## Introduction

Skyrmions - topologically stable three-dimensional (3D) vector field configurations confined within a two-dimensional (2D) domain - have been prompting significant interest in a number of physical systems, including elementary particles1, Bose-Einstein condensates2, nematic liquid crystals3, and chiral magnets4,5. Beyond elementary skyrmion, nested multiple skyrmions have also been demonstrated in magnetic materials, such as the skyrmionium6,7,8 and target skyrmions (TSs)9,10,11, offering large tunable topological degrees of freedom. As compact and topologically-robust information carriers, skyrmions have been proposed for promising applications such as high-density data storage and transfer12,13. These advances have recently motivated the exploration of optical and plasmonic analogues to skyrmions14,15,16,17,18. 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 light14 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 illuminations14,15,16. Such interference-based approaches require stringent external excitation conditions, which can only construct regular-shaped single-mode 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 space-coiling meta-structure, manifesting additional flexibility and robustness provided by the space-coiling guiding mechanism. The observed LSP magnetic field profile manifests a texture of multiple π-twist concentric features with alternating positive and negative topological charges, which resemble TSs9,10,11 previously reported in magnetic materials. Our design supports a train of near-equidistant needle-like sharp modes in contrast to conventional localized plasmon resonance spectra with irregular peaks and linewidths19,20,21,22. 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 multi-ring 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 meta-structure comprises a single-armed spiral metallic stripe tightly coiled on itself, with gap width a, spiral pitch d, radius R, thickness h, and spiral turn number nr, as shown in the upper panel of Fig. 1. The continuous air gap forms a space-coiling region that confines the EM fields at a deeply subwavelength scale. Resonant spectra and field configurations of this space-coiling meta-structure are analyzed using finite element simulations and vectorial near-field measurements in the microwave regime (see “Materials and Methods”), showing a train of deeply subwavelength resonances at equally-spaced frequencies, f0, f0 + Δf, f0 + 2Δf, …, where, f0 and Δf are the fundamental resonance frequency and free spectral range (FSR), respectively (Middle panel of Fig. 1), determined by the fineness nr of the space-coiling meta-structure (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 π-twist along the radial direction. As we demonstrate below, the fundamental (π-twist) mode corresponds to an elementary skyrmion, with topological charge 1, and the second (2π-twist) mode forms a skyrmionium6,7,8, with topological charge 0. Higher-order modes are multiple-π-twist TSs9,10, 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 space-coiling meta-structure. Hence, they do not require carefully tailored external illuminations and can be excited by various near-field or far-field 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 space-coiling guiding mechanism, the skyrmion multi-resonances remain stable even when the meta-structure is continuously deformed into arbitrary shapes.

To gain insights into the origin of the scattering response, we first examine a 2D space-coiling cylinder, i.e., metallic stripe spiral of the infinite thickness (h → ∞), under transverse-magnetic (TM) illumination with magnetic field H pointing along the z-direction (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 metal26,27,28. On the contrary, our structure is formed by a single connected PEC groove arranged in a space-coiling fashion, supporting purely magnetic LSP modes with only radial lobes (Fig. 2a). Its scattering cross-section (SCS) exhibits a multi-resonant spectrum with equally-spaced needle-like sharp peaks at the deeply subwavelength scale (feature size: R~λ/1200 at nr = 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 half-ended 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 space-coiling cylinder can be modeled as a homogeneous rod with extremely anisotropic properties (Supplementary Fig. S3), with an anisotropic permittivity tensor with infinite azimuthal- and z-oriented components while staying finite in the radial direction. This extreme form of anisotropy supports quasi-static modal distributions, having a radially polarized electric E-field and an azimuthal surface current J, in contrast to conventional magnetic Mie resonators where E is typically parallel to J29. Such exotic resonant mode supports extreme field enhancement (Fig. 2b), as high as 105 in the geometry of Fig. 2a (inset). Figure 2c shows the linear dependence of the resonant frequency (k0R) and resonant linewidths (Δk0R) 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 nr, indicating that the resonator compactness can be enhanced by densifying the space-coiling fineness. Figure 2e shows that increasing the meta-structure 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 finite-thickness space-coiling meta-structures to identify the skyrmion nature of their vectorial field configurations. In the 2D geometry, the LSP field profiles only exhibit scalar (H-field) or 2D vectorial (E-field) properties. However, for the 3D space-coiling meta-structure with finite thickness, both H-field and E-field distributions manifest 3D vectorial configurations at the interface between the meta-structure and the surrounding background (Supplementary Fig. S5). With decreasing the 3D space-coiling meta-structure 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 number14

$$S=\frac{1}{4\pi }\iint {{{{{\bf{h}}}}}}\cdot \left(\frac{\partial {{{{{\bf{h}}}}}}}{\partial x}\times \frac{\partial {{{{{\bf{h}}}}}}}{\partial y}\right)dxdy,$$
(1)

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 ultra-thin space-coiling meta-structure (h = 0.016 mm, a = 1 mm, d = 1.5 mm, R = 30 mm, nr = 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 multi-resonant 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 eigen-resonances of the meta-structure, we observe a strong excitation of the skyrmions with the near-equidistant multi-resonant spectrum for various near-fields or far-field sources (Supplementary Fig. S7), in stark contrast with previous skyrmions based on the interference of carefully tailored propagating surface plasmons14,15,16.

Because these modes resonate at a deeply subwavelength scale with large radial wavevectors kr k0, the z-component 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 dm = λ/100, and a half vertical size hm = λ/400 (Fig. 3d, left-lower panel), leading to an extremely subwavelength mode volume Vm = 2πhm(dm/2)2 = π(λ/2)3/106. We stress that such tiny mode volume is obtained with just nr = 20, limited by our fabrication and measurement setup. Further squeezing may be achieved by increasing nr. We experimentally observe a relatively high Q-factor of 165 (Fig. 3c), yielding a Purcell factor exceeding 107, promising for various applications requiring strong light-matter interactions. Both out-of-plane (Fig. 3d) and in-plane (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

$${S}_{i}=\frac{1}{4\pi }\int _{0}^{2\pi }d\varphi \int _{{\rho }_{i}}^{{\rho }_{i+1}}d\rho \frac{d\varTheta (\rho )}{d\rho }\sin \,\varTheta (\rho )=-\frac{1}{2}\,\cos \,\varTheta (\rho )\bigg|\begin{array}{c}{\rho }_{i+1}\\ {\rho }_{i}\end{array},$$
(2)

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 multiple-π-twist TS constructed by multiple elementary skyrmions9, with rich possibilities to implement various topological configurations of different orders10.

The vectorial nature of the skyrmion modes can be observed in our real-space measurements of all three magnetic field components, shown in Fig. 4. The in-plane fields are along the radial direction, Hx and Hy reveal a nodal-line profile along their perpendicular axis (y axis and x axis, respectively). At the same time, the out-of-plane component (Hz) 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 hedgehog-like vector configuration (bottom panel of Fig. 4), which is the direct signature of Neel-type skyrmions31. 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 H-field, the E-field profile shows a skyrmion configuration with an extra-π/2-twist, yielding a total skyrmion number of 1/2 (Supplementary Fig. S9)32,33,34. The H-field and E-field are parallel due to the extreme anisotropy of the space-coiling 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. Time-varying 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 well-defined skyrmion topology nature14.

### Arbitrarily shaped LSP skyrmions

One unique feature of the skyrmions supported by this space-coiling meta-structure 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 higher-order 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 meta-structure with the highly sought property of shape-independent resonance features envisioned in zero-index metamaterials35.

To experimentally demonstrate the robustness, we fabricated space-coiling meta-structures with different shapes: elliptical, polygonal, and heart-shaped, keeping constant the effective coil length. We show the measured spectra in the left-most panels of Fig. 5a–c. Even though the geometries are drastically different, their EM response manifests similar near-equidistant multi-resonant spectra. The spatial profiles of all vectorial magnetic field components of these modes for different shapes are directly measured by our near-field scanning technique (as shown in Supplementary Figs. S15S17), 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 multiple-π-twists from the geometry center to the periphery. Such arbitrarily shaped skyrmion configurations have been long-sought in various systems5, 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 space-coiling meta-structure. The skyrmions stem from the eigen-resonances of the tailored meta-structure, and hence they do not require external interference in the illumination. We observe multiple-π-twist TS vectorial configurations in real-space through near-field scanning, yielding extremely subwavelength features, down to λ3/106. The magnetic LSP skyrmion shows large topology stability against continuous deformations of the space-coiling geometry, and manifests an overall stable multi-resonant spectrum and flexible skyrmionic textures with arbitrary shape. Although our proof-of-principle has been demonstrated in the microwave regime, we envision exciting opportunities in various frequency ranges, from near-DC 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 space-coiling cylinder and the EM spectral response of the space-coiling 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 space-coiling cylinder. The SCS was extracted by integrating the out-going Poynting vector along a closed surface enclosing the structure. EM spectral responses of the ultra-thin space-coiling metasurface were calculated by probing the magnetic field at its central point above the structure. All field components within a square area enclosing the space-coiling meta-structure were extracted to calculate the skyrmion densities and skyrmion numbers.

### Sample preparation and field measurements

The space-coiling meta-structure samples were fabricated with printed circuit board technology by printing 0.016-mm-thick copper space-coiling patterns with air gap a = 1 mm, spiral pitch d = 1.5 mm, and turn number nr = 20 on top of a 0.2-mm-thick dielectric substrate with relative permittivity εr = 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 (Hx, Hy, Hz) 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 point-by-point measurements.