PHYSICAL REVIEW APPLIED 10, 034060 (2018)

Boundary Effects of Weak Nonlocality in Multilayered Dielectric Metamaterials

Giuseppe Castaldi,1 Andrea Alù,2,3,4 and Vincenzo Galdi1,*1Fields & Waves Lab, Department of Engineering, University of Sannio, 82100 Benevento, Italy

2Photonics Initiative, Advanced Science Research Center, City University of New York, New York, New York

10031, USA3Physics Program, Graduate Center, City University of New York, New York, New York 10026, USA

4Department of Electrical Engineering, City College of New York, New York, New York 10031, USA

(Received 22 December 2017; revised manuscript received 26 May 2018; published 26 September 2018)

Nonlocal (spatial-dispersion) effects in multilayered metamaterials composed of periodic stacks of alter-nating, deeply subwavelength dielectric layers are known to be negligibly weak. Counterintuitively, undercertain critical conditions, weak nonlocality may build up strong boundary effects that are not captured byconventional (local) effective-medium models based on simple mixing formulas. Here we show that thisphenomenon can be fruitfully studied and understood in terms of error propagation in the iterated maps ofthe trace and antitrace of the optical transfer matrix of the multilayer. Our approach effectively parame-terizes these peculiar effects via remarkably simple and insightful closed-form expressions, which enabledirect identification of the critical parameters and regimes. We also show how these boundary effects canbe captured by suitable nonlocal corrections.

DOI: 10.1103/PhysRevApplied.10.034060

I. INTRODUCTION

Away from the quantum regime, the macroscopicelectromagnetic response of material media is typicallymodeled via constitutive relationships featuring a set ofintensive properties, such as dielectric permittivity, electri-cal conductivity, and magnetic permeability [1]. Althoughthese quantities clearly depend on the fine (atomic andmolecular) structure of the medium, they do not punctuallydescribe the strong field fluctuations on such fine scales,but describe only some suitably averaged behavior.

Besides being a cornerstone of the electrodynamics ofcontinuous media [1], the above-mentioned homogeniza-tion concept is also heavily applied to the descriptionof “metamaterials” (i.e., composite materials made ofsubwavelength-sized (dielectric or metallic) inclusions in ahost medium), which can be purposely designed to exhibitspecific desired properties [2,3].

In their arguably simplest conceivable form, homoge-nized models are based on mixing formulas (e.g., MaxwellGarnett formula) that essentially depend on the inclu-sions’ material constituents as well as their shapes, orien-tations, and filling fractions, but not on their specific sizesand spatial arrangement [4]. Such effective-medium the-ory (EMT) is known to work especially well for dielectricstructures featuring electrically small inclusions, whereasit may become significantly inaccurate in the presence

*vgaldi@unisannio.it

of metallic constituents and or inclusions with moderateelectrical sizes. In these last cases, nonlocal corrections (inthe form of spatial derivatives of the fields or, equivalently,wavevector dependence in the constitutive relationships)are typically needed to account for the arising spatialdispersion (see, e.g., Refs. [5–9]).

Contrary to the conventional wisdom above, HerzigSheinfux et al. [10] recently pointed out a deceptively sim-ple example of an all-dielectric multilayered metamaterialfeaturing deeply subwavelength layers that may exhibitpeculiar boundary effects that are not captured by stan-dard (local) EMT approaches. More specifically, undercertain critical illumination conditions, the optical trans-mission (and reflection) of a finite-thickness slab of suchmetamaterial may differ substantially from the local EMTprediction, and may become ultrasensitive to the spatialorder and/or size of the layers as well as to the additionor removal of a very thin layer (see also Ref. [11]). Thesecounterintuitive effects, experimentally demonstrated byZhukovsky et al. [12], have been attributed to the pecu-liar (interface-dominated) phase-accumulation mechanismin the structure [10], and have been shown to be potentiallycaptured by suitable (possibly nonlocal and bianisotropic)corrections [13,14].

Besides the inherent academic interest in the homoge-nization aspects, the ultrasensitivity phenomenon describedabove may open intriguing avenues in applications suchas optical sensing and switching [10,11]. Moreover, sim-ilar mechanisms have also been shown to play a key

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CASTALDI, ALÙ, and GALDI PHYS. REV. APPLIED 10, 034060 (2018)

role in establishing Anderson localization in disorderednanophotonic structures [15,16].

Against the background above, in this paper, we proposea simple and physically incisive modeling of the boundaryeffects that can be induced by weak nonlocality in multilay-ered dielectric metamaterials. Our approach, based on thetrace-and-antitrace-map formalism [17], directly relatesthe geometrical and constitutive parameters of interest to aset of meaningful observables through simple closed-formexpressions. Besides elucidating the underlying mecha-nisms, this directly enables the identification of criticalparameters and regimes, and naturally suggests possiblenonlocal corrections capable of capturing the effects.

Accordingly, the rest of the paper is structured as fol-lows. In Sec. II, we introduce the problem geometry andbackground, and outline the mathematical formulation ofour approach. In Sec. III, we apply the approach to the ana-lytical modeling of the boundary effects induced by weaknonlocality, and identify two distinct mechanisms. More-over, we illustrate some representative results, and alsoexplore possible nonlocal corrections. Finally, in Sec. IV,we provide some concluding remarks and perspectives.Ancillary technical derivations are detailed in four appen-dices.

II. BACKGROUND AND MATHEMATICALFORMULATION

A. Geometry

Referring to the schematic in Fig. 1, as in Ref. [10],we consider a multilayered metamaterial consisting of aperiodic arrangement of alternating dielectric layers (ofinfinite extent in the x-y plane, and stacked along thez direction), with relative permittivities εa and εb andthicknesses da and db, respectively. More specifically, weconsider a structure of finite thickness L made of n unitcells (bilayers) embedded in a homogeneous medium withrelative permittivity εe, illuminated by a time-harmonic[exp(−iωt)], transverse-electric (TE) polarized plane wavewith y-directed electric field, impinging from the exte-rior medium at an angle θ with respect to the z axis.Accordingly, the relevant components of the impingingwavevector ke can be expressed as

kze = k√εe cos θ , kx = k√εe sin θ , (1)

with k = ω/c denoting the vacuum wavenumber (and c thecorresponding wave speed), and kx subject to momentumconservation [18]. In what follows, we assume there arepropagating fields in the exterior medium (i.e., real-valuedincidence angles θ ), lossless dielectric materials (i.e., real-valued, positive εa, εb, εe), and deeply subwavelength layerthicknesses (i.e., da, db � λ, with λ = 2π/k denoting thevacuum wavelength).

yz

x

EH

ka b

e

da db

d

Unit cell

e

M

Hxi Hx

o

Eyo Ey

i

e

L nd

FIG. 1. Problem geometry. A multilayered metamaterial com-posed of alternating dielectric layers, with relative permittivitiesεa and εb and thicknesses da and db, respectively. The structureis assumed to be of infinite extent in the x-y plane and of finitethickness (L = nd) along the z direction, and is embedded ina homogeneous medium with relative permittivity εe. The twoinsets on the left illustrate the transfer-matrix formalism [top, seeEq. (4)] and the TE-polarized plane-wave illumination (bottom).

As will be clearer hereafter, our assumption of consid-ering the same exterior medium at the two ends of themetamaterial slab, although slightly less general than thatin previous studies [10,13,14], allows an effective descrip-tion of the underlying physical mechanisms in terms of aminimal number of parameters. Dealing with a more gen-eral scenario featuring two different exterior media impliesonly formal complications, but it does not add significantphysical insight into the phenomenon.

B. Local EMT formulation

Following the standard (local) EMT formulation [4], theabove-mentioned multilayered metamaterial can be mod-eled in terms of a homogenized, uniaxially anisotropicmedium. For the assumed TE polarization, the relevant(in-plane) component of the resulting relative-permittivitytensor is given by [4]

ε̄‖ = faεa + fbεb, (2)

with fa = da/d and fb = db/d = 1 − fa denoting the fillingfractions of the two material constituents, and d = da + dbthe unit-cell thickness. Here and henceforth, the overbaris used to tag EMT-based quantities. In such an effectivemedium, the longitudinal wavenumber is given by

k̄z =√

k2ε̄‖ − k2x , (3)

with the (conserved) transverse component kx alreadydefined in Eq. (1).

C. Transfer-matrix formalism

Our approach exploits as a rigorous reference solutionthe well-known transfer-matrix method (see Chap. 1 inRef. [18]). As illustrated in the inset in Fig. 1, for the

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assumed TE polarization, the tangential field componentsat two interfaces of a layer can be related via

[E(i)y

iZeH (i)x

]= M ·

[E(o)y

iZeH (o)x

], (4)

where the superscripts (i) and (o) denote the input andoutput interfaces, respectively, M is a unimodular, dimen-sionless 2 × 2 transfer matrix, and

Ze = ωμ0kze (5)

represents the TE wave impedance in the exterior medium,with μ0 denoting the vacuum magnetic permeability andkze already defined in Eq. (1). When multiple layers arecascaded, by iterating the above representation, we canobtain the resulting transfer matrix via the chain productof the matrices representing the single layers (see Chap.1 in Ref. [18]). Thus, for example, the transfer matrixpertaining to a unit cell as in Fig. 1 is given by

Mab

= Ma· M

b, (6)

with the expressions of the matrices Ma

and Mb

explic-itly given in Appendix A. Likewise, the transfer matrixpertaining to a multilayer composed of n unit cells isstraightforwardly obtained via a nth power: viz.,

Mn

= Mnab

. (7)

From the transfer matrices in Eqs. (6) and (7), the opti-cal response of the multilayered metamaterial can be fullycharacterized, in terms of both a Bloch-type dispersionrelationship (in the infinite periodic limit) and transmissionand/or reflection (for a finite number of layers) (see Chap.1 in Ref. [18]).

D. Trace and antitrace maps

For the study of some fundamental aspects of the opti-cal response of a multilayered metamaterial, two suitablecombinations of the transfer-matrix elements, known as“trace” and “antitrace,” are sufficient [17]. For a generic2 × 2 matrix

M =[M11 M12M21 M22

], (8)

they are defined as [19]

Tr(M)

≡ M11 + M22, (9a)

Atr(M)

≡ M21 − M12, (9b)

respectively. For the periodic multilayer of interest here, ifwe let

χn ≡ Tr(Mn

ab

), υn ≡ Atr

(Mn

ab

), (10)

it can be shown [17] that the Bloch-type dispersion rela-tionship (infinite number of layers) is given by

cos (kzd) = χ12 . (11)

Moreover, in the assumed scenario featuring the sameexterior medium at the two ends (see Fig. 1), the trans-mission coefficient of a finite-size metamaterial slab com-prising n unit cells can be expressed as [17] (see alsoAppendix B for details)

τn = 2χn + iυn . (12)

It is worth highlighting that, in view of the assumed loss-less condition, the trace and antitrace in Eq. (10) are realvalued [17]. If the number n of unit cells is varied, theirevolution is governed by two iterated maps [17] (see alsoAppendix B for details),

χn = χ1χn−1 − χn−2, (13a)

υn = χ1υn−1 − υn−2, n ≥ 2, (13b)which are coupled through the initial (unit-cell) trace χ1.Remarkably, the above iterated maps admit a closed-formanalytical solution as [17] (see also Appendix B for details)

χn = Un−1(χ1

2

)χ1 − 2Un−2

(χ12

)

= Un(χ1

2

)− Un−2

(χ12

)= 2Tn

(χ12

),

(14a)

υn = Un−1(χ1

2

)υ1, (14b)

with Tn and Un denoting Chebyshev polynomials of thefirst and second kind, respectively (see Chap. 22 in [20]).

In spite of the different formalism adopted, the resultsin Eq. (12) [with Eqs. (14a) and (14b)] are exactly equiva-lent to those obtained via the conventional transfer-matrixmethod.

In what follows, we restrict our attention to the caseχ1 ≤ 2, which, recalling the dispersion relationship inEq. (11), corresponds to the propagating condition (real-valued kz) for the infinite multilayer. For finite-size struc-tures, of direct interest in our study, by recalling Eq. (14a)

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and that |Tn(ξ)| ≤ 1 for |ξ | ≤ 1 (see Eq. 22.14.4 in Ref.[20]), this also implies that

|χn| ≤ 2, n ≥ 1. (15)

We stress that our assumption of identical exterior mediaat the two ends of the metamaterial does not imply a sig-nificant loss of generality in the description of the basicphenomena. If two different exterior media were assumed,as in Refs. [10,13,14], it would no longer be possible toexpress the transmission coefficient solely in terms of thetrace and antitrace. Aside from the arising formal com-plications, this would not hinder the applicability of ourapproach, since it is possible to derive iterated maps for allthe terms of the transfer matrix [17].

Moreover, it is worth highlighting that the above trace-and-antitrace-map formalism is not restricted to periodicmultilayers, and can be extended to deal with rather gen-eral classes of aperiodically ordered structures generatedby two-letter substitutional sequences (e.g., Fibonacci andThue-Morse sequences) [17,21–23], although the arisingiterated maps do not generally admit analytical, closed-form solutions.

III. BOUNDARY EFFECTS OF WEAKNONLOCALITY

A. Background

In Ref. [10], for the case εe > ε̄‖, it was observed that theagreement between the exact optical response of a finite-thickness multilayered metamaterial and the local EMTprediction would strongly deteriorate for incidence anglesapproaching the critical angle

θc = arcsin(√

ε̄‖εe

), (16)

which characterizes the total-internal-reflection conditionbetween the exterior and effective media or, equivalently,the vanishing of the EMT-based longitudinal wavenum-ber k̄z in Eq. (3). Assuming θ � θc, in view of Eq. (2),this regime implies that the field is propagating in thehigher-permittivity layers and is evanescent in the lower-permittivity ones. In Ref. [14], an additional, independentmechanism was identified, leading to the breakdown of thestandard EMT. This latter mechanism is not restricted tothe critical-angle incidence above, but it is rather relatedto a phase mismatch at the interface separating the lastlayer and the exterior medium, and becomes particularlysignificant for εe = ε̄‖.

The breakdown of the local EMT model in theabove scenarios is rather counterintuitive, as fully dielec-tric structures with deeply subwavelength inclusions areknown to exhibit negligibly weak nonlocal effects. In fact,

such breakdown appears to be attributable to boundaryeffects, as it is manifested only in the transmission (andreflection) response of finite-size structures, whereas thebulk properties (dispersion relationship) are still accuratelycaptured by the local EMT prediction [10,12].

B. Connection with the trace-and-antitrace-mapformalism

To effectively model the phenomena described above, itis expedient to interpret the EMT homogenized structure asa fictitious multilayer composed of homogeneous unit cellsof thickness d and relative permittivity ε̄‖ in Eq. (2). Bycomparing the traces and antitraces of the transfer matri-ces pertaining to the actual and homogenized unit cells, weobtain (see Appendix C for details)

�χ1 = χ1 − χ̄1 ≈ − (kd)4(εa − εb)2f 2a f 2b

12, (17a)

�υ1 = υ1 − ῡ1

≈ k(kd)3(εa − εb)fafb[(εa + εb − 2εe)fb − εa + εe]

6kze,

(17b)

where, following the previously introduced notation,EMT-based quantities are indicated by an overbar. Theerrors in Eqs. (17a) and (17b) are manifestations ofthe inherent nonlocality of multilayered metamaterials.In structures such as hyperbolic metamaterials made ofmetallodielectric multilayers, these errors can be of siz-able magnitude even for deeply subwavelength layers [6],thereby leading to strong bulk effects, such as additionalextraordinary waves [24]. Conversely, for the case of fullydielectric, deeply subwavelength layers (kd � 1) of inter-est here, these errors are usually quite small, and areconsequently expected to yield second-order effects. Thisis especially true for the trace error in Eq. (17a), whereasEq. (17b) indicates that the antitrace error may become notso small for kze � k (i.e., for near-grazing incidence fromthe exterior medium).

We recall, from Eq. (11), that the dispersion relation-ship depends solely on the initial (unit-cell) trace χ1; thisexplains why, in the assumed conditions of deeply sub-wavelength dielectric layers, the EMT prediction of thebulk properties remains uniformly accurate even in thecritical regimes observed in Refs. [10,12,14]. This is notnecessarily the case for the transmission coefficient ofa finite-thickness structure [cf. Eq. (12)], which insteaddepends on both the trace map and the antitrace map. Evenin the case of negligibly weak nonlocality, resulting invery accurate approximations at the unit-cell level (i.e.,|�χ1|, |�υ1| � 1), there is no guarantee that the errorswill remain negligibly small as the iterations proceed (i.e.,

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BOUNDARY EFFECTS OF WEAK NONLOCALITY... PHYS. REV. APPLIED 10, 034060 (2018)

Electrical thickness, L/λ0 20 40 60 80

Δχn

−4

−2

0

2

4Δυ

n

−10

−5

0

5

10

|Δτ n

|

0

0.5

1

1.5

2

No. of unit cells, n0 500 1000 1500 2000

(a)

(b)

(c)

FIG. 2. (a),(b) Trace and antitrace error maps [Eqs. (18a) and(18b), respectively] for a multilayer with εa = 1, εb = 5, da =db = 0.02λ (i.e., d = λ/25, fa = fb = 0.5), an exterior mediumwith relative permittivity εe = 4, and near-critical incidenceangle θ = 59◦. The solid curves indicate the rigorous refer-ence solutions, whereas the dashed curves indicate the slowlyvarying envelopes from the leading terms in Eqs. (19a) and(19b). (c) Corresponding transmission-coefficient difference map[Eq. (18c)] computed via the rigorous reference solution. Theresults are shown as a function of the number of unit cellsand corresponding electrical thickness L/λ (shown on the upperhorizontal axis).

as the number of unit cells increases). This is the keyobservation behind our approach, from which it emergesthat the boundary effects observed in Refs. [10,12,14] canbe incisively interpreted and parameterized as an error-propagation problem in the trace and antitrace iteratedmaps. In particular, we show that the propagation effectsof the initial (unit-cell) trace and antitrace errors (�χ1and �υ1, respectively) are directly associated with the twodistinct mechanisms identified in Refs. [10,14].

C. Analytical modeling

We define the error maps

�χn ≡ χn − χ̄n, (18a)

�υn ≡ υn − ῡn, (18b)�τn ≡ τn − τ̄n, (18c)

which describe the propagation of the initial errors in Eqs.(17a) and (17b). In particular, for a metamaterial composedof n bilayers, �τn quantifies the departure of the EMT-based approximation of the transmission coefficient fromthe exact (transfer-matrix-based) prediction. By assuming|�χ1|, |�υ1| � 1 and exploiting the analytical solutionsin Eqs. (14a) and (14b) (see Appendix C for details), weobtain

�χn ≈ X sin(nκ) sin(n), (19a)�υn ≈ ϒ cos(nκ) sin(n) + O(�χ1, �υ1), (19b)

where O denotes the Landau “big-O” symbol, and

X = −4, ϒ = 2(

k̄zkze

+ kzek̄z

), (20)

κ = k̄zd + O (�χ1) , (21a)

= − �χ12√

4 − χ̄21+ O (�χ21 ) ≈ (kd)

4 (εa − εb)2 f 2a f 2b48k̄zd

.

(21b)

We observe from Eqs. (19a), (19b), (20), (21a), and(21b) that the leading terms in the error maps depend onthe initial (unit-cell) trace error �χ1, whereas the initialantitrace error �υ1 affects only the higher-order correc-tion O (�χ1, �υ1) in Eq. (19b). Throughout the paper, werefer to these boundary effects as “type I” and “type II,”respectively.

D. Type-I boundary effects

We start by considering the type-I boundary effects;that is, ignoring the higher-order correction O(�χ1, �υ1)in Eq. (19b) and focusing on the �χ1-dependent leadingterms. Under this assumption, we observe that the trace andantitrace errors propagate with oscillatory laws character-ized by two scales (κ and , identical for both) and twodifferent amplitudes. Remarkably, both amplitudes do notdepend on the initial (unit-cell) errors. More specifically,the trace error is always ≤ 4 in absolute value, irrespectiveof the incidence conditions and exterior medium, whichis a direct consequence of the inherent trace bound inEq. (15). Conversely, it is readily verified from Eq. (20)that the amplitude ϒ of the antitrace error oscillations isalways ≥ 4 and critically depends on the incidence con-ditions and exterior medium. In particular, it can becomearbitrarily large in the two (opposite) limits k̄z/kze �1 and kze/k̄z � 1. The former limit corresponds to the

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Electrical thickness, L/λ0 100 200 300 400

Δχn

−4

−2

0

2

4Δυ

n

−6−4−2

0246

|Δτ n

|

0

0.5

1

1.5

2

No. of unit cells, n0 2000 4000 6000 8000 104

(a)

(b)

(c)

FIG. 3. As in Fig. 2 but with εe = 2 and θ = 70◦. The shadedareas are representative of the very fast oscillations, which arenot individually distinguishable on the scale of the plots.

critical-angle-incidence condition [see Eq. (16)] exploredin Ref. [10]. The latter limit, instead, becomes relevantwhen εe < ε̄‖ and, to the best of our knowledge, has notbeen observed and explored before. The above observa-tions imply that, in their evolutions, the trace and antitraceerror maps can exhibit peaks that are well beyond unity,which, via Eq. (12), translate into significant departuresof the optical response from its EMT-based prediction(i.e., transmission-coefficient errors �τn on the scale ofunity).

Looking at the two scales that characterize the erroroscillations, we observe from Eqs. (21a) and (21b) thatκ essentially accounts for the phase accumulation in theEMT-homogenized medium, with a small correction onthe order of �χ1, whereas directly depends on �χ1,with a higher-order correction on the order of (�χ1)2 (seeAppendix C for details). For typical parameter ranges ofinterest in this study, the two scales may be markedlydifferent and, in particular, � κ . This yields a slowlyvarying envelope (ruled by , and identical for both thetrace error and the antitrace error modulated by fast oscil-lations (in quadrature, and ruled by κ). In this case, the

FIG. 4. Nonlocal correction [see Eq. (23)] on the conventionalEMT model in Eq. (2) as a function of the (normalized) trans-verse wavenumber for a multilayer with εa = 1, εb = 5, da =db = 0.02λ (i.e., d = λ/25, fa = fb = 0.5). Note the 103 scalefactor on the vertical axis. The magnified view in the inset showsthe parameter ranges of main interest (note the broken horizontalaxis), with the circle, triangle, square, and diamond markers indi-cating the exterior-medium and incidence conditions consideredin the various examples.

slow scale is particularly meaningful to understandand parameterize the type-I boundary effects, and we canestimate from Eq. (21b) the critical size [i.e., the numberof unit cells (apart from periodicities)]

np = π2 ≈24π k̄zd

(kd)4(εa − εb)2f 2a f 2b(22)

around which the errors assume their peak valuesand, hence, the transmission-coefficient error (magnitude)|�τn| is maximum. Likewise, it is apparent that for sizesof approximately 2np (plus periodicities) the errors tend tovanish.

We observe from Eq. (20) that the antitrace peak error ϒdepends on the effective and exterior relative permittivities(ε̄‖ and εe, respectively) as well as on the incidence angle.However, it does not depend on how the metamaterial issynthesized—that is, the actual multilayer parameters (εa,εb, d/λ, fa, and fb)—and hence the degree on nonlocal-ity. The degree on nonlocality, instead, plays a key role inestablishing the critical size in Eq. (22). In other words,however weak the degree of nonlocality (i.e., howeversmall �χ1), once the desired effective and exterior per-mittivities and incidence conditions are set, a maximum

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Electrical thickness, L/λ0 20 40 60 80

Δχn×

103

−2

0

2Δυ

n

−0.05

0

0.05

0.1

|Δτ n

|×10

3

0

2

4

6

810

No. of unit cells, n0 500 1000 1500 2000

(a)

(b)

(c)

FIG. 5. As in Fig. 2 but considering the nonlocal effectivemodel in Eq. (23). Note the 103 scale factors on the vertical axesin (a),(c).

potential strength of the type-I boundary effects is inher-ently established, which can become significantly largein certain critical conditions. The degree of nonlocalityaffects only the spatial scale for which the effects aremanifested.

As an illustrative example, similarly to the study in Ref.[10], we consider a multilayered metamaterial with param-eters εa = 1, εb = 5, and da = db = 0.02λ (i.e., fa = fb =0.5, d = λ/25). We start by considering a scenario fea-turing an exterior medium with εe = 4 and an incidenceangle θ = 59◦ close to the critical angle [θc = 60◦, fromEq. (16)]. We observe that for this parameter configurationthe initial (unit-cell) trace and antitrace errors are both verysmall (�χ1 = −3.32 × 10−4 and �υ1 = −5.14 × 10−3),and we therefore expect the leading terms in Eqs. (19a)and (19b) (i.e., type-I effects) to be dominant. Accordingly,since k̄z/kze = 0.240 (i.e., ϒ ≈ 8.82), we expect sensi-ble departures from the EMT predictions. Moreover, since

= 1.34 × 10−3 � κ = 6.21 × 10−2, we expect the fast-scale-slow-scale interpretation to hold.

Figure 2 illustrates the results pertaining to the errormaps in Eqs. (18a) and (18b). As expected, the errormaps exhibit a two-scale oscillatory behavior, in very goodagreement with the predictions in Eqs. (19a) and (19b).In particular, although it takes about 1200 unit cells (i.e.,

Electrical thickness, L/λ0 100 200 300 400

−0.05

0

0.05

−0.1

−0.05

0

0.05

0.1

0

0.01

0.02

0.03

0.040.05

0 2000 4000 6000 8000 104Δχ

nΔυ

n|Δτ n

|

(a)

(b)

(c)

No. of unit cells, n

FIG. 6. As in Fig. 3 but considering the nonlocal effectivemodel in Eq. (23). Also in this case, the shaded areas are repre-sentative of the very fast oscillations, which are not individuallydistinguishable on the scale of the plots.

L ≈ 50λ) to reach the peak error, sensible departures fromthe EMT predictions may be observed also for hundredsof unit cells (i.e., L ≈ 5λ). As shown in Fig. 2(c), thiscorresponds to transmission-coefficient (magnitude) errorson the scale of unity. For the same multilayer parame-ters as above, Fig. 3 shows some results pertaining toεe = 2 and θ = 70◦, representative of the somehow oppo-site regime (k̄z/kze = 2.30), which was not considered inprevious studies. We observe that the initial trace error �χ1is the same as in the previous example (since it dependsonly on the multilayer parameters, which are not changed),whereas the antitrace error is different (�υ1 = −1.09 ×10−2), but is still sufficiently small for the type-I effectsto be dominant. Although total reflection is not possiblein this scenario (since εe < ε̄‖), the field is still propagat-ing in the higher-permittivity (“b”-type) layers and is stillevanescent in the lower-permittivity (“a”-type) ones, andthe type-I boundary effects still become visible for suffi-ciently large sizes (np = 5275). Since the fast scale is nowalmost 3 orders of magnitude larger than the slow scale(κ = 937.7), the fast oscillations are not individually dis-tinguishable on the scale of the plots, and therefore we canobserve only some shaded areas, whose envelopes are in

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(a) Electrical thickness, L/λ0 20 40 60 80

Δχn

−5−2.5

0

2.5

5

(b)

Δυn

−10

0

10

20

(c)

|Δτ n

|

0

0.5

1

1.5

2

0 500 1000 1500 2000No. of unit cells, n

FIG. 7. As in Fig. 2 but with θ = 89◦ and εe = 3. The dashedlines in (a),(b) indicate the amplitude bounds [see Eq. (20)]predicted by the leading terms in Eqs. (19a) and (19b).

very good agreement with our approximate modeling inEqs. (19a) and (19b). Other than that, the same observa-tions as for the previous example hold.

E. Nonlocal corrections

The results above naturally suggest that the type-Iboundary effects can be captured by a suitable nonlocalcorrection providing a more accurate approximation of theinitial (unit-cell) trace error �χ1.

To this aim, along the lines of the approach pro-posed in Ref. [6], we derive a nonlocal effective model(see Appendix D for details) in terms of a wavenumber-dependent relative permittivity

ε̂‖(kx)

= 6 + k2x d

2 −√36 + 12d2(k2x − k2ε̄‖) + d4(k2x − αak2)(k2x −αbk2)k2d2

,

(23)

withαa = f 2a εa + f 2b εb + 2fafbεa, (24a)αb = f 2a εa + f 2b εb + 2fafbεb. (24b)

(a) Electrical thickness, L/λ0 20 40 60 80

Δχn×

105

−20−10

01020

(b)

Δυn

−5

0

5

10

(c)

|Δτ n

|0

0.2

0.4

0.6

0.8

0 500 1000 1500 2000No. of unit cells, n

FIG. 8. As in Fig. 7 but considering the nonlocal effectivemodel in Eq. (23). Note the 105 scale factor on the vertical axisin (a).

Here and henceforth, the caret is used to tag quanti-ties related to the nonlocal effective model. It can beverified that in the limit kd → 0 the nonlocal model inEq. (23) consistently reduces to the conventional (local)EMT model in Eq. (2). As illustrated in Fig. 4, for thefinite but small values of kd of interest here, the actualnonlocal corrections are very small (on the third decimalfigure), thereby confirming the anticipated weak character.As a consequence, strong-nonlocality-induced effects suchas additional extraordinary waves should not be expected,and there is no need to enforce additional boundary con-ditions when one is solving the boundary-value problem.For our specific case, it can be shown (see Appendix D fordetails) that the initial (unit-cell) trace error arising fromEq. (23) is given by

�χ1 = χ1 − χ̂1 ∼ O[(kd)6(εa − εb)2f 2a f 2b ]. (25)

By comparison with the EMT-based counterpart inEq. (17a), we observe a much-faster decrease (sixth power,instead of fourth) of the error with the unit-cell electrical

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BOUNDARY EFFECTS OF WEAK NONLOCALITY... PHYS. REV. APPLIED 10, 034060 (2018)

(a) Electrical thickness, L/λ0 100 200 300

Δχn

−4

−2

0

2

4

(b)

Δυn

−100

−50

0

50

100

(c)

|Δτ n

|

0

0.5

1

1.5

2

No. unit cells, n0 2000 4000 6000 8000

FIG. 9. As in Fig. 3 but with εe = 2 and θ = 89◦. Also in thiscase, the shaded areas are representative of the very fast oscilla-tions, which are not individually distinguishable on the scale ofthe plots.

thickness kd, which, in the regime of interest, poten-tially translates [via Eqs. (21b) and (22)] into orders-of-magnitude increases of the critical size for the boundaryeffects to manifest themselves.

As an illustrative example, Fig. 5 shows the resultscorresponding to the configuration in Fig. 2. For theseparameters, the nonlocal effective model in Eq. (23) yieldsa 4-orders-of-magnitude-smaller initial trace error (�χ1 =8.55 × 10−8). As a consequence, on the same spatial scaleof Fig. 2, both the trace error [Fig. 5(a)] and the antitraceerror [Fig. 5(b)] are now strongly reduced, thereby yieldingvery small errors in the transmission coefficient [Fig. 5(c)].

Qualitatively similar results can be observed in Fig. 6for the parameter configuration in Fig. 3.

Clearly, higher-order nonlocal corrections could bederived [6], which would further reduce the initial traceerror �χ1, and hence more accurately capture the arisingtype-I boundary effects. Alternatively, more-sophisticatednonlocal effective models could be used [9] that enforce theexact matching of the Bloch-type and effective dispersiverelationships.

(a) Electrical thickness, L/λ0 100 200 300

Δχn

−0.05

−0.025

0

0.025

0.05

(b)

Δυn

−1

0

1

(c)

|Δτ n

|0

0.1

0.2

0.3

0.40.5

0 2000 4000 6000 8000No. of unit cells, n

FIG. 10. As in Fig. 9 but considering the nonlocal effectivemodel in Eq. (23). Also in this case, the shaded areas are repre-sentative of the very fast oscillations, which are not individuallydistinguishable on the scale of the plots.

F. Type-II boundary effects

We now move on to consider the type-II boundaryeffects, which become relevant in scenarios where theinitial (unit-cell) antitrace error �υ1 and, hence, thehigher-order correction term in Eq. (19b) are no longernegligibly small. From Eq. (17b), it is clear that this mayhappen, for instance, when kze � k (i.e., for near-grazingincidence from the exterior medium). This is exempli-fied in Figs. 7 and 8 for a scenario with εe = ε̄‖ = 3and θ = 89◦, for which �χ1 remains negligibly smallbut �υ1 = −0.18 is no longer negligible. More specif-ically, Fig. 7 shows the error maps pertaining to thestandard (local) EMT model. Unlike the previous exam-ples (see Figs. 2 and 3), we can no longer interpret theoscillatory behaviors of the trace and antitrace errors interms of fast oscillations and a slow envelope, since thetwo scales are actually inverted (κ < ) and rather closein value (κ = 7.60 × 10−3, = 1.09 × 10−2). Neverthe-less, the trace error [Fig. 7(a)] still satisfies the amplitudebound implied by Eq. (19a). Conversely, we observe fromFig. 7(b) that the antitrace error may exceed (by morethan a factor of 2) the amplitude bound predicted by ϒ

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CASTALDI, ALÙ, and GALDI PHYS. REV. APPLIED 10, 034060 (2018)

in Eq. (20). As expected, these boundary effects are nolonger accurately captured by the leading terms in Eqs.(19a) and (19b).

Figure 8 shows instead the corresponding resultsobtained by our considering the nonlocal effective model inEq. (23). We observe that the trace error [Fig. 8(a)] is nowsubstantially reduced, but the antitrace error [Fig. 8(b)]still exhibits moderately large (> 5) peak values. As aresult, the transmission-coefficient error [Fig. 8(c)] canstill reach values of approximately 0.6 in magnitude, evenfor relatively small sizes, thereby indicating that theseboundary effects are not even captured very accurately bythe nonlocal correction. This should not be surprising, asthe nonlocal effective model in Eq. (23) was derived withthe aim of reducing the initial trace error (�χ1) only, andit does not affect the initial antitrace error �υ1 responsiblefor the type-II boundary effects.

Another insightful example is illustrated in Figs. 9 and10 for a scenario with εe = 2 and θ = 89◦. In this casethe initial antitrace error �υ1 = −0.21, albeit not smallin absolute terms, remains negligible by comparison withthe quite large amplitude of the leading term (ϒ = 81). Inother words, in this regime the type-I boundary effects arestill dominant. As a result, we observe from Fig. 9 that theleading terms in Eqs. (19a) and (19b) still provide an accu-rate modeling. However, as clearly exemplified in Fig. 10,when the nonlocal effective model is applied, and hencethe type-I effects are accurately captured, the residual type-II effects become clearly visible. The two examples aboveclearly indicate the different and independent nature of thetype-II boundary effects. For the scenario εe ≥ ε̄‖, a similaradditional mechanism was identified in Ref. [14]. Such aphenomenon, not restricted to the critical-angle incidenceand particularly significant for εe = ε̄‖, was explained interms of a phase mismatch at the interface separating thelast layer and the exterior medium. Remarkably, it wasconcluded that the effect could not be captured by localand nonlocal homogenization in terms of a single effectivelayer, and the addition of an artificial matched layer wasrequired.

Our results indicate that such boundary effects mayalso become dominant in scenarios with εe < ε̄‖. More-over, they also indicate that a single-parameter nonlo-cal effective model is generally not sufficient to capturethese effects. More complex (multiparameter) extensionsare required to more accurately approximate the initial(unit-cell) values of both the trace and antitrace. How-ever, since the antitrace (unlike the trace) inherentlydepends on the exterior medium, the simplest conceivableextension, entailing the introduction of a (wavenumber-dependent) effective magnetic permeability would alsoinherit such dependence, thereby leading to a model incon-sistency. A self-consistent model, with effective parame-ters independent of the exterior medium, should correctly

approximate all the transfer-matrix terms. This wouldinevitably entail some magnetoelectric coupling, alongthe lines of the approach in Ref. [13]. We regard thisfurther development as beyond the scope of the presentinvestigation.

IV. CONCLUSIONS AND OUTLOOK

We apply the trace-and-antitrace-map formalism tomodel the boundary effects induced by weak nonlocalityin multilayered metamaterials made of periodic stacks ofalternating, deeply subwavelength dielectric layers.

Our approach naturally identifies two distinct boundaryeffects, associated with different error-propagation effectsin the trace and antitrace maps. Moreover, it leads tosome analytical models that naturally highlight the criticalparameters and conditions (including some not consid-ered in previous studies), and provide very useful insightsin the development of nonlocal corrections. Overall, webelieve that our results nicely complement previous stud-ies on these phenomena [10,13,14] by offering a differentperspective and paving the way to intriguing extensions.

For instance, our approach can be fruitfully extendedto aperiodically ordered multilayers [17,21–23] so as toexplore the effects induced by the spatial arrangement.Because of the more-complex character of the arisingmaps, the trace bound in Eq. (15) would not necessar-ily hold in these scenarios, thereby giving rise to richerdynamics involving transitions between propagating andevanescent regimes. Within a related framework, also ofgreat interest are possible applications to the study ofAnderson localization effects in random multilayers, alongthe lines of Refs. [15,16].

Finally, also worthy of mention are possible exten-sions to non-Hermitian scenarios featuring balanced lossand gain, which are experiencing a surging interest inoptics and photonics (see, e.g., Ref. [25] for a recentreview). Within this framework, we observe that under thespecial conditions of parity-time symmetry [26] for theunit cell (εa = ε∗b , da = db), the trace and antitrace mapsremain real valued, thereby greatly simplifying the studyand interpretation of the phenomena. Moreover, sincethe conventional EMT model of a parity-time-symmetricmultilayer would yield a lossless (and gainless) effectivemedium, the phenomena studied here are expected to beclosely related to distinctive effects that can occur in non-Hermitian systems, such as the emergence of exceptionalpoints [27].

APPENDIX A: DETAILS ON THETRANSFER-MATRIX FORMALISM

Following Chap. 1 in Ref. [18], the transfer matri-ces pertaining to the single layers can be expressed as

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BOUNDARY EFFECTS OF WEAK NONLOCALITY... PHYS. REV. APPLIED 10, 034060 (2018)

Ma

=

⎡⎢⎣

cos (kzafad)kzekza

sin (kzafad)

−kzakze

sin (kzafad) cos (kzafad)

⎤⎥⎦ , (A1a)

Mb

=

⎡⎢⎣

cos (kzbfbd)kzekzb

sin (kzbfbd)

−kzbkze

sin (kzbfbd) cos (kzbfbd)

⎤⎥⎦ , (A1b)

where kza =√

k2εa − k2x and kzb =√

k2εb − k2x denotethe longitudinal wavenumbers in the two correspondingmedia, and all other symbols are defined in the main text.By multiplication of the two matrices above, the unit-cell trace and antitrace are straightforwardly obtained as[see Eq. (10) with n = 1]

χ1 = 2 cos(kzafad) cos(kzbfbd)

−(

kzakzb

+ kzbkza

)sin(kzafad) sin(kzbfbd), (A2a)

υ1 = −(

kzakz

+ kzkza

)sin(kzafad) cos(kzbfbd)

−(

kzbkz

+ kzkzb

)cos(kzafad) sin(kzbfbd). (A2b)

APPENDIX B: DETAILS ON THE TRACE ANDANTITRACE MAPS

Assuming we have a generic transfer matrix M (see theinset in Fig. 1) and unit-amplitude, TE-polarized plane-wave incidence, the input and output electric fields can bewritten as

E(i)y = exp (ikxx) [exp (ikzez) + � exp (−ikzez)] , (B1a)

E(o)y = τ exp {i [kxx + kze (z − L)]} , (B1b)where � and τ denote the reflection and transmissioncoefficients, respectively. By computing the correspond-ing magnetic field tangential components via the relevantMaxwell’s curl equation, and substituting them in Eq. (4),we obtain the linear system

[1 + �

−i(1 − �)]

= M ·[

τ

−iτ]

, (B2)

which, solved with respect to τ , yields

τ = 2M11 + M22 + i(M21 − M12) . (B3)

Equation (12) directly follows from Eq. (B3) by our partic-ularizing M = Mn

aband recalling the trace and antitrace

definitions in Eq. (9).

As a consequence of the Cayley-Hamilton theorem [19],the square of a 2 × 2 unimodular matrix can be expressedas [22]

M2 = Tr(M)M − I , (B4)

with I denoting the 2 × 2 identity matrix. The trace andantitrace maps in Eq. (13) are obtained from Eq. (B4),particularized for M = M

ab, by our multiplying both

sides by Mn−2 (with n ≥ 2) and calculating the trace andantitrace, respectively.

Via recursive application of Eq. (B4), we alsoobtain [22]

Mn = Un−1[

12

Tr(M)]

M − Un−2[

12

Tr(M)]

I ,(B5)

from which the analytical solutions in Eqs. (14a) and(14b) directly follow by our particularizing M = M

aband calculating the trace and antitrace.

APPENDIX C: DERIVATION OF EQS. (17a), (17b),(19a), AND (19b)

First, we derive the approximations in Eqs. (17a) and(17b) for the initial (unit-cell) trace and antitrace errors.By expanding Eq. (A2a) in a Maclaurin series with respectto d (up to the fourth order), we obtain

χ1 ≈ 2 − (k̄zd)2 + d4

12[k2x − k2

(f 2a εa + f 2b εb + 2fafbεa

)][k2x − k2

(f 2a εa + f 2b εb + 2fafbεb

)]. (C1)

Recalling the transfer matrix of the EMT-homogenizedmedium,

M̄ =

⎡⎢⎢⎣

cos(k̄zd)kzek̄z

sin(k̄zd)

− k̄zkze

sin(k̄zd) cos(k̄zd)

⎤⎥⎥⎦ , (C2)

we obtain likewise (for k̄zd � 1)

χ̄1 = 2 cos(k̄zd) ≈ 2 − (k̄zd)2 +

(k̄zd)4

12. (C3)

The approximation in Eq. (17a) follows by our subtractingEqs. (C1) and (C3) and recalling Eq. (3).

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CASTALDI, ALÙ, and GALDI PHYS. REV. APPLIED 10, 034060 (2018)

In a similar fashion, via Maclaurin expansions of υ1 andῡ1 (up to the third order in d), we obtain

υ1 ≈ d6kze{fa[k2 (εe + εa) − 2k2x

] (k2zad

2f 2a − 6)

+ 3fb[k2 (εe + εb) − 2k2x

] (k2zad

2f 2a − 2)

+ 3k2zbd2faf 2b[k2 (εe + εa) − 2k2x

]+ k2zbd2f 3b

[k2 (εe + εb) − 2k2x

]}, (C4)

ῡ1 = −(

k̄zkze

+ kzek̄z

)sin(k̄zd)

≈ d[(

ε̄‖ + εe)

k2 − 2k2x] (

k̄2z d2 − 6)

6kze, (C5)

which, after some algebra, yield the approximation inEq. (17b).

From Eq. (18a), by recalling the trigonometric formof the Chebyshev polynomials of the first kind (seeEq. 22.3.15 in Ref. [20]), we obtain

�χn = 2{

cos[

n arccos(

χ̄1 + �χ12

)]

− cos[

n arccos(

χ̄1

2

)]}

= −4 sin{

n2

[arccos

(χ̄1 + �χ1

2

)

+ arccos(

χ̄1

2

)]}

× sin{

n2

[arccos

(χ̄1 + �χ1

2

)

− arccos(

χ̄1

2

)]}, (C6)

where a sum-to-product identity (see Eq. 4.3.37 inRef. [20]) was used in the second equality. Next, under theassumption �χ1 � 1, we apply the following approxima-tions:

arccos(

χ̄1 + �χ12

)+ arccos

(χ̄1

2

)

≈ 2 arccos(

χ̄1

2

)− �χ1√

4 − χ21, (C7a)

arccos(

χ̄1 + �χ12

)− arccos

(χ̄1

2

)

≈ − �χ1√4 − χ21

− χ1 (�χ1)2

2√(

4 − χ21)3 , (C7b)

which, substituted in Eq. (C6), yield the parameterizationin Eq. (19a), with the expression of κ and in Eqs. (21a)and (21b) obtained by our recalling [from Eq. (17a)] that

χ̄1 = 2 cos(k̄zd)

, (C8)

and that, for k̄zd � 1,√

4 − χ̄21 = 2 sin(k̄zd) ≈ 2k̄zd. (C9)

Moving on to the antitrace error map in Eq. (18b), byrecalling the trigonometric form of the Chebyshev poly-nomials of the second kind [see Eq. 22.3.16 in Ref. [20]),we obtain

�υn =2 (ῡ1 + �υ1) sin

[n arccos

(χ̄1 + �χ1

2

)]√

4 − (χ̄1 + �χ1)2

−2ῡ1 sin

[n arccos

(χ̄1

2

)]√

4 − χ̄21. (C10)

Then, in the limit |�χ1|, |�υ1| � 1, we use the approxi-mation

2 (ῡ1 + �υ1)√4 − (χ̄1 + �χ1)2

≈ 2ῡ1√4 − χ̄21

+ 2�ῡ1√4 − χ̄21

+ 2χ̄1ῡ1�χ1√(4 − χ̄21

)3 , (C11)

which, substituted in Eq. (C10), yields the parameteriza-tion in Eq. (19b) by or applying the same approximationsas in Eqs. (C7a) and (C7) and recalling that [from Eqs. (C5)and (C7)]

2ῡ1√4 − χ̄21

= −(

k̄zkze

+ kzek̄z

). (C12)

APPENDIX D: DERIVATION OF EQ. (23)

Our nonlocal homogenization strategy is inspired by theapproach put forward in Ref. [6]. First, similarly to thederivation of Eq. (C3), we expand the trace of the transfermatrix of the nonlocally homogenized unit cell as

χ̂1 = 2 cos(

k̂zd)

≈ 2 −(

k̂zd)2

+(

k̂zd)4

12, (D1)

where

k̂z =√

k2ε̂‖ (kx) − k2x , (D2)

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BOUNDARY EFFECTS OF WEAK NONLOCALITY... PHYS. REV. APPLIED 10, 034060 (2018)

and, following the notation introduced in the main text, thecaret denotes quantities based on nonlocal homogeniza-tion. The basic idea is to choose the function ε̂‖(kx) so asto enforce the matching between Eqs. (D1) and (C1) upto the fourth order in d. After some algebra, this yields aquadratic equation in ε̂2‖; viz.,

k2d2ε̂2‖(kx) − 2(6 + k2x d2)ε̂‖(kx) + 2(6 + k2x d2)ε̄‖− k2d2αaαb = 0, (D3)

with the coefficients αa and αb defined in Eqs. (24a) and(24b). Of the two possible solutions, one turns out to bephysically inconsistent, as it diverges in the limit kd → 0.We are therefore left with the solution in Eq. (23), whichconsistently reduces to the standard (local) EMT model (2)in the above limit. This yields an initial (unit-cell) traceerror

�χ1 = χ1 − χ̂1 ≈ (kd)6(εa − εb)2f 2a f 2b

360× {3εa + fb [εb − 5εa + 2fb (εa + εb)]

+(

kxk

)2(4fafb − 3)

}, (D4)

as compactly indicated in Eq. (25).

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https://doi.org/10.1103/PhysRevB.75.115104https://doi.org/10.1063/1.2737935https://doi.org/10.1103/PhysRevB.84.075153https://doi.org/10.1103/PhysRevB.84.115438https://doi.org/10.1103/PhysRevB.86.115420https://doi.org/10.1103/PhysRevLett.113.243901https://doi.org/10.1088/0957-4484/26/18/184001https://doi.org/10.1103/PhysRevLett.115.177402https://doi.org/10.1103/PhysRevB.94.085428https://doi.org/10.1103/PhysRevB.96.035439https://doi.org/10.1038/ncomms12927https://doi.org/10.1126/science.aah6822https://doi.org/10.1103/PhysRevB.62.14020https://doi.org/10.1103/PhysRevB.42.1062https://doi.org/10.1103/PhysRevA.42.7112https://doi.org/10.1103/PhysRevB.87.235116https://doi.org/10.1103/PhysRevB.84.045424https://doi.org/10.1038/s41566-017-0031-1https://doi.org/10.1103/PhysRevLett.80.5243https://doi.org/10.1088/1751-8113/45/44/444016

I. INTRODUCTIONII. BACKGROUND AND MATHEMATICAL FORMULATIONA. GeometryB. Local EMT formulationC. Transfer-matrix formalismD. Trace and antitrace maps

III. BOUNDARY EFFECTS OF WEAK NONLOCALITYA. BackgroundB. Connection with the trace-and-antitrace-map formalismC. Analytical modelingD. Type-I boundary effectsE. Nonlocal correctionsF. Type-II boundary effects

IV. CONCLUSIONS AND OUTLOOKA. APPENDIX A: DETAILS ON THE TRANSFER-MATRIX FORMALISMB. APPENDIX B: DETAILS ON THE TRACE AND ANTITRACE MAPSC. APPENDIX C: DERIVATION OF EQS. (17a), (17b), (19a), AND (19b)D. APPENDIX D: DERIVATION OF EQ. [d23](23). References

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