Second-Order EM-Space Meron Topology

• Second-order EM-space meron is a higher-order half-skyrmion texture characterized by a 2-to-1 mapping of a spatial region onto one hemisphere of the energy symmetry sphere.
• It manifests in various realizations such as focused vortex beams, liquid-crystal microcavities, and photonic-crystal slabs with distinct electric–magnetic and polarization properties.
• Experimental reconstructions using polarimetry, digital holography, and photoemission electron microscopy reveal its double winding, topological invariants, and unique mapping behavior.

A second-order EM-space meron is a higher-order half-skyrmion texture of an electromagnetic field, but the precise meaning of “EM-space” depends on the state space in which the texture is defined. In "Topologies of light in electric-magnetic space" (Vernon, 22 Jul 2025), the term is used explicitly for a texture of the combined electric–magnetic field in an abstract two-dimensional electric–magnetic space, represented by the normalized vector $\mathbf{w}=\mathbf{W}/|\mathbf{W}|$ on the energy symmetry sphere. In related optical literature, second-order merons also denote double-winding polarization textures of light, with vorticity $v=\pm 2$ and topological charge $Q=\pm 1$ in a pseudospin or Stokes field (Król et al., 2020). Across these usages, the common structure is a meron-like hemisphere mapping with double winding or double coverage, realized in real space, momentum space, or momentum–energy space.

1. Definition in electric–magnetic space

In the explicit electric–magnetic-space construction, the electromagnetic field is described by a combined electric–magnetic state rather than by $\mathbf{E}$ or $\mathbf{H}$ in isolation. The relevant local invariants are the Stokes-like quantities

$$\begin{aligned} W_0 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 + \mu_0|\mathbf{H}|^2\Big),\ W_1 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 - \mu_0|\mathbf{H}|^2\Big),\ W_2 &= -\frac{1}{2c}\,\Re\{\mathbf{E}\cdot\mathbf{H}^*\},\ W_3 &= -\frac{1}{2c}\,\Im\{\mathbf{E}\cdot\mathbf{H}^*\}, \end{aligned}$$

with $\mathbf{W}=(W_1,W_2,W_3)$ constrained inside the energy symmetry sphere. The EM ellipse at each point is encoded by $\mathbf{W}$; its handedness is given by $\mathrm{sign}(W_3)$, while its geometry reflects electric–magnetic energy imbalance, reactive helicity, and helicity (Vernon, 22 Jul 2025).

Within this framework, singular optics acquires an EM-space counterpart. EM-space C lines satisfy $W_2=W_3=0$, EM-space L surfaces satisfy $v=\pm 2$0, and isolated point defects occur where $v=\pm 2$1. These structures are intrinsic to the full electromagnetic field and are not tied to a detector that is sensitive only to the electric or magnetic sector (Vernon, 22 Jul 2025).

The normalized field

$v=\pm 2$2

defines a map from a two-dimensional domain to the unit sphere. In this language, a meron is a 1-to-1 mapping from the domain to one hemisphere of the sphere, whereas a second-order meron is a 2-to-1 mapping to one hemisphere (Vernon, 22 Jul 2025).

2. Topological characterization

The topological invariant used throughout this literature is the skyrmion or winding number

$v=\pm 2$3

or the same formula with $v=\pm 2$4 replaced by a normalized Stokes or pseudospin field on a different state space. When the image covers the entire sphere once, the texture is a skyrmion; when it covers one hemisphere, it is a meron (Vernon, 22 Jul 2025).

A second-order usage appears in optical microcavities, where the polarization pseudospin field $v=\pm 2$5 is written in the standard skyrmion form and the charge is expressed as

$v=\pm 2$6

Here $v=\pm 2$7 is the vorticity of the in-plane pseudospin and $v=\pm 2$8 is the polarity set by the core circular polarization. First-order merons have $v=\pm 2$9 and $Q=\pm 1$0, whereas second-order merons have $Q=\pm 1$1 and $Q=\pm 1$2 (Król et al., 2020).

A closely related momentum-space formulation is used for photonic-crystal slabs hosting bound states in the continuum. There the normalized Stokes vector $Q=\pm 1$3 satisfies

$Q=\pm 1$4

and the second-order character is carried by $Q=\pm 1$5, with the hemisphere mapped twice (Rao et al., 21 May 2025).

This suggests a shared operational criterion across distinct formulations: “second-order” denotes double winding of the in-plane field or double coverage of a hemisphere, even though the underlying order parameter may be an EM ellipse field, a polarization pseudospin, or a momentum-space Stokes texture.

3. Principal realizations

Several distinct electromagnetic realizations now exist, and they differ primarily in the base space and order parameter.

Setting	Order parameter	Second-order feature
Focused nonparaxial vortex beam	$Q=\pm 1$6 on the ESS	Double wrapping of one hemisphere
Liquid-crystal optical microcavity	Polarization pseudospin $Q=\pm 1$7	$Q=\pm 1$8, $Q=\pm 1$9
Photonic-crystal slab with BIC	Momentum-space Stokes vector $\mathbf{E}$0	Hemisphere mapped twice, $\mathbf{E}$1
Plasmonic degenerate orbitals	Electric or magnetic near-field textures	Second-order meron–antimeron pair with $\mathbf{E}$2

In the focused-vortex-beam example, the field is monochromatic, topological charge $\mathbf{E}$3, linearly polarized, and strongly focused with beam waist $\mathbf{E}$4. In the focal plane, $\mathbf{E}$5 and $\mathbf{E}$6 form four-lobed patterns with relative $\mathbf{E}$7 rotation, while $\mathbf{E}$8 is negative at the center, crosses zero inside the bright ring, and becomes positive further out. Inside the radius of the zero rings of $\mathbf{E}$9 and $\mathbf{H}$0, the winding number approaches $\mathbf{H}$1; over the full plane it becomes $\mathbf{H}$2 because the outer region unwraps the upper hemisphere. The inner texture is therefore identified as a particle-like second-order EM-space meron (Vernon, 22 Jul 2025).

In the liquid-crystal microcavity realization, light confined in a birefringent cavity supports a polarization pseudospin texture on the Poincaré sphere. The second-order meron has a circularly polarized core, linear polarization away from the core, and a linear-polarization axis that rotates by $\mathbf{H}$3 around the center. The effective Hamiltonian contains momentum-dependent $\mathbf{H}$4 and $\mathbf{H}$5 terms, and the texture is observed directly through Stokes-parameter imaging (Król et al., 2020).

In photonic-crystal slabs, a bound state in the continuum with topological charge $\mathbf{H}$6 generates a momentum-space vortex topology. Under circularly polarized illumination, the cross-polarized component acquires orbital angular momentum $\mathbf{H}$7, so the in-plane Stokes vector winds with $\mathbf{H}$8. The resulting momentum-space meron is second-order because the meron domain is mapped onto a hemisphere of the Poincaré sphere twice (Rao et al., 21 May 2025).

A different higher-order plasmonic realization appears in doubly degenerate orbitals of a metallic ring resonator. For the $\mathbf{H}$9 irrep, the electric or magnetic near field contains two merons and two antimerons, giving a second-order meron–antimeron pair with absolute skyrmion number $$\begin{aligned} W_0 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 + \mu_0|\mathbf{H}|^2\Big),\ W_1 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 - \mu_0|\mathbf{H}|^2\Big),\ W_2 &= -\frac{1}{2c}\,\Re\{\mathbf{E}\cdot\mathbf{H}^*\},\ W_3 &= -\frac{1}{2c}\,\Im\{\mathbf{E}\cdot\mathbf{H}^*\}, \end{aligned}$$0 (Yang et al., 11 Apr 2025).

4. Relation to electromagnetic spin merons in real space

The phrase “second-order EM-space meron” is not used in "Spatio-temporal topology of plasmonic spin meron pairs revealed by polarimetric photo-emission microscopy" (Dreher et al., 2024), but that work is central to the broader concept because it establishes a full electromagnetic-spin formulation in real space and time. There the relevant field is the spin angular momentum density

$$\begin{aligned} W_0 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 + \mu_0|\mathbf{H}|^2\Big),\ W_1 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 - \mu_0|\mathbf{H}|^2\Big),\ W_2 &= -\frac{1}{2c}\,\Re\{\mathbf{E}\cdot\mathbf{H}^*\},\ W_3 &= -\frac{1}{2c}\,\Im\{\mathbf{E}\cdot\mathbf{H}^*\}, \end{aligned}$$1

and the topology on the metal surface is classified by the Chern number

$$\begin{aligned} W_0 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 + \mu_0|\mathbf{H}|^2\Big),\ W_1 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 - \mu_0|\mathbf{H}|^2\Big),\ W_2 &= -\frac{1}{2c}\,\Re\{\mathbf{E}\cdot\mathbf{H}^*\},\ W_3 &= -\frac{1}{2c}\,\Im\{\mathbf{E}\cdot\mathbf{H}^*\}, \end{aligned}$$2

In that plasmonic setting, a meron is a half-skyrmion of the electromagnetic spin field with $$\begin{aligned} W_0 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 + \mu_0|\mathbf{H}|^2\Big),\ W_1 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 - \mu_0|\mathbf{H}|^2\Big),\ W_2 &= -\frac{1}{2c}\,\Re\{\mathbf{E}\cdot\mathbf{H}^*\},\ W_3 &= -\frac{1}{2c}\,\Im\{\mathbf{E}\cdot\mathbf{H}^*\}, \end{aligned}$$3, characterized by polarity and vorticity. The experiment realizes a first-order meron pair, each with $$\begin{aligned} W_0 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 + \mu_0|\mathbf{H}|^2\Big),\ W_1 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 - \mu_0|\mathbf{H}|^2\Big),\ W_2 &= -\frac{1}{2c}\,\Re\{\mathbf{E}\cdot\mathbf{H}^*\},\ W_3 &= -\frac{1}{2c}\,\Im\{\mathbf{E}\cdot\mathbf{H}^*\}, \end{aligned}$$4, for total $$\begin{aligned} W_0 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 + \mu_0|\mathbf{H}|^2\Big),\ W_1 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 - \mu_0|\mathbf{H}|^2\Big),\ W_2 &= -\frac{1}{2c}\,\Re\{\mathbf{E}\cdot\mathbf{H}^*\},\ W_3 &= -\frac{1}{2c}\,\Im\{\mathbf{E}\cdot\mathbf{H}^*\}, \end{aligned}$$5. The in-plane singularity structure is constrained by the Poincaré–Hopf theorem on a disk-like domain with Euler characteristic $$\begin{aligned} W_0 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 + \mu_0|\mathbf{H}|^2\Big),\ W_1 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 - \mu_0|\mathbf{H}|^2\Big),\ W_2 &= -\frac{1}{2c}\,\Re\{\mathbf{E}\cdot\mathbf{H}^*\},\ W_3 &= -\frac{1}{2c}\,\Im\{\mathbf{E}\cdot\mathbf{H}^*\}, \end{aligned}$$6, which requires a compensating amplitude vortex between the two meron cores (Dreher et al., 2024).

The same work explicitly cites Król et al. on second-order meron polarization textures in optical microcavities and notes that full 3D EM spin imaging, L-line boundaries, Chern-number evaluation, and topological constraints from Poincaré–Hopf together provide the machinery needed to identify higher-order merons in electromagnetic spin fields. This suggests a direct route from first-order EM-spin merons to second-order EM-space merons by engineering higher winding or more complex composite textures, although the experiment itself realizes a first-order meron pair rather than a second-order single-core object (Dreher et al., 2024).

An additional real-space contrast is provided by "Intrinsic Meron Spin Textures in Generic Focused Fields" (Liu et al., 30 Dec 2025), where generic high-NA focusing yields a first-order meron with $$\begin{aligned} W_0 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 + \mu_0|\mathbf{H}|^2\Big),\ W_1 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 - \mu_0|\mathbf{H}|^2\Big),\ W_2 &= -\frac{1}{2c}\,\Re\{\mathbf{E}\cdot\mathbf{H}^*\},\ W_3 &= -\frac{1}{2c}\,\Im\{\mathbf{E}\cdot\mathbf{H}^*\}, \end{aligned}$$7, polarity $$\begin{aligned} W_0 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 + \mu_0|\mathbf{H}|^2\Big),\ W_1 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 - \mu_0|\mathbf{H}|^2\Big),\ W_2 &= -\frac{1}{2c}\,\Re\{\mathbf{E}\cdot\mathbf{H}^*\},\ W_3 &= -\frac{1}{2c}\,\Im\{\mathbf{E}\cdot\mathbf{H}^*\}, \end{aligned}$$8, and vorticity $$\begin{aligned} W_0 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 + \mu_0|\mathbf{H}|^2\Big),\ W_1 &= \frac{1}{4}\Big(\varepsilon_0|\mathbf{E}|^2 - \mu_0|\mathbf{H}|^2\Big),\ W_2 &= -\frac{1}{2c}\,\Re\{\mathbf{E}\cdot\mathbf{H}^*\},\ W_3 &= -\frac{1}{2c}\,\Im\{\mathbf{E}\cdot\mathbf{H}^*\}, \end{aligned}$$9. That work explicitly frames higher-order generalization in terms of increasing the winding number of the in-plane spin. A plausible implication is that the double-winding criterion used elsewhere for second-order merons is compatible with this first-order focused-field baseline.

5. Experimental reconstruction and observables

The measurement strategies depend on which EM-space is under consideration. In electric–magnetic space, the central observable is the EM ellipse defined from $\mathbf{W}=(W_1,W_2,W_3)$0 and $\mathbf{W}=(W_1,W_2,W_3)$1 through the $\mathbf{W}=(W_1,W_2,W_3)$2. Because $\mathbf{W}=(W_1,W_2,W_3)$3 is proportional to $\mathbf{W}=(W_1,W_2,W_3)$4, detecting the full EM-space topology is not equivalent to measuring conventional electric polarization alone; the work introducing the EM ellipse notes that access to $\mathbf{W}=(W_1,W_2,W_3)$5 in particular requires magnetoelectric or nonreciprocal particles (Vernon, 22 Jul 2025).

In real-space plasmonics, polarimetric photoemission electron microscopy reconstructs the full 3D surface-plasmon field and then the electromagnetic spin density. The photoelectron yield is generated by a second-order two-photon process, Fourier decomposition isolates the fundamental SPP frequency and wavevector, and four probe polarizations provide enough equations to recover the in-plane electric field. The method then reconstructs the full 3D $\mathbf{W}=(W_1,W_2,W_3)$6, computes $\mathbf{W}=(W_1,W_2,W_3)$7, and finally computes $\mathbf{W}=(W_1,W_2,W_3)$8. The reported spatial resolution is about $\mathbf{W}=(W_1,W_2,W_3)$9 nm and the temporal sampling is $\mathbf{W}$0 fs (Dreher et al., 2024).

In momentum space, the BIC platform uses Fourier-plane polarimetry to reconstruct $\mathbf{W}$1, $\mathbf{W}$2, and $\mathbf{W}$3 over $\mathbf{W}$4 and then numerically integrates the skyrmion density. The second-order character is inferred from the $\mathbf{W}$5 phase winding of the cross-polarized component and the resulting $\mathbf{W}$6 Stokes texture (Rao et al., 21 May 2025).

In momentum–energy space, ultrafast space–time merons are synthesized by combining chirped volume Bragg gratings, spectral conformal mapping with spatial light modulators, log–polar coordinate transformation, and a large-area birefringent metasurface. Digital holography and spatiotemporal Stokes reconstruction are then used to recover the spectral-surface polarization texture and the real-space 3D localized spin structure (Yessenov et al., 15 Mar 2025).

In optical microcavities, the diagnostic is full Stokes tomography of the transmitted field. The second-order meron is identified from a circularly polarized core, linear polarization away from the core, and a polarization axis that rotates twice around the center (Król et al., 2020).

6. Scope, terminology, and common ambiguities

A persistent ambiguity is that “second-order meron” is not defined identically across all subfields. In electric–magnetic space, the explicit definition is a 2-to-1 mapping of a real-space region to one hemisphere of the energy symmetry sphere (Vernon, 22 Jul 2025). In microcavity and BIC settings, the same phrase refers to double winding of a polarization or Stokes field, typically encoded by $\mathbf{W}$7 and often yielding $\mathbf{W}$8 or $\mathbf{W}$9 depending on convention (Król et al., 2020, Rao et al., 21 May 2025). These are notationally different but topologically aligned in the sense that both describe double winding of a meron-like hemisphere mapping.

Another common misconception is to equate a second-order EM-space meron with an ordinary optical polarization vortex. The electric–magnetic-space formulation was introduced precisely because a topological picture of the electric field or magnetic field in isolation cannot capture the topology that resides in the spatially dependent relationship between the two fields, including parity, duality, and time-reversal symmetry breaking (Vernon, 22 Jul 2025).

A further source of confusion is the meaning of “EM-space” itself. In optics, it can mean the abstract electric–magnetic mixing space of the EM ellipse, the Poincaré-sphere space of a polarization pseudospin, the real-space electromagnetic spin field, or a momentum–energy spectral surface. Outside optics, the term meron also appears in magnetic and gauge-theoretic contexts where the underlying space is instead a spin order-parameter manifold or a non-Abelian gauge configuration space (Kim et al., 2023, Diez et al., 15 Apr 2026). Those usages are mathematically related at the level of half-skyrmion topology, but they are not the same object as the electric–magnetic-space meron of light.

Taken together, the literature defines the second-order EM-space meron not as a single universally fixed entity, but as a family of higher-order electromagnetic half-skyrmion textures distinguished by double winding, double hemisphere coverage, or second-order meron–antimeron composition. The most explicit modern formulation is the EM-ellipse texture on the energy symmetry sphere (Vernon, 22 Jul 2025), while optical microcavities, BIC photonic crystals, plasmonic near fields, and time-dependent electromagnetic spin textures provide experimentally concrete realizations and closely related higher-order analogues (Król et al., 2020, Rao et al., 21 May 2025, Yang et al., 11 Apr 2025, Dreher et al., 2024).