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

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

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 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].

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

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

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]

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

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 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.

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 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 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

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

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

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 Δγ_s 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).

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 Δγ_s 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).

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).