-space linking number provides a unified topological framework for aperiodic Helmholtz-decomposable vector wave fields, supported by analytic, numerical, and experimental evidence across electromagnetic and hydrodynamic systems.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central claim constructs a momentum-space linking number as a Berry-phase topological invariant for aperiodic Helmholtz-decomposable vector wave fields using analytic definitions, numerical examples, and experimental observations on electromagnetic and hydrodynamic waves. This identification follows from standard properties of Berry phases and linking numbers applied to the wave equation under the stated periodicity and decomposability conditions, without reducing any prediction to a fitted parameter, self-definition, or load-bearing self-citation. The derivation remains independent of its own outputs and is externally benchmarked against known topological invariants in wave physics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the wave fields admit a Helmholtz decomposition and that the linking number is well-defined and invariant in momentum space.

axioms (1)

domain assumption Wave fields are Helmholtz-decomposable
Explicitly required in the abstract for the existence of the momentum-space topological invariant.

pith-pipeline@v0.9.0 · 9548 in / 981 out tokens · 63296 ms · 2026-05-07T12:42:58.393242+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages

[1]

M. J. Gilbert, Topological electronics.Communications Physics4(1), 70 (2021)

work page 2021
[2]

B. A. Bernevig,et al., Fractional quantization in insulators from Hall to Chern.Nature Physics 21(11), 1702–1713 (2025)

work page 2025
[3]

Wen, Colloquium: Zoo of quantum-topological phases of matter.Reviews of Modern Physics89(4), 041004 (2017)

X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter.Reviews of Modern Physics89(4), 041004 (2017)

work page 2017
[4]

Senthil, Symmetry-Protected Topological Phases of Quantum Matter.Annual Review of Condensed Matter Physics6, 299–324 (2015)

T. Senthil, Symmetry-Protected Topological Phases of Quantum Matter.Annual Review of Condensed Matter Physics6, 299–324 (2015)

work page 2015
[5]

L. Lu, J. D. Joannopoulos, M. Solja ˇci´c, Topological photonics.Nature Photonics8(11), 821– 829 (2014)

work page 2014
[6]

Bogdanov, A

A. Bogdanov, A. Hubert, Thermodynamically stable magnetic vortex states in magnetic crys- tals.Journal of Magnetism and Magnetic Materials138(3), 255–269 (1994)

work page 1994
[7]

U. K. R ¨oßler, A. N. Bogdanov, C. Pfleiderer, Spontaneous skyrmion ground states in magnetic metals.Nature442(7104), 797–801 (2006)

work page 2006
[8]

M¨ uhlbauer,et al., Skyrmion lattice in a chiral magnet.Science323(5916), 915–9 (2009)

S. M¨ uhlbauer,et al., Skyrmion lattice in a chiral magnet.Science323(5916), 915–9 (2009)

work page 2009
[9]

A. Fert, N. Reyren, V. Cros, Magnetic skyrmions: advances in physics and potential applica- tions.Nature Reviews Materials2(7), 17031 (2017)

work page 2017
[10]

Everschor-Sitte, J

K. Everschor-Sitte, J. Masell, R. M. Reeve, M. Kl ¨aui, Perspective: Magnetic skyrmions—Overview of recent progress in an active research field.Journal of Applied Physics 124(24), 240901 (2018)

work page 2018
[11]

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

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

work page 2018
[12]

L. P. Du, A. P. Yang, A. V. Zayats, X. C. Yuan, Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum.Nature Physics 15(7), 650 (2019). 14

work page 2019
[13]

T. J. Davis,et al., Ultrafast vector imaging of plasmonic skyrmion dynamics with deep sub- wavelength resolution.Science368(6489), eaba6415 (2020)

work page 2020
[14]

Gao,et al., Paraxial skyrmionic beams.Physical Review A102(5), 053513 (2020)

S. Gao,et al., Paraxial skyrmionic beams.Physical Review A102(5), 053513 (2020)

work page 2020
[15]

Shen,et al., Optical skyrmions and other topological quasiparticles of light.Nature Photonics 18(1), 15–25 (2024)

Y. Shen,et al., Optical skyrmions and other topological quasiparticles of light.Nature Photonics 18(1), 15–25 (2024)

work page 2024
[16]

D. A. Smirnova, F. Nori, K. Y. Bliokh, Water-Wave Vortices and Skyrmions.Physical Review Letters132(5), 054003 (2024)

work page 2024
[17]

Wang,et al., Topological water-wave structures manipulating particles.Nature638(8050), 394–400 (2025)

B. Wang,et al., Topological water-wave structures manipulating particles.Nature638(8050), 394–400 (2025)

work page 2025
[18]

Verbeeck, H

J. Verbeeck, H. Tian, P. Schattschneider, Production and application of electron vortex beams. Nature467(7313), 301–304 (2010)

work page 2010
[19]

B. J. McMorran,et al., Electron Vortex Beams with High Quanta of Orbital Angular Momen- tum.Science331(6014), 192–195 (2011)

work page 2011
[20]

A. A. Abrikosov, On the Magnetic Properties of Superconductors of the Second Group.Sov. Phys. JETP5, 1174–1182 (1957)

work page 1957
[21]

Essmann, H

U. Essmann, H. Tr ¨auble, The direct observation of individual flux lines in type II supercon- ductors.Physics Letters A24(10), 526–527 (1967)

work page 1967
[22]

Blatter, M

G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin, V. M. Vinokur, Vortices in high-temperature superconductors.Reviews of Modern Physics66(4), 1125–1388 (1994)

work page 1994
[23]

K. Y. Bliokh, H. Punzmann, H. Xia, F. Nori, M. Shats, Field theory spin and momentum in water waves.Science Advances8(3), eabm1295 (2022)

work page 2022
[24]

K. Y. Bliokh, Momentum, spin, and orbital angular momentum of electromagnetic, acoustic, and water waves.Contemporary Physics65(4), 219–238 (2024)

work page 2024
[25]

J. F. Nye, M. V. Berry, Dislocations in wave trains.Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences336(1605), 165–190 (1974). 15

work page 1974
[26]

Allen, M

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes.Physical Review A45(11), 8185–8189 (1992)

work page 1992
[27]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, Helical-wavefront laser beams produced with a spiral phaseplate.Optics Communications112(5), 321–327 (1994)

work page 1994
[28]

Kim,et al., Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens.Nano Lett10(2), 529–36 (2010)

H. Kim,et al., Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens.Nano Lett10(2), 529–36 (2010)

work page 2010
[29]

A. M. Yao, M. J. Padgett, Orbital angular momentum: origins, behavior and applications. Advances in Optics and Photonics3(2), 161–204 (2011)

work page 2011
[30]

Spektor,et al., Revealing the subfemtosecond dynamics of orbital angular momentum in nanoplasmonic vortices.Science355(6330), 1187–1191 (2017)

G. Spektor,et al., Revealing the subfemtosecond dynamics of orbital angular momentum in nanoplasmonic vortices.Science355(6330), 1187–1191 (2017)

work page 2017
[31]

Dai,et al., Plasmonic topological quasiparticle on the nanometre and femtosecond scales

Y. Dai,et al., Plasmonic topological quasiparticle on the nanometre and femtosecond scales. Nature588(7839), 616–619 (2020)

work page 2020
[32]

Dai,et al., Ultrafast microscopy of a twisted plasmonic spin skyrmion.Applied Physics Reviews9(1) (2022)

Y. Dai,et al., Ultrafast microscopy of a twisted plasmonic spin skyrmion.Applied Physics Reviews9(1) (2022)

work page 2022
[33]

Ghosh, S

A. Ghosh, S. Yang, Y. Dai, H. Petek, The Spin Texture Topology of Polygonal Plasmon Fields. ACS Photonics10(1), 13–23 (2023)

work page 2023
[34]

Dreher,et al., Spatiotemporal topology of plasmonic spin meron pairs revealed by polari- metric photo-emission microscopy.Advanced Photonics6(6), 066007 (2024)

P. Dreher,et al., Spatiotemporal topology of plasmonic spin meron pairs revealed by polari- metric photo-emission microscopy.Advanced Photonics6(6), 066007 (2024)

work page 2024
[35]

Tian,et al., N ´eel-type optical target skyrmions inherited from evanescent electromagnetic fields with rotational symmetry.Nanoscale15, 13224–13232 (2023)

B. Tian,et al., N ´eel-type optical target skyrmions inherited from evanescent electromagnetic fields with rotational symmetry.Nanoscale15, 13224–13232 (2023)

work page 2023
[36]

M. V. Berry, M. R. Dennis, Phase singularities in isotropic random waves.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences456(2001), 2059–2079 (2000). 16

work page 2001
[37]

Berry, M

M. Berry, M. Dennis, Polarization singularities in isotropic random vector waves.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences457(2005), 141–155 (2001)

work page 2005
[38]

M. V. Berry, Quantal Phase Factors Accompanying Adiabatic Changes.Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences392(1802), 45–57 (1984)

work page 1984
[39]

K. Y. Bliokh, Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium.Journal Of Optics A: Pure And Applied Optics11(9), 094009 (2009)

work page 2009
[40]

Materials and methods are available as supplementary material

work page
[41]

Wildeman, Real-time quantitative Schlieren imaging by fast Fourier demodulation of a checkered backdrop.Experiments in Fluids59(6), 97 (2018)

S. Wildeman, Real-time quantitative Schlieren imaging by fast Fourier demodulation of a checkered backdrop.Experiments in Fluids59(6), 97 (2018)

work page 2018
[42]

Bauer,et al., Ultrafast Time Dynamics of Plasmonic Fractional Orbital Angular Momentum

T. Bauer,et al., Ultrafast Time Dynamics of Plasmonic Fractional Orbital Angular Momentum. ACS Photonics10(12), 4252 – 4258 (2023)

work page 2023
[43]

M. V. Berry, W. Liu, No general relation between phase vortices and orbital angular momentum. Journal of Physics A: Mathematical and Theoretical55(37), 374001 (2022)

work page 2022
[44]

Davis, F.-J

T. Davis, F.-J. Meyer zu Heringdorf, H. Giessen, Topology of Surface Plasmon Polaritons with Integer and Fractional Orbital Angular Momentum.ACS Photonics10(10), 3772 – 3780 (2023)

work page 2023
[45]

N. B. Simpson, K. Dholakia, L. Allen, M. J. Padgett, Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner.Optics Letters22(1), 52–54 (1997)

work page 1997
[46]

Leach, E

J. Leach, E. Yao, M. J. Padgett, Observation of the vortex structure of a non-integer vortex beam.New Journal of Physics6(1), 71 (2004)

work page 2004
[47]

M. V. Berry, Optical vortices evolving from helicoidal integer and fractional phase steps. Journal of Optics A: Pure and Applied Optics6(2), 259 (2004)

work page 2004
[48]

Gbur, Fractional vortex Hilbert’s Hotel.Optica3(3), 222–225 (2016)

G. Gbur, Fractional vortex Hilbert’s Hotel.Optica3(3), 222–225 (2016). 17

work page 2016
[49]

J. B. G ¨otte,et al., Light beams with fractional orbital angular momentum and their vortex structure.Optics Express16(2), 993–1006 (2008)

work page 2008
[50]

Tsesses,et al., Four-dimensional conserved topological charge vectors in plasmonic qua- sicrystals.Science387(6734), 644–648 (2025)

S. Tsesses,et al., Four-dimensional conserved topological charge vectors in plasmonic qua- sicrystals.Science387(6734), 644–648 (2025)

work page 2025
[51]

Bucher,et al., Superluminal correlations in ensembles of optical phase singularities.Nature 651(8107), 920–926 (2026)

T. Bucher,et al., Superluminal correlations in ensembles of optical phase singularities.Nature 651(8107), 920–926 (2026)

work page 2026
[52]

Schwab,et al., Skyrmion bags of light in plasmonic moir ´e superlattices.Nature Physics 21(6), 988 – 994 (2025)

J. Schwab,et al., Skyrmion bags of light in plasmonic moir ´e superlattices.Nature Physics 21(6), 988 – 994 (2025)

work page 2025
[53]

Guo,et al., Topological robustness of classical and quantum optical skyrmions in atmo- spheric turbulence.Nature Communications(2026)

Z. Guo,et al., Topological robustness of classical and quantum optical skyrmions in atmo- spheric turbulence.Nature Communications(2026)

work page 2026
[54]

Z. Wang, Y. Chong, J. D. Joannopoulos, M. Soljaci ´c, Observation of unidirectional backscattering-immune topological electromagnetic states.Nature461(7265), 772–775 (2009)

work page 2009
[55]

Mei,et al., Robust and High-Capacity Phononic Communications through Topological Edge States by Discrete Degree-of-Freedom Multiplexing.Physical Review Applied12(5) (2019)

J. Mei,et al., Robust and High-Capacity Phononic Communications through Topological Edge States by Discrete Degree-of-Freedom Multiplexing.Physical Review Applied12(5) (2019)

work page 2019
[56]

A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, Entanglement of the orbital angular momentum states of photons.Nature412(6844), 313–316 (2001)

work page 2001
[57]

Hamadou Ibrahim, F

A. Hamadou Ibrahim, F. S. Roux, M. McLaren, T. Konrad, A. Forbes, Orbital-angular- momentum entanglement in turbulence.Physical Review A88(1), 012312 (2013), pRA

work page 2013
[58]

Trichili,et al., Optical communication beyond orbital angular momentum.Scientific Reports 6(1), 27674 (2016)

A. Trichili,et al., Optical communication beyond orbital angular momentum.Scientific Reports 6(1), 27674 (2016)

work page 2016
[59]

Radha, M

B. Radha, M. Arif, R. Datta, T. K. Kundu, G. U. Kulkarni, Movable Au microplates as fluorescence enhancing substrates for live cells.Nano Research3(10), 738–747 (2010)

work page 2010
[60]

Radha, G

B. Radha, G. U. Kulkarni, A Real Time Microscopy Study of the Growth of Giant Au Mi- croplates.Crystal Growth & Design11(1), 320–327 (2011). 18

work page 2011
[61]

Spektor,et al., Mixing the Light Spin with Plasmon Orbit by Nonlinear Light-Matter Interaction in Gold.Physical Review X9(2), 021031 (2019)

G. Spektor,et al., Mixing the Light Spin with Plasmon Orbit by Nonlinear Light-Matter Interaction in Gold.Physical Review X9(2), 021031 (2019)

work page 2019
[62]

Neuhaus,et al., Polarimetric interferometer with attosecond stability for time-resolved pump-probe photoemission microscopy.Ultramicroscopy279, 114248 (2026)

A. Neuhaus,et al., Polarimetric interferometer with attosecond stability for time-resolved pump-probe photoemission microscopy.Ultramicroscopy279, 114248 (2026)

work page 2026
[63]

A. Kubo, N. Pontius, H. Petek, Femtosecond microscopy of surface plasmon polariton wave packet evolution at the silver/vacuum interface.Nano Lett7(2), 470–5 (2007)

work page 2007
[64]

L. I. Chelaru, F.-J. Meyer zu Heringdorf, In situ monitoring of surface plasmons in single- crystalline Ag-nanowires.Surface Science601(18), 4541–4545 (2007)

work page 2007
[65]

Schmidt,et al., SPELEEM: Combining LEEM and Spectroscopic Imaging.Surface Review and Letters05(06), 1287–1296 (1998)

T. Schmidt,et al., SPELEEM: Combining LEEM and Spectroscopic Imaging.Surface Review and Letters05(06), 1287–1296 (1998)

work page 1998
[66]

Janoschka,et al., Implementation and operation of a fiber-coupled CMOS detector in a low energy electron Microscope.Ultramicroscopy221, 113180 (2021)

D. Janoschka,et al., Implementation and operation of a fiber-coupled CMOS detector in a low energy electron Microscope.Ultramicroscopy221, 113180 (2021)

work page 2021
[67]

Dreher,et al., Quantitative determination of the electric field strength in a plasmon focus from ponderomotive energy shifts.Nanophotonics(2022)

P. Dreher,et al., Quantitative determination of the electric field strength in a plasmon focus from ponderomotive energy shifts.Nanophotonics(2022)

work page 2022
[68]

Burresi,et al., Observation of Polarization Singularities at the Nanoscale.Phys

M. Burresi,et al., Observation of Polarization Singularities at the Nanoscale.Phys. Rev. Lett. 102, 033902 (2009)

work page 2009
[69]

Belinfante, On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields.Physica7(5), 449–474 (1940)

F. Belinfante, On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields.Physica7(5), 449–474 (1940). Acknowledgments We thank Julian Schwaab and Laurens Kuipers for discussions and valuable input. Funding:We acknowledge support from the ERC (grant no. 321331-Complexplas (B.F. and H...

work page 1940

[1] [1]

M. J. Gilbert, Topological electronics.Communications Physics4(1), 70 (2021)

work page 2021

[2] [2]

B. A. Bernevig,et al., Fractional quantization in insulators from Hall to Chern.Nature Physics 21(11), 1702–1713 (2025)

work page 2025

[3] [3]

Wen, Colloquium: Zoo of quantum-topological phases of matter.Reviews of Modern Physics89(4), 041004 (2017)

X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter.Reviews of Modern Physics89(4), 041004 (2017)

work page 2017

[4] [4]

Senthil, Symmetry-Protected Topological Phases of Quantum Matter.Annual Review of Condensed Matter Physics6, 299–324 (2015)

T. Senthil, Symmetry-Protected Topological Phases of Quantum Matter.Annual Review of Condensed Matter Physics6, 299–324 (2015)

work page 2015

[5] [5]

L. Lu, J. D. Joannopoulos, M. Solja ˇci´c, Topological photonics.Nature Photonics8(11), 821– 829 (2014)

work page 2014

[6] [6]

Bogdanov, A

A. Bogdanov, A. Hubert, Thermodynamically stable magnetic vortex states in magnetic crys- tals.Journal of Magnetism and Magnetic Materials138(3), 255–269 (1994)

work page 1994

[7] [7]

U. K. R ¨oßler, A. N. Bogdanov, C. Pfleiderer, Spontaneous skyrmion ground states in magnetic metals.Nature442(7104), 797–801 (2006)

work page 2006

[8] [8]

M¨ uhlbauer,et al., Skyrmion lattice in a chiral magnet.Science323(5916), 915–9 (2009)

S. M¨ uhlbauer,et al., Skyrmion lattice in a chiral magnet.Science323(5916), 915–9 (2009)

work page 2009

[9] [9]

A. Fert, N. Reyren, V. Cros, Magnetic skyrmions: advances in physics and potential applica- tions.Nature Reviews Materials2(7), 17031 (2017)

work page 2017

[10] [10]

Everschor-Sitte, J

K. Everschor-Sitte, J. Masell, R. M. Reeve, M. Kl ¨aui, Perspective: Magnetic skyrmions—Overview of recent progress in an active research field.Journal of Applied Physics 124(24), 240901 (2018)

work page 2018

[11] [11]

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

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

work page 2018

[12] [12]

L. P. Du, A. P. Yang, A. V. Zayats, X. C. Yuan, Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum.Nature Physics 15(7), 650 (2019). 14

work page 2019

[13] [13]

T. J. Davis,et al., Ultrafast vector imaging of plasmonic skyrmion dynamics with deep sub- wavelength resolution.Science368(6489), eaba6415 (2020)

work page 2020

[14] [14]

Gao,et al., Paraxial skyrmionic beams.Physical Review A102(5), 053513 (2020)

S. Gao,et al., Paraxial skyrmionic beams.Physical Review A102(5), 053513 (2020)

work page 2020

[15] [15]

Shen,et al., Optical skyrmions and other topological quasiparticles of light.Nature Photonics 18(1), 15–25 (2024)

Y. Shen,et al., Optical skyrmions and other topological quasiparticles of light.Nature Photonics 18(1), 15–25 (2024)

work page 2024

[16] [16]

D. A. Smirnova, F. Nori, K. Y. Bliokh, Water-Wave Vortices and Skyrmions.Physical Review Letters132(5), 054003 (2024)

work page 2024

[17] [17]

Wang,et al., Topological water-wave structures manipulating particles.Nature638(8050), 394–400 (2025)

B. Wang,et al., Topological water-wave structures manipulating particles.Nature638(8050), 394–400 (2025)

work page 2025

[18] [18]

Verbeeck, H

J. Verbeeck, H. Tian, P. Schattschneider, Production and application of electron vortex beams. Nature467(7313), 301–304 (2010)

work page 2010

[19] [19]

B. J. McMorran,et al., Electron Vortex Beams with High Quanta of Orbital Angular Momen- tum.Science331(6014), 192–195 (2011)

work page 2011

[20] [20]

A. A. Abrikosov, On the Magnetic Properties of Superconductors of the Second Group.Sov. Phys. JETP5, 1174–1182 (1957)

work page 1957

[21] [21]

Essmann, H

U. Essmann, H. Tr ¨auble, The direct observation of individual flux lines in type II supercon- ductors.Physics Letters A24(10), 526–527 (1967)

work page 1967

[22] [22]

Blatter, M

G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin, V. M. Vinokur, Vortices in high-temperature superconductors.Reviews of Modern Physics66(4), 1125–1388 (1994)

work page 1994

[23] [23]

K. Y. Bliokh, H. Punzmann, H. Xia, F. Nori, M. Shats, Field theory spin and momentum in water waves.Science Advances8(3), eabm1295 (2022)

work page 2022

[24] [24]

K. Y. Bliokh, Momentum, spin, and orbital angular momentum of electromagnetic, acoustic, and water waves.Contemporary Physics65(4), 219–238 (2024)

work page 2024

[25] [25]

J. F. Nye, M. V. Berry, Dislocations in wave trains.Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences336(1605), 165–190 (1974). 15

work page 1974

[26] [26]

Allen, M

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes.Physical Review A45(11), 8185–8189 (1992)

work page 1992

[27] [27]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, Helical-wavefront laser beams produced with a spiral phaseplate.Optics Communications112(5), 321–327 (1994)

work page 1994

[28] [28]

Kim,et al., Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens.Nano Lett10(2), 529–36 (2010)

H. Kim,et al., Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens.Nano Lett10(2), 529–36 (2010)

work page 2010

[29] [29]

A. M. Yao, M. J. Padgett, Orbital angular momentum: origins, behavior and applications. Advances in Optics and Photonics3(2), 161–204 (2011)

work page 2011

[30] [30]

Spektor,et al., Revealing the subfemtosecond dynamics of orbital angular momentum in nanoplasmonic vortices.Science355(6330), 1187–1191 (2017)

G. Spektor,et al., Revealing the subfemtosecond dynamics of orbital angular momentum in nanoplasmonic vortices.Science355(6330), 1187–1191 (2017)

work page 2017

[31] [31]

Dai,et al., Plasmonic topological quasiparticle on the nanometre and femtosecond scales

Y. Dai,et al., Plasmonic topological quasiparticle on the nanometre and femtosecond scales. Nature588(7839), 616–619 (2020)

work page 2020

[32] [32]

Dai,et al., Ultrafast microscopy of a twisted plasmonic spin skyrmion.Applied Physics Reviews9(1) (2022)

Y. Dai,et al., Ultrafast microscopy of a twisted plasmonic spin skyrmion.Applied Physics Reviews9(1) (2022)

work page 2022

[33] [33]

Ghosh, S

A. Ghosh, S. Yang, Y. Dai, H. Petek, The Spin Texture Topology of Polygonal Plasmon Fields. ACS Photonics10(1), 13–23 (2023)

work page 2023

[34] [34]

Dreher,et al., Spatiotemporal topology of plasmonic spin meron pairs revealed by polari- metric photo-emission microscopy.Advanced Photonics6(6), 066007 (2024)

P. Dreher,et al., Spatiotemporal topology of plasmonic spin meron pairs revealed by polari- metric photo-emission microscopy.Advanced Photonics6(6), 066007 (2024)

work page 2024

[35] [35]

Tian,et al., N ´eel-type optical target skyrmions inherited from evanescent electromagnetic fields with rotational symmetry.Nanoscale15, 13224–13232 (2023)

B. Tian,et al., N ´eel-type optical target skyrmions inherited from evanescent electromagnetic fields with rotational symmetry.Nanoscale15, 13224–13232 (2023)

work page 2023

[36] [36]

M. V. Berry, M. R. Dennis, Phase singularities in isotropic random waves.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences456(2001), 2059–2079 (2000). 16

work page 2001

[37] [37]

Berry, M

M. Berry, M. Dennis, Polarization singularities in isotropic random vector waves.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences457(2005), 141–155 (2001)

work page 2005

[38] [38]

M. V. Berry, Quantal Phase Factors Accompanying Adiabatic Changes.Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences392(1802), 45–57 (1984)

work page 1984

[39] [39]

K. Y. Bliokh, Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium.Journal Of Optics A: Pure And Applied Optics11(9), 094009 (2009)

work page 2009

[40] [40]

Materials and methods are available as supplementary material

work page

[41] [41]

Wildeman, Real-time quantitative Schlieren imaging by fast Fourier demodulation of a checkered backdrop.Experiments in Fluids59(6), 97 (2018)

S. Wildeman, Real-time quantitative Schlieren imaging by fast Fourier demodulation of a checkered backdrop.Experiments in Fluids59(6), 97 (2018)

work page 2018

[42] [42]

Bauer,et al., Ultrafast Time Dynamics of Plasmonic Fractional Orbital Angular Momentum

T. Bauer,et al., Ultrafast Time Dynamics of Plasmonic Fractional Orbital Angular Momentum. ACS Photonics10(12), 4252 – 4258 (2023)

work page 2023

[43] [43]

M. V. Berry, W. Liu, No general relation between phase vortices and orbital angular momentum. Journal of Physics A: Mathematical and Theoretical55(37), 374001 (2022)

work page 2022

[44] [44]

Davis, F.-J

T. Davis, F.-J. Meyer zu Heringdorf, H. Giessen, Topology of Surface Plasmon Polaritons with Integer and Fractional Orbital Angular Momentum.ACS Photonics10(10), 3772 – 3780 (2023)

work page 2023

[45] [45]

N. B. Simpson, K. Dholakia, L. Allen, M. J. Padgett, Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner.Optics Letters22(1), 52–54 (1997)

work page 1997

[46] [46]

Leach, E

J. Leach, E. Yao, M. J. Padgett, Observation of the vortex structure of a non-integer vortex beam.New Journal of Physics6(1), 71 (2004)

work page 2004

[47] [47]

M. V. Berry, Optical vortices evolving from helicoidal integer and fractional phase steps. Journal of Optics A: Pure and Applied Optics6(2), 259 (2004)

work page 2004

[48] [48]

Gbur, Fractional vortex Hilbert’s Hotel.Optica3(3), 222–225 (2016)

G. Gbur, Fractional vortex Hilbert’s Hotel.Optica3(3), 222–225 (2016). 17

work page 2016

[49] [49]

J. B. G ¨otte,et al., Light beams with fractional orbital angular momentum and their vortex structure.Optics Express16(2), 993–1006 (2008)

work page 2008

[50] [50]

Tsesses,et al., Four-dimensional conserved topological charge vectors in plasmonic qua- sicrystals.Science387(6734), 644–648 (2025)

S. Tsesses,et al., Four-dimensional conserved topological charge vectors in plasmonic qua- sicrystals.Science387(6734), 644–648 (2025)

work page 2025

[51] [51]

Bucher,et al., Superluminal correlations in ensembles of optical phase singularities.Nature 651(8107), 920–926 (2026)

T. Bucher,et al., Superluminal correlations in ensembles of optical phase singularities.Nature 651(8107), 920–926 (2026)

work page 2026

[52] [52]

Schwab,et al., Skyrmion bags of light in plasmonic moir ´e superlattices.Nature Physics 21(6), 988 – 994 (2025)

J. Schwab,et al., Skyrmion bags of light in plasmonic moir ´e superlattices.Nature Physics 21(6), 988 – 994 (2025)

work page 2025

[53] [53]

Guo,et al., Topological robustness of classical and quantum optical skyrmions in atmo- spheric turbulence.Nature Communications(2026)

Z. Guo,et al., Topological robustness of classical and quantum optical skyrmions in atmo- spheric turbulence.Nature Communications(2026)

work page 2026

[54] [54]

Z. Wang, Y. Chong, J. D. Joannopoulos, M. Soljaci ´c, Observation of unidirectional backscattering-immune topological electromagnetic states.Nature461(7265), 772–775 (2009)

work page 2009

[55] [55]

Mei,et al., Robust and High-Capacity Phononic Communications through Topological Edge States by Discrete Degree-of-Freedom Multiplexing.Physical Review Applied12(5) (2019)

J. Mei,et al., Robust and High-Capacity Phononic Communications through Topological Edge States by Discrete Degree-of-Freedom Multiplexing.Physical Review Applied12(5) (2019)

work page 2019

[56] [56]

A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, Entanglement of the orbital angular momentum states of photons.Nature412(6844), 313–316 (2001)

work page 2001

[57] [57]

Hamadou Ibrahim, F

A. Hamadou Ibrahim, F. S. Roux, M. McLaren, T. Konrad, A. Forbes, Orbital-angular- momentum entanglement in turbulence.Physical Review A88(1), 012312 (2013), pRA

work page 2013

[58] [58]

Trichili,et al., Optical communication beyond orbital angular momentum.Scientific Reports 6(1), 27674 (2016)

A. Trichili,et al., Optical communication beyond orbital angular momentum.Scientific Reports 6(1), 27674 (2016)

work page 2016

[59] [59]

Radha, M

B. Radha, M. Arif, R. Datta, T. K. Kundu, G. U. Kulkarni, Movable Au microplates as fluorescence enhancing substrates for live cells.Nano Research3(10), 738–747 (2010)

work page 2010

[60] [60]

Radha, G

B. Radha, G. U. Kulkarni, A Real Time Microscopy Study of the Growth of Giant Au Mi- croplates.Crystal Growth & Design11(1), 320–327 (2011). 18

work page 2011

[61] [61]

Spektor,et al., Mixing the Light Spin with Plasmon Orbit by Nonlinear Light-Matter Interaction in Gold.Physical Review X9(2), 021031 (2019)

G. Spektor,et al., Mixing the Light Spin with Plasmon Orbit by Nonlinear Light-Matter Interaction in Gold.Physical Review X9(2), 021031 (2019)

work page 2019

[62] [62]

Neuhaus,et al., Polarimetric interferometer with attosecond stability for time-resolved pump-probe photoemission microscopy.Ultramicroscopy279, 114248 (2026)

A. Neuhaus,et al., Polarimetric interferometer with attosecond stability for time-resolved pump-probe photoemission microscopy.Ultramicroscopy279, 114248 (2026)

work page 2026

[63] [63]

A. Kubo, N. Pontius, H. Petek, Femtosecond microscopy of surface plasmon polariton wave packet evolution at the silver/vacuum interface.Nano Lett7(2), 470–5 (2007)

work page 2007

[64] [64]

L. I. Chelaru, F.-J. Meyer zu Heringdorf, In situ monitoring of surface plasmons in single- crystalline Ag-nanowires.Surface Science601(18), 4541–4545 (2007)

work page 2007

[65] [65]

Schmidt,et al., SPELEEM: Combining LEEM and Spectroscopic Imaging.Surface Review and Letters05(06), 1287–1296 (1998)

T. Schmidt,et al., SPELEEM: Combining LEEM and Spectroscopic Imaging.Surface Review and Letters05(06), 1287–1296 (1998)

work page 1998

[66] [66]

Janoschka,et al., Implementation and operation of a fiber-coupled CMOS detector in a low energy electron Microscope.Ultramicroscopy221, 113180 (2021)

D. Janoschka,et al., Implementation and operation of a fiber-coupled CMOS detector in a low energy electron Microscope.Ultramicroscopy221, 113180 (2021)

work page 2021

[67] [67]

Dreher,et al., Quantitative determination of the electric field strength in a plasmon focus from ponderomotive energy shifts.Nanophotonics(2022)

P. Dreher,et al., Quantitative determination of the electric field strength in a plasmon focus from ponderomotive energy shifts.Nanophotonics(2022)

work page 2022

[68] [68]

Burresi,et al., Observation of Polarization Singularities at the Nanoscale.Phys

M. Burresi,et al., Observation of Polarization Singularities at the Nanoscale.Phys. Rev. Lett. 102, 033902 (2009)

work page 2009

[69] [69]

Belinfante, On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields.Physica7(5), 449–474 (1940)

F. Belinfante, On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields.Physica7(5), 449–474 (1940). Acknowledgments We thank Julian Schwaab and Laurens Kuipers for discussions and valuable input. Funding:We acknowledge support from the ERC (grant no. 321331-Complexplas (B.F. and H...

work page 1940