# Simulation Study of Dynamic Rotation and Deformation for Plasmonic Electric Field-Skyrmions
## 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 \(\theta\) 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. \[E_{Z}(d,t)=\sum_{m=1}^{6}E_{Z}^{m,(m+\phi)}(d,t) \tag{1}\]  
where x, y, and z are three Cartesian coordinates. Meanwhile, \(\beta\) signifies the angular coordinates of the column coordinate system, \(\omega\) represents the angular frequency, and \(E_{SPP}\) is the amplitude of the electric field, where the expression of \(E_{0}\) as shown in Equation (2) is an important part of the final expression of the Z-axis component of the SPP electric field. \[E_{0}=E_{SPP}\int\limits_{-\pi/15}^{\pi/15}\exp\biggl{(}\frac{-x\cos\beta-y\sin \beta}{2L}\biggr{)}\exp[i(k_{SPP}x\cos\beta+k_{SPP}y\sin\beta-\omega t)]d\beta \tag{2}\]  
Considering the phase \(\mathbf{\Phi_{\mathrm{m}}}\) of SPP waves generated by each arc-shaped slit [26], the relative polarization directions are \(\mathbf{\Psi_{\mathrm{m}}}\), where the value of \((\mathbf{\Psi_{\mathrm{m}}}-\mathbf{\Psi_{\mathrm{m+6}}})\) is \(-\pi\). In addition, each pair of arc-shaped slit in the symmetric excitation structure carries an additional phase difference \(\pi\)[27]. Therefore, the phase difference \((\mathbf{\Phi_{\mathrm{m}}}-\mathbf{\Phi_{\mathrm{m+6}}})\) of SPP in Equation (3) can be represented by the phase difference \((\mathbf{\varphi_{\mathrm{m}}}-\mathbf{\varphi_{\mathrm{m+6}}})\) of the incident source. \[\phi_{SPP_{\mathrm{m}}}-\phi_{SPP_{(m+6)}}=\varphi_{m}-\varphi_{m+6}+\psi_{m} -\psi_{m+6}+\pi \tag{3}\]  
Thus, the SPP wave propagates over a distance of \(\mathbf{d}\), and the Z-component of the electric field \(\mathbf{E_{\mathrm{Z}}}\) that excites the localized surface at the center of the structure can be expressed as Equation (4):  
\[E_{Z}(d,t)=2\sum_{m=1}^{6}E_{0}\cos\biggl{[}\frac{\phi_{SPP_{\mathrm{m}}}-\phi _{SPP_{(m+6)}}}{2}\biggr{]}\exp\left\{i\left[\frac{\phi_{SPP_{\mathrm{m}}}+ \phi_{SPP_{(m+6)}}}{2}\right]\right\} \tag{4}\]  
We found that the superposition of the SPP electric field depends on the value of \(\mathbf{E_{0}}\) and the phase difference of symmetric pairs of incident light spots in Equation (4).