# Periodic skyrmionic textures via conformal cartographic projections

David Marco1, 2 Isael Herrera1 Sophie Brasselet1 Miguel A. Alonso1, 3, 4, 5,
<EMAIL_ADDRESS>1Aix Marseille Univ., CNRS, Centrale Marseille,
Institut Fresnel, UMR 7249, 13397 Marseille Cedex 20, France
2Instituto de Bioingeniería, Universidad Miguel Hernández de Elche, 03202
Elche, Spain
3The Institute of Optics, University of Rochester, Rochester, NY14627, USA
4Center for Coherence and Quantum Optics, University of Rochester, Rochester,
NY14627, USA
5Laboratory for Laser Energetics, University of Rochester, Rochester, NY14627,
USA

###### Abstract

We find periodic skyrmionic textures via conformal cartographic projections
that map either an entire spherical parameter space or a hemisphere onto every
regular polygon that provides regular tessellations of the plane. These maps
preserve the sign of the Skyrme density throughout the entire space. We
implement these textures in the polarization state of a laser beam, and
demonstrate that paraxial fields where a periodic texture preserving the sign
of the Skyrme density is implemented in the polarization state distribution
unavoidably exhibit zeros.

††preprint: APS/123-QED

Introduction. — Skyrmions were introduced by Tony Skyrme in 1961 [1] as
particle-like solutions to the pion field to describe nucleons. They represent
localized regions in the field where an entire 3D spherical parameter space is
covered. Their 2D analogues, also known as baby skyrmions, have led to
intensive investigation, particularly in magnetic systems [2]. Mathematically,
we refer to a skyrmion as a distribution (or texture) of a vector field in
space that completely spans once a spherical parameter space, such that the
Skyrme density (the Jacobian between the parameter space and the spatial
coordinates in the region occupied by the skyrmion) does not change sign. Each
point in this parameter space corresponds to a different possible state of the
vector. By definition, the integral of the Skyrme density over the region
occupied by the skyrmion must equal $\pm 1$.

Let us focus on field distributions on a plane $(x,y)$ that span a 2D
spherical parameter space. In this context, merons are defined as “half-
skyrmions” corresponding to the mapping of one hemisphere, north or south,
their Skyrme number then being $\pm 1/2$. Note, though, that sometimes an
extra requirement is imposed on half-skyrmions to qualify as merons: a meron
must be enclosed by points corresponding to the equator [2]. Skyrmions and
other 2D textures have been observed in a plethora of physical systems, such
as magnetic materials [2, 3, 4], sound waves [5, 6], superfluids [7, 8] and
optical fields [9]. In nonparaxial optical fields, many skyrmionic textures
have been identified, including isolated skyrmions whose vector parameter is
the normalized photonic spin angular momentum, both for evanescent [10] and
propagating fields [11]. Periodic textures, such as skyrmion lattices in the
instantaneous orientation of the electric field in evanescent waves [12] and
spin optical meron lattices [13, 14, 15, 16], have also been observed.

In paraxial optics, 2D skyrmionic textures were identified in the distribution
of polarization at the transverse plane of monochromatic beams, where the
parameter space spanned corresponds to the Poincaré sphere. These
distributions are often referred to as Stokes textures [9], since they span
all normalized values of the Stokes vector. Full Poincaré beams [17, 18, 19],
which display a spatially variant polarization pattern in a transverse section
that is the stereographic projection of the Poincaré sphere, can be regarded
as isolated skyrmions. Recent work has extended beyond traditional skyrmions,
including Stokes bimerons, skyrmioniums, or multimerons [20, 21].

Regarding periodic textures, optical polarization lattices formed by tiles
that separately map each Poincaré sphere hemisphere can be recognized as
Stokes meron lattices. For example, the superposition of three plane waves
with appropriate polarizations and wavevectors results in a triangular lattice
[22]. Note that the Skyrme density in this lattice oscillates in sign, as does
that in nonparaxial spin optical meron lattices [13, 14, 15, 16]. In contrast,
other periodic textures preserve the Skyrme density’s sign over all space,
such as optical nonparaxial skyrmion lattices [12], meron lattices in
superfluid 3He-A [7], or textures in the velocity field of sound waves [6].

Our goal is to find and characterize periodic lattices whose Skyrme density
has uniform sign, and where the mapping between the spherical parameter space
and the flat space is conformal (except at specific points). This is achieved
by using mappings developed in nineteenth- and twentieth-century cartography,
where the sphere is mapped onto specific regular polygons while preserving
local angles. Physical space is then tessellated with these polygons without
reflections, only translations and rotations. We implement these skyrmion
lattices as Stokes textures in paraxial monochromatic fields, for which we
find a prescription that generates well-behaved fields presenting the desired
textures. Therefore, we map the Poincaré sphere as if it were the Earth (Fig.
1(a)). These maps apply to other spherical parameter spaces as well.

$y,s_{2}$
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Figure 1:  (a) The Earth and a spherical parameter space spanned in the
textures, for example the Poincaré sphere. (b-e) Periodic skyrmionic textures
resulting from conformal cartographic mappings that provide regular
tessellations: (b) Peirce’s quincuncial projection, (c) Adams’ world in a
hexagon, (d) Lee’s conformal world in a tetrahedron (where a mapping
singularity at the South pole makes Antarctica’s coastal features appear
twice), and (e) Wray’s variation of Lee’s projection. In all cases, the entire
sphere is mapped onto a regular polygon (outlined in black on the Earth maps).
One of these polygons is highlighted in darker blue in each case, which forms
a tessellation of the plane with rotated duplicates of itself. For Wray’s
projection (e), a rectangle is emphasized since the triangle onto which the
sphere is mapped can be rearranged into this shape. For all cases, a northern
and a southern hemispheres are also highlighted in red and purple,
respectively.

Only three tiles provide regular tessellations: (i.e., vertex-to-vertex
tilings of the plane with identical copies of a regular polygon [23]):
squares, equilateral triangles and regular hexagons. Here, we explore
conformal maps that yield regular tessellations where either the entire sphere
or a hemisphere is mapped onto each of these three regular polygons. The only
case left out is that in which a hemisphere is mapped onto a hexagon since it
is impossible to tile the plane without edge conflicts using only rotated
copies of the hexagonal hemispheres.

Cartographic mappings. — We now present the conformal cartographic
transformations and their implementation as skyrmionic textures (Fig. 1). We
begin with Peirce’s quincuncial projection [24], which maps the whole sphere,
and also each hemisphere, onto a square. The vertices of each square tile
mapping a hemisphere are equally spaced along the sphere’s equator, as
illustrated in Fig. 1(b). The map is conformal everywhere except at these four
singular points. The skyrmionic texture that results from this map is
topologically equivalent to a nonconformal texture in the context of
superfluid 3He-A [7].

The second mapping we investigate is Adams’ world in a hexagon projection
[25], which maps the sphere onto a regular hexagon, while also mapping each
hemisphere onto an equilateral triangle. This map exhibits singular points at
the vertices of each triangular tile that spans a hemisphere. These singular
points are again equally spaced along the equator, as depicted in Fig. 1(c).

Finally, we explore a projection where the entire sphere is mapped onto an
equilateral triangle, known as Lee’s world in a tetrahedron projection [26].
Here, the sphere is divided into four equal spherical equilateral triangles,
each of which is subsequently projected onto the corresponding face of a
regular tetrahedron. Unfolding the tetrahedron yields an equilateral triangle
that tessellates the plane with rotated copies of itself. As shown in Fig.
1(d), one of the tetrahedron’s vertices is placed at the south pole. Upon the
unfolding of the tetrahedron, the vertices of the resulting equilateral
triangle map precisely the south pole. This texture is topologically
equivalent to one generated recently in the instantaneous velocity field of
sound waves [6]. Wray proposed a modified version of Lee’s mapping, depicted
in Fig. 1(e), that offers a more democratic representation for the poles [27].
It is achieved through a rotation, so that each pole is positioned at the
midpoints of two opposing edges of the tetrahedron. In contrast to Lee’s
original projection, Wray’s transformation results in the equator forming not
closed loops but meandering lines. Upon unfolding the tetrahedron, the
resulting triangle can be rearranged into a rectangle (with aspect ratio
$\sqrt{3}:1$), yielding a tessellation composed of rectangles. In both
versions of Lee’s projection (Figs. 1(d,e)), the singular points are placed at
the vertices of the tetrahedron.

A way to understand these mappings is the following. Suppose that the sphere
is an inflated rubber balloon, which is then deflated while holding it from
the points corresponding to the vertices (three equidistant equatorial points
for Adams’ mapping, four uniformly spaced equatorial points for Peirce’s, and
four equidistant non-coplanar points for Lee’s or Wray’s). The resulting shape
is a two-layered triangle, a two-layered square, or a tetrahedron, which when
unfolded can tile space. Note that some of these constructions are more robust
to deformation than others: for Peirce’s projection, tiling would remain
possible if the four vertices defined a rectangle (preserving right angles)
rather than a square, while Adams’ mapping does require the three vertices to
coincide exactly with the corners of an equilateral triangle. Lee’s (or
Wray’s) tetrahedron projection can also be deformed continuously, as long as
the four facets can still be unfolded into a triangle, and in fact it can be
deformed continuously onto Pierce’s mapping.

Mathematically, the realization of these maps involves the sequential
concatenation of two conformal transformations: a planar map defined by an
analytic function denoted as $w(z)$, where $z=x+{\rm i}y$ ($x$ and $y$
representing the Cartesian coordinates in physical space), and a stereographic
projection from the north pole. These operations result in a map from the
plane to the azimuthal and polar coordinates on the sphere as
$\phi=\mathrm{arg}\left[w(z)\right]$ and $\theta=2\arctan|w(z)|$,
respectively. The 3D Cartesian coordinates in the ambient space of the sphere
are $s_{1}=\sin\theta\cos\phi$, $s_{2}=\sin\theta\sin\phi$ and
$s_{3}=\cos\theta$.

For Peirce’s and Adams’ projections, $w(z)$ is the inverse function of the
Schwarz-Christoffel transformation [28] that maps the unit disk onto a square
[29] and an equilateral triangle [27], respectively. The expressions for
$w(z)$ for each of these projections (Peirce [29], Adams, Lee, and Wray [27])
are:

$\displaystyle w_{\mathrm{P}}(z)$ $\displaystyle=\frac{e^{{\rm
i}(\phi_{0}+\alpha)}}{\sqrt{2}}\mathrm{sd}\left(\tilde{z}_{1},\frac{1}{\sqrt{2}}\right),$
(1a) $\displaystyle w_{\mathrm{A}}(z)$ $\displaystyle=e^{{\rm
i}(\phi_{0}+\alpha)}\mathrm{sm}(\tilde{z}_{2}),$ (1b) $\displaystyle
w_{\mathrm{L}}(z)$ $\displaystyle=2^{1/6}e^{{\rm
i}(\phi_{0}+\alpha)}\mathrm{sm}(\tilde{z}_{2})\mathrm{cm}(\tilde{z}_{2}),$
(1c) $\displaystyle w_{\mathrm{W}}(z)$ $\displaystyle=\frac{e^{{\rm
i}(\phi_{0}+\alpha)}}{2}\mathrm{sd}\left(\tilde{z}_{3},\frac{\sqrt{3}-1}{2\sqrt{2}}\right),$
(1d)

where sd is the ratio between the Jacobi elliptic functions sn and dn, while
sm and cm correspond to the Dixon elliptic functions, which can be expressed
in terms of the Weierstrass elliptic function denoted as $\wp(z)$ and its
derivative $\wp^{\prime}(z)$: $\mathrm{sm}(z)=6\wp(z)/(1-3\wp^{\prime}(z))$
and $\mathrm{cm}(z)=(3\wp^{\prime}(z)+1)/(3\wp^{\prime}(z)-1)$ [30]. The three
complex numbers $\tilde{z}_{i}$ are defined as
$\tilde{z}_{1,2,3}=\gamma_{1,2,3}\;e^{-{\rm i}\alpha}(z-z_{0})/d$, with
$\gamma_{1}=\Gamma^{2}(1/4)/(2\sqrt{2\pi})$,
$\gamma_{\mathrm{2}}=\Gamma^{3}(1/3)/(2\pi)$, and
$\gamma_{\mathrm{3}}=3^{1/4}\Gamma^{3}(1/3)/(2^{1/3}\pi)$, where $\Gamma$
denotes the gamma function. The angle $\alpha$ rotates the polygon onto which
the sphere is mapped, while $\phi_{0}$ sets the initial value from which
$\phi$ is swept within this polygon. Note that $\phi_{0}$ influences the type
of meron that appears in the textures (e.g., Neel- or Bloch-type [9]). The
real and imaginary parts of the complex number $z_{0}$ are the $x$ and $y$
coordinates defining the center of the map. Finally, $d$ sets the length of
the edge of either the square or the equilateral triangle spanning a
hemisphere in the case of Peirce’s and Adams’ mappings, while for Lee’s and
Wray’s mappings it corresponds to the length of the edge of the triangle
spanning the sphere. Note that, due to the periodicity of the Jacobi and Dixon
elliptic functions, these mappings work over the whole plane and not only
within a central tile. The periodicity of the tiling is then natural. The
inverse transformations to each of the mappings in Eqs. (1) are given in the
Supplemental Information, Sec. 1.

The Skyrme number $N_{\textrm{S}}$ quantifies the number of times the unit
vector $\mathbf{s}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ wraps
around the sphere, and the sense of wrapping determines its sign. Within a
given region $\sigma$, the Skyrme number is
$N_{\textrm{S}}=(4\pi)^{-1}\iint_{\sigma}\rho_{\textrm{S}}(x,y)\;\mathrm{d}x\,\mathrm{d}y$,
where
$\rho_{\textrm{S}}(x,y)=\mathbf{s}(x,y)\cdot\left[\partial_{x}\mathbf{s}(x,y)\times\partial_{y}\mathbf{s}(x,y)\right]$
is the Skyrme density, which is the Jacobian of the transformation, and can be
expressed as $\rho_{\mathrm{S}}=\rho_{\mathrm{SP}}\rho_{w}$, where
$\rho_{\mathrm{SP}}=4(1+|w(z)|^{2})^{-2}$ is the Jacobian of the stereographic
projection, and
$\rho_{w}=\left[\partial_{x}\mathrm{Re}(w(z))\right]^{2}+\left[\partial_{y}\mathrm{Re}(w(z))\right]^{2}$
is the Jacobian of the transformation $w(z)$. The last expression was
simplified using the Cauchy-Riemann conditions. These equations imply that
$\rho_{\mathrm{S}}$ is always positive, as shown in Fig. 2. Within the context
of cartography, this means that the well-known shapes of the continents are
never reversed. Note that a different convention for the stereographic
projection could be used that reverses its sign, leading to skyrmionic
textures with negative Skyrme density.

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Figure 2:  Skyrme density $\rho_{\mathrm{S}}$ for (a) Peirce, (b) Adams, and
(c) Lee/Wray maps within the area displayed in Fig. 1. The unit cell of each
texture is outlined by dashed lines.

For all projections in Fig. 1, it holds that $N_{\textrm{S}}=1$ within the
polygons that map the entire sphere. Peirce’s and Adams’ projections lead to
lattices of square and triangular merons, respectively, in which any two
neighboring merons map different hemispheres. Within each meron,
$N_{\textrm{S}}=1/2$. On the other hand, in the first version of Lee’s
projection (Fig. 1(d)), there are merons that span twice the southern
hemisphere, leading to $N_{\textrm{S}}=1$. In fact, both in Lee’s and Wray’s
versions of the tetrahedron projection, different regions covering the same
hemisphere are in direct contact and not separated by a line corresponding to
the equator, so these textures do not classify as meron lattices according to
some definitions [2]. This is due to the fact that some or all of the
mapping’s singular points are not along the equator.

Periodic skyrmionic textures in monochromatic paraxial optical fields. — We
now explore the implementation of these skyrmionic textures in the spatial
distribution of polarization for a monochromatic optical beam of frequency
$\omega$. The transverse electric field at any point $(x,y)$ and time $t$ is
given by $\bm{\mathcal{E}}=\mathrm{Re}\left[\mathbf{E}\;\mathrm{exp}(-{\rm
i}\omega t)\right]$, where $\mathbf{E}$ is the complex field, which can be
parametrized as

$\displaystyle\mathbf{E}$
$\displaystyle=E_{0}\left[\cos(\theta/2)\mathbf{l}+e^{{\rm
i}\phi}\sin(\theta/2)\mathbf{r}\right]=E_{\mathbf{l}}\mathbf{l}+E_{\mathbf{r}}\mathbf{r},$
(2)

with $E_{0}$ being a complex function of $x$ and $y$ that provides an overall
amplitude and phase and that does not affect the polarization (defined by the
expression in brackets). Here, the unit vectors
$\mathbf{l},\mathbf{r}=(\mathbf{x}\pm{\rm i}\mathbf{y})/\sqrt{2}$ represent
left- and right-circular polarization, where $\mathbf{x}$ and $\mathbf{y}$ are
unit vectors along the $x$ and $y$ axes, respectively. The circular
polarization components $E_{\mathbf{l},\mathbf{r}}$ are complex functions of
$x$ and $y$. The relative phase between them is
$\phi=\phi_{\mathbf{r}}-\phi_{\mathbf{l}}$ (with
$\phi_{\mathbf{r},\mathbf{l}}=\mathrm{arg}[E_{\mathbf{r},\mathbf{l}}]$). At
each point, $\bm{\mathcal{E}}$ traces an ellipse within the transverse plane
over each optical cycle. The ellipticity and handedness of this path is
determined by $\theta$, while $\phi$ is twice the orientation angle of the
ellipse’s major axis with respect to $x$. Each possible polarization ellipse
then corresponds to a point on the Poincaré sphere (Fig.1(a)) described by a
normalized Stokes vector
$\mathbf{s}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$. Spanning the
Poincaré sphere then means achieving every paraxial polarization. The northern
and southern hemispheres correspond to left- and right-handed ellipses,
respectively, with the two poles corresponding to the two circular
polarizations, while linearly polarized ellipses are positioned along the
equator.

We now derive equations for the optical fields that implement these textures,
independently of the mapping function $w(z)$. Since
$w(z)=\tan(\theta/2)e^{{\rm i}\phi}$, Eq. (2) can be written as
$\mathbf{E}=E_{\mathbf{l}}(z)[\mathbf{l}+w(z)\mathbf{r}]$, so any reasonable
choice for $E_{\mathbf{l}}(z)$ leads to the desired polarization texture as
long as $E_{\mathbf{r}}(z)=E_{\mathbf{l}}(z)w(z)$. However, it is desirable to
choose a functional form that avoids singularities in the field, and that
treats the two hemispheres similarly. Note that, in our convention, the
hemisphere with left(right)-handed ellipses corresponds to $|w(z)|<(>)\,1$,
with the pole representing left(right)-circular polarization corresponding to
$w(z)=0\,(\infty)$. It is then desirable to find an expression for
$E_{\mathbf{l}}(z)$ such that $E_{\mathbf{r}}(z)=E_{\mathbf{l}}(z)w(z)$ takes
a similar functional form except for the replacement $w\to 1/w$. One such
solution is

$\displaystyle E_{\mathbf{l}}(z)$ $\displaystyle=$ $\displaystyle{\cal
E}\sqrt{-w_{0}}\,\frac{w^{*}(z)-w_{0}^{*}}{|w(z)|^{2}+|w_{0}|^{2}},$ (3a)
$\displaystyle E_{\mathbf{r}}(z)$ $\displaystyle=$ $\displaystyle{\cal
E}\sqrt{\frac{1}{-w_{0}}}\,\frac{1/w^{*}(z)-1/w_{0}^{*}}{|1/w(z)|^{2}+|1/w_{0}|^{2}},$
(3b)

where ${\cal E}$ is a constant amplitude factor and $w_{0}$ is a dimensionless
constant. Note that both polarization components vanish at points where
$w(z)=w_{0}$, meaning that this construction forces zeros in the field at
points where the polarization is such that $\tan(\theta/2)e^{{\rm
i}\phi}=w_{0}$. As we discuss later, this is not a problem of the construction
but a feature with a topological origin. It is easy to show that the
expression for the intensity becomes simpler when $w_{0}$ is chosen such that
$|w_{0}|=1$, giving

$\displaystyle I(z)=\frac{|w(z)-w_{0}|^{2}}{|w(z)|^{2}+1}.$ (4)

In this case the field zeros coincide with points of linear polarization with
orientation angle equal to $\mathrm{arg}(w_{0})/2$. In what follows we choose
$w_{0}=-1$ to simplify the expressions, so that the zeros coincide with points
of vertical linear polarization. Here we focus on Peirce’s texture as a
representative example, shown in Fig. 3: intensity and phase of
$E_{\mathbf{l},\mathbf{r}}$ (a), and polarization and intensity distributions
(b). The figures for the other maps are shown in the Supplemental Material,
Sec. 2, which also includes plots of the fields’ discrete Fourier spectra for
the four maps, where the intensity of the diffraction orders is small for high
spatial frequencies.

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C-point
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Scalar vortex
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$|E_{\mathbf{r}}|^{2}$
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-1
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Figure 3:  Unit cell of the optical beam implementing in its polarization
state Peirce’s texture according to Eqs. (1) with $\alpha=\pi/4$,
$\phi_{0}=7\pi/6$, and $z_{0}=(-1+{\rm i})/2$. (a) Theoretical intensity and
phase distributions of the circular polarization components
$E_{\mathbf{l},\mathbf{r}}$. (b) Polarization and intensity distributions.
Phase vortices result in either C-points or scalar vortices.

The textures described here can be regarded as optical polarization lattices
[31], exhibiting polarization singularities such as L-lines (lines of linear
polarization) and C-points (points of circular polarization) [32]. C-points
emerge in regions where a vortex in $\phi_{\mathbf{l},\mathbf{r}}$ with a
topological charge of $m_{\mathbf{l},\mathbf{r}}$ is superimposed onto a
region with no vortex in $\phi_{\mathbf{r},\mathbf{l}}$, rendering a vortex in
$\phi$ with a topological charge of $m_{\phi}=\mp m_{\mathbf{l},\mathbf{r}}$.
The polarization ellipse undergoes a rotation of $m_{\phi}\pi$ radians around
the singularity in $\phi$. For $m_{\phi}>0$, the ellipse performs a
counterclockwise rotation, while for $m_{\phi}<0$ it rotates clockwise. In the
proximity of C-points with $m_{\phi}=\pm 1$, the ellipses exhibit distinct
singular patterns known as lemons ($m_{\phi}=1$) and stars ($m_{\phi}=-1$)
[32].

The polarization distribution in Fig. 3(b) shows an important feature: For a
texture where $\rho_{\mathrm{S}}>0$ everywhere, all left(right)-handed
C-points originate from a $\phi$ vortex with $m_{\phi}>(<)\,0$, and then all
C-points with $m_{\phi}=\pm 1$ manifest as left-handed lemons and right-handed
stars. (For a texture where $\rho_{\mathrm{S}}<0$ everywhere the situation
would reverse.) A way to understand this fact comes from considering mapping a
small patch of the sphere. Let the polar coordinates of the local planar map
be $(r,\varphi)$. When mapping the north pole,
$r\approx\kappa\theta,\varphi=\phi$, where $\kappa$ is a positive constant.
The Skyrme density has the same sign as the Jacobian between $(\theta,\phi)$
and $(r,\varphi)$, which equals $1/\kappa>0$. Now consider transporting
continuously the map to the south pole without reversing local directions
(i.e., $\rho_{\mathrm{S}}$ does not change sign). At the south pole the
coordinates can be set to $r\approx\kappa(\pi-\theta),\varphi=-\phi$, which
yield the same positive Jacobian. However, $m_{\phi}=\partial_{\varphi}\phi$
is positive at the north pole (left-circular polarization) and negative at the
south pole (right-circular polarization).

Together with a well-established result from index theory, the conclusion
about C-points just mentioned implies that paraxial fields displaying periodic
skyrmionic Stokes textures that maintain the sign of $\rho_{\mathrm{S}}$ must
inevitably present zeros. This is because, within a unit cell of a periodic
complex scalar field, the topological charges of its vortices must add up to
zero. However, as just discussed, the vortices for each circular component
that lead to C-points all have the same charge. Each component must then
include extra vortices with the opposite charge, and the only way for these
not to produce extra C-points is if their locations coincide for the two
circular components, hence giving rise to field zeros. The field construction
in Eqs. (3) naturally introduces these zeros, and places them at points with a
specific polarization.

We generated the textures experimentally by implementing the circular
polarization distributions in Eqs. (3) in a laser beam. We used a vector beam
generation technique [33] based on a spatial light modulator (SLM), with an
algorithm [34] that allows both amplitude and phase modulation with a phase-
only SLM. We then measured the intensities of various polarization projections
and used them to compute the skyrmionic texture (details in Supplemental
Information, Sec. 3). Figure 4 shows, for Peirce’s projection, the measured
Stokes vector distribution (a) and $\rho_{\mathrm{S}}$ (b). Similar results
for the other three projections are given in the Supplemental Material, Sec.
4. For all cases, small regions of highly negative $\rho_{\mathrm{S}}$ emerge
near the points where Eqs. (3) predict the field’s zeros (corresponding to
vertical polarization). This is because these zeros are unstable: any small
misalignment of the zeros of both polarization components causes the intensity
not to vanish exactly and the polarization to vary rapidly, covering the
entire sphere (in the opposite sense) within a small region. Possible sources
of zero misalignment in the experiment are defocus or the clipping of
diffraction orders by the aperture. Excluding some of these orders carrying
low intensities has little effect on the overall pattern but leads to rapid
polarization variation in the low-intensity regions (see Supplemental
Information, Sec. 5), similar to that in the measured data.

(a)
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1
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$s_{3}$
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-1
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Figure 4:  Unit cell of the measured (a) Stokes vector distribution and (b)
Skyrme density for Peirce’s field.

Conclusions. — The skyrmionic lattices presented here serve as examples
representing different topologies that allow preserving the sign of the Skyrme
density. Note that rotated versions of each of these cartographic projections
can also be considered, as we did for two forms of Lee’s tetrahedron
projection. Similarly, as suggested earlier, some of these mappings accept
deformations that remain conformal and still tessellate the plane but not with
regular polygons. Further, other conformal projections that tessellate the
plane can be explored [27, 29], such as projections of the sphere onto a
triangle or a square using Lagrange’s projection of the sphere onto a disk
[35] instead of the stereographic projection. An interesting case is Adams’
world in a square projection [36], which renders merons spanning a hemisphere
four times for both hemispheres. Another texture with merons spanning a
hemisphere six times for one of the two hemispheres would result from Cox’
world in a triangle projection [37]. It would be interesting to know if any of
these conformal textures emerges naturally in some physical context, where the
fields span a different spherical parameter space. In any case, these lattices
serve as models for others that appear naturally in physical situations (e.g.,
[6, 7]) and that are topologically equivalent to these even if their mapping
is not conformal.

While the topological number $N_{\rm S}$ over many cells of a periodic
skyrmionic texture can accumulate arbitrarily, the sum of the topological
charges of the vortices of a periodic complex field must vanish within each
unit cell. The optical implementations considered here for these skyrmionic
textures are based on the polarization of a complex two-component vector
field. Achieving a Skyrme density with uniform sign requires the introduction
of zeros in the vector field, making the texture unstable to perturbations
such as propagation. It is an open question whether there exist topologically
equivalent textures that might not be conformal but where these instabilities
are reduced (e.g., by achieving propagation invariance). This will be the
subject of follow-up work. Note that one should not speak of strict
topological protection of skyrmions in implementations where the spanned
spherical parameter space results from normalizing a vector that can vanish,
since transition between two topologically different states is then possible.

Finally, it would be interesting to study if periodic textures with uniform
sign of the Skyrme density (conformal or not) exist for higher
dimensionalities. Recently, optical implementations of skyrmionic structures
spanning 3-spheres [38] and 4-spheres [39] were proposed, the latter
corresponding to a space-time periodic texture spanning all states of
monochromatic nonparaxial polarization, but whose Skyrme density oscillates in
sign.

###### Acknowledgements.

This research received funding from the Agence Nationale de Recherche (ANR)
through the project 3DPol, ANR-21-CE24-0014-01. D. M. acknowledges Ministerio
de Universidades, Spain, Universidad Miguel Hernández and the European Union
(Next generation EU fund) for a Margarita Salas grant from the program Ayudas
para la Recualificación del Sistema Universitario Español.

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## Supplemental Information

###  1 Inverse mappings

We provide here the inverse maps of the transformations $w_{i}(z_{i})$ in Eqs.
(1) in the main text, where $i=\mathrm{P},\mathrm{A},\mathrm{L},\mathrm{W}$
stand for the Peirce (P), Adams (A), Lee (L) and Wray (W) projections:

$\displaystyle z_{\mathrm{P}}(w_{\mathrm{P}})$
$\displaystyle=z_{0}+\frac{\sqrt{2}d}{\gamma_{1}}e^{{\rm
i}\alpha}F\left(\arcsin\tilde{w}_{\mathrm{P}},{\rm i}\right),$ (5a)
$\displaystyle z_{\mathrm{A}}(w_{\mathrm{A}})$
$\displaystyle=z_{0}+\frac{d}{\gamma_{2}}\tilde{w}_{\mathrm{A}}e^{{\rm
i}\alpha}{}_{2}F_{1}\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};\tilde{w}^{3}_{\mathrm{A}}\right),$
(5b) $\displaystyle z_{\mathrm{L}}(w_{\mathrm{L}})$
$\displaystyle=z_{0}+\frac{d}{\gamma_{2}}f(\tilde{w}_{\mathrm{L}})e^{{\rm
i}\alpha}{}_{2}F_{1}\left[\frac{1}{3},\frac{2}{3};\frac{4}{3};f^{3}(\tilde{w}_{\mathrm{L}})\right],$
(5c) $\displaystyle z_{\mathrm{W}}(w_{\mathrm{W}})$
$\displaystyle=z_{0}+\frac{d}{\gamma_{3}}e^{{\rm
i}\alpha}\mathrm{sd}^{-1}\left(2\tilde{w}_{\mathrm{W}},\frac{\sqrt{3}-1}{2\sqrt{2}}\right),$
(5d)

with

$\displaystyle
f^{3}(\tilde{w}_{\mathrm{L}})=\frac{1}{2}\left(1-\sqrt{1-2\sqrt{2}\tilde{w}^{3}_{\mathrm{L}}}\right),$
(6)

and

$\displaystyle\tilde{w}_{\mathrm{P},\mathrm{A},\mathrm{L},\mathrm{W}}=e^{-{\rm
i}(\phi_{0}+\alpha)}w_{\mathrm{P},\mathrm{A},\mathrm{L},\mathrm{W}},$ (7)

where $F$ represents the incomplete elliptic integral of the first kind,
${}_{2}F_{1}$ stands for the ordinary hypergeometric function, and
$\mathrm{sd}^{-1}$ denotes the inverse of the Jacobi elliptic function sd, and
the parameters $\gamma_{1,2,3}$ are given in the main text, as is a
description of the effect of the parameters $\alpha$, $\phi_{0}$, $z_{0}$ and
$d$.

S. P
---
I. S. P
---
$\mathrm{Im}(w),s_{2}$
---
$\mathrm{Re}(w),s_{1}$
---
$\mathrm{Im}(w_{\mathrm{P}}),s_{2}$
---
$\mathrm{Re}(w_{\mathrm{P}}),s_{1}$
---
(a)
---
$z_{\mathrm{P}}(w_{\mathrm{P}})$
---
$w_{\mathrm{P}}(z_{\mathrm{P}})$
---
(b)
---
(c)
---
$z_{\mathrm{A}}(w_{\mathrm{A}})$
---
$w_{\mathrm{A}}(z_{\mathrm{A}})$
---
$s_{1}$
---
$s_{2}$
---
$s_{3}$
---
$\mathrm{Im}(z_{\mathrm{P}}),s_{2}$
---
$\mathrm{Re}(z_{\mathrm{P}}),s_{1}$
---
$\mathrm{Im}(z_{\mathrm{A}}),s_{2}$
---
$\mathrm{Re}(z_{\mathrm{A}}),s_{1}$
---
$\mathrm{Im}(w_{\mathrm{A}}),s_{2}$
---
$\mathrm{Re}(w_{\mathrm{A}}),s_{1}$
---
Figure 5:  (a) Depiction of the stereographic projection (S. P) and of the
inverse stereographic projection (I. S. P) between a meron within the unit
disk and the unit sphere’s northern hemisphere. (b,c) Schwarz maps and their
inverse transformations between a meron within the unit disk and a (b) squared
and a (c) equilateral triangular meron.

Equations (1) in the main text define complex quantities, $w_{i}(z_{i})$,
which, via an inverse stereographic projection, yield values for the azimuthal
and polar spherical angles, $\phi=\mathrm{arg}\left[w_{i}(z_{i})\right]$ and
$\theta=2\arctan|w_{i}(z_{i})|$, respectively, for any point on the plane. The
$x$ and $y$ Cartesian coordinates of this point correspond to the real and
imaginary parts of the complex number $z_{i}$, respectively. On the other
hand, the real and imaginary parts of the inverse transformations
$z_{i}(w_{i})$ provide a pair of Cartesian coordinates on the plane where a
specific value of $\phi$ and $\theta$ is mapped. The spherical coordinates are
related to $w_{i}$ through a stereographic projection as
$w_{i}=\tan(\theta/2)e^{{\rm i}\phi}$.

Equations (5a) and (5b) are well-known Schwarz maps. The interior of the unit
disk can be mapped conformally onto the interior of a regular polygon of $n$
sides through the Schwarz integral [28]:

$\displaystyle z=\int_{0}^{w}\frac{1}{\left(1-w^{n}\right)^{2/n}}\mathrm{d}w,$
(8)

where the real and imaginary parts of $w$ are the Cartesian coordinates within
the disk, and the real and imaginary parts of $z$ are the Cartesian
coordinates within the polygon. The first two equations result from this
integral for a square ($n=4$) and for an equilateral triangle ($n=3$) [30],
but we introduced the additional parameters $\alpha$, $\phi_{0}$, $z_{0}$ and
$d$ to control certain properties of the map, as explained in the main text.

Figure 5(a) illustrates the transformation between a meron within the unit
disk and the northern hemisphere of the unit sphere through the stereographic
projection and its inverse. Figures 5(b,c) illustrate the Schwarz maps,
$z_{i}(w_{i})$, and their inverse functions, $w_{i}(z_{i})$, between a meron
within the unit disk, and a meron within a square ($i=\mathrm{P}$) and within
an equilateral triangle ($i=\mathrm{A}$), respectively. When these maps are
extended to the whole plane, they yield the entire Peirce and Adams textures
shown in Fig. 1 of the main document.

The inverse map of $w_{\mathrm{L}}(z_{\mathrm{L}})$ (namely
$z_{\mathrm{L}}(w_{\mathrm{L}})$ in Eq. (5c)) was derived by expressing the
Dixon elliptic function $\mathrm{cm}z$ in the transformation
$w_{\mathrm{L}}(z_{\mathrm{L}})$ (Eq. (1c) in the main text) in terms of the
Dixon elliptic function $\mathrm{sm}z$ using the fact that
$\textrm{cm}^{3}z+\textrm{sm}^{3}z=1$. We then use Eq.(5b), which is the
inverse function of $\mathrm{sm}z$ up to certain multiplicative and additive
factors, to calculate the inverse of $w_{\mathrm{L}}(z_{\mathrm{L}})$.

###  2 Additional theoretical distributions of the textures and fields

Here, we show additional results for the textures and present the optical
fields implementing the Adams, Lee and Wray textures. The parameters in Eqs.
(1) in the main text are set as follows. For Adams’ projection: $\alpha=0$,
$\phi_{0}=\pi/2$ and $z_{0}=-1/\sqrt{3}$. For Lee’s projection: $\alpha=0$,
$\phi_{0}=-5\pi/6$ and $z_{0}=1/\sqrt{3}$. For Wray’s projection:
$\alpha=\pi/2$, $\phi_{0}=-7\pi/12$ and $z_{0}=\sqrt{3}/4$. We show a
representation of the parameters $\phi$ and $s_{3}$ for every field in Fig. 6.

$2\pi$
---
0
---
-1
---
1
---
$\phi$
---
$s_{3}$
---
(c)
---
(d)
---
(a)
---
(b)
---
$y$
---
$x$
---
Figure 6:  Distributions of $\phi$, $s_{3}$ for the textures resulting from
(a) Peirce’s, (b) Adams’, (c) Lee’s and (d) Wray’s projections. The plot shown
in (a) constitutes the entire unit cell of Peirce’s texture. The unit cells in
(b-d) are delimited by dashed lines.

The intensity and phase distributions of the circular polarization components,
$E_{\mathbf{l},\mathbf{r}}$, as well as the polarization distributions, for
Adams, Lee and Wray fields are presented in Fig. 7.

(a)
---
(b)
---
$x$
---
$\mathrm{arg}(E_{\mathbf{l}})$
---
$|E_{\mathbf{l}}|^{2}$
---
$\mathrm{arg}(E_{\mathbf{r}})$
---
$|E_{\mathbf{r}}|^{2}$
---
(c)
---
(d)
---
$\mathrm{arg}(E_{\mathbf{l}})$
---
$|E_{\mathbf{l}}|^{2}$
---
$\mathrm{arg}(E_{\mathbf{r}})$
---
$|E_{\mathbf{r}}|^{2}$
---
(e)
---
(f)
---
$\mathrm{arg}(E_{\mathbf{l}})$
---
$|E_{\mathbf{l}}|^{2}$
---
$\mathrm{arg}(E_{\mathbf{r}})$
---
$|E_{\mathbf{r}}|^{2}$
---
$y$
---
$\small{2\pi}$
---
0
---
0
---
-1
---
1
---
max
---
Phase
---
Intensity
---
0
---
max
---
Intensity
---
C-point
---
Scalar vortex
---
$s_{3}$
---
Figure 7:  (a,c,e) Theoretical intensity and phase distributions of the
circular polarization components, and (b,d,f) polarization and total intensity
distributions, for (a,b) Adams, (c,d) Lee and (e,f) Wray fields. We indicate
which vortices in the circular components render either C-points or scalar
vortices.

As pointed out in the main text, the fields display a discrete spatial Fourier
spectrum. The intensity and phase of the first diffraction orders of the
spectrum for each field, which contain most of the energy, are presented in
Fig. 8.

(a)
---
max
---
0
---
0
---
$\small{2\pi}$
---
$\mathrm{arg}(E_{\mathbf{l}})$
---
$\mathrm{arg}(E_{\mathbf{r}})$
---
$|E_{\mathbf{l}}|^{2}$
---
$|E_{\mathbf{r}}|^{2}$
---
$\mathrm{arg}(E_{\mathbf{l}})$
---
$\mathrm{arg}(E_{\mathbf{r}})$
---
$|E_{\mathbf{l}}|^{2}$
---
$|E_{\mathbf{r}}|^{2}$
---
$\mathrm{arg}(E_{\mathbf{l}})$
---
$\mathrm{arg}(E_{\mathbf{r}})$
---
$|E_{\mathbf{l}}|^{2}$
---
$|E_{\mathbf{r}}|^{2}$
---
$\mathrm{arg}(E_{\mathbf{l}})$
---
$\mathrm{arg}(E_{\mathbf{r}})$
---
$|E_{\mathbf{l}}|^{2}$
---
$|E_{\mathbf{r}}|^{2}$
---
(b)
---
(c)
---
(d)
---
Figure 8:  Intensity and phase of the first diffraction orders of the spatial
spectrum of the circular polarization components of (a) Peirce, (b) Adams, (c)
Lee and (d) Wray fields.

###  3 Experimental setup

As explained in the main text, the generation of the skyrmionic textures
requires the control of the amplitude and phase of the transverse polarization
components. There are several experimental proposals to generate complex
vector fields based on the modulation of two orthogonal polarization
components. Here, we adapt the approach of Maurer et al. [33] combined with
the algorithm introduced by Bolduc et al. [34] for encoding amplitude and
phase in a phase-only hologram. The setup for the generation and measurement
of the skyrmionic textures is shown in Fig. 9(a).

(a)
---
Laser:
---
$\lambda=532$ nm
(b)
---
L1
---
L2
---
L3
---
L4
---
HWP2
---
QWP
---
LP
---
CAMERA
---
HWP1
---
SLM
---
BS
---
WP
---
M1
---
M2
---
0
---
$2\pi$
---
SF
---
Figure 9:  (a) Experimental setup (not to scale). A linearly polarized
continuous-wave (CW) laser beam ($\lambda=532$ nm) passes through a half-wave
plate (HWP1) that controls the direction of polarization. The beam is then
redirected by a 50:50 beamsplitter (BS) to a Wollaston prism (WP) that
separates it into two orthogonal linearly polarized beams by 1∘. The two beams
are magnified by a telescope composed of lenses L1 and L2 with focal lengths
of $f_{1}=100$ cm and $f_{2}=500$ cm, respectively. Due to the fact that the
SLM can only modulate the horizontal polarization, a HWP2 rotates the vertical
polarization by 90∘. When the two beams are reflected by the SLM, the HWP2
rotates by 90∘ the beam polarization again. The two orthogonal polarized beams
are then recombined by the WP and are separated from the initial beam by the
BS. With a second telescope (composed of lenses L3 and L4 with focal lengths
of $f_{3}=300$ cm and $f_{4}=400$ cm, respectively) the SLM plane is imaged
onto a CMOS camera. A spatial filter (SF) is placed at the Fourier plane of
L4. In order to measure the polarization projections, a retractable quarter
wave plate (QWP) and a retractable linear polarizer (LP) are placed in front
of the camera (enclosed within dashed lines). (b) Example of two holograms
displayed on the SLM in order to modulate the horizontal (top) and vertical
(bottom) components of the Wray field. Each circular area has a diameter of
260 pixels.

To modulate independently the two transverse polarization components of a
laser beam, the SLM screen is divided into two regions. In each of these
regions, a circular area is defined where the holograms are displayed. For
simplicity, we modulate the horizontal and vertical components of the
skyrmionic texture. These components are computed from Eqs. (4) of the main
text as

$E_{\textbf{x},\textbf{y}}=\frac{c_{\textbf{x},\textbf{y}}}{\sqrt{2}}\left(E_{\textbf{r}}\pm
E_{\textbf{l}}\right),$ (9)

where x, y are the horizontal and vertical lineal polarization components,
$c_{\textbf{x}}=1$ and $c_{\textbf{y}}={\rm i}$.

According to Ref. [34], if $A$ and $\Phi$ are the desired amplitude and phase
modulated by the SLM, then the displayed hologram is given by
$\Psi=M\mathrm{Mod(}F\mathrm{,}2\pi{)}$ (assuming an incoming beam with
constant amplitude and phase), where $M=1+\frac{1}{\pi}\mathrm{sinc}^{-1}(A)$
and $F=\Phi-\pi M$. In order to perfectly recombine the two components with a
Wollaston prism (WP), a blazed phase $\Phi_{B}$ of opposite sign must be
displayed in each area. In addition, a correction phase $\Phi_{C}$ (provided
by the manufacturer) must be used to account for the intrinsic aberrations of
the SLM, so that the final desired amplitude is given by
$A_{\textbf{x},\textbf{y}}=|E_{\textbf{x},\textbf{y}}|$ and the final desired
phase is
$\Phi_{\textbf{x},\textbf{y}}=\mathrm{Mod}(\mathrm{arg}[E_{\textbf{x},\textbf{y}}]\pm\Phi_{B}+\Phi_{C},2\pi)$.

As shown in Fig. 9(a), a telescopic system projects the SLM plane onto the
camera, and a spatial filter (SF) is used to remove spurious contributions. By
rotating a linear polarizer (LP), the intensities of the components in x, y, p
and m (the latter two at $\pm 45^{\circ}$ from x, respectively) are measured.
By placing a quarter wave plate (QWP) with fast axis at $\pm 45^{\circ}$ with
respect to x, the intensities of the r and l components are measured. Let us
refer to these intensities as $I_{i}$ with
$i=\mathbf{x},\mathbf{y},\mathbf{p},\mathbf{m},\mathbf{l},\mathbf{r}$, (see
Fig. 10). We also measure the total intensity $I$ by removing the LP and the
QWP. We can then calculate the Stokes parameters according to:

$\displaystyle\quad S_{0}=I,$ (10) $\displaystyle\quad
S_{1}=I_{\textbf{x}}-I_{\textbf{y}},$ $\displaystyle\quad
S_{2}=I_{\textbf{p}}-I_{\textbf{m}},$ $\displaystyle\quad
S_{3}=I_{\textbf{l}}-I_{\textbf{r}}.$

We compute the normalized Stokes vector as
$\textbf{s}^{\prime}=(S_{1},S_{2},S_{3})/S_{0}$. Ideally this is a unit vector
given that the field is fully polarized. However, due to noise in the
measurements and the spatial extent of the pixels, the norm of this vector can
be smaller (and in some cases higher) than unity. We therefore renormalize it
as $\textbf{s}=\textbf{s}^{\prime}/|\textbf{s}^{\prime}|$.

$I_{\mathbf{x}}$
---
$I_{\mathbf{p}}$
---
$I_{\mathbf{l}}$
---
0.676 mm
---
max
---
$I$
---
$I_{\mathbf{x}}$
---
$I_{\mathbf{y}}$
---
$I_{\mathbf{p}}$
---
$I_{\mathbf{m}}$
---
$I_{\mathbf{l}}$
---
$I_{\mathbf{r}}$
---
$I$
---
$I$
---
$I$
---
$I_{\mathbf{x}}$
---
$I_{\mathbf{p}}$
---
$I_{\mathbf{l}}$
---
$I_{\mathbf{y}}$
---
$I_{\mathbf{m}}$
---
$I_{\mathbf{r}}$
---
(a)
---
(c)
---
(b)
---
(d)
---
$I_{\mathbf{y}}$
---
$I_{\mathbf{m}}$
---
$I_{\mathbf{r}}$
---
$I_{\mathbf{y}}$
---
$I_{\mathbf{m}}$
---
$I_{\mathbf{r}}$
---
$I_{\mathbf{x}}$
---
$I_{\mathbf{p}}$
---
$I_{\mathbf{l}}$
---
0
---
Figure 10:  Measurements of the total intensity and of the intensities of each
of the six polarization projections: x, y, p, m, r and l for (a) Pierce, (b)
Adams, (c) Lee and (d) Wray fields. All the images are normalized with respect
to their maximum. The circular area has a diameter of 261 pixels, with a pixel
size of 5.20 µm. The exposure time for each image is 2.6 s.

The components of $\mathbf{s}=(s_{1},s_{2},s_{3})$ are used to compute the
experimental values of the azimuthal and polar angles on the Poincaré sphere,
$\phi$ and $\theta$ respectively, as

$\displaystyle\phi$ $\displaystyle=\mathrm{atan2}(s_{2},s_{1}),$ (11)
$\displaystyle\theta$ $\displaystyle=\arccos s_{3}.$ (12)

The semi-major and semi-minor axes of the polarization ellipses are obtained
from

$\displaystyle a$
$\displaystyle=\left[1+\frac{\tan(\theta-\pi/4)}{2}\right]^{-1/2},$ (13)
$\displaystyle b$ $\displaystyle=\sqrt{1-a^{2}}.$ (14)

###  4 Additional experimental results

The measured Stokes vector distribution and the Skyrme density for Adams, Lee
and Wray fields are shown in Fig. 11. Figure 12 shows the experimental
distributions of $\phi$ and $s_{3}$ for all the fields. The measured
polarization map and the intensity distribution for the Peirce, Adams, Lee,
and Wray fields are depicted in Fig. 13. Like for the Peirce field, the Skyrme
density for the other three fields exhibits highly negative regions near
points where Eqs. (3) in the main text predict the field’s zeros.

-1
---
1
---
0
---
0
---
max
---
min
---
max
---
min
---
max
---
min
---
(a)
---
(b)
---
(c)
---
(d)
---
(e)
---
(f)
---
$\rho_{\mathrm{S}}$
---
0.1 mm
---
0.1 mm
---
0.1 mm
---
$\rho_{\mathrm{S}}$
---
$\rho_{\mathrm{S}}$
---
$y$
---
$x$
---
$s_{3}$
---
$x,s_{1}$
---
$x,s_{1}$
---
$x,s_{1}$
---
$y,s_{2}$
---
$y,s_{2}$
---
$y,s_{2}$
---
0
---
Figure 11: Measured (a,c,e) Stokes vector distribution, and (b,d,f) Skyrme
density for (a,b) Adams, (c,d) Lee and (e,f) Wray fields.  $2\pi$
---
0
---
1
---
-1
---
$\phi$
---
$s_{3}$
---
(a)
---
(b)
---
(c)
---
(d)
---
$y$
---
$x$
---
Figure 12: Experimental $\phi,s_{3}$ distributions for (a) Peirce, (b) Adams,
(c) Lee and (d) Wray fields. -1
---
1
---
$s_{3}$
---
0
---
max
---
(a)
---
(b)
---
(d)
---
(c)
---
Intensity
---
$y$
---
$x$
---
Figure 13: Experimental polarization maps for (a) Peirce, (b) Adams, (c) Lee
and (d) Wray fields.

###  5 Instability of the zeros in the field

In Fig. 4(b) in the main text, we demonstrate that Peirce’s optical field
exhibits regions of highly negative Skyrme density, $\rho_{\mathrm{S}}$, at
the positions where the zeros of the field are predicted to be. In fact, this
occurs for all fields, as shown in Fig. 11. As explained in the main text, the
clipping of higher orders of the fields’ spectra gives rise to
$\rho_{\mathrm{S}}<0$ in these regions. Figure 14 shows the polarization map
and the intensity of the field obtained considering only the diffraction
orders we observed in the spectrum of our experimental field, which are
illustrated in Fig. 8(a). Note that this polarization distribution is closer
to the measured polarization distribution shown in Fig. 13(a), especially
around the points where zeros were expected had the full spectrum been
considered. It is also shown in Fig. 14 that a right-handed lemon/left-handed
star pair is generated near these points, heralding zones of highly negative
Skyrme density. Right at the points where the zeros were expected,
$\rho_{\mathrm{S}}$ can take arbitrarily large values following a small
perturbation.

$x$
---
$y$
---
$\rho_{\mathrm{S}}$
---
0
---
max
---
-1
---
0
---
1
---
$s_{3}$
---

Intensity
---

$y$
---
$x$
---
Figure 14:  Illustration of the impact of aperture size limitations for
Peirce’s field. When considering the entire spatial frequency spectrum of the
field, the polarization state distribution matches the one depicted in Fig.
3(b) in the main text. However, when only the diffraction orders in Fig. 8(a)
are taken into account, the polarization state distribution changes to the one
shown here. Near the zeros of the field in Fig. 3(b) in the main text, now a
pair of C-points emanates from them. The behavior of both the polarization
state and the Skyrme density near these points is illustrated here.