ceramic substrate is about 5.91 dB; however, the out-of-bandattenuation values were below 24 dB in the lower-frequencyregion, and below 10 dB in the higher-frequency region. Thecompactness of the circuit size makes the cross-coupled filtersusing microstrip miniaturized hairpin resonators and high-permit-tivity ceramic substrates an attractive design for further develop-ment and applications in modern communication systems.

ACKNOWLEDGMENT

This work was supported by the National Science Council of theRepublic of China under grant no. NSC 91-2213-E-006-119.

REFERENCES

1. C.C. You, C.L. Huang, and C.C. Wei, Single-block ceramic micro-wave bandpass filters, Microwave J 37 (1994), 24–35.

2. S.B. Cohn, Parallel-coupled transmission line resonator filters, IRETrans Microwave Theory Tech (1958), 223–231.

3. S. Caspi and J. Adelman, Design of combline and interdigital filterwith tapped-line input, IEEE Trans Microwave Theory Tech 36(1988), 759–763.

4. G.L. Matthaei, Comb-line bandpass filter of narrow or moderate band-width, IEEE Trans Microwave Theory Tech 6 (1963), 82–91.

5. U.H. Gysel, New theory and design for hairpin-line filters, IEEE TransMicrowave Theory Tech 22 (1974), 523–531.

6. E.G. Cristal and S. Frankel, Hairpin-Line and Hybrid Hairpin-Line/Half-Wave Parallel-Coupled-line Filters, IEEE Trans Microwave The-ory Tech 20 (1972), 719–728.

7. J.S. Hong and M.J. Lancaster, Cross-Coupled Microstrip Hairpin-Resonator Filters, IEEE Trans Microwave Theory Tech 46 (1998),118–122.

8. G.L. Matthaei, N.O. Fenzi, R.J. Forse, and S.M. Rohlfing, Hairpin-comb filters for HTS and other narrowband applications, IEEE TransMicrowave Theory Tech 45 (1997), 1226–1231.

9. M. Sagawa, K. Takahashi, and M. Makimoto, Miniaturized hairpinresonator filters and their application to receiver front-end MICs, IEEETrans Microwave Theory Tech 37 (1989), 1991–1997.

10. H. Yabuki, M. Sagawa, and M. Makimoto, Voltage controlled push-push oscillators using miniaturized hairpin resonators, IEEE MTT-SInt Microwave Symp Dig (1991), Boston, MA; 1175–1178.

11. R. Levy, Filters with single transmission zeros at real or imaginaryfrequencies, IEEE Trans Microwave Tech 24 (1976), 172–181.

12. R.M. Kurzok, General four-resonator filters at microwave frequencies,IEEE Trans Microwave Tech 14 (1966), 295–296.

13. C.S. Hsu, C.L. Huang, J.F. Tseng, and C.Y. Huang, Improved high-Qmicrowave dielectric resonator using CuO-doped MgNb2O6 ceramics,Mater Res Bull 38 (2003), 1091–1099.

14. H. Yabuki, Y. Endo, M. Sagawa, and M. Makimoto, Miniaturizedhairpin resonators and their application to push-push oscillators, Proc3rd Asia-Pacific Microwave Conf, 1990, pp. 263–266.

15. J.S. Hong and M.J. Lancaster, Microstrip filters for RF/microwaveapplications, Wiley, New York, 2001.

16. Zeland Software, Inc., IE3D 6.0, New York, 1999.

© 2004 Wiley Periodicals, Inc.

NOVEL COMBINED MOD-P MULTIBANDANTENNA STRUCTURES INSPIRED BYFRACTAL GEOMETRIES

Jordi Soler, Daniel Garcia, Carles Puente, and Jaume AngueraTechnology Department, Fractus S.A.c/ Alcalde Barnils 64–68Edifici Testa–Modul C, 3a plantaParc Empresarial Sant Joan08190 Sant Cugat del Valles, Spain

Received 20 November 2003

ABSTRACT: A novel set of multiband antenna structures, which com-bine different geometries based on fractal elements, are presented. Thenew combined radiating elements profit from the multifrequency proper-ties of the noncombined geometries, thus providing a mechanism to ad-just the antenna parameters. © 2004 Wiley Periodicals, Inc. MicrowaveOpt Technol Lett 41: 423–426, 2004; Published online in Wiley Inter-Science (www.interscience.wiley.com). DOI 10.1002/mop.20159

Key words: multiband monopoles; fractal-shaped antennas; mod-p Sier-pinski structures

1. INTRODUCTION

One of the main challenges in the field of multiband antennas is thedifficulty of finding an antenna which can operate at more than two orthree frequency bands. Most antenna engineers prefer to have anantenna design which will work, for example, in GSM900, DCS,UMTS, BLUETOOTH™, and WLAN systems. However, such amultifrequency antenna design not only needs to feature severalmatched bands, but also needs to be easy to tune; that is, the geometryshould be easily reshaped in order to add new bands and, also, thespacing between bands and the input resistance need to be easilychanged by using simple variations on the antenna geometry. Multi-level-shaped antennas have proved to be a good and efficient solutionfor the design of multiband antennas [1]. The classical Sierpinskiantenna represents a clear example of multiband antenna geometry,where, for instance, a new resonance can be simply introduced byadding a new fractal iteration. However, since the spacing betweenbands is closely linked to its self-similar structure, the spacing be-tween bands is fixed at 2 [2]. Alternatively, top-loading can be addedto the Sierpinski monopole in order to control the spacing betweenbands and the input impedance at the lower resonances of the antenna[3]. In addition, by modifying the Sierpinski monopole flare angle, theantenna pattern at the different resonances can be tailored, as de-scribed in [4]. Further investigations on geometries inspired by fractalgeometries show that the classical Sierpinski gasket is in fact a specialcase of a wider set of structures, which can be referred to as Pascal–Sierpinski gaskets [5]. In [6], the authors showed how the so-calledmod-p Sierpinski gaskets can be used as a very efficient solution forthe design of multiband antennas. In detail, mod-p structures permitmultiband antennas to be designed with log periods larger than 2. Asan extension of that work, several strategies to combine differentmod-p structures were analyzed in order to find a technique to controlthe spacing and number of bands of a unique single-fed multibandantenna element. In the present work, several antenna designs com-bining two different mod-p structures, also with different iterations,are studied. In this paper, two combined mod-p Sierpinski multibandantennas are presented.

2. GENERATION OF COMBINED MOD-p ANTENNASTRUCTURES

The geometry of a single mod-p structure can be extracted fromthe Pascal triangle. In particular, by removing from the Pascal

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 41, No. 5, June 5 2004 423

triangle those coefficients which are divisible by a factor p, themod-p structure can be recovered [7]. Taking advantage of theself-similarity of these mod-p structures, each geometry can begenerated by using an IFS scheme, as detailed in [6]. Figure 1shows the already reported three-iteration mod-3 Sierpinski mono-pole and its measured input parameters [6]. The monopole wasprinted over a thin dielectric substrate (�r � 3.38, h � 0.8 mm),mounted over a 800 � 800 mm2 square-conductor ground planeand fed using a coaxial probe. The overall height of the antennawas h � 178 mm and the flare angle � � 60°. Looking at antennareturn loss, the resonant frequencies are log-periodically spaced bya factor which matches the scale factor p, which in this case is 3.The three bands are marked with different colored zones. Also,inside each band two resonances can be observed; they are 0.92and 1.46 GHz for the first band, 2.74 and 4.32 GHz for the secondband, and 7.46 and 12.37 GHz for the third band. In general, forthe mod-p Sierpinski antennas, the number of resonances insideeach band is directly linked to the number of scale levels in thestructure, as shown in Figure 1. With the criteria selected in Figure1, the mod-3 antenna features two higher-scale levels; that is,above the basic level two additional levels with the same structure,although with more copies, are present. Then, since there are twoscale levels, two resonances can be found inside each band. Thischaracteristic is also repeated for the rest of the mod-p monopoleantennas. The spacings between these resonances is fixed, depend-ing upon the mod-p structure. For the mod-3 monopole, thespacing is 1.6. Also, as occurs with other multiband antennadesigns inspired by fractal geometries, the number of bands isrelated to the number of fractal iterations. The similarity of theradiation patterns is demonstrated in [6].

From these results, the proper combination of two mod-p geom-etries inside a unique structure may be interesting if it permits us tocombine different bands with different log periods and scale factors.Figure 2 shows two mod-p combined antenna geometries; they are atwo-iteration mod-2 structure inserted into a one-iteration mod-3structure, and a one-iteration mod-2 structure inserted into a one-iteration mod-5 Sierpinski gasket. Rather than being based on math-ematical concepts, these combinations are designed so as to obtainspacings of two and three within the same radiating element for thefirst design, and of two and five for the second monopole design.

3. ANALYSIS OF RESULTS

The input and radiation parameters of both combined mod-p struc-tures described in Figure 2 are simulated using a method of moments(MoM) code. For the computations, the antennas are mounted over aninfinite perfectly conducting ground plane and fed using a coaxialprobe. Both monopoles are 200-mm high and with a flare angle � �60°. The input-reflection coefficient relative to 50�, and input resis-tance and reactance of both monopoles are shown in Figure 3. Thisfirst band, 0.22 GHz, is the fundamental mode associated with the firstmode of a solid bow-tie monopole with the same height and flareangle as the analyzed mod-p Sierpinski gaskets. For this analysis, thefirst resonance is not considered because, due to the lack of fractaliterations, the current cannot propagate as in the higher modes andthus the current distribution is similar to the currents in the classicalbow-tie monopole. This effect, referred to as the truncation effect, wasfurther analyzed in [2]. Therefore, as shown in Figure 3, three bandsare marked. The first one is directly linked to the one-iteration mod-3structure included in the antenna geometry. As the original mod-3Sierpinski gasket, two resonances are present within the band. Onlyone band appears, as one mod-3 iteration is included in the geometry.The upper two bands are achieved by using two iterations of a mod-2structure. By the same token, the combined mod-5 and mod-2 geom-etry presents two bands; they are the first with four resonances inside,which are a consequence of using one iteration with a factor of 5. Theupper band is linked to the one-iteration mod-2 structure. The spac-ings between bands are shown and analyzed in Figure 4.

Figure 4 depicts two diagrams which permit us to clearlyidentify and associate the properties of the combined mod-p Sier-pinski gaskets with the log periods of both monopoles. In detail, inthe diagrams the resonances are expressed in GHz and severallines indicate the frequency spacings between them. For the com-bined mod-2 and mod-3 monopole, log periods of two and threeare shown. Similarly, for the monopole including mod-2 andmod-5 structures, spacings around five and two are identified.

The 3D radiation patterns of both monopoles were computedusing the same MoM code. Figure 5 shows two of the main cuts(� � 90° and � � 90°) of the total pattern for the one-iteration

Figure 1 Mod-3 Sierpinski monopole antenna built after three iterations(top) and measured return loss relative to 50� and input resistance andreactance as functions of the frequency (bottom)

Figure 2 Two-iteration mod-2 structure inserted into a one-iterationmod-3 monopole structure (left) and a one-iteration mod-2 structure in-serted into a one-iteration mod-5 Sierpinski monopole (right)

424 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 41, No. 5, June 5 2004

mod-2 structure inserted into a one-iteration mod-5 Sierpinskimonopole at different resonances of 0.88, 1.53, 2.15, 2.62, and4.77 GHz. The cuts at 0.22 GHz are not included, since they followthe classical toroidal pattern of a monopole operating at its fun-damental mode. A certain degree of similarity is observed through-out all the bands. The pattern at 1.53 GHz corresponds to theshape, which also provides the noncombined original mod-3 Sier-pinski gasket at the second resonance of each of its bands. Forthese results, one of the advantages of computing the patternsinstead of measuring them is that the pattern data are not depen-dant upon the ground-plane effects. Then, back-radiation anddiffraction effects are avoided. The measured results for the orig-inal noncombined mod-p Sierpinski gaskets are included in [6].

5. CONCLUSION

A novel combined set of structures based on the mod-p Sierpinskigaskets, which permits us to design and control the properties ofthese multiband monopoles, has been presented. In addition tocontrolling the number of bands, the combination of different scalefactors in the same geometry permits us to adjust the spacing

Figure 3 Computed return loss relative to 50� and input resistance andreactance for both the two-iteration mod-2 structure inserted into a one-iteration mod-3 monopole structure (top) and the one-iteration mod-2structure inserted into a one-iteration mod-5 Sierpinski monopole (bottom)

Figure 4 Diagram of resonant frequencies and spacing between them forboth the two-iteration mod-2 structure inserted into a one-iteration mod-3monopole structure (top) and the one-iteration mod-2 structure insertedinto a one-iteration mod-5 Sierpinski monopole (bottom)

Figure 5 Computed total component of the radiation pattern for thehorizontal (� � 90°) and vertical (� � 90°) planes for the one-iterationmod-2 structure inserted into a one-iteration mod-5 Sierpinski monopole

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 41, No. 5, June 5 2004 425

between bands. The obtained results can be generalized for othermod-p structures.

ACKNOWLEDGMENT

This work has been supported by the company FRACTUS S.A.

REFERENCES

1. C. Puente, J. Romeu, C. Borja, J. Anguera, and J. Soler, MultilevelAntennae, invention patent WO0122528.

2. C. Puente, J. Romeu, R. Pous, X. Garcia, and F. Benıtez, Fractalmultiband antenna based on the Sierpinski gasket, IEE Electron Lett 32(1996), 1–2.

3. J. Soler and J. Romeu, Dual-band Sierpinski fractal monopole antenna,IEEE Antennas Propagat Soc Int Symp, Salt Lake City, UT, 2000.

4. J. Soler, C. Puente, and A. Puerto, A dual-band bidirectional multilevelmonopole antenna, Microwave Opt Technol Lett 34 (2002), 445–448.

5. N.S. Holter, A. Lakhtakia, V.K. Varadan, V.V. Varadan, and R. Mess-ier, On a new class of fractals: the Pascal-Sierpinski gaskets, J Phys AMath Gen 19 (1986) 1753–1759.

6. J. Soler, J. Romeu, and C. Puente, Mod-p Sierpinski fractal multibandantenna, AP2000 Millen Conf Antennas Propagat, Davos, Switzerland,2000.

7. H.O. Peitgen, H. Jurgens, and D. Saupe, Chaos and fractals: Newfrontiers in science, Springer-Verlag, New York, 1992.

© 2004 Wiley Periodicals, Inc.

A STUDY OF WAVEGUIDES FILLEDWITH ANISOTROPIC METAMATERIALS

Yansheng XuCentre de recherches avancees en microondes et en electroniquespatiale(POLY-GRAMES)Departement de genie electriqueEcole Polytechnique de MontrealC. P. 6079, succursalle “centre-ville”Montreal, Quebec, Canada, H3C 3A7

Received 30 October 2003

ABSTRACT: In this paper, a rectangular waveguide and a nonraditivedielectric (NRD) waveguide, filled with anisotropic negative permittivityand permeability, are studied in detail. A general condition of the exis-tence of TE and TM modes in these waveguides is derived. The uniquecharacteristics of these waveguides are explored. Both fully- and partially-filled cases of rectangular waveguides are examined. Characteristics ofNRD waveguides with anisotropic metamaterial cores are studied. © 2004Wiley Periodicals, Inc. Microwave Opt Technol Lett 41: 426–431, 2004;Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/mop.20160

Key words: rectangular waveguide; nonraditive dielectric waveguide;negative permittivity; negative permeability; anisotropic medium; uniax-ial medium

1. INTRODUCTION

Recently, an intensive study of the propagation of electromagneticwaves in media with negative permittivity � and permeability �has been presented in the literature [1–6]. Some unique properties,such as a negative index of refraction, supporting backward waves,and so forth, have been shown. In [3, 5, 6], waves in metamaterialslabs were studied and in [4], slabs of single negative metamate-rials in parallel-plate waveguides are examined. At the same time,the methods of realizing these materials are also examined anddemonstrated [7–9]. In [7], the split-ring resonator (SRR) wassuccessfully developed to created materials with negative perme-ability. Subsequently, 3D electromagnetic artificial dielectrics(metamaterials), composed of arrays of resonant cells consisting ofthin wire strips and SRRs were realized to synthesize double-negative materials. On this basis, materials with a negative-refrac-tion index were developed and reversed refraction was demon-strated [8–10]. However, the presented metamaterials are madeusing 3D constructions which are complicated, bulky, and difficultto fabricate. It will be much simpler and easier to use a planarstructure with a square (or circular) array of resonators, such assplit rings, to realize an anisotropic material with only one or twonegative components of � and � instead. It is noted in [7] that anisotropic metamaterial is made by adding additional arrays ofresonators (SRRs) to its anisotropic counterpart; hence, it is muchmore complicated. On the other hand, it will be shown in the nextsections that waveguides containing anisotropic negative materialshave unique propagation characteristics and can provide valuableapplications in practice. The approach of using anisotropic meta-materials is more flexible and simple, and it can offer more choicesand simplify the fabrication procedure drastically. In this paper,the general rules of TE- and TM-wave decomposition in anisotro-pic metamaterials will be obtained and waves in waveguidescontaining anisotropic metamaterials will be studied in detail.

2. CONDITIONS OF THE EXISTENCE OF TE AND TMMODES IN ANISOTROPIC METAMATERIALS

In this paper, a general case of material with six different compo-nents of permittivity and permeability in rectangular coordinateswill be studied. All six components, �x, �y, �z, �x, �y, and �z canbe positive or negative and can be equal to or different from eachother. A time dependence of the form ej�t is assumed and thevariation of the field components in the axial direction z is givenby e�j�z. Following the approach of [11] and separating the fieldcomponents to be longitudinal and transversal (Et, Ez, Ht, andHz), from Maxwell’s equations we have in rectangular coordinates

�Hx Hy Ex Ey�T � A� �1 � ��

Ez

y

Ez

x

Hz

y�

Hz

x �T

, (1)

where T denotes the transposition of the matrix and the followingequation:

A� �1 �

��j��y��

2�x�y �2� 0 0 j���2�x�y �2�0 �j��x��

2�y�x �2� �j���2�y�x �2� 00 �j���2�y�x �2� �j��y��

2�y�x �2� 0j���2�x�y �2� 0 0 �j��x��

2�x�y �2��

��2�x�y �2���2�y�x �2�.

426 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 41, No. 5, June 5 2004