Int. J. Electron. Commun. (AEÜ) 62 (2008) 349–355

www.elsevier.de/aeue

Electromagnetic properties of inverted � dielectric waveguides

José R. García∗, Susana F. Fernández, Diego P. Ayuso, Miguel G. Granda

Facultad de Ciencias, Applied Physics Department, University of Oviedo, C/ Calvo Sotelo, s/n. 33007 Oviedo, Spain

Received 28 July 2006; accepted 9 May 2007

Abstract

We present the inverted � dielectric waveguide as an alternative for designing microwave devices as well as electromagneticband gap structures. Inverted � dielectric waveguides in cascade, and with high-index contrast, are enclosed by metallic rect-angular waveguides. The generalized scattering matrix concept, together with the generalized telegraphist equations (GTE)formulism and the Modal-Matching Technique (MMT) were implemented for theoretical analysis. Numerical and experi-mental results confirm the possibilities of the new dielectric waveguide for designing single and low-cost microwave devices.� 2007 Elsevier GmbH. All rights reserved.

Keywords: Dielectric waveguides; Discontinuities; Microwave devices; Band-gap structures

1. Introduction

At microwave frequencies, dielectric waveguides wereextensively used for designing passive devices such as di-rectional couplers, Y-junctions and transitions [1–6]. Singleand multiple abrupt discontinuities in dielectric waveguideswith rectangular cross-section were studied in depth [7–12].For simplicity and design criteria, the image and slab dielec-tric waveguides were taken as canonical structure; however,microwave devices combining image dielectric waveguides,directed � dielectric waveguides and metallic walls weresuccessfully designed and implemented [1,2]. Besides, di-electric rods and dielectric posts were proposed for electro-magnetic guiding and resonance proposals [13,14].

In the present paper, a new dielectric waveguide is intro-duced as a good candidate for microwave devices design.We denote it as inverted � dielectric waveguide. Inverted �dielectric waveguides, showing high contrast in the per-mittivity of the core, were connected in cascade and ana-lyzed theoretically and experimentally. For this purpose, the

∗ Corresponding author. Tel.: +3485 103301; fax: +3485 103324.E-mail address: jose@uniovi.es (J.R. García).

1434-8411/$ - see front matter � 2007 Elsevier GmbH. All rights reserved.doi:10.1016/j.aeue.2007.05.005

inverted � dielectric waveguides, with low- and high-corepermittivity, were connected abruptly and alternatively in-side metallic rectangular waveguides. The theoretical anal-ysis was carried out by means of the Generalized ScatteringMatrix Method (GSMM). In order to obtain the General-ized Scattering Matrix (GSM) of any cascaded set of abruptdiscontinuities, the Generalized Telegraphist Equations(GTE) formulation [15] and the Modal-Matching Technique(MMT) [16–18] were applied. Different periodical configu-rations were tested theoretically and experimentally. Resultsfor the reflection and transmission coefficients (moduliand phase) of the fundamental proper mode, as well asfor higher-order modes, are given. The power conservationendorses the good performance of the theoretical analysis.

2. Theory

Basically, the inverted � dielectric waveguide consists ofa square dielectric rod (permittivity �1), with a symmetricaland longitudinal channel. The channel may be air or anydielectric (permittivity �2), which can be removed and/orslided of a free way inside of the channel. Fig. 1 representsan inverted � dielectric waveguide (a) and the cross-section

350 J.R. García et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 349–355

Fig. 1. Inverted � dielectric waveguide (a) and cross-section ofa shielded inverted � dielectric waveguide (b). Geometrical andelectrical parameters are shown.

Fig. 2. Multiple discontinuities set in inverted � dielectric waveg-uides connected abruptly. Geometrical and electrical parametersare shown.

of an inverted � dielectric waveguide inside a metallicrectangular waveguide (b).

We are interested in solving the scattering electro-magnetic problem inside metallic rectangular waveguidespartially filled with n inverted � dielectric waveguides

Fig. 3. An abrupt discontinuity between two inverted � dielectricwaveguides, a and b, connected abruptly and enclosed by perfectlyconducting walls.

connected in cascade and abruptly, as it is shown inFig. 2. The complete mathematical algorithm is divided intwo steps:

(A) dielectric waveguides proper modes analysis(B) GSM evaluation: single and multiple discontinuities

2.1. Dielectric waveguides proper modes analysis

The method applies the MMT to analyze each single dis-continuity. Fig. 3 is a side view of an abrupt discontinuitybetween two inverted � dielectric waveguides enclosed byperfectly conducting walls. The application of the MMT re-quires the evaluation of the proper modes in the shielded di-electric waveguides. Slab and planar dielectric waveguides(1D) propagate TE and TM proper modes; however, rect-angular and channel dielectric waveguides (2D) propagatehybrid modes, which can be approximated by E

ypq and Ex

pq

proper modes for media with diagonal dielectric tensors[19–25]. This is the case of the inverted � dielectric wave-guide.

Assuming waveguides free from losses, small variationsof the permittivity, and following the procedure shown in[26–29], the proper modes solutions in each shielded in-verted � dielectric waveguide are calculated. Surface, fast,evanescent and complex wave proper mode solutions can beobtained.

2.2. GSM evaluation: single and multiplediscontinuities

Regarding Fig. 3, we denote as waveguide a and waveg-uide b the shielded inverted � dielectric waveguides locatedat z < 0 and z > 0, respectively, of the abrupt discontinuity,at z = 0.

Once the proper modes of each inverted � dielectricwaveguide were calculated, the application of the MMT

J.R. García et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 349–355 351

Fig. 4. General representation of a cascaded set of N abrupt dis-continuities in shielded dielectric waveguides.

procedure at each discontinuity provides its single GSM[26–29]. This process is applied to each discontinuity sothat, finally, we have the same number of single GSMs asdiscontinuities.

In our case, the GSM of a single discontinuity symbol-izes a matrix of electromagnetic ports (proper modes) cor-responding to physical ports; besides, we are interested inmultiple discontinuities in cascade.

Fig. 4 is a general representation of a cascaded set ofN abrupt discontinuities in arbitrary dielectric waveguidesenclosed by perfectly conducting walls.

We denote the shielded dielectric waveguides using thelabels: a, b, c, . . . , n, n + 1. The capitals: M, N, P, Q, . . . ,

R, T , U represent the number of proper modes to betaken in the successive waveguides, respectively. So, n + 1different waveguides give N cascaded abrupt discontinu-ities, labelled as: I, II, III, . . . , N − 1, N . The disconti-nuities are separated by the dielectric waveguide lengths:l1, l2, l3, . . . , ln−1. The successive single GSMs are denotedas [SI], [SII], [SIII], . . . , [SN−1], [SN ]. To determine thetotal GSM, [ST], we join the N GSMs, two by two andcorrelatively [27,30,31]. Finally, we obtain the total GSM,[ST], corresponding to the N discontinuities. It has the form

[ST] =[ [ST

11(M × M)] [ST12(M × U)]

[ST21(U × M)] [ST

22(U × U)]]

, (1)

where M and U represent, respectively, the number of propermodes in the empty metallic rectangular waveguides locatedbefore and after the complete shielded dielectric structure.

3. Results

In order to demonstrate the efficiency of the method foranalyzing inverted � dielectric waveguides, as well as thepossible application of this new dielectric waveguide for mi-crowave devices design purpose, we have obtained experi-mental results which are compared with the theoretical ones.

As we are interested in the electromagnetic performanceof the shielded dielectric structures, the input and output

Fig. 5. Top and cross section views of three periodical configura-tions built in inverted � dielectric waveguides connected abruptly.In all cases: �1 = 2.1 and �2 = 10. Dimensions in mm.

physical ports (i = 1, 2; j = 1, 2) are always empty metal-lic rectangular waveguides. Consequently, our scattering re-sults of interest refer to the fundamental and higher-ordermodes solutions supported by the empty rectangular metal-lic waveguides enclosing the dielectric waveguides. Evi-dently, the number of proper modes in our empty metallicwaveguides depends on the cross-section dimensions andfrequency range under analysis.

The inverted � dielectric waveguides were built in teflon(�r1 = �1 = 2.1), and fixed to the wider side of the rectan-gular metallic waveguide with double adhesive tape, sym-metrically, as shown in Fig. 1(b). The dielectric block in thechannel was alumina (�r2=�2=10). The cross dimensions ofthe rectangular metallic waveguide were 22.86×10.16 mm2,typically used in the X-band frequencies. The experimentalresults were taken with the HP-8510 network analyzer. In allcases, results of the reflection and transmission coefficients,both moduli and phase, for the fundamental proper modewere measured and compared with the theoretical ones. Forthis reason, although the complete generalized scatteringmatrix was calculated, only the first column of the subma-trices [ST

11, ST21] and [ST

12, ST22] is relevant because we are

interested, specifically, in the fundamental proper mode. Wewill denote by R and T the reflection, S11, and transmission,S21, coefficients for the fundamental proper mode.

The results presentation starts for the dielectric configu-rations shown in Fig. 5: three inverted � dielectric waveg-uides in cascade with air–alumina–air in the channel (a),one inverted � dielectric waveguide with alumina filling the

352 J.R. García et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 349–355

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-40

-35

-30

-25

-20

-15

-10

-5

0

R, T

(dB

)

F (GHz)

R, experiment

R, theory

T, experiment

T, theory

R, experiment

R, theory

T, experiment

T, theory

(a)

0

20

40

60

80

100

Reflecte

d,R

, and tra

nsm

itte

d, T

, p

ow

er

(%)

R (%)

T (%)

R+T (%)

(b)

0

50

100

150

200

250

300

350

400

Phase (

degre

es)

(c)

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

F (GHz)

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

F (GHz)

Fig. 6. Theoretical and experimental results for structure 5(a).Reflection, R, and transmission, T, coefficients, versus frequency,for the fundamental proper mode: moduli (a), power conservation(b) and phase (c). Basic modes number: 15. Proper modes numberin all waveguides: 10.

channel (b) and the same length as in case (a) and15 inverted � dielectric waveguides in cascade withalumina–air–alumina, . . . as dielectric in the channel (c).

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-25

-20

-15

-10

-5

0

R, T

(dB

)

F (GHz)

R, experimental R, theory T, experimental T, theory

Fig. 7. Theoretical and experimental results for structure 5(b).Reflection, R, and transmission, T, coefficients, versus frequency,for the fundamental proper mode: moduli. Basic modes number:15. Proper modes number in all waveguides: 10.

(a)

18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0

0

20

40

60

80

100

Hig

her

ord

er

modes p

ow

er

(%)

F (GHz)

R1

, 1st

mode

R2

, 2nd

mode

T1

, 1st

mode

T2

, 2nd

mode

Total power

(b)

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-25

-30

-40

-35

-20

-15

-10

-5

0

R, T

(dB

)

F (GHz)

R, experimental

R, theory

T, experimental

T, theory

Fig. 8. Theoretical and experimental results for structure 5(c).Reflection, R, and transmission, T, coefficients, versus frequency,for the fundamental proper mode: moduli (a) and higher-ordermodes power conservation (b). Basic modes number: 30. Propermodes number in all waveguides: 20.

J.R. García et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 349–355 353

Fig. 9. Top and side views of five inverted � dielectric waveguidesin cascade. Cross-section dimensions are shown (mm).

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-25

-20

-15

-10

-5

0

R, T

(d

B)

F (GHz)

RT EBG

Fig. 10. Theoretical results for the periodical structure in Fig. 9when li (i=1, 2, 3, 4, 5)=30 mm. Reflected, R, and transmitted, T,coefficients, versus frequency, for the fundamental proper mode.

In Fig. 6(a)–(c), theoretical results are compared with theexperimental ones for the reflection, R, and transmission, T,coefficients of the fundamental proper mode for the structurein Fig. 5(a). Both moduli and phase are shown, as well asthe power conservation characteristics.

Fig. 7 compares the theoretical and experimental resultsfor the reflection, R, and transmission, T, coefficients of thefundamental proper mode for the structure in Fig. 5(b). Theresults for the 15 inverted � dielectric waveguides connectedin cascade (Fig. 5(c)) are shown in Fig. 8(a)–(b).

In order to explore in depth the possibilities of the inverted� dielectric waveguide for microwave design proposes, ad-ditional numerical results were obtained. We have analyzeda periodical structure consisting of five inverted � dielec-tric waveguides connected in cascade, as shown in Fig. 9.Fig. 10 shows the reflected, R, and transmitted, T, coeffi-cients of the fundamental proper mode, for the periodicalstructure in Fig. 9.

According to the previous results, EBG structures canbe obtained at microwave frequencies by using shieldedinverted � dielectric waveguides, appropriately designedand connected in cascade. In our case, several differ-ent periodical configurations were designed and tested to

Fig. 11. Top view and cross-section of a dielectric periodicalconfiguration, in inverted � dielectric waveguides, analyzed atmicrowave frequencies: 23 inverted � dielectric waveguides incascade (�2 = 1 (air) in odd-order waveguides and �2 = 10 ineven-order waveguides).

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-80

-70

-60

-50

-40

-30

-20

-10

0

R, T

(dB

)

F (GHz)

R

TEBG

Fig. 12. Theoretical results for the periodical structure in Fig. 11.Reflected, R, and transmitted, T, coefficients, versus frequency, forthe fundamental proper mode.

demonstrate this behavior. As an example, Fig. 11 shows aperiodic structure and the cross section dimensions. In thiscase, the physical parameters were as follows: 23 inverted� dielectric waveguides; conducting walls cross section di-mensions (mm); a =22.86, b=10.16; basic modes number:20; proper modes number: M = N = P = · · · = 20; �2 = 2.1(teflon), �2 = 10 (alumina) in odd-order inverted � dielec-tric waveguides and �2 = 1 (air) in even-order inverted �dielectric waveguides.

Fig. 12 presents the moduli of the reflection, R, and trans-mission, T, coefficients for the fundamental proper mode.An EBG, from 9.125 to 10 GHz, was obtained.

The authors continue exploring the possibilities of theshielded inverted � dielectric waveguide for designing fil-ters and electromagnetic band gaps devices at microwavefrequencies. Recent results confirm very interesting perfor-mances and will be published in the future.

354 J.R. García et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 349–355

4. Conclusions

The procedure described in this paper provides good ac-curacy in finding the proper modes in arbitrary dielectricwaveguides bounded by perfectly conducting walls withrectangular cross section. Surface, fast, evanescent and com-plex proper modes can be evaluated. By using these wavesolutions, the method combines the MMT technique withthe GSM concept, resulting in a successful and powerful toolto evaluate the electromagnetic scattering caused by abruptdiscontinuities in cascade between dielectric waveguides.The inverted � dielectric waveguide was introduced as agood candidate for microwave devices design. Different pe-riodical configurations in inverted � dielectric waveguides,showing high contrast in the permittivity of the core, wereconnected in cascade inside a metallic rectangular waveg-uide, and analyzed theoretically and experimentally. In allcases, an excellent agreement between theory and experi-ment was noticed. Additional numerical results have shownthe possibility of designing low-cost microwave passive de-vices, by using rectangular metallic waveguides enclosinginverted � dielectric waveguides in cascade.

Acknowledgments

This work was supported by the Projects from MEC(TIC2002-02300), (TEC2005-05541) and from FICY IB05-151C1. Special thanks to Prof. A. Mediavilla, from Univer-sity of Cantabria, for taking measurements.

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José Rodríguez García is a Professorat the University of Oviedo (Spain).He received the Teaching degree fromthe University of Oviedo, and the BScExtraordinary PhD degrees in physicsfrom the University of Santander(Spain). From 1982 to 1988 he workedin electromagnetic field analysis ondielectric waveguides in the Electron-ics Department of the University of

Santander. As member of the COST-216 European Project, hehas worked at the ETH (Zurich) as well as for the CTNE andNESTLE Companies. Since 1988 he has been a Professor at theUniversity of Oviedo and since 1993 Supervisor and Coordinatorof the Secondary School. He is a member of the ElectromagnetismAcademy (USA) and member of the SPIE society. He was selectedas Man of the Year 1997 by the ABI and he was included in:Who’s Who in Electromagnetics, Who’s Who in the World andWho’s Who in Contemporary Achievements. He conducts researchin the following areas: electromagnetic field theory, modelling,characterization and evaluation of integrated optical waveguidesand devices.

Diego Francisco Pozo Ayuso receivedthe degree in Physics from the Uni-versity of Oviedo, Spain. Since 2003he is working in optical waveguidefabrication and characterization in thePhysics Department of the Universityof Oviedo. In 2005 he received a grantto doctoral courses from this Univer-sity. Moreover, this year he receivedthe pedagogic aptitude for teachers of

Secondary Education, obtained a master in management of the en-vironment from Asturias Business School and worked in an envi-ronmental control company. He is currently working as researcherfor a coordinated project between the inmunoelectroanalisis groupand the integrated optic group of the University of Oviedo. Hisresearch interests are fabrication and characterization of opti-cal waveguides, capillary electrophoresis microchips (MCE) andbiosensors microdevices based on Mach–Zehnder interferometer.

Miguel García Granda received hisdegree in Physics in 2003 from the Uni-versity of Oviedo (Spain). During 2003he joined the Hanh-Meitner Institut inBerlin (Germany) where he workedin chalcopyrite solar cells fabricationtechniques and structural characteriza-tion. In 2004 he received a doctoralgrant from the Spanish Ministry of Ed-ucation and Science. Currently, he is

pursuing his PhD which is directed towards the study of opti-cal waveguides design, fabrication and characterization, as wellas Mach–Zehnder optical modulators, within a cooperation frame-work between the Universities of Oviedo (Spain) and Paderborn(Germany).