Polarization and related spin properties are important characteristics of electromagnetic waves and their manipulation is crucial in almost all photonic applications. Magnetic materials are often used for controlling light polarization through the magneto-optical Kerr or Faraday effects. Recently, complex topological structures of the optical spin have been demonstrated in the evanescent light field, which in the presence of the spin–orbit coupling may form photonic skyrmions. Here, we investigate the optical spin–orbit coupling in the presence of magnetization and the interaction between photonic skyrmions and magnetic domains. We demonstrate that the magnetization is responsible for the modulation of the optical spin distribution, resulting in twisted Neel-type skyrmions. This effect can be used for the visualization of magnetic domain structure with both in plane and polar orientation of magnetization, and in turn for creation of complex optical spin distributions using magnetization patterns. The demonstrated interplay between photonic skyrmions and magneto-optical effects may also provide novel opportunities for investigation and manipulation of magnetic skyrmions using optical spin–orbit coupling.
Magnetic materials and their nanostructures can influence the polarization state of light upon reflection or transmission due to the Kerr or Faraday effects, respectively, and widely used in photonics for polarization rotators and optical isolators. They are also a cornerstone for data storage applications which are traditionally based on manipulation of magnetic domains. Magnetic skyrmions [1, 2], which are topological nanostructures of magnetization, have attracted recently significant attention for magnetic memory development with low energy requirements. Various magneto-optical (MO) techniques are used to study, characterize, and visualize the magnetic materials and domains using magnetization-dependent rotation of the polarization plane of light on reflection or transmission , , , , . Such Kerr or Faraday rotation from a ferromagnetic film is weak and require a significant macroscopic propagation length to be detected. To enhance the MO activity, metallic nanostructures with plasmonic effects can be employed , , , , , , , , .
Prominent optical analogies of magnetic topological structures have been recently demonstrated, such as photonic skyrmions and merons , , . Photonic spin-skyrmions rely on the spin–orbit coupling in the evanescent field of guided modes. Spin angular momentum (SAM), which is related to circular polarization of light, and orbital angular momentum (OAM) are intrinsic properties of light waves and play a critical role in light–matter interactions , , , . This dynamic mutual conversion between SAM and OAM is enhanced on the nanoscale , ,  and manifests itself in numerous unusual physical phenomena, such photonic analogue of the quantum spin Hall effect [27, 28], unidirectional coupling to the guided modes , chiral detection , lateral optical forces , as well as photonic topological insulators , many of which rely on the transverse spin carrying by the evanescent field of waveguiding modes , , . Spin–orbit coupling provides a new, unexplored degree of freedom for engineering magneto-optical interactions with complex topological structures of light polarization, such as photonic skyrmions.
In this article, we study optical spin–orbit interactions in the presence of magnetization and magneto-optical interactions of photonic skyrmions. We analyze the near-field response of their MO activity for different magnetization orientation. By examining the electromagnetic field in a magneto-plasmonic system, we demonstrate that a photonic skyrmion, which is formed by the spin–orbit coupling in evanescent vortex beam, is modulated in the presence of the MO response as the orientation of magnetization changes. This provides an opportunity for the visualization of a magnetic domain structure by observing the spin states of the skyrmion. The superposed spin states of a pair of opposite photonic skyrmions, generated by opposite optical vortices, yield distinct domain contrast with a resolution below 120 nm. Furthermore, due to the different response of photonic skyrmions with opposite topological charges to each magnetization orientation, the spin-based domain observation eliminates the magnetization orientation constrains of conventional Kerr microscopy. The results provide new understanding of optical spin and magnetization interactions and offer novel opportunities for study and manipulation of magnetic structures, including magnetic skyrmions, using optical spin–orbit coupling.
2 Results and discussion
2.1 Magneto-optical interactions in the near-field
Spin-skyrmions are formed due to spin–orbit coupling in the evanescent field of guided waves, such as surface plasmon polaritons (SPPs) . Therefore, we consider a general multilayered structure comprising a waveguiding (generally nonmagnetic) layer adjacent to a magnetic film or waveguiding in a magnetic layer itself (Figure 1(A)). The proposed framework describes the properties of such a multilayer for any position of the magnetic film in the multilayered stack and any illumination direction. The permittivity tensor of a magnetic layer is given by
where ε i is the relative permittivity of the magnetic material and the off-diagonal elements g x,y,z are the gyration vector constants, which are linear in magnetization. By applying the Maxwell’s equations, the wave equation inside a magnetic layer takes the form
where k 0 is the wave vector in vacuum. For a surface wave propagating along x axis, the electric field is characterized by , with β denoting the propagation constant and the longitudinal wave vector k z can be obtained from the nontrivial solutions of electric field E 0 in Equation (2), which depends on the magnetization orientation.
Transversal magnetization (g x = g z = 0) couples the x and z components of the electric field, maintaining the independence of transverse electric (TE) and transverse magnetic (TM) polarized guided modes. From Equation (2), the wavevector inside the magnetic layer takes the relation for a TE mode or for a TM mode. For a TE polarized surface guided mode, the mode dispersion equation and propagation constant β are the same as for nonmagnetic or demagnetized systems (no off-diagonal components present in Equation (1)). While for a TM polarized surface mode, the coupling between E x and E z results in a modulation on the propagation constant β, which scales linearly with the gyration vector constant [10, 36, 37].
For polar (g x = g y = 0) or longitudinal (g y = g z = 0) magnetization, the separation between TE and TM polarization is not possible because of the coupling of magnetization to E y . The longitudinal wave vector inside the magnetic layer can be represented as , where and is the magneto-optical contribution (See Supplementary Note 1). This birefringence inside the magnetic layer implies a 4 × 4 transfer matrix to maintain the boundary conditions at each interface, which has been developed in many formalisms to deal with the propagation and reflection problems in stratified anisotropic media , , , , . Taking into account the smallness of g, the propagation constant β of a surface wave in a multilayer structure (Figure 1(A)) is practically not influenced by magnetization, while the electric field ratio between the in plane TE and TM components, η = E y /E x , in the upper medium is approximately proportional to g (See Supplementary Note 1).
The near-field MO effect is prominent in the SPP modes supported by a structure containing ferromagnetic layers with the large gyration despite the high Ohmic losses. In view of this, we consider in the following the SPPs at an interface between dielectric (medium 1) and ferromagnetic plasmonic metal (medium 2). For polar magnetization (magnetic field is in z direction, normal to the interface), η can be obtained as
where , and ε 1 and ε 2 are the relative permittivities of a dielectric and a ferromagnetic metal, respectively (See Supplementary Note 1). While for longitudinal magnetization (magnetic field is in x direction, along the interface and parallel to the SPP propagation direction), the ratio
where . For a fixed gyration vector constant g, the ratio of η for the two types of magnetization configurations satisfies , which is attributed to the fact that for a polar magnetization, static magnetic field couples to E x and E y , while for longitudinal magnetization to E z and E y with the same coupling constant. For plasmonic ferromagnetic metal [Re (ε 2) << −1], η for the longitudinal magnetization is much smaller than that for the polar configuration.
Simulated transverse electric fields at the air/Co interface under the excitation at a wavelength of 633 nm (Figure 1(B) and (C)) confirm prediction of Equation (3), showing the coupling to the TE-polarized electric field component E y with magnitude depending on the magnetization orientation. Note that in the absence of magnetization in Co, simulated E y is zero and E x is the same as in Figure 1(B) (See Supplementary Note 2 for details of the simulations).
2.2 Photonic skyrmions in the presence of magnetization
The spin–orbit interaction in a guided TM or TE polarized electromagnetic field can be introduced by considering an evanescent optical vortex (eOV) in a source free, homogeneous and isotropic medium, which can be described by a Hertz vector potential with a helical phase term in the cylindrical coordinates (r, φ, z) as 
where A is a constant, k r and k z are the transverse and longitudinal wavevector components satisfying with k 0 denoting the wave vector in vacuum (in the case of SPPs, k r = k SPP), l is an integer corresponding to the topological charge of the eOV, and J l is the Bessel function of the first kind of order l. For a TM (H z = 0) or TE (E z = 0) polarized eOV, the electric fields can be derived from an electric or magnetic Hertz vector potential as
where ε and μ are the absolute permittivity and permeability of the medium, and are the electric and magnetic Hertz vector potentials with ω denoting the angular frequency of the wave, and and take the form of Equation (4). The electric field distributions for the TM and TE polarized eOVs with topological charge l = 1 are shown in Figure 2(A) and (B) respectively. For a TM-polarized evanescent vortex beam, the dominant E z field forms a doughnut-shaped profile. While for the TE-polarized one, the electric field has maximum in the center.
The resulting spatial distribution of a SAM can be described by the topological structure similar to a magnetic skyrmion: optical spin-skyrmion. Corresponding skyrmion number, which is calculated as  with n = S/|S| denoting the unit spin vector, is determined by the sign of topological charge l. For l > 0, the spin vectors tilt progressive from the ‘up’ state in the center to the opposite ‘down’ state along the radial direction, typical for the Neel-type skyrmions with positive skyrmion number (Figure 2(C)). Skyrmions with negative skyrmion number can be formed as well for l < 0, with spin down in the center (Figure 2(D)). The appearance of this photonic spin topological structure is due to the spin–orbit coupling governed by the spin-momentum locking in the evanescent field of a SPP [17, 47].
In the presence of a magneto-optical effect, the separation between TM and TE modes is restrained and a superposition of electric and magnetic Hertz vector potentials is required to unveil the spin–orbit interaction. From the near-field MO effect discussed above, the Hertz potentials are linked through the relation
where η is the in-plane electric field ratio defined in Equation (3) and denotes the wave impendence. The coupling between these Hertz potentials modulates the skyrmions formed by either pure electric or pure magnetic Hertz potential with each SAM component affected, so that the modulated skyrmion is transformed into twisted Neel-type skyrmion [48, 49] due to the nonzero azimuthal component of the SAM S φ (Figure 2(C)–(F)). The modulation of the skyrmion contains the information on the magnetization of the layer.
which shows a weak modulation of the spin texture compared to a non magnetized photonic skyrmion descried by . By comparing two skyrmions with opposite skyrmion numbers (l = ±1, Figure 2(C) and (D)), the magnetization effect can be obtained as
where sign is determined by the imaginary part of η, in other words, the orientation of magnetization. This gives rise to an opportunity to characterize the magnetization of the various domain structures. The spin state modulation can be normalized to the field intensity as so that
which can be measured in experiment. In the polar configuration, the skyrmion number is preserved in the presence of magnetization (see Supplementary Note 3).
In the numerical simulations, a tightly focused (NA = 1.49) radially polarized (RP) beam (a wavelength of 633 nm) with a spiral phase of topological charge l = ±1 is used to illuminate a sample consisting of a 50-nm-thick cobalt film sandwiched by a silica substrate and air (Figure 3(A)) (see Supplementary Note 2 for details of the simulations). This provides a SPP excitation and generation of a photonic skyrmion at an air/cobalt interface. For the magnetization oriented in a positive z direction, the spin states of opposite skyrmions generated by the RP beams carrying opposite topological charge are shown in Figure 3(B) and (C). This results in the Δγ s distribution with non negative values (Figure 3(D)), which coincides well with the theoretical calculations using Equation (9) (inset in Figure 3(D)). Corresponding superimposed spin state for the magnetization oriented in a negative z direction shows the opposite sign (Figure 3(E)). It is worth noting that Δγ s is not attenuated away from the center of the beam despite the high Ohmic losses in a ferromagnetic metal, since the loss influences only the intensity of the wave but not the spin. As predicted by Equation (9), the sign of Δγs is determined by the orientation of magnetization and can act as an effective indicator for magnetic domain observation.
As an example, we consider the magnetization of a cobalt film consisting of two individual domains of opposite magnetization with the domain wall located at x = 0 and x = −1 (Figure 3(F) and (G)). The magnetic contrast is clearly observable through the sign of Δγ s . The cross-sectional profiles of Δγ s marked on Figure 3(F) and (G) with green dashed lines demonstrate a sharp MO contrast with a lateral resolution below 120 nm (light green area in Figure 3(H)). It should be noted that the domain contrast will be smeared at the position where J 0(k r r) ∼ 0 or J 2(k r r) ∼ 0 since, according to Equation (9), Δγ s for either positive or negative magnetization (Figure 3(D) and (E)) is approaching zero as J 0(k r r) or J 2(k r r) is close to zero, resulting in an overlap between a domain wall and zero points of Δγ s . Nevertheless, the domain wall is still visible in the Δγ s maps. Since the resolution depends on the in-plane wave vector of the SPP generated at a Co/air interface, higher resolution can be realized by decreasing the incident wavelength, or adding gap layers between cobalt film and air to increase the SPP wave vector.
For in-plane magnetization (transversal and longitudinal cases are not defined separately here for the cylindrical symmetry of illumination), η is inhomogeneous along the surface plane and depends on the projection of the local in-plane wave vector onto the magnetization direction, resulting in an angle dependent Δγs as
where φ 0 is the magnetization orientation with respect to x-axis and η lon denotes the electric field ratio calculated for the longitudinal magnetization from Equation (3b). In this case, the resulting spin structure cannot be assigned a skyrmion number in the presence of magnetization. Simulated superposed spin states of a pair of opposite skyrmions generated by plasmonic vortices with topological charges l = ±1 (γ s for opposite skyrmions separately are shown in Figure S1) exhibit a strong dependence on the magnetization direction (Equation (10)), which is determined by φ 0 (Figure 4(A)–(C)). Although the ‘central zero line’, which is normal to the magnetization direction, in the Δγ s distributions is of similar appearance to a domain wall in a polar magnetization case in Figure 3(F), its origin is completely different. First of all, from Equation (10), Δγ s reaches extreme values as J 0(k r r) or J 2(k r r) approaches zero for in-plane magnetization (Figure 4(A)–(C)), while for polar magnetization, Δγ s is approaching zero as J 0(k r r) or J 2(k r r) ∼ 0. This results in the distinguishable spin state patterns between polar and longitudinal magnetizations (cf Figures 3(F) and 4(A)). The modulation of the spin states for in-plane magnetization is much smaller than that in a polar case due to the different ratio η, as discussed in Section 2.1. In addition, a cosine function varies slowly near zero points, making the ‘central zero line’ for in-plane magnetization much wider in Figure 4(A) than the sharp domain wall in Figure 3(F). As shown in Figure 4(D)–(I), the magnetic domain configuration and magnetization direction is observed in the variation of the spin states. It is worth noting that the sign of the oscillations at the central areas in each of the Δγ s map in Figure 4 is due to the TE and TM polarized transmission coefficients of the focused beams in the presence of longitudinal magnetization (See Supplementary Note 4 and Figures S2 and S3 for details). The proposed technique can provide 2D domain observation capabilities and, in turn, control of the optical spin patterns in a similar manner (Figure 5).
3 Conclusion and outlook
We have demonstrated and investigated the optical spin–orbit coupling in the presence of magnetization, as well as the interaction between photonic skyrmions and magnetic domains. We showed that the spin–orbit coupling in the intrinsic near-field magneto-optical activity is responsible for the magnetization-modulated photonic skyrmions with resized and twisted spin vectors, resulting in the so-called twisted Neel-type skyrmions. The superimposed spin vectors of a pair of skyrmions with opposite skyrmion numbers can act as a straightforward and flexible indicator for the magnetization orientation, which gives rise to the visualization of magnetic domain structure. By employing plasmonic vortices formed at the surface of a thin cobalt film, we demonstrated the resolution for the magnetic domain contrast below 120 nm in a case of polar magnetization. Due to different response of photonic skyrmions for each magnetization orientation, same domain observation configuration can be applied for both in plane and out of plane magnetization. The studied magnetization-induced spin–orbit coupling may be important for investigations and applications of novel magneto-optical effects in waveguiding and nanophotonic geometries, as well as manipulation of magnetic skyrmions.
Funding source: UK Engineering and Physical Sciences Research Council
Funding source: Leadership of Guangdong province program
Award Identifier / Grant number: 00201505
Funding source: Natural Science Foundation of Guangdong Province
Award Identifier / Grant number: 2016A030312010
Funding source: Guangdong Major Project of Basic Research
Award Identifier / Grant number: 2020B0301030009
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 61427819
Award Identifier / Grant number: 61622504
Award Identifier / Grant number: 61705135
Award Identifier / Grant number: 61935013
Award Identifier / Grant number: 62075139
Award Identifier / Grant number: U1701661
Funding source: Shenzhen Peacock Plan
Award Identifier / Grant number: KQTD2015071016560101
Award Identifier / Grant number: KQTD20170330110444030
Funding source: Science and Technology Innovation Commission of Shenzhen
Award Identifier / Grant number: JCYJ20200109114018750
Award Identifier / Grant number: RCJC20200714114435063
Funding source: Guangdong Special Support Program
Author contributions: X.L and A.V.Z. conceived the idea for this research. X.L. developed theory and performed numerical simulations under the guidance of L.D., X.Y. and A.V.Z. All the authors contributed to the writing of the manuscript.
Research funding: This work was supported by EPSRC (UK) under the Reactive Plasmonics Programme grant (EP/M013812/1). European Research Council iCOMM project 789340, National Natural Science Foundation of China grants U1701661, 61935013, 62075139, 61427819, 61622504 and 61705135, Guangdong Major Project of Basic Research No. 2020B0301030009, Leadership of Guangdong province program grant 00201505, Natural Science Foundation of Guangdong Province grant 2016A030312010, Science and Technology Innovation Commission of Shenzhen grants RCJC20200714114435063, JCYJ20200109114018750, and Shenzhen Peacock Plan KQTD2015071016560101 and KQTD20170330110444030. L.D. acknowledges the support from the Guangdong Special Support Program.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
 X. Z. Yu, Y. Onose, N. Kanazawa, et al.., “Real-space observation of a two-dimensional skyrmion crystal,” Nature, vol. 465, pp. 901–904, 2010, https://doi.org/10.1038/nature09124.Search in Google Scholar
 N. Romming, C. Hanneken, M. Menzel, et al.., “Writing and deleting single magnetic skyrmions,” Science, vol. 341, pp. 636–639, 2013, https://doi.org/10.1126/science.1240573.Search in Google Scholar
 F. Schmidt and A. Hubert, “Domain observations on CoCr-layers with a digitally enhanced Kerr-microscope,” J. Magn. Magn. Mater., vol. 61, pp. 307–320, 1986, https://doi.org/10.1016/0304-8853(86)90044-2.Search in Google Scholar
 W. Dickson, S. Takahashi, R. Pollard, R. Atkinson, and A. V. Zayats, “High-resolution optical imaging of magnetic-domain structures,” IEEE Trans. Nanotechnol., vol. 4, pp. 229–237, 2005, https://doi.org/10.1109/tnano.2004.837850.Search in Google Scholar
 R. Schäfer, “Investigation of domains and dynamics of domain walls by the magneto‐optical Kerr‐effect,” in Handbook of Magnetism and Advanced Magnetic Materials, Hoboken, Wiley, 2007.10.1002/9780470022184.hmm310Search in Google Scholar
 J. McCord, “Progress in magnetic domain observation by advanced magneto-optical microscopy,” J. Appl. Phys., vol. 48, 333001, 2015, https://doi.org/10.1088/0022-3727/48/33/333001.Search in Google Scholar
 A. Hubert and R. Schäfer, Magnetic Domains: The Analysis of Magnetic Microstructures, Berlin, Springer, 2008.Search in Google Scholar
 C. Hermann, V. Kosobukin, G. Lampel, J. Peretti, V. Safarov, and P. Bertrand, “Surface-enhanced magneto-optics in metallic multilayer films,” Phys. Rev. B, vol. 64, 235422, 2001.10.1103/PhysRevB.64.235422Search in Google Scholar
 A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett., vol. 91, 183901, 2003, https://doi.org/10.1103/physrevlett.91.183901.Search in Google Scholar PubMed
 J. B. González-Díaz, A. García-Martín, G. Armelles, et al.., “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B, vol. 76, p. 153402, 2007.10.1103/PhysRevB.76.153402Search in Google Scholar
 B. Fan, M. E. Nasir, L. H. Nicholls, A. V. Zayats, and V. A. Podolskiy, “Magneto‐optical metamaterials: nonreciprocal transmission and Faraday effect enhancement,” Adv. Opt. Mater., vol. 7, 1801420, 2019, https://doi.org/10.1002/adom.201801420.Search in Google Scholar
 B. Sepulveda, J. B. Gonzalez-Diaz, A. Garcia-Martin, L. M. Lechuga, and G. Armelles, “Plasmon-induced magneto-optical activity in nanosized gold disks,” Phys. Rev. Lett., vol. 104, 147401, 2010, https://doi.org/10.1103/physrevlett.104.147401.Search in Google Scholar
 V. I. Belotelov, I. A. Akimov, M. Pohl, et al.., “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol., vol. 6, pp. 370–376, 2011, https://doi.org/10.1038/nnano.2011.54.Search in Google Scholar PubMed
 G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater., vol. 1, pp. 10–35, 2013, https://doi.org/10.1002/adom.201200011.Search in Google Scholar
 L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, et al.., “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X, vol. 3, p. 041019, 2013.10.1103/PhysRevX.3.041019Search in Google Scholar
 N. Maccaferri, X. Inchausti, A. García-Martín, et al.., “Resonant enhancement of magneto-optical activity induced by surface plasmon polariton modes coupling in 2D magnetoplasmonic crystals,” ACS Photonics, vol. 2, pp. 1769–1779, 2015, https://doi.org/10.1021/acsphotonics.5b00490.Search in Google Scholar
 L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys., vol. 15, pp. 650–654, 2019, https://doi.org/10.1038/s41567-019-0487-7.Search in Google Scholar
 S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science, vol. 361, pp. 993–996, 2018, https://doi.org/10.1126/science.aau0227.Search in Google Scholar PubMed
 X. Lei, A. Yang, P. Shi, et al.., “Photonic spin lattices: symmetry constraints for skyrmion and meron topologies,” arXiv preprint arXiv:2103.15366, 2021.10.1103/PhysRevLett.127.237403Search in Google Scholar PubMed
 A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett., vol. 88, 053601, 2002, https://doi.org/10.1103/PhysRevLett.88.053601.Search in Google Scholar PubMed
 A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt., vol. 13, 053001, 2011, https://doi.org/10.1088/2040-8978/13/5/053001.Search in Google Scholar
 K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics, vol. 9, pp. 796–808, 2015, https://doi.org/10.1038/nphoton.2015.201.Search in Google Scholar
 J. Petersen, J. Volz, and A. Rauschenbeutel, “Chiral nanophotonic waveguide interface based on spin-orbit interaction of light,” Science, vol. 346, pp. 67–71, 2014, https://doi.org/10.1126/science.1257671.Search in Google Scholar PubMed
 D. O’Connor, P. Ginzburg, F. J. Rodriguez-Fortuno, G. A. Wurtz, and A. V. Zayats, “Spin-orbit coupling in surface plasmon scattering by nanostructures,” Nat. Commun., vol. 5, 5327, 2014, https://doi.org/10.1038/ncomms6327.Search in Google Scholar PubMed
 F. J. Rodriguez-Fortuno, G. Marino, P. Ginzburg, et al.., “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science, vol. 340, pp. 328–330, 2013, https://doi.org/10.1126/science.1233739.Search in Google Scholar PubMed
 L. Du, S. S. Kou, E. Balaur, et al.., “Broadband chirality-coded meta-aperture for photon-spin resolving,” Nat. Commun., vol. 6, 10051, 2015, https://doi.org/10.1038/ncomms10051.Search in Google Scholar PubMed PubMed Central
 L. Wei, A. V. Zayats, and F. J. Rodriguez-Fortuno, “Interferometric evanescent wave excitation of a nanoantenna for ultrasensitive displacement and phase metrology,” Phys. Rev. Lett., vol. 121, 193901, 2018, https://doi.org/10.1103/physrevlett.121.193901.Search in Google Scholar PubMed
 A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics, vol. 9, pp. 789–795, 2015, https://doi.org/10.1038/nphoton.2015.203.Search in Google Scholar
 V. V. Temnov, G. Armelles, U. Woggon, et al.., “Active magneto-plasmonics in hybrid metal–ferromagnet structures,” Nat. Photonics, vol. 4, pp. 107–111, 2010, https://doi.org/10.1038/nphoton.2009.265.Search in Google Scholar
 D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, et al.., “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett., vol. 97, 183114, 2010, https://doi.org/10.1063/1.3512874.Search in Google Scholar
 D. W. Berreman, “Optics in stratified and anisotropic media: 4× 4-matrix formulation,” J. Opt. Soc. Am., vol. 62, pp. 502–510, 1972, https://doi.org/10.1364/josa.62.000502.Search in Google Scholar
 M. Vassell, “Structure of optical guided modes in planar multilayers of optically anisotropic materials,” J. Opt. Soc. Am., vol. 64, pp. 166–173, 1974, https://doi.org/10.1364/josa.64.000166.Search in Google Scholar
 J. Zak, E. Moog, C. Liu, and S. Bader, “Universal approach to magneto-optics,” J. Magn. Magn. Mater., vol. 89, pp. 107–123, 1990, https://doi.org/10.1016/0304-8853(90)90713-z.Search in Google Scholar
 Š. Višňovský, R. Lopušník, M. Bauer, J. Bok, J. Fassbender, and B. Hillebrands, “Magnetooptic ellipsometry in multilayers at arbitrary magnetization,” Opt. Express, vol. 9, pp. 121–135, 2001, https://doi.org/10.1364/oe.9.000121.Search in Google Scholar PubMed
 P. Johnson and R. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B, vol. 9, p. 5056, 1974, https://doi.org/10.1103/physrevb.9.5056.Search in Google Scholar
 N. Nagaosa and Y. Tokura, “Topological properties and dynamics of magnetic skyrmions,” Nat. Nanotechnol., vol. 8, pp. 899–911, 2013, https://doi.org/10.1038/nnano.2013.243.Search in Google Scholar PubMed
 P. Shi, L. Du, C. Li, A. V. Zayats, and X. Yuan, “Transverse spin dynamics in structured electromagnetic guided waves,” Proc. Natl. Acad. Sci. USA, vol. 118, e2018816118, 2021.10.1073/pnas.2018816118Search in Google Scholar PubMed PubMed Central
 F. N. Rybakov, A. B. Borisov, and A. N. Bogdanov, “Three-dimensional skyrmion states in thin films of cubic helimagnets,” Phys. Rev. B, vol. 87, 094424, 2013.10.1103/PhysRevB.87.094424Search in Google Scholar
 S. L. Zhang, G. van der Laan, W. W. Wang, A. A. Haghighirad, and T. Hesjedal, “Direct observation of twisted surface skyrmions in bulk crystals,” Phys. Rev. Lett., vol. 120, 227202, 2018, https://doi.org/10.1103/physrevlett.120.227202.Search in Google Scholar
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