Metamaterials 3 (2009) 10–27

Available online at www.sciencedirect.com

Invited paper

Metamaterial made of paired planar conductors: Particleresonances, phenomena and properties

G. Donzelli a,d, A. Vallecchi a, F. Capolino b,∗, A. Schuchinsky c

a Department of Information Engineering, University of Siena, Via Roma 56, 53100 Siena, Italyb Department of Electrical Engineering and Computer Science, University of California, Irvine, CA 92697, USA

c Queen’s University Belfast, Belfast BT3 9DT, United Kingdomd Now with Altran Italia, 20154 Milano, Italy

Received 4 April 2008; received in revised form 11 November 2008; accepted 16 December 2008Available online 24 January 2009

Abstract

The properties and characteristics of a recently proposed anisotropic metamaterial based upon layered arrays of tightly coupledpairs of “dogbone” shaped stripe conductors have been explored in detail. It has been found that a metamaterial composed of suchstacked layers exhibits artificial magnetism and may support backward wave propagation. The equivalent network models of theconstitutive conductor pairs arranged in the periodic array have been devised and applied to the identification of the specific typesof resonances, and to the analysis of their contribution into the effective dielectric and magnetic properties of the artificial medium.The proposed “dogbone” configuration of conductor pairs has the advantage of being entirely realizable and assemblable in planar

technology. It also appears more prospective than simple cut-wire or metal-plate pairs because the additional geometrical parametersprovide an efficient control of separation between the electric and magnetic resonances that, in turn, makes it possible to obtain afairly broadband left-handed behaviour of the structure at low frequencies.© 2009 Published by Elsevier B.V.

Metama

Keywords: Artificial magnetism; Coupled conductors; Dogbones;Transmission lines

1. Introduction

Artificial magnetism is considered as one of the majorattributes of metamaterials along with negative refractiveindex (NRI). Periodic arrays of magnetic dipoles emu-lated by resonant loops are often realized with the aidof split ring resonators (SRRs) [1–3]. Combined with

wires, SRRs are used as the constituent particles of arti-ficial media in which effective NRI can be observedfor plane waves of fixed polarization incident at certain

∗ Corresponding author.E-mail address: capolino@dii.unisi.it (F. Capolino).

1873-1988/$ – see front matter © 2009 Published by Elsevier B.V.doi:10.1016/j.metmat.2008.12.001

terials; Microwaves; Negative refractive index; Planar technology;

angles [4,5]. In numerous publications, such media aredescribed as materials with negative permeability andpermittivity.

At optical frequencies, materials made of pairs ofnanorods have been suggested in Ref. [6] to create cur-rent loops resonating in the transmission line mode, i.e.,each pair of rods constitutes a waveguide resonator com-prised of two conductors. It has been shown that a pairof coupled conducting wires may possess two types ofresonances, named electric and magnetic resonances,

and could represent the basic building block of a NRImedium. Although the SRRs and coupled rods havedemonstrated feasibility of producing the NRI responseof the medium, search of novel physical structures for

G. Donzelli et al. / Metamaterials 3 (2009) 10–27 11

Fig. 1. Wire medium (a) and cut-wire medium (b). The frequency dependent effective permittivity at the shown polarization of the incident electricfi n electri( uced mai

tlh[ss

rdpgmasmtnpcatbfipblaaet

eld is characterised by a plasma-like resonance frequency fp and by ac) One layer of SRRs, polarization of incident magnetic field and indn the x–y plane, the SRRs are aligned parallel to the x–z plane.

he constituent particles still remains an acute prob-em. Alternative geometries based on the same principlesave been proposed for microwave frequencies in Refs.7–10], and the planar arrangement of the “dogbone”haped tightly coupled conductor stripes [11–14] is theubject of this paper.

In this work we perform a detailed study of theesonances in the dogbone structure, their types andependences on the geometrical parameters. We alsoropose an accurate transmission line model of a sin-le dogbone particle, which enables us to predict theagnetic resonance frequencies and provides simple

pproximate formulas useful for design purposes. Aingle layer of dogbone pairs is analyzed first. Thenultilayered structures are investigated, and we show

he existence of a frequency band where artificial mag-etism and backward wave propagation exist in theeriodic arrangements with lattice constants signifi-antly smaller than the wavelength. Using the simplelgorithm based on the observation of the transmit-ed wave phase (see, e.g. [15,16]), the metamaterialackward wave propagation characteristics are retrievedrom the scattering parameters of a wave normallyncident on a few stacked layers of arrayed dogboneairs. The metamaterial properties are retrieved alsoy observing the Bloch propagation constant betweenayers that clearly identifies the backward wave prop-

gation bands. We address the readers to [17] forcomprehensive discussion of metamaterial homog-

nization. The results of these analyses provide theheoretical demonstration that the dogbone pair struc-

c resonance frequency fe associated with a standard stop band of FSS.gnetic moments m at the magnetic resonance fm. The layer of SRR is

ture can exhibit a fairly broadband NRI behavior at lowfrequencies.

The paper is organized as follows. In Section 2 weintroduce the main concepts and recall the effectivemedium description of metamaterials made of wires,cut-wires and SRRs. In Section 3 we carry out anextensive parametric analysis and investigate the trans-mission properties of a single layer of dogbone pairson a dielectric substrate and in free-space. In Section4, a transmission line model is introduced and appliedto prediction of the magnetic resonance frequency. InSection 5 we show that metamaterials composed ofseveral layers of dogbone pairs may exhibit backwardwave propagation. Major findings are summarized inSection 6.

2. Background. Electric and magneticresonances in wire and SRR materials

The wire medium, the cut-wire medium, and theperiodic array of SRRs shown in Fig. 1 represent thecanonical types of metamaterials, which are usuallycharacterized by effective permittivity and permeabilityexpressed in terms of the associated plasma frequency,and electric and magnetic resonances. Since the 50’sand 60’s the wire medium shown in Fig. 1(a) has

been described for an incident electric field parallelto the wires as an artificial dielectric [18–20] withan isotropic effective plasma-like relative permittivityεeffr = 1 − f 2

P/f 2, where f is the signal frequency, fp

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12 G. Donzelli et al. / M

is the plasma frequency which depends on the wirediameter and interspacing, see [21–24] (such mediumis assumed to be lossless, which is a reasonable approx-imation at microwave frequencies). Spatial dispersionshould be additionally taken into account when the wavestravel at a slant angle to the wire axes [25,26]. For wavespropagating normally to the wires, and in the absence oflosses, a homogenized wire medium exhibits negativeeffective permittivity at f < fp (see Fig. 1(a)), and suchan artificial dielectric is opaque, whereas for f > fp theeffective permittivity is positive and the artificial dielec-tric is translucent. When the wires are cut, see Fig. 1(b),the wire medium is still described by an effective per-mittivity but εr has different frequency dependence, asexplained in Ref. [24].

The medium made of SRRs, Fig. 1(c), can bealso described by a homogeneous effective perme-ability which is represented in the general formμ

effr = 1 − Af 2/(f 2 − f 2

m − jΓf ) where fm and Γ

stand for the resonance and damping frequencies, respec-tively, and A denotes the resonance amplitude [1,27].

When wires and SRRs are combined in a single struc-ture, the resulting composite medium may exhibit NRI[4,5] entailed by both effective permittivity and perme-ability being negative, and thus supports backward wavepropagation. Indeed, as shown in Fig. 1(c), the SRRsare periodically offset in the x–y plane whilst each SRRlies in the x–z plane, orthogonal to the incident magneticfield H. However, physical realization of such materialinvolves a complex manufacturing process. Though itis not impractical at microwaves, the fabrication andassembling of the SRRs and wire arrays for THz andoptical frequencies, where the most exciting applicationsare expected, face major challenges. In order to allevi-ate these difficulties, an alternative arrangement of theconstitutive resonant particles fully compatible with theplanar technology is proposed in this paper to realizecomposite metamaterials very well-suited for applica-tions at microwave and millimeter-wave frequencies andalso having the potential for expanding to the opticalrange.

This investigation has been inspired by some recentworks [6–10] which suggested that pairs of finite lengthwires could be used as magnetic resonators insteadof SRRs. The wires could also give simultaneously anegative permittivity in the same frequency range andtherefore a NRI, without the need for continuous wires.However, achieving simultaneously the negative perme-

ability and permittivity with pairs of metallic wires issubject to some restrictions, cf. [7,11]. Indeed, the diffi-culty of obtaining NRI with cut-wire pairs rests in the factthat the electric resonance is generally stronger and wider

rials 3 (2009) 10–27

than the magnetic one. Thus, if the electric resonanceof the cut-wires is well above the magnetic resonancefrequency, simultaneous negative permittivity and per-meability cannot be attained. As a consequence, oneusually needs to locate the magnetic resonance within thenegative permittivity region (unless the region of nega-tive permeability can be made wide enough to overlap,at least partially, the negative permittivity region). How-ever, in a simple cut-wire pair, an independent tuning ofthese two resonances is impossible because the frequen-cies of both the electric and the magnetic resonancesare controlled by a single geometrical parameter—thelength of the wires (both frequencies are approximatelyproportional to the inverse of wire length) [8].

A convenient approach to realize NRI response ofshort-strip pairs is to combine them with continuouswires. This provides a large negative permittivity banddue to the plasmonic response of the wire medium, andone can exploit only the negative effective permeabilityproduced by the strip pairs [7].

Another possibility to accomplish an independentvariation of the electric and magnetic resonances is tostrengthen the interaction of the pairs in the adjacentunit cells along the electric field direction. This can beachieved by reducing the distance between the contigu-ous pairs or increasing the strip widths at their ends [11].In general, an additional inter-pair capacitance, resultingfrom this enhanced interaction, influences the electricresonance frequency stronger than the magnetic one.For very closely spaced conductor pairs, the magneticresonance is substantially affected by the capacitancebetween the top and bottom conductors in the pair. Whenthe spacing between the top and bottom conductors in thepair is comparable with the distance between the con-tiguous pairs, the capacitive coupling between the pairshas strong impact on the magnetic resonance, as shownin this paper. Thus modification of the cut-wire topologytoward the dogbone shape, whose lateral arms contributeto both the self- and mutual capacitances, provides anew means for controlling the locations of the electricand magnetic resonance frequencies, so as to make themeven completely overlap and display the NRI behaviourover a broad frequency band [11].

It is necessary to note that the concept of strong cou-pling inside the conductor pairs also underlies operationof the so-called fishnet structure, which was originallyintroduced in Ref. [28]. Indeed, the fishnet design canbe seen as the evolution of the strip-pairs and continuous

wires [7] when the width of the strips is increased untilthey join the infinite wires [8] thus producing a continu-ous connected mesh. The paired fishnets also provide theadditional parameters to control the positions of electric

etamaterials 3 (2009) 10–27 13

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G. Donzelli et al. / M

nd magnetic resonances, and the fishnet proved to ben effective structure for implementation of NRI char-cteristics [28–32]. However, fully isotropic and deepub-wavelength scale designs (required, for example, forhe full exploitation of the superlensing capabilities of

etamaterials) is difficult to achieve with the fishnet con-guration which has a unit cell size typically of an orderf half a wavelength and whose optimal operation haseen demonstrated for normal incidence.

The arrays of dogbone shaped conductors discussedn this paper are free from the aforementioned limitationsnd exhibit the attractive feature of a reduced unit cellize, due to the additional capacitive loading provided byhe dogbone lateral arms. Furthermore, the dogbone pairsepresent the basic blocks of more intricate arrangementshich can handle arbitrary polarizations and oblique

ncident fields. For example, a completely symmetri-al configuration is realized by simply combining tworthogonal dogbone particles to form tightly couplederusalem crosses, which are invariant to linear polar-zation of a wave at normal incidence. Therefore all theesults obtained in this paper for the dogbone pairs atsingle incident polarization can be readily extended to

he Jerusalem cross pairs, and the polarization-dependentRI behaviour of the dogbones enables an isotropic

ffective NRI response of the Jerusalem crosses [33].Finally, it is noteworthy that the NRI properties of

he dogbone pairs are theoretically demonstrated in thisaper for microwave frequencies. But the ease of fabri-ation of this kind of planar metamaterial, using a singleielectric sheet, is expected to facilitate also its scal-ng to THz and optical frequencies, provided that thehange of the metal properties from lossy conductorst microwave frequencies to lossy dielectrics with neg-tive permittivity in the THz range is adequately takennto account. Indeed, these dogbone structures presenthe same level of fabrication complexity as simple cut-ire pairs, which have already been manufactured and

tudied experimentally at optical wavelengths [6].

. Metamaterial in planar technology: thedogbone” particle and its properties

Objective of this work is to investigate a planareriodic structure which has the potential to act as anffective NRI medium. A parametric study has been per-ormed to gain physical insight into the mechanisms ofhe resonance formation and field interactions in the pla-

ar doubly periodic arrays of resonant particles madef pairs of tightly coupled dogbone shaped conductors.he properties of such structures illuminated by a nor-ally incident (along the z-axis) plane wave, as shown in

Fig. 2. Dogbone particle made of a pair of tightly coupled planar con-ductors separated by an insulating dielectric layer. The incident electricfield is polarized along the x-direction.

Fig. 2, are discussed for different geometries of the unitcells. The incident electric field is polarized along thex-direction, parallel to the central main segment of thedogbone. In this paper we aim at investigating the basicproperties of dogbone-based arrays such as resonancefrequency locations and NRI possibilities, and thereforeconductors are assumed lossless, which is a reasonableapproximation at microwave frequencies.

3.1. Dogbone pair of conductors separated by adielectric layer

The properties of an array formed by the dogboneconductor pairs shown in Fig. 2 are investigated throughsimulations of the transmission coefficient |T| for aset of structure parameters which are varied one at atime. The dogbone pairs are assumed to be formed byinfinitesimally thin perfect electric conductors. The sim-ulation results at the following default values (in mm):A = B = 7.5, C = 8, B1 = 4, A2 = 7.4, B2 = 0.8, A1 = 1,H = 0.2, εr = 16, are presented in Figs. 3–11 and dis-cussed in this section. The main purpose of this detailedanalysis is to identify and evaluate the impact of specificgeometric features on the array resonance characteris-tics.

At first, the cell size A (the period) along the x-direction was varied. The plots in Fig. 3(a) illustratethe first two resonances in this structure. In order to

assess the accuracy of the obtained results, the struc-ture with the default parameters was simulated usingboth Ansoft HFSS and CST Microwave Studio, and thetwo sets of simulated data have shown a good agree-

14 G. Donzelli et al. / Metamaterials 3 (2009) 10–27

Fig. 3. Transmission coefficient vs. size A of a unit cell in a doubly periodic planar array composed of dogbone paired conductors (a) shown in Fig. 2,and (b) conductors with short-circuited “arms” (see insert). The electric resonances for both the structures are at fe ≈ 3.9, 6.4, 7.5 GHz, whereas themagnetic resonance at fm ≈ 3.1 GHz in case (a) is suppressed when the lateral arms of the dogbones are short-circuited in case (b).

Fig. 4. Transmission coefficient vs. unit cell size B for a doubly peri-odic planar array composed of dogbone pairs shown in Fig. 2. Theelectric resonance is at fe ≈ 3–4 GHz. The magnetic resonance is atfm ≈ 3.1 GHz.

Fig. 5. Transmission coefficient vs. dogbone length A2 in a doublyperiodic planar array composed of the dogbone pairs shown in Fig. 2.The electric resonances are at fe ≈ 6.8, 8.2, 9.3 GHz. The magneticresonances fm vary between 3 and 4 GHz.

Fig. 6. Transmission coefficient vs. lateral arm length B1 in a dou-bly periodic planar array composed of the dogbone pairs shown inFig. 2. The electric resonance varies in the range fe ≈ 3.2–3.9 GHz.The magnetic resonance varies in the range fm ≈ 2.5–3.1 GHz.

Fig. 7. Transmission coefficient vs. width B2 of the dogbone centralpart in a doubly periodic planar array composed of the dog-bone pairs shown in Fig. 2. The electric resonance varies in therange fe ≈ 4–4.6 GHz. The magnetic resonance varies in the rangefm ≈ 3.3–3.7 GHz.

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ent. The lowest resonance at f ≈ 3.1 GHz appears to beractically unaffected by variations of A, thus indicat-ng that the fields are confined to the close proximity ofhe dogbone pair and are loosely coupled to the adjacentells. This resonance, usually referred to as the mag-etic resonance, is of particular interest in this papers elaborated later. Its frequency fm mainly depends onhe dogbone particle dimensions and is associated withhe so-called transmission line (TL) mode which createsn equivalent magnetic dipole moment. The other reso-ance, denoted as electric resonance, is associated withdipole-like symmetric mode, which creates an electricipole moment. It exhibits a much wider stopband, andts frequency fe (≈3.9, 6.4, 7.5 GHz) strongly depends onhe lattice constant A. This implies that the electric res-nance is rendered by the capacitive coupling betweenhe dogbone particles in the adjacent cells, similarly tohat in the capacitive type of frequency selective surfacesFSS), for example, of dipoles or Jerusalem crosses [34].

In order to verify such a designation of the resonantodes, the same dogbone particle has been simu-

ated when its lateral arms were short-circuited, seeig. 3(b). Comparison of the corresponding plots inig. 3(a) and (b) shows that the magnetic resonancet fm ≈ 3.1 GHz is suppressed in the structure with thehort-circuited arms whilst the electric resonances at feemain nearly unchanged. This confirms that the res-nance at fm ≈ 3.1 GHz in Fig. 3(a) is indeed of theagnetic type, which corresponds to the antisymmetricode with oppositely directed currents flowing on the

op and bottom conductors of each dogbone pair, andhese currents are strongly affected when the arms arehort-circuited. Conversely, the electric resonance at fes weakly perturbed by the short circuits because it isroduced by a symmetric mode whose currents flow inhe same direction on both the top and bottom conductorsf the dogbone particle.

The proposed interpretation of the resonance modes islso supported by the simulation results for variable cellize B (the period along the y-axis). Fig. 4 shows that theagnetic resonance at fm ≈ 3.1 GHz remains unchangedhile the electric resonance slightly shifts in the range

e ≈ 3.4–3.9 GHz because the particle mutual couplingn y-direction is weakly influenced by the size B as longs (B − B1) � H.

Once the resonance types have been identified and theattice constant effect is understood, we can evaluate howhe resonance frequencies are affected by the geometry

f the dogbone itself. The dogbone length A2 along the-direction was varied first, and its effect is illustrated inig. 5 at the fixed cell size A. Both magnetic and electricesonances are displaced now. A smaller shift of the mag-

rials 3 (2009) 10–27 15

netic resonance indicates that its frequency is attributedto the incremental increase of the total coupling betweentop and bottom conductors and self-inductance of thedogbone particles. This effect will be further discussedin connection with variations of other dimensions andlater interpreted with the aid of the TL model describedin Section 4. The electric resonance fe proved to bemore sensitive to the size A2. However it is necessaryto emphasize that the fe shift is predominantly deter-mined by the change of capacitive coupling between theadjacent cells, and to a smaller degree by the dogbonelength. Therefore, when increasing A2, not only does theelectrical length increase (causing a decrease of fm andfe), but also the capacitive coupling between contiguouscells (inversely proportional to A − A2) increases, andthis latter effect is the main cause of the fe shift towardslower frequencies. Indeed, comparison of Figs. 3 and 5shows that the rate of fe changes with A2 and A is nearlythe same (cf. A = 8 and 8.5 mm and A2 = 6.4 and 6.9 mm).

As mentioned in Section 2, the lateral arms of thedogbone particles represent one of the major distinc-tive features of this structure. Therefore the effect of thedogbone “arms” length B1 should be analyzed to gaininsight into the unique properties of the dogbone parti-cles. The simulation results in Fig. 6 demonstrate thatboth the magnetic and electric resonances shift towardslower frequencies at larger B1. However different mech-anisms appear to be responsible for this phenomenon.The lower frequency of the magnetic resonance at thelonger arms’ length B1 is associated with the increaseof the capacitance between the top and bottom parts ofthe dogbone arms (similarly to the case of A2 variationsin Fig. 5). Indeed, it is evident now and will be furtherillustrated in the rest of this section that the magnetic res-onance can be attributed to the resonant TL mode (alsocalled antisymmetric mode) as elaborated in Section 4.The electric resonance, on the other hand, is governed bythe coupling between the particles in the adjacent cellssimilarly to that in the dipole-based FSS [24]. Thereforeincremental increase of B1 has about the same impacton fm as on fe, though the former phenomenon is mainlyassociated with the dogbone elongation, whilst the lat-ter is caused by a larger capacitance to the adjacentcells.

The discussion of the dogbone particle resonances hasso far been concerned with the effects of the dogbone armlengths and the unit cell size. While being instrumental inidentifying the resonance types and features, we could

not distinguish the coupling mechanisms involved. Inorder to gain deeper insight in the nature of these reso-nance interactions, the effect of the stripe widths B2 andA1 should be assessed.

16 G. Donzelli et al. / Metamaterials 3 (2009) 10–27

Fig. 10. Transmission coefficients for the doubly periodic planar arrays

Fig. 8. Transmission coefficient vs. lateral arm width A1 in a dou-bly periodic planar array composed of the dogbone pairs shown inFig. 2. The electric resonance varies in the range fe ≈ 3.5–4.5 GHz.The magnetic resonance varies in the range fm ≈ 2.8–4 GHz.

The characteristics simulated at several widths B2 ofthe dogbone central part are shown in Fig. 7. Inspectionof these plots shows that variation of B2 incurs simulta-neous shift of both electric and magnetic resonances inthe same direction, similarly to that in the case of variableB1, cf. Fig. 6. When the stripe width increases, the induc-tive response of the central stripes weakens (for bothantisymmetric and symmetric modes), and both mag-netic and electric resonances occur at higher frequencies.The qualitative interpretation of this phenomenon is fur-ther corroborated using the TL model of the dogboneparticle in Section 4.

In contrast to B2 variations, the stripe width A1 of

the dogbone lateral arms, at A2 kept unchanged, has dis-tinctively different effect on the magnetic and electricresonances as illustrated in Fig. 8. Namely, when thelateral stripes become narrower, the fm increases and fe

Fig. 9. Transmission coefficient for a doubly periodic planar array composed o(H) of the dielectric spacer separating the top and bottom conductors in a dogbfixed to H = 0.2 mm. (b) Thickness is variable (H = 0.2, 0.6, 1.4, 3) while permat fe ≈ 3.3–3.7 GHz and fm ≈ 2.5–3.1 GHz, respectively.

of dogbone pairs with isolated conductors (Fig. 2) and short-circuitedarms (closed dogbones in insert) in free-space. The electric resonanceoccurs at fe ≈ 9.1 GHz, and the magnetic resonance is at fm ≈ 11 GHz.

decreases. Therefore, they move in opposite directionsand even can interchange their positions, i.e., the mag-netic resonance may occur either below or above theelectric resonance. Also, these resonances may degener-ate as shown in Fig. 8 for A1 = 0.5 mm. It is necessary tonote that width A1 has only marginal impact on fe whichslowly varies due to variations of the dogbone centralpart (A2 − 2 × A1). Conversely, the magnetic resonanceis strongly influenced by the width A1 when A1 becomescommensurable with thickness H of the dielectric spacerseparating a pair of conductors in the dogbone particleof Fig. 2. Then, when the stripes become narrower, thecapacitance between the top and bottom lateral arms of

f the dogbone pairs shown in Fig. 2 vs. permittivity (εr) and thicknessone pair. (a) Permittivity is variable (εr =1, 2, 4, 16) while thickness isittivity is fixed to εr = 16: the electric and magnetic resonances occur

tightly coupled dogbones decreases and fm rapidly movestowards higher frequencies. Thus the use of thin arms(small A1) is also a way to achieve the NRI effect whenthe dielectric spacer is present (see Section 5.2). Such

G. Donzelli et al. / Metamaterials 3 (2009) 10–27 17

F uctors:e GHz nec (in the

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ig. 11. Current distributions on a pair of tightly coupled dogbone condlectric resonance fe; (b) antisymmetric current distribution at f = 10.9larity of the plots, the spacing between the conductors is not to scale

ehaviour of the magnetic resonance is also consistentith the predictions of the TL model discussed later inection 4.

As mentioned in the preceding discussion, the mag-etic resonance appeared to be particularly sensitive tohe stripe width commensurate to the thickness H of thepacer separating the pairs of conductors in the dogbonearticle in Fig. 2. Therefore, it is important to examineow the layer parameters affect the resonance frequen-ies, which are presented in Fig. 9. The transmissionharacteristics in Fig. 9(a) simulated for different val-es of the spacer permittivity εr reveal that variation ofr causes interchange of the magnetic and electric res-nance positions with respect to each other. A similareature was observed earlier in connection with the effectf A1 variations. However in contrast to the latter case,oth types of resonances shift towards lower frequenciest higher εr, albeit fe and fm vary with substantially dif-erent rates. Indeed, the magnetic resonance should beore sensitive to εr because the fields are predominantly

onfined to the dielectric spacer between the dogboneonductors, cf. Section 3.2. The electric resonance is lessffected because the electric field of the symmetric modes mainly located outside the thin dielectric spacer.

Finally, the effect of separation H between the top andottom dogbone conductors is demonstrated in Fig. 9(b).he magnetic resonance fm barely varies with H, whileremains small. However at larger H, fm decreases and

he electric resonance undergoes considerable changesoo. The latter phenomena cannot be described by theL line model of a single particle presented in Section 4ince the fringing field effects at bends and stripe edges

(a) symmetric current distribution at f = 9.1 GHz near the frequency ofar the magnetic resonance frequency fm. Note that in order to improvesimulations H = 0.2 mm).

play a significant role at large ratios H/w, where w isthe stripe width, either A1 or B2. Furthermore, whenH > (A − A2) the capacitive coupling between contigu-ous dogbone particles starts to affect significantly themagnetic resonance frequency.

With reference to the default dimensions of the dog-bone particle, it is important to note that the lateral sizesof the unit cell (A × B) at the magnetic resonance fre-quency fm ≈ 3 GHz equal A = B = λ/13.3, i.e., they aresubstantially smaller than the wavelength. This signifi-cant cell size miniaturization is due to the capacitancebetween the top and bottom parts of the dogbone parti-cle, and to the permittivity of the dielectric spacer, bothcontributing to lower the magnetic resonance frequency.Therefore the induced magnetic dipole density is approx-imately equal to 176 effective magnetic dipoles per λ2.

3.2. Dogbone pair of conductors in free-space

The mechanisms of the resonance formation in thedogbone particles (Fig. 2) are explored in this section inmore detail. For the sake of simplicity, we assume thepermittivity of the spacer between the conductors in thedogbone pair to be equal to εr = 1 while the structuredimensions (in mm) are held the same as in the preced-ing section: A = 7.5, B = 7.5, B1 = 4, A2 = 7.4, B2 = 0.8,A1 = 1.0, H = 0.2. Let us now examine the transmissioncoefficients |T| shown in Fig. 10 for a doubly periodic

planar array of dogbone pairs with isolated conductorsand with short-circuited arms (see insert, Fig. 10).

Comparison of the plots for these two arrangementsshows that in the frequency range shown in Fig. 10, only

18 G. Donzelli et al. / Metamate

Fig. 12. Transmission coefficient |T| at variable H in the doubly peri-

thinner or lower permittivity dielectric spacers betweenthe top and bottom conductors in the dogbone particle(Figs. 9 and 12).

odic planar arrays of dogbone pairconductors in free-space. The unitcell dimensions (in mm), A = 7.5, B = 7.5, B1 = 4, A2 = 7.4, B2 = 0.8,A1 = 1.0, are the same as in Fig. 3.

the resonance at fm ≈ 10.9 GHz is suppressed by short-circuiting the dogbone arms whilst the wide dip aboutfrequency fe ≈ 9.2 GHz remains practically unaffected.The stopband response about fe is typical of frequencyselective surfaces, and the currents on the pair of dog-bone conductors are in-phase (symmetric), as illustratedin Fig. 11(a). Therefore the pair of closely spaced short-circuited conductors essentially behaves like a singleconductor, and no artificial magnetism is observed at fe.

The narrow band resonance at frequencyfm ≈ 10.9 GHz is associated with the antisymmet-ric current distributions in the pair of isolated dogboneconductors as demonstrated in Fig. 11(b). These oppo-sitely directed currents form a magnetic dipole pointingat the y-direction and produce artificial magnetism nearthe frequency fm. The properties of the antisymmetricmodes in the dogbone pairs are comprehensivelydescribed by the TL model further elaborated in Section4.

As already mentioned in the earlier discussion ofFig. 9, variations of the spacing permittivity between thedogbone conductors much stronger affect the magneticresonance fm than the electric resonance fe. Comparisonof Figs. 3 and 10 confirms this trend and suggests thatsuch behaviour of fm is related to a closer confinement ofthe antisymmetric TL mode to the interior of the dogboneparticle. This implies that the capacitance between thetop and bottom conductors in the dogbone pair stronglyinfluences the magnetic resonance. To further examinethis conjecture, let us consider how the conductor sepa-ration H affects fm in the case of εr = 1. The simulation

results in Fig. 12 show that while fm decreases withincreasing H, as was also observed in Fig. 9(b), the reso-nance curves significantly differ here. Namely, at smallvalues of H ∼ 0.2–0.6 mm, fm > fe, and the magnetic res-

rials 3 (2009) 10–27

onance is readily identifiable. But when H increasesfurther to 1.4–2.6 mm, not only becomes fm < fe but alsothe Q-factor considerably decreases. The latter qualita-tive changes in the magnetic resonance behaviour cannotbe explained within a single mode TL model becausethe capacitive coupling between the top and bottom con-ductors in a pair may become weaker than that betweenthe pairs in the adjacent cells. In these circumstances,interaction between contiguous dogbone particles affectsalso the magnetic resonance as the distance betweenthe contiguous conductors (A − A2) = 0.1 mm becomesconsiderably smaller than the separation H between thetop and bottom conductors in the pair (in this geome-try A = 7.5 mm and A2 = 7.4 mm). Moreover, at larger H,the fringing field effects are more significant as the con-ductor separation (H = 1.4 mm) exceeds the stripe widthsA1 = 1 mm or B2 = 0.8 mm.

For the free-space arrangement, the particle size atthe magnetic resonance frequency of fm ≈ 10.9 GHz isas small as A = B = λ/3.67. Though in the absence of adielectric spacer the size reduction, with respect to thewavelength, is less than that in Section 3.1, considerableminiaturisation has been achieved, owing to the strongcapacitive coupling between the top and bottom parts ofthe dogbone particle.

The presented analysis has demonstrated that thedogbone particles provide a means for controlling thepositions of the electric and magnetic resonances. Thisproperty is of particular importance for the implemen-tation of an artificial medium with NRI. In Section 5we will show that dogbone pairs can be used to real-ize a medium supporting backward wave propagationat frequencies near fm. If homogenization is allowed,backward wave propagation occurs when both the effec-tive permittivity and permeability are negative. Thisdouble-negative requirement is generally satisfied nearfm, as long as fm > fe, because the magnetic resonanceis usually much narrower than the electric one1. Tofulfill the condition fm > fe, it is necessary either toincrease fm or to decrease fe. The fe can be reducedeither by increasing the capacitance between the con-tiguous dogbones or by making the dogbone longer, cf.Figs. 3 and 5. Alternatively fm can be increased in thedogbone with narrower lateral arms (Fig. 8), or with

1 As shown in Fig. 14, the magnetic resonance becomes broader atlarger H. In this case the condition fm > fe may not be strictly requiredfor realizing NRI.

G. Donzelli et al. / Metamaterials 3 (2009) 10–27 19

F a sings a dieleca

gths

4m

cclNcwagtawlcFt

wtir

ig. 13. (a) TL model for antisymmetric distribution of currents oneparation H and width w = B2 or w = A1. They are be separated bypproximating the magnetic resonance for a dogbone particle.

In order to gain a deeper insight in the effects of theeometrical parameters on the fundamental properties ofhe dogbone particles, a TL model of the dogbone pairsas been developed and will be discussed in the nextection.

. Approximate transmission line model foragnetic resonances

In the antisymmetric mode, an isolated dogboneonductor pair (we neglect coupling between adjacentells) can be approximately described by an equiva-ent circuit composed of interconnected sections of TLs.amely, the dogbone “arms” are represented as open-

ircuit (OC) stubs with characteristic admittance Ycy andavenumber βy of the quasi-TEM (transverse electric

nd magnetic) or pure TEM (at εr = 1) wave propa-ating along the y-axis (see Figs. 2 and 13) with theime dependence exp(jωt). These stubs provide the shuntdmittances Yst = jYcy tan(B1·βy/2) at the reference portshere they are connected to the TL central section of

ength D = (A2 − A1) along the x-direction which has theharacteristic admittance Ycx and wavenumber βx, seeig. 13. The length D is defined as the distance between

he two mid-points in the dogbone “arms”.To evaluate the characteristic admittances Y and

c

avenumbers β of the TLs and stubs, it is conveniento express them in terms of the capacitance Cl and thenductance Ll per unit length (subscripts x, y are tempo-ary omitted here because the same expressions are used

le dogbone particle. (b) Cross-section of the TL. The stripes havetric slab with dielectric constant εr. (c) Simplified LC resonant circuit

for both the TL central section and stubs):

Yc =√

Cl

Ll

; β = ω√

ClLl = ω

vp

(1)

where ω is the angular frequency and vp = 1/√

ClLl isthe phase velocity of the propagating wave.

Cl and Ll for the TL with the cross-section shown inFig. 13(b) can be evaluated by using the standard approx-imations for microstrip lines with quasi-TEM waves[35]. For the quasi-TEM wave, vp is alternatively defined

as vp = c/

√εeffr where c = 1/

√ε0μ0 is the speed of

light in free-space, ε0 and μ0 are the permittivity andpermeability of free-space, and ε

effr is a relative effective

permittivity of the equivalent homogeneous medium fill-ing the TL cross-section, which may differ for a TL alongthe x- and y-direction. For a quasi-TEM mode on the TLshown in Fig. 13(b), εeff

r can be approximated as follows[35]:

εeffr ≈ εr + 1

2+ εr − 1

2

1√1 + 6/r

(2)

where r = w/H is the TL aspect ratio; w is the width ofstripes: either B2 (for a TL along x) or A1 (for a TL alongy) and H is the separation between them, Fig. 13(b). Mak-

ing use of the effective permittivity ε

effr , the capacitance

per unit length Cl is represented as follows:

Cl ≈ εeffr ε0C [F/m] (3)

etamate

20 G. Donzelli et al. / M

where C̄ is the capacitance per unit length of the pair ofstripes in free-space, normalized to ε0. Several approxi-mations are available to calculate C̄, see [35,36], and thefollowing closed form expressions with relative error lessthan ±1% at any aspect ratio r is used here:

C̄ ≈

⎧⎪⎪⎨⎪⎪⎩

r + 1.21 − 0.11

r+1

2

(1 − 1

2r

)6

, 2r ≥ 1

π

ln((4/r)+(r/2)), 2r≤1.

(4)

The inductance per unit length Ll is readily expressedin terms of C̄ with the aid of (1) and (3):

Ll = 1

v2pCl

= μ0

C̄. (5)

Then the characteristic admittance Yc takes the form:

Yc =√

Cl

Ll

≈√

εeffr

η0C̄. (6)

where η0 = √μ0/ε0 = 120π Ω is the free-space

impedance and the parameters εeffr , C̄ are given by (2)

and (4).In the case when the whole transmission line is filled

with a homogeneous medium (for example, air) withpermittivity εr, one shall have ε

effr = εr, vp = c/

√εr,

β = k0√

εr, and Yc ≈ √εr C̄/η0. These expressions are

applicable to the TL segments along both the x- and y-directions. If all TLs are identical or εr = 1 (free-space)βx = βy, whereas the characteristic admittances Ycx, Ycy

are equal only if the TLs are identical along both axes.In the case of free-space, βx = βy = ω

√μ0ε0 = ω/c.

The resonance frequency fm for the TL circuit shownin Fig. 13 is obtained by evaluating the total input admit-

tance↔Y = �Y + Y at a reference port of the equivalent

circuit, where �Y and �Y are the admittances of the TLlooking toward opposite directions from this port. Theresonances of the dogbone particle occur at frequencies

that render either↔Y = ∞ or

↔Y = 0. When the admit-

tance↔Y is evaluated at the central part of the dogbone,

the symmetry conditions apply ( �Y = �Y ). Here we areinterested in the lowest resonance fm, whose mode hasa maximum current at the center of the dogbone thatmaximizes the strength of the magnetic dipole moment.

Therefore, at this resonance it is necessary that↔Y = ∞,

viz:

�Y = Y = Ycx

2Yst + Ycx s

Ycx + 2Yst s= ∞ (7)

where s = j tan(βxD/2), Yst = jYcy tan(B1 · βy/2) and Ycx,Ycy are calculated with the aid of (6). Here we have

rials 3 (2009) 10–27

assumed that the dogbone particle may comprise the TLsections of dissimilar shapes along the x- an y-axes thatresults in different characteristic admittances Ycx, andYcy and wavenumbers βx and βy given by (1).

The condition↔Y = ∞ requires the denominator of

(7) to vanish, i.e., the resonance occurs at

Ycx + 2Yst s = 0. (8)

The solutions of (8) can be obtained numericallyor approximately as detailed below. Note that in thismodel we have neglected fringing capacitances of theTL in open circuit stubs, and reactances at the TL bendsand junctions. In most cases such approximations arefound to be satisfactory and give adequate qualitative andapproximate quantitative results. Alternatively, correc-tions can be introduced by means of effective elongationof the respective TL sections.

4.1. Approximate resonance condition

The above equations are simplified here to determineapproximate values of fm. In the case of βxD/2 � π/2 andB1·βy/2 � π/2, with D = A2 − A1, we can use the follow-ing approximations: s ≈ jβxD/2 and Yst = jYcyB1·βy/2.Accordingly, the resonance condition (8) takes the form

Ycx − Ycy · βy · βx · B1 · D/2 = 0, (9)

which gives the magnetic resonance angular frequency

ωm = √vp,xvp,y

√Ycx

Ycy

√2

B1 · D

= c√εeffr,y

√C̄x

C̄y

√2

B1 · D. (10)

It is interesting to note that the same result canbe obtained from the electrostatic and magnetostaticapproximations for the capacitance and inductance of thedogbone. The total capacitance of the dogbone is asso-ciated with the charge accumulated on its lateral arms,each of which approximately contributes

Ceff = ε0εeffr,y

A1 B1

H, (11)

where we have neglected the fringing effects (this isa satisfactory approximation for w � H). A similarapproximation for the total inductance of the central sec-

tion of the dogbone, where the current is maximal, isgiven by

Leff = μ0H

B2D. (12)

G. Donzelli et al. / Metamaterials 3 (2009) 10–27 21

Fig. 14. Magnetic resonance frequency f vs. permittivity (ε ) and thickness (H) of the dielectric spacer supporting the dogbone particle. Datao ic strucb variablep

oo

ω

wiv

4

aeidipc

peCetd8ratcto

m r

btained from the rigorous numerical simulations (assuming a periodased on the TL model of the dogbone particle. (a) Permittivity isermittivity εr = 16 is fixed.

Then, taking into account the capacitive contributionsf both dogbone arms (cf. Fig. 13(c)), the magnetic res-nance ωm equals

m = 1√Leff Ceff /2

= c√εeffr,y

√2 · B2√

B1 · A1 · D(13)

hich is identical to (10) when the characteristicmpedances are equal Ycx = Ycy (and thus B2 = A1) andp,x = vp,y.

.2. Accuracy and limitations of the TL model

In this section we assess the relative accuracy andpplicability of the TL approximations developed here tovaluate fm of the dogbone particle. The results presentedn Fig. 14 summarize the magnetic resonance depen-ence on the parameters of the dielectric spacer, namely,ts permittivity εr and thickness H, to which fm was foundarticularly sensitive in the preceding parametric study,f. Figs. 9 and 12.

In Fig. 14(a), fm was calculated at several values ofermittivity εr, while the dogbone geometrical param-ters assumed the default values (in mm): A = B = 7.5,= 8, B1 = 4, A2 = 7.4, B2 = 0.8, A1 = 1, H = 0.2. The ref-

rence data for fm (black curve) have been retrieved fromhe transmission response of a doubly periodic array ofogbone particles simulated numerically at εr = 1, 2, 4,, 12, 16 as described in Section 3. The other curvesepresent the fm estimates obtained from Eqs. (8), (10),nd (13). It appears that for the considered configura-

ion with a small separation between the top and bottomonductors in the dogbone particle, all the approxima-ions based on a TL model provide an accurate predictionf the magnetic resonance frequency. Furthermore, they

ture) are compared with the approximate formulas (8), (10) and (13)while thickness H = 0.2 mm is fixed. (b) Thickness H varies while

exhibit the same trend observed earlier in Fig. 9 that fmrapidly decreases with εr, thus reconfirming the fact thatthe fields of the antisymmetric mode are tightly confinedto the dielectric spacer separating the pair of dogboneconductors.

The TL approximations have also been assessed inFig. 14(b) for a variable thickness H of the dielectricspacer with permittivity εr = 16 between the top andbottom conductors of the dogbone pair. The fm valuesdeduced from the rigorous numerical simulations of adoubly periodic array of dogbone particles are com-pared again with the corresponding approximate valuesobtained from Eqs. (8), (10), and (13). The plots inFig. 14(b) show that fm stays practically constant at smallvalues of H, and the fm approximations are in reason-able agreement with the results of numerical simulations.However, when H grows, the numerical simulationsshow an appreciable decrease of fm, as was also observedin Figs. 9(b) and 12. However, the TL formulas no longerprovide a satisfactory approximation for fm at larger H.This indeed is not surprising since these approximateformulas completely neglect the fringing capacitancesat the TL ends and bends, and the capacitive couplingbetween the adjacent cells, which become even morepronounced at larger H. The latter interaction affects notonly the electric but also the magnetic resonance whenthe separation H between the top and bottom conductorsin the pair exceeds the distance (A − A1)/2 between thecontiguous particles. Therefore, the rapid variation of fmat large values of H cannot be described by a simple TLmodel for a single isolated particle, and the collective

response of the array will be modeled more accuratelyin a future investigation.

The results presented in Fig. 14 allow one to concludethat in spite of their limitation, the developed analytical

22 G. Donzelli et al. / Metamate

Fig. 15. Metamaterial composed of stacked layers of the arrayed dog-

bone particles (Fig. 2). In order to identify the existence of backwardwave propagation, phase variations of the waves transmitted throughthe stack are analyzed at the reference ports P and Q when increasingthe number of stacked layers.

models provide an adequate qualitative description for abroad range of dogbone configurations. These approx-imations give an insight into the effect of the structureparameters on the magnetic resonance and are instru-mental in the initial design of the constituent dogboneparticles for the specified frequency response.

5. Backward wave propagation in a mediumformed by stacked layers of dogbone particles

In this section we show that metamaterials composedof stacked layers of dogbone particles comprised oftightly coupled pairs of conductors as shown in Fig. 2,can support backward wave propagation in certain fre-quency bands. To illustrate this phenomenon, we adopttwo methods based on numerical simulations. The firstapproach is rather qualitative and may be affected by theinterface effects, while the second one is based on therigorous analysis of the Brillouin dispersion diagram foran infinite stack of dogbone layers periodic along thez-direction.

In the first method, the magnitude and phase progres-sion/regression in the transmitted waves is evaluated byvarying the numbers of stacked layers. The phase varia-tion of the scattering parameter SQP (S21 in the standardterms) between the reference interfaces P and Q (Fig. 15)equals

ϕ = ϕQ − ϕP ≈ −kML (14)

where kM is the wavenumber in the metamaterial and Lis the stack thickness. For the time convention exp(jωt)used, ϕ < 0 corresponds to forward wave propagating

with kM > 0, whereas ϕ > 0 and kM < 0 are associated tobackward wave propagation. This approach gives accu-rate results when reflection at the air/material interfacesis negligible in the analyzed frequency band.

rials 3 (2009) 10–27

A simple approximate procedure similar to that usedin Ref. [16] is applied first to determine the kM sign.Namely, we increase the number m of the stacked layersof dogbone pairs spaced apart in z-direction for dis-tance C, which corresponds to the unit cell height inFig. 2 or equivalent “layer” thickness. Then for the totalstack thickness L = mC, where m = 1, 2, . . ., we evaluatethe phase difference ϕ at the interfaces Q and P. Tomake the method less dependent of the interface proper-ties, we analyze the incremental phase difference ϕm+1

m

between the cases of m and (m + 1) layer stacks. In par-ticular, for backward wave propagation in a multilayeredmetamaterial, the phase ϕm+1 at the (m + 1)th layer inter-face is larger than the phase ϕm at the mth layer interfacethat gives

ϕm+1m = ϕm+1 − ϕm = −kMC > 0. (15)

A number of recent studies are concerned with thehomogenization of the artificial medium, though thistopic is still extensively debated in the context ofmetamaterials. Homogenization aspects of the dogbone-based layered medium are beyond the scope of this paper,therefore we will not discuss the frequency ranges wherethis is permitted. Nevertheless, the relevant basic con-cepts will be used here with the sole purpose of providingthe coherent definitions of the characteristics adopted inour approach. We will not retrieve the effective mediumpermeability and permittivity parameters, but we willobtain an approximation for an effective refractive indexn. Accordingly, for a wave traveling along the z-directionwith electric field polarized along the x-direction, thewavenumber can be defined as kM = nk0, where k0 isthe free-space wavenumber. Therefore in this paper, nis used only as a derived parameter strictly related tothe phase velocity of a wave with fixed polarization andincidence angle. Then we can retrieve n by using themethod employed in Ref. [16] for SRR metamaterialsand determine n from the incremental phase differenceas follows:

n ≈ −ϕm+1m

k0C. (16)

The second method adopted here is based upon a rig-orous analysis of wave propagation in infinite periodiclattices. We evaluate numerically the dispersion char-acteristics (frequency vs. wavenumber kM along z) forthe doubly periodic (in x–y plane) planar arrays of dog-bone pairs stacked along the z-direction with period C

(Fig. 15). The dispersion diagrams are obtained by theeigenmode solver in CST Microwave Studio with peri-odic boundary conditions applied to the unit cell with thedogbone conductor pair shown in Fig. 2. The simulated

G. Donzelli et al. / Metamate

Fig. 16. Transmission coefficient for a few layers of arrayed dogboneparticles stacked along the z-direction with period C = 2.6 mm (similarrwim

swem

5

lpt(Bpctll

Fwp

esults for a single layer of dogbone particles are shown in Fig. 12ith the only difference that here H = 0.4 mm and several layers are

ncluded). The electric resonance fe is in the band 9.4–9.8 GHz, theagnetic resonance fm is in the band 10.5–10.65 GHz.

tructure is periodic along the x-, y- and z-directions,ith the lattice constants A, B, and C, respectively. The

xamples below illustrate applications of the two afore-entioned approaches.

.1. Example: dogbone pairs in free-space

Let us consider a slab composed of several stackedayers of arrayed dogbone particles in free-space (i.e., theermittivity of a spacer between the paired conductors inhe dogbone particle is εr = 1). The geometric parametersin mm) are the same as in Section 3.1: A = 7.5, B = 7.5,1 = 4, A2 = 7.4, B2 = 0.8, A1 = 1.0, and the interlayereriod is C = 2.6 mm. The separation between the paired

onductors in the dogbone particle is chosen H = 0.4 mmo realize fm > fe (according to the data in Fig. 12, theatter inequality is held at any H ≤ 0.6 mm). The simu-ation results in Fig. 16 show that a transmission peak

ig. 17. Magnitude (a) and phase (b) of the transmission coefficient for a feith the interlayer period C = 2.6 mm. Within the passband at 10.5–10.65 GHropagation exists because of the phase difference ϕm+1

m > 0.

rials 3 (2009) 10–27 23

exists between 10.5 and 10.65 GHz, which is associatedwith the artificial magnetism at the magnetic resonance,as already discussed in Section 3.2 (cf. current distri-bution in the antisymmetric mode shown in Fig. 11 forH = 0.2 mm).

The transmission coefficient T magnitude and phasefor this structure is zoomed in Fig. 17 around the trans-mission peaks corresponding to the magnetic resonance.From the phase plots in Fig. 17(b), we can readily inferthat the phase ϕ of the transmission coefficient Tincreases with the number of layers, thus showing that aphase difference ϕm+1

m > 0 can be realized. Hence thestructure supports backward wave propagation along thez-direction in the frequency band 10.5 − 10.65 GHz.

According to Eq. (16), the refractive index n forthe case shown in Fig. 17 varies with frequency in therange n ≈ −2 to −3.5. The results are not completelystable when the number of layers is increased becauseof the interlayer couplings and reflections at the twoair-metamaterial interfaces. Nevertheless, the qualitativetrend of the n behaviour remains consistent when thenumber of stacked layers increases.

An alternative proof of the existence of backwardwave propagation has been obtained from the dis-persion characteristics for the infinite periodic (alongthe z-direction with period C = 2.6 mm) arrangement ofstacked layers of the arrayed free-space dogbone parti-cles with the same parameters as in Figs. 16 and 17. Theone-dimensional dispersion diagrams for the spacingH = 0.2, 0.4, 0.6 mm between the top and bottom con-ductors in the dogbone particle are presented in Fig. 18.The three branches shown in Fig. 18(a) correspond to

the low-frequency region of the dispersion diagram ofthe dogbone particle lattice. The lower and upper fre-quency branches correspond to the usual low-frequencypassband that is split in two portions by the resonant stop-

w layers of arrayed dogbone particles, stacked along the z-directionz occurring in the vicinity of the magnetic resonance, backward wave

24 G. Donzelli et al. / Metamaterials 3 (2009) 10–27

Fig. 18. Dispersion diagrams for an infinite periodic (along the z-direction with period C = 2.6 mm) arrangement of stacked layers of the arrayedmm. Ththe NRf. Figs.d with

printed on the opposite faces of a dielectric substratewith εr = 2.2 and thicknesses H = 0.25, 0.5, 1 mm arepresented in Fig. 20 (the other dimensions of thestructure are the same as in Fig. 19). While the fun-

free-space dogbone particles with spacer thicknesses H = 0.2, 0.4, 0.6dispersion diagram of the dogbone particle lattice. (b) Enlargement ofIf the lateral arms of the pair of dogbones would be short-circuited (cwould be replaced by a single dogbone conductor, the second passban

band occurring far below the Bragg resonance due to alattice of resonant electric scatterers. A narrow passbandin the middle arises as the result of the superpositionof the electric and magnetic resonances of the dogboneparticles. We have numerically verified that it disap-pears when the lateral arms of the dogbone pair areshort-circuited (see also Figs. 3 and 10) or the dogboneparticle contains only a single dogbone-shaped conduc-tor. Thus, the middle branches of the dispersion diagram,which are shown in more detail in Fig. 18(b), existonly in the presence of both the magnetic and electricresonances of the dogbone particles and represent back-ward waves with oppositely directed phase and groupvelocities. In this frequency band the lattice of dogboneparticles can be homogenized and characterized as a NRImedium.

The backward wave propagation for all three thick-nesses of spacing between the dogbone pair conductorsis confirmed by the negative slope of the dispersioncurves in Fig. 18(b). These plots also show that for smallvalues of H the NRI passband is narrow, as alreadydemonstrated in Figs. 16 and 17, but a broader NRIband can be achieved at larger H, which provides abetter overlap of the electric and magnetic resonances.These results are in full agreement with what we haveobserved in Fig. 12, and also show that the fm decreases atlarger H.

5.2. Example: dogbone pairs with low permittivitydielectric spacers

In this example, we consider stacked layers ofthe arrayed dogbone particles printed on the oppositefaces of a dielectric substrate (spacer) with permit-

e structure parameters are the same as in Fig. 17. (a) Low-frequencyI passband (the negative slopes indicate backward wave propagation).3 and 10), or, in an analogous manner, the dogbone pairs in each cellnegative slop would disappear.

tivity εr = 2.2. The structure dimensions (in mm) areA = B = 7.5, B1 = 4, A2 = 7.4, A1 = B2 = 0.5.

The simulated magnitude of the transmission coef-ficient |T| through a single layer of the planar array ofdogbone particles is shown in Fig. 19, for several thick-nesses H of the dielectric substrate separating the topand bottom conductors. A narrow passband, which cor-responds to the magnetic resonance of the dogbone pairand strongly depends on H, is readily observable in theplots.

The dispersion diagrams for an infinite periodic(along the z-direction with the period C = 3 mm) arrange-ment of stacked layers of the arrayed dogbone particles

Fig. 19. Transmission coefficient |T| for a single layer of arrayeddogbone particles made of a pair of conductors printed on a low permit-tivity dielectric substrates (εr = 2.2) of several thicknesses: H = 0.25,0.5, 1 mm. The structure dimensions (in mm) are: A = B = 7.5, B1 = 4,A2 = 7.4, A1 = B2 = 0.5.

G. Donzelli et al. / Metamaterials 3 (2009) 10–27 25

F on withp ermittivd f the dop

dtssciNlsMHbitowm

q

Fpds

ig. 20. Dispersion diagrams for an infinite periodic (along the z-directiarticles printed on the opposite faces of dielectric substrates with pispersion diagram of the dogbone particle lattice. (b) Enlargement oropagation).

amental properties of a NRI medium discussed inhe preceding example for the free-space arrangementtill hold in this case, the presence of the dielectricubstrate adds new features. This is particularly con-erned with the width of the NRI passband. Indeed,nspection of Figs. 18(b) and 20(b) shows that theRI fractional bandwidth (Df/fm, where Df is the abso-

ute bandwidth) increases in the case of the dielectricubstrate, and Df is broader for thicker substrates.

oreover, comparison of the dispersion curves for= 1 mm in Figs. 18(b) and 20(b) shows that the Df

ecomes broader in the latter case as well. These resultsndicate that the use of the high permittivity dielec-ric substrates can lead not only to miniaturisationf the dogbone particles but also to a broader band-

idth of the NRI performance of such an artificialedium.It is important to note that in the NRI fre-

uency bands shown in Fig. 20(b), the ratio of the

ig. 21. Dispersion diagrams for an infinite periodic (along the z-direction wairs (degenerated dogbone particles with B1 = B2 = 0.5 mm and other dimenielectric substrates with permittivity εr = 2.2 and thicknesses H = 0.25, 0.5, 1trip lattice. (b) Enlargement of the NRI passband.

period C = 3 mm) arrangement of stacked layers of the arrayed dogboneity εr = 2.2 and thicknesses H = 0.25, 0.5, 1 mm. (a) Low-frequencyuble-resonant passband (the negative slopes indicate backward wave

wavelength to the lattice constant A is of an orderλ/A ≈ 6. Therefore the low-frequency NRI bands arenot related to the higher order Brillouin zones of thephotonic crystals where the condition A � λ is not sat-isfied.

The above dispersion analyses have been repeated inFig. 21 when B1 = B2 = 0.5 mm and the dogbones con-ductors are reduced to simple short strips (cut-wires). Allother parameters are as in Fig. 19, i.e., A = B = 7.5 mm,A2 = 7.4 mm, and the particles are printed on the oppositefaces of dielectric substrates with εr = 2.2 and thick-nesses H = 0.25, 0.5, 1 mm. It is found that althoughalso a strip-pair medium can provide backward waves,the NRI transmission band for this structure occurs atsignificantly higher frequencies, which limits its appli-

cability as a NRI medium. It is important to note that inthe NRI frequency bands shown in Fig. 21(b), the ratioof the wavelength to the lattice constant A is of an orderλ/A ≈ 3.5.

ith period C = 3 mm) arrangement of stacked layers of arrayed stripsions as in Fig. 19). The strips are located on the opposite faces of

mm as in Fig. 20. (a) Low-frequency dispersion diagram of the paired

26 G. Donzelli et al. / Metamaterials 3 (2009) 10–27

of thection w19.

Fig. 22. Central frequency fm (a) and fractional bandwidth Df/fm (b)supporting the dogbone pairs in the infinite periodic (along the z-direparticles. The structure dimensions and layout are the same as in Fig.

To assess the concurrent effect of the dielectric spacerthickness H and the lattice constant C (along the prop-agation direction) on the NRI centre frequency fm andfractional bandwidth Df/fm these characteristics havebeen retrieved from the preceding dispersion analysisand are shown in Fig. 22. The analyzed dogbone struc-ture is the same as that in Fig. 19, except that now H andC are variable. As apparent from Fig. 22(a), fm progres-sively moves towards lower frequencies at larger H untilit reaches its minimum at H ∼= C/2, and then it starts togrow. Furthermore, the plots in Fig. 22(b) demonstratethat at the fm minimum the NRI bandwidth Df/fm reachesits maximum and can exceed 10%. Thus we can con-clude that arrays of dogbone pairs with a proper aspectratio can provide a reasonable bandwidth of NRI per-formance at low frequencies. Combined with the trulysub-wavelength size of the dogbone particles, the pre-sented structures offer new opportunities for the practicalrealization of a homogenized NRI medium.

6. Conclusions

The properties of a recently proposed metamaterialarrangement based upon periodic arrays of tightly cou-pled pairs of dogbone shaped planar conductors havebeen explored in detail. A comprehensive parametricanalysis was carried out to evaluate the impact of indi-vidual characteristic dimensions of the constituent unitcell and the dogbone particle onto the electromagneticresponse of the metamaterial. It has been shown thatthis structure supports backward wave propagation ina rather broad band of the low-frequency region. We

have conclusively demonstrated that these properties areassociated with the electric (symmetric) and magnetic(antisymmetric) resonances of the tightly coupled con-ductor pairs, and the effective negative refractive index

backward wave passband vs. thickness H of the dielectric substrateith period C) arrangements of the stacked planar arrays of dogbone

can be achieved, when the frequency of magnetic res-onance is above the electric resonance. The origins ofthe artificial magnetism in the dogbone particles wereexplored, and it was shown that a magnetic dipole iscreated by a current loop on the pair of tightly coupledconductors with the antisymmetric current distribution.

We have developed an approximate model of the con-stitutive dogbone particles, based on a transmission lineanalysis. The model provides an accurate prediction ofthe magnetic resonance frequency and gives physicalinsight into the mechanisms of the resonance formation.The conditions of the magnetic resonance existence wereobtained and the effect of the dogbone parameters on theresonance frequency was assessed. It has been shownthat the lateral arms in the dogbone particles play a cru-cial role in controlling both the electric and magneticresonances. The model suggested a means for controllingthe magnetic resonance frequency and enabled an effi-cient design of the dogbone particles with the specifiedproperties.

It has been demonstrated that backward wavepropagation along the z-direction is supported in a meta-material made of the stacked layers of the arrayeddogbone particles. This feature has been confirmed bymonitoring the phase variation of a plane wave at theoutput interface of the layers’ stack, when increasingthe number of layers. The existence of backward wavepropagation has also been inferred from the dispersiondiagrams for the infinite periodic arrangement of thestacked layers of the arrayed dogbone particles. Com-parison with a metamaterial made of layers of pairedcut-wires has demonstrated that the dogbone shaped

conductor pairs provide better control of the NRI bandposition.

The studied metamaterial has the advantage that it canbe fabricated and assembled in planar technology. The

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losely spaced pairs of dogbone shaped planar conduc-ors naturally couple to the magnetic field of normallyncident plane waves and readily exhibit the magneticesonance of the antisymmetric mode.

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