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D. R. Smith and J. B. Pendry Vol. 23, No. 3 /March 2006/J. Opt. Soc. Am. B 391

Homogenization of metamaterialsby field averaging (invited paper)

David R. Smith

Department of Electrical and Computer Engineering, Duke University, Box 90291, Durham, North Carolina 27708

John B. Pendry

The Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK

Received July 19, 2005; accepted August 21, 2005; posted September 28, 2005 (Doc. ID 63535)

Over the past several years, metamaterials have been introduced and rapidly been adopted as a means ofachieving unique electromagnetic material response. In metamaterials, artificially structured—often periodi-cally positioned—inclusions replace the atoms and molecules of conventional materials. The scale of these in-clusions is smaller than that of the electromagnetic wavelength of interest, so that a homogenized descriptionapplies. We present a homogenization technique in which macroscopic fields are determined via averaging thelocal fields obtained from a full-wave electromagnetic simulation or analytical calculation. The field-averagingmethod can be applied to homogenize any periodic structure with unit cells having inclusions of arbitrary ge-ometry and material. By analyzing the dispersion diagrams and retrieved parameters found by field averaging,we review the properties of several basic metamaterial structures. © 2006 Optical Society of America

OCIS codes: 160.0160, 160.1190, 260.2110, 350.5500.

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. INTRODUCTIONhere has been a surge of interest over the past severalears in the development of artificially structured peri-dic materials whose electromagnetic properties can beuccinctly expressed in terms of homogenized materialarameters. Unlike photonic crystal structures, whoseypically complex electromagnetic modes are best de-cribed using the language of band structure and similaroncepts associated with photonic crystals, the detailedescription of an artificially structured composite can fre-uently be replaced by a small set of averaged param-ters, such as the electric permittivity, magnetic perme-bility, refractive index, and wave impedance.It might seem that this simplified description of an oth-

rwise complex medium could result in all the interestinglectromagnetic features of the medium being lost. On theontrary, work on bianisotropic and chiral materials overhe past decade has shown the important role that artifi-ial materials can play in terms of achieving unusual elec-romagnetic phenomena never before realized or difficulto obtain in conventional materials.1–3 So successful haveeen the recent advances in artificial materials researchhat a new term has emerged and been adopted by theommunity to describe these novel structures:etamaterials.4

What constitutes a metamaterial is still not well de-ned, or at least the term is not uniformly applied by alluthors; other well-established terms such as effectiveedia and artificial dielectrics—even photonic crystals—

ould be considered to be referring to the sametructures.5 As conceived, the term metamaterial carried

somewhat different implication relative to previouserms for artificial structures: the derived electromag-

0740-3224/06/030391-13/$15.00 © 2

etic response of a metamaterial is unachievable or un-vailable in conventional materials.Although the term metamaterial remains nebulous, the

rtificial structures that have become associated with theetamaterials typically have several features in common:etamaterials are described with homogenized material

arameters; they are often based on conducting, resonantnclusions; the inclusions are positioned periodically; andhe scale of periodicity is smaller—usually by a factor of 5o 10 times—than that of the free-space wavelength of ex-itation. Structures that bear these features do not fit eas-ly into the existing categories of artificial materials. Forxample, although the unit-cell dimension of the recentlyntroduced metamaterials is less than the wavelength, its only marginally so; the more relevant parameter0d—the free-space wavenumber multiplied by the unit-ell dimension—is of the order of unity. Homogenizationheories are typically valid when the unit-cell size is in-ignificant with respect to the wavelength (the zero-requency limit) and thus might be expected to result in aoor description of metamaterials.Artificial structures possessing negative index and all

he features noted above have become perhaps the bestnown examples of metamaterials. The possibility of aaterial with negative refractive index was hypothesized

n 1968 by Veselago,6 who showed that a negative refrac-ive index would be a key property of a hypothetical com-osite whose electric permittivity and magnetic perme-bility were both less than zero. Although such a materialondition is, in principle, achievable in a naturally occur-ing material, the fundamental atomic or molecular pro-esses that lead to the electromagnetic response of mate-ials do not appear to favor the overlapping electric and

006 Optical Society of America

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392 J. Opt. Soc. Am. B/Vol. 23, No. 3 /March 2006 D. R. Smith and J. B. Pendry

agnetic resonances that would be necessary to achieveegative index.7

In 2000, a metamaterial, based on conducting wiresnd split-ring resonators (SRRs), was demonstrated toave a negative refractive index over a region of micro-ave frequencies.8 Given the absence of any conventionalaterial with this property, the negative-index composite

s consistent with the original and most strict definition ofmetamaterial.4 Since the introduction of this structure,

he conducting wire9 and SRR10 inclusions and theirariations have become widely used as the fundamentaluilding blocks in artificial materials to provide tailoredrtificial electric and magnetic responses.11–17

Conductors are especially useful in the construction ofrtificial materials, as they strongly scatter electromag-etic fields. Moreover, conductors have an inherently in-uctive response due to the currents that flow withinhem, a response absent in structures composed of dielec-ric materials only. Thus, in metallic and metallodielectricetamaterials, the existence of both inductive and ca-

acitive elements within the unit cell can result in strongesonances that couple either electrically or magneticallyo the electromagnetic field. Being related to the localharge and current distributions rather than the overalleometry, these resonances can occur in structures muchmaller than the free-space wavelength. Conducting in-lusions are thus ideal as the basis of artificial structuresith unique homogenized electromagnetic properties.Given the recent successes that metamaterials have

chieved and the hope that metamaterials will lead to en-anced electromagnetic devices and new applications,here is a need for accurate and efficient methods of de-ermining the electromagnetic parameters for a giventructure. The goal of assigning electromagnetic param-ters to a composite or mixture of two or more componentaterials has long been the aim of traditional analytic ho-ogenization theories, which have resulted in many well-

nown approximate mixing formulas, such as thelausius–Mossotti and Maxwell–Garnett formulas, or theoherent potential approximation.18 All known analyticalethods, however, are valid under certain limitations and

articular geometries or classes of structures. Foretamaterials comprising conducting and possibly reso-

ant elements, and for which the periodicity is not neces-arily negligible relative to the free-space wavelength,nalytical homogenization techniques are unreliable orot applicable.Given that Maxwell’s equations can now be solved rap-

dly and efficiently—at least for domains relatively smallith respect to the wavelength—with custom or commer-

ial software running on modern personal computers, theocal electric and magnetic fields, current distributions,nd other quantities of interest can be found easily forirtually any type of structure. A purely numerical ap-roach to the homogenization of artificial structures ishus feasible and avoids the limitations associated withrior analytic homogenization models.One approach to the assignment of electromagnetic pa-

ameters to a simulated structure is via S-parameteretrieval.19 In the S-parameter retrieval method, the com-lex reflection and transmission coefficients are calcu-ated for a finite thickness of metamaterial, usually one

nit cell, and compared with analytical formulas for theeflection and transmission coefficients of a homogeneouslab with the same thickness. The retrieval method cane applied to both simulated or measured S-parameterata, although thin samples (in the propagation direction)re best, and the retrieved electromagnetic parametersre not unique.Despite its versatility and ease of use, S-parameter re-

rieval does not provide significant physical insight intohe nature of the artificial material. The S-parameterbservables—reflection and transmission coefficients—re not easily connected to analytical homogenizationheories, which instead usually involve analysis of theocal-field and current distributions.

In deriving the effective-medium properties of severaleriodic artificially magnetic structures, Pendry et al. pre-ented a straightforward method of homogenizationased on averaging the local fields according to10

Hi = d−1� H · dxi,

Ei = d−1� E · dxi,

Di = d−2� D · dsi,

Bi = d−2� B · dsi. �1�

ere, E and H are obtained from line integrals over theocal E and H fields, and D and B are obtained from in-egrals of the local D and B fields over the surfaces of thenit cell. The length of a side of the unit cell, assumed cu-ic for this example, is d.Equations (1) imply that a discrete set of averaged

elds are to be defined at each unit cell, replacing the lo-al fields that vary throughout the unit cell. In this sense,he averaging scheme of Eqs. (1) resembles a finite differ-ncing of Maxwell’s equations, with a resulting spatialiscretization analogous to those used in finite-differenceime-domain and similar algorithms.20,21 In the case that��, where � is the wavelength in the effective mediumefined by the periodic structure, then we expect Eqs. (1)o provide an appropriate macroscopic description, fromhich we can ascertain the effective electromagnetic pa-

ameters.The condition that d—effectively the scale of the

nhomogeneity—be negligible with respect to the wave-ength is equivalent to the statement that we are in thelectrostatic or magnetostatic limit. In the static limit,ne can formulate Eqs. (1) directly by considering thelectrostatic or magnetostatic potentials.22 In either case,he validity of Eqs. (1) would appear to be limited to thetatic regime. However, for most of the metamaterialsemonstrated to date, d is not negligible with respect tohe wavelength; yet, in these cases, the description of thetructure as a homogeneous medium has produced sen-ible and useful results, implying that it should be pos-

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D. R. Smith and J. B. Pendry Vol. 23, No. 3 /March 2006/J. Opt. Soc. Am. B 393

ible to modify or extend the homogenization conceptsmbodied in Eqs. (1). In 2000, such a modification wasuggested and shown to produce self-consistent results forhe permittivity and permeability of wire and SRRedia.23

Our intent here is to justify a numerically based ho-ogenization scheme based on Eqs. (1), in which the local

elds computed for one unit cell of a periodic structurere averaged to yield a set of macroscopic fields. Onceaving computed the macroscopic fields, we can then de-ermine the constitutive relationships between the mac-oscopic fields, arriving at the effective electromagneticarameters. We will see that there will be virtually no re-trictions on the contents of the unit cell, nor will the unitell necessarily need to be small in comparison with theavelength.

. FIELD-AVERAGING METHODo facilitate our development of a generalized metamate-ial homogenization theory, we must first solve Maxwell’squations subject to periodic boundary conditions to findhe local fields associated with each propagating mode.e will restrict our discussion to the transverse modes, so

hat we need consider only the Maxwell curl equations:

� � E = i�B,

� � H = − i�D. �2�

ere, E ,B ,D, and H are all local fields. Locally, D and Hre related to E and B by the appropriate constitutive re-ationship for a particular homogeneous material. We as-ume the unit cell contains two or more such homoge-eous materials such that the unit cell is inhomogeneous.n general, for a periodic medium for which a is a primi-ive lattice vector, the solutions to Eqs. (2) are Bloch func-ions having the form

f�x� = u�x�exp�iq · x�, �3�

here f�x� is any of the four field components in Eqs. (2)nd u�x+a�=u�x� is a periodic function. Note from Eq. (3)hat the Bloch wave acquires a phase advance of q ·a as itropagates across the unit cell, where q is the Bloch oreciprocal-lattice vector.

As can be seen from Eq. (3), the boundary condition onhe fields has the form

f�x + a� = f�x�exp�iq · a�. �4�

alues of q that lie outside the range −��q ·a� +� canlways be related to a q value within this range havinghe same boundary condition. Frequency solutions for all

values can thus be folded back into this first Brillouinone, with the resulting band structure providing a speci-cation of the electromagnetic mode structure of the com-uted unit cell.We can compute the exact frequencies and local fields of

he electromagnetic modes of a periodic structure by first

pecifying a value of q (or q ·a) and solving Eqs. (2) sub-ect to the boundary conditions of Eq. (4). This method ofolution is known as the eigenfrequency method, as theave equation assumes the form of an eigenvalue equa-

ion once a value of q has been selected. For each value ofthe eigensolver will return an infinite set of modes, each

f which is continuous as a function of q.Note from the form of the solutions in Eqs. (3) and (4)

hat if we are uninterested in the details of the fieldsithin a unit cell, then the Bloch-wave solutions through-ut the periodic medium are similar to plane-wave solu-ions if we restrict our evaluation of the fields to a set ofrid points that lie, for example, at the vertices or edges ofhe unit cells. By averaging Maxwell’s equations for theocal fields over a unit cell, we arrive at a modified versionf Maxwell’s equations in which the averaged fields areefined only at specific grid points. Thus, the exact elec-romagnetic response and geometric details of the inclu-ions within a unit cell are eliminated, encapsulated inhe newly defined averaged (or macroscopic) fields¯ ,B ,D, and H. If we assume these macroscopic fields sat-sfy a constitutive relationship, then electromagnetic pa-ameters such as � and � can be defined. We expect theomogenized material parameters to be consistent withther properties of the medium, such as the dispersion re-ation, �q�=neff� /c=���� /c.

To illustrate the method by which we define field aver-ges, consider the (unshaded) cube shown in Fig. 1, eachide of which has a length 2d. The origin of the coordinateystem is assumed to be at the center of the sphere, whichepresents the contents of one repeated unit cell of the in-nite structure. Within the unit cell there is no restrictionn the materials or geometry of the inclusions, so that wean generally allow for such properties as finite or infiniteonductivity (perfect electric conductor) as well as arbi-rary values of the local � and �. Within the cube, and ex-erior to the inclusions, the local Ex ,Ey, and Bz fields areelated by the first of the Maxwell curl equations in Eqs.2):

ig. 1. Definition of the averaged fields relative to a periodicnit cell. Maxwell’s curl equations in a cubic geometry suggestshe averaged electric fields lie on the edges of a cubic lattice,hile the averaged magnetic fields lie on the edges of a second

nterpenetrating cubic lattice, illustrated as the shaded cube.

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394 J. Opt. Soc. Am. B/Vol. 23, No. 3 /March 2006 D. R. Smith and J. B. Pendry

�Ez

�y−

�Ey

�z= i�Bx,

�Ex

�z−

�Ez

�x= i�By,

�Ex

�y−

�Ey

�x= i�Bz. �5�

Integrating the first of Eqs. (5) over the x= +d surface,he second over the y= +d surface, and the third over the= +d surface of the unit cell and applying Stokes’s theo-em to the left-hand sides, we obtain

�−d

d

Ez�d,d,z�dz −�−d

d

Ez�d,− d,z�dz −�−d

d

Ey�d,y,d�dy

+�−d

d

Ey�d,y,− d�dy = i��−d

d

dy�−d

d

dz Bx�d,y,z�,

�−d

d

Ex�x,d,d�dx −�−d

d

Ex�x,d,− d�dx −�−d

d

Ez�d,d,z�dz

+�−d

d

Ez�− d,d,z�dz = i��−d

d

dx�−d

d

dz By�x,d,z�,

�−d

d

Ex�x,d,d�dx −�−d

d

Ex�x,− d,d�dx −�−d

d

Ey�d,y,d�dy

+�−d

d

Ey�− d,y,d�dy = i��−d

d

dx�−d

d

dy Bz�x,y,d�. �6�

method of averaging now presents itself naturally; wentroduce, for example, the following definitions:

Ex�0, ± d,d� = �2d�−1�−d

d

Ex�x, ± d,d�dx,

Ey�±d,0,d� = �2d�−1�−d

d

Ey�±d,y,d�dy,

Bz�0,0, ± d� = �2d�−2�−d

d

dx�−d

d

dyBz�x,y, ± d�. �7�

he first two integrals are line integrals over the edges ofhe z= +d face of the cubic cell, and the third integral is aurface integral over the face. Two more sets of similarquations can be defined for the x= +d and y= +d faces.ote that Eqs. (6) relate line integrals of the electric fieldround the periphery of a given face of the unit cell to aurface integral of the magnetic field over the same face.f we view the integrals as averages of the local fields,hen Eqs. (7) represent the relationship of the micro-copic, or local fields, to the macroscopic fields. We haveefined the averages for the line integrals to be taken athe midpoints of the edges, and the surface integral aver-ge is taken at the midpoint of the face.

Using the definitions in Eqs. (7), we can replace theomponents of the curl equation, Eqs. (6), with

Ez�d,d,0� − Ez�d,− d,0� − Ey�d,0,d� + Ey�d,0,− d�

= i2d�Bx�d,0,0�,

Ex�0,d,d� − Ex�0,d,− d� − Ez�d,d,0� + Ez�− d,d,0�

= i2d�By�0,d,0�,

Ex�0,d,d� − Ex�0,− d,d� − Ey�d,0,d� + Ey�− d,0,d�

= i2d�Bz�0,0,d�. �8�

he exact Maxwell curl equation is thus replaced by anite-difference equation relating the electric field line in-egral averages to the magnetic field surface integral av-rage. Presumably, we need no longer consider the locallectric and magnetic field distributions that varyhroughout the unit cell but may rather consider only the2 electric field values evaluated at the edge centers andhe six magnetic field averages evaluated at the face cen-ers of the unit cell. Note that the averaged electric fieldsre defined at the edges of a cubic lattice, and the aver-ged magnetic fields are defined similarly on a second cu-ic lattice interlaced with the first, as indicated in Fig. 1.he resultant scheme is identical to that used in standardnite-differencing techniques, such as introduced by Yee20

or the finite-difference time domain or Pendry andacKinnon21 for the transfer-matrix method.To solve for the electromagnetic fields in a structure, weust obtain a second equation for the averaged fields.ollowing the path that led to Eqs. (8), we start with theecond Maxwell equation relating the curl of the magneticeld to the electric field, or

� � H = − i�D. �9�

riting down the components of Eq. (9) explicitly, weave

� �Hz

�y−

�Hy

�z � = − i�Dx,

� �Hx

�z−

�Hz

�x � = − i�Dy,

� �Hx

�y−

�Hy

�x � = − i�Dz. �10�

s before, we integrate Eqs. (10) over the faces of the unitell, which yields

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D. R. Smith and J. B. Pendry Vol. 23, No. 3 /March 2006/J. Opt. Soc. Am. B 395

�0

2d

Hz�0,2d,z�dz −�0

2d

Hz�0,0,z�dz −�0

2d

Hy�0,y,2d�dy

+�0

2d

Hy�0,y,0�dy = i��0

2d

dy�0

2d

dz Dx�0,y,z�,

�0

2d

Hx�x,0,2d�dx −�0

2d

Hx�x,0,0�dx −�0

2d

Hz�2d,0,z�dz

+�0

2d

Hz�0,0,z�dz = i��0

2d

dx�0

2d

dz Dy�x,0,z�,

�0

2d

Hx�x,2d,0�dx −�0

2d

Hx�x,0,0�dx −�0

2d

Hy�2d,y,0�dy

+�0

2d

Hy�0,y,0�dy = i��0

2d

dx�0

2d

dy Dz�x,y,0�. �11�

efining field averages as in Eq. (7), we obtain

Hx�d,d ± d,0� = �2d�−1�0

2d

Hx�x,d ± d,0�dx,

Hy�d ± d,d,0� = �2d�−1�0

2d

Hy�d ± d,y,0�dy,

Dz�d,d,0� = �2d�−2�0

2d

dx�0

2d

dy Dz�x,y,0�. �12�

sing Eqs. (12) with Eqs. (11), we arrive at an additionalet of equations relating the averaged components of H tohose of D as follows:

Hz�0,2d,d� − Hz�0,0,d� − Hy�0,d,2d� + Hy�0,d,0�

= i2d�Dx�0,d,d�,

Hx�d,0,2d� − Hx�d,0,0� − Hz�2d,0,d� + Hz�0,0,d�

= i2d�Dy�d,0,d�,

Hx�d,2d,0� − Hx�d,0,0� − Hy�2d,d,0� + Hy�0,d,0�

= i2d�Dz�d,d,0�. �13�

o solve Eqs. (8) and (13), we must assume a constitutiveelationship between the averaged fields E ,D and B ,H.or a reciprocal medium we can assume the following

orm:

D = �0�E − i��0�0�H,

B = i��0�0�TE + �0�H, �14�

here we have added a bar above the electromagnetic pa-ameters to indicate that they represent an average overhe unit cell. In Eqs. (14) we have allowed for the possi-ility of magnetoelectric coupling, which introduces the

agnetoelectric tensor �.24 We will discuss belowetamaterials for which � cannot be neglected; however,

y one’s choosing the proper symmetry of the unit cell, �an usually be made negligible.

Equations (8) and (13) form a set of finite-differencequations that can be solved to yield plane-wave solu-ions. To illustrate the nature of these solutions, considerplane wave propagating along the y axis, polarized such

hat the electric field is directed along the z axis while theagnetic field is directed along the x axis. All fields are

xpected to have the spatial dependence exp�iq ·x�, whereis the position vector. From Eqs. (8) we find

Ex sin�qyd� = d�Bz, �15�

nd from Eqs. (13) we have

Hz sin�qyd� = d�Dx. �16�

sing the linear constitutive relationship of Eqs. (14)ith Eqs. (15) and (16) and setting the magnetoelectric

ensor components to zero for now, we arrive at the fol-owing dispersion relationship:

sin2�qyd� = �2d2�x�z. �17�

s the cell size becomes much smaller than the wave-ength within the material �qyd→0�, the dispersion of the

edium approaches that of free space, as expected.It is instructive to consider the explicit form for �x and

¯ z when the unit cell is empty. Using the constitutive re-ation between E and D, for example, we relate the mac-oscopic fields to �x and �z as follows:

�x =Dx�0,d,d�

Ex�0,d,d�= �0

�2d�−2�0

2d

dz�0

2d

dy Ex�d,y,z�

�2d�−1�0

2d

dx Ex�x,d,d�

.

�18�

or the empty cell, the local field varies as Ez�x ,y ,z�E0 exp�iqyy�, and we find

�x = �0

sin�qyd�

qyd. �19�

similar calculation for the permeability yields

�z = �0

sin�qyd�

qyd. �20�

Equations (19) and (20) reveal that the material pa-ameters found from the field-averaging method exhibitpatial dispersion; that is, the material parameters areunctions of the propagation vector. Inserting Eqs. (19)nd (20) into Eq. (17), we recover the correct dispersionelation for free space, or

� = qy��0�0. �21�

n short, those terms dependent on the wave vector in theffective material parameters are exactly canceled byimilar terms in the finite-difference dispersion relationf Eq. (17).

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396 J. Opt. Soc. Am. B/Vol. 23, No. 3 /March 2006 D. R. Smith and J. B. Pendry

Spatially dispersive medium parameters are less intui-ive and less convenient to apply. However, the terms de-endent on q in Eqs. (19) and (20) are seen to be a resultf the finite differencing or discretization of Maxwell’squations and have no other physical origin. Since we re-over the correct free-space dispersion relation, Eq. (21),ia this scheme there is the hint that the averaged mate-ial parameters may have validity if we simply removehe term sin�qyd� /qyd. When the unit cell is not empty,his procedure will obviously not lead to an exact resultut may still be a reasonable approximation.Consider the most general form for the local fields of

he periodic system, as in Eq. (3). Assuming the same po-arization and direction of propagation as above, the elec-ric field for the periodic system has the form

Ex�x,y,z� = E0u�y�exp�iqyy�, �22�

here u�y� is a periodic function that satisfies u�y+2d�u�y�. Using Eq. (22) in Eq. (18), we have

�x = �0

�0

2d

dy u�y�exp�iqyy�

2d. �23�

ecause u�y� is a periodic function, it can be expanded inFourier series as

u�y� = �m

um exp�imy�, �24�

here m=�m /d. Using Eq. (24) in Eq. (23), we find

�x =�0

2d�m

um�0

2d

dy expi�qz + m�y

= �0u0

sin�qyd�

qyd+

�0

d �m�0

um

sin�qy + m�d

qy + m, �25�

here we have separated the m=0 term from the rest ofhe summation. If we neglect all higher-order terms, thene see that

�x = �0�x�sin�qyd�

qyd. �26�

e have defined �x�=u0 in Eq. (26). Comparing Eq. (26)ith Eq. (24), we see that u0 corresponds to an average

ver the face of the unit cell of the field periodic in y. In aimilar manner,

�z = �0�z�sin�qyd�

qyd. �27�

nserting Eqs. (26) and (27) into Eq. (17), we find

� = cqy��x��z�, �28�

n which we have used c=1/��0�0. Equation (28) showshat the values for �x� and �z� defined in Eqs. (26) and (27)ppear as would be expected in the dispersion equationor a homogeneous medium. It is thus reasonable to ac-ept these quantities, found by one’s first performing fieldverages and then removing the terms due to finite-ifferencing effects, as the definition of the macroscopic

ermittivity and permeability. The approximation of re-aining only the m=0 term will be seen to provide quiteood agreement with other retrieval methods.

. NUMERICAL EXAMPLES. Field Averaging Using Eigensolutionsny method by which expressions or values for the localelds are obtained can be used in conjunction with theeld averages [Eqs. (7) and (12), for example] to obtainhe effective electromagnetic material parameters. To il-ustrate the method here, we make use of the eigensolvern HFSS (Ansoft), a commercial finite-element-based elec-romagnetic simulation software package, to solve for theocal fields in a unit cell of arbitrary composition. In theigensolver module, periodic boundary conditions are ap-lied to the bounding faces of the unit cell to simulate annfinite structure. The surfaces along one of the axes areelated by periodic boundary conditions with a phase ad-ance that varies between 0° and 180°. At each phase ad-ance, a set of eigenvalues (mode frequencies) and eigen-odes (field distributions) are generated.When the unit cell possesses a mirror symmetry, elec-

ric or magnetic boundary conditions can be substitutedor the periodic boundary conditions. For example, if weeek the propagating modes corresponding to a unit cellith mirror symmetries about the planes perpendicular

o the propagation direction, then applying electricoundaries along one of the axes and magnetic bound-ries along the other will correctly simulate the effectiveedium, having the added benefit that the polarization of

he solutions will be selected.The advantage of the eigensolver method is that, by

ne’s examining the modes as a function of phase ad-ance, the electromagnetic properties of the medium cane understood in the context of its photonic band struc-ure. The disadvantage of the eigensolver is that it doesot easily provide information in the bandgap regions—here � or � (but not both) are negative, for example, orhere material losses are large. These difficulties can bevoided by one’s applying other numerical computationalpproaches, such as frequency- or time-domain simula-ors.

To obtain the field averages from the simulation data,e define integration lines along the edges of the cell to

ompute E and H and integration surfaces over the facesf the cell to compute D and B. The integrals are per-ormed using the macro capabilities of HFSS in the post-rocessor. For the flux integrals (D and B) taken over sur-aces whose normals are perpendicular to the propagationirection, a factor of sin�qd� /qd is removed from the av-rage. For structures with no magnetoelectric coupling,e can relate the effective electromagnetic material pa-

ameters directly to the appropriate ratios of the aver-ged fields.In the absence of losses, the fields and field ratios

hould be real; however, depending on the convention ofhe software used, the line integrals may contain a factorf exp�iq ·x0− i�t�, where x0 is the position in the celllong the propagation direction where the integrationines are defined and t is the time. To accommodate for

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D. R. Smith and J. B. Pendry Vol. 23, No. 3 /March 2006/J. Opt. Soc. Am. B 397

his possibility, it is necessary to perform the field aver-ges at two phases 90° apart, which can arbitrarily beepresented as the real and imaginary parts of the fields.n the absence of loss in the system and with no magne-oelectric coupling, the 0° and 90° field averages shouldaintain a constant ratio as a function of phase advance,

uch that multiplying the complex fields by a phase factorill result in purely real fields.In certain geometries, such as a medium of continuous

ires, a conducting element may pass continuously fromne unit cell to the next. For the flux averages to be cor-ect in this case, the formulation of the effective-mediumarameters given above would need to be modified so aso include the contribution of the current flowing betweenhe cells. As an alternative, a thin slice could be made inhe conductor at the surface so that the current is noonger continuous; if the slice is very small, the electro-

agnetic properties of the simulated structure will not beffected. The field averaging would then include the depo-arizing fields within the gap, effectively accounting forhe contribution of the current in the conductor.

To avoid having to account for elements that transitionetween unit cells when we compute the effective mate-ial parameters, we can often apply an alternative ap-roach that avoids the flux averages. Because the eigen-olver provides mode frequencies as a function of thehase advance , or wavenumber k= /d, it is straightfor-ard to determine the effective index n as the ratio of aiven eigenfrequency to the wavenumber, or n=� /k. Onceaving determined the index, we can characterize the ma-erial by computing the wave impedance z= E /H. In thebsence of magnetoelectric coupling the relations

�� = n/z, �� = nz �29�

ield the effective-medium tensor components along theiven axis. Since the quantities E and H are line integralslong the edges of the unit cell, the averaging over thenit-cell faces through which metamaterial elementscontaining currents) might pass is avoided.

. Dielectric Sphere Mediumefore considering metallic metamaterial structures, it isseful to first consider a system that can also be analyzedsing analytical effective-medium theories. Such a sys-em is an array of dielectric spheres, as shown in Fig. 2left). When the spheres are small relative to their spac-ng, the Maxwell–Garnett equation should yield the effec-ive dielectric of the composite, or18

ig. 2. Cubic lattice of dielectric spheres. (right) Close-up of aingle unit cell of the array. The dielectric constant of the spheress �=4.

�MG =1 + 2�1� �−1

�+2�1 − �1� �−1

�+2�. �30�

e test this by performing several simulations in HFSS us-ng the unit cell shown in Fig. 2 (right), varying the ra-ius of the dielectric ��=4� sphere relative to the cubic cellength. As the Maxwell–Garnett formula is strictly validn the static limit, we expect agreement only at d /�=0the zero-frequency limit).

The computed permittivity and permeability corre-ponding to four different sphere radii are shown in Fig. 3s a function of d /� and reveal several interesting fea-ures. At zero d /�, the recovered permittivity is in exactgreement with that predicted by Eq. (30), shown as theorizontal dashed lines. The permeability approachesnity at zero frequency, as would be expected in the elec-rostatic limit. At finite wavelengths, however, the com-uted values of the permittivity and the permeability de-iate characteristically from the static limit, with theomposite permeability taking values less than unity andhe composite permittivity taking values greater than thetatic value. That the material parameters disperse as aunction of frequency should not be considered surprising,ince a bandgap is approached (by definition) when thehase advance approaches 180°. At the band edge the in-ex approaches a fixed value set by the lattice constant,=c� / �d��, but the permittivity and permeability canave resonant forms. (Note that we now define the unit-ell length as d rather than 2d.) This phenomenon haseen reported in studies of the effective-medium param-ters of metamaterials25 and will be seen to play an evenreater role in the metamaterials to be discussed.

. Wire Mediums a first example of a conducting metamaterial, we con-ider a one-dimensional array of wires with square crossection, as illustrated in the inset to Fig. 4. When the elec-ric field is polarized along the wire and the wave propa-ates in the plane perpendicular to the wire axis, then the

ig. 3. Effective permittivity and permeability of a cubic arrayf dielectric spheres, as a function of sphere radius. The dielectriconstant of the spheres is �=4. The permeability values all startrom zero frequency at unity and disperse downward with in-reasing frequency. The permittivity values start from zero fre-uency in accordance with the Maxwell–Garnett formula andisperse upward with increasing frequency.

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398 J. Opt. Soc. Am. B/Vol. 23, No. 3 /March 2006 D. R. Smith and J. B. Pendry

ire medium can be reasonably described—at least atower frequencies—by an effective permittivity havinghe form

���� = 1 −�p

2

�2 . �31�

he effective plasma frequency, �p, is a constant relatedo the lattice parameters, such as the unit-cell length ofhe wire medium and the cross-sectional area of the wire.osses, ignored here, since the simulations assume per-

ect conductors, add a damping term in Eq. (31).Equation (31) was reported in 19969 and has since been

onfirmed theoretically by numerous other authors.26–30

he striking feature shown by Eq. (31) is that the effec-ive permittivity is negative for frequencies below fp�p / �2��. An eigenmode simulation of the wire mediumroduces the dispersion curve plotted in Fig. 4, which re-eals a single propagating band that starts at fp, in gen-ral qualitative agreement with Eq. (31).

The simple plasmonic form of Eq. (31) excludes effectsue to the periodicity of the medium. In particular, theropagating band shown in Fig. 4 terminates at a secondand of forbidden frequencies at the band edge (where k� /d). Because the effects due to the higher-frequencyand structure are not included in Eq. (31), we expect theroperties of the actual wire-medium structure to divergerom the simple model as the higher frequencies are ap-roached. The material parameters as determined via theeld-averaging technique are presented in Fig. 5 (solidurves). For comparison, the permittivity for the homoge-eous plasmonic medium [Eq. (31)] is plotted as theashed curve and can be seen to deviate significantly fromhe wire medium at frequencies above fp. The presence ofhe upper-frequency bandgap forces the permittivity ofhe wire structure toward zero, with the permeability ap-roaching a resonance. Clearly, since the wavenumber isxed at the band edge to be k=� /d, the effective indexust also be fixed as n=ck /�; thus, if one of the material

omponents approaches a resonance, the other must ap-roach zero such that the square root of their product isqual to the effective index at the band edge.25

ig. 4. Dispersion diagram for the wire lattice. (inset) The wireattice consists of an array of conducting wires with square crossection. The ratio of the side of the (square) wire to the unit-cellength is 0.03.

The dispersion curves exhibited by the actual wiretructure, as shown in Fig. 5, are nonintuitive, so it is rea-onable to question whether the effective material param-ters produced are in any way meaningful. The effectiveaterial parameters should, for example, describe theanner in which a plane wave is reflected and transmit-

ed through a finite section of the composite. We can as-ess this by comparing the material parameters retrievedia field averaging with those retrieved by an-parameter inversion. Using an algorithm described pre-iously to invert the transmission and reflection coeffi-ients obtained from an S-parameter simulation on theame wire structure,19 we obtained the results for the per-ittivity and permeability shown as the gray or black

ots in Fig. 5. As the figure reveals, there is excellentgreement between the field averaging and the-parameter retrieval methods, with any discrepancyithin the numerical error for these calculations. Al-

hough we do not show similar comparisons in the subse-uent figures, we find generally that the material param-ters retrieved by the two methods consistently agree toithin the numerical accuracy of our simulations.The results of Figs. 4 and 5 suggest a relatively simple

icture of the homogenized properties of the anisotropicire medium. In particular, Eq. (31) is a valid descriptionf the medium so long as incident waves are restricted toropagate perpendicular to the axes of the wires, as mod-led in the simulations presented here. For a wave whoseropagation vector makes an arbitrary angle relative tohe wire axes and that has a component of its electric fieldlong the wire axes, the resulting mode structure is con-iderably more complex. For such a wave, Eq. (31) noonger holds; the transverse permittivity displays strongpatial dispersion, and additional longitudinal modesay also be excited.26–30 Although we do not discuss the

nalysis of such modes here, we note that the boundaryonditions can be adjusted in the eigensolver to simulateropagation along nonprincipal axes of the wire medium,ith the field-averaging method subsequently applied.

ig. 5. Retrieved values of permittivity (black curve) and per-eability (gray curve) for the wire lattice. Because there are no

ropagating modes below the plasma frequency, the curves begint the plasma frequency. The dots correspond to an S-parameteretrieval performed on an S-parameter calculation of the unitell. The dashed curve corresponds to the ideal form for a plas-onic medium, Eq. (31).

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D. R. Smith and J. B. Pendry Vol. 23, No. 3 /March 2006/J. Opt. Soc. Am. B 399

. Split-Ring Resonator Mediumhe wire medium provides a conceptually straightfor-ard means of achieving a basic Drude or Drude–Lorentz

ype of electric response. Because of this, the wire me-ium in various incarnations has achieved prominence inetamaterial designs. In the same way, the conducting

plit-ring-resonator (SRR) medium introduced in 1999 byendry et al. has become the archetypical magnetic com-onent in metamaterials.10 Applying an effective-mediumheory based on Eqs. (1), Pendry et al. showed that thelectromagnetic response of the SRR medium–an examplef which is shown in Fig. 6(a)–could be approximately de-cribed by the resonant form

���� = 1 −F�2

�2 − �02 , �32�

here �0 is a resonant frequency set by the intrinsic ca-acitance and inductance of the geometry and F is a con-tant relating to the filling factor of the unit cell. Becausehe SRR is composed of conductors with no inherentlyagnetic components, the magnetic response of the SRR

an be considered artificial, the result of induced circulat-ng currents.

To illustrate the properties of the SRR medium, we con-ider the structure shown in Fig. 6(a) and in the insets toigs. 7–9. To facilitate the discussion of the SRR proper-ies, we assume a set of axes fixed to the SRRs as shown

ig. 6. (a) Diagram of the SRR and axes used in the simula-ions. w=0.8d , t=g=0.04d, and the linewidths are 0.08d, where ds the unit-cell length. (b) SRRs arranged asymmetrically. Atructure composed as shown exhibits electric, magnetic, andagnetoelectric resonant responses. (c) SRRs arranged sym-etrically. The restoration of mirror plane symmetry along the z

xis eliminates the magnetoelectric coupling. (d) This arrange-ent of SRRs reduces the resonant electric and magnetoelectric

esponses.

n Fig. 6(a). Analytical calculations have revealed that thelectromagnetic material parameters of the SRR mediumave the approximate form

�zz = �,

�yy = � + �2

��m02 − �2�

,

�xx = 1 +��2

��m02 − �2�

,

�xy =���m0

2

��m02 − �2�

, �33�

ith all other diagonal elements equal to unity, all otherff-diagonal elements equal to zero, and the constants, ,�, and � related to the particular SRR geometry.31

Although introduced as a means of obtaining a mag-etic response, Eqs. (33) show that the SRR is actually aighly anisotropic, electromagnetically complex struc-ure, exhibiting not only a resonant magnetic responseut also resonant electric and magnetoelectricesponses.32 The resonant magnetoelectric coupling, inarticular, results from the lack of a mirror plane symme-ry about the axis lying in the SRR plane and perpendicu-ar to the ring gaps. A medium arranged as shown in Fig.(b) thus exhibits bianisotropy. One can form mediumith the magnetoelectric coupling term equal to zero bylternating the SRRs so that gaps alternately face eachther and away from each other, as shown in Fig. 6(c). Toemove the resonant electric response as well, one mustlter the inherent symmetry of the SRR; a structure thatinimizes both the resonant electric and the magneto-

lectric couplings is proposed in Fig. 6(d).Consider first the SRR medium of Fig. 6(b), oriented as

hown in the leftmost inset to Fig. 7. The wave is as-umed to propagate along the y axis, with the electriceld polarized along the z axis. Under this circumstancehe macroscopic fields are related by

ig. 7. Dispersion curves for the SRR medium, with the SRRsriented for magnetic response, solid black curve (inset, left);lectric response, dashed black curve (inset, middle); and magne-oelectric response, solid gray curve (inset, right).

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400 J. Opt. Soc. Am. B/Vol. 23, No. 3 /March 2006 D. R. Smith and J. B. Pendry

Bx = �0�xxHx + i��0�0�yxEy,

Dz = �zzEz. �34�

owever, since an averaged electric field component inhe direction of propagation must vanish as required byhe divergence equation, only the values of �xx and �zz areeeded to characterize waves propagating along this axisf the SRR medium.

The mode frequency as a function of phase advance ishown as the black curve in Fig. 7, which reveals to goodpproximation the resonant form predicted by assuminghat the SRR medium of Fig. 6 is approximated by a per-eability like that of Eq. (32). The lower branch rises

rom zero frequency to the band-edge frequency ��m0�, af-er which a frequency gap occurs, followed by a secondropagation band. The computed dispersion curve devi-tes from that predicted by Eq. (32) owing to the presencef the second frequency gap. The retrieved permeability,hown in Fig. 8 as the black curve, reveals the expectedesonant form: unity at zero frequency; large positive val-es near the resonant frequency; no propagating modesound in the gap region (where the permeability is nega-ive); and zero at the start of the next propagating bandnd increasing with frequency toward the next gap.The effective-medium arguments leading to Eqs. (32)

nd (33) also predict that the SRR medium, for the sameolarization of incident wave, will have a nontrivial elec-ric response. Pendry et al. argued that an approximateorm for the �zz, valid at low frequencies, should be/�1−F, with F�1.10 This correction is necessary, as itaintains the light line at or below c. The permittivity,

lotted as the gray curve in Fig. 8, is indeed approxi-ately constant at low frequencies with magnitude sig-

ificantly larger than unity. Somewhat surprising, how-ver, is that the permittivity approaches zero at theesonance and appears to have an antiresonant form ineneral. As described above, the antiresonant nature ofhe permittivity in this case has been discussed as a con-equence of the underlying periodicity of the medium.25

Figure 8 reveals the expected result that the SRR me-ium can respond with a resonant permeability and, in

ig. 8. Retrieved permittivity and permeability for the SRR me-ium, oriented for primarily magnetic response.

articular, can provide usable negative values of the per-eability from which negative-index materials can be

ormed. However, this interpretation implies a particularelationship between the direction and the polarization ofhe exciting wave and the SRR. In Fig. 7, a second disper-ion curve is shown, for which the exciting fields are suchhat a predominantly electric response is expected. In thisase, the relevant constitutive relationship is

Dy = �0�yyEy − i��0�0�yxHx,

Bz = �zzHz. �35�

s before, the magnetic field component along the direc-ion of propagation (the x axis in this case) must vanish,nd only the values of �yy and �zz are needed to charac-erize the SRR medium.

Figure 9 shows the retrieved material parameters forhe SRR oriented as in the inset to Fig. 9. It is found thatyy exhibits a strong resonance, as expected. Although thenalytic theory leading Eqs. (33) predicts no response forzz, we see there is actually an antiresonance in �zz, againwing to the finite size of the unit cell.

. Calculation of the Bianisotropy Parametersn the two orientations of the SRR relative to the incidentave considered in Subsection 3.D, consideration of theagnetoelectric tensor elements could be avoided. Thereay be structures designed to exhibit magnetoelectric

oupling, however, where it is desired to calculate theagnitude of the coupling. If we orient the SRR as shown

n the inset to Fig. 10, the constitutive relations have theorm

Dy = �0�yyEy − i��0�0�yxHx,

Bx = �0�xxHx + i��0�0�yxEy. �36�

ecause of the presence of the magnetoelectric terms, theffective index for waves propagating along this axis hashe form

neff = ��yy�xx − �yx2 . �37�

ewriting Eqs. (36) as

ig. 9. Permittivity and permeability for the SRR medium, ori-nted for primarily electric response.

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D. R. Smith and J. B. Pendry Vol. 23, No. 3 /March 2006/J. Opt. Soc. Am. B 401

�yx = iz� 1

�0

Dy

Ey

− �yy� = iz�� − �yy�,

�yx = − i1

z� 1

�0

Bx

Hx

− �xx� = − i1

z� − �xx�, �38�

here z /z0= Ey /Hx, we find that the combination of Eqs.37) and (38) provide enough information, in principal, toecover all the relevant tensor elements to characterizehe bianisotropic SRR medium of Fig. 6(b). In particular,he three equations can be combined to yield

�yx = − i�� − neff

2 �

�z� −

z�, �39�

here �=Dy / Ey and �= Bx /Hx. �yy and �xx can then be de-ermined from Eqs. (38).

For a lossless system, the material parameters and theagnetoelectric coupling terms are expected to be real-

alued functions. An inspection of Eqs. (38) thus showshat the field ratios, denoted by � ,, and z, will now beomplex-valued functions. The various products and ra-ios in Eqs. (38) and (39) must therefore be performed us-ng the complex values obtained from the 0° and 90° fieldverages, as described earlier. In general, we find thattability is much more difficult to achieve for structureshat possess magnetoelectric coupling, especially near theesonant frequency of the composite structure.

In Fig. 10 we compare the significant material param-ters of the SRR shown in Fig. 6(a) for frequencies up tohe bandgap. Equations (38) were used to calculate �yx,ith � ,, and z being calculated from field averages with

he SRR in the bianisotropic configuration and �yy and �xxaken from the field averages obtained for the SRR in thelectric and magnetic configurations, respectively. The be-avior of all terms is consistent with analyticredictions.31,32

ig. 10. Material parameters �yy and �xx and the magnetoelec-ric coupling term �yx computed as a function of frequency (innits of d /c). The shaded region indicates the onset of the band-ap for the SRR oriented in the bianisotropic configurationrightmost inset, Fig. 7).

. Negative-Index Mediums described above, the wire medium possesses a fre-uency region over which ��0 for waves propagating per-endicular to the wire axes and polarized such that theirlectric field vectors lie in the direction of the wires. Like-ise, the SRRs possess a frequency region over which �0, for the SRR medium properly oriented as described

bove. If the two structures are combined such that theRR-medium magnetic resonance overlaps with the ��0

requency region of the wire medium, then the compositeedium should exhibit negative index. The structure is

imilar in concept to the negative-index composite first re-orted in 2000.8

The inset to Fig. 11 shows a combined wire-and-SRRnit cell. The wire has a square cross section, with side

ength 0.04 of the unit-cell length. The SRR and unit cellave the same dimensions as described above in Subsec-ion 3.D. The lower computed dispersion curve in Fig. 11hows the familiar signature of negative index, with thehase and group velocities having opposite sign.Applying the field-averaging procedure on the eigen-

elds corresponding to the passbands, we recover theffective-medium parameters shown in Fig. 12. For modesn the lowest passband, the relevant components of � and

are seen to be negative, as is the recovered index. Also,s expected, the recovered material parameters for thepper band are all positive.

. CONCLUSIONield averaging provides a useful means of characterizing

he effective-medium parameters of an arbitrary metama-erial. It is particularly useful for complex structures inhich the transmitted and reflected amplitudes are notasily interpreted; in such cases it can be difficult tohoose the appropriate solution (out of the available set ofolutions) that matches the expected effective-mediumheory. In contrast, the material parameters obtained byhe field-averaging method are not subject to the multipleranches that occur in the S-parameter method. Whenombined with dispersion curve calculations, field averag-

ig. 11. Dispersion curves for a combination wire–SRR me-ium. Shaded regions indicate frequency bandgaps. (inset) De-iction of the unit cell.

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402 J. Opt. Soc. Am. B/Vol. 23, No. 3 /March 2006 D. R. Smith and J. B. Pendry

ng provides a wealth of information distinct and comple-entary to that obtained by S-parameter analysis and re-

rieval.The calculations presented here were performed usingscript written in the Visual Basic Scripting language

hat runs within the HFSS software. The script stepshrough phase advances between 0° and 180°, computinghe eigenmodes and eigenfields and performing the fieldverages at each phase step. Steps in intervals of 20°ere used in all the simulations presented above. Inputarameters are provided in an EXCEL spreadsheet read byhe script, with the computed mode frequencies and fieldverages being output to the same EXCEL spreadsheet. Inhis way, the process of characterizing a metamaterial cane automated, allowing also for sophisticated iterative de-ign cycles.

The examples presented are typical of metamaterials,n that the size of the unit cell is generally smaller but byo means negligible compared with the wavelength, as in-icated by the normalized frequency scale in all the fig-res. Still, the field-averaging method provides physicallyeasonable retrieved parameters, which are also consis-ent with those obtained by S-parameter retrieval. Theesults of both methods indicate that metamaterials cane usefully—even if only approximately—described byffective-medium parameters. It can be readily seen fromhe examples presented that somewhat less intuitive ma-erial parameters are obtained as the higher-order bandtructure (above the first Brillouin zone) is approached.lthough the field-averaging method appears to provideonsistent results at these frequencies, it can be expectedo be unreliable at frequencies where multiple passbandsre present.Finally, we should note that we have presented ex-

mples for which the unit cell is relatively simple, consist-ng of only one or two elements, generally designed toave desired properties in only one or two spatial dimen-ions. When the unit cell becomes more complicated, hav-ng multiple elements, additional bands are found in theigenmode simulation that make the analysis more diffi-ult. These bands may not be excited in a plane-waveransmission–reflection experiment but do introduce ad-

ig. 12. Retrieved material parameters (�zz ,�xx, and n) as aunction of normalized frequency for the unit cell depicted in Fig.1.

itional complexity into the eigenmode spectrum. Forhese more intricate structures, the field-averagingethod can, in principle, be applied, but the analysis be-

omes more involved.

CKNOWLEDGMENTShe authors acknowledge informative discussions with S.retyakov and R. Marqués. This work was supported by aultidisciplinary University Research Initiative, spon-

ored by Defense Advanced Research Projects Agency, Of-ce of Naval Research (grant N00014-01-1-0803).Corresponding author D. R. Smith may be reached by

-mail at drsmith@ee.duke.edu.

EFERENCES1. M. M. I. Saadoun and N. Engheta, “A reciprocal phase

shifter using novel pseudochiral or omega-medium,”Microwave Opt. Technol. Lett. 5, 184–188 (1992).

2. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J.Viitanen, Electromagnetic Waves in Chiral and Bi-IsotropicMedia (Artech House, 1994).

3. W. S. Weiglhofer and A. Lakhtakia, eds., Introduction toComplex Mediums for Optics and Electromagnetic (SPIE,2003).

4. R. M. Walser, “Electromagnetic metamaterials,” inComplex Mediums II: Beyond Linear Isotropic Dielectrics,A. Lakhtakia, W. S. Weiglhofer, and I. J. Hodgkinson, eds.,Proc. SPIE 4467, 1–15 (2001).

5. A. Sihvola, “Electromagnetic emergence in metamaterials,”in Advances in Electromagnetics of Complex Media andMetamaterials, S. Zouhdi, A. Sihvola, and M. Arsalane,eds., Vol. 89 of NATO Science Series II: Mathematics,Physics, and Chemistry (Kluwer Academic, 2003), pp. 3–17.

6. V. G. Veselago, “The electrodynamics of substances withsimultaneously negative values of � and �,” Sov. Phys. Usp.10, 509–514 (1968).

7. J. B. Pendry and D. R. Smith, “Reversing light withnegative refraction,” Phys. Today 57, 37–43 (2004).

8. D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser,and S. Schultz, “A composite medium with simultaneouslynegative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).

9. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs,“Extremely low frequency plasmons in metallicmesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).

0. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J.Stewart, “Magnetism from conductors and enhancednonlinear phenomena,” IEEE Trans. Microwave TheoryTech. 47, 2075–2084 (1999).

1. P. Markoš and C. M. Soukoulis, “Numerical studies ofleft-handed materials and arrays of split ring resonators,”Phys. Rev. E 65, 036622 (2002).

2. M. Bayindir, K. Aydin, E. Ozbay, P. Markoš, and C. M.Soukoulis, “Transmission properties of compositemetamaterials in free space,” Appl. Phys. Lett. 81, 120–122(2002).

3. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah,and M. Tanielian, “Experimental verification andsimulation of negative index of refraction using Snell’slaw,” Phys. Rev. Lett. 90, 107401 (2003).

4. P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford,and J. Schelleng, “Electromagnetic waves focused by anegative-index planar lens,” Phys. Rev. E 67, 025602(2003).

5. A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimentalobservations of a left-handed material that obeys Snell’slaw,” Phys. Rev. Lett. 90, 137401 (2003).

1

1

1

1

2

2

2

2

2

2

2

2

2

2

3

3

3

D. R. Smith and J. B. Pendry Vol. 23, No. 3 /March 2006/J. Opt. Soc. Am. B 403

6. R. W. Ziolkowski, “Design, fabrication, and testing ofdouble negative metamaterials,” IEEE Trans. AntennasPropag. 51, 1516–1529 (2003).

7. F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J.Bonache, M. Beruete, R. Marques, F. Martin, and M.Sorolla, “Babinet principle applied to the design ofmetasurfaces and metamaterials,” Phys. Rev. Lett. 93,197401 (2004).

8. V. M. Shalaev, “Electromagnetic properties of small-particle composites,” Phys. Rep. 272, 61–137 (1996).

9. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis,“Determination of effective permittivity and permeabilityof metamaterials from reflection and transmissioncoefficients,” Phys. Rev. B 65, 195104 (2002).

0. K. S. Yee, “Numerical solution of initial boundary valueproblems involving Maxwell’s equations in isotropicmedia,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).

1. J. B. Pendry, “Calculating photonic band structure,” J.Phys. Condens. Matter 8, 1085–1108 (1996).

2. D. J. Bergman, “The dielectric constant of a compositematerial—a problem in classical physics,” Phys. Rep. 9,377–407 (1978).

3. D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, “Directcalculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett. 77, 2246–2248(2000).

4. J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley,1990), Sec. 2.6.

5. T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis,“Resonant and antiresonant frequency dependence of theeffective parameters of metamaterials,” Phys. Rev. E 68,065602 (2003).

6. S. I. Maslovski, S. A. Tretyakov, and P. A. Belov, “Wiremedia with negative effective permittivity: a quasi-staticmodel,” Microwave Opt. Technol. Lett. 35, 47–51 (2002).

7. P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M.Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strongspatial dispersion in wire media in the very largewavelength limit,” Phys. Rev. B 67, 113103 (2003).

8. A. L. Pokrovsky, “Analytical and numerical studies of wire-mesh metallic photonic crystals,” Phys. Rev. B 69, 195108(2004).

9. C. R. Simovski and P. A. Belov, “Low-frequency spatialdispersion in wire media,” Phys. Rev. E 70, 046616 (2004).

0. M. G. Silveirinha and C. A. Fernandes, “Homogenization of3-D-connected and nonconnected wire metamaterials,”IEEE Trans. Microwave Theory Tech. 53, 1418–1430(2005).

1. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role ofbianisotropy in negative permeability and left-handedmetamaterials,” Phys. Rev. B 65, 144440 (2002).

2. R. Marqués, F. Mesa, J. Martel, and F. Medina,“Comparative analysis of edge- and broadside–coupled splitring resonators for metamaterial design—theory andexperiments,” IEEE Trans. Antennas Propag. 51,2572–2581 (2003).