Simulation Study of Dynamic Rotation and Deformation for Plasmonic Electric Field-Skyrmions

Abstract

The topological properties of optical skyrmions in confined electromagnetic fields are perfectly presented through spin vectors and electric-field vectors. However, currently, electric-field optical skyrmions in surface plasmon polaritons are mostly presented in the form of a Néel type. Most control strategies involve linear directional movement, and topological manipulation methods are monotonous. We specifically propose a multi-arc symmetric slit array, which generates skyrmions from the surface plasmon polariton (SPP) field under excitation of a linearly polarized Gaussian light-source array and exhibits strong dependence processes on the rotation, deformation, and phase distribution of the incident light source. We also discuss the independence and synthesis of deformation and rotation related to phase difference and positions of regulation, respectively, which provide the possibility for rich deformations under different rotation states. Our work extends new ideas for the dynamic control of plasmonic skyrmions, which is of great significance to fields such as spin photonics and nano-positioning.

1. Introduction

Skyrmions, as a topological defect, were described by Tony Skyrme [1]. In the last ten years, the magnetic skyrmion has rapidly developed due to its valuable properties (possibilities for application in spintronic devices [2]; high speed [3]; and low-current drives [4]). The integer and semi-integer of stable and topological spin structures [5] have been proposed for the Dzyaloshinskii−Moriya interaction [6]. Since evanescent-field optical skyrmions were generated by plasmonic gratings [7], the research on optical skyrmions in structured light fields [8] has begun.

Recently, the research on optical skyrmions has mainly focused on the theoretical level [9]. In addition to electric field vectors [10], stokes vectors [11] and transverse spin vectors [12] can be used similarly as objects to construct robust skyrmion structures and even to realize high-order forms such as hopfions [13]. When considering the spin−orbit interaction [14], the transverse spin texture of the plasmonic vortex can form a meron-like skyrmion [15]. However, such an optical skyrmion mostly exists in the form of a Néel type accompanied by a skyrmion number of 1 when the SOI (spin−orbit interaction) disappears [16]. For example, skyrmion arrays in SPP (surface plasmon polariton) fields have been excited by gratings [17], a high-spatiotemporal-resolution electric field vector distribution obtained through the use of 2PEEM (two-photon emission electron microscopy) technology [10], and skyrmions regulated in the field of graphene surfaces [18].

SPP is the metal/dielectric surface’s evanescent field and electromagnetic composite field [19]. Through Maxwell’s equations, spin vectors perpendicular to the propagation direction are generated in the energy flow [20]. The electric field of SPP always undergoes polarity reversal in the plane where SPP waves propagate due to the presence of the imaginary component in the vertical direction [21], which is the cause of the Néel-type skyrmion configuration of the interference field. Therefore, to construct stable arrays, skyrmion regulation is limited to linear directional movements.

Considering the possibilities of SPP-field skyrmion topological transformation [22] and the ability to achieve angular rotation and even more and free deformations for new topological types, we propose a structure, which has 12 arc-shaped slits hollowed out on the surface of the metal Ag; each slit is vertically excited with focused Gaussian, linearly polarized focused spots and is given the flexible phase. Through our simulation in FDTD (FDTD Solutions 2016a), we combined the rotations and deformations of skyrmions in the local plane, and the results showed two types of regulation that have independence and synthesizability, which are related to the positions and phase difference of incident Gaussian spots, respectively. This work provides powerful supplements to the field of optical skyrmions and improves the topological properties of electric fields under the tight coupling conditions of SPP fields. Significant application prospects have been demonstrated in fields such as high-density storage [23] and quantum communication [24].

2. Simulation and Methods

We assume that the 12 pairs of sources are incident on the Ag sheet along the Z-axis direction, each pair of sources with a symmetry distance of “2d”. The arc-shaped slit can be approximated as the superposition of multiple linear slits along the tangential direction. Specifically, the vertical views in the X-Y plane of linear slits were shown in Figure 1a−c. Each slit generates an SPP field independently on the surface, and SPP waves interfere, corresponding to the generation of quadrangular arrays, meron arrays, and hexagonal arrays of skyrmions. The arc-shaped slits are shown in black dashed circles in Figure 1d,e, which have the special symmetry. We take a segment of the arc with a central angle θ on the circular slit in Figure 1d,e, and first use six linear slits to equivalently replace the arc-shaped slit along the tangent direction of the arc in Figure 1d. More generally, as shown in Figure 1e, we assume that the number of linear slits is K. When the value of K is large, we extend the superposition by multiple linear slits. In theory, according to the microelement analysis method, the arc is equivalent to the concatenation of infinite linear segments. We use integration in Equation (2) to obtain the sum of the SPP field generated by all linear slits.

We provide a rigorous mathematical expression for the Z-axis component of the SPP electric field E_Z in equation (1), which was generated by an arc-shaped slit array, as shown in Figure 2. We label the slit at the highest position in the positive Y-axis as 1 and sequentially label the 12 slits as n (n = 1, … 12; m is the number of pairs of symmetric slits, m = 1, … 6.) in a clockwise direction. K_SPP is the wave vector of SPP [25]. Here, we mainly consider its imaginary part, and the reciprocal of this imaginary part is positively correlated with the propagation length L of SPP.

$$E_Z\left(d,t\right)={\displaystyle \sum _{m=1}^6{E}_Z^{m,\left(m+6\right)}\left(d,t\right)}$$

(1)

where x, y, and z are three Cartesian coordinates. Meanwhile, β signifies the angular coordinates of the column coordinate system, ω represents the angular frequency, and E_SPP is the amplitude of the electric field, where the expression of E₀ as shown in Equation (2) is an important part of the final expression of the Z-axis component of the SPP electric field.

$$E_0=E_{SPP}{\displaystyle \underset{-\pi /15}{\overset{\pi /15}{\int }}\exp \left(\frac{-x\cos \beta -y\sin \beta }{2L}\right)}\exp \left[i\left(k_{SPP}x\cos \beta +k_{SPP}y\sin \beta -\omega t\right)\right]d\beta$$

(2)

Considering the phase Φ_m of SPP waves generated by each arc-shaped slit [26], the relative polarization directions are Ψ_m, where the value of (Ψ_m − Ψ_m+6) is −π. In addition, each pair of arc-shaped slit in the symmetric excitation structure carries an additional phase difference π [27]. Therefore, the phase difference (Φ_m − Φ_m+6) of SPP in Equation (3) can be represented by the phase difference (φ_m − φ_m+6) of the incident source.

$${\varphi }_{SP{P}_m}-{\varphi }_{SP{P}_{\left(m+6\right)}}={\phi }_m-{\phi }_{m+6}+{\psi }_m-{\psi }_{m+6}+\pi$$

(3)

Thus, the SPP wave propagates over a distance of d, and the Z-component of the electric field E_Z that excites the localized surface at the center of the structure can be expressed as Equation (4):

$$E_Z\left(d,t\right)=2{\displaystyle \sum _{m=1}^6{E}_0}\cos \left[{\displaystyle \frac{{\varphi }_{SP{P}_m}-{\varphi }_{SP{P}_{\left(m+6\right)}}}{2}}\right]\exp \left\{i\left[{\displaystyle \frac{{\varphi }_{SP{P}_m}+{\varphi }_{SP{P}_{\left(m+6\right)}}}{2}}\right]\right\}$$

(4)

We found that the superposition of the SPP electric field depends on the value of E₀ and the phase difference of symmetric pairs of incident light spots in Equation (4). In addition, according to the expression of E₀, the integration result of E₀ depends on the upper and lower limitations (the integration variable is β).

The simulation parameters are as follows: The wavelength of the Gaussian linearly polarized source is 632.8 nm, located at the Z-axis coordinate of −105 nm. The diameter of focusing spots is assumed to be 360 nm, and the multi-arc symmetric structure is vertically incident along the Z-axis, as shown in Figure 2a. The distance d from the inner side of each arc-shaped slit to the center of the structure is 0.95 μm, and the distance D from the outer side of each arc-shaped slit to the center of the structure is 1.05 μm. The central angle θ of the arc-shaped slit is (2/15)π, and the width of the slit (D–d) is 100 nm. The phase of the excitation light source at each position of the slit is independently controlled. The SPP field is generated on the surface of metal Ag with a thickness of 120 nm, and SiO₂ with a thickness of 100 nm is used as the substrate for the simulation experiment; in addition, boundary conditions are PML. Moreover, the two-dimensional frequency-domain field has a width of 1.4 μm and power monitors have Z-axis coordinates of 0.5 μm, and 0.2 μm, respectively, which were used to present the full view of the SPP electric-field.

3. Results and Discussions

3.1. Dynamic Rotation for Skyrmions

Stable topological optical skyrmions can be constructed by plasmonic standing wave interference. Considering the sub-wavelength characteristics of electric field skyrmions, implementing their rotational properties leads to more interesting discoveries and broader prospects for applications in nano-photonics. We control the phase difference of the light sources as a variable, and by adjusting the phase difference in different directions, we have achieved a gradual counterclockwise rotation of the skyrmion from horizontal to small-acute-angle tilted, further increasing the inclination, and finally fully rotating it to a right-angled tilted position. Throughout the entire process, the skyrmion rotates precisely within the subwavelength local range, maintaining its original topological shape (with the same deformation).

Figure 3 shows the process of dynamic rotation of the skyrmion. From the perspective of the squared electric field strength, the middle bright spot has a higher electric field strength, while the surrounding bright spot has a lower electric field strength. The complete morphology of the skyrmion and topological domain walls is inside the blue dashed lines. The direction of the yellow dashed arrow indicates that the rotation characteristics are sensitive to the positions of the adjusted slits. We did not consider magnetic field vectors here because they do not meet the conditions for SPP to form skyrmions.

The detailed parameters corresponding to skyrmion rotation states are presented in

Table 1. The incident light phase value and slit position correspond to different skyrmion rotation states.

Rotation State	Adjusted Values of Phase	Positions of Slits
Horizontal state $({\theta }_r=0rad)$	$\begin{array}{l}{\phi }_4=0.5\pi ,{\phi }_n=0,\\ \left(n=1,\dots ,12,n\ne 4\right)\end{array}$	4
Small-acute-angle tilted state $({\theta }_r={\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.}rad)$	$\begin{array}{l}{\phi }_3=0.5\pi ,{\phi }_n=0,\\ \left(n=1,\dots ,12,n\ne 3\right)\end{array}$	3
large-acute-angle tilted state $({\theta }_r={\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$3$}\right.}rad)$	$\begin{array}{l}{\phi }_2=0.5\pi ,{\phi }_n=0,\\ \left(n=1,\dots ,12,n\ne 2\right)\end{array}$	2
Right-angled tilted state $({\theta }_r={\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}rad)$	$\begin{array}{l}{\phi }_1=0.5\pi ,{\phi }_n=0,\\ \left(n=1,\dots ,12,n\ne 1\right)\end{array}$	1

. It is worth noting that the parameter θ_r of rotation states indicates the angle between the yellow dashed arrow and the green dashed arrow in Figure 3a–d. The angle between the blue dashed arrow in Figure 3e–h and the direction of the positive X-axis is θ_p, which is the parameter of the direction of the position vector of the regulated slits. We selected slits in different orientations to control the phase difference, which will be crucial. In addition, according to our simulation results, rotations occur in all 12 directions in the two-dimensional plane. Even considering the strong dependence of rotation direction on orientation control, we can further synthesize rotation direction to generate more directions, which provides an interesting discovery for manipulating topological structures in the sub-wavelength scale.

3.2. Dynamic Deformation for Skyrmions

We have discovered that while controlling the rotation direction of the skyrmion, we can independently deform the skyrmion. As shown in Figure 4, we can deform the skyrmion under different rotations by changing the phase difference of the incident light source. This deformation can be explained from the perspective of the skyrmions’ number: skyrmions gradually become standard circles with skyrmions no longer equal to 1 or, more precisely, real numbers less than 1. The detailed results and discussions on the three-dimensional characteristics of the electric field vector of standard skyrmions and the skyrmion number S under the initialization conditions of the simulation experiment are presented in the Appendix A.

It is precisely because of the independence of deformation and rotation that we combined these two methods in Figure 4b and discovered the unique topological structure. At this point, the skyrmion is simultaneously affected by both horizontal and vertical rotations, exhibiting a composite rotation state of inclinations. At the same time, by gradually increasing the phase difference between the two adjusted positions, we found that the skyrmions in the synthesized rotational state were still deformed. To further analyze the rotations in

Table 2. The incident light phase value and slit position correspond to different skyrmion deformation states.

Rotation State	Deformations	Adjusted Values of Phase	Positions of Slits
Horizontal state ${\theta }_r=0rad$	Standard initial-state skyrmions	${\phi }_n=0,\left(n=1,\dots ,12\right)$	/
	Obviously deformed to the right	$\begin{array}{l}{\phi }_1=\pi ,{\phi }_n=0,\\ \left(n=1,\dots ,12,n\ne 1\right)\end{array}$	4
Acute angle tilted state ${\theta }_r={\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.}rad$	Slightly deformed in the upward-right tilt direction	$\begin{array}{l}{\phi }_1={\phi }_4={\displaystyle \frac{1}{6}}\pi ,{\phi }_n=0,\\ \left(n=1,\dots ,12,n\ne 1,4\right)\end{array}$	1, 4
	Deformation in the upward-right tilt direction	$\begin{array}{l}{\phi }_1={\phi }_4={\displaystyle \frac{1}{3}}\pi ,{\phi }_n=0,\\ \left(n=1,\dots ,12,n\ne 1,4\right)\end{array}$	1, 4
	Obvious deformation in the upward-right tilt direction	$\begin{array}{l}{\phi }_1={\phi }_4={\displaystyle \frac{1}{2}}\pi ,{\phi }_n=0,\\ \left(n=1,\dots ,12,n\ne 1,4\right)\end{array}$	1, 4
	Severe deformation in the upward-right tilt direction	$\begin{array}{l}{\phi }_1={\phi }_4=\pi ,{\phi }_n=0,\\ \left(n=1,\dots ,12,n\ne 1,4\right)\end{array}$	1, 4
Right angle tilted state ${\theta }_r={\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}rad$	Obvious upward tilt deformation	$\begin{array}{l}{\phi }_1=\pi ,{\phi }_n=0,\\ \left(n=1,\dots ,12,n\ne 1\right)\end{array}$	1

, especially the acute-angle tilted state, corresponding to Figure 4b, we added the detailed instructions in the Appendix B.

The detailed simulation parameters of the results, as shown in Figure 4, are shown in Table 2. As the phase difference increases, the degree of deformation in Figure 4a,b increases (the topological domain walls gradually open, and breaks appear). By comparing Figure 4a,c, we assign precise rotational states to the skyrmion while completing its deformation. It can be seen that the skyrmion transforms from Figure 4a (the left figure) to Figure 4a (the right figure), and then rotates to Figure 4c. However, this process utilizes the principle of coordinated regulation of deformation and rotation, and we only need one step.

With the gradual changes of phase difference, there were continuous deformations of skyrmions. Assuming that the value of θ_p is (2/3)π rad, more detailed deformations are presented in the Appendix C.

4. Conclusions

The optical skyrmion, as an active new field in nano-photonics, has enormous application prospects in multiple directions, such as polarization sensing [28], spin photonics [9], and high-order topological morphology [29]. Specifically, plasmonic field skyrmions have an application in deformation sensing, possessing the ability to achieve a high sensitivity (>4.82%) and extreme accuracy (~99.99%) in a deep-subwavelength area [30].

We propose a multi-arc symmetric structure that enables the rotation and deformation of SPP electric-field-vector skyrmions. Based on the strong dependence of SPP interference field morphology on the electric field, we discovered and demonstrated the independent and controllable properties of skyrmion deformation and skyrmion rotation state through the linkage control of phase difference and the positions of incident spots in the simulation. Specifically, our results not only enable the deformation of skyrmions in the linear direction, but also endow the deformation process with free rotational states, achieving more diverse patterns. It is worth noting that our rotational states are not limited to the 12 directions, and it can be inferred that more diverse rotation states can be achieved under the condition that more different directions act at the same time, which provides a new idea for freely and flexibly manipulating complex topological forms of skyrmions in the future.