PHYSICAL REVIEW APPLIED 21, 024025 (2024) Topologically controlled multiskyrmions in photonic gradient-index lenses Yijie Shen ,1,2,* Chao He ,3,† Zipei Song ,3 Binguo Chen,4 Honghui He ,4 Yifei Ma ,3 Julian A.J. Fells,3 Steve J. Elston,3 Stephen M. Morris,3 Martin J. Booth ,3 and Andrew Forbes5 1 Centre for Disruptive Photonic Technologies, School of Physical and Mathematical Sciences & The Photonics Institute, Nanyang Technological University, Singapore 639798, Singapore 2 School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore 3 Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, United Kingdom 4 Guangdong Engineering Center of Polarization Imaging and Sensing Technology, Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen, 518055, China 5 School of Physics, University of the Witwatersrand, Private Bag 3, Wits, 2050, South Africa (Received 24 September 2023; revised 21 November 2023; accepted 19 January 2024; published 13 February 2024) Skyrmions are topologically protected quasiparticles, originally studied in condensed-matter systems and recently in photonics, with great potential in high-density information storage. Despite the recent atten- tion, most optical solutions require complex systems yet produce limited topologies. Here we demonstrate an extended family of quasiparticles beyond normal skyrmions that are controlled in compact photonic gradient-index media, extending to higher-order members such as multiskyrmions and multimerons, with increasingly complex topologies. We introduce multiple topological numbers (centrality, radiality, vorticity, and polarity) in addition to the skyrmion number to describe these photonic quasiparticles. Our compact creation system lends itself to integrated and programmable solutions of complex parti- cle textures, with potential impacts on both photonic and condensed-matter systems for revolutionizing topological informatics and logic devices. DOI: 10.1103/PhysRevApplied.21.024025 I. INTRODUCTION In the current age of information explosion, the pursuit of next-generation information carriers and data storage is endless, eminently benefiting our daily lives. Since Skyrme [1] proposed a model to unify a large class of fundamen- tal particles by methods of topology in the 1960s, known today as skyrmions, these have emerged as one of the highest-potential information carriers due to their topolog- ically robust spin textures localized in ultrasmall regions, in particular, which has revolutionized the high-density- data-storage technologies in magnetic materials in recent years [2–5]. To further expand the capacity in informatic applications, it is highly desired to manipulate complex quasiparticles with higher-order topological textures in addition to the fundamental skyrmions [6]. For instance, the meron textures with fractional skyrmion numbers can be controlled in chiral magnets [7], and recently in antifer- romagnets at room temperature [8]. Novel quasiparticles such as skyrmion bags [9], skyrmion bundles [10], and skyrmion braids [11] with large range control of skyrmion *yijie.shen@ntu.edu.sg †chao.he@eng.ox.ac.uk numbers were created in magnetic materials. Moreover, exotic spatially knotted quasiparticles of hopfions [12–14], torons [15,16], heliknotons [17], and polyskyrmionomers [18] characterized by complex topologies were designed in magnets, colloids, and chiral liquid crystals. The real- ization of novel quasiparticles in diversified condensed- matter systems has triggered the development of static or quasistatic ultrahigh-density-data-storage techniques. Optical or photonic skyrmions, which were recently realized, provide new degrees of freedom in terms of the construction of topological quasiparticles [19]. Following the initial demonstrations of the formation of photonic skyrmions in surface plasma by evanescent electromag- netic fields [20,21], as well as optical spins [22,23], pho- tonic skyrmions have also been generated in, for example, optical polarization Stokes vector fields in free space [24– 27], electromagnetic fields in space-time [28–30], and pseudospins in nonlinear crystals [31]. However, previous studies have demonstrated a very limited number of topo- logical states, often requiring the use of complex, bulky, and expensive systems. Here we demonstrate both experimentally and theoret- ically an extended class of photonic quasiparticles con- structed by vectorial structured light in gradient-index 2331-7019/24/21(2)/024025(7) 024025-1 © 2024 American Physical Society https://orcid.org/0000-0002-6700-9902 https://orcid.org/0000-0001-9654-830X https://orcid.org/0009-0003-9345-0739 https://orcid.org/0000-0001-7369-7433 https://orcid.org/0000-0003-0521-5687 https://orcid.org/0000-0002-9525-8981 https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevApplied.21.024025&domain=pdf&date_stamp=2024-02-13 http://dx.doi.org/10.1103/PhysRevApplied.21.024025 YIJIE SHEN et al. PHYS. REV. APPLIED 21, 024025 (2024) (GRIN) lenses. The quasiparticles that are formed with use of vectorial structured light exhibit novel forms of higher-order configurations beyond elementary skyrmions with increasingly complex topology and geometry. These complex quasiparticles can be characterized by multiple topological numbers and their diversified topological states can be controlled by cascaded GRIN lenses, as a gener- ation method with a highly symmetric and high-quality beam profile [32]. The compact generator of high-quality photonic quasiparticles offers the potential to be inte- grated with on-chip solid-state storage devices and the application of ultrahigh-capacity information coding and transfer [33]. II. RESULTS A. Vectorial optics of GRIN lenses GRIN lenses are a kind of optical element that pos- sesses a spatially varying refractive-index profile as well as birefringence, which enable focusing, imaging, and shaping of light through a rodlike structure [32,34–36], as shown in Fig. 1. GRIN lenses can be fabricated with diameters on the submillimeter scale, which means that GRIN-lens arrays can be formed with dimensions on the order of millimeters [Fig. 1(a2)]. Each GRIN lens can be composed of different cascaded segments with differ- ent refractive-index-gradient designs, together with seg- ments of quarter-wave plates (QWPs) and half-wave plates (HWPs). If we input a plane wave into one side of a cascaded GRIN lens, the output will be light with spatially varying polarization patterns [32,36]. Here we show that through judicious assembly of the cascades of GRIN lenses combined with tuning of the polarization of the input light, we can precisely generate a wide variety of vectorial light fields as high-order photonic skyrmions with com- plex topologies, which can be arranged in a very compact volume. Note that the formation of a vector beam, such as a skyrmionic beam, in a GRIN lens is not similar to the abrupt topological phase transition in condensed matter. It is a continue transition (without abrupt change) pro- cess to form a polarization pattern fulfilling topological skyrmion mapping, i.e., unwrapping the Poincaré sphere (S2) to a localized real space (R2) [19]. For instance, when uniform circularly polarized light goes into the GRIN-lens end surface, the polarization states except that at the center will gradually change into various elliptical polarization states covering all the states on a Poincaré sphere. There is a distinction between such continuous deformation of the optical field and the smooth deformation preserving its topology. For the photonic skyrmion topology, it has just been shown that this is the smooth case if the defor- mation can be written as a coordinate transformation [37]. When the mapping function is not smoothly deformed by some system, the skyrmion number will change, even if the field changes continuously, as has been shown even in free space with mixed-order modes [38]. Our GRIN system is evidently such a system too. <5 mm ~40 mm GRIN-lens array <500 μm Fast and slow axis 0 10–5 1.45 1.55 Refractive index Second-order skyrmionium Néel-type skyrmion Bloch-type skyrmion Antiskyrmion Birefringence sN = 2 tanPolarization RCP LCP zs x–1 s ys sN = 2 sN = 1 sN = 1 sN = sN = 0 sN = 4 sN = –1 Skyrmion Meron (a1) (a2) (a3) (b1) (b2) (b3) (b4) (c1) (e1) (e2) (e3) (c2) (c3) (c4) (d)(b)(a) (e) (c) ½ 1 –1–� � cN = 1 cN = 4 FIG. 1. GRIN lenses for photonic skyrmion generation. (a) Schematics of (a1) the GRIN lens and (a2) the GRIN-lens array, with radial and azimuthal distributions of the fast and slow axes at their cross section, the radius of which can be less than 500 µm, and (a3) a picture of a real GRIN lens. (b),(c) Diverse polarization vectorial light fields corresponding to the different complex photonic quasiparticles generated can be controlled by cascades of GRIN lenses, represented by (b) polarization-ellipse distributions and (c) Stokes-vector distributions, which include (b1),(c1) a skyrmion of Ns = 2, (b1),(c1) a skyrmion of Ns = 0, (b2),(c2) a skyrmionium of Ns = 0, (b3),(c3) a quadruskyrmion of Ns = 4 comprising four elementary skyrmions of Ns = 1, and (b2),(c2) a quadrumeron of Ns = 2 comprising four elementary merons of Ns = 1/2. (d) Conceptual schematic of the vector texture of a second-order skyrmionium (Nr = 2, Ns = 0), which includes a skyrmion (Nr = 1, Ns = 2) and meron (Nr = 1/2, Ns = 1) structure in its subspace. (e) Topological protection of the photonic skyrmions, where (e1) a Néel-type skyrmion can be transformed into (e2) a Bloch-type skyrmion on prop- agation and is resilient to perturbation under the same skyrmion number of Ns = 1. It is, however, impossible for it to be transformed into (e3) an antiskyrmion of different skyrmion number Ns of −1. 024025-2 TOPOLOGICALLY CONTROLLED MULTISKYRMIONS... PHYS. REV. APPLIED 21, 024025 (2024) B. Topological photonic quasiparticles A “skyrmion” refers to a three-component real vector or spin texture mapped from a two-sphere to a localized 2D real space, denoted as s(x, y) = [sx(x, y), sy(x, y), sz(x, y)], with basic topology characterized by the skyrmion num- ber Ns (which counts how many times the spins can wrap around a two-sphere with full azimuth): Ns = 1 4π ∫∫ σ s · ( ∂s ∂x × ∂s ∂y ) dxdy, (1) where σ represents the region in which the particle is confined. Photonic skyrmions or quasiparticles can be con- structed by the polarization Stokes vectors, i.e., the vector defined by the three Stokes parameters, of the obtained structured light, analogously to the magnetic spin used to construct magnetic skyrmions [19,38,39]. Figure 1(b1) shows a theoretical result for the polarization-ellipse dis- tribution of a skyrmionic light field generated by a cus- tomized GRIN lens, the Stokes vector field of which shows a fundamental skyrmion of Ns = 2 [Fig. 1(c1)]. Note that for the fundamental skyrmion, the signal of the skyrmion number was determined by its central spin with a spin- up state (+) or a spin-down state (−), which is defined as the polarity, Np = sgn(Ns), and the absolute value of the skyrmion number determines the vortex charge of the transverse- (x, y) component distribution, defined as the vorticity, Nv = |Ns|. The result in Fig. 1(c1) shows a central spin with spin-up state and vortex charge of the transverse distribution of 2, thus resulting in Ns = 2. In addition to the fundamental skyrmion, cascaded GRIN lens can also generate generalized quasiparticles, for instance, the skyrmionium [41], a skyrmion radially nested with another skyrmion with opposite topological number, resulting in a skyrmion number Ns of (+2) + (−2) = 0 [Figs. 1(b2) and 1(c2)]. If the radially nested number, Nr, is larger than 2, the extended configura- tions were previously called “Nrπ skyrmions” or “target skyrmions” [42,43], which, we argue, can also be easily realized in our cascaded GRIN-lens system. The versatil- ity of GRIN lenses that are cascaded to form arrays allows us to access novel forms of photonic quasiparticles that are challenging to observe in other systems, for instance, the exotic multiskyrmions and multimerons, comprising multiple elementary skyrmions and merons in subspace with high symmetry. Figures 1(b3) and 1(c3) demonstrate a quadruskyrmion with four elementary skyrmions of unit skyrmion number, resulting in total skyrmion number Ns = 4. Figures 1(b4) and 1(c4), on the other hand, demon- strate a quadrumeron with four elementary merons of half skyrmion number, resulting in total skyrmion number Ns = 2. To describe the symmetry in multiskyrmions and multi- merons, we define a number of centrality, Nc, which counts how many spin-up or spin-down center points there are. The centrality can also be interpreted as the numbers of sin- gularities of the transverse-component distribution of the multiskyrmionic field, which thus can be detected by an algorithm for counting multiple singularities of a complex light field [44]. The quadruskyrmion and quadrumeron are both of centrality Nc = 4 [Figs. 1(c3) and 1(c4)]. On the other hand, the skyrmions, skyrmioniums, and target skyrmions are always of centrality Nc = 1 [Figs. 1(c1) and 1(c2)]. Figure 1(d) conceptually demonstrates the ele- mentary relationship among a skyrmionium, a skyrmion, and a meron. Demonstrated by prior work, such photonic Stokes skyrmions possess the property of topological stability, meaning that the skyrmion number remains invariant on propagation evolution [39,45], and is robust with regard to environmental perturbations [46]. Once a skyrmion is generated by a GRIN lens, its topology will be protected on further propagation in isotropic media or coupled in free space [39,45]. For instance, a Néel-type skyrmion with hedgehoglike texture [Fig. 1(e1)] can evolve into a Bloch-type skyrmion with vortexlike texture [Fig. 1(e2)], provided they are of the same skyrmion number Ns = 1; however, it is impossible for such a skyrmion to be trans- formed into an antiskyrmion of opposite skyrmion number Ns = −1 [Fig. 1(e3)]. The multiskyrmions composed of multiple elementary skyrmions can still possess the poten- tial of topological stability, which can also be coupled with free-space paraxial propagation; see Supplemental Material [40]. C. Experimental generation and topological control We have designed a series of GRIN-lens cascades to experimentally generate a diverse set of complex photonic quasiparticles with controlled multiple topological num- bers in addition to skyrmion number Ns, which include polarity Np , vorticity Nv , radiality Nr, and centrality Nc. Firstly, we show the topological control of the fundamen- tal skyrmions. Using a GRIN-lens segment with input light of left-handed circular polarization (LCP), we can create a second-order skyrmion of Np = −1 and Nv = 2; see the left side of Fig. 2(a). To control the polarity Np , we tune the input light polarization from LCP to linear polariza- tion and then to right-handed circular polarization (RCP); correspondingly, the skyrmion evolves into biskyrmion and quadrumeron configurations as intermediate states and finally into the skyrmion with opposite polarity [Np = 1 and Nv = 2; see the right side of Fig. 2(a)]. Note that for the case of linear-polarization input, the output quasiparti- cle will exhibit a singular symmetry inducing a skyrmion number of zero, Ns = 0, where the polarity, Np = sgn(Ns), cannot be defined. To control radiality, we apply a cascade of two identical GRIN-lens segments, which results in a skyrmionium; see the left side of Fig. 2(b). Polarity control by tuning input 024025-3 YIJIE SHEN et al. PHYS. REV. APPLIED 21, 024025 (2024) Second-order skyrmion Plane-wave input GRIN lens GRIN lens GRIN lens GRIN lens GRIN lens GRIN lens HWP HWPQWP 4× Skyrmion output (a) (b) (c) (d) (e) Second-order skyrmionBiskyrmion biskyrmion 4�-target quadrumeron tan (E /E ) Input polarization 4�-target quadrumeron Quadrumeron Octuskyrmion Second-order skyrmionium Second-order skyrmioniumQuadrumeroniumBiskyrmionium biskyrmionium Quadruskyrmion 0 π/4–π/4 –π/8 –π/8 QuadruskyrmionQuadrubimeron Octuquarteron pN = 1sN = 0 rN = 1 vN = 2 rN = 2 vN = 0 vN = 2 rN = 1 vN = 4 rN = 1 vN = 2 rN = 4 v y x –1 N = 0 pN = –1 Quadrumeron Quadrumeron vN = 2 vN = 2 vN = 0 FIG. 2. Experimental results for a wide variety of topologically transformed quasiparticles obtained with use of cascaded GRIN lenses of different configurations (see corresponding simulated results in Supplemental Material [40]): (a) a skyrmion of Nr = 1 and Nv = 2 is transformed into a skyrmion of the same radiality and vorticity but opposite polarity, (b) a skyrmionium of Nr = 2 and Nv = 0 is transformed into a skyrmionium of the same radiality and vorticity but opposite polarity, (c) a quadruskyrmion of Nr = 2 and Nv = 4 is transformed into a quadruskyrmion of the same radiality and vorticity but opposite polarity, (d) a quadrumeron of Nr = 1 and Nv = 2 is transformed into a quadrumeron of the same radiality and vorticity but opposite polarity, and (e) a target quadrumeron of Nr = 4 and Nv = 0 is transformed into a target quadrumeron of the same radiality and vorticity but opposite polarity by our tuning the input light polarization from LCP to RCP. The insets show the layouts of optical element cascades for the generation of the diverse topological quasiparticle states in (a)–(e). polarization works for this case as well, and the corre- sponding results are similar to those presented in Fig. 2(a) but with a layer of radially nested structure, resulting in a skyrmion number of zero. To obtain more-complex quasiparticles with increased skyrmion number, we show experimental results for a quadruskyrmion obtained by our applying a cascaded con- figuration of a GRIN lens followed by an HWP followed by another GRIN lens; see the left side of Fig. 2(c), which comprises four skyrmions with the same unit vorticity, with centrality Nc = 4. Polarity control can also be applied to the quadruskyrmion. In addition, by our switching to a cascade involving a QWP followed by an HWP and then a GRIN lens, the quadrumeron can be generated, comprising four elementary merons of the same half charge, but with the same centrality Nc of 4; see Fig. 2(d) [distinct from the quadrumeron in the middle of Fig. 2(a)]. By varying the design of the cascaded GRIN-lens array, we can access increasingly complex topologies. With cas- cading multiple times in the style of a QWP followed by an HWP and then a GRIN lens, we can generate quasi- particles of a multi-radially-nested quadrumeronium and a target quadrumeron. Figure 2(e) shows an experimen- tal quadrumeronium and target quadrumeron with radiality Nr = 4 obtained by a multiple of four cascades of the con- figuration described previously, i.e., a four-times nested matryoshka-like structure based on a basic central struc- ture of a quadrumeron as in Fig. 2(d). All the distributions 024025-4 TOPOLOGICALLY CONTROLLED MULTISKYRMIONS... PHYS. REV. APPLIED 21, 024025 (2024) of the experimental quasiparticle fields and all the topo- logical numbers can be detected by the methods of Stokes polarimetry of structured light, and details are provided in Supplemental Material [40] and Refs. [44,47,48]. On the basis of the principle outlined above, we can summarize a systematic rule for guiding the prac- tical generation of these topological quasiparticles. We note that Ncascade is the number of combinations to be respectively cascaded together, and NGRIN, NQWP, and NHWP are the numbers of GRIN lenses, QWPs, and HWPs in each combination, respectively. For the case of pure GRIN-lens cascading, i.e., each combination includes only one GRIN lens (NGRIN = 1), the relation connecting the topological number and the cascading number can be expressed as (Nr, Nv , Nc) = (Ncascade, 2 × (Ncascade mod 2), 1). For the cases of QWP-HWP–GRIN- lens cascading, the relation connecting the topologi- cal numbers and the elements in the GRIN system can be expressed as (Nr, Nv , Nc) = (Ncascade, 2 × (NGRIN + NQWP) × (Ncascade mod 2), 1). Also, the polarity Np = ±1 and the sign are determined by RCP or LCP input. On the basis of this systematic rule, we can customize quasipar- ticles with on-demand topological numbers with related element cascading design. Moreover, the quasiparticles that can be accessed are not limited to those demonstrated here. For example, we could easily generate and control the following pho- tonic skyrmions: quadruskyrmioniums, octuskyrmions, octumerons, octuskyrmioniums, octumeroniums, etc. Fur- thermore, there is still a huge space to explore in terms of the creation of more-exotic and more-complex quasiparti- cles that have not existed before. III. DISCUSSION In summary, we have created a novel family of topologi- cal quasiparticles controlled by a very compact system, i.e., photonic-GRIN-lens cascades, extending the fundamental skyrmions to exotic radially nested skyrmioniums, multi- skyrmions, and multimerons, which possess sophisticated spin textures characterized by multiple topological num- bers. The scheme presented here could be further extended in terms of diversity of the photonic quasiparticles by tak- ing advantage of the versatility of potential designs of the cascades of the GRIN lenses, which, importantly, can always be integrated and assembled within a very com- pact and confined transverse region. Therefore, this work opens a direction to control skyrmionic topologies of light in more-flexible and more-compact optical systems in the future; for instance, optical skyrmions in fibers. The successful generation of skyrmionic beams from a GRIN system can also motivate more-extended studies on topological beam formation and topological light- matter interaction. One can create an optical field that is skyrmionic by changes that are continuous with respect to the field when it is propagating in the GRIN lenses, where the skyrmion number also changes, but that do not smoothly deform the mapping function. This might beg the question of why one should bother with such photonic topology. Here one must appreciate that care must be taken to ensure that once created, any change is a smooth defor- mation of the map, and here the rules of the game must still be developed—the topic is still very new. But the oppor- tunity is huge: just as we know that matter responds to light with an enhanced effect when the light is structured (chiral light for chiral matter, orbital angular momentum (OAM) light for rotational probes, etc.), so too can we anticipate that topological matter will respond to topologi- cal light. The interaction and exchange of topologies from light to matter, not limited by GRIN lenses, and back holds exciting prospects for many fields. Finally, the photonic quasiparticles proposed and dis- covered here in photonic fields are also highly desired for other communities and in other condensed-matter sys- tems. On the one hand, quasiparticles were transferred to photonics from magnetic and condensed-matter systems to create interdisciplinary studies; on the other hand, pho- tonic methods can inspire new forms of quasiparticles that have not existed before in any system, leading to new fun- damental physical effects that could be explored and new applications such as topological optoelectronic devices. IV. METHODS A. Stokes polarimetry of structured light The experimental reconstruction of the optical quasi- particles is achieved through Stokes polarimetry, more specifically, through the reconstruction of the distributions of three normalized Stokes parameters s1, s2, and s3, which were computed from a set of polarization projection and intensity measurements. Figure 3 demonstrates the setup for Stokes-vector-field measurements. When the gener- ated skyrmionic beams from a GRIN lens or GRIN-lens cascades pass through a QWP and a polarizer, spatial- variance intensity patterns are recorded by a CCD camera. We used the QWP, polarizer, and CCD camera to form a Stokes polarimeter to measure the polarization distribu- tion by rotating the QWP to four different angles. This is a well-known process that was reported in previous technical reviews [47,48]. The principal equations for calculation of … GRIN optics and 4F systemsCCD P QWP FIG. 3. The configuration setup used for Stokes polarimetry. The GRIN-lens cascades are combined with different types of GRIN lens and interstitial optical elements (such as a 4f system). Although the GRIN lens itself can be an imaging device, we use a 4f system for extended imaging purposes. P, polarizer. 024025-5 YIJIE SHEN et al. PHYS. REV. APPLIED 21, 024025 (2024) the polarization field are as follows: Sn out = MP · M n QWP · Sin = MP · M n QWP · (A−1 · I), (2) where Sin is the Stokes vector of the incident light field, MP and M n QWP are Mueller matrices of the corresponding polar- izer and QWP, Sn out is the output Stokes vector for the nth fast-axis orientation state of the QWP, A is an instrument matrix, which is derived from MPM n QWP, and I = A · Sin is the intensity information recorded by the CCD camera. Note that a 2D distributed Stokes vector field Sin(x, y) con- sists of [1, s1(x, y), s2(x, y), s3(x, y)], where s1, s2, and s3 are the vector components of the Stokes vector [47]. B. Detection of quasiparticle topological numbers After reconstruction of the Stokes field of a quasipar- ticle, s(x, y) = [s1(x, y), s2(x, y), s3(x, y)], the number Nr can be easily determined by counting the radially nested structures in the vector field. For the case of Nr > 1, we always care about the basic structure in the center, which we refer to here as the “core structure,” and the outer layers are just the same as the core structure only with staggered polarity. Then we care about the number Nc of the core structure, i.e., how many spin-up or spin-down center points in the core region exist, which is equiva- lent to how many singularities of the transverse compo- nent [s1(x, y), s2(x, y)] are contained in the core region. This singularity-counting problem can be solved by an algorithm for singularity searching of a complex multisin- gularity light field [44]. To further characterize structural details, the skyrmion density distribution of the quasi- particle is calculated by ρs(x, y) = s(x, y) · [∂xs(x, y) × ∂ys(x, y)]. If Nr > 1, the skyrmion density distribution will also correspondingly show a radially nested structure. 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Appl. 10, 1 (2021). 024025-7 https://doi.org/10.1038/s41566-023-01325-7 https://doi.org/10.1126/science.aau0227 https://doi.org/10.1126/science.aba6415 https://doi.org/10.1038/s41567-019-0487-7 https://doi.org/10.1038/s41586-020-3030-1 https://doi.org/10.1021/acsphotonics.1c01703 https://doi.org/10.1038/s41467-021-26171-5 https://doi.org/10.1117/1.AP.5.1.015001 https://doi.org/10.1364/OPTICA.487989 https://doi.org/10.1038/s41467-021-26037-w https://doi.org/10.1038/s41566-022-01028-5 https://doi.org/10.1088/2040-8986/ace4dc https://doi.org/10.1038/s41467-021-21250-z https://doi.org/10.1038/s41467-019-12286-3 https://doi.org/10.1021/acsphotonics.2c01640 https://doi.org/10.1038/nmeth.1339 https://doi.org/10.1038/nprot.2012.078 https://doi.org/10.1038/s41377-019-0228-9 https://doi.org/10.1038/s41566-023-01360-4 https://doi.org/10.1103/PhysRevA.102.053513 https://doi.org/10.1364/OL.431122 http://link.aps.org/supplemental/10.1103/PhysRevApplied.21.024025 https://doi.org/10.1103/PhysRevB.94.094420 https://doi.org/10.1088/1367-2630/ab348e https://doi.org/10.1103/PhysRevLett.119.197205 https://doi.org/10.1515/nanoph-2021-0489 https://doi.org/10.1103/PhysRevA.104.049901 https://doi.org/10.1038/s41566-022-01023-w https://doi.org/10.1117/1.AP.4.2.026001 https://doi.org/10.1038/s41377-021-00639-x I. INTRODUCTION II. RESULTS A. Vectorial optics of GRIN lenses B. Topological photonic quasiparticles C. Experimental generation and topological control III. DISCUSSION IV. METHODS A. Stokes polarimetry of structured light B. Detection of quasiparticle topological numbers ACKNOWLEDGMENTS . 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