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BY 4.0 license Open Access Published by De Gruyter August 24, 2021

Optical spin–orbit coupling in the presence of magnetization: photonic skyrmion interaction with magnetic domains

Xinrui Lei ORCID logo, Luping Du, Xiaocong Yuan and Anatoly V. Zayats
From the journal Nanophotonics

Abstract

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.

1 Introduction

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 [3], [4], [5], [6], [7]. 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 [8], [9], [10], [11], [12], [13], [14], [15], [16].

Prominent optical analogies of magnetic topological structures have been recently demonstrated, such as photonic skyrmions and merons [17], [18], [19]. 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 [20], [21], [22], [23]. This dynamic mutual conversion between SAM and OAM is enhanced on the nanoscale [24], [25], [26] 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 [29], chiral detection [30], lateral optical forces [31], as well as photonic topological insulators [32], many of which rely on the transverse spin carrying by the evanescent field of waveguiding modes [33], [34], [35]. 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) [17]. 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

(1) ϵ ˆ i = ( ϵ i i g z i g y i g z ϵ i i g x i g y i g x ϵ i ) ,

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

(2) ( E ) 2 E = k 0 2 ϵ ˆ i E ,

where k 0 is the wave vector in vacuum. For a surface wave propagating along x axis, the electric field is characterized by E = E 0 e i β x ± k z z , 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.

Figure 1: 
(A) Geometry of surface wave propagation in a multilayer consisting of both magnetic and nonmagnetic layers. Red and green arrows represent the magnetization orientation corresponding to longitudinal and polar magnetization, respectively. (B,C) Simulated in-plane electric fields of the surface wave propagating in x direction at an interface of air and magnetized semi-infinite medium (Co): (B) E

x
 and (C) E

y
 field components for different magnetization orientations. The difference between the E

x
 fields in the case of polar and longitudinal magnetizations is negligible. The permittivity ε
2 and the gyration vector g of Co are taken from the experimental data as ε
2 = −12.5 + 18.46i [43] and g = 0.59 − 0.48i [44].

Figure 1:

(A) Geometry of surface wave propagation in a multilayer consisting of both magnetic and nonmagnetic layers. Red and green arrows represent the magnetization orientation corresponding to longitudinal and polar magnetization, respectively. (B,C) Simulated in-plane electric fields of the surface wave propagating in x direction at an interface of air and magnetized semi-infinite medium (Co): (B) E x and (C) E y field components for different magnetization orientations. The difference between the E x fields in the case of polar and longitudinal magnetizations is negligible. The permittivity ε 2 and the gyration vector g of Co are taken from the experimental data as ε 2 = −12.5 + 18.46i [43] and g = 0.59 − 0.48i [44].

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 k z i = β 2 ϵ i k 0 2 for a TE mode or k z i = β 2 ϵ i k 0 2 ( 1 g y 2 / ϵ i 2 ) 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 k z i ± = k z i 0 ± Δ k ( g ) , where k z i 0 = β 2 ϵ i k 0 2 and Δ k ( g ) 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 [38], [39], [40], [41], [42]. 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

(3a) η z i ϵ 1 2 k 0 2 2 ( ϵ 1 ϵ 2 ) ϵ 2 k z 1 2 g z  ,

where k z 1 = β 2 ϵ 1 k 0 2 , 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

(3b) η x ϵ 1 2 k 0 2 2 ( ϵ 1 ϵ 2 ) ϵ 2 k z 1 2 β k z 2 0 g x  ,

where k z 2 0 = β 2 ϵ 2 k 0 2 . For a fixed gyration vector constant g, the ratio of η for the two types of magnetization configurations satisfies η x / η z = i β / k z 2 0 = E z / E x , 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 [45]

(4) Ψ = A J l ( k r r ) e i l φ e k z z ,

where A is a constant, k r and k z are the transverse and longitudinal wavevector components satisfying k r 2 k z 2 = k 0 2 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

(5a) E T M = ( π e ) μ ϵ 2 π e t 2 = ( k z r , k z r φ , k r 2 ) Ψ e  ,

(5b) E T E = μ × π m t = i ω μ ( 1 r φ , r , 0 ) Ψ m  ,

where ε and μ are the absolute permittivity and permeability of the medium, π e = Ψ e e i ω t e z and π m = Ψ m e i ω t e z are the electric and magnetic Hertz vector potentials with ω denoting the angular frequency of the wave, and Ψ e and Ψ m take the form of Equation (4). The electric field distributions for the TM and TE polarized eOVs with topological charge = 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.

Figure 2: 
(A, B) Electric field profile of (A) TM polarized evanescent vortex induced by an electric Hertz vector potential and (B) TE polarized evanescent vortex induced by a magnetic Hertz vector potential, both with topological charge l = 1 in the absence of magnetization. (C) Distribution of the photonic spin orientation formed by the evanescent vortex beam with topological charge l = 1. The opaque arrows indicate the direction of the unit spin vector in (A), while the semitransparent arrows represent the modulated spin vectors in the presence of the polar MO effect. (D) The same as (C) for a topological charge l = −1. (E, F) Top view of the spin vectors in the presence of MO effect in (C) and (D), respectively. The in plane wavevector (k

r
) of the evanescent vortex for the simulations is set to 1.012k, where k is the wavevector in the medium, and η is set to 0.3 + 0.03i. In (A) and (B), x and y axis scales are normalized to the in-plane wavelength of the guided mode k
r.

Figure 2:

(A, B) Electric field profile of (A) TM polarized evanescent vortex induced by an electric Hertz vector potential and (B) TE polarized evanescent vortex induced by a magnetic Hertz vector potential, both with topological charge l = 1 in the absence of magnetization. (C) Distribution of the photonic spin orientation formed by the evanescent vortex beam with topological charge l = 1. The opaque arrows indicate the direction of the unit spin vector in (A), while the semitransparent arrows represent the modulated spin vectors in the presence of the polar MO effect. (D) The same as (C) for a topological charge l = −1. (E, F) Top view of the spin vectors in the presence of MO effect in (C) and (D), respectively. The in plane wavevector (k r ) of the evanescent vortex for the simulations is set to 1.012k, where k is the wavevector in the medium, and η is set to 0.3 + 0.03i. In (A) and (B), x and y axis scales are normalized to the in-plane wavelength of the guided mode k r.

The resulting spatial distribution of a SAM S Im ( E × E ) can be described by the topological structure similar to a magnetic skyrmion: optical spin-skyrmion. Corresponding skyrmion number, which is calculated as [46] Q = ( 1 / 4 π ) n ( x n × y n ) d x d y = ± 1 with 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

(6) Ψ m = η k z i Z k 0 Ψ e ,

where η is the in-plane electric field ratio defined in Equation (3) and Z = μ / ϵ 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.

In the case of polar magnetization, η is homogeneous for each polar angle φ along the plane. By substituting Equation (6) into Equation (5), the SAM normal to the interface can be obtained as

(7) S z = 2 Im ( E r E φ ) = A 2 k r 2 k z 2 e 2 k z z 2 [ ( 1 + | η z | 2 + 2 Im η z ) J l 1 2 ( k r r ) ( 1 + | η z | 2 2 Im η z ) J l + 1 2 ( k r r ) ]  ,

which shows a weak modulation of the spin texture compared to a non magnetized photonic skyrmion descried by S z 0 = A 2 k r 2 k z 2 e 2 k z z 2 [ J l 1 2 ( k r r ) J l + 1 2 ( k r r ) ] . By comparing two skyrmions with opposite skyrmion numbers (l = ±1, Figure 2(C) and (D)), the magnetization effect can be obtained as

(8) Δ S z = S z , 1 + S z , 1 = 2 Im η z [ J 0 2 ( k r r ) + J 2 2 ( k r r ) ] A 2 k r 2 k z 2 e 2 k z z  ,

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 I x y = | E r | 2 + | E φ | 2 as γ s = S z I x y so that

(9) Δ γ s = γ s , 1 + γ s , 1 Im η z 16 J 0 2 ( k r r ) J 2 2 ( k r r ) [ J 0 2 ( k r r ) + J 2 2 ( k r r ) ] 2  ,

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.

Figure 3: 
(A) Schematic diagram for the generation of a photonic skyrmion in a ferromagnetic material on the example of the SPP excited on a surface of a thin (50 nm) Co film by a tightly focused (NA = 1.49) radially polarized beam. Incident angles below 40° are blocked to stabilize the generation of photonic skyrmions. The permittivity and the gyration vector constant of cobalt at wavelength of 633 nm are the same as in a Figure 1. (B,C) Spin states of (B) positive and (C) negative skyrmions generated with l = ±1 RP beam, respectively, with the Co magnetization in a positive z direction. (D) Spatial distribution of Δγ

s
 for the skyrmions in (B) and (C). Corresponding theoretical distributions of Δγ

s
 obtained from Equation (9) is depicted in the bottom left inset. (E) The same as (D) for the magnetization oriented in negative z direction. (F,G) Spatial distributions of Δγ

s
 for the magnetic structure consisting of two domains with opposite magnetization orientation: the domain wall is at (F) x = 0 and (G) x = −1 μm. (H, I) Cross-sections marked in (F, G) with a green dashed line. Inset in (H) is a zoomed view of the spin structure of the area indicated with light green box in (H).

Figure 3:

(A) Schematic diagram for the generation of a photonic skyrmion in a ferromagnetic material on the example of the SPP excited on a surface of a thin (50 nm) Co film by a tightly focused (NA = 1.49) radially polarized beam. Incident angles below 40° are blocked to stabilize the generation of photonic skyrmions. The permittivity and the gyration vector constant of cobalt at wavelength of 633 nm are the same as in a Figure 1. (B,C) Spin states of (B) positive and (C) negative skyrmions generated with l = ±1 RP beam, respectively, with the Co magnetization in a positive z direction. (D) Spatial distribution of Δγ s for the skyrmions in (B) and (C). Corresponding theoretical distributions of Δγ s obtained from Equation (9) is depicted in the bottom left inset. (E) The same as (D) for the magnetization oriented in negative z direction. (F,G) Spatial distributions of Δγ s for the magnetic structure consisting of two domains with opposite magnetization orientation: the domain wall is at (F) x = 0 and (G) x = −1 μm. (H, I) Cross-sections marked in (F, G) with a green dashed line. Inset in (H) is a zoomed view of the spin structure of the area indicated with light green box in (H).

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

(10) Δ γ s cos ( φ φ 0 ) Re η lon [ J 0 2 ( k r r ) J 2 2 ( k r r ) ] 2 [ J 0 2 ( k r r ) + J 2 2 ( k r r ) ] 2  ,

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).

Figure 4: 
(A–C) Simulated distributions of Δγ

s
 for different magnetization orientations in the plane of the film (xy plane): (A) φ
0 = 0, (B) φ
0 = π/4, (C) φ
0 = π/2. (D–I) The same as (A–C) for the domain structure with the domain wall located at (D) x = 0, (E–F) y = 0, (G) x = −1 μm, (H,I) y = −1 μm. The magnetic domain can be visualized by the features of Δγ

s
 distributions, while the magnetization orientation can be unveiled from the ‘central zero line’, which is normal to the magnetization direction. Inserts show the magnetization orientation in the domains. The illumination configuration is the same as in Figure 3(A).

Figure 4:

(A–C) Simulated distributions of Δγ s for different magnetization orientations in the plane of the film (xy plane): (A) φ 0 = 0, (B) φ 0 = π/4, (C) φ 0 = π/2. (D–I) The same as (A–C) for the domain structure with the domain wall located at (D) x = 0, (E–F) y = 0, (G) x = −1 μm, (H,I) y = −1 μm. The magnetic domain can be visualized by the features of Δγ s distributions, while the magnetization orientation can be unveiled from the ‘central zero line’, which is normal to the magnetization direction. Inserts show the magnetization orientation in the domains. The illumination configuration is the same as in Figure 3(A).

Figure 5: 
Simulated spatial distributions of Δγ

s
 for the domain structures with two orthogonal domain walls located at x = 0 and y = 0: (A) Polar magnetization and (B) in-plane magnetization. Inserts show the magnetization orientation in the domains. The illumination configuration is the same as in Figure 3(A).

Figure 5:

Simulated spatial distributions of Δγ s for the domain structures with two orthogonal domain walls located at x = 0 and y = 0: (A) Polar magnetization and (B) in-plane magnetization. Inserts show the magnetization orientation in the domains. The illumination configuration is the same as in Figure 3(A).

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.


Corresponding authors: Luping Du and Xiaocong Yuan, Nanophotonics Research Centre, Shenzhen Key Laboratory of Micro-Scale Optical Information Technology, Shenzhen University, Shenzhen, Guangdong, 518060, China; and Anatoly V. Zayats, Department of Physics and London Centre for Nanotechnology, King’s College London, Strand, London, WC2R 2LS, UK, E-mail: , , (A. V. Zayats)

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

  1. 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.

  2. 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.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2021-0201).


Received: 2021-04-29
Accepted: 2021-07-26
Published Online: 2021-08-24

© 2021 Xinrui Lei et al., published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

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