Deep-subwavelength engineering of stealthy hyperuniformity

2.1 Deep-subwavelength engineering of multilayers

As the simplest example of deep-subwavelength engineered disorder, we investigate a one-dimensional (1D) multilayer composed of two material phases with high and low refractive indices denoted as n _H and n _L, respectively, as n _H > n _L. The multilayer is surrounded by a homogeneous material with a refractive index n _ext, and we assume the transverse electric (TE) planewave incidence from the surrounding material. To examine the impact of deep-subwavelength microstructures, we focus on a set of multilayers characterized by the identical effective refractive index n _EMT, defined as n _EMT = [f _H n _H ² + f _L n _L ²]^1/2, where f _H and f _L denote the volume fractions of high- and low-index materials, respectively, both set at f _H = f _L = 0.5. It is worth mentioning that multilayers in this material set are indistinguishable with identical wave responses according to EMT [37], although it has already been disproven at two extremes – crystals [16], [21] and uncorrelated disorder [20], [21] – of microstructural phases, even under deep-subwavelength conditions.

In classifying materials regarding their wave properties, it is critical to employ suitable statistical parameters that properly extract the microstructural information [38], such as the perturbation strength, orders of correlations, and the clustering of constituents. In our study, we utilize the structure factor S(k) – the reciprocal-space representation of the two-point probability function [38], [39] (Supplementary Note S1 for the calculation of structure factors). Although S(k) directly determines light scattering under the first-order Born approximation, we extend our discussion beyond this approximation, as addressed later.

Despite the simplicity of multilayers considered, S(k) offers various ways of exploring microstructural phases (Figure 1a–e). At one end of the microstructural phase diagram, a 1D crystal is illustrated by the Bragg peaks [40] at S(k) (Figure 1a). At the opposite end, there is the Poisson uncorrelated disorder [38], which possesses the constant S(k) in the thermodynamic limit (Figure 1e). To explore the intermediate phases between these two extremes, we employ the concept of SHU [1], [29], [30], [31], [39], [41], [42], the suppression of long-wavelength density fluctuation as S(|k| < K) ≈ 0 for a specific positive value K. Notably, the SHU material exhibits the length-scale dependent degree of disorder: the crystal-like long-range (or small |k|) order and the Poisson-like short-range (or large |k|) order in terms of S(k) (Figure 1b). Therefore, we can envisage engineering disorder in terms of two distinct transitions from crystals to the Poisson disorder: degraded long-range order with broken SHU (Figure 1c) and degraded short-range order while maintaining SHU (Figure 1d).

Figure 1:

Deep-subwavelength engineered disorder. (a–e) Schematics for 1D multilayers with different degrees of disorder: crystal (a), SHU with larger K (b), deformed SHU with broken long-range order (c), SHU with smaller K (d), and Poisson uncorrelated disorder (e). The top and bottom subfigures in (a–e) illustrate the schematics of exemplified microstructures and their averaged S(k), respectively. The flow of the orange arrows represents the order-to-disorder transitions of interest in this work. (f) Schematic of the proposed experimental setup for measuring angular transmission. In the proposed setup, a silicon prism is employed to implement the critical angle condition in the excitation and modify the incident direction of an E _x -polarized light to the multilayer. The light then passes through the multilayer and exits the prism for detection of transparency. (g) Classification of angular responses: FP (θ < θ _L), GH (θ _L ≤ θ < θ _C), EE (θ _C ≤ θ < θ _H), and FE (θ _H ≤ θ) regimes. In (g), we assume n _H = 3.6, n _L = 1.03, and n _ext = 4.3, which lead to θ _L = 14°, θ _C = 38°, and θ _H = 57°. k _z,H and k _z,L denote the wavenumbers along the z-axis in high- and low-index materials, respectively. The details of materials for practical implementation are discussed in Section 4.

To investigate the EMT breakdown in relation to phase evolutions across the interfaces, we examine oblique incidence at angle θ to engineered multilayers (Figure 1f), which allows for angle-dependent combinations of propagating and evanescent modes within a multilayer owing to the distinct critical angle of each material phase n _H and n _L (Figure 1g). Specifically, when n _ext > n _H,L, three distinct angles characterize the wave properties of multilayers: θ _H for the critical angle from n _ext to n _H, θ _L for the critical angle from n _ext to n _L, and θ _C for the critical angle from n _ext to n _EMT. Given the relationship θ _L < θ _C < θ _H, the angular responses of θ are classified into four regimes [20] (Figure 1g): the fully propagating (FP) regime (θ < θ _L), the Goos–Hänchen (GH) regime (θ _L ≤ θ < θ _C), the EMT evanescent (EE) regime (θ _C ≤ θ < θ _H), and the fully evanescent (FE) regime (θ _H ≤ θ). We explore wave behaviours in each angular regime in engineered disorder, extending beyond previous studies on crystals [16] and uncorrelated disorder [20], [21].

2.2 Deep-subwavelength SHU

As the first step of exploring the intermediate regime in microstructural phases, we investigate the uniqueness of the SHU multilayers compared to crystals and uncorrelated disorder. For the incidence at a free-space wavelength of λ = 500 nm, we analyse the wave localization properties of multilayers with thickness L across various material microstructures. The localization of each multilayer is quantified by the localization length ξ, which is defined statistically for disordered materials [43], as follows:

where T(θ) denotes the incident-angle-dependent transmission through a realization of multilayers calculated from the scattering matrix method [34], and ⟨…⟩ represents the ensemble average for random realizations of multilayers to examine the thermodynamic limit of disordered materials under the ergodic condition [38]. We focus on the angular range near the GH regime from θ _L = 14° to θ _C = 38°, where the substantial breakdown of EMT was observed in crystals [16] and uncorrelated disorder [20], [21].

Figure 2a–c shows the structure factors S(k) of a crystal and the statistically designed SHU and Poisson uncorrelated disorder, which are calculated for the multilayers of the thickness from L = 2λ to L = 6λ. While the crystal has a periodicity of 20 nm with the first-order Bragg peaks at |k| = K = 50π/λ (Figure 2a), the thickness of the high-index (n _H) layers in both the SHU and uncorrelated disorder is set to 2 nm. These layers are iteratively placed in a perturbative manner inside the low-index (n _L) background to achieve the target S(k), while maintaining f _H = f _L = 0.5 (see Section 4 and Supplementary Note S2 for the inverse design process). The design process successfully provides the target structure factors illustrated in Figure 1a–e; the SHU satisfies S(|k| < K) ≈ 0 and S(|k| ≥ K) ≈ S ₀ (Figure 2b), while S(|k| > 0) ≈ S ₀ in uncorrelated disorder (Figure 2c), where S ₀ is the constant determined by the averaged S(k) in uncorrelated disorder.

Figure 2:

Deep-subwavelength SHU. (a–c) S(k) and (d–f) log(ξ(θ, L)) for a crystal (a, d), SHU (b, e), and uncorrelated disorder (c, f). L = 4λ and K = 50π/λ in (a–c). Black points and lines represent each realization and ensemble averages, respectively, in (a–c). (g) Normalized electric field amplitude along the x-axis within a crystal of L = 4λ for θ = θ _L = 14° and θ = θ _C = 38°. (h, i) Logarithmic plots of the localization lengths ξ as a function of L for θ = θ _L = 14° (h) and θ = θ _C = 38° (i). The upper horizontal axes in (h, i) represent the order m of the Fabry–Pérot resonance, which is determined by the Bloch wavenumber of a crystal having 20 nm periodicity. The fluctuation for crystals in (h) originates from the discretized sampling of L with crystal periodicity of 20 nm. Transparent and solid lines denote each realization and their ensemble averages, respectively. In analyzing SHU (b, e, h) and uncorrelated disorder (c, f, i), ensembles of 10³ realizations are examined. All the other parameters are the same as those in Figure 1.

Localization lengths with respect to incident angles are shown in Figure 2d–f for each microstructural phase. Under the EMT, all the given deep-subwavelength multilayers are modelled as a homogeneous layer with the same effective index, n _EMT = [(n _H ² + n _L ²)/2]^1/2. This modelling leads to angular transmissions that exhibit Fabry–Perot resonance patterns (Figure 2d–f), where the effective wavelength is determined by the incident angles (Figure 2g). However, although the identical ξ(θ, L) regardless of microstructures is expected in the EMT modelling, the EMT breakdown leads to the substantial decrease of ξ(θ, L) in uncorrelated disorder (Figure 2f) compared to that of crystals (Figure 2d).

The importance of correlation in a deep-subwavelength regime is clearly demonstrated by the localization property of SHU multilayers. Although the overall localization behaviours of the SHU look very similar to those of a crystal (Figure 2d and e), a thorough analysis of the incident angle θ unveils the uniqueness of SHU (Figure 2h and i). Near the boundary between the FP and GH regimes (θ ≈ θ _L = 14°, Figure 2h), the evolution of localization around varying thicknesses shows similarities in both a crystal and the SHU in terms of the widths and peak positions of ξ(L). However, when the effect of the Goos–Hänchen interface phase shifts becomes more pronounced, which occurs near the boundary between the GH and EE regimes (θ ≈ θ _C = 38°, Figure 2i), localization behaviours in the SHU no longer follow those in a crystal, and instead, resemble the broadening shown in uncorrelated disorder. This result reveals that the length-scale-dependent order of the SHU – characterized by crystal-like long-range order and Poisson-like short-range order – directly affects its angular responses in the GH regime – exhibiting crystal-like localization near the FP regime and Poisson-like localization near the EE regime. Notably, because the discrepancy between the FP and EE regimes stems from the presence of evanescent modes inside multilayers (Figure 1g), the uniqueness in the angular responses of each microstructure – a crystal, SHU, and uncorrelated disorder – originates from the distinct distributions of evanescent modes. Therefore, we can envisage more selective and deterministic engineering of optical angular functionalities near the EE regime, by independently manipulating short- and long-range order and the corresponding evanescent-mode distributions.

2.3 Nontrivial transitions from SHU to uncorrelated disorder

To employ the concept of engineered disorder [1] to impose optical functionalities on deep-subwavelength microstructures, we investigate microstructural phase transitions between the SHU and uncorrelated disorder and their impact on localization (Figure 3). When we apply the inclusion of uncorrelated perturbations to the SHU state, the value of S(k) increases across the entire range of k, leading to the trivial transition from SHU to uncorrelated disorder and the consequent gradual changes in wave behaviours. Instead of such a trivial transition, we devise the nontrivial transitions that are clearly differentiated by their short-range and long-range order, as well as by the maintenance of SHU. The first transition is characterized by increasing S(|k| < K), which corresponds to the breakdown of long-range order (Figure 3a, b, and d). At the same time, the multilayers are no longer SHU because S(|k| < K) ≠ 0. In contrast, the second transition is designed with the decrease of K, which degrades the short-range order of multilayers, while maintaining their SHU with S(|k| < K) ≈ 0 (Figure 3a, c, and d). Notably, the range of engineered length scales maintains far below the wavelength of light: controlling S(k) below λ/25 in Figure 3b and across λ/10 (Figure 3c-1), λ/4 (Figure 3c-2), and λ/3 (Figure 3c-3). Examples of designed multilayer patterns are shown in Supplementary Note S3.

Figure 3:

Transitions across different degrees of disorder through differentiated tailoring of length-scale order. (a–d) Localization lengths and the corresponding structure factors as functions of the incident angle θ: SHU (a), breakdown of long-range order (b: b-1, b-2, and b-3), decrease of K (c: c-1, c-2, and c-3), and uncorrelated disorder (d). In the left panels of (a–d), points and lines represent the realizations of log(ξ) and their ensemble averages log(⟨ξ⟩), respectively, where ⟨ξ⟩ denotes the average of each realization of ξ. In the right panels of (a–d), partially transparent and solid lines for S(k) represent each realization and ensemble averages, respectively. In all cases, ensembles of 10² realizations are examined. We set the entire thickness as L = 2λ for the free-space wavelength λ = 500 nm, same as Figure 1. All the other parameters are the same as those in Figure 1.

Figure 3 illustrates that these two nontrivial transitions deliver distinct evolutions in maintaining Fabry–Perot resonances. At the initial SHU state (Figure 3a), we observe distinct Fabry–Perot resonances as already depicted in Figure 2b, while the peak positions and angular widths vary across the FP, GH, and EE regimes due to changes in effective wavelengths. Through the first-type transition with increasing S(|k| < K), the breakdown of SHU leads to a highly sensitive annihilation of Fabry–Perot resonances, with the resonance peaks decreasing by an order of magnitude from a minor perturbation level of S(|k| < K) ≈ S ₀/20 for S(|k| ≥ K) ≈ S ₀ (from Figure 3a and b).

In contrast, the second-type transition, which involves decreasing K while maintaining SHU, enables the engineering of the angular range where Fabry–Perot resonances are preserved. Notably, an abrupt change in the Fabry–Perot resonances begins to occur near the SHU state at K = 10π/λ (Figure 3c-2 and c-3), initiating the annihilation of the highest-order of resonance due to disrupted short-range order. The further decrease in K sequentially annihilates lower-order resonances until the annihilation of the zeroth-order resonance as shown in Figure 3d.

2.4 Angle-selective engineering of localization

Based on the results shown in Figure 3, which demonstrate the strong connection between the density fluctuation characterized by S(k) and wave localization, we further implement angle-selective engineering of localization. We employ the transition of material phases at a target length scale, focusing on initial states of SHU and uncorrelated disorder. Considering the angular range of interest depicted in Figure 3, we explore the manipulations of the SHU state near K = 10π/λ. Initially, the SHU multilayers exhibit Fabry–Perot resonances (Figure 4a and b, red lines), while uncorrelated multilayers show angularly flattened strong localization (Figure 4c and d, blue lines). To manipulate the density fluctuation at a specific length scale, we design an ensemble of disorder realizations generated by the structure factors S(k) of which the value at the target k increases (black line in Figure 4a) or decreases (black line in Figure 4c).

Figure 4:

Angular-selective localization. (a, b) Selective annihilation and (c, d) selective creation of the target Fabry–Perot resonances: the designed S(k) transitions (a, c) and incident-angle-dependent localization lengths (b, d). In (a, c), the black arrows illustrate the transition for each case. All the other parameters and plots are the same as those in Figure 3.

Through this length-scale-specific engineering of deep-subwavelength microstructures, the resonance at a specific angle can be selectively controlled over a few orders of magnitude (black lines in Figure 4b and d). This capability to fine-tune the wave localization length can be extended to other incident angles, even allowing for precise control across multiple angles (Supplementary Note S4). Notably, when considering L = 2λ and n _EMT = 2.648, the overall structures lie in the regime of the broken Born approximation. The results in Figures 3 and 4 confirm that S(k) serves as an excellent design tool for engineering disorder in deep-subwavelength microstructures, even when the entire system exhibits strong scattering. When considering the scaling theory of light [44], which states the ultimate wave localization in 1D disordered structures, our approach corresponds to a method of tailoring the degree of localization based on deep-subwavelength-scale engineering.

4.1 Deep-subwavelength multilayers

The refractive indices of multilayers are assumed to be n _L = 1.03 for silica aerogel layers and n _H = 3.6 for gallium phosphide (GaP) layers. The multilayers are surrounded by a homogeneous medium composed of silicon with n _ext = 4.3. We assume the lossless condition for these materials.

4.2 Inverse design process

To achieve the target profile of S(k) in the designed material, we employ the following iterative optimization process. First, we determine the parameters for a crystal and calculate its S(k) (Figures 1a and 2a) by following Supplementary Note S1. We also set the target structure factor S ₀(k) for the microstructural phase of interest (Figure 1b–e). Second, we obtain an initial state by perturbing 10⁴ randomly sampled n _H layers with replacement to enhance the convergence of the optimization method. The perturbation in preparing the initial state is uniformly random and its statistical range is set to be maximized while ensuring the hard particle condition of n _H layers. We then apply 2 × 10⁵ iterative optimization processes to the initial state.

In each iteration, we utilize the cost function defined by the mean squared error (MSE). To enhance the convergence performance, we calculate the weighted MSE for each discretized range of the reciprocal axis k. First, for the reciprocal space of interest K _I, we divide K _I into the subspaces κ _j (j = 1, 2, 3, and 4). The MSE for the jth subspace is defined as follows:

where |κ _j | is the length of the subspace κ _j . Second, to reflect the distinct importance of each subspace in the structure factor, the total cost function is defined as:

where the values of the weighting factor w _j are determined empirically for each microstructural phase. Finally, with the designed cost function in Eq. (3), we apply the uniformly random perturbation to n _H layers with the maximum statistical range within the hard particle condition. To hinder the local minima issue, we apply the modified Metropolis acceptance rule [38] of which the acceptance probability is determined by empirical hyperparameters according to simulated annealing method [33]. The proposed optimization process shows an excellent performance as shown in Figures 2–4. An example of the optimization process is described in Supplementary Note S2.