PHYSICAL REVIEW B 86, 115420 (2012)

Nonlocal effective parameters of multilayered metal-dielectric metamaterials

A. V. Chebykin,1 A. A. Orlov,1 C. R. Simovski,1,2 Yu. S. Kivshar,1,3 and P. A. Belov1,4

1National Research University of Information Technologies, Mechanics, and Optics (ITMO), St. Petersburg 197101, Russia2School of Electric and Electronic Engineers, Aalto University, Aalto 00076, Finland

3Nonlinear Physics Center, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia4Queen Mary College, University of London, Mile End Road, London E1 4NS, United Kingdom

(Received 4 July 2012; revised manuscript received 3 September 2012; published 14 September 2012)

We consider multilayered metal-dielectric metamaterials composed of alternating nanolayers of two types andcalculate the components of their effective dielectric permittivity tensors as functions of both frequency and wavevector. We demonstrate that such structures can be described as strongly nonlocal uniaxial effective media, andwe analyze how the nonlocal permittivity tensor components are related to other manifestations of strong spatialdispersion in such structures, and how the resonance of permittivity depends on the propagation direction.

DOI: 10.1103/PhysRevB.86.115420 PACS number(s): 78.67.Pt, 78.20.Bh

I. INTRODUCTION

Multilayered metamaterials (MLMMs), which can bedefined as one-dimensional structures of two periodicallyalternating optically thin layers with sufficient optical contrastbetween them, are known to have unusual electromagneticproperties, being promising for many applications, especiallyin the optical frequency range. Optical MLMMs have layerthickness as small as a few tens of nanometers, and thedielectric layers alternate with silver or gold layers having thethickness of the same order. Such MLMMs can transport sub-wavelength images,1–4 and this effect makes them promisingfor nanolithography.5 Also, MLMM-based optical cloaks havebeen proposed (see, e.g., Refs. 6 and 7). In a certain frequencyrange, such metal-dielectric MLMMs can be modeled asmedia with the hyperbolic dispersion surfaces in the spaceof wave vectors. Such media are usually characterized byan uniaxial permittivity tensor whose axial and transversecomponents have different signs and therefore these media arecalled indefinite materials.8 With indefinite materials used assubstrates or/and superstrates of nanoemitters such as quantumdots or nanoantennas it is possible to obtain huge values forthe Purcell factor.9–14

The study of MLMMs was probably initiated in the classicalpaper by Rytov15 who derived the dispersion equations for TM-and TE-polarized waves and also analyzed the correspondingeigenmodes. An attempt to interpret the refraction indicesand wave impedances of these eigenmodes in terms ofeffective material parameters resulted in retrieved values ofε and μ which were characterized by Rytov as physicallymeaningless parameters since they violated the causality andpassivity limitations even in the case when both alternatinglayers were dielectric and had subwavelength thickness. Rytovconcluded that the local effective medium concept can beapplied to MLMMs only with finite accuracy and only in somespecial cases when the quasistatic model of MLMMs is alsoapplicable.

A more advanced theory of MLMMs was developedin Ref. 16. This effective medium model generalized thequasistatic model taking into account the retardation effectsof the first order with respect to the optical thickness of alayer. However, this theory was concentrated mainly on the

normal propagation, and the limits of its validity remainednot well defined. Later, it was shown17,18 that the spatialdispersion related to surface plasmon polaritons excited atmetal-dielectric interfaces of such structures plays a key rolein optical properties of MLMMs. Spatial dispersion impliesthat the effective permittivity tensor components depend onthe wave vector.19,20 The existence of strong spatial dispersioneffects were revealed for different types of metamaterials, e.g.,for arrays of split-ring resonators21 and wire metamaterials,22

and it is also inherent to MLMMs. New physics brought tomultilayered structures by nonlocality includes such effects asemission control,13,14 single polarization beam-splitting,26 andnonlocal transformation optics.23 Since the effective materialparameters proposed in Refs. 15 and 16 did not take intoaccount the effects of strong spatial dispersion which actuallypresent in MLMMs, in order to describe them correctly oneneeds to generalize the homogenization theory taking intoaccount nonlocal effects.

A nonlocal homogenization theory that results in thetensorial permittivity ε calculated for all possible valuesof the wave vector k and frequency ω was suggested inRef. 24. Possible bianisotropic and magnetic responses ofthe effective medium are comprised, respectively, in thefirst- and second-order terms of the k-power expansion ofε(k). When this ε is complemented by additional boundaryconditions, it allows solving a boundary value problem for ahalf space and even for finite-thickness layers of the medium.Unlike other homogenization models, this theory does notimply any approximation. It is a strict approach based on theexact knowledge of microscopic fields in an infinite regularlattice composed of electromagnetically linear and reciprocaldielectric or conducting inclusions. However, for a majority oflattices this homogenization procedure is numerical. In otherwords, as a rule the components of the nonlocal tensor ε cannotbe derived analytically beyond certain approximations, andthey have to be simulated numerically. To our knowledge,only the case of MLMMs due to its geometrical simplicityallows the exact analytical calculation of ε. This calculationwas described in our recent work25 and presented there asan iterative set of rigorous expressions explicitly dependingon ω and k. To compute the components of the nonlocalpermittivity tensor, these expressions require performing some

115420-11098-0121/2012/86(11)/115420(8) ©2012 American Physical Society

CHEBYKIN, ORLOV, SIMOVSKI, KIVSHAR, AND BELOV PHYSICAL REVIEW B 86, 115420 (2012)

dd

Nonlocalhomogenization

k

FIG. 1. (Color online) Original structure and its homogenizedrepresentation.

matrix algebra. In this paper, we make the next step toward theclosed-form expressions. However, these expressions turnedout to be very involved, and for some components of ε it isreasonable to keep their matrix form.

Thus, this paper aims to analyze the dependencies ofthe tensor ε on ω and k and to find a link between thesedependencies and the effects of strong spatial dispersionsobserved in MLMMs; this analysis extends further the resultsof our earlier publications.25,26 A study of eigenmodes ofmultilayered metamaterials considered in this paper andcorresponding field distributions can be found, for example, inthe recent Ref. 27.

II. QUASISTATIC AND NONLOCALEFFECTIVE MEDIUM MODELS

In the static limit, an infinite multilayered structure com-posed of two alternating layers with thicknesses d1, d2 andpermittivities ε1, ε2 can be considered as a uniaxial mediumhaving the following permittivity tensor:

εloc =⎛⎝ ε⊥ 0 0

0 ε‖ 00 0 ε‖

⎞⎠ , (1)

where

ε‖ = ε1d1 + ε2d2

d1 + d2, (2)

ε⊥ =(

ε−11 d1 + ε−1

2 d2

d1 + d2

)−1

. (3)

This model is employed for the optical frequency region, andit results in the representation of a metal-dielectric MLMM asa local indefinite medium since the permittivity of metals isnegative in this range (more exactly, it has a negative real part).However, in Ref. 26 it was revealed that such a description ofplasmonic MLMMs is not adequate. The first-order correctionsof the quasistatic model obtained in Ref. 16 are not helpful inthis case since the spatial dispersion in such structures is strong.Even if the dependence ε(k) can be replaced by a power series,the series convergence is not rapid enough to be restricted bysecond- or third-order terms. Moreover, in the case of strongspatial dispersion the dependence ε(k) can be resonant [i.e., inthe lossless case the function ε(k) can have singularities].

Similar to our previous work,25 we choose the coordinatesystem in such a way that the axis x is normal to the layers, andthe Bloch wave with the wave vector k propagates in the plane(x,y) (kz = 0). Then, a homogenized MLMM (see Fig. 1) can

be described by the permittivity tensor:25

ε(ω,k) =⎛⎝εxx(ω,k) εxy(ω,k) 0

εyx(ω,k) εyy(ω,k) 00 0 εzz(ω,k)

⎞⎠ . (4)

From the reciprocity condition, we have εxy(ω,k) =εyx(ω,k), so that there are four independent components ofε. In the Appendix we present the rigorous equations forcalculating these four scalar values.

A quite unexpected result of our earlier study25 is thepresence of the nonzero value of the off-diagonal componentsεxy = εyx . It means that in order to transform the permittivitytensor to a diagonal form, one has to rotate the coordinatesystem XYZ around the axes z. The tensor becomes diagonal;i.e., the medium behaves as a uniaxial medium when the opticalaxis x ′ is tilted to the normal axis x by a certain angle θ . Thisangle depends on both frequency ω and propagation direction.For any ω it vanishes for both normal ky = 0 and in-planekx = 0, and it is maximal for nearly bisector propagation(ky = kx). The nonzero value of θ is an important parametercharacterizing the spatial dispersion effect.

Another feature of the spatial dispersion is the nonequiv-alence of two components εyy �= εzz both referring to theplane of layers. If the spatial dispersion effects are negligibleboth these components would be equivalent and equal tothe transversal permittivity of an effective uniaxial medium(whose optical axis would be then normal to the layers). Inthe quasistatic limit D ≡ d1 + d2 → 0 (more exactly whenk0D → 0, where k0 = ω/c is the free space wave number),the nonlocal permittivity tensor (4) numerically turns intoEqs. (1)–(3). This tendency is pointwise (not uniform for all k).

We have checked our calculations of ε(ω,k) in the fol-lowing way. Eigenmodes in an arbitrary anisotropic materialdescribed by the permittivity tensor ε are solutions of knowndispersion equations. For TE waves this dispersion equation isknown,19

k2x + k2

y = εzz

(ω

c

)2

. (5)

For TM waves we have the following equation:19

(εxxεyy − ε2

xy

)(ω

c

)2

− εyyk2y − εxxk

2x − 2kxkyεxy = 0. (6)

For an arbitrary MLMM, the rigorous dispersionequations15 are the following (for TE and TM waves, respec-tively):

�TE = sin(kx1d1) sin(kx2d2)(k2x1 + k2

x2

) + 2kx1kx2[cos(kxD)

− cos(kx1d1) cos(kx2d2)] = 0, (7)

�TM = sin(kx1d1) sin(kx2d2)(k2x1 + k2

x2

) + 2kx1kx2[cos(kxD)

− cos(kx1d1) cos(kx2d2)] = 0. (8)

Substituting these expressions for the components ofε(ω,k), we find that Eqs. (5) and (6) are equivalent to Eqs. (7)and (8), respectively. Dispersion diagrams of Ref. 26 wereobtained by using (8), and for metal-dielectric MLMMswith permittivities ε1 = 4.6 and ε2 = 1 − λ2/λ2

p, where

115420-2

NONLOCAL EFFECTIVE PARAMETERS OF MULTILAYERED . . . PHYSICAL REVIEW B 86, 115420 (2012)

0 1 2 3 4

40

80

120

160

0.3

0.6

0.9

1.2

20

40

60

-1

-0.6

0

0.6

1

10

20

30

-6

-4

-2

0

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

2

4

0 0.5 1 1.5 2 2.5 3 3.5 40.08

0.085

0.09

0.095

0.1

0.105

0.11 12 3

4 5

D/

(a)

(b)

(c)

6

0.5

0.7

0.9

1.1

1.3

20

40

60

80

100

6

0 1 2 3 4 0 1 2 3 4

(d)

1

3

5

a

d

c

b

xx y y( )= /( /c)k k 2 2yy y( )=0k

yy

yy

yy

yyxx

ky2 2/( /c)

xx

ky2 2/( /c)

xx

ky2 2/( /c)

xx

ky2 2/( /c)

ky

~ky

~

ky

~ ky

~

ky

~

ky

~

ky

~

ky

~

ky

~

FIG. 2. (Color online) Dispersion diagram of MLMMs andgraphical solutions of the dispersion equations (10) for d1 = 1.5d2:(a) D/λ = 0.108, (b) D/λ = 0.107, (c) D/λ = 0.1, and (d) D/λ =0.085.

λp = 250 nm. The dispersion diagrams we discuss are cal-culated by solving Eq. (6), and they turned out to be exactlythe same as those obtained in Ref. 26.

III. DISPERSION AND NONLOCAL PERMITTIVITY

We consider the special case of the in-plane propagationkx = 0 when εxy = 0. Then Eq. (8) simplifies to become

εyy

[εxx

(ω

c

)2

− k2y

]= 0, (9)

and it is equivalent to the system of two dispersion equations,

εyy(ky) = 0, εxx(ky)

(ω

c

)2

− k2y = 0. (10)

The first equation in the system (10) corresponds to longitudi-nal modes, and the second equation, to transverse modes.

In Fig. 2 (top) we show the dispersion curves of the structurewith the same values of ε1 and ε2 as above and thicknesses

d1 = 1.5d2 (namely, d1 = 37.5 nm and d2 = 25 nm). On thefrequency axis, we mark four example points and obtainsix values of the normalized wave number k̃y = kyD/π

corresponding to the selected frequencies. These results canbe also obtained by the graphical solutions of the system (10),and we illustrate them with the plots in Figs. 2(a)–2(d). It isalso interesting to analyze the selected cases of the values ofεxx and εyy which correspond to these six dispersion states(the dispersion state is a pair of ω, k).

At the normalized frequency ωa ≡ D/λ = 0.108, we haveεxx ≈ 60 and εyy ≈ 0.85. The ratio εxx/εyy ≈ 70 is high butit turns out not to be sufficiently large for strong spatialdispersion at this frequency. There is only one wave whichin the lossless approximation has zero group velocity (point1). The presence of such a wave can be explained asa competition of the forward and backward SPP modes.At the normalized frequency ωb ≡ D/λ = 0.107, we haveεxx ≈ 40 and εyy ≈ 0.95 (point 2). Then εxx/εyy ≈ 45 andanisotropy is again not sufficient for the spatial dispersion.However, at point 3 we have εxx ≈ 130 and εyy ≈ 0.6. Thenεxx/εyy ≈ 220, and this amazing anisotropy turns out to belarge enough for the manifestation of spatial dispersion. Point3 corresponds to the backward wave which exists in spite ofall positive components of the permittivity and permeability.This wave clearly originates from the medium discreteness,and it is the feature of strong spatial dispersion. This strongspatial dispersion corresponds to the large ratio εxx/εyy > 200which is hardly compatible with the concept of a continuousmedium.

The strong spatial dispersion is observed also at the nor-malized frequency ωc ≡ D/λ = 0.1 where at point 4 we haveεxx ≈ 20 and εyy ≈ 0.05. This dispersion state correspondsto the usual (forward transverse) wave. However, at thesame frequency there appears point 5, where εxx ≈ 24 andεyy = 0. The anisotropy in the last case is infinitely large,and the wave corresponding to point 5 is longitudinal. Thepresence of the longitudinal wave in the lattice is obviously afeature of strong spatial dispersion.21 Finally, at the frequencyωd ≡ D/λ = 0.085 (point 6) where εxx ≈ 14 and εyy ≈ −0.4,we have a rather modest ratio |εxx/εyy | ≈ 35, and there are nofeatures of the strong spatial dispersion (only one forwardtransverse wave).

The plots presented in Fig. 3 for the case d2 = 1.5d1 areanalogous of the plots in Fig. 2. Here at ωa ≡ D/λ = 0.115(point 1) εxx ≈ −5 and εyy = 0. The wave is not onlylongitudinal, it is also backward, and this is the frequencywhere the spatial dispersion is strong. At ωb ≡ D/λ = 0.099(point 2) we have εxx ≈ 120 and εyy ≈ −1. Then |εxx/εyy | ≈120, and this ratio turns out to be not sufficient to justifythe spatial dispersion. At this frequency, there exists only onetransverse forward wave.

For the case of a thin metal (d1 = 1.5d2), we revealthat in the vicinity of the plasmon resonance frequencythe dispersion diagram possesses a fine structure shown inFig. 4. The branch for which two waves coexist at the samefrequency demonstrates even more unusual behavior: Threewaves instead of one can be excited at the same frequency atthe single branch. Assuming that there exists also the secondbranch, we have four waves in the region of the fine structureof MLMM dispersion.

115420-3

CHEBYKIN, ORLOV, SIMOVSKI, KIVSHAR, AND BELOV PHYSICAL REVIEW B 86, 115420 (2012)

0

1 2 3 4-500

50100150200250300

0.08

0.09

0.1

1

a

b

xx

ky2 2/( /c)

-150-100-50

50100150

1 2 3 40

0xx

ky2 2/( /c)

-0.5

0

0.5

1

1.5

1 2 3 40

yy

1 2 3 40

yy

-1.3

-1.2

-1.1

-1

-0.9

1

2

1 2 3 40

0.11

0.12

0.13D

/ 2

(a)

(b)

xx y y( )= /( /c)k k 2 2yy y( )=0k

ky

~

ky

~

ky

~ky

~

ky

~

FIG. 3. (Color online) Dispersion diagram of MLMMs andgraphical solutions of dispersion equations (10) for d2 = 1.5d1 and(a) D/λ = 0.115, (b) D/λ = 0.099.

From this analysis, one can conclude that for the structureunder study there is a direct correspondence between itsanisotropy and the spatial dispersion. Of course, the existenceof the spatial dispersion effect follows from the dependenceof the permittivity tensor on the wave vector. However,

D/

2

ky

~0 2 4 6 8 100.1053

0.1054

0.1055

0.1056

0.1057

41 3

(a)

xx y y( )= /( /c)k k 22

yy y( )=0k

xx

ky22

/( /c)

0 2 4 6 8 10k

y

~

200

400

600

800

1000

yy

0 2 4 6 8 10k

y

~0

-0.2

0.20.40.60.8

11.2

FIG. 4. (Color online) Fine structure of the dispersion curves ofMLMMs in the vicinity of the plasmon resonance frequency andgraphical solutions of dispersion equations (10) for d1 = 1.5d2 andD/λ = 0.10545.

0123400.2

0.40.6

0.81

kx

~

ky

~

0

-20

-40

-60

xx

ky

~ 0 0.2 0.4 0.6 0.8 1

kx

~1234

ky

~1

1.52

2.53

3.5

yy

01234

ky

~ 0.20.40.60.8

1

kx

~0

2.5

5

xyyx

=

0123400.2

0.40.6

0.81

kx

~

ky

~

0

-20

-40

-60

xx

01234 0.20.4

0.60.8

1

ky

~kx

~

zz

1.7

1.9

2.1

(a)

(c)

(b)

(d)

FIG. 5. Nonlocal permittivity tensor components: (a) εxx , (b)εyy , (c) εxy,εyx , (d) εzz vs the normalized wave vector componentsk̃x = kxD/π and k̃y = kyD/π for metal-dielectric MLMMs at thewavelength 480 nm (D/λ = 0.1302).

the presence or absence of the spatial dispersion featuresin the eigenwaves at a given frequency are also linked tothe anisotropy of the corresponding permittivity tensor. Ifanisotropy of ε(ω,k) for a certain pair (ω,k) is sufficiently large(for the present structure, for |εxx/εyy | > 200), one shouldexpect a strong spatial dispersion at the frequency ω.

IV. ANALYSIS OF SPATIALLY DISPERSIVEPERMITTIVITY

Next, we analyze how the permittivity tensor of a metal-dielectric MLMM depends on the wave vector at differentfrequencies. In Figs. 5–8 these dependencies are shown forMLMMs with the parameters ε1 = 4.6, ε2 = 1 − λ2/λ2

p, d1 =37.5 nm, and d2 = 25 nm, λp = 250 nm at the wavelengths of480 nm, 580 nm, 630 nm, and 780 nm, respectively.

As can be seen from Figs. 5(a) to 8(a), at the highestnormalized frequency D/λ = 0.1302 [see Fig. 5(a)] thecomponent εxx grows with k̃y . At slightly lower frequencyD/λ = 0.1078, this component becomes a resonant function

00.5

1

00.5

1

1234

1 2 3 40

-6

0

3

-3

6

300200100

0-100-200-300

ky

~

ky~

k~~~

kx~

x~

x~

yyxx

00.5

1

1 2 3 4

10

-10

20

-20

30

-30

0

ky

~kx~

x

xy

00.5

11234 0

1

1.2

1.4

ky~kx

~x

~x

~

zz

(a) (b)

(d)(c)

FIG. 6. The same as in Fig. 5 but at the wavelength 580 nm(D/λ = 0.1078).

115420-4

NONLOCAL EFFECTIVE PARAMETERS OF MULTILAYERED . . . PHYSICAL REVIEW B 86, 115420 (2012)

01234

ky

~0.5

1

kx

~18

22

24

20

xxxx

01234

ky

~0.5 1

kx

~

0

-3.5

-7

xyyx

=

012340.5

1

ky

~kx

~0.6

0.8

1zz

012340.5

1

ky

~ kx

~-1.05-0.35

0.35

1.051.75

yy

(a)

(c)

(b)

(d)

FIG. 7. The same as in Fig. 5 but at the wavelength 630 nm(D/λ = 0.0992).

of the wave vector [see Fig. 6(a)], and at D/λ = 0.0992it has a maximum [see Fig. 7(a)]. At the lowest frequencyD/λ = 0.0801 [see Fig. 8(a)], this component decreases withk̃y . Resonant behavior of the tensor components can bedescribed in terms of the permittivity pole,(

εxx εxy

εyx εyy

)= 1

[ωres(k) − ω]

(Axx(ω,k) Axy(ω,k)Ayx(ω,k) Ayy(ω,k)

),

(11)

where Aαβ are analytical functions. In the theory of continuousmedia this representation of the permittivity tensor (without itsdependence on k) has been obtained in the approximation oflow optical losses and ωres is the frequency of the resonantabsorption.19 Our study also neglects losses, and we caninterpret our ω̃res = D/λres as the presumable frequency ofresonant absorption if some small losses are introduced. Thisresonant absorption is of course related to the collectiveplasmon resonance of metal layers. However, due to thespatial dispersion this frequency depends on k. We cansee in Figs. 6(a)–6(c) that at the wavelength 580 nm threeindependent components of ε are resonant (εxx, εyy , and εxy =εyx). Such behavior has been found in a rather wide rangearound 580 nm for all these components. In Fig. 9, we showthe dispersion surface ω̃res = D/λres. A crossing of this surfacewith the dispersion diagram would deliver the frequencies

0123400.20.4

0.60.8

1

kx

~

12

8

4

xx

01234

ky

~0.2

0.40.6

0.81

kx

~-12-60yy

01

23

4 ky

~0

0.20.4

0.60.8

kx

~

1-4-20

xyyx

=

01234

-0.7-0.5-0.3

0.20.4

0.60.8

1

ky

~kx

~

zz

ky

~

(a) (b)

(c) (d)

FIG. 8. The same as in Fig. 5 but at the wavelength 780 nm(D/λ = 0.0801).

12340.5

0

0

10.1

0.11

0.12

0.13

ky

~kx

~

res

~

FIG. 9. Normalized resonance frequency ω̃res = ωresD/2πc as afunction of the components k̃x = kxD/π and k̃y = kyD/π .

and propagation directions at which the resonant absorptionholds. Briefly, we claim that the resonant frequency in metal-dielectric MLMMs depends on the propagation direction.

The only nonresonant component of the permittivity tensoris εzz. As can be seen in Figs. 5(d) to 8(d), there are noqualitative changes in the behavior of this component in respectto the wave vector at different frequencies: The component εzz

decreases with an increase of k̃y , but it grows with the increaseof k̃x .

Next, we calculate the variation of εxy(ω,k) when thecoordinate system rotates around z. The purpose of thiscalculation is to eliminate the off-diagonal component of thepermittivity tensor transforming the latter to the form of atensor of a uniaxial medium. This is possible to do for any ky ata given frequency ω (whereas the wave vector component kx isrelated to ky via the dispersion equation). For the semi-infiniteinterval k0 < ky < ∞ at every frequency, there exists a finitevalue ky at which the needed rotation angle θ is maximal. Atthe wavelength 480 nm the maximum of the function θ (ky)for this semi-infinite interval of ky equals 8◦56′. At 630 nmθmax = 15◦40′, and θmax = 13◦7′ at 780 nm. For a half spaceof our MLMMs whose boundary is parallel to the layers, theeigenmodes with k0 < ky < ∞ can be excited only by incidentevanescent waves. Eigenmodes with ky < k0 can be excited bypropagating plane waves, and for these eigenmodes the opticalaxis rotation turned out to be much smaller. For ky < k0, thevalues of θmax are equal to 46′ at the wavelength 480 nm, to30′ at 580 nm, to 21′ at 630 nm, and to 17′ at 780 nm. Wecan conclude that for incident propagating waves this effectof spatial dispersion can be practically negligible. However,this refers only to one special geometry of the boundary. Ifthe interface of the MLMM half space is orthogonal to thelayers the eigenmodes k0 < ky < ∞ can be excited by incidentplane waves kx < k0, as well. In this case the componentkx of the incident wave vector is preserved in the excitedeigenmode and eigenmodes with kx < k0 and ky > k0 exist inthe metal-dielectric MLMMs. For this geometry, the opticalaxis rotation is an important effect of spatial dispersion whichholds for propagating incident waves as well.

V. LAYERED DIELECTRIC STRUCTURES

For comparison, we apply the model of the nonlocaleffective medium to the case of all-dielectric MLMMs. Asan example, we consider an all-dielectric structure created bythe layers of two dielectric materials with ε1 = 1 and ε2 = 4.6

115420-5

CHEBYKIN, ORLOV, SIMOVSKI, KIVSHAR, AND BELOV PHYSICAL REVIEW B 86, 115420 (2012)

01234

ky

~0.5

1

kx

~1.8

2.3

2.8

xxxx

2.2

2.6

3

3.4

0

0.5

1

kx

~

1234

ky

~

yy

3.16

3.2

3.24

0

0.5

1

kx

~

1234

ky

~

-0.15

0.15

0

01

23

4 ky

~

xyyx

=

-0.15

0.15

0

01

23

4 ky

~

xyyx

=

-0.15

0.15

0

01

23

4 ky

~

xyyx

=

-0.15

0.15

0

01

23

4 ky

~

xyyx

=

0.5

1

kx

~

ky

~ky

~

zz

(a) (b)

(c) (d)

FIG. 10. Nonlocal permittivity tensor components: (a) εxx ,(b) εyy , (c) εxy,εyx , (d) εzz vs the normalized wave vector componentsk̃x = kxD/π and k̃y = kyD/π for all-dielectric layered structureswith ε1 = 4.6 and ε2 = 1 at the wavelength of 580 nm (d1 = 1.5d2).

at wavelength 580 nm, also taking the layers’ thicknesses asd1 = 37.5 nm and d2 = 25 nm, respectively. The results arebased on the analysis stated above, and they are presented inFig. 10. From those results we observe that the components ofthe permittivity tensor vary noticeably along with variationsof the wave vector even for this all-dielectric structure.Therefore, the spatial dispersion effects are still not negligiblethough they are not so strong as in the case of metal-dielectricMLMMs.

The component εxx varies within 32% from its maximumvalue, and the component εyy changes in the range of 31%.In the case of all-dielectric multilayers, off-diagonal tensorcomponents are also present, but their values range from −0.15to 0.13, depending on the wave vector. The smallest variation(no more than 2%) is found for the component εzz.

Off-diagonal components εxy and εyx do not vanish, andthey are quite noticeable. We calculate also the value of theangle θmax for the all-dielectric structure under consideration.At the wavelength 580 nm we obtain θmax = 7◦10′, for k0 <

ky < ∞, and θmax = 31′ for ky < k0.

VI. CONCLUSIONS

We have studied the effects of spatial dispersion on theproperties of nanostructured metal-dielectric metamaterials.We have verified the general concepts of the nonlocal ho-mogenization theory earlier developed for such structures25,26

and also analyzed some novel effects associated with themanifestation of the nonlocal response. In particular, wehave revealed that in nanostructured media the optical axisbecomes tilted with respect to its position in the homogeneousdielectric medium, and this tilt depends on both frequency andpropagation direction. We have shown numerically that thiseffect is significant not only for metal-dielectric metamaterialsbut also for all-dielectric nanostructured materials. We havealso pointed out the correspondence between the extremeanisotropy in the nonlocal permittivity tensor and the spatialdispersion. Multilayered metamaterials provide a remarkableexample of nanostructured materials for which the nonlocalhomogenization theory can be developed and implementedanalytically.

ACKNOWLEDGMENTS

This work has been supported by the Ministry of Educationand Science of the Russian Federation, the Dynasty Founda-tion, EPSRC (UK), and the Australian Research Council.

APPENDIX

After some algebra, we rewrite the relations of the ap-pendices of Ref. 25 in a more explicit form. The componentorthogonal to the propagation plane can be written as follows:

εzz(ω,k) = [(ε1d1

(ε2k

20 − k2

) + ε2d2(ε1k

20 − k2

))(ε1k

20 − k2

)(ε2k

20 − k2

)�TE − 2α(ε1 − ε2)2k2

0k2]

× [(d1

(ε2k

20 − k2

) + d2(ε1k

20 − k2

)(ε1k

20 − k2

))(ε2k

20 − k2

)�TE − 2α(ε1 − ε2)2k4

0

]−1,

α = 2A + B(k2x1 + k2

x

) + C(k2x2 + k2

x

),

A = kx1kx2kx [cos(kx2d2) sin(kxd1) + cos(kx1d1) sin(kxd2) − sin(kxD)] , (A1)

B = kx2 sin(kx1d1) [cos(kx2d2) − cos(kxd2)] ,

C = kx1 sin(kx2d2) [cos(kx1d1) − cos(kxd1)] ,

�TE = sin(kx1d1) sin(kx2d2)(k2x1 + k2

x2

) + 2kx1kx2 [cos(kxD) − cos(kx1d1) cos(kx2d2)] ,

where k is the wave vector in the metamaterial, kx1 =√

ε1k20 − k2

y , kx2 =√

ε2k20 − k2

y .The relations for other components look more compact in the matrix form:(

εxx(ω,k) εxy(ω,k)

εyx(ω,k) εyy(ω,k)

)= 〈E〉−1〈D〉. (A2)

115420-6

NONLOCAL EFFECTIVE PARAMETERS OF MULTILAYERED . . . PHYSICAL REVIEW B 86, 115420 (2012)

Components of the matrices 〈E〉 and 〈D〉 can be written in the following form:

〈E〉xx = [k2x

(k2

0

(ε2

1d2 + ε22d1

) − k2(ε1d2 + ε2d1)) + ε1ε2k

20

(k2D − k2

0(ε1d2 + ε2d1))]

(ε1ε2)−1

+ Pk2y

ε1ε2

{2Aε1ε2k

20

[k2

0(ε1 + ε2) − k2] − Bε1

[ε2

2k40

(ε1k

20 − k2

y

) + k2x

(k2 − k2

0(ε1 + ε2))2]

−Cε2[ε2

1k40

(ε2k

20 − k2

y

) + k2x

(k2 − k2

0(ε1 + ε2))2]}

,

〈E〉xy = 〈E〉yx = −kxky

[k2(ε2d1 + ε1d2) − k2

0

(ε2

2d1 + ε21d2

)](ε1ε2)−1

− Pky

ε1ε2

{A

ε1ε2k20

kx

[ε1ε2k

40 + (

k2y − k2

x

)(k2 − (ε1 + ε2)k2

0

)] − Bε1kx

[ε2

2k40

(ε1k

20 − k2

y

)+ [

ε1ε2k40 + k2

y

(k2 − (ε1 + ε2)k2

0

)][(ε1 + ε2)k2

0 − k2)]] − Cε2kx

[ε2

1k40

(ε2k

20 − k2

y

) + ((ε1 + ε2)k2

0 − k2)

× (k4

0ε1ε2 + k2y

(k2 − (ε1 + ε2)k2

0

))]},

〈E〉yy = [k2y

(k2

0

(ε2

1d2 + ε22d1

) − k2(ε1d2 + ε2d1)) + ε1ε2k

20

(k2D − k2

0(ε1d2 + ε2d1))]

(ε1ε2)−1

− P

ε1ε2

{2Aε1ε2k

20

[k2y

((ε1 + ε2)k2

0 − k2) − k4

0ε1ε2] + Bε1

[(ε1ε2k

40 + k2

y

(k2 − (ε1 + ε2)k2

0

))2

+ ε22k

40k

2x

(ε1k

20 − k2

y

)] + Cε2[ε2

1k40k

2x

(ε2k

20 − k2

y

) + (ε1ε2k

40 + k2

y

(k2 − (ε1 + ε2)k2

0

))2]},

〈D〉xx = k20

[k2x(ε2d1 + ε1d2) + k2(ε1d1 + ε2d2)

] − D(k4

0ε1ε2 + k2xk

2) − Pk2y

{A

[ − ε1ε2k40 + (ε1 + ε2)k4 − k2k2

0

]+Bk2

0

[ε1k

2x

(ε1k

20 − k2

) + ε2(ε1k

20

(2k2

x + k2y

) − k2yk

2)] + Ck2

0

[ε2k

2x

(ε2k

20 − k2

) + ε1(ε2k

20

(2k2

x + k2y

) − k2yk

2)]}

,

〈D〉xy = kxky

(k2

0(ε2d1 + ε1d2) − k2D) − Pky

{A

kx

[k2yk

2(k2 − ε2k

20

) + ε1(ε2k

40

(k2 + k2

x

) − k2k20k

2y

)]−Bk2

0kx

[ε1k

2y

(k2 − ε1k

20

) + ε2(ε1k

20

(k2x + ε1k

20

) − k2yk

2)] − Ck20kx

[ε2k

2y

(k2 − ε2k

20

)+ ε1

(ε2k

20

(k2x + ε2k

20

) − k2yk

2)]}

,

〈D〉yx = kxky

(k2

0(ε2d1 + ε1d2) − k2D) + Pkxky

{A

[−ε1ε2k40 + (ε1 + ε2)k4 − k2k2

0

]+Bk2

0

[ε1k

2x

(ε1k

20 − k2

) + ε2(ε1k

20

(2k2

x + k2y

) − k2yk

2)] + Ck2

0

[ε2k

2x

(ε2k

20 − k2

)+ ε1

(ε2k

20

(2k2

x + k2y

) − k2yk

2)]}

,

〈D〉yy = k20

[k2y(ε2d1 + ε1d2) + k2(ε1d1 + ε2d2)

] − D(ε1ε2k

40 + k2

yk2)

+Pkx

{A

kx

[k2yk

2(k2 − ε2k

20

) + ε1(ε2k

40

(k2 + k2

x

) − k2k20k

2y

)] − Bk20kx

[ε1k

2y

(k2 − ε1k

20

)+ ε2

(ε1k

20

(k2x + ε1k

20

) − k2yk

2)] − Ck2

0kx

[ε2k

2y

(k2 − ε2k

20

) + ε1(ε2k

20

(k2x + ε2k

20

) − k2yk

2)]}

,

where

P = 2(ε1 − ε2)2�−1TM(

k2 − ε1k20

)(k2 − ε2k

20

) . (A3)

The physcial meaning of 〈E〉 and 〈D〉 is the following: These two matrices describe, respectively, the components of theelectric field and electric displacement field, normalized by the factor i[ωε0(k2 − ε1k

20)(k2 − ε2k

20)]−1 and averaged over the

period of the structure, D = d1 + d2.

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