1. INTRODUCTION

Disordered hyperuniform many-body systems [1–3] are an emerging class of amorphous states of matter that are endowed with novel wave and other transport properties with advantages over their periodic counterparts [3–23]. Such hyperuniform two-phase composites are characterized by an anomalous suppression of volume-fraction fluctuations in the infinite-wavelength limit [2,3], i.e., the local volume-fraction variance $\sigma _V^2(R)$ inside a spherical observation window of radius $R$ decays faster than ${R^{- d}}$ in $d$ dimensions in the large-$R$ limit, $\mathop {\lim}\nolimits_{R \to \infty} {R^d}\sigma _V^2(R) = 0.$ Equivalently, its associated spectral density ${\tilde \chi _{_{V}}}({\textbf k})$ vanishes as the wavenumber $| {\textbf k} |$ tends to zero, i.e., $\mathop {\lim}\nolimits_{| {\textbf k} | \to 0} {\tilde \chi _{_{V}}}({\textbf k}) = 0$. One important class of such hyperuniform media is the disordered stealthy varieties in which ${\tilde \chi _{_{V}}}({\textbf k}) = 0$ for $0 \lt | {\textbf k} | \lt K$ [24–27], meaning that they completely suppress single scattering of incident radiation for these wave vectors [3,25]. The degree of stealthiness $\chi$ is the ratio of the number of the constrained wave vectors in the reciprocal space to the total number of degrees of freedom. Recent studies showed that such exotic disordered media exhibit novel electromagnetic wave transport properties, including high transparency in the optically dense regime, maximized absorption, and complete photonic bandgap formation [4,5,7,8,10,12–17,20–23,28–30] in two and three spatial dimensions. For example, previous work on light transparency of stealthy hyperuniform systems considered two-dimensional (2D) point scatterers [7] and 2D and three-dimensional (3D) two-phase dielectric composites [10,31]. Here, we undertake the first study of the electromagnetic properties of one-dimensional (1D) disordered stealthy hyperuniform layered media.

The problem of wave propagation in a two-phase (or multi-phase) layered medium has been extensively studied because of its simplicity and its ease of fabrication [32,33], including surface plasmons [34,35], Anderson localization [36–41], and deep-subwavelength disorder [42], as well as practical applications [18,43–45]. An important wave characteristic is the effective dynamic dielectric constant tensor ${{\boldsymbol \varepsilon}_e}({{{\textbf k}_q},\omega})$ for the incident radiation of frequency $\omega$ and wave vector ${{\textbf k}_q}$. This complex-valued quantity determines the effective wavenumber ${k_e}$ and extinction mean free path ${\ell _e}$. While there have been many theoretical/numerical treatments to estimate ${{\boldsymbol \varepsilon}_e}({{{\textbf k}_q},\omega})$ of layered media [46–53], previous approximations are applicable to disordered media in the quasistatic or long-wavelength regime, i.e., $| {{{\textbf k}_q}} |\xi \ll 1$, where $\xi$ is a characteristic inhomogeneity length scale.

Torquato and Kim [31] recently derived the general nonlocal strong-contrast expansion for the effective dielectric constant tensor ${{\boldsymbol \varepsilon}_e}\!({{{\textbf k}_q},\omega})$ that can be applied to two-phase media with various symmetries. This expansion exactly treats multiple scattering to all orders beyond the quasistatic regime (i.e., $0 \le | {{{\textbf k}_q}} |\xi \lesssim 1$) as a series involving functionals of the $n$-point correlation functions ${S^{(i)}}({{{\textbf x}_1}, \ldots ,{{\textbf x}_n}})$ for all $n$ (Section 2). Here, the quantity ${S^{(i)}}({{{\textbf x}_1}, \ldots ,{{\textbf x}_n}})$ gives the probability of finding $n$ points at positions ${{\textbf x}_1}, \ldots ,{{\textbf x}_n}$ all in phase $i\;(= 1,2)$. Because of the fast-convergence property of this series (or the linear fractional form of this series), truncating it at the $n$-point level yields multiple-scattering approximations that still accurately capture multiple scattering to all orders for a wide class of microstructures, including statistically anisotropic media. The second-order truncations already provide accurate approximations for 2D and 3D statistically isotropic two-phase media [31]. However, and importantly, analogous approximations for statistically anisotropic media have yet to be extracted and applied.

Here, we theoretically and numerically investigate the tensor ${{\boldsymbol \varepsilon}_e}({{{\textbf k}_q},\omega})$ of 3D anisotropic layered media consisting of infinite parallel dielectric slabs of phases 1 and 2 whose thicknesses are derived from 1D disordered stealthy hyperuniform packings at various $\chi$ values. For simplicity, we focus on normally incident waves of transverse polarization, where ${{\textbf k}_q} = {k_q}{\hat{\textbf z}}$ (see Fig. 1); thus, the wavenumber ${k_q}$ is the independent variable of the effective dielectric constant. From the exact strong-contrast expansion, we derive, for the first time, formulas for ${{\boldsymbol \varepsilon}_e}({{k_q},\omega})$ for 3D anisotropic layered media that accurately account for multiple scattering in terms of the spectral density ${\tilde \chi _{_{V}}}({\textbf k})$, enabling us to probe a wide range of wavenumbers. The quantity ${\tilde \chi _{_{V}}}({\textbf k})$ is the Fourier transform of the autocovariance function ${\chi _{_{V}}}({\textbf r}) \equiv S_2^{(i)}({\textbf r}) - {\phi _i}^2$ [54], where ${\textbf r} \equiv {{\textbf x}_2} - {{\textbf x}_1}$, and it can be measured from scattering experiments [55]. To our knowledge, this expression is the first closed-form formula of ${{\boldsymbol \varepsilon}_e}({{k_q},\omega})$ for general 1D media that applies well beyond the quasistatic regime.

 

Fig. 1. Schematic of three-dimensional disordered anisotropic stratified media consisting of infinite parallel slabs of phases 1 (cyan) and 2 (dark blue). A plane electromagnetic wave of transverse polarization is normally incident into the medium, and its wave vector is shown as a red arrow.

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We numerically verify that our derived formula can accurately capture multiple scattering effects due to correlated disorder beyond the quasistatic regime by using finite-difference time-domain (FDTD) simulations. For this purpose, we numerically generate stealthy hyperuniform stratified dielectric two-phase media via a modified collective-coordinate procedure described in [16,31] (Section 3). For dimensionless wavenumbers up to ${k_1}/\rho \lesssim 1.5$, our predictions indeed show excellent agreement with the real and imaginary parts of the effective dielectric constant as well as transmittance found from FDTD simulations (Sections 4 and 5 of Supplement 1). Notably, our formula predicts that stealthy hyperuniform layered media are perfectly transparent (defined as ${\mathop{\rm Im}\nolimits} [{\varepsilon _e}({{k_q},\omega})] = 0$) up to a finite wavenumber ${K_T} = K/(2\sqrt {{\phi _1}{\varepsilon _1} + {\phi _2}{\varepsilon _2}})$ (i.e., no Anderson localization) in the infinite-volume limit; see Eq. (12). This result is especially remarkable because extended states in 1D disordered systems are more difficult to achieve than in higher dimensions [17,38–41].

We also show that a perfect transparency interval cannot exist in disordered 1D non-stealthy media, hyperuniform or not, and thus Anderson localization can be present at all wavenumbers. Our results, combined with the methods to generate media with a prescribed spectral density [13,24–26], provide a new inverse-design approach [56] to engineer and fabricate multilayered dielectric media with novel wave properties.

2. THEORY

A. Exact Strong-Contrast Expansion

Here, we briefly summarize the general nonlocal strong-contrast-expansion formalism of the effective dynamic dielectric constant tensor ${{\boldsymbol \varepsilon}_e}({{{\textbf k}_q},\omega})$ for 3D two-phase media with arbitrary symmetries [31]. (The strong-property-fluctuation theory [57,58] corresponds to a special case of our strong-contrast formalism, as detailed in Section 1 of Supplement 1.) We consider a macroscopically large two-phase composite specimen in three dimensions embedded inside an infinitely large reference phase $q$ [31,59]. For simplicity, we take the phase $q$ to be the matrix phase (i.e., $q = 1,2)$ and assume that phase 1 and 2 are nonmagnetic and dielectrically isotropic with real-valued and frequency-independent dielectric constants. These assumptions imply the linear dispersion relation in the reference phase [i.e., ${k_q}(\omega) \equiv | {{{\textbf k}_q}(\omega)} | =\def\LDeqbreak{} \sqrt {{\varepsilon _q}} \omega /c$], where $c$ is the speed of light in vacuum; thus, we henceforth do not explicitly indicate the $\omega$ dependence.

The general nonlocal strong-contrast expansion is a series expansion of the linear fractional form of the tensor ${{\boldsymbol \varepsilon}_e}({{{\textbf k}_q}})$, given as

B. Multiple-Scattering Approximations for Layered Media

We can now extract from the exact strong-contrast expansion [Eq. (1)] accurate multiple-scattering approximations for layered media by truncating the expansion at the $n$-point level. We focus here on such a formula at the two-point level that depends on the spectral density ${\tilde \chi _{_{V}}}({\textbf k})$ because it is still accurate and easy to compute. For simplicity, we restrict ourselves to normally incident waves (Fig. 1); thus, the effective dielectric constant now depends on the wavenumber ${k_q}$. We outline the derivation here and provide details in Section 2 of Supplement 1.

Since layered media have rotational symmetry about the $z$ axis and translational symmetry in the $x \!-\! y$ plane, the spectral density can be expressed as

Applying Eq. (4) and Eq. (5) into the second-order truncation of the series [Eq. (1)] yields

Note that $\varepsilon _e^ \bot ({{k_q}})$ is dependent on the incident wavenumber ${k_q}$ and is complex-valued if ${\tilde \chi _{_{V}}}(0) + {\tilde \chi _{_{V}}}({2{k_q}}) \gt 0$, implying that the media can be lossy due to forward scattering and backscattering from inhomogeneties in the local dielectric constant. By contrast, $\varepsilon _e^z({{k_q}})$ is independent of ${k_q}$, reflecting the fact that a traveling longitudinal wave cannot exist under our assumptions. Hence, we focus on $\varepsilon _e^ \bot ({{k_q}})$ in the rest of this work. In the static limit, Eqs. (8) and (9) reduce to the arithmetic and harmonic means of the dielectric constants, respectively,

Interestingly, these static results are exact for any microstructure [54]. Renormalization of the reference phase for the optimal convergence (Section 2 of Supplement 1), equivalent to using the effective Green’s function in [31], yields a scaled strong-contrast approximation for disordered layered media,

In the scaled approximation [Eq. (11)], the quantity ${F^{{\rm (1D)}}}({{k_q}})$ is generally complex-valued at a given incident wavenumber ${k_q}$, producing a corresponding $\varepsilon _e^ \bot ({k_q})$ with a nonnegative imaginary part. Following conventional usage, a composite attenuates waves at a given wavenumber if the imaginary part of the effective dielectric constant is positive. Such attenuation occurs here only because of multiple scattering effects (not absorption).

Importantly, our strong-contrast approximation [Eq. (11)] predicts that stealthy hyperuniform layered media can be perfectly transparent (i.e., ${\mathop{\rm Im}\nolimits} [\varepsilon _e^ \bot] = 0$) in the infinite-volume limit for a finite range of wavenumbers. Since any stealthy hyperuniform systems completely suppress both forward scattering and backscattering [i.e., ${\tilde \chi _{_{V}}}(0) + {\tilde \chi _{_{V}}}({2{k_1}}) = 0$] for ${k_1} \lt K/2$, ${\mathop{\rm Im}\nolimits} [{F^{{\rm (1D)}}}({{k_1}})] = 0$, and substituting this condition into the scaled formula [Eq. (11)] yields a perfect transparency interval,

3. MODEL MICROSTRUCTURES

Stealthy hyperuniform particle systems are defined by a spectral density that vanishes in a finite range of wavenumbers that includes the origin [${\tilde \chi _{_{V}}}({\textbf k}) = 0$ for $0 \lt | {\textbf k} | \le K$]. The degree of stealthiness $\chi$ is measured by the ratio of the number of the constrained wave vectors in the reciprocal space to the total degrees of freedom, i.e., in one dimension, $\chi = K/(2\pi \rho)$, where $\rho$ is the number density of particles. For $\chi \lt 0.5$ in two and three dimensions or $\chi \lt 1/3$ in one dimension, these stealthy hyperuniform systems are highly degenerate and disordered [26]. Thus, we consider 1D cases in which $\chi$ takes the following values: $\chi = 0.1,0.2,0.3$. Henceforth, we take the characteristic inhomogeneity length scale $\xi$ to be the mean particle separation $1/\rho$, which is of the order of the mean nearest-neighbor distance ${\ell _P}$, (i.e., ${\ell _P} \sim 1/\rho$). This choice means that the range of validity of our nonlocal theory is ${k_1}/\rho \lesssim 1.5$ for the current models, as shown later in Section 5.

We numerically generate 1D packings of packing fraction ${\phi _2} = 0.2$ in the following two-step procedure. First, we generate point configurations of $N$ particles in a periodic fundamental cell $\mathfrak{F}$ via the collective-coordinate optimization technique [24–26], which numerically generates ground states with very high-precision of the following potential energy:

The soft repulsion [Eq. (14)] is to get the stealthy hyperuniform ground states whose nearest-neighbor distances are larger than the length scale $\sigma$ [16,31]. Finally, we create packings with dielectric constant ${\varepsilon _2}$ by circumscribing the points by identical rods of radius $a \lt \sigma /2$ without overlaps [8]; see Fig. 2(a). The parameters used to generate these systems are listed in Section 3 of Supplement 1.

 

Fig. 2. Disordered stealthy hyperuniform layered media of packing fraction ${\phi _2} = 0.2$ and unit number density $\rho = 1$ at three values of $\chi = 0.1,0.2,0.3$. (a) Representative images at $\chi = 0.1$ (top), $\chi = 0.2$ (middle), and $\chi = 0.3$ (bottom). The particle phases are shown in different colors. (b) Spectral densities ${\tilde \chi _{_{V}}}(k)$ as functions of dimensionless wavenumber $k/\rho$.

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We compute the spectral density ${\tilde \chi _{_{V}}}(k)$ from the generated packings as shown in Fig. 2(b). From the long- to intermediate-wavelength regimes ($k/\rho \lesssim 10$), we clearly see that stealthy hyperuniform packings exhibit a higher degree of correlations as $\chi$ increases. In the small-wavelength regime ($k/\rho \gg 10$ or, equivalently, $ka \gg 1$), however, the curves tend to collapse onto a single curve, reflecting the fact that these three systems consist of identical hard rods.

4. SIMULATIONS

We validate the accuracy of our predictions on the effective dielectric constant $\varepsilon _e^ \bot ({{k_1}})$ and the transmittance $T$ by comparing primarily to full-waveform simulations [67] via an open-source FDTD package MEEP [68], although we use the transfer matrix method to compute $T$, as explained in Supplement 1. We take the matrix to be the reference phase (i.e., $q = 1$) and the particles to be the polarized phase (i.e., $p = 2$) and set the phase contrast ratio as ${\varepsilon _2}/{\varepsilon _1} = 4$. We measure the transmittance spectra $T$ through the disordered stealthy hyperuniform layered media, which are then compared to the predictions from Eq. (11). We also directly extract $\varepsilon _e^ \bot ({{k_1}})$ from the nonlocal constitutive relation $\varepsilon _e^ \bot ({{k_1}}) = \langle {\tilde D({{k_e},\omega})} \rangle /\langle {\tilde E({{k_e},\omega})} \rangle$ at a given frequency $\omega$, as was done in [31]; see Fig. S6 of Supplement 1. Here, $\langle {\tilde D({{k_e},\omega})} \rangle$ and $\langle {\tilde E({{k_e},\omega})} \rangle$ are the spatial Fourier transforms of the ensemble averages of dielectric displacement field $\langle {D({x,\omega})} \rangle$ and electric field $\langle {E({x,\omega})} \rangle$ at the complex-valued effective wavenumber ${k_e}$, respectively; see details in Section 3 of Supplement 1.

5. RESULTS

We now show how our multiple-scattering approximation [Eq. (11)] enables us to accurately predict the real and imaginary parts of the effective dielectric constant tensor ${{\boldsymbol \varepsilon}_e}({{k_1}})$ for disordered stealthy hyperuniform layered media for $\chi = 0.1,0.2,$ and 0.3. We begin by computing the nonlocal attenuation function ${F^{{\rm (1D)}}}(k)$ given in Eq. (6) from ${\tilde \chi _{_{V}}}(k)$; see Fig. 3. Stealthy hyperuniform layered media can achieve ${\mathop{\rm Im}\nolimits} [{F^{{\rm (1D)}}}(k)] = 0$ up to a finite wavenumber $k$, leading to the prediction of perfect transparency interval; see Eq. (12).

 

Fig. 3. Real (upper) and imaginary (lower) parts of the nonlocal attenuation functions ${F^{{\rm (1D)}}}(k)$ as functions of dimensionless wavenumber $k/\rho$ for 1D disordered stealthy hyperuniform layered media of packing fraction ${\phi _2} = 0.2$ at three values of $\chi = 0.1,0.2,0.3$. The functions are computed from the spectral densities in Fig. 2 and Eq. (6).

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Using the values of ${F^{{\rm (1D)}}}(k)$ in Fig. 3, we then compute the scaled approximation [Eq. (11)] for $\varepsilon _e^ \bot ({{k_1}})$; see Fig. 4. Both real and imaginary parts of these predictions show excellent agreement with the results from FDTD simulations up to ${k_1}/\rho \lesssim 1.5$ (see Fig. S6 of Supplement 1). In Fig. 4, the real part of our formula increases with ${k_1}$ (normal dispersion) within the perfect transparency interval and then decreases with ${k_1}$ (anomalous dispersion) outside of those intervals. Such a spectral dependence of ${\mathop{\rm Re}\nolimits} [\varepsilon _e^ \bot ({k_1})]$ comes from the fact that the strong-contrast formula satisfies the Kramers–Kronig relations [65] (see Section 6 of Supplement 1), as does the corresponding approximation for 3D statistically isotropic media [31]. Equation (11) also shows qualitatively accurate dielectric responses even beyond the intermediate-wavelength regime (i.e., ${k_1}/\rho \gtrsim 1.5$) because the Kramers–Kronig relations closely relate the high-frequency predictions of ${\mathop{\rm Re}\nolimits} [\varepsilon _e^ \bot]$ to the accurate predictions of ${\mathop{\rm Im}\nolimits} [\varepsilon _e^ \bot]$ in a finite spectral range and vice versa [69].

 

Fig. 4. Predictions of the scaled strong-contrast approximation [Eq. (11)] of the effective dynamic dielectric constant $\varepsilon _e^ \bot ({{k_1}})$ as a function of the dimensionless incident wavenumber ${k_1}/\rho$ for disordered stealthy hyperuniform layered media of ${\phi _2} = 0.2$ and ${\varepsilon _2}/{\varepsilon _1} = 4$ at $\chi = 0.1,0.2,0.3$. The lower panel is a semilog plot of the imaginary part ${\mathop{\rm Im}\nolimits} [\varepsilon _e^ \bot ({{k_1}})]$. For the effective dielectric constants, our theory is accurate up to ${k_1}/\rho \lesssim 1.5$; see Fig. S6 of Supplement 1.

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As shown in Fig. 4, these composites are perfectly transparent (i.e., ${\mathop{\rm Im}\nolimits} [\varepsilon _e^ \bot] = 0$) for a wide range of frequencies, as predicted by Eq. (12). At the edges of these transparency intervals, a discontinuous change occurs in ${\mathop{\rm Im}\nolimits} [\varepsilon _e^ \bot]$ with ${k_1}$ because the imaginary part is directly proportional to ${\tilde \chi _{_{V}}}({2{k_1}})$; see ${\mathop{\rm Im}\nolimits} [{F^{{\rm (1D)}}}(k)]$ given in Eq. (6). In higher dimensions, however, such abrupt transitions become increasingly more difficult to achieve, as observed in [31]; see Section 2 of Supplement 1 for detail. Figure 5 depicts how the predicted perfect transparency interval from Eq. (12) varies with the phase contrast ratio ${\varepsilon _2}/{\varepsilon _1}$ for given values of $\chi$, ${\phi _2}$, and $\rho$. We numerically demonstrate that these predicted intervals are valid for $1 \lt {\varepsilon _2}/{\varepsilon _1} \lt 10$; see Fig. S3 of Supplement 1.

 

Fig. 5. Prediction [Eq. (12)] of the upper bound ${K_T}$ of the perfect transparency intervals of 1D stealthy hyperuniform layered media with ${\phi _2} = 0.2$ and $\rho = 1$ as a function of the phase contrast ratio ${\varepsilon _2}/{\varepsilon _1}$. We consider three $\chi$ values: $\chi = 0.1,0.2,$ and 0.3.

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From our scaled approximation [Eq. (11)], we also predict the normal transmittance $T$ through a layered medium at ${k_1}$ by assuming that the system is a homogeneous slab of thickness $L$ with an effective dielectric constant $\varepsilon _e^ \bot ({{k_1}})$ and is optically thin so that waves inside it can interfere coherently. To estimate $T$, we use an Airy formula [46] of transmittance for a lossy homogeneous slab with absorption:

We now compare our theoretical predictions for transmission to the corresponding results obtained from FDTD simulations; see Fig. 6. Remarkably, our theory very accurately predicts the perfect transparency intervals [Eq. (12)], i.e., no Anderson localization (green regions in Fig. 6), because they correctly incorporate multiple scattering at finite wavelengths via the spectral density. This observation is noteworthy because extended states in 1D disordered systems are much more difficult to achieve than in higher dimensions [17,38,39,41]. Within these transparency intervals (${k_1} \lt {K_T}$), our theory accurately predicts the small-amplitude periodic oscillations in $T$ around unity, which come from coherent interference of the multiply reflected waves due to the finite system thickness $L$ and, thus, reduce to a constant close to unity when $L$ is much larger than the coherence length of light [46,70]. Outside of the perfect transparency intervals, where scattering attenuation is strong, $T$ from the Airy formula [Eq. (15)] provides a lower bound on the simulation results for the reason explained in Section 4 of Supplement 1. However, we confirm that our theory still accurately predicts, qualitatively, the spectral dependence of $T$ because Eq. (11) yields a physically feasible dielectric response due to the Kramers–Kronig relations. Within these strong-attenuation intervals, $T$ is increasingly suppressed as $L$ increases and becomes virtually zero for sufficiently large $L$ since ${\mathop{\rm Im}\nolimits} [\varepsilon _e^ \bot ({{k_1}})] \gt 0$ if ${k_1} \gt {K_T}$. Stealthy hyperuniformity is required for disordered layered media to possess perfect transparency for a finite range of wavenumbers. Indeed, we show that such a perfect transparency interval cannot exist for “non-stealthy” hyperuniform 1D media, even though these correlated disordered systems anomalously suppress large-scale volume-fraction fluctuations; see the example in Section 7 of Supplement 1.

 

Fig. 6. Transmittance spectra $T$ as a function of the dimensionless wavenumber ${k_1}/\rho$ for disordered stealthy hyperuniform layered media of packing fraction ${\phi _2}\;(\equiv 2\rho a) = 0.2$ and phase-contrast ratio ${\varepsilon _2}/{\varepsilon _1} = 4$ at three values of (a) $\chi = 0.1$, (b) $\chi = 0.2$, and (c) $\chi = 0.3$. The predictions are computed from Eq. (15) using the scaled approximation [Eq. (11)]. The green-shaded area indicates the predicted transparency intervals [Eq. (12)]. Within these intervals, our predictions show excellent agreement with simulations, since there is no absorption.

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6. CONCLUSION AND DISCUSSION

In summary, we have theoretically and numerically investigated the effective wave properties, including the effective dynamic dielectric constant tensor ${{\boldsymbol \varepsilon}_e}({{k_q}}) = \varepsilon _e^ \bot ({{k_q}})({\boldsymbol I} - {{\hat{\boldsymbol z}}{\hat{\boldsymbol z}}}) + \varepsilon _e^z({{k_q}}){{\hat{\boldsymbol z}}{\hat{\boldsymbol z}}}$ and transmittance $T$, of 3D statistically anisotropic two-phase layered media made of 1D disordered stealthy hyperuniform packings. To predict ${{\boldsymbol \varepsilon}_e}({{k_q}})$ of such exotic disordered models, we derived for the first time a multiple-scattering approximation [Eq. (11)] for statistically anisotropic media from the strong-contrast-expansion formalism. Predictions of both the effective dielectric constant $\varepsilon _e^ \bot ({{k_q}})$ and transmittance $T$ are in excellent agreement with corresponding results obtained from the FDTD simulations up to ${k_1}/\rho \lesssim 1.5$. Remarkably, our predictions of $T$ are virtually identical to the simulation results within the perfect transparency ranges. Our multiple-scattering approximation is the first closed-form formula that provides a simple but accurate relation between the effective wave properties of 3D layered media and their spectral density that applies well beyond the quasistatic regime. Beyond the valid range $({k_1}/\rho \gtrsim 1.5)$, Eq. (11) can still provide qualitatively accurate and physically realistic predictions of dielectric responses since it satisfies the Kramers–Kronig relations.

We applied this newly derived formula [Eq. (11)] to disordered stealthy hyperuniform layered media at $\chi = 0.1,0.2$, and 0.3. It is noteworthy that our multiple-scattering formula [Eq. (11)] predicts that these disordered systems are perfectly transparent in the infinite-volume limit for a finite ratio $\varepsilon_2/\varepsilon_1$, implying no Anderson localization up to a finite wavenumber proportional to $\chi$, as given by Eq. (12). This observation is remarkable in that such extended states in 1D disordered systems are more difficult to achieve than in higher dimensions [17,38–41]. If the localization length is actually finite, we expect that it will be extremely large for sufficiently small $\varepsilon_2/\varepsilon_1$ compared to any practically large sample size in the transparency interval in such stratified media, as will be reported elsewhere [71]. In contrast, for disordered non-stealthy layered media, hyperuniform or not, our theory shows that there is no spectral range of perfect transparency, implying that localization emerges as the system size grows.

Our findings also have important practical implications. For example, we clearly demonstrate that disordered stealthy hyperuniform layered media can be employed as low-pass filters that transmit waves up to a selected wavenumber. Furthermore, combining our theory with the capabilities to generate media with a prescribed spectral density [13,24–26] enables an inverse-design approach [56] to engineer and fabricate layered dielectric materials with novel wave properties. One possible design is a layered medium satisfying ${\tilde \chi _{_{V}}}(k) = 0$ for $k \lt \varepsilon$ and $2{k_L} \lt k \lt 2{k_U}$, which transmits waves within a narrow spectrum ${k_L} \lt {k_1} \lt {k_U}$. Notably, such computationally designed layered media as well as other 1D disordered models can be readily fabricated via vacuum deposition [72], spin-coating [73], and 3D printing techniques [74]. Thus, our results offer promising prospects for engineering novel optoelectronic devices.

We note that our formalism can be extended to cases of obliquely incident wave vectors ${{\textbf k}_q}$, in which both effective dielectric constants $\varepsilon _e^ \bot ({{{\textbf k}_q}})$ and $\varepsilon _e^z({{{\textbf k}_q}})$ become important. Moreover, it also will be interesting to extend our theory to lossy dielectric or metallic phases whose dielectric constants are now frequency-dependent and complex-valued.