Skyrmions are topologically stable quasi-particles originally introduced by Skyrme to describe the topological structure of nucleons in high-energy and condensed matter physics [1]. These quasi-particles have been predicted and observed in a wide range of fields, including Bose-Einstein condensates [2,3], spintronics [4], and magnetic media [5–7]. The discretized and particle-like properties of magnetic skyrmions render them robust against perturbations, making them ideal candidates for high-density data storage and low-energy magnetic memory applications [8–10].

As counterparts of magnetic skyrmions in photonics, optical skyrmions have been observed in surface plasmon polaritons (SPPs) or propagating beams in free space recently [11–17], emerging as a frontier topic in optics and photonics [18–23]. With the thriving development of topological photonics, various types of skyrmion textures have been proposed and experimentally generated and manipulated by various means [15]. Notable achievements include topological textures constructed by electromagnetic fields [11], spin angular momentums [12], pseudospin [24], Stokes [25–27], and energy flows [22]. Among them, the Stokes skyrmion demonstrates unique characteristics: the capacity to easily achieve all types of quasiparticles with tunable order and to also propagate in free space, which provides additional degrees of freedom for applications [15,28,29]. These optical quasi-particles, with a variety of topological characteristics, have demonstrated potential applications in modern spin optics, metrology, and topological technologies [12,13,30–33], and offer exciting possibilities for applications in communication and precision photonics [15]. In particular, Stokes skyrmions, with their stable topological properties, have opened up new prospects for applications such as optical communications [16,34], imaging [35], and light-matter interaction [36,37].

The topological stability of skyrmions is critical to their resilience against noise and defects in complex environments, forming a foundation for ultrastable information transfer and storage. However, previous studies have mainly focused on the generation and control of optical skyrmions, while the underlying mechanisms governing their topological stability remain largely unexplored. In particular, most existing analyses rely solely on the surface integral value, the skyrmion number (Nsk), as a global topological indicator [17,20], without sufficiently characterizing the spatial distribution and the resilience of the skyrmion texture under perturbations. Research on the topological stability of optical skyrmions remains in its infancy, yet it holds significant theoretical and practical importance. Moreover, current approaches for generating optical skyrmions often depend on complex and bulky optical setups, like spatial light modulators (SLMs) or digital micromirror devices (DMDs) [14,26], which constrains their integration and scalability. This highlights the need for alternative approaches that enable more compact and tunable generation schemes.

In this work, we first propose the concept of the topological protection degree (TPD) which refers to the robustness of the spatial distribution of topological texture, under the condition that the global skyrmion number (Nsk) remains invariant in external perturbations. To quantitatively assess this robustness, we propose a metric termed the topological similarity degree (TSD) as local topological features, based on the comparison of skyrmion density (ρsk) distributions before and after perturbations. A beam is considered to exhibit strong TPD when TSD>0.8, and weak TPD when TSD≤0.8. Furthermore, we design and demonstrate an electrically controlled generator capable of producing various types of optical skyrmions on demand. Unlike bulky optical setups, we can flexibly achieve the manipulation of different types of skyrmions with only compact conventional devices (e.g., liquid-crystal phase retarders and Q-plates). Finally, we experimentally validate the TPD of paraxial skyrmion beams against complex aberrations. Our work not only provides a foundation for deeper exploration of topological protection of optical skyrmions but also paves the way for their application in information processing and data storage [38]. Since our framework relies solely on the measured or simulated vector field, it is readily transferable to other systems such as magnetic skyrmions [5–7], ferroelectrics [39,40], and even acoustic or water-wave analogues [41–44]. Thus, the TPD concept can serve as a general design principle for robust information carriers and reconfigurable topological devices across photonics, spintronics, and beyond [15].

Optical skyrmions are quasi-particle states of light with unique topological vector textures that can be constructed by customizing the polarization Stokes vectors. A typical class of such Stokes skyrmionic beams can be defined as [25–27] |ψ⟩=α1LG0,0|e1⟩+α2eiΔφLGp,ℓ|e2⟩,(1)where |e1⟩=(1−icos2θ,−isin2θ)T/2 and |e2⟩=(sin(δ)−isin(δ+2θ),cos(δ)+icos(δ+2θ))T/2 are Jones vectors and represent polarization states. Here, δ∈[0,π/2] controls the transition between orthogonal and non-orthogonal polarization states, and θ controls the transition between different orthogonal polarization states, e.g., horizontal/vertical (H/V) to right/left circular (R/L) polarization. LGp,ℓ is the LG mode, with p and l being the radial and azimuthal indices, respectively. Δφ is the phase difference between the orthogonal modes, controlling the helicity textures of the topological quasi-particles. α1 and α2 are constants that control the amplitude ratio of the orthogonal modes. We define γ=⟨e1|e2⟩ as the inner product of the polarization modes (|e1⟩ and |e2⟩).

The topological properties of skyrmions with topological textures can be characterized by the skyrmion number Nsk [6,27]: Nsk=14π∬σs·(∂s∂x×∂s∂y)dxdy.(2)Here, s=[s1,s2,s3]T is the spatial distribution of the normalized Stokes vector, and σ denotes the region to confine the skyrmion, which generally has a finite range. The skyrmion number is an integer representing the number of times the Stokes vectors wrap around the unit sphere. The two-dimensional integral is simplified to a product of polarity and vorticity: Nsk=q·m=−12[cosβ(r)]r=0r=rσ·12π[α(ϕ)]ϕ=0ϕ=2π, where q is the polarity, describing the direction change of the out-of-plane Stokes vector, which corresponds to Stokes vector s3. m is the vorticity, controlling the distribution of the transverse Stokes vector direction. The topological structure of the skyrmion is determined jointly by the polarity, vorticity, and phase difference Δφ of the modes, and it is worth mentioning that the relative mode amplitudes α2/α1 also change the structure. The structural details of topological quasi-particles are characterized by the skyrmion density distribution [15,45]: ρsk(x,y)=s(x,y)·[∂xs(x,y)×∂ys(x,y)].(3)

According to Eqs. (2) and (3), Nsk reflects global information about the topological texture, whereas ρsk describes local topological features. While Nsk typically remains stable under parameter variations, ρsk may undergo significant changes. Therefore, we define the TPD of a skyrmion based on the stability of ρsk, assuming that Nsk remains invariant. The structural similarity index (Section 5 and Ref. [46]), originally developed for image similarity analysis, is here repurposed to characterize the similarity of skyrmion density and is referred to as the topological similarity degree (TSD) throughout this work. A beam under parameter variations is considered to exhibit strong topological protection when the TSD exceeds 0.8, indicating that the skyrmion density remains robust and largely unaffected. TSD=1 signifies no change at all in the spatial features of skyrmion density. Conversely, TSD below 0.8 corresponds to weak topological protection, as illustrated in Fig. 1. The empirical cut-off TSD=0.8 follows common practice in image-quality research, where SSIM>0.8 is generally interpreted as high-fidelity similarity between the test and reference images (see Appendix A for details).

Next, we theoretically analyze the TPDs of skyrmions by introducing three types of parameter variations to the skyrmion beam: amplitude, phase, and polarization. As the reference state for TSD evaluation, we choose the parameters Δφ=0, α2/α1=1, θ=π/4 (R/L), γ=0, p=0, l=−1 in Eq. (1). The phase-type variations are implemented by changing the orthogonal-mode phase difference Δφ in Eq. (1). As Δφ increases from 0 to π/2 and then to π, the skyrmion configuration transitions from Néel-I (△) to Bloch (○) and then to Néel-II (☆). Despite this transformation in texture, both the skyrmion number (red line, Nsk≈0.96) and skyrmion density (blue line, TSD=1) remain unchanged, as illustrated in Fig. 2(a). This indicates that phase-type variations alter only the internal skyrmion structure without affecting its global or local topological features. Therefore, the optical skyrmion exhibits a strong TPD against phase-type variations.

The amplitude-type variations are introduced by changing the orthogonal-mode amplitude ratio (α2/α1) in Eq. (1). We examine the variation of Nsk and TSD with the α2/α1, and the ρsk distributions for three specific amplitude ratios are presented, as shown in Fig. 2(b). As the amplitude ratio α2/α1 approaches zero, the skyrmion beam degenerates into a fundamental Gaussian mode with right circular polarization (non-skyrmion state), where Nsk tends to zero and TSD=0. Raising α2/α1 to 0.5, Nsk rises to 0.87, and the effective region of ρsk expands (△), accompanied by an increase in the TSD to 0.15, marking the gradual recovery of skyrmion topology. As α2/α1 increases further, the global information Nsk quickly saturates (≈1), signaling that the overall topological number is well protected in this region. However, the spatial distribution of ρsk undergoes continuous changes. Notably, at α2/α1=1 (the reference state), the TSD reaches a peak value of one, and the ρsk distribution (○) is centrally concentrated. Pushing α2/α1 still higher causes the ρsk to shrink (☆), so TSD falls and then levels off, as illustrated in Fig. 2(b). Hence, when α2/α1 is in the range of 0.8–1.3, the simultaneous conditions of Nsk>0.9 and TSD>0.8 indicate the formation of strong TPDs. This indicates that a strong TPD only forms in the local area, and the rest are weak TPDs. Moreover, Nsk exhibits higher stability when the vortex-mode component (|e2⟩) is enhanced, compared to variations that amplify the non-vortex mode (|e1⟩).

For polarization-type variations, we analyze the TPD by considering both orthogonal and non-orthogonal polarization-type variations. The orthogonal polarization-type variations are implemented by changing the polarization correlation angle θ from Eq. (1) (H/V→R/L). As θ changes from 0 to π/4, the optical quasiparticle type changes from bimeron (▵, ○) to skyrmion (☆). Notably, this transition is abrupt, with no intermediate topological states. Despite the transformation in topological texture, both the skyrmion number (red line, Nsk≈0.96) and skyrmion density (blue line, TSD=1) remain unchanged, as shown in Fig. 2(c). This indicates that orthogonal polarization-type variations only change the internal texture of the quasiparticles without affecting their global or local topological features. From another perspective, this variation corresponds to a coordinate transformation on the Poincaré sphere. Therefore, optical quasiparticles have a strong TPD against orthogonal polarization-type variations [26,47].

The non-orthogonal polarization-type variations are introduced by changing the inner product of the polarization modes (γ) in Eq. (1). We analyze the variation of Nsk and TSD with the γ, and the ρsk distributions for three specific amplitude ratios are presented, as shown in Fig. 2(d). Because γ alters the degree of non-orthogonality, Nsk remains essentially constant, whereas TSD drops steadily. In this process, the spatial distribution of ρsk continues to change, and the spatial distribution of ρsk varies from the center to off-center, and the effective region gradually contracts (▵ → ☆). When γ<0.6, the joint conditions Nsk>0.9 and TSD>0.8 denote a strong TPD window; outside this interval only weak protection persists. Moreover, the extreme case of non-orthogonal polarization-type variations, i.e., γ=1, degenerates into a uniformly polarized state (non-skyrmion state).

For the above three types of parameter variations, we can clarify that for phase-type and orthogonal polarization-type variations, skyrmions always exhibit strong TPDs, whereas for amplitude-type and non-orthogonal polarization-type variations, strong TPDs are exhibited only in the local domain, with weak TPDs in most of the region. The proposed framework is not limited to conventional skyrmions; it can be extended to other topological quasi-particles such as anti-skyrmions and higher-order skyrmions. Moreover, it provides a general strategy for analyzing various types of perturbations in skyrmion fields [17,20]. This study provides a basis for the use of optical skyrmions in optical communications and photonic computing by analyzing the TPD of optical skyrmions [38].

To experimentally validate the robustness predicted under parameter variations, we built a compact, all-electrical skyrmion generator that combines a liquid-crystal phase retarder with a Q-plate. Compared with conventional methods relying on complex and bulky systems, such as SLM or DMD, our method enables dynamic control and ease of system integration. Based on this, we experimentally analyze the topological protection degree of optical skyrmions by applying phase- and amplitude-type perturbations through electrical control, as shown in Figs. 3 and 4 (detailed in Section 5).

Using this platform, we first generate multiple types of optical skyrmions (Néel-, intermediate-, Bloch-, and anti-type skyrmions), demonstrating true on-demand versatility. For quantitative tests, we then prepare a reference beam with q=1, m=1 and an initial inter-modal phase offset Δφ, as given: |ψ1i⟩=LG0,0|R⟩+eiΔφLG0,−1|L⟩.(4)Δφ=0 serves as the base state for subsequent perturbation studies. Phase-type perturbations are introduced by tuning the voltage of a liquid-crystal phase retarder (LCPR), which directly modulates Δφ between the two orthogonal modes. We realized the initial Néel-type skyrmion with Δφ=0 and then continuously adjusted the voltage of the LCPR to apply a phase-type perturbation to the skyrmion beam. Starting from the Néel-type (Δφ=0), the voltage is swept continuously; this drives the Stokes vectors S1 and S2 to rotate by π/2 in the S1−S2 plane while S3 remains unchanged. The optical texture therefore evolves smoothly from Néel-type through an intermediate-type (Δφ=π/4) to the Bloch configuration (Δφ=π/2), as illustrated in Fig. 3. This controlled evolution allows us to verify the predicted strong topological protection under phase-type perturbations.

Then, in order to analyze the degrees of topological protection of the optical skyrmion under amplitude-type perturbations, we construct the experimental setup shown in Fig. 4(a), which leads to the following skyrmion configuration (ideal Nsk=2) [48–50]: |ψφ⟩=cosφ2LG0,0|R⟩+sinφ2eiπ/2LG0,−2|L⟩,(5)where φ is the phase retardation of the LCPR. The amplitude ratio of two orthogonal modes can be precisely modulated by the phase retardation φ of the LCPR, where the amplitude ratio of the orthogonal modes α2/α1=tan(φ/2). By applying different voltages to the LCPR via the signal generator, a transition of φ from 0 to π can be realized, which gives an amplitude-type perturbation to the skyrmion beam. Under the amplitude-type perturbation, the Stokes vector does not undergo any rotation; only the relative intensity changes, and the skyrmion type does not change, but the size of the skyrmion center region is altered, as shown in Fig. 4. It could be noted that the small difference in phase curvature between the two modes is causing the spiraled lobes in the Stokes parameters seen in the results [51].

Having established controlled generation and tunable perturbations in Figs. 3 and 4, we now quantify how robust the skyrmion textures remain. Post-processing the measured Stokes data yields the global skyrmion number (Nsk) and the local similarity metric (TSD) as functions of the applied phase- and amplitude-type perturbations; the resulting curves are compiled in Fig. 5. With pure phase-type perturbations (Fig. 3), both the skyrmion number (red line, Nsk≈0.92) and skyrmion density (blue line, TSD≈0.87) remain stable, as shown in Fig. 5(a). Both values exceed the Nsk>0.9 and TSD>0.8 benchmarks, placing the beam squarely in the strong TPD regime—exactly as predicted by the simulations in Fig. 2(a). Amplitude-type perturbations (Fig. 4) produce a distinctly different trend. As the vortex-to-non-vortex ratio α2/α1 is raised from zero, both metrics climb: at α2/α1≈0.6 the skyrmion number reaches Nsk≈1.76 while TSD is still modest (≈0.65). With a further increase to α2/α1=1 (the reference state), the TSD reaches a peak of 1, Nsk≈1.90. As α2/α1 continues to increase, it causes TSD to decrease (finally stabilized), and Nsk remains nearly constant. In the experiment, when α2/α1 is in the range of 0.77–1.4, the conditions of Nsk>1.8 and TSD>0.8 indicate the formation of strong TPDs windows. Therefore, amplitude-type perturbations (Fig. 4) yield strong protection only in the narrow window. Outside this interval, TSD drops below the 0.8 threshold even though Nsk remains high. This measured evolution is plotted in Fig. 5(b) and mirrors the theoretical curve of Fig. 7 (see Appendix B for details).

We experimentally verify the TPD of optical skyrmions against perturbations, both phase- and amplitude-type perturbations. These results confirm the theoretical prediction that skyrmions exhibit strong TPD under phase-type perturbations, whereas amplitude-type perturbations support strong TPD only within a limited parameter window. Our work provides new insights into the topological robustness of optical skyrmions that may be important for the skyrmion to move towards applications. The compact, voltage-controlled platform can offer a practical route toward integrating topologically resilient skyrmion beams into signal routing, high-density data storage, and other photonic applications. It is also worth noting that beyond the scheme adopted in this work, there are also many other emerging directions for exploring compact Stokes texture generators, such as gradient index lenses [45], liquid-crystal flat-optics [52], structured light solid-state lasers [53,54], spin-orbit wave plates [55], and on-chip microlaser and microring emitter technologies [56,57].

We propose a comprehensive framework for characterizing and controlling the topological protection of optical skyrmions. By introducing the concept of TPD and the corresponding quantitative metric, TSD, we establish a new perspective for evaluating the robustness of local topological textures under parameter variations or external perturbations. Through both theoretical modeling and experimental validation, we demonstrate that skyrmions can exhibit distinct protection behaviors, classified as strong TPDs or weak TPDs. It is worth mentioning that to enable flexible control and real-world implementation, we develop an electrically tunable skyrmion generator based on compact optical components, overcoming the limitations of bulky diffractive systems. Beyond optical skyrmions, the proposed framework can be extended to other topological photonic structures (e.g., anti-skyrmions, higher-order skyrmions), making it a broadly applicable tool for probing and harnessing topological robustness in complex optical fields.

The emerging field of free-space topological optical textures opens new directions of topologically protected information transfer [28]. In this paper, we do not claim universal robustness of optical skyrmions to every perturbation; rather, we define their topological protection and establish a corresponding framework that can be further expanded [15,17]. Table 1 summarizes the parameter variations corresponding to the actual perturbations and their effects on the skyrmion topology. (i) Modulating the phase difference Δφ merely rotates the entire Stokes vector components, changing the topological texture while leaving both ρsk and Nsk intact. (ii) Adjusting the mode amplitude ratio will result in a change in the Stokes vector magnitude, reshaping the texture and ρsk while Nsk remains unaffected. (iii) Under an orthogonal polarization basis transformation, the transposition of Stokes vectors [e.g., from (S1, S2, S3) to (S3, S1, S2)] leads to polarization texture transition without affecting topological properties, ρsk, and Nsk. (iv) Under the transformation of the non-orthogonal polarization basis, this leads to a complete change in the Stokes vector, thus causing a change in the polarization texture and ρsk, but the Nsk is similarly protected. We emphasize that Nsk, as the most universal global topological quantity, can effectively reflect the topological stability of the system [15,17,20]. For more precise cases (e.g., characterizing structural details), it is a better scheme to study the skyrmion density, which provides a more comprehensive metric for evaluating topological protection during transformations or perturbations.Table 1.

Parameter Variations Corresponding to Different Types of Perturbations and Their Effects on Skyrmions

Our results provide both theoretical and experimental foundations for advancing the robustness, tunability, and functionalization of topological skyrmion light and open new possibilities for applications in optical information processing, storage, and topological photonic devices [15,28]. We validate our proposed topological protection degree theory using skyrmions generated by LG modes, but our theory can be extended to skyrmions generated by Bessel [47,58], Airy [59,60], and perfect vortex modes [61]. Because the method relies solely on the measured (or simulated) vector field, it can be ported to magnetic skyrmions [5–7], ferroelectric [39,40], or even for acoustic and water-wave analogues [41–44], wherever topological robustness must be quantified. We therefore expect the TPD concept to guide the design of durable information carriers and reconfigurable topological devices across photonics, spintronics, and beyond.

The state of protection under parameter variation or external perturbation can be quantified as follows: reference image (the original skyrmion density map, ρnorm) and tested image (the disturbed skyrmion density map, ρsk). The luminance, contrast, and structure of the two images are compared using the SSIM algorithm [46]: SSIM(ρsk,ρnorm)=(2μskμnorm+C1)(2ρsk,norm+C2)(μsk2+μnorm2+C1)(σsk2+σnorm2+C2),(6)where μsk and μnorm are the local means of the perturbed and unperturbed skyrmion density images, respectively, σsk and σnorm represent their standard deviations, ρsk,norm is the cross-covariance, and C1 and C2 are constants to stabilize the SSIM calculation by mitigating noise oversensitivity.

We can construct skyrmions by modulating and combining two orthogonal basis state LG0,0 and LGp,ℓ modes with relative phases. The experimental setups shown in Figs. 3(a) and 4(a) enable electrically driven skyrmion generation. The experimental setup for phase-type variations consists of two optical paths. First, a He–Ne laser generates a Gaussian beam with a wavelength of 632.8 nm. The intensity and polarization state of the beam are adjusted using a half-wave plate (HWP) and a Glan laser polarizer (GLP), as shown in Fig. 3(a). Subsequently, the two beams passing through the polarizing beamsplitter are modulated to produce Gaussian and LG modes with orthogonal polarizations (either linear or circular) and relative phase differences. Different voltages applied to the liquid-crystal phase retarder (LCPR) control the phase difference between the two modes. The quarter-wave plate (QWP) is used to achieve mutual conversion between linear and circular polarization, while the Q-plate is utilized for generating LG modes, which can be given by the Jones matrix $$\begin{gathered} M=MQWP(θ)MQP(ϕ)MQWP(θ) \\ =[−2icos(2θ−ϕ)−cos(4θ−ϕ)+cos(ϕ)−sin(4θ−ϕ)+sin(ϕ)−sin(4θ−ϕ)+sin(ϕ)−2icos(2θ−ϕ)+cos(4θ−ϕ)−cos(ϕ)]. \end{gathered}$$(7)To generate the skyrmion beam, we set QWP1 and QWP2 at 45° relative to their fast axes. This configuration enables the superposition of a pair of orthogonal linearly polarized LG modes, which are subsequently converted into opposite circular polarization states via QWP3, resulting in the formation of a skyrmion state.

The experimental setup for amplitude-type variations consists of one optical path, as shown in Fig. 4(a). The Gaussian beam is output through the He–Ne laser. Next, the GLP and the QWP convert the linearly polarized beam into a circularly polarized one. The beam then passes through the first Q-plate, the LCPR, and the second Q-plate (similar to Ref. [48]), thus forming skyrmion states.

Finally, the spatial distribution of the Stokes vectors S0, S1, S2, and S3 is obtained by capturing the linearly and circularly polarized components of the combined QWP and GLP rotations with a charge-coupled device (CCD). When the optical axes of the QWP and GLP make up the following combinations, what we get in the CCD are exactly the parameters [62]: $$\begin{gathered} S0=I(0,0)+I(π/2,π/2), \\ S1=I(0,0)−I(π/2,π/2), \\ S2=I(π/4,π/4)−I(−π/4,−π/4), \\ S3=I(π/4,0)−I(−π/4,0), \end{gathered}$$(8)where I(α,β) represents the intensity obtained when the optical axis of the QWP and the polarization direction of the GLP are α and β relative to the x-axis, respectively. Other dynamic Stokes polarimetry methods can also be applied to achieve higher precision and convenience in measurements [51].