Generation of optical skyrmionic beams, topological textures of polarization Stokes vector in the transverse plane and propagating at the longitudinal axis provide a versatile platform for topologically nontrivial optical informatics and light–matter interactions. However, their acceleration along curved trajectories is to be studied. In this study, we experimentally demonstrate the first accelerating skyrmion lattices conveyed by Airy structured light, characterized by topologically stable skyrmion textures with self-acceleration along parabolic trajectories. We quantify topological stability using the Skyrme number Nsk, where |Nsk| = 1 represents an ideal skyrmion. Our experiments demonstrate that the skyrmion unit cell can maintain a Skyrme number |Nsk| > 0.9 within a propagation range of ±1.22 zR (zR is the Rayleigh length) upon parabolic acceleration. Notably, the meron structure remains |Nsk| stable within 0.5 ± 0.02 over a significantly extended range of ±3.06 zR. Our work provides a new potential carrier for topologically robust information distribution, particle sorting, and manipulation.

Topics
Informatics, Topological properties, Optical devices, Polarization, Skyrmions, Orbital angular momentum, Merons

The concept of topological solitons, as stable particle-like field configurations, traces its roots to Lord Kelvin’s 19th-century hypothesis of knotted vortices.1–3 This idea was revived in 1961 by Tony Skyrme, who proposed a topological field theory where atomic nuclei emerge as 3D continuous-field solitons—later termed skyrmions—characterized by an integer Skyrme number quantifying their mapping from real space to the three-sphere.1,4,5 Initially studied in high-energy physics,6 skyrmions gained prominence in optics through their analogy to structured light fields with nontrivial polarization topologies.7–9 Especially, it becomes a mainstream to extend more complex topological structures controlled in higher dimensions of free space,10 such as conformal skyrmions, meron lattices,11,12 bimerons,13,14 multiskyrmions and multimerons,15,16 and hopfions,1,17,18 with more flexible control and resilient propagation.

Unlike the condensed-matter-based skyrmions localized in materials for potential data storage or local data processing,6–9 the optical skyrmions possess the ability to propagate in free space, which is invaluable for practical information transfer applications and nontrivial light–matter interactions.19–22 Recently, exploiting advanced technologies of structured light modulation,23–25 both individual skyrmions and skyrmion lattices can be not only generated26–29 but also propagated in free space with controlled topologies.30–35 However, all the experimentally observed free-space skyrmions can only propagate straightly. Controlling topological skyrmions that spatially accelerate along curved trajectories is crucial for higher-level information sorting or transfer but remains a challenge.

Hence, the primary goals of this study are (1) to develop and experimentally validate a method for generating accelerating skyrmions and (2) to quantitatively assess their topological stability through the evolution of Nsk during propagation. It is important to note that these optical skyrmions (or baby skyrmions)36 represent a reduced-dimensional special case of the 3D particle-like skyrmionic hopfion,1 consisting of 2D topological textures in Stokes space. They are characterized by a 3D Stokes vector field S(x,y)=[S1(x,y),S2(x,y),S3(x,y)]T⁠. These textures exist on the transverse plane of the light field and are constructed via stereographic projection of a 3D Poincaré sphere, exhibiting continuous radial polarization rotation from the core outward [Fig. 1(a)]. To achieve such structures, we generate a vector beam, specifically a Poincaré beam, through the non-separable superposition of orthogonally polarized Laguerre–Gaussian (LG) beams.30,37,38 Subsequently, accelerating skyrmion lattices are obtained via Fourier transform after cubic phase modulation. Here, the term “skyrmion lattice” refers to a periodic lattice-like array of skyrmion unit cells embedded within a single structured beam, distinct from both (i) an ensemble of independent beams and (ii) ideal lattices with perfect periodicity.11,29,39 Through theoretical simulations and experimental verification, we show that the periodic Gaussian- and vortex-like intensity profiles of Airy and vortex Airy-like beams naturally form periodic skyrmion structures. Notably, these skyrmion lattices maintain robust topological properties within the Rayleigh length of the Airy beams during free-space propagation, confirming their stability. These results not only expand the fundamental understanding of skyrmion dynamics but also open new avenues for practical applications in topological information distribution, particle sorting, and manipulation in nontrivial light–matter interactions.

FIG. 1.
Concept of periodically accelerating skyrmion lattices. (a) Schematic of periodic skyrmion lattice generation via cubic phase modulation. The original Bloch-type skyrmion texture (Nsk = −1) transforms into square skyrmion lattices, each unit cell retaining Nsk = −1. The color map represents the Stokes vector orientation mapped to the HSL color space, with maximum saturation. Hue encodes transverse orientation ψ = arg(S1 + iS2), and lightness corresponds to S3. (b) Airy and vortex Airy-like beams, exhibiting periodic intensity lattices that are complementary to each other (highlighted in red and blue regions), accelerate along curved trajectories.
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Concept of periodically accelerating skyrmion lattices. (a) Schematic of periodic skyrmion lattice generation via cubic phase modulation. The original Bloch-type skyrmion texture (Nsk = −1) transforms into square skyrmion lattices, each unit cell retaining Nsk = −1. The color map represents the Stokes vector orientation mapped to the HSL color space, with maximum saturation. Hue encodes transverse orientation ψ = arg(S1 + iS2), and lightness corresponds to S3. (b) Airy and vortex Airy-like beams, exhibiting periodic intensity lattices that are complementary to each other (highlighted in red and blue regions), accelerate along curved trajectories.

FIG. 1.
Concept of periodically accelerating skyrmion lattices. (a) Schematic of periodic skyrmion lattice generation via cubic phase modulation. The original Bloch-type skyrmion texture (Nsk = −1) transforms into square skyrmion lattices, each unit cell retaining Nsk = −1. The color map represents the Stokes vector orientation mapped to the HSL color space, with maximum saturation. Hue encodes transverse orientation ψ = arg(S1 + iS2), and lightness corresponds to S3. (b) Airy and vortex Airy-like beams, exhibiting periodic intensity lattices that are complementary to each other (highlighted in red and blue regions), accelerate along curved trajectories.
View largeDownload slide

Concept of periodically accelerating skyrmion lattices. (a) Schematic of periodic skyrmion lattice generation via cubic phase modulation. The original Bloch-type skyrmion texture (Nsk = −1) transforms into square skyrmion lattices, each unit cell retaining Nsk = −1. The color map represents the Stokes vector orientation mapped to the HSL color space, with maximum saturation. Hue encodes transverse orientation ψ = arg(S1 + iS2), and lightness corresponds to S3. (b) Airy and vortex Airy-like beams, exhibiting periodic intensity lattices that are complementary to each other (highlighted in red and blue regions), accelerate along curved trajectories.

Close modal
To better understand the optical skyrmion lattices described in this study, we first revisit the fundamental theory of skyrmions. The classic Néel-type skyrmion texture, as depicted in the inset of Fig. 1(a), is formed by unwrapping the vectors of a 3D parametric sphere onto a 2D transverse plane, resulting in a skyrmion structure represented by a 3D vector distribution S(x, y). Similarly, unwrapping hemispheres of the sphere forms meron textures, which are also analyzed in this study. Generally, for both skyrmion and meron textures, the topological invariant (the number of times the vector S wraps around the sphere) is quantified by the Skyrme number,4,40,41
(1)
where the integrand ρsk=S⋅∂xS×∂yS represents the Skyrme density. Theoretically, Nsk should be an integer for a perfect skyrmion structure. However, in practice, factors such as the integration region or propagation disturbances can cause deviations from this ideal value.42 Consequently, the experimentally obtained Skyrme number is often used to evaluate the quality of the measured skyrmion structure.26,28,34
Typically, optical skyrmions are created by the coherent combination of orthogonally polarized Laguerre–Gaussian (LG) beams with different topological charges.43 The field distribution is given by
(2)
where α ∈ [0, 1] defines the relative amplitude of the two modes, and LGℓ,p represents the LG mode with azimuthal and radial indices ℓ and p, respectively (see the supplementary material, S1, for detailed expressions). Here, ℓ1≠ℓ2⁠, and eiθ determines the intramodal phase, which governs the skyrmion state, and êR and êL are unit vectors of right-circular polarization (RCP) and left-circular polarization (LCP). By modulating the parameters in Eq. (2), we can construct the distribution of the normalized Stokes vector S(x,y)=S1x,y,S2x,y,S3(x,y)T⁠, which describes the 3D vector distribution of optical topological textures31 in the transverse plane of the light field. For example, if ℓ1 = 1, ℓ2 = 0, and θ = (−π)/2, we can generate a 2D Bloch-type skyrmion structure, whose Skyrme number Nsk = −1, as shown in Fig. 1(a).

Returning to the main focus, a natural route to construct accelerating skyrmion lattices leverages the unique properties of Airy beams, which exhibit non-diffracting, self-accelerating intensity distributions with periodic lattice structures in the transverse plane.44,45 While ideal Airy beams require infinite energy, only finite-energy Airy beams can be realized through the Fourier transform of a Gaussian beam modulated by a cubic phase. Introducing vortex phases during this process results in so-called vortex Airy-like beams. Although it is accepted to call such beams Airy as they retain self-accelerating properties due to their cubic phase,46–48 it should be noted that they are actual spiral-phase-contrast Airy beams. As a result, periodic vortex lattices emerge, while the exact Airy function is no longer preserved. Thereafter, the combination of these beams results in the formation of periodic skyrmion lattices (see the supplementary material, S2, for more details).

The corresponding mathematical process can be described as
(3)
where Aℓ⋅ is vortex Airy-like beam converted by topological charge ℓ, and b is the scaling factor of the Airy function determining the rate of transverse acceleration. Hence, the Bloch-type skyrmion texture mentioned earlier evolves into periodic skyrmion lattices at the Fourier plane, displaying square symmetry in their unit cells. To highlight these structures, a three-by-three grid area is selected. Although the size of the unit cells decreases toward the Airy tails due to the intensity profile of Airy beams, all cells display Néel-type skyrmion textures (see inset) with a consistent Skyrme number of Nsk = −1. The transition between Bloch-type and Néel-type skyrmion textures at different propagation planes (z0 to zf) is attributed to the accumulation of Gouy phase, which varies with the spatial mode order.49 

For ideal Airy beams, the topologically stable skyrmion textures self-accelerate along parabolic trajectories. However, for the finite-energy Airy beams used in this study, topological stability is compromised. As shown in Fig. 1(b), while both simulated Airy and vortex Airy-like beams accelerate along parabolic trajectories, transferring their self-accelerating behavior to skyrmion lattices, significant deformations in the periodic intensity distributions (particularly for vortex Airy-like) occur during propagation. As a result, on the Fourier plane zf, the perfect skyrmion lattices, which exist in the periodic intensity cells of Airy beams as shown in the inset, will inevitably decay during propagation due to these intensity deformations (see the supplementary material, S2, for more details).30 

In this context, we further consider the meron structure, as depicted in the inset of Fig. 1(a). This structure, located within the red ring line (the equatorial projection of the Poincaré sphere) of the skyrmion, is characterized by its half-wrapping of the Poincaré sphere. The Skyrme number for the meron, Nsk=0.5⁠, is not a standard integer topological invariant but rather a numerical index that describes the degree of coverage of the Poincaré sphere. Of particular interest is the potentially enhanced structural stability of meron structures upon propagation, which is localized in skyrmion core regions compared to decaying outer skyrmion domains.

To further demonstrate the accelerating properties of skyrmion lattices, the experimental setup is depicted in Fig. 2. A self-locking polarization Mach–Zehnder interferometer, incorporating two beam displacers (BD-1 and BD-2), forms the core of the system. A continuous 532 nm TEM00 laser beam (CNI Laser MSL-III-532) passes through BD-1, splitting into two orthogonally polarized beams directed to separate regions of a spatial light modulator (SLM, HOLOEYE ERIS-NIR-153). These beams are individually modulated using complex amplitude modulation (CAM) holograms corresponding to the Fourier transforms of the desired Airy and vortex Airy-like beams.50 The beam radius w0 and scaling factor b are set as 1 mm and 3/113⁠, respectively. A half-wave plate (HWP2, fixed at 45°) ensures horizontal polarization for both beams before modulation by the SLM. After modulation, BD-2 recombines the beams. A quarter-wave plate (QWP1, fixed at 45°) and a Fourier lens (L) with a focal length of 200 mm are used to reconstruct the Airy vector beam at the Fourier plane. The propagation tomographies of the Airy beams are recorded via a digital propagation technique integrated into the CAM holograms, eliminating the need for physical translation of the CMOS camera (Allied Vision Alvium 1800 U-240m) along the z axis.51,52 Furthermore, the camera, combined with polarizers (QWP2, HWP3, and PBS), performs spatial Stokes tomography, capturing the structures of skyrmion lattices in Stokes space.

FIG. 2.
Experimental setup. The key components include beam displacer prisms (BD), a spatial light modulator (SLM), a camera (CMOS), a half-wave plate (HWP), and a quarter-wave plate (QWP). The upper-right corner inset displays the holograms used in the experiment. The holograms within the dashed box are used to generate the Airy beams, while the hologram outside the box is coherently superimposed with those inside to perform digital propagation measurements. The lower-right corner inset shows the polarization-dependent intensity profiles obtained via spatial Stokes tomography at the Fourier plane.
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Experimental setup. The key components include beam displacer prisms (BD), a spatial light modulator (SLM), a camera (CMOS), a half-wave plate (HWP), and a quarter-wave plate (QWP). The upper-right corner inset displays the holograms used in the experiment. The holograms within the dashed box are used to generate the Airy beams, while the hologram outside the box is coherently superimposed with those inside to perform digital propagation measurements. The lower-right corner inset shows the polarization-dependent intensity profiles obtained via spatial Stokes tomography at the Fourier plane.

FIG. 2.
Experimental setup. The key components include beam displacer prisms (BD), a spatial light modulator (SLM), a camera (CMOS), a half-wave plate (HWP), and a quarter-wave plate (QWP). The upper-right corner inset displays the holograms used in the experiment. The holograms within the dashed box are used to generate the Airy beams, while the hologram outside the box is coherently superimposed with those inside to perform digital propagation measurements. The lower-right corner inset shows the polarization-dependent intensity profiles obtained via spatial Stokes tomography at the Fourier plane.
View largeDownload slide

Experimental setup. The key components include beam displacer prisms (BD), a spatial light modulator (SLM), a camera (CMOS), a half-wave plate (HWP), and a quarter-wave plate (QWP). The upper-right corner inset displays the holograms used in the experiment. The holograms within the dashed box are used to generate the Airy beams, while the hologram outside the box is coherently superimposed with those inside to perform digital propagation measurements. The lower-right corner inset shows the polarization-dependent intensity profiles obtained via spatial Stokes tomography at the Fourier plane.

Close modal

Figure 3(a) shows the observed 3D profiles of vortex Airy-like (RCP) and Airy (LCP) beams (reconstructed from 26 slices) with a Rayleigh length of zR = 32.701 mm, where both components propagate along diagonal curved trajectories (see the supplementary material, S2, for more details). Transverse intensity profiles below the 3D profiles reveal periodic lattice structures that degrade with propagation, more prominently for vortex Airy-like beams. At z = 100 mm, the vortex lattices even become indistinguishable. Unlike skyrmion beams that propagate straight and form skyrmion tubes along the propagation direction, the skyrmion lattices in this study extend as parallel and curved skyrmion tubes in 3D space, forming curved skyrmion density tubes, as shown in Fig. 3(b). Hence, although 3D polarization distributions have been measured, no 3D topological features like hopfions exist in this study due to the presence of parallel skyrmion tubes.

FIG. 3.
Experimental results for periodically accelerating skyrmion lattices. (a) 3D intensity profiles of vortex Airy-like (RCP) and Airy (LCP) beams, reconstructed from experimental measurements, with selectively sampled transverse intensity lattices at different propagation distances (0, 32, 64, and 100 mm). (b) 3D skyrmion density distributions corresponding to the self-accelerating intensity lattices. (c) Detailed distributions of the Stokes vector and skyrmion density at different z planes. As the propagation distance increases, skyrmion textures degrade, and positive ρsk values (red) gradually emerge alongside the initially negative ρsk lattices (blue).
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Experimental results for periodically accelerating skyrmion lattices. (a) 3D intensity profiles of vortex Airy-like (RCP) and Airy (LCP) beams, reconstructed from experimental measurements, with selectively sampled transverse intensity lattices at different propagation distances (0, 32, 64, and 100 mm). (b) 3D skyrmion density distributions corresponding to the self-accelerating intensity lattices. (c) Detailed distributions of the Stokes vector and skyrmion density at different z planes. As the propagation distance increases, skyrmion textures degrade, and positive ρsk values (red) gradually emerge alongside the initially negative ρsk lattices (blue).

FIG. 3.
Experimental results for periodically accelerating skyrmion lattices. (a) 3D intensity profiles of vortex Airy-like (RCP) and Airy (LCP) beams, reconstructed from experimental measurements, with selectively sampled transverse intensity lattices at different propagation distances (0, 32, 64, and 100 mm). (b) 3D skyrmion density distributions corresponding to the self-accelerating intensity lattices. (c) Detailed distributions of the Stokes vector and skyrmion density at different z planes. As the propagation distance increases, skyrmion textures degrade, and positive ρsk values (red) gradually emerge alongside the initially negative ρsk lattices (blue).
View largeDownload slide

Experimental results for periodically accelerating skyrmion lattices. (a) 3D intensity profiles of vortex Airy-like (RCP) and Airy (LCP) beams, reconstructed from experimental measurements, with selectively sampled transverse intensity lattices at different propagation distances (0, 32, 64, and 100 mm). (b) 3D skyrmion density distributions corresponding to the self-accelerating intensity lattices. (c) Detailed distributions of the Stokes vector and skyrmion density at different z planes. As the propagation distance increases, skyrmion textures degrade, and positive ρsk values (red) gradually emerge alongside the initially negative ρsk lattices (blue).

Close modal

To further evaluate topological stability, Fig. 3(c) illustrates four slices of skyrmion lattices and their corresponding skyrmion densities, focusing on the same three-by-three grid area as in Fig. 1, at different z positions. At the Fourier plane (z = 0), perfectly periodic Stokes vector and skyrmion density distributions are observed, consistent with the theoretical results in Fig. 1(a). However, as propagation distance increases, vortex lattice degradation leads to skyrmion cell deformation, a reduction in the S3 = 1 area [white region of S in Fig. 3(c)], and the emergence of abnormal positive ρsk values (theoretical references for these results are provided in Fig. S3). These results provide an intuitive understanding that the topological quality of the skyrmion lattices gradually degrades during propagation.

To further quantitatively analyze the stability, the Skyrme numbers of three cells of skyrmion lattices (labeled 1, 2, and 3) were calculated, as shown in Fig. 4(a). As the lattices degrade, the Skyrme number Nsk gradually decreases with distance, with cell 2 showing the fastest decline due to its smaller integration area, which becomes insufficient for maintaining accuracy with increasing lattice complexity. Despite these challenges, skyrmion lattices maintain stable acceleration over a range of ±40 mm (±1.22 zR), with Nsk>0.9⁠. Beyond this range, Nsk drops to approximately −0.6.

FIG. 4.
Quantitative analysis of accelerating skyrmion and meron lattices. (a) Skyrme number for skyrmion lattices, showing a gradual decrease with propagation distance. (b) Skyrme number for meron lattices, remaining stable over propagation distance. Point data correspond to experimental observations, and solid lines represent theoretical results.
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Quantitative analysis of accelerating skyrmion and meron lattices. (a) Skyrme number for skyrmion lattices, showing a gradual decrease with propagation distance. (b) Skyrme number for meron lattices, remaining stable over propagation distance. Point data correspond to experimental observations, and solid lines represent theoretical results.

FIG. 4.
Quantitative analysis of accelerating skyrmion and meron lattices. (a) Skyrme number for skyrmion lattices, showing a gradual decrease with propagation distance. (b) Skyrme number for meron lattices, remaining stable over propagation distance. Point data correspond to experimental observations, and solid lines represent theoretical results.
View largeDownload slide

Quantitative analysis of accelerating skyrmion and meron lattices. (a) Skyrme number for skyrmion lattices, showing a gradual decrease with propagation distance. (b) Skyrme number for meron lattices, remaining stable over propagation distance. Point data correspond to experimental observations, and solid lines represent theoretical results.

Close modal

Interestingly, meron lattices, which occupy the core regions of skyrmion structures, demonstrate a greater level of robustness. As shown in Fig. 4(b), meron lattices maintain stability across the entire measured range (±3.06 zR), with Nsk ≈ −0.5. This stability is achieved at the expense of reduced S3 = 1 regions, as the inner topological structures are better preserved.

Overall, Skyrmion lattices in this work not only inherit the stable acceleration and self-healing properties of Airy beams53–55 (see the supplementary material, S3, for details of self-healing) but also introduce an additional degree of freedom due to their intricate topological structures. These features not only unlock new opportunities for exploring their dynamic behavior but also show promise for practical applications.

We have experimentally demonstrated a versatile family of optical skyrmion lattices capable of accelerating along curved trajectories in free space. These lattices are generated by imprinting cubic phase modulation onto traditional skyrmion beams. The transverse acceleration rate of these trajectories can be flexibly controlled by adjusting the scaling factor b. Importantly, only LG beams that form skyrmion structures with odd topological charge differences can generate skyrmion lattices.

Leveraging the non-diffracting and self-accelerating nature of Airy beams, skyrmion lattices exhibit stable transverse acceleration and robust topological properties over a propagation range of ±1.22 zR, even under propagation-induced lattice deformation. Greater robustness is observed for Meron lattices, which occupy the core regions of skyrmion structures and remain stable over an extended range of ±3.06 zR.

These unique features establish skyrmion lattices as a promising platform for exploring advanced applications such as particle sorting,56–58 advanced microscopy,57–62 plasma channel induction,57,63 electron beam manipulation,57,58 and guiding electric discharges,64 thereby opening new frontiers in structured light research.

See the supplementary material for more details of Laguerre–Gaussian modes, Airy, and vortex Airy beams.

Singapore Ministry of Education (MOE) AcRF Tier 1 (Grant No. RG157/23), MoE AcRF Tier 1 Thematic (Grant No. RT11/23), Imperial-Nanyang Technological University Collaboration Fund (Grant No. INCF-2024-007), Singapore Agency for Science, Technology and Research (A*STAR) MTC Individual Research Grants (Grant No. M24N7c0080), Nanyang Assistant Professorship Start Up Grant, and National Natural Science Foundation of China (Grant Nos. 12474324, 62075050, and 11934013).

The authors have no conflicts to disclose.

Haijun Wu: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Weijie Zhou: Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Zhihan Zhu: Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Yijie Shen: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
D.
Sugic
,
R.
Droop
,
E.
Otte
,
D.
Ehrmanntraut
,
F.
Nori
,
J.
Ruostekoski
,
C.
Denz
, and
M. R.
Dennis
, “
Particle-like topologies in light
,”
Nat. Commun.
12
,
6785
(
2021
).
https://doi.org/10.1038/s41467-021-26171-5
Google Scholar
Crossref
Search ADS
PubMed
 
2.
W.
Thomson
, “
4. On vortex atoms
,”
Proc. R. Soc. Edinburgh
6
,
94
–
105
(
1869
).
https://doi.org/10.1017/s0370164600045430
Google Scholar
Crossref
Search ADS
 
3.
T. H. R.
Skyrme
, “
A non-linear field theory
,”
Proc. R. Soc. London, Ser. A
260
,
127
–
138
(
1961
).
https://doi.org/10.1098/rspa.1961.0018
Google Scholar
Crossref
Search ADS
 
4.
Y.
Shen
,
Q.
Zhang
,
P.
Shi
,
L.
Du
,
X.
Yuan
, and
A. V.
Zayats
, “
Optical skyrmions and other topological quasiparticles of light
,”
Nat. Photonics
18
,
15
–
25
(
2024
).
https://doi.org/10.1038/s41566-023-01325-7
Google Scholar
Crossref
Search ADS
 
5.
B. A.
Bernevig
,
C.
Felser
, and
H.
Beidenkopf
, “
Progress and prospects in magnetic topological materials
,”
Nature
603
,
41
–
51
(
2022
).
https://doi.org/10.1038/s41586-021-04105-x
Google Scholar
Crossref
Search ADS
PubMed
 
6.
A. N.
Bogdanov
 and
C.
Panagopoulos
, “
Physical foundations and basic properties of magnetic skyrmions
,”
Nat. Rev. Phys.
2
,
492
–
498
(
2020
).
https://doi.org/10.1038/s42254-020-0203-7
Google Scholar
Crossref
Search ADS
 
7.
A.
Fert
,
N.
Reyren
, and
V.
Cros
, “
Magnetic skyrmions: Advances in physics and potential applications
,”
Nat. Rev. Mater.
2
,
17031
(
2017
).
https://doi.org/10.1038/natrevmats.2017.31
Google Scholar
Crossref
Search ADS
 
8.
L.
Han
,
C.
Addiego
,
S.
Prokhorenko
,
M.
Wang
,
H.
Fu
,
Y.
Nahas
,
X.
Yan
,
S.
Cai
,
T.
Wei
,
Y.
Fang
 et al, “
High-density switchable skyrmion-like polar nanodomains integrated on silicon
,”
Nature
603
,
63
–
67
(
2022
).
https://doi.org/10.1038/s41586-021-04338-w
Google Scholar
Crossref
Search ADS
PubMed
 
9.
S.
Chen
,
J.
Lourembam
,
P.
Ho
,
A. K. J.
Toh
,
J.
Huang
,
X.
Chen
,
H. K.
Tan
,
S. L. K.
Yap
,
R. J. J.
Lim
,
H. R.
Tan
 et al, “
All-electrical skyrmionic magnetic tunnel junction
,”
Nature
627
,
522
–
527
(
2024
).
https://doi.org/10.1038/s41586-024-07131-7
Google Scholar
Crossref
Search ADS
PubMed
 
10.
Y.
Shen
,
H.
Wang
, and
S.
Fan
, “
Free-space topological optical textures: Tutorial
,”
Adv. Opt. Photonics
17
,
295
–
374
(
2025
).
https://doi.org/10.1364/aop.547634
Google Scholar
Crossref
Search ADS
 
11.
D.
Marco
,
I.
Herrera
,
S.
Brasselet
, and
M. A.
Alonso
, “
Propagation-invariant optical meron lattices
,”
ACS Photonics
11
,
2397
–
2405
(
2024
).
https://doi.org/10.1021/acsphotonics.4c00292
Google Scholar
Crossref
Search ADS
 
12.
D.
Marco
,
I.
Herrera
,
S.
Brasselet
, and
M. A.
Alonso
, “
Periodic skyrmionic textures via conformal cartographic projections
,”
APL Photonics
9
,
110803
(
2024
).
https://doi.org/10.1063/5.0230959
Google Scholar
Crossref
Search ADS
 
13.
Y.
Shen
, “
Topological bimeronic beams
,”
Opt. Lett.
46
,
3737
–
3740
(
2021
).
https://doi.org/10.1364/ol.431122
Google Scholar
Crossref
Search ADS
PubMed
 
14.
J.
Berškys
 and
S.
Orlov
, “
Accelerating Airy beams with particle-like polarization topologies and free-space bimeronic lattices
,”
Opt. Lett.
48
,
1168
–
1171
(
2023
).
https://doi.org/10.1364/ol.483339
Google Scholar
Crossref
Search ADS
PubMed
 
15.
Y.
Shen
,
C.
He
,
Z.
Song
,
B.
Chen
,
H.
He
,
Y.
Ma
,
J. A. J.
Fells
,
S. J.
Elston
,
S. M.
Morris
,
M. J.
Booth
 et al, “
Topologically controlled multiskyrmions in photonic gradient-index lenses
,”
Phys. Rev. Appl.
21
,
024025
(
2024
).
https://doi.org/10.1103/physrevapplied.21.024025
Google Scholar
Crossref
Search ADS
 
16.
A.
McWilliam
,
C. M.
Cisowski
,
Z.
Ye
,
F. C.
Speirits
,
J. B.
Götte
,
S. M.
Barnett
, and
S.
Franke-Arnold
, “
Topological approach of characterizing optical skyrmions and multi-skyrmions
,”
Laser Photonics Rev.
17
,
2300155
(
2023
).
https://doi.org/10.1002/lpor.202300155
Google Scholar
Crossref
Search ADS
 
17.
Y.
Shen
,
B.
Yu
,
H.
Wu
,
C.
Li
,
Z.
Zhu
, and
A. V.
Zayats
, “
Topological transformation and free-space transport of photonic hopfions
,”
Adv. Photonics
5
,
015001
(
2023
).
https://doi.org/10.1117/1.ap.5.1.015001
Google Scholar
Crossref
Search ADS
 
18.
R.
Tamura
,
S. R.
Allam
,
N. M.
Litchinitser
, and
T.
Omatsu
, “
Three-dimensional projection of optical hopfion textures in a material
,”
ACS Photonics
11
,
4958
–
4965
(
2024
).
https://doi.org/10.1021/acsphotonics.4c01547
Google Scholar
Crossref
Search ADS
 
19.
Z.
Wan
,
H.
Wang
,
Q.
Liu
,
X.
Fu
, and
Y.
Shen
, “
Ultra-degree-of-freedom structured light for ultracapacity information carriers
,”
ACS Photonics
10
,
2149
–
2164
(
2023
).
https://doi.org/10.1021/acsphotonics.2c01640
Google Scholar
Crossref
Search ADS
 
20.
R.
Wang
,
P.-Y.
Bao
,
X.
Feng
,
J.
Wu
,
B.-Z.
Wang
, and
Y.
Shen
, “
Single-antenna super-resolution positioning with nonseparable toroidal pulses
,”
Commun. Phys.
7
,
356
(
2024
).
https://doi.org/10.1038/s42005-024-01850-z
Google Scholar
Crossref
Search ADS
 
21.
D.
Habibović
,
K. R.
Hamilton
,
O.
Neufeld
, and
L.
Rego
, “
Emerging tailored light sources for studying chirality and symmetry
,”
Nat. Rev. Phys.
6
,
663
–
675
(
2024
).
https://doi.org/10.1038/s42254-024-00763-8
Google Scholar
Crossref
Search ADS
 
22.
R.
Tamura
,
P.
Kumar
,
A. S.
Rao
,
K.
Tsuda
,
F.
Getzlaff
,
K.
Miyamoto
,
N. M.
Litchinitser
, and
T.
Omatsu
, “
Direct imprint of optical skyrmions in azopolymers as photoinduced relief structures
,”
APL Photonics
9
,
046104
(
2024
).
https://doi.org/10.1063/5.0192239
Google Scholar
Crossref
Search ADS
 
23.
A.
Forbes
,
M.
de Oliveira
, and
M. R.
Dennis
, “
Structured light
,”
Nat. Photonics
15
,
253
–
262
(
2021
).
https://doi.org/10.1038/s41566-021-00780-4
Google Scholar
Crossref
Search ADS
 
24.
C.
He
,
Y.
Shen
, and
A.
Forbes
, “
Towards higher-dimensional structured light
,”
Light: Sci. Appl.
11
,
205
(
2022
).
https://doi.org/10.1038/s41377-022-00897-3
Google Scholar
Crossref
Search ADS
PubMed
 
25.
C.
Rosales-Guzmán
 and
V.
Rodríguez-Fajardo
, “
A perspective on structured light’s applications
,”
Appl. Phys. Lett.
125
,
200503
(
2024
).
https://doi.org/10.1063/5.0236477
Google Scholar
Crossref
Search ADS
 
26.
S.
Tsesses
,
E.
Ostrovsky
,
K.
Cohen
,
B.
Gjonaj
,
N. H.
Lindner
, and
G.
Bartal
, “
Optical skyrmion lattice in evanescent electromagnetic fields
,”
Science
361
,
993
–
996
(
2018
).
https://doi.org/10.1126/science.aau0227
Google Scholar
Crossref
Search ADS
PubMed
 
27.
L.
Du
,
A.
Yang
,
A. V.
Zayats
, and
X.
Yuan
, “
Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum
,”
Nat. Phys.
15
,
650
–
654
(
2019
).
https://doi.org/10.1038/s41567-019-0487-7
Google Scholar
Crossref
Search ADS
 
28.
T. J.
Davis
,
D.
Janoschka
,
P.
Dreher
,
B.
Frank
,
F.-J.
Meyer zu Heringdorf
, and
H.
Giessen
, “
Ultrafast vector imaging of plasmonic skyrmion dynamics with deep subwavelength resolution
,”
Science
368
,
eaba6415
(
2020
).
https://doi.org/10.1126/science.aba6415
Google Scholar
Crossref
Search ADS
PubMed
 
29.
Y.
Dai
,
Z.
Zhou
,
A.
Ghosh
,
R. S. K.
Mong
,
A.
Kubo
,
C.-B.
Huang
, and
H.
Petek
, “
Plasmonic topological quasiparticle on the nanometre and femtosecond scales
,”
Nature
588
,
616
–
619
(
2020
).
https://doi.org/10.1038/s41586-020-3030-1
Google Scholar
Crossref
Search ADS
PubMed
 
30.
S.
Gao
,
F. C.
Speirits
,
F.
Castellucci
,
S.
Franke-Arnold
,
S. M.
Barnett
, and
J. B.
Götte
, “
Paraxial skyrmionic beams
,”
Phys. Rev. A
102
,
053513
(
2020
).
https://doi.org/10.1103/physreva.102.053513
Google Scholar
Crossref
Search ADS
 
31.
Y.
Shen
,
E. C.
Martínez
, and
C.
Rosales-Guzmán
, “
Generation of optical skyrmions with tunable topological textures
,”
ACS Photonics
9
,
296
–
303
(
2022
).
https://doi.org/10.1021/acsphotonics.1c01703
Google Scholar
Crossref
Search ADS
 
32.
V.
Hakobyan
,
Y.
Shen
, and
E.
Brasselet
, “
Unitary spin-orbit optical-skyrmionic wave plates
,”
Phys. Rev. Appl.
22
,
054038
(
2024
).
https://doi.org/10.1103/physrevapplied.22.054038
Google Scholar
Crossref
Search ADS
 
33.
T.
He
,
Y.
Meng
,
L.
Wang
,
H.
Zhong
,
N.
Mata-Cervera
,
D.
Li
,
P.
Yan
,
Q.
Liu
,
Y.
Shen
, and
Q.
Xiao
, “
Optical skyrmions from metafibers with subwavelength features
,”
Nat. Commun.
15
,
10141
(
2024
).
https://doi.org/10.1038/s41467-024-54207-z
Google Scholar
Crossref
Search ADS
PubMed
 
34.
W. R.
Kerridge-Johns
,
A. S.
Rao
, and
T.
Omatsu
, “
Optical skyrmion laser using a wedged output coupler
,”
Optica
11
,
769
–
775
(
2024
).
https://doi.org/10.1364/optica.521901
Google Scholar
Crossref
Search ADS
 
35.
W.
Lin
,
Y.
Ota
,
Y.
Arakawa
, and
S.
Iwamoto
, “
On-chip optical skyrmionic beam generators
,”
Optica
11
,
1588
–
1594
(
2024
).
https://doi.org/10.1364/optica.540469
Google Scholar
Crossref
Search ADS
 
36.
C. D.
Parmee
,
M. R.
Dennis
, and
J.
Ruostekoski
, “
Optical excitations of skyrmions, knotted solitons, and defects in atoms
,”
Commun. Phys.
5
,
54
(
2022
).
https://doi.org/10.1038/s42005-022-00829-y
Google Scholar
Crossref
Search ADS
 
37.
A. M.
Beckley
,
T. G.
Brown
, and
M. A.
Alonso
, “
Full Poincaré beams
,”
Opt. Express
18
,
10777
–
10785
(
2010
).
https://doi.org/10.1364/oe.18.010777
Google Scholar
Crossref
Search ADS
PubMed
 
38.
S.
Donati
,
L.
Dominici
,
G.
Dagvadorj
,
D.
Ballarini
,
M.
De Giorgi
,
A.
Bramati
,
G.
Gigli
,
Y. G.
Rubo
,
M. H.
Szymańska
, and
D.
Sanvitto
, “
Twist of generalized skyrmions and spin vortices in a polariton superfluid
,”
Proc. Natl. Acad. Sci. U. S. A.
113
,
14926
–
14931
(
2016
).
https://doi.org/10.1073/pnas.1610123114
Google Scholar
Crossref
Search ADS
PubMed
 
39.
J. M.
Hickmann
,
E. J. S.
Fonseca
,
W. C.
Soares
, and
S.
Chávez-Cerda
, “
Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum
,”
Phys. Rev. Lett.
105
,
053904
(
2010
).
https://doi.org/10.1103/physrevlett.105.053904
Google Scholar
Crossref
Search ADS
PubMed
 
40.
B.
Göbel
,
I.
Mertig
, and
O. A.
Tretiakov
, “
Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles
,”
Phys. Rep.
895
,
1
–
28
(
2021
).
https://doi.org/10.1016/j.physrep.2020.10.001
Google Scholar
Crossref
Search ADS
 
41.
N.
Nagaosa
 and
Y.
Tokura
, “
Topological properties and dynamics of magnetic skyrmions
,”
Nat. Nanotechnol.
8
,
899
–
911
(
2013
).
https://doi.org/10.1038/nnano.2013.243
Google Scholar
Crossref
Search ADS
PubMed
 
42.
D.
Raftrey
,
S.
Finizio
,
R. V.
Chopdekar
,
S.
Dhuey
,
T.
Bayaraa
,
P.
Ashby
,
J.
Raabe
,
T.
Santos
,
S.
Griffin
, and
P.
Fischer
, “
Quantifying the topology of magnetic skyrmions in three dimensions
,”
Sci. Adv.
10
,
eadp8615
(
2024
).
https://doi.org/10.1126/sciadv.adp8615
Google Scholar
Crossref
Search ADS
PubMed
 
43.
L.
Allen
,
M. W.
Beijersbergen
,
R. J. C.
Spreeuw
, and
J. P.
Woerdman
, “
Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes
,”
Phys. Rev. A
45
,
8185
(
1992
).
https://doi.org/10.1103/physreva.45.8185
Google Scholar
Crossref
Search ADS
PubMed
 
44.
N. K.
Efremidis
,
Z.
Chen
,
M.
Segev
, and
D. N.
Christodoulides
, “
Airy beams and accelerating waves: An overview of recent advances
,”
Optica
6
,
686
–
701
(
2019
).
https://doi.org/10.1364/optica.6.000686
Google Scholar
Crossref
Search ADS
 
45.
Z.
Shi
,
J.
Xue
,
X.
Zhu
,
Y.
Xiang
, and
H.
Li
, “
Interaction of Airy-Gaussian beams in photonic lattices with defects
,”
Phys. Rev. E
95
,
042209
(
2017
).
https://doi.org/10.1103/physreve.95.042209
Google Scholar
Crossref
Search ADS
PubMed
 
46.
H. T.
Dai
,
Y. J.
Liu
,
D.
Luo
, and
X. W.
Sun
, “
Propagation dynamics of an optical vortex imposed on an Airy beam
,”
Opt. Lett.
35
,
4075
–
4077
(
2010
).
https://doi.org/10.1364/ol.35.004075
Google Scholar
Crossref
Search ADS
PubMed
 
47.
X.
Qiu
,
F.
Li
,
W.
Zhang
,
Z.
Zhu
, and
L.
Chen
, “
Spiral phase contrast imaging in nonlinear optics: Seeing phase objects using invisible illumination
,”
Optica
5
,
208
–
212
(
2018
).
https://doi.org/10.1364/optica.5.000208
Google Scholar
Crossref
Search ADS
 
48.
J.
Pan
,
H.
Wang
,
Y.
Shen
,
X.
Fu
, and
Q.
Liu
, “
Airy coherent vortices: 3D multilayer self-accelerating structured light
,”
Appl. Phys. Lett.
121
,
141102
(
2022
).
https://doi.org/10.1063/5.0104188
Google Scholar
Crossref
Search ADS
 
49.
R.-Y.
Zhong
,
Z.-H.
Zhu
,
H.-J.
Wu
,
C.
Rosales-Guzmán
,
S.-W.
Song
, and
B.-S.
Shi
, “
Gouy-phase-mediated propagation variations and revivals of transverse structure in vectorially structured light
,”
Phys. Rev. A
103
,
053520
(
2021
).
https://doi.org/10.1103/physreva.103.053520
Google Scholar
Crossref
Search ADS
 
50.
C.
Rosales-Guzmán
 and
A.
Forbes
,
How to Shape Light with Spatial Light Modulators
(SPIE, Bellingham, 2017).
https://doi.org/10.1117/3.2281295
Crossref
Search ADS
 
51.
W.
Osten
,
A.
Faridian
,
P.
Gao
,
K.
Körner
,
D.
Naik
,
G.
Pedrini
,
A. K.
Singh
,
M.
Takeda
, and
M.
Wilke
, “
Recent advances in digital holography (invited)
,”
Appl. Opt.
53
,
G44
–
G63
(
2014
).
https://doi.org/10.1364/ao.53.000g44
Google Scholar
Crossref
Search ADS
PubMed
 
52.
C.
Schulze
,
D.
Flamm
,
M.
Duparré
, and
A.
Forbes
, “
Beam-quality measurements using a spatial light modulator
,”
Opt. Lett.
37
,
4687
–
4689
(
2012
).
https://doi.org/10.1364/OL.37.004687
Google Scholar
Crossref
Search ADS
PubMed
 
53.
X.
Chu
,
G.
Zhou
, and
R.
Chen
, “
Analytical study of the self-healing property of Airy beams
,”
Phys. Rev. A
85
,
013815
(
2012
).
https://doi.org/10.1103/physreva.85.013815
Google Scholar
Crossref
Search ADS
 
54.
J.
Broky
,
G. A.
Siviloglou
,
A.
Dogariu
, and
D. N.
Christodoulides
, “
Self-healing properties of optical Airy beams
,”
Opt. Express
16
,
12880
–
12891
(
2008
).
https://doi.org/10.1364/oe.16.012880
Google Scholar
Crossref
Search ADS
PubMed
 
55.
L.
Zhang
,
F.
Ye
,
M.
Cao
,
D.
Wei
,
P.
Zhang
,
H.
Gao
, and
F.
Li
, “
Investigating the self-healing property of an optical Airy beam
,”
Opt. Lett.
40
,
5066
–
5069
(
2015
).
https://doi.org/10.1364/ol.40.005066
Google Scholar
Crossref
Search ADS
PubMed
 
56.
J.
Baumgartl
,
M.
Mazilu
, and
K.
Dholakia
, “
Optically mediated particle clearing using Airy wavepackets
,”
Nat. Photonics
2
,
675
–
678
(
2008
).
https://doi.org/10.1038/nphoton.2008.201
Google Scholar
Crossref
Search ADS
 
57.
R.
Schley
,
I.
Kaminer
,
E.
Greenfield
,
R.
Bekenstein
,
Y.
Lumer
, and
M.
Segev
, “
Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories
,”
Nat. Commun.
5
,
5189
(
2014
).
https://doi.org/10.1038/ncomms6189
Google Scholar
Crossref
Search ADS
PubMed
 
58.
N.
Voloch-Bloch
,
Y.
Lereah
,
Y.
Lilach
,
A.
Gover
, and
A.
Arie
, “
Generation of electron Airy beams
,”
Nature
494
,
331
–
335
(
2013
).
https://doi.org/10.1038/nature11840
Google Scholar
Crossref
Search ADS
PubMed
 
59.
S.
Jia
,
J. C.
Vaughan
, and
X.
Zhuang
, “
Isotropic three-dimensional super-resolution imaging with a self-bending point spread function
,”
Nat. Photonics
8
,
302
–
306
(
2014
).
https://doi.org/10.1038/nphoton.2014.13
Google Scholar
Crossref
Search ADS
PubMed
 
60.
T.
Vettenburg
,
H. I. C.
Dalgarno
,
J.
Nylk
,
C.
Coll-Lladó
,
D. E. K.
Ferrier
,
T.
Čižmár
,
F. J.
Gunn-Moore
, and
K.
Dholakia
, “
Light-sheet microscopy using an Airy beam
,”
Nat. Methods
11
,
541
–
544
(
2014
).
https://doi.org/10.1038/nmeth.2922
Google Scholar
Crossref
Search ADS
PubMed
 
61.
J.
Nylk
,
K.
McCluskey
,
M. A.
Preciado
,
M.
Mazilu
,
Z.
Yang
,
F. J.
Gunn-Moore
,
S.
Aggarwal
,
J. A.
Tello
,
D. E. K.
Ferrier
, and
K.
Dholakia
, “
Light-sheet microscopy with attenuation-compensated propagation-invariant beams
,”
Sci. Adv.
4
,
eaar4817
(
2018
).
https://doi.org/10.1126/sciadv.aar4817
Google Scholar
Crossref
Search ADS
PubMed
 
62.
J.
Wang
,
X.
Hua
,
C.
Guo
,
W.
Liu
, and
S.
Jia
, “
Airy-beam tomographic microscopy
,”
Optica
7
,
790
–
793
(
2020
).
https://doi.org/10.1364/optica.389894
Google Scholar
Crossref
Search ADS
PubMed
 
63.
P.
Polynkin
,
M.
Kolesik
,
J. V.
Moloney
,
G. A.
Siviloglou
, and
D. N.
Christodoulides
, “
Curved plasma channel generation using ultraintense Airy beams
,”
Science
324
,
229
–
232
(
2009
).
https://doi.org/10.1126/science.1169544
Google Scholar
Crossref
Search ADS
PubMed
 
64.
M.
Clerici
,
Y.
Hu
,
P.
Lassonde
,
C.
Milián
,
A.
Couairon
,
D. N.
Christodoulides
,
Z.
Chen
,
L.
Razzari
,
F.
Vidal
,
F.
Légaré
 et al, “
Laser-assisted guiding of electric discharges around objects
,”
Sci. Adv.
1
, e1400111 (2015).
https://doi.org/10.1126/sciadv.1400111
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