PHYSICAL REVIEW B 91, 174201 (2015)

Effective-medium theory for one-dimensional gratings

Shiwei Tang,1 Baocheng Zhu,1 Min Jia,1 Qiong He,1,2 Shulin Sun,3 Yongfeng Mei,4 and Lei Zhou1,2,*

1State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education),Fudan University, Shanghai 200433, China

2Collaborative Innovation Center of Advanced Microstructures, Fudan University, Shanghai 200433, China3Department of Optical Science and Engineering, and Shanghai Ultra-Precision Optical Manufacturing Engineering Center, Fudan

University, Shanghai 200433, China4Department of Materials Science, Fudan University, Shanghai 200433, China

(Received 3 December 2014; revised manuscript received 4 May 2015; published 22 May 2015)

Based on a mode-expansion theory under single-mode approximation, we derived the scatterings parametersfor a general one-dimensional photonic grating composed of two different materials, and then establishedan effective-medium theory for such a composite by equating the obtained scattering parameters to thoseof a homogeneous medium. Our effective-medium theory well describes the grating structures with generalmaterial and geometrical parameters, and recovers two previous formulas, which are valid only at certain limitingconditions. The theory is justified by full-wave simulations and microwave experiments.

DOI: 10.1103/PhysRevB.91.174201 PACS number(s): 78.67.Pt, 81.05.Xj, 78.20.Ci, 77.22.Ch

I. INTRODUCTION

Metamaterials are electromagnetic composites constructedby subwavelength-sized microstructures, and thus can ex-hibit tailored effective permittivity ε and permeability μ.Metamaterials have attracted much attention recently due totheir strong abilities to manipulate electromagnetic waves,leading to many fascinating physical effects unattainablein naturally existing materials [1–7]. In this research field,effective-medium theory plays a crucial role since it servesas a bridge to link theories, which are frequently conductedon model hypothetical systems, and experiments, which arealways performed on realistic metamaterial systems withcomplex microstructures. Therefore, a good effective-mediumtheory, which can precisely predict the effective parameters ofcomplex metamaterials, is always highly desired.

As a particular class of metamaterial, grating structuresconsisting of alternative stacking of two layers with subwave-length thicknesses and distinct electromagnetic properties[see Fig. 1(a)] have been widely studied in recent years.Many interesting physical effects were discovered based onsuch structures, such as hyperlensing [8–10], focusing [11],cloaking [12,13], slow-light transport [14], and radiationlift-time engineering [15–18]. To describe the electromagneticproperties of these systems, one frequently used the standardformulas in which the effective ε, or the inverse of effectiveε, is written as a volume average of local permittivity or theinverse of local permittivity, depending on the polarization[19,20]. However, since these formulas were derived in thelong wavelength limit, they are valid only for grating structuresstrictly satisfying the long-wavelength-limit condition, thatis, the optical thicknesses of both layers should be muchsmaller than the input wavelength. Whenever the permittivityand/or permeability of a single layer exhibits very large values,such long-wavelength-limit formulas become invalid, eventhough the whole system still satisfies the subwavelengthcondition (i.e., the lattice constant of the grating is smaller

*Corresponding author: phzhou@fudan.edu.cn

than wavelength). On the other hand, based on the mode-expansion theory, a delicate approach [21,22] was proposedto homogenize such AB systems, which does not requirea strict long-wavelength-limit condition. Unfortunately, thedeveloped theory is limited to the special case that one layermust be a perfect electric conductor (PEC) [21]. Given so manyinteresting applications/phenomena already discovered basedon these systems [8–18], and given considerable efforts madeon developing appropriate effective-medium descriptions forsuch systems [23–28], we feel that an effective-medium theoryfor such grating structures, which is valid under more generalconditions, is highly desired.

In this work, we establish a theory to study the effective-medium properties of one-dimensional gratings with arbitraryconstitutional and geometrical parameters. Our effective-medium theory not only recovers those previously derivedformulas valid only at certain limiting conditions, but workswell far beyond these limitations to nearly cover the wholeparameter spaces as long as the subwavelength condition issatisfied. Finite-difference-time-domain (FDTD) simulationsand microwave experiments are performed on realistic struc-tures to justify our theory.

Our paper is organized as follows. We first derive theeffective-medium theory in Sec. II, and then justify its validityin Sec. III by full-wave simulations. Section IV is devoted toa detailed comparison between our effective-medium theoryand two previous formulas. After presenting the experimentalconfirmation of our effective-medium theory in Sec. V, weconclude the paper in Sec. VI.

II. EFFECTIVE-MEDIUM THEORY

Figure 1 schematically depicts the geometry of our system,whose unit cell contains two layers with thicknesses dA anddB , and electromagnetic parameters εA,μA and εB,μB . Weare primarily interested in the effective permittivity εxx

eff of thestructure, since it is this component that caused strong debatesand confusions in previous studies [24–26]. To study εxx

eff , weassume that a transverse-magnetic (TM) polarized light (with�E||x̂) is normally coupled into the grating through the xy

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TANG, ZHU, JIA, HE, SUN, MEI, AND ZHOU PHYSICAL REVIEW B 91, 174201 (2015)

FIG. 1. (Color online) Schematic picture of (a) the grating struc-ture under study and (b) its effective medium model.

interface [Fig. 1(a)], and study the corresponding light scat-tering problem. Obviously, light scatterings at this interfacecannot be studied by a simple 2 × 2 transfer-matrix method(TMM), since higher-order diffractions are inevitable [seeFig. 1(a)]. Our strategy to determine the effective parametersεxx

eff ,μyy

eff of the given system is described as follows. We firstemploy the rigorous mode-expansion theory to calculate thescattering coefficients of the system [Fig. 1(a)], and then deriveanalytical formulas for the zero-order transmission/reflectioncoefficients based on a single-mode approximation. Finally,the effective-medium parameters are fixed by equating theobtained zero-order scattering coefficients to those of a ho-mogeneous medium under the same illuminations [Fig. 1(b)].Without causing confusion, in what follows we neglect thesuperscripts xx and yy in εxx

eff and μyy

eff .According to the mode-expansion theory, we need to first

expand the electromagnetic fields to linear combinations ofeigenmodes in different regions. For the present configuration,electromagnetic eigenmodes in region I (air region above thegrating, see Fig. 1) are just diffraction modes, which are TM-polarized plane waves taking parallel wave vectors k(I)

x,n = n ·2π/d (n = 0, ± 1, ± 2, . . .) with d = dA + dB being the peri-odicity of our system. The corresponding field patterns are de-

noted by [ �E(I)n (x), �H (I)

n (x)]e±ik(I)z,n·z where k(I)

z,n =√

k20 − (k(I)

x,n)2

with k0 = ω/c and ± denote forward and backward modes,respectively. In region II, which is the semi-infinite space occu-pied by the grating, the relevant electromagnetic eigenmodesare a series of Bloch waves characterized by Bloch wavevectors Kx,q = q · 2π/d with q = 0, ± 1, ± 2, . . . . Sinceregion II exhibits translation-invariant symmetry along the z

axis, the eigen-wave-functions in this region can be written ingeneral as [ �E(II)

q (x), �H (II)q (x)]e±ik

(II)z,q ·z with perpendicular wave

vectors k(II)z,q being unknown parameters to be determined.

Meanwhile, eigenmodes in such AB periodic systems can bestudied by a standard TMM imposing the Bloch conditionalong the x direction, i.e., �E(II)

q (x + d) = eiKx,qd �E(II)q (x) and

�H (II)q (x + d) = eiKx,qd �H (II)

q (x). A straightforward calculationshows that, given a frequency ω and thus k0 = ω/c, k(II)

z,q arecorrelated with the Bloch wave vector Kx,q by the followingequation [29] (see Appendix A),

2 cos(kAx dA + kB

x dB

) −(

εBkAx

εAkBx

+ εAkBx

εBkAx

− 2

)

× sin(kAx dA

)sin

(kBx dB

) = 2 cos(Kx,qd), (1)

where

kAx =

√εAμAk2

0 − (k

(II)z,q

)2, kB

x =√

εBμBk20 − (

k(II)z,q

)2. (2)

Solving Eq. (1) yields a series of k(II)z,q for all eigenmodes.

The corresponding wave functions [ �E(II)q (x), �H (II)

q (x)] can beeasily obtained by the TMM calculations.

With eigenmodes in both regions known, we can formallyexpress the total E fields in different regions as

�E(I)total(x,z) = �Einc(x)eik0·z +

∑n

rn�E(I)

n (x)e−ik(I)z,n·z

(3)�E(II)total(x,z) =

∑q

tq �E(II)q (x)eik

(II)z,q ·z,

where �Einc(x)eik0·z = �E(I)0 (x)eik0·z is the incident field, rn

denotes the reflection coefficient for the nth mode in region Iand tq the transmission coefficient for the qth mode in regionII. H fields satisfy two similar equations. We note that E and Hcomponents of these eigenmodes are not totally independent,but rather correlated with each other via

E(I)n,x(x) = k(I)

z,n

ωε0H (I)

n,y(x)(4)

E(II)q,x(x) = k(II)

z,q

ωε0ε(x)H (II)

q,y (x),

where we have defined ε(x) = εA for x located inside layer Aand ε(x) = εB for x located inside layer B. According to thefield continuity conditions at the z = 0 interface, we get fromEq. (3) that

E(I)0,x(x) +

∑n

rnE(I)n,x(x) =

∑q

tqE(II)q,x(x)

(5)H

(I)0,y(x) +

∑n

rnH(I)n,y(x) =

∑q

tqH(II)q,y (x).

We can use the orthonormal properties of those eigenmodesto derive the equations satisfied by the expansion parameters{rn,tq}. In region I, it is easy to verify that eigenmodes (diffrac-tion modes) satisfy the following orthonormal conditions∫

u.c.

[E(I)

m,x(x)]∗ · E(I)

n,x(x)dx = δm,n, (6)

where the integrals are performed within a unit cell. Theorthonormal conditions for H (I)

n,y(x) can be derived fromEqs. (4) and (6). However, things are quite nontrivial for theeigenmodes inside region II, where the orthonormal conditionsshould now be written as∫

u.c.

(1

ε(x)H (II)

q,y (x)∗ · H(II)q ′,y(x)

)dx = δqq ′ , (7)

in which a metric term 1/ε(x) must be added [30] (seeAppendix B). Again, the orthonormal conditions for E(II)

q,x(x)can be derived from Eqs. (7) and (4) [30]. In the limitingcase of one layer being PEC (i.e., let εA(x) → −∞), we notethat both E and H fields vanish inside layer A so that Eq. (7)recovers the conventional orthonormal conditions satisfied bythe waveguide modes in layer B sandwiched by two PEC walls(e.g., Eq. (A7) in Ref. [31]).

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Employing the two orthonormal conditions Eqs. (6)–(7)[32], we can derive from Eq. (5) a set of equations to determinethe expansion coefficients,

(1 − r0)(SH

0q

)∗ +∑n�=0

[rn

(SH

nq

)∗] = tq

(8)δn0 + rn =

∑q

tqSEnq,

where the S parameters are defined by

SEnq =

∫u.c.

(E(I)

n,x

)∗ · E(II)q,xdx

(9)SH

nq =∫

u.c.

(H (I)

n,y

)∗ · 1

ε(x)H (II)

q,y dx,

which represent the overlapping between different modes intwo regions.

In principle, the scattering coefficients {tq ,rn} can beobtained by numerically solving Eq. (8) with enough modesconsidered. To retrieve the effective-medium properties ofthe system, we need to derive the analytical formulas forthe zero-order scattering coefficients, which can be donebased on a single-mode approximation [i.e., retaining onlythe fundamental modes (labeled by indexes n = 0,q = 0)in both regions]. Such an approximation is justified whenthe effective overlapping integral between two fundamentalmodes (S00)2 = SE

00 · (SH00)∗ is the dominant term among all

overlapping integrals (Snq)2 = SEnq · (SH

nq)∗. Under such anapproximation, we can solve Eq. (8) rigorously to get thereflection coefficient for the specular mode as

r0 = (S00)2 − 1

(S00)2 + 1. (10)

Mapping our system to a homogeneous medium [seeFig. 1(b)] with effective parameters εeff,μeff , and noting thatthe reflection coefficient at the surface of such a medium isr0 = (Zeff − 1)/(Zeff + 1) with Zeff = √

μeff/√

εeff , we foundthat the effective impedance of our grating structure is

Zeff = (S00)2. (11)

On the other hand, since we only consider the fundamentalmode inside the grating, which is characterized by a wavevector k

(II)z,0 , mapping our medium to a homogeneous effective

medium means that the effective refraction index of the gratingmust be

neff = √εeff · √

μeff = k(II)z,0

/k0. (12)

Solving Eqs. (11)–(12), we finally get the followinganalytical formulas

εeff = k(II)z,0

k0 · (S00)2

(13)

μeff = k(II)z,0 · (S00)2

k0

to determine the effective parameters of our grating medium.Two involved physical quantities, k(II)

z,0 ,(S00)2, can be computedby solving Eqs. (1)–(2) and (9).

At this point, it is helpful to discuss under what conditionsthe single-mode approximation is valid. In most cases wherethe subwavelength condition d λ is satisfied, such anapproximation is valid since all high-order diffraction modesin air are evanescent waves and the high-order modes inside thegrating take imaginary wave vectors due to the subwavelengthconfinement in lateral directions. However, we will showthat in some particular situations, such an approximationbecomes invalid even though the subwavelength condition isstill satisfied.

III. VALIDATIONS BY NUMERICAL SIMULATIONS

We employ full-wave simulations to verify the newly estab-lished effective-medium theory based on two representativeexamples. The first example is a dielectric/air grating withparameters εA = 40,μA = 1,εB = μB = 1, and dA = 3dB .For any frequency satisfying the subwavelength condition,i.e., k0d/2π 1, we can employ our theory to determinethe effective parameters of the structure. εeff and μeff thusobtained are plotted in Fig. 2(a) as functions of frequency. Inthe long wavelength limit where k0d/2π → 0, we find thatthe calculated εeff and μeff approach to those values obtainedby standard volume average method [20], i.e., ε̄ = 3.721and μ̄ = 1. However, deviations from these long-wavelength-limit values become non-negligible as frequency increases,especially for μeff . These results appear quite counterintuitiveat first glance since the constitutional materials constructingour grating are all purely dielectric. Such a magnetic response,although very weak here, must be contributed by displacementcurrents excited inside the dielectric layers.

FIG. 2. (Color online) (a) Effective parameters of a dielectric/airgrating calculated by our effective-medium theory. (b) Transmissionand reflection spectra of a 15d-thick slab of such grating structure,calculated by FDTD simulations on realistic system (symbols) andour effective-medium theory on model system (lines). Here theconstitutional parameters are εA = 40,μA = 1,εB = μB = 1, anddA = 3dB .

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FIG. 3. (Color online) Transmittance and reflectance for twoslabs of AB gratings with different thicknesses h, calculated by FDTDsimulations on realistic systems (symbols) and our effective-mediumtheory on model systems (lines). Here, the working wavelength isfixed as λ = 25d . In these AB structures, the B layer is always fixedas air while the A layer is a dielectric with (a) εA = 40 and a metalwith (b) εA = −100.

To justify our effective-medium theory, we employ FDTDsimulations to calculate the transmission/reflection spectra ofelectromagnetic waves through a slab of such AB gratingwith thickness h = 15d, and then compare in Fig. 2(b) theFDTD results with those calculated by a standard TMM on ahomogeneous slab of the same thickness with electromagneticparameters given by εeff,μeff as shown in Fig. 2(a). Excellentagreement is noted between FDTD and the effective-medium-theory results in the whole frequency region considered. Wenext fix the wavelength at λ = 25d to compute the trans-mittance and reflectance of electromagnetic waves througha series of slabs of our gratings with different thicknessesh. Comparison between FDTD and effective-medium theoryshown in Fig. 3(a) indicates that our theory can well reproducethe full-wave simulations on realistic structures.

The second example we study is a metal/air grating,which was frequently studied in the literature as a hyperbolicmetamaterial [21,27,33]. The unit cell is the same as that of thefirst example, only with the A layer replaced by a metal layerdescribed by a Drude-like dielectric function εA = 1 − λ2/λ2

p

with λp = 1.875d denoting the plasmon wavelength. Weemploy our effective-medium theory to calculate εeff,μeff

of this system and plot the obtained results in Fig. 4(a).Compared with the case of dielectric AB grating [see Fig. 2(a)],we find the present metal/air grating exhibits a diamagneticresponse with μeff < 1. We also note that even at very lowfrequencies, the calculated μeff does not converge to the long-wavelength-limit value μ̄ = 1, indicating the failure of thenaı̈ve long-wavelength-limit formulas in studying this system.Such a strong diamagnetic response has been noted in previousstudies [34]. Again, to justify our effective-medium theory,we perform two series of FDTD simulations similar to thosefor the dielectric/air systems, and compare the results with

FIG. 4. (Color online) (a) Effective parameters of a metal/airgrating calculated by the effective-medium theory. (b) Transmissionand reflection spectra of a 15d-thick slab of such grating structure,calculated by FDTD simulations on realistic system (symbols) andthe effective-medium theory on model system (lines). Here the con-stitutional parameters are εA = 1 − λ2/λ2

p with λp = 1.875d,μA =1,εB = μB = 1, and dA = 3dB .

those calculated by our effective-medium theory. Comparisonsshown in Fig. 3(b) and Fig. 4(b) justified our effective-mediumtheory for such a system.

IV. COMPARISONS WITH PREVIOUS THEORIES

Our effective-medium theory can cover two previouslydeveloped formulas under certain limiting conditions. Con-sider first the limit of εA → −∞ when layer A becomesa PEC so that both the E and H field inside layer A areexactly 0. In such a case, the fundamental mode in theAB structure is just a transverse electromagnetic waveguidemode trapped only inside layer B with wave vector given byk

(II)z,0 = k0

√εBμB . Meanwhile, it is straightforward to use the

normalized wave function of such a transverse electromagnetic

mode to demonstrate that SE00 = k0

ωε0

√μBdB

d, SH

00 = ωε0k0

√dB

εBd,

and thus

(S00)2 =√

μB

εB

dB

d. (14)

Putting the above results into Eq. (13), we find the effectiveparameters are now given by

εeff = εBd/dB(15)

μeff = μBdB/d,

which are just the results obtained by the PEC-based effective-medium theory for such systems [21].

Consider the second limiting case that λ0/√

εAμA dA,λ0/

√εBμB dB, which represents the true long wavelength

limit. In this limit, noting that k(II)z,0 is of the same order with k0,

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EFFECTIVE-MEDIUM THEORY FOR ONE-DIMENSIONAL . . . PHYSICAL REVIEW B 91, 174201 (2015)

we thus expect that both kAx dA and kB

x dB are small parameters[see Eq. (2)]. Expanding Eq. (1) in terms of kA

x dA and kBx dB ,

and only retaining the leading-order terms, we get

(kAx dA

)2 + (kBx dB

)2 +(εBkA

x

εAkBx

+ εAkBx

εBkAx

) (kAx dA

)(kBx dB

)= 0.

(16)

Putting Eq. (2) into Eq. (16), we solve the resultant equationto get (

k(II)z,0

)2 = μAdA + μBdB

εA−1dA + εB

−1dB

k20 . (17)

Let us now check the overlapping functions SE00 and SH

00.Since kA

x dA → 0 and kBx dB → 0, we expect that Hy does not

vary too much within both layer A and layer B. In addition,considering that Hy should be continuous across the ABboundary, it is a good zero-order approximation to assumethat Hy keeps at a constant value within the entire unit cell.Adopting the orthonormal condition Eq. (7), we find that thewave function of the fundamental mode inside region II canbe approximately written as

H(II)0,y ≈ 1√

ε−1A dA + ε−1

B dB

. (18)

The corresponding wave function for E(II)0,x can be obtained

with Eqs. (18) and (4). Put the above results into Eq. (9),we find from simple computations that the two overlappingintegrals are

SE00 = k

(II)z,0

ωε0

√ε−1A dA+ε−1

B dB

d(19)

SH00 = ωε0

k0

√ε−1A dA+ε−1

B dB

d.

Therefore, the two effective parameters can be derived fromEqs. (19) and (13), which are

εeff = d

ε−1A dA + ε−1

B dB,

(20)μeff = μAdA + μBdB

d.

Equation (20) recovers the well-known formulas derived inthe long wavelength limit for this polarization [19].

We now compare our effective-medium theory with thesetwo previous formulas under more general conditions. Inparticular, we will check the validity regions of these dif-ferent versions of effective-medium theory. To quantitativelydescribe how good an effective-medium theory is, we define aphysical quantity

δT = 〈|tEMT − tFDTD|〉 , (21)

which measures the difference between the (complex)transmission coefficient tEMT calculated by a particulareffective-medium theory on a homogeneous model systemand the FDTD-calculated complex value tFDTD on a realisticinhomogeneous AB grating system. To suppress theaccidental fluctuations caused by Fabry-Perot resonances

FIG. 5. (Color online) (a) δT for a series of AB gratings withvarying εA, calculated by the present effective-medium theory(circles), the PEC-based formula Eq. (15) (triangles), and the long-wavelength-limit formula Eq. (20) (squares). (b) Effective parametersof the systems calculated by the present effective-medium theory, withinset depicting how the two overlapping integrals vary against εA.

through a finite-thickness slab, we average |tEMT − tFDTD|over samples with thicknesses varying from 0.25d–15d. Theeffective-medium theory can be our theory developed here,and can also be the two previous formulas [i.e, Eq. (15)and Eq. (20)]. By computing δT for different versions ofeffective-medium theory, we can quantitatively check howgood these effective-medium theories are in describing theinhomogeneous medium under study.

We now compare these theories by studying a series ofAB gratings with B layer fixed as air and with εA varying from−∞ to ∞. Fix the working wavelength as λ = 25d, we employdifferent versions of effective-medium theory to compute theircorresponding δT ’s and compare the results in Fig. 5(a) asfunctions of εA. The effective parameters calculated by thepresent theory are plotted in Fig. 5(b) for the sake of easy com-parison. Figure 5(a) shows clearly that, while the PEC-basedformulas [i.e., Eq. (15)] can only work in the limiting casewith εA → −∞ and the long-wavelength-limit formulas [i.e.,Eq. (20)] can only work in two small regions −40 < εA < −15and 0 < εA < 20, our effective-medium theory works verywell in nearly the entire parameter space studied, includingthose regions where previous formulas do not work at all.

However, in a small shaded region centered at εA = −3, allthree effective-medium theories do not work well, as shown bythe large values of δT calculated by different theories. A simpleexplanation of such failure is that the effective εeff is so largenear the point of εA = −3 that the effective wavelength insidethe medium, defined as λeff = λ0/

√εeff · μeff , is significantly

enhanced resulting in the failure of the single-mode approxi-mation. Such a simple picture is reinforced by quantitativelycalculating the overlapping integrals involving higher-ordermodes. The inset to Fig. 5(b) depicts how the two mostimportant overlapping integrals, namely (S00)2 and (S01)2, varyagainst εA. Clearly, the coupling between the first high-ordermode inside the grating and the fundamental mode in airbecomes significantly enhanced in the shaded region, and itcan even compete with the coupling between two fundamental

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TANG, ZHU, JIA, HE, SUN, MEI, AND ZHOU PHYSICAL REVIEW B 91, 174201 (2015)

modes in the vicinity of εA → −3. In fact, the fundamentalmode in region II, which is defined as the mode that hasthe largest overlapping integral with the fundamental mode inregion I, changes its wave-function characteristics at the pointεA → −3. Therefore, in such a parameter region, the innerproperties inside the grating are no longer solely dominatedby a single fundamental mode, so that it is impossible tofind an effective-medium theory to homogenize the complexstructure. We note that a similar conclusion has also beendrawn from a different theoretical approach [24,26]. However,in all other parameter regions as long as the AB grating can behomogenized as an effective medium, our effective-mediumtheory is always the best one to describe the complex system.

Although here we did not consider the nonlocal responsesin developing our effective-medium theory, we note that thetheory can be easily extended to include such effects by simplyconsidering the oblique-incidence situations. In fact, we findthat our theory, even without considering the nonlocal effects,can still well describe the optical responses of the gratingsunder oblique incidences (see Appendix C). In addition,we note that the homogenization approach described in thispaper is so general that it can also be employed to developeffective-medium theories for more complex structures withtwo-dimensional microstructures.

V. EXPERIMENTAL VERIFICATIONS

We now perform microwave experiments to verifythe newly developed effective-medium theory for one-dimensional gratings. To highlight the importance of oureffective-medium theory, we purposely choose a parameterregion where the previous two formulas do not work well.Figure 5(a) shows that the region where εA takes a moderatelylarge negative value is a desired region. However, metal doesnot exhibit a negative ε in the GHz regime, but rather behaveslike a PEC. To realize a negative-εA slab, we design andfabricate a three-dimensional metallic mesh structure withsubwavelength square openings. Figure 6 (a) shows the pictureof the fabricated mesh structure, with the inset depicting acomputer-generated zoom-in view of the structure. In thewavelength regime where λ a with a being the latticeconstant of the mesh, our system is an excellent metamaterialto mimic a negative-εA slab, as desired.

By computing the transmission/reflection spectra for a slabof the designed mesh structure, we successfully retrieved theeffective permittivity and permeability of the mesh metama-terial by the standard S-parameter retrieval method [35]. Thesimulated results are plotted in Fig. 6(b), where indeed wefind that εmesh takes a desired negative value in the frequencyregion of interest. It is worth noting that μmesh is significantlysmaller than 1, due to the non-negligible diamagnetic effectsin such three-dimensional mesh structures.

Having obtained an appropriate material to represent layerA, we next take air as layer B to form an AB grating, andthen perform microwave experiments to measure the trans-mission/reflection spectra of the AB grating. The widths oflayer A and layer B are dA = 3 mm,dB = 2 mm, respectively,and the total thickness of the constructed AB grating is h =60 mm. The lateral dimension of the fabricated AB structureis (400 mm × 400 mm, see Appendix D for sample picture).

FIG. 6. (Color online) (a) Picture of the fabricated metallic mesh,with inset depicting a computer-generated zoom-in view of thestructure. (b) Effective-medium parameters of the mesh sample,retrieved from FDTD simulations. (c) Effective-medium parametersof the air/mesh grating calculated by our effective-medium theorywith dmesh = 3 mm and dair = 2 mm. (d) Transmission spectra of theair/mesh grating with thickness h = 60 mm, obtained by measure-ments (star), FDTD simulations on realistic structures (black lines),and model calculations based on our effective-medium theory (redline) and the long-wavelength-limit formulas (blue line).

Open stars in Fig. 6(d) are the measured transmittance spec-trum of the fabricated sample, which is in good agreement withFDTD simulations on realistic structures. In our experiments,we have adopted the time-domain gating technique to avoidmultiple reflections between the source and receiver horns (seeAppendix D for experimental details). Meanwhile, knowingthe electromagnetic and geometric properties of the construc-tional A and B layers, we employ our effective-medium theory

FIG. 7. (Color online) Reflectance for two slabs of AB gratingsunder different incident angles, calculated by FDTD simulationson realistic systems (symbols) and our effective-medium theory onmodel systems (lines). Here, the working wavelength is fixed asλ = 25d . In these AB structures, the B layer is always fixed as airwhile the A layer is a dielectric with (a) εA = 40 and a metal with (b)εA = −100.

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to compute the effective parameters of the resultant AB gratingstructure, and depict the results in Fig. 6(c). Using theseeffective parameters, we employ the standard TMM to cal-culate the transmission spectra through a model homogeneousslab with thickness h = 60 mm. Figure 6(d) shows that oureffective-medium theory can well reproduce both measuredand FDTD simulated transmittance spectra. In contrast, if wederive the effective parameters based on the long-wavelength-limit formulas [i.e., Eq. (20)], we find that the correspondingtransmission spectra calculated by such a model show signif-icant deviations from the experimental results. Results calcu-lated by PEC-based effective-medium theory [i.e., Eq. (15)]show even stronger deviations from the experimental andFDTD results and are not included in Fig. 6(d). Such a compar-ison unambiguously validates the newly established effective-medium theory and highlights the importance of adopting ourtheory to describe such a grating in general cases, especiallyin those cases where previous theories do not work well.

VI. CONCLUSIONS

In summary, we established an effective-medium theory tohomogenize a grating structure consisting of two differentlayers with distinct electromagnetic properties. Our theorycan recover two previously developed formulas, which arevalid only at certain limiting conditions, and more importantly,still works well in parameter spaces where previous formulasfail. We performed microwave experiments and full-wavesimulations to successfully validate our theory.

ACKNOWLEDGMENTS

This work was supported by the National Natural ScienceFoundation China (Grants No. 11474057, No. 11174055,No. 11204040, No. 51322201, and No. 11404063), andShanghai Science and Technology Committee (Grant No.14PJ1401200).

APPENDIX A: DERIVATION OF EQ. (1)

Consider the propagation of a transverse-magnetic (TM)mode (with nonzero field components Ex,Ez,Hy) in such anAB structure [see Fig. 1(a)]. Matching boundary conditionsand using the transfer-matrix method (TMM) [29], we findthat the evolution of Hy(x) field across a unit cell in an ABstructure can be written as(

H+B (d)

H−B (d)

)= TBMBATAMAB

(H+

B (0)H−

B (0)

)

= Q

(H+

B (0)H−

B (0)

), (A1)

where

TA =(

eiPA 00 e−iPA

), TB =

(eiPB 00 e−iPB

), (A2)

MBA = 12

((1 + �) (1 − �)(1 − �) (1 + �)

),

(A3)

MAB = 12

((1 + �−1) (1 − �−1)(1 − �−1) (1 + �−1)

)

in which PA = kAx dA,PB = kB

x dB, � = εBkAx

εAkBx

, and d = dA + dB

is the length of a unit cell. Here + and − denote the fieldcomponents for waves propagating to +x and −x directions.Imposing the Bloch condition, we find that eigenmodes (withTM polarization) in region II must satisfy

Tr[Q] = 2 cos Kd (A4)

with K being the Bloch wave-vector. Putting Eqs. (A1)–(A3)into (A4), we get an equation that is just Eq. (1) in the maintext.

APPENDIX B: DERIVATION OF EQ. (7)

As discussed in Ref. [30], the most general form of asecond-order linear differential operator is

L̂ = p(x)d2

dx2+ p0(x)

d

dx+ q(x). (B1)

If p(x),p0(x),q(x) are all real functions and the condition

d

dxp(x) = p0(x) (B2)

is satisfied, L̂ is called a self-adjoint operator and can berewritten as

L̂ = d

dx

[p(x)

d

dx

]+ q(x). (B3)

The most remarkable property of a self-adjoint operator isthat it satisfies

〈v|L̂u〉 = 〈L̂v|u〉, (B4)

where u(x),v(x) are two arbitrary complex functions definedin a bounded region [a,b], and 〈v|L̂u〉 = ∫ b

av∗(x) · L̂u(x)dx,

〈L̂v|u〉 = ∫ b

a(L̂v(x))

∗ · u(x)dx define the inner products be-tween two relevant functions. Note that either periodic orvanishing boundary condition should be specified at thedomain boundary in deriving Eq. (B4).

In general, a self-adjoint operator can have the followingeigenfunction equation

L̂u(x) − λw(x)u(x) = 0, (B5)

where λ and u(x) are the eigenvalue and associated eigen-function, respectively, and w(x) is a real function called theweight function (metric factor). For any two different eigen-values λi,λj and their associated eigenfunctions ui(x),uj (x),according to Eq. (B4), we must have

L̂ui(x) = λiw(x)ui(x), L̂uj (x) = λjw(x)uj (x). (B6)

Meanwhile, the self-adjoint property Eq. (B4) ensures that

〈uj |L̂ui〉 = 〈L̂uj |ui〉. (B7)

Therefore, putting Eq. (B5) into Eq. (B7), we get

(λi − λj )〈ui |wuj 〉 = 0. (B8)

Since λi − λj �= 0, we have the following generalized orthog-onal condition for any two different modes,

〈ui |wuj 〉 = δij . (B9)

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TANG, ZHU, JIA, HE, SUN, MEI, AND ZHOU PHYSICAL REVIEW B 91, 174201 (2015)

FIG. 8. (Color online) (a) Picture of the AB grating; (b) schematics of the experimental setup.

We now use the above general knowledge to study ourproblem. In our case, we consider a TM mode propagatinginside the AB structure described by relative permittivity ε(x)and permeability μ(x). Based on Maxwell’s equations, weobtain the wave equation satisfied by the Hy component, whichcan be rewritten as the following form

∂

∂x

(∂Hy(x)

ε(x)∂x

)+

(k2

0μ(x) − k2z

ε(x)

)Hy(x) = 0, (B10)

with k20 = ω2/c2. Comparing Eq. (B10) with Eqs. (B5) and

(B3), we find that k2z and Hy(x) are the eigenvalue and eigen-

wave-function of a self-adjoint operator L̂ = ddx

[ 1ε(x)

ddx

] +k2

0μ(x), respectively. In addition, Eq. (B10) implies that theweight function in our case is

w(x) = 1

ε(x). (B11)

Put the metric Eq. (B11) into Eq. (B9), we get the generalizedorthogonal condition for two different wave functions Hy(x),which is precisely Eq. (7) in the main text.

The above discussion enables us to find the appropriatemetric functions in more general cases. For example, still inthe TM case, if one rather chooses the Ex field as the wavefunctions, then it is easy to prove that the metric functionshould be w(x) = ε(x). On the other hand, if the polarization isTE instead, then the metric functions are completely differentand should be carefully restudied.

APPENDIX C: VALIDATION OF OUR THEORYUNDER OBLIQUE INCIDENCE: EFFECTS

OF NONLOCAL RESPONSE

In the main text, we have verified our effective-mediumtheory under normal-incidence condition. Here, we furthervalidate our theory under oblique incidences. Such a justifica-tion also proves that the nonlocal responses of the systems arenot significant, so that our theory, even without consideringthe nonlocal effects in its present form, can still work verywell.

In general, the grating structure should be considered as auniaxial anisotropic medium with effective permittivity tensor

given as

↔εeff =

⎛⎝ε⊥ 0 0

0 ε‖ 00 0 ε‖

⎞⎠ , (C1)

where ε⊥ has been given by our theory. The value of ε‖, on thehand, can also be easily obtained by extending our theory tothe TE polarization (still under normal incidence), which candone by noticing the symmetries between �E and �H fields inTM and TE cases.

With↔εeff known from our theory, we next take two

representative grating structures to validate our theory underoblique incidences. Fixing the working wavelength at λ =25d, we employed both our theory and the FDTD methodto compute the transmittance/reflectance of two grating struc-tures (with thicknesses h = 10d) under different incidenceangles. Comparison between FDTD and the effective-mediumtheory (Fig. 7) indicates that our theory works well underoblique incidences.

APPENDIX D: EXPERIMENTAL DETAILS

Figure 8(a) shows a picture of the AB grating sample that wefabricated and experimentally characterized. The microwavetransmittance spectra were measured by a vector networkanalyzer (Agilent E8362C PNA) and the experimental setup isshown in Fig. 8(b). Two identical horns, separated by a distanceof 106 cm and connected to the network analyzer, were used

FIG. 9. (Color online) Transmittance spectra measured with andwithout using the time-domain gating.

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to generate and receive the microwave signals. The microwavegenerated from the horn is polarized with �E||x̂, and is normallycoupled into the AB grating. The sample was placed on astage, 50 cm away from the receiving horn. The transmittanceis normalized to that measured without the samples. In ourexperiments, we used a time-domain gating technique [36]to cut in the time domain the high-order Fabry-Perot signals,

contributed by multiple reflections between two horns. Such atechnique, widely used in literature [37,38], can help us filterout the pure transmission/reflection signals contributed by thesample alone, and thus significantly improve the precisionand stability of the measurement. The transmittance spectrameasured with and without using the time-domain gating arecompared in Fig. 9.

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