1. Introduction

It is increasingly being recognized that one can engineer a spectrum of correlated disordered patterns in composites/metamaterials and devices to achieve novel photonic and phononic characteristics with advantages over their periodic counterparts [1–24]. The concept of disordered hyperuniformity has changed our understanding of the nature of randomness and order [25,26] and has provided a means to quantify the degree of order/disorder across length scales [27]. Hyperuniform systems have attracted great attention over the last decade because of their deep connections to a wide range of topics that arise in physics [11,26,28–38], materials science [4,7,10,39,40], mathematics [41–44], and biology [45,46] as well as for their emerging technological importance in the case of the disordered varieties [1,3–6,8,11,17,22,47–49]. A hyperuniform two-phase composite in $d$-dimensional Euclidean space $\mathbb {R}^d$ is characterized by an anomalous suppression of volume-fraction fluctuations in the infinite-wavelength limit, i.e., the spectral density $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({k}\right )}}$ obeys the condition [26,50]:

Hyperuniform two-phase media encompass all periodic systems, many quasiperiodic media, and exotic disordered ones and thus generalizes our notions of long-range order to include exotic disordered varieties; see Ref. [26] and references therein. Disordered hyperuniform systems lie between liquids and crystals; they are like liquids in that they are statistically isotropic without any Bragg peaks, and yet behave like crystals in the manner in which they suppress the large-scale density fluctuations [25,26,50]. Equivalently, a hyperuniform two-phase medium is characterized by a local volume-fraction variance $\mathinner {\sigma _V^2\mathopen {\left (R\right )}}$ associated with a spherical window of radius $R$ that goes to zero asymptotically more rapidly than the inverse of the window volume, i.e., faster than $R^{-d}$. There are several classes of hyperuniform two-phase media, distinct from nonhyperuniform media in which the variance goes to zero like $R^{-d}$ or slower (see section 2). One important subclass of such hyperuniform media is the disordered stealthy varieties in which $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({{\mathbf{k}}}\right )}}=0$ for $0<|{{\mathbf{k}}}|<K$ [28,51–53], meaning that they completely suppress single scattering of incident radiation for these wavevectors [26,52]. These exotic disordered systems exhibit novel electromagnetic wave transport properties, including high transparency in the optically dense regime, maximized absorption, and complete photonic band-gap formation [1,3,5,8,10–12,14,19–22,39,54–56].

A vital electromagnetic wave characteristic is the effective dynamic dielectric constant tensor $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left ({\mathbf{k}}_q, \omega \right )}}$ for the incident radiation of wave vector ${\mathbf{k}}_q$ and frequency $\omega$. Calculation of $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left ({\mathbf{k}}_q, \omega \right )}}$ is crucial for a wide range of applications, including remote sensing of terrain or vegetation [57–60], understanding wave propagation through turbulent atmospheres [61], and designing optical metamaterials [62]. While there have been many numerical and theoretical treatments to predict $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left ({\mathbf{k}}_q, \omega \right )}}$ for both statistically anisotropic [62–71] and isotropic [60,72–74] media, the preponderance of previous approximations are either microstructure independent or applicable in the long-wavelength (or quasistatic) regime, i.e., $|{{\mathbf{k}}_q}\vert \xi \ll 1$, where $\xi$ is a characteristic inhomogeneity length scale.

Torquato and Kim [75] recently derived exact general nonlocal strong-contrast expansions that are rational functions of the effective dielectric constant tensor $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left ({\mathbf{k}}_q, \omega \right )}}$ and which depend of two-phase composites on the microstructure via functionals of the $n$-point correlation functions $\mathinner {S_n ^{(i)}\mathopen {\left ({\mathbf{x}}_1,\ldots,{\mathbf{x}}_n\right )}}$ for all $n$ (see section 2) and exactly treat multiple scattering to all orders beyond the long-wavelength regime up through the intermediate-wavelength regime (i.e., $ 0 \leq\left|\mathbf{k}_q\right| \xi \lesssim 1 $). Our strong-contrast formulas assume the phases are dissipationless, and hence, attenuation can only occur due to multiple-scattering effects. While standard perturbation expansions of $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left ({\mathbf{k}}_q, \omega \right )}}$ [72,76,77] do not converge rapidly for large contrast ratios, this linear fractional expansion rapidly converges well beyond the long-wavelength regime even when the phase contrast ratio is large, which is analytically demonstrated in Sec. S5 of Supplement 1. Thus, truncation of this expansion at low order yields highly accurate estimates of $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left ({\mathbf{k}}_q, \omega \right )}}$ because higher-order contributions are negligibly small, implying that this resulting formula very accurately approximates multiple scattering through all orders [75,78]. Specifically, second-order truncations already provide accurate approximations beyond the long-wavelength regime for TE polarization in transversely isotropic media [75] and transverse polarizations in 3D layered [78] and 3D fully isotropic media [75].

In a more recent paper [78], we focused on the application of the strong-contrast formalism to layered media, consisting of infinite parallel dielectric slabs of phases 1 and 2 whose thicknesses are derived from one-dimensional (1D) disordered media. There, we showed, among other results, that the imaginary part of the effective dielectric constant at the two-point level is proportional to a sum involving both forward scattering and backscattering contributions, i.e.,

 

Fig. 1. Schematic of 3D statistically anisotropic disordered two-phase media with two types of symmetries. We assume that both phases (shown in cyan and dark blue) of composites are dissipationless, and attenuation in these composites occurs only due to scattering. (a) Layered media consisting of infinite parallel slabs of phases 1 (cyan) and 2 (dark blue). A plane electromagnetic wave of the transverse polarization is normally incident to the medium, and its wave vector is shown as a red arrow. (b) Transversely isotropic media with cylindrical symmetry derived from 2D isotropic two-phase media. A plane electromagnetic wave of transverse magnetic (TM) polarization is normally incident to the medium, and its wave vector is shown as a red arrow.

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This paper aims to demonstrate the versatility of the nonlocal strong-contrast expansion derived in Ref. [75] to provide accurate approximations suited for various structural symmetries by tuning the expansion parameter $\boldsymbol {L}_p^{(q)}$ (see Sec. S1 of Supplement 1). This task is carried out by choosing an appropriate exclusion volume shape around the singularity of the dyadic Green’s function in the general expansion and then truncating the series at the $n$-point level, as briefly discussed in Ref. [75]. We, for the first time, report the key predictive formulas of $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left ({\mathbf{k}}_q\right )}}$ extracted from the third-order truncations of the expansion for two types of statistically anisotropic two-phase composites: 3D layered media and 3D transversely isotropic media; see section 3. For layered media, we focus on normally incident waves of transverse polarization, where ${\mathbf{k}}_q = k_q \hat {{\mathbf{z}}}$; see Fig. 1(a). Waves propagating in such anisotropic 3D media can be rigorously considered to be waves propagating in 1D ($d=1$) systems, and so we sometimes refer to them as 1D media. The resulting third-order formula [Eq. (12)] for layered media is a higher-order variant of the second-order formula derived in Ref. [78]. Transversely isotropic media consist of infinite parallel dielectric cylinders of phase 2 embedded in phase 1 whose cross-section is derived from two-dimensional (2D) statistical isotropic media. Here, we consider normally incident waves (i.e., ${\mathbf{k}}_q = k_q \hat {{\mathbf{y}}}$) and focus on transverse magnetic (TM) polarization; see Fig. 1(b). Such waves propagating in transversely isotropic 3D media can be rigorously considered to be waves propagating in 2D ($d=2$) systems and so we sometimes refer to them as 2D media. The resulting formula [Eq. (18)] is for TM polarization, whereas the 2D formula in Ref. [75] is for TE polarization. In both layered and transversely isotropic media, the wave number $k_q$ in the reference phase is the independent variable of the effective dielectric constant.

In the present work, we prove the perfect transparency interval predicted by the second-order formula [Eq. (2)] of stealthy hyperuniform layered media is also exact through third-order terms; see section 4. We also prove that transversely isotropic stealthy hyperuniform media are perfectly transparent through the third-order terms for TM polarization within the interval given in Eq. (3). We observe, moreover, that the high predictive power of the second-order formula that predicts Eq. (3) trivially implies that there can be no Anderson localization within the predicted perfect transparency interval in 1D and 2D stealthy hyperuniform media in practice because the localization length, associated with only possibly negligibly small higher-order contribution, would be very large compared to any practically large sample size.

We also apply our strong-contrast formulas to estimate $\mathinner {\varepsilon _e\mathopen {\left (k_q\right )}}$ for two models of disordered hyperuniform layered and transversely isotropic media: nonstealthy hyperuniform polydisperse packings in a matrix and stealthy hyperuniform packings in a matrix (see section 5). By contrast, Ref. [78] exclusively studied stealthy hyperuniform layered media. For all models considered here, we numerically construct their realizations at $\phi _2=0.2$ and $0.25$ for layered media and transversely isotropic media, respectively. For each model, we obtain the spectral density and then evaluate the nonlocal attenuation function, which is a crucial quantity in our second-order formulas. Realizations of all models are later employed for the full-waveform simulations.

Subsequently, in section 6, we corroborate the accuracy of the second-order formulas for both layered ($d=1$) and transversely isotropic ($d=2$) hyperuniform models for wave numbers ($k_1 \rho ^{-1/d} \lesssim 1.5$) well beyond the long-wavelength regime by showing excellent agreement with the finite-difference time-domain (FDTD) simulations. Furthermore, these approximations can provide qualitatively accurate predictions even beyond the intermediate-wavelength regime (i.e., $k_1 \rho ^{-1/d} \gtrsim 1.5$) since they meet the Kramers-Kronig relations [79] that relate the accurate low-frequency predictions to the high-frequency predictions. Our second-order formulas predict that for hyperuniform models across dimensions, the real parts of the effective dielectric constants, closely related to the effective wave speed, similarly increase with $k_1$ for small wave numbers (i.e., $k_1\rho ^{-1/d}<0.1$). By contrast, these formulas predict that the two disordered models have qualitatively different attenuation behaviors, i.e., the imaginary parts of the effective dielectric constants. However, they cannot achieve the finite perfect transparency intervals that stealthy hyperuniform models exhibit. Since such accurate second-order formulas depend solely on the spectral density $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({{\mathbf{k}}}\right )}}$, combining them with the methods to construct two-phase media with a targeted spectral density [10,24,51–53] would facilitate inverse approaches [80] to engineer and fabricate anisotropic dielectric media with tailored novel wave properties.

2. Background

2.1 $n$-point correlation functions

A two-phase random medium is a domain of space $\mathcal {V} \subseteq \mathbb {R}^d$ that is partitioned into two disjoint regions that make up $\mathcal {V}$: a phase 1 region $\mathcal {V}_1$ of volume fraction $\phi _1$ and a phase 2 region $\mathcal {V}_2$ of volume fraction $\phi _2$ [81]. The phase indicator function $\mathinner {\mathcal {I}^{(i)}\mathopen {\left ({\mathbf{x}}\right )}}$ of phase $i$ for a given realization is defined as

The $n$-point correlation function $S_n^{(i)}$ for phase $i$ is defined as [81,82] $\mathinner {S_n^{(i)}\mathopen {\left (\mathbf {x}_1, \ldots, \mathbf {x}_n\right )}} = \left \langle \prod _{j=1}^n {\cal I}^{(i)}(\mathbf {x}_j)\right \rangle,$ where $\left \langle \cdot \right \rangle$ denotes an ensemble average over realizations. The function $S_n^{(i)}(\mathbf {x}_1, \ldots, \mathbf {x}_n)$ has a probabilistic interpretation: It gives the probability of finding the ends of the vectors $\mathbf {x}_1, \ldots, \mathbf {x}_n$ all in phase $i$. For statistically homogeneous media, $\mathinner {S_n^{(i)}\mathopen {\left ({\mathbf{x}}_1, \ldots, {\mathbf{x}}_n\right )}}$ is translationally invariant and, in particular, the one-point function is position-independent, i.e., $S_1^{(i)}(\mathbf {x_1}) = \phi _i$.

The autocovariance function $\chi _{_V}(\mathbf{r})$, which is directly related to the two-point correlation function $\mathinner {S_2^{(i)}\mathopen {\left ({\mathbf{r}}\right )}}$, is defined by

2.2 Packings

We call a packing in $\mathbb {R}^d$ a collection of nonoverlapping particles [84]. For a packing in a periodic simulation box $\mathfrak {F}$, which consists of $N$ spheres of radii $a_1,\ldots,a_N$ and centers ${\mathbf{r}}_1,\ldots, {\mathbf{r}}_N$, its spectral density can be written as [84,85]

2.3 Classification of hyperuniform and nonhyperuniform media

The hyperuniformity concept generalizes the traditional notion of long-range order in many-particle systems to include not only all perfect crystals and perfect quasicrystals but also exotic amorphous states of matter, according to Refs. [25,26]. For disordered two-phase systems, the spectral density frequently exhibits a power-law scaling behavior in the small-wave number limit:

The value of a positive exponent $\alpha$ relates to the three different scaling regimes (classes) associated with the large-$R$ behaviors of the volume-fraction variance [26,50]:

By contrast, for any nonhyperuniform two-phase system, the variance has the following large-$R$ scaling behaviors [86]:

3. Theory

Here, we report the key formulas for the effective dynamic dielectric constant tensor of 3D layered media and 3D transversely isotropic media, which are extracted from the general strong-contrast expansion derived in Ref. [75]. Herein, we consider a plane electric wave of an angular frequency $\omega$ and a wavevector ${\mathbf{k}}_q$ in the reference phase $q$, which is taken as one phase (i.e., $q = 1$ or $2$) of the composite unless otherwise stated. We make three assumptions about both phases: (a) they have constant real-valued dielectric constants, (b) they are dielectrically isotropic, and (c) they are nonmagnetic. Thus, a dielectric composite that we consider here cannot absorb waves but attenuate them solely due to multiple scattering from fluctuations in the local dielectric constant. Since these assumptions give a simple relation between the wave number $k_q$ in the reference phase $q$ and angular frequency $\omega$ [i.e., $ k_q(\omega) \equiv\left|\mathbf{k}_q(\omega)\right|=\sqrt{\varepsilon_q} \omega / c $], where $c$ is the speed of light in vacuum, we henceforth do not explicitly indicate the $\omega$ dependence, i.e., $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left ({\mathbf{k}}_q, \omega \right )}}=\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left ({\mathbf{k}}_q\right )}}$. The linear fractional form of the strong-contrast expansion results in a rapidly converging series whose lower-order truncations lead to accurate approximation formulas for $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left ({\mathbf{k}}_q, \omega \right )}}$, even for large contrast ratios. This is to be contrasted with standard weak-contrast expansions that do not converge rapidly for large contrast ratios; see Sec. S5 of Supplement 1 for such a quantitative explanation. Indeed, its truncation at the two-point level yields a highly accurate estimation of $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left ({\mathbf{k}}_q, \omega \right )}}$ due to the fact that higher-order contributions are negligibly small, implying that this resulting formula very accurately approximates multiple scattering through all orders [75,78]. Furthermore, similar to the strong-fluctuation theory [59,91], the expansion parameter of this series varies with the shape of the infinitesimal exclusion volume associated with the singularity of the dyadic Green’s function. We exploit the fact that an appropriate choice of the exclusion volume can lead the series to be valid even for large values of phase contrast ratio $\varepsilon _2/\varepsilon _1$ and various microstructural symmetries. We provide detailed derivations in Secs. S2 and S3 of Supplement 1.

3.1 Multiple-scattering approximations for layered media

We consider our multiple-scattering approximations for the effective dynamic dielectric constant tensor of 3D layered media when waves are normally incident [see Fig. 1(a)], i.e., ${\mathbf{k}}_q = k_q \hat {{\mathbf{z}}}$, where $\hat {{\mathbf{z}}}$ is a unit vector along the $z$-direction. Thus, these formulas depend on the wave number $k_q$. They are extracted from exact strong-contrast series by choosing a disk-like exclusion volume normal to $\hat {{\mathbf{z}}}$ that involves the singularity of the dyadic Green’s function and then truncating this series at the three-point level. We outline the derivation in Sec. S2 of Supplement 1.

Due to the symmetries of layered media, one can decompose the effective dielectric constant tensor into two orthogonal components $\mathinner {\varepsilon _e ^\perp \mathopen {\left (k_q\right )}}$ and $\mathinner {\varepsilon _e ^z\mathopen {\left (k_q\right )}}$ for the transverse and longitudinal polarizations, respectively, as follows: $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left (k_q\right )}} = \mathinner {\varepsilon _e ^\perp \mathopen {\left (k_q\right )}} (\boldsymbol {I}-\hat {{\mathbf{z}}}\hat {{\mathbf{z}}}) + \mathinner {\varepsilon _e^z\mathopen {\left (k_q\right )}}\hat {{\mathbf{z}}}\hat {{\mathbf{z}}}$. Thus, we derive two independent approximations by truncating the strong-contrast expansion and solving for $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left (k_q\right )}}$ from this linear fractional form [Eq. (S42) of Supplement 1]. Renormalization of the resulting formulas with the reference phase for the optimal convergence (Sec. S4 of Supplement 1), equivalent to using the effective Green’s function in Ref. [75], yields scaled strong-contrast approximations for disordered layered media at the three-point level:

Note that $\mathinner {\varepsilon _e^\perp \mathopen {\left (k_q\right )}}$ given in Eq. (12) is complex-valued, implying that the media can be lossy due to forward scattering and backscattering from fluctuations in the local dielectric constant. By contrast, $\mathinner {\varepsilon _e^z\mathopen {\left (k_q\right )}}$ is independent of $k_q$, reflecting the fact that a traveling longitudinal wave cannot exist under our assumptions. Hence, we focus on $\mathinner {\varepsilon _e^\perp \mathopen {\left (k_q\right )}}$ in the rest of this work. In the static limit, Eq. (12) and Eq. (13) reduce to the arithmetic and harmonic means of the local dielectric constants, respectively:

Interestingly, these static results are exact for any microstructure [81].

In the long-wavelength regime ($k_q /\rho \ll 1$), the imaginary part of Eq. (12) is determined by the asymptotic behavior of $\mathinner {A_2^\perp \mathopen {\left (k_q;\varepsilon _q\right )}}$. Thus, assuming that the spectral density has a power-law scaling [Eq. (8)], one immediately obtains

It is seen that hyperuniform media attenuate less than nonhyperuniform ones in the long-wavelength regime. Furthermore, while all typical nonhyperuniform systems exhibit a similar attenuation behavior [i.e., $\textrm{Im} [\varepsilon _e]\propto {k_1}$ for small $k_1$], hyperuniform systems can exhibit a wide range of behaviors by tuning the exponent $\alpha$. The attenuation behavior of stealthy hyperuniform models is elaborated in section 4.

3.2 Multiple-scattering approximations for transversely isotropic media

We obtain our multiple-scattering approximations for the effective dynamic dielectric constant tensor of 3D transversely isotropic media for the situation in which waves are normally incident [see Fig. 1(b)], i.e., ${\mathbf{k}}_q = k_q \hat {{\mathbf{y}}}$, where $\hat {{\mathbf{y}}}$ is a unit vector along the $y$-direction. These formulas depend on the wave number $k_q$ in the reference phase $q$. We extract them from the exact strong-contrast series by choosing a needle-like exclusion volume aligned along $\hat {{\mathbf{y}}}$ that involves the singularity of the dyadic Green’s function and then truncating this series at the three-point level. We outline the derivation in Sec. S3 of Supplement 1.

Due to the symmetries of the problems, one can decompose the effective dielectric constant tensor into two orthogonal components $\mathinner {\varepsilon _e ^{TM}\mathopen {\left (k_q\right )}}$ and $\mathinner {\varepsilon _e ^{TE}\mathopen {\left (k_q\right )}}$ for TM and TE polarizations, respectively, as follows: $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left (k_q\right )}} = \mathinner {\varepsilon _e ^{TM}\mathopen {\left (k_q\right )}} \hat {{\mathbf{z}}}\hat {{\mathbf{z}}} + \mathinner {\varepsilon _e^{TE}\mathopen {\left (k_q\right )}}(\boldsymbol {I}-\hat {{\mathbf{z}}}\hat {{\mathbf{z}}})$. Thus, we extract two independent approximations of $\mathinner {{\varepsilon _e^{TE}}\mathopen {\left ({k_q}\right )}}$ and $\mathinner {{\varepsilon _e^{TM}}\mathopen {\left ({k_q}\right )}}$ by truncating the strong-contrast series at third-order terms; see Eq. (S78) and Eq. (S79) of Supplement 1. Solving for the effective dielectric constants from these linear fractional forms and then renormalizing them with the optimal reference phase [59,75,91] (Sec. S4 of Supplement 1), we finally obtain the scaled strong-contrast approximations at the three-point level for disordered transversely isotropic media:

The second- and third-order coefficients of each polarization are defined as

In the long-wavelength regime ($k_q /\rho \ll 1$), the imaginary part of Eq. (18) is determined by the asymptotic behavior of $\mathinner {{A_{2}^{TM}}\mathopen {\left ({k_*^{TM}; \left \langle \varepsilon \right \rangle }\right )}}$. Thus, assuming that the spectral density has a power-law scaling [Eq. (8)], one can immediately obtain

4. Perfect transparency intervals of stealthy hyperuniform systems

We have previously shown that stealthy hyperuniform two-phase composites with $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({k}\right )}}=0$ for $k<K$ can be perfectly transparent or, equivalently, have a zero imaginary part of effective dielectric constant, at the two-point level in a finite range of wave numbers for TE polarization in transversely isotropic media [75] and transverse polarization in 3D fully isotropic media [75] and layered media [78]. Across space dimensions, the predictions of such perfect transparency intervals have a similar form:

Moreover, we also show that the interval [Eq. (26)] for layered media still applies to the three-point level.

We now prove the perfect transparency interval for TM polarization in transversely isotropic media [Eq. (26) with Eq. (30)] at the two-point level. Using the second-order formula of $\mathinner {{\varepsilon _e^{TM}}\mathopen {\left ({k_q}\right )}}$ in Eq. (18) and the nonlocal attenuation function [Eq. (21)], one obtains

 

Fig. 2. (a) Schematic of the on-shell scattering contributions to the nonlocal attenuation function for transversely isotropic stealthy hyperuniform media. The white and gray areas show the exclusion region of stealthy hyperuniform media and the rest of Fourier space, respectively. The colored circles indicate the on-shell wavevectors satisfying $|{k_*\hat {{\mathbf{y}}}+{\mathbf{q}}}|=k_*$, in which elastic single-scattering event can occur. At the two-point level, since only on-shell scattering $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({{\mathbf{q}}}\right )}}$ on such a circle contributes to attenuation, if the circle lies inside the exclusion region entirely (i.e., $k_*<K/2$), no attenuation arises. (b) Normalized sizes $K_T/K$ of the perfect transparency interval [Eq. (26)] for both layered and transversely isotropic stealthy hyperuniform media as a function of contrast ratio $\varepsilon _p/\varepsilon _q$. We consider five values of $\phi _p = \delta ^+, 0.1,0.5,0.9,1-\delta ^+$, where $\delta ^+$ denotes an infinitesimally small positive number.

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In the present work, we also sketch a proof that the perfect transparency interval [Eq. (26)] for layered media is still valid at the three-point level. Accounting for the perfect transparency at the two-point level [see Eq. (2)], from the third-order formula of $\mathinner {\varepsilon _e^\perp \mathopen {\left (k_q\right )}}$ [Eq. (12)], we obtain $\textrm{Im} [\mathinner {\varepsilon _e^\perp \mathopen {\left (k_q\right )}}] \propto (\varepsilon _p-\varepsilon _q)^3 \textrm{Im} [\mathinner {A_3^\perp \mathopen {\left (k_*, \left \langle \varepsilon \right \rangle \right )}}]$ for $0\leq k_*(= k_q \sqrt {\left \langle \varepsilon \right \rangle /\varepsilon _q}) < K/2$. Thus, it is sufficient to show that

We outline the proof of Eq. (32) here (see Ref. [94] and Sec. S2.D of Supplement 1 for details). We first decompose $\mathinner {A_3^\perp \mathopen {\left (k_*;\left \langle \varepsilon \right \rangle \right )}}$ into three terms as

We also sketch a proof that the perfect transparency interval for TM polarization in transversely isotropic media [Eq. (26) with Eq. (30)] is still valid through the third-order terms. Since the formula of $\mathinner {{\varepsilon _e^{TM}}\mathopen {\left ({k_q}\right )}}$ [Eq. (18)] for transversely isotropic media has the form identical to Eq. (12) for layered media, it is sufficient to show that

Its proof is analogous to the proof of Eq. (32) for layered media. Thus, we similarly decompose $\mathinner {{A_{3}^{TM}}\mathopen {\left ({k_*, \left \langle \varepsilon \right \rangle }\right )}}$ into three terms as

We now discuss the implications of the finite perfect transparency interval (trivially implying no Anderson localization [2,95–99], in principle) for stealthy hyperuniform media across dimensions predicted in Eq. (26). As noted in the Introduction, the rapid convergence properties of strong-contrast expansions yield second-order truncations that already have high predictive power, as validated via FDTD simulations discussed in section 6 and in Ref. [78]. These outcomes imply that third- and higher-order contributions are negligibly small for relatively large contrast ratios ($\varepsilon _2/\varepsilon _1 \lesssim 10$), i.e., $\sum _{n=4}^\infty (\varepsilon _p-\varepsilon _q)^nA_n \approx 0$. Therefore, the localization length associated with only possibly negligibly small higher-order contributions should be very large compared to any practically large sample size, and thus, there can be no Anderson localization in practice within the predicted perfect transparency interval in 1D and 2D stealthy hyperuniform media. Indeed, for both layered and transversely isotropic stealthy hyperuniform media, our prediction of the finite perfect transparency interval [Eq. (26)] is even stronger, since we have shown they are exact through third-order terms. The prediction for layered media is remarkable because the traditional understanding is that localization must occur for any type of disorder in one dimension [2,14,97–99].

We now show how the size $K_T$ of perfect transparency interval given in Eq. (26) varies with a contrast ratio $\varepsilon _p/\varepsilon _q$ and the volume fraction $\phi _p$ of phase with the higher dielectric constant. In Fig. 2(b), we see such dependence of the size $K_T$ for layered and transversely isotropic stealthy hyperuniform media, given by Eq. (26) in conjunction with Eq. (27) or, equivalently, Eq. (30). The size $K_T$ decreases with the contrast ratio $\varepsilon _p/\varepsilon _q$ because a higher contrast ratio strengthens scattering. Similarly, as $\phi _p$ increases without changing the exclusion-region radius $K$, $K_T$ also decreases because $\phi _p$ is proportional to the density of scatterers or the frequency of scattering events. One can also observe such a tendency that $K_T$ decreases with $\varepsilon _p/\varepsilon _q$ and $\phi _p$ in TE polarization in transversely isotropic media and 3D fully isotropic media because $\varepsilon _*$ given Eqs. (27)–(30 commonly increase with $\varepsilon _p/\varepsilon _q$. For fixed values of $\phi _p$, $\varepsilon _p/\varepsilon _q$, and $K$, the size $K_T$ is the smallest for TM polarization in transversely isotropic media (and layered media), followed by 3D fully isotropic ones, and is the largest for TE polarization in transversely isotropic media. This ranking is also inversely related to the strength of reflectance (i.e., scattering) on an interface between two dielectric materials depending on polarizations, governed by the Fresnel equations [79]. For example, in the transversely isotropic media, TM-polarized waves are always reflected to a greater degree than TE-polarized waves. We note that Eq. (26) is no longer valid at three trivial limits (i.e., $\varepsilon _p/\varepsilon _q=1$, $\phi _p=0$, and $\phi _p=1$) since the medium becomes homogeneous and perfectly transparent at any wave number $k_q$.

5. Model microstructures

We consider two models of 1D and 2D disordered hyperuniform media that enable us to generate 3D model microstructures of layered and transversely isotropic media, respectively. For each symmetry, these models include nonstealthy hyperuniform polydisperse sphere packings in a matrix and stealthy hyperuniform sphere packings in a matrix. Since these models are particulate composites, we henceforth take the characteristic inhomogeneity length scale $\xi$ to be the mean particle separation $\rho ^{-1/d}$, which is of the order of the mean nearest-neighbor distance $\ell _P$, (i.e., $\xi = \rho ^{-1/d} \sim \ell _P$). The particles of dielectric constant $\varepsilon _2$ are distributed throughout a matrix of dielectric constant $\varepsilon _1$. We take the particle volume fraction $\phi _2 =0.2, 0.25$, respectively, in one and two dimensions.

We numerically generate the configurations of these models (see Sec. S6 of Supplement 1). Figure 3 shows representative images of these configurations of all 1D and 2D models considered here. For all models, we compute the spectral density (see Fig. 4), as described in section 5.3, and then use them to evaluate their nonlocal attenuation functions (see Fig. 5). We employ the configurations of hyperuniform models in FDTD simulations to extract their effective dynamic dielectric constants (see section 6).

 

Fig. 3. Top row shows representative images of configurations of two models of layered media with $\phi _2=0.2$: nonstealthy hyperuniform polydisperse sphere packings in a matrix (a) and stealthy hyperuniform sphere packings in a matrix (b). Bottom row shows representative images of configurations of two models of transversely isotropic media with $\phi _2=0.25$: nonstealthy hyperuniform polydisperse sphere packings in a matrix (c) and stealthy hyperuniform sphere packings in a matrix (d).

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5.1 Nonstealthy Hyperuniform Polydisperse Sphere Packings in a Matrix

Nonstealthy hyperuniform (NSHU) packings of spheres with a polydispersity in size in a matrix can be constructed from nonhyperuniform progenitor point patterns via a tessellation-based procedure [40,100]. Specifically, the progenitor point patterns are the centers of 1D and 2D equilibrium packings of identical spheres with packing fraction $\phi _b$. One begins with the Voronoi tessellation [81] of these progenitor point patterns. We then move the particle center in a Voronoi cell to its centroid and rescale the particle such that the packing fraction inside this cell is identical to a prescribed value $\phi _2<1$. The same process is repeated over all cells. The final packing fraction is $\phi _2 = \sum _{j=1}^N \mathinner {v_1\mathopen {\left (a_j\right )}}/V_\mathfrak {F} = \rho \mathinner {v_1\mathopen {\left (a\right )}}$, where $\rho$ is the number density of particle centers and $a$ represents the mean sphere radius. In the thermodynamic limit, the spectral densities of the resulting NSHU media exhibit a power-law scaling $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({{\mathbf{k}}}\right )}} \sim |{{\mathbf{k}}}|^4$ for small wave numbers [100], and thus these systems are of class I.

 

Fig. 4. Semi-log plots of spectral densities $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({k}\right )}}$ as functions of dimensionless wave number for the two hyperuniform models in (a) one and (b) two dimensions: NSHU and SHU models. The particle-phase volume fractions are $\phi _2=0.2$ in (a) and $\phi _2=0.25$ in (b).

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5.2 Stealthy hyperuniform sphere packings in a matrix

Stealthy hyperuniform (SHU) two-phase media have $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({{\mathbf{k}}}\right )}}=0$ for the finite range $0<|{{\mathbf{k}}}|\leq K$, called the exclusion region. We specifically consider $d$-dimensional SHU sphere packings in a matrix with packing fraction $\phi _2$. For such SHU media, the degree of stealthiness $\chi$ is measured by the ratio of the number of the wave vectors within the exclusion region in the Fourier space to the total degrees of freedom, i.e., $\chi = K / (2\pi \rho )$ in one dimension, and $\chi =K^2 / (16\pi \rho )$ in two dimensions. These SHU systems are highly degenerate and disordered if $\chi <1/3$ in one dimension or $\chi <1/2$ in two and three dimensions [53]. Thus, we consider 1D and 2D cases for $\chi =0.3$ in the present work.

We numerically generate such 1D and 2D SHU media in the following two-step procedure. First, we generate point configurations consisting of $N$ particles in a fundamental cell $\mathfrak {F}$ under periodic boundary conditions via the collective-coordinate optimization technique [51–53], which finds numerically the ground-state configurations of the following potential energy;

 

Fig. 5. Real (upper) and imaginary (lower) parts of the nonlocal attenuation functions versus dimensionless wave number for the two models in (a) one and (b) two dimensions: NSHU and SHU media. The packing fractions in (a) and (b) are $\phi _2=0.2$ and $0.25$, respectively. Values in (a) are computed from the spectral densities in Fig. 4(a) and Eq. (14). In (b), negative values of $\mathinner {{F^\mathrm {(2D)}}\mathopen {\left ({k}\right )}}$ are computed from the spectral densities in Fig. 4(b) and Eq. (21). In the lower panels, each inset is in a log-log scale, where the dashed lines with numbers depict the small-$k$ scalings for NSHU systems predicted in Eq. (17) and Eq. (25).

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5.3 Spectral densities for the hyperuniform models

Here, we plot the radial spectral density $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({k}\right )}}$ of 1D and 2D hyperuniform models; see Fig. 4. We take the particle-phase volume fraction $\phi _2=0.2$ and $\phi _2=0.25$ for 1D and 2D cases, respectively. For all models, we numerically obtain $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({k}\right )}}$ by using Eq. (6) from numerically generated configurations.

Across all length scales, the spectral densities of these two models are considerably different from one another. In the long-wavelength regime ($k/\rho ^{1/d} \ll 1$), SHU media suppress volume-fraction fluctuations [i.e., $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({k}\right )}}=0$] to a greater degree than the NSHU media over a wider range of wavelengths. In the small-wavelength regime ($k/\rho ^{1/d} \gg 10$), the oscillations of $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({k}\right )}}$ reflect the particle-size distribution. Specifically, the spectral densities of the SHU model exhibit strong oscillations because this model consists of particles of identical size, whereas those of the NSHU model exhibit weaker oscillations because of a broad disparity in particle sizes.

6. Results

We now show how our multiple-scattering approximations for layered media [Eq. (12)] and transversely isotropic media [Eq. (18)] enable us to accurately predict the real and imaginary parts of the effective dynamic dielectric constants for anisotropic disordered two-phase media. For simplicity, we consider the second-order approximations by neglecting the third-order terms $A_3^\perp$ and $A_3^{TM}$ in Eq. (12) and Eq. (18), respectively. These formulas depend on the nonlocal attenuation functions that incorporate microstructural information via the spectral density $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({k}\right )}}$ only. We begin by computing the nonlocal attenuation functions $\mathinner {{F^\mathrm {(1D)}}\mathopen {\left ({k}\right )}}$ [given in Eq. (14)] and $\mathinner {{F^\mathrm {(2D)}}\mathopen {\left ({k}\right )}}$ [given in Eq. (21)] of the two hyperuniform models from $\mathinner {{\tilde {\chi }_{_V}}\mathopen {\left ({k}\right )}}$; see Fig. 5. For SHU media, the imaginary parts $\textrm{Im} [\mathinner {{F^\mathrm {(1D)}}\mathopen {\left ({k}\right )}}]$ and $\textrm{Im} [\mathinner {{F^\mathrm {(2D)}}\mathopen {\left ({k}\right )}}]$ are exactly zero up to a finite wave number $k$, leading to the prediction of perfect transparency interval; see Eq. (26). By contrast, for the NSHU media, the imaginary parts are close but not equal to zero for small wave numbers, as shown in the insets of Fig. 5.

To validate the accuracy of our two second-order approximations, we show that the predicted effective dielectric constants are in excellent agreement with those extracted from FDTD simulations. We make such comparisons in Fig. 6 for the disordered SHU media. (We present analogous comparisons for the NSHU media in Sec. S7 of Supplement 1.) This model can provide stringent tests of the predictive power of the approximations at finite wave numbers because they are characterized by nontrivial spatial correlations at intermediate length scales. For both layered media and transversely isotropic ones, the real and imaginary parts of these predictions show excellent agreement with the results from FDTD simulations up to $k_1 \rho ^{-1/d} \lesssim 1.5$, which is well beyond the long-wavelength regime. It is noteworthy that our second-order approximation formulas [Eq. (12) and Eq. (18)] can be used to accurately estimate the effective dielectric response for higher contrast ratios (e.g., $9\lesssim \varepsilon _2/\varepsilon _1\lesssim 12$); see, for example, the predicted transparency interval as a function of contrast ratio $\varepsilon _2/\varepsilon _1$ as shown in Fig. 2(b). For the imaginary parts, both approximations and simulations consistently show that $\textrm{Im} [\mathinner {\varepsilon _e\mathopen {\left (k_1\right )}}]=0$ within the finite perfect transparency intervals predicted in Eq. (26). We also observe that the real parts $ \operatorname{Re}\left[\varepsilon_e\left(k_1\right)\right] $ increase with $k_1$ (i.e., normal dispersion) within the perfect transparency intervals and then decrease with $k_1$ (i.e., anomalous dispersion) outside those intervals. Such a spectral dependence of the real parts comes from the fact that our two second-order strong-contrast formulas satisfy the Kramers-Kronig relations [79], as shown in Refs. [75,78]. We analytically and numerically show that our second-order formulas for both layered and transversely isotropic media meet the Kramers-Kronig relations in Sec. S8 of Supplement 1.

 

Fig. 6. Comparison of the predictions of the second-order strong-contrast approximations for the effective dynamic dielectric constant as a function of the dimensionless wave number for SHU systems with various symmetries to FDTD simulations. We consider (a) layered media with $\phi _2=0.20$, $\chi =0.30$, and $\varepsilon _2/\varepsilon _1=4$, and (b) transversely isotropic media with $\phi _2=0.25$, $\chi =0.30$, and $\varepsilon _2/\varepsilon _1=4$. Values of approximations in (a) and (b) are computed from Eq. (12) and Eq. (18) with $A_3(k)=0$, respectively.

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Having established the accuracy of the two second-order strong-contrast approximations [Eq. (12) and Eq. (18)] for anisotropic disordered media in Fig. 6 and Fig. S3 of Supplement 1, we apply them to estimate $\mathinner {\varepsilon _e\mathopen {\left (k_q\right )}}$ for models of disordered hyperuniform layered and transversely isotropic media: the NSHU and SHU models Fig. 7 shows the predictions at a fixed contrast ratio $\varepsilon _2/\varepsilon _1=4$. For these two disordered models of layered media and transversely isotropic media, the real parts of the effective dielectric constants $\mathrm{Re} [\varepsilon _e]$, associated with the effective wave speed, similarly increase with $k_1$ for small wave numbers ($k_1\rho ^{-1/d}\lesssim 1.0$). We see that the imaginary parts of the effective dielectric constants have distinctly different small-wave number behaviors of the two models of both layered and transversely isotropic media. The NSHU media exhibit $\textrm{Im} [\varepsilon _e]\sim {k_1}^{4+d}$ in the limit $k_1\to 0$, as predicted in both Eq. (17) and Eq. (25). While the imaginary part of such a NSHU medium is expected to be much smaller than that of typical nonhyperuniform ones, it cannot be exactly zero at any wave number, unlike a SHU medium, as discussed in section 4.

 

Fig. 7. Predictions of the scaled second-order strong-contrast approximations for the effective dynamic dielectric constant as a function of the dimensionless incident wave number for the two hyperuniform models of (a) layered media and (b) transversely isotropic media with a contrast ratio $\varepsilon _2/\varepsilon _1 = 4$: SHU and NSHU media. We consider the packing fractions $\phi _2=0.2$ and $0.25$ in (a) and (b), respectively. Values in (a) and (b) are computed from Eq. (12) and Eq. (18), respectively, at the two-point level. In the lower panels, each inset is in a log-log scale, where the dashed lines with numbers depict the small-$k$ scaling for the NSHU systems, predicted in Eq. (17) and Eq. (25).

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7. Conclusion and discussion

We have demonstrated that the second- or third-order truncations of the rapidly converging strong-contrast expansion [75] of the effective dielectric constant tensor $\mathinner {\boldsymbol {\varepsilon }_e\mathopen {\left ({\mathbf{k}}_q, \omega \right )}}$ provide a powerful theoretical tool to extract accurate multiple-scattering approximations suited for various microstructural symmetries. This task was accomplished by judiciously tuning the tensor expansion parameter $\boldsymbol {L}_p^{(q)}$ (see Sec. S1 of Supplement 1), which is determined by choosing the exclusion volume shape with an appropriate symmetry around the singularity of the dyadic Green’s function [75,78,102]. We have shown that the third-order truncations of such series yield closed-form formulas applicable to layered media [Eq. (12) and Eq. (13)] and to transversely isotropic media [Eq. (18) and Eq. (19)], respectively; see section 3. These formulas incorporate microstructural information through two- and three-point correlation functions. Including the formulas for 3D fully isotropic composites derived in Ref. [75], we have obtained a family of multiple-scattering approximations applicable to three distinct types of microstructural symmetries. In the present work, since Eq. (13) is purely static, and Eq. (19) is similar to what was studied in Ref. [75], we focused on two key formulas Eq. (12) and Eq. (18), which are for transverse polarization in layered media and TM polarization in transversely isotropic media, respectively.

Using these formulas, we proved that the finite perfect transparency interval of SHU media, predicted in Eq. (26), is exact through the third-order terms for both transversely isotropic media (TM case) and layered media. The latter result further validates the accuracy of our previous prediction from the second-order approximation in Ref. [78]; see Eq. (2). The rapid convergence properties of strong-contrast expansions (see Sec. S5 of Supplement 1) and the high predictive power of their second-order truncations validated via FDTD simulations imply that higher-order contributions are negligibly small. This trivially implies that, in principle, there can be no Anderson localization within the predicted perfect transparency interval in SHU media. In practice, this means that the localization length, associated with only possibly negligibly small higher-order contributions, should be very large compared to any practically large sample size. Indeed, for SHU layered and transversely isotropic media, our predictions of the perfect transparency interval Eq. (26) are even stronger, since we have shown they are exact through third-order terms, and thus, there can be no localization rigorously through the third-order terms in the thermodynamic limit when the contrast ratio is sufficiently small. As noted in Ref. [78], this prediction is remarkable because the traditional understanding is that localization must occur for any type of disorder in one dimension [2,14,97–99]. In future work, we will systematically study the localization length of 1D SHU media [103].

We have applied our formulas to estimate $\mathinner {\varepsilon _e\mathopen {\left (k_q\right )}}$ for two models of disordered hyperuniform layered and transversely isotropic media: NSHU and SHU models. For all hyperuniform models considered here, we corroborated that the second-order formulas for layered media [Eq. (12), where $A_3^\perp =0$] and transversely isotropic media [Eq. (18), where $A_3^{TM}=0$] are already very accurate well beyond the long-wavelength regime (i.e., $k_1 \rho ^{-1/d} \lesssim 1.5$) by showing excellent agreement with FDTD simulations; see section 6 and Sec. S7 of Supplement 1. The real parts $\mathrm{Re} [\varepsilon _e]$ of the effective dielectric constants exhibit similar behavior (i.e., $\mathrm{Re} [\varepsilon _e]$ increases with $k_1$) in the long-wavelength regime, implying the effective wave speed is largely insensitive to microstructures; see Fig. 7. However, our predictions reveal that the two models exhibit distinctly different effective attenuation behaviors; see the insets of Fig. 7. While NSHU models do not have a finite perfect transparency interval, they can exhibit a wide range of attenuation behaviors by tuning the exponent $\alpha$ in a power-law scaling of the spectral density, as predicted in Eq. (17) and Eq. (25).

Our findings also have important practical implications. For example, combining our accurate second-order formulas depending solely on the spectral density with the methods to construct two-phase media with a prescribed spectral density [10,24,51–53] enables one to take an inverse-design approach [80] to engineering anisotropic dielectric materials with novel wave properties. Notably, such computationally designed composite media can be readily fabricated via vacuum deposition [104], spin-coating [105], 2D photolithographic [106], and 3D printing techniques [107]. As demonstrated in Ref. [55], such anisotropic properties can be employed to design broadband polarizers that transmit waves up to selected wave numbers, depending on the polarization. Thus, our results offer promising prospects for engineering novel optoelectronic materials and devices across space dimensions by engineering the spectral density of the systems [10].