Skyrmions, initially introduced by the physicist from Britain, Tony Skyrme, are quasiparticles in condensed matter physics with topological characteristics [1]. Typically found in magnetic materials, they can be envisioned as localized, vortex-like spin textures within the material. Skyrmions have attracted considerable interest due to their potential applications in high-density magnetic information storage, transmission, and spintronic devices [2–8]. Additionally, there is a growing field of research into optical skyrmions, which are linked to light manipulation and have implications for various optical technologies. Recent advances in topological optics have spurred rapid growth in the field of optical skyrmions, providing new perspectives and techniques for achieving topological characteristics in structured light, space-time light, and light–matter interaction [9–12]. The applications encompass spin optics, imaging and metrology, optical power, structured light, topology, and quantum technologies [13–15]. However, the control of various topologies of optical skyrmions remains a nascent and evolving area of study.

The concept of optical skyrmions initially emerged in the context of electric fields of evanescent waves. In 2018, Tsesses et al. expanded this concept to electromagnetic fields through the utilization of hexagonal resonant cavity interference [16]. Du et al. demonstrated the generation of spin vector optical Neel skyrmions by harnessing the spin-orbit coupling of a tightly focused vector beam to create evanescent waves on metal surfaces [17]. Subsequent reports [18–20] have described mapping and manipulation efforts related to plasmonic skyrmions. Additionally, various techniques, including Dirac monopoles in momentum space [21], microcavities [22], pseudospins in 3D nonlinear photonic crystals [23], Raman transitions [24], and electric or magnetic field vectors in spatiotemporal optical pulses [25], have been employed to construct optical skyrmions. Analogous quasiparticles were later observed to exist in the momentum space of liquid crystal-filled microcavities [26], as well as in evanescent optical vortex lattices governed by symmetry [27], photonic crystals [28], and SPPs in an Archimedean coupling structure [29]. These quasiparticles are also known as merons or half-skyrmions. However, it should be noted that when these skyrmions propagate, they are unable to maintain topological stability.

One distinct characteristic of Stokes skyrmions is their topological stability throughout propagation. In homogeneous media, the vector texture can transition from Bloch type to Néel type and vice versa while maintaining the skyrmion number [30]. Conversely, the vector field texture remains constant throughout propagation when the texture is anti-skyrmion [31]. Specifically, adjustable Stokes skyrmions with Néel, Bloch, and anti-types textures can be created using the digital holographic approach [32]. Theoretically, it is possible to generate Stokes skyrmions from a microlaser when the skyrmion levels are controlled by the microresonator grating [22]. Optical Stokes skyrmions in the Bessel profile have the non-diffracting property [33]. The topological method is used to calculate Stokes skyrmion numbers that avoid products of polarization gradients and significantly increase precision and accuracy [34]. The direct imprint of the topologically protected polarization textures of optical Stokes skyrmions onto a material is demonstrated in the experiment [35]. Non-local quantum Stokes skyrmions are observed in two entangled photons with engineered quantum states [36]. However, the skyrmion is immobile transversely. The hydrodynamic Magnus force acting on the skyrmionic topological charge also induces a transverse skyrmion deflection, commonly known as the skyrmion Hall effect (SkHE) [5,37]. If the repulsive force produced at the edge is sufficiently strong, the skyrmion will smoothly travel along the edge. However, it can be destroyed if the current is strong enough to push the skyrmion close to the edge. Therefore, precise control over SkHE is essential before skyrmions can be extensively utilized in spintronic applications. One proposed approach to address this issue is the use of artificial antiferromagnets, or antiferromagnets, in which two linked skyrmions with opposing charges have a net topological number of zero [38,39]. However, these methods are limited to manipulating the zero topological number, making it difficult to effectively alter the SkHE. One potential method of accurately controlling the SkHE is using temperature variations to regulate the angular momentum of ferrimagnets [40]. Meanwhile, the optical skyrmion Hall effect, which can precisely modify optical SkHE and has potential applications in logic, synaptic computing, and analog memory, has received less attention. The symmetric and asymmetric manipulation of skyrmions offers an extra degree of freedom for their generation and control, thereby paving the way for the development of novel skyrmion-controlled nanophotonic devices.

In this study, we utilized a hybrid multi-zone filter to achieve Hall effect-like splitting of optical Stokes skyrmions (HESSs), enabling effective separation and manipulation of these quasiparticles. We employed azimuthally polarized light with uniform amplitude and phase as the incident vector beam. The incident phase was shaped using two multi-zone filters and two spiral phase plates with opposite topological charges. By adjusting the horizontal parameters, we controlled the skyrmion’s position, inducing symmetric or asymmetric HESSs. Through simulation and experimental validation, we successfully demonstrated both symmetric and asymmetric skyrmion Hall-like splitting. This work highlights the emergence of complex vector field structures representing optical quasiparticles with diverse topological characteristics. These advancements bear significant implications across various fields, including modern spin optics, structured light, and topological and quantum technologies.

Figure 1 illustrates the generation of the Hall effect-like splitting of optical Stokes skyrmions. After modulation by multi-ring phase, azimuthally polarized light generates two split high-order skyrmions. In fact, azimuthally polarized light can be seen as a superposition of azimuthally polarized light with a topological charge of $+1$ and azimuthally polarized light with a topological charge of $-1$. These two optical fields respectively generate high-order skyrmions with skyrmion numbers of $-2$ and $+2$. Therefore, the incident optical field can be considered as two overlapping high-order skyrmions with opposite skyrmion numbers. The multi-ring phase is composed of two types of multi-zone filters, which carry helical phases with topological charges of $+1$ and $-1$, respectively. If these two ring phases are coaxial, they will lead to the annihilation of skyrmions with opposite skyrmion numbers. To prevent the disappearance of skyrmions, we superimpose different gratings onto the two types of multi-ring phases, causing the skyrmions to be symmetrically or asymmetrically deflected to different positions. This enables the coexistence of the two types of skyrmions. Therefore, the incident optical field, which contains two overlapping skyrmions with opposite skyrmion numbers, will lead to the splitting of skyrmions after passing through the ring phase. We refer to this phenomenon as the HESS.

The tightly focused incident light field is azimuthally polarized light with uniform amplitude and phase $\phi$. $\phi$ is based on two multi-zone filters (MZFs), two spiral phase plates with reversed topological charges, and two translation factors, which can be calculated by Eq. (1). Transmissions of MZF1, MZF2, and circular filter (CF) are calculated by Eqs. (2)–(4), respectively: $$\phi =(l_1\theta +{\beta }_{1}kx/f)·f_{\mathrm{MZF}1}+(l_2\theta +{\beta }_{2}kx/f)·f_{\mathrm{MZF}2},$$(1)$$f_{\mathrm{MZF}1}=\{\begin{array}{cc}1& (i-1)R/N\le r\le (i-1+\alpha )R/N\\ 0& \mathrm{else}\end{array},$$(2)$$f_{\mathrm{MZF}2}=f_{\mathrm{CF}}-f_{\mathrm{MZF}1},$$(3)$$f_{\mathrm{CF}}=\{\begin{array}{cc}1& 0\le r\le R\\ 0& \mathrm{else}\end{array},$$(4)where the parameters $l_1$ and ${\beta }_1$ correspond to the topological charge of the spiral phase plate and the horizontal adjusted parameter representing the multi-zone width in the MZF1 region of the translation factor, respectively. $k=2\pi /\lambda$, $x$ is the horizontal coordinate, and $f$ is the focal length of the high-NA objective lens and set as 2 mm. The parameters $l_2$ and ${\beta }_2$ correspond to the topological charge of the spiral phase plate and the horizontal adjusted parameter representing the multi-zone width in the MZF2 region of the translation factor, respectively. $R$ is the radius of the phase region. $i$ represents the serial number of the transmission area in the MZF. $N$ represents the number of transparent zones in the MZF. In this paper, $l_1$, $l_2$, and $N$ are set as 1, $-1$, and 4, respectively. $\alpha$ adjusts the width of the transparent zone to adjust the intensities of left and right tight focused foci. To achieve two equally intense tightly focused foci, we employ the intensity non-uniformity $\epsilon =\sum _{i=1}^2|I_{\mathrm{X}\mathrm{P}i}-\overline{I_{\mathrm{X}\mathrm{P}}}|/\overline{I_{\mathrm{X}\mathrm{P}}}$ of the $x$ polarized light components as the evaluation criterion. Here, $I_{\mathrm{X}\mathrm{P}i}$ represents the intensity peak of the $i$th $x$- polarized light component, and $\overline{I_{\mathrm{X}\mathrm{P}}}$ is the average intensity of the left and right $x$-polarized light components. Through simulations, we find that when $\alpha$ is set to 0.568, the two horizontal polarized light components exhibit equal intensities. This indirectly confirms the balanced strength of the two tightly focused focal points. Furthermore, in this paper, we set the following parameter values: $l_1=1$, ${\beta }_1=9$, $l_2=-1$, and ${\beta }_2=-9$.

Based on the Richards–Wolf formulas, the calculation expression of the electric and magnetic fields in close proximity to the focus is as follows: $$\begin{gathered} \mathbf{U}(\rho ,\psi ,z)=-\frac{if}{\lambda }{\int }_0^{\alpha }{\int }_0^{2\pi }B(\theta ,\phi )T(\theta )\mathbf{P}(\theta ,\phi ) \\ \exp \{ik[\rho \sin \theta \cos (\phi -\psi )+z\cos \theta ]\}\sin \theta \mathrm{d}\theta \mathrm{d}\phi . \end{gathered}$$(5)

The variables are denoted as follows: $\mathbf{U}(\rho ,\psi ,z)$ represents the electrical or magnetic field at the $z$ plane, and $(\rho ,\psi )$ shows the polar coordinates at this focusing plane; $f$ signifies the focal length; $\lambda$ represents the wavelength; $\alpha$ is determined by the numerical aperture of the lens ($\mathrm{NA}=\sin \alpha$); $B(\theta )=\exp (\frac{-{\gamma }^2{\sin }^2\theta }{{\sin }^2\alpha })$ in Eq. (5) signifies the amplitude of the input field, which is the Gaussian beam in the paper; $\theta$ and $\phi$ represent the polar angle and azimuthal angle at the incident plane, respectively; $T(\theta )$ denotes the apodization function; $\mathbf{P}(\theta ,\phi )$ signifies the polarization matrix for the electric and magnetic fields; and $k=2\pi /\lambda$ represents the wavenumber: $$\begin{gathered} \mathbf{P}(\theta ,\phi )=\left[\begin{array}{c}1+{\cos }^2\phi (\cos \theta -1)\\ \sin \phi \cos \phi (\cos \theta -1)\\ -\sin \theta \cos \phi \end{array}\right]a(\theta ,\phi ) \\ +\left[\begin{array}{c}\sin \phi \cos \phi (\cos \theta -1)\\ 1+{\sin }^2\phi (\cos \theta -1)\\ -\sin \theta \sin \phi \end{array}\right]b(\theta ,\phi ), \end{gathered}$$(6)where $a(\theta ,\phi )$ and $b(\theta ,\phi )$ represent the $x$- and $y$-components of the incident beam, respectively.

The incident light field $\mathbf{E}$ is an $m$th order azimuthally polarized vortex Gaussian beam with a topological charge $l$, $$\mathbf{E}=B(\theta ,\phi )\exp (il\phi )\left[\begin{array}{c}\cos (m\phi +\frac{\pi }{2})\\ \sin (m\phi +\frac{\pi }{2})\end{array}\right].$$(7)

The $x$, $y$, and $z$ components $E_x$, $E_y$, and $E_z$ of the electric field in the tightly focused field are as follows: $$\begin{gathered} E_x=\frac{i^{l-m-1}}{2}{e}^{i(l\phi -m\phi -\frac{\pi }{2})}(I_{0,l-m}+e^{i2\phi }{I}_{2,l-m+2}) \\ +\frac{i^{l+m-1}}{2}{e}^{i(l\phi +m\phi +\frac{\pi }{2})}(I_{0,l+m}+e^{-i2\phi }{I}_{2,l+m-2}), \\ {E}_y=\frac{i^{l-m}}{2}{e}^{i(l\phi -m\phi -\frac{\pi }{2})}(I_{0,l-m}-e^{i2\phi }{I}_{2,l-m+2}) \\ +\frac{i^{l+m}}{2}{e}^{i(l\phi +m\phi +\frac{\pi }{2})}(-I_{0,l+m}+e^{-i2\phi }{I}_{2,l+m-2}), \\ {E}_z=-i^{l-m}{e}^{i(l\phi -m\phi -\frac{\pi }{2})}(e^{i\phi }{I}_{1,l-m+1}) \\ -i^{l+m}{e}^{i(l\phi +m\phi +\frac{\pi }{2})}(-e^{-i\phi }{I}_{1,l+m-1}). \end{gathered}$$(8)

The Stokes parameter $S_3$ is calculated by the difference in intensity between left and right circularly polarized electric fields $E_L$ and $E_R$: $$\begin{gathered} E_L=\frac{1}{\sqrt{2}}(E_x+i{E}_y), \\ {E}_R=\frac{1}{\sqrt{2}}(E_x-i{E}_y), \\ {S}_3={|E_L|}^2-{|E_R|}^2 \\ =\frac{1}{2}({|I_{0,l+m}+I_{2,l-m+2}|}^2-{|I_{0,l-m}+I_{2,l+m-2}|}^2), \end{gathered}$$(9)where $$\begin{gathered} I_{0,\nu }=\left(\frac{\pi f}{\lambda }\right){\int }_0^{\alpha }\sin \theta {\cos }^{1/2}\theta (1+\cos \theta )\times A(\theta )e^{ikz\cos \theta }{J}_{\nu }(x)\mathrm{d}\theta , \\ {I}_{1,\nu }=\left(\frac{\pi f}{\lambda }\right){\int }_0^{\alpha }{\sin }^2\theta {\cos }^{1/2}\theta A(\theta )e^{ikz\cos \theta }{J}_{\nu }(x)\mathrm{d}\theta , \\ {I}_{2,\nu }=\left(\frac{\pi f}{\lambda }\right){\int }_0^{\alpha }\sin \theta {\cos }^{1/2}\theta (1-\cos \theta )\times A(\theta )e^{ikz\cos \theta }{J}_{\nu }(x)\mathrm{d}\theta . \end{gathered}$$(10)

In Eq. (10), $x=kr\sin \theta$, $r$ is the polar radius, $J_{\nu }(x)$ is the $\nu$th order Bessel function, and $A(\theta )=B(\theta ,\phi )$. When ($l=1$, $m=1$), there is $S_3=\frac{1}{2}({|I_{0,2}+I_{2,2}|}^2-{|I_{0,0}+I_{2,0}|}^2)$. Since $J_0(0)\text{=}1$ and $J_2(0)\text{=}0$, for $r=0$, $S_3<0$. When ($l=-1$, $m=1$), there is $S_3=\frac{1}{2}({|I_{0,0}+I_{2,0}|}^2-{|I_{0,-2}+I_{2,-2}|}^2)$. Since $J_0(0)\text{=}1$ and $J_{-2}(0)\text{=}0$, for $r=0$, $S_3>0$. Therefore, in these two cases, the value of $S_3$ is opposite at the center point when $r=0$. Since the direction of a single vector in skyrmion is determined by the Stokes parameters $S_1$, $S_2$, and $S_3$, at the center point where $S_1$ and $S_2$ are both 0, the vector direction of the skyrmion is only determined by $S_3$. Therefore, the skyrmions in these two cases are antiparallel, resulting in opposite skyrmion numbers. If the centers of the two types of skyrmions coincide, they would annihilate. To generate the HESS, we achieve the coexistence of the two types of skyrmions by superposing different offset gratings to shift the centers of the skyrmions away from the focal plane center to different positions.

We will demonstrate the symmetric HESS. ${\beta }_1$ and ${\beta }_2$ are assigned values of 9 and $-9$, respectively, to generate the symmetric skyrmions. The experimental setup is presented in Appendix A. Figures 2(a1)–2(e1) present the related information of the left sections depicted in Figs. 5(a1)–5(f1) (see Appendix B). It should be noted that Appendix C offers magnified views of the intensity profiles depicted in Appendix B. Figures 2(a1)–2(c1) show the numerical Stokes parameters $S_1$, $S_2$, and $S_3$, respectively. The reconstruction of the optical skyrmions was achieved through Stokes polarimetry, more specifically, through the reconstruction of the Stokes parameters $S_0$, $S_1$, $S_2$, and $S_3$, which were computed from a set of intensity measurements. A CMOS camera allowed measurement of the required intensities, from which the Stokes parameters were reconstructed using the relations $S_1=I_{\mathrm{X}\mathrm{P}}-I_{\mathrm{Y}\mathrm{P}}$, $S_2=I_{\mathrm{L}+45\mathrm{P}}-I_{\mathrm{L}-45\mathrm{P}}$, and $S_3=I_{\mathrm{LCP}}-I_{\mathrm{RCP}}$. Figure 2(d1) presents the 3D vector field, and Fig. 2(e1) presents the 2D top view. All arrows are represented with the color-labeled $S_3$ component. The topological properties of the skyrmion texture can be characterized by the skyrmion number $n$, with $$n=\frac{1}{4\pi }\iint \mathbf{M}·(\frac{\partial \mathbf{M}}{\partial x}\times \frac{\partial \mathbf{M}}{\partial y})\mathrm{d}x\mathrm{d}y,$$(11)where $\mathbf{M}$ describes the normalized Stokes vector field, and the integral is taken over the region where it is confined. For a skyrmion texture, $n$ is an integer that describes the number of times the vector wraps around a unit sphere. It can be found that when the radius is $r_0=438.75\mathrm{nm}$, $n$ reaches the maximum value of 1.99267 close to 2. This matches the same radius of the dashed circles in Figs. 2(a1)–2(c1) and 2(e1). Thus, the left skyrmion is designated as the higher-order skyrmion with the positive skyrmion number. Figures 2(a2)–2(e2) present the related information of the left sections depicted in Figs. 5(a2)–5(f2). Figures 2(a2)–2(c2) show the corresponding experimental Stokes parameters $S_1$, $S_2$, and $S_3$, respectively. Figure 2(d2) presents the 3D vector field of the experimental Stokes skyrmion, and Fig. 2(e2) presents the corresponding 2D top view. It can be found that when the radius is $r_0=172.5\mathrm{\mu m}$, the measured $n$ reaches the maximum value of 1.91905 close to 2. This matches the same radius of the dashed circles in Figs. 2(a2)–2(c2) and 2(e2). Thus, we prove in the experiment that the left skyrmion is designated as the higher-order skyrmion with the positive skyrmion number. Figures 2(a3)–2(e3) present the related information of the right sections depicted in Figs. 5(a1)–5(f1). Figures 2(a3)–2(c3) show the corresponding numerical Stokes parameters $S_1$, $S_2$, and $S_3$, respectively. Figure 2(d3) presents the 3D vector field of the numerical Stokes skyrmion, and Fig. 2(e3) presents the corresponding 2D top view. When the radius is $r_0=495.625\mathrm{nm}$, the measured $n$ reaches the minimum value of $-1.99436$ close to $-2$. This matches the same radius of the dashed circles in Figs. 2(a3)–2(c3) and 2(e3). Thus, we prove in the simulation that the right skyrmion is designated as the higher-order skyrmion with the negative skyrmion number. Figures 2(a4)–2(e4) present the related information of the right sections depicted in Figs. 5(a2)–5(f2). Figures 2(a4)–2(c4) show the corresponding experimental Stokes parameters $S_1$, $S_2$, and $S_3$, respectively. Figure 2(d4) presents the 3D vector field of the experimental Stokes skyrmion, and Fig. 2(e4) presents the corresponding 2D top view. It can be found that when the radius is $r_0=186.3\mathrm{\mu m}$, the measured $n$ reaches the minimum value of $-1.84496$ close to $-2$. This matches the same radius of the dashed circles in Figs. 2(a4)–2(c4) and 2(e4). Thus, we prove in the experiment that the right skyrmion is designated as the higher-order skyrmion with the negative skyrmion number. Therefore, both our experiments and simulations have shown that the left and right symmetric skyrmions have opposite skyrmion numbers, thereby forming the symmetric HESS.

Next, we will demonstrate the asymmetric HESS. ${\beta }_1$ and ${\beta }_2$ are assigned values of 9 and $-15$, respectively, to generate the asymmetric skyrmions. Figures 3(a1)–3(e1) present the related information of the left sections depicted in Figs. 6(a1)–6(f1) (see Appendix B). Figures 3(a1)–3(c1) show the numerical Stokes parameters $S_1$, $S_2$, and $S_3$, respectively. The reconstruction of the optical skyrmions was achieved through Stokes polarimetry, more specifically, through the reconstruction of the Stokes parameters $S_0$, $S_1$, $S_2$, and $S_3$, which were computed from a set of intensity measurements. Figure 3(d1) presents the 3D vector field, and Fig. 3(e1) presents the 2D top view. All arrows are represented with the color-labeled $S_3$ component. It can be found that when the radius is $r_0=438.75\mathrm{nm}$, $n$ reaches the maximum value of 1.99334 close to 2. This matches the same radius of the dashed circles in Figs. 3(a1)–3(c1) and 3(e1). Thus, the left skyrmion is designated as the higher-order skyrmion with the positive skyrmion number. Figures 3(a2)–3(e2) present the related information of the left sections depicted in Figs. 6(a2)–6(f2). Figures 3(a2)–3(c2) show the corresponding experimental Stokes parameters $S_1$, $S_2$, and $S_3$, respectively. Figure 3(d2) presents the 3D vector field of the experimental Stokes skyrmion, and Fig. 3(e2) presents the corresponding 2D top view. It can be found that when the radius is $r_0=169.05\mathrm{\mu m}$, the measured $n$ reaches the maximum value of 1.92691 close to 2. This matches the same radius of the dashed circles in Figs. 3(a2)–3(c2) and 3(e2). Thus, we prove in the experiment that the left skyrmion is designated as the higher-order skyrmion with the positive skyrmion number. Figures 3(a3)–3(e3) present the related information of the right sections depicted in Figs. 6(a1)–6(f1). Figures 3(a3)–3(c3) show the corresponding numerical Stokes parameters $S_1$, $S_2$, and $S_3$, respectively. Figure 3(d3) presents the 3D vector field of the numerical Stokes skyrmion, and Fig. 3(e3) presents the corresponding 2D top view. When the radius is $r_0=495.625\mathrm{nm}$, the measured $n$ reaches the minimum value of $-1.99332$ close to $-2$. This matches the same radius of the dashed circles in Figs. 3(a3)–3(c3) and 3(e3). Thus, we prove in the simulation that the right skyrmion is designated as the higher-order skyrmion with the negative skyrmion number. Figures 3(a4)–3(e4) present the related information of the right sections depicted in Figs. 6(a2)–6(f2). Figures 3(a4)–3(c4) show the corresponding experimental Stokes parameters $S_1$, $S_2$, and $S_3$, respectively. Figure 3(d4) presents the 3D vector field of the experimental Stokes skyrmion, and Fig. 6(e4) presents the corresponding 2D top view. It can be found that when the radius is $r_0=182.85\mathrm{\mu m}$, the measured $n$ reaches the minimum value of $-1.86148$ close to $-2$. This matches the same radius of the dashed circles in Figs. 3(a4)–3(c4) and 3(e4). Thus, we prove in the experiment that the right skyrmion is designated as the higher-order skyrmion with the negative skyrmion number. Therefore, both our experiments and simulations have shown that the left and right asymmetric skyrmions have opposite skyrmion numbers, thereby forming the asymmetric HESS.

Appendix D contains information about the polarization and Stokes vector distributions before and after splitting. Appendix E presents the polarization ellipses distribution for each beam spot shown in Figs. 2 and 3. Two tilted phase profiles with ${\beta }_1$ and ${\beta }_2$ create two distinct separation channels for the incident beam. The topological charges of the spiral phases in the two channels are $+1$ and $-1$, respectively. Conventional azimuthally polarized light can be considered as a superposition of azimuthally polarized light with topological charges of $+1$ and $-1$. Hence, azimuthally polarized light with topological charges of $+1$ and $-1$ can generate high-order skyrmions with skyrmion numbers of $-2$ and $+2$, respectively. For the separation channel with a spiral phase of topological charge $+1$, when azimuthally polarized vortex beams carrying topological charges of $+1$ and $-1$ pass through this channel, the topological charge of the azimuthally polarized vortex beam with charge $-1$ will be neutralized to 0. As a result, after passing through the spiral phase with a topological charge of $+1$, the azimuthally polarized vortex beams with topological charges of $+1$ and $-1$ are transformed into azimuthally polarized vortex beams with topological charges of $+1$ and 0. Although the azimuthally polarized vortex beam with a topological charge of 0 can be seen as a superposition of two high-order skyrmions with opposite skyrmion numbers, when these two skyrmions are not separated, the skyrmions with opposite skyrmion numbers will cancel each other out, resulting in no visible skyrmion structure overall. Therefore, the azimuthally polarized vortex beam with topological charges of $+1$ and 0 will only produce high-order skyrmions with skyrmion numbers of $-2$. In summary, high-order skyrmions with skyrmion numbers of $-2$ and $+2$, after passing through the spiral phase with a topological charge of $+1$, will yield high-order skyrmions with skyrmion numbers of $-2$. This means that the spiral phase with a topological charge of $+1$ will filter out high-order skyrmions with skyrmion numbers of $+2$. Similarly, the spiral phase with a topological charge of $-1$ will filter out high-order skyrmions with skyrmion numbers of $-2$. Multi-zone filters are used to adjust the intensity of the Stokes parameters that construct the two types of high-order skyrmions.

In conclusion, we have introduced a technique for Hall effect-like splitting of optical Stokes skyrmions within a tightly focused system. The vector beam utilized in our approach consists of azimuthally polarized light. By integrating two multi-zone filters and two spiral phase plates, each with reversed topological charges, we successfully created a hybrid multi-zone filter capable of realizing Hall effect-like splitting of two optical Stokes skyrmions with opposite skyrmion numbers. Adjusting the horizontal parameter enabled precise manipulation of the skyrmion positions. Our simulations and experiments confirmed both symmetric and asymmetric manifestations of Hall effect-like splitting of optical Stokes skyrmions. Unlike previous studies on optical skyrmions, we are the first, to the best of our knowledge, to discover that azimuthally polarized light can carry both positive and negative skyrmions. Furthermore, we have achieved the symmetric and asymmetric separation of these skyrmions using multi-zone filters for the first time. Asymmetric separation, in particular, enables the arbitrary manipulation of positive and negative skyrmions, offering an additional degree of freedom for their generation and control. This advancement opens the door to the development of novel skyrmion-controlled nanophotonic devices. The emergence of complex vector field structures, characterizing optical quasiparticles with diverse topological characteristics, underscores the significance of our approach across various applications including modern spin optics, imaging, structured light, and topological and quantum technologies.