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# Q -Plates: From Optical Vortices to Optical Skyrmions

Vagharshak Hakobyan, Etienne Brasselet

► To cite this version:

Vagharshak Hakobyan, Etienne Brasselet. Q -Plates: From Optical Vortices to Optical Skyrmions. Physical Review Letters, 2025, 134 (8), pp.083802. 10.1103/PhysRevLett.134.083802. hal-05337412

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Distributed under a Creative Commons Attribution 4.0 International License
Q-Plates: From Optical Vortices to Optical Skyrmions

Vagharshak Hakobyan and Etienne Brasselet

University of Bordeaux, CNRS, Laboratoire Ondes et Matière d'Aquitaine, F-33400 Talence, France

We report on the extension of the q-plate concept, a hallmark of spin-orbit optical vortex generation since its introduction in 2006, to the generation of optical skyrmions. Stokes skyrmions of arbitrary order with polarization-controlled Skyrme number and reconfigurable multiskyrmions are obtained. This is done by endowing q-plates with a winding number associated with the radial degree of freedom that adds to the usual one associated with the azimuthal degree of freedom.

DOI: 10.1103/PhysRevLett.134.083802

Structured light [1] has become a topic that continues to diversify and offer new fundamental and applied opportunities [2–4]. A typical example is the case of optical vortex beams—beams carrying phase singularities. A phase singularity is associated with an integer winding number $l$ equal to the circulation of the phase along a closed circuit surrounding the singularity divided by $2\pi$, which tells about the amount of orbital angular momentum carried by a vortex beam [5]. In 2006, Marrucci and coworkers took advantage of a purely geometric phase shaping approach [6] to generate optical vortex beams from space-variant liquid crystal half-wave plates with a slow axis orientation being a linear function of the azimuthal angle in the polar coordinate system $(r, \phi)$, $\Psi(\phi) = q_{\phi}\phi + \Psi_0$ [7], where $\Psi_0$ is a constant and

$$q_{\phi} = \frac{1}{2\pi} \int_{0}^{2\pi} \nabla\Psi(\phi) \cdot d\mathbf{l} \quad (1)$$

is the azimuthal winding number, defined as a circulation along a closed contour around the origin, and gave its name to such devices widely known as q-plates. Basically, a q-plate allows generating an optical vortex beam with topological charge $l = 2\sigma q_{\phi}$, where $\sigma = \pm 1$ refers to the helicity of an incident circularly polarized light beam. For almost two decades, q-plates fostered a vast amount of scientific progress from classical to quantum perspectives [8] and its technology is now mature [9]. Notably, the q-plate concept has been extended beyond liquid crystal optics into metasurfaces and 3D laser printing technologies.

Here we extend the original approach by adding a winding number linked to the radial degree of freedom,

$$q_r = \frac{1}{2\pi} \int_{0}^{\infty} \nabla\Delta(r) \cdot d\mathbf{l}, \quad (2)$$

which is associated with a radially dependent smooth birefringent phase retardation $\Delta(r)$ between the slow and the fast axis, hence $q_r \ge 0$. Such “second quantization” thus acts not only on a different spatial degree of freedom but also on a different property of the optically anisotropic medium. Within the usual restriction $2q_{\phi} \in \mathbb{Z}$ leading to the generation of optical vortices with integer topological charge, we also adopt in our experiments the condition $2q_r \in \mathbb{N}$, which leads isolated optical skyrmions embedded in a uniform polarization state background.

For the sake of illustration, see Fig. 1, here we opt for a design defined as $\Psi(\phi) = q_{\phi}\phi$ and $\Delta(r)/2\pi = q_r(r/r_0)$ for $r < r_0$ and $\Delta(r)/2\pi = q_r$ for $r > r_0$. The demonstration is made in the visible domain, at 532 nm wavelength, for two types of optical elements having a disk-shaped structural footprint of radius 3 mm made from femtosecond-laser nanostructured silica glass (see Supplemental Material [10]) and designed for $(q_{\phi}, q_r) = (1/2, 1/2)$ and $(q_{\phi}, q_r) = (1, 1/2)$. Their experimental characterization is

<figure><img src="image_3.png" /><figcaption>FIG. 1. Optical characterization of optical elements with $(q_{\phi}, q_r) = (1/2, 1/2)$ and $(1, 1/2)$, and $r_0 = 1$ mm. (a) Combined image of the birefringent phase retardation and optical axis orientation, where the brightness refers to retardance and the hue colormap refers to orientation as indicated by the color bar. Scale bar: 1 mm. (b) Azimuthal dependence of $\Psi$. (c) Radial dependence of azimuthally averaged $\Delta$. Solid black lines refer to the prescribed design.</figcaption></figure>

<footnote>*Contact author: etienne.brasselet@u-bordeaux.fr</footnote>
performed by spatially resolved polarimetry and summarized in Fig. 1. We refer to Refs. [10–12] for details on the fabrication and to Ref. [13] for characterization protocols. This makes it possible to fully control how the output polarization state covers the Poincaré sphere of polarization, every point P of it being uniquely associated with a given polarization state described by the 3D reduced Stokes vector $\mathbf{s} = (s_1, s_2, s_3)$,

$$ \mathbf{s} = (\cos 2\chi \cos 2\psi, \cos 2\chi \sin 2\psi, \sin 2\chi), \quad (3) $$

which is parametrized by the ellipticity angle $-\pi/4 \le \chi \le \pi/4$ and the azimuth angle $0 \le \psi \le \pi$ of the polarization ellipse [14], see Fig. 2(a). Said differently, optical skyrmions [15]—the optical analogue of topologically nontrivial field textures that have been extensively studied in magnetism [16] in the context of spintronics, but which extends to all branches of physics, becomes accessible for the reduced Stokes vector field. Such fields are characterized by the Skyrme number $N$,

$$ N = \frac{1}{4\pi} \iint \mathbf{s} \cdot \left(\frac{\partial \mathbf{s}}{\partial x} \times \frac{\partial \mathbf{s}}{\partial y}\right) dxdy, \quad (4) $$

whose value can be calculated on experimental grounds once the Stokes vector field $\mathbf{s}$ is measured.

The generation of skyrmions is studied in a configuration that corresponds to common tabletop optical experiments using an incident fundamental Gaussian paraxial laser beam with millimeter size waist radius $w_0 \approx r_0$. The Stokes vector field is spatially resolved after propagation distances of the order of a meter, as depicted in Fig. 2(b). This is done by recording the intensity profiles ($I_X$) of the horizontal ($X = H$), vertical ($X = V$), diagonal ($X = D$), anti-diagonal ($X = A$), right-circular ($X = R$), and left-circular ($X = L$) components of the output light field owing to usual polarization optics. Indeed, this gives access to Stokes maps $\mathbf{s}(x, y)$ according to $s_1 = (I_H - I_V)/(I_H + I_V)$, $s_2 = (I_D - I_A)/(I_D + I_A)$, and $s_3 = (I_L - I_R)/(I_L + I_R)$. The results are displayed in Fig. 2(c) with and without the presence of a relay lens for the two possible incident circular polarization states ($\chi_{in} = \pm 45^\circ$, that is, $\sigma = \pm 1$). The corresponding vector representation of the observed Stokes textures is shown in Fig. 3, which offers a better visual perception of the generation of skyrmions.

In all cases, first-order and second-order skyrmions are recovered quantitatively for $q_\phi = 1/2$ and for $q_\phi = 1$, respectively, for which we find Skyrme numbers very close to the expected values $N = 2\sigma q_\phi$. The latter formula as well as simulations of the generated Stokes textures can be obtained by making use of the relationship between the electric field components and the reduced Stokes vector,

<figure><img src="image_4.png" /><figcaption>FIG. 2. (a) Poincaré sphere of polarization and reduced Stokes vector $\mathbf{s}$. (b) Setup for the observation of the generated skyrmion beam (inset: intensity distribution), revealed by the projection of the light field on various polarization state owing to a polarization controller (PC) and the recording the corresponding intensity profiles with a camera (Cam) placed at a distance $D = 1.2$ m. An optional relay lens (L) with focal length $f = 30$ cm is placed at mid-distance between the optical element and the camera. The incident beam is a Gaussian beam with waist radius 1.5 mm. (c) Experimental Stokes vector textures for $(q_\phi, q_r) = (1/2, 1/2)$ and $(1, 1/2)$ that correspond to skyrmions with Skyrme number ideally equal to $N = 2\sigma q_\phi$, respectively.</figcaption></figure>

$$ \mathbf{s} = \frac{(|E_x|^2 - |E_y|^2, 2 \operatorname{Re}(E_x^* E_y), 2 \operatorname{Im}(E_x^* E_y))}{(|E_x|^2 + |E_y|^2)} \quad (5) $$

Let us consider the general case of a uniformly polarized incident field described by the Jones vector $\mathbf{e}_{in} = (\cos\psi_{in} \cos\chi_{in} - i \sin\psi_{in} \sin\chi_{in}, \sin\psi_{in} \cos\chi_{in} + i \cos\psi_{in} \sin\chi_{in})^\top$ in the (H, V) linear polarization basis, where $^\top$ is the matrix transpose. From Jones calculus, we find that the latter is transformed into

$$ \mathbf{e}_{\text{out}} = \sum_{\sigma=\pm 1} (\mathbf{e}_{\text{in}} \cdot \mathbf{e}_{\sigma}^*) \left( \cos \frac{\Delta}{2} \mathbf{e}_{\sigma} + i \sin \frac{\Delta}{2} e^{i2\sigma\Psi} \mathbf{e}_{-\sigma} \right), \quad (6) $$

where the complex weighting factors $\mathbf{e}_{in} \cdot \mathbf{e}_{\sigma}^*$ are the components of the incident Jones vector in the circular polarization basis $\mathbf{e}_{\pm} = (\mathbf{x} \pm i\mathbf{y})/\sqrt{2}$ (see Supplemental Material [10]). The corresponding reduced Stokes vector is
<figure><img src="image_5.png" /><figcaption>FIG. 3. Experimental vector representation of the Stokes vector textures in the plane (x, y), which corresponds to the data presented in Fig. 2(c) when using the relay lens. Scale bar: 1 mm. See Fig. 2(a) for the color map and **s** vector definition.</figcaption></figure>

then obtained by applying $E_x \rightarrow e_{\text{out}} \cdot x$ and $E_y \rightarrow e_{\text{out}} \cdot y$ in Eq. (5).

More generally, Stokes textures with polarity $\sigma = \pm 1$, order $2|q_{\phi}|$, and Skyrme number $N = \sigma q_{\phi} (1 - (-1)^{2q_r})$ are obtained for circularly polarized incident Gaussian beam with the special cases $q_r = 1$ corresponding to a skyrmionium and $q_r > 1$ corresponding to $n\pi$-skyrmion with $n\pi = 2q_r\pi$ radial twist (see Fig. 4), respectively, which are some of the possible textures beyond fundamental skyrmions [17]. Also, the Skyrme number can be continuously controlled by changing the incident polarization state. Indeed, from Eq. (6), one gets $N = 2s_3^{\text{in}}q_{\phi}$ for elliptically polarized incident beam as demonstrated in Fig. 5 for $(q_{\phi}, q_r) = (1/2, 1/2)$, see also Supplemental Material [10] videos 1 and 2 for animations.

<figure><img src="image_6.png" /><figcaption>FIG. 4. Numerical simulation of $n\pi$ skyrmions either for optical elements with $(q_{\phi}, q_r) = (1/2, n/2)$ or when cascading in series $n$ elements with $(q_{\phi}, q_r) = (1/2, 1/2)$. Inset: experimental data recorded after 50-cm-long free propagation from the sample. The incident beam is left-handed circularly polarized and box size is $2w_0 \times 2w_0$. See Fig. 2(a) for the color map.</figcaption></figure>

Next, inspired by the use of only fundamental $q$-plates with $|q_{\phi}| = 1/2$ in order to obtain an effective $q$-plate of arbitrary order via a cascading strategy [18] we propose to use low-order optical elements in series in order to achieve high-order polarization textures. As an example, $n\pi$-skyrmions of order $m$ and arbitrary integer $n$ can be generated by placing in series $n$ optical elements defined by $(q_{\phi}, q_r) = (m/2, 1/2)$ with $m$ integer, whose polarity is controlled by the handedness of the incident circularly polarized beam. One can also consider the additional use of homogeneous or inhomogeneous waveplates with arbitrary retardance, which can be tested by iterating the single-stage spin-orbit optical process described by Eq. (6). Without going into the zoology of possible options, we just illustrate with a couple of examples how powerful the principle of placing elements in series can be. First, we show that inserting a half-wave plate between two successive optical elements of the same type allows generating multiskyrmions as shown in Fig. 6(a). Such a combination allows generating multiple copies of the skyrmion generated by a single optical element. These linear arrays of multiskyrmions are similar to those produced in a recent work by Shen *et al.* dealing with gradient-index lenses [19], though here the structured optical elements are compact and robust wave plates instead of several-centimeter-long fragile devices. Circular arrays can also be obtained by using other sequences of optical elements as illustrated in the insets of Fig. 6(a). Second, we show that multiskyrmions can be manipulated by rotating the half-wave plate around the beam propagation direction; see Fig. 6(b).

<figure><img src="image_7.png" /><figcaption>FIG. 5. Experimental Skyrme number as the incident Stokes vector goes from South to North pole on the Poincaré sphere along a meridian, for $(q_{\phi}, q_r) = (1/2, 1/2)$ with and without use of the relay lens, see Fig. 2(b). Solid line: linear behavior. Inset panels illustrate the experimental Stokes vector textures in the plane (x, y) for $s_3^{\text{in}} = (-1/2, 0, 1/2)$ and $\psi_{\text{in}} = 45^{\circ}$. See Fig. 2(a) for the color map.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>FIG. 6. Numerical simulation of the Stokes vector textures in the plane (x, y) when cascading optical elements with half-wave plates (HWP) in between, in order to generate (a) and manipulate (b) multiskyrmlons. The indices A and B, respectively, refer to $(q_\phi, q_r) = (1/2, 1/2)$ and $(q_\phi, q_r) = (1, 1/2)$, and HWP$_\theta$ refers to half-wave plate having its slow axis oriented at an angle $\theta$ from the x axis. Inset 1 of panels (a): the sequence of elements is $Q_B/HWP_0/Q_A$. Inset 2 of panels (a): the sequence of elements is $Q_B/HWP_0/Q_B$. Inset of panels (b): experimental data recorded after 50-cm-long free propagation from the sample The incident beam is left-handed circularly polarized. Box size is $2w_0 \times 2w_0$ except for the panels referring to the cascading schemes $[Q_A/HWP_0]^2/Q_A$ and $[Q_A/HWP_0]^3/Q_A$, for which it is $w_0 \times w_0$. See Fig. 2(a) for the color map.</figcaption></figure>

Having a compact, robust, and reconfigurable approach to the preparation of skyrmionic fields should foster the development of novel applications making use of optical skyrmions; see, for instance, recent works on laser processing of materials [20], on-demand optical generation of magnetic fields [21], or optical trapping and manipulation [22].

An important property of optical skyrmions that should not be eluded here is their topological stability in the absence of perturbations. Stokes textures indeed evolve during free space propagation, only marginally after distances of the order of a meter for millimeter-sized beam [see Figs. 2(c) and (3)], but this is no longer the case in the far field. This is retrieved by using a Fourier lens with focal length $f' = 15$ cm placed at a distance $f'$ from the optical elements and resolving the Stokes texture in the plane located at a distance $f'$ from the lens, which corresponds to the reciprocal space. The “truncated” Skyrme number $N_{\text{trunc}}$ is then calculated over a disk with radius $\kappa$ centered on the beam axis, as a function of $\kappa$,

$$N_{\text{trunc}}(\kappa) = \frac{1}{4\pi} \iint \sqrt{\kappa_x^2 + \kappa_y^2} d\kappa_x d\kappa_y. \quad (7)$$

<figure><img src="image_9.png" /><figcaption>FIG. 7. (a) Skyrme number over a disk of increasing radius in the Fourier plane of an optical element with $(q_\phi, q_r) = (1/2, 1/2)$ for right-handed and left-handed circularly polarized incident beam. (b) Corresponding experimental and simulated Stokes vector textures. See Fig. 2 for the color map.</figcaption></figure>

The experimental results are shown in Fig. 7 for $(q_\phi, q_r) = (1/2, 1/2)$ and $\chi_{\text{in}} = \pm 45^\circ$, and compared with numerical simulations in two situations: first, accounting for the radial modulation of the dynamic phase prefactor $\exp(i\epsilon\Delta)$ in the right-hand side of Eq. (6) associated with the inevitable change in the average refractive index of glass when artificial birefringence is encoded by laser nanostructuring, using $\epsilon = -0.65$ [13]; and then, assuming that an effective compensation strategy has been developed, that is $\epsilon = 0$. A degradation is observed in both cases, which requires further work to develop of a strategy to recover the topological stability, for instance, based on the determination of an optimal retardance profile using methods developed in another context [23]. The role of propagation has also been discussed in the case of q-plates (uniform retardance, $\Delta_0 = \pi$) for which the vortex beam profile evolves as diffraction develops [24], whereas detuned q-plates ($\Delta_0 \neq \pi$) can lead to a Stokes skyrmion after propagation to the far field [25], however, without the versatility offered by exploiting the radial degree of freedom at the design level.

Dealing here with the generation and manipulation of only one type of optical skyrmion, the Stokes skyrmion, this arguably limits the scope of the proposed optical elements. In fact, skyrmions have also been found for the electric field vector [26], the spin vector [27], and the Poynting vector [28]. Given that the longitudinal component of the electric field ($E_z$) is at work in these types of skyrmions, the study of the topological properties of the structured 3D electromagnetic field, as well as the various quantities that can be derived from it, that emerges from optical element after being focused by a lens is thus worthy of further investigation.

We also notice that the use of doubly inhomogeneous (optical axis orientation and retardance) wave plates was reported several years ago. Early works were done in the context of the generation of single [29] or arrays [30] of optical vortices based on the spin-orbit interaction of light,
however, without dealing with polarization textures. This
was addressed by Beckley *et al.* [31], who introduced the
concept of full Poincaré beams—a class of beams that
cover the entire surface of the Poincaré sphere—and drew
on earlier work on mechanically stressed glass rods [29] to
propose an experimental implementation. By doing so,
Beckley *et al.* retrospectively reported on what appears as
the first experimental attempt of an optical Stokes skyrmion
and despite the existing overlaps and differences in the
definitions of full Poincaré and Stokes skyrmion beams in
the literature, the two are definitely closely related [31–34].
In addition, we note that the Spilman *et al.* [29] device
exhibits a retardance profile that is neither axisymmetric
nor monotonic (although the retardance varies linearly with
$r$ sufficiently close to the revolution axis of the rod)
and, more importantly, without saturation away from the
center that prevents the definition of a radial winding
number from Eq. (2). The case of linear radial dependence
of the retardance was recently reexamined theoretically by
Radhakrishna *et al.* [35] in the context of full Poincaré
beams with finite extent, which corresponds to the clipping
of the beam if an experiment were carried out. Conse-
quently, and notwithstanding the fact that optical skyrmions
were not discussed as such, none of these works allows the
production of a smooth beam carrying an isolated skyrmion
embedded in a uniformly polarized background, as has
been achieved here.

Summarizing, the proposed generalization of q-plates provides a simple and systematic way to generate optical skyrmions from inhomogeneous anisotropic wave plates, whose topology can be controlled by design. In particular, similarly to the use of $2q_φ \notin \mathbb{Z}$ behaving as fractional optical vortex masks, polarization texture diversification can be obtained using $2q_r \notin \mathbb{N}$. For instance, merons with polarity $σ = ±1$, order $2|q_φ|$, and Skyrme number $N = σq_φ$ are expected for $2q_φ \in \mathbb{Z}$ and $q_r = 1/4$. In terms of manufacturing, optical elements with arbitrary set of $(q_φ, q_r)$ can be produced from laser materials processing technologies as that used here. More interestingly, the fact that effective devices with arbitrary doubly inhomogeneous properties can be designed from a stack of wave plates structured with uniform delay, as discussed theoretically in [36], makes it possible to envision the fabrication of the proposed optical elements from any technology capable of producing q-plates. All this makes the present concept, which applies to any wavelength, worth exploring further toward the advent of optical skyrmionics—just as q-plates have done and kept doing for optical vortices both in the classical and quantum regimes for almost two decades.

**Acknowledgments**—We acknowledge the support of French National Research Agency project ANR-21-CE24-0014-01, Wits-CNRS 2021 joint PhD programme, and IdEx University of Bordeaux / Grand Research Program GPR LIGHT.

[1] A. Forbes, M. De Oliveira, and M. R. Dennis, Structured light, Nat. Photonics 15, 253 (2021).

[2] Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities, Light Sci. Appl. 8, 90 (2019).

[3] K. Y. Bliokh, E. Karimi, M. J. Padgett, M. A. Alonso, M. R. Dennis, A. Dudley, A. Forbes, S. Zahedpour, S. W. Hancock, H. M. Milchberg et al., Roadmap on structured waves, J. Opt. 25, 103001 (2023).

[4] C. Rosales-Guzmán and V. Rodríguez-Fajardo, A perspective on structured light’s applications, Appl. Phys. Lett. 125, 200503 (2024).

[5] 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).

[6] R. Bhandari, Polarization of light and topological phases, Phys. Rep. 281, 1 (1997).

[7] L. Marrucci, C. Manzo, and D. Paparo, Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media, Phys. Rev. Lett. 96, 163905 (2006).

[8] L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications, J. Opt. 13, 064001 (2011).

[9] A. Rubano, F. Cardano, B. Piccirillo, and L. Marrucci, Q-plate technology: A progress review, J. Opt. Soc. Am. B 36, D70 (2019).

[10] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.134.083802 for the fabrication of the optical elements and the derivation of Eq. (6).

[11] M. Sakakura, Y. Lei, L. Wang, Y.-H. Yu, and P. G. Kazansky, Ultralow-loss geometric phase and polarization shaping by ultrafast laser writing in silica glass, Light Sci. Appl. 9, 1 (2020).

[12] Y. Lei, H. Wang, G. Shayeganrad, and P. G. Kazansky, Ultrafast laser nanostructuring in transparent materials for beam shaping and data storage, Opt. Mater. Express 12, 3327 (2022).

[13] V. Hakobyan, K. Singh, Y. Lei, P. Kazansky, D. Coursault, A. Forbes, and E. Brasselet, Single-stage spin-orbit Laguerre-Gaussian modal beam shaping from silica optics, Phys. Rev. Appl. 21, 064003 (2024).

[14] M. Born and E. Wolf, *Principles of Optics* (Cambridge University Press, Cambridge, England, 2019).

[15] 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 (2024).

[16] A. N. Bogdanov and C. Panagopoulos, Physical foundations and basic properties of magnetic skyrmions, Nat. Rev. Phys. 2, 492 (2020).

[17] B. Göbel, I. Mertig, and O. A. Tretiakov, Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles, Phys. Rep. 895, 1 (2021).

[18] L. Marrucci, C. Manzo, and D. Paparo, Pancharatnam-berry phase optical elements for wave front shaping in the visible domain: Switchable helical mode generation, Appl. Phys. Lett. 88, 221102 (2006).
[19] Y. Shen, C. He, Z. Song, B. Chen, H. He, Y. Ma, J. A. 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).

[20] 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 photo-induced relief structures, APL Photonics 9, 046104 (2024).

[21] V. Karakhanyan and T. Grosjean, Optomagnetism with a plasmonic skyrmion, Opt. Lett. 49, 3440 (2024).

[22] M. J. O'Donnell and S. Hanna, Optical forces and torques in skyrmionic beams, in Optical Trapping and Optical Micro-manipulation XXI (SPIE, 2024), Vol. 13112, pp. 74–81, 10.1117/12.3029919.

[23] A. Vella and M. A. Alonso, Optimal birefringence distributions for imaging polarimetry, Opt. Express 27, 36799 (2019).

[24] G. Vallone, A. Sponselli, V. D'Ambrosio, L. Marrucci, F. Sciarrino, and P. Villoresi, Birth and evolution of an optical vortex, Opt. Express 24, 16390 (2016).

[25] V. Hakobyan, Y. Shen, and E. Brasselet, Unitary spin-orbit optical-skyrmionic wave plates, Phys. Rev. Appl. 22, 054038 (2024).

[26] S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. Lindner, and G. Bartal, Optical skyrmion lattice in evanescent electromagnetic fields, Science 361, 993 (2018).

[27] H. Wang and S. Fan, Photonic spin hopfions and monopole loops, Phys. Rev. Lett. 131, 263801 (2023).

[28] S. Wang, Z. Zhou, Z. Zheng, J. Sun, H. Cao, S. Song, Z.-L. Deng, F. Qin, Y. Cao, and X. Li, Topological structures of energy flow: Poynting vector skyrmions, Phys. Rev. Lett. 133, 073802 (2024).

[29] A. K. Spilman and T. G. Brown, Stress birefringent, space-variant wave plates for vortex illumination, Appl. Opt. 46, 61 (2007).

[30] E. Brasselet, Tunable optical vortex arrays from a single nematic topological defect, Phys. Rev. Lett. 108, 087801 (2012).

[31] A. M. Beckley, T. G. Brown, and M. A. Alonso, Full Poincaré beams, Opt. Express 18, 10777 (2010).

[32] 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).

[33] 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).

[34] L. Shi, Z. Che, and Y. Kivshar, Embarking on a skyrmion odyssey, Photonics Insights 3, C02 (2024).

[35] B. Radhakrishna, G. Kadiri, and G. Raghavan, Polarimetric method of generating full Poincaré beams within a finite extent, J. Opt. Soc. Am. A 39, 662 (2022).

[36] B. Radhakrishna, G. Kadiri, and G. Raghavan, Realization of doubly inhomogeneous waveplates for structuring of light beams, J. Opt. Soc. Am. B 38, 1909 (2021).