## Generating a lattice of optical skyrmions

The topological properties of systems can be robust to defects and imperfections. Skyrmions are one such topological entity, which resemble “hedgehog-like” structures. Tsesses *et al.* controlled the interference of plasmon polaritons on a patterned metallic surface to generate a lattice of optical skyrmions. Such control could potentially optically mimic complex condensed matter systems or help in the development of robust optical communications and circuits.

*Science*, this issue p. 993

## Abstract

Topological defects play a key role in a variety of physical systems, ranging from high-energy to solid-state physics. A skyrmion is a type of topological defect that has shown promise for applications in the fields of magnetic storage and spintronics. We show that optical skyrmion lattices can be generated using evanescent electromagnetic fields and demonstrate this using surface plasmon polaritons, imaged by phase-resolved near-field optical microscopy. We show how the optical skyrmion lattice exhibits robustness to imperfections while the topological domain walls in the lattice can be continuously tuned, changing the spatial structure of the skyrmions from bubble type to Néel type. Extending the generation of skyrmions to photonic systems provides various possibilities for applications in optical information processing, transfer, and storage.

Topological defects are field configurations that cannot be deformed to a standard, smooth shape. They are at the core of many fascinating phenomena in hydrodynamics, aerodynamics, exotic phases of matter (*1*, *2*), cosmology (*3*), and optics (*4*) and, in many cases, are of importance to practical applications. The intricate dynamics of a multitude of topological defects and the efforts to control them are of key importance in high-temperature superconductivity (*5*) and topological phase transitions such as the Berezinskii-Kosterlitz-Thouless transition (*6*).

One type of topological defect is a skyrmion (*7*), a topologically stable configuration of a three-component vector field in two dimensions. Skyrmions were initially developed theoretically in elementary particles and have since been demonstrated in Bose-Einstein condensates (*8*) and nematic liquid crystals (*9*) and as a phase transition in chiral magnets (*10*, *11*). The skyrmion lattice phase and single magnetic skyrmions (*12*, *13*) are considered a promising route toward high-density magnetic information storage and transfer (*14*, *15*), as they are very robust to material defects and can be driven by low applied currents (*12*, *16*, *17*). A skyrmion may take on various shapes, which are all topologically equivalent. Bloch-type (*11*) and Néel-type (*18*) skyrmions exhibit a smoothly varying field configuration, with the derivatives of the vector field spread out in space. In bubble-type skyrmions (*19*, *20*), the variations of the vector field are confined to line-like areas, known as topological domain walls, which separate two domains in which the field vectors are opposite.

In optics, topological phenomena have been researched intensely in the past decade. Since the first observation of photonic topological insulators (*21*), optical topological phenomena have been rigorously studied, both theoretically and experimentally (*22*–*25*), with applications such as topologically protected lasing (*26*–*28*). Indeed, topological defects in optics were first extensively explored via phase and polarization singularities, both in free-space propagating light (*29*–*31*) and in two-dimensionally confined light (*32*–*34*). However, only recently has there been any experimental investigation of optical topological domain walls (*35*), whereas the field of optical skyrmions has so far remained untapped.

## Formation of optical skyrmion lattices

The configuration of a three-component real vector field on a two-dimensional (2D) space provides a smooth mapping of that space to the unit sphere. The topological invariant that identifies skyrmions counts the number of times that the field configuration covers the entire sphere. This topological invariant, which we denote by *S*, is known as the skyrmion number and takes integer values. For the skyrmion to be topologically robust, the space on which it is defined cannot have a boundary. This condition is indeed satisfied by a periodic field configuration, in which a skyrmion is obtained in each unit cell of a lattice. For such a “skyrmion lattice” configuration, the skyrmion number can be written in an integral form(1)where the area *A* covers one unit cell of the lattice and is the skyrmion number density; is a real, normalized, three-component field; and *x* and *y* are directions in the 2D plane. The skyrmion number *S*, being an integer, is robust to deformations of the field as long as remains nonsingular and maintains the periodicity of the lattice.

We find that the electric field vector of electromagnetic waves can be structured so as to meet all the requirements needed to create a skyrmion lattice, similar to the magnetization vector of the skyrmion lattices in chiral magnets. Being a 3D field in a 2D space, optical skyrmions must be formed by electromagnetic waves confined to two dimensions, such as in guided waves. A more thorough explanation as to why free-space electromagnetic waves do not exhibit skyrmions can be found in the supplementary text.

Consider an electric field comprised of six interfering transverse-magnetic (TM) guided waves with equal amplitudes, freely propagating in the transverse (*x*-*y*) plane and evanescently decaying in the axial direction (*z*). The six waves are directed toward each other in the transverse plane and possess transverse wave vectors of similar magnitude, such that they create three standing waves oriented about 0°, 60°, and 120°. The axial (out-of-plane) field component in the frequency (ω) domain can therefore be expressed as a sum of three cosine functions(2)where *E*_{0} is a real normalization constant and and are the transverse and axial components of the wave vector ( is real and is imaginary), such that ( is the free-space wave number). The transverse (in-plane) electric field components in the frequency domain can be readily derived from Maxwell’s equations:(3)Namely, for waves evanescently decaying in one dimension, all electric field components are entirely real (up to a global phase), allowing the definition of a real unit vector associated with the electric field and enabling a description of the field configuration as an optical skyrmion lattice. This is a direct consequence of the phase added in the spin-momentum locking process of evanescent electromagnetic fields (*36*) and hence does not hold for propagating waves [see (*37*), section S.1].

Figure 1 depicts the electric field described by Eqs. 2 and 3 for . The axial electric field has the form of a hexagonally symmetric lattice (Fig. 1A). The transverse field follows the same symmetry yet possesses pronounced polarization singularities, which are expressed by zero-amplitude points at the center of each lattice site (Fig. 1B), at which the field direction is ill-defined (Fig. 1C). The normalized 3D electric field (Fig. 1D) confirms the formation of a skyrmion lattice; each lattice site exhibits the distinct features of a Néel-type skyrmion (*18*), and the calculated skyrmion number of each site, using Eq. 1, is *S* = 1.

Formation of the optical skyrmion lattice can be represented in the momentum space as a transition from free-space propagation in 3D () to guided mode propagation in 2D (). To rigorously describe this transition, we must expand the skyrmion number definition to complex electromagnetic fields by defining in Eq. 1 as the real part of the local unit vector of the electric field. This new definition is consistent with the prior description of optical skyrmions in the ideal case, which is determined by Eqs. 2 and 3 in a lossless medium. Figure 2 presents the transition by showing the skyrmion number in a single lattice site as a function of using the above definition for .

Figure 2 also shows the skyrmion number density contrast [ψ = (*s*_{max} + *s*_{min})/(*s*_{max} − *s*_{min})] in a single lattice site as a function of , which is a parameter providing a quantitative measure of the spatial confinement of the skyrmion density. For slightly larger than , the skyrmions are spatially confined (density contrast close to 1), with clear domain walls separating between two specific field states, effectively creating bubble-type skyrmions (point i). As grows, the domain walls start to smear (point ii), creating skyrmions with increasingly uniform skyrmion number density and converging to the Néel-type field formation shown in Fig. 1 (point iii in Fig. 2, density contrast of 0.5). This stems directly from the scaling factor , which can be tuned by varying the effective index (normalized propagation constant) of a guided mode.

## Observation of an optical skyrmion lattice

Observing the optical skyrmion lattice has several prerequisites: First, it requires a physical system that allows 2D guided waves to propagate in six specific directions while interfering them with carefully controlled phase differences. Second, it necessitates a measurement apparatus that will enable the phase-resolved imaging of such electromagnetic waves at a resolution far better than the optical diffraction limit. Additionally, a nonideal physical system—for example, one that is finite and with inherent losses—provides an excellent platform to examine the robustness of the topological properties of the optical skyrmions.

Such losses distort the unit vector associated with the electric field and create an unwanted phase difference between its components. However, in the regime of small losses, the configuration of the real part of the field still yields a well-defined skyrmion lattice, while being sufficiently larger than the imaginary part such that the latter is negligible [see full derivation in (*37*), section S.3]. In this regime, the skyrmion number *S* deviates slightly from its quantized value at zero losses owing to the weak breaking of the lattice translation symmetry, increasingly more so for unit cells that are farther away from the point of origin. Figure 3 gives a quantitative connection between the amount of loss and the robustness of the skyrmion number, showing the number of unit cells exhibiting *S* > 0.99. For example, a configuration in which 49 sites exhibit *S* > 0.99 is obtained if the electromagnetic waves creating it persist for roughly 400 periods before decaying—an easily achieved goal in many photonic systems.

A physical system capable of fulfilling the above-mentioned requirements consists of surface plasmon polaritons (SPPs) (*38*)—electromagnetic surface waves existing at the interface between metallic and dielectric materials. SPPs only exist in TM polarization, and the phase difference between SPPs propagating along different directions can be easily controlled (*39*). Furthermore, they are, by design, an imperfect system, owing to the ohmic losses generated by the metal.

In our system (see Fig. 4), SPPs are excited at the interface of air and gold, resulting in both a small propagation decay and a transverse wave vector just slightly larger than the free-space wave number []. Circularly polarized light impinges on a specially designed, hexagonally shaped coupling slit, exciting SPPs from each slit edge toward its center. The slit provides the same phase to the SPPs created by all edges, yet not exactly the same amplitude (owing to a different propagation length of the SPPs generated by two of the edges and their propagation loss). This results in a distortion of the skyrmion lattice, which, together with the finite structure, helps in examining its robustness.

In the experimental setup used to detect optical skyrmions [see (*37*) and fig. S1], a scattering near-field scanning optical microscope (Neaspec neaSNOM) enables phase-resolved measurement of the electric field normal to the surface by means of pseudoheterodyne interferometric detection (*40*). The ability to detect phase information is crucial, as it not only provides the full axial electric field but also allows us to perform a spatial Fourier transform (and its inverse) to filter out noises. The phase information, filtering ability, and high spatial resolution of the measurement enable the correct extraction of the transverse field components and, thus, of the skyrmion number.

The plasmonic wave number and applied boundary conditions create a plasmonic skyrmion lattice in the central part of the sample with characteristics typical of bubbles (Fig. 5). These characteristics are a result of the relatively small transverse wave vector, leading to an axial field component five times larger than the transverse ones. Although the real electric field is slightly distorted, as expected (Fig. 5, A to C), it still shows similarity to the electric field configuration presented in Fig. 1. The extracted skyrmion number density at the center of the lattice (Fig. 5E) resembles that of the bubble-type skyrmion lattice shown in the inset of Fig. 2 (point i). Calculating the skyrmion number in each lattice site, we reach the result *S* = 0.997 ± 0.058, thus demonstrating the robustness of the optical skyrmions.

Although not demonstrated experimentally in this work, the generation of a single skyrmion in the electric field of evanescent electromagnetic waves is possible as well [see (*37*), section S.5]. A single optical skyrmion may be achieved by engineering finite boundary conditions while still preserving a closed manifold, in a similar manner to schemes already suggested and observed in magnetic skyrmions (*41*, *42*).

## Discussion

We show that a skyrmion lattice can be obtained as a solution to a linear plasmonic system by proper engineering of the boundary conditions and the transverse momentum of the electromagnetic waves. The topological invariant classifying the skyrmions in the lattice is protected by the lattice symmetry, and the lattice itself may be realized in any photonic system with evanescent waves—for example, planar waveguide modes and waves undergoing total internal reflection.

The generation of an optical skyrmion lattice paves the way toward inducing skyrmion lattices “on demand” in matter systems (e.g., cold atoms or dielectric particles in a fluid) through light-matter interactions. Namely, this will allow stimulated creation of skyrmions in matter, as opposed to skyrmions in all other systems [including optically excited skyrmions in chiral magnets (*43*)], which are spontaneously created.

Furthermore, optical nonlinearity in systems supporting evanescent waves (e.g., SPPs in gold or nonlinear dielectric waveguides) could give rise to soliton-like skyrmion states exhibiting a topologically protected skyrmion number, which will be robust against external perturbation.

Here we focused on TM-polarized electromagnetic waves, which exhibited a skyrmion lattice in the electric field. Arranging transverse-electric polarized waves in a similar way, by using a dielectric waveguide or by total internal reflection, would create skyrmions in the magnetic field of the electromagnetic wave, with the potential to stimulate a skyrmion lattice not only electrically but also magnetically by using either femtosecond pulsed excitations or submillikelvin temperatures.

Optical skyrmion lattices could also bring about new optical effects, highlighting their potential applications in optical information processing, storage, and transfer. For example, in light-emission processes such as fluorescence or harmonic generation, the specific field configuration of optical skyrmions would result in the simultaneous emission of all possible polarizations in a structured, phase-locked, and coherent way. Alternatively, probe beams could nonlinearly interact with the skyrmion lattice via Kerr nonlinearity and be affected by a complex, polarization-dependent Berry curvature.

## Supplementary Materials

www.sciencemag.org/content/361/6406/993/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S5

Reference (*44*)

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We thank S. Dolev for his help in fabrication of the measured samples. S.T. also acknowledges the generous support of the Jacobs Foundation and I. Khanonkin’s help in performing simulations.

**Funding:**This research was supported by “Circle of Light,” Israeli Centers for Research Excellence (I-CORE), through the Israel Science Foundation (ISF), grant no. 1802/12; and by the European Research Council (Horizon 2020 program), grant no. 639172.

**Author contributions:**S.T., B.G., and G.B. conceived the project; S.T. and K.C. patterned the samples; E.O., S.T., and K.C. performed the measurements; and S.T., N.L., and G.B. performed simulations and analytical calculations. All authors took part in preparing the manuscript.

**Competing interests:**The authors declare no competing interests.

**Data and materials availability:**All data are available in the main text or the supplementary materials.