Photonics Research, Volume. 13, Issue 4, 995(2025)

Spin angular momentum engineering within highly localized focal fields: from simple orientation to complex topologies

Yongxi Zeng1,2,3, Yanzhong Yu3, Jian Chen2, Houan Teng2, Musheng Chen1,3, Pinghui Wu3, and Qiwen Zhan1,2、*
Author Affiliations
  • 1College of Physics and Information Engineering, Quanzhou Normal University, Quanzhou 362000, China
  • 2School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
  • 3Fujian Provincial Key Laboratory for Advanced Micro-nano Photonics Technology and Devices & Key Laboratory of Information Functional Material for Fujian Higher Education, Quanzhou 362000, China
  • show less

    Optical skyrmions, as quasiparticles with non-trivial topological structures, have garnered significant attention in recent years. This paper proposes a method for customized spin angular momentum (SAM) distribution in highly localized focal fields, thereby enabling the generation of SAM skyrmion and bimeron topologies. The skyrmionic SAM textures can be flexibly controlled, such as polarity, vorticity, and helicity. In addition, the two-dimensional projection plane can be arbitrarily oriented within three-dimensional space. By utilizing time-reversal techniques, we obtain the required illumination fields of the 4π-focusing system and subsequently evaluate the tightly focused field using vector Debye integral theory. Our results show that the SAM orientation within the focal field is controlled by the orientation of orthogonal dipole pairs. Using the radiation field of a multi-concentric array of orthogonal dipole pairs, the distribution of SAM orientation in the target plane can be tailored to generate SAM topological structures such as skyrmions and bimerons. Highly localized and tunable SAM engineering holds great potential for applications in optical manipulation, light–matter interactions, optical information processing, transmission, and storage.

    1. INTRODUCTION

    Light, as an electromagnetic radiation, carries linear momentum and angular momentum (AM), which contribute to the optical pressure [1,2] and rotational force [3], respectively. It is now well known that there are two types of AM: spin angular momentum (SAM), associated with the circularly polarized state and first proposed by Poynting in 1909 [4], and orbital angular momentum (OAM) [5]. Rotating a particle around its axis is crucial in optical manipulation and is typically achieved by controlling the SAM. The rotation axis induced by the SAM is generally parallel to the direction of light propagation and is referred to as the longitudinal SAM [68]. Researchers later revealed the unusual transverse spin of surface-polariton evanescent waves caused by circular polarization in the propagation plane [9]. This discovery attracted significant attention and has been widely studied in contexts such as evanescent waves [10], two-wave interference [11], and tightly focused beams [12]. In the tightly focused optical field, a longitudinal component along the light propagation direction, together with the transverse component, can form a circular polarization state that produces transverse SAM [13]. If the orientation of the SAM can be adjusted arbitrarily, it will greatly improve the flexibility of particle rotation manipulation around its axis and extend the application space [14,15].

    Skyrmions, a class of quasiparticles with topological protection, were initially proposed in 1961 by Skyrme during his study on the unified field theory of meson and baryon interactions [16,17]. Subsequently, skyrmions have been investigated and experimentally confirmed to exist in various physical systems, including high-energy physics [18], Bose–Einstein condensates [19], liquid-crystal materials [20], and magnetic materials [2124]. Of particular significance are the magnetic skyrmions formed by spin textures in chiral magnets, which have paved the way for novel approaches to achieving high-capacity information encoding and low-energy magnetic storage [22]. Magnetic skyrmions are nanoscale vortices of electron spins, emerging as a significant consequence of spin–orbit coupling in magnetic materials. Similarly, light, as an electromagnetic wave, also carries both SAM and OAM [4,5]. In evanescent fields or highly localized fields, the spin–orbit coupling of light gives rise to various unconventional effects in physics, optics, and quantum technologies [2527]. In this context, it was not until 2018 that optical skyrmions—defined through variations in the electric-field vector—were first reported, marking a pivotal moment in photonics topologies research [28]. Scholars have subsequently reported various types of optical skyrmions, constructed using different optical vectors, including SAM vectors of confined free-space waves [2931], Stokes vectors in paraxial beams [32], and magnetic-field vectors of pulse-propagating light [33].

    To the best of our knowledge, there have been few reports on the construction of SAM skyrmion and bimeron topologies with prescribed characteristics in highly localized focal fields, particularly on a prescribed two-dimensional (2D) projection plane. In this paper, we present a simpler and more direct method for controlling the SAM orientation distribution in a sub-diffraction-limited focal field, compared to those in Refs. [14,15]. Building on this, we propose a method for controlling the topological forms of highly localized SAM skyrmions and bimerons on a prescribed 2D plane. The method is implemented by integrating time-reversal techniques, antenna radiation theory, and Debye vector diffraction integral theory. The designed focal field has potential applications in fields such as high-resolution microscopy, precision metrology, and optical micromanipulation [25,34]. The multi-degree-of-freedom tunability of these topological structures shows great potential for enabling light–matter interactions at the microscale and nanoscale, offering opportunities to explore novel physical phenomena [35].

    2. SUB-DIFFRACTION-LIMITED FOCAL FIELD WITH ARBITRARY SAM ORIENTATION

    A. Methods

    An inverse method was adopted for the SAM engineering of focal fields. The focal field distribution was simulated using the radiation fields of dipoles. Based on electromagnetic radiation theory [36,37], the radiation fields of dipoles along the three principal axes (x, y, z) in the global Cartesian coordinates can be described by [38] [Edipolex(θ,φ)Edipoley(θ,φ)Edipolez(θ,φ)]=C[cosθcosφeθ+sinφeφcosθsinφeθcosφeφsinθeθ],where C=jωμ0I0l0ejkf/4πf is a constant independent of the radiation pattern, ω is the angular frequency, μ0 is the permeability, I0 represents the current magnitude, l0 is the dipole length, k is the wavenumber, and f denotes the focal length of the lens.

    An orthogonal dipole pair (ODP) can be formed by positioning a pair of dipoles along the x and y axes with feeding currents that differ by 90° in phase. The coherent superposition of their radiation fields in the xy plane then produces circular polarization [14,15,39]. The handedness of the circular polarization depends on the sign of the phase difference between their feeding currents. When the y-dipole current lags by 90°, the radiation field in the xy plane exhibits right-handed circular polarization with spin along the z axis. Conversely, a 90° phase lead in the y-dipole current produces left-handed circular polarization, with spin directed along the negative z axis.

    To adjust the spin orientation of the far-field radiation of the aforementioned ODP in its plane, a simpler and more direct method compared to Refs. [14,15] is employed. We rotate the global Cartesian coordinate system by an angle θ0 along the φ=φ0 plane to form a local Cartesian coordinate. In the local coordinate, the spatial orientation of the z axis (i.e., the normal vector n of the ODP plane) is characterized by (θ0, φ0), as shown in the inset of Fig. 1.

    Diagram of a 4π-focusing system, with the inset illustrating the spatial rotation of an orthogonal dipole pair.

    Figure 1.Diagram of a 4π-focusing system, with the inset illustrating the spatial rotation of an orthogonal dipole pair.

    The radiation fields of dipoles oriented along the three principal axes (x, y, z) of the local Cartesian coordinate are given by [Edipolex(θ,φ)Edipoley(θ,φ)Edipolez(θ,φ)]=R(θ0,φ0)[Edipolex(θ,φ)Edipoley(θ,φ)Edipolez(θ,φ)],where R(θ0,φ0) is the direction cosine matrix corresponding to the spatial orientation (θ0, φ0), and can be expressed as R(θ0,φ0)=[cosαxcosβxcosγxcosαycosβycosγycosαzcosβzcosγz]=[cosθ0cos2φ0+sin2φ0(cosθ01)sinφ0cosφ0sinθ0cosφ0(cosθ01)sinφ0cosφ0cosθ0sin2φ0+cos2φ0sinθ0sinφ0sinθ0cosφ0sinθ0sinφ0cosθ0].

    Here, ti (t=α, β, γ and i=x, y, z) represents the angle between the rotated i axis and the three principal axes of the global coordinate. When the global coordinate system is rotated, the dipole pair originally aligned along the x and y axes of the global system is now aligned along the x and y axes of the local Cartesian coordinate. If the current in the y-axis dipole lags 90° behind the x-axis dipole, their radiation field is given by Epair(θ,φ)=Edipolex(θ,φ)+jEdipoley(θ,φ)=C[Mθ(θ,φ)eθ+Mφ(θ,φ)eφ],where Mθ(θ,φ) and Mφ(θ,φ) represent the radial and angular components of the radiation field from the ODP, respectively. The resultant radiation field in the xy plane exhibits right-handed circular polarization, with its spin orientation along the z axis, corresponding to the spatial orientation (θ0, φ0). Therefore, we can flexibly control the SAM orientation of the ODP’s radiation field by adjusting the rotation parameters (θ0, φ0).

    In order to converge the radiation field, a 4π confocal optical system was employed [40,41]. This system consists of two identical high numerical aperture (NA) objectives arranged in a confocal configuration. If the objective lenses satisfy the sine condition, the analytical expression for the light field at the entrance pupil is given by [38] Epupil(ρ,φ)=C[Mθ(θ,φ)eρ+Mφ(θ,φ)eφ]/cosθ.

    Notably, taking into account the bending effects of the objectives on the radiation field, eθ in the image region has been converted to eρ in the object region. Utilizing time-reversal theory, the reverse propagation and convergence of the entrance pupil field can be quantitatively evaluated through Richards–Wolf theory [42,43], which facilitates the evaluation of the localized focal field characteristics produced by the 4π system: Ef(r,ϕ,z)=jλ0θmax02πEΩ(θ,φ)e[jkrsinθcos(φϕ)jkzcosθ]sinθdθdφ+jλππ+θmax02πEΩ(θ,φ)ejπe[jkrsinθcos(φϕ)jkzcosθ]sinθdθdφ.

    EΩ(θ,φ), the approximate spherical wavefront used for convergence in Eq. (6), is given by EΩ(θ,φ)=C{[Mθ(θ,φ)cosθcosφMφ(θ,φ)sinφ]ex+[Mθ(θ,φ)cosθsinφ+Mφ(θ,φ)cosφ]ey+Mθ(θ,φ)sinθez}.

    The SAM of the focal field can be calculated by [25,44] SIm(Ef*×Ef),where Ef and Ef* are the focused electric field and its conjugate component, respectively. The spatial orientation is determined by the three components of S, which can be characterized by direction angles (α, β, γ), representing the angles between the SAM and the global coordinate x, y, and z axes.

    B. Results of Tailoring Spin Orientation within the Focal Field

    To fully converge the entire radiation field of the ODP and completely restore the propagation components of the radiation field across the entire focal region, the NA of the objective lenses is set to 1, resulting in a convergence angle of 90°. The objective lenses are selected to satisfy the sine condition.

    When the ODP is aligned along the x and y axes of the global coordinate, with rotation parameters (θ0, φ0) set to (0°, 0°), the resulting focal field exhibits a typical purely longitudinal spin, as shown in Fig. 2. Figure 2(a) displays the normalized intensity and polarization distribution of the entrance pupil field calculated by Eq. (5). It can be observed that the intensity is strongest at the periphery of the pupil, gradually weakening toward the center. The central region shows a circularly polarized distribution, with the ellipticity decreasing slightly toward the outer region. The intensity and polarization distribution in the three principal planes of the focal field are shown in Fig. 2(b), where the intensity profiles in the xz and yz planes are elliptical, with x and y polarization, respectively. The intensity profile in the xy plane forms a perfect circle, with the central lobe region exhibiting circular polarization. The three-dimensional (3D) contour of the focal field clearly forms an oblate ellipsoid symmetric around the z axis, with SAM directed along the z axis, indicating longitudinal SAM. Line scans of the focal field intensity along the three principal axes are shown in Fig. 2(c), with full width at half-maximum (FWHM) values of 0.48λ, 0.48λ, and 0.41λ along the x, y, and z axes, respectively. The focal spot volume is 0.0494λ3, representing a sub-diffraction-limited spot, significantly smaller than the spot volumes of 0.21λ3 and 0.3λ3 reported in Ref. [15] and Ref. [45], respectively. One of the primary reasons for the smaller spot size in this method is the use of a 4π focusing system, which greatly enhances axial resolution. Figures 2(d)–2(f) display the three components of the SAM in the xy plane. It is evident that the longitudinal spin focal spot is dominated by the Sz component, with the Sx and Sy components being nearly zero. Additionally, Figs. 2(g)–2(i) depict the directional angle distributions. It can be observed that the angles α and β maintain a value of π/2, while in the central region of the focal spot, the angle γ is zero, and it undergoes a periodic reversal from 0 to π along the radial direction. Their direction angles within the focal region are (89.99°, 89.99°, 0°), closely matching the theoretical values (90°, 90°, 0°), indicating that the spin is oriented along the optical axis.

    Sub-diffraction-limited focal field with purely longitudinal SAM. (a) Intensity and polarization distribution of the required incident field. (b) Intensity and polarization distribution of the focus field in the three principal planes, and the opposite rotation directions of polarization are represented by white and black ellipses. (c) Line scanning of the focal field along the x, y, and z axes. (d)–(f) Sx, Sy, and Sz distributions in the focal plane. (g)–(i) Directional angles of the SAM vectors in the focal plane.

    Figure 2.Sub-diffraction-limited focal field with purely longitudinal SAM. (a) Intensity and polarization distribution of the required incident field. (b) Intensity and polarization distribution of the focus field in the three principal planes, and the opposite rotation directions of polarization are represented by white and black ellipses. (c) Line scanning of the focal field along the x, y, and z axes. (d)–(f) Sx, Sy, and Sz distributions in the focal plane. (g)–(i) Directional angles of the SAM vectors in the focal plane.

    When the rotation parameter θ0=90°, meaning one dipole is oriented along the z direction and the other dipole is positioned in the xy plane, a purely transverse SAM focal field can be generated. The direction of the transverse SAM depends on φ0. Details of the corresponding pupil field and focal field can be seen in Figs. 8 and 9 (Appendix A).

    When θ0 is neither 0° nor 90°, a focal spot with non-pure longitudinal or non-pure transverse SAM can be customized. Without loss of generality, we take the rotation parameters (30°, 120°) as an example, and the results are shown in Fig. 3. Figure 3(a) shows the entrance pupil field required to construct a focal spot with the desired spin direction (30°, 120°). It can be seen that this field distribution is very complex, with modulation of both amplitude and polarization. In this case, the linearly polarized regions are no longer distributed in a straight line, but in a curved pattern. As the distance from both sides of the curve increases, the ellipticity of the pupil field also increases. Similarly, the polarization handedness on either side of the curve is opposite. For more details on how the pupil field varies with changes in θ0 and φ0, please refer to Visualization 1 and Visualization 2. The entrance pupil fields in Figs. 2(a) and 3(a), Figs. 8 and 9 (Appendix A) can be generated by modulating the four degrees of freedom—amplitude, phase, polarization ratio, and retardation—of the incident light field using our developed vectorial optical field generator [13,46]. Figure 3(b) shows the intensity and polarization distribution in the three principal planes of the focal field, while Figs. 3(d)–3(f) and 3(g)–3(i) display the three components of the SAM and the direction angles in the xy plane. Unlike the previous examples, all three components of the SAM are non-zero. Additionally, the polarizations of main lobe regions in the three principal planes exhibit different ellipticity and handedness. Figures 3(g)–3(i) show that the SAM direction angles are measured as (104.51°, 64.39°, 29.97°), which are very close to the theoretical values of (104.48°, 64.34°, 30°). These instances demonstrate that by adjusting the rotation parameters (θ0, φ0), it is possible to flexibly and inversely construct sub-diffraction-limited focal fields with arbitrary SAM orientations.

    Sub-diffraction-limited focal field with prescribed SAM, exemplified by (θ0,φ0)=(30°,120°). (a) Intensity and polarization distribution of the required incident field. (b) Intensity and polarization distribution of the focus fields in the three principal planes. (c) Line scanning of the focal field along the x, y, and z axes. (d)–(f) Sx, Sy, and Sz distributions in the x−y plane. (g)–(i) Directional angles of the SAM vectors in the x−y plane.

    Figure 3.Sub-diffraction-limited focal field with prescribed SAM, exemplified by (θ0,φ0)=(30°,120°). (a) Intensity and polarization distribution of the required incident field. (b) Intensity and polarization distribution of the focus fields in the three principal planes. (c) Line scanning of the focal field along the x, y, and z axes. (d)–(f) Sx, Sy, and Sz distributions in the xy plane. (g)–(i) Directional angles of the SAM vectors in the xy plane.

    3. HIGHLY LOCALIZED SAM SKYRMIONS AND BIMERONS

    Based on successful flexible control of SAM orientation in a highly localized focal field, we delve deeper into custom-engineering SAM skyrmion-like topologies within microscale and nanoscale focal field regions. It is worth noting that the spin topology is steady-state and time-invariant, which distinguishes it from skyrmions driven by time-dependent electric or magnetic field vectors [28,29,38].

    A. Methods

    Highly localized optical skyrmions formed by SAM vectors can be realized through a complete topological evolution by mapping a two-parameter sphere onto a 2D plane using a stereographic projection [35,47]. The topological properties of a skyrmion are determined by its polarity p, vorticity v, and helicity φr. Polarity describes the variation of the vector’s out-of-plane orientation, while vorticity characterizes the in-plane orientation. Helicity describes the initial phase of the in-plane vector. When p=1, the longitudinal component of the vector points vertically downward at the center and upward at the periphery, whereas when p=1, it points upward at the center and downward at the edges. For v=1, the in-plane transverse vector components form a symmetric vortex, while for v=1, they form an anti-vortex texture (or saddle texture), corresponding to an anti-skyrmion. Helicity, which defines the initial azimuthal orientation of the transverse component, leads to a Néel-type skyrmion when φr=0° or 180°, a Bloch-type skyrmion when φr=90° or 270°, and an intermediate state skyrmion for other values of φr.

    A bimeron, as a homeomorphic transformation of a skyrmion, consists of two merons with opposite polarity, and it can be obtained by rotating the unit vector sphere and then projecting it onto a 2D plane using stereographic projection [32,48,49]. If the rotation parameters of the unit sphere are set as (θb, φb), the vectors on the sphere are rotated accordingly, and a similar projection process yields the bimeron topological structure. A common scenario is to lay the unit vector sphere horizontally, which corresponds to θb=90°, as depicted in Box 1, panel d of Ref. [35]. In our method, we introduce additional degrees of freedom for controlling the bimeron by adjusting the rotation parameters (θb, φb) of the unit vector sphere. Simultaneously, by tuning the parameters (θ0, φ0), we can construct SAM skyrmions and bimerons on a specified plane.

    To obtain the pupil field required for the target focal field, we precisely arrange a multiple concentric circle array of orthogonal dipole pairs (MCAODPs) in the prescribed plane (xy plane in Fig. 1) to mimic the topology form by SAM vectors, as shown in Fig. 4.

    Arrangement of MCAODP.

    Figure 4.Arrangement of MCAODP.

    The MCAODP is composed of M concentric ring arrays, with the angular spacing between elements in each ring denoted as δi (i=1,2,,M). Nm is the number of ODPs on the Mth ring. An ODP is positioned at the center. For elements on the same ring, the polar angle of the ODP normals is identical, and the azimuthal angle depends on the ODP’s positions. The normals of all ODPs are symmetrically distributed with respect to the array center. The total radiation field of the arrays is a coherent superposition of the radiation fields of each ODP, as expressed by Emcaodp(θ,φ)=E0+m=1Mn=1NmAFmnEmn,where E0=Edipolex(θ,φ)j·p·Edipoley(θ,φ) represents the radiation pattern of the central element (indexed as the 0th element). When p=1, the normal of the 0th element is oriented at (θ0, φ0), with local spatial parameters of (0°, 0°), meaning the spin vector aligns along the z axis. AFmnEmn represents the radiation field of the mnth element. Emn=Edipolex(θ,φ)j·p·Edipoley(θ,φ) denotes the radiation field of the mnth ODP when located at the origin. Edipolex and Edipoley are the radiation fields of the ODP along the x and y axes, respectively, after applying an additional rotation with parameters (θm, φmn) to the local Cartesian coordinates (x, y, z). AFmn is the position factor for the radiation field of the mnth element, arising from its spatial displacement relative to the 0th element, and derived as AFmn=exp{jkrm[(cosαxcosφp_mn+cosαysinφp_mn)sinθcosφ+(cosβxcosφp_mn+cosβysinφp_mn)sinθsinφ+(cosγxcosφp_mn+cosγysinφp_mn)cosθ]},where φp_mn represents the azimuthal angle of the mnth element in the xy plane, and rm is the radius of the mth ring.

    By carefully designing the positions and rotation parameters of each element in the MCAODP, the SAM orientation of the focal field on the xy plane can be precisely mimicked, thereby controlling the spin topological texture on the prescribed plane. Due to the central symmetry of the topological vector texture, the polar angle in the rotation parameters of elements on the same ring remains the same. The polar angle varies from 0° at the center to 180° at the outermost ring, ensuring a complete reversal of the longitudinal component of the SAM. The azimuthal angle in the rotation parameters is determined by the spatial position of the ODP φp_mn, as well as the polarity p, vorticity v, and helicity φr. The relationship between the polar and azimuthal angles in the rotation parameters and the topological structure of the focal field is expressed by {θm=mπ/M(m=0,1,,M)φmn=v·φp_mn+φr+(1+p)·π/2.

    If the unit mapping sphere is first rotated using the rotation parameters (θb, φb), the vectors on the unit sphere undergo corresponding rotations. Then, by mapping the vectors on the sphere onto a 2D plane through stereographic projection, the vector topological structure of the bimeron can be constructed. When calculating the radiation field of each element in the MCAODP, the dipole along the reference axis requires the operation R(θb,φb)1[Edipolex(θ,φ)Edipoley(θ,φ)Edipolez(θ,φ)]T, where R(θb,φb)1 is the inverse matrix of the direction cosine matrix in Eq. (3) when the parameters are (θb, φb).

    To validate the stability of the topologies, their skyrmion numbers can be calculated. Geometrically, the skyrmion number is used to compute the number of times the vectors rotate around the unit sphere, and is given by [28,50] SN=14πASfoc_u·(Sfoc_ux×Sfoc_uy)dxdy.

    Here, Sfoc_u is the unit SAM vector of the focal field, and A is the circular integral region of a complete topological structure. Due to the radial symmetry of the SAM, the skyrmion number calculation can be simplified to SN=p·v. Theoretically, the skyrmion number for both skyrmions and bimerons is 1.

    B. Results of Tailored Highly Localized SAM Skyrmions and Bimerons

    According to the methods described in Section 3.A, we now present the generation of skyrmions and bimerons formed by SAM vectors in the focal field. The optimized parameters for the MCAODP are as follows: M=5; the ring spacing is 0.24λ; and the angular spacing of the rings from the innermost to the outermost is denoted as π/(3i)(i=1,2,3,4,5), resulting in the number of ODPs in each ring being 1, 6, 12, 18, 24, and 30, respectively. Subsequently, we will adjust the topological plane using (θ0, φ0), and modify the types of skyrmions and bimerons by (θb, φb) as well as (p, v, φr).

    We first present the SAM topologies located on the focal plane. The normal of the xy plane (θ0, φ0) is set to (0°, 0°), and the MCAODP is arranged in the xy plane of the global Cartesian coordinate. By setting (θb, φb) to (0°, 0°) and (90°, 0°), along with (p=1, v=1, φr=0°), we can construct Néel-type skyrmions and bimerons formed by SAM vectors, whose characteristics of the input field and focal field are illustrated in Fig. 5.

    Néel-type SAM skyrmion (a)–(f) and bimeron (g)–(l) with negative polarity in the x−y plane. (a) and (g) are the intensity and polarization distribution of the required pupil field for the SAM skyrmion and bimeron, respectively. (b) and (h) are the intensity and polarization distribution of the SAM skyrmion and bimeron in the focal plane, respectively. (c) and (i) are the 3D distribution of SAM in the focal plane; the arrows indicate the orientation of the normalized SAM and color coded with its z component. (d)–(f) and (j)–(l) are the directional angles of their SAM vectors in the x−y plane.

    Figure 5.Néel-type SAM skyrmion (a)–(f) and bimeron (g)–(l) with negative polarity in the xy plane. (a) and (g) are the intensity and polarization distribution of the required pupil field for the SAM skyrmion and bimeron, respectively. (b) and (h) are the intensity and polarization distribution of the SAM skyrmion and bimeron in the focal plane, respectively. (c) and (i) are the 3D distribution of SAM in the focal plane; the arrows indicate the orientation of the normalized SAM and color coded with its z component. (d)–(f) and (j)–(l) are the directional angles of their SAM vectors in the xy plane.

    Figures 5(a) and 5(g) depict the intensity and polarization distributions of the pupil field necessary to construct the desired SAM skyrmion and bimeron. The intensity and polarization patterns are symmetrical with respect to the origin and y axis, respectively. In Fig. 5(a), the intensity profile displays a bright ring with a slightly weaker center, where the central region exhibits circular polarization. This is associated with the 0th element in the MCAODP whose normal is along the z axis. The bright ring region exhibits elliptical polarization, which is linked to the concentric ring structure in our model. In Fig. 5(g), the vector optical field dominates in the upper half-space (y>0), which correlates with the horizontal placement of the mapping unit sphere along the x axis. It is also evident that the polarization distribution is symmetric about the y axis, with the region near the y axis primarily exhibiting linear polarization. If the unit vector sphere is placed horizontally and φb is adjusted, the intensity and polarization distribution patterns of the pupil field are identical to Fig. 5(g), and rotate with φb. Notably, the intensity rotates clockwise with increasing φb, while the polarization rotates counterclockwise (see Visualization 3 for details). Figures 5(b) and 5(h) show the intensity and polarization distributions of the constructed SAM skyrmion and bimeron on the focal planes. Both fields display a bright spot surrounded by a weaker bright ring. The center of the skyrmionic focal plane features circular polarization, while the peripheral ring exhibits slightly elliptical polarization. In contrast, the focal plane polarization of the bimeron predominantly shows linear polarization along the y axis. Figures 5(c) and 5(i) show the 3D SAM distribution pattern of the SAM skyrmion and bimeron, respectively. In these figures, the direction of the arrows indicates the SAM orientation, while the color represents the magnitude of the longitudinal SAM component Sz. As shown in Fig. 5(c), the Néel-type skyrmion exhibits a hedgehog-like topological texture. In contrast, Fig. 5(i) shows that the focal plane of the Néel-type bimeron is divided along x=0 into two regions, the left side displays a hedgehog texture, while the right side features an anti-vortex texture. Figures 5(d)–5(f) illustrate the directional angles of the Néel-type SAM skyrmion depicted in Fig. 5(c) within the focal plane, and the dashed circles indicate the size of this topological structure. It can be observed that the distributions of the α and β are similar, with their patterns offset by a π/2 rotation. In the x=0 region, α is approximately 90°, and in the y=0 region, β is also approximately 90°. At the center, γ is 0°, and it gradually changes to 180° at a radius of 1.06λ, completing a full spin vector reversal. The distribution pattern of the SAM vector matches the arrangement of the MCAODP. Figures 5(j)–5(l) illustrate the directional angles of the SAM bimeron on the focal plane, as depicted in Fig. 5(i). The distributions of the β and γ are similar, with a π/2 rotational offset. Along the y=0 axis, β is approximately 90°, while γ is also 90° along the x=0 axis. Notably, α transitions from 0° at the center to 180° at a radius of 1.06λ, completing a full reversal along the x axis. The skyrmion numbers of the Néel-type SAM skyrmion and bimeron in the focal plane, calculated using Eq. (13), are 0.9921 and 0.9920, respectively. This confirms the successful construction of highly localized skyrmion and bimeron topologies in the focal plane. By adjusting the polarity, vorticity, and helicity, the topological textures of the SAM skyrmion and bimeron in the focal plane can be adjusted (see Appendix B for details).

    Using the aforementioned model, specific SAM topologies can be generated on prescribed planes by adjusting the normal of the xy plane. We next demonstrate the generation of SAM skyrmions and bimerons in the yz and xz planes, where the normal parameters of the xy plane, (θ0, φ0), are set to (90°, 0°) and (90°, 90°), respectively. The resulting partial focal fields are shown in Fig. 6, where Fig. 6(a) shows a Bloch-type SAM skyrmion in the yz plane, Fig. 6(b) depicts a Néel-type SAM skyrmion in the xz plane, and Figs. 6(c) and 6(d) depict Néel-type bimerons in the yz and xz planes, respectively.

    SAM skyrmions and bimerons in the y−z and x−z planes. (a) is a Bloch-type SAM skyrmion on the y−z plane with p=1, v=1. (b) is a Néel-type SAM skyrmion on the x−z plane with p=1, v=1. (c) and (d) are Néel-type SAM bimerons on the y−z and x−z planes with p=1, v=1, respectively.

    Figure 6.SAM skyrmions and bimerons in the yz and xz planes. (a) is a Bloch-type SAM skyrmion on the yz plane with p=1, v=1. (b) is a Néel-type SAM skyrmion on the xz plane with p=1, v=1. (c) and (d) are Néel-type SAM bimerons on the yz and xz planes with p=1, v=1, respectively.

    Finally, we demonstrate the comprehensive engineering capability of the focused SAM topology within the proposed model, enabling the construction of diverse SAM topological textures in non-primary planes. Without loss of generality, four distinct examples of SAM topologies located in different non-primary planes are provided in Fig. 7, showcasing the model’s versatility and precision in controlling SAM texture configurations. Figure 7(a) shows a Néel-type SAM skyrmion with p=1, v=1, and (θ0,φ0)=(40°,60°). Figure 7(b) depicts a Néel-type SAM skyrmion with p=1, v=1, and (θ0,φ0)=(45°,45°). Figure 7(c) is a Néel-type SAM bimeron with p=1, v=1, (θ0,φ0)=(50°,60°), and (θb,φb)=(90°,0°). Figure 7(d) is a Bloch-type SAM bimeron with p=1, v=1, (θ0,φ0)=(45°,45°), and (θb,φb)=(90°,90°). The skyrmion numbers of Fig. 7 are 0.9867, 0.9898, 0.9753, and 0.9654, respectively.

    SAM skyrmions and bimerons in arbitrary prescribed planes. (a) is a Néel-type SAM skyrmion with p=−1, v=1, and (θ0,φ0)=(40°,60°). (b) is a Néel-type SAM skyrmion with p=−1, v=1, and (θ0,φ0)=(45°,45°). (c) is a Néel-type SAM bimeron with p=1, v=1, (θ0,φ0)=(50°,60°), and (θb,φb)=(90°,0°). (d) is a Bloch-type SAM bimeron with p=−1, v=1, (θ0,φ0)=(45°,45°), and (θb,φb)=(90°,90°).

    Figure 7.SAM skyrmions and bimerons in arbitrary prescribed planes. (a) is a Néel-type SAM skyrmion with p=1, v=1, and (θ0,φ0)=(40°,60°). (b) is a Néel-type SAM skyrmion with p=1, v=1, and (θ0,φ0)=(45°,45°). (c) is a Néel-type SAM bimeron with p=1, v=1, (θ0,φ0)=(50°,60°), and (θb,φb)=(90°,0°). (d) is a Bloch-type SAM bimeron with p=1, v=1, (θ0,φ0)=(45°,45°), and (θb,φb)=(90°,90°).

    4. CONCLUSIONS

    In summary, we have successfully demonstrated the customized engineering of SAM in highly localized focal fields. By using the radiation field of a single ODP, we can construct a sub-diffraction-limited focal spot with arbitrarily controlled SAM orientation. Building upon this, we have employed ODPs as array elements within a meticulously arranged MCAODP configuration, enabling the creation of highly localized SAM skyrmions and bimerons with multi-degree-of-freedom control in the focal region of a 4π system. Our method allows for effective manipulation of the spatial orientation of the SAM by adjusting the normals of individual ODPs. Furthermore, by modifying the normal parameters (θ0, φ0) of the MCAODP arrangement, we can prescribe the SAM topology plane. The topological structure of the focal field can transition between skyrmions (θb=φb=0) and bimerons, with variations controlled through parameters (p, v, φr). Our approach significantly enhances the flexibility in controlling SAM topology in tightly focused fields. This comprehensive control over highly localized SAM skyrmions and bimerons opens avenues for investigating optical topological properties at the microscale/nanoscale and exploring novel phenomena arising from the interactions between topological quasiparticles, such as skyrmions and bimerons, and various materials.

    APPENDIX A: MORE RESULTS OF SUB-DIFFRACTION-LIMITED FOCAL FIELD WITH PURELY TRANSVERSE SAM

    When the rotation parameter θ0 is set to 90°, a focal field with purely transverse SAM can be constructed. Taking φ0=0° as an example, a focal spot with SAM aligned along the x axis is obtained, as shown in Fig. 8. The entrance pupil field is depicted in Fig. 8(a). The normalized intensity and polarization distributions are symmetric about the y axis, with higher intensity on either side of the x axis and weaker intensity in the central region. At x=0, the entrance pupil field is linearly polarized along the y axis. As the x coordinate increases, the ellipticity progressively increases, exhibiting opposite handedness on either side of x=0, as indicated by the white and black regions in the figure. Figures 8(b)–8(i) provide detailed characteristics of the x-axis SAM focal field. Figure 8(b) reveals that the focal field is circularly polarized in the yz plane and linearly polarized in the xy and xz planes. Figure 8(c) presents line scans of the focal field intensity along the x, y, and z axes, where the FWHM values are measured as 0.41λ, 0.48λ, and 0.48λ, respectively. The focal spot volume is calculated as 0.0494λ3, confirming a sub-diffraction-limited focal spot. Figures 8(d)–8(f) and 8(g)–8(i) illustrate the three components of the SAM and their corresponding direction angles. It is evident that the SAM is predominantly composed of the Sx component, while the Sy and Sz components are nearly zero. Within the focal field region, the direction angles are measured as (0°, 89.99°, 89.99°), which closely align with the theoretical values of (0°, 90°, 90°). This agreement confirms that the spin direction is aligned along the x axis.

    Sub-diffraction-limited focal field with x direction SAM. (a) Intensity and polarization distribution of the required incident field. (b) Intensity and polarization distribution of the focus field in the three principal planes. (c) Line scanning of the focal field along the x, y, and z axes. (d)–(f) Sx, Sy, and Sz distributions in the y−z plane. (g)–(i) Directional angles of the SAM vectors in the y−z plane.

    Figure 8.Sub-diffraction-limited focal field with x direction SAM. (a) Intensity and polarization distribution of the required incident field. (b) Intensity and polarization distribution of the focus field in the three principal planes. (c) Line scanning of the focal field along the x, y, and z axes. (d)–(f) Sx, Sy, and Sz distributions in the yz plane. (g)–(i) Directional angles of the SAM vectors in the yz plane.

    Sub-diffraction-limited focal field with purely transverse SAM, exemplified by (θ0,φ0)=(90°,45°). (a) Intensity and polarization distribution of the required incident field. (b) Intensity and polarization distribution of the focus field in the three principal planes. (c) Line scanning of the focal field along the x, y, and z axes. (d)–(f) Sx, Sy, and Sz distributions in the x−y plane. (g)–(i) Directional angles of the SAM vectors in the x−y plane.

    Figure 9.Sub-diffraction-limited focal field with purely transverse SAM, exemplified by (θ0,φ0)=(90°,45°). (a) Intensity and polarization distribution of the required incident field. (b) Intensity and polarization distribution of the focus field in the three principal planes. (c) Line scanning of the focal field along the x, y, and z axes. (d)–(f) Sx, Sy, and Sz distributions in the xy plane. (g)–(i) Directional angles of the SAM vectors in the xy plane.

    APPENDIX B: MORE RESULTS OF SAM SKYRMIONS AND BIMERONS WITH TUNABLE TOPOLOGICAL TEXTURES IN THE FOCAL PLANE

    By adjusting the polarity p, vorticity v, and helicity φr, the types of SAM skyrmion and bimeron can be tailored.

    For instance, when the polarity p is changed to 1 while keeping the other parameters in Fig. 5 unchanged, the spatial distribution of the transverse component of the SAM skyrmion remains consistent. However, the longitudinal component undergoes a reversal [compare Figs. 5(f) and 10(e)], switching from vertically downward at the center to vertically upward at the periphery. A similar scenario is observed for the Néel-type bimeron in Fig. 5, where reversing the polarity to a negative value leaves the SAM vector in the yz plane unchanged but causes an inversion of the x component [compare Figs. 5(j) and 10(f)]. The spatial distributions of the SAM skyrmion and bimeron after polarity adjustment are shown in Figs. 10(a) and 10(b). For the Néel skyrmion, the spin vector is flipped, transitioning from downward at the center to upward at the periphery. Similarly, the Sx component of the Néel-bimeron is also flipped, with an anti-vortex texture on the left of the x axis and a hedgehog texture on the right. The skyrmion numbers for these adjusted structures are calculated to be 0.9921 and 0.9920, respectively. This demonstrates that altering the sign of p allows for effective polarity manipulation of the constructed SAM topologies.

    Néel-type SAM skyrmion and bimeron with positive polarity in the x−y plane. (a) and (c)–(e) are the 3D distribution of the SAM vector and its directional angles in the focal plane for the SAM skyrmion, respectively. (b) and (f)–(h) are the 3D distribution of the SAM vector and its directional angles in the focal plane for the SAM bimeron, respectively.

    Figure 10.Néel-type SAM skyrmion and bimeron with positive polarity in the xy plane. (a) and (c)–(e) are the 3D distribution of the SAM vector and its directional angles in the focal plane for the SAM skyrmion, respectively. (b) and (f)–(h) are the 3D distribution of the SAM vector and its directional angles in the focal plane for the SAM bimeron, respectively.

    SAM anti-skyrmion and anti-bimeron with positive polarity in the x−y plane. (a) and (c) are the 3D distribution of the SAM vector in the focal plane for the two focused field topologies, and (b) and (d) are the 2D distribution of in-plane components.

    Figure 11.SAM anti-skyrmion and anti-bimeron with positive polarity in the xy plane. (a) and (c) are the 3D distribution of the SAM vector in the focal plane for the two focused field topologies, and (b) and (d) are the 2D distribution of in-plane components.

    SAM bimeron with p=v=1, φr=0°, θb=90°, and varying φb. (a)–(d) φb is 45°, 90°, 135°, and 180°, respectively.

    Figure 12.SAM bimeron with p=v=1, φr=0°, θb=90°, and varying φb. (a)–(d) φb is 45°, 90°, 135°, and 180°, respectively.

    Néel-type SAM bimeron with p=v=1, φr=0°, φb=45°, and varying θb. (a)–(d) θb is 45°, 70°, 110°, and 135°, respectively.

    Figure 13.Néel-type SAM bimeron with p=v=1, φr=0°, φb=45°, and varying θb. (a)–(d) θb is 45°, 70°, 110°, and 135°, respectively.

    SAM skyrmion and bimeron with p=−1, v=1, and varying vorticity φr. (a)–(d) and (e)–(h) are the 3D distributions of the SAM and its in-plane component distribution. (a), (c) φr=45°; (b), (d) φr=90°.

    Figure 14.SAM skyrmion and bimeron with p=1, v=1, and varying vorticity φr. (a)–(d) and (e)–(h) are the 3D distributions of the SAM and its in-plane component distribution. (a), (c) φr=45°; (b), (d) φr=90°.

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    Yongxi Zeng, Yanzhong Yu, Jian Chen, Houan Teng, Musheng Chen, Pinghui Wu, Qiwen Zhan, "Spin angular momentum engineering within highly localized focal fields: from simple orientation to complex topologies," Photonics Res. 13, 995 (2025)

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    Paper Information

    Category: Physical Optics

    Received: Dec. 8, 2024

    Accepted: Jan. 22, 2025

    Published Online: Mar. 31, 2025

    The Author Email: Qiwen Zhan (qwzhan@usst.edu.cn)

    DOI:10.1364/PRJ.550414

    CSTR:32188.14.PRJ.550414

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