incremental and di•erential maxwell garnett formalisms for bi-anisotropic...

Incremental and dierential Maxwell Garnett formalisms forbi-anisotropic composites

Bernhard Michel a, Akhlesh Lakhtakia b, Werner S. Weiglhofer c,*, Tom G. Mackay c

aScienti®c Consulting, Kirchenstraûe 13, D-90537 Feucht-Moosbach, GermanybCATMAS Ð Computational & Theoretical Materials Sciences Group, Department of Engineering Science and Mechanics,

Pennsylvania State University, University Park, PA 16802-6812, USAcDepartment of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, UK

Received 14 December 1999; received in revised form 1 June 2000; accepted 27 June 2000

Abstract

We present, compare and contextualize two approaches to the homogenization of bi-anisotropic-in-bi-anisotropic particulate

composite medias: (i) the incremental Maxwell Garnett (IMG) formalism, in which the composite medium is built incrementally byadding the inclusions in N discrete steps to the host medium; and (ii) the dierential Maxwell Garnett (DMG) formalism, which isobtained from the IMG in the limit N!1 . Both formalisms are applicable to arbitrary inclusion concentration and are well-suited for computational purposes. Either of the two formalisms may be used as an alternative to the well-known Bruggemanformalism. Numerical results for the homogenization of a uniaxial dielectric composite medium and of a chiroferrite are presented.# 2000 Elsevier Science Ltd. All rights reserved.

Keywords: A. Particle-reinforced composites; B. Electrical properties; B. Magnetic properties; B. Modelling; C. Anisotropy

1. Introduction

Discrete random media Ð comprising electricallysmall particles of a certain material dispersed randomlyin some host medium Ð have been considered in theelectromagnetics literature for about two centuries ashomogeneous material continua. Several homogeniza-tion formalisms exist to connect the eective electro-magnetic response properties of a homogenizedcomposite medium (HCM) to those of its constituentmaterial phases. Initial elementary formalisms such asthat of Biot and Arago [1] were succeeded by morepowerful ones, such as the strong-property-¯uctuationtheory [2±4]. Yet, as even cursory glances at a recentanthology of milestone papers [5] and a handbook ofmaterials [6] reveal, simple homogenization formalismsremain in great demand. That may be because the reachof electromagnetics is vast, so that the results of homo-genization formalisms are commonly applied in a vari-ety of areas by researchers who do not necessarilyspecialize in electromagnetics [7].

Perhaps the most widely used homogenization form-alism is the Maxwell Garnett (MG) formalism, which Ðin one guise or another Ð predates Maxwell Garnett'sseminal 1904 paper [8] by several decades [9]. Thisattraction may be due to its simplicity, which lends to itsderivation from two dierent routes: (i) heuristicallyfrom a physical model [10], and (ii) rigorously from anintegral equation [11]. Most recently, it was set up for bi-anisotropic-in-bi-anisotropic composite media [12,13],thereby covering a large domain of electromagneticapplications in the materials sciences. Its simplicity,however, is also a handicap in that the MG formalismcan be used only for dilute composite mediums [14,15].The present communication enlarges upon an earlier

attempt [16] to overcome this handicap of the MGformalism. The incremental Maxwell Garnett (IMG)formalism is applicable to dense composite mediums. Ithas an iterative ¯avour, being based on the repeated useof the MG formalism for certain intermediate dilutecomposite mediums. As the IMG formalism had beenamply illustrated in detail for isotropic dielectric-in-dielectric composites, but its application for bi-aniso-tropic-in-bi-anisotropic composite mediums had beenmerely sketched out, in an earlier publication [16], we

0266-3538/01/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.PI I : S0266-3538(00 )00149-4

Composites Science and Technology 61 (2001) 13±18

www.elsevier.com/locate/compscitech

* Corresponding author.

embark here on a general application and exempli®ca-tion of this new formalism. Furthermore, we show thatthe IMG formalism leads to a dierential MaxwellGarnett (DMG) formalism that is based on the numer-ical solution of a system of dierential equations.

2. Theoretical developments

Suppose electrically small inclusions made of a med-ium labelled b are randomly dispersed in a host mediumlabelled a. Both mediums are linear and bianisotropic,their frequency-domain constitutive relations being spe-ci®ed as [13,16]:

DB

� � C�� E

H

� �; � a; b : 1

The 6�6 constitutive dyadic C� is composed of 3�3dyadics in the following way:

C� ��

��

!; � a; b ; 2

where �� and �� are the permittivity and permeabilitydyadics, whereas �� and �� are the two magnetoelectricdyadics. The volumetric proportion of the inclusionmedium equals fb, 04fb41 , while that of the hostmedium is fa 1ÿ fb. Although the inclusions can havenonspherical shapes [12,16], we con®ne ourselves here tospherical inclusions for simplicity of exposition.Our objective is to present the following estimates of

the eective 6�6 constitutive dyadic CHCM of the HCM:

1. CIMG from the IMG formalism, and2. CDMG from the DMG formalism.

Both of these estimates are compared with the esti-mate CMG from the straightforward MG formalism aswell as with the estimate CBr from the Bruggeman form-alism [12,13]. In the sequel, an exp ÿi!t time-depen-dence is implicit, where ! is the angular frequency.

2.1. Maxwell Garnett and Bruggeman formalisms

We begin by de®ning the 6�6 polarizability dyadic

a�0 in � C�0 ÿ C�

� �� I i!D~ �� C�0 ÿ C�

� �h iÿ13

of an electrically small sphere of medium �0 embeddedin medium �, where I is the 6�6 identity dyadic. The6�6 depolarization dyadic D~ can be decomposed into3�3 dyadics according to

D~ � D~ �

ee D~ �

em

D~ �me D~ �

mm

!: 4

The 3�3 dyadics D~ �ll0 can be calculated for a sphericalexclusion region via the integral representation [17]

D~ �ll0 1

4�i!

2��0

d�

��0

d�sin�

�r̂ ��ll0 �r̂� �

r̂r̂

r̂ �� r̂� �

r̂ �� r̂� �

ÿ r̂ �� r̂� �

r̂ �� r̂� � ;

l; l0 e;mÿ �;5

where r̂ is the unit radial vector in a spherical coordinatesystem, and

�ee ��; �em ÿ��; �me ÿ��; �mm ��: 6

We note parenthetically that generalizations of theforegoing results to ellipsoidal exclusion volumes arestraightforward [17].The Maxwell Garnett estimate CMG of the con-

stitutive dyadic of the HCM is then given by [12]

CMG Ca fbabina � Iÿ i!fbD~ a �abina� �ÿ1

: 7

The calculation of the left side of (7) is simple andstraightforward, using matrix algebra.The Bruggeman estimate CBr is obtained by solving

the equation [12,13]

faaa in Br fbaa in Br 0: 8

As both the inclusion and the host media are suppo-sedly dispersed as particulates in the HCM itself in thisformalism, (8) must be solved numerically for CBr

except for the simplest cases.

2.2. Incremental Maxwell Garnett formalism

In the IMG formalism, the actual composite medium isbuilt incrementally by adding the inclusions not all atonce, but in N stages. After each increment, the compo-site medium is homogenized using the MG formalism.Let C n , 14n4N , denote the eective constitutive

dyadic of the composite medium after the nth stage.Inclusions are embedded in a homogeneous medium ofconstitutive dyadic C nÿ1 , in the nth stage, in the ratio�b: 1ÿ �b , where 04�b41. Assuming �b to be su-ciently small, we apply the MG formalism to this inter-mediate composite medium. In this fashion, the iterativescheme

14 B. Michel et al. / Composites Science and Technology 61 (2001) 13±18

C n1 C n �bab in n

� Iÿ i!�bD~ n �ab in n h iÿ1

; n 0; 1; 2; . . . ; 9

emerges, beginning with C 0 Ca.Let �n denote the volumetric proportion of the inclu-

sion material in the nth stage. As a result of the incre-mental process, the relation

�n 1ÿ 1ÿ �b n 10

follows. Our desire to terminate the iterative scheme inN stages implies the relation �N fb, which ®xes theincremental proportion

�b 1ÿ 1ÿ fb 1=N: 11

The ®nal result of implementing (9) N times is the IMGestimate CIMG C N . It conforms to the following rea-sonable expectations: limfa ! 0 CIMG Cb andlimfa ! 1C

IMG Ca. Typically, �b40:2 for adequateresults, although a stronger contrast between the con-stitutive properties of the constituent media may reducethe upper value of �b to less than 0.1. Clearly, highervalues of fb entail larger values of N, although the ade-quate value of N for a given set of parameters may notbe large anyhow.

2.3. Dierential Maxwell Garnett formalism

The DMG formalism arises from the IMG formalismas follows: We rewrite (10) as

�b �n1 ÿ �n1ÿ �n ; 12

and insert this result into (9). In the limit N!1, thedierence equation (9) is converted into the ordinarydierential equation

@

@�C � 1

1ÿ � ab in �; 13

with initial value C 0 Ca. Eq. (13) can be solved e-ciently by standard ordinary dierential equation sol-vers [18]. We also remind the reader that ab in � on theright side of (13) depends explicitly (and in a non-trivialway) on C � through (3). Here, � is a continuous vari-able representing the inclusion volumetric proportion,and Cb in � is the 6�6 polarizability dyadic of an inclu-sion sphere embedded in medium with C � as its con-stitutive dyadic. The particular solution of (13) forwhich � fb is the DMG estimate; i.e.

CDMG C fb : 14

For an isotropic composite medium made with two iso-tropic dielectric materials, Eq. (13) reduces exactly toEq. (49) of Bruggeman's original paper [19].

2.4. Numerical implementation

Whereas the MG formalism is a single-step proce-dure, the IMG and the DMG formalisms require a cer-tain computational step to be repeated several times:namely, the evaluation of the right sides of (9) and (13)for the IMG and DMG formalisms, respectively. Eachiteration step requires the calculation of a depolariza-tion dyadic and for certain classes of media this can bedone analytically. These are bi-anisotropic media inwhich the symmetric parts of ��, ��, ��, and �� are uni-axial [17], and biaxial dielectric/magnetic media [20].However, in general, two-dimensional integrations haveto be performed as per (5). In addition, simple matrix-algebraic manipulations are needed.Numerical implementation of the Bruggeman formal-

ism is even more complicated [13]. This is because (8)comprises 36 coupled non-linear equations for the scalarcomponents of CBr, and iterative techniques have to beused. The iteration scheme employed by us [12,13] con-verged rapidly in most cases tested. However, no prac-tical stability/convergence criteria are available for anarbitrary choice of the constitutive parameters of mediaa and b, so that the success of any particular algorithmcannot be predicted in advance.The situation is completely dierent for the IMG and

the DMG formalisms. The number of iteration steps forthe IMG formalism is ®nite, because N is ®nite, and theissue of convergence is therefore trivial. For imple-menting the DMG formalism, one can rely on well-tes-ted algorithms in numerical libraries so that noconvergence problems are to be expected either. Thus,from the point of view of numerical implementation, theIMG and the DMG formalisms are simpler to use thanthe Bruggeman formalism and constitute attractivealternatives to the latter.Finally, even though the implementation of both the

IMG and the DMG formalisms requires iteration overseveral intermediate values of the 6�6 constitutive dya-dic of the HCM, the former formalism is more compu-tationally intensive than the latter. This can be easilyexplained as follows: Eq. (9) shows that the depolariza-tion dyadic D~ n is needed three times per step in theIMG formalism, while (13) makes it clear that thedepolarization dyadic is needed only once per step in theDMG formalism.

3. Numerical results and discussion

We ®rst illustrate the IMG and the DMG formalismin relation to the MG and the Bruggeman formalisms.

B. Michel et al. / Composites Science and Technology 61 (2001) 13±18 15

In all of our calculations, we noticed that CIMG con-verged to CDMG, as the iteration number N wasincreased, regardless of the complexity of the compo-nent media a and b.In Fig. 1, the DMG and IMG estimates are compared

with the MG and the Bruggeman (Br) estimates in thesimple case of a composite medium consisting of a uni-axial dielectric host medium with spherical isotropicdielectric inclusions. That is,

�a �0 �aI �au ÿ �aÿ �

uuh i

; �b �0�bI; 15

�a;b �0I; �a;b �a;b 0; 16

where �0 and �0 are the permittivity and permeability offree space, u is a unit vector parallel to the optical axisof the uniaxial medium, and I is the 3�3 unit dyadic.The permittivity dyadic of the HCM is written as

�HCM �0 �HCMI �HCMu ÿ �HCMÿ �

uuh i

; 17

(trivially, �HCM �0I, �HCM �HCM 0), and wasestimated using the following values: �au 4, �a 1 and�b 10. The inclusion volumetric proportion fb is anindependent variable in the presented graphs. The orderof the IMG calculations was set to N 5 to keep thedierences with the DMG appreciable on the graphspresented. The IMG and DMG estimates agree wellover the whole range of volumetric proportions fb, thedierences being much smaller than those between theMG and the Bruggeman estimates. The comparisonclearly shows the rapid convergence of the IMG form-alism: the MG estimate, which is identical to the IMGestimate with N 1, is still far o the fully convergedDMG result if fb > 0:3. However, already N 5 itera-tions (the curve labelled ``IMG'' in Fig. 1) are sucient

to push the IMG estimate very close to the DMG esti-mate. Both the DMG and IMG estimates are boundedby the MG and the Bruggeman estimates for 04fb41.Let us now turn to a full-¯edged bi-anisotropic com-

posite medium, viz., a chiroferrite conceptualized as arandom deposition of electrically small, isotropic chiralspheres in a ferrite host. Although chiroferrites wereconceptualized thus by Engheta et al. [21] in 1992, theconstitutive relations given by them are inadequate. Thede®ciencies were eliminated by Weiglhofer and Lakhta-kia [22] and Weiglhofer et al. [23] in 1998.

Fig. 1. Estimated relative permittivity scalars of a composite formed

by a random dispersion of isotropic dielectric spheres in an uniaxial

dielectric host medium, when �a 1, �au 4 and �b 10, as functionsof the inclusion volumetric proportion fb. The IMG algorithm was

implemented with N 5.

Fig. 2. Estimated constitutive parameters of a chiroferrite HCM as

functions of the inclusion volumetric proportion fb: (a) �HCM, �HCMu ,

�HCMg ; (b) �HCM, �HCMu , �

HCMg ; (c)�

HCM,�HCMu ,�HCMg . See the text for the

input parameters. The IMG algorithm was implemented with N 5.


The constitutive dyadics are denoted as

�� 0 ��Iÿ i��gu� I ��u ÿ ��ÿ �

uuh i

; 18

�� i �0�0 1=2 ��Iÿ i��gu� I ��u ÿ ��ÿ �

uuh i

; 19

�� i �0�0 1=2 ��Iÿ i��gu�� I ��u ÿ ��

ÿ �uu

h i; 20

�� 0 ��Iÿ i��gu� I ��u ÿ ��ÿ �

uuh i

; 21

where � can stand for any one of a, b, MG, Br, IMG,DMG. The chiroferrite combines gyrotropic anisotropy(through �� and ��) with gyrotropic-like bi-anisotropy(through �� and ��). We used the following parametersfor the calculations reported in Fig. 2: �au �a 5; �ag 0; �� 0; �au �a 1:1; �ag 1:3 for medium a;and �bu �b 4; �bg 0; � ÿ�; �bu �b 1; �bg 0; �bu �b 1:5; �bg 0 for medium b.Fig. 2 shows the IMG, the DMG and the Bruggeman

estimates of the three scalar components of the con-stitutive dyadics �HCM, �HCM, and �HCM, plotted asfunctions of fb. There is no need to display results for�HCM, since �HCM ÿ�HCM follows numerically for thechosen composite medium from all three formalisms.We ®nd an almost perfect agreement of the Bruggeman,IMG and DMG estimates for all constitutive para-meters of the HCM, except for the gyrotropic compo-nent �HCMg of the permittivity (Fig. 2c). Even so, thedierences between the three estimates �HCMg are notsigni®cant, because �HCMg

�� is lower than both �HCM and�HCMu in magnitude by three orders.

4. Conclusion

In this paper, two extensions of the Maxwell Garnettformalism for bi-anisotropic composites to arbitraryinclusion volume fractions were studied:

1. the incremental Maxwell Garnett (IMG) formalism,in which the composite medium is built incremen-tally by adding the inclusions in discrete steps tothe host medium, and

2. the dierential Maxwell Garnett (DMG) formal-ism, a limiting case of the IMG formalism obtainedby adding inclusions continuously.

Together with the Bruggeman formalism, there are nowthree approaches available to treat bianisotropic com-posites with intermediate inclusion volume fractions.The microscopic model underlying the IMG and the

DMG formalisms are discrete inclusions embedded in aseries of simply connected host media. In contrast, both

component media are treated symmetrically in theBruggeman formalism. If there is a strong contrastbetween the constitutive properties of the two compo-nent media (as in Fig. 1), the predictions of all threeformalisms dier signi®cantly. Therefore, that homo-genization formalism should be chosen which is moreappropriate for the microstructure of the compositemedium. The number of incremental steps N cannot be®xed using this argument, but may eventually serve as atunable parameter to obtain agreement with experi-mental results.If the contrast is comparatively weak (as in Fig. 2),

however, all three formalisms predict similar results andcan be used interchangeably. The simplicity androbustness of the 9 numerical implementation is then aclear advantage for the IMG/DMG formalisms over theBruggeman formalism. The DMG formalism is lesscomputationally intensive than its predecessor formal-ism. Current interest in the homogenization of the opti-cal properties of photonic band-gap materials [24] andof the mechanical properties of microstructured materi-als [25] suggest that the underlying basis of the IMG/DMG formalisms can be applied widely.

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BRUGGEMAN APPROACH FORISOTROPIC CHIRAL MIXTURESREVISITED

Benjamin M. Ross and Akhlesh LakhtakiaCATMAS—Computational & Theoretical Materials Sciences GroupDepartment of Engineering Science and MechanicsPennsylvania State University, University Park, PA 16802-6812

Received 6 September 2004

ABSTRACT: Two interpretations of the Bruggeman approach for thehomogenization of isotropic chiral mixtures are shown to lead to differ-ent results. Whereas the standard interpretation is shown to yield theaverage polarizability density approach, a recent interpretation turnsout to deliver a null excess polarization approach. The difference be-tween the two interpretations arises from differing treatments of the lo-cal field. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett44: 524–527, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20685

Key words: Bruggeman approach; chiral materials; composite materi-als; excess polarization; homogenization; local field; polarizability den-sity

1. INTRODUCTION

Homogenization of particulate composite materials is at least atwo-centuries-old theoretical problem; yet, it retains its freshnessto this day. Indeed, it can be argued that, as theoretical approachescan—at best—only estimate the effective constitutive parametersof a mixture of two or more component materials (but still viewedas being homogeneous), homogenization is unlikely to lose itscharm for theorists for the foreseeable future [1, 2].

This viewpoint rose to the fore recently when we had occasionto look at the Bruggeman approach for the homogenization of anisotropic mixture of two isotropic chiral materials. In this ap-proach, the volume fractions of both component materials aretaken into account, but the particulate dimensions are effectivelynull valued, as explicated by Kampia and Lakhtakia [3]. In theextended Bruggeman approach, the particulate dimensions areconsidered as electrically small but finite, as exemplified byShanker [4] for the chosen mixture. However, we found thatShanker’s interpretation of the Bruggeman approach differs in anessential point from that of Kampia and Lakhtakia—in addition tothe differing treatments of the particulate dimensions. Our rumi-nations on the newly discovered difference led to this paper.

2. THEORY IN BRIEF

Let us consider an isotropic mixture of two isotropic chiral mate-rials labeled a and b. Their frequency-domain constitutive rela-tions are stated as

D � �0�p�E � �p� � E�B � �0�p�H � �p� � H��, �p � a, b�, (1)

where �0 and �0 are the respective permittivity and permeability offree space (that is, vacuum); �a,b are the relative permittivityscalars, �a,b are the relative permeability scalars, and �a,b are thechirality pseudoscalars in the Drude–Born–Fedorov representation[5]; and an exp(�j�t) time-dependence is implicit. The volumet-ric fractions of the two component materials are denoted by fa andfb � 1 � fa. The aim of any homogenization exercise is to predictthe quantities �HCM, �HCM, and �HCM appearing in the consti-tutive relations as follows:

D � �0�HCM�E � �HCM� � E�B � �0�HCM�H � �HCM� � H��, (2)

which presumably hold for the homogenized composite material(HCM). The exercise is well-founded only if the particles of bothcomponent materials can be considered to be electrically small [6].

The Bruggeman approach for homogenization was initiated forisotropic mixtures of isotropic dielectric materials, but has beensubsequently extended to far more complex situations [1, 2]. Thegeneral formulation of the approach is as follows: suppose thecomposite material has been homogenized, and it obeys Eq. (2).Disperse in it, homogeneously and randomly, a small numberdensity of particles of both types of component materials in thevolumetric ratio fa : fb; and then homogenize. The properties ofthe HCM could not have been altered as a consequence.

All particles of type p, ( p � a, b), are identical and areequivalent to the electric and magnetic dipole moments, pp andmp, when immersed in the HCM. The standard interpretation ofthe Bruggeman approach then requires the solution of the follow-ing two equations [7]:

fapa � fbpb � 0fama � fbmb � 0�. (3)

In the present context, Kampia and Lakhtakia [3] solved (3) for�HCM, �HCM, and �HCM.

An alternative interpretation is that the dispersal of particles ofcomponent material p is equivalent to the creation of excesspolarization and excess magnetization, Pp and Mp, ( p � a, b), inthe HCM. But the total excess polarization and magnetization mustbe null valued. Then the two equations,

Pa � Pb � 0Ma � Mb � 0�, (4)

can be solved to determine �HCM, �HCM, and �HCM. AlthoughEq. (4) were stated by Kampia and Lakhtakia [3], these equationswere not solved by them; indeed, expressions for Pp and Mp werenot even provided by them. However, Shanker [4] did presentexpressions for Pp and Mp, and then solved Eq. (4).

3. NUMERICAL RESULTS

We decided to compare the implementations of Eqs. (3) and (4).All particles of both component materials were treated as spheresof radius R. Expressions for the polarizability densities (relatingelectric- and magnetic-dipole moments to exciting the electric andmagnetic fields) and polarization densities (relating excess polar-ization and excess magnetization to electric and magnetic fields)were obtained from Shanker’s paper [4].

Computed values of �HCM, �HCM, and �HCM as functions offb are shown in Figures 1 and 2 for k0R 3 0 and k0R � 0.2,where k0 � �(�0�0)

1/ 2 is the free-space wavenumber. Theconstitutive properties of the component materials for the twofigures are the same as chosen by Shanker [4].

Quite clearly, Figures 1 and 2 show that the incorporation of thefinite size of the particles gives rise to a dissipative HCM, evenwhen both component materials are nondissipative. This conclu-sion is true whenever a nonzero length-scale is considered in ahomogenization approach—whether as the particle size [4, 8], or acorrelation length for the particle-distribution statistics [2], or both[9]. The incorporation of the length scale appears to account, insome manner, for the scattering loss.

524 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 44, No. 6, March 20 2005

More importantly, whether the length scale is neglected (Fig. 1)or considered (Fig. 2), estimates of �HCM, �HCM, and �HCM fromEqs. (3) and (4) do not coincide. There seems to be a basicdifference between (3) and (4), which persists even when �a,b �1, �a,b � 0, and R 3 0. An explanation of this difference, in thatsimple context for the sake of clarity, is provided in the nextsection.

4. EXPLANATION

4.1 PreliminariesWe begin with the derivation of an important equation. Let allspace be occupied by a homogeneous dielectric material withrelative permittivity �h at the frequency of interest; thus, its rele-vant frequency-domain constitutive relation is given by

D � �0�hE. (5)

Suppose that an electrically small sphere made of a dielectricmaterial with relative permittivity �i were to be introduced. Thisparticle would act as an electric dipole moment,

p � v�0�i/hẽ, (6)

where v is the volume of the particle, ẽ is the electric field at thelocation of the particle if the particle were to be removed and theresulting hole filled with the host material, and the product of �0and �i/h is the polarizability density of the particle embedded inthe specific host material. The exact expression of �i/h does notmatter for our purpose here [10]; but we note that it is independentof R for the Bruggeman approach, and dependent on R for theextended Bruggeman approach [11].

Let many identical particles be randomly dispersed in the hostmaterial, such that their number density N is macroscopicallyuniform. Then, the particles can be replaced by an excess polar-ization,

P � Nv�0�i/hẼ, (7)

where

Ẽ � E � P/3�0�h (8)

is the local electric field [10]. The qualifier excess is used herebecause this P is in addition to the polarization �0(�h � 1)E thatindicates the presence of the host material.

By virtue of Eqs. (7) and (8), the excess polarization is given by

P � �0f�i/h

1 f�i/h/3�hE, (9)

where f � Nv is the volumetric fraction of the particulate material.Hence, the constitutive relation of the HCM is given by

D � �0�hE � P � �0��h � f�i/h1 f�i/h/3�h�E � �0�HCME, (10)so that

�HCM � �h �f�i/h

1 f�i/h/3�h(11)

is the estimated relative permittivity of the HCM at the frequencyof interest. The first rigorous derivation of the foregoing equationcan be attributed to Faxén [12].

Parenthetically, a Maxwell–Garnett formula for �HCM can bederived by setting �h � �a and �i � �b in Eq. (11), which is quiteappropriate if fb � fa; otherwise, the choice {�h � �b, �i � �a}should be made. These two Maxwell–Garnett estimates also con-stitute the so-called Hashin–Shtrikman bounds on �HCM [13].

4.2 Standard Interpretation of the Bruggeman Approach: Eq. (3)As stated in section 2, let us imagine that the composite materialhas already been homogenized. Into this HCM, let spherical par-ticles of both component materials be randomly dispersed. Thecombined volumetric fraction of the particles introduced into theHCM is f �� 1, with ffa and ffb being the respective volumetricfractions of the two component materials in the particles. Hence,

�i/h � fa�a/HCM � fb�b/HCM (12)

is the polarizability density of a material-averaged particle embed-ded in a material with �h � �HCM. Eq. (9) then yields

P � �0f� fa�a/HCM � fb�b/HCM�

1 f� fa�a/HCM � fb�b/HCM�/3�HCME (13)

Figure 1 Estimated values of �HCM, �HCM, and �HCM/�a as functionsof fb, when �a � 1, �a � 1, �a � 10

�3 m, �b � 2, �b � 1.5, �b �2.37 10�3 m, and k0R 3 0. Solid lines represent data computed usingEq. (3), while dashed lines join datapoints obtained using Eq. (4)

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 44, No. 6, March 20 2005 525

for the excess polarization.But the introduction of the material-averaged particles must not

change the HCM’s constitutive properties, as the relative propor-tion of the component materials remains unchanged; accordingly,the excess polarization of Eq. (13) is null valued, and the solutionof the equation,

0 � fa�a/HCM � fb�b/HCM, (14)

yields an estimate of �HCM. Thus, the standard interpretation ofthe Bruggeman approach leading to (3) is designated as the aver-age-polarizability-density approach.

4.3 Shanker’s Interpretation of the Bruggeman Approach: Eq. (4)Once again, suppose that the composite material has been homog-enized into a HCM with relative permittivity �HCM. Suppose, next,that particles of materials a and b are randomly dispersed theHCM and that their respective volumetric fractions in the newcomposite material are fa and fb. Following Shanker [4], we findthat the excess polarizations due to the two types of particles addup to

P � �0� fa�a/HCM1 fa�a/HCM/3�HCM � fb�b/HCM1 fb�b/HCM/3�HCM�E, (15)by virtue of Eq. (9).

The introduction of the particles into the HCM amounts simplyto the complete replacement of the HCM by itself; hence, Eq. (11)leads to

�HCM � �HCM �fa�a/HCM

1 fa�a/HCM/3�HCM�

fb�b/HCM1 fb�b/HCM/3�HCM

,

(16)

which yields the formula

0 �fa�a/HCM

1 fa�a/HCM/3�HCM�

fb�b/HCM1 fb�b/HCM/3�HCM

(17)

for an estimate of �HCM. Thus Shanker’s interpretation of theBruggeman approach leading to Eq. (4) is designated as the nullexcess polarization approach.

4.4 Comparison of the Two InterpretationsEq. (15) differs from Eq. (13) in a very significant way: Whereasparticles of the two component materials were amalgamated intomaterial-averaged particles whose polarizability density was usedto estimate the excess polarization as per Eq. (13), material aver-aging was not done for Eq. (15); instead, particles of both materialswere kept apart and two separate contributions were made to theestimate (15) of the excess polarization.

This difference can be understood also in terms of the differenttreatments of the local field. For (13), the local field pertains tomaterial-averaged particles, which is quite reasonable. In contrast,(15) contains two different local fields. The first local field pertainsonly to particles of material a embedded in the HCM, and leads tothe first term in the sum on the right side of (15); while the secondlocal field pertains only to particles of material b embedded in theHCM, and leads to the second term in the sum on the right side of

Figure 2 Same as Fig. 1, but for k0R � 0.2

526 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 44, No. 6, March 20 2005

(15). Accordingly, (15) lacks rigor in comparison to (13), and theformer can be considered simply as an empirical formula.

In closing, if Pa and Pb could somehow be separately estimatedin Shanker’s interpretation with the same local field, the twointerpretations could very possibly yield identical estimates of theconstitutive parameters of the HCM.

ACKNOWLEDGMENT

The second author appreciates several discussions with Dr. Bern-hard Michel.

REFERENCES

1. B. Michel, Recent developments in the homogenization of linearbianisotropic composite materials, Electromagnetic fields in uncon-ventional materials and structures, O.N. Singh and A. Lakhtakia (Ed-itors), Wiley–Interscience, New York, 2000.

2. T.G. Mackay, Homogenization of linear and nonlinear compositematerials, Introduction to complex mediums for optics and electro-magnetics, W.S. Weiglhofer and A. Lakhtakia (Editors), SPIE Press,Bellingham, WA, 2003.

3. R.D. Kampia and A. Lakhtakia, Bruggeman model for chiral particu-late composites, J Phys D Appl Phys 25 (1992), 1390–1394.

4. B. Shanker, The extended Bruggeman approach for chiral-in-chiralmixtures, J Phys D Appl Phys 29 (1996), 281–288.

5. A. Lakhtakia, Beltrami fields in chiral media, World Scientific, Sin-gapore, 1994.

6. A. Lakhtakia (Editor), Selected papers on linear optical compositematerials, SPIE Press, Bellingham, WA, 1996.

7. W.S. Weiglhofer, A. Lakhtakia, and B. Michel, Maxwell, Garnett, andBruggeman formalisms for a particulate composite with bianisotropichost medium, Microwave Opt Technol Lett 15 (1997), 263–266;corrections 22 (1999), 221.

8. W.T. Doyle, Optical properties of a suspension of metal spheres, PhysRev B 39 (1989), 9852–9858.

9. T.G. Mackay, Depolarization volume and correlation length in thehomogenization of anisotropic dielectric composites, Waves in Ran-dom Media 14 (2004), 485–498.

10. A. Lakhtakia, Size-dependent Maxwell–Garnett formula from an in-tegral equation formalism, Optik 91 (1992), 134–137. [The correctversion of Eq. (17) of this paper is: P � [3f��0/(3v�0 � f�)]E.]

11. M.T. Prinkey, A. Lakhtakia, and B. Shanker, On the extended Max-well–Garnett and the extended Bruggeman approaches for dielectric-in-dielectric composites, Optik 96 (1994), 25–30.

12. H. Faxén, Der Zusammenhang zwischen de Maxwellschen Gleichun-gen für Dielektrika und den atomistischen Ansätzen von H.A. Lorentzu.a., Z Phys 2 (1920), 219–229 (in German).

13. Z. Hashin and S. Shtrikman, A variational approach to the theory ofthe effective magnetic permeability of multiphase materials, J ApplPhys 33 (1962), 3125–3131.

© 2005 Wiley Periodicals, Inc.

APPLICATION OF LEFT-HANDEDMEDIA IN DISTRIBUTED AMPLIFIERS

K. W. EcclestonDepartment of Electrical and Computer EngineeringNational University of Singapore10 Kent Ridge CrescentSingapore 119260

Received 31 August 2004

ABSTRACT: A combination of left-handed and conventional right-handed media transmission lines is used in place of conventionalright-handed media transmission lines in a single-ended dual-fed dis-tributed amplifier. This approach results in increased bandwidth andcircuit size reduction. © 2005 Wiley Periodicals, Inc. Microwave OptTechnol Lett 44: 527–530, 2005; Published online in Wiley Inter-Science (www.interscience.wiley.com). DOI 10.1002/mop.20686

Key words: distributed amplifier; dual-fed distributed amplifier; micro-wave amplifier; left-handed transmission lines; meta-materials

1. INTRODUCTION

It is often necessary to combine the output power of severaltransistors to obtain high output power. By integrating powercombining and amplification in a single circuit, one can eliminatemultiway power combiners and the transmission lines that inter-connect the amplifier modules to the combiners, thereby reducingcircuit size and reducing power losses. The single-ended dual-feddistributed amplifier (SE-DFDA) has been shown to integratepower combining and efficient power amplification and eliminatethe need for multiway power combiners [1, 2].

The SE-DFDA is a half-circuit version of the dual-fed distrib-uted amplifier (DFDA) [3]. A 3-FET SE-DFDA, along with aninput generator and load, is shown in Figure 1. The circuit usesconventional right-handed transmission lines (RH-TLs) whosecharacteristic impedances (ZoG and ZoD) are indicated. The inser-tion phase of each transmission-line section at the amplifier centerfrequency is indicated and is either or /2. In practice, a pair ofidentical SE-DFDAs is combined using a pair of 90° hybrids toform a balanced amplifier [1–3].

If we assume that the effects of FET input and outputcapacitance is negligible, then when the insertion-phase param-eter is in odd-integer multiples of 180°, the FETs haveidentical load lines, drain voltages, and drain currents [2], andhence identical output power [4]. With identical loadlines, allFETs can be made to operate with maximum efficiency and befully utilized [4]. If the short-circuit stubs are removed, thenthis optimum operating condition can be achieved with equalto 0° and integer multiples of 180° [5]. The case of equal to0° (or parallel connection) is impractical at microwave frequen-cies. Therefore, in both cases, must be at least 180° whenusing RH-TLs and, hence, the optimum SE-DFDA will bephysically large and have limited bandwidth.

Artificial media with negative refractive index have attractedmuch interest lately [6] and offer new options to solve microwave-transmission problems [7]. Okabe et al. replaced the 270° RH-TLof a rat-race hybrid with essentially a left-handed transmission line(LH-TL), resulting in significant reduction in circuit size andincreased bandwidth [7]. A left-handed transmission line (LH-TL)has a positive insertion phase as opposed to negative insertionphase, as in the case of the RH-TL [7]. In this work, we use both

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 44, No. 6, March 20 2005 527

Optics Communications 234 (2004) 35–42

www.elsevier.com/locate/optcom

A limitation of the Bruggeman formalism for homogenization

Tom G. Mackay a,*, Akhlesh Lakhtakia b,1

a School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh EH9 3JZ, UKb CATMAS – Computational and Theoretical Materials Sciences Group, Department of Engineering Science and Mechanics,

Pennsylvania State University, University Park, PA, 16802–6812, USA

Received 1 December 2003; received in revised form 27 January 2004; accepted 4 February 2004

Abstract

The Bruggeman formalism provides an estimate of the effective permittivity of a particulate composite medium

comprising two component mediums. The Bruggeman estimate is required to lie within the Wiener bounds and the

Hashin–Shtrikman bounds. Considering the homogenization of weakly dissipative component mediums characterized

by relative permittivities with real parts of opposite signs, we show that the Bruggeman estimate may not be physically

reasonable when the component mediums are weakly dissipative; furthermore, both the Wiener bounds and the Ha-

shin–Shtrikman bounds exhibit strong resonances.

� 2004 Elsevier B.V. All rights reserved.

PACS: 42.25.Dd; 42.70.)a; 05.40.)a

Keywords: Homogenization; Negative permittivity; Bruggeman formalism; Maxwell Garnett formalism; Hashin–Shtrikman bounds;

Wiener bounds

1. Introduction

Metamaterials in the form of particulate com-

posite mediums are currently of considerable sci-

entific and technological interest [1]. Provided thatwavelengths are sufficiently long compared with

the length scales of inhomogeneities, such a

metamaterial may be envisaged as a homogenized

* Corresponding author. Tel.: +44-131-650-5058; fax: +44-

131-650-6553.

E-mail addresses: [email protected] (T.G. Mackay),

[email protected] (A. Lakhtakia).1 Fax: +1-814-865-9974.

0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reservdoi:10.1016/j.optcom.2004.02.007

composite medium (HCM), arising from two ho-

mogeneous component mediums [2,3]. HCMs with

especially interesting properties may be conceptu-

alized if the real parts of the relative permittivities

(and/or relative permeabilities) of the two com-ponent mediums have opposite signs [4]. This

possibility arises for metal-in-insulator dielectric

composites [5,6] and has recently become feasible

with the fabrication of dielectric–magnetic mate-

rials displaying a negative index of refraction in

the microwave frequency range [7,8].

Over many years, several theoretical formalisms

have been developed in order to estimate the ef-fective constitutive parameters of particulate

ed.

mail to: [email protected]

36 T.G. Mackay, A. Lakhtakia / Optics Communications 234 (2004) 35–42

composite mediums [2]. In particular, the Maxwell

Garnett and the Bruggeman homogenization for-

malisms have been widely used [9]. Generally, the

Maxwell Garnett formalism is seen to hold only

for dilute composite mediums [10]. 2 More widely

applicable is the Bruggeman formalism that wasinitially founded on the intuition that the total

polarization field is zero throughout the HCM

[16]. A rigorous basis for the Bruggeman formal-

ism is also available, within the framework of the

strong-permittivity-fluctuation theory (SPFT)

[17,18].

Estimates of HCM constitutive parameters

generated by homogenization formalisms may berequired to lie within certain bounds. In particular,

the Wiener bounds [19,20] and the Hashin–

Shtrikman bounds [21] are often invoked. The

Hashin–Shtrikman bounds coincide with the con-

stitutive parameter estimates of the Maxwell

Garnett homogenization formalism [20]. The ap-

plicability of theoretical bounds on the HCM

permittivity has recently been the focus of atten-tion for composites specified by relative permit-

tivities with positive-valued real parts [22].

In this communication, we consider the appli-

cation of the Bruggeman formalism, together with

the Wiener and Hashin–Shtrikman bounds, to

isotropic dielectric HCMs which arise from com-

ponent mediums characterized by complex-valued

relative permittivities whose real parts have op-posite signs. This is scenario is typical of metal-in-

insulator HCMs [20,23], for example. By duality,

our analysis extends to isotropic magnetic HCMs.

It also extends to isotropic dielectric–magnetic

2 The restriction on the applicability of the Maxwell Garnett

formalism to dilute composite mediums generally emerges from

comparison with experimental data [9]. As the particulate

volume fraction increases, the distribution of particles begins to

lose the randomness which is inherent to the theory of the

Maxwell Garnett formalism [11]. However, the restriction could

be bypassed if the distribution of particles in a composite

medium continues to lack order even under densification, which

thought underlies the random unit cell approach developed by

Smith and colleagues [12–14]. An anonymous reviewer has

suggested that self-assembly techniques [15] can yield random-

ness even at large particulate volume fractions, and could

therefore extend the applicability of the Maxwell Garnett

formalism.

HCMs, because the permeability and the permit-

tivity are then independent of each other in the

Bruggeman formalism [24] (as also in the Maxwell

Garnett formalism [25]). Therefore, our findings

are very relevant to the application of homogeni-

zation formalisms [4] to mediums displaying neg-ative index of refraction [26], for example.

Furthermore, the implications of our mathemati-

cal study extend beyond the Bruggeman formalism

to the SPFT as well [3].

A note on notation: an expð�ixtÞ time-depen-dence is implicit in the following sections; and the

real and imaginary parts of complex-valued

quantities are denoted by Reð�Þ and Imð�Þ,respectively.

2. Theory

2.1. Bruggeman equation

Consider the homogenization of two isotropicdielectric component mediums labelled a and b.Let their relative permittivities be denoted by eaand eb, respectively. For later convenience, wedefine

d ¼eaeb

if ea; eb 2 R;Re eað ÞRe ebð Þ if ea; eb 2 C:

(ð1Þ

The Bruggeman estimate of the HCM relative

permittivity, namely eBrHCM, is provided implicitlyvia the relation [9]

eBrHCM ¼faea eb þ 2eBrHCM

� �þ fbeb ea þ 2eBrHCM

� �fa eb þ 2eBrHCMð Þ þ fb ea þ 2eBrHCMð Þ

; ð2Þ

wherein fa and fb are the respective volume frac-tions of component mediums a and b, and theparticles of both component mediums are assumed

to be spherical. The Bruggeman equation (2)

emerges naturally within the SPFT framework [3].

A rearrangement of (2) gives the quadratic

equation

2 eBrHCM� �2þ eBrHCM ea fbð½ �2faÞþ eb fað �2fbÞ�� eaeb ¼ 0:

ð3Þ

Only those eBrHCM-solutions of (3) are valid underthe principle of causality as encapsulated by the

T.G. Mackay, A. Lakhtakia / Optics Communications 234 (2004) 35–42 37

Kramers–Kronig relations [28] which conform to

the restriction ImðeBrHCMÞP 0.Let D be the discriminant of the quadratic

equation (3); i.e.

D ¼ ea fbð½ � 2faÞ þ eb fað � 2fbÞ�2 þ 8eaeb: ð4ÞSince fb ¼ 1� fa, we may express (4) as

D ¼ 9f 2a eað � ebÞ2 � 6fa eað � ebÞ eað � 2ebÞ

þ eað þ 2ebÞ2: ð5Þ

An insight into the applicability of the Bruggeman

formalism may be gained by considering the fa-roots of the equation D ¼ 0; these are as follows:

fajD¼0 ¼ea � 2eb � 2

ffiffiffi2

p ffiffiffiffiffiffiffiffiffiffiffi�eaebp3ðea � ebÞ

: ð6Þ

On restricting attention to nondissipative compo-

nent mediums (i.e., ea;b 2 R), it is clear that fajD¼0are complex-valued if d > 0. Consequently, D > 0which implies that eBrHCM 2 R. On the other hand,fajD¼0 are real-valued if d < 0. Thus, the Brugg-eman estimate eBrHCM for d < 0 may be complex-valued with nonzero imaginary part, even though

neither component medium is dissipative.

2.2. Bounds on the HCM relative permittivity

Various bounds on the HCM relative permit-

tivity have been developed. Two of the most

widely used are the Wiener bounds [19,20]

Wa ¼ faea þfbeb

� ��1Wb ¼ faea þ fbeb

); ð7Þ

and the Hashin–Shtrikman bounds [21]

HSa ¼ eb þ 3faeb ea�ebð Þeaþ2eb�fa ea�ebð ÞHSb ¼ ea þ 3fbea eb�eað Þebþ2ea�fb eb�eað Þ

): ð8Þ

While both the Wiener bounds and the Hashin–

Shtrikman bounds were originally derived for real-

valued constitutive parameters, generalizations to

complex-valued constitutive parameters have been

established [23].The Hashin–Shtrikman bound HSa is equiva-

lent to the Maxwell Garnett estimate of the HCM

relative permittivity eMGHCM=a based on sphericalparticles of component medium a embedded in the

host component medium b. Similarly, HSb isequivalent to the Maxwell Garnett estimate of

the HCM relative permittivity eMGHCM=b based onspherical particles of component medium b em-bedded in the host component medium a. The es-timate eMGHCM=a is valid for fa K 0:3, whereas theestimate eMGHCM=b is valid for fb K 0:3; but see thefootnote in Section 1.

To gain insights into the asymptotic behaviour

of the Wiener and Hashin–Shtrikman bounds, let

us again restrict attention to the case of nondissi-

pative component mediums (i.e., ea;b 2 R). From(7), we see that Wb remains finite for all values of d,but Wa may become infinite for d < 0 since

Waj ! 1 as d ! �fafb: ð9Þ

In a similar vein, from (8) we find that

jHSaj ! 1 as d !fb � 3fb

; ð10Þ

thus, for all values of d < �2 there exists a valueof fb 2 ð0; 1Þ at which HSa is unbounded.Analogously,

jHSbj ! 1 as d !fa

fa � 3; ð11Þ

so we can always find a value of fa 2 ð0; 1Þ atwhich HSb is unbounded, provided that d 2ð� 1

2; 0Þ.

3. Numerical results

Let us now present calculated values of the

HCM relative permittivity eBrHCM, along with thecorresponding values of the bounds Wa;b and HSa;b,for some representative examples. Both nondissi-pative and dissipative HCMs are considered for

d ¼ �3.

3.1. Nondissipative component mediums

The effects of dissipation may be very clearly

appreciated through first considering the idealized

situation wherein the components mediums are

nondissipative [27]. Furthermore, although the

absence of dissipation is unphysical due to the

Fig. 2. The real (top) and imaginary (bottom) parts of the

Bruggeman estimate eBrHCM (solid line) plotted against fa forea ¼ �6 and eb ¼ 2. Also plotted are the real parts of theWiener bounds, Wa (thick dashed line) and Wb (thin dashedline), and the Hashin–Shtrikman bounds, HSa (thick broken

dashed line) and HSb (thin broken dashed line). The imaginary


dictates of causality [28], weak dissipation in a

particular spectral regime is definitely possible and

is then often ignored [9, Section 2.5].

Thus, it is instructive to begin with the com-

monplace scenario wherein both ea > 0 and eb > 0.For example, let ea ¼ 6 and eb ¼ 2. In Fig. 1, eBrHCMis plotted against fa, along with the correspondingWiener bounds Wa;b and Hashin–Shtrikmanbounds HSa;b. The latter bounds are stricter than

the former bounds in the sense that

Wa < HSa < eBrHCM < HSb < Wb: ð12ÞThe close agreement between eBrHCM and the lowerHashin–Shtrikman bound HSa at low volume

fractions fa is indicative of the fact thatHSa � eMGHCM=a. Similarly, eBrHCM agrees closely withthe upper Hashin–Shtrikman bound HSb at high

values of fa since HSb � eMGHCM=b.A markedly different situation develops if the

real-valued ea and eb have opposite signs. For ex-ample, the values of eBrHCM calculated for ea ¼ �6and eb ¼ 2 are graphed against fa in Fig. 2, to-gether with the corresponding Wiener and Ha-

shin–Shtrikman bounds. The Bruggeman estimateeBrHCM is complex-valued with nonzero imaginarypart for fa K 0:82. This estimate is not physicallyreasonable. The Bruggeman homogenization for-

malism – unlike the SPFT which is its natural

generalization – has no mechanism for taking co-

Fig. 1. The Bruggeman estimate eBrHCM (solid line) plottedagainst fa for ea ¼ 6 and eb ¼ 2. Also plotted are the Wienerbounds, Wa (thick dashed line) and Wb (thin dashed line), andthe Hashin–Shtrikman bounds, HSa (thick broken dashed line)

and HSb (thin broken dashed line).

parts of Wa;b and HSa;b are null-valued.

herent scattering losses into account. Furthermore,

no account has been taken in the Bruggeman

equation (2) for the finite size of the particles[11,29,30]. Therefore, the Bruggeman estimate of

the HCM relative permittivity is required to

be real-valued if the component mediums are

nondissipative.

While eBrHCM in Fig. 2 is complex-valued, theWiener bounds and the Hashin–Shtrikman

bounds are both real-valued. In accordance with

(9), we see that jWaj ! 1 as fa ! 34. Similarly,jHSaj ! 1 in the limit fa ! 14, as may be antici-pated from (10). Furthermore, since HSa � eMGHCM=a,the Maxwell Garnett formalism is clearly inap-

propriate here. We also observe that the inequali-

ties (12) which hold for d > 0, do not hold ford < 0.


3.2. Weakly dissipative component mediums

Let us now investigate eBrHCM and its associatedbounds when the component mediums are dissi-

pative; i.e., ea;b 2 C. We begin with those cases forwhich d > 0: for example, we take ea ¼ 6þ 0:3iand eb ¼ 2þ 0:2i. In Fig. 3, eBrHCM is plotted againstfa, and the associated Wiener bounds Wa;b and theHashin–Shtrikman bounds HSa;b are also pre-

sented. The behaviour of the real parts of eBrHCM,Wa;b and HSa;b closely resembles that displayed inthe nondissipative example of Fig. 1. In fact, the

following generalization of (12) holds:

Re Wað Þ < Re HSað Þ < Re eBrHCM� �

< Re HSb� �

< Re Wb� �

: ð13Þ

However, this ordering (13) does not extend to the

imaginary parts of eBrHCM, Wa;b and HSa;b.

Fig. 3. The real (top) and imaginary (bottom) parts of the

Bruggeman estimate eBrHCM (solid line) plotted against fa forea ¼ 6þ 0:3i and eb ¼ 2þ 0:2i. Also plotted are the Wienerbounds, Wa (thick dashed line) and Wb (thin dashed line), andthe Hashin–Shtrikman bounds, HSa (thick broken dashed line)

and HSb (thin broken dashed line).

Turning to the cases for d < 0, we let ea ¼�6þ 0:3i and eb ¼ 2þ 0:2i, for example. Thecorresponding Bruggeman estimate eBrHCM isgraphed as function of fa, along with the Wienerbounds Wa;b and the Hashin–Shtrikman boundsHSa;b in Fig. 4. Since Im ea;bð Þ 6¼ 0, the real parts ofWa and HSa remain finite, unlike in the corre-sponding nondissipative scenario presented in

Fig. 2.

However, the real and imaginary parts of Waand HSa exhibit strong resonances in the vicinity

of fa ¼ 34 (for Wa) and fa ¼ 14 (for HSa). These res-onances become considerably more pronounced if

the degree of dissipation exhibited by the compo-nent mediums is reduced. For example, in Fig. 5

the graphs corresponding to Fig. 4 are reproduced

for ea ¼ �6þ 0:003i and eb ¼ 2þ 0:002i. We ob-serve in particular that ImðeBrHCMÞ > 1 for0:05K fa K 0:8. Thus, the Bruggeman estimateeBrHCM vastly exceeds both the Wiener bounds Wa;band the Hashin–Shtrikman bounds HSa;b for a

Fig. 4. As Fig. 3 but for ea ¼ �6þ 0:3i and eb ¼ 2þ 0:2i.

Fig. 6. As Fig. 3 but for ea ¼ �6þ 3i and eb ¼ 2þ 2i.Fig. 5. As Fig. 3 but for ea ¼ �6þ 0:003i and eb ¼ 2þ 0:002i.


wide range of fa. Since Imðea;bÞ6 0:003, the esti-mates of ImðeBrHCMÞ are clearly unreasonable.

Furthermore, since the real and imaginary parts

of HSa � eMGHCM=a exhibit sharp resonances atfa ¼ 14, we may infer that the Maxwell Garnettformalism is inapplicable for d < 0.

3.3. Highly dissipative component mediums

On comparing Figs. 4 and 5, we conclude that

the Bruggeman formalism, the Weiner bounds and

the Hashin–Shtrikman bounds become increasinginappropriate as the degree of dissipation de-

creases towards zero. This means that all three

could be applicable rather well when the dissipa-

tion is not weak.

Therefore, let us examine the scenario wherein

the real and imaginary parts of the relative per-

mittivities of the component medium are of the

same order of magnitude; i.e., we take ea ¼�6þ 3i and eb ¼ 2þ 2i. The corresponding plotsof the Bruggeman estimate eBrHCM together with the

Wiener bounds Wa;b and the Hashin–Shtrikmanbounds HSa;b are presented in Fig. 6. The real and

imaginary parts of the Bruggeman estimate are

physically plausible, and both lie within the Ha-shin–Shtrikman bounds. The Hashin–Shtrikman

bounds themselves do not exhibit resonances, and

the Weiner bounds do not exhibit strong reso-

nances. Accordingly, we conclude that many pre-

viously published results are not erroneous, but

caution is still advised.

4. Discussion

The Bruggeman homogenization formalism is

well-established in the context of isotropic dielectric

HCMs, as well as more generally [2]. However, this

formalism was shown in Section 3.2 to be inappli-

cable for HCMs which arise from two isotropic

dielectric component mediums, characterized byrelative permittivities ea and eb, with:


(i) ReðeaÞ and ReðebÞ having opposite signs; and(ii) jReðea;bÞj � jImðea;bÞj.

Since the Bruggeman formalism provides the

comparison medium which underpins the SPFT, it

may be inferred that the SPFT is likewise not ap-plicable to the scenarios of (i) with (ii).

It is also demonstrated in Section 3.2 that both

the Wiener bounds and the Hashin–Shtrikman

bounds can exhibit strong resonances when the

component mediums are characterized by (i) with

(ii). In the vicinity of resonances, these bounds

clearly do not constitute tight bounds on the HCM

relative permittivity. As a direct consequence, theMaxwell Garnett homogenization formalism, like

the Bruggeman homogenization formalism, is in-

applicable to the scenarios of (i) with (ii). This

limitation also extends to the recently developed

incremental [31] and differential [32] variants of the

Maxwell Garnett formalism.

If the component mediums are sufficiently dissi-

pative then the Bruggeman formalism and the Ha-shin–Shtrikman bounds (and therefore also the

Maxwell Garnett formalism) provide physically

plausible estimates, despite the real parts of the

component medium relative permittivities having

opposite signs – as shown in Section 3.3. The explicit

delineation of the appropriate parameter range(s) for

the Bruggeman formalism and the Hashin–Shtrik-

man bounds is a matter for future investigation.Bounds can, of course, be violated by a for-

malism if the underlying conditions for the for-

malism are in conflict with those used for deriving

the bounds. Sihvola [22] has catalogued the fol-

lowing conflicts:

(a) Bounds derived for nondissipative component

mediums can be invalid for the real parts of ei-

ther eMGHCM=a or eMGHCM=b for a composite medium

containing dissipative component mediums.

(b) Percolation cannot be captured by the Max-

well Garnett formalism [5,33]. Hence, the Ha-

shin–Shtrikman bounds, being based on the

Maxwell Garnett formalism, can be violated

by the Bruggeman estimate eBrHCM for a percola-tive composite medium.

(c) The derivations of bounds generally assumethat the particles in a composite medium have

simple shapes. If the particle shapes are com-

plicated, the composite medium may display

properties not characteristic of the either of

the component mediums. For instance, mag-

netic properties can be displayed when the par-

ticles in a composite medium have complex

shapes [34,35], even though the componentmediums are nonmagnetic. Clearly, the mag-

netic analogs of Wa, Wb, HSa and HSb are theninapplicable.

(d) Wa, Wb, HSa, and HSb as well as their magneticanalogs are also invalid prima facie when the

component mediums exhibit magnetoelectric

properties [3,11,36].

(e) Bounds derived for electrically small particlesbecome inapplicable with increasing frequency,

due to the emergence of finite-size effects [29].

Even the concept of homogenization becomes

questionable with increasing electrical size [2,

p. xiii].

In contrast, the bounds and the homogeniza-

tion formalisms studied in this paper share the

same premises; yet, a conflict arises in certain sit-uations because the bounds exhibit resonance

while the homogenization estimates do not.

5. Concluding remarks

As several conventional approaches to homog-

enization are not appropriate to the HCMs arisingfrom component mediums characterized by (i)

with (ii), there is a requirement for new theoretical

techniques to treat this case. This requirement is

all the more pressing, given the growing scientific

and technological importance of new types of

metamaterials [1,26].

Acknowledgements

We thank two anonymous reviewers for com-

ments that led to the improvement of this paper.

References

[1] R.M. Walser, in: W.S. Weiglhofer, A. Lakhtakia (Eds.),

Introduction to Complex Mediums for Optics and Elec-

tromagnetics, SPIE, Bellingham, WA, 2003, p. 295.


[2] A. Lakhtakia (Ed.), Selected Papers on Linear Optical

Composite Materials, SPIE, Bellingham, WA, 1996.

[3] T.G. Mackay, in: W.S. Weiglhofer, A. Lakhtakia (Eds.),


tromagnetics, SPIE, Bellingham, WA, 2003, pp. 317–345.

[4] A. Lakhtakia, Int. J. Infrared Millim. Waves 23 (2002) 813.

[5] J.A. Sherwin, A. Lakhtakia, B. Michel, Opt. Commun. 178

(2000) 267.

[6] T.G. Mackay, A. Lakhtakia, W.S. Weiglhofer, Opt.

Commun. 197 (2001) 89.

[7] R.A. Shelby, D.R. Smith, S. Schultz, Science 292 (2001) 77.

[8] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser,

S. Schultz, Phys. Rev. Lett. 84 (2000) 4184.

[9] L. Ward, The Optical Constants of Bulk Materials and

Films, second ed., Adam Hilger, Bristol, UK, 1995

(Chapter 8).

[10] J.C. Maxwell Garnett, Phil. Trans. R. Soc. Lond. A 203

(1904) 385 (reproduced in [2]).

[11] A. Lakhtakia, B. Shanker, Int. J. Appl. Electromag. Mater.

4 (1993) 65.

[12] G.B. Smith, J. Phys. D: Appl. Phys. 10 (1977) L39.

[13] G.A. Niklasson, C.G. Granqvist, O. Hunderi, Appl. Opt.

20 (1981) 26.

[14] G.B. Smith, in: W.S. Weiglhofer, A. Lakhtakia (Eds.),


tromagnetics, SPIE, Bellingham, WA, 2003, pp. 421–446.

[15] S. Zhang, Mater. Today 6 (5) (2003) 20.

[16] D.A.G. Bruggeman, Ann. Phys. Lpz. 24 (1935) 636

(reproduced in [2]).

[17] L. Tsang, J.A. Kong, Radio Sci. 16 (1981) 303 (reproduced

in [2]).

[18] T.G. Mackay, A. Lakhtakia, W.S. Weiglhofer, Phys. Rev.

E 62 (2000) 6052, Erratum 63 (2001) 049901.

[19] O. Wiener, Abh. Math.-Phys. Kl. S€achs. 32 (1912) 507

(Reproduced in [2]).

[20] D.E. Aspnes, Am. J. Phys. 50 (1982) 704 (Reproduced in

[2]).

[21] Z. Hashin, S. Shtrikman, J. Appl. Phys. 33 (1962) 3125.

[22] A.H. Sihvola, IEEE Trans. Geosci. Remote Sens. 40 (2002)

880.

[23] G.W. Milton, Appl. Phys. Lett. 37 (1980) 300.

[24] R.D. Kampia, A. Lakhtakia, J. Phys. D: Appl. Phys. 25

(1992) 1390.

[25] A. Lakhtakia, Int. J. Electron. 73 (1992) 1355.

[26] A. Lakhtakia, M.W. McCall, W.S. Weiglhofer, in: W.S.

Weiglhofer, A. Lakhtakia (Eds.), Introduction to Complex

Mediums for Optics and Electromagnetics, SPIE, Belling-

ham, WA, 2003, pp. 347–363.

[27] H.C. van de Hulst, Light Scattering by Small Particles,

Dover, New York, 1981 (Chapter 10).

[28] C.F. Bohren, D.R. Huffman, Absorption and Scattering of

Light by Small Particles, Wiley, NewYork, 1983, Sec. 2.3.2.

[29] M.T. Prinkey,A. Lakhtakia, B. Shanker,Optik 96 (1994) 25.

[30] B. Shanker, J. Phys. D: Appl. Phys. 29 (1996) 281.

[31] A. Lakhtakia, Microw. Opt. Technol. Lett. 17 (1998) 276.

[32] B. Michel, A. Lakhtakia, W.S. Weiglhofer, T.G. Mackay,

Compos. Sci. Technol. 61 (2001) 13.

[33] A. Sihvola, S. Saastamoinen, K. Heiska, Remote Sens.

Rev. 9 (1994) 39.

[34] A. Lakhtakia, Beltrami Fields in Chiral Media, World

Scientific, Singapore, 1994 (Chapter 3).

[35] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart,

IEEE Trans. Microw. Theory Tech. 47 (1999) 2075.

[36] B. Michel, in: O.N. Singh, A. Lakhtakia (Eds.), Electro-

magnetic Fields in Unconventional Materials and Struc-

tures, Wiley, New York, 2000, p. 39.

DC � 10 logPinduced (victim)Pradiated (source)

� 20 logIinduced (victim)Iradiated (source)

�in matched condition� (1)

The decoupling coefficient for the upright arrangement is �33.902dB. The maximally reduced decoupling levels for each arrange-

ment type are as follows; Type A—8th arrangement (Case 8):�36.503 dB; Type B—7th arrangement (Case 17): �65.849 dB;and Type C—9th arrangement (Case 29): �49.771 dB. Decou-pling coefficients for each case are shown in Figure 4. Type B isthe most coupling-reducible arrangement. If the acceptable cou-pling margin is set to �45 dB, case 15, 16, 17, 18, 19, 20, 27, 28,29, or 30 is preferred. As remarked before, the more decouplinglevels are decreased, the more their radiation patterns distort.Selected antenna radiation patterns for each type are shown inFigure 5.

Decoupling levels are calculated under the conjugate-matchedconditions. Input impedances of the source (of the stern side)antenna are shown in Figure 6. Load impedances of the victim (ofthe bow side) antenna are shown in Figure 7.

To show the reliability of the simulation code, some measure-ments of antenna radiation patterns are shown in Figure 8. Radi-ation-pattern measurements are performed in POSTECH anechoicchamber. There are some mismatches in the stern side because ofthe presence of connecting BNC cable on it.

CONCLUSIONS

To reduce the coupling levels between the source and victimantennas, 30 cases of the slanted antenna arrangement are analyzedwith the moment-method code. Each arrangement produces thecoupling coefficient, antenna radiation pattern, and antenna imped-ances for the conjugate-matching condition. For Type A, thedegree of radiation pattern distortion is quite large, and the cou-pling reduction effect is low. For Type B, the degree of radiationpattern distortion is small, and the coupling reduction effect isextremely large. For Type C, the degree of radiation patterndistortion is small, and the coupling reduction effect is good.

If a large coupling reduction is required, an antenna arrange-ment like that of cases 16, 17, or 18 is suggested. If a moderatequantity of coupling reduction with a negligible radiation patterndistortion is required, an antenna arrangement likr that of case 28,29, or 30 is suggested.

REFERENCES

1. E.K. Miller, G.J. Burke, Low-frequency computational electromagnet-ics for antenna analysis, Proc IEEE 80 (1992).

2. C.W. Trueman, S.J. Kubina, Fields of complex surfaces using wire gridmodeling, IEEE Trans Magn MAG-27 (1991).

3. F. Schlagenhaufer, T. Klook, Investigation of the coupling betweenHF-antennas on complex structures, 1998 IEEE Int Symp on Electro-magnetic Compatibility, Vol. 2, 1998.


BRAGG–PIPPARD FORMALISM FORBIANISOTROPIC PARTICULATECOMPOSITES

Joseph A. Sherwin and Akhlesh LakhtakiaCATMAS—Computational & Theoretical Materials Sciences GroupDepartment of Engineering Science and MechanicsPennsylvania State UniversityUniversity Park, Pennsylvania 16802-6812

Received 4 October 2001

ABSTRACT: Here the Bragg-Pippard (BP) formalism is extended inorder to homogenize linear bianisotropic-in-bianisotropic particulatecomposites, assuming the inclusion particles to be ellipsoidal. The BP

Figure 8 Comparison between simulation and experimental results

40 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 33, No. 1, April 5 2002

formalism yields satisfactory results as the inclusion volume fractionf 3 1, unlike the Maxwell Garnett (MG) formalism but like the Brugge-man formalism. However, the BP formalism is asymmetric with respectto the two component mediums in the same way as the MG formalismis, but the Bruggeman formalism can be symmetric. © 2002 Wiley Peri-odicals, Inc. Microwave Opt Technol Lett 33: 40–44, 2002; DOI10.1002/mop.10225

Key words: bianisotropic composites; homogenization; Maxwell Gar-nett formalism; Bruggeman formalism; Bragg–Pippard formalism

1. INTRODUCTION

Two depolarization dyadics enter the Maxwell Garnett (MG) for-malism for the homogenization of dielectric-in-dielectric particu-late composites [1]: the first for an exclusion region that is con-formal with the particulate inclusions, and the second for aspherical Lorentzian cavity. Both depolarization dyadics involvethe dielectric properties of the host medium. When the inclusionsare spherical, the Maxwell Garnett formalism yields well-behaved

results as the inclusion volume fraction f 3 0, 1. When theinclusions are not spherical, the formalism fails in the limit f 3 1.This shortcoming was remedied by Bragg and Pippard [2], whomade the second depolarization identical to the first. The Bragg–Pippard (BP) formalism has its dedicated followers, chiefly amongthin-film modelers—see, for example, the recent book of Hodgkin-son and Wu [3] on columnar thin films.

In this communication, the Bragg–Pippard formalism is ex-tended to homogenize very general bianisotropic-in-bianisotropicparticulate composites, assuming the inclusion particles to beellipsoidal. A comparison with the results yielded by the MG andthe Bruggeman formalisms [4] is also presented for two examplecomposites. An exp(�i�t) time dependence is implicit through-out, where � is the angular frequency. A remark about notation: 3vectors (6 vectors) are in normal (bold) face and underlined,whereas 3 � 3 dyadics (6 � 6 dyadics) are in normal (bold) faceand underlined twice.

2. THEORY

Let the chosen composite consist of identical, parallel, ellipsoidalinclusions made of a homogeneous medium with constitutiverelations

Figure 1 Estimated components of �� p obtained with the use of theBragg–Pippard ( p � BP), the Maxwell Garnett ( p � MG), and theBruggeman ( p � Br) formalisms. For the MG and the BP formalisms,the inclusions are made of a ferrite and the host medium is isotropic chiral;see (10) and (11) for the constitutive properties. The inclusion shape dyadicU�� s is given by (12), whereas U�� v � U�� s for the Bruggeman estimates

Figure 2 Same as Figure 1, but for the estimated components of ��p

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 33, No. 1, April 5 2002 41

D� � �� s � E� � ��s � H� ,

B� � ��s � E� � ��

s � H� , (1)

randomly embedded in a host medium with constitutive relations

D� � �� v � E � ��v � H�

B� � ��v � E� � ��

v � H� . (2)

Here, �� p and �� p ( p � s, v), are the permittivity and the perme-ability 3 � 3 dyadics, whereas ��

p and ��p are the two magneto-

electric 3 � 3 dyadics, all functions of the angular frequency �. Itis convenient to define the 6 � 6 constitutive dyadics

C��p � �� p �� p�� p �� p� , p � s, v, (3)

when dealing with bianisotropic mediums.Each ellipsoidal inclusion is described with the help of a shape

dyadic U��s, which is real and of rank 3 [5]. The 6 � 6 polarizability

dyadic of the inclusion in the host medium is delineated per unitvolume by [4]

a�� s/v � �C�� s � C�� v� � �I�� i�D�� s/v � �C�� s � C�� v��1, (4)

where I�� is the 6 � 6 identity dyadic. D�� s/v is the 6 � 6 depolar-ization dyadic of an infinitesimally small exclusion region in thehost medium, the shape of this region being given by U��

s also; seeWeiglhofer, Lakhtakia, and Michel [4] for a detailed exposition ofD�� s/v.

The Maxwell Garnett estimate of the 6 � 6 constitutive dyadicof the homogenized composite turns out to be as follows:

C�� MG � C�� v � fa�� s/v � �I�� i�f D̃��s/v � a�� s/v��1. (5)

Here, the depolarization dyadic D̃��s/v is computed after replacing

U��s by the 3 � 3 identity dyadic I�� in the expression of D�� s/v.

Figure 3 Same as Figure 1, but for the estimated components of ��p Figure 4 Estimated components of ��p obtained with the use of the

Bragg–Pippard ( p � BP), the Maxwell Garnett ( p � MG), and theBruggeman ( p � Br) formalisms. For the MG and the BP formalisms, theinclusions are made of an isotropic chiral medium and the host medium isa ferrite; see (10) and (11) for the constitutive properties. The inclusionshape dyadic U�� s is given by (12), whereas U�� v � U�� s for the Bruggemanestimates


In an extension of the idea first laid out by Bragg and Pippard[2] the Bragg–Pippard estimate of the 6 � 6 constitutive dyadic ofthe homogenized composite is defined as

C�� BP � C�� v � fa�� s/v � �I�� i�fD�� s/v � a�� s/v��1. (6)

Finally, the Bruggeman estimate C�� Br of the 6 � 6 constitutivedyadic of the homogenized composite is computed by solving the6 � 6 dyadic equation [4, 6]

fa�� s/Br � � f � 1�a��v/Br, (7)

where

a�� s/Br � �C�� s � C�� Br� � �I�� i�D�� s/Br � �C�� s � C�� Br��1, (8)

a�� v/Br � �C�� v � C�� Br� � �I�� i�D�� v/Br � �C�� v � C�� Br��1, (9)

whereas D�� s/Br and D�� v/Br are defined in analogy with D�� s/v. A shapedyadic U��

v has to be decided upon for calculating D�� v/Br, the resultsbeing dependent on the choice made [7]. In the absence of anyother guiding principle, U��

v � U��s here.

3. NUMERICAL RESULTS AND DISCUSSION

In order to illustrate our extension of the BP formalism, (6) waschosen to be implemented on chi-roferrites realized as compositescomprising ferrite and isotropic chiral mediums [8]. The BP esti-mates were compared with the MG and the Bruggeman (Br)estimates.

For both examples, a ferrite medium was chosen with thefollowing constitutive dyadics:

��ferr � 4.8�0I��, ��

ferr � ��ferr � 0�� ,

��ferr � �0�4I�� 2.2u�3u�3 � 1.4i��u�1u�2 � u�2u�1��. (10)

The properties of the chosen isotropic chiral medium are as fol-lows:

��chi � 3.7�1 � 0.2i��0I��, ��

chi � 1.7�0I��,

��chi � ��

chi � 1.4��0.01 � i��0�0I��. (11)

Here, u� 1,2,3 are unit vectors in a Cartesian coordinate system,whereas �0 and �0 are the permittivity and the permeability of freespace (i.e., vacuum).

U�� s � u�1u�1 � 3u�2u�2 � 10u�3u�3 (12)

was chosen as the inclusion shape dyadic. As a result, the forms ofthe constitutive dyadics of the homogenized composite are asfollows:

��p � �0��11

p u� 1u� 1 � �22p u� 2u� 2 � �33

p u� 3u� 3 � �12p �u� 1u� 2 � u� 2u� 1��, (13)

��p � �0��11

p u� 1u� 1 � �22p u� 2u� 2 � �33

p u� 3u� 3 � �12p �u� 1u� 2 � u� 2u� 1��, (14)

��p � ��

p

� i��0�0 �� 11p u�1u�1 � �22p u�2u�2 � �33p u�3u�3 � �12p �u�1u�2 � u�2u�1��, (15)

where p � BP, MG, Br.For the first of two examples, ferrite inclusions embedded in an

isotropic chiral host medium were chosen. Therefore, ��s � ��

ferr, etcetera, and ��

v � ��chi, et cetera. The real and imaginary parts of the

estimated components, ��p, ��

p, and ��p ( p � BP, MG, Br) are

presented in Figures 1–3. For the second example, the roles of thecomponent mediums were reversed for the MG and the BP for-malisms; thus, ��

s � ��chi, et cetera, and ��

v � ��ferr, et cetera. The

real and imaginary parts of the estimated components, ��p, ��

p, and��

p ( p � BP, MG, Br), for isotropic chiral inclusions embedded ina ferrite host medium are presented in Figures 4–6.

The BP formalism clearly preserves the character of the indi-vidual components as f 3 0, 1, as illustrated by Figures 1–6. TheMG formalism only yields this behavior as f 3 0. In general, asf 3 1, the MG formalism diverges considerably from the correctendpoint behavior. Hence, the particular troublesome aspect of theMG formalism that was overcome by Bragg and Pippard fordielectric-in-dielectric composites for nonspherical inclusions issurmountable by the application of their formalism for bianisotro-pic-in-bianisotropic composites also.

Even for spherical inclusions, the MG formalism is asymmetricwith respect to the volume fractions of the component mediums


MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 33, No. 1, April 5 2002 43

[1]: When the inclusion and the host mediums are interchanged butthe respective volume fractions remain invariant, the estimatedconstitutive properties of the homogenized composite change. Thesame asymmetry is highly pronounced when the inclusions arenonspherical. In contrast, (7) clearly shows that the Bruggemanestimates are unaffected by the interchange of the two componentmediums. Comparing Figures 1–3 with Figures 4–6, we observethat the BP formalism is similar to the MG formalism than theBruggeman formalism, in this respect. However, U��

v � U��s been

used, the Bruggeman estimates would have been asymmetric too.

REFERENCES

1. T.C. Choy, Effective medium theory: Principles and applications, Ox-ford University Press, Oxford, 1999; pp. 7–9.

2. W.L. Bragg and A.B. Pippard, The form birefringence of macromole-cules, Acta Crystallogr. 6 (1953), 865–867.

3. I.J. Hodgkinson and Q.H. Wu, Birefringent thin films and polarizingelements, World Scientific, Singapore, 1997, pp. 149–152.

4. W.S. Weiglhofer, A. Lakhtakia, and B. Michel, Maxwell Garnett andBruggeman formalisms for a particulate composite with bianisotropichost medium, Microwave Opt Technol Lett 15 (1997), 263–266 [Erra-tum: 22 (1999), 221].

5. A. Lakhtakia, Orthogonal symmetries of polarizability dyadics of bi-

anisotropic ellipsoids. Microwave Opt Technol Lett 27 (2000), 175–177.

6. T.G. Mackay and W.S. Weiglhofer, Homogenization of biaxial com-posite materials: Bianisotropic properties. J Opt A: Pure Appl Phys 3(2001), 45–52.

7. A. Lakhtakia, B. Michel, and W.S. Weiglhofer, Bruggeman formalismfor two models of uniaxial composite media: Dielectric properties,Comp Sci Technol 57 (1997), 185–196.

8. W.S. Weiglhofer, A. Lakhtakia, and B. Michel, On the constitutiveparameters of a chiroferrite composite medium, Microwave Opt Tech-nol Lett 18 (1998), 342–345.


MICROSTRIP ANTENNA CONTROLLEDBY p-i-n DIODES: INFLUENCE OF THEBIAS CURRENT ON THE ANTENNAEFFICIENCY

J. M. Laheurte, H. Tosi, and J. L. DubardLaboratoire d’ElectroniqueAntennes et TélécommunicationsUniversité de Nice-Sophia Antipolis250 rue Albert Einstein06560 Valbonne, France

Received 5 October 2001

ABSTRACT: This Letter points out the drop in radiation efficiencyobserved in active microstrip antennas with switching capabilities forinadequate values of the bias current. For a frequency agile antennabased on p-i-n diodes, a trade-off between the dc consumption of thediodes and the antenna gain is demonstrated. It is also shown that thistype of integrated active antenna can be reasonably predicted by a sim-ple resistor model of the diode in a TLM analysis. © 2002 Wiley Peri-odicals, Inc. Microwave Opt Technol Lett 33: 44–47, 2002; DOI10.1002/mop.10226

Key words: active antenna, radiation efficiency, p-i-n diodes, TLMmodeling

1. INTRODUCTION

Active integrated antennas (AIAs) can be classified into severalgroups depending on the analog functions inserted in the antenna(oscillation, amplification, mixing, switching, active filtering, etc.[1]) or the application under consideration (e.g., quasi-opticalpower combining, beam-steering array, transponder, Doppler sen-sor, RF front end [2, 3]). An AIA is best defined by the lack oftransmitting line between the radiating element and the activecomponent. As a result, added noise and ohmic losses due to theline are suppressed and the resulting structure is more compact.The first goal of the designer is then to find the proper location forthe component inside the antenna in terms of impedance matchingand electromagnetic perturbation. AIAs are generally based on thehybrid and crafty combination of a component with an originalantenna, mainly for low-cost applications (e.g., identification tags,Doppler sensors) where simplicity and compactness are requiredbut monolithic construction on silicon or GaAs substrates are alsodeveloped [4].

However, when frequency tunable antennas are considered, thenonzero resistance of the active components is often omitted andcan drastically affect the overall performance. The sensitivity ofthe return loss to the diode resistance in the ON state is firststressed in a microstrip antenna showing switching capabilities. A



Int. J. Electron. Commun. (AEÜ) 59 (2005) 348–351

www.elsevier.de/aeue

Size-dependent Bruggeman approach for dielectric–magnetic compositematerials

Akhlesh Lakhtakiaa, Tom G. Mackayb,∗aCATMAS—Computational and Theoretical Materials Science Group, Department of Engineering Science and Mechanics,Pennsylvania State University, University Park, PA 16802-6812, USAbSchool of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, UK

Abstract

Expressions arising from the Bruggeman approach for the homogenization of dielectric–magnetic composite materials,without ignoring the sizes of the spherical particles, are presented. These expressions exhibit the proper limit behavior. Theincorporation of size dependence is directly responsible for the emergence of dielectric–magnetic coupling in the estimatedrelative permittivity and permeability of the homogenized composite material.� 2004 Elsevier GmbH. All rights reserved.

Keywords:Bruggeman approach; Dielectric–magnetic material; Homogenization; Maxwell Garnett approach; Particulate compositematerial; Size dependence

1. Introduction

The objective of this communication is to introduce a size-dependent variant of the celebrated Bruggeman approach[1, Eq. 32]and thereby couple the dielectric and magneticproperties of a particulate composite material (PCM) withisotropic dielectric–magnetic constituent materials.

Homogenization of PCMs has been a continuing themein electromagnetism for about two centuries[2]. The mostpopular approaches consider the particles to be vanis

incremental and di•erential maxwell garnett formalisms for bi-anisotropic...

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