Fig. 1. Schematic illustration of how skyrmion number (Nsk, global information) and skyrmion density (ρsk, local texture) respond differently to external perturbations. First row: under perturbation, the topological texture of the skyrmion beam changes, but Nsk remains unchanged. Second row: at this point, the distribution of ρsk may or may not change, leading to either weak or strong topological protection. Throughout this paper, Stokes fields are depicted using hue to signify azimuthal angle tan μ=s2/s1 and saturation to represent height s3 [15].

Fig. 2. Effects of three types of parameter variations on skyrmions: (a) phase variations, (b) amplitude variations, and (c), (d) polarization variations. Here, Δφ denotes the mode phase difference, α2/α1 represents the mode amplitude ratio, θ is the polarization correlation angle (achieving orthogonal polarization conversion, H/V→R/L), and γ is the inner product of the polarization modes [γ=⟨e1|e2⟩; here |e1⟩=R and |e2⟩=sin(δ)R+cos(δ)L]. The red line represents the change in Nsk under parameter variation, and the blue dashed line represents the change in TSD (SSIM value).

Fig. 3. Experimental measurement setup and measured Stokes vector fields for optical skyrmions under phase-type perturbations. (a) The He–Ne laser (21 mW) generates a Gaussian beam at 632.8 nm. HWP, half-wave plate; GLP, Glan laser polarizer; PBS, polarizing beamsplitter; QWP, quarter-wave plate; QP, Q-plate; M, mirror; LCPR, liquid-crystal phase retarder (LBTEK LCVR-H10C-A), with photograph inset in top right corner; BS, beamsplitter; CCD, charge-coupled device. (b)–(d) Experimental measurements (first column) of the generated skyrmions and their corresponding theoretical forms (second column). The Néel- (b), intermediate- (c), and Bloch- (d) type skyrmions were obtained. The first and second columns show the corresponding four unnormalized Stokes vector components (S0, S1, S2, and S3), as well as the polarization ellipses of the right (red) and left (blue) handedness overlaid on the S0 component (intensity), respectively; and the third column shows the corresponding skyrmionic textures, Nsk≈0.92.

Fig. 4. (a) Experimental measurement setup. (b), (c) Measured Stokes vector field for optical skyrmion under amplitude-type perturbation: spatial distribution of the experimental Stokes vectors, polarization ellipses, and the theoretical skyrmion textures for two amplitude ratios (α2/α1≈0.6 and α2/α1≈1.7), respectively. The corresponding experimental Nsk values for these cases are 1.79 and 1.88.

Fig. 5. Experimental curves of Nsk and ρsk similarity; the data for (a) and (b) are from Figs. 3 and 4, respectively. (a) Variation curves of Nsk and similarity of ρsk under phase-type perturbations, inserted with the vector distribution of the experimental results: Néel- and Bloch-type skyrmions, respectively. (b) Variation curves of Nsk and similarity of ρsk under amplitude-type perturbations, inserted with the vector distribution of the experimental results: α2/α1≈0.58 and α2/α1≈1.73, respectively.

Fig. 6. Local SSIM maps for theoretically calculated ρsk under parameter variation. (a) Reference image of the skyrmion density under unperturbed conditions. (b)–(e) Local SSIM maps under phase-type, amplitude-type, and polarization-type variations, where A/D is 45°/135° linear polarization.

Fig. 7. Effect of amplitude-type variations on the skyrmions for Nsk=2.

Fig. 8. Experimental skyrmion density (a), (c) and local SSIM maps (b), (d) under phase-type and amplitude-type perturbations.

Fig. 9. Calculation regions for (a) Nsk=1 and (b) Nsk=2 in the experiment.