Abstract
Topological wave structures, such as vortices1,2,3,4,5,6, polarization textures7,8,9,10,11 and skyrmions12,13,14,15,16,17,18,19, appear in various quantum and classical wave fields, including optics and acoustics. In particular, optical vortices have found numerous applications20,21, ranging from quantum information to astrophysics. Furthermore, both optical and acoustic structured waves are crucial in the manipulation of small particles22,23,24,25, from atoms to macroscopic biological objects. Recently, there has been a surge of interest in structured water surface waves, which can be notable analogues of quantum, optical and acoustic wave systems26,27,28,29. However, topological water-wave forms, especially their ability to manipulate particles, have not yet been demonstrated. Here we describe the controllable generation of topological structures, namely wave vortices, skyrmions and polarization Möbius strips, in gravity water waves. Most importantly, we demonstrate the efficient manipulation of subwavelength and wavelength-order floating particles with topologically structured water waves. This includes trapping the particles in the high-intensity field zones and controlling their orbital and spinning motion due to the orbital and spin angular momenta of the water waves. Our results reveal the water-wave counterpart of optical and acoustic manipulation, which paves the way for applications in hydrodynamics and microfluidics.
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Acknowledgements
B.W., Z.C. and L.S. thank W. Liu for helpful discussions. This work was partially supported by the National Key Research and Development Program of China (Grant Nos. 2023YFA1406900 and 2022YFA1404800), the National Natural Science Foundation of China (Grant Nos. 12234007, 12321161645 and 12221004), a Major Program of the National Natural Science Foundation of China (Grant Nos. T2394480 and T2394481), the Science and Technology Commission of Shanghai Municipality (Grant Nos. 22142200400, 21DZ1101500, 2019SHZDZX01 and 23DZ2260100), the China Postdoctoral Science Foundation (Grant Nos. 2022M720810, 2022TQ0078, 2023M741024 and 2024T170218), a Nanyang Technological University Start-Up Grant, a Singapore Ministry of Education (MoE) AcRF Tier 1 grant (Grant No. RG157/23), a MoE AcRF Tier 1 Thematic grant (Grant No. RT11/23), the Imperial–Nanyang Technological University Collaboration Fund (Grant No. INCF-2024-007), Ikerbasque (Basque Foundation of Science), the Marie Skłodowska-Curie COFUND Programme of the European Commission (Project HORIZON-MSCA-2022-COFUND-101126600-SmartBRAIN3), the International Research Agendas Programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund and the Teaming Horizon 2020 programme of the European Commission (ENSEMBLE3 Project No. MAB/2020/14), and the project of the Ministry of Science and Higher Education (Poland) ‘Support for the activities of Centers of Excellence established in Poland under the Horizon 2020 programme’ (Contract MEiN/2023/DIR/3797).
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Y.S., L.S., K.Y.B. and J.Z. conceived the basic idea behind the work and supervised its development. Z.C. and L.S. designed the experiments. Z.C. performed the experiments. B.W. and Z.C. analysed the experimental data. K.Y.B. prepared the first version of the manuscript with input from all authors. All authors took part in discussions, interpreting the results and revising the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Reconstruction of the vertical displacement field \(\boldsymbol{\mathscr{Z}}\)(x,y,t) via the FCD technique.
Here we show a hexagonal structure for the three-wave interference, Fig. 2, where only one source side produces an x-propagating near-plane wave (upper row), and the 24-gonal structure for the Bessel-vortex generation, Fig. 3, where only one source generates a wave propagating towards the center of the structure (lower row). a, Reference images without waves. b, Distorted images in the presence of the wave at t = 0. c, Reconstructed field distributions \({\mathcal{Z}}(x,y,t=0)\).
Extended Data Fig. 2 Topologically unstable spin merons in three-wave interference.
Shown are: distributions of the unit spin-density vectors \(\bar{{\bf{S}}}(x,y)\), represented by brightness (\({\bar{S}}_{z}\)) and color (\({\tan }^{-1}({\bar{S}}_{y}/{\bar{S}}_{x})\)), the meron boundaries \({S}_{z}=0\) (red curves), and mappings of the unit spin vectors in the merons onto the unit sphere, with the corresponding topological numbers \({Q}_{S}\). a, An ideal theoretically-predicted three-wave interference exhibiting a lattice of triangular alternating merons with \({Q}_{S}=\pm 1/2\). b, Perturbed three-wave interference field observed in our experiment (see Fig. 2). The curves \({S}_{z}=0\) do not form closed meron areas. Closing the gaps with black lines and mapping the unit spin vectors onto the unit sphere results in ‘quasi-merons’ with non-half-integer numbers \({Q}_{S}\).
Extended Data Fig. 3 Structures in the interference of four water waves.
a, Same as in Fig. 2 but for the interference of four plane waves with the angles of propagation \(\varphi =(0,{\rm{\pi }}/2,{\rm{\pi }},3{\rm{\pi }}/2)\) and the corresponding phases delays \(\phi =\varphi \). This field is equivalent (up to the global spatial shifts) to the interference of two orthogonal standing waves phase-shifted by \({\rm{\pi }}/2\)28,34,35. b, The complex vertical-displacement field Z(x,y) exhibits a square lattice of \({\ell }=\pm 1\) vortices. c, The normalized 3D displacement field \(\bar{\boldsymbol{\mathscr{R}}}(x,y,t=0)\) does not contain skyrmions or merons (the black curves correspond to \({\mathcal{Z}}(x,y,t=0)=0\)). d, The distribution of the normalized spin density S(x,y) exhibits ‘quasi-merons’, similar to the three-wave case in Extended Data Fig. 2. The black curves correspond to \({S}_{z}(x,y)=0\). In contrast to the three-wave case, the polarization ellipses around centers of these quasi-merons do not form polarization Möbius strips. This because the C-points in the centers are double-degenerate due to the higher symmetry of the four-wave configuration.
Extended Data Fig. 4 Skyrmions in the Bessel modes.
Distributions of the normalized instantaneous displacement vectors \(\bar{\boldsymbol{\mathscr{R}}}(x,y,t=0)\) (encoded by the brightness and colors) in the generated Bessel modes with \({\ell }=0\) and \({\ell }=2\). a, The \({\ell }=0\) non-vortex mode exhibits a skyrmion with Q = 1 in the center (the boundary is shown by the green solid curve). The topological charge integrated inside the dashed contour vanishes: Q = 0. b, Vortex Bessel modes with \({\ell }\ne 0\) (here \({\ell }=2\)) do not contain \(\bar{\boldsymbol{\mathscr{R}}}\)-skyrmions.
Extended Data Fig. 5 Unstable trapping of floating particles.
Dynamics of a floating particle with radius a = 4.75 mm in the Bessel-vortex wave with \({\ell }=2\). Unlike Fig. 4c in the main text, here the particle is attracted to the intensity-maximum ring, orbits there for some time, and then escapes, because the radial centrifugal force prevails the trapping gradient force.
Extended Data Fig. 6 Schematics of monopole-like and dipole-like water-wave excitations.
a, Vertical oscillations of a small particle produce a monopole-like water-wave field. b, Horizontal oscillations of a small particle generate a dipole wavefield.
Supplementary information
Supplementary Video 1 (download MP4 )
Temporal evolution of the measured water-surface elevation \({\mathcal{Z}}\)(x, y, t) in the three-wave interference experiment. The t = 0 distribution is shown in Fig. 2a.
Supplementary Video 2 (download MP4 )
Temporal evolution of the measured water-surface elevation \({\mathcal{Z}}\)(x, y, t) in the Bessel-vortex generation experiments with topological vortex charges ℓ = 0, 1, 2 and 8. The t = 0 distribution for ℓ = 2 is shown in Fig. 3a, and the corresponding distributions of the complex field Z(x, y) in the central area are shown in Fig. 3b,c.
Supplementary Video 3 (download MP4 )
Temporal evolution of the measured water-surface elevation \({\mathcal{Z}}\)(x, y, t) in the Bessel-vortex generation experiments with topological vortex charges ℓ = 0, 1, 2 and 8. The t = 0 distribution for ℓ = 2 is shown in Fig. 3a, and the corresponding distributions of the complex field Z(x, y) in the central area are shown in Fig. 3b,c.
Supplementary Video 4 (download MP4 )
Temporal evolution of the measured water-surface elevation \({\mathcal{Z}}\)(x, y, t) in the Bessel-vortex generation experiments with topological vortex charges ℓ = 0, 1, 2 and 8. The t = 0 distribution for ℓ = 2 is shown in Fig. 3a, and the corresponding distributions of the complex field Z(x, y) in the central area are shown in Fig. 3b,c.
Supplementary Video 5 (download MP4 )
Temporal evolution of the measured water-surface elevation \({\mathcal{Z}}\)(x, y, t) in the Bessel-vortex generation experiments with topological vortex charges ℓ = 0, 1, 2 and 8. The t = 0 distribution for ℓ = 2 is shown in Fig. 3a, and the corresponding distributions of the complex field Z(x, y) in the central area are shown in Fig. 3b,c.
Supplementary Video 6 (download MP4 )
Dynamics of a particle of radius a = 2.4 mm in the field of the Bessel mode ℓ = 0. See the snapshots in Fig. 4b.
Supplementary Video 7 (download MP4 )
Dynamics of a particle of radius a = 3.1 mm in the field of the Bessel mode ℓ = 2. See the snapshots in Fig. 4c.
Supplementary Video 8 (download MP4 )
Dynamics of a particle of radius a = 4.75 mm in the field of the Bessel mode ℓ = 2. See the snapshots in Extended Data Fig. 5.
Supplementary Video 9 (download MP4 )
Dynamics of a ping-pong ball of radius a = 20 mm in the field of the Bessel mode ℓ = 8. See the snapshots in Fig. 4d.
Supplementary Video 10 (download MP4 )
Dynamics of a particle of radius a = 6.35 mm near the ℓ = ±1 vortices or C points in the three-wave interference field (Fig. 2). See the snapshots in Fig. 5.
Supplementary Video 11 (download MP4 )
Dynamics of a particle of radius a = 6.35 mm near the ℓ = ±1 vortices or C points in the three-wave interference field (Fig. 2). See the snapshots in Fig. 5.
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Wang, B., Che, Z., Cheng, C. et al. Topological water-wave structures manipulating particles. Nature 638, 394–400 (2025). https://doi.org/10.1038/s41586-024-08384-y
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DOI: https://doi.org/10.1038/s41586-024-08384-y
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