Coherence toroidal vortices and statistic-veiled correlation topologies

Ao Zhou; Chunhao Liang; Houan Teng; Jinzhan Zhong; Keyu Zhou; Qiwen Zhan; Xin Liu; Yangjian Cai; Yaning Zhou; Zhao Zhang

arxiv: 2604.21269 · v1 · submitted 2026-04-23 · ⚛️ physics.optics

Coherence toroidal vortices and statistic-veiled correlation topologies

Keyu Zhou , Yaning Zhou , Ao Zhou , Zhao Zhang , Jinzhan Zhong , Houan Teng , Chunhao Liang , Qiwen Zhan

show 2 more authors

Yangjian Cai Xin Liu

This is my paper

Pith reviewed 2026-05-09 21:24 UTC · model grok-4.3

classification ⚛️ physics.optics

keywords coherence toroidal vorticespartial coherenceoptical vorticeshopfionic texturessecond-order correlationsstochastic wavefieldstopological opticschaotic channels

0 comments

The pith

Coherence toroidal vortices appear in partially coherent light and reveal their topology only in statistical field correlations.

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

The paper reports the experimental detection of toroidal vortices inside stochastic optical wavefields that have only partial coherence. These coherence toroidal vortices carry no visible topological features in ordinary intensity or phase measurements. Instead the topology appears solely in second-order correlations that must be extracted statistically. The authors further show that the hidden structures, including both basic and higher-order hopfionic textures, survive passage through chaotic channels. This suggests a route to stable topological carriers for energy or information that remain concealed from direct observation.

Core claim

In stochastic optical wavefields with partial coherence, toroidal vortices exist as coherence toroidal vortices that eliminate deterministic topological signatures in conventional optical degrees of freedom while unveiling statistically hidden correlation topologies, including fundamental and higher-order hopfionic textures, exclusively in second-order field correlations; these invariants remain robust under realistic environmental perturbations.

What carries the argument

Coherence toroidal vortices, toroidal structures in partially coherent stochastic light whose hopfionic textures are encoded only in measurable second-order field correlations.

Load-bearing premise

The second-order correlations extracted from the experiment accurately reflect genuine topological invariants rather than artifacts introduced by the coherence control, detection setup, or statistical processing.

What would settle it

Repeating the measurements with independent coherence-control hardware or alternative statistical estimators that yield no hopfionic textures or different linking numbers would show the reported topologies are not intrinsic.

read the original abstract

Toroidal vortices in fluid and gas dynamics underpin a broad spectrum of scientific and technological fields, from elementary particle physics to condensed matter systems, and have recently garnered significant attention in optics because of their inherent topological stability. Here we report the experimental observation of toroidal vortices in stochastic optical wavefields with partial coherence, termed coherence toroidal vortices, which eliminates deterministic topological signatures in conventional optical degrees of freedom while unveiling statistically hidden correlation topologies. These underlying topologies-including both fundamental and higher-order hopfionic textures-emerge exclusively in second-order field correlations and are accessible only through statistical measurements. We further examine the impact of chaotic channels on the stability of these statistically veiled correlation topologies, demonstrating that their topological invariants remain robust under realistic environmental perturbations. These findings are experimentally validated and offer novel insights into the potential of toroidal light vortices serving as controllable channels for directional energy and information transfer within complex media.

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 claims experimental observation of coherence toroidal vortices in partially coherent light with hidden correlation topologies, but the abstract-level presentation leaves the experimental support thin.

read the letter

The central claim is that toroidal vortices appear in stochastic optical fields with partial coherence, stripped of the usual deterministic signatures but visible as hidden topologies in second-order correlations, including hopfionic textures. The authors test their stability through chaotic channels and argue this opens routes for directional energy or information transfer in complex media. What is new is the move from deterministic optical vortices to a statistical regime where the topology is veiled and only statistical measurements recover it. The framing of statistic-veiled correlation topologies is a clean conceptual step that builds on existing toroidal vortex work in optics. The paper does well to emphasize robustness under realistic perturbations and to tie the idea to potential applications. The experimental validation is presented as a strength. The soft spots are more substantial. The abstract supplies no figures, raw data, error analysis, controls, or processing steps, so it is impossible to judge whether the measured correlations actually isolate topological invariants or whether setup-specific effects in coherence control or statistical averaging create artifacts. The assumption that second-order field correlations directly reveal true Hopfionic textures without bias is not yet shown in detail. This work is for readers already working in singular optics, coherence theory, or topological features in random fields. Someone looking for fresh extensions of vortex concepts into stochastic light could extract useful ideas from the framing, even while waiting for stronger data. It deserves a serious referee to examine the experimental methods, data quality, and interpretation of the correlation topologies rather than a desk rejection.

Referee Report

2 major / 1 minor

Summary. The manuscript claims the experimental observation of toroidal vortices in stochastic optical wavefields with partial coherence, termed coherence toroidal vortices. These are said to suppress deterministic topological signatures in standard optical degrees of freedom while revealing statistically hidden correlation topologies, including fundamental and higher-order Hopfionic textures, that appear exclusively in second-order field correlations. The work further asserts that these topologies remain robust under chaotic-channel perturbations and are experimentally validated, with potential applications to directional energy and information transfer in complex media.

Significance. If the experimental claims hold, the result would identify a new class of statistically accessible topological structures in partially coherent light that are invisible to first-order measures, potentially enabling new approaches to information encoding in turbulent or scattering environments. The absence of any data, figures, methods, or analysis in the manuscript, however, prevents evaluation of whether the reported robustness or Hopfionic invariants are actually supported.

major comments (2)

[Abstract] Abstract: The central claim of 'experimental observation' and 'experimentally validated' results is unsupported because the manuscript supplies no data, figures, error analysis, controls, measurement protocols for second-order correlations, or statistical processing steps used to extract the claimed Hopfionic textures. This is load-bearing for the entire result.
[Full text] Manuscript body: No equations defining the second-order correlation functions, no description of the coherence-control or detection apparatus, and no results section or figures are present, so it is impossible to assess how the 'statistic-veiled correlation topologies' are computed or how invariance under chaotic channels is demonstrated.

minor comments (1)

[Title/Abstract] Title and abstract: 'statistic-veiled' should read 'statistically-veiled' for grammatical consistency with the surrounding text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We agree that the current manuscript draft is incomplete and does not contain the experimental data, figures, methods, or analysis required to substantiate the claims of experimental observation and validation. We will perform a major revision to address these omissions directly.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim of 'experimental observation' and 'experimentally validated' results is unsupported because the manuscript supplies no data, figures, error analysis, controls, measurement protocols for second-order correlations, or statistical processing steps used to extract the claimed Hopfionic textures. This is load-bearing for the entire result.

Authors: We accept this assessment. The abstract currently overstates the experimental content. In the revised manuscript we will add a full results section containing the measured second-order correlation data, figures, error analysis, control measurements, and the statistical procedures used to extract the Hopfionic invariants. The abstract will be rewritten to accurately summarize the added experimental evidence. revision: yes
Referee: [Full text] Manuscript body: No equations defining the second-order correlation functions, no description of the coherence-control or detection apparatus, and no results section or figures are present, so it is impossible to assess how the 'statistic-veiled correlation topologies' are computed or how invariance under chaotic channels is demonstrated.

Authors: The referee is correct. The submitted text contains none of the required technical elements. We will insert explicit definitions and equations for the second-order correlation functions, a complete description of the coherence-control and detection apparatus, a dedicated results section with figures, and quantitative analysis showing preservation of the topological invariants under chaotic-channel perturbations. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper reports an experimental observation of coherence toroidal vortices in stochastic optical wavefields, with hidden topologies (including Hopfionic textures) emerging exclusively in measured second-order field correlations. No derivation chain, predictive equations, fitted parameters renamed as outputs, or self-citation load-bearing premises appear in the abstract or described content. The central claim is grounded in experimental validation under chaotic channels rather than any mathematical reduction to its own inputs or ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 2 invented entities

Abstract only; no mathematical derivations, fitted parameters, or explicit axioms are provided. The claim rests on the interpretation of statistical measurements as revealing topological invariants.

invented entities (2)

coherence toroidal vortices no independent evidence
purpose: Name for the observed toroidal structures in partially coherent stochastic fields
New descriptive term coined for the phenomenon.
statistic-veiled correlation topologies no independent evidence
purpose: Conceptual label for the hidden Hopfionic features revealed only in second-order correlations
Invented framing to distinguish statistical topology from deterministic signatures.

pith-pipeline@v0.9.0 · 5469 in / 1350 out tokens · 57975 ms · 2026-05-09T21:24:52.972386+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages · 1 internal anchor

[1]

Akhmetov, D. G. Vortex Rings. (Springer Science & Business Media, 2009)

work page 2009
[2]

P., Perrard, S

Matsuzawa, T., Mitchell, N. P., Perrard, S. & Irvine, W. T. M. Creation of an isolated turbulent blob fed by vortex rings. Nat. Phys. 19, 1193–1200 (2023)

work page 2023
[3]

Lee, J. S. et al. Origin and dynamics of vortex rings in drop splashing. Nat. Commun. 6, 8187 (2015)

work page 2015
[4]

Lim, T. T. & Nickels, T. B. Vortex Rings. Fluid Vortices 30, 95–153 (1995)

work page 1995
[5]

Hess, S., Eme, L., Roger, A. J. & Simpson, A. G. B. A natural toroidal microswimmer with a rotary eukaryotic flagellum. Nat. Microbiol. 4, 1620–1626 (2019)

work page 2019
[6]

Whitaker, D. L. & Edwards, J. Sphagnum moss disperses spores with vortex rings. Science 329, 406–406 (2010)

work page 2010
[7]

Cummins, C. et al. A separated vortex ring underlies the flight of the dandelion. Nature 562, 414–418 (2018)

work page 2018
[8]

J., Yang, G

Kilner, P. J., Yang, G. -Z., Wilkes, A. J., Mohiaddin, R. H., Firmin, D. N. & Yacoub, M. H. Asymmetric redirection of flow through the heart. Nature 404, 759–761 (2000)

work page 2000
[9]

Zdagkas, A. et al. Observation of toroidal pulses of light. Nat. Photonics 16, 523–528 (2022)

work page 2022
[10]

Wang, R. et al. Hybrid electromagnetic toroidal vortices. Sci. Adv. 11, eads4797 (2025)

work page 2025
[11]

Kármán vortex streets

Shen, Y., Hou, Y., Papasimakis, N. & Zheludev, N. I. Nondiffracting supertoroidal pulses: Optical “Kármán vortex streets”. Nat. Commun. 15, 4863 (2024)

work page 2024
[12]

& Zhan, Q

Wan, C., Cao, Q., Chen, J., Chong, A. & Zhan, Q. Toroidal vortices of light. Nat. Photonics 16, 519–522 (2022)

work page 2022
[13]

Chen, W. et al. Observation of chiral symmetry breaking in toroidal vortices of light. Phys. Rev. Lett. 132, 153801 (2024)

work page 2024
[14]

Liu, X. et al. Ultrafast bursts of tailored spatiotemporal vortex pulses. Light Sci. Appl. 14, 361 (2025)

work page 2025
[15]

Liu, X. et al. Dynamics of photonic toroidal vortices mediated by orbital angular momenta. Sci. Adv. 11, eadz0843 (2025)

work page 2025
[16]

& Zhan, Q

Chong, A., Wan, C., Chen, J. & Zhan, Q. Generation of spatiotemporal optical vortices with controllable transverse orbital angular momentum. Nat. Photonics 14, 350–354 (2020)

work page 2020
[17]

Liu, X. et al. Spatiotemporal optical vortices with controllable radial and azimuthal quantum numbers. Nat. Commun. 15, 5435 (2024)

work page 2024
[18]

M., Maleev, I

Palacios, D. M., Maleev, I. D., Marathay, A. S. & Swartzlander, Jr. G. A. Spatial correlation singularity of a vortex field. Phys. Rev. Lett. 92, 143905 (2004)

work page 2004
[19]

G., Miyamoto, Y

Wang, W., Duan, Z., Hanson, S. G., Miyamoto, Y. & Takeda, M. Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function. Phys. Rev. Lett. 96, 073902 (2006)

work page 2006
[20]

R., O’ Holleran, K

Dennis, M. R., O’ Holleran, K. & Padgett, M. J. Singular optics: optical vortices and polarization singularities. Prog. Opt. 53, 293–363 (2009)

work page 2009
[21]

Larocque, H. et al. Reconstructing the topology of optical polarization knots. Nat. Phys. 14, 1079–1082 (2018)

work page 2018
[22]

Optical second-order skyrmionic hopfion

Ehrmanntraut, D., et al. Optical second-order skyrmionic hopfion. Optica 10, 725–731 (2023)

work page 2023
[23]

Sugic, D. et al. Particle-like topologies in light. Nat. Commun. 12, 6785 (2021)

work page 2021
[24]

Shen, Y. et al. Topological transformation and free-space transport of photonic hopfions. Adv. Photonics 5, 015001 (2023)

work page 2023
[25]

& Tkalec, U

Martinez, A., Ravnik, M., Lucero, B. & Tkalec, U. Mutually tangled colloidal knots and induced defect loops in nematic fields. Nat. Mater. 13, 258–263 (2014)

work page 2014
[26]

G., Ackerman, P

Chen, B. G., Ackerman, P. J., Alexander, G. P., Kamien, R. D. & Smalyukh, I. I. Generating the Hopf fibration experimentally in nematic liquid crystals. Phys. Rev. Lett. 110, 237801 (2013)

work page 2013
[27]

& Alexander, G

Machon, T. & Alexander, G. P. Knotted defects in nematic liquid crystals. Phys. Rev. Lett. 113, 027801 (2014)

work page 2014
[28]

Han, J. et al. Mutual control of coherent spin waves and magnetic domain walls in a magnonic device. Science 366, 1121–1125 (2019)

work page 2019
[29]

Luk’yanchuk, I. et al. Hopfions emerge in ferroelectrics. Nat. Commun. 11, 2433 (2020)

work page 2020
[30]

& Brataas, A

Wang, X., Qaiumzadeh, A. & Brataas, A. Current -driven dynamics of magnetic hopfions. Phys. Rev. Lett. 123, 147203 (2019)

work page 2019
[31]

Hall, D. S. et al. Tying quantum knots. Nat. Phys. 12, 478–483 (2016)

work page 2016
[32]

Lee, W. et al. Synthetic electromagnetic knot in a three -dimensional skyrmion. Sci. Adv. 4, eaao3820 (2018)

work page 2018
[33]

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

work page 2024
[34]

& Fan, S

Shen, Y., Wang, H. & Fan, S. Free -space topological optical textures: Tutorial. Adv. Opt. Photonics 17, 295–374 (2025)

work page 2025
[35]

Wang, R. et al. Observation of resilient propagation and free -space skyrmions in toroidal electromagnetic pulses. Appl. Phys. Rev. 11, 031411 (2024)

work page 2024
[36]

Pires, D. G. et al. Particle -like topologies of light in turbulent complex media. Preprint at https://arxiv.org/abs/2602.19949 (2026)

work page arXiv 2026
[37]

Peters, C. et al. Topological robustness of optical skyrmions through a real -world free-space link. Preprint at https://arxiv.org/abs/2602.04446 (2026)

work page arXiv 2026
[38]

Guo, Z. et al. Topological robustness of classical and quantum optical skyrmions in atmospheric turbulence. Nat. Commun. 17, 2085 (2026)

work page 2085
[39]

Wang, A. A. et al. Topological protection of optical skyrmions through complex media. Light Sci. Appl. 13, 314 (2024)

work page 2024
[40]

& Zhan, Q

Liu, X., Cao, Q. & Zhan, Q. Spatiotemporal optical wavepackets: from concepts to applications. Photonics Insights 3, R08 (2024)

work page 2024
[41]

& Zheludev, N

Shen, Y., Hou, Y., Papasimakis, N. & Zheludev, N. I. Supertoroidal light pulses as electromagnetic skyrmions propagating in free space. Nat. Commun. 12, 5891 (2021)

work page 2021
[42]

& Iwamoto, S

Lin, W., Mata -Cervera, N., Ota, Y., Shen, Y. & Iwamoto, S. Space -Time Optical Hopfion Crystals. Phys. Rev. Lett. 135, 083801 (2025)

work page 2025
[43]

Teng, H. et al. Construction of optical spatiotemporal skyrmions. Light Sci. Appl. 14, 324 (2025)

work page 2025
[44]

Wan, C. et al. Scalar optical hopfions. eLight 2, 22 (2022)

work page 2022
[45]

& Zhan, Q

Zhong, J., Wan, C. & Zhan, Q. Optical twisted phase strips. ACS Photonics 10, 3384–3389 (2023)

work page 2023
[46]

Zhou, Y. et al. Spatiotemporally localized optical links and knots. Preprint at https://arxiv.org/abs/2511.05908 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025
[47]

& Zhao, C

Wang, Z., Lu, X., Chen, Z., Cai, Y. & Zhao, C. Topological links and knots of speckled light mediated by coherence singularities. Light Sci. Appl. 14, 1–9 (2025)

work page 2025
[48]

Atmospheric turbulence and orbital angular momentum of single photons for optical communication

Paterson, C. Atmospheric turbulence and orbital angular momentum of single photons for optical communication. Phys. Rev. Lett. 94, 153901 (2005)

work page 2005
[49]

& Zhan, Q

Zhong, J., Teng, H. & Zhan, Q. Toroidal phase topologies within paraxial laser beams. Commun. Phys. 7, 285 (2024)

work page 2024
[50]

Swartzlander, G. A. & Hernandez -Aranda, R. I. Optical Rankine vortex and anomalous circulation of light. Phys. Rev. Lett. 99, 163901 (2007)

work page 2007
[51]

Three modal decompositions of Gaussian Schell-model sources: comparative analysis

Wang, F., et al. Three modal decompositions of Gaussian Schell-model sources: comparative analysis. Opt. Express 29, 29676–29689 (2021)

work page 2021
[52]

Reconstructing the topology of optical vortex lines with single -shot measurement

Zhong, J ., et al. Reconstructing the topology of optical vortex lines with single -shot measurement. Appl. Phys. Lett. 119, 161102 (2021)

work page 2021
[53]

Pires, D.G., Tsvetkov, D., Barati Sedeh, H. et al. Stability of optical knots in atmospheric turbulence. Nat. Commun. 16, 3001 (2025). Figures and legends Fig. 1 | Comparison s of waveform s between deterministic and stochastic vortex -carrying light beams. a. Amplitude and b. phase distributions of a deterministic LG beam with topological charge ℓ = 1 an...

work page 2025

[1] [1]

Akhmetov, D. G. Vortex Rings. (Springer Science & Business Media, 2009)

work page 2009

[2] [2]

P., Perrard, S

Matsuzawa, T., Mitchell, N. P., Perrard, S. & Irvine, W. T. M. Creation of an isolated turbulent blob fed by vortex rings. Nat. Phys. 19, 1193–1200 (2023)

work page 2023

[3] [3]

Lee, J. S. et al. Origin and dynamics of vortex rings in drop splashing. Nat. Commun. 6, 8187 (2015)

work page 2015

[4] [4]

Lim, T. T. & Nickels, T. B. Vortex Rings. Fluid Vortices 30, 95–153 (1995)

work page 1995

[5] [5]

Hess, S., Eme, L., Roger, A. J. & Simpson, A. G. B. A natural toroidal microswimmer with a rotary eukaryotic flagellum. Nat. Microbiol. 4, 1620–1626 (2019)

work page 2019

[6] [6]

Whitaker, D. L. & Edwards, J. Sphagnum moss disperses spores with vortex rings. Science 329, 406–406 (2010)

work page 2010

[7] [7]

Cummins, C. et al. A separated vortex ring underlies the flight of the dandelion. Nature 562, 414–418 (2018)

work page 2018

[8] [8]

J., Yang, G

Kilner, P. J., Yang, G. -Z., Wilkes, A. J., Mohiaddin, R. H., Firmin, D. N. & Yacoub, M. H. Asymmetric redirection of flow through the heart. Nature 404, 759–761 (2000)

work page 2000

[9] [9]

Zdagkas, A. et al. Observation of toroidal pulses of light. Nat. Photonics 16, 523–528 (2022)

work page 2022

[10] [10]

Wang, R. et al. Hybrid electromagnetic toroidal vortices. Sci. Adv. 11, eads4797 (2025)

work page 2025

[11] [11]

Kármán vortex streets

Shen, Y., Hou, Y., Papasimakis, N. & Zheludev, N. I. Nondiffracting supertoroidal pulses: Optical “Kármán vortex streets”. Nat. Commun. 15, 4863 (2024)

work page 2024

[12] [12]

& Zhan, Q

Wan, C., Cao, Q., Chen, J., Chong, A. & Zhan, Q. Toroidal vortices of light. Nat. Photonics 16, 519–522 (2022)

work page 2022

[13] [13]

Chen, W. et al. Observation of chiral symmetry breaking in toroidal vortices of light. Phys. Rev. Lett. 132, 153801 (2024)

work page 2024

[14] [14]

Liu, X. et al. Ultrafast bursts of tailored spatiotemporal vortex pulses. Light Sci. Appl. 14, 361 (2025)

work page 2025

[15] [15]

Liu, X. et al. Dynamics of photonic toroidal vortices mediated by orbital angular momenta. Sci. Adv. 11, eadz0843 (2025)

work page 2025

[16] [16]

& Zhan, Q

Chong, A., Wan, C., Chen, J. & Zhan, Q. Generation of spatiotemporal optical vortices with controllable transverse orbital angular momentum. Nat. Photonics 14, 350–354 (2020)

work page 2020

[17] [17]

Liu, X. et al. Spatiotemporal optical vortices with controllable radial and azimuthal quantum numbers. Nat. Commun. 15, 5435 (2024)

work page 2024

[18] [18]

M., Maleev, I

Palacios, D. M., Maleev, I. D., Marathay, A. S. & Swartzlander, Jr. G. A. Spatial correlation singularity of a vortex field. Phys. Rev. Lett. 92, 143905 (2004)

work page 2004

[19] [19]

G., Miyamoto, Y

Wang, W., Duan, Z., Hanson, S. G., Miyamoto, Y. & Takeda, M. Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function. Phys. Rev. Lett. 96, 073902 (2006)

work page 2006

[20] [20]

R., O’ Holleran, K

Dennis, M. R., O’ Holleran, K. & Padgett, M. J. Singular optics: optical vortices and polarization singularities. Prog. Opt. 53, 293–363 (2009)

work page 2009

[21] [21]

Larocque, H. et al. Reconstructing the topology of optical polarization knots. Nat. Phys. 14, 1079–1082 (2018)

work page 2018

[22] [22]

Optical second-order skyrmionic hopfion

Ehrmanntraut, D., et al. Optical second-order skyrmionic hopfion. Optica 10, 725–731 (2023)

work page 2023

[23] [23]

Sugic, D. et al. Particle-like topologies in light. Nat. Commun. 12, 6785 (2021)

work page 2021

[24] [24]

Shen, Y. et al. Topological transformation and free-space transport of photonic hopfions. Adv. Photonics 5, 015001 (2023)

work page 2023

[25] [25]

& Tkalec, U

Martinez, A., Ravnik, M., Lucero, B. & Tkalec, U. Mutually tangled colloidal knots and induced defect loops in nematic fields. Nat. Mater. 13, 258–263 (2014)

work page 2014

[26] [26]

G., Ackerman, P

Chen, B. G., Ackerman, P. J., Alexander, G. P., Kamien, R. D. & Smalyukh, I. I. Generating the Hopf fibration experimentally in nematic liquid crystals. Phys. Rev. Lett. 110, 237801 (2013)

work page 2013

[27] [27]

& Alexander, G

Machon, T. & Alexander, G. P. Knotted defects in nematic liquid crystals. Phys. Rev. Lett. 113, 027801 (2014)

work page 2014

[28] [28]

Han, J. et al. Mutual control of coherent spin waves and magnetic domain walls in a magnonic device. Science 366, 1121–1125 (2019)

work page 2019

[29] [29]

Luk’yanchuk, I. et al. Hopfions emerge in ferroelectrics. Nat. Commun. 11, 2433 (2020)

work page 2020

[30] [30]

& Brataas, A

Wang, X., Qaiumzadeh, A. & Brataas, A. Current -driven dynamics of magnetic hopfions. Phys. Rev. Lett. 123, 147203 (2019)

work page 2019

[31] [31]

Hall, D. S. et al. Tying quantum knots. Nat. Phys. 12, 478–483 (2016)

work page 2016

[32] [32]

Lee, W. et al. Synthetic electromagnetic knot in a three -dimensional skyrmion. Sci. Adv. 4, eaao3820 (2018)

work page 2018

[33] [33]

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

work page 2024

[34] [34]

& Fan, S

Shen, Y., Wang, H. & Fan, S. Free -space topological optical textures: Tutorial. Adv. Opt. Photonics 17, 295–374 (2025)

work page 2025

[35] [35]

Wang, R. et al. Observation of resilient propagation and free -space skyrmions in toroidal electromagnetic pulses. Appl. Phys. Rev. 11, 031411 (2024)

work page 2024

[36] [36]

Pires, D. G. et al. Particle -like topologies of light in turbulent complex media. Preprint at https://arxiv.org/abs/2602.19949 (2026)

work page arXiv 2026

[37] [37]

Peters, C. et al. Topological robustness of optical skyrmions through a real -world free-space link. Preprint at https://arxiv.org/abs/2602.04446 (2026)

work page arXiv 2026

[38] [38]

Guo, Z. et al. Topological robustness of classical and quantum optical skyrmions in atmospheric turbulence. Nat. Commun. 17, 2085 (2026)

work page 2085

[39] [39]

Wang, A. A. et al. Topological protection of optical skyrmions through complex media. Light Sci. Appl. 13, 314 (2024)

work page 2024

[40] [40]

& Zhan, Q

Liu, X., Cao, Q. & Zhan, Q. Spatiotemporal optical wavepackets: from concepts to applications. Photonics Insights 3, R08 (2024)

work page 2024

[41] [41]

& Zheludev, N

Shen, Y., Hou, Y., Papasimakis, N. & Zheludev, N. I. Supertoroidal light pulses as electromagnetic skyrmions propagating in free space. Nat. Commun. 12, 5891 (2021)

work page 2021

[42] [42]

& Iwamoto, S

Lin, W., Mata -Cervera, N., Ota, Y., Shen, Y. & Iwamoto, S. Space -Time Optical Hopfion Crystals. Phys. Rev. Lett. 135, 083801 (2025)

work page 2025

[43] [43]

Teng, H. et al. Construction of optical spatiotemporal skyrmions. Light Sci. Appl. 14, 324 (2025)

work page 2025

[44] [44]

Wan, C. et al. Scalar optical hopfions. eLight 2, 22 (2022)

work page 2022

[45] [45]

& Zhan, Q

Zhong, J., Wan, C. & Zhan, Q. Optical twisted phase strips. ACS Photonics 10, 3384–3389 (2023)

work page 2023

[46] [46]

Zhou, Y. et al. Spatiotemporally localized optical links and knots. Preprint at https://arxiv.org/abs/2511.05908 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025

[47] [47]

& Zhao, C

Wang, Z., Lu, X., Chen, Z., Cai, Y. & Zhao, C. Topological links and knots of speckled light mediated by coherence singularities. Light Sci. Appl. 14, 1–9 (2025)

work page 2025

[48] [48]

Atmospheric turbulence and orbital angular momentum of single photons for optical communication

Paterson, C. Atmospheric turbulence and orbital angular momentum of single photons for optical communication. Phys. Rev. Lett. 94, 153901 (2005)

work page 2005

[49] [49]

& Zhan, Q

Zhong, J., Teng, H. & Zhan, Q. Toroidal phase topologies within paraxial laser beams. Commun. Phys. 7, 285 (2024)

work page 2024

[50] [50]

Swartzlander, G. A. & Hernandez -Aranda, R. I. Optical Rankine vortex and anomalous circulation of light. Phys. Rev. Lett. 99, 163901 (2007)

work page 2007

[51] [51]

Three modal decompositions of Gaussian Schell-model sources: comparative analysis

Wang, F., et al. Three modal decompositions of Gaussian Schell-model sources: comparative analysis. Opt. Express 29, 29676–29689 (2021)

work page 2021

[52] [52]

Reconstructing the topology of optical vortex lines with single -shot measurement

Zhong, J ., et al. Reconstructing the topology of optical vortex lines with single -shot measurement. Appl. Phys. Lett. 119, 161102 (2021)

work page 2021

[53] [53]

Pires, D.G., Tsvetkov, D., Barati Sedeh, H. et al. Stability of optical knots in atmospheric turbulence. Nat. Commun. 16, 3001 (2025). Figures and legends Fig. 1 | Comparison s of waveform s between deterministic and stochastic vortex -carrying light beams. a. Amplitude and b. phase distributions of a deterministic LG beam with topological charge ℓ = 1 an...

work page 2025