Smith et al

Electromagnetic parameter retrieval from inhomogeneous metamaterials

D. R. Smith,1,2,* D. C. Vier,2 Th. Koschny,3,4 and C. M. Soukoulis3,41Department of Electrical and Computer Engineering, Duke University, Box 90291, Durham, North Carolina 27708, USA

2Department of Physics, University of California, San Diego, La Jolla, California 92093, USA3Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA

4Foundation for Research and Technology Hellas (FORTH), 71110 Heraklion, Crete, GreecesReceived 21 November 2004; published 22 March 2005d

We discuss the validity of standard retrieval methods that assign bulk electromagnetic properties, such as theelectric permittivity and the magnetic permeability m, from calculations of the scattering sSd parameters forfinite-thickness samples. S-parameter retrieval methods have recently become the principal means of charac-terizing artificially structured metamaterials, which, by nature, are inherently inhomogeneous. While the unitcell of a metamaterial can be made considerably smaller than the free space wavelength, there remains asignificant variation of the phase across the unit cell at operational frequencies in nearly all metamaterialstructures reported to date. In this respect, metamaterials do not rigorously satisfy an effective medium limitand are closer conceptually to photonic crystals. Nevertheless, we show here that a modification of the standardS-parameter retrieval procedure yields physically reasonable values for the retrieved electromagnetic param-eters, even when there is significant inhomogeneity within the unit cell of the structure. We thus distinguish ametamaterial regime, as opposed to the effective medium or photonic crystal regimes, in which a refractiveindex can be rigorously established but where the wave impedance can only be approximately defined. Wepresent numerical simulations on typical metamaterial structures to illustrate the modified retrieval algorithmand the impact on the retrieved material parameters. We find that no changes to the standard retrieval proce-dures are necessary when the inhomogeneous unit cell is symmetric along the propagation axis; however, whenthe unit cell does not possess this symmetry, a modified procedurein which a periodic structure isassumedis required to obtain meaningful electromagnetic material parameters.

DOI: 10.1103/PhysRevE.71.036617 PACS numberssd: 41.20.2q, 42.70.2a

I. INTRODUCTION

A. Effective mediaIt is conceptually convenient to replace a collection of

scattering objects by a homogeneous medium, whose elec-tromagnetic properties result from an averaging of the localresponding electromagnetic fields and current distributions.Ideally, there would be no distinction in the observed elec-tromagnetic response of the hypothetical continuous materialversus that of the composite it replaces. This equivalence canbe readily achieved when the applied fields are static or havespatial variation on a scale significantly larger than the scaleof the local inhomogeneity, in which case the composite issaid to form an effective medium.

The electromagnetic properties of an inhomogeneouscomposite can be determined exactly by solving Maxwellsequations, which relate the local electric and magnetic fieldsto the local charge and current densities. When the particulardetails of the inhomogeneous structure are unimportant tothe behavior of the relevant fields of interest, the local field,charge, and current distributions are averaged, yielding themacroscopic form of Maxwells equations f1g. To solve thisset of equations, a relationship must be assumed that relatesthe four macroscopic field vectors that arise from theaveragingor homogenizationprocedure. It is here thatthe electric permittivity sd and the magnetic permeability

smd tensors are typically defined, which encapsulate the spe-cific local details of the composite medium f25g.

Depending on the symmetry and complexity of the scat-tering objects that comprise the composite medium, the and m tensors may not provide sufficient information to ob-tain a solution from Maxwells equations, and additionalelectromagnetic material parameters must be introduced f6g.Such media, including chiral and bianisotropic, can couplepolarization states and are known to host a wide array ofwave propagation and other electromagnetic phenomena f7g.We exclude these more complicated materials from our dis-cussion here, however, focusing our attention on linear, pas-sive media whose electromagnetic properties can be entirelyspecified by the and m tensors. Moreover, we restrict ourcharacterization of the material to one of its principal axes,and furthermore assume the probing wave is linearly polar-ized; thus, only the two scalar components and m are rel-evant.

The analytical approaches used in effective mediumtheory offer important physical insight into the nature of theelectromagnetic response of a composite; however, analyticaltechniques become increasingly difficult to apply in caseswhere the scattering elements have complex geometry. As analternative for such composites, a numerical approach is fea-sible in which the local electromagnetic fields of a structureare calculated by direct integration of Maxwells equations,and an averaging procedure applied to define the macro-scopic fields and material parameters f2,8,9g. Such an ap-proach is feasible for simulations, but does not extend to*Email address: [email protected]

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experimental measurements, where retrieval techniques areof prime importance for characterization of structures.

Distilling the complexity of an artificial medium into bulkelectromagnetic parameters is of such advantage that homog-enization methods have been extended beyond the traditionalregime where effective medium theories would be consid-ered valid f10,11g. The derived electromagnetic parameters,even if approximate, can be instrumental in the design ofartificial materials and in the interpretation of their scatteringproperties. Indeed, much of the recent activity in electromag-netic composites has demonstrated that scattering elementswhose dimensions are an appreciable fraction of a wave-length s.l /10d can be described in practice by the effectivemedium parameters and m. In particular, artificial materi-als, or metamaterials, with negative refractive index havebeen designed and characterized following the assumptionthat some form of effective medium theory applies f12g.These metamaterials have successfully been shown to inter-act with electromagnetic radiation in the same manner aswould homogeneous materials with equivalent material pa-rameters f13g.

B. Periodic structures

While the metamaterials introduced over the past severalyears have been mostly represented in the literature as ex-amples of effective media, their actual design typically con-sists of scattering elements arranged or patterned periodi-cally. This method of forming a metamaterial is convenientfrom the standpoint of design and analysis, because a com-plete numerical solution of Maxwells equations can be ob-tained from consideration of one unit cell of a periodic struc-ture f14,15g. The electromagnetic structure associated withperiodic systemsor photonic crystalspossesses numer-ous modes, especially toward higher frequencies, that aretypically summarized in band diagrams. The band diagramenumerates Bloch waves rather than plane waves, as theBloch wave represents the solution to Maxwells equationsfor systems with and m having spatial periodicity f14g.

The photonic crystal description of a periodic structure isvalid for all ratios of the wavelength relative to the scale ofinhomogeneity. But when the wavelength is very large rela-tive to the inhomogeneity, then it can be expected that aneffective medium description should also be valid. It mightalso be expected that there should exist a transitional regimebetween the effective medium and the photonic crystal de-scriptions of a periodic structure, where definitions of andm can be applied that are approximate but yet have practicalutility. It is this regime into which most of the recently dem-onstrated metamaterial structures fall. The metamaterial usedto demonstrate negative refraction in 2001, for example, hada unit cell dimension d that was roughly one-sixth the freespace wavelength f13g, meaning that the physically relevantoptical path length kd=2pd /l,1. The validity of ascribingvalues of and m to periodic systems for which kd is notsmall relative to unity was explored by Kyriazidou et al.,who concluded that the effective response functions for pe-riodic structures can be extended significantly beyond thetraditional limits of effective medium theory, even to the

point where the free space wavelength is on the order of theunit cell dimensions f11g.

Similar to f11g, we present here an algorithmic approachfor the assignment of effective medium parameters to a pe-riodic structure. Our approach makes use of the scatteringparameters determined for a finite-thickness, planar slab ofthe inhomogeneous structure to be characterized. Knowingthe phases and amplitudes of the waves transmitted and re-flected from the slab, we can retrieve values for the complexrefractive index n and wave impedance z. We find that avalid effective refractive index can always be obtained forthe inhomogeneous, periodic structure. A wave impedancecan also be assigned to the composite; however, the imped-ance value will depend on the termination of the unit cell. Inparticular, for structures that are not symmetric along thewave propagation direction, two different values of imped-ance are retrieved corresponding to the two incident direc-tions of wave propagation. This ambiguity in the impedanceleads to a fundamental ambiguity in the definitions of andm, which increases as the ratio of unit cell dimension towavelength increases. Nevertheless, our analysis shows thatthe methods used to date to characterize metamaterials areapproximately accurate and, with slight modification, aresufficient to be used to generate initial designs of metamate-rial structures.

II. RETRIEVAL METHODS

A. S-parameter retrieval

If an inhomogeneous structure can be replaced conceptu-ally by a continuous material, there should be no differencein the scattering characteristics between the two. A proce-dure, then, for the assignment of effective material param-eters to an inhomogeneous structure consists of comparingthe scattered waves si.e., the complex transmission and re-flection coefficients, or S parametersd from a planar slab ofthe inhomogeneous material to those scattered from a hypo-thetical continuous material. Assuming the continuous mate-rial is characterized by an index n and an impedance z, rela-tively simple analytic expressions can be found relating nand z of a slab to the S parameters. The inversion of S pa-rameters is an established method for the experimental char-acterization of unknown materials f16,17g. An S-parametermeasurement is depicted in Fig. 1sad.

The thickness of the slab, L, is irrelevant to the retrievedmaterial parameters, which are intrinsic properties of the ma-terial. It is thus advantageous when characterizing a materialto use as thin a sample as possibleideally, to be in the limitkL!1. Because metamaterials are formed from discrete ele-ments whose periodicity places a strict minimum on thethickness of a sample, we assume that the thickness usedhereon represents both one unit cell of the sample as well asthe total slab thickness. That is, we set L=d. Note that whilemeasurements on a single thickness of sample cannotuniquely determine the material parameters of the sample, itis usually the case that some knowledge regarding the ex-pected sample properties is available beforehand to helpidentify the most appropriate material parameter set.

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The procedure of S-parameter retrieval applied initially tometamaterial structures resulted in material parameters thatwere physically reasonable f18g, and have been shown to beconsistent with experiments on fabricated samples f19g. Yet avariety of artifacts exist in the retrieved material parametersthat are related to the inherent inhomogeneity of the metama-terial. These artifacts are a result of approximating a functionsuch as sv ,kd with effsvd; we thus expect anomalies mayexist for effsvd. The artifacts in the retrieved material pa-rameters are particularly severe for metamaterials that makeuse of resonant elements, as large fluctuations in the indexand impedance can occur, such that the wavelength withinthe material can be on the order of or smaller than the unitcell dimension. An example of this behavior can be seen inthe retrieved imaginary parts of and m, which typicallydiffer in sign for a unit cell that has a magnetic or an electricresonance. In homogeneous passive media, the imaginarycomponents of the material parameters are restricted to posi-tive values f1g. This anomalous behavior vanishes as the unitcell size approaches zero f20g.

A more vexing issue for metamaterials is the finite size ofthe unit cell. As we will show in detail, metamaterial struc-tures are often not conceptually reducible to the model ofFig. 1sad. It is typically the case that we must consider unitcells whose properties are more like those shown in Figs.1sbd and 1scd in which the equivalent unit cell consists of twosor mored distinct materials each of whose material proper-ties differ; that is, the equivalent one-dimensional s1Ddmodel of the material is inherently inhomogeneous, althoughwith the assumption that the unit cell will be repeated toform a periodic medium. To understand the limitations inusing the retrieval procedure on such inhomogeneous mate-

rials, we must therefore understand how the effective me-dium and photonic crystal descriptions of periodic structuresrelate to the S-parameter retrieval procedure.

B. The retrieval technique

We outline here the general approach to the retrieval ofmaterial parameters from S parameters for homogeneous ma-terials. For the sake of generality, it is useful to first definethe one-dimensional transfer matrix, which relates the fieldson one side of a planar slab to the other. The transfer matrixcan be defined from

F8 = TF , s1d

where

F = S EHredD . s2dE and Hred are the complex electric and magnetic field am-plitudes located on the right-hand sunprimedd and left-handsprimedd faces of the slab. Note that the magnetic field as-sumed throughout is a reduced magnetic field f15g having thenormalization Hred= s+ivm0dH. The transfer matrix for a ho-mogeneous 1D slab has the analytic form

T =1 cossnkdd z

ksinsnkdd

kz

sinsnkdd cossnkdd 2 , s3dwhere n is the refractive index and z is the wave impedanceof the slab f21g. n and z are related to and m by the rela-tions

= n/z, m = nz . s4d

The total field amplitudes are not conveniently probed inmeasurements, whereas the scattered field amplitudes andphases can be measured in a straightforward manner. A scat-tering sSd matrix relates the incoming field amplitudes to theoutgoing field amplitudes, and can be directly related to ex-perimentally determined quantities. The elements of the Smatrix can be found from the elements of the T matrix asfollows f22g:

S21 =2

T11 + T22 + SikT12 + T21ik D, s5d

S11 =T11 T22 + SikT12 T21ik DT11 + T22 + SikT12 + T21ik D

,

S22 =T22 T11 + SikT12 T21ik DT11 + T22 + SikT12 + T21ik D

,

FIG. 1. S-parameter measurements on sad a homogeneous 1Dslab; sbd an inhomogeneous asymmetric 1D slab; and scd a symmet-ric inhomogeneous 1D slab. The parameter d represents the thick-ness of a single unit cell of the structure. The different shadedregions represent different homogeneous materials, each with dis-tinct values of material parameters.

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S12 =2 detsTd

T11 + T22 + SikT12 + T21ik D.

For a slab of homogeneous material, such as that shown inFig. 1sad, Eq. s3d shows that T11=T22=Ts and detsTd=1, andthe S matrix is symmetric. Thus,

S21 = S12 =1

Ts +12SikT12 + T21ik D

, s6d

S11 = S22 =

12ST21ik ikT12D

Ts +12SikT12 + T21ik D

.

Using the analytic expression for the T-matrix elements inEq. s3d gives the S parameters

S21 = S12 =1

cossnkdd i2Sz + 1z Dsinsnkdd

s7d

and

S11 = S22 =i2S1z zDsinsnkdd . s8d

Equations s7d and s8d can be inverted to find n and z in termsof the scattering parameters as follows:

n =1kd

cos1F 12S21s1 S112 + S212 dG , s9dz =s1 + S11d2 S212

s1 S11d2 S212 , s10d

which provide a complete material description for a slabcomposed of a homogeneous material. In practice, however,the multiple branches associated with the inverse cosine ofEq. s10d make the unambiguous determination of the mate-rial parameters difficult unless it is known that the wave-length within the medium is much larger than the slab lengthf18g. The application of Eqs. s9d and s10d to metamaterials iscomplicated by the minimum sample thickness set by theunit cell size. Moreover, for metamaterials based on resonantstructures, there is always a frequency region over which thebranches associated with the inverse cosine in Eq. s9d be-come very close together, and where it becomes difficult todetermine the correct solution. Methods that make use of theanalyticity of and m can be applied to achieve a morestable and accurate retrieval algorithm for these structuresf23g.

When the fundamental unit cell is inhomogeneous, thevalidity of Eqs. s9d and s10d is not clear. In particular, if the1D material is modeled as shown in Fig. 1sbd, where there isan asymmetry along the propagation direction, then S11 andS22 will differ, and the retrieval procedure will produce dif-

ferent material parameters depending on which direction theincoming plane wave is directed. Even if the unit cell can besymmetrized, as illustrated in Fig. 1scd, the question remainsas to whether the retrieval process recovers meaningful val-ues for the material parameters. We will address these issuesdirectly in Sec. II E below.

C. Effective medium theory using S-parameter inversion

We expect that when the scale of the unit cell is smallwith respect to the phase advance across it, a homogeniza-tion scheme should be applicable that will lead to averagedvalues for the material parameters. Since we are deriving thematerial parameters based on the S-parameter retrieval pro-cess, it is useful to demonstrate that homogenization doesindeed take place within the context of this model.

Based on the model of 1D slabs, we note that the T matrixcan be expanded in orders of the optical path length sf=nkdd as follows:

T = I + T1w + T2w2 fl , s11dwhere I is the identity matrix and T1 can be found from Eq.s3d as

T1 =10 z

kkz

0 2 . s12dUsing Eqs. s11d and s12d in Eq. s6d, and keeping only termsto first order, we find

S21 =1

1 i2Sz + 1z Dw

, s13d

S11 =i2S1z zDwS21.

Equations s13d can be inverted to find approximate expres-sions for n and z analogous to Eqs. s9d and s10d.

To model the effect of nonuniform unit cells, we considera unit cell that is divided into N slab regions, each withdifferent material properties si.e., each having an optical pathlength f j and impedance zjd, as shown in Figs. 1sbd and 1scd.The total thickness of the unit cell is then d=d1+d2+ fl+dN. The transfer matrix for the composite can be found bymultiplying the transfer matrix for each constituent slab,yielding

T = TNTN1 fl T2T1, s14dwhich, to first order, is

T = I + oj=1

N

T j1njkdj , s15d

with


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T j1

=1 0 zjk

kzj

0 2 . s16dUsing Eq. s16d in Eq. s6d, we find

S21 =1

1 i2o j=1

N Szj + 1zjDnjkdj , s17d

S11 =i2oj=1

N S 1zj

zjDnjkdjS21.Comparing Eqs. s17d with Eqs. s13d, we see that, to firstorder, we can define averaged quantities for the multicellstructure as

Szav + 1zav

Dnav = oj=1

N Szj + 1zjDnj djd , s18d

Szav 1zav

Dnav = oj=1

N Szj 1zjDnj djd .

Neither n nor z average in a particularly simple manner;however, Eqs. s18d can be written in the physically satisfyingform

mav = zavnav = oj=1

N

zjnjdjd

= oj=1

N

m jdjd

, s19d

av =nav

zav= o

j=1

Nnjzj

djd

= oj=1

N

jdjd

,

which shows that the first order equations average to theexpected effective medium values, in which the average per-mittivity is equal to the volume average of the permittivity ofeach component, and the average permeability is equal to thevolume average of the permeability of each component. Notethat in the context of this model, the homogenization limit isequivalent to requiring that the ordering of the slabs withinthe unit cell make no difference in the properties of the com-posite. From Eq. s15d we see that fTi ,T jg=Oswiw jd, whichprovides a measure of the error in applying Eqs. s19d to afinite unit cell.

As we have noted, the optical path length over a unit cellfor metamaterial samples is seldom such that Eqs. s19d arevalid. However, if the metamaterial is composed of repeatedcells, then an analysis based on periodic structures can beapplied. From this analysis we may then extract some guid-ance as to the degree to which we might consider a periodicstructure as a homogenized medium.

D. Periodic inhomogeneous systems

For periodic systems, no matter what the scale of the unitcell relative to the wavelength, there exists a phase advance

per unit cell that can always be defined based on the period-icity. This phase advance allows the periodic structure in onedimension to be described by an indexeven if effectiveatall scales. We can determine the properties of a periodicstructure from the T matrix associated with a single unit cell.The fields on one side of a unit cell corresponding to a peri-odic structure are related to the fields on the other side by aphase factor, or

Fsx + dd = eiadFsxd , s20d

where F is the field vector defined in Eq. s2d and a is thephase advance per unit cell f21g. Equation s20d is the Blochcondition. Using Eq. s20d with Eq. s1d, we have

F8 = TF = eiadF . s21d

Equation s21d allows us to find the dispersion relation for theperiodic medium from knowledge of the transfer matrix bysolving

uT eiadIu = 0, s22d

from which we find

T11T22 jsT11 + T22d + j2 T12T21 = 0, s23d

where we have written j=expsiadd. By using detsTd=1, Eq.s23d can be simplified to

j +1j

= T11 + T22, s24d

or

2 cossadd = T11 + T22. s25d

Note that for a binary system, composed of two repeatedslabs with different material properties, T=T1T2, and wefind

cossadd = cossn1kd1dcossn2kd2d

12S z1z2 + z2z1Dcossn1kd1dcossn2kd2d , s26d

the well known result for a periodic 1D photonic crystal f21g.Equation s26d and, more generally, Eq. s25d provide a defi-nition of index that is valid outside of the homogenizationlimit.

E. Material parameter retrieval for inhomogeneous systems

If the optical path length across the unit cell of a structureis not small, then the effective medium limit of Eqs. s19d isnot applicable. Since the S-parameter retrieval procedure ispredicated on the assumption that the analyzed structure canbe replace by a continuous material defined by the param-eters n and z, it is clear that the retrieved parametersif atall validwill not be simply related to the properties of theconstituent components. Moreover, if the unit cell is notsymmetric in the propagation direction fe.g., szdszdg,then the standard retrieval procedure fails to provide aunique answer for n. Depending on the direction of propaga-tion of the incident plane wave with respect to the unit cell,we find the index is defined by either


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cossnkdd =1

2S21s1 S11

2 + S212 d s27d

or

cossnkdd =1

2S21s1 S22

2 + S212 d , s28d

which give different values since S11 and S22 are not thesame for asymmetric structures.

However, if the unit cell is repeated infinitely, Eq. s25dshows that a unique value of index can be recovered. Tocompare the form of Eq. s25d with Eqs. s27d and s28d, wefirst express the S-matrix elements in terms of the T-matrixelements as follows f22g:

T11 =s1 + S11ds1 S22d + S21S12

2S21,

T12 =s1 + S11ds1 + S22d S21S12

2S21, s29d

T21 =s1 S11ds1 S22d S21S12

2S21,

T22 =s1 S11ds1 + S22d + S21S12

2S21.

Using Eqs. s29d in Eq. s25d yields

cossadd =1 S11S22 + S21

2

2S21. s30d

Equation s30d shows that, regardless of the wavelength tounit cell ratio, an effective index can be recovered from amodified S-parameter retrieval procedure that utilizes all el-ements of the S matrix. A comparison of Eq. s30d with Eqs.s27d and s28d shows that the standard retrieval procedure canbe applied to find the index for an inhomogeneous structureusing an averaged Sav of

Sav = S11S22. s31dThe analysis that leads to Eq. s30d does not require any

assumption that the unit cell size be negligible in comparisonto the optical path length. As such, the retrieved index will bevalid well beyond the regime where traditional effective me-dium theory would be valid, in agreement with f11g. Thisresult explains the apparent success in assigning bulk mate-rial parameters to metamaterial structures. Note that the re-trieval procedures for both homogeneous and symmetric in-homogeneous unit cells are identical, since S11=S22.

To obtain values for and m, it is also necessary to de-termine the impedance zred for the inhomogeneous medium.For a continuous material, zred is an intrinsic parameter thatdescribes the ratio of the electric to the magnetic fieldssE /Hredd for a plane wave. Clearly, for an inhomogeneousmaterial this ratio will vary throughout the unit cell leadingto an unavoidable ambiguity in the definition of zred. zred canbe found from Eq. s21d as

zred =T12

T11 js32d

or, equivalently,

zred =T22 j

T21. s33d

Solving Eq. s23d for j yields

j =sT11 + T22d

2sT22 T11d2 + 4T12T21

4. s34d

On substitution of Eq. s34d into Eq. s33d for Eq. s32dg weobtain two results for zred corresponding to the two eigenval-ues of Eq. s23d:

zred =sT22 T11d 7 sT22 T11d2 + 4T12T21

2T21. s35d

The two roots of this equation correspond to the two direc-tions of wave propagation. Inspection of Eq. s35d reveals thatfor a reciprocal structure, T11=T22 and the impedance has theform

zred2

=

T12T21

. s36d

Using the values for the T-matrix elements, Eqs. s29d, werecover the same form for zred as in Eq. s10d.

Although an inhomogeneous periodic structure does nothave a well defined impedance, as the ratio of E /Hred willvary periodically throughout the structure, this variation be-comes negligible for very small unit cell sizes relative to thewavelength. Nevertheless, the lack of a unique definition forzred indicates that the values of and m retrieved are notgenerally assignable, although they can be applied in an ar-tificial manner if the metamaterial is always to be terminatedin the same location of the unit cell. The surface terminationhas an increasing influence on the scattering properties of thestructure as the scale of inhomogeneity increases relative tothe wavelength.

III. NUMERICAL EXAMPLES

A. S-parameter retrieval for a symmetric structureWe illustrate some of the issues associated with the re-

trieval procedure by using S parameters obtained from nu-merical simulations on negative index metamaterials. Nega-tive index metamaterials, which have been of recent interest,pose a significant challenge to retrieval methods becausethey utilize resonant elements and exhibit both an electricand magnetic response. A single unit cell of a typical sym-metric sin the propagation directiond metamaterial structureis shown in Fig. 2. This particular structure is composed oftwo types of conducting elementsa split ring resonatorsSRRd and a wirethat have been designed to yield a bandof negative refractive index at microwave frequencies. Simi-lar structures have now been analyzed using S-parameter re-trieval methods by numerous authors f1820,23g, with thegeneral observation that physically reasonable results can beobtained.


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We note in passing that the SRR generally exhibits bi-anisotropy, since an applied magnetic field induces both anelectric as well as a magnetic response. This material re-sponse should properly be accounted for in a complete char-acterization procedure. However, for the orientation of theSRR shown in Fig. 2 sring gaps in the direction of the elec-tric fieldd, previous work has indicated that the cross-coupling terms can be negligible, so that we are justified herein pursuing a description of the composite in terms of only and m.

The details of the retrieval procedure for metamaterialshave been outlined previously in f18g, with a more stableimplementation described in f23g. The geometry and dimen-sions for the present example were chosen to be roughlyconsistent with previous studies. The unit cell is cubic, witha cell dimension of d=2.5 mm. A 0.25 mm thick substrate ofFR4 s=4.4, loss tangent of 0.02d is assumed. A copper SRRand wire are positioned on opposite sides of the substrate,modeling the structures typically produced by lithographiccircuit board techniques. The copper thickness is 0.017 mm.The width of the wire is 0.14 mm, and it runs the length ofthe unit cell. The outer ring length of the SRR is 2.2 mm andboth rings have a linewidth of 0.2 mm. The gap in each ringis 0.3 mm, and the gap between the inner and outer rings is0.15 mm.

The S parameters for the unit cell of Fig. 2 are computedusing HFSS sAnsoftd, a commercial finite-element-based elec-tromagnetic mode solver. Both the S parameters and the re-trieved material parameters are presented in Fig. 3. Figures3sad and 3sbd show the magnitude and phase of the computedS parameters. Note the dip in the phase of S21, which indi-cates the presence of a negative index band.

The retrieved index in Fig. 3scd confirms the negativeindex band that lies between roughly 9 and 12 GHz. Thestructure was designed so as to be roughly impedancematched sz=1d where n=1, as this material condition isdesirable for many applications f24,25g. The retrieved im-pedance, shown in Fig. 3sdd, shows that the structure is in-deed roughly matched at the frequency where n=1. Usingthe impedance and index in Eq. s4d, we find functional forms

for the permittivity and permeability, as shown in Figs. 3sedand 3sfd.

While the retrieval procedure leads to generally satisfyingresultsan expected resonance function for m, forexampleunphysical artifacts occur in the recovered mate-

FIG. 2. sLeftd A single unit cell of a negative index metamate-rial. The split ring resonator sSRRd responds magnetically and thewire responds electrically to electromagnetic fields. sRight, topdPlanar view of the unit cell. Wave propagation is from left to right.sRight, bottomd An equivalent model of the unit cell, schematicallyindicating that it is a symmetric but inhomogeneous structure.

FIG. 3. sad Magnitude and sbd phase of the simulated S param-eters for the unit cell in Fig. 2. Retrieved index scd, impedance sdd,permittivity sed, and permeability sfd are also shown.


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rial parameters suggesting that a homogenization limit hasnot been reached. Note, for example, that there is a slightantiresonance in the real part of near 9 GHz fFig. 3sedg,which is consistent with the imaginary part of being nega-tive at the same frequencies. This artifact can be recoveredby replacing the actual SRR-wire unit cell with the inhomo-geneous unit cell indicated in Fig. 2scd. The inhomogeneousunit cell consists of three slabs, the center slab of whichcontains a material with idealized frequency-dependent andm. The outer slabs are uniform material svacuum or dielec-tric, for exampled. The details of how this type of inhomo-geneity leads to artifacts and distortions in the retrieved ma-terial parameters have been recently described f20g, and willnot be repeated here, as a more detailed analysis is forthcom-ing f26g. For the structure presented in Fig. 2, the anomaliesare relatively minor, and the retrieved material parametersprovide somewhat accurate descriptions of the electromag-netic material properties.

Note that while we appeal to the slab model of Sec. II inour analysis here and in the following sections, we do notattempt to solve for the effective slab parameters or thick-nesses, as this would constitute a nontrivial inverse problem,and is not of direct relevance here as we are searching for ahomogenized description of the composite medium.

B. S-parameter retrieval for a nonsymmetric structure

The unit cell in Fig. 2 can be made asymmetric by movingthe wire, for example, off of the symmetry axis. The unit cellshown in Fig. 4 is identical to that of Fig. 2, except that thewire has been shifted a distance of 0.75 mm along the propa-gation direction. Figures 5sad and 5sbd show the S parameterscomputed for an infinitely repeated asymmetric unit cell, onecell thick in the direction of propagation. While the differ-ences between the magnitudes of S11 and S22 are modest,there is a great contrast in the phases of S11 and S22, implyingvery different impedance properties for the structure depend-ing on which side of the unit cell faces the incoming wavesboth the phases and magnitudes of S12 and S21 are identicald.The standard retrieval procedure to find the index fEqs. s27d

and s28dg yields the unphysical functional forms shown bythe solid and dashed gray lines in Fig. 5scd. Using the modi-fied S-parameter retrieval procedure fEq. s30dg, however,yields a frequency dependent form for the index nearly iden-tical to that of the symmetric unit cell, as shown by the solidblack line in Fig. 5scd.

The retrieved indices for the symmetric and asymmetricunit cells are compared in Fig. 5sdd, illustrating that, asidefrom a shift in frequency, the refractive properties of the two

FIG. 4. An asymmetric unit cell. The parameters of the SRR andwire are the same as in the unit cell depicted in Fig. 2, except thatthe wire has been shifted 0.75 mm away from its centered position.The equivalent model sbottom, rightd depicts the asymmetry of thestructure.

FIG. 5. sad Magnitude and sbd phase of the simulated S param-eters for the unit cell in Fig. 4. scd The retrieved refractive index,using the standard retrieval method sgray curvesd and the full Sparameters ssolid black curved. sdd A comparison of the refractiveindices for the symmetric and asymmetric unit cells. sed The twoimpedance functions z1 and z2 corresponding to the two differentpropagation directions.


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structures are very similar. The frequency shift is not surpris-ing, as there is likely an interaction between the SRR andwire that leads to somewhat different material properties de-pending on their relative positions. The effect of this inter-action, as evidenced from Fig. 5sdd, is relatively minor.

While the index is nearly the same for the structures hav-ing symmetric and asymmetric unit cells, the impedance isclearly different, as indicated in Fig. 5sed fcf. Fig. 3sddg. Sodifferent are the two solutions for z for the asymmetric struc-ture that in general the assignment of values of and m tothe composite becomes counterproductive. Thus, although awell-defined refractive index exists for the composite, themanner in which a wave scatters from this material can de-pend strongly on how the surface is terminated.

Figure 5sed shows that the two solutions for the imped-ance found from Eq. s35d, although very different, can actu-ally have similar values at frequencies removed from eitherthe resonance or the effective plasma frequency whereResd=0 sat ,14 GHzd. Given that the refractive index iswell defined, at frequencies where the two impedance solu-tions are the same the asymmetric unit cell also mimics ahomogenized material, although the material properties aresomewhat different from those retrieved from the symmetricunit cell. The asymmetric structure, for example, is not im-pedance matched where n=1 as is the symmetric structure.

C. Higher-dimensional metamaterial structures

The issue of symmetry does not arise for most 1D struc-tures, since the unit cell can usually be made symmetricalong the propagation axis. However, for structures in whichit is desired to tune the electromagnetic properties in morethan one dimension, the resulting unit cell may be unavoid-ably asymmetric. While higher-dimensional symmetric struc-tures have been proposed that can serve as the basis for iso-tropic negative index metamaterials f27g, construction ofsuch structures may not always be feasible for the fabricationmethods most readily available.

A simple method of making the SRR-wire unit cell ap-proximately isotropic within a plane of propagation sand fora given polarizationd is to add a second SRR-wire structure

within the unit cell, oriented perpendicular to the first SRR.An example unit cell of this type is shown in Fig. 6. Thismethod of approximating isotropy has been used in previ-ously demonstrated negative index metamaterials based oncircuit board fabrication, in which strips of patterned circuitboard are assembled together in a wine crate structure f13g.Note, however, that this sample must be considered as inho-mogeneous, given the wavelengths of operation relative tosize of the unit cell, and must be analyzed in the same man-ner as the asymmetric unit cell above.

FIG. 6. A higher-dimensional structure, designed to have a fre-quency where n=1 and z=1. The unit cell is asymmetric in anydirection of propagation within the plane. An equivalent model ofthe unit cell is shown sbottom, rightd.

FIG. 7. sad Magnitude and sbd phase of the simulated S param-eters for the unit cell in Fig. 2. scd The retrieved index using thestandard retrieval method sgray curvesd and the full S parametermethod ssolid black curved. sdd The impedance values are shown forboth directions of incidence on the unit cell, as are the permittivityand permeability sed.


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The unit cell for this structure is the same size as in theprevious examples s2.5 mmd, but has different parametersfor the SRRs, substrate, and wires. A 0.25 mm substrate ofDuroid circuit board sRogers, =2.2, loss tangent of 0.0009dis assumed. A copper SRR and wire are positioned on oppo-site sides of both perpendicularly positioned circuit boards.The copper thickness is 0.017 mm. The width of the wire is0.10 mm, and it runs the length of the unit cell. The outerring length of the SRR is 2.2 mm and both rings have alinewidth of 0.35 mm. The gap in each ring is 0.35 mm, andthe gap between the inner and outer rings is 0.15 mm. Thedimensions of the rings and wires of this structure were care-fully tuned to the above values, so that the structure wouldbe matched where n=1.

The retrieved parameters for the 2D unit cell of Fig. 6 areshown in Fig. 7. The frequency dependent forms for Resmdand Resd are shown in Fig. 7sed. At frequencies above15 GHz, the material parameters retrieved for waves travel-ing in opposite directions are fairly similar, even though theydisagree in general over the broader frequency range. Near15 GHz, both retrieved values of z are nearly equal to unitywhile the refractive index is near 1, so that the structureresponds as if it were approximately homogeneous at thatfrequency. Note that for this asymmetric 2D structure, simi-lar to the example of Fig. 5scd, the refractive index deter-mined from the modified retrieval method is quite differentfrom that determined from the standard retrieval method.This example further illustrates the necessity of using themodified full S-parameter retrieval method for asymmetricstructures.

IV. CONCLUSION

We have shown that metamaterials based on periodicstructures occupy conceptually a special niche between ef-fective media and photonic crystals. The S-parameter re-trieval techniques that have been utilized recently to charac-terize metamaterials have been shown to be valid formetamaterials having symmetric unit cells, even when theoptical path length is on the order of the unit cell size. Forasymmetric unit cells, a modified retrieval method leads to aunique value of the refractive index, but two generally dis-tinct values for the wave impedance that are assigned to thestructure depending on the direction of wave propagationwith respect to the unit cell. The general lack of uniquenessof z complicates but does not obviate the design of structuresor devices that are necessarily asymmetric, since frequencyregions can be found where the two values of z are nearlyequal. Nevertheless, the expanded analysis presented here isrequired to determine whether or not an inhomogeneousstructure can be approximated as a homogeneous metamate-rial.

ACKNOWLEDGMENTS

This work was supported by DARPA sContract No.MDA972-01-2-0016d and DARPA/ONR through a MURIprogram sContract No. N00014-01-1-0803d. The work ofC.M.S. is further supported by the Ames Laboratory sCon-tract No. W-7405-Eng-82d and EU-FET project DALHM.The authors are grateful to Dr. Tomasz Grzegorczyk for ex-tensive comments and discussion.

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