Introduction

When two identical layers of graphene are superposed with a controlled twist angle between them, moiré superlattices can form, giving rise to a range of nontrivial electronic properties1. For example, at a magic angle of approximately 1.08°, flat bands were found around the electronic Fermi energy levels, leading to a correlated insulating phase2. Shortly thereafter, other magic angles were found to induce additional exotic properties within moiré superlattices of bilayer graphene, such as unconventional superconductivity3, higher-order topological insulating states4, tunable spin-polarized phases5, and Chern insulators6. This approach has catalyzed the rapid development of twistronics, in which the simple superposition of twisted bilayer 2D materials can yield new moiré periodic structures with fundamentally altered properties compared to the individual monolayer materials. Twistronics has since expanded to other materials, including various van der Waals optoelectronic materials7, mechanical systems8, acoustic lattices9, and magnetic skyrmion materials10, thereby transforming multiple domains within physics and materials science.

Twistronics has also been implemented in the optical domain11. By generating moiré patterns in optical lattices, it is now possible to excite flat bands within laser-written materials, enabling control over the localization—and delocalization—of light fields12,13, optical soliton formation14, Thouless pumping15, reconfigurable moiré nanolaser16, and skyrmion bags topology17,18. Twisted bilayer photonic materials with van der Waals coupling have proven effective in exciting topological polaritons19,20. Additionally, twisting photonic crystal slabs have been used to control flat-band phenomena21,22,23,24, and vortex beams generation25. Metasurface twisting has emerged as a powerful tool for beam steering and non-diffractive control26. Extending twistronics into various photonic fields holds exciting potential for producing intriguing physical effects.

In this work, we bring the concept of twistronics into topological lattices based on surface plasmonic polariton (SPP) waves, thereby introducing a combination of twistronics and topological spin photonics. Topological lattices have been constructed on SPP waves, which have led to the development of various forms of topological quasiparticles27,28,29,30, enabling the formation of skyrmion, meron and their mixing topologies through symmetry control at specific symmetries30,31,32. By incorporating twistronics, we demonstrate that a bilayer of photonic topological lattices can produce diverse moiré spin superlattices at moiré angles under spin-orbit coupling (SOC) of light, resulting in quasiparticle topologies including real-space skyrmion lattices and meron clusters, multilayer fractal patterns and extreme slow-light control, unattainable through conventional spin photonics. In contrast to previous consideration of twistronics in optical domain12,13,14,15,16,17,18,19,20,21,22,23,24,25,26, our framework includes intense SOC, using the total angular momentum (TAM) as an additional degree of freedom to control light-matter interactions and offering potential for applications in TAM-based optical communications, high-capacity data storage and chiral manipulations and light-matter interactions.

Results

Spin-orbit couplings in photonic spin-twistronics

Moiré superlattices are formed by superimposing two identical layers of these topological lattices with a controllable twist angle, illustrated in the bottom panels of Figs. 1(a), 1(c). It is worth noting that, there have been several literatures33,34 that report the novel topologies and physics in photonic quasicrystalline lattices35. However, the moiré spin lattices investigated in our work are formed in the instances that the rotational and translational symmetries must be satisfied simultaneously. Therefore, the underlying mechanisms to form the moiré spin lattices and spin quasicrystallines are different, and particularly, the phenomena and properties are also different.

Fig. 1: Concept of photonic moiré spin superlattice formation.
Fig. 1: Concept of photonic moiré spin superlattice formation.
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a Superposition of two individual C4-rotational symmetric spin lattices (top and middle) into a spin-bilayer (bottom). Insets on the side show vector diagrams of (top) the meron-like texture in C4 rotational symmetry and (bottom) the moiré lattice obtained at a twist angle 2ϑC4 = arctan(3/4) and with a total angular momentum (TAM) l = 7. b Experimentally reconstructed spin textures in Sz and (right) skyrmion number densities. Each C4-symmetric unit cell of the moiré spin lattice can be considered as a combination of skyrmion topologies with skyrmion number nsk ≈ ±1. c, d Analogous results to panels (a, b), obtained in the case of C3-rotational symmetric sublattices, with the moiré superlattice obtained at a twist angle 2ϑC3 = arccos(1/7) and a TAM l = 8.

To investigate the SOC properties in moiré superlattices, the angular momentum of light was decomposed into three categories34,36,37: intrinsic SAM related to optical helicity, intrinsic OAM depended on vortex phase and extrinsic OAM. Note that the extrinsic OAM that is related to the trajectory of light can be reconsidered as the momentum of light. We consider the TAM flux (J = r × P) in a closed surface σ, i.e.,

$${\int }\!\!\!\!{\int }_{\!\!\!\!\sigma }{{\bf{J}}}\cdot d{{\boldsymbol{\sigma }}}={\int }\!\!\!\!{\int }\!\!\!\!{\int }_{\!\!\!\!V}\nabla \cdot {{\bf{J}}}dV=-{\int }\!\!\!\!{\int }\!\!\!\!{\int }_{\!\!\!\!V}{{\bf{r}}}\cdot (\nabla \times {{\bf{P}}})dV$$
(1)

Here, P is the kinetic momentum proportional to the Poynting vector36. This expression indicates the variation of TAM flux across a closed surface is equal to the r·(∇×P) in a connected space, and thus r·(∇×P) represents obviously the SOC in a general optical system. Curiously, this term was demonstrated equally to the three-dimensional SAM vector for arbitrary SPP waves by the spin-momentum locking relations34: \({{\bf{S}}}=\frac{1}{2{k}^{2}}\nabla \times {{\bf{P}}}\). This is logical because from Eqs. (4–6) in Method, the formation of sublattices and superlattices are determined by the rotational symmetry associated conservation of TAM, and the arrangement of the local distribution of SAM density is closed determined by the SOCs of light.

On the other hand, in classical EM theory, the Ez is proportional to the Hertz potential Ψ, i.e., Ψ ∝ Ez. In this instance, the kinetic momentum density can be given by the Dirac inner product of Hertz potential as follows \({{\bf{P}}}=\frac{\varepsilon {k}^{2}}{2\omega {\beta }^{2}}{\mathrm{Re}}\{{\varPsi }^{\ast }(-i\nabla )\varPsi \}=\frac{\varepsilon {k}^{2}}{2\hslash \omega {\beta }^{2}}\langle \varPsi |\hat{{{\bf{p}}}}|\varPsi \rangle\). Here, ℏ is the reduced Plank constant. Then, the SAMs can be re-expressed as

$${{\bf{S}}}=\frac{\varepsilon }{4\omega {\beta }^{2}}{\mbox{Im}}\{\nabla {\varPsi }^{\ast }\times \nabla \varPsi \}=-\frac{\varepsilon }{4\omega {\beta }^{2}}\langle \nabla \varPsi |\times i|\nabla \varPsi \rangle$$
(2)

Therefore, the local distribution of SAM density can be considered as the Berry curvature of the Hertz potential Ψ representing the SOCs of light. This analysis is also consistent with the conclusion in Eq. (1), because the local assigning of TAM flux into OAM and SAM components are determined by the geometric phase (integral of the Berry curvature of the Hertz potential) of light.

Photonic moiré spin superlattices

After understanding the SOCs in photonic topological lattices, we now introduce the theoretical details about the creation of photonic moiré spin lattices by superimposing two topological sublattices, such as skyrmion or meron-like lattices, with an inclination angle of ±ϑ, as shown in the top panels of Figs. 1(a) and 1(c).

At a twist angle of 2ϑC4 = arctan(3/4) and a TAM quantum number l = 7, the moiré superlattices of C4-symmetric sublattices exhibits a skyrmion configuration with a skyrmion number nsk = ±1. Here, the skyrmion number evaluating the topological geometry is defined by \({n}_{sk}=\frac{1}{4\pi }{\iint }_{cell}{{\bf{m}}}\cdot (\frac{\partial {{\bf{m}}}}{\partial x}\times \frac{\partial {{\bf{m}}}}{\partial y}){d}^{2}{{{\bf{r}}}}_{\perp }\) 30, which quantifies how many times the normalized spin vector m(r⊥) = S(r⊥)/|S(r⊥)| wraps a unit sphere. Although the skyrmion lattices have been verified with electric field vector on a SPP platform27,28, only the fractal skyrmion spin lattices whose skyrmion number of sublattice is nsk = ±1 have been discovered under the SOCs of light31,32. This is because the local distribution of SAM is depended on the intensely couplings between the electric and magnetic ellipticities and the position of light (the extrinsic OAM), which leads to the two-layer fractal pattern of lattices in C6 rotational symmetry and the skyrmion number of unit cell of fractal skyrmion lattice is nsk = 0, which is a manifestation of trivial topology. Moreover, when the twist angle is 2ϑC4 = arctan(8/15) and the TAM quantum number is l = 2, 4, 6, 10, meron cluster-like spin lattices emerge within the C4 moiré superlattices. This method enables the construction of a variety of topological quasiparticles in moiré superlattices. Distinct from previously studied moiré superlattices of EM fields, the moiré spin lattices in this approach can be finely tuned by the TAM quantum number, which we will explore in detail in our work.

From the above analysis, the moiré spin lattices are influenced intensely by the TAM quantum number from the rotational symmetry associated conservative property of TAM, in contrast to most of previous works on photonic bilayer moiré lattices based on the specific moiré angles corresponding to those of condensed matter physics38,39. Consequently, the conditions for the formation of moiré spin lattices are stricter compared to those for conventional moiré lattices. This is one of key theoretical innovation point of work.

Topologies in photonic moiré spin superlattices

Another intriguing phenomenon is the formation of disversified topologies in moiré spin superlattices, as predicted in electronics38,39. In Figs. 1(b), 1(d) the unit cells comprise a combination of topological skyrmions with skyrmion number nsk = ±1 generated by meron sublattices in C3 or C4 rotational symmetries. Previously, only meron lattices could be formed under C3 or C4 rotational symmetries, while fractal skyrmion lattices were associated with C6 rotational symmetry. However, the situations are different in plasmonic spin-twistronics, because the symmetry constraints are a failure herein. Here, we name the spin lattices in C6 rotational symmetry as fractal skyrmion lattices because no regions exists where the integral of the skyrmion number density yields a nonzero integer, indicating that the topology of the SAM is trivial only when considering specific extracted sets of these fractal lattices31,32.

Moiré superlattices introduce extra degrees-of-freedom for constructing multiple types of topological lattices, such as twisted angles and the TAM quantum number. For example, for the moiré lattices formed by C4-symmetric sublattices with a twisted angle 2ϑC4 = arctan(3/4) and a TAM quantum number l = 7, the unit cell can be viewed as a combination of skyrmions with skyrmion number nsk = ±1 in C4 rotational symmetry, as shown in Fig. 1(a). In Fig. 1(b), we present the reconstructed spin texture alongside the experimentally measured distribution of the Sz component of SAM for both the sublattice (left panel) and the moiré lattice (right panel) at the twisted angle 2ϑC4 = arctan(3/4). The inset images display the stereographic projections and the skyrmion number densities for both the sublattices and the superlattices. The calculated skyrmion numbers for the unit cells of experimental sublattices are 0.481 and −0.492, reflecting the presence of meron topologies. In contrast, the calculated skyrmion numbers of the unit cells of the experimental superlattices are −0.923 and 0.903, indicating skyrmion topologies. The calculated skyrmion number approaches the corresponding theoretical skyrmion number, because the resolution of our experimental measurements is about several tens of nanometres while the fine structures of topological spin textures are about several nanometres and even in the sub-nanometre scale.

Similarly, for the moiré spin lattices constructed from C3-symmetric sublattices with a twisted angle 2ϑC3 = arccos(1/7) and a quantum number of TAM l = 8, the unit cell can also be regarded as a combination of skyrmions with skyrmion number nsk = ±1 in C3 rotational symmetry, as shown in Fig. 1(c). In Fig. 1(d), we display the reconstructed spin textures and the experimentally measured distributions of Sz for the sublattice (left panel) and the moiré lattice (right panel) at the twisted angle 2ϑC3 = arccos(1/7), respectively. The inset images reveal the stereographic projections and the skyrmion number densities for both the sublattices and the superlattices. The calculated skyrmion numbers for the unit cells of the experimental sublattices are 0.451 and −0.489, indicative of meron topologies. In comparison, the calculated skyrmion numbers for the unit cells of the experimental superlattices are −0.914 and 0.867, confirming the presence of skyrmion topologies.

Skyrmion cluster states refer to stable, high-degree multi-skyrmion configurations in which an arbitrary number of skyrmions are contained within a larger skyrmion. These states have been widely observed in experiments within the realm of condensed matter physics40,41. In this context, we demonstrate that meron clusters are formed in the photonic moiré superlattice. A meron cluster can be described as a collection of meron topologies arranged in arbitrary geometries. By tuning the quantum number l of the TAM, the geometric arrangement of these meron topologies can be controlled. For example, in the moiré lattices constructed from C4-symmetric sublattices with a twisted angle 2ϑC4 = arctan(8/15) and TAM quantum number l = 2, 4, 8, and 10, the meron cluster-like textures can be generated. As the quantum number varies from l = 2 to l = 4, the orientations of the photonic meron topologies can be effectively tuned, as illustrated in Fig. 2(a-b). The left panels present the reconstructed spin textures along with the experimentally measured distributions of Sz, with inset images showing the corresponding theoretical comparisons. The middle panels display vector diagrams of the meron textures. From the stereographic projections in the right panel, we observe that the calculated skyrmion numbers for the unit cells of the experimental superlattices are −0.443 and 0.445, respectively, indicating the presence of meron topologies. Experimental data for the quantum numbers of TAM l = 2, 4, 6 and 10 can be found in Supplementary Fig. 18 to Supplementary Fig. 21 of SIs.

Fig. 2: Experimental results of meron cluster-like spin textures in photonic moiré spin lattices with C4 rotational symmetry.
Fig. 2: Experimental results of meron cluster-like spin textures in photonic moiré spin lattices with C4 rotational symmetry.
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Moiré spin lattices are obtained experimentally for a twist angle of 2ϑC4 = arctan(8/15) and quantum numbers of TAM (a) l = 2 and (b) l = 4. The obtained texture is meron cluster, intended as a collection of meron topologies arranged in arbitrary geometries. The (left) reconstructed spin textures are shown alongside with the (left) experimentally reconstructed Sz components and the (middle) corresponding vector diagrams. The integral skyrmion number densities of the meron topologies are found to be −0.443 and 0.445 in (a) and −0.426 in (b), respectively, as evidenced by the stereographic projections in the right panels.

Multilayer fractal patterns in photonic moiré spin superlattices

Fractal (self-similarity) properties are prevalent in the photonic moiré spin lattices. For instance, consider the moiré spin superlattices constructed by C3 rotational symmetric sublattices, as shown in Fig. 3(a). The Sz component can be expressed as follows:

$${S}_{z}\propto \left\{\begin{array}{l}+\,\sin \left[\sqrt{3}\beta (x\,\sin \vartheta -y\,\cos \vartheta )-l\frac{4\pi }{3}\right] \hfill \\ -\,\sin \left[\sqrt{3}\beta (x\,\sin \left[\vartheta+\frac{\pi }{3}\right]-y\,\cos \left[\vartheta+\frac{\pi }{3}\right])-l\frac{2\pi }{3}\right]\hfill \\ -\,\sin \left[\sqrt{3}\beta (x\,\sin \left[\vartheta -\frac{\pi }{3}\right]-y\,\cos \left[\vartheta -\frac{\pi }{3}\right])-l\frac{2\pi }{3}\right]\end{array} \right.$$
(3)
Fig. 3: Fractals in photonic moiré superlattices.
Fig. 3: Fractals in photonic moiré superlattices.
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a A moiré superlattice is formed with twist angle 2ϑC3 = arccos(−1/26) and total angular momentum (TAM) l = 10. b Measured Sz, with the inset showing its theoretical distribution, and (e) the corresponding wavenumber representation in Fourier space. Four distinct sets of wavenumbers are observed, as outlined by the colored, dashed circles in (e). Spin textures corresponding to each subset of wavenumbers are shown in (c, d, f, g). The spin texture obtained between the green and blue dashed circles (f) can be further divided into two sublattices, as illustrated in (h, i). In each panel, inset shows the corresponding wavenumbers in Fourier space. Detailed theoretical results can be found in Supplementary Fig. 11. The scale is indicated by the gray line in (a).

For a twisted angle of 2ϑC3 = arccos(−1/26) and a quantum number of TAM l = 10, the experimentally measured Sz component (with the corresponding theoretical comparison shown in the inset) can be found in Fig. 3(b). To investigate the fractal properties of the moiré spin superlattices, we perform a Fourier transformation on the experimentally measured SAM density Sz. This analysis reveals that there are four sets of wavenumbers in Fourier space, as illustrated in Fig. 3(e). In contrast, there are only two sets of wavenumbers in Fourier space for conventional topological spin lattices31,32. By extracting these distinct groups of wavenumbers and performing an inverse Fourier transformation, we can obtain three sets of triangular lattices, presented in Figs. 3(c), (d), (g), respectively. The secondary outer set shown in Fig. 3(f) contains a pair of sublattices, a result of the moiré properties of the lattice. By carefully selecting the angles of inclination, we can extract the two sublattices that satisfy C3 rotational symmetry, shown in Figs. 3(h), 3(i). Both sublattices are observed to be inclined, confirming our earlier analysis. Furthermore, all lattices exhibit C3 rotational symmetry, underscoring the fractal properties of the moiré lattices.

Notably, in optical systems, the moiré superlattices are inherently limited by the diffraction limit. Thus, the maximum wavenumber cannot exceed 2k, where k is the wavenumber in free space. This means that the scale of photonic fractal lattices cannot be made indefinitely small, highlighting a distinction between optical phenomena and natural fractals.

In summary, we demonstrated theoretically and experimentally that the unprecedented multilayer of photonic fractal lattices that have not been observed in conventional optical system until now. Most of previous work only reported that the number of layers for optical fractal were two31,32, whereas in our work, the number of layers is 4 and the information of moiré angles are also included in the fractal, which can be considered as a well candidate for information security in optical communications and data storage.

Inherent slow light controls in photonic moiré spin superlattices

Slow light is an interesting phenomenon and it was widely found in integrated photonics. Whereas in moiré lattices, widespread flat-bands also result in the slow light effects12,22,23,24. The underlying mechanisms of these slow light effects can be considered as the multiply scattering of light in nanostructures. However, the photonic moiré superlattices are excited at the air/metal interface with no structure in the metal layer and only the photonic moiré lattices at the air half-space are considered. The mechanisms of inherent slow light effects in photonic moiré lattices are different from those of nanostructures. Moreover, the dispersion relation of SPP wave determines that the group velocity of a SPP plane wave in air is given by vgp = kc/β < c42, but it cannot move the group velocity of SPP wave into the slow light region.

However, by examining the group velocity vg = |vg | = |P/W| that is defined as the ratio of the Poynting vector p (which is proportional to the kinetic Poynting momentum P = p/c2)42 to the energy density W via the moiré angles and TAM quantum number, the group velocity can be greatly lower than the group velocity of the SPP plane wave vgp for certain photonic moiré spin superlattices. We present the calculated the log10 of mean group velocity (log10(ṽg/vgp)) as a function of the moiré angles described in Eq. 7 and Eq. 9 in Method, as well as the integer quantum number l. The results are shown in Fig. 4(a, b). For instance, in the moiré lattice formed by the C4 rotational symmetric sublattices with a twisted angle of 2ϑC4 = arctan(5/12) and a TAM quantum number l = 16, the maximum local group velocity (ṽg) of this moiré lattice is found to be less than one-tenth of the vgp, as depicted in the top panel of Fig. 4(c). The corresponding 1D contour can be seen in the bottom panel. From Fig. 4(a), 4(b), one can find that when the quantum number of TAM is zero, the slow light can reach 10−5 of group velocity of SPP plane wave, and whereas if the quantum number of TAM is not zero, the slow light can reach 10−2 of group velocity of SPP plane wave.

Fig. 4: Inherent slow light in photonic moiré spin lattices.
Fig. 4: Inherent slow light in photonic moiré spin lattices.
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a The logarithm of the mean group velocity (log10(ṽg/vgp)) is shown as a function of the twist angle 2ϑ and the total angular momentum (TAM) quantum number, for both (a) C4 and (b) C6 rotational symmetries. The mean group velocity is up to five orders of magnitude smaller than the group velocity of SPP in vacuum (vgp = kc/β). Note that as the TAM quantum number is zero, in C4 and C6 rotational symmetry, each pair of oppositely propagating surface waves will form a standing wave, in which the energy flux and angular momentum are null. This case is, however, out of the scope of this paper, whose primary focus is on spin-orbit coupling. c 2D spatial distribution of the local group velocity in a moiré superlattice with a twist angle of 2ϑC4 = arctan(5/12) and TAM l = 16 and its one-dimensional profile along the x-axis (indicated by the blue dashed line in (c)). The maximum group velocity reached is about one-tenth of vgp. This phenomenon is closely related to the presence of off-axis vortex-antivortex flux and the optical super-oscillation effect, as illustrated by the spatial distribution of TAM (d) and the enlarged figures in the bottom panel, corresponding to the region outlined by a dashed green square in (c). In this diagram, the white ⊕ symbol represents the vortex, while the black ⊖ symbol denotes the antivortex. All velocities are normalized by vgp and the mean group velocity is calculated as ṽg = Σ(vg ∙ W)/ΣW over a region of 10 μm × 10 μm with a grid size of 0.005 μm.

When the TAM quantum number is zero, in C4 and C6 rotational symmetry, each pair of oppositely propagating surface waves will form a standing wave, in which the energy flux and angular momentum is absent. Whereas if the TAM quantum number is nonzero, it is general that the interference of waves can be regarded as a conventional explanation to generate the inherent slow light. However, the slow light effect is not an average effect because the group velocity varies everywhere, as shown in Fig. 4(c). Therefore, understanding the underlying mechanism of why and where the extreme slow light occurs is important for designing the moiré superlattices for optical manipulations, the enhanced light-matter interactions and further applications. In the instance, this inherent slow light phenomenon arises due to the formation of local vortex-antivortex flux in the moiré superlattice (Fig. 4(d) and the enlarged figures in the bottom panel) and this off-axis vortex-antivortex flux is intricately related to the optical super-oscillation effect43 and can facilitate the creation of deep-subwavelength fine spin structures in confined optical fields30, which is a mechanism that has never been uncovered. Further details of the experimental results and their corresponding theoretical comparisons can be found in Supplementary Fig. 23 of SIs. More importantly, because the inherent slow light is present with no nanostructure, the underlying mechanism may offer unprecedented optical manipulation capabilities for trapping particles and atoms, enhancing light-matter interactions, and achieving energy transfers beyond what conventional optical waves allow.

Discussion

The integration of twistronics into spin photonics enhances the flexibility in controlling topological spin superlattices. Our findings demonstrate that photonic bilayer spin superlattices can generate complex periodicities, tunable skyrmion topologies and multilayer fractal patterns, as well as facilitate inherent slow light phenomena. Nonetheless, substantial opportunities remain for further exploration in designing spin superlattices, including, for instance, studies of trilayer spin lattices18. Moreover, introducing nonlinear effects into photonic spin lattices could provide a platform to emulate nonlinear interlayer coupling, potentially leading to the emergence of flat bands12,22,23,24 and the manifestation of van der Waals forces20.

Beyond enabling advanced manipulations of electromagnetic fields with diverse topological configurations, photonic spin-twistronics could be set to unlock a broad spectrum of applications. Photonic moiré spin lattices, governed by SOCs of light and intricately interacting with material chirality, would offer promising opportunities in TAM-based optical data storage44, chiral manipulations45, atomic-scale chiral light localization13, chiral sorting46, and chiral laser emissions47. The precisely controllable photonic spin superlattices could be implemented as an additional feature in optical tweezers48, for quantum simulations using ultracold atoms via optical trapping technologies to higher-dimensional applications49, and enhance super-resolution precision in optical sensing and metrology43. Additionally, the realization of diverse, particle-like topologies in photonic spin superlattices would also hold promises for high-density optical information storage and retrieval50. The methodologies developed in photonic spin-twistronics can be extended beyond photonics, to encompass more broadly the field of spin physics, including acoustics51, water waves48, and other wave-like physical systems.

Methods

Theoretical model

SPP is a well-established platform for the realization of topological spin photonics. Owing to the fact that only the transverse magnetic (TM) electromagnetic (EM) wave mode can be excited in a SPP platform at an air/metal interface40, the vector Helmholtz equation can be simplified into a scalar Helmholtz equation ∇2Ez + k2Ez = 0, which can be solved to generate diverse topological lattices by considering the strictly translational and rotational symmetries and the associated conservative property of TAM [Section I of SIs], originating from two mechanisms including the intrinsic orbital angular momentum (OAM) of light and the spin-to-orbital angular momentum conversion of light. Here, Ez is the electric field component normal to the interface, and k = 2π/λ is the wavenumber in the air half-space with λ the wavelength (632.8 nm in our work). For example, by engineering the SOCs through the adjusting of TAM quantum number l and considering the rotational symmetry \({\hat{R}}_{z}(\varphi )\{{E}_{z}[{\hat{R}}_{z}(-\varphi ){{\bf{r}}}]\hat{{{\bf{z}}}}\}={e}^{il\varphi }\{{E}_{z}({{\bf{r}}})\hat{{{\bf{z}}}}\}\), lattices with C6-symmetry yield fractal, skyrmion topological spin textures, whereas C4 or C3-symmetries generate meron spin textures37,38, as shown in the top panels of Figs. 1(a), 1(c). These topological lattices are served as the sublattices for the construction of photonic moiré spin lattices. Here, \({\hat{R}}_{z}\left(\varphi \right)\) is the rotating operator on a coordinate r along the z-axis by an azimuthal angle φ, and eilφ represents the vortex phase with respect to the TAM quantum number. Crucially, in the N-fold rotational symmetry (denoted as CN) with n = 1, …, N, the solution of Ez can be re-expressed as

$${E}_{z}=A{e}^{-{k}_{z}z}\sum _{n=1}^{N}{e}^{il{\theta }_{n}}{e}^{i\beta {{{\bf{r}}}}_{\perp }\cdot {{{\bf{e}}}}_{n}}.$$
(4)

where θn = 2nπ/N and en = (cosθn, sinθn); β and ikz are the wavenumbers in the transverse (propagation constant) and longitudinal (evanescent component) directions, respectively; β2 – kz2 = k2. The transverse position vector r⊥ = ρ(cosφ, sinφ) is defined with ρ representing the radial coordinate in the cylindrical coordinate system, and i is the imaginary unit. The spin textures of sublattices arranged in threefold (C3), fourfold (C4) and sixfold (C6) rotational symmetries are provided in Supplementary Fig. 1 to Supplementary Fig. 3 of SIs. From Eq. 4, one can use the superposition of a series of SPP plane waves to approach the desired photonic topological lattices. However, the superposition of SPP plane waves is not the underlying mechanism for the generation of topological spin lattices but the rotational and translational symmetries and the associated conservative property.

Unlike the moiré superlattices of 2D materials in which the sample should be prepared carefully1,2,3,4,5,6, the photonic moiré superlattices can be constructed by directly rotating the sublattices by a half moiré angle ϑ at a single SPP platform, i.e.,

$${E}_{z}(+\vartheta )={\hat{R}}_{z}(+\vartheta )\{{E}_{z}[{\hat{R}}_{z}(-\vartheta ){{\bf{r}}}]\hat{{{\bf{z}}}}\}=A\sum _{n=1}^{N}{e}^{il({\theta }_{n}+\vartheta )}{e}^{i\beta {{{\bf{r}}}}_{\perp }(+\vartheta )\cdot {{{\bf{e}}}}_{n}}{e}^{-{k}_{z}z},$$
(5a)
$${E}_{z}(-\vartheta )={\hat{R}}_{z}(-\vartheta )\{{E}_{z}[{\hat{R}}_{z}(+\vartheta ){{\bf{r}}}]\hat{{{\bf{z}}}}\}=A\sum _{n=1}^{N}{e}^{il({\theta }_{n}-\vartheta )}{e}^{i\beta {{{\bf{r}}}}_{\perp }(-\vartheta )\cdot {{{\bf{e}}}}_{n}}{e}^{-{k}_{z}z},$$
(5b)

and the superimposed Ez can be expressed as

$${E}_{z}={E}_{z}(+\vartheta )+{E}_{z}(-\vartheta ),$$
(6)

Here, the transverse position vector r⊥(±ϑ) = ρ(cos(φ ± ϑ), sin(φ ± ϑ)) is defined and technically, the spin angular momentum (SAM) density can be obtained by calculating S = Im{ε(E* × E) + μ(H* × H)}/4ω with ω the angular frequency of the monochromatic time-harmonic EM field, ε and μ the permittivity and permeability of air, respectively.

TAM-regulated moiré angles for the formation of spin superlattices

When forming moiré spin lattices from C4 rotational symmetric sublattices, the twisted angle must satisfy the following relation [Section II of SIs]:

$$2{\vartheta }_{C4}=\arctan \frac{2(2{m}_{1}+1)(2{m}_{2}+1)}{[(2{m}_{1}+1)+(2{m}_{2}+1)][2{m}_{1}-2{m}_{2}]},$$
(7)

where m1 and m2 can be any integers. From Eq. (7), it can be noted that the denominator is an even integer, while the numerator is an odd integer. For example, when m1 = 1 and m2 = 0, we find that 2ϑC4 = arctan(3/4). This specific twisted angle results in the formation of the moiré spin lattice, as illustrated in the right panel of Fig. 1(a).

In the case of sublattices exhibiting C3 and C6 rotational symmetries, the twisted angles must satisfy the following conditions [Section II of SIs]:

$$2{\theta }_{C3}=\arccos \frac{{m}_{1}^{2}-3{m}_{2}^{2}}{{m}_{1}^{2}+3{m}_{2}^{2}},$$
(8)

and

$$2{\theta }_{C6}=\arccos \frac{{({m}_{1}-{m}_{2})}^{2}-3{({m}_{1}+{m}_{2})}^{2}}{{({m}_{1}-{m}_{2})}^{2}+3{({m}_{1}+{m}_{2})}^{2}},$$
(9)

respectively. In fact, these two conditions are coincident. For example, from Eq. (9), when m1 = 2 and m2 = 1, we obtain that 2ϑC3 = arccos(1/7). This specific twisted angle ensures the formation of the moiré spin lattice, as shown in the right panels of Fig. 1(c).

The additional degree-of-freedom associated with the TAM quantum number imposes an extra restriction on the formation of moiré spin lattices. However, this also introduces interesting phenomena within the moiré spin lattices.

The first feature of the moiré spin lattices is their periodicity with respect to the TAM quantum number l. For moiré spin lattices constructed from sublattices exhibiting CN rotational symmetry (where N = 3, 4, 6) and a twisted angle 2ϑCN, the period p is determined by the conditions p mod N = 0 and p = mπ/ϑ, where m is an arbitrary integer. Generally speaking, the quantity mπ/ϑ is not strictly an integer. However, if p = mπ/ϑ can be approximated as an integer, it represents the period of the photonic moiré lattices in relation to the TAM quantum number. For example, for photonic moiré lattices formed by C4 rotational symmetric sublattices with a moiré angle of 2ϑC4 = arctan(5/12), we find that p = mπ/ϑ ≈ 15.915 ≈ 16 when m = 1, which obviously satisfies 16 mod 4 = 0. Thus, p = 16 can be considered the period of these photonic moiré lattices, as shown in Supplementary Fig. 4 and Supplementary Fig. 7 of SIs. Similarly, for photonic moiré lattices constructed from C6 rotational symmetric sublattices with a moiré angle of 2ϑC6 = arccos(11/14), we find that p = mπ/ϑ ≈ 65.946 ≈ 66 when m = 7, satisfying 66 mod 6 = 0. Therefore, p = 66 can be regarded as the period of these photonic moiré lattices, as depicted in Supplementary Fig. 5 and Supplementary Fig. 8 of SIs.

A similar condition applies to the moiré lattices formed by C3 rotational symmetric sublattices. In summary, based on the specific moiré angles and SOCs of light, we find the periodicity of photonic moiré lattices with respect to TAM quantum number and the specific moiré angles. Therefore, the introduction of SOCs into the photonic moiré lattices not only provide extra degrees-of-freedom to manipulate the topological spin lattices and is potential for the applications such as angular momentum detection in OAM-based optical communications.

SAM measurement on SPP platform

In our work, the experimental setup used to demonstrate the photonic moiré lattices is shown in Supplementary Fig. 15(a) of SIs, utilizing SPPs as the example. Note that in condensed matter physics, most of skyrmionic topologies are stable in low temperature environment, whereas in optics, the topological spin texture can survive in room temperature and our experiments are also performed in room temperature. In addition, we construct the moiré superlattice by rotating and superimposing the photonic topological lattices in the SPP platform instead of rotating and superimposing the metal film. Therefore, the distributions of moiré superlattices can be detected at a single metal film by an in-house near-field scanning optical microscopy (NSOM) system. A linearly polarized beam with a wavelength of 632.8 nm, sourced from a He-Ne laser, served as the excitation source. The incident beam first passed through a linear polarizer (LP) and a half-wave plate (HWP) to achieve the desired linear polarization. Following this, the beam was modulated by a spatial light modulator (SLM), which controlled the phase of the beam according to the expression:

$$\psi={{\mbox{arg}}}\left[{\sum }_{n=1}^{N}\begin{array}{l}{e}^{i{k}_{t}(x\,\cos ({\theta }_{n}+\vartheta )+y\,\sin ({\theta }_{n}+\vartheta ))}{e}^{il({\theta }_{n}+\vartheta )}\\+{\sum }_{n=1}^{N}{e}^{i{k}_{t}(x\,\cos ({\theta }_{n}-\vartheta )+y\,\sin ({\theta }_{n}-\vartheta ))}{e}^{il({\theta }_{n}-\vartheta )}\end{array}\right]$$
(10)

to generate the desired pair of plane waves. The parameter kt was carefully chosen to match the beam size with the pupil of the objective lens. The integer l was introduced to produce the desired vortex phase. The beams subsequently passed through a HWP and 1st-order vortex-wave plate (VWP) to create radially polarized light. This light was then tightly focused using an oil-immersion objective lens (Olympus, NA = 1.49, 100×), exciting the SPP plane waves on a gold film (50 nm). The surface EM field was measured using the in-house NSOM system. A polystyrene (PS) nanoparticle probe with a diameter of 300 nm was positioned on the gold film. The position of the PS particle was precisely controlled by a piezo-stage (Physik Instrumente, P-545). The near-field signal scattered by the PS particle was directed to an objective lens (Olympus, NA = 0.7, 60×), which acted as a low-pass filter. This signal was then split and analyzed using a combination of a QWP and a LP to extract the individual circular polarization component (ILCP and IRCP) of the scattering signal. Using dipole theory, the horizontal components of the electric field could be reconstructed. These components were subsequently directed to two photomultiplier tubes (PMTs) for measuring the intensity information of the two signals. This setup enabled the characterization of the out-of-plane SAM component (i.e., along the optical axis) of the focused beams, as described by the relation37,43:

$${S}_{z}=\frac{\varepsilon }{4\omega }\frac{{\beta }^{2}}{{k}_{z}^{2}}({I}_{{\rm{RCP}}}-{I}_{{\rm{LCP}}})\propto {I}_{{\rm{RCP}}}-{I}_{{\rm{LCP}}}$$
(11)

The transverse SAM component can be reconstructed by the Maxwell’s EM theory38, as introduced in Section V in SIs.

By comparing the theoretical and experimental skyrmion number in Fig. 1(a), 1(b), respectively, one can find a small difference between them. The difference are owing to the inaccuracies of experiments in NSOM imaging of the SAM of topological spin lattices. There are two primary limitations in the experimental measurement:

First, to measure the SAM of the topological lattices, especially the out-of-plane component, we should detect the smaller transverse electric field components at the metal interface. This is a great challenge, because from the electric Gauss’ law ∇·E = 0 the intensity of transverse components is about an order of magnitude smaller than that of the out-of-plane electric field component. Therefore, to increasing the scattering cross section, we use the PS sphere as a nanoprobe with diameter 300 nm. This huge probe would increase the contact area between the sphere and metal film and thus downgrade the resolution of the NSOM imaging system, because we can only measure the average field under the PS sphere. The resolution of NSOM imaging system is about 10 nm to 20 nm and we cannot measure the distribution of spin angular momentum at nanometer scale. Therefore, the experimental skyrmion number hardly reaches an integer.

Second, to measure the SAM of the topological lattices, we build an in-house NSOM imaging system. However, to detect this sub-wavelength information of light, we need to image the spin angular momentum of the topological lattices by scanning techniques. If we image a region of 4 μm × 4 μm by 20 nm interval, we will cost about 5 minutes to obtain an imaging. This long-time cost will result the extra disturbance or uncertainty of measurement in the imaging process, and thus it can affect the experimental results and make that the experimental skyrmion number hardly reaches an integer.

Third, to generate the moiré superlattices, we load hologram into a spatial light modulator in Supplementary Fig. 15(b). Because the size of pixel of a spatial light modulator is about 10μm, the discontinuities caused by the staircasing of loaded phase would lead to the distortion of hologram, and result in the disturbance in the formation of photonic topological sublattices and moiré superlattices and thus the non-integer of skyrmion number.

Robustness of moiré spin superlattices

We check the robustness of moiré spin superlattices against the moiré angle theoretically. Without loss of generality, we take the disruption of the deviation angle as an example, and the theoretical results are shown in Supplementary Fig. 10 and Supplementary Fig. 11 of SIs, respectively. For a photonic moiré spin superlattice formed by superimposing C4-symmetric sublattices at a twist angle 2ϑC4 = arctan(3/4) (≈36.87°) and with a TAM quantum number l = 7, the moiré spin skyrmions show robustness against twist angle deviations and demonstrate well-defined topology in the range of 34.87° ~ 37.87°. Whereas for a photonic moiré spin superlattice formed by superimposing C3-symmetric sublatticesat a twist angle 2ϑC3 = arccos(1/7) (≈81.79°) and with a TAM quantum number l = 8, The moiré spin skyrmions show robustness against twist angle deviations and demonstrate well-defined topology in the range of 79.79° ~ 83.79°. The similar results can be obtained by investigating the robustness against fractional TAM quantum number. The robustness of topological lattice in moiré superlattices against disturbances can offer potential applications in high-capacity data storage and communications, etc.