Abstract
Optical skyrmions have recently been constructed by tailoring vectorial nearfield distributions through the interference of multiple surface plasmon polaritons, offering promising features for advanced information processing, transport and storage. Here, we provide experimental demonstration of electromagnetic skyrmions based on magnetic localized spoof plasmons (LSP) showing large topological robustness against continuous deformations, without stringent external interference conditions. By directly measuring the spatial profile of all three vectorial magnetic fields, we reveal multiple πtwist target skyrmion configurations mapped to multiresonant nearequidistant LSP eigenmodes. The realspace skyrmion topology is robust against deformations of the metastructure, demonstrating flexible skyrmionic textures for arbitrary shapes. The observed magnetic LSP skyrmions pave the way to ultracompact and robust plasmonic devices, such as flexible sensors, wearable electronics and ultracompact antennas.
Introduction
Skyrmions  topologically stable threedimensional (3D) vector field configurations confined within a twodimensional (2D) domain  have been prompting significant interest in a number of physical systems, including elementary particles^{1}, BoseEinstein condensates^{2}, nematic liquid crystals^{3}, and chiral magnets^{4,5}. Beyond elementary skyrmion, nested multiple skyrmions have also been demonstrated in magnetic materials, such as the skyrmionium^{6,7,8} and target skyrmions (TSs)^{9,10,11}, offering large tunable topological degrees of freedom. As compact and topologicallyrobust information carriers, skyrmions have been proposed for promising applications such as highdensity data storage and transfer^{12,13}. These advances have recently motivated the exploration of optical and plasmonic analogues to skyrmions^{14,15,16,17,18}. Unlike topological photonic crystals, where the topological invariants protecting the unusual features of these systems are defined in reciprocal space, optical skyrmions manifest topological properties in real space, offering a topological state of light^{14} with promising applications in optical information processing, metrology, transfer and storage. The experimental realization of optical skyrmions has been exclusively conducted based on electric or synthetic fields by interfering or tightly focused propagating surface plasmons within a smooth metallic film using carefully controlled external illuminations^{14,15,16}. Such interferencebased approaches require stringent external excitation conditions, which can only construct regularshaped singlemode skyrmions at a given frequency.
In this work, we demonstrate that localized plasmon skyrmions can provide a unique way to build arbitrarily shaped skyrmionic textures, promising high flexibility and robustness for applications in information processing and metrology. Unlike previous optical skyrmion configurations based on propagating surface plasmon wave interference, we realize electromagnetic (EM) skyrmions based on magnetic localized spoof plasmons (LSPs) sustained by a wisely designed spacecoiling metastructure, manifesting additional flexibility and robustness provided by the spacecoiling guiding mechanism. The observed LSP magnetic field profile manifests a texture of multiple πtwist concentric features with alternating positive and negative topological charges, which resemble TSs^{9,10,11} previously reported in magnetic materials. Our design supports a train of nearequidistant needlelike sharp modes in contrast to conventional localized plasmon resonance spectra with irregular peaks and linewidths^{19,20,21,22}. Remarkably, the skyrmion number S, a topological invariant defined in real space, of the observed skyrmions is unperturbed when the geometry is continuously deformed in arbitrary shapes, leading to robust vectorial field topologies with multiring profiles, even in the presence of sharp corners and irregular shapes. Such magnetic LSP skyrmions provide a unique way to build arbitrarily shaped skyrmionic textures unattainable with previous interference approaches, promising for many flexible and robust applications based on skyrmions.
Results
Theoretical modeling and design of LSP skyrmions
Our employed metastructure comprises a singlearmed spiral metallic stripe tightly coiled on itself, with gap width a, spiral pitch d, radius R, thickness h, and spiral turn number n_{r}, as shown in the upper panel of Fig. 1. The continuous air gap forms a spacecoiling region that confines the EM fields at a deeply subwavelength scale. Resonant spectra and field configurations of this spacecoiling metastructure are analyzed using finite element simulations and vectorial nearfield measurements in the microwave regime (see “Materials and Methods”), showing a train of deeply subwavelength resonances at equallyspaced frequencies, f_{0}, f_{0} + Δf, f_{0} + 2Δf, …, where, f_{0} and Δf are the fundamental resonance frequency and free spectral range (FSR), respectively (Middle panel of Fig. 1), determined by the fineness n_{r} of the spacecoiling metastructure (Supplementary Note 1)^{23,24,25}.
Each resonant mode supports a magnetic field profile with axial symmetry, and its unit vector spatially rotates integer multiples of πtwist along the radial direction. As we demonstrate below, the fundamental (πtwist) mode corresponds to an elementary skyrmion, with topological charge 1, and the second (2πtwist) mode forms a skyrmionium^{6,7,8}, with topological charge 0. Higherorder modes are multipleπtwist TSs^{9,10}, which possess topological charge 1 for odd modes and 0 for even modes due to the accumulated cancellation from adjacent opposite twists. Unlike previous realizations of optical skyrmions, these LSP skyrmion vectorial field profiles correspond to eigenmodes of the spacecoiling metastructure. Hence, they do not require carefully tailored external illuminations and can be excited by various nearfield or farfield sources. In addition, the skyrmion field topology is robust to arbitrary shape deformations, as illustrated in the lower panel of Fig. 1. Together with the spacecoiling guiding mechanism, the skyrmion multiresonances remain stable even when the metastructure is continuously deformed into arbitrary shapes.
To gain insights into the origin of the scattering response, we first examine a 2D spacecoiling cylinder, i.e., metallic stripe spiral of the infinite thickness (h → ∞), under transversemagnetic (TM) illumination with magnetic field H pointing along the zdirection (Fig. 2a). It has been shown that a textured perfect electric conductor (PEC) surface with multiple disconnected grooves supports LSPs, even if there is no field penetrating the metal^{26,27,28}. On the contrary, our structure is formed by a single connected PEC groove arranged in a spacecoiling fashion, supporting purely magnetic LSP modes with only radial lobes (Fig. 2a). Its scattering crosssection (SCS) exhibits a multiresonant spectrum with equallyspaced needlelike sharp peaks at the deeply subwavelength scale (feature size: R~λ/1200 at n_{r} = 100). This spectrum can be explained by considering the coil as a meanderline waveguide: the resonant peak position and FSR can be predicted by the halfended effective waveguide model (Supplementary Fig. S1). For feature sizes much smaller than the wavelength, the electric modes do not contribute to the SCS (Supplementary Note 1 and Fig. S2), and the observed LSP resonant features are purely magnetic.
The spacecoiling cylinder can be modeled as a homogeneous rod with extremely anisotropic properties (Supplementary Fig. S3), with an anisotropic permittivity tensor with infinite azimuthal and zoriented components while staying finite in the radial direction. This extreme form of anisotropy supports quasistatic modal distributions, having a radially polarized electric Efield and an azimuthal surface current J, in contrast to conventional magnetic Mie resonators where E is typically parallel to J^{29}. Such exotic resonant mode supports extreme field enhancement (Fig. 2b), as high as 10^{5} in the geometry of Fig. 2a (inset). Figure 2c shows the linear dependence of the resonant frequency (k_{0}R) and resonant linewidths (Δk_{0}R) with the mode index, confirming its equidistant response. Figure 2d shows that the field confinement factor (resonant wavelength over cavity radius) of the fundamental mode linearly increases with n_{r}, indicating that the resonator compactness can be enhanced by densifying the spacecoiling fineness. Figure 2e shows that increasing the metastructure fineness also increases field enhancement and quality factor, following a quadratic trend. These features are fundamentally limited by the considered finite conductivity of the involved materials, implying a tradeoff between quality factors and material losses studied in Supplementary Figs. S4a, b. Our study shows that realistic metals can support these resonant responses over a wide frequency range, spanning GHz and THz frequencies, as shown in Supplementary Fig. S4c, d.
Now, we consider finitethickness spacecoiling metastructures to identify the skyrmion nature of their vectorial field configurations. In the 2D geometry, the LSP field profiles only exhibit scalar (Hfield) or 2D vectorial (Efield) properties. However, for the 3D spacecoiling metastructure with finite thickness, both Hfield and Efield distributions manifest 3D vectorial configurations at the interface between the metastructure and the surrounding background (Supplementary Fig. S5). With decreasing the 3D spacecoiling metastructure thickness, the resonant behavior and the localization of the field pattern are preserved, with a small frequency shift compared to the 2D scenario (Supplementary Fig. S6)^{30}. The topological properties of the 3D vectorial field configuration can be quantitatively evaluated by the skyrmion number^{14}
where \({{{{{\bf{h}}}}}}=\{{H}_{x},{H}_{y},{H}_{z}\}/{{{{{\bf{H}}}}}}\) is the local unit vector of the field, and the integrand \({{{{{\bf{h}}}}}}\cdot (\frac{\partial {{{{{\bf{h}}}}}}}{\partial x}\times \frac{\partial {{{{{\bf{h}}}}}}}{\partial y})\) is the skyrmion density. The skyrmion number is a topological invariant that characterizes the order of topological knots formed by field vectors, i.e., the number of times the field wraps around the unit sphere. The skyrmion number of the magnetic field profile calculated at the air interface (Supplementary Figs. S5a–c and S6b) is equal to 1, confirming a pure skyrmion field configuration.
Experimental observation of LSP skyrmions
To experimentally observe the skyrmion vectorial field configurations, we fabricated an ultrathin spacecoiling metastructure (h = 0.016 mm, a = 1 mm, d = 1.5 mm, R = 30 mm, n_{r} = 20) over a printed circuit board (Fig. 3a). We scanned the near field of the resonant modes with a 3D scanning platform connected to a vector network analyzer (Fig. 3b). The measured response indeed manifests a multiresonant spectrum with nearly equidistant sharp peaks (blue curve in Fig. 3c and Supplementary Fig. S18). This feature is consistent with the simulation results with the same geometry parameters (red curve in Fig. 3c). Since these modes are eigenresonances of the metastructure, we observe a strong excitation of the skyrmions with the nearequidistant multiresonant spectrum for various nearfields or farfield sources (Supplementary Fig. S7), in stark contrast with previous skyrmions based on the interference of carefully tailored propagating surface plasmons^{14,15,16}.
Because these modes resonate at a deeply subwavelength scale with large radial wavevectors k_{r} ≫ k_{0}, the zcomponent wavevector \({k}_{z}=i\sqrt{{k}_{r}^{2}{k}_{0}^{2}}\) is mostly imaginary, yielding strong field confinement far beyond the diffraction limit. The measured elementary skyrmion has a lateral size d_{m} = λ/100, and a half vertical size h_{m} = λ/400 (Fig. 3d, leftlower panel), leading to an extremely subwavelength mode volume V_{m} = 2πh_{m}(d_{m}/2)^{2} = π(λ/2)^{3}/10^{6}. We stress that such tiny mode volume is obtained with just n_{r} = 20, limited by our fabrication and measurement setup. Further squeezing may be achieved by increasing n_{r}. We experimentally observe a relatively high Qfactor of 165 (Fig. 3c), yielding a Purcell factor exceeding 10^{7}, promising for various applications requiring strong lightmatter interactions. Both outofplane (Fig. 3d) and inplane (Supplementary Fig. S8) magnetic fields along the radial direction have been simulated and measured, yielding excellent agreement, except around the center of the sample, where we observe additional field nulls for the lower modes m = 1, 2, due to strong coupling between localized EM fields and the magnetic loop probe.
Due to the axial symmetry of the magnetic field profile, the unit vector can be written as \({{{{{\bf{h}}}}}}(x,y,z)={\{\sin \varTheta (\rho )\cos \varphi ,\sin \varTheta (\rho )\sin \varphi ,\cos \varTheta (\rho )\}}^{T}\), where ρ and φ are coordinates in the polar system and \(\varTheta (\rho )\) is the orientation angle of the unit vector. The skyrmion number of the ith radial mode lobe can be calculated in closed form as
showing that the skyrmion number only depends on the initial and final states of \(\varTheta (\rho )\). Figure 3e shows the unit vectors and \(\cos \,\varTheta (\rho )\) distributions along the radial direction. According to Eq. (2), each radial lobe of the mode profile has a skyrmion number +1 or −1, representing an elementary skyrmion polarized in opposite directions. The accumulated total skyrmion number is 1 for odd modes and 0 for even modes, respectively, building a multipleπtwist TS constructed by multiple elementary skyrmions^{9}, with rich possibilities to implement various topological configurations of different orders^{10}.
The vectorial nature of the skyrmion modes can be observed in our realspace measurements of all three magnetic field components, shown in Fig. 4. The inplane fields are along the radial direction, H_{x} and H_{y} reveal a nodalline profile along their perpendicular axis (y axis and x axis, respectively). At the same time, the outofplane component (H_{z}) has only radial variations, in good agreement with our simulations. The mode lobes are distributed purely along the radial direction, in stark contrast to conventional WGM modes with multiple azimuthal modes. The three field components form a hedgehoglike vector configuration (bottom panel of Fig. 4), which is the direct signature of Neeltype skyrmions^{31}. This outcome is also confirmed by the skyrmion density and skyrmion number extraction from the field patterns (Supplementary Figs. S11 and S13a).
In contrast to the full skyrmion supported by the Hfield, the Efield profile shows a skyrmion configuration with an extraπ/2twist, yielding a total skyrmion number of 1/2 (Supplementary Fig. S9)^{32,33,34}. The Hfield and Efield are parallel due to the extreme anisotropy of the spacecoiling structure and have a π/2 relative phase shift (Supplementary Fig. S10 and Movie S1) due to the standing wave nature (along the radial direction) of the resonant mode. Timevarying properties of these skyrmion configurations are shown in Supplementary Movie S1 and S2, indicating that the topological profiles are preserved throughout the entire oscillation period of the EM field, with welldefined skyrmion topology nature^{14}.
Arbitrarily shaped LSP skyrmions
One unique feature of the skyrmions supported by this spacecoiling metastructure is its inherent topological features in real space, which remains robust against continuous geometrical deformations. As the geometry is stretched into an ellipse or deformed to polygonal or even an asymmetric heart shape (Supplementary Fig. S12a), the elementary skyrmion field configuration adapts itself to the new geometrical shape (Supplementary Fig. S12b) without affecting its topology. The skyrmion density distribution is modified, with maxima accumulating in regions with sharp curvatures (Supplementary Fig. S12c), but the skyrmion number is strictly preserved, revealing its robust topological features against shape deformations. All higherorder skyrmion modes show similar robustness, as their skyrmion number is always 1 for odd modes and 0 for even modes, independent of the geometrical shape (Supplementary Figs. S13, S14a). Meanwhile, the resonance frequencies are weakly affected by these geometrical changes mainly due to the preserved meanderline waveguiding mechanism (Supplementary Fig. S14b), realizing in a practical metastructure with the highly sought property of shapeindependent resonance features envisioned in zeroindex metamaterials^{35}.
To experimentally demonstrate the robustness, we fabricated spacecoiling metastructures with different shapes: elliptical, polygonal, and heartshaped, keeping constant the effective coil length. We show the measured spectra in the leftmost panels of Fig. 5a–c. Even though the geometries are drastically different, their EM response manifests similar nearequidistant multiresonant spectra. The spatial profiles of all vectorial magnetic field components of these modes for different shapes are directly measured by our nearfield scanning technique (as shown in Supplementary Figs. S15–S17), which agree well with the simulation results, manifesting arbitrarily shaped skyrmion textures indicated by the extracted skyrmion number from the arbitrarily shaped mode profile (Supplementary Fig. S14a). Based on the three field components, we reconstruct their vectorial field configurations in the right panels of Fig. 5a–c, showing multiple nested rings adapting their shapes, with multipleπtwists from the geometry center to the periphery. Such arbitrarily shaped skyrmion configurations have been longsought in various systems^{5}, providing a unique way to realize flexible skyrmions for applications in various technological areas. In addition to continuous shape deformations, we also studied the effect of abrupt defects introduced within the meanderline geometry (Supplementary Note 7 and Fig. S19), particularly in the forms of gaps or shorts in the line. These results show that, as long as the defect does not abruptly modify the field continuity, the structure’s skyrmion nature is preserved.
Discussion
In this work we show magnetic LSP skyrmions in a spacecoiling metastructure. The skyrmions stem from the eigenresonances of the tailored metastructure, and hence they do not require external interference in the illumination. We observe multipleπtwist TS vectorial configurations in realspace through nearfield scanning, yielding extremely subwavelength features, down to λ^{3}/10^{6}. The magnetic LSP skyrmion shows large topology stability against continuous deformations of the spacecoiling geometry, and manifests an overall stable multiresonant spectrum and flexible skyrmionic textures with arbitrary shape. Although our proofofprinciple has been demonstrated in the microwave regime, we envision exciting opportunities in various frequency ranges, from nearDC to THz regimes. Our findings offer an ideal platform to support the next revolution of information processing with inherent advantages in compactness, stability, and precision for potential applications, including miniaturized spectroscopy, THz sources and microwave photonics.
Methods
Numerical simulations
The SCS spectra of the 2D spacecoiling cylinder and the EM spectral response of the spacecoiling metasurface were calculated with finite element simulations implemented by COMSOL Multiphysics. In simulations, a plane wave with TM polarization was used to illuminate the PEC spacecoiling cylinder. The SCS was extracted by integrating the outgoing Poynting vector along a closed surface enclosing the structure. EM spectral responses of the ultrathin spacecoiling metasurface were calculated by probing the magnetic field at its central point above the structure. All field components within a square area enclosing the spacecoiling metastructure were extracted to calculate the skyrmion densities and skyrmion numbers.
Sample preparation and field measurements
The spacecoiling metastructure samples were fabricated with printed circuit board technology by printing 0.016mmthick copper spacecoiling patterns with air gap a = 1 mm, spiral pitch d = 1.5 mm, and turn number n_{r} = 20 on top of a 0.2mmthick dielectric substrate with relative permittivity ε_{r} = 3.5. The EM spectral response was measured with an integrated microwave system (Linbou NFS03) composed of a vector network analyzer (Agilent PNA N5222A) and a 3D scanning platform. Two magnetic (loop) antennas were connected to the two ports of the vector network analyzer. One antenna was placed at the bottom central point of the sample as a point source. The other antenna was set up on a 3D moving platform to probe the spatial magnetic field configuration. Measurement of all vectorial components (H_{x}, H_{y}, H_{z}) of the magnetic fields were realized by the probe antenna facing in the corresponding directions (x, y, z). By mounting the loop antenna on a 3D scanning platform, the spatial field patterns over a given area were obtained through pointbypoint measurements.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
A.A. acknowledges the Air Force Office of Scientific Research, the Simons Foundation, and the National Science Foundation; Z.L.D. acknowledges the National Natural Science Foundation of China (NSFC) (Grant 62075084), Guangdong Basic and Applied Basic Research Foundation (2020A1515010615), the Fundamental Research Funds for the Central Universities (21620415), Guangzhou Science and Technology Program (202102020566) and the China Scholarship Council (Grant 201906785011). X.L. acknowledges the Guangdong Provincial Innovation and Entrepreneurship Project (Grant 2016ZT06D081).
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Z.L.D., A.K., X.L. and A.A. conceived the idea. Z.L.D. carried out the theoretical design and simulation of the spacecoiling metastructure. Z.L.D. and T.S. performed the experimental measurements. Z.L.D., A.A., X.L., and A.K. proposed the physical concept and analyzed the data. Z.L.D. initiated the draft with inputs from A.A., X.L. and A.K. All authors contributed to discussions about the manuscript.
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Deng, ZL., Shi, T., Krasnok, A. et al. Observation of localized magnetic plasmon skyrmions. Nat Commun 13, 8 (2022). https://doi.org/10.1038/s4146702127710w
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DOI: https://doi.org/10.1038/s4146702127710w
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