Nonlocal Effective Electromagnetic Wave Characteristics of Composite Media: Beyond the Quasistatic Regime

PHYSICAL REVIEW X 11, 021002 (2021)

Salvatore Torquato * Department of Physics, Princeton University, Princeton, New Jersey 08544, USA; Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA; Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA; and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA

Jaeuk Kim Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

(Received 3 July 2020; revised 29 November 2020; accepted 21 January 2021; published 2 April 2021)

We derive exact nonlocal homogenized constitutive relations for the effective electromagnetic wave properties of disordered two-phase composites and metamaterials from first principles. This exact formalism enables us to extend the long-wavelength limitations of conventional homogenization estimates of the effective dynamic dielectric constant tensor εeðkI; ωÞ for arbitrary microstructures so that it can capture spatial dispersion well beyond the quasistatic regime (where ω and kI are the frequency and wave vector of the incident radiation). We accomplish this task by deriving nonlocal strong-contrast expansions that exactly account for complete microstructural information (infinite set of n-point correlation functions) and hence multiple scattering to all orders for the range of wave numbers for which our extended homogenization theory applies, i.e., 0 ≤ jkIjl ≲ 1 (where l is a characteristic heterogeneity length scale). Because of the fast-convergence properties of such expansions, their lower-order truncations yield accurate closed-form approximate formulas for εeðkI; ωÞ that apply for a wide class of microstructures. These nonlocal formulas are resummed representations of the strong-contrast expansions that still accurately capture multiple scattering to all orders via the microstructural information embodied in the spectral density, which is easy to compute for any composite. The accuracy of these microstructure-dependent approximations is validated by comparison to full-waveform simulation computations for both 2D and 3D ordered and disordered models of composite media. Thus, our closed-form formulas enable one to predict accurately and efficiently the effective wave characteristics well beyond the quasistatic regime for a wide class of composite microstructures without having to perform full-blown simulations. We find that disordered hyperuniform media are generally less lossy than their nonhyperuniform counterparts. We also show that certain disordered hyperuniform particulate composites exhibit novel wave characteristics, including the capacity to act as low-pass filters that transmit waves “isotropically” up to a selected wave number and refractive indices that abruptly change over a narrow range of wave numbers. Our results demonstrate that one can design the effective wave characteristics of a disordered composite by engineering the microstructure to possess tailored spatial correlations at prescribed length scales. Thus, our findings can accelerate the discovery of novel electromagnetic composites.

DOI: 10.1103/PhysRevX.11.021002 Subject Areas: Metamaterials, Soft Matter

I. INTRODUCTION one and dates back to work by some of the luminaries of science, including Maxwell [1], Lord Rayleigh [2], and The theoretical problem of estimating the effective Einstein [3]. The preponderance of previous theoretical properties of multiphase composite media is an outstanding studies have focused on the determination of static effective properties (e.g., dielectric constant, elastic moduli, and *[email protected]; http://torquato.princeton.edu fluid permeability) using a variety of methods, including approximation schemes [1,4–6], bounding techniques Published by the American Physical Society under the terms of [7–12], and exact series-expansion procedures [13–16]. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to The latter set of investigations teaches us that an exact the author(s) and the published article’s title, journal citation, determination of an effective property, given the phase and DOI. properties of the composite, generally requires an infinite

2160-3308=21=11(2)=021002(26) 021002-1 Published by the American Physical Society SALVATORE TORQUATO and JAEUK KIM PHYS. REV. X 11, 021002 (2021)

FIG. 1. Left panel: The preponderance of previous theoretical treatments of the effective dynamic dielectric constant are local in nature and restricted to dispersions of well-defined dielectric scatterers (inclusions) in a matrix, as illustrated in the leftmost panel. In contrast to the nonlocal strong-contrast formalism presented here, earlier studies cannot treat the more general two-phase microstructures shown in the middle panel (spinodal decomposition pattern [28]) and rightmost panel (Debye random medium [29,30]), both of which have “phase-inversion” symmetry [11]. Our formalism can, in principle, treat two-phase media of arbitrary microstructures. set of correlation functions that characterizes the composite expansion are explicitly given in terms of integrals over microstructure. products of Green’s functions and the n-point correlation Our focus in this paper is the determination of the effective functions of the random two-phase medium (defined in dynamic dielectric constant tensor εeðkI; ωÞ of a two-phase Sec. II A) to infinite order. This representation exactly treats dielectric composite, which depends on the frequency ω or multiple scattering to all orders for the range of wave wave vector kI of the incident radiation [17,18]. From this numbers for which our extended homogenization theory effective property, one can determine the corresponding applies, i.e., 0 ≤ jkIjl ≲ 1. effective wave speed ce and attenuation coefficient γe. It is noteworthy that the strong-contrast formalism is a The preponderance of previous homogenization theories significant departure from standard multiple-scattering of εeðkI; ωÞ apply only in the quasistatic or long-wavelength theory [17,20,33,34], as highlighted in Sec. III. Moreover, regime [19–23], i.e., applicablewhen jkIjl ≪ 1, where l is a as we show there, our strong-contrast formalism has a variety characteristic heterogeneity length scale. This spectral range of “tuning knobs” that enable one to obtain distinctly is the realm of nonresonant dielectric behavior. Virtually all different expansions and approximations designed for earlier formulas εeðkI; ωÞ apply to very special micro- different classes of microstructures. structures, namely, dielectric scatterers that are well-defined Because of the fast-convergence properties of strong- inclusions, e.g., nonoverlapping spheres in a matrix (see contrast expansions elaborated in Sec. III B,theirlower- Fig. 1). Examples of such popular closed-form approxima- order truncations yield accurate closed-form approximate tion formulas devised for spherical scatterers in a matrix formulas for the effective dielectric constant that apply for a include the Maxwell-Garnett [24,25] and quasicrystalline wide class of microstructures over the aforementioned broad [18,26,27] estimates, among others [17]. range of incident wavelengths, volume fractions, and con- In the present investigation, we derive nonlocal homog- trast ratios (Sec. IV). Thus, we are able to accurately account enized constitutive relations from Maxwell’s equations to for multiple scattering in the resonant realm (e.g., Bragg obtain exact expressions for the effective dynamic dielectric diffraction for periodic media), in contrast to the Maxwell- constant tensor εeðkI; ωÞ of a macroscopically anisotropic Garnett and quasicrystalline approximations, which are two-phase medium of arbitrary microstructure that is valid known to break down in this spectral range. These nonlocal well beyond the quasistatic regime, i.e., from the infinite- strong-contrast formulas can be regarded as approximate wavelength limit down to intermediate wavelengths resummations of the expansions that still accurately capture (0 ≤ jkIjl ≲ 1). This task is accomplished by extending multiple-scattering effects to all orders via the nonlocal the strong-contrast expansion formalism, which has been attenuation function FðQÞ (Sec. VI). The key quantity Q χ˜ Q used in the past exclusively for the static limit [11] and Fð Þ is a functional of the spectral density V ð Þ quasistatic regime [31], and establishing that the resulting (Sec. II B), which is straightforward to determine for general homogenized constitutive relations are nonlocal in space microstructures either theoretically, computationally, or via (Sec. III); i.e., the average polarization field at position x scattering experiments. We employ precise full-waveform depends on the average electric field at other positions simulation methods (Sec. VII) to show that these micro- around x. (Such nonlocal relations are well known in the structure-dependent approximations are accurate for both context of crystal optics in order to account for “spatial two-dimensional (2D) and three-dimensional (3D) ordered dispersion,” i.e., the dependence of dielectric properties on and disordered models of particulate composite media a wave vector [32].) The terms of the strong-contrast (Sec. VIII). This validation means that they can be used to

021002-2 NONLOCAL EFFECTIVE ELECTROMAGNETIC WAVE … PHYS. REV. X 11, 021002 (2021) predict accurately the effective wave characteristics well findings, we show that disordered hyperuniform media are beyond the quasistatic regime for a wide class of composite generally less lossy than their nonhyperuniform counter- microstructures (Sec. IX) without having to perform full- parts. We also demonstrate that disordered stealthy hyper- blown simulations. This broad microstructure class includes uniform particulate composites exhibit singular wave particulate media consisting of identical or polydisperse characteristics, including the capacity to act as low-pass particles of general shape (ellipsoids, cubes, cylinders, filters that transmit waves “isotropically” up to a selected polyhedra) that may or not overlap, and cellular networks wave number. They also can be engineered to exhibit as well as media without well-defined inclusions (see Sec. III refractive indices that abruptly change over a narrow range B for details). Thus, our nonlocal formulas can be employed of wave numbers by tuning the spectral density. Our results to accelerate the discovery of novel electromagnetic com- demonstrate that one can design the effective wave char- posites by appropriate tailoring of the spectral densities acteristics of a disordered composite, hyperuniform or not, [35,36] and then generating the microstructures that achieve by engineering the microstructure to possess tailored spatial them [36]. correlations at prescribed length scales. Although our strong-contrast formulas for the effective dynamic dielectric constant apply to periodic two-phase II. BACKGROUND media, the primary applications that we have in mind are A. n-point correlation functions correlated disordered microstructures because they can provide advantages over periodic ones with high crystallo- A two-phase random medium is a domain of space graphic symmetries [37,38] which include perfect isotropy V ⊆ Rd that is partitioned into two disjoint regions that and robustness against defects [39,40]. We are interested in make up V: a phase 1 region V1 of volume fraction ϕ1 and a both “garden-variety” models of disordered two-phase phase 2 region V2 of volume fraction ϕ2 [11]. The phase media [11] as well as exotic hyperuniform forms [41–43]. indicator function IðiÞðxÞ of phase i for a given realization Hyperuniform two-phase systems are characterized by an is defined as anomalous suppression of volume-fraction fluctuations in – 1; x ∈ Vi; the infinite-wavelength limit [41 43], i.e., the spectral IðiÞðxÞ¼ ð2Þ χ˜ Q 0 x ∉ V density V ð Þ obeys the condition ; i: χ˜ Q 0 ðiÞ lim V ð Þ¼ : ð1Þ The n-point correlation function Sn for phase i is defined jQj→0 by [11,74] Such two-phase media encompass all periodic systems, Yn ðiÞ x … x I ðiÞ x many quasiperiodic media, and exotic disordered ones; see Sn ð 1; ; nÞ¼ ð jÞ ; ð3Þ Ref. [43] and references therein. Disordered hyperuniform j¼1 systems lie between liquids and crystals; they are like where angular brackets denote an ensemble average over liquids in that they are statistically isotropic without any ðiÞ x … x Bragg peaks, and yet behave like crystals in the manner in realizations. The function Sn ð 1; ; nÞ has a probabilistic interpretation: It gives the probability of finding the which they suppress the large-scale density fluctuations x … x [41–43]. Hyperuniform systems have attracted great atten- ends of the vectors 1; ; n all in phase i. For statistically ðiÞ x … x tion over the last decade because of their deep connections homogeneous media, Sn ð 1; ; nÞ is translationally to a wide range of topics that arise in physics [28,37, invariant and, in particular, the one-point function is – – ðiÞ 43 53], materials science [36,54 57], mathematics position independent, i.e., S1 ðx1Þ¼ϕi. [58–60], and biology [43,61] as well as for their emerging technological importance in the case of the disordered B. Two-point statistics varieties [40,43,52,54,62–66]. For statistically homogeneous media, the two-point We apply our strong-contrast formulas to predict the real correlation function for phase 2 is simply related to that and imaginary parts of the effective dielectric constant for ð2Þ r ð1Þ r − 2ϕ 1 model microstructures with typical disorder (nonhyperuni- for phase 1 via the expression S2 ð Þ¼S2 ð Þ 1 þ , form) as well as those with exotic hyperuniform disorder and hence, the autocovariance function is given by (Sec. V). We are particularly interested in exploring the χ r ≡ ð1Þ r − ϕ 2 ð2Þ r − ϕ 2 dielectric properties of a special class of hyperuniform V ð Þ S2 ð Þ 1 ¼ S2 ð Þ 2 ; ð4Þ composites called disordered stealthy hyperuniform media, which are defined to be those that possess zero-scattering which we see is the same for phase 1 and phase 2. Thus, χ r 0 ϕ ϕ intensity for a set of wave vectors around the origin V ð ¼ Þ¼ 1 2 and, assuming the medium possesses no χ r 0 [36,44,67–69]. Such materials have recently been shown long-range order, limjrj→∞ V ð Þ¼ . For statistically χ r to be endowed with novel optical, acoustic, mechanical, homogeneous and isotropic media, V ð Þ depends only and transport properties [31,55,56,63,70–73]. Among other on the magnitude of its argument r ¼jrj, and hence is a

021002-3 SALVATORE TORQUATO and JAEUK KIM PHYS. REV. X 11, 021002 (2021) radial function. In such cases, its slope at the origin is πd=2ad v1ðaÞ¼ ð13Þ directly related to the specific surface s (interface area per Γð1 þ d=2Þ unit volume); i.e., asymptotically, we have [11] is the d-dimensional volume of a sphere of radius a. χ r ϕ ϕ − β r O r 2 V ð Þ¼ 1 2 ðdÞsj jþ ðj j Þ; ð5Þ D. Hyperuniformity and volume-fraction fluctuations where Originally introduced in the context of point configura- Γðd=2Þ tions [41], the hyperuniformity concept was generalized to βðdÞ¼ pffiffiffi ; ð6Þ treat two-phase media [42]. Here the phase volume fraction 2 πΓ(ðd þ 1Þ=2) fluctuates within a spherical window of radius R, which can and ΓðxÞ is the gamma function. be characterized by the local volume-fraction variance σ2 The nonnegative spectral density χ˜ ðQÞ, which is propor- VðRÞ. This variance is directly related to integrals involv- V χ r tional to scattering intensity [75], is the Fourier transform of ing either the autocovariance function V ð Þ [78] or the χ˜ Q χ r spectral density V ð Þ [42]. V ð Þ, i.e., Z For typical disordered two-phase media, the variance σ2 −d − Q r VðRÞ for large R goes to zero like R [11,78,79]. χ˜ Q χ r i · r ≥ 0 Q 2 V ð Þ¼ V ð Þe d ; for all ; ð7Þ σ Rd However, for hyperuniform disordered media, VðRÞ goes to zero asymptotically more rapidly than the inverse of the where Q represents the momentum-transfer wave vector. window volume, i.e., faster than R−d, which is equivalent to For statistically homogeneous media, the spectral density the condition (1) on the spectral density. Stealthy hyper- must obey the following sum rule [76]: uniform two-phase media are a subclass of hyperuniform χ˜ Q Z systems in which V ð Þ is zero for a range of wave vectors 1 χ˜ Q Q χ r 0 ϕ ϕ around the origin, i.e., d V ð Þd ¼ V ð ¼ Þ¼ 1 2: ð8Þ ð2πÞ Rd χ˜ Q 0 0 ≤ Q ≤ V ð Þ¼ for j j QU; ð14Þ For isotropic media, the spectral density depends only on Q ¼jQj and, as a consequence of Eq. (5), its large-k where QU is some positive number. behavior is controlled by the following power-law form: As in the case of hyperuniform point configurations [41–43,80], there are three different scaling regimes γðdÞs (classes) that describe the associated large-R behaviors χ˜ ðQÞ ∼ ;Q→ ∞; ð9Þ V Qdþ1 of the volume-fraction variance when the spectral density χ˜ Q ∼ Q α Q → 0 goes to zero as a power law V ð Þ j j as j j : where 8 − 1 −1 2 > ðdþ Þ α 1 γðdÞ¼2dπðd Þ= Γ(ðd þ 1Þ=2)βðdÞð10Þ < R ; > ðClass IÞ; σ2 ∼ −ðdþ1Þ α 1 VðRÞ > R ln R; ¼ ðClass IIÞ; ð15Þ is a d-dimensional constant and βðdÞ is given by Eq. (6). :> R−ðdþαÞ; 0 < α < 1 ðClass IIIÞ;

C. Packings where the exponent α is a positive constant. Thus, the We call a packing in Rd a collection of nonoverlapping characteristics length of the representative elementary particles [77]. In the case of a packing of identical spheres volume for a hyperuniform medium will depend on the ρ χ˜ Q of radius a at number density , the spectral density V ð Þ hyperuniformity class (scaling). Class I is the strongest is directly related to the structure factor SðQÞ of the sphere form of hyperuniformity, which includes all perfect peri- centers [11,35] odic packings as well as some disordered packings, such as disordered stealthy packings described in Sec. VD. χ˜ Q ϕ α˜ ; Q V ð Þ¼ 2 2ðQ aÞSð Þ; ð11Þ E. Popular effective-medium approximations where Here we explicitly state the specific functional forms of 1 2π d an extended Maxwell-Garnett approximation and quasi- α˜ ; a 2 2ðQ aÞ¼ Jd=2ðQaÞ; ð12Þ v1ðaÞ Q crystalline approximation for the effective dynamic dielectric constant εeðk1Þ of isotropic media composed of JνðxÞ is the Bessel function of the first kind of order ν, identical spheres of dielectric constant ε2 embedded in a ϕ2 ¼ ρv1ðaÞ is the packing fraction (fraction of space matrix phase of dielectric constant ε1. In Sec. VIII,we covered by the spheres), and compare the predictions of these formulas to those of our

021002-4 NONLOCAL EFFECTIVE ELECTROMAGNETIC WAVE … PHYS. REV. X 11, 021002 (2021) nonlocal approximations. The small-wave-number expan- 2. Quasicrystalline approximations sions of these popular approximations are provided in the The quasicrystalline approximation (QCA) for the Supplemental Material [81]. There we also provide the quasistatic effective dynamic dielectric constant εeðk1Þ corresponding asymptotic behaviors of our strong-contrast employs the “effective” Green’s function of spherical approximations. scatterers up to the level of the pair correlation function g2ðrÞ [18,26,27]. However, the QCA accounts only for the 1. Maxwell-Garnett approximation structure factorR in the infinite-wavelength limit [27] [i.e., S 0 1 ρ g2 r − 1 dr], and consequently, spatial Maxwell-Garnett approximations (MGAs) [24,25] are ð Þ¼ þ ð ð Þ Þ correlations at finite wavelengths are ignored. The QCA derived by substituting the dielectric polarizability of a 3 single dielectric sphere into the Clausius-Mossotti equation for d ¼ can be explicitly written as follows [18]: [25], which consequently ignores the spatial correlations −1 2 εeðk1Þ − ε1 between the particles. In three dimensions, we utilize the ϕ2 β21 ε ðk1Þþ2ε1 following extended MGA that makes use of the exact e 2 electric dipole polarizability αeðk1Þ of a single dielectric 3 ϕ2 − i ϕ2S 0 k1a sphere of radius a [82]: ¼ 3 ð Þð Þ −1 2 3 ε − ε × 1 i k1a S 0 β21; 18 eðk1Þ 1 3 þ 3 1 − β ϕ ð Þ ð Þ ð Þ ¼ ϕ2αeðk1Þ=a ; ð16Þ ð 21 2Þ εeðk1Þþ2ε1 where β21 is defined in Eq. (33). Interestingly, the QCA where predicts that the hyperuniform composites [Sð0Þ¼0] will be transparent for all wave numbers, which cannot be true 3 ψ ψ 0 − ψ ψ 0 for stealthy hyperuniform media [cf. Eq. (14)], since the α i m 1ðmk1aÞ 1ðk1aÞ 1ðk1aÞ 1ðmk1aÞ eðk1Þ¼ 3 0 0 ; transparency interval must be finite; see Sec. VIII. Note that 2k1 mψ 1ðmk1aÞξ1ðk1aÞ − ξ1ðk1aÞψ 1ðmk1aÞ Eq. (18) is the complex conjugate of the one given in pffiffiffiffiffiffiffiffiffiffiffi ð1Þ Ref. [18] so that it is consistent with the sign convention for m ≡ ε2=ε1, ψ 1ðxÞ ≡ xj1ðxÞ, ξ1ðxÞ ≡ xh1 ðxÞ, the prime the imaginary part Im½εe used here. 0 ð1Þ symbol ð Þ denotes the derivative of a function, h1 ðxÞ is the spherical Hankel function of the first kind of order 1, III. EXACT STRONG-CONTRAST and j1ðxÞ is the spherical Bessel function of the first kind of EXPANSIONS AND NONLOCALITY order 1. The 2D analog of Eq. (16) can be obtained by using the The original strong-contrast expansions for the effective dynamic dielectric polarizability αe of a dielectric cylinder dynamic dielectric constant obtained by Rechtsman and of radius a given in Ref. [83]: Torquato [31] were derived from homogenized constitutive relations that are local in space and hence are strictly valid only in the quasistatic regime. In the present work, we follow εeðk1Þ − ε1 ϕ2 2 ¼ αeðk1Þ=a ; ð17Þ the general strong-contrast formalism of Torquato [11] that ε ðk1Þþε1 2π e was devised for the purely static problem [11] and show that it naturally leads to exact homogenized constitutive where relations for the averaged fields that are nonlocal in space. The crucial consequences of this development are exact 4 ε − ε α ð 2 1Þ 0 Hð1Þ expressions for the effective dynamic dielectric constant eðk1Þ¼ 2 J1ðmk1aÞ½J1ðmk1aÞ 1 ðk1aÞ ik1 mε1 tensor εeðkIÞ for a macroscopically anisotropic medium of ð1Þ0 −1 arbitrary microstructure into which a plane wave of wave − mJ1ðmk1aÞH1 ðk1aÞ ; vector kI is incident. These expressions for εeðkIÞ are valid from the infinite-wavelength limit down to wavelengths Hð1Þ where ν ðxÞ is the Hankel function of the first kind of (λ ¼ 2π=jkIj) on the order of the heterogeneity length order ν. Here, only transverse-electric (TE) polarization is scale l. We then briefly explain how our theory departs considered; i.e., the electric field is perpendicular to the axis substantially from standard multiple-scattering theory of the cylinder. [17,20,33,34]. We explicitly show they necessarily require As with all MGA theories, formulas (16) and (17) complete microstructural information, as embodied in the neglect spatial correlations between the particles and infinite set of n-point correlation functions (Sec. II A)ofthe hence are only valid for low inclusion packing fractions. composite. We also describe the fast-convergence properties In the static limit, Eqs. (16) and (17) reduce to the Hashin- of strong-contrast expansions and their consequences for ε ε k Shtrikman estimates HS; see relation (72). extracting accurate approximations for eð IÞ.

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A. Strong-contrast expansions is the local dielectric constant, and IðpÞðxÞ is the indicator Consider a macroscopically large ellipsoidal two-phase function for phase p [cf. Eq. (2)]. The vector PðxÞ is the statistically homogeneous but anisotropic composite speci- induced flux field relative to reference phase q due to the men in Rd embedded inside an infinitely large reference presence of phase p, and hence is zero in the reference “ ” ≠ phase I with a dielectric constant tensor εI. The micro- phase q and nonzero in the polarized phase p (p q). structure is perfectly general, and it is assumed that a Using the Green’s function formalism, the local electric characteristic heterogeneity length scale l is much smaller field can be expressed in terms of the following integral than the specimen size, i.e., l ≪ L. The shape of this equation [11,31]: specimen is purposely chosen to be nonspherical since any Z rigorously correct expression for the effective property ðqÞ 0 0 0 EðxÞ¼E0ðxÞþ G ðx − x Þ · Pðx Þdx ; ð23Þ must ultimately be independent of the shape of the composite specimen in the infinite-volume limit. It is ’ ðqÞ r assumed that the applied or incident electric field E0ðxÞ where the second-rank-tensor Green s function G ð Þ is a plane wave of an angular frequency ω and wave vector associated with the reference phase q is given by [84] kI in the reference phase, i.e., GðqÞðrÞ¼−DðqÞδðrÞþHðqÞðrÞ; ð24Þ ˜ E0ðxÞ¼E0 exp (iðkI · x − ωtÞ): ð19Þ DðqÞ is a constant second-rank tensor that arises when one Our interest is the exact expression for the effective excludes an infinitesimal region around the position of the ε k ω dynamic dielectric constant tensor eð I; Þ. Without loss singularity x0 ¼ x in the Green’s function, and HðqÞðrÞ of generality, we assume a linear dispersionffiffiffiffi relation in the represents the contribution outside of the “exclusion” ≡ k pε ω reference phase, i.e., kI j Ij¼ I =c, where c is the region: speed of light in vacuum, and thus, we henceforth do not 2 explicitly indicate the dependence of functions on ω. π k d= 1 1 HðqÞðrÞ¼i q f½k rHð Þ ðk rÞ − Hð Þ ðk rÞδ The composite is assumed to be nonmagnetic, implying ij 2ε 2πr q d=2−1 q d=2 q ij that the phase magnetic permeabilities are identical, i.e., q μ μ μ μ Hð1Þ rˆ rˆ 1 ¼ 2 ¼ 0, where 0 is the magnetic permeability of the þ kqr d=2þ1ðkqrÞ i jg; ð25Þ vacuum. For simplicity, we assume real-valued, frequency- independent isotropic phase dielectric constants ε1 and ε2. ð1Þ where rˆ ≡ r= r is a unit vector directed to r, and Hν x is ε j j ð Þ Nonetheless, the composite can be generally lossy (i.e., e the Hankel function of the first kind of order ν. The Fourier is complex valued) due to scattering from the inhomoge- transform of Eq. (24) is particularly simple and concise: neities in the local dielectric constant. It is noteworthy that our results can be straightforwardly extended to phase 1 k 2δ − k k dielectric constants that are complex valued (dissipative G˜ ðqÞ ðkÞ¼ q ij i j : ð26Þ ij ε 2 − 2 media), but this is not done in the present work. q k kq Here we present a compact derivation of strong-contrast expansions. It follows the general formalism of Torquato Note that Eq. (26) is independent of the shape of the [11] closely but departs from it at certain key steps when exclusion region, which stands in contrast to the shape- ðqÞ r establishing the nonlocality of the homogenized constitu- dependent Fourier transform of Hij ð Þ; see Supplemental tive relation. (A detailed derivation is given in the Material [81] for details. Supplemental Material [81].) For simplicity, we take the The use of Eqs. (21) and (23) leads to an integral reference phase I to be phase q (equal to 1 or 2). Under equation for the generalized cavity intensity field FðxÞ: the aforementioned assumptions, the local electric field Z E x ð Þ solves the time-harmonic Maxwell equation [31]: ðqÞ 0 0 0 FðxÞ¼E0ðxÞþ H ðx − x Þ · Pðx Þdx ; ð27Þ ϵ ω 2 ∇ × ∇ × EðxÞ − k 2EðxÞ¼ PðxÞ; ð20Þ q c where the integral subscript ϵ indicates that the integral is to be carried out by omitting the exclusion region and then where PðxÞ is the polarization field given by allowing it to uniformly shrink to zero. Here, FðxÞ is related directly to EðxÞ via PðxÞ ≡ ½εðxÞ − εqEðxÞ; ð21Þ ðqÞ FðxÞ¼fI þ D ½εðxÞ − εqg · EðxÞ: ð28Þ and Using the definitions (21) and (28), we obtain a linear ε x ε − ε IðpÞ x ε ð Þ¼ð p qÞ ð Þþ q ð22Þ constitutive relation between PðxÞ and FðxÞ:

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PðxÞ¼LðqÞðxÞ · FðxÞ; ð29Þ E0ðxÞ that is nonlocal in space. It is more convenient at this stage to utilize a compact linear operator notation, which where enables us to express the integral equation (27) as

ðqÞ ðqÞ −1 F E HP L ðxÞ¼½εðxÞ − εq · fI þ D ½εðxÞ − εqg : ð30Þ ¼ 0 þ ; ð35Þ

It is noteworthy that one is free to choose any convenient where we temporarily drop the superscript q. A combina- exclusion-region shape, provided that its boundary is tion of this equation with Eq. (29) yields the following sufficiently smooth. The choice of the exclusion-region integral equation for the polarization field: shape is crucially important because it determines the type of modified electric field that results as well as the P ¼ LE0 þ LHP: ð36Þ corresponding expansion parameter in the series expansion for the effective dynamic dielectric constant tensor. For The desired nonlocal relation is obtained from Eq. (36) by example, when a spheroidal-shaped exclusion region at a successive substitutions: position x in Rd is chosen to be aligned with the polarization vector PðxÞ at that position, we have P ¼ SE0; ð37Þ

A where DðqÞ ¼ I; ð31Þ εq S ¼½I − LH−1L ð38Þ where I is the second-rank identity tensor, and A ∈ ½0; 1 is the depolarization factor for a spheroid [11] with the is a generalized scattering operator that has superior aforementioned alignment. In the special cases of a sphere, mathematical properties compared to the scattering oper- disklike limit, and needlelike limit, A ¼ 1=d; 1; 0, respec- ator T that arises in standard multiple-scattering theory tively. Thus, for these three cases, Eq. (30) yields the [17,20,33], as we elaborate below in Remark viii. More following shape-dependent tensor: explicitly, the nonlocal relation (37) can be expressed as Z ðqÞ x; L ð A Þ Pð1Þ¼ d2Sð1; 2Þ · E0ð2Þ; ð39Þ 8 ϵ > ε β 1 <> d q pq;A¼ =d ðsphericalÞ; ðpÞ where boldface numbers 1, 2 are shorthand notations for ¼ I ðxÞI εqð1 − εq=εpÞ;A¼ 1 ðdisklikeÞ; > position vectors r1, r2. Ensemble averaging Eq. (39) and : ε ε ε − 1 0 qð p= q Þ;A¼ ðneedlelikeÞ; invoking statistical homogeneity yields the convolution ð32Þ relation Z β where pq is the dielectric polarizability defined by hPið1Þ¼ d2hSið1 − 2Þ · E0ð2Þ; ð40Þ ϵ ε − ε β ≡ p q pq : ð33Þ S ε þðd − 1Þε where the operator h i depends on relative positions, i.e., p q hSið1; 2Þ¼hSið1 − 2Þ, and angular brackets denote an Moreover, in these three cases, the generalized cavity ensemble average. Formally, the nonlocal relation (40) is intensity field Fðx; AÞ reduces to the same as the one given in Torquato [11] for the static problem, but nonlocality was not explicitly invoked there. 8 E x PðxÞ 1 Taking the Fourier transform of Eq. (40) yields a compact > ð Þþ ε ;A¼ =d ðsphericalÞ; < d q Fourier representation of this nonlocal relation, namely, Fðx; AÞ → DðxÞ 1 ð34Þ > ε ;A¼ ðdisklikeÞ; > q f f ˜ : hPiðkÞ¼hSiðkÞ · E0ðkÞ; ð41Þ EðxÞ;A ¼ 0 ðneedlelikeÞ; R f k ≡ x − k x x respectively, where DðxÞ is the displacement field. where hfið Þ hfið Þ expð i · Þd . From Eq. (19), E˜ k E˜ δ k − k k Importantly, among these three cases, the original and 0ð Þ¼ 0 ð qÞ, implying that the wave vector in modified strong-contrast expansions arise only when the Eq. (41) must be identical to kq. exclusion region is taken to be a sphere, as we elaborate As in the static case [11,15] and quasistatic regime [31], below. the ensemble-averaged operator hSiðrÞ, which is given We now show that our formalism yields an exact relation explicitly in the Supplemental Material [81] in terms of the between the polarization field PðxÞ and the applied field n-point correlation functions and products of the tensor

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hFi¼½hSi−1 þ HhPi: ð42Þ Inverting this expression leads to the following nonlocal constitutive relation: Z ðqÞ hPið1Þ¼ d2Le ð1 − 2Þ · hFið2Þ; ð43Þ

ðqÞ where Le ðrÞ is a kernel that is derived immediately below and explicitly given by Z ðqÞ 0 0 0 0 Le ðrÞ ≡ dr ½εeðr Þ − εqIδðr Þ · fIδðr − r Þ

ðqÞ 0 0 −1 þ D · ½εeðr − r Þ − εqIδðr − r Þg : ð44Þ We are not aware of any previous work that derives such an exact nonlocal homogenized constitutive relation (43) from first principles. ðqÞ Note that the support ls of the kernel Le ðrÞ relative to the incident wavelength λ determines the degree of spatial dispersion. When λ is finite, the relation between hPiðxÞ and hFiðxÞ in Eq. (43) is nonlocal in space. In the regime l ≪ λ s , the nonlocal relation (43) canR be well approximated ðqÞ 0 0 by the local relation hPiðxÞ ≈ ½ Le ðx Þdx · hFiðxÞ. ðqÞ Indeed, in the static limit, Le ðrÞ tends to a Dirac delta function δðrÞ, expression (43) becomes the position- independent local relation

ðqÞ hPi¼Le · hFið45Þ FIG. 2. (a) Schematic of a large d-dimensional ellipsoidal, macroscopically anisotropic two-phase composite medium em- derived earlier [11]. bedded in an infinite reference phase of dielectric constant tensor The nonlocal constitutive relation in direct space, Eq. (43), can be reduced to a linear product form in εI (gray regions) under an applied electric field E0ðxÞ¼ ˜ Fourier space by taking the Fourier transform of Eq. (43): E0 exp (iðkI · x − ωtÞ) of a frequency ω and a wave vector kI at infinity. The wavelength λ associated with the applied field can fP k ðqÞ k fF k span from the quasistatic regime (2πl=λ ≪ 1) down to the h ið qÞ¼Le ð qÞ · h ið qÞ: ð46Þ 2πl λ ≲ 1 l intermediate-wavelength regime ( = ), where is a ðqÞ k characteristic heterogeneity length scale. (b) After homogeniza- The wave-vector-dependent effective tensor Le ð qÞ is tion, the same ellipsoid can be regarded as a homogeneous postulated (see discussion in the Supplemental Material specimen with an effective dielectric constant εeðkI; ωÞ, which [81]) to be given by depends on ω and kI. As noted in the main text, we omit the ω ðqÞ k ≡ ε k − ε ðqÞ ε k − ε −1 dependence of εe because (without loss of generally) we assume a Le ð qÞ ½ eð qÞ qI · fI þ D · ½ eð qÞ qIg : linear dispersion relation between jk j and ω. I ð47Þ

ðqÞ HðrÞ, depends on the shape of the macroscopic ellipsoidal This linear fractional form for Le ðkÞ is consistent with composite specimen (see Fig. 2). This shape-dependence the one derived for the static limit [11] and for the arises because H decays like r−d for large r, and hence, quasistatic regime [31]. Taking the inverse Fourier trans- hSiðrÞ involves conditionally convergent integrals [11].To form of Eq. (47) yields its corresponding direct-space avoid such conditional convergence issues, we follow representation (44). Taking the Fourier transform of previous strong-contrast formulations by seeking to elimi- Eq. (42) yields nate the applied field E0 in Eq. (40) in favor of the average F r f f −1 ˜ f cavity field h ið Þ in order to get a corresponding nonlocal hFiðkqÞ¼½hSiðkqÞ þ HðkqÞ · hPiðkqÞ: ð48Þ homogenized constitutive relation between hPiðrÞ and hFiðrÞ or vice versa. Thus, solving for E0 in Eq. (40) Comparing Eq. (46) to Eq. (48), and specifically and substituting into the ensemble average of Eq. (35) choosing a spherical exclusion region, as discussed in yields Eq. (32), yields the desired exact strong-contrast

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ðpÞ expansions for general macroscopically anisotropic two- where An ðkqÞ is a wave-vector-dependent second-rank phase media: tensor that is a functional involving the set of correlation ðpÞ ðpÞ ðpÞ functions S1 ;S2 ; …;Sn (defined in Sec. II A) and ϕ 2β2 ε k d − 1 ε I · ε k − ε I −1 p pq½ eð qÞþð Þ q ½ eð qÞ q products of the second-rank-tensor field HðqÞðrÞ, which X∞ ðpÞ n is explicitly given as [see also Eq. (25)] ¼ ϕpβpqI − An ðkqÞβpq ; ð49Þ n¼2

8 h i 1 1 2 1 < i 2Hð Þ − kq Hð Þ δ ikq Hð Þ rˆ rˆ 2 4 kq 0 ðkqrÞ r 1 ðkqrÞ ij þ 4 2 ðkqrÞ i j;d¼ ; ðqÞ r H ijð Þ¼: ð50Þ expðikqrÞ 2 2 ˆ ˆ 3 f½−1 þ ikqr þðkqrÞ δij þ½3 − 3ikqr − ðkqrÞ rirjg;d¼ 3; εq4πr where kq ≡ jkqj. Specifically, for n ¼ 2 and n ≥ 3, these n-point tensors associated with the polarized phase p are, respectively, given by Z ðpÞ − k r k ε r ðqÞ r i q· χ r A2 ð qÞ¼d q d H ð Þe V ð Þ; ð51Þ ϵ Z − ε n−2 p d q ð Þ ðqÞ −ikq·ðr1−r2Þ ðqÞ −ikq·ðr2−r3Þ An ðkqÞ¼dεq dr1 drn−1H ðr1 − r2Þe · H ðr2 − r3Þe · ϕp ϵ p ðqÞ −ikq·ðrn−1−rnÞ ð Þ × H ðrn−1 − rnÞe Δn ðr1; …; rnÞ; ð52Þ R R r ≡ r ΔðpÞ where ϵ d limϵ→0þ jrj>ϵ d , and n is a position-dependent determinant involving correlation functions of the polarized phase p up to the n-point level:

ðpÞ ðpÞ S2 ðr1; r2Þ S1 ðr1Þ 0

ðpÞ ðpÞ S3 ðr1; r2; r3Þ S2 ðr2; r3Þ 0 ΔðpÞ r … r n ð 1; ; nÞ¼ . . . . : ð53Þ ......

ðpÞ r … r ðpÞ r … r ðpÞ r r Sn ð 1; ; nÞ Sn−1ð 2; ; nÞ S2 ð n−1; nÞ

For macroscopically isotropic media, the effective constant tensor itself. The series expansion in dielectric tensor is isotropic, i.e., εeðkqÞ¼εeðkqÞI. The powers of the polarizability βpq of this particular ε k corresponding strong-contrast expansion for eð qÞ is rational function of εeðkqÞ has important conse- obtained by taking the trace of both sides of Eq. (49): quences for the predictive power of approximations derived from the expansion, as detailed in 2 2 −1 ϕp βpq½εeðkqÞþðd − 1Þεq½εeðkqÞ − εq Sec. III B. X∞ (ii) The fact that the exact expansion (49) extracted from ðpÞ n ¼ ϕpβpq − An ðkqÞβpq ; ð54Þ our nonlocal relation (46) is explicitly given in terms n¼2 of integrals over products of the relevant Green’s functions and the n-point correlation functions to ðpÞ ðpÞ where εeðkqÞ¼Tr½εeðkqÞ=d, An ðkqÞ¼Tr½An ðkqÞ=d infinite order implies that multiple scattering to all for n ≥ 2 and Tr½ denotes the trace operation. Furthermore, orders is exactly treated for the range of wave for statistically isotropic media, the effective dielectric numbers for which our extended homogenization constant becomes independent of the direction of the wave theory applies, i.e., 0 ≤ jkqjl ≲ 1. vector, i.e., εeðkqÞ¼εeðkqÞ. (iii) Note that Eq. (49) represents two different series Remarks: expansions: one for q ¼ 1 and p ¼ 2 and the other (i) Importantly, the strong-contrast expansion (49) is a for q ¼ 2 and p ¼ 1. series representation of a linear fractional trans- (iv) The exact expansions represented by Eq. (49) are formation of the variable εeðkqÞ (left-hand side independent of the reference phase q and hence of the equation), rather than the effective dielectric independent of the wave vector kq.

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(v) For d ¼ 2, the strong-contrast expansion applies for B. Convergence properties and accuracy TE polarization only. This implies that the electric of truncated series field and wave vector are parallel to the plane or The form of the strong-contrast expansion parameter βpq transverse to an axis of symmetry in a 3D system in Eq. (49) is a direct consequence of the choice of a whose cross sections are identical. spherical region excluded from the volume integrals in (vi) Formally, the original strong-contrast expansions Eq. (49) due to singularities in the Green’s functions [31]. that apply in the quasistatic regime [31] can be It is bounded by obtained from the nonlocal strong-contrast expansions (49) by simply replacing the exponential 1 εp − εq functions that appear in the expressions for the − ≤ βpq ≡ ≤ 1; ð57Þ ðpÞ d − 1 εp þðd − 1Þεq second-rank tensors An ðkqÞ, defined by Eq. (52), by unity. (vii) For statistically isotropic media, the effective phase which implies that the strong-contrast expansion (49) speed c ðk Þ and attenuation coefficient γ ðk Þ are can converge rapidly, even for infinite contrast ratio e q e q ε ε → ∞ determined by the scalar effective dielectric constant p= q . Other choices for the shape of the exclusion εeðkqÞ: region will lead to different expansion parameters that will generally be bounded but can lead to expansions with h qffiffiffiffiffiffiffiffiffiffiffiffiffii significantly different convergence properties [11].In −1 ceðkqÞ=c ¼ neðkqÞ ¼ Re 1= εeðkqÞ ; ð55Þ Appendix A, we present the corresponding expansions for disklike and needlelike exclusion regions, which are exceptional cases that lead to slowly converging h qffiffiffiffiffiffiffiffiffiffiffiffiffii −1 weak-contrast expansions with expansion parameter γeðkqÞ=c ¼ κeðkqÞ ¼ Im 1= εeðkqÞ ; ð56Þ ðεp − εqÞ=εq, and thus are unbounded when εp=εq → ∞. Importantly, in the purely static case, the expansion (49) becomes identical to one derived by Sen and Torquato [15] where neðkqÞ and κeðkqÞ are the effective refractive and extinction indices, respectively. The quantity and its truncation after second-order terms (i.e., setting ðpÞ 0 ≥ 3 expð−2πγe=ceÞ is the factor by which the incident- An ¼ for all n ) yields the generalized Hashin- wave amplitude is attenuated for a period 2π=ω. Shtrikman bounds [85] derived by Willis [86] that are (viii) The strong-contrast formalism is a significant depar- optimal since they are realized by certain statistically ture from perturbative expansions obtained from anisotropic composites in which there is a disconnected, standard multiple-scattering theory [17,20,33,34]. dispersed phase in a connected matrix phase [87]. In the The operator S defined in Eq. (38) is a generalization case of an isotropic effective dielectric constant εe, the of the standard scattering operator T ¼½I − VG−1V optimal Hashin-Shtrikman upper and lower bounds for any ε ε [17,20,33], where V ≡ ½εðrÞ − εqI is the scattering phase-contrast ratio 2= 1 are exactly realized by the potential. Thus, the perturbation series resulting from multiscale “coated-spheres” model, which is depicted in T is a weak-contrast expansion with the inherent Fig. 3 in two dimensions. Affine transformations of the limitations that it converges only for small contrast coated spheres in the d orthogonal directions lead to ratios (see also Sec. III B). By contrast, due to the oriented coated ellipsoids that are optimal for the macro- different possible choices for the exclusion regions scopically anisotropic case. The lower bound corresponds and reference phases, there is an infinite variety of to the case when the high-dielectric-constant phase is the series expansions that result from the strong-contrast dispersed, disconnected phase, and the upper bound formalism with generally fast-convergence proper- corresponds to the instance in which the high-dielectric- ties. The particular strong-contrast expansion can be constant phase is the connected matrix. Thus, Torquato designed for different classes of microstructures [11,88] observed that the strong-contrast expansions (49) in (see Appendixes A and D). The corresponding the static limit can be regarded as ones that perturb around strong-contrast “self-energy” Σ is a linear fractional such optimal composites, implying that the first few terms transform of Le [see Eq. (48)], namely, Σ ¼ of the expansion can yield accurate approximations of the −1 −1 −1 Le ½I − DLe , which is a generalization of the effective property for a class of particulate composites as self-energy in standard multiple-scattering theory well as more general microstructures, depending on [17,20,33,34]. Thus, it is highly nontrivial to relate whether the high-dielectric phase percolates or not. For diagrammatic expansions of the strong-contrast for- example, even when ε2=ε1 ≫ 1, the dispersed phase 2 can malism to those of multiple-scattering theory. An consist of identical or polydisperse particles of general elaboration of how strong-contrast expansion gen- shape (ellipsoids, cubes, cylinders, polyhedra) with pre- eralizes those from multiple-scattering theory is scribed orientations that may or not overlap, provided presented in the Supplemental Material [81]. that the particles are prevented from forming large

021002-10 NONLOCAL EFFECTIVE ELECTROMAGNETIC WAVE … PHYS. REV. X 11, 021002 (2021) ε k eð qÞ ϕ 2β ϕ 1 − ϕ β ε ¼ I þ d p pq pð p pqÞI q X∞ −1 ðpÞ n−1 − An ðkqÞβpq : ð58Þ n¼2

Expanding Eq. (58) in powers of the scalar polarizability βpq yields the series ∞ ε ðk Þ X e q ðpÞ k β n ε ¼ Bn ð qÞ pq ; ð59Þ q n¼0

ðpÞ where the first several functionals Bn ðkqÞ are explicitly ðpÞ ðpÞ ðpÞ given in terms of A0 ; A1 ; …; An as

ðpÞ B0 ðkqÞ¼I; ðpÞ B1 ðkqÞ¼dϕpI; ðpÞ ðpÞ 2 B2 ðkqÞ¼d½A2 ðkqÞþϕpI; FIG. 3. Schematic of the optimal multiscale coated-spheres model that realizes the isotropic Hashin-Shtrikman bounds on ε ðpÞ d ðpÞ 2 ðpÞ e B3 ðkqÞ¼ ½A2 ðkqÞ þ ϕpA3 ðkqÞ [85]. Each composite sphere is composed of a spherical inclusion ϕp of one phase (dispersed phase) that is surrounded by a concentric 2ϕ2 ðpÞ k ϕ4 spherical shell of the other phase such that the fraction of þ pA2 ð qÞþ pI; space occupied by the dispersed phase is equal to its overall d 3 BðpÞ k AðpÞ k 2ϕ AðpÞ k AðpÞ k phase volume fraction. The composite spheres fill all space, 4 ð qÞ¼ϕ2 ½ 2 ð qÞ þ p 2 ð qÞ · 3 ð qÞ implying that their sizes range down to the infinitesimally small. p When phase 2 is the disconnected inclusion (dispersed) phase, 2 ðpÞ 2 2 ðpÞ þ 3ϕpA2 ðkqÞ þ ϕpA4 ðkqÞ this two-phase medium minimizes and maximizes the effective ε 3 ðpÞ 4 ðpÞ 6 static dielectric constant e for prescribed volume fraction þ 2ϕpA3 ðkqÞþ3ϕpA2 ðkqÞþϕpI; and contrast ratio, when ε2=ε1 > 1 and ε2=ε1 < 1, respectively. It has recently been proved that these highly degenerate where Tn stands for n successive inner products of a optimal Hashin-Shtrikman multiscale distributions of spheres second-rank tensor T. are hyperuniform [57,89]. Let us now compare the exact expansion (59) to the one that results when expanding the truncation of the exact −1 expression (49) for ½εe þðd − 1Þεq ⋅ ½εe − εq at the clusters compared to the specimen size. Moreover, when two-point level: ε2=ε1 ≪ 1, the matrix phase can be a cellular network [56]. Finally, for moderate values of the contrast ratio ε2=ε1, ε k eð qÞ ≈ ϕ β 1 − ϕ β − ðpÞ k β ϕ −1 even more general microstructures (e.g., those without I þ d p pq½ð p pqÞI A2 ð qÞ pq= p εq well-defined inclusions) can be accurately treated. Importantly, we show that for the dynamic problem under ð60Þ consideration, the first few terms of the expansion (49) X∞ ε k ðpÞ n yield accurate approximations of eð qÞ for a similar wide ¼ Cn ðkqÞβpq ; ð61Þ class of two-phase media (see Sec. VIII). Analogous n¼0 approximations were derived and applied for the quasistatic ðpÞ k regime [31,36]. where the nth-order functional Cn ð qÞ for any n is given ðpÞ We now show how lower-order truncations of the series in terms of the volume fraction ϕp and A2 ðkqÞ, which has (49) can well approximate higher-order functionals (i.e., the following diagrammatic representation: higher-order diagrams) of the exact series to all orders in terms of lower-order diagrams. Such truncations of strong- ð62Þ contrast expansions are tantamount to approximate but resummations of the strong expansions, which enables multiple scattering and spatial dispersion effects to be Here the solid and wavy lines joining two nodes represent χ r accurately captured to all orders. Solving the left-hand the spatial correlation via V ð Þ and a wave-vector- ðqÞ −ik ·r side of Eq. (49) for εe yields the rational function in βpq: dependent Green’s function H ðrÞe q between the

021002-11 SALVATORE TORQUATO and JAEUK KIM PHYS. REV. X 11, 021002 (2021) nodes, respectively. The black node indicates a volume ε k d − 1 ε I ϕ 2β2 eð qÞþð Þ q ϕ β − ðpÞ k β 2 integral and carries a factor of dε . The first several p pq ¼ p pqI A2 ð qÞ pq ; q εeðkqÞ − εqI ðpÞ k functionals Cn ð qÞ are explicitly given as ð65Þ ðpÞ k C0 ð qÞ¼I; ε ðk Þþðd − 1Þε I ϕ 2β2 e q q ¼ ϕ β I − ½AðpÞðk Þβ 2 ðpÞ p pq ε k − ε p pq 2 q pq C1 ðkqÞ¼dϕpI; eð qÞ qI ðpÞ ðpÞ 2 ðpÞ k β 3 C2 ðkqÞ¼d½A2 ðkqÞþϕp I; þ A3 ð qÞ pq : ð66Þ d ðpÞ k ðpÞ k ϕ 2 2 Compared to the quasistatic approximation [31], these C3 ð qÞ¼ϕ ½A2 ð qÞþ p I ; p nonlocal approximations substantially extend the range 0 ≤ k l ≲ 1 ðpÞ k d ðpÞ k ϕ 2 3 of applicable wave number, namely, j qj . C4 ð qÞ¼ 2 ½A2 ð qÞþ p I : ϕp B. Macroscopically isotropic media Thus, comparing Eq. (59) to Eq. (61), we see that trunca- All of the applications considered in this paper focus tion of the expansion (49) for ½ε þðd − 1Þε ⋅ ðε − ε Þ−1 e q e q on the case of macroscopically isotropic media; i.e., they at the two-point level actually translates into approxima- are described by the scalar effective dielectric constant tions of the higher-order functionals to all orders in terms of ε ðk Þ¼Tr½ε ðk Þ=d but depend on the direction of the the first-order diagram ϕ and the second-order diagram e q e q p wave vector k . (62). This two-point truncation can be thought of as an q approximate but accurate resummed representation of the exact expansion (59). Note that the approximate expansion 1. Strong-contrast approximation at the two-point level (61) is exact through second order in βpq. Clearly, trunca- Solving Eq. (54) for the effective dielectric constant tion of Eq. (49) at the three-point level (see Appendix C) εeðkqÞ yields the strong-contrast approximation for mac- will yield even better approximations of the higher-order roscopically isotropic media: functionals. ε ðk Þ dβ ϕ 2 e q ¼ 1 þ pq p IV. STRONG-CONTRAST ε ðpÞ q ϕpð1 − βpqϕpÞ − βpqA2 ðkqÞ APPROXIMATION FORMULAS dβ ϕ 2 ¼ 1 þ pq p ; ð67Þ Here we describe lower-order truncations of the strong- ðd−1Þπβpq ϕ ð1 − β ϕ Þþ 2 Fðk Þ contrast expansions that are expected to yield accurate p pq p 2d= Γðd=2Þ q closed-form formulas for εeðkqÞ that apply over a broad l ≲ 1 ðpÞ ðpÞ range of wavelengths (kq ), volume fractions, and where βpq is defined in Eq. (33), A2 ðkqÞ≡Tr½A2 ðkqÞ=d, contrast ratios for a wide class of microstructures. and FðQÞ is what we call the nonlocal attenuation function of a composite for reasons we describe below. The direct- A. Macroscopically anisotropic media and Fourier-space representations of FðQÞ are given as For the ensuing treatment, it is convenient to rewrite the Z 2d=2Γðd=2Þ i Q d=2−1 expansions (49), valid for macroscopically anisotropic FðQÞ ≡ − Q2 media in Rd, in the following manner: π ϵ 4 2πr 1 Hð Þ Qr e−iQ·rχ r dr ε k d − 1 ε I × d=2−1ð Þ V ð Þ ð68Þ ϕ 2β2 eð qÞþð Þ q p pq ε ðk Þ − ε I Z e q q Γðd=2Þ χ˜ ðQÞ ¼ − Q2 V dq: ð69Þ XM 2d=2πdþ1 q Q 2 − 2 ðpÞ n j þ j Q ¼ ϕpβpqI − An ðkqÞβpq þ RMðkqÞ; ð63Þ 2 n¼ The exponential expð−iQ · rÞ in Eq. (68) arises from the where the Mth-order remainder term is defined as phase difference associated with the incident waves at positions separated by r. In the quasistatic regime, this X∞ ðpÞ phase factor is negligible, and Eq. (68) reduces to the local R ðk Þ ≡ A ðk Þβ n: ð64Þ M q n q pq attenuation function FðQÞ (derived in Ref. [31] and sum- n¼Mþ1 marized in the Supplemental Material [81]) because it is Truncating the exact nonlocal expansion (63) at the barely different from unity over the correlation length R k 0 χ two- and three-point levels, i.e., setting 2ð qÞ¼ and associated with the autocovariance function V ðrÞ. The R3ðkqÞ¼0, respectively, yields strong-contrast approximation (67) was postulated in

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Ref. [73] on physical grounds. By contrast, the present work ε ðk Þ dβ ϕ 2 e q ¼ 1 þ pq p qffiffiffiffiffi : ð73Þ derives it as a consequence of our exact nonlocal formalism ðd−1Þπβ ε εq ϕ 1 − β ϕ pq HS pð pq pÞþ d=2 F ε kq (Sec. III). 2 Γðd=2Þ q For statistically isotropic media, the effective dielectric constant as well as the attenuation function are indepen- We now show that the scaled approximation (73) indeed dent of the direction of the incident wave vector kq, provides good estimates of leading-order corrections of and thus, they can be considered as functions of the R2ðkqÞ in powers of kq. To do so, we employ the concept of ðqÞ wave number, i.e., εeðkqÞ¼εeðkqÞ and FðkqÞ¼FðkqÞ. the averaged (effective) Green’s function hG ðqÞi of an Then, the real and imaginary parts of Eq. (68) can be inhomogeneous medium which in principle accounts for simplified as the all multiple-scattering events 8 2 2 R π 2 ω < − Q = χ˜ 2 ϕ ϕ 2 ðqÞ q 2 − ω 2 − qq −1 π2 0 V ð Q cos Þd ;d¼ ; hG ð Þi ¼ f½q keð Þ I g ; ð74Þ Im½FðQÞ ¼ R ð70Þ c : Q 2Q − 3 2 qχ˜ q dq; d 3; 2ð2πÞ = 0 V ð Þ ¼ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where keðωÞ ≡ εeðωÞω=c ¼ εeðωÞ=εqkq is the effec- Z ω ε ω 2 2 ∞ 1 tive wave number at a frequency , and eð Þ is the exact − Q Re½FðQÞ ¼ p:v: dq 2 2 Im½FðqÞ; ð71Þ effective dynamic dielectric constant, assuming a well- π 0 qðQ − q Þ defined homogenization description [17,90]. Since exact complete microstructural information is, in principle, where Eq. (71) is valid for d ¼ 2,3,andp:v. stands for accounted for with the effective Green’s function (74), the Cauchy principle value. Following conventional the exact strong-contrast expansion can be approximately usage, we say that a composite attenuates waves at a equated to the one truncated at the two-point level with an given wave number if the imaginary part of the effective attenuation function given in terms of the effective Green’s dielectric constant is positive. Recall that attenuation in function, i.e., the present study occurs only because of multiple- scattering effects (not absorption). While it is the ε − 1 ε 2 2 eðkqÞþðd Þ q imaginary part of FðQÞ that determines directly the ϕp βpq εeðkqÞ − εq degree of attenuation or, equivalently, Im½εe,wesee from Eq. (71) that the real part of FðQÞ is directly ðpÞ 2 ¼ ϕpβpq − A2 ðkqÞβpq þ R2ðkqÞ; ð75Þ related to its imaginary part. It is for this reason that we refer to the complex function FðQÞ as the (nonlocal) ðpÞ 2 ≈ ϕ β − A (k ðωÞ)β : ð76Þ attenuation function. p pq 2 e pq

ðpÞ When the functional form of A2 ðQÞ or, equivalently, 2. Modified strong-contrast approximation FðQÞ is available, it is possible to solve Eq. (76) for εeðkqÞ at the two-point level in a self-consistentpffiffiffiffiffiffiffiffiffiffiffiffiffiffi manner. Instead, we show that by ω ≈ ε ε Here we extend the validity of the strong-contrast assuming keð Þ HS= qkq, which results in Eq. (73), approximation (67) so that it is accurate at larger wave we obtain good estimates of the leading-order corrections numbers and hence better captures spatial dispersion. This of R2ðkqÞ in powers of kq. Thus, the scaled approximation is done by an appropriate rescaling of the wave number in (73) provides better estimates of the higher-order three- the reference phase, kq, which we show is tantamount to point approximation (given in Appendix C) than the approximately accounting for higher-order contributions in unmodified strong-contrast approximation (60). This can R the remainder term 2ðkqÞ. Given that the strong-contrast be easily confirmed in the quasistatic regime from the expansion for isotropic media perturbs around the Hashin- ðpÞ ðpÞ small-k expansions of A ðk Þ and A ðk Þ given in Shtrikman structures (see Fig. 3) in the static limit, it is q 2 q 3 q pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ref. [31]. We also confirm the improved predictive natural to use the scaling ε =ε k , where ε is the HS q q HS capacity of the scaled strong-contrast approximation to Hashin-Shtrikman estimate, i.e., better capture dispersive characteristics for ordered and disordered models via comparison to finite-difference time- dϕ β ε ≡ ε 1 p pq domain (FDTD) simulations; see Figs. 8 and 9, and HS q þ 1 − ϕ β ; ð72Þ p pq Sec. VIII in the Supplemental Material [81]. which gives the Hashin-Shtrikman lower bound and V. MODEL MICROSTRUCTURES upper bound if εp > εq and εp < εq, respectively. This scaling yields the following scaled strong-contrast approxi- We consider four models of 2D and 3D disordered media mation for statistically isotropic media: to understand the effect of microstructure on the effective

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FIG. 4. Representative images of configurations of the four models of 2D disordered particulate media described in this section. These include (a) overlapping spheres, (b) equilibrium packings, (c) class I hyperuniform polydisperse packings, and (d) stealthy hyperuniform packings. For all models, the volume fraction of the dispersed phase (shown in black) is ϕ2 ¼ 0.25. Note that (a) and (b) are not hyperuniform. While these models consist of distributions of particles, both overlapping and nonoverlapping, the formulas derived in Sec. IV can be applied to any two-phase microstructure. Indeed, Appendix D describes applications to media with phase-inversion symmetry. dynamic dielectric constant, two of which are nonhyperuni- A. Overlapping spheres form (overlapping spheres and equilibrium hard-sphere Overlapping spheres (also called the fully-penetrable- packings) and two of which are hyperuniform (hyperuni- sphere model) refer to an uncorrelated (Poisson) distribu- form polydisperse packings and stealthy hyperuniform tion of spheres of radius a throughout a matrix [11].For packings of identical spheres). The particles of dielectric such nonhyperuniform models at number density ρ in ε constant 2 are distributed throughout a matrix of dielectric d-dimensional Euclidean space Rd, the autocovariance ε constant 1. We also compute the spectral density for each function is known analytically [11]: model, which is the required microstructural information to evaluate the nonlocal strong-contrast approximations dis- 2 χVðrÞ¼exp ( − ρv2ðr; aÞ) − ϕ1 ; ð77Þ cussed in Sec. VB. Representative images of configurations of the four where ϕ1 ¼ exp ( − ρv1ðaÞ) is the volume fraction of the aforementioned models of 2D disordered particulate media matrixphase(phase1),v1ðaÞ isgivenbyEq.(13),andv2ðr; aÞ are depicted in Fig. 4. Note that the degree of volume- represents the union volume of two spheres whose centers are fraction fluctuations decreases from the leftmost image to separated by a distance r. In two and three dimensions, the the rightmost one. latter quantity is explicitly given, respectively, by

( 2Θðx − 1Þþ2 ½π þ xð1 − x2Þ1=2 − cos−1ðxÞΘð1 − xÞ;d¼ 2; v2ðr; aÞ π ¼ 3 3x x v1ðaÞ 2Θðx − 1Þþ 1 þ 2 − 2 Θð1 − xÞ;d¼ 3; where x ≡ r=2a, and ΘðxÞ (equal to 1 for x>0 and zero by the Percus-Yevick solution [11,93], which is analytically otherwise) is the Heaviside step function. For d ¼ 2 and solvable for odd values of d.Ford ¼ 3, the Percus-Yevick d ¼ 3, the particle phase (phase 2) percolates when ϕ2 ≈ solution gives the following expression for the structure 0.68 (Ref. [91]) and ϕ2 ≈ 0.29 (Ref. [92]), respectively. The factor SðQÞ [11]: corresponding spectral densities are easily found numeri- 16π 3 cally by performing the Fourier transforms indicated in 1 − ρ a 24 ϕ − 12 2 ϕ 2 SðQÞ¼ 6 f½ a1 2 ða1 þ a2Þ 2q Eq. (7). In this work, we apply this model for ϕ2’s well q 4 below the percolation thresholds. þð12a2ϕ2 þ 2a1 þ a2ϕ2Þq cosðqÞ 3 þ½24a1ϕ2q − 2ða1 þ 2a1ϕ2 þ 12a2ϕ2Þq sinðqÞ B. Equilibrium packings −1 2 Another disordered nonhyperuniform model we treat is − 24ϕ2ða1 − a2q Þg ; ð78Þ distributions of equilibrium (Gibbs) of identical hard 2 4 spheres of radius a along the stable fluid branch [11,93]. where q ¼ 2Qa, a1 ¼ð1 þ 2ϕ2Þ =ð1 − ϕ2Þ , and a2 ¼ 2 4 The structure factors of such packings are well approximated −ð1þ0.5ϕ2Þ =ð1−ϕ2Þ . Using this solution in conjunction

021002-14 NONLOCAL EFFECTIVE ELECTROMAGNETIC WAVE … PHYS. REV. X 11, 021002 (2021) with Eq. (11) yields the corresponding spectral density stealthy, and hyperuniform, and their nearest-neighbor χ˜ 2 σ V ðQÞ.Ford ¼ , we obtain the spectral density from distances are larger than the length scale due to the Monte Carlo generated disk packings [11]. soft-core repulsion uðrÞ. Finally, to create packings, we follow Ref. [70] by circumscribing the points by identical C. Hyperuniform polydisperse packings spheres of radius a<σ=2 under the constraint that they Class I hyperuniform packings of spheres with a poly- cannot overlap (see the Supplemental Material [81] for “ ” dispersity in size can be constructed from nonhyperuniform certain results concerning stealthy nonhyperuniform 0 progenitor point patterns via a tessellation-based procedure packings in which QL > .) The parameters used to [57,89]. Specifically, we employ the centers of 2D and 3D generate these disordered stealthy packings are summa- configurations of identical hard spheres in equilibrium at a rized in the Supplemental Material [81]. packing fraction 0.45 and particle number N ¼ 1000 as the progenitor point patterns. One begins with the Voronoi E. Spectral densities for the four models tessellation [11] of these progenitor point patterns. We then χ˜ Here, we plot the spectral density V ðQÞ for the four rescale the particle in the jth Voronoi cell Cj without models at the selected particle-phase volume fraction of changing its center such that the packing fraction inside this ϕ2 ¼ 0.25; see Fig. 5. From the long- to intermediate- cell is identical to a prescribed value ϕ2 < 1. The same wavelength regimes (Qa ≲ 4), their spectral densities are process isP repeated over all cells. The final packing fraction considerably different from one another. Overlapping ϕ N ρ ρ is 2 ¼ j¼1 v1ðajÞ=VF ¼ v1ðaÞ, where is the number spheres depart the most from hyperuniformity, followed density of particle centers and a represents the mean sphere by equilibrium packings. Stealthy packings suppress radius. In the small-jQj regime, the spectral densities of the volume-fraction fluctuations to a greater degree than hyper- resulting particulate composites exhibit a power-law scal- uniform polydisperse packings over a wider range of wave- χ˜ Q ∼ Q 4 Qa ≫ 4 ing V ð Þ j j , which are of class I. lengths. In the small-wavelength regime ( ), all four curves tend to collapse onto a single curve, reflecting the fact D. Stealthy hyperuniform packings that all four models are composed of spheres of similar sizes. Stealthy hyperuniform particle systems, which are also class I, are defined by the spectral density vanishing around VI. RESULTS FOR THE NONLOCAL χ˜ Q 0 0 Q ≤ ATTENUATION FUNCTION the origin, i.e., V ð Þ¼ for < j j QU; see Eq. (14). We obtain the spectral density from realizations of dis- We report some general behaviors of the nonlocal 2 ordered stealthy hyperuniform packings for d ¼ , 3 that attenuation function FðQÞ [cf. Eqs. (68) or (69)] for are numerically generated via the following two-step nonhyperuniform and hyperuniform media for long and procedure. Specifically, we first generate such point con- intermediate wavelengths. We also provide plots of both the F figurations consisting of N particles in a fundamental cell real and imaginary parts of FðQÞ for the four models of under periodic boundary conditions via the collective- disordered two-phase media considered in this work, which coordinate optimization technique [67–69], which amounts depends on wave number Q. to finding numerically the ground-state configurations for The function FðQÞ depends on the microstructure via the following potential energy: χ˜ the spectral density V ðQÞ. Thus, assuming that the latter quantity has the power-law scaling χ˜ ðQÞ ∼ Qα as Q → 0, 1 X X V N Φðr Þ¼ v˜ðQÞSðQÞþ uðrijÞ; ð79Þ the asymptotic behavior of FðQÞ in the long-wavelength VF Q i

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0.8 Overlapping spheres (a) 0.08 Overlapping spheres Equilibrium packing Equilibrium packing )] 0.06 Stealthy hyperuniform; Q a = 1.5 Q U 0.6 Hyperuniform polydisperse packing ( F 0.04 Hyperuniform polydisperse packing Stealthy hyperuniform; Q a = 1.5

) U

Q 0.02 -Re[ (

V 0.4 χ ~ 0 -1 (b) 0.08 10 0.2 )] -4

Q 10 ( -7 F 0.04 10 0 0.1 1

0246 810 -Im[ 0 Qa 0 1 2 Qa FIG. 5. The spectral density χ˜V ðQÞ as a function of dimensionless wave number Qa for the four models of 3D disordered FIG. 6. The negative values of (a) the real and (b) imaginary media: overlapping spheres, equilibrium packings, class I hyper- parts of the nonlocal attenuation function FðQÞ as a function of uniform polydisperse packings, and stealthy hyperuniform pack- the dimensionless wave number Qa [defined in Eq. (68)] for the ings. In all cases, the volume fraction of the dispersed phase is four models of 3D disordered composite media considered in this ϕ2 ¼ 0.25. For hyperuniform polydisperse packings, a is the paper. The inset in (b) is the log-log plot of the larger panel. The mean sphere radius. The other three models consist of identical volume fraction of the dispersed phase for each model is spheres of radius a. Corresponding graphs of the spectral ϕ2 ¼ 0.25. The first three models consist of identical spheres densities for the 2D models are provided in the Supplemental of radius a. For class I hyperuniform polydisperse particulate Material [81]. media, a is the mean sphere radius.

Im½εeðkqÞ ∼ Im½FðkqÞ one obtained from previous simulation results for stealthy d kq ; nonhyperuniform hyperuniform “point” scatterers [63], not the finite-sized ∼ as kq → 0: k dþα; hyperuniform particles considered here. q Figure 6 shows FðQÞ for the four distinct models of ð84Þ disordered particulate media in R3: overlapping spheres, equilibrium packings, stealthy hyperuniform packings, Thus, hyperuniform media are less lossy than their non- and hyperuniform polydisperse packings. (Its 2D counter- hyperuniform counterparts in the quasistatic regime. part is provided in the Supplemental Material [81].) We χ˜ In the case of stealthy hyperuniform media [i.e., V ðQÞ¼ clearly see that these attenuation functions exhibit common 0 0 ≤ for Q q, since (phase 1) and the particles to be the polarized phase ε ε HS > q, the scaled approximation accurately predicts a (phase 2). narrower transparency interval than the unscaled variant, as The general simulation setup is schematically illustrated verified in Sec. VIII. Interestingly, the transparency interval in Fig. 7. The simulation procedure for macroscopically obtained from the less accurate formula (67) agrees with the isotropic media is carried out in three steps:

021002-16 NONLOCAL EFFECTIVE ELECTROMAGNETIC WAVE … PHYS. REV. X 11, 021002 (2021) (a) D˜ k ; ω 2 ε ≡ yð e Þ ε ≡ ke ðk1Þ ˜ ; ðk1Þ ω ; Eyðke; ωÞ =c

PML PML ke is a complex-valued effective wave number, and (b) Z 1 D˜ ðq; ωÞ ≡ D ðr; ωÞe−iqxˆ·rdr; y V y j j ZV 1 ˜ −iqxˆ·r Eyðq; ωÞ ≡ Eyðr; ωÞe dr; jVj V

where V is a rectangular parallelepiped subregion within the composite (shown in gray in Fig. 6) that is PML PML slightly smaller than the simulation box and is used to reduce undesired boundary effects. The homog- FIG. 7. Schematic of the general simulation setup for either (a) periodic or (b) nonperiodic composites consisting of N enization task is carried out by numerically finding ε − ε 2 spheres of radius a in a matrix. In both cases, Gaussian pulses the minimizer of j j with an initial guess ε ε of electric fields propagate from the planar sources (shown in red ¼ HS via the Broyden-Fletcher-Goldfarb-Shanno ε lines) to the packings (shown in black circles). The wave number nonlinear optimization algorithm [99], where HS is (spectrum) of the pulses spans between min½k1 and max½k1. the Hashin-Shtrikman estimate given by Eq. (72). Periodic boundary conditions are applied along all directions, Details of step 2 are provided in the Supplemental except for the propagation direction xˆ. The perfectly matched Material [81]. layers (PML shown in blue) of thickness LPML are placed at both (3) Steps 1 and 2 are repeated for a sufficient number of ends of the simulation box to absorb any reflected and transmitted configurations for disordered media. Then, we com- waves. To estimate the effective dielectric constant as described pute the effective dielectric constant at a given k1 by in step 2 below, we consider the subregion V (shown in gray) that ε ε ε excludes from the composite within the simulation box two ensemble averaging ðk1Þ, i.e., eðk1Þ¼h ðk1Þi. It is important to note that we ensure that the run times relatively thin slabs of thickness Lboundary along the propagation direction. employed in step 1 are sufficiently long such that the computed effective dielectric constants achieve stable and accurate steady-state values. We emphasize that the result (1) For a given medium, we obtain the steady-state ε ðk1Þ obtained from Eq. (87) in step 2 is nonlocal in space r ω spatial distributions of electric field Eyð ; Þ and because it is calculated from the nonlocal constitutive electric displacement field D r; ω at a given ˜ ˜ yð Þ relation D ðk ; ωÞ¼ε ðk1ÞE ðk ; ωÞ. For periodic media, ω y e y e frequency (or the corresponding wave number step 3 is unnecessary because all configurations are k1). Specifically, the planar source generates Gaus- identical. sian pulses of electric fields that propagate along the xˆ direction with wave number k1 that spans between VIII. COMPARISON OF SIMULATIONS min½k1 and max½k1. Using the aforementioned OF εeðk1Þ TO VARIOUS APPROXIMATIONS MEEP package [98], we compute time evolution of FORMULAS electric field E ðr;tÞ and electric displacement field y pffiffiffiffiffi D ðr;tÞ for a period of time 6π ε1=fc min½k1g, In this section, we compare our simulations of the y ε k where c is the speed of light in vacuum. We then effective dynamic dielectric constant eð 1Þ for various compute the temporal Fourier transforms of these 2D and 3D ordered and disordered model micro- fields inside packings. The values of the simulation structures to the predictions of the strong-contrast formulas parameters (indicated in Fig. 7) for the 2D and 3D as well as to conventional approximations, such as ordered and disordered models studied in this article MGA (17) and QCA (18). Most of these models provide are summarized in the Supplemental Material [81]. stringent tests of the predictive power of the approximations at finite wave numbers because they are characterized (2) At each value of k1, we postprocess E ðr; ωÞ and y by nontrivial spatial correlations at intermediate length Dyðr; ωÞ to estimate the effective dielectric constant ε ω scales. ðk1; Þ of a single configuration by solving the following self-consistent equation: A. 2D and 3D periodic media ε ε We first carry out our FDTD simulations for the effective ðk1Þ¼ ðk1Þ; ð87Þ dynamic dielectric constant εeðk1Þ of 2D and 3D periodic packings (square and simple-cubic lattice packings), which where necessarily depends on the direction of the incident wave k1.

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2 While these periodic packings are macroscopically isotropic, Simple cubic lattice packing ]

1 φ = 0.25, ε /ε = 4 due to their cubic symmetry, they are statistically anisotropic ε 1.8 2 2 1 )/ (see Supplemental Material [81] for details). For simplicity, 1 k ( k e we consider only the case where 1 is aligned with one of the ε 1.6 Γ minimal lattice vectors, i.e., the -X direction in the first Re[ Brillouin zone. Such periodic models enable us to validate 1.4 FDTD simulations our simulations because εeðk1Þ also can be accurately ] Maxwell-Garnett approx. 1 extracted from the lowest two photonic bands that are ε Quasicrystalline approx.

)/ 0.2 1 Strong-contrast approx. calculated via MPB, an open-source software package k ( e Scaled strong-contrast approx. [100]. The results from the band-structure calculations ε 0.1 and our FDTD simulations show excellent agreement. In Im[ particular, our simulations accurately predict two salient 0 0123 dielectric characteristics that must be exhibited by periodic k L packings: transparency up to a finite wave number associated 1 (a) with the edge of the first Brillouin zone (i.e., Im½εe¼0 for 0 ≤ jk1j ≲ π), and resonancelike attenuation due to Square lattice packing Bragg diffraction within the photonic band gap [101] ] 1 1.6 φ = 0.25,ε /ε = 4 (i.e., a peak in Im½εe or, equivalently, a sharp transition 2 2 1 ) / ε 1 in Re½ e [102]). Thus, our numerical homogenization k ( e scheme is valid down to intermediate wavelengths (see εε 1.4

the Supplemental Material [81] for comparison of the band- Re[ structure and FDTD computations). 1.2 FDTD simulations

Importantly, while our strong-contrast approximations ] 1 Maxwell-Garnett approx. ε Eqs. (67) and (73) account for directionality of the incident 0.2 Strong-contrast approx. ) /

1 Scaled strong-contrast approx.

waves, the MGA and QCA are independent of the direction k ( k e of 1. In Fig. 8, the FDTD simulation results are compared ε 0.1 2

with the MGAs for d ¼ , 3 [Eqs. (17) and (16)], QCA (18) Im[ for d ¼ 3, as well as the unscaled and scaled strong- 0 contrast approximations (67) and (73) for d ¼ 2, 3. While 0123 k L all approximations agree with the FDTD simulations in the 1 quasistatic regime, the MGA and QCA fail to capture (b) properly two key features: no loss of energy up to a finite wave number and resonancelike attenuation in the band FIG. 8. Comparison of the predictions of the strong-contrast gaps. Each strong-contrast approximation captures both of formulas Eqs. (67) and (73) to the Maxwell-Garnett [Eqs. (17) and (16)] and QCA (18) approximations for the effective dynamic these salient characteristics. However, it is noteworthy that dielectric constant ε ðk1Þ as a function of the dimensionless wave the scaled strong-contrast approximation [Eq. (73)] agrees e number k1L of periodic packings to our corresponding computer very well with the FDTD simulations. For contrast ratios simulation results. We consider (a) 3D simple cubic lattice and ε ε 1 2= 1 < , FDTD simulations are also in very good agree- (b) 2D square lattice of packing fraction ϕ2 ¼ 0.25 and contrast ment with the predictions of strong-contrast approxima- ratio ε2=ε1 ¼ 4. Here, k1 is the wave number in the reference tions for a wide range of wave numbers, as detailed in the (matrix) phase along the Γ-X direction, and L is the side length of Supplemental Material [81]. a unit cell.

B. Disordered nonhyperuniform any microstructural information, except for the particle and hyperuniform media shape, and thus cannot account for long-range correlations, To test the predictive capacity of approximation formulas such as the lossless property of stealthy hyperuniform for εeðk1Þ for disordered media as measured against media. By contrast, while the QCA formula yields better simulations, we choose to study two distinctly different estimates of Im½εe for nonhyperuniform systems, it cannot models: disordered nonhyperuniform packings (equilib- generally capture the correct transparency characteristics rium packings) and disordered stealthy hyperuniform dis- of hyperuniform systems, e.g., it incorrectly predicts χ˜ 0 0 ≤ 1 5 ε 0 ordered packings [i.e., V ðQÞ¼ for Qa < . ] for Im½ eðk1Þ ¼ for all wave numbers, regardless of whether both 2D and 3D. Again, we compare our simulation results the medium is stealthy hyperuniform or nonstealthy hyper- to the MGAs [Eqs. (17) and (16)], QCA (18), strong- uniform; see Fig. 9(b). contrast approximation (67), and the scaled counterpart On the other hand, the scaled strong-contrast approxi- (73). The conventional approximations fail to capture mation provides excellent estimates of εeðk1Þ for spatial dispersion effects. Specifically, the MGA neglects both disordered models, even for large wave numbers

021002-18 NONLOCAL EFFECTIVE ELECTROMAGNETIC WAVE … PHYS. REV. X 11, 021002 (2021)

1.6 IX. PREDICTIONS OF STRONG-CONTRAST ] 3D equilibrium packing 1 ε φ = 0.25, ε /ε = 4 APPROXIMATIONS FOR DISORDERED 1.55 2 2 1 ) /

1 PARTICULATE MEDIA k ( e 1.5 ε Having established the accuracy of the scaled strong-

Re[ 1.45 contrast approximation (73) for ordered and disordered media in the previous section, we now apply it to the four 0.15 FDTD simulations ]

1 Maxwell-Garnett approx. different disordered models discussed in Sec. V in order to ε Quaiscrystalline approx. study how ε ðk1Þ varies with the microstructure. We first ) / 0.1 Strong-contrast approx. e 1

k Scaled strong-contrast approx. study how ε ðk1Þ varies with k1 at a fixed contrast ratio ( e

e ε 0.05 ε2=ε1 ¼ 10 for the four models; see Fig. 10. According

Im[ to Eq. (84), nonhyperuniform and hyperuniform media in 0 the quasistatic regime have the different scalings, i.e., 0 0.2 0.4 0.6 0.8 1 ε ∼ d ε ∼ dþα k a Im½ eðk1Þ k1 and Im½ eðk1Þ k1 , respectively, 1 α 0 (a) where > for hyperuniform systems. This implies that hyperuniform media are less lossy than their nonhyperuni- 1.6 form counterparts as k1 tends to zero, as seen in the insets of

] 3D stealthy hyperuniform packing

1 Fig. 10. Moreover, beyond the quasistatic regime, each ε φ = 0.25, ε /ε = 4 1.55 2 2 1 ) /

1 2.1 k d = 3, φ = 0.25, ε/ε = 10 ( 1.5 221 e ] ε 1 1.8 ε

/ Re[ 1.45 Overlapping spheres e ε 1.5 Equilibrium packing 0.15 FDTD simulations Stealthy hyperuniform; Q a = 1.5 ]

Re[ U 1 Maxwell-Garnett approx. ε 1.2 Hyperuniform polydisperse packing Quasicrystalline approx.

) / 0.1 Strong-contrast approx. 1 k

Scaled strong-contrast approx. ] (

e 1 ε -1 ε 0.05 0.4 10 / 3 e ε

Im[ -4 0 0.2 10

Im[ 7 0 0.2 0.4 0.6 0.8 1 -7 10 0.1 1 k a 0 1 0 0.5 1 1.5 2 2.5 (b) ka1 FIG. 9. Comparison of the predictions of the strong-contrast (a) formulas Eqs. (67) and (73) to the MGA (16) and QCA (18) 1.6 d = 2, φ = 0.25, ε/ε= 10 approximations for the effective dynamic dielectric constant 221 ε ] eðk1Þ as a function of the dimensionless wave number k1a 1 ε

of 3D disordered sphere packings to our corresponding / e Overlapping disks computer simulation results. We consider (a) equilibrium pack- ε Equilibrium packing χ˜ 0 1.4 Stealthy hyperuniform; Q a = 1.3 ings and (b) stealthy hyperuniform packings [ V ðQÞ¼ for Re[ U Hyperuniform polydisperse packing 0 ≤ Qa < 1.5] of sphere radius a, packing fraction ϕ2 ¼ 0.25, and phase-contrast ratio ε2=ε1 ¼ 4. Here, k1 is the wave number

] 0.2

in the reference (matrix) phase, and the error bars in the FDTD 1 ε -1 10 simulations represent the standard errors over independent / 2 e ε 0.1 -4 configurations. 10

Im[ -7 6 10 0 0.01 0.1 (0 ≤ k1a ≤ 1); see Fig. 9. Moreover, the predictions of both 0 0.5 1 1.5 2 strong-contrast approximations accurately capture the ka salient microstructural differences between the nonhyper- 1 uniform and hyperuniform models because they incorpo- (b) rate spatial correlations at finite wavelengths via the χ˜ FIG. 10. Predictions of the scaled strong-contrast approxima- spectral density V ðQÞ. For example, they properly predict tion (73) for the effective dynamic dielectric constant ε ðk1Þ as a that stealthy hyperuniform media are lossless up to a finite e function of the dimensionless wave number k1a of the four wave number, even if at different cutoff values; see models of disordered media at volume fraction ϕ2 ¼ 0.25 and Eq. (86). Corresponding 2D results are presented in the contrast ratio ε2=ε1 ¼ 10: (a) three dimensions and (b) two Supplemental Material [81] because they are qualitatively dimensions. The inset in the lower panel is the log-log plot of the same as the 3D results. the larger panel.

021002-19 SALVATORE TORQUATO and JAEUK KIM PHYS. REV. X 11, 021002 (2021)

1/2 Stealthy hyperuniform packings, ε/ε = 4 d = 3, φ = 0.25, (ε /ε ) k a = 0.7 0.85 2 1 2 HS 1 1 0.16 1 Overlapping spheres c

Equilibrium packings ) / 0.8 φ = 0.40, Q a = 1.5 1 2 U

Q a k ] Stealthy hyperuniform packings; = 1.5 φ = 0.25, U ( Q a = 1.33

0.12 e 2 U 1

Hyperuniform polydisperse packings c ε 0.75 / e ε 0.08 0.001 1

c 0.06 Im[

) / 0

0.04 1 k (

e 0.03 γ 0.5 0.6 0.7 0.8 - 0 0 0246 810 012 ε/ε k a 21 1 (a) FIG. 12. Predictions of the scaled strong-contrast approxima-

1/2 tion (73) for the effective wave speed ce and the negative of the 0.15 d = 2, φ = 0.25, (ε /ε ) k a = 0.6 2 HS 1 1 attenuation coefficient γe as a function of the dimensionless wave Overlapping disks number k1a for 3D stealthy hyperuniform sphere packings Equilibrium packings of contrast ratio ε2=ε1 ¼ 4 at two different packing fractions: ] Stealthy hyperuniform dispersions; QU a = 1.3 1 0.1 ϕ2 ¼ 0.4 with Q a ¼ 1.5 and ϕ2 ¼ 0.25 with Q a ≈ 1.33. The ε Hyperuniform polydisperse packings U U inset is a magnification of the lower panel. / e ε

Im[ 0.05 We now examine how the imaginary part Im½εe varies with the contrast ratio ε2=ε1 for the disordered models for a given large wave number k1 inside the transparency interval 0 for 2D and 3D stealthy hyperuniform systems. These 024 6 810 results are summarized in Fig. 11. The disparity in the ε/ε attenuation characteristics across microstructures widens 21 (b) significantly as the contrast ratio increases. Clearly, overlapping spheres are the lossiest systems. Hyperuniform FIG. 11. Predictions of the strong-contrast approximation (73) polydisperse packings can be nearly as lossless as stealthy for the effective dynamic dielectric constant εeðk1Þ as a function hyperuniform ones. of the dielectric-contrast ratio ε2=ε1 for the four disordered We also study the effect of packing fraction ϕ2 on the ϕ 0 25 models, as perp Fig.ffiffiffiffiffiffiffiffiffiffiffiffiffiffi10, at packing fraction 2 ¼ . and wave effective phase speed ceðk1Þ and effective attenuation coef- ε ε 0 7 γ numberpffiffiffiffiffiffiffiffiffiffiffiffiffiffi (a) HS= 1k1a ¼ . in three dimensions and ficient eðk1Þ, as defined by Eqs. (55) and (56), respectively. ε ε 0 6 (b) HS= 1k1a ¼ . in two dimensions. For concreteness, we focus on 3D stealthy hyperuniform packings. We first generate such packings at a packing fraction ϕ2 ¼ 0.4 and QUa ¼ 1.5, as described in Sec. VD. model exhibits effective transparency for a range of wave Without changing particle positions, we then shrink particle numbers that depends on the microstructure. For 2D and 3D radii to attain a packing fraction ϕ2 ¼ 0.25, whose stealthy models, hyperuniform polydisperse packings tend to be regime is now Q a ≈ 1.33. The coefficients c ðk1Þ and effectively transparent for a wide range of wave numbers U e γeðk1Þ for these packings with ε2=ε1 ¼ 4 are estimated from compared to the nonhyperuniform ones, while the stealthy the scaled approximation (73); see Fig. 12. It is seen that the hyperuniform systems are perfectly transparent for thewidest waves propagate significantly more slowly through the range of wave numbers, as established in Eq. (86) and denser medium due to an increase in multiple-scattering Sec. VIII. For each model, the effective transparency spectral events. Moreover, the transparency intervals (wave-number range must be accompanied by normal dispersion [i.e., an ranges where the effective attenuation coefficients vanish) ε increase in Re½ eðk1Þ with k1] [32] because our strong- are larger for the packing with the higher stealthy cutoff value contrast approximation is consistent with the Kramers- Q a ¼ 1.5a (ϕ2 ¼ 0.4), as predicted by Eq. (86). Kronig relations (see Appendix B). Moreover, we see that U anomalous dispersion [i.e., a decrease in Re½ε ðk1Þ with k1] e X. DISCUSSION occurs at wave numbers larger but near the respective transition between the effective transparency and appreciable All previous closed-form homogenization estimates of the attenuation, which again is dictated by the Kramers-Kronig effective dynamic dielectric constant apply only at long relations. The specific anomalous dispersion behavior is wavelengths (quasistatic regime) and for very special mac- microstructure dependent. roscopically isotropic disordered composite microstructures,

021002-20 NONLOCAL EFFECTIVE ELECTROMAGNETIC WAVE … PHYS. REV. X 11, 021002 (2021) namely, nonoverlapping spheres or spheroids in a matrix. In computationally expensive full-blown simulations. Thus, our this work, we lay the theoretical foundation that enables us to nonlocal formulas can be used to accelerate the discovery of substantially extend previous work in both its generality and novel electromagnetic composites by appropriate tailoring of applicability. First, we derive exact homogenized constitutive the spectral densities and then constructing the corresponding relations for the effective dynamic dielectric constant tensor microstructures by using the Fourier-space inverse methods εeðkqÞ that arenonlocal in space from first principles.Second, [36]. For example, from our findings in the present study, it is clear that stealthy disordered particulate media can be our strong-contrast representation of εeðkqÞ exactly accounts for complete microstructural information (infinite set of n- employed as low-pass filters that transmit waves isotropically point correlation functions) for arbitrary microstructures and up to a selected wave number. Moreover, using the spectral hence multiple scattering to all orders for the range of wave densities of the type found by Chen and Torquato [36] for numbers for which our extended homogenization theory stealthy hyperuniform packings (characterized by a peak value at Q Q with intensities that rapidly decay to zero for larger applies, i.e., 0 ≤ jk jl ≲ 1 (where l is a characteristic ¼ U q wave numbers) and formula (73), one can design materials heterogeneity length scale). Third, we extract from the exact with refractive indices that abruptly change over a narrow expansions accurate nonlocal closed-form approximate for- ε k range of wave numbers. Of course, one could also explore the mulas for eð qÞ, relations (67) and (73), which areresummed design space of effective wave properties of nonhyperuniform representations of the exact expansions that incorporate disordered composite media for potential applications. microstructural information through the spectral density Previously, disordered media were often described using χ˜ Q V ð Þ, which is easily ascertained for general microstructures cluster expansions [17,20,33], while ordered media were either theoretically, computationally, or via scattering experi- often studied through dispersion relations and band-struc- ments. Depending on whether the high-dielectric phase tures calculations. Accordingly, it has been of primary percolates or not, the wide class of microstructures that we importance to bridge the gap between the treatments of can treat includes particulate media consisting of identical or ordered and disordered to better understand the optical polydisperse particles of general shape (ellipsoids, cubes, properties of correlated media. Thus, our work represents cylinders, polyhedra) with prescribed orientations that may or an initial step toward a unified theory to describe the not overlap, cellular networks, as well as media without well- effective optical properties of both ordered and disordered defined inclusions (Sec. III B). Our approximations account microstructures over a wide range of incident wavelengths. for multiple scattering across a range of wave numbers. There are a variety of directions for future research. First, Fourth, we carry out precise full-waveform simulations for our formalism can be straightforwardly extended to hetero- various 2D and 3D models of ordered and disordered media to geneous materials composed of more than two phases or validate the accuracy of our nonlocal microstructure-depen- continuous media. Second, it is also of interest to extend dent approximations for wave numbers well beyond the our formalism to applications and phenomena (e.g., mag- quasistatic regime. netic effects) relevant to the smaller wavelengths noted for Having established the accuracy of the scaled strong- metamaterials [103–105]. contrast approximation (73), we then apply it to four models of 2D and 3D disordered media (both nonhyperuniform and ACKNOWLEDGMENTS hyperuniform) to investigate the effect of microstructure on the effective wave characteristics. Among other findings, we We thank Z. Ma, M. Klatt, Y. Chen, and L. Dal Negro for show that disordered hyperuniform media are generally less very helpful discussions. The authors gratefully acknowl- lossy than their nonhyperuniform counterparts. We also find edge the support of Air Force Office of Scientific Research that our scaled formula (73) accurately predicts that disor- Program on Mechanics of Multifunctional Materials and dered stealthy hyperuniform media possess a transparency Microsystems under Grant No. FA9550-18-1-0514. 0 0 5 ε ε −1=2 wave-number interval ½ ; . QUð HS= qÞ [cf. Eq. (86)], where most nonhyperuniform disordered media are opaque. APPENDIX A: DIFFERENT EXPANSIONS AS A RESULT OF DIFFERENT Note that, using multiple-scattering simulations, Leseur et al. EXCLUSION-REGION SHAPES [63] were the first to show that stealthy hyperuniform systems should exhibit a transparency interval, but for To get a sense of how the resulting expansions change due “point” scatterers, not the finite-sized scatterers considered to the choice of the exclusion-region shape, we consider the here. Interestingly, their transparency-interval prediction aforementioned oriented spheroidal exclusion region in the coincides with the one predicted by our less accurate two limiting disklike and needlelike cases. Comparing the strong-contrast formula [cf. Eq. (86)]. expansion parameters in the limit cases given in Eq. (32) to The accuracy of our nonlocal closed-form formulas has the strong-contrast expansion with a spherical exclusion important practical implications, since one can now use them region given in Eq. (54), one can obtain the counterparts of to accurately and efficiently predict the effective wave Eq. (54) with disklike and needlelike exclusion regions. characteristics well beyond the quasistatic regime of a wide Specifically, we replace the parameters in Eq. (54) class of composite microstructures without having to perform according to the following mappings:

021002-21 SALVATORE TORQUATO and JAEUK KIM PHYS. REV. X 11, 021002 (2021)

εeðkqÞþðd − 1Þεq dεeðkqÞ βpq → ðεp − εqÞ=ðdεpÞ; → ; disklike; εeðkqÞ − εq εeðkqÞ − εq

εeðkqÞþðd − 1Þεq dεq βpq → ðεp − εqÞ=ðdεqÞ; → ; needlelike; εeðkqÞ − εq εeðkqÞ − εq resulting in the following expansions, respectively,

∞ ε −ε 2 dε ðk Þ ε −ε X ε −ε n ϕ 2 p q e q ¼ ϕ p q − AðpÞðk ;A ¼ 1Þ p q ; ðA1Þ p dε ε k −ε p dε n q dε p eð qÞ q p n¼2 p ε − ε 2 dε ε − ε X∞ ε − ε n ϕ 2 p q q ¼ ϕ p q − AðpÞðk ; A ¼ 0Þ p q : ðA2Þ p dε ε k − ε p dε n q dε q eð qÞ q q n¼2 q

ðpÞ Here the functionals An ðkq; A Þ are identical to Eqs. (51) the two-point level, yielding Eq. (67). Here we analytically and (52), except for the exclusion-region shape. show that the effective dielectric constant εeðkqÞ for iso- In the Supplemental Material [81], we discuss how to tropic media obtained from either the unscaled or scaled obtain the analogs of Eqs. (A1) and (A2) that apply to strong-contrast formulas [see Eqs. (67) and (73)] also macroscopically anisotropic media. The corresponding satisfies the Kramers-Kronig relations. series expansions involve tensorial expansion parameters We begin by rewriting either strong-contrast approxi- −1 that have rapid convergence properties for stratified and mation as εeðkqÞ ≈ εq þ½a þ bFðkqÞ , where a and b are transversely isotropic media, respectively. nonzero real numbers. The general analytic properties of the nonlocal attenuation function FðQÞ (detailed in the APPENDIX B: KRAMERS-KRONIG RELATIONS Supplemental Material [81]) induce εeðkqÞ to have the Kramers-Kronig relations connect the real and imaginary following three properties necessary to satisfy the Kramers- ε parts of any complex function that is analytic in the upper Kronig relations (B1) and (B2): (i) eðkqÞ is an analytic half-plane and meets mild conditions [106,107]. Since function in the upper half-plane of kq, (ii) εðkqÞ − εq causality in a dielectric response function of a homo- vanishes like 1=jkqj as jkqj goes to infinity, and geneous material implies such analyticity properties, the (iii) Re½εeðkqÞ and Im½εðkqÞ are even and odd functions Kramers-Kronig relations enable one to directly link the of kq, respectively. Property (i) is valid if a þ bFðkqÞ ≠ 0, real part of a response function to its imaginary part or vice which is met for all disordered systems considered here. versa, even if the real or imaginary parts are only available The fact that the strong-contrast approximations satisfy in a finite frequency range [106,107]. Thus, when a Eqs. (B1) and (B2) makes physical sense since FðQÞ heterogeneous material can be treated as a homogeneous involves GðqÞðx; x0Þ [cf. Eq. (24)], which is the temporal ε k material with a dynamic effective dielectric constant eð qÞ, Fourier transform of the retarded Green’s function Kramers-Kronig relations immediately apply to the exact GðqÞ x;t;x0;t0 [106] that accounts for causality. We also strong-contrast expansion (63), i.e., ð Þ numerically show in the Supplemental Material [81] that Z 2 ∞ ε our approximations obey Eqs. (B1) and (B2). ε ε ∞ qIm½ eðqÞ Re½ eðkqÞ ¼ eð Þþπ p:v: dq 2 2 ; ðB1Þ 0 q − kq Z APPENDIX C: STRONG-CONTRAST 2k ∞ Re½ε ðqÞ − ε ð∞Þ ε k − q : : dq e e ; APPROXIMATION AT THE Im½ eð qÞ ¼ π p v 2 − 2 ðB2Þ 0 q kq THREE-POINT LEVEL where we assume a linear dispersion relation in the reference Here we present the strong-contrast approximation at the pffiffiffiffiffi phase (i.e., kq ¼ εqω=c) and limω→∞ εeðωÞ¼εq is real three-point level for a spherical exclusion region. It is ðpÞ 0 ≥ 4 valued. The Kramers-Kronig relations may or may not be obtained from Eq. (54) by setting An ¼ for n and by ε k obeyed when the strong-contrast expansion is truncated at solving it in eð qÞ:

ε ðk Þ dβ ϕ 2 e q ¼ 1 þ pq p ; ðC1Þ ε d=2 2 ðpÞ q ϕpð1 − βpqϕpÞþðd − 1Þπ=½2 Γðd=2ÞβpqFðkqÞ − βpq A3 ðkqÞ

021002-22 NONLOCAL EFFECTIVE ELECTROMAGNETIC WAVE … PHYS. REV. X 11, 021002 (2021) where the local three-point parameter is given as Z ε 2 p ðd qÞ 1 p ð Þ ðqÞ −ikq·ðx1−x2Þ ðqÞ −ikq·ðx2−x3Þ ð Þ A3 ðkqÞ ≡ − dx1dx2 Tr½H ðx1 − x2Þe · H ðx2 − x3Þe Δ3 ðx1; x2; x3ÞðC2Þ ϕp ϵ d Z 1 1 1 − q q −1 2 4 2 2 qˆ qˆ 2 −1 Δ˜ ðpÞ q k q k ¼ 2d d 1d 2 2 2 2 2 fðd Þ kq þq1 q2 ½dð 1 · 2Þ g 3 ð 1 þ q; 2 þ qÞ; ðC3Þ ϕpð2πÞ q1 −kq q2 −kq where, due to the statistical homogeneity, Z

˜ ðpÞ −iq1·r1 −iq2·r2 ðpÞ ðpÞ ðpÞ Δ3 ðq1; q2Þ ≡ dr1dr2e e ½S2 ðr1ÞS2 ðr2Þ − ϕpS3 ðr1; r2Þ:

Note that Eq. (C3) is obtained from Eq. (C2) via Parseval’s theorem. The static limit of Eq. (C2) for statistically isotropic media is given by [11] ZZ 2 r s ðpÞ 0 − 1 ϕ 1 − ϕ ζ d d d rˆ sˆ 2 − 1 ðpÞ − ðpÞ ðpÞ ϕ A3 ð Þ¼ðd Þ pð pÞ p ¼ d d ½dð · Þ ½S3 ðr; s; tÞ S2 ðrÞS2 ðsÞ= p; ðC4Þ Ωd r s

d where Ωd is the surface area of a unit sphere in R , t ≡ jr − sj and the parameter ζp lies in the closed interval [0, 1].

APPENDIX D: STRONG-CONTRAST FORMULA ε1 þ ε2 1 εeðωÞ¼ þ ð−dðε1ϕ2 þ ε2ϕ1Þ FOR PHASE-INVERSION SYMMETRIC MEDIA 2 2A2ðωÞ

To further illustrate the flexibility and power of the þf4A2ðωÞ½d − A2ðωÞε1ε2 strong-contrast formalism, we apply it to treat media with 2 1=2 þ½ðε1 þ ε2ÞA2ðωÞ − dðϕ2ε1 þ ϕ1ε2Þ g Þ; ðD2Þ phase-inversion symmetry. A two-phase medium possesses phase-inversion symmetry if the morphology of phase 1 at volume fraction ϕ1 is statistically identical to that of phase ð1Þ ð2Þ where A2ðωÞ ≡ d − 1 þ A2 ðk2Þ=ϕ1 þ A2 ðk1Þ=ϕ2, and 2 in the system when the volume fraction of phase 1 is we choose the physically meaningful solution from the 1 − ϕ1 [11]. In particular, we follow the same procedure quadratic equation. Note that in the static limit, A2ðωÞ used by Torquato [11] to derive strong-contrast expansions designed to apply to media with phase-inversion symmetry φ ε /ε in the static limit and which can be regarded as expansions ] = 0.25, = 4 1 221 ε that perturb around the microstructures corresponding to 1.5 ) / the “self-consistent” formula. All we need to do is add the 1 k ( expansion (54) with p ¼ 2 and q ¼ 1 to that with p ¼ 1 e ε and q ¼ 2: 1.45 Re[

ε ðωÞþðd − 1Þε1 ε ðωÞþðd − 1Þε2 ϕ e e ] 0.09 2 þ ϕ1 1 3D Debye random media ε ω − ε ε ω − ε ε eð Þ 1 eð Þ 2 3D equilibrium packing

∞ 2 1 ) / 0.06 X ð Þ ð Þ 1 An ðk1Þ −2 An ðk2Þ −2 k 2 − − β n β n ( ¼ d 21 þ 12 ; ðD1Þ e ϕ2 ϕ1 ε n¼2 0.03 Im[ where the term (2 − d) on the right-hand side is obtained 0 β −1 β −1 0 0.2 0.4 0.6 0.8 1 from 21 þ 12 . Since the exact expansion (54) is k a independent of choice of reference phase and because 1 the phase-inversion symmetry places each phase on the FIG. 13. Evaluation of the effective dielectric constant ε as a same footing, we can write the effective dielectric e function of the dimensionless wave number k1a for 3D equilib- ε ω constant independent of the reference phase, i.e., eð Þ¼ rium packings and 3D Debye random media of volume fraction ε ( ω ω) ε ( ω ω) pffiffiffiffiffi e k1ð Þ; ¼ e k2ð Þ; . Truncating Eq. (D1) at the ϕ2 ¼ 0.25 and contrast ratio ε2=ε1 ¼ 4. Here, k1 ≡ ε1ω=c is two-point level and solving it in εeðωÞ yields the following the wave number in the matrix (reference) phase 1, and a is approximation: particle radius.

021002-23 SALVATORE TORQUATO and JAEUK KIM PHYS. REV. X 11, 021002 (2021) converges to d − 1, and thus, Eq. (D2) reduces to the well- [14] B. U. Felderhof, G. W. Ford, and E. G. D. Cohen, Cluster known self-consistent formula [11]. Expansion for the Dielectric Constant of a Polarizable Examples of phase-inversion symmetric media include Suspension, J. Stat. Phys. 28, 135 (1982). the d-dimensional random checkerboard [11] and what has [15] A. K. Sen and S. Torquato, Effective Conductivity of been called Debye random media [29,30,75]. Here we Anisotropic Two-Phase Composite Media, Phys. Rev. B 39, 4504 (1989). apply the approximation (D2) to Debye random media, [16] S. Torquato, Exact Expression for the Effective Elastic which are defined by their autocovariance function [29,30] Tensor of Disordered Composites, Phys. Rev. Lett. 79, 681 (1997). χ ϕ ϕ −r=r0 V ðrÞ¼ 1 2e ; ðD3Þ [17] P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic Press, New York, 1995). where a positive quantity r0 represents a characteristic 2 [18] A. Sihvola, Electromagnetic Mixing Formulas and length scale. Here, we take r0 ¼ a= . Applications (IET Digital Library, London, 1999). Figure 13 compares the effective dynamic dielectric [19] J. B. Keller, Stochastic Equations and Wave Propagation constant for 3D Debye random media to that of 3D in Random Media, Proc. Symp. Appl. Math. 16, 145 equilibrium hard spheres as predicted from the strong- (1964). contrast approximations devised, respectively, for phase- [20] U. Frisch, in Probabilistic Methods in Applied Mathemat- inversion symmetric media [Eq. (D2)] and dispersions of ics, 1st ed., edited by A. T. Bharucha-Reid (Academic particles [Eqs. (67)], which do not have such a symmetry. Press, New York, 1968), Vol. 1, pp. 75–198. We see that Debye random media are more lossy than [21] L. Tsang and J. A. 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