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Electromagnetic parameter retrieval from inhomogeneous metamaterials

D. R. Smith,1,2,* D. C. Vier,2Th. Koschny,3,4and C. M. Soukoulis3,4

1Department 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, USA

3Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA

4Foundation for Research and Technology Hellas (FORTH), 71110 Heraklion, Crete, Greece

?Received 21 November 2004; published 22 March 2005?

We discuss the validity of standard retrieval methods that assign bulk electromagnetic properties, such as the

electric permittivity ? and the magnetic permeability ?, from calculations of the scattering ?S? parameters for

finite-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 unit

cell of a metamaterial can be made considerably smaller than the free space wavelength, there remains a

significant variation of the phase across the unit cell at operational frequencies in nearly all metamaterial

structures reported to date. In this respect, metamaterials do not rigorously satisfy an effective medium limit

and are closer conceptually to photonic crystals. Nevertheless, we show here that a modification of the standard

S-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 a

metamaterial regime, as opposed to the effective medium or photonic crystal regimes, in which a refractive

index can be rigorously established but where the wave impedance can only be approximately defined. We

present numerical simulations on typical metamaterial structures to illustrate the modified retrieval algorithm

and 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, when

the unit cell does not possess this symmetry, a modified procedure—in which a periodic structure is

assumed—is required to obtain meaningful electromagnetic material parameters.

DOI: 10.1103/PhysRevE.71.036617PACS number?s?: 41.20.?q, 42.70.?a

I. INTRODUCTION

A. Effective media

It is conceptually convenient to replace a collection of

scattering objects by a homogeneous medium, whose elec-

tromagnetic properties result from an averaging of the local

responding electromagnetic fields and current distributions.

Ideally, there would be no distinction in the observed elec-

tromagnetic response of the hypothetical continuous material

versus that of the composite it replaces. This equivalence can

be readily achieved when the applied fields are static or have

spatial variation on a scale significantly larger than the scale

of the local inhomogeneity, in which case the composite is

said to form an effective medium.

The electromagnetic properties of an inhomogeneous

composite can be determined exactly by solving Maxwell’s

equations, which relate the local electric and magnetic fields

to the local charge and current densities. When the particular

details of the inhomogeneous structure are unimportant to

the behavior of the relevant fields of interest, the local field,

charge, and current distributions are averaged, yielding the

macroscopic form of Maxwell’s equations ?1?. To solve this

set of equations, a relationship must be assumed that relates

the four macroscopic field vectors that arise from the

averaging—or homogenization—procedure. It is here that

the electric permittivity ??? and the magnetic permeability

??? tensors are typically defined, which encapsulate the spe-

cific local details of the composite medium ?2–5?.

Depending on the symmetry and complexity of the scat-

tering objects that comprise the composite medium, the ?

and ? tensors may not provide sufficient information to ob-

tain a solution from Maxwell’s equations, and additional

electromagnetic material parameters must be introduced ?6?.

Such media, including chiral and bianisotropic, can couple

polarization states and are known to host a wide array of

wave propagation and other electromagnetic phenomena ?7?.

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 entirely

specified by the ? and ? tensors. Moreover, we restrict our

characterization 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 ? are rel-

evant.

The analytical approaches used in effective medium

theory offer important physical insight into the nature of the

electromagnetic response of a composite; however, analytical

techniques become increasingly difficult to apply in cases

where the scattering elements have complex geometry. As an

alternative for such composites, a numerical approach is fea-

sible in which the local electromagnetic fields of a structure

are calculated by direct integration of Maxwell’s equations,

and an averaging procedure applied to define the macro-

scopic fields and material parameters ?2,8,9?. Such an ap-

proach is feasible for simulations, but does not extend to

*Email address: drsmith@ee.duke.edu

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experimental measurements, where retrieval techniques are

of prime importance for characterization of structures.

Distilling the complexity of an artificial medium into bulk

electromagnetic parameters is of such advantage that homog-

enization methods have been extended beyond the traditional

regime where effective medium theories would be consid-

ered valid ?10,11?. The derived electromagnetic parameters,

even if approximate, can be instrumental in the design of

artificial materials and in the interpretation of their scattering

properties. Indeed, much of the recent activity in electromag-

netic composites has demonstrated that scattering elements

whose dimensions are an appreciable fraction of a wave-

length ???/10? can be described in practice by the effective

medium parameters ? and ?. In particular, artificial materi-

als, or metamaterials, with negative refractive index have

been designed and characterized following the assumption

that some form of effective medium theory applies ?12?.

These metamaterials have successfully been shown to inter-

act with electromagnetic radiation in the same manner as

would homogeneous materials with equivalent material pa-

rameters ?13?.

B. Periodic structures

While the metamaterials introduced over the past several

years 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 convenient

from the standpoint of design and analysis, because a com-

plete numerical solution of Maxwell’s equations can be ob-

tained from consideration of one unit cell of a periodic struc-

ture ?14,15?. The electromagnetic structure associated with

periodic systems—or photonic crystals—possesses numer-

ous modes, especially toward higher frequencies, that are

typically summarized in band diagrams. The band diagram

enumerates Bloch waves rather than plane waves, as the

Bloch wave represents the solution to Maxwell’s equations

for systems with ? and ? having spatial periodicity ?14?.

The photonic crystal description of a periodic structure is

valid for all ratios of the wavelength relative to the scale of

inhomogeneity. But when the wavelength is very large rela-

tive to the inhomogeneity, then it can be expected that an

effective medium description should also be valid. It might

also be expected that there should exist a transitional regime

between the effective medium and the photonic crystal de-

scriptions of a periodic structure, where definitions of ? and

? can be applied that are approximate but yet have practical

utility. It is this regime into which most of the recently dem-

onstrated metamaterial structures fall. The metamaterial used

to demonstrate negative refraction in 2001, for example, had

a unit cell dimension d that was roughly one-sixth the free

space wavelength ?13?, meaning that the physically relevant

optical path length kd=2?d/??1. The validity of ascribing

values of ? and ? to periodic systems for which kd is not

small 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 the

traditional limits of effective medium theory, even to the

point where the free space wavelength is on the order of the

unit cell dimensions ?11?.

Similar to ?11?, we present here an algorithmic approach

for the assignment of effective medium parameters to a pe-

riodic structure. Our approach makes use of the scattering

parameters determined for a finite-thickness, planar slab of

the inhomogeneous structure to be characterized. Knowing

the phases and amplitudes of the waves transmitted and re-

flected from the slab, we can retrieve values for the complex

refractive index n and wave impedance z. We find that a

valid effective refractive index can always be obtained for

the inhomogeneous, periodic structure. A wave impedance

can also be assigned to the composite; however, the imped-

ance value will depend on the termination of the unit cell. In

particular, for structures that are not symmetric along the

wave propagation direction, two different values of imped-

ance are retrieved corresponding to the two incident direc-

tions of wave propagation. This ambiguity in the impedance

leads to a fundamental ambiguity in the definitions of ? and

?, which increases as the ratio of unit cell dimension to

wavelength increases. Nevertheless, our analysis shows that

the methods used to date to characterize metamaterials are

approximately accurate and, with slight modification, are

sufficient 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 difference

in the scattering characteristics between the two. A proce-

dure, then, for the assignment of effective material param-

eters to an inhomogeneous structure consists of comparing

the scattered waves ?i.e., the complex transmission and re-

flection coefficients, or S parameters? from a planar slab of

the 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 n

and 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 ?16,17?. An S-parameter

measurement is depicted in Fig. 1?a?.

The thickness of the slab, L, is irrelevant to the retrieved

material parameters, which are intrinsic properties of the ma-

terial. It is thus advantageous when characterizing a material

to use as thin a sample as possible—ideally, to be in the limit

kL?1. Because metamaterials are formed from discrete ele-

ments whose periodicity places a strict minimum on the

thickness of a sample, we assume that the thickness used

hereon represents both one unit cell of the sample as well as

the total slab thickness. That is, we set L=d. Note that while

measurements on a single thickness of sample cannot

uniquely determine the material parameters of the sample, it

is usually the case that some knowledge regarding the ex-

pected sample properties is available beforehand to help

identify the most appropriate material parameter set.

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The procedure of S-parameter retrieval applied initially to

metamaterial structures resulted in material parameters that

were physically reasonable ?18?, and have been shown to be

consistent with experiments on fabricated samples ?19?. Yet a

variety of artifacts exist in the retrieved material parameters

that are related to the inherent inhomogeneity of the metama-

terial. These artifacts are a result of approximating a function

such as ???,k? with ?eff???; we thus expect anomalies may

exist for ?eff???. The artifacts in the retrieved material pa-

rameters are particularly severe for metamaterials that make

use of resonant elements, as large fluctuations in the index

and impedance can occur, such that the wavelength within

the material can be on the order of or smaller than the unit

cell dimension. An example of this behavior can be seen in

the retrieved imaginary parts of ? and ?, which typically

differ in sign for a unit cell that has a magnetic or an electric

resonance. In homogeneous passive media, the imaginary

components of the material parameters are restricted to posi-

tive values ?1?. This anomalous behavior vanishes as the unit

cell size approaches zero ?20?.

A more vexing issue for metamaterials is the finite size of

the unit cell. As we will show in detail, metamaterial struc-

tures are often not conceptually reducible to the model of

Fig. 1?a?. It is typically the case that we must consider unit

cells whose properties are more like those shown in Figs.

1?b? and 1?c? in which the equivalent unit cell consists of two

?or more? distinct materials each of whose material proper-

ties differ; that is, the equivalent one-dimensional ?1D?

model of the material is inherently inhomogeneous, although

with the assumption that the unit cell will be repeated to

form a periodic medium. To understand the limitations in

using the retrieval procedure on such inhomogeneous mate-

rials, we must therefore understand how the effective me-

dium and photonic crystal descriptions of periodic structures

relate to the S-parameter retrieval procedure.

B. The retrieval technique

We outline here the general approach to the retrieval of

material parameters from S parameters for homogeneous ma-

terials. For the sake of generality, it is useful to first define

the one-dimensional transfer matrix, which relates the fields

on one side of a planar slab to the other. The transfer matrix

can be defined from

F? = TF,

?1?

where

F =?

E

Hred?.

?2?

E and Hredare the complex electric and magnetic field am-

plitudes located on the right-hand ?unprimed? and left-hand

?primed? faces of the slab. Note that the magnetic field as-

sumed throughout is a reduced magnetic field ?15? having the

normalization Hred=?+i??0?H. The transfer matrix for a ho-

mogeneous 1D slab has the analytic form

T =?

where n is the refractive index and z is the wave impedance

of the slab ?21?. n and z are related to ? and ? by the rela-

tions

cos?nkd?

−z

ksin?nkd?

k

zsin?nkd?

cos?nkd? ?

,

?3?

? = n/z,

? = nz.

?4?

The total field amplitudes are not conveniently probed in

measurements, whereas the scattered field amplitudes and

phases can be measured in a straightforward manner. A scat-

tering ?S? matrix relates the incoming field amplitudes to the

outgoing field amplitudes, and can be directly related to ex-

perimentally determined quantities. The elements of the S

matrix can be found from the elements of the T matrix as

follows ?22?:

S21=

2

T11+ T22+?ikT12+T21

T11− T22+?ikT12−T21

T11+ T22+?ikT12+T21

T22− T11+?ikT12−T21

T11+ T22+?ikT12+T21

ik?

ik?

ik?

ik?

ik?

,

?5?

S11=,

S22=,

FIG. 1. S-parameter measurements on ?a? a homogeneous 1D

slab; ?b? an inhomogeneous asymmetric 1D slab; and ?c? a symmet-

ric inhomogeneous 1D slab. The parameter d represents the thick-

ness of a single unit cell of the structure. The different shaded

regions represent different homogeneous materials, each with dis-

tinct values of material parameters.

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S12=

2 det?T?

T11+ T22+?ikT12+T21

ik?

.

For a slab of homogeneous material, such as that shown in

Fig. 1?a?, Eq. ?3? shows that T11=T22=Tsand det?T?=1, and

the S matrix is symmetric. Thus,

S21= S12=

1

Ts+1

2?ikT12+T21

2?T21

ik

Ts+1

ik?

,

?6?

S11= S22=

1

− ikT12?

2?ikT12+T21

ik?

.

Using the analytic expression for the T-matrix elements in

Eq. ?3? gives the S parameters

S21= S12=

1

cos?nkd? −i

2?z +1

z?sin?nkd?

?7?

and

S11= S22=i

2?1

z− z?sin?nkd?.

?8?

Equations ?7? and ?8? can be inverted to find n and z in terms

of the scattering parameters as follows:

kdcos−1?

2S21

z =??1 + S11?2− S21

?1 − S11?2− S21

which provide a complete material description for a slab

composed of a homogeneous material. In practice, however,

the multiple branches associated with the inverse cosine of

Eq. ?10? 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 length

?18?. The application of Eqs. ?9? and ?10? to metamaterials is

complicated by the minimum sample thickness set by the

unit cell size. Moreover, for metamaterials based on resonant

structures, there is always a frequency region over which the

branches associated with the inverse cosine in Eq. ?9? be-

come very close together, and where it becomes difficult to

determine the correct solution. Methods that make use of the

analyticity of ? and ? can be applied to achieve a more

stable and accurate retrieval algorithm for these structures

?23?.

When the fundamental unit cell is inhomogeneous, the

validity of Eqs. ?9? and ?10? is not clear. In particular, if the

1D material is modeled as shown in Fig. 1?b?, where there is

an asymmetry along the propagation direction, then S11and

S22will differ, and the retrieval procedure will produce dif-

n =

11

?1 − S11

2+ S21

2??,

?9?

2

2,

?10?

ferent material parameters depending on which direction the

incoming plane wave is directed. Even if the unit cell can be

symmetrized, as illustrated in Fig. 1?c?, the question remains

as to whether the retrieval process recovers meaningful val-

ues for the material parameters. We will address these issues

directly in Sec. II E below.

C. Effective medium theory using S-parameter inversion

We expect that when the scale of the unit cell is small

with respect to the phase advance across it, a homogeniza-

tion scheme should be applicable that will lead to averaged

values for the material parameters. Since we are deriving the

material parameters based on the S-parameter retrieval pro-

cess, it is useful to demonstrate that homogenization does

indeed take place within the context of this model.

Based on the model of 1D slabs, we note that the T matrix

can be expanded in orders of the optical path length ??

=nkd? as follows:

T = I + T1? + T2?2¯ ,

where I is the identity matrix and T1can be found from Eq.

?3? as

T1=?

z

?11?

0 −z

k

k

0?

.

?12?

Using Eqs. ?11? and ?12? in Eq. ?6?, and keeping only terms

to first order, we find

S21=

1

1 −i

2?z +1

z??

,

?13?

S11=i

2?1

z− z??S21.

Equations ?13? can be inverted to find approximate expres-

sions for n and z analogous to Eqs. ?9? and ?10?.

To model the effect of nonuniform unit cells, we consider

a unit cell that is divided into N slab regions, each with

different material properties ?i.e., each having an optical path

length ?jand impedance zj?, as shown in Figs. 1?b? and 1?c?.

The total thickness of the unit cell is then d=d1+d2+¯

+dN. The transfer matrix for the composite can be found by

multiplying the transfer matrix for each constituent slab,

yielding

T = TNTN−1¯ T2T1,

?14?

which, to first order, is

T = I +?

j=1

N

Tj

1njkdj,

?15?

with

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Tj

1=?

0

−zj

k

k

zj

0?

.

?16?

Using Eq. ?16? in Eq. ?6?, we find

S21=

1

1 −i

2?j=1

N?zj+1

zj?njkdj

,

?17?

S11=i

2?

j=1

N?1

zj

− zj?njkdjS21.

Comparing Eqs. ?17? with Eqs. ?13?, we see that, to first

order, we can define averaged quantities for the multicell

structure as

?zav+

j=1

?zav−

j=1

1

zav?nav=?

N?zj+1

zj?nj

dj

d,

?18?

1

zav?nav=?

N?zj−1

zj?nj

dj

d.

Neither n nor z average in a particularly simple manner;

however, Eqs. ?18? can be written in the physically satisfying

form

?av= zavnav=?

j=1

N

zjnj

dj

d=?

j=1

N

?j

dj

d,

?19?

?av=nav

zav

=?

j=1

N

nj

zj

dj

d=?

j=1

N

?j

dj

d,

which shows that the first order equations average to the

expected effective medium values, in which the average per-

mittivity is equal to the volume average of the permittivity of

each component, and the average permeability is equal to the

volume average of the permeability of each component. Note

that in the context of this model, the homogenization limit is

equivalent to requiring that the ordering of the slabs within

the unit cell make no difference in the properties of the com-

posite. From Eq. ?15? we see that ?Ti,Tj?=O??i?j?, which

provides a measure of the error in applying Eqs. ?19? to a

finite unit cell.

As we have noted, the optical path length over a unit cell

for metamaterial samples is seldom such that Eqs. ?19? are

valid. However, if the metamaterial is composed of repeated

cells, then an analysis based on periodic structures can be

applied. From this analysis we may then extract some guid-

ance as to the degree to which we might consider a periodic

structure as a homogenized medium.

D. Periodic inhomogeneous systems

For periodic systems, no matter what the scale of the unit

cell 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 one

dimension to be described by an index—even if effective—at

all scales. We can determine the properties of a periodic

structure 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 a

phase factor, or

F?x + d? = ei?dF?x?,

?20?

where F is the field vector defined in Eq. ?2? and ? is the

phase advance per unit cell ?21?. Equation ?20? is the Bloch

condition. Using Eq. ?20? with Eq. ?1?, we have

F? = TF = ei?dF.

?21?

Equation ?21? allows us to find the dispersion relation for the

periodic medium from knowledge of the transfer matrix by

solving

?T − ei?dI? = 0,

?22?

from which we find

T11T22− ??T11+ T22? + ?2− T12T21= 0,

where we have written ?=exp?i?d?. By using det?T?=1, Eq.

?23? can be simplified to

?23?

? +1

?= T11+ T22,

?24?

or

2 cos??d? = T11+ T22.

?25?

Note that for a binary system, composed of two repeated

slabs with different material properties, T=T1T2, and we

find

cos??d? = cos?n1kd1?cos?n2kd2?

−1

2?z1

z2

+z2

z1?cos?n1kd1?cos?n2kd2?,

?26?

the well known result for a periodic 1D photonic crystal ?21?.

Equation ?26? and, more generally, Eq. ?25? provide a defi-

nition of index that is valid outside of the homogenization

limit.

E. Material parameter retrieval for inhomogeneous systems

If the optical path length across the unit cell of a structure

is not small, then the effective medium limit of Eqs. ?19? is

not applicable. Since the S-parameter retrieval procedure is

predicated on the assumption that the analyzed structure can

be replace by a continuous material defined by the param-

eters n and z, it is clear that the retrieved parameters—if at

all valid—will not be simply related to the properties of the

constituent components. Moreover, if the unit cell is not

symmetric in the propagation direction ?e.g., ??z????−z??,

then the standard retrieval procedure fails to provide a

unique 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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