Correlated-Disordered Metasurfaces

• Correlated-disordered metasurfaces are planar arrays that use intentional spatial correlations, exclusion rules, and hyperuniform patterns to tailor optical responses.
• They employ controlled displacement and topological descriptors to tune key attributes like specular reflection, resonant linewidths, and far-field scattering.
• Advanced metrics, including persistent homology and reciprocal-space analysis, enable inverse design and predictive modeling of complex electromagnetic interactions.

Correlated-disordered metasurfaces are planar assemblies of meta-atoms or scatterers that lack global periodicity but retain engineered spatial correlations, so that pair separations, local exclusion rules, reciprocal-space constraints, or connectivity statistics become design variables rather than fabrication byproducts. In this setting, disorder is not restricted to Poisson-like randomness: it can be imposed through minimum inter-particle distances, Gaussian-correlated lattice perturbations, hyperuniform or stealthy-hyperuniform point patterns, shuffled lattices, or connected morphologies near a critical packing threshold. Across these realizations, correlated disorder is used to control specular and diffuse scattering, collective resonances, photon density of states, localization, directional emission, broadband absorption, and visual appearance in ways that are not accessible with either perfect periodicity or purely uncorrelated disorder (Madeleine et al., 2023, Lalanne et al., 2023).

1. Definitions, disorder classes, and structural control

A central distinction is between uncorrelated and correlated positional disorder. In perturbed lattices, uncorrelated disorder is generated by independently displacing each lattice site by a random vector, whereas correlated disorder introduces spatial smoothing so that nearby sites move coherently (Madeleine et al., 2023). In one generative model for metasurfaces of pitch $P$, each nanostructure at position $\mathbf r_i$ is displaced by $\Delta \mathbf r_i$ with components uniformly drawn in $[-S_d P, S_d P]$, and correlations are added through a Gaussian weight

$$C_{ij}=\exp\!\left\{-\left[\frac{r_{ij}}{2L_cP}\right]^2\right\},$$

with full width at half maximum $2\sqrt{2\ln 2}\,L_cP$, so that the total perturbation is

$$\mathbf r_i'=\mathbf r_i+\Delta \mathbf r_i+\sum_{j\neq i}\Delta \mathbf r_j\,C_{ij}.$$

Here $S_d$ controls displacement amplitude and $L_c$ the correlation length and smoothness of the distortion (Madeleine et al., 2023).

A second broad class arises from exclusion constraints. Random sequential addition, Poisson-disk sampling, and related hard-core processes enforce a minimum spacing and therefore generate short-range repulsion and short-range order rather than independent point placement. In the optics literature this correlated-disorder class is frequently used to suppress diffuse scattering near the specular direction or to regularize fabrication-robust layouts (Vynck et al., 2022, Lalanne et al., 2023). In shuffled lattices, such as randomly jittered silicon nanopillar arrays of nominal period $P_0=310\,\mathrm{nm}$, short-range positional correlations remain centered around the original lattice spacing while long-range order and Bragg peaks are progressively smeared as the disorder parameter increases; at large jitter, close, far, and even overlapping arrangements can occur because no strict hard-core exclusion is imposed (Chen et al., 16 Mar 2025).

Hyperuniform and stealthy-hyperuniform patterns constitute a reciprocal-space definition of correlated disorder. With mean density $\mathbf r_i$0, pair correlation $\mathbf r_i$1, and $\mathbf r_i$2, the static structure factor is

$\mathbf r_i$3

Hyperuniformity requires $\mathbf r_i$4, equivalently a number variance scaling slower than Poisson, while stealthy hyperuniformity imposes $\mathbf r_i$5 for $\mathbf r_i$6 (Madeleine et al., 2023, Monsarrat et al., 2021, Castro-Lopez et al., 2017). In metasurface design this suppresses long-wavelength density fluctuations and concentrates scattering into selected momentum channels.

A further structural regime appears when the notion of isolated meta-atoms begins to break down. In ultrathin dielectric metasurfaces patterned from square posts, a “critical packing regime” occurs when a significant fraction of metaatoms become physically connected; in the reported scanning electron micrographs this corresponds to about half of the metaatoms touching, with short chains and loops coexisting with isolated posts (Chen et al., 4 May 2025). This regime is neither a dilute particulate topology nor a fully semi-continuous aggregate, and its optical response cannot be reduced to either limit.

2. Structural descriptors beyond conventional disorder parameters

For metasurfaces with weak or uncorrelated positional disorder, conventional descriptors such as $\mathbf r_i$7, nearest-neighbor distributions, and Fourier-based metrics are often sufficient. Under strong correlations, however, these descriptors become less reliable as universal disorder coordinates. In strongly correlated Gaussian-distorted lattices, collective point displacements obscure the original lattice, broaden the accessible configuration space, and make the generative parameter $\mathbf r_i$8 a poor proxy for “how disordered” a sample actually is; finite-size windows and measurement noise further degrade statistical descriptors (Madeleine et al., 2023).

A topology-inspired alternative uses persistent homology on the point coordinates. A Vietoris–Rips filtration is built by drawing balls of radius $\mathbf r_i$9, connecting points whose balls intersect, and tracking topological features as $\Delta \mathbf r_i$0 grows. In two dimensions, the main features are connected components $\Delta \mathbf r_i$1 and loops $\Delta \mathbf r_i$2, summarized by Betti numbers $\Delta \mathbf r_i$3 and $\Delta \mathbf r_i$4 as functions of $\Delta \mathbf r_i$5, or by the birth–death pairs $\Delta \mathbf r_i$6 forming a persistence diagram $\Delta \mathbf r_i$7 (Madeleine et al., 2023). On this basis, two disorder descriptors were introduced:

$\Delta \mathbf r_i$8

and

$\Delta \mathbf r_i$9

with $[-S_d P, S_d P]$0 and $[-S_d P, S_d P]$1 the number of $[-S_d P, S_d P]$2 features. In the reported interpretation, $[-S_d P, S_d P]$3 is a proxy for the typical nearest-neighbor scale, whereas $[-S_d P, S_d P]$4 is minimal for ordered lattices, invariant under global rescaling, independent of dataset size $[-S_d P, S_d P]$5, and orthogonal to $[-S_d P, S_d P]$6 (Madeleine et al., 2023).

The computational pipeline used these descriptors on 1203 perturbed square-lattice patterns with $[-S_d P, S_d P]$7, $[-S_d P, S_d P]$8, and $[-S_d P, S_d P]$9, each containing $$C_{ij}=\exp\!\left\{-\left[\frac{r_{ij}}{2L_cP}\right]^2\right\},$$0 points. Vietoris–Rips filtrations were computed with Ripser; Wasserstein distances between persistence diagrams were computed with GUDHI and embedded by classical multidimensional scaling. The reported behavior was that uncorrelated disorder yielded clear clustering by $$C_{ij}=\exp\!\left\{-\left[\frac{r_{ij}}{2L_cP}\right]^2\right\},$$1, intermediate correlation widened and overlapped clusters, and strong correlation merged them, making the generative parameters unreliable predictors. Coloring the same embeddings by $$C_{ij}=\exp\!\left\{-\left[\frac{r_{ij}}{2L_cP}\right]^2\right\},$$2 recovered the spacing scale across $$C_{ij}=\exp\!\left\{-\left[\frac{r_{ij}}{2L_cP}\right]^2\right\},$$3, while coloring by $$C_{ij}=\exp\!\left\{-\left[\frac{r_{ij}}{2L_cP}\right]^2\right\},$$4 recovered disorder strength across $$C_{ij}=\exp\!\left\{-\left[\frac{r_{ij}}{2L_cP}\right]^2\right\},$$5 and $$C_{ij}=\exp\!\left\{-\left[\frac{r_{ij}}{2L_cP}\right]^2\right\},$$6 (Madeleine et al., 2023).

Topological descriptors are not the only nontrivial structural measures in the field. In morphogenetically generated correlated media, a translational order metric

$$C_{ij}=\exp\!\left\{-\left[\frac{r_{ij}}{2L_cP}\right]^2\right\},$$7

was used to follow the evolution from correlated disorder to crystallization. The reported normalized values were $$C_{ij}=\exp\!\left\{-\left[\frac{r_{ij}}{2L_cP}\right]^2\right\},$$8 for early correlated disorder, $$C_{ij}=\exp\!\left\{-\left[\frac{r_{ij}}{2L_cP}\right]^2\right\},$$9 for a transitional regime, and $2\sqrt{2\ln 2}\,L_cP$0 for a hexagonal crystal (Chehami et al., 2023). The broader methodological implication is that correlated-disordered metasurfaces are often better characterized by descriptors that remain reference-free under changes of pitch, correlation length, or morphology than by a single generative disorder amplitude.

3. Scattering theory, collective electrodynamics, and reciprocity-space design

The electromagnetic response of disordered metasurfaces is commonly decomposed into coherent and diffuse contributions. For the scattered field,

$2\sqrt{2\ln 2}\,L_cP$1

and the averaged intensity separates as

$2\sqrt{2\ln 2}\,L_cP$2

Within an independent scattering approximation, the radiant intensity is often written as a single-particle form factor multiplied by a structure factor, while effective-field and quasi-crystalline closures incorporate collective interactions more self-consistently (Vynck et al., 2022, Lalanne et al., 2023). For coherent specular reflection and transmission by a particle monolayer, the analytical expressions in the independent scattering approximation and the effective field approximation depend explicitly on the one-point density and the single-particle scattering amplitude; pair correlations enter the coherent channel only through more advanced closures such as QCA or through full-wave simulations (Vynck et al., 2022).

In dense or resonant metasurfaces, discrete-element electrodynamics becomes essential. For asymmetrically split-ring arrays, each resonant arc is modeled as a damped oscillator and all recurrent scattering processes are retained through a many-body coupling matrix $2\sqrt{2\ln 2}\,L_cP$3, so that the collective amplitudes obey

$2\sqrt{2\ln 2}\,L_cP$4

with collective eigenvalues

$2\sqrt{2\ln 2}\,L_cP$5

This microscopic description captures radiative and non-radiative decay, retardation, electric–electric and magnetic–magnetic dipole interactions, and electric–magnetic cross-coupling (Jenkins et al., 2018). The central physical point is that the response of a disordered metasurface is mapped from pair separations and multiple scattering sequences into collective linewidths, frequency shifts, mode localization, and far-field resonances.

For arrays supporting surface lattice resonances, periodic references remain useful. In a square lattice of pitch $2\sqrt{2\ln 2}\,L_cP$6 and effective refractive index $2\sqrt{2\ln 2}\,L_cP$7, the Rayleigh anomaly for diffraction order $2\sqrt{2\ln 2}\,L_cP$8 satisfies

$2\sqrt{2\ln 2}\,L_cP$9

Correlated disorder can emulate effective lattice vectors through short-range order and peaks in $$\mathbf r_i'=\mathbf r_i+\Delta \mathbf r_i+\sum_{j\neq i}\Delta \mathbf r_j\,C_{ij}.$$0, thereby supporting SLR-like diffractive coupling without perfect periodicity (Madeleine et al., 2023).

A distinct analytical framework emerges when disorder itself is designed in reciprocal space. For lattices perturbed by correlated random displacements, the disorder statistics define three scattering components: a diffuse background, Bragg-like diffraction orders, and correlation halos. The halo term is absent for uncorrelated disorder, depends on the increment distributions $$\mathbf r_i'=\mathbf r_i+\Delta \mathbf r_i+\sum_{j\neq i}\Delta \mathbf r_j\,C_{ij}.$$1, and can be positive or negative. In consequence, correlation halos are not broadened diffraction peaks; they are independent features whose positions depend on the correlation range and can remain visible after ordinary diffraction orders vanish (Langevin et al., 19 Feb 2026). This result expands the standard view in which disorder merely broadens reciprocal-lattice peaks into a broadband background.

4. Resonances, localization, and experimentally observed optical regimes

The reported optical consequences of correlated disorder are diverse because the relevant mechanisms differ across metasurface classes. In plasmonic nanoparticle lattices designed with topological descriptors, the strength of surface lattice resonances was correlated with $$\mathbf r_i'=\mathbf r_i+\Delta \mathbf r_i+\sum_{j\neq i}\Delta \mathbf r_j\,C_{ij}.$$2 while keeping $$\mathbf r_i'=\mathbf r_i+\Delta \mathbf r_i+\sum_{j\neq i}\Delta \mathbf r_j\,C_{ij}.$$3 approximately fixed. In theory, for gold nanocylinders of height $$\mathbf r_i'=\mathbf r_i+\Delta \mathbf r_i+\sum_{j\neq i}\Delta \mathbf r_j\,C_{ij}.$$4 and diameter $$\mathbf r_i'=\mathbf r_i+\Delta \mathbf r_i+\sum_{j\neq i}\Delta \mathbf r_j\,C_{ij}.$$5 in a medium of refractive index $$\mathbf r_i'=\mathbf r_i+\Delta \mathbf r_i+\sum_{j\neq i}\Delta \mathbf r_j\,C_{ij}.$$6, three metasurfaces selected at the same $$\mathbf r_i'=\mathbf r_i+\Delta \mathbf r_i+\sum_{j\neq i}\Delta \mathbf r_j\,C_{ij}.$$7 but different $$\mathbf r_i'=\mathbf r_i+\Delta \mathbf r_i+\sum_{j\neq i}\Delta \mathbf r_j\,C_{ij}.$$8 showed progressively weaker resonances as $$\mathbf r_i'=\mathbf r_i+\Delta \mathbf r_i+\sum_{j\neq i}\Delta \mathbf r_j\,C_{ij}.$$9 increased, with reported quality factors $S_d$0, $S_d$1, and $S_d$2 from lowest to highest $S_d$3. In experiment, focused-ion-beam-fabricated elongated gold nanodisks coated with IC1-200 and measured at normal incidence showed that in 5 out of 6 correlated comparisons the metasurface with lower $S_d$4, even when $S_d$5 was larger, exhibited higher $S_d$6 (Madeleine et al., 2023).

Pattern	TD	Reported $S_d$7
periodic, $S_d$8	0.000	10.1, 11.5
$S_d$9	0.030	4.0, 5.2
$L_c$0	0.012	9.3, 6.8
$L_c$1	0.025	6.7, 4.4
$L_c$2	0.005	7.8, 7.0
$L_c$3	0.026	8.0, 11.5
$L_c$4	0.002	10.1, 11.5

The main exception occurred for $L_c$5 and perpendicular polarization, where both correlated samples had $L_c$6 because large $L_c$7 smooths short-range disorder while preserving long-scale distortions; $L_c$8 detects the latter, whereas SLRs remain robust to it (Madeleine et al., 2023).

In metamaterial arrays with strong radiative interactions, positional disorder drives a different transition. In a regular $L_c$9 array of asymmetrically split rings, approximately $P_0=310\,\mathrm{nm}$0 of the driven response concentrates into a single spatially extended many-body subradiant eigenmode with magnetic dipoles in phase and collective decay rate $P_0=310\,\mathrm{nm}$1. A gradual increase of positional disorder rapidly localizes the mode and red-shifts the far-field transmission resonance through a cooperative Lamb shift; for one representative realization, $P_0=310\,\mathrm{nm}$2 increased only weakly to $P_0=310\,\mathrm{nm}$3 at $P_0=310\,\mathrm{nm}$4, showing that localization and subradiance can coexist at moderate disorder (Jenkins et al., 2018).

Correlated disorder also modifies the density of states and localization in two-dimensional resonant media. For vector TE waves in stealth-hyperuniform point patterns, localization occurs at moderate density in the same window where the density of states exhibits a pseudo-gap; the reported localization island appears around $P_0=310\,\mathrm{nm}$5–$P_0=310\,\mathrm{nm}$6 and $P_0=310\,\mathrm{nm}$7, whereas no signature of localization is found for white-noise disorder. For scalar TM waves, localization occurs at high density irrespective of correlations (Monsarrat et al., 2021). The proposed microscopic origin is destructive interference between independent scattering and recurrent loop scattering weighted by the short-range peak of $P_0=310\,\mathrm{nm}$8 near $P_0=310\,\mathrm{nm}$9.

Near the critical packing threshold, connected dielectric metasurfaces display abrupt far-field changes tied to a redistribution of quasi-normal modes in the complex-frequency plane. In particulate arrays, the photon density of states contains clustered clouds and bandgap-like voids; at critical packing these voids begin to vanish and new collective resonances shift to lower frequencies; in aggregate regimes the PDoS becomes broad and nearly uniform across the visible, with $\mathbf r_i$00 factors generally below $\mathbf r_i$01 (Chen et al., 4 May 2025). In the far field, correlated particulate arrays show a blue shift of the diffuse BRDF maximum with increasing density, while transitions into critical and aggregate regimes produce broadband diffuse whitening. For $\mathbf r_i$02, $\mathbf r_i$03, and $\mathbf r_i$04, the reported diffuse brightness approaches $\mathbf r_i$05 across the visible even though the structure is a single $\mathbf r_i$06 poly-Si layer, and at critical packing a broadband specular “mirror” peak reaches $\mathbf r_i$07 near $\mathbf r_i$08 with $\mathbf r_i$09 (Chen et al., 4 May 2025).

Hyperuniform reciprocal-space engineering leads to yet another regime: isotropic annular scattering and emission. Gold metasurfaces derived from stealth-hyperuniform point sets with stealthiness $\mathbf r_i$10 and 4000 points exhibit a single broad isotropic diffraction maximum, and both scattering and fluorescence measurements show rotationally symmetric rings at in-plane momentum

$\mathbf r_i$11

so that $\mathbf r_i$12. The opening of the ring therefore scales with the inverse correlation length parameter $\mathbf r_i$13, and in fluorescence the ring closes near $\mathbf r_i$14 because of band folding into the light cone (Castro-Lopez et al., 2017).

A separate application to infrared silicon photodetection uses a shuffled-lattice disordered metasurface integrated with upconversion nanoparticles. In that work the layout is generated by independent uniform jitter of Si nanopillars around a square lattice, and the paper does not report explicit correlation descriptors such as $\mathbf r_i$15 or $\mathbf r_i$16; nevertheless, the highly disordered configuration with $\mathbf r_i$17 is reported to increase infrared absorption by $\mathbf r_i$18-fold and the near field by $\mathbf r_i$19-fold relative to the ordered structure, while the measured responsivity at $\mathbf r_i$20 reaches $\mathbf r_i$21 at room temperature, corresponding to an external quantum efficiency of $\mathbf r_i$22 (Chen et al., 16 Mar 2025). The stated mechanism is disorder-induced mode packing of hybrid Mie–plasmonic cavities together with field enhancement that boosts Er$\mathbf r_i$23 upconversion and hot-electron generation.

5. Design methodologies, fabrication routes, and inverse design

A practical design workflow for correlated-disordered metasurfaces has been articulated most explicitly for topological learning. The sequence is: specify a target optical property, choose a disorder model such as minimum-distance-constrained patterns, Gaussian-correlated displacements, or hyperuniform and stealthy-hyperuniform patterns, generate candidate point sets over ranges of $\mathbf r_i$24, $\mathbf r_i$25, pitch, or minimum spacing, compute Vietoris–Rips persistence diagrams and the descriptors $\mathbf r_i$26 and $\mathbf r_i$27, select patterns at matched $\mathbf r_i$28 and desired $\mathbf r_i$29, predict the optical response with a dipole approximation or a learned regression, and then iterate until the target metrics are met before fabrication and validation (Madeleine et al., 2023). In this formulation, $\mathbf r_i$30 fixes local spacing while $\mathbf r_i$31 ranks positional disorder.

Morphogenetic design provides a different route to correlated disorder by replacing global optimization with local reaction–diffusion rules. In the Gray–Scott model,

$\mathbf r_i$32

the reported parameter set $\mathbf r_i$33, $\mathbf r_i$34, $\mathbf r_i$35, $\mathbf r_i$36 generates self-replicating spots whose early-time patterns are stealthy-hyperuniform-like and isotropic in $\mathbf r_i$37, whose transitional states develop weak local crystallites, and whose late-time states converge toward a compact hexagonal crystal (Chehami et al., 2023). After thresholding at $\mathbf r_i$38, the reported supercells yield spot diameters of about $\mathbf r_i$39 and average center-to-center spacing of about $\mathbf r_i$40. Extruded into dielectric rods of $\mathbf r_i$41 in air, these patterns produce isotropic microwave TM bandgaps with reported normalized widths of approximately $\mathbf r_i$42 in the correlated-disordered state, $\mathbf r_i$43 in the transitional state, and $\mathbf r_i$44 in the crystal (Chehami et al., 2023). The method is notable because the paper states that it eliminates cost-function minimization and is natively scalable to large domains.

Reciprocal-space engineering begins from a target $\mathbf r_i$45 rather than a real-space motif. In hyperuniform gold metasurfaces, a point pattern was chosen so that $\mathbf r_i$46 had a pronounced exclusion region near $\mathbf r_i$47 and a single broad isotropic maximum at $\mathbf r_i$48; pillar-type and network-type layouts were then produced while preserving the dominant isotropic resonance (Castro-Lopez et al., 2017). In a more general correlated-noise framework, the averaged far-field intensity is expressed as the sum of a diffuse term, a Bragg term, and a correlation-halo term, with the displacement probability density function $\mathbf r_i$49 and the correlation length $\mathbf r_i$50 as independent design knobs. The paper gives an explicit constructive example,

$\mathbf r_i$51

whose Fourier transform preserves only selected diffraction orders; for $\mathbf r_i$52, the retained orders are $\mathbf r_i$53 with reported intensities $\mathbf r_i$54 at specular and $\mathbf r_i$55 at $\mathbf r_i$56 (Langevin et al., 19 Feb 2026). The same paper describes the resulting framework as a practical method for inverse design, namely finding the disorder that produces desired scattering patterns.

Fabrication routes reflect the breadth of the field. Reported examples include focused-ion-beam fabrication of gold nanodisks on $\mathbf r_i$57 areas followed by spin-coating with IC1-200 (Madeleine et al., 2023), electron-beam lithography and reactive-ion etching of $\mathbf r_i$58 poly-Si on fused silica with tunable connectivity (Chen et al., 4 May 2025), electron-beam lithography and ICP etching of Si nanopillars followed by $\mathbf r_i$59 Al deposition and spin-coated NaYF$\mathbf r_i$60:Er$\mathbf r_i$61 core–shell nanoparticles (Chen et al., 16 Mar 2025), electron-beam lithography of hyperuniform gold pillars and networks on glass followed by a $\mathbf r_i$62 PMMA layer doped with DCM dye (Castro-Lopez et al., 2017), and colloidal deposition of approximately $\mathbf r_i$63 silver nanocubes above a SiO$\mathbf r_i$64/Si reflector (Agreda et al., 2022). Large-area and bottom-up routes are likewise emphasized in the broader review literature because correlated disorder can be more fabrication-resilient than tightly periodic phase profiles (Lalanne et al., 2023).

6. Misconceptions, limitations, and research directions

A recurrent misconception is that disorder in metasurfaces is synonymous with uncontrollable whitish diffuse scattering. Several results contradict that reduction. Correlated disorder can support sharp or quasi-sharp collective resonances, isotropic momentum-space rings, specular suppression near $\mathbf r_i$65, critical-packing mirror peaks, prescribed diffuse whitening, or Morpho-like correlation halos (Madeleine et al., 2023, Castro-Lopez et al., 2017, Chen et al., 4 May 2025, Langevin et al., 19 Feb 2026). Another misconception is that a single scalar disorder amplitude is always a sufficient disorder coordinate. Under strong positional correlations, $\mathbf r_i$66 can become ambiguous, and topological descriptors or reciprocal-space statistics become more informative than the generative parameter itself (Madeleine et al., 2023).

Model validity is strongly regime-dependent. In coherent specular theory, ISA is accurate for very dilute monolayers, and EFA remains quantitatively useful up to about $\mathbf r_i$67 coverage and large angles for high-index dielectric particles, but for plasmonic particles at comparable coverage strong near-field coupling and hot spots degrade both approximations (Vynck et al., 2022). In critical-packing dielectric metasurfaces, extended Maxwell–Garnett modeling matches particulate regimes but underestimates the strong reflection peak near the connectivity threshold because connected-cluster physics and collective electric–magnetic dipolar response are not captured by a local effective medium (Chen et al., 4 May 2025). In SLR design, global $\mathbf r_i$68 can overestimate disorder that does not materially affect the resonance quality factor because long-scale distortions matter less than short-range coupling; a local $\mathbf r_i$69 computed in sliding windows was proposed as a better match to near-field interaction ranges (Madeleine et al., 2023).

The physical role of correlations is also not monotonic. Large $\mathbf r_i$70 can improve short-range order and preserve high-$\mathbf r_i$71 resonances even while increasing global topological disorder; hyperuniformity can reduce low-angle scattering and broaden absorption; connected morphologies can whiten diffuse light but also erase spectral selectivity by filling PDoS voids (Madeleine et al., 2023, Chen et al., 4 May 2025). Polarization dependence remains important in several platforms, from elongated plasmonic nanodisks to TE/TM localization windows in resonant point patterns (Madeleine et al., 2023, Monsarrat et al., 2021). In particular, the localization results in correlated two-dimensional media show that vector TE waves localize in a correlated moderate-density regime but exhibit no localization signature in white-noise disorder, whereas scalar TM waves localize at high density regardless of correlation class (Monsarrat et al., 2021).

Current research directions accordingly emphasize descriptors and models that preserve structural universality while remaining tied to physically relevant coupling scales. Reported directions include localized topological descriptors aligned with coupling radii and integration with inverse design, topology optimization, and graph neural networks using topological features as priors or constraints (Madeleine et al., 2023); nonlocal effective models and polarization control for critical-packing topologies (Chen et al., 4 May 2025); and broader k-space design of hyperuniform, quasi-periodic, or multilayer correlated-disordered metasurfaces for broadband antireflection, structural color, transparent displays, chiral films, light trapping, and wavefront manipulation (Lalanne et al., 2023). The field’s unifying idea is that disorder becomes most useful when it is specified statistically, topologically, or in reciprocal space rather than treated as an uncontrolled deviation from a crystal.