Dispersion Analysis of Periodic Structures by Solving ...Dispersion Analysis of Periodic Structures...

Frequenz, Vol. 65 (2011), pp. 247–252 Copyright © 2011 De Gruyter. DOI 10.1515/FREQ.2011.034

Dispersion Analysis of Periodic Structuresby Solving Corresponding Excitation Problems

Yvonne Weitsch,1;� Huanlei Chen1 andThomas F. Eibert1

1 Lehrstuhl für Hochfrequenztechnik, Technische Univer-sität München, Germany

Abstract. Eigenproblems of periodic structures with com-plicated material distribution are solved efficiently by solv-ing corresponding excitation problems. A unit cell with pe-riodic boundary conditions is discretised and handled bya doubly periodic hybrid finite element boundary integraltechnique, which even considers complex propagation con-stants. Instead of solving algebraic eigenproblems, the anal-ogy to resonance problems is exploited which gives im-proved physical insight into the problem. Moreover, openproblem types can be handled. Numerical results for com-posite right/left-handed waveguides in one and three dimen-sions confirm the presented approach.

Keywords. Waveguides, periodic problems, eigenprob-lems, resonators.

PACS®(2010). 84.40.Az, 02.60.Lj, 02.70.Dh.

1 Introduction

Linear systems are completely characterised by their eigen-solutions and eigenvalues. With regard to electromagneticdevices, the eigensolutions and eigenvalues reveal the per-formance of these devices immediately, which promotes anefficient design procedure. By knowing the eigensolutionsand eigenvalues, the response of a system to any kind of ex-citation can be predicted by a mere projection of the eigen-solutions onto the excitation. Hence, the determination ofeigensolutions and eigenvalues is of main scientific inter-est. Any electromagnetic field problem can be consideredas an eigenproblem by excluding any form of excitation.Presumably every numerical field solving procedure can beemployed for solving eigenproblems if the underlying dis-cretised operator, which describes the field problem withrespect to the boundary conditions, is considered without

Corresponding author: Yvonne Weitsch, Lehrstuhl fürHochfrequenztechnik, Technische Universität München, Arcisstr. 21,80333 München; E-mail: [email protected].

Received: June 19, 2011.

exciting sources. A collection of solution methods for nu-merical eigenproblems is provided in [9,17]. The numericaleigenproblem solvers included in most of the commercialsoftware packages handle eigenproblems in form of a lin-ear equation system, which becomes large if many discreti-sation unknowns are present. Consequently, eigenproblemsoften result as expensive or even impossible to solve. In par-ticular the convergence behaviour of numerical solvers ap-plied to nonlinear eigenproblems is mostly not satisfying.Nonlinear eigenproblems arise from integral equation for-mulations or problems including a perfectly matched layer(PML). The physical cause for convergence failure is of-ten not even obvious. Furthermore, unphysical solutionstogether with PML are often detected but which are desiredto avoid. An attempt is started in [2], where a-priori knowl-edge of the field pattern shall help to distinguish wantedfrom unphysical modes already during the solution process.The purpose is to improve the efficiency of the iterativeJacobi–Davidson algorithm built-in in the software packageof CST MWS [1]. The application area is yet still restrictedto simple structures such as dielectric substrates with airholes without metal. Likewise, many eigenproblem com-putations are limited to relatively small eigenproblem sizesas for instance in the analysis of the dispersion behaviourof waveguides, mostly meaning to solve two-dimensionaleigenproblems. However, periodic configurations in one,two or three dimensions have gained notable interest in re-cent time especially in context with artificial materials ormetamaterials. Although the periodic nature allows a re-duction of the solution process to an elementary periodicunit cell, the numerical description and the correspondingdiscretisation of the 3D unit cell can fast lead to a huge num-ber of unknowns. The structures normally exhibit elaboratematerial and geometry variations within small dimensions.

Nonetheless, one possibility to obtain a solution for largeeigenproblems is to reduce the number of unknowns. If thedispersion analysis of periodic configurations is demandedthe fields in the periodic boundaries can be expanded by amodal series. The modes are typically the eigenmodes ofthe host uniform waveguide, which is periodically loaded.The unit cell is excited mode-wise and the scattering or S -parameters reliably result from the driven full-wave simula-tion. The S -matrix is related to the transfer or T -matrix andthe final eigenproblem is constituted by the usually smallnumber of T -parameters. The procedure works well if asmall number of port modes is sufficient to accurately for-mulate the field problem. If systems are concerned which

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248 Y. Weitsch, H. Chen and T. F. Eibert

are in particular operated in the fundamental mode to en-sure a stable and predictable performance, the method isvery well suited. Yet for open problems, the discrete and fi-nite set of modal solutions becomes continuous and extendsto infinity. The series expansion needs to be replaced by anintegral formulation. However, the expansion method candeliver approximate results [18, 20], where the waveguideis well separated from the open half space.

Another way to facilitate eigenproblems is a specificform of the so-called Trefftz method [10, 12] relying on themethod of fundamental solutions [8, 11, 15]. The particularsolutions are there generated by exciting the solution do-main by sources from outside. The procedure is extendedto generalised Sturm–Liouville eigenproblems [16] and tothe use of internally placed sources [14,15]. The advantageof these methods is that the algebraic solution of the eigen-problem can be circumvented by instead solving a corre-sponding excitation problem repeated times.

Similar procedures have been known in electrical engi-neering for a long time. The resonance behaviour of filtersand resonators is investigated by measurements and equiva-lent circuits by applying an excitation and by observing thefrequency-dependent response of the system to the excita-tion. From the course of the reaction to the excitation, theresonance frequency identical with the eigenvalue can bedirectly inferred. This analogy is exploited in the methodpresented in this article. An eigenproblem is solved as aresonator problem by finding the singular point !res whilean excitation is defined [5, 6]. Internal excitations are ap-plied, e.g. by impressed currents or in form of (discrete)ports, which are adequately coupled to the field problem.Thus, circuit related considerations are possible, which arealso usable for problem regularisations as well as for energyand power inspections. A further benefit is that the excita-tion can be specifically placed and the refining search inthe iterative solution process is localised. Problematic PMLmodes are not excited at all.

The approach explained above is implemented in a hy-brid finite element boundary integral (FEBI) solver for dou-bly periodic structures, which is well suited for one- andtwo-dimensional artificial waveguide structures. In the FEpart of the code, an edge-based periodic boundary condi-tion (PBC) is defined while a Floquet-mode based peri-odic Green’s function serves for the spectral domain inte-gral equation. Open problems can perfectly be manageddue to the BI part. Moreover, the solver is upgraded inthe PBCs in that the solver becomes suitable for complexpropagation constants and thus evanescent and leaky moderepresentations are supported. These modes are typicallynot promoted by standard numerical solvers. As alterna-tive, the proposed technique can be combined with otherelectromagnetic field solvers, e.g. CST MWS [1].

First, a composite right/left-handed hollow waveguide isinvestigated, where the right-handed and left-handed dis-persion behavior is clearly verified. A three-dimensional

synthesized artificial material composition follows wherethe solution method easily presents the eigenvalues of thewaveguide modes.

2 Periodic Eigenproblem Formulation

The analysis of periodic configurations reduces to the anal-ysis of one unit cell subject to periodic boundary conditionsaccording to the Bloch-Floquet theorem. Eigensolutions arecharacterised as functions which do not change except fora multiplicative constant, the eigenvalue, if the correspond-ing operator is applied to them. Concerning a unit cell aselement of a periodic array as illustrated by Figure 1, theeigenvector v D .: : : ; a0m; : : : ; b0m; : : :/ contains the fieldquantities, e.g. the incident a and reflected b modes, on oneside of the unit cell. It is mapped by a factor, the propaga-tion constant or the transfer matrix T , onto the field quanti-ties on the opposite side according to

e�pv D T v or.T � e�pI/v D 0: (1)

The length of the unit cell is p. Equation (1) can also beexpressed by

L.!; �/v D 0; (2)

which is an operator L depending on discrete pairs of fre-quency ! and propagation constant � . Various pairs of !; �eventually render the dispersion behaviour. Thus, the corre-sponding eigensolutions result as null space of the operator.In discretised versions, the operator is a matrix, which mustbecome singular. Considering the equations for shunt andseries resonant circuits

Y.!/U D�

j

�!C � 1

!L

�CGl

�U D 0; (3)

Z.!/I D�

j

�!L � 1

!C

�CRl

�I D 0; (4)

respectively, they become singular, too and yield the eigen-values as resonant frequencies !res. The circuits are pic-tured in Figure 2 showing a capacitor C , an inductance Land a loss conductance Gl or a loss resistance Rl , respec-tively. For lossy resonators, !res will become a complex

Ma

T

'1b1b 1a

'1a

Mb 'Mb'Ma

Figure 1. The T -matrix acts as mapping factor for a unitcell.



Dispersion Analysis of Periodic Structures 249

C L lG GiG

Resonator

0I

0U∼

iR C L lR

R

Resonator

Shunt:

Series:

Figure 2. Equivalent circuit of shunt and series resonatorswith excitation and load.

iR iRlR

1I 2I

Port 1 Port 2

C L

Figure 3. Excitation of general eigenproblem by ports ac-cording to circuit equivalence.

quantity, which in turn is real for vanishing Gl or Rl , re-spectively. In case of circuits, it is common practice to ap-ply an excitation and to monitor the reaction (voltage or cur-rent) to the excitation. In addition, the generator or load re-sistance can serve as regularisation tool for the singular res-onance behaviour without affecting the resonant frequency.Following this strategy, the boundary value problems areexcited for various .!; �/ combinations as illustrated in Fig-ure 3 for an excitation by ports. In a numerical implementa-tion, a pair of ports as shown in Figure 3 excite the problemand the S -parameters are to observe. However, the cou-pling of the ports may introduce parasitic reactive couplingelements. The coupling elements influence the curve of theS -parameters. As one measure, the parasitic elements canbe removed by knowing the coupling model. As anothermeasure, another observable can be chosen e.g. the electricfield because the field in the solution domain will still reli-ably indicate the resonance meaning the eigenvalue.

PBCPBCBI

BI

FEx

yz

a

b

Figure 4. The combination of FE method and BI part in thehybrid FEBI technique.

3 Numerical Implementation

In this work, a periodic hybrid finite element boundary in-tegral (FEBI) method according to [4] and [3] is employedand applied to doubly periodic array configurations. Due tothe Bloch–Floquet theorem, the field solution of the struc-ture is to factor in a periodic and a non-periodic part accord-ing to

E.r Cm�a C n�b/ D E.r/ e�jkt00�.m�aCn�b/; (5)

H .r Cm�a C n�b/ D H .r/ e�jkt00�.m�aCn�b/: (6)

�a and �b are the lattice vectors in x- and y-direction, re-spectively, as illustrated in Figure 4. kt00 corresponds to �in Equation (2) and is considered as complex with respect tothe PBCs. The subscript t00 specifies that the wavevectoris in parallel to xy-plane and indicates the .00/th Floquetmode within the 2D Floquet mode series [3, 4]. Accordingto the Floquet theorem, only one periodic unit cell is dis-cretised and computed by the FE technique. The planar topand bottom surfaces of the unit cell mesh are incorporatedby the BI part, where appropriate spectral domain 2D peri-odic Green’s functions are implemented such that complexkt00 are possible. The Green’s functions account for the in-fluence of possible homogeneous material layers above andbelow the FE unit cell.

According to the above described solution procedure, theFE unit cell is to excite correspondingly which can be inform of impressed volume currents or impressed electricalfield strengths. This is realisable by discrete ports in a full-wave driven simulation. The electric field strength as wellas any other kind of integral measure can be observed atindividual locations. To accelerate the search of appropri-ate pairs of .!;kt00/, i.e. the eigenvalue, bisectional searchalgorithms in 1D together with start vector estimation fromprevious solutions within the iterative linear equation solverare utilised.




Figure 5. A unit cell of the metallic rectangular waveguidewith dielectric-filled corrugations according to [7] meshedwith tetrahedra.

Port 1[1]Port 2[1]

Ports

Figure 6. The currents for the discrete ports are positionedin the corrugation as shown in the inset.

4 Numerical Results

As first reference model to validate the proposed eigenvaluecomputation technique serves a dielectric-filled corrugatedhollow waveguide, which has already been computed in[7, 19]. A unit cell of this periodical composition is dis-cretised with tetrahedral mesh as visualised in Figure 5 tobe analysed by the hybrid FEBI method according to [3, 4]as described in the previous section. Since the problem isclosed only the FE portion is relevant for the solution. Cur-rents can be impressed in some of the tetrahedra. Equiva-lently, it is possible to excite the unit cell model by discreteports in CST MWS [1]. The discrete ports can exemplarilyplaced in the corrugation. Due to the high-impedance envi-ronment of the ports, the port resistors should exhibit a highvalue e.g. 5000 � not to disturb the authentic field distribu-tion. At resonance, identical port currents without losses areregistered, as shown in Figure 6, which also demonstratesthe total power loss to be equally distributed over the twoport resistors. The course of the discrete port currents addi-tionally discloses the resonance behaviour. By changing theposition of the discrete ports the resonance behaviour mightbe slightly shifted because of parasitic effects. However, by

7 7.5 8 8.5 9 9.5-2

-1.5

-1

-0.5

0

0.5

Frequency in GHz

/

in 1

/m

Scattering matrix approachEshrah et al.FEBI with current excitation

attenuation/bandgap

left-handedpropagation

right-handedpropagation

Figure 7. Dispersion diagram of hollow waveguide withdielectric-filled corrugations.

Port 1

Port 2

xy

z

Figure 8. The metallic structure of the 3D unit cell accord-ing to the structure presented in [21] but modified in orderto suppress the distorted plane wave mode.

choosing another observable, e.g. the electric field strength,the true resonance behaviour can be maintained. When-ever the electric field strength exhibits peaks the eigenvalues.!;kt00/ are determined. Eventually, they can be plotted inthe dispersion diagram as displayed in Figure 7. The resultsobtained by employing the FEBI method is compared to[19] and [7], where the latter technique only reveals a solu-tion for the frequency region where left-handed wave prop-agation takes place. The FEBI method moreover providesthe attenuation constant and a right-handed phase constantas well as the scattering matrix approach [13] does. Theresults of both methods agree well with each other.

The next problem is a 3D periodic unit cell as givenin Figure 8. Two dielectric substrates (suppressed in theFigure) are contained in the metallic structure which is ametamaterial unit cell. It reveals left-handed dispersion be-haviour at some frequencies as described in more detail in[21]. Because of the complexity of the model and the large



Dispersion Analysis of Periodic Structures 251

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-90

-80

-70

-60

-50

-40

-30

-20

-10

0

px=0°px=20°px=40°px=60°px=80°px=100°px=120°px=140°px=160°px=180°px=200°

Frequenzy in GHz

|S21

| in

dB

Figure 9. Maximum indicates eigensolution.

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

Periodic phase shift in x in degree

Freq

uenc

y in

GH

z

left-handed band

Figure 10. Dispersion diagram of 3D metamaterial unit cellshown in Figure 8.

number of unknowns resulting from the discretisation, themodel would be a challenge for many eigenproblem solvers.Moreover, a series expansion of the fields in mode functionsas required for the SMA [13, 19] becomes difficult becausehardly any homogeneous waveguide regions can be foundand rather an endless number of waveguide modes wouldbe necessary for an accurate description of the field prob-lem. Nevertheless, the proposed eigenproblem formulationprovides reliable results. Observing the transmission pa-rameter, which is displayed in Figure 9 for a varying peri-odic phase difference, an eigensolution is found whenevera maximum is reached. The results for a swept periodicphase difference are presented in the dispersion diagram inFigure 10 for the first modes propagating in the transversex-direction.

5 Conclusions

Eigenproblems of periodically constituted configurationshave been successfully solved by corresponding excitationproblems subject to periodic boundary conditions. More-over, the resulting resonance problems deliver a physicalunderstanding of the dispersion behaviour of the periodicstructures. The excitation can be placed selectively whatallows a more efficient solution procedure. Even complexpropagation constants can be solved for. Commercial fieldsolvers can be used for a convenient solution of the excita-tion problems, which opens up the possibility for finding theattenuation constants of propagating modes by perturbationmethods.

Acknowledgments

The work of Huanlei Chen has been supported by DeutscheForschungsgesellschaft (DFG) through the TUM Inter-national Graduate School of Science and Engineering(IGSSE). Yvonne Weitsch is a member of the TUM Gradu-ate School.

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