Paraxial Light Skyrmions Overview

• Paraxial light skyrmions are optical quasiparticles formed by paraxial vector beams with nontrivial transverse polarization textures defined via normalized Stokes vectors.
• They are generated through coherent superpositions of orthogonally polarized spatial modes, where tunable orbital angular momentum differences yield Néel, Bloch, or anti-skyrmionic states.
• Their robust topological invariants and tailored polarization textures enable advancements in optical manipulation, communication, and structured light engineering.

Paraxial light skyrmions are optical quasiparticles in which a paraxial vector beam carries a nontrivial transverse polarization texture whose normalized local Stokes or Poincaré vector wraps the Poincaré sphere. In the paraxial regime, the skyrmionic order parameter is therefore typically not the full electric-field vector $(E_x,E_y,E_z)$, but the local polarization-state vector reconstructed from a two-component optical state. This distinguishes paraxial Stokes skyrmions from evanescent electric-field skyrmions in plasmonics and from other nonparaxial free-space constructions based on spin density or Poynting vector fields (Shen et al., 2022, Gao et al., 2019).

1. Definition and historical positioning

The modern optical-skyrmion literature began with evanescent electromagnetic fields. In “Optical skyrmions in evanescent electromagnetic fields” (Tsesses et al., 2018), the skyrmion variable is the normalized electric-field direction of interfering TM surface plasmon polaritons, and the construction is explicitly guided, evanescent, and non-paraxial. That work is foundational for optical-skyrmion topology, but it is not a paraxial free-space realization.

The paraxial turn came from vector-vortex-beam theory. “Paraxial Skyrmionic beams” (Gao et al., 2019) formulated skyrmions as a topological property of freely propagating paraxial vector beams, with the relevant order parameter given by the local Poincaré or Bloch vector of a two-component spinor field. This established that ordinary paraxial beams can support skyrmionic topology when the spatial variation of polarization and modal amplitude is treated jointly.

Experimental free-space realization followed quickly. “Synthesis and observation of optical skyrmionic structure in free space” (Zhu et al., 2021) implemented skyrmionic optical structures in a linear free-space system using orthogonally polarized paraxial transverse modes, while “Generation of tunable optical skyrmions on Skyrme-Poincaré sphere” (Shen et al., 2021) demonstrated tunable free-space skyrmions whose realized texture is explicitly a Stokes-vector texture built from fundamental Gaussian and first-order Laguerre–Gaussian modes. The review “Optical skyrmions and other topological quasiparticles of light” (Shen et al., 2022) subsequently codified the now-standard distinction: paraxial optical skyrmions are accepted primarily as Stokes skyrmions of vector beams, not as strict electric-field skyrmions of a purely transverse propagating field.

2. Order parameter, topology, and classification

For paraxial light skyrmions, the central object is a normalized three-component polarization order parameter. In the vector-beam formulation of (Gao et al., 2019), a local normalized state

$$|\psi(\mathbf r)\rangle = \frac{|0\rangle+e^{i\theta_0}v(\mathbf r)|1\rangle}{\sqrt{1+|v(\mathbf r)|^2}}$$

defines a local Poincaré/Bloch vector

$$\mathbf M(\mathbf r)=\langle \psi(\mathbf r)|\boldsymbol{\sigma}|\psi(\mathbf r)\rangle,$$

which, for light, is the normalized Stokes vector. In the free-space experiment of (Zhu et al., 2021), the analogous polarization pseudospin is written

$$\bm M=\langle\Phi(\bm r)|{\bm \sigma}|\Phi(\bm r)\rangle.$$

The topological invariant is the standard skyrmion number. In the Stokes-vector language used across the paraxial literature,

$$s=\frac{1}{4\pi }\iint_\sigma \mathbf{S}\cdot \left( \frac{\partial \mathbf{S}}{\partial x}\times \frac{\partial \mathbf{S}}{\partial y} \right)\,dx\,dy,$$

or equivalently, in the notation of (Zhu et al., 2021),

$$N=\frac{1}{4\pi}\int \Sigma_z\,dxdy, \qquad \Sigma_i=\frac{1}{2}\epsilon_{ijk}\epsilon_{pqr}M_p\frac{\partial M_q}{\partial x_j}\frac{\partial M_r}{\partial x_k}.$$

For generic skyrmion textures one may write

$$\mathbf{S}=(\cos\alpha(\phi)\sin\beta(r),\sin\alpha(\phi)\sin\beta(r),\cos\beta(r)),$$

so that the skyrmion number factorizes as

$s=p\cdot m,$

with polarity

$p=\frac{1}{2}[\cos\beta (r)]_{r=0}^{r=r_\sigma},$

vorticity

$$m=\frac{1}{2\pi }[\alpha (\phi )]_{\phi =0}^{\phi =2\pi },$$

and helicity introduced through

$$|\psi(\mathbf r)\rangle = \frac{|0\rangle+e^{i\theta_0}v(\mathbf r)|1\rangle}{\sqrt{1+|v(\mathbf r)|^2}}$$0

Within this taxonomy, $$|\psi(\mathbf r)\rangle = \frac{|0\rangle+e^{i\theta_0}v(\mathbf r)|1\rangle}{\sqrt{1+|v(\mathbf r)|^2}}$$1 with $$|\psi(\mathbf r)\rangle = \frac{|0\rangle+e^{i\theta_0}v(\mathbf r)|1\rangle}{\sqrt{1+|v(\mathbf r)|^2}}$$2 gives Néel type, $$|\psi(\mathbf r)\rangle = \frac{|0\rangle+e^{i\theta_0}v(\mathbf r)|1\rangle}{\sqrt{1+|v(\mathbf r)|^2}}$$3 with $$|\psi(\mathbf r)\rangle = \frac{|0\rangle+e^{i\theta_0}v(\mathbf r)|1\rangle}{\sqrt{1+|v(\mathbf r)|^2}}$$4 gives Bloch type, and $$|\psi(\mathbf r)\rangle = \frac{|0\rangle+e^{i\theta_0}v(\mathbf r)|1\rangle}{\sqrt{1+|v(\mathbf r)|^2}}$$5 gives an anti-skyrmion (Shen et al., 2021, Shen et al., 2022).

A key point in the paraxial setting is that the topology is defined on the polarization manifold, not on the instantaneous electric vector in real space. This is why paraxial beams, though transverse in non-absorbing media, can still support a genuine skyrmion invariant in Stokes space (Shen et al., 2022).

3. Canonical paraxial constructions and texture families

The canonical paraxial construction is a coherent superposition of two orthogonally polarized spatial modes,

$$|\psi(\mathbf r)\rangle = \frac{|0\rangle+e^{i\theta_0}v(\mathbf r)|1\rangle}{\sqrt{1+|v(\mathbf r)|^2}}$$6

with nontrivial topology arising from the spatially varying ratio $$|\psi(\mathbf r)\rangle = \frac{|0\rangle+e^{i\theta_0}v(\mathbf r)|1\rangle}{\sqrt{1+|v(\mathbf r)|^2}}$$7. For Laguerre–Gaussian superpositions,

$$|\psi(\mathbf r)\rangle = \frac{|0\rangle+e^{i\theta_0}v(\mathbf r)|1\rangle}{\sqrt{1+|v(\mathbf r)|^2}}$$8

so the azimuthal winding is controlled by $$|\psi(\mathbf r)\rangle = \frac{|0\rangle+e^{i\theta_0}v(\mathbf r)|1\rangle}{\sqrt{1+|v(\mathbf r)|^2}}$$9 and the radial sphere covering is controlled by the amplitude ratio $$\mathbf M(\mathbf r)=\langle \psi(\mathbf r)|\boldsymbol{\sigma}|\psi(\mathbf r)\rangle,$$0. In the simplest $$\mathbf M(\mathbf r)=\langle \psi(\mathbf r)|\boldsymbol{\sigma}|\psi(\mathbf r)\rangle,$$1 case, the beams realize optical analogues of hedgehog and spiral skyrmions with $$\mathbf M(\mathbf r)=\langle \psi(\mathbf r)|\boldsymbol{\sigma}|\psi(\mathbf r)\rangle,$$2; for $$\mathbf M(\mathbf r)=\langle \psi(\mathbf r)|\boldsymbol{\sigma}|\psi(\mathbf r)\rangle,$$3, the corresponding skyrmion number is $$\mathbf M(\mathbf r)=\langle \psi(\mathbf r)|\boldsymbol{\sigma}|\psi(\mathbf r)\rangle,$$4 when center and infinity correspond to orthogonal polarization states (Gao et al., 2019).

The free-space realization of (Zhu et al., 2021) uses a Gaussian mode and an LG-like vortex mode in orthogonal linear polarizations, with the ideal relation

$$\mathbf M(\mathbf r)=\langle \psi(\mathbf r)|\boldsymbol{\sigma}|\psi(\mathbf r)\rangle,$$5

The experimentally realized family is Bloch-type, and the skyrmion number is tuned by varying the OAM difference. The practical beam generated by an SLM is a hypergeometric Gaussian beam rather than an ideal LG mode, but the paraxial pseudospin texture remains the relevant topological object.

The tunable construction of (Shen et al., 2021) starts from a generalized skyrmion template and transfers it to the Stokes field of a vector beam

$$\mathbf M(\mathbf r)=\langle \psi(\mathbf r)|\boldsymbol{\sigma}|\psi(\mathbf r)\rangle,$$6

Although the theoretical template is SPP-inspired, the experimentally accessible free-space realization replaces zero- and first-order Bessel modes with the fundamental Gaussian and first-order Laguerre–Gaussian modes. The resulting skyrmions are therefore effectively paraxial Stokes skyrmions. Their texture classes are organized on the Skyrme–Poincaré sphere, whose equator connects Néel and Bloch states and whose poles correspond to anti-skyrmions.

Beyond isolated first-order examples, the paraxial taxonomy has broadened considerably. “Building ideal paraxial optical skyrmions using rational maps” (Cisowski et al., 2022) gives an exact topology-first construction based on

$$\mathbf M(\mathbf r)=\langle \psi(\mathbf r)|\boldsymbol{\sigma}|\psi(\mathbf r)\rangle,$$7

with $$\mathbf M(\mathbf r)=\langle \psi(\mathbf r)|\boldsymbol{\sigma}|\psi(\mathbf r)\rangle,$$8 generating Néel-type skyrmions, $$\mathbf M(\mathbf r)=\langle \psi(\mathbf r)|\boldsymbol{\sigma}|\psi(\mathbf r)\rangle,$$9 anti-skyrmions, target-space rotations generating bimerons, and products of shifted factors generating multi-skyrmions and lattices. The 2025 review “Optical Quasiparticles: Skyrmions, Bimerons and Skyrmionic Hopfions in Paraxial Laser Beams” (Allam et al., 18 Jan 2025) situates skyrmions, bimerons, skyrmioniums, Bessel skyrmions, and skyrmionic hopfions within the broader paraxial vector-beam landscape.

4. Experimental generation and polarimetric reconstruction

The first clear free-space experimental route used standard paraxial optics. In (Zhu et al., 2021), a continuous-wave 780 nm semiconductor laser is coupled into a single-mode fiber, a PBS and a half-wave plate prepare $$\bm M=\langle\Phi(\bm r)|{\bm \sigma}|\Phi(\bm r)\rangle.$$0, a liquid-crystal-on-silicon SLM imprints the vortex only on the horizontal component, and a polarization-state analyzer projects onto

$$\bm M=\langle\Phi(\bm r)|{\bm \sigma}|\Phi(\bm r)\rangle.$$1

The Poincaré-vector components are reconstructed from six spatially resolved projections through

$$\bm M=\langle\Phi(\bm r)|{\bm \sigma}|\Phi(\bm r)\rangle.$$2

For $$\bm M=\langle\Phi(\bm r)|{\bm \sigma}|\Phi(\bm r)\rangle.$$3, the reconstructed skyrmion numbers were reported as

$$\bm M=\langle\Phi(\bm r)|{\bm \sigma}|\Phi(\bm r)\rangle.$$4

with deviations attributed to imperfect LG generation, polarization-optics imperfections, and finite CCD resolution (Zhu et al., 2021).

The tunable free-space realization of (Shen et al., 2021) uses a horizontally polarized HeNe laser at $$\bm M=\langle\Phi(\bm r)|{\bm \sigma}|\Phi(\bm r)\rangle.$$5 nm, an SLM split into two independently encoded regions, and a common-path triangular Sagnac interferometer. The two scalar components are shaped digitally, recombined into a vector beam, and verified by Stokes polarimetry with

$$\bm M=\langle\Phi(\bm r)|{\bm \sigma}|\Phi(\bm r)\rangle.$$6

This experiment reconstructs the Stokes texture and local polarization ellipses for Néel, Bloch, anti-skyrmion, and intermediate states on the Skyrme–Poincaré sphere.

A separate advance concerns characterization rather than generation. “Topological approach of characterizing optical Skyrmions and Skyrmion lattices” (McWilliam et al., 2022) recasts the skyrmion number in terms of polarization singularities and the Stokes phase

$$\bm M=\langle\Phi(\bm r)|{\bm \sigma}|\Phi(\bm r)\rangle.$$7

leading to

$$\bm M=\langle\Phi(\bm r)|{\bm \sigma}|\Phi(\bm r)\rangle.$$8

This avoids derivative-sensitive direct integration of noisy experimental Stokes maps and makes basis rotation on the Poincaré sphere an explicit tool for improving measurement robustness.

5. Propagation, invariance, and conceptual boundaries

Paraxial light skyrmions are often described as topological and propagation robust, but the literature imposes specific conditions on that statement. The 2022 review (Shen et al., 2022) states that, to be classified as a skyrmion, the vector texture must be protected upon propagation, and that in paraxial vector beams the two modal components must have a fixed ratio of the Gouy phase and be aligned at the beam waist. If the Gouy phases are misaligned, the beam is not topologically protected and may show variant topological charge upon propagation.

The original paraxial theory already emphasized that the skyrmion number on a transverse plane need not be globally conserved in arbitrary propagation scenarios. In (Gao et al., 2019), the divergence-free Skyrmion field $$\bm M=\langle\Phi(\bm r)|{\bm \sigma}|\Phi(\bm r)\rangle.$$9 permits flux to escape radially to infinity, so a change in plane-integrated skyrmion number can occur when the asymptotic boundary polarization changes. This is not a contradiction of the topological picture; it is a statement about the importance of compactification and boundary conditions.

“Theory of paraxial optical Skyrmions” (Ye et al., 2024) sharpened this distinction. For regular paraxial textures,

$$s=\frac{1}{4\pi }\iint_\sigma \mathbf{S}\cdot \left( \frac{\partial \mathbf{S}}{\partial x}\times \frac{\partial \mathbf{S}}{\partial y} \right)\,dx\,dy,$$0

and one may define a Skyrmion vector potential $$s=\frac{1}{4\pi }\iint_\sigma \mathbf{S}\cdot \left( \frac{\partial \mathbf{S}}{\partial x}\times \frac{\partial \mathbf{S}}{\partial y} \right)\,dx\,dy,$$1 through

$$s=\frac{1}{4\pi }\iint_\sigma \mathbf{S}\cdot \left( \frac{\partial \mathbf{S}}{\partial x}\times \frac{\partial \mathbf{S}}{\partial y} \right)\,dx\,dy,$$2

The same work also identifies a specifically optical class of non-integer skyrmions: when the on-axis field becomes singular, $$s=\frac{1}{4\pi }\iint_\sigma \mathbf{S}\cdot \left( \frac{\partial \mathbf{S}}{\partial x}\times \frac{\partial \mathbf{S}}{\partial y} \right)\,dx\,dy,$$3 is no longer divergence-free there, and the plane-integrated $$s=\frac{1}{4\pi }\iint_\sigma \mathbf{S}\cdot \left( \frac{\partial \mathbf{S}}{\partial x}\times \frac{\partial \mathbf{S}}{\partial y} \right)\,dx\,dy,$$4 can vary continuously with propagation.

A persistent misconception is that all optical skyrmions are equivalent. They are not. The plasmonic skyrmions of (Tsesses et al., 2018) are electric-field-direction skyrmions in evanescent guided fields; the Poynting-vector skyrmions of (Wang et al., 2023) are strongly nonparaxial focal-volume textures in a $$s=\frac{1}{4\pi }\iint_\sigma \mathbf{S}\cdot \left( \frac{\partial \mathbf{S}}{\partial x}\times \frac{\partial \mathbf{S}}{\partial y} \right)\,dx\,dy,$$5 geometry with $$s=\frac{1}{4\pi }\iint_\sigma \mathbf{S}\cdot \left( \frac{\partial \mathbf{S}}{\partial x}\times \frac{\partial \mathbf{S}}{\partial y} \right)\,dx\,dy,$$6; paraxial light skyrmions, by contrast, are primarily Stokes-vector or polarization-pseudospin textures in ordinary propagating beams. The distinction is not semantic; it concerns the order parameter, dimensionality, and admissible topology.

6. Analytical generalizations and enlarged state spaces

Recent theory has pushed paraxial skyrmions beyond isolated Gaussian-plus-vortex examples. The general field-theoretic framework of (Ye et al., 2024) interprets the skyrmion field lines as lines of constant polarization and imports a Mermin–Ho-type vector-potential structure from $$s=\frac{1}{4\pi }\iint_\sigma \mathbf{S}\cdot \left( \frac{\partial \mathbf{S}}{\partial x}\times \frac{\partial \mathbf{S}}{\partial y} \right)\,dx\,dy,$$7He-A. In this formulation, integer and non-integer paraxial skyrmions belong to the same general Stokes-field theory but differ in whether the Skyrmion field remains source free.

The rational-map program of (Cisowski et al., 2022) makes the compactified transverse-plane topology explicit. It distinguishes genuine paraxial optical skyrmions from generic Poincaré beams by requiring full polarization, compactifiability, and an integer sphere-covering degree. This framework also provides explicit ideal constructions for Néel-type, Bloch-type, anti-skyrmions, bimerons, multi-skyrmions, and skyrmion lattices, thereby extending the subject from single-beam exemplars to analytic skyrmion field design.

An even larger enlargement appears in “Skyrmionic SU(6) structured light” (Saito, 19 Mar 2025). There the paraxial structured-light basis consists of six spin–orbit states

$$s=\frac{1}{4\pi }\iint_\sigma \mathbf{S}\cdot \left( \frac{\partial \mathbf{S}}{\partial x}\times \frac{\partial \mathbf{S}}{\partial y} \right)\,dx\,dy,$$8

and the skyrmion sector is embedded in a full $$s=\frac{1}{4\pi }\iint_\sigma \mathbf{S}\cdot \left( \frac{\partial \mathbf{S}}{\partial x}\times \frac{\partial \mathbf{S}}{\partial y} \right)\,dx\,dy,$$9 manifold with $$N=\frac{1}{4\pi}\int \Sigma_z\,dxdy, \qquad \Sigma_i=\frac{1}{2}\epsilon_{ijk}\epsilon_{pqr}M_p\frac{\partial M_q}{\partial x_j}\frac{\partial M_r}{\partial x_k}.$$0 generators. Conventional Néel and Bloch states then become particular $$N=\frac{1}{4\pi}\int \Sigma_z\,dxdy, \qquad \Sigma_i=\frac{1}{2}\epsilon_{ijk}\epsilon_{pqr}M_p\frac{\partial M_q}{\partial x_j}\frac{\partial M_r}{\partial x_k}.$$1 submanifolds, while continuous unitary transformations connect skyrmions, anti-skyrmions, and other generalized textures, including dipolar and antidipolar states, on higher-order Poincaré spheres and a skyrmionic torus. A direct implication, explicitly drawn in that work, is that topological stability depends on the class of allowed deformations: within a fixed two-mode subspace the texture behaves as a robust skyrmion, but broader $$N=\frac{1}{4\pi}\int \Sigma_z\,dxdy, \qquad \Sigma_i=\frac{1}{2}\epsilon_{ijk}\epsilon_{pqr}M_p\frac{\partial M_q}{\partial x_j}\frac{\partial M_r}{\partial x_k}.$$2 operations can connect it to other states.

7. Emerging directions and applications

Paraxial light skyrmions have expanded from static isolated beams to actively engineered propagating structures. “Skyrmions with customized intensity distribution and trajectory” (Tian et al., 13 Aug 2025) introduces angular-spectrum synthesis for Stokes-vector skyrmions constructed from orthogonally circularly polarized Bessel beams,

$$N=\frac{1}{4\pi}\int \Sigma_z\,dxdy, \qquad \Sigma_i=\frac{1}{2}\epsilon_{ijk}\epsilon_{pqr}M_p\frac{\partial M_q}{\partial x_j}\frac{\partial M_r}{\partial x_k}.$$3

with independent control of the longitudinal intensity profile via an amplitude compensation function $$N=\frac{1}{4\pi}\int \Sigma_z\,dxdy, \qquad \Sigma_i=\frac{1}{2}\epsilon_{ijk}\epsilon_{pqr}M_p\frac{\partial M_q}{\partial x_j}\frac{\partial M_r}{\partial x_k}.$$4, trajectory control via displacement factors $$N=\frac{1}{4\pi}\int \Sigma_z\,dxdy, \qquad \Sigma_i=\frac{1}{2}\epsilon_{ijk}\epsilon_{pqr}M_p\frac{\partial M_q}{\partial x_j}\frac{\partial M_r}{\partial x_k}.$$5, and array synthesis via a displacement phase factor. The reported texture classes include Néel-type, Bloch-type, anti-type, and 2nd-order skyrmions, together with arrays of customized shape, intensity distribution, and trajectory.

The ultrafast extension is given by “Attosecond Light Skyrmion Pulses via High Harmonic Generation” (Marco et al., 23 Sep 2025). There, a linearly polarized vector-beam driver with fractional orbital angular momentum generates XUV harmonics whose far-field polarization textures all map onto nearly identical skyrmions. Using 1.2 $$N=\frac{1}{4\pi}\int \Sigma_z\,dxdy, \qquad \Sigma_i=\frac{1}{2}\epsilon_{ijk}\epsilon_{pqr}M_p\frac{\partial M_q}{\partial x_j}\frac{\partial M_r}{\partial x_k}.$$6m driving fields and experimentally realistic Al/Zr filtering, the coherent superposition of five harmonics centered at about $$N=\frac{1}{4\pi}\int \Sigma_z\,dxdy, \qquad \Sigma_i=\frac{1}{2}\epsilon_{ijk}\epsilon_{pqr}M_p\frac{\partial M_q}{\partial x_j}\frac{\partial M_r}{\partial x_k}.$$7 eV yields $$N=\frac{1}{4\pi}\int \Sigma_z\,dxdy, \qquad \Sigma_i=\frac{1}{2}\epsilon_{ijk}\epsilon_{pqr}M_p\frac{\partial M_q}{\partial x_j}\frac{\partial M_r}{\partial x_k}.$$8 attosecond pulse trains while preserving skyrmion numbers close to unity.

Application claims in the paraxial literature are broad but consistent. The 2021 free-space experiment (Zhu et al., 2021) emphasizes manipulation of tiny objects and propagation over long distances. The tunable Skyrme–Poincaré-sphere work (Shen et al., 2021) highlights information storage, communication, and cryptography. The 2025 review (Allam et al., 18 Jan 2025) adds structured fabrication, optical communications, quantum communication using skyrmionic states, and the special utility of Bessel skyrmions through non-diffraction and self-healing. These claims remain application-oriented rather than device-level standards, but they define the main technological horizon of the field.

Taken together, the literature establishes paraxial light skyrmions as a distinct optical-topological regime: a family of polarization quasiparticles in paraxial vector beams, rigorously described by Stokes-space topology, experimentally accessible with standard beam-shaping optics, and now extending into analytical field theory, generalized spin–orbit state spaces, programmable beam trajectories, and attosecond structured light.