Photonic Spin-Meron Tube Lattice

• Photonic spin-meron tube lattice is a 3D arrangement of half-skyrmionic spin textures formed via square-block diffraction and non-paraxial spin-orbit coupling.
• The lattice leverages engineered interference and symmetry constraints, such as C4 and D4, to maintain coherent meronic patterns across successive transverse planes.
• Practical realizations in free-space optics and plasmonic systems demonstrate sustained topological charge conservation over propagation lengths of at least 25λ.

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 $25\lambda$ (Hegde et al., 20 Aug 2025). Closely related earlier work established square photonic spin-meron lattices in guided or evanescent fields (Lei et al., 2021) 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 (Hegde et al., 3 Feb 2026).

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 (Hegde et al., 3 Feb 2026). 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 (Hegde et al., 20 Aug 2025).

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 $Q=\pm \tfrac12$ (Lei et al., 2021). 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 (Hegde et al., 20 Aug 2025).

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 (Hegde et al., 3 Feb 2026). This distinction becomes central once a lattice is extended along $25\lambda$0: 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 $25\lambda$1 symmetry producing a staggered-flux square meron pattern and $25\lambda$2 symmetry producing a hexagonal skyrmion pattern (Lei et al., 2021). For the square case, the spin field is explicitly written as

$25\lambda$3

which makes the alternating core polarity and half-skyrmionic character explicit (Lei et al., 2021).

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 $25\lambda$4-symmetric field with an edge-imposed $25\lambda$5 phase ladder, encoded by the edge coefficients

$25\lambda$6

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$7 (Hegde et al., 20 Aug 2025). In the same framework, a triangular $25\lambda$8 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 (Hegde et al., 20 Aug 2025).

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 (Lei et al., 2021). 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 $25\lambda$9, and the diffracted field in $z$0 is computed by Stratton-Chu theory together with full-vectorial FDTD (Hegde et al., 20 Aug 2025). The analytical model treats the square element as a strongly absorbing block with

$z$1

while FDTD validation also considers a realistic silicon block of size $z$2 and thickness $z$3, with $z$4 and $z$5 absorption (Hegde et al., 20 Aug 2025). 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 (Hegde et al., 20 Aug 2025).

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 $z$6. The field above the metal splits into an evanescent SPP sector and a propagating diffracted sector,

$z$7

with the first decaying exponentially and the second persisting to larger $z$8 (Hegde et al., 3 Feb 2026). 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

$z$9

and for the reported structure $$\mathbf{s}=\frac{1}{4\omega}\,\mathrm{Im}\!\left(\varepsilon\,\mathbf{E}^\ast\times\mathbf{E}+\mu\,\mathbf{H}^\ast\times\mathbf{H}\right), \qquad \mathbf{S}=\frac{\mathbf{s}}{|\mathbf{s}|},$$0, corresponding to $$\mathbf{s}=\frac{1}{4\omega}\,\mathrm{Im}\!\left(\varepsilon\,\mathbf{E}^\ast\times\mathbf{E}+\mu\,\mathbf{H}^\ast\times\mathbf{H}\right), \qquad \mathbf{S}=\frac{\mathbf{s}}{|\mathbf{s}|},$$1 and $$\mathbf{s}=\frac{1}{4\omega}\,\mathrm{Im}\!\left(\varepsilon\,\mathbf{E}^\ast\times\mathbf{E}+\mu\,\mathbf{H}^\ast\times\mathbf{H}\right), \qquad \mathbf{S}=\frac{\mathbf{s}}{|\mathbf{s}|},$$2 (Hegde et al., 3 Feb 2026). 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 (Lei et al., 2021).

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 $$\mathbf{s}=\frac{1}{4\omega}\,\mathrm{Im}\!\left(\varepsilon\,\mathbf{E}^\ast\times\mathbf{E}+\mu\,\mathbf{H}^\ast\times\mathbf{H}\right), \qquad \mathbf{S}=\frac{\mathbf{s}}{|\mathbf{s}|},$$3 to $$\mathbf{s}=\frac{1}{4\omega}\,\mathrm{Im}\!\left(\varepsilon\,\mathbf{E}^\ast\times\mathbf{E}+\mu\,\mathbf{H}^\ast\times\mathbf{H}\right), \qquad \mathbf{S}=\frac{\mathbf{s}}{|\mathbf{s}|},$$4, and the authors state that the full 3-D spin-meron tubes remain coherent for at least $$\mathbf{s}=\frac{1}{4\omega}\,\mathrm{Im}\!\left(\varepsilon\,\mathbf{E}^\ast\times\mathbf{E}+\mu\,\mathbf{H}^\ast\times\mathbf{H}\right), \qquad \mathbf{S}=\frac{\mathbf{s}}{|\mathbf{s}|},$$5, with the skyrmion charge of each lattice site conserved during propagation (Hegde et al., 20 Aug 2025). 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 $$\mathbf{s}=\frac{1}{4\omega}\,\mathrm{Im}\!\left(\varepsilon\,\mathbf{E}^\ast\times\mathbf{E}+\mu\,\mathbf{H}^\ast\times\mathbf{H}\right), \qquad \mathbf{S}=\frac{\mathbf{s}}{|\mathbf{s}|},$$6 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 $$\mathbf{s}=\frac{1}{4\omega}\,\mathrm{Im}\!\left(\varepsilon\,\mathbf{E}^\ast\times\mathbf{E}+\mu\,\mathbf{H}^\ast\times\mathbf{H}\right), \qquad \mathbf{S}=\frac{\mathbf{s}}{|\mathbf{s}|},$$7-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 (Hegde et al., 3 Feb 2026). 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 $$\mathbf{s}=\frac{1}{4\omega}\,\mathrm{Im}\!\left(\varepsilon\,\mathbf{E}^\ast\times\mathbf{E}+\mu\,\mathbf{H}^\ast\times\mathbf{H}\right), \qquad \mathbf{S}=\frac{\mathbf{s}}{|\mathbf{s}|},$$8, 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 (Putley et al., 2024). 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

$$\mathbf{s}=\frac{1}{4\omega}\,\mathrm{Im}\!\left(\varepsilon\,\mathbf{E}^\ast\times\mathbf{E}+\mu\,\mathbf{H}^\ast\times\mathbf{H}\right), \qquad \mathbf{S}=\frac{\mathbf{s}}{|\mathbf{s}|},$$9

and singularities are assigned winding number

$$\mathbf{n}(\mathbf r)=\mathbf S(\mathbf r)/|\mathbf S(\mathbf r)|$$0

The corresponding generalized site charge is

$$\mathbf{n}(\mathbf r)=\mathbf S(\mathbf r)/|\mathbf S(\mathbf r)|$$1

which reduces to a $$\mathbf{n}(\mathbf r)=\mathbf S(\mathbf r)/|\mathbf S(\mathbf r)|$$2-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 (Hegde et al., 3 Feb 2026).

The complementary field-theoretic measure is the skyrmion density

$$\mathbf{n}(\mathbf r)=\mathbf S(\mathbf r)/|\mathbf S(\mathbf r)|$$3

In the near-field SPP regime, opposite site charges nearly cancel inside a unit cell, so $$\mathbf{n}(\mathbf r)=\mathbf S(\mathbf r)/|\mathbf S(\mathbf r)|$$4 is close to zero. In the crossover interval, $$\mathbf{n}(\mathbf r)=\mathbf S(\mathbf r)/|\mathbf S(\mathbf r)|$$5 deviates from zero as the half-integer cancellation breaks down (Hegde et al., 3 Feb 2026). The paper also introduces the spin-dominance factor

$$\mathbf{n}(\mathbf r)=\mathbf S(\mathbf r)/|\mathbf S(\mathbf r)|$$6

whose $$\mathbf{n}(\mathbf r)=\mathbf S(\mathbf r)/|\mathbf S(\mathbf r)|$$7 contours mark the meron boundaries and reveal boundary deformation during the crossover (Hegde et al., 3 Feb 2026).

In the free-space tube lattice, diagnostics are formulated more directly in terms of $$\mathbf{n}(\mathbf r)=\mathbf S(\mathbf r)/|\mathbf S(\mathbf r)|$$8, $$\mathbf{n}(\mathbf r)=\mathbf S(\mathbf r)/|\mathbf S(\mathbf r)|$$9, and $x$0 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 (Hegde et al., 20 Aug 2025). 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 $x$1, not real-space tube lattices (Guo et al., 2019, Rao et al., 21 May 2025). 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 $x$2 through the common evanescent factor $x$3, yet the papers do not formulate these objects as bona fide 3D tube lattices (Putley et al., 2024). 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 (Shi et al., 2024).

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 (Liu et al., 30 Dec 2025). 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 (Yang et al., 11 Apr 2025).

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 (Hegde et al., 20 Aug 2025). 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 $x$4 (Hegde et al., 3 Feb 2026). 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.