mapping method. The results of the obtained closed-form analyt-ical formulations have been compared with the results of availablestudies in the literature and Sonnet simulations. Both of thesecomparisons demonstrate and verify the correctness of the methodpresented in this paper. Based on the low dispersive features of theSC and SBC CPWs, these transmission lines can be utilized inmicrowave frequencies for related applications. Also, the obtainedanalytical formulations are fast and accurate for use in CAD-oriented microwave circuit design.

ACKNOWLEDGMENT

This work was supported by the Scientific and TechnologicalResearch Council of Turkey (TUB-TAK) under grant no. 105 E022.

REFERENCES

1. S.S. Bedair and I. Wolff, Fast and accurate analytic formulas forcalculating the parameters of a general broadside-coupled coplanarwaveguide for (M)MIC applications, IEEE Trans Microwave TheoryTech 37 (1989), 843–850.

2. M. Tran and C. Nguyen, Modified broadside-coupled microstrips linessuitable for MIC and MMIC applications and a new class of broadside-coupled band-pass filters, IEEE Trans Microwave Theory Tech 41(1993), 1336–1342.

3. C. Nguyen, Broadside-coupled coplanar waveguides and their end-coupled band-pass filter applications, IEEE Trans Microwave TheoryTech 40 (1992), 2181–2189.

4. C. Nguyen, Dispersion characteristics of the broadside-coupled copla-nar waveguide, IEEE Trans Microwave Theory Tech 41 (1993), 1630–1633.

5. N. Yuan, C. Ruan, W. Lin, J. He, and C. He, Coplanar coupled lines:The effects of the presence of the lateral ground planes, upper and lowerground planes and the V-Shaped microshield ground walls, Proc IEE142 (1995), 63–66.

6. G.G. Gentili and G. Macchiarella, Quasistatic analysis of shieldedplanar transmission lines with finite metallization thickness by a mixedspectral-space domain method, IEEE Trans Microwave Theory Tech 42(1994), 249–255.

7. D. Homentcovschi, A. Manolescu, A.M. Manolescu, and L. Kreindler,An analytical solution for the coupled striplinelike microstrip lineproblem, IEEE Trans Microwave Theory Tech 36 (1988), 1002–1007.

8. N.I. Dib and L.P.B. Katehi, Impedance calculation for the microshieldline, IEEE Microwave Guided Wave Lett 2 (1992), 406–408.

9. K.K.M. Cheng and I.D. Robertson, Quasi-TEM study of microshieldlines with practical cavity sidewall profiles, IEEE Trans MicrowaveTheory Tech 43 (1995), 2689–2694.

© 2006 Wiley Periodicals, Inc.

SCATTERING AND RADIATIONCHARACTERISTICS OF STEPDISCONTINUITY IN A DOUBLE-NEGATIVE (DNG) SLAB WAVEGUIDEOPERATING IN THE EVANESCENTSURFACE MODE

Meng Huang and Shanjia XuDepartment of Electronics Engineering and Information ScienceUniversity of Science and Technology of ChinaHefei, Anhui 230027, People’s Republic of China

Received 19 November 2005

ABSTRACT: The scattering and radiation characteristics of a step dis-continuity in a double-negative (DNG) slab waveguide operating in theevanescent surface mode are analyzed by a method which combines therigorous mode-matching method with multimode network theory. In theanalysis, the radiation problem is transformed into a propagation prob-lem so as to tremendously simplify the calculation procedure. The dis-persion properties of the grounded planar DNG slab waveguide for or-dinary and evanescent surface modes are examined, from which the left-hand property of the operating mode is verified. The relative scatteredpower and radiation pattern of the step discontinuity are given for thefirst time, from which the backward radiation of the stepped DNG guideis clearly demonstrated. © 2006 Wiley Periodicals, Inc. Microwave OptTechnol Lett 48: 1085–1088, 2006; Published online in Wiley Inter-Science (www.interscience.wiley.com). DOI 10.1002/mop.21607

Key words: DNG slab; step discontinuity; evanescent surface wave;backward radiation

1. INTRODUCTION

A homogeneous isotropic electromagnetic material with negativepermittivity and permeability referred to as a “left-handed (LH)”medium has been theoretically analyzed by V.G. Veselago in 1967[1], and the anomalous negative refraction was indicated in [2]. Inrecent years, different fundamental issues related with the mediumhave been investigated, including the dispersion of guided wavesin LH material [3, 4] and the “backward” nature of wave propa-gation in such kind of material. It has been indicated that theeigenmodes of DNG slabs surrounded by a right-handed medium[4] consist of two kinds of surface modes: one is the ordinarysurface wave and the other is the evanescent surface wave. Also,the anomalous dispersion and energy-flux properties have beenreported.

In this paper, the scattering and radiation characteristics of stepdiscontinuity in a DNG slab waveguide operating in an evanescentsurface wave are studied using a method which combines therigorous mode-matching method with the multimode network the-ory. The dispersion curves of both ordinary and evanescent surfacewaves with TM-mode polarization are given in section 2. Theexistence of the evanescent surface mode is justified from thedispersion curves in the left-hand region.

The propagation and radiation characteristics of the step dis-continuity in the longitudinal direction are given in section 3.Unlike the conventional method to treat the radiation as a “source-field” problem, in the present approach, the radiation problem hasbecome the propagation problem of a series of surface and spacewaves from the viewpoint of scattering. As a result, the radiationanalysis procedure is tremendously simplified. Based on this anal-ysis, the numerical results, including the relative scattered power

This work is supported by the National Natural Science Foundation ofChina (nos. 60471037 and 60371010).

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 6, June 2006 1085

and radiation pattern of the step, are presented and discussed insection 4.

2. DISPERSION PROPERTIES OF SURFACE WAVES IN DNGSLAB

Figure 1(a) shows the structure under consideration, which is agrounded DNG slab with negative permittivity � � �0�r andpermeability � � �0�r. Figure 1(b) is its transverse equivalentnetwork presentation.

Without loss of generality, in this paper the TM mode isconsidered. The expressions of the relevant characteristic imped-ances in the air and slab regions respectively with subscripts 0 ands are given by

Z0TM �

kx0

��0, Zx

TM �kxs

��0�r, (1)

where

kx0 � �k02�a�a � kz

2�1/ 2,

kxs � �k02�r�r � kz

2�1/ 2,

with

k02 � �2�0�0. (2)

The dispersion equation can be obtained with the transverseresonance condition as:

jZstan�kxsd� � Z0 � 0. (3)

The present analysis verifies that there are two kinds of eigensurface waves for the DNG structure, the ordinary surface wave,and the evanescent surface wave. The former has real transversewavenumber kxs � �xs and is propagating inside the slab; thelatter has positive imaginary transverse wavenumber kxs � j�xs

and is increasing inside the slab, while both of them have negativepurely imaginary transverse wavenumber kx0 � �j�x0, and henceare transversely attenuating in the air region.

Figure 2 shows the dispersion curves of the TM mode in thegrounded DNG slab. In order to guarantee the DNG slab anoptically denser medium, the value of �r�r is taken to be largerthan �a�a.

It can be seen from the dispersion curves that in the region ofd/ 0.674, all the ordinary surface waves are below cutoff,only evanescent surface wave propagates along the z direction.Furthermore, there are no cutoff frequencies for the evanescentsurface wave in this region. This reveals that the evanescent

surface mode is the fundamental mode in the DNG slab waveguideinstead of the ordinary surface-wave mode.

The curves in Figure 2 are actually the �–� diagram, fromwhich the group velocity (�g � ��/��) and phase velocity (�p ��/�) can be inferred. It is shown that �g and �p are alwaysantiparallel (�p�g � 0) for the evanescent surface wave, whichindicates that the mode is of the left-hand property.

3. SCATTERING CHARACTERISTICS OF THE STEPDISCONTINUITY IN DNG SLAB

In the analysis, a technique of discretizing the continuous spectruminto a complete set of discrete modes is employed by introducingan oversize metallic plate far above the slab. With such an approx-imation, the mathematical analysis and the physical interpretationon wave phenomena can be kept simple and clear. It should bestressed that although the discretization of modes does introducesome degree of approximation as far as the surface waves areconcerned, the presence of an oversize metallic plate does notchange the physics of the problem.

Figure 3 depicts the step discontinuity in the DNG slabwaveguide in which the regions in the two sides of the disconti-nuity are respectively denoted as regions I and II. Referring toFigure 3, it is observed that the tangential components of the fieldsat the discontinuity are the x and the y components; therefore, onlythose components shall be considered explicitly. It is well knownthat the general-field solution in each constituent region may beexpressed in terms of the superposition of a complete set of theeigenmode functions. For region I, they can be expressed asfollows:

Figure 1 (a) Grounded DNG slab structure under consideration and (b)its transverse equivalent network presentation

Figure 2 DNG slab TM-mode dispersion curve (solid lines: ordinarysurface waves; dashed lines: evanescent surface waves)

Figure 3 DNG slab step discontinuity and discretization of continuousspectrum

1086 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 6, June 2006 DOI 10.1002/mop

Ex� x, z� � �n�1

�1

�� x� n� x�Vn� z�,

Hy� x, z� � �n�1

�

n� x�In� z�. (4)

Here, n is the nth mode function and Vn and In are respectivelythe voltage and current of the nth mode. A similar set of tangentialfield components may be written for region II, but it is omitted herefor simplicity.

At the step discontinuity (at z � 0), the tangential fieldcomponents must be continuous:

�n�1

�1

�� x� n� x�Vn� z� � �

n�1

�1

�� � x� � n� x�V� n� z�,

�n�1

�

n� x�In� z� � �n�1

�

� n� x�I�n� z�. (5)

The quantities with a superbar indicate those in region II. Scalarmultiplying these equations with m and making use of the fol-lowing orthogonality relation of the eigenfunctions,

� m( x)� 1

�( x)� n( x)� � �mn. (6)

We then obtain the following matrix equations:

V � PV� , I � QI�, (7)

where V and I are column vectors with the transmission-linevoltage and current of the TM mode, Vn and In are, respectively,the nth components; a similar definition holds for those vectorswith a superbar. P and Q are matrices characterizing the couplingof modes at the step discontinuity, and their elements are definedby scalar products of the mode functions on the two sides of thediscontinuity as follows:

Pmn � � m( x)� 1

�� ( x)� � n( x)� ,

Qmn � � m( x)� 1

�( x)� � n( x)� . (8)

According to the multimode network theory, we obtain thevoltage and current relations:

V�0�� � Z�0��I�0��,

V�0�� � Z�0��I�0��. (9)

Using Eq. (7) and the above matrix identities, we obtain

Z�0�� � PZ�0��Q�1. (10)

The reflection coefficient matrix � at the z � 0� plane lookingto the right of the discontinuity can be easily obtained as follows:

Z�0�� � Z� 0,

Z� � Z�0��,

� � �Z� � Z0��1�Z� � Z0�, (11)

where Z0 and Z� 0 are the characteristic impedance matrix in regionsI and II, respectively.

The voltage and current of the modes can be derived by theincident wave matrix:

V� � �1 � ��Ainc,

I� � �1 � ��Ainc,

I� � Q�1I�,

V� � Z�I�. (12)

Thus, power carried by the nth TM mode can be derived by thevoltage and current column vectors on the right side:

Pn �1

2Re�Vn

��In��*�. (13)

The radiation angles can be calculated by the following formula-tion:

�n � cos�1��n/k0�. (14)

Figure 4 Comparison of scattering characteristics calculated with dif-ferent methods

Figure 5 Variations of TM mode scattered power with d2/, �r ��0.6, �r � �2.55

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 6, June 2006 1087

Then the radiation pattern can be determined by the power of thenth mode and the corresponding radiation angle.

4. NUMERICAL RESULTS

To demonstrate the scattering and radiation characteristics of stepdiscontinuity in the grounded DNG slab for TM-mode incidence,some numerical results are given here. The geometrical and phys-ical parameters are shown in Figure 3.

Figure 4 shows a comparison of the scattering characteristicsfor a step discontinuity in double-positive (DPS) slab waveguide,with all parameters of the structure the same as in Rozzi [6] forconvenience of comparison. It is found that the present resultsagree very well with those given in [6]; thus, the effectiveness andaccuracy of the present analysis are justified.

According to the analysis in section 2, in the region d/ 0.674, all the ordinary surface waves are below cutoff, and onlythe evanescent surface wave propagates along the longitudinaldirection. In order to guarantee the structure working in the left-hand region, d1/ � 0.45 is chosen since in this case there is onlyevanescent surface wave in region I.

Figure 5 shows the variation of the reflected, radiated, andtransmitted power of the fundamental TM mode in the DNG slabwith the thickness d2 of the slab in region II, while the thicknessd1 � 0.45 is fixed. When d2/ � 0, there is no surface wave inregion (II) so does the transmission power. And about 68% of theincident power reflects while 32% radiates. As d2 increases, therelative transmitted power is gradually enhanced and the reflectedpower and radiated power decrease; this is because the disconti-nuity is weakened as d2 increases. When d2 reaches to 0.45 �d1, the step discontinuity disappears, all the incident power trans-mits into region II, and the reflected and the radiated powerdecrease to zero.

Figure 6 shows the radiation patterns for TM-mode incidence atdifferent frequencies. According to the analysis in section 2, thefundamental mode of the structure is left-hand property in naturebecause it is operating in the left-hand region. Therefore, thestepped structure should radiate backward, which is clearly dem-onstrated in this figure. As the frequency increases from 45 to 55GHz, the maximum radiation angle increases from about 130.7° to149.8°, and the 3-dB beamwidth of the pattern decreases.

5. CONCLUSION

In this paper, the scattering and radiation characteristics of a stepdiscontinuity in DNG slab operating in evanescent surface waveshave been analyzed using the mode-matching method combinedwith multimode network theory. The TM-mode dispersion curvesof the grounded DNG slab have been presented, including ordinaryand evanescent surface waves. The scattering characteristics of thediscontinuity structure have been carefully investigated. Differing

from the conventional “source-field” viewpoint on the radiationproblem, the present approach changes the radiation problem to thepropagation problem; thus, the analysis procedure is tremendouslysimplified without jeopardizing the accuracy. Furthermore, theradiation patterns of the step have been given and discussed. It wasverified that the evanescent surface mode in the present structure isoperating in the left-hand region, so that it has the left-handproperty, as shown by the dispersion curves and the backward-radiation patterns presented.

REFERENCES

1. V.G. Veselago, The electrodynamics of substances with simultaneouslynegative values of � and �, Usp Fiz Nauk 92 (1967), 517–526.

2. A. Shelby, D.R. Smith, and S. Schultz, Experimental verification of anegative index of refraction, Science 292 (2001), 77–79.

3. A. Alu and N. Engheta, Guided modes in a waveguide filled with a pairof single-negative (SNG), double-negative (DNG), and/or doubleposi-tive (DPS) layers, IEEE Trans MTT 52 (2004), 199–210.

4. P. Baccarelli, et al., Fundamental modal properties of surface waves onmetamaterial grounded slabs. IEEE Trans MTT 53 (2005).

5. S.T. Peng and A.A. Oliner, Guidance and leakage properties of a classof the open dielectric waveguides, Part I: Mathematical formulation,IEEE Trans MTT MTT-29 (1981), 843–855.

6. T.E. Rozzi, Rigorous analysis of the step discontinuity in a planardielectric waveguide, IEEE Trans MTT MTT-26 (1978), 738–746.

© 2006 Wiley Periodicals, Inc.

USE OF EXTENDED GROUND PLANETO IMPROVE THE PERFORMANCE OFA SLOT-LOOP ANTENNA WITHNARROW OUTER CONDUCTOR

Yuming Song,1 Joseph C. Modro,2 Pauline O’Riordan,2 andZhipeng Wu1

1 Electromagnetics CentreSchool of Electrical and Electronic EngineeringUniversity of ManchesterManchester M60 1QD, United Kingdom2 TDK Electronics Ireland Ltd.3022 Lake DriveCitywest, Dublin 24, Ireland

Received 18 November 2005

ABSTRACT: The performance of a slot-loop antenna with a very nar-row outer conductor can be significantly improved by connecting to anextended ground plane whose length is a quarter wavelength. The an-tenna has a reduced size and wider impedance bandwidth. Simulationshave been carried out in the one-wavelength and two-wavelength modesand confirmed by experiments. © 2006 Wiley Periodicals, Inc.Microwave Opt Technol Lett 48: 1088–1092, 2006; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21608

Key words: slot loop antenna; bandwidth improvement; antenna minia-turisation

1. INTRODUCTION

Slot-loop antennas have received much attention recently for theirpotential application in wireless communications because they areeasy to integrate with monolithic microwave integrated circuits.The slot-loop antenna has low Q-factor thus provides wide im-pedance bandwidth [1]. By varying the slot width, the bandwidthcan be further improved [2]. It is easy to generate several modes of

Figure 6 TM-mode radiation patterns at different frequencies d1 � 3mm, d2 � 2.367 mm, �r � �0.6, and �r � �2.55

1088 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 6, June 2006 DOI 10.1002/mop