Skyrmions are topologically nontrivial spin textures appearing in a wide variety of physical systems, including two-dimensional electron gases,1^,2 Bose–Einstein condensates,3^,4 liquid crystals,5 superfluid,6 and chiral magnets.7^,8 Since the first observation in magnetic materials with broken space-inversion symmetry,9 magnetic skyrmions characterized by a magnetization swirl have attracted research attention, as they possess ultracompact size and great stability stemming from topological protection,10^–14 which is promising for data storage and spintronic applications.15^–19 Depending on the anisotropy of magnetic material and Dzyaloshinskii–Moriya interaction (DMI) induced by the spin–orbit coupling (SOC) of electrons, different types of skyrmions are formed and stabilized.20^–24

Photonic counterparts of magnetic skyrmions were observed recently,25^–27 with topological textures constructed by spin angular momentums (SAMs),25^,28^–32 electromagnetic fields,26^,33^–36 Stokes pseudospin vectors,37^–40 or electromagnetic energy flow.41 On the one hand, Stokes skyrmions were formed in paraxial beams, where the topological feature is embedded in the polarization texture of structured light and manipulated by the selection of Poincaré beams. On the other hand, the high inhomogeneity of light provides a versatile platform to sculpture electromagnetic fields, resulting in the formation of field and spin skyrmions in evanescent fields. As spin-1 particles, photons exhibit skyrmionic features in both real and momentum spaces.42^,43 Similar to the DMI in magnetic skyrmions, the SOC of light, which is enhanced within evanescent fields and at the subwavelength scale,44^–48 plays a significant role in constructing spin skyrmions, where the field’s symmetry determines the spin topology.30 The local spin twisting in skyrmions has been used in advanced applications in superresolution imaging28 and nanoscale metrology.49^,50 Of the many skyrmionic objects proposed in optical systems, they are formed either in paraxial beams where the transverse wave vector is negligible or in evanescent waves where the mirror symmetry is broken. This leaves a gap: skyrmionic structures in nonparaxial propagating waves have remained a challenge, despite their significant fundamental and practical importance. Although skyrmion-like spin structures were constructed in tightly focused vortex beams,25^,32 these textures are defined only in a restricted area and are not valid across the entire plane.

In this work, we verified the existence of topological spin textures in a nonparaxial free-space beam in the presence of a hybrid SOC. By introducing the interaction between the Hertz potentials to break the dual symmetry of light, we demonstrate a continuous rotation of local spin vectors, giving rise to the formation of Bloch-type skyrmioniums, which is stretching across the entire plane [Fig. 1(a)]. The skyrmionium would condense into Bloch-type skyrmion lattices and meron lattices as the rotational symmetry of the field is broken. The topological spin texture was experimentally demonstrated on the example of a tightly focused vortex beam, where the complete three-dimensional spin distributions were measured with a dual-mode waveguide probe. By extending the hybrid concept of optical vortices to the evanescent case, we constructed a topologic diagram of photonic spin skyrmions encompassing both evanescent and propagating waves. Similar to magnetic skyrmions where Néel or Bloch types are stabilized by the interfacial or bulk DMI, we show that the SOC with evanescent or propagating field is responsible for Néel or Bloch type photonic skyrmions. An Euler-like polyhedron formula is established in the hierarchical skyrmion lattice to show the constraints among symmetry, spatial dimensionality, and sublattice configurations. These results demonstrate the electron–photon correspondence in constructing skyrmions and extending spin topology to free space. This provides new insights into topological photonics and paves the way for advances in spin optics, sensing, and metrology.

To reveal the topological property of a nonparaxial beam, we start with a single polarized electromagnetic field [transverse electric (TE) or transverse magnetic (TM)], which can be described by an electric or magnetic Hertz vector potential. The SAM density51 for a single polarized wave propagating along the z axis can be calculated as (see Note I in the Supplementary Material for details) S=14ωIm(εE*×E+μH*×H)∝−kr2Im(∇Ψ*×∇Ψ)+2k2ezez·Im(∇Ψ*×∇Ψ),(1)where Ψ is the Hertz potential, ω is the angular frequency of the wave, ε and μ are the absolute permittivity and permeability of the medium, k is the wave vector with kr denoting the transverse component, and ez is the unit vector along the z axis. Equation (1) indicates that the Berry curvature of the Hertz potential ⟨∇Ψ|×i|∇Ψ⟩ plays a key role in constructing topological spin textures, where a nontrivial spin texture arises from the winding of the Hertz potential. Therefore, the orbital angular momentum of light, embodied in the helical phase of the Hertz potential, is essential in generating topological textures of SAM, which is a manifestation of SOC. We consider an optical vortex (OV) in a source-free, homogeneous, and isotropic medium, which can be described in the cylindrical coordinate (r,φ,z) as Ψ=AJl(krr)eilφeikzz, where A is a constant, kr and kz are the transverse and longitudinal wave-vector components, l is an integer corresponding to the topological charge of the OV, and Jl is the Bessel function of the first kind of order l. By applying a single polarized OV into Eq. (1), the radial component of SAM density vanishes and the spin vectors rotate progressively from the core to the periphery toward the azimuthal direction. However, the formation of topological texture is prevented because the longitudinal and azimuthal components of SAM density are in phase [Fig. 1(b)], leading to a discontinuous change of the local spin orientation n=S/|S| within a cycle (see Note I and Fig. S1 in the Supplementary Material for details).

To eliminate the discontinuity in local SAM and generate well-defined topological domains, we introduce hybrid polarized OVs comprising both TE and TM modes. In view of the contribution of electric (Ψe) and magnetic Hertz potentials (Ψm), the SAM density for a hybrid mode can be decomposed as Sh=Se+Sm+Sc, where Se and Sm are SAMs contributed by individual Ψe and Ψm, and the third term Sc represents the coupling between the electric and magnetic Hertz potentials (see Note I in the Supplementary Material for details). Without loss of generality, we assume Ψm=−ηΨe /Z, where η is a complex coefficient and Z=μ/ε denotes the wave impedance. The spin vectors Se and Sm contributed by each Hertz potential take the form of Eq. (1), and the coupling SAM Sc can be expressed in terms of the time-averaged Poynting vector P as Sc=Imη2kω2P∝kr2Im(Ψ*∇Ψ)+ezkz(∇Ψ*·∇Ψ−k2Ψ*Ψ).(2)

In Eq. (2), the coupling SAM depends on both the relative phase between the Hertz potentials and the electromagnetic energy flow. Because the spin vectors of single polarized OVs possess incomplete domains, the coupling SAM, which is proportional to the Poynting vector, plays a crucial role in constructing topological spin textures.

Similar to the spin vectors, the Poynting vector of OV rotates perpendicularly to the radial direction with Pr=0 while forming different patterns than the SAM density [Fig. 1(c)]. On the one hand, the azimuthal and longitudinal components of the Poynting vector are locked to the topological charge and propagation direction of OV, respectively, which maintain the same sign during propagation and manifest a directional energy flow. On the other hand, the zero points of Hertz potential pin the zero points of azimuthal energy flow. Consequently, the local spin orientations would rotate continuously with the import of coupling SAM [Fig. 1(d)], which possess well-defined domains at the zero points of Hertz potential (r=ri, i=1,2,3…). The spin vectors tilt progressively along the azimuthal direction with a 2π radial twist of nz within each domain, leading to the formation of Bloch-type skyrmioniums [top panel of Fig. 1(a)]. Because the polarity denoting the spin reversal from the center to the boundary is zero, the skyrmion number characterizing the topological invariant can be obtained as Q=(1/4π)∬n·(∂xn×∂yn)dxdy=0, where n=S/|S| represents the unit spin vector. Due to the continuous rotation of local spin vectors, the skyrmionic texture stretches across the xy plane.

The SOC in a hybrid polarized vortex beam gives rise to the formation of individual photonic spin skyrmioniums. As a superposition lattice of multiple optical vortices is introduced, skyrmioniums induced by each vortex would interact and condense into topological spin lattices. The spin topology under different symmetries of vortex lattice can be revealed by considering superpositions of hybrid Hertz vector potentials feeding on each lattice point. Only hexagonal and square lattices exist with equal lattice constants in two dimensions, which are investigated in the following. In view of the translational and rotational symmetry imposed by a lattice possessing 2N-fold symmetry (N=2, 3), the total Hertz potential can be obtained as Ψ2N∝∑n=12Neilθneikrr·eneikzz, where θn=nπ/N, en=(cosθn,sinθn). From the Berry curvature of the Hertz potential, N sets of sublattice can be observed in the spin texture, which is constrained by the fold of lattice symmetry (see Note III and Fig. S2 in the Supplementary Material for details). For a hexagonal lattice with N=3, the Hertz potential and SAM distribution exhibit C6 symmetry [Figs. 2(a) and 2(b)]. From the frequency domain spectrum of spin texture [inset of Fig. 2(b)], three sublattices are observed with wave vectors K61=kr, K62=3kr, K63=2kr. In each sublattice, the local spin vectors vary progressively from the central “up” state to the edge “down” state toward the azimuthal direction, manifesting a Bloch-type skyrmion with skyrmion number Q=1 [Figs. 2(c) and 2(d)]. For square lattice with N=2, C4 symmetry is exhibited in the Hertz potential and SAM distribution [Figs. 2(e) and 2(f)], and two sublattices are obtained in the frequency domain spectrum of spin texture [inset of Fig. 2(f)] with wave vectors K41=2kr, K42=2kr. The spin orientations possess distinct domains in each sublattice, and merons are confined in a unit cell with the spin vector rotating from the central “up” or “down” state to the edge Sz=0, manifesting alternating “core-up” and “core-down” Bloch-type spin merons with skyrmion number Q=±1/2 [Figs. 2(g) and 2(h)]. The polarity denoting the spin variation from the central to the edge changes from zero to nonzero with the import of vortex lattice (e.g., 0 for individual vortex, 1 for hexagonal lattice, and ±1/2 for square lattice), transforming topologically trivial skyrmioniums to nontrivial spin textures, which is attributed to the restricted size in a unit cell of the spin lattice for respective symmetries. Depending on the size of the unit cell, the spin vectors rotate to different states at the edge for hexagonal and square lattices, leading to the symmetry constraints for skyrmion and meron topologies.

We experimentally demonstrated the topological spin texture in free space on the example of a tightly focused vortex beam [Fig. 3(a)]. By employing a spatial light modulator (SLM) to generate the vortex phase and combining a quarter-wave plate with a vortex wave plate to modulate the polarization of the incident light, a superposition of radially and azimuthally polarized laser beams (π/2 phase delayed, wavelength λ=532nm) with helical wavefront is tightly focused onto the glass substrate. The input beam can be expressed as Ein=(er+iγeφ)eilφ, where er and eφ denote unit vector along the radial and azimuthal directions, respectively, γ is a real number, and l is an integer representing the topological charge of the vortex beam. The incident beam was modulated by the SLM to limit the azimuthal angle within 10 deg and maintain a fixed transverse wave vector, consistent with the theoretical model (see Note IV in the Supplementary Material for details). A spin-resolved near-field scanning optical microscope was used with a nanosphere-waveguide structure as a probe to characterize the spin texture. The nanosphere waveguide is designed to support the resonances of TE and TM modes with different wave vectors simultaneously. This dual-mode waveguide probe can separate the scattering light into TE and TM components, radiating them into the far field at different angles. The in-plane and longitudinal SAM densities are reconstructed through the directional intensity distributions of TM and TE modes, respectively. This allows for the measurement of the complete three-dimensional SAM distributions (see Note V in the Supplementary Material for details).

For circularly symmetric incident beams, a topological charge l=2 is used to improve the visibility of spin vectors. The in-plane spin vector predominantly rotates along the azimuthal direction [Figs. 3(b) and 3(c)], and the longitudinal SAM exhibits an annular pattern [Fig. 3(d)]. The local spin orientations form individual Bloch skyrmioniums in the vicinity of the focal plane [Fig. 3(e)]. As the polar angle of the incident beam was restricted to exhibit fourfold or sixfold symmetry, OV lattices would be generated, as shown in Fig. 4. The interference of optical vortices in the presence of the SOC gives rise to the formation of interleaved Bloch-type spin meron or spin skyrmion textures. A circularly polarized beam is used as incidence directly because it can be decomposed as a superposition of radially and azimuthally polarized vortex beams with π/2 phase delay. The measured in-plane SAM densities of the generated vortex lattice demonstrate the hierarchical wave vector values in the Fourier spectra for respective symmetries [Figs. 4(c) and 4(g)], revealing both the symmetry and dimension constraints for skyrmion sublattices. The local spin orientations in the inner sublattices in Figs. 4(c) and 4(g) form Bloch-type spin meron and skyrmion lattices [Figs. 4(d) and 4(h)]. The skyrmionic texture exhibits dynamic behavior upon propagation and with varying numerical aperture (NA) during focusing. Specifically, its helicity can be modulated by the longitudinal distance, whereas the size of the skyrmionium is influenced by the NA (see Note IV in the Supplementary Material for details).

The Berry curvature of the Hertz potential is essential for constructing topological spin textures, where the rotation direction of spin vectors is determined by the spatial dimensionality. In a single polarized free-space OV, the azimuthal and longitudinal components of the electric field are in phase, whereas the radial component is out of phase. This results in the local spin vector rotating in the φ−z plane [Fig. 5(a)]. By contrast, for a single polarized evanescent OV, an additional π/2 phase shift is introduced into the transverse electric fields due to the localization of light along the z axis, leading to a π/2 rotation of spin vectors [Fig. 5(b)].

The broken mirror symmetry in evanescent waves gives rise to the formation of Néel-type skyrmions. The construction of topological spin textures in free space depends on the dual symmetry breaking of electric and magnetic Hertz potentials due to their coupling, which necessitates the use of a hybrid polarized vortex beam. The decomposition of SAM density for hybrid mode is also applicable to evanescent optical vortices. In this case, only the azimuthal component is present in the coupling SAM (see Note II in the Supplementary Material for details), transforming the Néel-type skyrmions into twisted ones.

From the above, the spin topology of individual photonic skyrmions formed due to the SOC depends on both the polarization and spatial dimensionality of light. This can be represented by a topologic diagram, as shown in Fig. 5(c). Néel-type skyrmions are located along the x axis with kr/k>1 for a single polarized OV. Tracking along the y axis, hybrid polarization is introduced, leading to the formation of twisted skyrmions with helicity γ=arctan(Sφ/Sr)=arctan(2Imη1+|η|2kkz). By reducing kr to the vicinity of k, the helicity of skyrmion γ approaches π/2 (kz∼0), twisting the skyrmion to a quasi-Bloch-type. In the absence of two-dimensional confinement of light where kr/k<1, a Bloch-type skyrmionium is constructed with a fixed helicity of π/2 due to the free-space propagating of the vortex. For a single polarized OV located along the x axis with kr/k<1, a discontinuous change of the local spin orientation prevents the formation of topological spin textures. The formation of Néel or Bloch-type photonic skyrmions in the presence of SOC in an evanescent or propagating field corresponds to magnetic skyrmions, where a Néel or Bloch type is stabilized by the interfacial or bulk DMI.20^–22

The topologic diagram of individual photonic skyrmions can be extended to skyrmion lattices. In free-space OV lattices, the sublattice configuration is constrained by the symmetry of the optical field. This results in two sets of Bloch-type meron lattices for fourfold symmetry and three sets of skyrmion lattices for sixfold symmetry, respectively. Similarly, N−1 sets of sublattices are observed in the spin texture of evanescent OV lattices due to the confinement of electromagnetic field along the z axis (see Note III in the Supplementary Material for details). This yields an Euler-like polyhedron formula as N+D−K=3,(3)where 2N represents the fold of symmetry, D denotes the spatial dimensionality, and K is the sets of sublattices for the spin texture denoted by the wave vector in the frequency domain. Equation (3) reveals that the hierarchical skyrmion lattice is constrained by both the symmetry and spatial dimensionality of the optical field, where the topology and sublattice configurations are determined. Apart from the skyrmion lattice with six- and fourfold symmetries, topological quasi-crystals can also be realized by utilizing other symmetries52^,53 that exhibit quasi-periodic spin textures. Although the absence of well-defined domains in topological quasi-crystals hinders the formation of conventional skyrmionic textures, the sublattice configurations within the quasi-crystalline lattices still adhere to the Euler-like polyhedron formula (see Note III and Fig. S3 in the Supplementary Material for details). In that case, Eq. (3) is extended to encompass both 2N and 2N+1 symmetries.

In magnetic materials, it is the interplay among magnetic field, material anisotropy, and temperature that determines the spin topology and forms the phase diagram of magnetic skyrmions.11^–13^,22^–24 By contrast, the topological state of photon spin is determined by polarization and spatial dimensionality, which are intrinsic properties of light. This enables greater flexibility in manipulating photonic skyrmions. For example, by varying the azimuthal angle of focusing to encompass the critical angle, the helicity denoting a Néel- or Bloch-type skyrmion can be switched. Skyrmion and meron topologies can be switched by modulating the polar angle of incidence. A properly designed reciprocal lattice also allows for wavelength-tuned transformation between different topologies.54

In summary, we have demonstrated the formation of Bloch-type spin skyrmionic textures in the presence of SOC of light in free space. By introducing the interplay between electric and magnetic Hertz vector potentials, we showed that the SOC in a hybrid polarized vortex beam gives rise to the formation of Bloch-type spin skyrmioniums. As the rotational symmetry of the field is broken, hierarchical skyrmion lattices are formed, which are constrained by both the symmetry and spatial dimensionality of the field, where the spin skyrmion or spin meron topology, sets of sublattices, and the helicity of skyrmions are determined. These topological spin textures were experimentally verified by employing a tightly focused vortex beam with limited azimuthal and polar angles. By bridging the gap in the skyrmionic group, we establish a topologic diagram of photonic skyrmions, illustrating how the SOC of photons governs the spin topology, analogous to the role of SOC of electrons in determining the topology of magnetic skyrmions. Although our reported results here focus on homogeneous cases where the electric and magnetic Hertz potentials are proportional, the decomposition of SAM density for a hybrid mode can be extended to more complicated systems. For example, as the orbital angular momenta in each electromagnetic mode differ and the Hertz potentials are noncoaxial, a peculiar spin texture is formed with linear variation along one axis, enabling picometric displacement sensing. In addition, more electromagnetic modes can engage in SOC, potentially inducing higher-order interactions between Hertz potentials. These couplings between various electromagnetic modes can manipulate the polarity, vorticity, and helicity of topological spin textures. The results create a new avenue for expanding the photonic skyrmion group and offer a fresh perspective on optical quasi-particles, which may find potential applications in quantum technologies, spin optics, and topological photonics.

Xinrui Lei received his PhD in physics from the University of Science and Technology of China. He is currently an associate professor at the University of Shanghai for Science and Technology. His research interests include spin topology of light and optical spin-orbit coupling.

Aiping Yang received her PhD from Shenzhen University in 2018 and her master’s degree from Nankai University in 2014. Her research interests include near-field optical information characterization and spin-orbit interaction in optics. She is currently focusing on the interaction between chiral light fields and chiral materials, as well as their applications.

Xusheng Chen is a PhD candidate at the Nanophotonics Research Center of Shenzhen University. He received his master’s degree in optical engineering from Shenzhen University in 2023. His research interests include optical spin characterization and near-field detection technologies for chiral materials.

Luping Du received his PhD in electrical and electronic engineering from the Nanyang Technological University. He is currently a professor at Shenzhen University. His research interests include nanophotonics, electromagnetic field topology, and optical spin-orbit coupling.

Peng Shi received his PhD in physics from the University of Science and Technology of China. He is currently an associate professor at Shenzhen University. His research interests include angular momentum of light and electromagnetic field topology.

Qiwen Zhan received his PhD in electrical and computer engineering from the University of Minnesota. He is currently the principal investigator of Nano-photonics Research Group at the University of Shanghai for Science and Technology. His research interests include spatiotemporally structured light and nanophotonics.

Xiaocong Yuan is Changjiang Scholar Professor and director of the Nanophotonics Research Center at Shenzhen University. He received his PhD in physics from King’s College London in 1994. He is a fellow of SPIE and Optica, and a board member of the Chinese Optical Society and the Chinese Optical Engineering Society. His research interests include singular optics, nanophotonics, plasmonics, advanced light manipulations, and their applications in communications, imaging, and optical tweezers.