Stealthy Hyperuniform Layered Media

• Disordered stealthy hyperuniform layered media are stratified two-phase composites using a stealthy hyperuniform point process to suppress low-wavenumber fluctuations.
• They exhibit a finite transparency band with zero scattering attenuation due to reciprocal-space constraints that eliminate low-k spectral weight.
• Their engineered transport and electromagnetic properties emerge from Fourier-space design, enabling optimized diffusion and controlled wave propagation.

Disordered stealthy hyperuniform layered media are stratified two-phase composites in which the layer sequence is disordered yet the relevant spectral density vanishes over a finite interval near the origin of reciprocal space. In the most explicit formulation, they are three-dimensional media made of infinite parallel slabs of two dielectric phases arranged along the $z$-direction, with the thickness sequence generated from a one-dimensional stealthy hyperuniform point process; more broadly, they belong to the larger class of anisotropic stealthy hyperuniform composites whose reciprocal-space constraints can generate stripe-like, lamellar-like, chain-like, or wavy banded morphologies without periodic order (Kim et al., 2023, Shi et al., 2023). Their distinctive significance is that the same suppression of long-wavelength fluctuations that defines hyperuniformity can, in the stealthy case, produce a finite interval of perfect transparency, strongly reduced attenuation, and accelerated transport despite the absence of crystalline periodicity (Kim et al., 2023, Kim et al., 2024).

1. Statistical definition and layered specialization

For statistically homogeneous two-phase media, the standard description begins with the phase-indicator function

$$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$

the two-point probability function $S_2^{(i)}(\mathbf r)$, and the autocovariance

$\chi_V(\mathbf r)=S_2^{(i)}(\mathbf r)-\phi_i^2,$

where $\phi_i$ is the volume fraction of phase $i$. The Fourier transform of $\chi_V$ is the spectral density $\tilde{\chi}_V(\mathbf k)$, which plays for two-phase media the same role that the structure factor plays for point configurations (Chen et al., 2017).

A two-phase medium is hyperuniform when

$$\lim_{|\mathbf k|\to 0}\tilde{\chi}_V(\mathbf k)=0,$$

equivalently when the real-space sum rule

$\int_{\mathbb R^d}\chi_V(\mathbf r)\,d\mathbf r=0$

holds. If

$$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$0

then the local-volume-fraction variance falls into the standard three classes,

$$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$1

Stealthiness is stronger: for two-phase media it means complete suppression of scattering in a finite exclusion region,

$$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$2

or, in the isotropic form most often used,

$$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$3

(Shi et al., 2023).

For layered media, translational invariance in the $$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$4-$$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$5 plane collapses the full three-dimensional spectral density to a one-dimensional object: $$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$6 The physically relevant disorder is therefore encoded entirely in $$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$7 along the layering direction. In the 1D stealthy-hyperuniform systems used to generate layered media, the stealthiness parameter is

$$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$8

where $$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$9 is the number density of rod centers; the disordered stealthy regime corresponds to $S_2^{(i)}(\mathbf r)$0, and explicit layered examples were studied for $S_2^{(i)}(\mathbf r)$1 (Kim et al., 2023).

2. Reciprocal-space design and quasi-layered morphologies

The foundational inverse-design idea is to prescribe a target spectral density and reconstruct a two-phase medium directly in Fourier space. In the isotropic construction procedure, a finite digitized medium is evolved by simulated annealing to minimize the mismatch energy

$S_2^{(i)}(\mathbf r)$2

with trial moves consisting of pixel swaps between phases and Metropolis acceptance. In that formulation, isotropy is built in by using the angularly averaged $S_2^{(i)}(\mathbf r)$3 rather than the full vector-dependent $S_2^{(i)}(\mathbf r)$4 (Chen et al., 2017).

The anisotropic extension removes that restriction and targets an arbitrary realizable $S_2^{(i)}(\mathbf r)$5. Its objective function is

$S_2^{(i)}(\mathbf r)$6

with $S_2^{(i)}(\mathbf r)$7 the set of independent constrained wavevectors, and for stealthy targets

$S_2^{(i)}(\mathbf r)$8

The simulated-annealing protocol starts from a binary trial microstructure at fixed phase fraction, exchanges pixels of opposite phase, accepts moves with

$S_2^{(i)}(\mathbf r)$9

cools via

$\chi_V(\mathbf r)=S_2^{(i)}(\mathbf r)-\phi_i^2,$0

and terminates when $\chi_V(\mathbf r)=S_2^{(i)}(\mathbf r)-\phi_i^2,$1 (Shi et al., 2023).

In this framework anisotropy is imposed entirely through the geometry of the exclusion region $\chi_V(\mathbf r)=S_2^{(i)}(\mathbf r)-\phi_i^2,$2. That point is central for layered media. Circular $\chi_V(\mathbf r)=S_2^{(i)}(\mathbf r)-\phi_i^2,$3 yields isotropic SHU composites, but elongated or lobe-selective regions generate directional organization. The paper reports that an elliptical-disk $\chi_V(\mathbf r)=S_2^{(i)}(\mathbf r)-\phi_i^2,$4 produces “necklace-like chains” horizontally at low $\chi_V(\mathbf r)=S_2^{(i)}(\mathbf r)-\phi_i^2,$5 and elongated bands at higher $\chi_V(\mathbf r)=S_2^{(i)}(\mathbf r)-\phi_i^2,$6; a rectangular $\chi_V(\mathbf r)=S_2^{(i)}(\mathbf r)-\phi_i^2,$7 yields “chain-like arrangements at low $\chi_V(\mathbf r)=S_2^{(i)}(\mathbf r)-\phi_i^2,$8” and “stripes at high $\chi_V(\mathbf r)=S_2^{(i)}(\mathbf r)-\phi_i^2,$9”; a butterfly-shaped $\phi_i$0 produces diagonal chains and stripe-like bands; and a lemniscate-shaped $\phi_i$1 generates horizontal chain-like and stripe-like structures with a “wavy” character (Shi et al., 2023). These morphologies are not periodic laminates, but they are the clearest computational prototypes of disordered layered or quasi-layered SHU media.

The earlier isotropic framework remains foundational because it established the direct relation between prescribed low-$\phi_i$2 suppression, anomalously small coarse-grained volume-fraction fluctuations, and unusual transport and electromagnetic properties in two-phase media (Chen et al., 2017). The anisotropic generalization is what converts that logic into a practical recipe for layered and directionally hyperuniform systems.

3. Effective electromagnetic theory beyond the quasistatic regime

The explicit layered theory is based on the nonlocal strong-contrast expansion for the effective dynamic dielectric tensor. For normally incident waves along the layering direction, the effective tensor decomposes as

$\phi_i$3

so the propagating response is governed by $\phi_i$4. The core two-point microstructural input is the nonlocal attenuation function

$\phi_i$5

whose real part renormalizes dispersion and whose imaginary part controls scattering attenuation (Kim et al., 2023).

The layered effective dielectric approximation derived from this formalism is valid well beyond the quasistatic regime and reduces exactly in the static limit to

$\phi_i$6

A closed-form Airy expression then gives the slab transmittance $\phi_i$7 in terms of $\phi_i$8, so the full frequency dependence of transmission is reduced to the spectral-density dependence of $\phi_i$9 (Kim et al., 2023).

For layered SHU media the decisive consequence is a finite perfect-transparency interval. Since hyperuniformity gives $i$0 and stealthiness gives $i$1 whenever $i$2, the imaginary part of $i$3 vanishes below a threshold. After the scaling built into the strong-contrast approximation, the resulting interval is

$i$4

Within this interval,

$i$5

so there is no scattering attenuation in the effective description. By contrast, nonstealthy hyperuniform layered media satisfy only $i$6 as $i$7; they reduce attenuation strongly, but they are not perfectly transparent on any finite interval (Kim et al., 2023).

The later three-point analysis shows that this transparency interval is exact through third order for layered SHU media. The general result is

$i$8

and the proof establishes

$i$9

The practical interpretation given is that SHU layered media are perfectly transparent within that interval and that there can be no Anderson localization in practice there because the localization length is much larger than any practically large sample size (Kim et al., 2023).

4. Attenuation, diffusion spreadability, and cross-property relations

The same spectral density that controls wave attenuation also controls long-time diffusive transport. For two-phase media with equal diffusion coefficient in both phases, the exact spreadability relation is

$\chi_V$0

If $\chi_V$1, then the excess spreadability decays algebraically with exponent $\chi_V$2; for SHU media it decays exponentially,

$\chi_V$3

(Kim et al., 2024).

In the layered case, the comparison among Debye random media, equilibrium hard rods, nonstealthy hyperuniform media, and SHU media is especially sharp. At $\chi_V$4, $\chi_V$5, and $\chi_V$6, the reported values are: Debye, $\chi_V$7 and excess spreadability $\chi_V$8; equilibrium rods, $\chi_V$9 and $\tilde{\chi}_V(\mathbf k)$0; NSHU, $\tilde{\chi}_V(\mathbf k)$1 and $\tilde{\chi}_V(\mathbf k)$2; SHU, $\tilde{\chi}_V(\mathbf k)$3 and $\tilde{\chi}_V(\mathbf k)$4 (Kim et al., 2024). The monotonic progression from nonhyperuniform to NSHU to SHU is the paper’s explicit cross-property relation: the stronger the suppression of low-$\tilde{\chi}_V(\mathbf k)$5 spectral weight, the smaller the attenuation and the faster the diffusive equilibration.

For broader two-phase SHU media obtained by decorating stealthy point configurations with spheres, the same Fourier-space logic also regularizes pore geometry and improves transport. The effective diffusion coefficient through the void phase increases with the stealthiness parameter $\tilde{\chi}_V(\mathbf k)$6 at fixed phase fraction, and in the high-$\tilde{\chi}_V(\mathbf k)$7 disordered regime these isotropic media can approach the Hashin–Shtrikman upper bound

$\tilde{\chi}_V(\mathbf k)$8

while remaining disordered and isotropic. The same study shows that overlap of the decorated spheres can destroy two-phase hyperuniformity even when the underlying point configuration remains stealthy hyperuniform, a point that is directly relevant to layered constructions with broadened or merged interfaces (Zhang et al., 2016).

5. Deep-subwavelength multilayers and experimental validation of the scattering mechanism

Deep-subwavelength multilayers show that SHU layering is not only a beyond-quasistatic effective-medium phenomenon. In a 1D multilayer study at free-space wavelength $\tilde{\chi}_V(\mathbf k)$9, the two phases were silica aerogel with $$\lim_{|\mathbf k|\to 0}\tilde{\chi}_V(\mathbf k)=0,$$0 and gallium phosphide with $$\lim_{|\mathbf k|\to 0}\tilde{\chi}_V(\mathbf k)=0,$$1, embedded in silicon with $$\lim_{|\mathbf k|\to 0}\tilde{\chi}_V(\mathbf k)=0,$$2, at filling fractions $$\lim_{|\mathbf k|\to 0}\tilde{\chi}_V(\mathbf k)=0,$$3, giving $$\lim_{|\mathbf k|\to 0}\tilde{\chi}_V(\mathbf k)=0,$$4. The SHU stacks used high-index inclusions of thickness $$\lim_{|\mathbf k|\to 0}\tilde{\chi}_V(\mathbf k)=0,$$5, i.e. $$\lim_{|\mathbf k|\to 0}\tilde{\chi}_V(\mathbf k)=0,$$6, and the angular regimes were organized by the critical angles

$$\lim_{|\mathbf k|\to 0}\tilde{\chi}_V(\mathbf k)=0,$$7

Despite identical EMT refractive index, crystal, SHU, and uncorrelated stacks exhibited markedly different localization behavior because interface-driven physics, especially in the Goos-Hänchen and EMT-evanescent regimes, made the wave sensitive to the correlation structure. By tailoring the target structure factor $$\lim_{|\mathbf k|\to 0}\tilde{\chi}_V(\mathbf k)=0,$$8, the study selectively annihilated or created resonances at chosen incident angles, establishing SHU as a classification framework for deep-subwavelength microstructural phases lying between periodic multilayers and uncorrelated disorder (Park et al., 2024).

A separate water-wave experiment provides direct transport-level validation of the SHU transparency mechanism in correlated disorder. In a two-dimensional planar array of cylindrical scatterers, the SHU pattern was generated with $$\lim_{|\mathbf k|\to 0}\tilde{\chi}_V(\mathbf k)=0,$$9, giving

$\int_{\mathbb R^d}\chi_V(\mathbf r)\,d\mathbf r=0$0

corresponding to about $\int_{\mathbb R^d}\chi_V(\mathbf r)\,d\mathbf r=0$1. The measured effective scattering coefficient showed a crossover exactly at the predicted threshold $\int_{\mathbb R^d}\chi_V(\mathbf r)\,d\mathbf r=0$2: below it, $\int_{\mathbb R^d}\chi_V(\mathbf r)\,d\mathbf r=0$3; above it, $\int_{\mathbb R^d}\chi_V(\mathbf r)\,d\mathbf r=0$4. The fluctuations of the effective scattering coefficient nearly vanished in the non-scattering region because each single SHU realization individually satisfied the constraint $\int_{\mathbb R^d}\chi_V(\mathbf r)\,d\mathbf r=0$5 for $\int_{\mathbb R^d}\chi_V(\mathbf r)\,d\mathbf r=0$6 (Campaniello et al., 5 Feb 2026). Although this experiment was not layered, it validates the same momentum-transfer logic used in layered SHU theory. A plausible implication is that a stratified medium with an appropriately engineered 1D or anisotropic exclusion interval should exhibit an analogous transition between scattering and non-scattering transport.

6. Extensions, caveats, and unresolved directions

Several distinctions are essential. First, perfect hyperuniformity is not the same as stealthiness. The tessellation-based construction guarantees

$\int_{\mathbb R^d}\chi_V(\mathbf r)\,d\mathbf r=0$7

and often yields small-$\int_{\mathbb R^d}\chi_V(\mathbf r)\,d\mathbf r=0$8 scaling such as $\int_{\mathbb R^d}\chi_V(\mathbf r)\,d\mathbf r=0$9, but it does not establish

$$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$00

Thus it is a real-space route to perfect hyperuniformity, not by itself a route to stealthy layered media (Kim et al., 2019).

Second, numerical exactness matters. Large-scale collective-coordinate studies define the distance to stealthiness by

$$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$01

and report that reducing residual deviations in the exclusion region from about $$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$02 to about $$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$03 changes qualitative conclusions about transparency, energy landscapes, and Anderson localization. Even very small leakage of spectral weight into the stealthy interval can mask the physical behavior associated with truly stealthy disorder (Morse et al., 2023).

Third, non-Hermitian generalizations extend the layered problem from passive dielectrics to complex gain-loss profiles. In that setting the generalized structure factor is

$$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$04

with

$$\mathcal I^{(i)}(\mathbf x)= \begin{cases} 1, & \mathbf x \in \mathcal V_i,\ 0, & \text{otherwise,} \end{cases}$$05

The key result is that real-imaginary cross-correlations are irrelevant for hyperuniformity but essential for stealthiness, and can generate unidirectional scattering phases inaccessible in Hermitian materials. Since the formalism is dimension-agnostic, a 1D specialization to layered gain-loss media is immediate. This suggests a route to non-Hermitian stealthy hyperuniform stacks whose real and imaginary index profiles are each hyperuniform, while their odd cross-correlation engineers directional or unidirectional stealthiness (Lee et al., 10 Feb 2026).

Finally, the explicit layered transparency theory remains restricted to normal incidence, transverse polarization, real frequency-independent local dielectric constants, and rapidly convergent second- and third-order strong-contrast truncations. Oblique incidence, lossy dielectrics, metallic phases, and direct localization-length calculations beyond the presently validated range remain open directions (Kim et al., 2023, Kim et al., 2023). The general research trajectory is nevertheless clear: layered SHU media are not merely disordered analogues of periodic laminates, but a distinct Fourier-space class of stratified matter in which long-wavelength fluctuations, attenuation, and transport can be engineered through the geometry of spectral exclusion.