on two different formulations of the coupled mode method: application to 3d rectangular...

Int. J. Electron. Commun. (AEÜ) 60 (2006) 690–704

www.elsevier.de/aeue

On two different formulations of the Coupled Mode Method: Application to3D rectangular chirowaveguides

Alvaro Gomeza, Angel Vegasb, Miguel A. Solanob,∗aDepartamento de Electricidad y Electronica, Universidad de Valladolid, Paseo Prado de la Magdalena s/n, Valladolid 47011, SpainbDepartamento de Ingenieria de Comunicaciones, Universidad de Cantabria, Avda. de los Castros s/n, 39005 Santander, Cantabria, Spain

Received 19 October 2005

Abstract

This paper develops two different formulations of the Coupled Mode Method (CMM), called EH-indirect and EH-directformulations, for rectangular waveguides nonhomogeneously loaded with bi-isotropic media. The results of the EH-indirectformulation are compared with the more usual EH-direct formulation, and its range of applicability is established. Whenthe EH-indirect or EH-direct formulations of the CMM are combined with the classical Mode Matching Method (MMM) ahybrid technique is obtained that is capable of solving three-dimensional (3D) discontinuities without restriction in number.The results of the generalized scattering matrix for the two formulations are compared and it is concluded that the indirectformulation gives more accurate results than the direct formulation.� 2006 Elsevier GmbH. All rights reserved.

Keywords: Bi-isotropic media; Coupled Mode Method; Discontinuities; Mode Matching Method; Nonhomogeneously loadedwaveguides; Rectangular waveguides

1. Introduction

During the past decades a great deal of theoretical workhas been done in order to characterize different electromag-netic guiding structures, which can be considered, gener-ally speaking, as composed of different types of disconti-nuities in waveguides that are partially or totally filled withdifferent dielectric media. From the point of view of thetype of analysis employed, the theoretical methods can beclassified as

• Analytical methods [1], in cases with simple geometries.• Intensive numerical methods, in both the time [2] and

frequency [3] domains.• Semi-analytical methods [4].

∗ Corresponding author. Tel.: +34 942 201538; fax: +34 942 201488.E-mail address: [email protected] (M.A. Solano).

1434-8411/$ - see front matter � 2006 Elsevier GmbH. All rights reserved.doi:10.1016/j.aeue.2006.01.005

From the point of view of the type of discontinuities,two large groups can be identified: one where the conduc-tor boundary is not uniform in the propagation direction [5]and the other where it is uniform and the discontinuities arecaused by the different media inside the boundary [6]. Obvi-ously, combinations of these two can also be found [7]. Fur-thermore, the kind of media can vary from simple isotropicdielectrics, to the more general bi-anisotropic media [8,9].

This paper is devoted to the Coupled Mode Method(CMM), which can be catalogued as a semi-numericalmethod, and its application to isotropic chiral media insidea rectangular waveguide. The CMM was first formulatedby S. Schelkunoff [10] and, afterwards, applied to differentstructures as, for example, open dielectric guides [11,12],waveguides with magnetized ferrites [13–15], bi-isotropicmedia [16–19] or bi-anisotropic media [20,21]. The maingoal of the CMM is to obtain the electromagnetic field andthe propagation constants in a uniform waveguide in the

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A. Gomez et al. / Int. J. Electron. Commun. (AEÜ) 60 (2006) 690–704 691

direction of energy propagation. The CMM is especiallyuseful when it is combined with the Mode Matching Method(MMM) [15] in order to characterize discontinuities betweenwaveguides partially filled with isotropic, anisotropic or bi-isotropic media.

In the first part of this paper we develop two differ-ent formulations of the CMM for a rectangular waveguidenonhomogeneously loaded with isotropic chiral media. Weuse the generic name “EH-formulations” to denote thesetwo formulations which, in fact, are two different strate-gies for obtaining some of the equations involved in theCMM [22]. The name EH for these formulations of theCMM comes from the fact that the fields �E and �H arethe fields expanded in terms of a set of base functions. Inthe second part we present some results of the scatteringmatrix for three-dimensional (3D) discontinuities betweenrectangular chirowaveguides and they are compared withsimilar results of other authors. Afterwards, some resultsof the propagation constants and the electromagnetic fieldfor rectangular chirowaveguides, when the chiral mediumis partially filling the rectangular waveguide, are shown.To our knowledge, these are the first results of propaga-tion constants and electromagnetic field for this type ofstructures.

2. Theory of the CMM

The CMM is based on expanding all components of theelectromagnetic field in the z-uniform guide under analysisas a linear combination of the T-potentials, which give thecomponents of the electromagnetic field of the modes (calledbasis modes) of an empty rectangular waveguide, as

Ex =ntm∑i=1

V(i)

�T(i)

�x+

nte∑l=1

V[l]�T[l]�y

, (1)

Ey =ntm∑i=1

V(i)

�T(i)

�y−

nte∑l=1

V[l]�T[l]�x

, (2)

Hx = −ntm∑i=1

I(i)

�T(i)

�y+

nte∑l=1

I[l]�T[l]�x

, (3)

Hy =ntm∑i=1

I(i)

�T(i)

�x+

nte∑l=1

I[l]�T[l]�y

, (4)

Ez =ntm∑i=1

kc(i)Vz(i)T(i), (5)

Hz =nte∑l=1

kc[l]Iz[l]T[l] + H0, (6)

where ntm and nte are, respectively, the number of TM andTE basis modes included in the expansions; V ’s and I ’s1

are the expansion coefficients; kc(i) and kc[l] are the cutoffwavenumbers of the basis modes. Parentheses refer to TM-modes and brackets to TE–modes. V ’s have dimensions ofvoltage and I ’s of current. Obviously, in order to imple-ment the method in a computer, the indices ntm and ntehave to be truncated to a finite number. The term H0 takesinto account the trivial solution for the TEnm modes in theexpansion for Hz (i.e., the “mode” TE00 as a part of theset of basis functions) which must be taken into accountin any medium except in an isotropic dielectric medium[13,15]. The expressions for T-potentials, (which depen-dence on x and y has been removed for simplicity), can beseen in Appendix A. Substituting (1)–(6) into the x- and y-components of Maxwell’s curl equations, and applying theGalerkin method, the Generalized Telegraphist’s Equationsare obtained as [10]

dV(p)

dz= j�

∫s

(Bx

�T(p)

�y− By

�T(p)

�x

)ds + kc(p)

V z(p), (7)

dV[q]dz

= −j�∫

s

(Bx

�T[q]�x

+ By

�T[q]�y

)ds, (8)

dI(p)

dz= −j�

∫s

(Dx

�T(p)

�x+ Dy

�T(p)

�y

)ds, (9)

dI[q]dz

= −j�∫

s

(Dx

�T[q]�y

−Dy

�T[q]�x

)ds+kc[q]I

z[q], (10)

where S is the cross-section of the waveguide, i.e., the inte-gration area. The indices p and q have the same meaning asthe indices i and l in the former equations. In [10] the termH0 is not included, but this has not influence in obtaining(7)–(10).

In order to obtain the desired equations giving the relationbetween the derivative with respect to the z-coordinate of theexpansion coefficients and the own coefficients, two taskshave to be done. One is to express the transverse componentsof the magnetic induction and the electric flux density interms of the T-potentials. This can be achieved by means ofthe constitutive relations; for an isotropic chiral medium thegeneral constitutive relations are [23,24]

�D = �r�0 �E − j�√

�0�0 �H , (11)

�B = �r�0�H + j�

√�0�0 �E, (12)

where �r is the relative permittivity, �0 is the vacuum per-mittivity (being the permittivity � = �r�0), �r is the relativepermeability, �0 is the vacuum permeability (being the per-meability � = �r�0) and � is the Pasteur parameter.

1 The coefficients V ’s and I ’s depend on z-coordinate as e−�z. Forsimplicity this dependence is removed throughout the paper.

692 A. Gomez et al. / Int. J. Electron. Commun. (AEÜ) 60 (2006) 690–704

Using (11) and (12) Dx , Dy , Bx , and By can be expressedin terms of Ex , Ey , Hx , and Hy , which are developed interms of the T-potentials by means of (1)–(4). Thus, (7)–(10)can be re-written as

dV(p)

dz= − k0

ntm∑i=1

V(i)

∫s

�

(�T(i)

�x

�T(p)

�y

− �T(i)

�y

�T(p)

�x

)ds − k0

nte∑l=1

V[l]

×∫

s

�

(�T[l]�x

�T(p)

�x+ �T[l]

�y

�T(p)

�y

)ds

− j��0

ntm∑i=1

I(i)

∫s

�r

(�T(i)

�x

�T(p)

�x

+ �T(i)

�y

�T(p)

�y

)ds + j��0

nte∑l=1

I[l]

×∫

s

�r

(�T[l]�x

�T(p)

�y− �T[l]

�y

�T(p)

�x

)ds

+ kc(p)V z

(p), (13)

dV[q]dz

= k0

ntm∑i=1

V(i)

∫s

�

(�T(i)

�x

�T[q]�x

+ �T(i)

�y

�T[q]�y

)ds + k0

nte∑l=1

V[l]

×∫

s

�

(�T[l]�y

�T[q]�x

− �T[l]�x

�T[q]�y

)ds

+ j��0

ntm∑i=1

I(i)

∫s

�r

(�T(i)

�y

�T[q]�x

− �T(i)

�x

�T[q]�y

)ds − j��0

nte∑l=1

I[l]

×∫

s

�r

(�T[l]�x

�T[q]�x

+ �T[l]�y

�T[q]�y

)ds, (14)

dI(p)

dz= − j��0

ntm∑i=1

V(i)

∫s

�r

(�T(i)

�x

�T(p)

�x

+ �T(i)

�y

�T(p)

�y

)ds − j��0

nte∑l=1

V[l]

×∫

s

�r

(�T[l]�y

�T(p)

�x− �T[l]

�x

�T(p)

�y

)ds

+ k0

ntm∑i=1

I(i)

∫s

�

(�T(i)

�y

�T(p)

�x

− �T(i)

�x

�T(p)

�y

)ds − k0

nte∑l=1

I[l]

×∫

s

�

(�T[l]�x

�T(p)

�x+ �T[l]

�y

�T(p)

�y

)ds, (15)

dI[q]dz

= − j��0

ntm∑i=1

V(i)

∫s

�r

(�T(i)

�x

�T[q]�y

− �T(i)

�y

�T[q]�x

)ds − j��0

nte∑l=1

V[l]

×∫

s

�r

(�T[l]�x

�T[q]�x

+ �T[l]�y

�T[q]�y

)ds

+ k0

ntm∑i=1

I(i)

∫s

�

(�T(i)

�x

�T[q]�x

+ �T(i)

�y

�T[q]�y

)ds − k0

nte∑l=1

I[l]

×∫

s

�

(�T[l]�x

�T[q]�y

− �T[l]�y

�T[q]�x

)ds

+ kc[q]Iz[q]. (16)

In (13)–(16) k0 is the vacuum wavenumber. Now, the othertask that must be done is to express the amplitudes of thelongitudinal Ez and Hz fields in terms of the transverse ones.Two strategies, or formulations, can be followed: direct andindirect.

2.1. EH-direct formulation (EHD)

The more intuitive way is to find Ez and Hz as a functionof Dz and Bz. Using (11) and (12) we get

Ez = 1

�

[�r

�0Dz + j�√

�0�0Bz

], (17)

Hz = 1

�

[j�√�0�0

Dz − �r

�0Bz

], (18)

where � = �r�r − �2 is a nondimensional parameter. Usingthe z-component of Maxwell’s curl equations, we have

Ez = 1

j��

[�r

�0

(�Hy

�x− �Hx

�y

)

− j�√�0�0

(�Ey

�x− �Ex

�y

)], (19)

Hz = − 1

j��

[j�√�0�0

(�Hy

�x− �Hx

�y

)

+ �r

�0

(�Ey

�x− �Ex

�y

)]. (20)

Now, using (1)–(6), we substitute the expansions for eachfield component into (19) and (20). Afterwards, the Galerkinmethod is applied to obtain the desired relations,

kc(p)V z

(p) = − 1

k0

nte∑l=1

V[l]k2c[l]k

2c(p)

∫S

�

�T[l]T(p) ds

− 1

j��0

ntm∑i=1

I(i)k2c(i)

k2c(p)

∫S

�r

�T(i)T(p) ds, (21)


kc[q]Iz[q] = 1

j��0

nte∑l=1

V[l]k2c[l]k

2c[q]

∫S

�r

�T[l]T[q] ds

− 1

k0

ntm∑i=1

I(i)k2c(i)

k2c[q]

∫S

�

�T(i)T[q] ds. (22)

If (21) is substituted into (13), and (22) into (16) the de-sired equations are obtained. As can be seen, in the directstrategy H0 has no influence on this process; however, H0has to be included in (6) and evaluated if Hz has to becomputed, except for isotropic dielectric media. Neverthe-less, no papers using the direct strategy consider this term[14,20,21,25,26].

Inspecting (21) and (22) it is obvious that the amplitudesof the longitudinal field components are obtained directlyin terms of the amplitudes of the transverse ones, and it isnot necessary to compute the inverse of any matrix (all thealgebraic process is manipulated in matrix form). But whatis important here is that (17) and (18) express, for partiallyfilled waveguides, a continuous field component (Ez andHz) as the product of discontinuous functions: one is com-posed of the different parameters of the medium inside therectangular waveguide, and the other of the discontinuousfield components (Dz and Bz). Analytically, this situationdoes not present any problem. However, all the field com-ponents are computed numerically as linear combinations ofcontinuous sinusoidal functions. Therefore, they are, strictlyspeaking, continuous functions even though some of themmust simulate functions that are analytically discontinuous.Thus, for example, (17) gives Ez (an analytically and nu-merically continuous function) as the product of one discon-tinuous function (parameters of the material) and Dz and Bz

(both analytically discontinuous but numerically continuousfunctions). This is not the ideal situation for obtaining theamplitude of Ez and is the main reason why the direct strat-egy gives poorer results than the indirect strategy for highvalues of the constitutive parameters. The same is true forHz in (18).

2.2. EH-indirect formulation (EHI)

In this formulation the constitutive relations (11) and (12),for the z-component, are used directly. Following the sameprocess as for the direct strategy we get

I(p) = − j��0

ntm∑i=1

kc(i)V z

(i)

∫S

�rT(i)T(p) ds

− k0

nte∑l=1

kc[l]Iz[l]

∫S

�T[l]T(p) ds

− k0h0

∫S

�T(p) ds, (23)

V[q] = − j��0

nte∑l=1

kc[l]Iz[l]

∫S

�rT[l]T[q] ds

− j��0h0

∫S

�rT[q] ds

+ k0

ntm∑i=1

kc(i)V z

(i)

∫S

�T(i)T[q] ds. (24)

It must be noted that now, unlike the direct strategy, inorder to obtain the final equations and it is necessary toobtain also the value of H0, which can be obtained from thez-component of (12) as

H0 = −[∫

S

�r ds

]−1{

nte∑l=1

kc[q]Iz[q]

∫S

�rT[q] ds

− j

Z0

ntm∑i=1

kc(i)Vz(i)

∫S

�T(i) ds

}, (25)

where Z0 = √�0/�0 = 120� �.

In this case, inspecting (23)–(25) the amplitudes of Ez andHz are not obtained directly and it is necessary to performthe inverse of one matrix. But the essential difference withrespect to the direct strategy is that now both equations (11)and (12) express the field components Dz and Bz, whichare numerically and analytically discontinuous, as productsof the numerically and analytically continuous field compo-nents Ez and Hz times the discontinuous functions of theconstitutive parameters. This is the ideal situation for obtain-ing good convergence in computing the amplitudes of thelongitudinal field components, although it has the drawbackof requiring the inverse of matrices.

3. Scattering in rectangular chirowaveguides

The first analysis of 3D discontinuities in chirowaveg-uides, presented in [25], was only valid for a structure com-posed of a section of chirowaveguide between two emptyrectangular waveguides (Fig. 1). The analysis was extendedby the same authors in [26]. They use a direct strategy ofthe CMM to obtain the propagation constants and the elec-tromagnetic field of each individual guide and, afterwards,

Fig. 1. Rectangular waveguide partially filled with a paral-lelepiped-shaped piece of an isotropic chiral medium.


0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

d (mm)

|S11

|

0 5 10 15 20-200

-100

0

100

200

d (mm)

Pha

se o

f S

11 (

deg

rees

)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

d (mm)

|S11

|

0 5 10 15 20-200

-100

0

100

200

d (mm)

Pha

se o

f S11

(de

gre

es)

EHI Formulation EHD Formulation [26]

(a) (b)

(c) (d)

Fig. 2. Results of the S11 parameter for the structure of Fig. 1 obtained with the indirect formulation (EHI) and the direct formulation(EHD) compared with the results of [26]. Dimensions in mm: a = 22.86, b = 10.16, w = 11, h = 5, f = 9 GHz. In Figs. 2a,b: �r = 3.78,�r = 1, � = 1.13. In Figs. 2c,d: �r = 4.32, � = 0.57.

following the technique proposed in [13], they uncouple thecoupled differential equation provided by CMM in order toobtain the generalized scattering matrix of the whole struc-ture. This method of analysis has two main drawbacks. First,they use a direct strategy2 which has been shown to be lessefficient than the indirect strategy when the values of theconstitutive parameters are high [19]. Second, the formula-tion for obtaining the generalized scattering matrix is onlyvalid for the structure analyzed. If the structure to be ana-lyzed has more discontinuities in the propagation directionthe formulation is not valid.

The method that we propose for solving 3D discontinu-ities avoids these two disadvantages, because the indirectstrategy is used for obtaining the propagation constants andthe electromagnetic field, and the scattering matrix are ob-tained by means of the MMM, without restriction on thenumber of discontinuities.

3.1. Mode Matching Method (MMM)

Once the electromagnetic field and the propagation con-stants of each individual guide are obtained, the application

2 Compare Eq. (7) of [26] with (17) and (18) of this paper which arethe same except for the constitutive parameters.

of the MMM does not depend on the kind of medium insidethe waveguide. The detailed formulation can be seen in [15]if the structure has the same metallic contour or in [27] if itis different. In the first case, the coupling integrals involvedin MMM are reduced to simply reordering the eigenvectors(coefficients of expansions (1)–(4)) and in the second onethey are reduced to a summation of products of the sameeigenvectors.

4. Results

There are no results in the literature about the propaga-tion constants and the electromagnetic field of rectangularwaveguides partially filled with chiral media. The only re-sults are for the scattering matrix of a finite length chi-rowaveguide between two empty waveguides [25,26]. Theonly difference between the method employed in [26] andour direct strategy of the CMM combined with the MMMis the technique used to obtain the generalized scatteringmatrix. In [26] a transmission line model, first presentedin [13], is used while we use the MMM. We have com-pared the results of the generalized scattering matrix of [13]for a ferrite obstacle in a rectangular waveguide to the re-sults obtained by using an indirect formulation of the CMM


0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

d (mm)

|S11

|

0 5 10 15 20

-200

-100

0

100

200

-200

-100

0

100

200

d (mm)

Ph

ase

of

S11

(d

egre

es)

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

d (mm)

|S11

|

0 5 10 15 20d (mm)

Ph

ase

of

S11

(d

egre

es)

EHI Formulation EHD Formulation [26]

(a) (b)

(c) (d)

Fig. 3. Results of the S11 parameter for the structure of Fig. 1 obtained with the indirect formulation (EHI) and the direct formulation(EHD) compared with the results of [26]. Dimensions in mm: a = 22.86, b = 10.16, w = 11, h = 5, f = 9 GHz. In Figs. 3a,b: �r = 1.32,�r = 1, � = 0.57. In Figs. 3c,d: �r = 2.54, � = 0.19.

(the same used in [13]) combined with the MMM. We foundthat the results are practically identical. Therefore, any ap-preciable difference between the results in [26] and the re-sults obtained using the hybrid technique CMM–MMM mustbe due to the use of the direct or the indirect strategies (orformulations) in the CMM.

All the results in [26] were obtained for relatively low val-ues of the permittivity and the chirality admittance �c. Thisparameter is related with � and �r by �c = (�/�r)

√�0/�0.

Therefore, few differences can be expected between the di-rect and indirect formulations, although they will be suf-ficient to distinguish between them. Only Fig. 5 of [26]presents results of the modulus of S11 for an isotropic di-electric obstacle with a high permittivity value (�r = 8.3)completely filling the height of the waveguide and placedsymmetrically with respect to the width of the waveguide.If the TE10-mode of the empty waveguide impinges on theobstacle, only LSEn0 modes, with n odd, are excited in thediscontinuities and, consequently, only TEn0 basis modes(with n odd) have to be used in the expansions of the field.The non-zero fields of these modes are Ey , Hx and Hz. Asthe only difference between the direct and indirect strate-gies is in the different treatment of the longitudinal compo-nents, in this case the possible difference is in Hz. But theobstacle is a non-magnetic material, so � = �0 and, con-sequently, the two strategies are identical. Thus, the results

of Fig. 5 in [26] are reproduced using only 5 basis modes(TEn0, n=1, 3, 5, 7, 9), irrespective of whether the direct orindirect strategy is used, and consequently, no conclusionsabout the accuracy of the direct and indirect formulationscan be drawn from that figure.

Before beginning with the comparisons, it is important topoint out that the authors of [26] use the constitutive relationsof [24]. These relations were first proposed by Boys and areknown as Boys-Post [28] constitutive relations. However,here we use the basic relations given by (11) and (12). Theconnections between the constitutive parameters of (11) and(12) and the constitutive parameters used in [24] are3

� = �BP + �BP�2c being �BP = �rBP�0,

� = �BP being �BP = �rBP�0,

� = �BP�c√�0�0

.

Note that for a chiral medium �BP �= �. The highest valuesin [26] are found in Fig. 8 (�rBP=2.5, �c=3 mS ⇒ �r =3.78,�=1.13) and in Fig. 11 (�rBP =4, �c =1.5 mS ⇒ �r =4.32,

3 We add the subscripts BP to the permittivity and the permeabilityparameters used in [26].


0 50 100 150 2002

2.1

2.2

2.3

2.4

2.5

Number of basis modes

β /k 0


β /k 0


β /k 0


β /k 0

0 50 100 150 2002.1

2.2

2.3

2.4

2.5

2.6

0 50 100 150 2002.2

2.3

2.4

2.5

2.6

2.7

0 50 100 150 2002.5

2.6

2.7

2.8

2.9

3

EHI FormulationEHD Formulation


EHI FormulationEHD Formulation EHI Formulation

EHD Formulation

(a) (b)

(c) (d)

Fig. 4. Propagation phase for the fundamental mode normalized to the vacuum wave number versus the number of basis TE and TMmodes (convergence diagram) for an infinite rectangular waveguide partially filled with a chiral image guide (the center guide of Fig. 1)taking � as a parameter. Dimensions in mm: a = 22.86, b = 10.16, w = 11, h = 5, f = 9 GHz, �r = 10, �r = 1. In (a) � = 0 (achiral case),in (b) � = 0.25, in (c) � = 0.5 and in (d) � = 1.

�=0.57) and they will be used for comparison in Fig. 2. Thehigher the values of the permittivity and chirality parameters(or equivalently the chirality admittance parameter in [26])the greater the differences in the results of the direct andindirect formulations.

Fig. 2 compares the results for the modulus and phaseof S11 [29] of the direct and indirect formulations with theresults presented in [26], for a structure composed of a fi-nite image chiral guide housed in a rectangular waveguidebetween two empty rectangular waveguides, as depicted inFig. 1. For the computation of results we have used 120modes, because, as will be shown later, this is a sufficientnumber of basis modes to achieve stable values. Unfortu-nately, in [26] it is not specified how many basis modes wereused in the expansions.

We can see that the results of [26] and the results of thedirect formulation are almost identical, confirming our pre-dictions, while those of the indirect formulation are some-what different. Fig. 3 shows a comparison for the structureof Fig. 1 but with low values of the parameters (�rBP = 1,�c = 1.5 mS ⇒ �r = 1.32, � = 0.57 and �rBP = 2.5, �c =0, 5 mS ⇒ �r = 2.54, � = 0.19). For these cases, the resultsof [26] are very similar to the results of the direct and indi-rect formulations. In conclusion, (i) the direct formulation

provides the same results as the formulation presented in[26] and they can be considered analogous; and (ii) when theparameters of the chiral medium are very low the resultsof the direct and indirect formulations are practically iden-tical, but when these values increase the results begin todiverge.

Now that we have shown that direct formulation is anal-ogous to the formulation presented in [26], we are going toshow how the phase constants provided by the direct and in-direct formulations for a uniform waveguide partially filledwith a chiral medium (i.e., the center waveguide of Fig. 1but of infinite length) behave against the number of basismodes. These basis modes are chosen by increasing the cut-off frequencies.

In Fig. 4, we show a convergence diagram for the funda-mental mode. The dimensions are shown in the figure andthe chiral medium has �r = 10 (a high value) and valuesof � from 0 (isotropic dielectric) to 1. As can be seen, thephase constant provided by the indirect formulation, for the“worst” case � = 1, reaches a stable value with 120 basismodes. The results of the direct formulation, on the otherhand, do not converge adequately unless nearly 180 basismodes are used. Analogous convergence diagrams are shownin Fig. 5 for the second mode. In the “worst” case (�r = 10



β /k 0


β /k 0


β /k 0


β /k 0

(a) (b)

(c)

(d)

0 50 100 150 2001.5

1.6

1.7

1.8

1.9

2

0 50 100 150 2001.5

1.6

1.7

1.8

1.9

2

0 50 100 150 2001.5

1.6

1.7

1.8

1.9

2

0 50 100 150 2001.6

1.7

1.8

1.9

2

2.1





Fig. 5. Propagation phase for the second mode normalized to the vacuum wave number versus the number of basis TE and TM modes(convergence diagram) for an infinite rectangular waveguide partially filled with a chiral image guide (the center guide of Fig. 1) taking� as a parameter. Dimensions in mm: a = 22.86, b = 10.16, w = 11, h = 5, f = 9 GHz, �r = 10, �r = 1. In (a) � = 0 (achiral case), in (b)� = 0.25, in (c) � = 0.5 and in (d) � = 1.

Fig. 6. Moduli of the three components of the electric field, normalized to their maximum values, as a function of the x- and y-coordinates.The plots at the top correspond to the mode of Fig. 4a and the plots at the bottom to the mode of Fig. 5a, i.e., the achiral case (� = 0).


Fig. 7. Moduli of the three components of the electric (top) and magnetic (bottom) fields, normalized to their maximum values, as afunction of the x- and y-coordinates. The plots correspond to the second mode of Fig. 5d, i.e., a chiral case with � = 1.

0 20 40 60 80 100 120 140 160 180 2003.5

4

4.5

5

5.5

6

6.5



α (d

B/m

)


0 20 40 60 80 100 120 140 160 180 2001.1

1.15

1.2

1.25

1.3

β /k 0


(a)

(b)

Fig. 8. Propagation constant for the fundamental mode versus the number of basis TE and TM modes (convergence diagram) for aninfinite rectangular waveguide partially filled with a chiral image guide (the center guide of Fig. 1). The attenuation constant (dB/m) isshown in Fig. 8(a) and the phase constant normalized to the vacuum wave number in Fig. 8(b). Dimensions in mm: a = 22.86, b = 10.16,w = 11, h = 5, f = 9 GHz, �r = 3.75 − j0.02, �r = 1.015 − j0.01 and � = 0.046 − j0.18. The constitutive parameters are extracted fromFig. 4 of [30] at 9 GHz.


8 9 10 11 120

1

2

3

4

5

8 9 10 11 121.6

1.7

1.8

1.9

2

jβ/k

0

8 9 10 11 1216

17

18

19

20

21

22

α (d

b/m

)

8 9 10 11 121.6

1.7

1.8

1.9

2

jβ/k

0

8 9 10 11 128

10

12

14

16

18

20

8 9 10 11 121.6

1.7

1.8

1.9

2

jβ/k

0

8 9 10 11 1225

30

35

40

45

α (d

b/m

)

α (d

b/m

)α

(db

/m)

8 9 10 11 121.6

1.7

1.8

1.9

2

jβ/k

0

Frequency (GHz)Frequency (GHz)

Frequency (GHz)Frequency (GHz)

εr = 3

κ =0.5

εr = 3 - j0.03

κ =0.5

εr = 3

κ =0.5 -j0.01

εr = 3 - j0.03

κ =0.5 - j0.01

(a) (b)

(c) (d)

Fig. 9. Dispersion diagram for the fundamental mode of an uniform rectangular waveguide filled with an isotropic chiral medium (i.e.the center waveguide of Fig. 1 but of infinite length). � is the attenuation constant (solid line) and is expressed in dB/m. is the phaseconstant (dashed line) and in the figure is normalized to the vacuum wavenumber k0. The propagation constant is � = � + j. (a) losslessisotropic chiral material, (b) isotropic chiral material with dielectric losses, (c) isotropic chiral material with chiral losses, and (d) isotropicchiral material with both dielectric and chiral losses. a = w = 22.86 mm, b = h = 10.16 mm, f = 9 GHz, �r = 1, �r = 10.

and �=1) both the indirect and direct formulations convergewith 120 basis modes. Similar convergence diagrams canbe plotted for low values of �r and �. In these cases bothformulations converge with 120 basis modes.

It is interesting to compare the convergence diagrams inFigs. 4a and 5a, which correspond to an isotropic guide(�= 0). In Fig. 4a, the direct and indirect formulations con-verge to different values, and in Fig. 5a the direct and indi-rect formulations converge to the same value.

The difference in the results of the two formulations is dueto the different ways in which the longitudinal field compo-nents are obtained, and to the fact that for an isotropic di-electric case only the electric field has influence. In Fig. 4 weare dealing with the fundamental mode, which is an EH-typemode, having an important Ez-component. This is shown inthe three plots at the top of Fig. 6 where the moduli of Ex , Ey

and Ez normalized to the maximum value of the electric fieldare plotted. The Ex component is almost zero, but the Ey andEz components are of the same order of magnitude. Conse-quently, the direct formulation does not give as good resultsas the indirect formulation [22] because the Ez-componentis not adequately represented. In Fig. 5 we are representingan HE-type mode which has a very small Ez-component.This can be seen in the three bottom plots of Fig. 6 which

refer to an HE mode. Now, the Ez-component is muchsmaller than the other electric field components. Hence,the accuracy of the representation of the Ez-component haspractically no influence on the final results, and the directand indirect formulations give practically the same results.As the value of the chirality parameter � is increased, theconvergence diagrams for the direct and indirect formula-tions become more and more different, which is especiallyevident in Figs. 4d and 5d for �=1. This is because in a chi-ral waveguide there are neither EH nor HE modes. As can beseen in Fig. 7, which shows the plots of the normalized mod-uli of the six components of the electromagnetic field for thesecond mode obtained by the indirect formulation, the modesare hybrid and no component is negligible compared to theothers.

It must be noted that the EH-formulations do not give,for the normal component of the magnetic field, the cor-rect boundary condition on a perfect conductor wall in con-tact with a chiral medium. This can be seen for the Hy-component of Fig. 7 on the portion of the wall y = 0 whichis contact with the chiral medium. In [18] an EH-directformulation, which is called in [18] EMF formulation, iscompared with another formulation of the CMM methodwhich exactly fulfills that boundary condition when they are


0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1



|S11

|

0 20 40 60 80 100 120 140 160 180 200-200

-100

0

100

200

Ph

ase

of

S11

(d

egre

es)



Fig. 10. Variation of modulus (a) and phase (b) of the S11 parameter as a function of the number of basis modes for the structure ofFig. 1 obtained using the indirect (EHI) and direct (EHD) formulations. Dimensions in mm: a = 22.86, b = 10.16, w = 11, h = 5, d = 8,f = 9.5 GHz, �r = 10, �r = 1, � = 1.5.

applied to a parallel-plate waveguide completely filled withan isotropic chiral medium. The advantages and disadvan-tages can be read in the conclusions of [18] and they can beextended to the present structure.

In Fig. 8 we show the convergence plot for the propaga-tion constant �=�+ j (� is the attenuation constant and isthe phase constant)4 of the fundamental mode for the struc-ture of Fig. 1, including losses in permittivity, permeabilityand chirality parameter (Pasteur parameter). The dimensionsand characteristics of the structure are shown in the figure.Inspecting the results we can infer similar conclusions tothe previous convergence results, which we can summarizeas: convergence for the phase and attenuation constants isreached, in the majority of the cases, with less basis modesfor the indirect formulation (normally 120–130 modes) thanfor the direct formulation (120–180 modes). However, it isdifficult to establish a number of basis modes valid for ageneral case. It is better to analyze each case separately.

Fig. 9 shows the propagation constant � = � + j versusfrequency for the fundamental mode of an uniform rectangu-lar waveguide filled with an isotropic chiral medium. Theseplots are called dispersion diagrams. It should be noted that

4 In a lossless waveguide a propagating mode has � = j and anevanescent mode has �=�. For a lossy waveguide any mode has �=�+j.

dispersion is related to the dependence of the constitutiveparameters on frequency. For example, a medium whosepermittivity varies with the frequency is dispersive. A lossymedium is also dispersive. But dispersion is also related tothe kind of electromagnetic wave (“mode”). Therefore, it isnot correct to assign the dispersion property to a mediumwithout considering the type of electromagnetic wave whichis propagating through it. Thus, could we say that a vacuumis not a dispersive medium? No, we could not because if weconsider, for instance, an ideal empty rectangular waveguide(perfect electric walls and a vacuum inside the waveguide,i.e. no losses at all) the fundamental mode TE10 propagat-ing through the vacuum, has a phase constant which is notlinearly related with the angular frequency “�”. As a con-sequence, the TE10 mode propagating through the vacuumhas a dispersive behavior, and we can assert that a rectangu-lar waveguide containing any kind of medium is dispersive.We are going to analyze this kind of dispersive behavior ina rectangular chirowaveguide, with the constitutive param-eters being independent of the frequency. In fact, this is thesituation when we work in a frequency range sufficientlyfar from the resonance of the constitutive parameters [30].Fig. 9a corresponds to a lossless isotropic chiral material.It can be seen that the attenuation constant is zero and thephase constant increases with frequency. Fig. 9b corresponds


8 9 10 11 120

0.2

0.4

0.6

0.8

1

Frequency (GHz) Frequency (GHz)


|S11

|

8 9 10 11 12-200

-100

0

100

200

Ph

ase

of

S11

(d

egre

es)

8 9 10 11 120

0.2

0.4

0.6

0.8

1

|S21

|

8 9 10 11 12-200

-100

0

100

200

Ph

ase

of

S21

(d

egre

es)


Fig. 11. Variation of the modulus and phase of the S11 and S21 parameters versus frequency for the structure of Fig. 1 obtained using theindirect (EHI) and direct (EHD) formulations. 120 basis modes are used in the computations. Dimensions in mm: a = 22.86, b = 10.16,w = 11, h = 5, d = 8, �r = 10, �r = 1, � = 1.5.

to the isotropic chiral material with dielectric losses. Nowthe attenuation constant is different from zero and increaseswith frequency. The same can be said of Fig. 9c which cor-responds to the isotropic chiral material with chiral losses.(The attenuation constant increases in a similar way to thecase with dielectric losses). Finally, Fig. 9d shows the caseof the isotropic chiral material with both dielectric and chi-ral losses. In this case, the attenuation constant � is practi-cally the sum of the attenuation constants obtained for thecases of dielectric losses and chiral losses alone. It is in-teresting to note that the phase constant remains unchang-ing although the losses are considered in both dielectric andisotropic chiral media. This is because the losses introducedare small. This result is identical to the situation of thepropagation of a Transverse ElectroMagnetic (TEM) modethrough a good dielectric medium or the propagation of theTE10 mode through a waveguide filled with a lossy dielectricmaterial with small losses [31]. If the dielectric or the chi-ral losses are increased, the phase constant depends on theselosses.

We have also analyzed what happens in the convergencefor the scattering parameters. In Fig. 10 we have plotted themodulus and phase of the S11 scattering parameter versusthe number of basis modes for a structure like that of Fig. 1

Fig. 12. Periodic structure in rectangular waveguide. Each unitcell is composed of a piece of chiral media and a section of emptywaveguide. There are 20 unit cells. The input and output guidesare semi-infinite empty rectangular waveguides. The total numberof discontinuities in the propagation direction is 42. Dimensionsin mm: a =22.86, b=10.16, w=11, h=5, u=8, v =10, �r =10,�r = 1, � = 1.5. d = u + v is the length of each unit cell.

with the dimensions indicated in the figure and with a chi-ral medium with high values of permittivity and chirality(�r = 10, � = 1.5). Stable values are reached by using 120


8 9 10 11 120

0.2

0.4

0.6

0.8

1



|S11

|

8 9 10 11 12-200

-100

0

100

200

Ph

ase

of

S11

(d

egre

es)

8 9 10 11 120

0.2

0.4

0.6

0.8

1

|S21

|

8 9 10 11 12-200

-100

0

100

200

Ph

ase

of

S21

(d

egre

es)


Fig. 13. Variation of the modulus and phase of the S11 and S21 parameters versus frequency for the periodic structure of Fig. 12. Theresults correspond to the indirect (EHI) and direct (EHD) formulations.

basis modes for both the indirect and direct formulations.Fig. 11 shows the plots versus frequency of the four scatter-ing parameters for the same structure as in Fig. 10. In thehigh-frequency range the differences between the direct andindirect formulations are evident.

The results presented here have demonstrated that the di-rect and indirect formulations give different results for highvalues of the constitutive parameters of the chiral mediuminside the waveguide. The crucial question is: Which of thetwo formulations “is right”? We have reached the conclu-sion that the only possible answer is the indirect formulation.Since no measurements are available, our conclusion must bebased on other sorts of reasons, which are summarized in thefollowing three points. Firstly, we have shown theoreticallyhow the indirect formulation uses a strategy for express-ing the longitudinal components of the electromagnetic fieldas a function of the transverse ones which ensures a goodtreatment of the continuity properties of the different fieldcomponents. Secondly, for waveguides containing isotropicchiral media with analytical solutions, we have shown thatthe indirect formulation gives more accurate results than thedirect formulation [19]. Thirdly, for other kinds of mate-rials, such as isotropic dielectrics and anisotropic magne-tized ferrites, the indirect formulation gives more accurateresults than the direct formulation as can be seen in [15,22].

Therefore, it seems obvious that the same will be true forchiral media in rectangular waveguides.

Finally, in order to illustrate the versatility of the tech-nique presented here, we have analyzed a periodic structurecomposed of 20 unit cells, each 18 mm long, housed in arectangular waveguide and bounded by two empty waveg-uides as is shown in Fig. 12. Each unit cell consists of anempty waveguide and a section of partially filled rectangularwaveguide partially filled with a chiral material. The resultsfor the scattering parameters as a function of frequency forboth EHI and EHD formulations are shown in Fig. 13.

5. Conclusions

In this paper we have developed direct and indirect for-mulations of the CMM that have been applied to rectangu-lar waveguides partially filled with isotropic chiral media.We have shown that for the propagation constants the di-rect formulation gives slower convergence than the indirectformulation and, when the constitutive parameters are high,the convergence is not reached with a reasonable numberof basis modes. When either one of these two formulationsof the CMM is combined with the classical Mode Match-ing Method (MMM), a hybrid technique is obtained which


is capable of analyzing 3D discontinuities without limit innumber. The results for the direct formulation are identicalto those presented in [26] but are somewhat different fromthose obtained using the indirect formulation. Based on theseand other results for other kind of media, we conclude thatfor isotropic chiral media too, the indirect formulation ismore robust than the direct formulation.

Acknowledgement

This work was supported by the DGI of the Spanish MEC,under Project TIC2003-09677-C03-01.

Appendix A

A.1. T-potentials

According to Schelkunoff’s notation [10], each TE or TMmode in a homogeneous isotropic waveguide, can be de-scribed by a function T (x, y) which verifies the habitualHelmholtz equation

∇2T (x, y) + kcT (x, y) = 0,

being kc the cut-off wave number of any mode TE or TM.These functions are called T-potentials [10] and give, excepta constant, ez for a TM mode or hz for a TE mode. As ez

and hz can be considered as potential functions [32] for ob-taining the rest of the electromagnetic field components, theT-functions are also called T-potentials. We add a subscriptwith parenthesis for the T-functions of the TM modes anda subscript with brackets for the TE modes. Also, we addthe habitual indices n, m of the TM and TE modes. Practi-cally any normalization condition can be applied to the T-functions. We use the following

∫s

[(�T〈k〉(x, y)

�x

)2

+(

�T〈k〉(x, y)

�y

)2]

ds = 1, (A.1)

where the subscripts 〈〉 represents parenthesis or brackets,k inside 〈〉 represents the pair n, m and the integration isperformed across the transverse section of the rectangularwaveguide. Then, the T-functions are easily obtained as

• TE modes

T[n,m] = [n,m]kc[n,m]

√ab

cos(n�

ax)

cos(m�

by)

,

n, m = 0, 1, 2, . . . but not n = m = 0, (A.2)

where

[n,m] ={

2 if n �= 0 and m �= 0√2 if n = 0 or m = 0

. (A.3)

• TM modes

T(n,m) = 2

kc(n,m)

√ab

sin(n�

ax)

sin(m�

by)

,

n, m = 1, 2, 3, . . . . (A.4)

In these expressions a and b are the width and the heightof the rectangular waveguide, respectively, and kc〈n,m〉 =√

(n�/a)2 + (m�/b)2 is the cut-off wavenumber of theeach mode.

References

[1] Sorrentino R. Exact analysis of rectangular wave-guides inhomogeneously filled with transversely magnetizedsemiconductor. IEEE Trans Microwave Theory Tech1976;MTT24:201–8.

[2] Yee KS. Numerical solution of the initial boundary valueproblem involving Maxwell’s equations in isotropic media.IEEE Trans Antennas Propag 1966;AP14:302–7.

[3] Albani M, Bernadi P. A numerical method based on thediscretization of Maxwell’s equations in the integral form.IEEE Trans Microwave Theory Tech 1974;MIT22:446–50.

[4] Mrozowski M, Mazur J. Lower bound on the eigenvaluesof the characteristic equation for an arbitrary multilayeredgyromagnetic structure with perpendicular magnetization.IEEE Trans Microwave Theory Tech 1989;37:640–3.

[5] Bornemann J, Arndt F. Modal-S-matrix design of optimumstepped ridged and finned waveguide transformers. IEEETrans Microwave Theory Tech 1987;35:561–7.

[6] Uher J, Arndt F, Bornemann J. Field theory design offerrite loaded waveguide nonreciprocal phase shifters withmultisection ferrite or dielectric slab impedance transformers.IEEE Trans Microwave Theory Tech 1987;35:552–60.

[7] Solano MA, Prieto A, Vegas A, Cagigal C, FontechaJL. Characterization of microwave devices consisting ofdifferent rectangular waveguides partially filled with dielectricmaterial. In: Proceedings of the 26th European MicrowaveConference. Prague, Czech Republic, 1996. p. 871–4.

[8] Alù A, Bilotti F, Vegni L. Generalized transmission lineequations for bianisotropic materials. IEEE Trans AntennasPropag 2003;51:3134–41.

[9] Weiglhofer WS. In: Weiglhofer WS, Lakhtakia A,editors. Introduction to complex mediums for optics andelectromagnetics. SPIE Press; 2003. p. 27–55.

[10] Schelkunoff SA. Generalized telegraphist’s equations forwaveguides. Bell Syst Tech J 1952;31:784–801.

[11] Ogusu K. Numerical analysis of the rectangular dielectricwaveguide and its modifications. IEEE Trans MicrowaveTheory Tech 1977;MTT25:874–85.

[12] Tao J-W, Atechian J, Ratovondrahanta R, Baudrand H.Transverse operator study of a large class of multidielectricwaveguides. IEE Proc Part H 1990;137:311–7.

[13] Chaloupka H. A coupled-line model for the scattering bydielectric and ferromagnetic obstacles in waveguides. AEÜ1980;34:145–51.

[14] Xu Y, Zhang G. A rigorous method for computation of ferritetoroidal phase shifters. IEEE Trans Microwave Theory Tech1988;36:929–33.


[15] Solano MA, Vegas A, Prieto A. Modelling multiplediscontinuities in rectangular waveguide partially filled withnon-reciprocal ferrites. IEEE Trans Microwave Theory Tech1993;41:797–802.

[16] Pelet P, Engheta N. Coupled-mode theory forChirowaveguides. J Appl Phys 1990;67:2742–5.

[17] Xu Y, Bosisio RG. Calculation of dielectric resonators withcomplicated constitutive parameters. IEE Proc MicrowaveAntennas Propag 1995;142:477–90.

[18] Xu Y, Bosisio RG. A study on the solutions ofchirowaveguides and bianisotropic waveguides with the useof coupled-mode analysis. Microwave Opt Technol Lett1997;14:308–11.

[19] Gómez A, Solano MA, Vegas A. New formulation of thecoupled mode method for the analysis of chirowaveguides.Complex Mediums, III: beyond Linear Isotropic Dielectrics,Seattle, Washington. Proc SPIE 2002;4806:290–301.

[20] Xu Y, Bosisio RG. An efficient method for study of generalbi-anisotropic waveguides. IEEE Trans Microwave TheoryTech 1995;43:873–9.

[21] Xu Y, Bosisio RG. A comprehensive study of generalbianisotropic waveguides with the use of the coupled-modeanalysis. Microwave Opt Technol Lett 1996;12:279–84.

[22] Vegas A, Prieto A, Solano MA. Optimisation of the coupled-mode method for the analysis of waveguides partially filledwith dielectrics of high permittivity: application to the studyof discontinuities. IEE Proc Part H 1993;140:401–6.

[23] Kong JA. Electromagnetic wave theory. New York: Wiley;1986.

[24] Engheta N, Jaggard DL. Electromagnetic chirality and itsapplications. IEEE Antennas Propag Soc Newslett 1988;10:6–12.

[25] Wu X, Jaggard DL. Three-dimensional discontinuities inchirowaveguides. Microwave Opt Technol Lett 1997;16:315–9.

[26] Wu X, Jaggard DL. A comprehensive study of discontinuitiesin chirowaveguides. IEEE Trans Microwave Theory Tech2002;50:2320–30.

[27] Solano MA, Prieto A, Vegas A. Theoretical and experimentalstudy of two port structures composed of differentsize rectangular waveguides partially filled with isotropicdielectrics. Int J Electron 1998;84:521–8.

[28] Weiglhofer WS. Electromagnetic theory of complexmaterials, engineered nanostructural films and materials. In:Lakhtakia A., Messier RF, editors. Proceedings of the SPIE,vol. 3790. WA, USA: Bellingham; 1999. p. 66–76.

[29] Collin RE. Foundations for microwave engineering. McGrawHill International Editions; 1992 [chapter 4, Section 4.7].

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[31] Balanis CA. Advanced engineering electromagnetics. NewYork: Wiley; 1989 [chapter 4, Section 4.3.1A and chapter 8,Section 8.2.5B].

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Álvaro Gómez was born in Santander,Spain. He received his Licenciado enCiencias Físicas degree in 2000 and hisPh.D. degree in 2005, both from theUniversity of Cantabria, Spain. From2000 to 2005, he was working with theCommunication Engineering Depart-ment at the University of Cantabria.Now he is working with Electrical andElectronic Department at the Univer-sity of Valladolid, Spain. His current

research activities include electromagnetic propagation in complexmedia and numerical methods in electromagnetics.

Angel Vegas was born in Santander,Spain. He received his degree of Li-cenciado in Physics in 1976 and theDoctor degree in 1983, both from theUniversity of Cantabria, Spain. From1977 to 1983 was been with the Depart-ment of Electronics at the Universityof Cantabria, where he became Asso-ciate Professor in 1984. He has workedin electromagnetic wave propagation inplasmas and microwave interferometry.

His current research and teaching interest include electromagnetictheory, computer methods in electromagnetism, and microwavemeasurements.

Miguel A. Solano received his Licen-ciado en Física degree in 1984 and hisPh.D. degree in 1991, both from theUniversity of Cantabria, Spain. Since1984, he was with the ElectronicsDepartment and, now he is with theCommunication Engineering Depart-ment at the University of Cantabria.During 1991, he was a Visitor Scholarat the University of Bristol (UK) andsince 1995 he is a Professor Titular

at the University of Cantabria. His current research activities in-clude electromagnetic propagation in complex media, numericalmethods in electromagnetics, biological effects of the electromag-netic field and e-learning.

on two different formulations of the coupled mode method: application to 3d rectangular...

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