Nanophotonics 2021; aop Research article Xinrui Lei, Luping Du*, Xiaocong Yuan* and Anatoly V. Zayats* Optical spin–orbit coupling in the presence of magnetization: photonic skyrmion interaction with magnetic domains https://doi.org/10.1515/nanoph-2021-0201 Keywords: magneto-optical effects; photonic skyrmion; Received April 29, 2021; accepted July 26, 2021; spin-orbit coupling. published online August 24, 2021 Abstract: Polarization and related spin properties are 1 Introduction important characteristics of electromagnetic waves and their manipulation is crucial in almost all photonic appli- Magnetic materials and their nanostructures can influence cations. Magnetic materials are often used for controlling the polarization state of light upon reflection or trans- light polarization through the magneto-optical Kerr or mission due to the Kerr or Faraday effects, respectively, Faraday effects. Recently, complex topological structures and widely used in photonics for polarization rotators and of the optical spin have been demonstrated in the optical isolators. They are also a cornerstone for data evanescent light field, which in the presence of the spin– storage applications which are traditionally based on orbit coupling may form photonic skyrmions. Here, we manipulation of magnetic domains. Magnetic skyrmions investigate the optical spin–orbit coupling in the presence [1, 2], which are topological nanostructures of magnetiza- of magnetization and the interaction between photonic tion, have attracted recently significant attention for skyrmions and magnetic domains. We demonstrate that magnetic memory development with low energy re- the magnetization is responsible for the modulation of the quirements. Various magneto-optical (MO) techniques are optical spin distribution, resulting in twisted Neel-type used to study, characterize, and visualize the magnetic skyrmions. This effect can be used for the visualization of materials and domains using magnetization-dependent magnetic domain structure with both in plane and polar rotation of the polarization plane of light on reflection or orientation of magnetization, and in turn for creation of transmission [3–7]. Such Kerr or Faraday rotation from a complex optical spin distributions using magnetization ferromagnetic film is weak and require a significant patterns. The demonstrated interplay between photonic macroscopic propagation length to be detected. To skyrmions and magneto-optical effects may also provide enhance the MO activity, metallic nanostructures with novel opportunities for investigation and manipulation of plasmonic effects can be employed [8–16]. magnetic skyrmions using optical spin–orbit coupling. Prominent optical analogies of magnetic topological structures have been recently demonstrated, such as photonic skyrmions and merons [17–19]. Photonic spin- skyrmions rely on the spin–orbit coupling in the evanes- *Corresponding authors: Luping Du and Xiaocong Yuan, cent field of guided modes. Spin angular momentum Nanophotonics Research Centre, Shenzhen Key Laboratory of Micro- (SAM), which is related to circular polarization of light, and Scale Optical Information Technology, Shenzhen University, orbital angular momentum (OAM) are intrinsic properties Shenzhen, Guangdong, 518060, China; and Anatoly V. Zayats, of light waves and play a critical role in light–matter Department of Physics and London Centre for Nanotechnology, King’s – College London, Strand, London, WC2R 2LS, UK, interactions [20 23]. This dynamic mutual conversion be- E-mail: [email protected] (L. Du), [email protected] (X. Yuan), tween SAM and OAM is enhanced on the nanoscale [24–26] [email protected] (A. V. Zayats). and manifests itself in numerous unusual physical phe- Xinrui Lei, Nanophotonics Research Centre, Shenzhen Key Laboratory nomena, such photonic analogue of the quantum spin Hall of Micro-Scale Optical Information Technology, Shenzhen University, effect [27, 28], unidirectional coupling to the guided modes Shenzhen, Guangdong, 518060, China; and Department of Physics [29], chiral detection [30], lateral optical forces [31], as well and London Centre for Nanotechnology, King’s College London, Strand, London, WC2R 2LS, UK, E-mail: [email protected]. https:// as photonic topological insulators [32], many of which rely orcid.org/0000-0001-5939-4276 on the transverse spin carrying by the evanescent field of Open Access. © 2021 Xinrui Lei et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 2 X. Lei et al.: Photonic skyrmion interaction with magnetic domains waveguiding modes [33–35]. Spin–orbit coupling provides where εi is the relative permittivity of the magnetic material a new, unexplored degree of freedom for engineering and the off-diagonal elements gx,y,z are the gyration vector magneto-optical interactions with complex topological constants, which are linear in magnetization. By applying structures of light polarization, such as photonic the Maxwell’s equations, the wave equation inside a skyrmions. magnetic layer takes the form In this article, we study optical spin–orbit interactions ∇(∇⋅E) − ∇2E = k2 ̂ϵ E, (2) in the presence of magnetization and magneto-optical 0 i interactions of photonic skyrmions. We analyze the near- where k0 is the wave vector in vacuum. For a surface wave field response of their MO activity for different magneti- propagating along x axis, the electric field is characterized iβx±kz z zation orientation. By examining the electromagnetic field by E = E0e , with β denoting the propagation constant in a magneto-plasmonic system, we demonstrate that a and the longitudinal wave vector kz can be obtained from photonic skyrmion, which is formed by the spin–orbit the nontrivial solutions of electric field E0 in Equation (2), coupling in evanescent vortex beam, is modulated in the which depends on the magnetization orientation. presence of the MO response as the orientation of magne- Transversal magnetization (gx = gz = 0) couples the x tization changes. This provides an opportunity for the and z components of the electric field, maintaining the visualization of a magnetic domain structure by observing independence of transverse electric (TE) and transverse the spin states of the skyrmion. The superposed spin states magnetic (TM) polarized guided modes. From Equation (2), of a pair of opposite photonic skyrmions, generated by the wavevector inside the magnetic layer takes the relation √̅̅̅̅̅̅̅̅ √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ opposite optical vortices, yield distinct domain contrast 2 2 2 2 2 2 kzi = β − ϵik for a TE mode or kzi = β − ϵik (1 − g /ϵ ) with a resolution below 120 nm. Furthermore, due to the 0 0 y i different response of photonic skyrmions with opposite for a TM mode. For a TE polarized surface guided mode, the β topological charges to each magnetization orientation, the mode dispersion equation and propagation constant are spin-based domain observation eliminates the magneti- the same as for nonmagnetic or demagnetized systems (no zation orientation constrains of conventional Kerr micro- off-diagonal components present in Equation (1)). While scopy. The results provide new understanding of optical for a TM polarized surface mode, the coupling between Ex spin and magnetization interactions and offer novel and Ez results in a modulation on the propagation constant β opportunities for study and manipulation of magnetic , which scales linearly with the gyration vector constant structures, including magnetic skyrmions, using optical [10, 36, 37]. spin–orbit coupling. For polar (gx = gy = 0) or longitudinal (gy = gz =0) magnetization, the separation between TE and TM polari- zation is not possible because of the coupling of magneti- zation to Ey. The longitudinal wave vector inside the ± 2 Results and discussion magnetic layer can be represented as k = k0 ± Δk(g), √̅̅̅̅̅̅̅̅ zi zi where k0 = β2 − ϵ k2 and Δk( g) is the magneto-optical 2.1 Magneto-optical interactions in the zi i 0 near-field contribution (See Supplementary Note 1). This birefrin- gence inside the magnetic layer implies a 4 × 4 transfer Spin-skyrmions are formed due to spin–orbit coupling in matrix to maintain the boundary conditions at each inter- the evanescent field of guided waves, such as surface face, which has been developed in many formalisms to fl plasmon polaritons (SPPs) [17]. Therefore, we consider a deal with the propagation and re ection problems in fi – general multilayered structure comprising a waveguiding strati ed anisotropic media [38 42]. Taking into account β (generally nonmagnetic) layer adjacent to a magnetic film the smallness of g, the propagation constant of a surface or waveguiding in a magnetic layer itself (Figure 1(A)). The wave in a multilayer structure (Figure 1(A)) is practically fl proposed framework describes the properties of such a not in uenced by magnetization, while the electric fi multilayer for any position of the magnetic film in the eld ratio between the in plane TE and TM components, η multilayered stack and any illumination direction. The = Ey/Ex, in the upper medium is approximately propor- permittivity tensor of a magnetic layer is given by tional to g (See Supplementary Note 1). The near-field MO effect is prominent in the SPP modes ϵ −ig ig ⎛⎜ i z y ⎞⎟ supported by a structure containing ferromagnetic layers ̂ϵ = ⎝⎜ ig ϵ −ig ⎠⎟, (1) i z i x with the large gyration despite the high Ohmic losses. In −ig ig ϵ y x i view of this, we consider in the following the SPPs at an X. Lei et al.: Photonic skyrmion