---
title: Photonic Spin-Meron Tube Lattice
url: https://www.emergentmind.com/topics/photonic-spin-meron-tube-lattice
type: topic
---

# Photonic Spin-Meron Tube Lattice

A photonic spin-meron tube lattice is an ordered three-dimensional continuation of meronic optical-spin textures, defined on successive transverse planes by the normalized local spin field and extended along a propagation or height coordinate as columnar topological objects. The term is used explicitly for a free-space geometry-driven system in which square-block diffraction and non-paraxial spin-orbit coupling generate finite-length spin-meron tubes with \(N_{\mathrm{sk}}\approx \pm 1/2\) that remain coherent for at least \(25\lambda\) [2508.14450]. Closely related earlier work established square photonic spin-meron lattices in guided or evanescent fields [2103.15366] and height-evolving plasmonic meron lattices above finite metallic squares, where laterally anchored cores persist while the in-plane helicity changes with \(z\), supporting a quasi-3D tube-like interpretation rather than a strictly invariant tube crystal [2602.03042].

## 1. Definition and topological content

In the photonic literature relevant to merons, the basic order parameter is not a scalar phase field but the local electromagnetic spin angular momentum density. A representative definition is  
\[
\mathbf{s}=\frac{1}{4\omega}\,\mathrm{Im}\!\left(\varepsilon\,\mathbf{E}^\ast\times\mathbf{E}+\mu\,\mathbf{H}^\ast\times\mathbf{H}\right),
\qquad
\mathbf{S}=\frac{\mathbf{s}}{|\mathbf{s}|},
\]
so the topology is carried by the unit spin vector rather than by intensity alone [2602.03042]. In the free-space tube-lattice formulation, the same logic is written as \(\mathbf{n}(\mathbf r)=\mathbf S(\mathbf r)/|\mathbf S(\mathbf r)|\), and the tube is understood as the longitudinal continuation of the same meronic spin pattern through successive \(x\)-\(y\) planes [2508.14450].

A meron is a half-skyrmion-like object in this normalized spin field. In the square photonic spin-meron lattice construction, the unit-cell topological number is evaluated as  
\[
Q=\frac{1}{4\pi}\iint \mathbf{n}\cdot\left(\partial_x\mathbf{n}\times \partial_y\mathbf{n}\right)\,dx\,dy,
\]
with meron cells carrying \(Q=\pm \tfrac12\) [2103.15366]. The same half-wrapping interpretation underlies the tube-lattice language in free space, where each localized site is a meron because its normalized spin vector covers half of the spin sphere, yielding \(N_{\mathrm{sk}}\approx \pm 1/2\) on transverse slices [2508.14450].

The meron texture is further classified by in-plane helicity. In the plasmonic height-evolution study, a texture is termed Néel-type when the in-plane spin is parallel or antiparallel to the local radial direction around an out-of-plane extremum, and Bloch-type when the in-plane spin circulates tangentially around that extremum [2602.03042]. This distinction becomes central once a lattice is extended along \(z\): a tube lattice need not preserve the same helicity on every plane, and in some realizations the longitudinal continuation is defined by persistent core positions rather than by rigidly invariant in-plane winding.

## 2. Symmetry selection and lattice archetypes

A foundational result for photonic spin-meron lattices is that, for guided or evanescent modes with spin-orbit coupling, field symmetry constrains the admissible real-space spin topology. In the periodic evanescent-vortex construction, the allowed lattices are restricted to hexagonal spin-skyrmion lattices and square spin-meron lattices, with \(D_4\) symmetry producing a staggered-flux square meron pattern and \(D_6\) symmetry producing a hexagonal skyrmion pattern [2103.15366]. For the square case, the spin field is explicitly written as
\[
\mathbf{S}\propto \big(k_z\sin(k_r x)\cos(k_r y),\;k_z\cos(k_r x)\sin(k_r y),\;k_r\cos(k_r x)\cos(k_r y)\big),
\]
which makes the alternating core polarity and half-skyrmionic character explicit [2103.15366].

The free-space tube-lattice realization inherits the square-meron branch of this symmetry logic but replaces guided-wave interference by polygonal diffraction. There, square-block diffraction produces a \(C_4\)-symmetric field with an edge-imposed \(\pi/2\) phase ladder, encoded by the edge coefficients
\[
h_m=i,\,1,\,-i,\,-1,
\]
and this discrete phase staircase locks the vortex lattice strongly enough that the full 3-D spin-meron tubes remain coherent for at least \(25\lambda\) [2508.14450]. In the same framework, a triangular \(C_3\) block yields a spin-skyrmion tube rather than a meron tube, so the free-space classification is also symmetry-selected, though in a distinct diffraction-based construction [2508.14450].

A broader implication, stated directly in the symmetry literature, is that spin-orbit coupling converts structured energy-flow or diffraction symmetries into structured spin topologies [2103.15366]. In that sense, the photonic spin-meron tube lattice is not an isolated object but the three-dimensional extension of a symmetry-constrained square meron lattice.

## 3. Physical realizations and generation mechanisms

The most explicit realization is free-space and geometry-driven. A circularly polarized plane wave is normally incident on a finite square dielectric absorber or high-index block placed at \(z=0\), and the diffracted field in \(z>0\) is computed by Stratton-Chu theory together with full-vectorial FDTD [2508.14450]. The analytical model treats the square element as a strongly absorbing block with
\[
\tilde n=n+i\kappa=4+5i,
\qquad
r=\frac{1-\tilde n}{1+\tilde n}=-0.8-0.2i,
\]
while FDTD validation also considers a realistic silicon block of size \(64\lambda\times 64\lambda\) and thickness \(2\lambda\), with \(n_{\mathrm{Si}}=3.48+0.006i\) and \(\sim 30\%\) absorption [2508.14450]. The essential mechanism is two-step: square-block diffraction creates a square-symmetric vortex lattice, and non-paraxial spin-orbit coupling, enabled by nonzero longitudinal field components, converts that diffraction field into a lattice of meronic spin textures [2508.14450].

A second realization is plasmonic and height dependent. In that system, a square slit or coupling structure patterned in a silver film is illuminated by a normally incident circularly polarized plane wave at \(\lambda=550\,\mathrm{nm}\). The field above the metal splits into an evanescent SPP sector and a propagating diffracted sector,
\[
\mathbf{E}(x,y,z)=\mathbf{E}_{\mathrm{SPP}}+\mathbf{E}_{\mathrm{diff}},
\]
with the first decaying exponentially and the second persisting to larger \(z\) [2602.03042]. Close to the surface the lattice is Néel-type; at large height it is Bloch-type; between them lies a narrow crossover region of rapid topology evolution. The lateral geometry is chosen through the commensurability condition
\[
L=m\lambda_{\mathrm{SPP}}=n\lambda,
\qquad m,n\in\mathbb N,
\]
and for the reported structure \(L=55\,\lambda_{\mathrm{SPP}}\), corresponding to \(m=55\) and \(n=53\) [2602.03042]. This yields a well-ordered meron lattice whose cores remain laterally anchored while the in-plane spin changes with height.

An earlier photonic precursor is the square spin-meron lattice generated by guided or evanescent modes with spin-orbit coupling and realized experimentally with SPPs on an air/silver interface. There the lattice is fundamentally a two-dimensional transverse spin texture rather than a tube lattice, but it establishes the square meron cell as the canonical photonic meron building block and provides the symmetry-based theory later extended into quasi-3D and free-space tube interpretations [2103.15366].

## 4. Longitudinal structure and the meaning of “tube”

In the free-space literature, “tube” does not denote an infinitely propagation-invariant Bessel-like object. It denotes a finite longitudinal segment over which the same topological spin texture remains identifiable from plane to plane. The square-block tube lattice was tracked from \(z=5\lambda\) to \(25\lambda\), and the authors state that the full 3-D spin-meron tubes remain coherent for at least \(25\lambda\), with the skyrmion charge of each lattice site conserved during propagation [2508.14450]. The lattice therefore consists of longitudinally registered meronic sites rather than a single observation-plane pattern.

The plasmonic height-evolution case is related but more restrictive. There, the lateral lattice of \(S_z\) extrema persists while the in-plane helicity changes continuously from Néel-like near the interface to Bloch-like at larger height. The system can therefore be regarded as a vertical stack, or continuous \(z\)-evolution, of two-dimensional meron slices; however, the paper explicitly states that it does not define a full 3D invariant and does not construct propagation-invariant vertical meron tubes in the free-space sense [2602.03042]. The resulting object is better described as a height-evolving plasmonic near-to-far-field tube-like texture.

A comparable caution applies to evanescent quasicrystals. In the SPP quasicrystal construction, every field component carries the same factor \(e^{-|k_z|z}\), so the lateral pattern is naturally extended into the normal direction with overall evanescent decay. This makes a columnar or tube-like reading plausible, but the work itself does not define a 3D lattice of tubes or a tube invariant [2409.03932]. Across the literature, therefore, “tube” ranges from rigorously tracked free-space longitudinal coherence to quasi-3D vertical continuation inferred from persistent transverse slices.

## 5. Defects, switching, and topological diagnostics

The most detailed diagnostics of spin-meron-lattice reorganization come from the height-sensitive plasmonic system. The in-plane spin phase is defined as
\[
\psi=\arg(S_x+iS_y),
\]
and singularities are assigned winding number
\[
v=\frac{1}{2\pi}\oint_{\mathcal C}d\psi.
\]
The corresponding generalized site charge is
\[
Q_{\mathrm{site}}=\frac12\sum_{j\in\mathrm{site}} p_j v_j,
\qquad
p_j=\mathrm{sgn}\!\left[S_z(\mathbf r_j)\right],
\]
which reduces to a \(\pm\tfrac12\)-like charge in the SPP-dominant regime and becomes height-dependent and fractional once additional off-boundary vortex-antivortex pairs are nucleated in the crossover zone [2602.03042].

The complementary field-theoretic measure is the skyrmion density
\[
D(x,y)=\frac{1}{4\pi}\,\mathbf{S}(x,y)\cdot\left(\frac{\partial\mathbf{S}(x,y)}{\partial x}\times \frac{\partial\mathbf{S}(x,y)}{\partial y}\right),
\qquad
N_{\mathrm{Sk}}=\iint_{\mathcal A}D(x,y)\,dx\,dy.
\]
In the near-field SPP regime, opposite site charges nearly cancel inside a unit cell, so \(N_{\mathrm{Sk}}\) is close to zero. In the crossover interval, \(N_{\mathrm{Sk}}\) deviates from zero as the half-integer cancellation breaks down [2602.03042]. The paper also introduces the spin-dominance factor
\[
\chi=|S_{\mathrm{out}}|^2-S_{\mathrm{in}}^2,
\]
whose \(\chi\approx 0\) contours mark the meron boundaries and reveal boundary deformation during the crossover [2602.03042].

In the free-space tube lattice, diagnostics are formulated more directly in terms of \(S_z/|\mathbf S|\), \(\mathbf S_\parallel/|\mathbf S|\), and \(\rho_{\mathrm{sk}}\) maps on successive transverse planes. The key claim is not topological switching but longitudinal persistence: repeated positive and negative meron cores remain in registry, the alternating pattern of half-integer winding numbers remains recognizable, and the skyrmion charge of each lattice site is conserved during propagation [2508.14450]. This difference in emphasis reflects the two main regimes in the field: defect-mediated topology conversion in height-evolving plasmonics, and defect-stabilized longitudinal coherence in free-space tube lattices.

## 6. Related photonic topologies and conceptual boundaries

The photonic spin-meron tube lattice should be distinguished from several nearby but nonidentical constructs. Momentum-space merons in photonic crystal slabs, whether produced by gapped Dirac valleys or by BIC-induced polarization vortices, are meronic textures of pseudospin or Stokes fields over \((k_x,k_y)\), not real-space tube lattices [1904.11516; 2505.15081]. They are directly relevant to meron topology and optical readout, but they do not form real-space longitudinal tubes.

Likewise, two-dimensional spin-meron quasicrystals in evanescent SPP fields contain ordered spin merons and skyrmions in the spin angular momentum itself and extend naturally along \(z\) through the common evanescent factor \(e^{-|k_z|z}\), yet the papers do not formulate these objects as bona fide 3D tube lattices [2409.03932]. Twisted bilayer plasmonic spin lattices add another layer of complexity: two 2D meron- or skyrmion-bearing SPP lattices, rotated relative to one another, generate Moiré spin superlattices with meron clusters and near-integer skyrmion unit cells, but these remain fundamentally interfacial bilayer textures rather than explicit tube crystals [2411.00645].

Single-site or localized meron mechanisms also lie adjacent to, but outside, the tube-lattice category. Intrinsic meron spin textures in generic focused fields establish that a half-skyrmionic spin texture can arise without wavefront engineering and remain robust against partial polarization and disorder, but no lattice or tube structure is constructed [2512.24084]. Localized plasmonic merons and meron-antimeron pairs in doubly degenerate orbitals of rotationally symmetric resonators provide symmetry-based charge rules and chirality locking for isolated sites, again without a lattice of tubes [2504.08558].

The central boundary of the term is therefore precise. A photonic spin-meron tube lattice is not simply any photonic meron, any spin lattice, or any vertical decay of an evanescent field. In the strictest current usage, it denotes a real-space lattice of longitudinally continued meronic spin textures whose transverse topological charge remains identifiable over a finite propagation interval [2508.14450]. In a broader but still defensible usage, it also encompasses quasi-3D systems in which a laterally organized meron lattice persists across height while its helicity or effective site charge evolves with \(z\) [2602.03042]. A plausible implication is that future cylindrical, stacked, or synthetic-dimensional implementations may merge these two notions, but such fully developed tube-lattice architectures are not yet the standard demonstrated case in the present literature.

Source: https://www.emergentmind.com/topics/photonic-spin-meron-tube-lattice