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Abstract
Photonic skyrmions and merons are topological quasiparticles characterized by nontrivial electromagnetic textures, which have received increasing research attention recently, providing novel degree of freedom to manipulate light-matter interactions and exhibiting excellent potential in deep-subwavelength imaging and nanometrology. Here, the topological stability of photonic spin meron lattices, which indicates the invariance of skyrmion number and robustness of spin texture under a continuous deformation of the field configuration, is demonstrated by inducing a perturbation to break the C4 symmetry in the presence spin-orbit coupling in an optical field. We revealed that amplitude perturbation would result in an amplitude-dependent shift of spin center, while phase perturbation leads to the deformation of domain walls, manifesting the metastability of photonic meron. Such spin topology is verified through the interference of plasmonic vortices with a broken rotational symmetry. The results provide new insights on optical topological quasiparticles, which may pave the way towards applications in topological photonics, optical information storage and transfer.
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1. Introduction
Topological spin textures in condensed matter physics such as skyrmions characterized by a magnetization swirl have attracted numerous research attention for the past decades [1–7], as they possess ultracompact size and great stability due to the topological nature of structure, promising for data storage and spintronic applications [8–11]. This topological feature was observed in optical system recently, with photonic counterpart of magnetic skyrmions constructed by either spin angular momentums (SAMs) [12–16], electromagnetic fields [17–20] or Stokes pseudospin vectors [21–24]. Among the numerous realizations, photonic spin skyrmions manifest the topological properties in real space through the spin-momentum locking in evanescent fields [25–27], and the deep-subwavelength features in local spin distributions have demonstrated advanced applications in nanoscale metrology [28] and magnetic domain imaging [29].
In addition to skyrmions carrying integer topological charge, exploration of topological quasiparticles with a half-integer topological charge known as merons, which originate from a singular solution of Yang-Mills theories [30], is another active topic in both condensed-matter physics and optics. In magnetic materials, merons are stabilized in the form of pairs or square lattices through easy-plane anisotropy [31–36]. Similarly, photonic spin merons were demonstrated to be constructed as square lattices in the presence of spin-orbit coupling in the environment with broken rotational symmetry very recently [37,38], where nontrivial winding in the impinging field gives rise to topological nontrivial spin textures with topology determined by the symmetry of field. Higher order merons and bimerons are also suggested in polarization textures [39,40].
While numerous skyrmionic objects have been proposed in optical community, the topological stability under a continuous deformation of the field configuration has not yet been investigated. In this paper, we demonstrate the metastability of photonic spin meron lattices by inducing a perturbation in the presence spin-orbit coupling. Theoretical model is established by breaking the C4 symmetry of the field, indicating that the gradient of perturbation is the key factor to determine the spin topology. By employing the generalized spin-momentum relation, we reveal that the central of spin texture in each unit cell will be shifted with domains unchanged under amplitude perturbation. While phase perturbation will deform the domain walls into trapezoid, and the skyrmion number is conserved, manifesting the metastability of square meron lattices. Finally, a proof-of-concept experiment is performed utilizing plasmonic vortices to demonstrate the spin topology under perturbation.
2. Results and discussion
2.1 Theoretical model
Photonic spin meron lattices are formed due to the spin–orbit coupling in evanescent optical vortex (eOV) under a C4 symmetry of the field, which can be described by an interfered Hertz vector potential in a source free and homogeneous medium as
The skyrmionic vector field is constructed from the unit vectors in the direction of a three-component spin in Eq. (3) as n = S/|S|, which possesses distinct domains where nz = 0. The topological invariant is characterized by the skyrmion number as [6,7]
For each unit cell of the spin lattice, the integral region Σ in Eq. (4) is |krx − mπ|<π/2, |kry − nπ|<π/2 with (m, n) denoting integer numbers, and the skyrmion number can be calculated as N = ± ½, manifesting the photonic spin meron topology.
The resultant C4 symmetry in the spin meron lattice is a consequence of C4 symmetry of Hertz potentials $\Psi [\hat{R}({\pi / 2}){\mathbf r}] = \Psi ({\mathbf r})\exp(i{\pi / 2})$, where $\hat{R}(\varphi )$ is the rotation matrix along z-axis. As a perturbation is imposed in the presence of spin-orbit coupling to break the C4 symmetry, additional term should be imported to the Hertz potential as Ψp = Ψ + Φ. The SAM can be obtained from Eq. (3) as
The first and third terms in Eq. (5) correspond to the Berry curvatures of original Hertz potential Ψ and perturbation Φ respectively, while the second term denotes the interaction between Ψ and Φ, indicating that the gradient of perturbation is the key factor to determine the variation on SAM. Amplitude and phase perturbations are considered to represent Φ in the following to analyze the topological property of spin vectors under perturbation.
2.2 Spin topology under amplitude and phase perturbations
By introducing an amplitude perturbation δ in Eq. (1) (assume A2 is changed to 1 + δ without loss of generality), the C4 symmetry is broken, and the Hertz potential is expressed as
The domains where the z component of the local spin orientation nz = 0 does not change under amplitude perturbation, since there is only a coefficient correction on Sz. To form a topological spin texture, the local spin orientation should rotate progressively from the central ‘up’ or ‘down’ state where nz = ± 1 to the edge, corresponding to a mapping onto the unit sphere from the north pole. Hence the in-plane SAM components should have zero points inside each unit cell, yielding |δ| < 2 from Eq. (8). And the central of the spin texture where nz = ± 1 would be shifted by |sin-1(δ/2)|/kr along y axis.
The Poynting vector and local SAM distributions at z = 0 plane for δ = 1 are shown in Figs. 1(a)–(c) as an example to demonstrate the spin topology. The zero points of Hertz potential correspond to the phase singularities (red and blue dots in Fig. 1(a)), which refer to the center of each Poynting vector vortex from Eq. (2). The generalized spin-momentum relationship in Eq. (3) indicates that the SAM originates from the vortices of the electromagnetic energy flow. According to right hand screw rule, only Sz is present in the center of Poynting vector vortex with sign determined by the rotation direction. The amplitude perturbation does not change the domains of each unit cell where nz = 0, while shifting the central of spin texture (Fig. 1(b)). In addition, the junction of Poynting vector vortices where P = 0 with nonzero Hertz potentials will be shifted by |cos-1(δ/2)|/kr along x axis (green dots in Figs. 1(a),(b)), resulting in a dislocation between the vortex junction and domain intersection of spin texture. This dislocation disunifies the transverse spin orientation around the vortex junction inside each unit cell, which is in analogy with an antivortex (see zoomed view of local spin orientation around the vortex junction in Fig. 1(c), that the spin vectors point oppositely along the two sides of vortex junction), hindering the formation of local Neel-type domain wall. Consequently, although the polarity of the spin texture in each unit cell is ± 1/2 as the local spin orientation rotates progressively from the central ‘up’ or ‘down’ state to the edge where nz = 0, the skyrmion number is not conserved, since the vorticity denoting the transverse spin orientations is not 1. The skyrmion number can be calculated through Eq. (4) as ± 0.1547, manifesting a destructive accumulation on vorticity.
Fig. 1. Spin topology under amplitude perturbation. (a) Poynting vector (arrows) and phase distribution of Hertz potential (background) at z = 0 plane for δ = 1. Red and blue dots denote the center of the anticlockwise (red arrows) and clockwise (blue arrows) Poynting vector vortices respectively, where the phase of the Hertz potential is singular. (b) Normalized local spin orientations n (arrows) with background representing nz for δ = 1. Red and blue dots denote the position where nz = ± 1. (c) Zoomed view of the yellow rectangle in (b), where the normalized local spin orientation is an antivortex. Green dots in (a)–(c) represent the junction of Poynting vector vortices where P = 0 with nonzero Hertz potentials. (d),(e) The same as in (a),(b) for δ = 3. No Poynting vector vortices are formed in (d) and the skyrmion number shrinks to zero in the spin textures in (e). (f) Skyrmion number (black) and figure of merit (red) dependences on δ (central unit cell of the spin lattice |x,y|<λr/4). The plot ranges in (a),(b) and (d),(e) are from −0.75 λr to 0.75 λr and the scale bar is λr/2 in (a), while the plot range in (c) is from −0.15 λr to 0.15 λr. The in plane wavevector kr of the evanescent field is set to 1.05 k.
As a contrast, the Poynting vector and SAM distributions at z = 0 plane for δ = 3 are shown in Figs. 1(d),(e). From Eq. (7), the x component of Poynting vector is negative around the whole plane, cancelling the circulation of electromagnetic energy flow (Fig. 1(d)). And the spin texture exhibits directional flux with the core shifted out of the domain (Fig. 1(e)), preventing the formation of topological objects. A figure of merit (FOM) is defined as Q = exp(−20|N−0.5|) to quantify the stability under perturbation, where N is the skyrmion number. For tiny amplitude perturbation, N is close to 0.5 and the figure of merit is approaching 1. The skyrmion number (black solid line) in the central unit cell (|x,y|<λr/4) and FOM (red solid line) for different value of δ are calculated in Fig. 1(f), demonstrating the perturbation tolerance as |δ| < 0.1 to limit the fluctuation of skyrmion number within 10% (Q > 1/e). It is worth noting that for |δ| ≥ 2, the skyrmion number shrinks to zero due to the absence of Poynting vector vortex.
As the amplitude perturbation δ is a spatial function, only periodic perturbation with period λr would preserve the topological spin texture. If the oscillation is along on the propagating direction of perturbated evanescent wave (x axis in our case as the perturbation is imposed on A2), the domain is kept as rectangle with the central of the spin texture shifted along y axis. While for a spatial perturbation with oscillation perpendicular to the propagating direction of evanescent wave, the domain for each unit cell would be reshaped to a curved surface.
As a phase perturbation φ is imposed in the presence of spin-orbit coupling (assume φ2 is changed to φ2 + φ in Eq. (1)), the Hertz potential, Poynting vector and SAM distributions can be calculated as
2.3 Numerical simulation
The spin topology of meron lattice under perturbation is numerically verified as follows. The spin–orbit coupling in evanescent optical vortex is realized by tightly focusing a radially polarized beam with helical wavefront (topological charge l = 1, wavelength λ = 633 nm) onto a thin silver film (thickness d = 50 nm, n1 = 0.135 + 3.99i) sandwiched by a silica substrate (n2 = 1.515) and air (n0 = 1), which provides an excitation of surface plasmon polaritons (SPPs) sustained at silver/air interface [41]. By modulating the incident beam with intensity mask comprised of C4 symmetry aperture, photonic spin meron lattice is formed due to interference of the SPPs, as shown in Fig. 3(a). According to the Richards-Wolf vectorial diffraction theory, the electromagnetic field near focus in air can be calculated as [42,43]
Fig. 2. Spin topology under phase perturbation. (a) Poynting vector (arrows) and phase distribution of Hertz potential (background) at z = 0 plane for φ = π/18. Red and blue dots denote the center of the anticlockwise and clockwise Poynting vector vortices respectively. (b) Normalized local spin orientations n (arrows) with background representing nz for φ = π/18. Red and blue dots denote the spin center in each unit cell. Green dots in (a),(b) represent the junction of Poynting vector vortices. (c) Side view of the normalized local spin vectors in (b), exhibiting a spin meron lattice. (d) Normalized z component of SAM for φ = π/6. (e) Skyrmion number (black) and figure of merit (red) dependences on φ (central unit cell of the spin lattice). The plot ranges in (a)–(c) are from −0.75 λr to 0.75 λr and the scale bar is λr/2 in (a). While the plot range in (d) is from −3 λr to 3 λr. The in plane wavevector kr of the evanescent field is set to 1.05 k.
Fig. 3. Numerical simulations of spin topology by tightly focused vector beam. (a) Schematic diagram of the proposed setup. A radially polarized beam with helical wavefront (topological charge l = 1) modulated by an intensity mask comprised of C4 symmetry apertures is tightly focused on a thin silver film to excited SPPs. Due to interference of the SPPs in the presence of spin-orbit interaction, the spin meron lattice is formed. (b)–(e) Normalized Sz distributions and local spin vector orientations for (b) Δφ2 = π/9 (c) φ2 = π + π/18 (d) Δφ1 = Dφ2 = π/9 and (e) φ1 = π/2 − π/18, φ2 = π + π/18, demonstrating the modulation on spin textures under perturbation. Insets in (b)–(e) denote the opening angle and orientation of each aperture in the intensity mask.
The electromagnetic field in Eq. (12) is a consequence of four interfering plasmonic fields propagating along different directions, which is in correspondence to the Hertz potential in Eq. (1). The opening angle Δφn determines the amplitude of SPPs along each direction, while the orientation of apertures φn represents the phase for each interfering SPP wave. By adjusting the opening angle or orientation of apertures, the C4 symmetry of aperture would be broken, which will induce amplitude and phase perturbation in the presence spin-orbit coupling. The normalized Sz distributions and local spin vector orientations for Δφ2 = π/9 corresponding to δ = 1 in Eqs. (6-8) are shown in Fig. 3(b), demonstrating a shifted spin center for each unit cell. On the other hand, by tuning the orientation of aperture as φ2 = π + π/18 (inset of Fig. 3(c)), the domain walls of spin texture are deformed into trapezoid, as shown in Fig. 3(c). These distributions correspond well to the theoretical predictions in Figs. 1(b) and 2(b), revealing spin topologies under respective perturbation. As multiple perturbations are imposed, the modulation on spin textures can be regarded as a superposition of individual variation, along with the interaction between perturbations. The spin texture for Δφ1 = D φ2 = π/9 is shown in Fig. 3(d), demonstrating a shift of spin center along both x and y axis. While the interaction between perturbations slightly compresses the domain wall. Similarly, under multiple phase perturbation as φ1 = π/2 − π/18 and φ2 = π + π/18, trapezoid domains with larger inner angle are formed in Fig. 3(e).
The aforementioned spin topology can be regarded as a consequence of imperfection of incident structured beam, where the amplitude and phase for the four interfering SPPs would be modified. That metastability of photonics meron lattice can also be verified through the interaction between circularly polarized light and fourfold metallic nanoslits [38]. By tuning the width or tilt angle of the nanoslits corresponding to amplitude and phase perturbation, same spin distribution can be obtained as in Figs. 3(b)–(d), which can be considered as an imperfection of nanostructures.
3. Conclusion
In conclusion, we have demonstrated the topological property of photonic spin meron lattice under both amplitude and phase perturbations, which are imposed by breaking the C4 symmetry in the presence of spin-orbit coupling, in correspondence to the imperfection of either impinging field or nanostructure. By employing the generalized spin-momentum relation, we show that amplitude perturbation would result in an amplitude-dependent shift of spin center in each unit cell, while keeping the location of domain walls. However, the perturbation tolerance is small to conserve the skyrmion number, since the dislocation between the vortex junction and domain intersection hinders the formation of local Neel-type domain walls and decreases the vorticity of spin texture. In contrast, phase perturbation leads to the deformation of domain walls with the skyrmion number merely changes, unveiling the metastability of photonic meron. In addition, the spin topology is verified through the focusing property of plasmonic vortices. These novel topological features of electromagnetic waves provide new insights on skyrmionic textures in the realm of optics, which may pave the way for spin-optics, sensing and metrology.
Funding
National Natural Science Foundation of China (12204309, 12274299, 92050202); Science and Technology Commission of Shanghai Municipality (19060502500); Shanghai Rising-Star Program (22YF1415200).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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