analysis of periodic structures under a waveguide...

UNIVERSIDAD POLITECNICA DE MADRID

ESCUELA TECNICA SUPERIOR DE INGENIEROS DE TELECOMUNICACION

ELECTROMAGNETIC ANALYSIS OFPERIODIC STRUCTURES UNDER A

WAVEGUIDE VIEWPOINT

TESIS DOCTORAL

Jose Enrique Varela CampeloIngeniero de Telecomunicacion

Master Universitario en Tecnologıas y Sistemas de Comunicaciones

Madrid, 2012

DEPARTAMENTO DEELECTROMAGNETISMO Y TEORIA DE CIRCUITOS

ESCUELA TECNICA SUPERIOR DE INGENIEROS DE TELECOMUNICACION

ELECTROMAGNETIC ANALYSIS OFPERIODIC STRUCTURES UNDER A

WAVEGUIDE VIEWPOINT

Autor:



Director:

Jaime Esteban MarzoDoctor Ingeniero de Telecomunicacion

Profesor Titular de Universidad

Madrid, 2012

Tesis Doctoral:

Electromagnetic Analysis of Periodic Structures

under a Waveguide Viewpoint

Autor:



Director:

Jaime Esteban MarzoDoctor Ingeniero de Telecomunciacion

Profesor Titular de Universidad

Departamento de Electromagnetismo y Teorıa de Circuitos

El Tribunal de Calificacion compuesto por:

PRESIDENTE:

Prof.

VOCALES:

Prof.

Prof.

Prof.

VOCAL SECRETARIO:

Prof.

VOCALES SUPLENTES:

Prof.

Prof.

Acuerda otorgarle la CALIFICACION de:

Madrid, a de de 2012.

Abstract

The fundamental objective of this Ph. D. dissertation is to demonstrate that, underparticular circumstances which cover most of the structures with practical interest, peri-odic structures can be understood and analyzed by means of closed waveguide theoriesand techniques.

To that aim, in the first place a transversely periodic cylindrical structure is consid-ered and the wave equation, under a combination of perfectly conducting and periodicboundary conditions, is studied. This theoretical study runs parallel to the classic analysisof perfectly conducting closed waveguides. Under the light shed by the aforementionedstudy it is clear that, under certain very common periodicity conditions, transversely peri-odic cylindrical structures share a lot of properties with closed waveguides. Particularly,they can be characterized by a complete set of TEM , TE and TM modes. As a result,this Ph. D. dissertation introduces the transversely periodic waveguide concept.

Once the analogies between the modes of a transversely periodic waveguide and theones of a closed waveguide have been established, a generalization of a well-known closedwaveguide characterization method, the generalized Transverse Resonance Technique, isdeveloped for the obtention of transversely periodic modes. At this point, all the neces-sary elements for the consideration of discontinuities between two different transverselyperiodic waveguides are at our disposal. The analysis of this type of discontinuities willbe carried out by means of another well known closed waveguide method, the ModeMatching technique. This Ph. D. dissertation contains a sufficient number of examples,including the analysis of a wire-medium slab, a cross-shaped patches periodic surface anda parallel plate waveguide with a textured surface, that demonstrate that the TransverseResonance Technique - Mode Matching hybrid is highly precise, efficient and versatile.Thus, the initial statement: ”periodic structures can be understood and analyzed by means

of closed waveguide theories and techniques”, will be corroborated.

Finally, this Ph. D. dissertation contains an adaptation of the aforementioned gener-alized Transverse Resonance Technique by means of which the analysis of laterally openperiodic waveguides, such as the well known Substrate Integrated Waveguides, can becarried out without any approximation. The analysis of this type of structures has sus-citated a lot of interest in the recent past and the previous analysis techniques proposedalways resorted to some kind of fictitious wall to close the structure.

vii

Resumen

El principal objetivo de esta tesis doctoral es demostrar que, bajo ciertas circunstanciasque se cumplen para la gran mayorıa de estructuras con interes practico, las estructurasperiodicas se pueden analizar y entender con conceptos y tecnicas propias de las guıas deonda cerradas.

Para ello, en un primer lugar se considera una estructura cilındrical transversalmenteperiodica y se estudia la ecuacion de onda bajo una combinacion de condiciones de con-torno periodicas y de conductor perfecto. Este estudio teorico y de caracter general, sigueel analisis clasico de las guıas de onda cerradas por conductor electrico perfecto. A la luzde los resultados queda claro que, bajo ciertas condiciones de periodicidad (muy comunesen la practica) las estructuras cilındricas transversalmente periodicas guardan multitud deanalogıas con las guıas de onda cerradas. En particular, pueden ser descritas mediante unconjunto completo de modos TEM , TE y TM . Por ello, esta tesis introduce el conceptode guıa de onda transversalmente periodica.

Una vez establecidas las similitudes entre las soluciones de la ecuacion de onda,bajo una combinacion de condiciones de contorno periodicas y de conductor perfecto,y los modos de guıas de onda cerradas, se lleva a cabo, con exito, la adaptacion de unconocido metodo de caracterizacion de guıas de onda cerradas, la tecnica de la Resonan-cia Transversal Generalizada, para la obtencion de los modos de guıas transversalmenteperiodicas. En este punto, se tienen todos los elementos necesarios para considerar dis-continuidades entre guıas de onda transversalmente periodicas. El analisis de este tipode discontinuidades se llevara a cabo mediante otro conocido metodo de analisis de es-tructuras cerradas, el Ajuste Modal. Esta tesis muestra multitud de ejemplos, como porejemplo el analisis de un wire-medium slab, una superficie de parches con forma de cruzo una guıa de placas paralelas donde una de dichas placas tiene cierta textura, en los quese demuestra que el metodo hıbrido formado por la Resonancia Transversal Generalizaday el Ajuste Modal, es tremendamente preciso, eficiente y versatil y confirmara la validezde el enunciado inicial: ”las estructuras periodicas se pueden analizar y entender con

conceptos y tecnicas propias de las guıas de onda cerradas”

Para terminar, esta tesis doctoral incluye tambien una modificacion de la tecnica dela Resonancia Transversal Generalizada mediante la cual es posible abordar el analisis deestructuras periodica abiertas en los laterales, como por ejemplo las famosas guıas de ondaintegradas en sustrato, sin ninguna approximacion. El analisis de este tipo de estructurasha despertado mucho interes en los ultimos anos y las tecnicas de analisis propuestas hasta

ix

el momento acostumbran a recurrir a algun tipo de pared ficticia para simular el caracterabierto de la estructura.

x

Contents

1 Introduction 11.1 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Text Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Transversely Periodic Waveguide Modes 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Eigenvalue Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Eigenfunction Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Variational Expression for the Eigenvalues . . . . . . . . . . . . . . . . . 10

2.5 Eigenfunction Completeness . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Transversely Periodic Waveguide Modes . . . . . . . . . . . . . . . . . . 25

2.6.1 Group IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6.2 Group I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6.3 Group II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.6.4 Group III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.7 General Theory of Waveguides . . . . . . . . . . . . . . . . . . . . . . . 45

2.8 Complex Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3 Characterization of Transversely Periodic Waveguides 613.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2 The Generalized Transverse Resonance Technique for TEz and TM z modes 63

3.2.1 Potential Expansions . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.2 Discontinuity Characterization . . . . . . . . . . . . . . . . . . . 67

3.2.3 Discontinuity Cascading . . . . . . . . . . . . . . . . . . . . . . 70

3.2.4 Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . 70

3.3 The Generalized Transverse Resonance Technique for TEM z modes . . . 73

3.3.1 Electrostatic Potential Expansion . . . . . . . . . . . . . . . . . 74

3.3.2 Discontinuity Characterization . . . . . . . . . . . . . . . . . . . 75


3.3.4 Inhomogeneous Linear System Resolution . . . . . . . . . . . . 80

3.3.5 The Special χy = 1 Case . . . . . . . . . . . . . . . . . . . . . . 82

3.3.6 Characteristic Impedance . . . . . . . . . . . . . . . . . . . . . . 83

3.4 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

xi

CONTENTS

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4 Analysis of Transversely Periodic Structures 914.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2 The Mode Matching Technique . . . . . . . . . . . . . . . . . . . . . . . 92

4.2.1 Transverse Electric Field Continuity . . . . . . . . . . . . . . . . 93

4.2.2 Transverse Magnetic Field Continuity . . . . . . . . . . . . . . . 97

4.2.3 Linear System of Equations - Matrix Formulation . . . . . . . . . 99

4.3 Analysis of Transversely Periodic Structures . . . . . . . . . . . . . . . . 101

4.3.1 The Periodic Rectangular Waveguide . . . . . . . . . . . . . . . 102


4.4 Numerical and Experimental Results . . . . . . . . . . . . . . . . . . . . 106

4.4.1 Wire-Medium Slab . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.4.2 Fishnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.4.3 Cross-Shaped Metal Patches . . . . . . . . . . . . . . . . . . . . 111

4.4.4 Rectangular Displaced Patches . . . . . . . . . . . . . . . . . . . 112

4.4.5 Periodic H-shaped Waveguide Simulator . . . . . . . . . . . . . 113

4.4.6 Parallel-plate with a bi-periodic textured surface . . . . . . . . . 115

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5 Analysis of Laterally Open Periodic Waveguides 1195.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2 Method of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2.1 Modifications to the Field Expansions . . . . . . . . . . . . . . . 123

5.2.2 Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . 124

5.3 Numerical and Experimental Results . . . . . . . . . . . . . . . . . . . . 125

5.3.1 Single-Row Waveguide . . . . . . . . . . . . . . . . . . . . . . . 127

5.3.2 Multiple-Row Waveguide . . . . . . . . . . . . . . . . . . . . . 127

5.3.3 Displaced-Row Waveguide . . . . . . . . . . . . . . . . . . . . . 130

5.3.4 Substrate Integrated Waveguides . . . . . . . . . . . . . . . . . . 131

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6 Conclusions 1396.1 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.2 Future Research Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.3 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.4 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

xii

CONTENTS

6.4.1 Journal Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416.4.2 Conference Proceedings . . . . . . . . . . . . . . . . . . . . . . 142

A Appendix A 143A.1 The Wave Equation for the Transverse Electric Field . . . . . . . . . . . 143A.2 Derivation of the Periodic Boundary Conditions for the Transverse Elec-

tric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145A.3 Some Mathematical Lemmas . . . . . . . . . . . . . . . . . . . . . . . . 147

Bibliography 149

xiii

1Introduction

1.1 Motivation and Objectives

The so-called meta-materials were, a couple of years ago, one of the most prolific researchlines as demonstrated by the nearly exponential growth of the journal and conference pa-pers, as well as the several books, [Eleftheriades and Balmain, 2005], [Caloz and Itoh,2006], [Engheta and Ziolkowski, 2006] and [Marques et al., 2008], that appeared sincethe first experimental demonstration of the left-handed behavior of an artificial struc-ture, [Smith et al., 2000]. Nowadays, after [Spielman et al., 2009] and the posteriordebate at IMS 2009, the term meta-materials has been somehow repudiated. Neverthe-less, it is evident the the meta-material viewpoint has given rise to some very interestingapplications, for instance [Sievenpiper et al., 1999], [Ziolkowski, 2009], [Mata-Contreraset al., 2009] and [Borja et al., 2010].

This Ph. D. dissertation is part of a bigger objective, the synthesis of the so-calledHigh Impedance Surfaces (HIS), [Sievenpiper et al., 1999]. A HIS is a periodic structuredesigned to be used in the frequency band where no surface waves can exists. Therefore,the reflection coefficient of a homogeneous plane wave impinging on such surface has a0 angle, as opposed to the 180 of the classic conductor surface. Of course, the synthesisand optimization of this type of surfaces requires a very efficient method of analysis.The analysis of layered surfaces is usually carried out by the integral equation method[Mittra et al., 1988], formulated either in the spatial or spectral domain, and making useof the Method of Moments (MoM) to solve the integral equation. This method dealseasily with excitations from arbitrary incidence angles, but requires some extra effort totake into account the finite metallization thickness [Webb et al., 1992], which is relevantat millimeter-wave frequencies. Layered surfaces, reflectarrays, and more complex bi-periodic structures (wire-slabs and mushroom-like structures) have also been analyzed by

1

CHAPTER 1. INTRODUCTION

hybrid MoM techniques [Bozzi and Perregrini, 2003], or by means of numerical CPU-intensive approaches, such as the Finite-Difference Time-Domain (FDTD) [Roden et al.,1998, Harms et al., 1994], the Finite Integral Technique, or the Finite Element Method(FEM) [Bardi et al., 2002] –in this last two cases sometimes with the use of commercialsoftware such as CST Microwave Studio R© [Kildal et al., 2011] or HFSSTM [Rajagopalanand Rahmat-Samii, 2010].

The objective of this Ph. D. dissertation is to demonstrate that periodic struc-tures can be analyzed and understood by means of closed waveguide theories andtechniques. As a result of this theoretical statement, an innovative method of analysis

for periodic structures will be proposed in this text. As opposed to the integral equa-tion approach, which searches for the current distribution on the metal patches (or for themagnetic currents at the apertures), this Ph. D. dissertation proposes a method of analy-sis that considers the field distribution between metallizations as the unknown magnitudeto be determined. Thus, the metallizations become boundary conditions for the fields inthe dielectric regions between them, rather than the carriers of the unknown currents. Theproposed approach will, not only be efficient, but also provide an alternative interpretationof the physical mechanisms that rule periodic structures.

1.2 Text Organization

This Ph. D. dissertation is organized as follows:

Chapter 1 serves as an introduction and reports the main motivation and objectiveof the work. In Chapter 2 transversely periodic cylindrical structures are studied undera waveguide viewpoint. It will be shown that, under certain circumstances, this typeof structures can be described by a complete set of TEM , TE and TM modes. Fur-thermore, by comparing these modes with the modes of any Perfect Electric Conductor(PEC) or Perfect Magnetic Conductor (PMC) closed waveguide a great number of analo-gies are found. As a result, it will be concluded that periodic structures can be analyzedand understood using closed waveguide theories and techniques. To sum up, Chapter 2

contains the study of the wave equation under a combination of conductor and periodicboundary conditions, a variational expression for the obtention of the eigenvalues, a com-pleteness demonstration, a thorough analysis of the three types of solutions and finally,some considerations regarding the complex power flow.

In Chapter 3 a semi-analytical method for the obtention of the aforementioned set ofmodes will be described. The proposed approach is an extension of the generalized Trans-

2

1.2. TEXT ORGANIZATION

verse Resonance Technique (TRT) which is highly efficient and versatile. This chaptercontains a detailed description of the potential expansions, the discontinuity characteriza-tion procedure and the generalized characteristic equation that are the building blocks onwhich the proposed technique relies. Finally, some results obtained by means of this TRTgeneralization will be presented.

Once an adequate method for obtaining the modes of transversely periodic structureshas been presented in Chapter 3, Chapter 4 proposes the Mode Matching technique as anefficient method for the characterization of discontinuities between transversely periodicstructures. It will be shown that the resulting equations are analogous to the ones ob-tained when considering a discontinuity between two closed waveguides. Besides, sincethe modes of the transversely periodic structures have been obtained by means of thegeneralized TRT described in Chapter 3, the integrals to be solved here are all analytic,therefore the overall MM efficiency is greatly enhanced. Finally, the analysis of six differ-ent structures will be presented in this chapter to show the pros of the proposed approach.

Chapter 5 presents a method for analyzing laterally open periodic waveguides. Byslightly modifying the generalized TRT described in Chapter 3, a very efficient methodfor analyzing structures such as the well known Substrate Integrated Waveguide (SIW) isobtained. This approach takes into account the open and periodic nature of the structurewithout any approximation. A detailed description of the required modifications are pre-sented in this chapter, as well as several application examples. It must be highlighted thatfour of these examples are measured breadboards, where the accuracy of the approach isdemonstrated.

Finally, Chapter 6 gives some conclusions in the form of the original contributions ofthis Ph. D. dissertation and the future research lines that it has given rise to. Furthermore,the framework in which this work has been carried out are reported together with thepublications this Ph. D. dissertation has generated.

3

2Transversely Periodic Waveguide Modes

2.1 Introduction

This chapter studies the eigenvalue problem that arises when considering a transverselyperiodic cylindrical structure with arbitrary cross section such as the one shown in fig-ure 2.1(a). Once the problem is appropriately defined, we will study the properties of theeigenvalues in section 2.2. Then, some comments concerning the orthogonality propertyof the eigenfunctions will be made and a suitable orthogonality relation will be obtainedin section 2.3. After that, a variational expression for the eigenvalues will be derivedin section 2.4 as a prerequisite for the study of the completeness of the eigenfunctions(section 2.5). Once the conditions to obtain a complete set of eigenfunctions are ob-tained, section 2.6 will study the properties of these eigenfunctions. Up to this point, theeigenvalue problem has been studied assuming a certain variation of the field with thelongitudinal component. Section 2.7 will remove this assumption. Finally, some veryinteresting properties regarding the behavior of the complex power flow will be analyzedin section 2.8.

Because of the particular geometry of a cylindrical structure extended in one direc-tion, henceforth the z-direction, it is possible to simplify the analysis by considering onlythe cross section of the structure (shown in figure 2.1(b)). Furthermore, it is convenient tofirst separate the electric and magnetic fields into their longitudinal and transverse com-ponents:

~E =(~Et + zEz

)e−jkzz; ~H =

(~Ht + zHz

)e−jkzz (2.1)

where the exponential variation with respect to z has been assumed and written ex-plicitly so that ~Et, ~Ht, Ez and Hz become independent of z. As aforementioned, howto eliminate the assumption on the exponential variation with z is subject of section 2.7.

5

CHAPTER 2. TRANSVERSELY PERIODIC WAVEGUIDE MODES

Since the structure is periodic in both transverse directions (x and y), if a periodicexcitation is assumed, the Floquet’s Theorem [Collin, 1960, Ch.9] may be used to greatlysimplify the analysis by considering only one period or unit cell. The green dash-dottedline in figure 2.1(b) shows the most commonly used unit cell when analyzing periodicstructures in the literature. However, the red dashed line on figure 2.1(b) will be used inthis chapter for convenience. Figure 2.1(c) shows the chosen unit cell in detail. The black-line contour, lc, can be either a Perfect Electric Conductor (PEC), a Perfect MagneticConductor (PMC), or any combination of both of them. The periodic contour is shownin dashed lines, and is made up of four segments denoted as lp,n, lp,s, lp,e and lp,w, wheresubscripts n, s, e and w stand for north, south, east and west, respectively. Additionally,we define l as the complete contour of the cross section, i.e. l = lc∪ lp,n∪ lp,s∪ lp,e∪ lp,w,and S as the surface enclosed by l. Finally, it is worth highlighting that figure 2.1(c) showsthe local coordinate system, (n, τ , z), that will be used henceforth along the contour l.

This chapter deals with the unit cell of figure 2.1(c) as the cross section of a ho-mogeneous waveguide, in order to study the normal modes encountered when periodicboundary conditions are part of the waveguide contour. The analogies and differencesbetween closed waveguide modes and the transversely periodic waveguide ones will behighlighted throughout the text.

lc

lp,e

lp,n

lp,s

lp,w

ε, µ

(c)

n

τ

(b)x

y

x

yz

(a)

S

Figure 2.1: (a) Generic bi-periodic cylindrical structure. The red dashed line shows a periodor unit cell. (b) Cross section of the bi-periodic homogeneous cylindrical structure shown in (a).The dash-dotted line shows the most commonly used unit cell. The dashed line shows the unit cellthat will be used in this dissertation. (c) Detail of a unit cell of the periodic cylindrical structure.The local coordinate system, (n, τ , z), that is used along the contour is shown in the upper-rightcorner.

6

2.2. EIGENVALUE STUDY

Table 2.1: Relevant Equations from Appendix A.1.

z ×∇Hz + jkz z × ~Ht = −jωε ~Et z ×∇Ez + jkz z × ~Et = jωµ ~Ht

∇× ~Ht = jωεzEz ∇× ~Et = −jωµzHz

∇ · ~Et = jkzEz ∇ · ~Ht = jkzHz

Table 2.2: PEC and PMC Boundary Conditions for the Transverse Electric Field

PEC n× ~Et = 0 ∇ · ~Et = 0

PMC n · ~Et = 0 n×∇× ~Et = 0

2.2 Eigenvalue Study

In order to study the cross section shown in figure 2.1(c) as an homogeneous waveguide,appropriate wave equation and boundary conditions have to be specified. The wave equa-tion can be written, for the transverse electric field, as [Kurokawa, 1969, Ch.3]:

∇×∇× ~Et −∇(∇ · ~Et

)− k2c ~Et = 0 (2.2)

wherek2c = k2 − k2z , with k2 = ω2µε , (2.3)

and kz was introduced in (2.1). The derivation of this equation from Maxwell’s equationscan be found in Appendix A.1. Furthermore, table 2.1 shows the most relevant equationsused in the aforementioned derivation process which will be of some use in this chapter.The PEC and PMC boundary conditions to be imposed along the lc contour of figure 2.1(c)for the transverse electric field are summarized in table 2.2. Additionally, let χx andχy be the complex periodicity constants linking contours lp,e with lp,w and lp,s with lp,n,respectively. The boundary conditions for the transverse electric field along the lp,n andlp,s periodic contours are shown in table 2.3. The boundary conditions along lp,w andlp,e are completely analogous to the ones of table 2.3, but using χx instead of χy. Thederivation of the periodic boundary conditions for the transverse electric field is detailedin Appendix A.2.

It is a well-known result that any homogeneous PEC closed cylindrical structure canbe described in terms of a complete set of TEM, TE and TM modes, which is also true for

7


Table 2.3: Periodic Boundary Conditions for the Transverse Electric Field

n× ~Et

∣∣∣lp,n

= χy n× ~Et

∣∣∣lp,s

n · ~Et∣∣∣lp,n

= χy n · ~Et∣∣∣lp,s

∇ · ~Et∣∣∣lp,n

= χy ∇ · ~Et∣∣∣lp,s

∇× ~Et

∣∣∣lp,n

= χy ∇× ~Et

∣∣∣lp,s

PMC closed waveguides. This section will follow [Kurokawa, 1969, Ch.3] to study theproperties of waveguide cross sections such as the one shown in figure 2.1(c) that includesperiodic boundaries.

Let ~Et,n be an eigenfunction of (2.2) and k2c,n its eigenvalue. By substituting thissolution in (2.2), scalar multiplying by ~E∗t,n and integrating over the cross section S, weobtain∫

S

~E∗t,n · ∇×∇× ~Et,ndS −∫S

~E∗t,n · ∇(∇· ~Et,n

)dS − k2c,n

∫S

~E∗t,n · ~Et,ndS = 0 (2.4)

By integrating by parts, the equation may be rewritten as follows:

k2c,n

∫S

∣∣∣ ~Et,n∣∣∣2 dS =

∫S

[∣∣∣∇× ~Et,n∣∣∣2 +∣∣∣∇· ~Et,n∣∣∣2] dS −

∮l

I•n,ndl −∮l

I×n,ndl (2.5)

where for any two eigenfunctions

I•n,m =(n· ~E∗t,n

)(∇· ~Et,m

)(2.6)

I×n,m =(n× ~E∗t,n

)·(∇× ~Et,m

)= −

[n×(∇× ~Et,m

)]· ~E∗t,n (2.7)

from which the expressions of I•n,n and I×n,n can be particularized (the general expressionswith m 6= n will be used further).

For homogeneous closed PEC waveguides, the line integrals of I•n,n and I×n,n in (2.5)become zero as it is readily deduced from table 2.2. Analogously, for homogeneousclosed PMC waveguides both line integrals are, once more, zero. As a result, for PECand PMC closed waveguides, k2c,n is real and positive. From this point, the well-knownproperties of waveguide modes, such as completeness or orthogonality, are derived and,therefore, these properties rely on the fact that the line integrals of I•n,m and I×n,m arezero for closed structures. A further step in the analysis of bi-periodical homogeneouscylindrical structures is the evaluation of these integrals over the complete contour of theFig. 2.1(c) cross section.

8

2.2. EIGENVALUE STUDY

Let us consider the lc contour. The line integrals of I•n,n and I×n,n over this section ofthe contour are zero as discussed above. Considering now the lp,n ∪ lp,s contour, the lineintegral of I•n,n is expanded, taking into account table 2.3, as:∫

lp,n∪lp,s

(n· ~E∗t,n

)(∇· ~Et,n

)dl =

(1− |χy|2

) ∫lp,s

(n· ~E∗t,n

)(∇· ~Et,n

)∣∣∣lp,s

dl (2.8)

Similarly, the line integral of I×n,n becomes:

∫lp,n∪lp,s

(n× ~E∗t,n

)·(∇× ~Et,n

)dl

=(1− |χy|2

) ∫lp,s

(n× ~E∗t,n

)·(∇× ~Et,n

)∣∣∣lp,s

dl (2.9)

Finally, considering the lp,w ∪ lp,e contour, analogous expressions are found for the lineintegrals of I•n,n and I×n,n. As a consequence,∮

l

In,ndl =(1− |χx|2

) ∫lp,w

In,ndl +(1− |χy|2

) ∫lp,s

In,ndl (2.10)

where In,m = I•n,m + I×n,m, and then (2.5) may be rewritten as:

k2c,n =

∫S

[∣∣∣∇× ~Et,n∣∣∣2 +∣∣∣∇· ~Et,n∣∣∣2] dS −

∮l

In,ndl∫S

∣∣∣ ~Et,n∣∣∣2 dS(2.11)

Thus, the k2c,n eigenvalue is, in general, complex. However, if the condition |χx| = |χy| =1 is imposed, (2.10) becomes zero and the eigenvalues, k2c,n, are real and positive asanalogous to the homogeneous PEC or PMC closed waveguides:

k2c,n =

∫S

[∣∣∣∇× ~Et,n∣∣∣2 +∣∣∣∇· ~Et,n∣∣∣2] dS∫

S

∣∣∣ ~Et,n∣∣∣2 dS(2.12)

9


2.3 Eigenfunction Orthogonality

The next step is to study the orthogonality between eigenfunctions. Let us consider~E∗t,m·(2.2) and its complex conjugate with the subscripts m and n interchanged:

~E∗t,m ·[∇×∇× ~Et,n

]− ~E∗t,m · ∇

(∇ · ~Et,n

)− k2c ~E∗t,m · ~Et,n = 0

~Et,n ·[∇×∇× ~E∗t,m

]− ~Et,n · ∇

(∇ · ~E∗t,m

)− k2c ~Et,n · ~E∗t,m = 0

by subtracting both of them, and integrating by parts over the cross section S,

(k2c,m − k2c,n

) ∫S

~Et,n · ~E∗t,mdS =

∮l

(Im,n − I∗n,m

)dl (2.13)

As opposed to the homogeneous PEC or PMC closed waveguides, this equation can notbe used to derive an orthogonality relation. However, by imposing the |χx| = |χy| = 1

condition, the right-hand side of the equation vanishes and the following orthogonalityrelation is obtained: ∫

S

~Et,n · ~E∗t,mdS = 0 (m 6= n) (2.14)

This chapter focuses on the |χx| = |χy| = 1 case in order to exploit the real-and-positive property of the eigenvalues k2c,n, and the orthogonality property of the eigen-functions. Note that these eigenfunctions, ~Et,n, cannot be assumed to be real without lossof generality (as opposed to the case of waveguides closed by PEC and PMC [Kurokawa,1969, Ch.3, Sec.4]), since the periodicity constants are complex.

2.4 Variational Expression for the Eigenvalues

This section provides a general variational expression, and it’s first order variation, forthe eigenvalues of the differential equation (2.2) when the contour is a combination ofconductor and periodic boundary conditions. It is important to highlight that the proposedvariational expression assumes the |χx| = |χy| = 1 condition. If lc,PEC and lc,PMC are thePEC and PMC parts of the lc contour respectively, and lp is the complete periodic contour,i.e., lp = lp,e∪ lp,w∪ lp,s∪ lp,n, then (2.15) can be used as a variational expression to derivean infinite series of eigenfunctions.

To verify that (2.15) is indeed a variational expression let δk2c be the first order vari-

10

2.4. VARIATIONAL EXPRESSION FOR THE EIGENVALUES

k2c

(~Et

)= Re

(∫S

[∣∣∣∇× ~Et∣∣∣2 +∣∣∣∇· ~Et∣∣∣2] dS −

∫lc,PEC

2(n× ~E∗t

)·(∇× ~Et

)dl

−∫lc,PMC

2(n· ~E∗t

)(∇· ~Et

)dl −

∫lp

[(n· ~E∗t

)(∇· ~Et

)+(n× ~E∗t

)·(∇× ~Et

)]dl

+

∫lp,s

χ∗y

[n× ~E∗t

∣∣∣lp,s· ∇× ~Et

∣∣∣lp,n

+ n· ~E∗t∣∣∣lp,s∇· ~Et

∣∣∣lp,n

]dl

−∫lp,s

χy

[n· ~E∗t

∣∣∣lp,n∇· ~Et

∣∣∣lp,s

+ n× ~E∗t∣∣∣lp,n· ∇× ~Et

∣∣∣lp,s

]dl

+

∫lp,w

χ∗x

[n× ~E∗t

∣∣∣lp,w· ∇× ~Et

∣∣∣lp,e

+ n· ~E∗t∣∣∣lp,w∇· ~Et

∣∣∣lp,e

]dl

−∫lp,w

χx

[n· ~E∗t

∣∣∣lp,e∇· ~Et

∣∣∣lp,w

+ n× ~E∗t∣∣∣lp,e· ∇× ~Et

∣∣∣lp,w

]dl

)/∫S

∣∣∣ ~Et∣∣∣2 dS (2.15)

ation of k2c corresponding to δ ~Et, a small variation from ~Et. In addition to δ ~Et, themagnitude of its derivatives∇· δ ~Et and∇×δ ~Et are also assumed to be small. It is impor-

tant to highlight that the forthcoming verification will obviate the x direction periodicity

since its treatment is completely analogous to the y direction one. In the first place let usmultiply (2.15) by the denominator of the right-hand side of the equation. Consider theleft-hand side of the resulting equation:

(k2c + δk2c

) ∫S

∣∣∣ ~Et + δ ~Et

∣∣∣2 dS = k2c

∫S

∣∣∣ ~Et∣∣∣2 dS

+ 2k2cRe

∫S

~Et · δ ~E∗t dS + δk2c

∫S

∣∣∣ ~Et∣∣∣2 dS (2.16)

where higher order terms have been neglected. Consider now the first integral on theright-hand side of (2.15):

∫S

[∣∣∣∇×( ~Et + δ ~Et

)∣∣∣2 +∣∣∣∇·( ~Et + δ ~Et

)∣∣∣2] dS =

∫S

[∣∣∣∇× ~Et∣∣∣2 +∣∣∣∇· ~Et∣∣∣2] dS

+ 2Re

∫S

[(∇× ~Et

)·(∇×δ ~E∗t

)+(∇· ~Et

)(∇·δ ~E∗t

)]dS (2.17)

11


The first order variation of the contour integral over lc,PEC of (2.15) is:∫lc,PEC

2(n×[~E∗t + δ ~E∗t

])·(∇×

[~Et + δ ~Et

])dl =

∫lc,PEC

2(n× ~E∗t

)·(∇× ~Et

)dl

+

∫lc,PEC

2[(n× ~E∗t

)·(∇×δ ~Et

)+(n×δ ~E∗t

)·(∇× ~Et

)]dl (2.18)

Furthermore, the contour integral over lc,PMC of (2.15) gives the following first ordervariation:∫

lc,PMC

2(n·[~E∗t + δ ~E∗t

])(∇·[~Et + δ ~Et

])dl =

∫lc,PMC

2(n· ~E∗t

)(∇· ~Et

)dl∫

lc,PMC

2[(n· ~E∗t

)(∇·δ ~Et

)+(n·δ ~E∗t

)(∇· ~Et

)]dl (2.19)

Next, consider the fourth integral of the right-hand side of (2.15), its first order variationis:∫

lp

[(n·[~E∗t + δ ~E∗t

])(∇·[~Et + δ ~Et

])+(n×[~E∗t + δ ~E∗t

])·(∇×

[~Et + δ ~Et

])]dl

=

∫lp

[(n· ~E∗t

)(∇· ~Et

)+(n× ~E∗t

)·(∇× ~Et

)+(n× ~E∗t

)·(∇×δ ~Et

)]dl

+

∫lp

[(n· ~E∗t

)(∇·δ ~Et

)+(n·δ ~E∗t

)(∇· ~Et

)+(n×δ ~E∗t

)·(∇× ~Et

)]dl (2.20)

The fifth integral gives:

∫lp,s

χ∗y

[n×[~E∗t + δ ~E∗t

]∣∣∣lp,s·∇×

[~Et + δ ~Et

]∣∣∣lp,n+ n·

[~E∗t + δ ~E∗t

]∣∣∣lp,s∇·[~Et + δ ~Et

]∣∣∣lp,n

]dl

=

∫lp,s

χ∗y

[n× ~E∗t

∣∣∣lp,s· ∇× ~Et

∣∣∣lp,n

+ n· ~E∗t∣∣∣lp,s∇· ~Et

∣∣∣lp,n

+ n× ~E∗t∣∣∣lp,s· ∇×δ ~Et

∣∣∣lp,n

]dl

+

∫lp,s

χ∗y

[n· ~E∗t

∣∣∣lp,s∇·δ ~Et

∣∣∣lp,n

+ n×δ ~E∗t∣∣∣lp,s· ∇× ~Et

∣∣∣lp,n

+ n·δ ~E∗t∣∣∣lp,s∇· ~Et

∣∣∣lp,n

]dl (2.21)

Additionally, the first order variation of the sixth integral of the right-hand side of (2.15)

12


is:∫lp,s

χy

[n·[~E∗t + δ ~E∗t

]∣∣∣lp,n∇·[~Et + δ ~Et

]∣∣∣lp,s+ n×

[~E∗t + δ ~E∗t

]∣∣∣lp,n· ∇×

[~Et + δ ~Et

]∣∣∣lp,s

]dl

=

∫lp,s

χy

[n· ~E∗t

∣∣∣lp,n∇· ~Et

∣∣∣lp,s

+ n× ~E∗t∣∣∣lp,n· ∇× ~Et

∣∣∣lp,s

+ n× ~E∗t∣∣∣lp,n· ∇×δ ~Et

∣∣∣lp,s

]dl

+

∫lp,s

χy

[n· ~E∗t

∣∣∣lp,n∇·δ ~Et

∣∣∣lp,s

+ n·δ ~E∗t∣∣∣lp,n∇· ~Et

∣∣∣lp,s

+ n×δ ~E∗t∣∣∣lp,n· ∇× ~Et

∣∣∣lp,s

]dl (2.22)

As aforementioned, the first order variation of the seventh and eight integrals are not con-sidered here since they are completely analogous to (2.21) and (2.22), but using χx insteadof χy and substituting the contours lp,s and lp,n by the contours lp,w and lp,e, respectively.

Let us now integrate by parts the second term of the right-hand side of (2.17):

Re

∫S

2[(∇× ~Et

)·(∇×δ ~E∗t

)+(∇· ~Et

)(∇·δ ~E∗t

)]dS

= 2Re

(∫S

δ ~E∗t

[∇×∇× ~Et −∇

(∇· ~Et

)]dS

+

∮l

[(n×δ ~E∗t

)·(∇× ~Et

)+(n·δ ~E∗t

)(∇· ~Et

)]dl

)(2.23)

where the contour integral can be rewritten as the sum of line integrals over the lc,PEC ,lc,PMC and lp. By using (2.16)-(2.22), simplifying the resulting expression by taking intoaccount (2.15) and substituting (2.23), the first order variation of the complete variationalexpression may be written as (2.24). Once more, (2.24) misses the x direction periodicityterms that would be analogous to the last four integrals of the equation.

In order to verify that (2.15) is indeed a variational expression there are still somealgebraic manipulations to be done in (2.24). First, the two integrals over the completeperiodic contour lp must be rewritten in terms of the integrals over each section of theaforementioned contour. Again, the x direction periodic contours are omitted here. There-

13


δk2c

∫S

∣∣∣ ~Et∣∣∣2 dS = Re

(∫S

δ ~E∗t ·[∇×∇× ~Et −∇

(∇· ~Et

)− k2c ~Et

]dS

+

∫lc,PEC

2[(n·δ ~E∗t

)(∇· ~Et

)−(n× ~E∗t

)·(∇×δ ~Et

)]dl

+

∫lc,PMC

2[δ ~E∗t ·

(n×∇× ~Et

)−(n· ~E∗t

)(∇·δ ~Et

)]dl

+

∫lp

[(n×δ ~E∗t

)·(∇× ~Et

)+(n·δ ~E∗t

)(∇· ~Et

)]dl

−∫lp

[(n× ~E∗t

)·(∇×δ ~Et

)+(n· ~E∗t

)(∇·δ ~Et

)]dl

+

∫lp,s

χ∗y

[n· ~E∗t


∣∣∣lp,n

+ n·δ ~E∗t∣∣∣lp,s∇· ~Et

∣∣∣lp,n

]dl

−∫lp,s

χy

[n· ~E∗t


∣∣∣lp,s

+ n·δ ~E∗t∣∣∣lp,n∇· ~Et

∣∣∣lp,s

]dl

+

∫lp,s

χ∗y

[n×δ ~E∗t

∣∣∣lp,s· ∇× ~Et

∣∣∣lp,n

+ n× ~E∗t∣∣∣lp,s· ∇×δ ~Et

∣∣∣lp,n

]dl

−∫lp,s

χy

[n×δ ~E∗t

∣∣∣lp,n· ∇× ~Et

∣∣∣lp,s

+ n× ~E∗t∣∣∣lp,n· ∇×δ ~Et

∣∣∣lp,s

]dl

)(2.24)

14


fore, the first term is rewritten as:∫lp

[(n×δ ~E∗t

)·(∇× ~Et

)+(n·δ ~E∗t

)(∇· ~Et

)]dl

=

∫lp,s

[n×δ ~E∗t

∣∣∣lp,n· ∇× ~Et

∣∣∣lp,n− n×δ ~E∗t

∣∣∣lp,s· ∇× ~Et

∣∣∣lp,s

]dl

+

∫lp,s

[n·δ ~E∗t

∣∣∣lp,n∇· ~Et

∣∣∣lp,n− n·δ ~E∗t

∣∣∣lp,s∇· ~Et

∣∣∣lp,s

]dl (2.25)

and the second:

−∫lp

[(n× ~E∗t

)·(∇×δ ~Et

)+(n· ~E∗t

)(∇·δ ~Et

)]dl

= −∫lp,s

[n× ~E∗t

∣∣∣lp,n· ∇×δ ~Et

∣∣∣lp,n− n× ~E∗t

∣∣∣lp,s· ∇×δ ~Et

∣∣∣lp,s

]dl

−∫lp,s

[n· ~E∗t


∣∣∣lp,n− n· ~E∗t


∣∣∣lp,s

]dl (2.26)

Merging now every line integral over the periodic contour lp,s, the terms may be rear-ranged to write:

∫lp,s

n×δ ~E∗t∣∣∣lp,n·[∇× ~Et

∣∣∣lp,n− χy ∇× ~Et

∣∣∣lp,s

]+ n·δ ~E∗t

∣∣∣lp,n

[∇· ~Et

∣∣∣lp,n− χy ∇· ~Et

∣∣∣lp,s

]dl

+

∫lp,s

n×δ ~E∗t∣∣∣lp,s·[χ∗y ∇× ~Et

∣∣∣lp,n− ∇× ~Et

∣∣∣lp,s

]+ n·δ ~E∗t

∣∣∣lp,s

[χ∗y ∇· ~Et

∣∣∣lp,n− ∇· ~Et

∣∣∣lp,s

]dl

+

∫lp,s

∇×δ ~Et∣∣∣lp,n·[χ∗y n× ~E∗t

∣∣∣lp,s− n× ~E∗t

∣∣∣lp,n

]+∇×δ ~Et

∣∣∣lp,s·[n× ~E∗t

∣∣∣lp,s− χy n× ~E∗t

∣∣∣lp,n

]dl

+

∫lp,s

∇·δ ~Et∣∣∣lp,n

[χ∗y n· ~E∗t

∣∣∣lp,s− n· ~E∗t

∣∣∣lp,n

]+∇·δ ~Et

∣∣∣lp,s

[n· ~E∗t

∣∣∣lp,s− χy n· ~E∗t

∣∣∣lp,n

]dl

This way, the complete (including the x-direction periodicity) first order variation of(2.15) is (2.27).

It follows from (2.27) that if ~Et is an eigenfunction (that satisfies both the differentialequation and the boundary conditions), the first order variation δk2c corresponding to anysmall δ ~Et vanishes (along the contour l, δ ~Et will be expanded as δEnn + δEτ τ ). Con-versely, if δk2c is equal to zero for every possible small variation δ ~Et from ~Et, ~Et is aneigenfunction for the following reasons:

15


δk2c

∫S

∣∣∣ ~Et∣∣∣2 dS = Re

(∫S

δ ~E∗t ·[∇×∇× ~Et −∇

(∇· ~Et

)− k2c ~Et

]dS

+

∫lc,PEC

2[(n·δ ~E∗t

)(∇· ~Et

)−(n× ~E∗t

)·(∇×δ ~Et

)]dl

+

∫lc,PMC

2[δ ~E∗t ·

(n×∇× ~Et

)−(n· ~E∗t

)(∇·δ ~Et

)]dl

+

∫lp,s

n×δ ~E∗t∣∣∣lp,n·[∇× ~Et

∣∣∣lp,n− χy ∇× ~Et

∣∣∣lp,s

]+ n·δ ~E∗t

∣∣∣lp,n

[∇· ~Et

∣∣∣lp,n− χy ∇· ~Et

∣∣∣lp,s

]dl

+

∫lp,s

n×δ ~E∗t∣∣∣lp,s·[χ∗y ∇× ~Et

∣∣∣lp,n− ∇× ~Et

∣∣∣lp,s

]+ n·δ ~E∗t

∣∣∣lp,s

[χ∗y ∇· ~Et

∣∣∣lp,n− ∇· ~Et

∣∣∣lp,s

]dl

+

∫lp,s

∇×δ ~Et∣∣∣lp,n·[χ∗y n× ~E∗t

∣∣∣lp,s− n× ~E∗t

∣∣∣lp,n

]+∇·δ ~Et

∣∣∣lp,n

[χ∗y n· ~E∗t

∣∣∣lp,s− n· ~E∗t

∣∣∣lp,n

]dl

+

∫lp,s

∇×δ ~Et∣∣∣lp,s·[n× ~E∗t

∣∣∣lp,s− χy n× ~E∗t

∣∣∣lp,n

]+∇·δ ~Et

∣∣∣lp,s

[n· ~E∗t

∣∣∣lp,s− χy n· ~E∗t

∣∣∣lp,n

]dl

+

∫lp,w

n×δ ~E∗t∣∣∣lp,e·[∇× ~Et

∣∣∣lp,e− χx ∇× ~Et

∣∣∣lp,w

]+ n·δ ~E∗t

∣∣∣lp,e

[∇· ~Et

∣∣∣lp,e− χx ∇· ~Et

∣∣∣lp,w

]dl

+

∫lp,w

n×δ ~E∗t∣∣∣lp,w·[χ∗x ∇× ~Et

∣∣∣lp,e− ∇× ~Et

∣∣∣lp,w

]+ n·δ ~E∗t

∣∣∣lp,s

[χ∗y ∇· ~Et

∣∣∣lp,n− ∇· ~Et

∣∣∣lp,s

]dl

+

∫lp,w

∇×δ ~Et∣∣∣lp,e·[χ∗x n× ~E∗t

∣∣∣lp,w− n× ~E∗t

∣∣∣lp,e

]+∇·δ ~Et

∣∣∣lp,e

[χ∗x n· ~E∗t

∣∣∣lp,w− n· ~E∗t

∣∣∣lp,e

]dl

+

∫lp,w

∇×δ ~Et∣∣∣lp,w·[n× ~E∗t

∣∣∣lp,w− χy n× ~E∗t

∣∣∣lp,e

]+∇·δ ~Et

∣∣∣lp,w

[n· ~E∗t

∣∣∣lp,w− χx n· ~E∗t

∣∣∣lp,e

]dl

(2.27)

16


1. If the differential equation (2.2) is not satisfied somewhere in S, δ ~Et can be madeparallel to ∇×∇× ~Et − ∇∇· ~Et − k2c

~Et, both n ·δ ~Et and ∇×δ ~Et can be madeequal to zero on lc,PEC , n×δ ~Et and ∇·δ ~Et can be made equal to zero on lc,PMC

and δ ~Et, ∇·δ ~Et and ∇×δ ~Et can be forced to fulfill the boundary conditions oftable 2.4 along the lp contour (the appropriate boundary conditions between thelp,e and lp,w contours are obtained from table 2.4 by substituting χy by χx and lp,nand lp,s by lp,e and lp,w, respectively). Please note that there is no incompatibilityin imposing both n ·δ ~Et and ∇×δ ~Et along the lc,PEC contour since the first oneimplies δEn = 0 and the second ∂δEτ/∂n = 0 Analogously, by simultaneouslyimposing both n×δ ~Et and ∇·δ ~Et along the lc,PMC contour no incompatibility isfound since the first one implies δEτ = 0 and the second one ∂δEn/∂τ = 0. Underthe aforementioned circumstances, the only non-zero term of the right-hand side of(2.27) is the first one, the surface integral. Therefore, δk2c would take a non-zerovalue, which contradicts the initial statement: δk2c is equal to zero for every possible

small variation δ ~Et from ~Et.

2. If the differential equation is satisfied everywhere on S but the boundary condition∇·~Et = 0 is not, then n·δ ~Et can be made to have the same sign as∇·~Et along lc,PEC ,∇×δ ~Et can be made equal to zero on lc,PEC , n×δ ~Et and∇·δ ~Et can be made equalto zero on lc,PMC and δ ~Et,∇·δ ~Et and∇×δ ~Et can be forced to fulfill the boundaryconditions of table 2.4 along the lp contour. Once more, a contradiction is obtainedsince under the aforementioned circumstances δk2c would take a non-zero value.

3. If the differential equation is satisfied everywhere on S, ∇· ~Et = 0 is satisfied onlc,PEC but n× ~Et = 0 is not, then ∇×δ ~Et can be made parallel to n× ~Et, n×δ ~Etand∇·δ ~Et can be made zero on lc,PMC and δ ~Et,∇·δ ~Et and∇×δ ~Et can be chosento fulfill table 2.4 along lp. Thus, a new contradiction is obtained.

4. If the differential equation is satisfied everywhere on S, the PEC boundary condi-tions are satisfied along lc,PEC , but n×∇× ~Et = 0 is not satisfied on lc,PMC , thenn×δ ~Et can be made parallel to ∇× ~Et and ∇·δ ~Et can be made equal to zero onlc,PMC and δ ~Et, ∇×δ ~Et and ∇·δ ~Et can be forced to fulfill table 2.4 along the lpcontour. Therefore, δk2c would be take a non-zero value according to (2.27) and,once more, a contradiction is obtained.

5. If the differential equation, the PEC boundary conditions and n×∇× ~Et = 0 onlc,PMC are satisfied but n· ~Et 6= 0 on lc,PMC , then ∇·δ ~Et can be made to have thesame sign as n·~Et and δ ~Et,∇·δ ~Et and∇×δ ~Et can be forced to fulfill the boundary

17


conditions of table 2.4 on lp. Therefore, δk2c would take a non-zero value accordingto (2.27), contradicting the original assumption.

6. If the differential equation and the PEC and PMC boundary conditions are fulfilledbut ∇× ~Et is not periodic in the y direction (not fulfilling the boundary conditions

on table 2.3), then n×δ ~E∗t∣∣∣lp,n

can be made parallel to[∇× ~Et

∣∣∣lp,n− χy ∇× ~Et

∣∣∣lp,s

],

n×δ ~Et∣∣∣lp,s

can be made equal to zero and n·δ ~Et,∇·δ ~Et and∇×δ ~Et can be chosen

to fulfill table 2.4 on the rest of lp. Once more, there is only one non-vanishing termon the right-hand side of (2.27) and a contradiction is obtained. Moreover, by in-terchanging n×δ ~Et

∣∣∣lp,n

with n×δ ~Et∣∣∣lp,s

above, a similar contradiction is obtained.

Furthermore, this same reasoning holds for the x direction boundary conditions byappropriately interchanging quantities.

7. If the differential equation, the PEC boundary conditions along the lc,PEC , the PMCboundary conditions along the lc,PMC and the periodic boundary condition for∇×~Et are fulfilled but ∇· ~Et is not periodic on the y direction, then n·δ ~E∗t

∣∣∣lp,n

can be

made to have the same sign as[∇· ~Et

∣∣∣lp,n− χy ∇· ~Et

∣∣∣lp,s

], n·δ ~E∗t

∣∣∣lp,s

can be made

to vanish and n ·δ ~Et, ∇·δ ~Et and ∇×δ ~Et can be made to fulfill table 2.4 on therest of lp. Therefore, a new contradiction is obtained. By interchanging n·δ ~E∗t

∣∣∣lp,n

with n·δ ~E∗t∣∣∣lp,s

an analogous contradiction occurs. Moreover, a similar reasoning

can be applied to the x direction periodicity.

8. If the differential equation, the PEC and PMC and periodic boundary condition for∇× ~Et and ∇· ~Et are fulfilled but n× ~Et is not periodic on the y direction, then

∇×δ ~Et∣∣∣lp,n

can be made to have the same sign as[χ∗y n× ~E∗t

∣∣∣lp,s− n× ~E∗t

∣∣∣lp,n

],

∇×δ ~Et∣∣∣lp,s

can be chosen to be zero and ∇·δ ~Et and ∇×δ ~Et can be chosen to

fulfill table 2.4 on the rest of lp. Once more, this would imply a non-zero δk2c thuscontradicting the original assumption. Of course, an analogous reasoning holds in-terchanging ∇×δ ~Et

∣∣∣lp,n

with ∇×δ ~Et∣∣∣lp,s

and for the x direction when appropriate

modifications are implemented.

9. Finally, if the differential equation, the PEC and PMC and periodic boundary condi-tion for∇×~Et,∇·~Et and n×~Et are fulfilled but n·~Et is not periodic on the y direction,

then ∇·δ ~Et∣∣∣lp,n

can be made to have the same sign as[χ∗y n· ~E∗t

∣∣∣lp,s− n· ~E∗t

∣∣∣lp,n

],

18

2.5. EIGENFUNCTION COMPLETENESS

Table 2.4: Periodic Boundary Conditions for δ ~Et

n× δ ~Et∣∣∣lp,n

= −χy n× δ ~Et∣∣∣lp,s

n · δ ~Et∣∣∣lp,n

= −χy n · δ ~Et∣∣∣lp,s

∇ · δ ~Et∣∣∣lp,n

= −χy ∇ · δ ~Et∣∣∣lp,s

∇× δ ~Et∣∣∣lp,n

= −χy ∇× δ ~Et∣∣∣lp,s

∇·δ ~Et∣∣∣lp,s

can be made to vanish and∇·δ ~Et can be forced to fulfill table 2.4 on the

remaining lp contour. Therefore, a contradiction occurs. An analogous procedurecan be used for the x direction periodicity.

2.5 Eigenfunction Completeness

Once the variational expression (2.15) is obtained an infinite series of eigenfunctions,~Et,n

, can be conceptually derived by successively adding orthogonality conditions.

Since k2c(~Et

)≥ 0 for all functions satisfying the boundary conditions, there must

be a function which minimizes k2c(~Et

). Let ~Et,1 be this function, then, for any small

variation δ ~Et from ~Et,1 satisfying the same boundary conditions, the first order variationδk2c from k2c

(~Et,1

)vanishes. It is worth highlighting that, when functions satisfying the

boundary conditions are considered, both the variational expression, (2.15), and its firstorder derivative, (2.27), simplify to:

k2c

(~Et

)=

∫S

∣∣∣∇× ~Et∣∣∣2 +∣∣∣∇· ~Et∣∣∣2 dS (2.28)

δk2c

∫S

∣∣∣ ~Et∣∣∣2 dS = Re

∫S

δ ~E∗t ·[∇×∇× ~Et −∇

(∇· ~Et

)− k2c ~Et

]dS (2.29)

respectively. As discussed in the preceding section, ~Et,1 thus chosen is an eigenfunction.Next, let ~Et,2 be a function that satisfies the boundary conditions and at the same timeminimizes k2c

(~Et

)with the additional condition that it is orthogonal to ~Et,1. Then, for

any small variation δ ~Et satisfying the boundary conditions, δk2c is again found to be zeroas follows. Let us first decompose δk2c into two parts, one due to the part of δ ~Et pro-portional to ~Et,1 and the other due to the part of δ ~Et orthogonal to ~Et,1 . Consider δk2c,‖,corresponding to the part of δ ~Et proportional to ~Et,1, by substituting δ ~E∗t = A~E∗t,1 (where

19


A is the proportionality constant) and ~Et = ~Et,2 in (2.29) we have:

δk2c,‖

∫S

∣∣∣ ~Et,2∣∣∣2 dS = Re

∫S

A~E∗t,1 ·[∇×∇× ~Et,2 −∇

(∇· ~Et,2

)− k2c,2 ~Et,2

]dS (2.30)

By integrating twice by parts the first term of the right-hand side,

Re

∫S

~E∗t,1 · ∇×∇× ~Et,2dS = Re

∫S

∇×∇× ~E∗t,1 · ~Et,2dS

−Re∮l

[(n× ~Et,2

)·(∇× ~E∗t,1

)+(n× ~E∗t,1

)·(∇× ~Et,2

)]dl

where, since both ~Et,1 and ~Et,2 satisfy the boundary conditions and assuming that the|χx| = |χy| = 1 condition holds, the line integral vanishes, (2.10). Repeating the sameprocess with the second term of the right-hand side of (2.30),

−Re∫S

~E∗t,1 · ∇(∇· ~Et,2

)dS = −Re

∫S

∇(∇· ~E∗t,1

)· ~Et,2dS

−Re∮l

[(∇· ~Et,2

)(n· ~E∗t,1

)−(∇· ~E∗t,1

)(n· ~Et,2

)]dl

where, once more, the contour integral vanishes provided that the |χx| = |χy| = 1 con-dition is satisfied. Substituting these last two equations in (2.30) and using the waveequation, (2.2),

δk2c,‖

∫S

∣∣∣ ~Et,2∣∣∣2 dS = Re

[A(k2c,1 − k2c,2

) ∫S

~E∗t,1 · ~Et,2dS]

= 0 (2.31)

since both ~Et,1 and ~Et,2 are orthogonal. Let us now consider, δk2c,⊥ corresponding to thepart of δ ~Et orthogonal to ~Et,1,

δ ~Et,⊥ = δ ~Et − ~Et,1

∫S

δ ~Et · ~E∗t,1dS∫S

~Et,1 · ~E∗t,1dS

Now that we have checked that δk2c,‖ vanishes, it is obvious, from the definition of ~Et,2,that δk2c,⊥ corresponding to that part of δ ~Et orthogonal to ~Et,1 also vanishes. As a result,δk2c = 0 for any small variation δ ~Et satisfying the boundary conditions. Applying thesame argument used for ~Et,1, ~Et,2 is found to be an another eigenfunction. Similarly, let~Et,3 be a function which satisfies the boundary conditions and minimizes k2c

(~Et

)under

20


the two additional conditions of being orthogonal to both ~Et,1 and ~Et,2. An argumentsimilar to the above shows that ~Et,3 is also an eigenfunction. Note that the ~Et,n’s thusobtained satisfy the orthogonality condition (2.14) even if k2c,n = k2c,m, as long as n 6= m.In this way, adding the orthogonality conditions one by one, one can find an infiniteseries of eigenfunctions ~Et,1, ~Et,2, ~Et,3. . . , with their corresponding eigenvalues satisfying0 ≤ k2c,1 ≤ k2c,2 ≤ k2c,3. . . , and k2c,n thus obtained increases indefinitely with n, i.e,

limn→∞

k2c,n =∞ (2.32)

A proof as to why k2c,n increases with n can be found in [Kurokawa, 1969, Appendix I].

Let us next prove the completeness of the set of eigenfunctions assuming the infinitegrowth of the eigenvalues. To do so, we first normalize all the eigenfunctions, i.e.,∫

S

∣∣∣ ~Et,n∣∣∣2 dS = 1 (2.33)

Let ~f be an arbitrary function which satisfies the boundary conditions and has derivatives∇· ~f and ∇× ~f which are square-integrable over S. Let ~fN be defined as:

~fN = ~f −N−1∑n=1

An ~Et,n , where An =

∫S

~f · ~E∗t,ndS (2.34)

and a2N by,

a2N =

∫S

∣∣∣~fN ∣∣∣2 dS (2.35)

Using the orthogonality and normalization conditions for the ~Et,n’s, this last equation canbe rewritten as:

a2N =

∫S

∣∣∣∣∣~f −N−1∑n=1

An ~Et,n

∣∣∣∣∣2

dS =

∫S

∣∣∣~f ∣∣∣2 dS −N−1∑n=1

|An|2 (2.36)

From the definition of ~fN , the normalized error function ~fN/aN is orthogonal to all the~Et,n’s for which n is smaller than N , i.e.

∫S

~fNaN· ~E∗t,ndS = 0 (n < N) (2.37)

Since ~Et,N gives the smallest value k2N of k2(~Et,n

)under the same orthogonality and

21


boundary conditions as for ~fN/aN , k2c(~fN/aN

)cannot be less than k2N . Noting that

~fN/aN is normalized by definition, we calculate k2c(~fN/aN

)as follows:

k2c

(~fN/aN

)= |aN |−2

∫S

∣∣∣∇× ~fN ∣∣∣2 +∣∣∣∇· ~fN ∣∣∣2dS

= |aN |−2∫S

∣∣∣∣∣∇× ~f −N−1∑n=1

An∇× ~Et,n

∣∣∣∣∣2

+

∣∣∣∣∣∇· ~f −N−1∑n=1

An∇· ~Et,n

∣∣∣∣∣2

dS (2.38)

By expanding both terms on the right-hand side of (2.38) and rearranging, the followingequation is obtained:

k2c

(~fN/aN

)= |aN |−2

∫S

∣∣∣∇× ~f ∣∣∣2 +∣∣∣∇· ~f ∣∣∣2dS

− 2 |aN |−2Re∑n

A∗n

[∫S

~f ·(∇×∇× ~E∗t,n −∇∇· ~E∗t,n

)dS

+

∮l

(n· ~f)(∇· ~E∗t,n

)+(n× ~f

)·(∇× ~E∗t,n

)dl

]+ |aN |−2

∑n

∑m

AnA∗m

[∫S

~Et,n ·(∇×∇× ~E∗t,m −∇∇· ~E∗t,m

)dS

+

∮l

(n· ~Et,n

)(∇· ~E∗t,m

)+(n× ~Et,n

)·(∇× ~E∗t,m

)dl

](2.39)

where both contour integrals vanish, since ~f , ~Et,n and ~Et,m satisfy the boundary condi-tions and the |χx| = |χy| = 1 condition is assumed to hold. By using the wave equation,(2.2), and the orthogonality and normalization properties of the eigenfunctions, (2.39),gives:

k2c

(~fN/aN

)= |aN |−2

∫S

∣∣∣∇× ~f ∣∣∣2 +∣∣∣∇· ~f ∣∣∣2dS − |aN |−2 N−1∑

n=1

k2c,n |An|2 (2.40)

Noting that this has to be larger than k2N and that the last term is positive, we have

|aN |2 ≤ k−2N

∫S

∣∣∣∇× ~f ∣∣∣2 +∣∣∣∇· ~f ∣∣∣2dS (2.41)

Since the integral is finite by hypothesis and k2N increases indefinitely, |aN |2 approaches

22


zero with increasing N . Thus, form the definition of aN , we have:

limn→∞

∫S

∣∣∣∣∣~f −N−1∑n=1

An ~Et,n

∣∣∣∣∣2

dS = 0 (2.42)

which shows that ~f can be expanded in terms of the eigenfunctions.

Let us call a function defined over a two-dimensional domain piecewise-continuouswhen it is continuous in the domain except along a finite number of lines each havinga finite length. Let ~F be a piecewise-continuous and square-integrable function definedin S, the cross section of the waveguide (figure 2.1(c)). The function ~F does not needto satisfy the boundary conditions, but it can be approximated by a function ~f , whichsatisfies the boundary conditions and has square-integrable derivatives, in the sense that:∫

S

∣∣∣~F − ~f∣∣∣2 dS < ε/4 (2.43)

where ε is an arbitrary small positive number. This is possible because a continuousfunction ~f can be constructed in such a way that ~f is equal to ~F outside S(ε) whileboth its magnitude and direction continuously change inside S(ε), where S(ε) indicatesthe small area which completely contains the lines of discontinuity of ~F , [Kurokawa,1969, p. 120]. It is always possible, therefore, to satisfy (2.43) by making S(ε) sufficientlysmall. On the other hand, from (2.42),

∫S

∣∣∣∣∣~f −N−1∑n=1

An ~Et,n

∣∣∣∣∣2

dS < ε/4 (2.44)

for a sufficiently large N . Therefore, we have

∫S

∣∣∣∣∣~F −N−1∑n=1

An ~Et,n

∣∣∣∣∣2

dS =

∫S

∣∣∣∣∣~F − ~f + ~f −N−1∑n=1

An ~Et,n

∣∣∣∣∣2

dS

≤ 2

∫S

∣∣∣~F − ~f∣∣∣2 dS + 2

∫S

∣∣∣∣∣~f −N−1∑n=1

An ~Et,n

∣∣∣∣∣2

dS ≤ ε

where Lemma A.3.2 and relations (2.43) and (2.44) have been used. Since ε is arbitrary,the above relation shows that a piecewise continuous and square-integrable function, but

23


otherwise arbitrary, function ~F can be expanded in terms of the ~Et,ns:

limn→∞

∫S

∣∣∣∣∣~F −N−1∑n=1

An ~Et,n

∣∣∣∣∣2

dS = 0 (2.45)

This property of the ~Et,n’s is called completeness. The above relation is usually writtenin the form

~F =∞∑n=1

An ~Et,n , where An =

∫S

~F · ~E∗t,ndS (2.46)

The reason that (2.34) can be rewritten in the form of (2.46) is easily seen from the fol-lowing:

∣∣∣∣∫S

~F · ~E∗t,ndS −∫S

~f · ~E∗t,ndS

∣∣∣∣2 =

∣∣∣∣∫S

(~F − ~f

)· ~E∗t,ndS

∣∣∣∣2≤∫S

∣∣∣(~F − ~f)∣∣∣2 dS

∫S

∣∣∣ ~Et,n∣∣∣2 dS ≤ ε/4

where use is made of Lemma A.3.1 and the normalization condition.

Additionally, let ~F =∑An ~Et,n and ~G =

∑Bn

~Et,n . If ~F is equal to ~G in the

sense that∫ ∣∣∣~F − ~G

∣∣∣2 dS = 0, then An = Bn for each n and vice versa. The proofof this assertions can be found in [Kurokawa, 1969, pp. 111-112] to which appropriatemodifications have to be introduced.

Now consider ~F × z, since it satisfies the conditions for the expansion to be possible,we have:

~F × z =∞∑n=1

~Et,n

∫S

(~F × z

)· ~E∗t,ndS (2.47)

After multiplying both sides by z×:

~F =∞∑n=1

z × ~Et,n

∫S

~F ·(z × ~E∗t,n

)dS (2.48)

This means that ~F can be expanded in terms of the z × ~Et,n’s. Later on, it will be shownthat z × ~Et,n is proportional to the transverse magnetic field, ~Ht,n.

24

2.6. TRANSVERSELY PERIODIC WAVEGUIDE MODES

2.6 Transversely Periodic Waveguide Modes

The objective of this section is to study the properties and characteristics of the set ofeigenfunctions obtained by means of (2.15). By analyzing (2.12):

k2c,n

∫S

∣∣∣ ~Et,n∣∣∣2 dS =

∫S

[∣∣∣∇× ~Et,n∣∣∣2 +∣∣∣∇· ~Et,n∣∣∣2] dS

it becomes apparent that the eigenvalues, k2c,n, can be classified, depending on the natureof the eigenvector, ~Et,n, into four different groups:

Group I . ∇× ~Et,n = 0 ∇· ~Et,n = 0

Group II . ∇× ~Et,n 6= 0 ∇· ~Et,n = 0

Group III . ∇× ~Et,n = 0 ∇· ~Et,n 6= 0

Group IV . ∇× ~Et,n 6= 0 ∇· ~Et,n 6= 0

Let us now study each of these groups. A complete set of eigenfunctions can be derivedsuch that each function in the set belongs to any of the first three groups. The proof ofthis statement is detailed in section 2.6.1. From table 2.1 let us highlight the two followingequations:

∇× ~Et = −jωµzHz ; ∇· ~Et = jkzEz (2.49)

The electromagnetic waves derived from the ~Et,n’s in group I have no longitudinal com-ponents, since Ez = 0 and Hz = 0 from (2.49) when kz 6= 0. These waves are calledtransverse electromagnetic modes or TEM z modes. Similarly, for the waves derived fromgroups II and III , we have Ez = 0 and Hz = 0, respectively. Thus, transverse electric,or TEz modes, are derived from group II and transverse magnetic, or TM z modes, arederived from group III .

2.6.1 Group IV .

The purpose of this section is to prove the above statement: ”A complete set of eigen-functions can be derived such that each function in the set belongs to any of the first threegroups”. Consider a ~Et,n belonging to group IV , two new functions can be defined:

~E ′t = A∇×∇× ~Et,n ~E ′′t = B∇(∇· ~Et,n

)25


where A and B are normalizing constants for ~E ′t and ~E ′′t , respectively. By means of thesenew functions, ~Et,n can be rewritten, using the wave equation (2.2), in the form:

~Et,n = k−2c,n

[A−1 ~E ′t −B−1 ~E ′′t

](2.50)

Since k2c,n 6= 0, (2.50) shows that ~Et,n can be expressed as a lineal combination of ~E ′t and~E ′′t in which neither of them vanishes.

If ~E ′t = 0, then:~Et,n = −k−2c,n∇

(∇· ~Et,n

)and ∇× ~Et,n would then be:

∇× ~Et,n = −k−2c,n∇×[∇(∇· ~Et,n

)]= 0

and a contradiction is obtained, since ∇× ~Et,n 6= 0 is a condition for a eigenfunction tobelong to group IV . Similarly, if ~E ′′t = 0 then:

~Et,n = k−2c,n∇×∇× ~E ′t,n

As a result, the divergence of ~Et,n is:

∇· ~Et,n = k−2c,n∇·(∇×∇× ~E ′t,n

)= 0

Once more, this contradicts the group IV assumption.

By substituting ~E ′t in∇×∇×(2.2) we have:

∇×∇×[∇×∇× ~Et

]−∇×∇×

[∇(∇ · ~Et,n

)]− k2c∇×∇× ~Et,n = 0

∇×∇× ~E ′t − k2c ~E ′t = 0

since ∇· ~E ′t is zero from the definition of ~E ′t, ∇∇· ~E ′t can be added to the right hand sideof the equation without changing its value, i.e.

∇×∇× ~E ′t −∇(∇ · ~E ′t

)− k2c ~E ′t = 0 (2.51)

which is exactly the same differential equation as (2.2). Therefore, if ~E ′t fulfills the sameboundary conditions as ~Et,n, then ~E ′t would be an eigenfunction of the same problem.

First, consider the PEC section, lc,PEC , of the contour of figure 2.1(c), both n×~E ′t = 0

26


and ∇· ~E ′t = 0 must be fulfilled. The second condition is automatically fulfilled bydefinition of ~E ′t. The first condition may be rewritten as:

n× ~E ′t = An×(∇×∇× ~Et,n

)= Ak2c,nn× ~Et,n + An×∇

(∇· ~Et,n

)The first term on the right-hand side is zero because of the boundary condition for ~Et,n.The second term may be rewritten as:

n×∇(∇· ~Et,n

)= n×

(n∂

∂n+ τ

∂

∂τ

)(∇· ~Et,n

)= z

∂

∂τ

(∇· ~Et,n

)since∇· ~Et,n is zero along lc,PEC its derivative with respect to τ must vanish. To sum up,~E ′t satisfies along the lc,PEC contour both:

n× ~E ′t = 0, ∇· ~E ′t = 0 (on lc,PEC) (2.52)

Let us now consider the lc,PMC contour. The PMC boundary conditions that shouldbe fulfilled by ~E ′t are n · ~E ′t = 0 and n×∇× ~E ′t = 0. The first one may be expanded asfollows:

n· ~E ′t = An·(∇×∇× ~Et,n

)= −A∇·

(n×∇× ~Et,n

)= −A

(n∂

∂n+ τ

∂

∂τ

)·(n×∇× ~Et,n

)where∇× ~Et,n is a vector on the z direction (see table 2.1), thus n×∇× ~Et,n is τ directedalong the lc,PMC contour and n· ~E ′t ends up being:

n· ~E ′t = −Aτ ∂∂τ·(n×∇× ~Et,n

)= 0

because of n×∇× ~Et,n is zero along the lc,PMC contour, its derivative with respect to τvanishes. Considering now the second boundary condition, n×∇× ~E ′t, let us expand thisexpression in the following way:

n×∇× ~E ′t = An×[∇×∇×

(∇× ~Et,n

)]= An×

[∇∇·

(∇× ~Et,n

)−∆

(∇× ~Et,n

)]= −An×∆

(∇× ~Et,n

)= −An× ∂2

∂z2

(∇× ~Et,n

)27


where the fact that∇×~Et,n is z directed has been used. Since, ~Et,n is function only of thetransverse components and, therefore, its derivative with respect to z is zero, n×∇× ~E ′tvanishes. To sum up, both

n· ~E ′t = 0, n×∇× ~E ′t = 0 (on lc,PMC) (2.53)

boundary conditions are fulfilled by ~E ′t.

Finally, let us consider the periodic contour, lp. As we already did when calculatingthe first order variation of (2.15), only the y direction periodic boundary conditions willbe considered here. The first periodic boundary condition that ~E ′t has to fulfill is:

n× ~E ′t∣∣∣lp,n

= χy n× ~E ′t∣∣∣lp,s

(2.54)

rewriting n× ~E ′t as before:

n× ~E ′t = Ak2c,nn× ~Et,n + Az∂

∂τ

(∇· ~Et,n

)where the first term is periodic because of the boundary conditions that ~Et,n has to fulfill(table 2.3) and the second term is also periodic since ∇· ~Et,n is periodic along lp andso must be its derivative with respect to τ . As a result, (2.54) is satisfied. The secondperiodic boundary condition to satisfy is:

n· ~E ′t∣∣∣lp,n

= χy n· ~E ′t∣∣∣lp,s

(2.55)

by rewriting n· ~E ′t in the following way:

n· ~E ′t = An·∇×(∇× ~Et,n

)= Az · ∂

∂τ

[∇× ~Et,n

]where the fact that ∇× ~Et,n is a z directed vector (table 2.1) has been exploited. Takinginto account that∇× ~Et,n satisfies the periodic boundary conditions on table 2.3 along lp,it is self-evident that its derivative with respect to τ will also satisfy those conditions and,therefore, (2.55) is fulfilled. The next boundary condition to be considered is:

∇× ~E ′t∣∣∣lp,n

= χy ∇× ~E ′t∣∣∣lp,s

(2.56)

28


where ∇× ~E ′t can be rewritten as:

∇× ~E ′t = A∇×[∇×∇× ~Et,n

]= A∇×

[∇∇· ~Et,n + k2c,n ~Et,n

]= Ak2c,n∇× ~Et,n

Since ∇× ~Et,n satisfies the periodic boundary conditions so does ∇× ~E ′t and (2.56) isfulfilled. Finally, the last boundary condition:

∇· ~E ′t∣∣∣lp,n

= χy ∇· ~E ′t∣∣∣lp,s

(2.57)

is automatically satisfied, since∇· ~E ′t vanishes along the whole cross section.

Considering the differential equation (2.51) and the boundary conditions (2.52)-(2.57)that ~E ′t has to fulfill, we conclude that ~E ′t is an eigenfunction. Since both ~E ′t and ~Et,n

satisfy the same differential equation and the same boundary conditions all of which arelinear, the linear combination of ~E ′t and ~Et,n, namely ~E ′′t , must satisfy the same equationand boundary conditions. Thus, ~E ′′t is another eigenfunction. The orthogonality relationbetween ~E ′t and ~E ′′t can be established as follows:

1

AB

∫S

~E ′t · ~E ′′∗t dS =

∫S

(∇×∇× ~Et,n

)·[∇(∇· ~E∗t,n

)]dS

=

∫S

∇·[(∇· ~E∗t,n

)(∇×∇× ~Et,n

)]dS =

∮l

(∇· ~E∗t,n

)(∇×∇× ~Et,n

)· ndl

Self-evidently, the line integral must vanish if the ~E ′t and ~E ′′t eigenfunctions are orthogo-nal. In the first place, the line integral along the lc,PEC contour vanishes since∇·~Et,n = 0

along lc,PEC . To solve the integral along lc,PMC let us rewrite the integrand as:

n·(∇×∇× ~Et,n

)(∇· ~E∗t,n

)= −∇·

(n×∇× ~Et,n

)(∇· ~E∗t,n

)= τ

∂

∂τ·(n×∇× ~Et,n

)(∇· ~E∗t,n

)where use has been made that ∇× ~Et,n is a z directed vector and, therefore, n×∇×~Et,n points in the τ direction. Since n×∇× ~Et,n is zero along the lc,PMC contour, itsderivative with respect to τ must vanish. Thus, the line integral along the lc,PMC contouralso vanishes. Finally, let us consider the lp contour. Once more, only the y directionperiodicity will be considered here. By directly applying the periodic boundary conditions

29


(table 2.3),

1

AB

∫S

~E ′t · ~E ′′∗t dS =(1− |χy|2

) ∫lp,s

z∂

∂τ·(∇× ~Et,n

)(∇· ~E∗t,n

)dl

where, again the fact that∇×~Et,n is periodic implies that z ∂∂τ·(∇× ~Et,n

)is also periodic

with the same periodicity. Finally, it becomes apparent that, if the |χx| = |χy| = 1

condition is satisfied, then ~E ′t and ~E ′′t are orthogonal.

Now suppose that eigenfunctions are obtained successively by adding an orthogonal-ity condition each time and we find that the nth eigenfunction happens to appear in groupIV for the first time. Instead of taking ~Et,n itself, let us take ~E ′t derived from it as the nth

eigenfunction and ~E ′′t as the n + 1st function. Then, ~E ′t and ~E ′′t are orthogonal to ~Et,m

(m < n). For instance, the proof for ~E ′t to be orthogonal to ~Et,m is as follows. If ~Et,mbelongs to group I or III , then∇×∇× ~Et,m vanishes; whereas, if ~Et,m belongs to groupII , then ∇×∇× ~Et,m = k2c,m

~Et,m. In either case, by twice integrating by parts and usingthe boundary conditions:∫S

~E∗t,m · ~E ′tdS = A

∫S

~E∗t,m ·(∇×∇× ~Et,n

)dS = A

∫S

(∇×∇× ~E∗t,m

)· ~Et,ndS = 0

The proof for ~E ′′t is similar. The n+ 2nd eigenfunction can be obtained so as to minimizethe variational expression for k2c under the condition that this function is orthogonal toall the functions up to n + 1st. In this way, without the loss of completeness, a set of

orthogonal eigenfunctions can be derived such that each function belongs to any of groups

I , II or III . Hereafter, we shall assume that this has been done.

2.6.2 Group I .

In the first place, consider TEM z modes. Since the transverse electric field is irrotational,it can be written as the gradient of a scalar function. Moreover, the transverse electric fieldmust also be solenoidal, thus,

~Et,n = ∇φ⇒ ∇· ~Et,n = ∇·∇φ = 0 (2.58)

Along the lc,PEC contour, the ∇· ~Et,n = 0 is automatically satisfied and the n× ~Et,n = 0

implies

n× ~Et,n = n×∇φ =∂φ

∂τ= 0 (2.59)

30


Therefore, the scalar potential φ must be constant along connected sections of the lc,PECcontour. The boundary condition for the scalar potential along the lc,PMC is

n· ~Et,n = n·∇φ =∂φ

∂n= 0 (2.60)

since the n×∇× ~Et,n = 0 is automatically satisfied. Finally, consider the boundaryconditions along the lp,n ∪ lp,s section of the periodic contour,

n× ~Et,n∣∣∣lp,n

= χy n× ~Et,n∣∣∣lp,s⇒ n×∇φ|lp,n = χy n×∇φ|lp,s

n· ~Et,n∣∣∣lp,n

= χy n· ~Et,n∣∣∣lp,s⇒ n·∇φ|lp,n = χy n·∇φ|lp,s

The first boundary condition can be rewritten, as implied by the derivative with respect toτ ,

φ|lp,n + A = χy

[φ|lp,s +B

](2.61)

where A and B are different constants that can be merged into one by taking, for instanceB to the left-hand side of the equation. However, this additional constant is zero aslong as both periodicity constants are not simultaneously one, χx 6= χy 6= 1, i.e. the

scalar potential will not be periodic when χx = χy = 1. No mathematical proof has beenobtained for this assertion. However, it is supported by the results of Chapters 3 and 4. Tosum up, the periodic boundary conditions for the scalar potential φ are,

φ|lp,n + const = χy φ|lp,s ;∂φ

∂n

∣∣∣∣lp,n

= χy∂φ

∂n

∣∣∣∣lp,s

(2.62)

lc

lp,n

lp,s

lp,elp,w

ε, µ

n

τ

V0

x

y

Figure 2.2: Cross-section of a generic transversely periodic cylindrical structure when the greendash-dotted unit cell of figure 2.1(b) is used instead of the red dashed unit-cell.

31


It is a well known fact that the number of TEM z modes in a PEC enclosed waveg-uide depends on the number of independent PEC boundaries. Let us now try to figureout the number of TEM modes in a transversely periodic waveguide. To that purpose,it will be advantageous to use the green dash-dotted unit-cell of figure 2.1(b) instead ofthe red dashed one. Therefore, consider the unit-cell shown in figure 2.2 where the con-tour l = lc ∪ lp is highlighted with a thin black dashed line and the vertical segmentsthat unite the lp and the lc contours are supposed to be infinitely close to each other. Thereader should note that the usage of the cross-section depicted in figure 2.1(c) or the onein figure 2.2 does not alter the results obtained to this point. Self-evidently, all the math-ematical expressions of this chapter hold in both cases without modification. In addition,it shall be highlighted that the following analysis is restricted to the more practical case

in which all the lc contour is PEC. Consider the following integral,∫S

|∇φ|2 dS =

∫S

∇φ · (∇φ)∗ dS =

∫S

∇·[φ∇φ∗] dS −∫S

φ∇·∇φ∗dS (2.63)

where the second integral vanishes by using (2.58). As long as χx 6= χy 6= 1, the aboveexpression can be rewritten as,∫

S

|∇φ|2 dS =

∫lc

φ∇φ∗ · ndl +

∫lp

φ∇φ∗ · ndl = V0

∫lc

∇φ∗ · ndl

+(1− |χx|2

)∫lp,w

φ (∇φ∗ · n) dl +(1− |χy|2

)∫lp,s

φ (∇φ∗ · n) dl (2.64)

since the scalar potential must be a constant (V0) along the lc contour. Therefore, if the|χx| = |χy| = 1 condition is satisfied,∫

S

|∇φ|2 dS = V0

∫lc

∇φ∗ · ndl 6= 0 (2.65)

When dealing with PEC closed waveguides with singly connected contours, the right-hand side of the above equation vanishes, [Kurokawa, 1969, p. 124]. Thus, this equationserves as a demonstration, in such case, that PEC closed waveguides with singly con-nected contours can not have a TEM mode. In the present case, despite the fact that theconsidered transversely periodic cylindrical structure has only one independent conduc-tor, the right hand side of the equation does not vanish as long as there is a conductorin the cross-section. Therefore, the fact that the considered cross-section has a TEMmode can be stated. If there are two or more independent PEC boundaries, then φ couldtake a different value on each boundary and there would beN independent ways to set the

32


potentials of the N conductors. Therefore, N independent solutions belonging to groupI would be found. In the particular case of a cross-section without a conductor insidethere will not be a TEM mode, this particular example will be further explained in sec-tion 4.3.1. On the other hand, when χx = χy = 1 the generic transversely periodiccylindrical structure of figure 2.2 has two TEM modes. Moreover, the aforementionedconductor-less cross-section has also two TEM modes, section 4.3.1. Since this state-ment has been obtained from the results of the following chapters, the number of TEMmodes for cross-sections with several independent PEC boundaries is difficult to predictin general.

Finally, let us consider the relation between the transverse electric and magnetic fields.From table 2.1 we have,

jωµ ~Ht,n = z ×∇Ez,n + jkz,nz × ~Et,n

by substituting∇· ~Et,n = jkz,nEz,n again from table 2.1 and rearranging we obtain:

~Ht,n =−1

ωµkz,nz ×

∇∇· ~Et,n − k2z,n ~Et,n

(2.66)

Since group I eigenfunctions are solenoidal and their eigenvalue vanishes (which impliesk2z = ω2µε),

~Ht,n =

√ε

µz × ~Et,n (2.67)

where we can define for TEM z modes the constant:

Zn =

√µ

ε(2.68)

which has impedance dimensions.

2.6.3 Group II .

Let us now consider the TEz modes, i.e., group II in (2.6). Self-evidently, no eigenvaluewill vanish, k2c,n 6= 0, as deduced from (2.12). By definition, ∇× ~Et,n is not equal to zerofor any ~Et,n from group II . Let us write this non-zero rotational in the form:

∇× ~Et,n = zkc,nφH,n (2.69)

33


this definition is similar to the left equation in (2.49). It shall be remarked that the constantkc,n in (2.69) ensures that by normalizing the φH,n’s, the ~Et,n’s will be automatically nor-malized too. This will be demonstrated later on, (2.76). Substituting (2.69) into∇×(2.2),we have

∇×∇×∇× ~Et,n −∇×∇(∇ · ~Et,n

)− k2c,n∇× ~Et,n = 0⇒

∇×∇× zφH,n − k2c,nzφH,n = 0

by expanding this last equation,

∆φH,n + k2c,nφH,n = 0 (in S ) (2.70)

Therefore, a differential equation that the φH,n’s (obtained through (2.69)) satisfy, hasbeen obtained.

Now that we have a differential equation, let us consider the boundary conditions thatthe φH,n’s fulfill. In the first place, consider the lc,PEC contour. In this section of thecontour, two boundary conditions are satisfied by ~Et,n. The first one, ∇· ~Et,n = 0, isautomatically satisfied by assumption, ~Et,n belongs to group II . The second boundarycondition, n× ~Et,n = 0, can be written in terms of φH,n as:

n× ~Et,n = n×[∇×∇× ~Et,n

]= kc,nn×[∇×zφH,n]

= kc,nn×[∇φH,n × z] = −kc,nz (n·∇φH,n) = 0

thus,

n·∇φH,n =∂φH,n∂n

= 0 (along lc,PEC) (2.71)

Consider now the lc,PMC contour. Once again, the transverse electric field satisfies twoboundary conditions in this type of contour. The first one, n· ~Et,n = 0 can be rewritten interms of φH,n as follows:

n· ~Et,n = n·[∇×∇× ~Et,n

]= kc,nn·[∇×zφH,n]

= kc,nn·[∇φH,n × z] = kc,nz · [n×∇φH,n]

remembering that φH,n is function only of the transverse components, (2.1), the first

34


boundary condition for the magnetic potential φH,n is:

z · [n×∇φH,n] =∂φH,n∂τ

= 0 (along lc,PMC) (2.72)

The second boundary condition that must be satisfied is n×∇× ~Et,n = 0, then:

φH,n = 0 (along lc,PMC) (2.73)

Self-evidently, if (2.73) is satisfied by φH,n then (2.72) is automatically fulfilled too.Therefore, along the lc,PMC contour the boundary condition that the magnetic potentialfulfills is (2.73). Finally, consider the lp,n∪lp,s section of the contour along which ~Et,n ful-fills the four periodic conditions of table 2.3. Let us now rewrite these four boundary con-ditions in terms of the magnetic potential φH,n. The first one, ∇· ~Et,n

∣∣∣lp,n

= χy ∇· ~Et,n∣∣∣lp,s

is automatically satisfied by the assumption of ~Et,n belonging to group II . The other threeboundary conditions can be rewritten as:


= χy n× ~Et,n∣∣∣lp,s⇒ n·∇φH,n|lp,n = χy n·∇φH,n|lp,s


= χy n· ~Et,n∣∣∣lp,s⇒ z · (n×∇φH,n)|lp,n = χy z · (n×∇φH,n)|lp,s

∇× ~Et,n∣∣∣lp,n

= χy ∇× ~Et,n∣∣∣lp,s⇒ φH,n|lp,n = χy φH,n|lp,s

where the second condition is automatically satisfied if the third identity holds, as anal-ogous to what happened with (2.72). Therefore, along periodic contours the magneticpotential φH,n fulfills:

φH,n|lp,n = χy φH,n|lp,s (2.74)

n·∇φH,n|lp,n = χy n·∇φH,n|lp,s (2.75)

The differential equation and the boundary conditions that the φH,n’s, obtained through(2.69), satisfy are summarized in table 2.5.

Suppose that φH,n and φH,m are two magnetic potentials corresponding to ~Et,n and

35


Table 2.5: Equivalent Eigenfunction problem for TEz modes.

∆φH,n + k2c,nφH,n = 0 (on S)

n·∇φH,n = 0 (along lc,PEC)

φH,n = 0 (along lc,PMC)

φH,n|lp,n = χy φH,n|lp,s (along lp,n ∪ lp,s)

n·∇φH,n|lp,n = χy n·∇φH,n|lp,s (along lp,n ∪ lp,s)

~Et,m, respectively. Then we have∫S

φH,nφ∗H,mdS = (kc,nkc,m)−1

∫S

∇× ~Et,n · ∇× ~E∗t,mdS

= (kc,nkc,m)−1[∫

S

~Et,n · ∇×∇× ~E∗t,mdS +

∮l

(I×n,m

)∗dS

]where the contour integral vanishes under the |χx| = |χy| = 1 assumption as stated by(2.10). Thus, ∫

S

φH,nφ∗H,mdS =

kc,mkc,n

∫S

~Et,n · ~E∗t,mdS (2.76)

Therefore, the φH,n’s obtained from the ~Et,n’s satisfy the orthogonality and normalizationconditions, i.e. they are orthonormal to each other. If n 6= m the right-hand side of (2.76)vanishes, if n = m it becomes unity.

It follows from this that a set of orthonormal eigenfunctions of table 2.5, can bederived from the ~Et,n’s belonging to group II. Conversely, a set of eigenfunctionsbelonging to group II can be derived from the eigenfunctions of table 2.5 using:

kc,n ~Et,n = ∇φH,n × z (2.77)

provided that kc,n 6= 0. In the first place, let us check that the ~Et,n’s defined through (2.77)are solenoidal,

∇· ~Et,n = k−1c,n∇·(∇φH,n × z) = k−1c,nz · (∇×∇φH,n) = 0

Next, we must check that the boundary conditions along the different sections of thewaveguide contour are satisfied. Consider the lc,PEC contour, since the ~Et,n’s are solenoidal

36


on S, the only boundary condition that is left to fulfill is n× ~Et,n = 0,

n× ~Et,n = k−1c,nn×(∇φH,n × z) = −k−1c,nz (n·∇φH,n) = 0

where (2.71) has been used. Consider now the lc,PMC contour. Two boundary conditionsmust be satisfied in this section of the contour. On the one hand,

n· ~Et,n = k−1c,nn·(∇φH,n × z) = k−1c,n∂φH,n∂τ

= 0

since φH,n vanishes along the lc,PMC contour as required by (2.73), its derivative withrespect to τ must vanish too. On the other hand,

n×∇× ~Et,n = k−1c,nn×∇×(∇φH,n × z) = k−1c,nn×(∇×∇×φH,nz)

= −k−1c,nn×z∆φH,n = −kc,nτφH,n = 0

because of (2.73). Finally, consider the periodic contour lp,n ∪ lp,s. The periodicity of∇· ~Et,n is automatically satisfied by assumption. By rewriting n× ~Et,n as:

n× ~Et,n = −k−1c,nz (n·∇φH,n)

we can ensure its periodicity since n·∇φH,n is periodic with the same periodicity, (2.75).Furthermore,

n· ~Et,n = k−1c,n∂φH,n∂τ

since φH,n is periodic, (2.74), its derivative with respect to τ must also be periodic and theperiodicity of n· ~Et,n can also be ensured. Next, the periodicity of∇× ~Et,n can be grantedby writing,

∇× ~Et,n = kc,nφH,nz

Finally, writing (2.77) as,∇φH,n = kc,nz × ~Et,n (2.78)

and substituting into∇(2.70):

∆(z × ~Et,n

)+ k2c,nz × ~Et,n = 0⇒

∇[∇·(z × ~Et,n

)]−∇×∇×

(z × ~Et,n

)+ k2c,n

(z × ~Et,n

)= 0

37


by substituting,

∇(z · ∇× ~Et,n

)= (z · ∇)∇× ~Et,n + z ×∇×∇× ~Et,n ⇒

∇(∇·[z × ~Et,n

])= −z ×∇×∇× ~Et,n

in the first term, and after checking that the second term vanishes:

∇×(z × ~Et,n

)= z

(∇· ~Et,n

)− (∇·z) ~Et,n = 0

since ~Et,n is assumed to belong to group II ,∇(2.70) can be rewritten as:

z ×[∇×∇× ~Et,n − k2c,n ~Et,n

]= 0 ⇒ ∇×∇× ~Et,n − k2c,n ~Et,n = 0 (2.79)

because both∇×∇×~Et,n and ~Et,n have no z components. Therefore, we have shown that~Et,n derived from (2.77) belong to group II . Furthermore, since∫

S

~Et,n · ~E∗t,mdS = (kc,nkc,m)−1∫S

∇φH,n · ∇φ∗H,mdS

= (kc,nkc,m)−1[−∫S

φH,n∆φ∗H,mdS +

∮l

φH,n(n·∇φ∗H,m

)dl

]consider the contour integral. On the one hand, along the lc,PEC and lc,PMC contoursboth n ·∇φ∗H,m and φH,n respectively vanish. On the other hand, along the lp,n ∪ lp,s thecontour integral does not vanish in general. Using (2.70) the orthogonality relation takesthe following form:∫

S

~Et,n · ~E∗t,mdS =kc,mkc,n

∫S

φH,nφ∗H,mdS +

1− |χy|2

kc,nkc,m

∫lp,s

φH,n(n·∇φ∗H,m

)dl (2.80)

Therefore, if the |χx| = |χy| = 1 condition is assumed, the above equation shows that, ifthe φH,n’s are orthonormal, so are the ~Et,n’s.

The above discussion shows that all the ~Et,n’s in group II can be derived fromthe independent solutions for the eigenvalue problem of table 2.5 with the conditionk2c,n 6= 0 and vice-versa. It should be noted here that the eigenfunctions of table 2.5can form a complete set of orthonormal functions. The proof for the completeness is

38


essentially the same as before if we use the following variational expression:

k2c (φH) =

[∫S

|∇φH |2 dS − 2Re

∫lc,PMC

φ∗H (n×∇φH) dl

−Re∫lp

φ∗H (n×∇φH) dl −Re∫lp,s

χ∗y φ∗H |lp,s n×∇φH |lp,n dl

+Re

∫lp,s

χy φ∗H |lp,n n×∇φH |lp,s dl

]/∫S

|φH |2 dS (2.81)

The first order variation of (2.81) is:

δk2c

∫S

|φH |2 dS = −2Re

∫S

δφ∗H[∆φH + k2cφH

]dS

+ 2Re

∫lc,PEC

δφ∗H (n·∇φH) dl − 2Re

∫lc,PMC

φ∗H (n·∇δφH) dl

+Re

∫lp,s

δφ∗H |lp,s[n·∇φH |lp,s − χ

∗y n·∇φH |lp,n

]dl

+Re

∫lp,s

δφ∗H |lp,n[χy n·∇φH |lp,s − n·∇φH |lp,n

]dl

−Re∫lp,s

n·∇δφH |lp,s[φ∗H |lp,s − χy φ

∗H |lp,n

]dl

−Re∫lp,s

n·∇δφH |lp,n[χ∗y φ

∗H |lp,s − φ∗H |lp,n

]dl (2.82)

which shows that if φH is an eigenfunction of the problem defined in table 2.5, thenthe first order variation δk2c corresponding to any small variation δφH from φH vanishes.Conversely, if the first order variation δk2c vanishes for every possible small variation,δφH , from φH , then φH is an eigenfunction.

It shall be remarked that in the above correspondence between eigenfunctions theonly missing one is that for k2c,n = 0. From the variational expression (2.81), ∇φH = 0

if the eigenvalue vanishes. Thus, a φH,n=constant eigenfunction seems to be missing inthe φH,n’s set when builded through (2.69). However, consider the periodic boundaryconditions,

φH,n|lp,n = χy φH,n|lp,s ; φH,n|lp,e = χx φH,n|lp,w

in general (χx, χy ∈ C), the only constant function that fulfills this boundary conditions is

39


φH,n = 0. Only if χx = χy = 1, this constant function could exist. But, even in that case,if the waveguide contour has some PMC section, then the constant functions vanishesonce more, since along the lc,PMC contour φH = 0. Therefore, we conclude that any

piecewise-continuous square-integrable scalar function defined in S can be expanded in

terms of the(k−1c,nz · ∇× ~Et,n

)’s except for a constant term needed when the periodicity

constants are simultaneously χx = χy = 1 and when the waveguide contour has no

PMC section.

Finally, let us, once more, consider the relation between the transverse electric andmagnetic fields. From the general relation, (2.66),

~Ht,n =−1

ωµkz,nz ×

∇∇· ~Et,n − k2z,n ~Et,n

and noting that eigenfunctions belonging to group II are solenoidal with non-vanishingeigenvalues, we write:

~Ht,n =kz,nωµ

z × ~Et,n (2.83)

where we can define for TEz modes:

Zn =ωµ

kz,n(2.84)

which has impedance dimensions.

2.6.4 Group III .

Let us now turn our attention to TM z modes, i.e., group III in (2.6). By defining theelectric potential of the nth eigenfunction, φE,n, through

∇· ~Et,n = kc,nφE,n (2.85)

Once more, this definition is similar to the second equation in (2.49) and the presence ofkc,n is due to normalization considerations. By substituting (2.85) in ∇·(2.2) we obtain,

∇·∇φE,n + k2c,nφE,n = 0 (in S) (2.86)

a differential equation in terms of the electric potential, φE,n, is obtained.

The next step is to consider the boundary conditions that the potential φE,n definedthrough (2.85) satisfies. Along the lc,PEC contour the transverse electric field satisfies

40


two boundary conditions,

∇· ~Et,n = 0⇒ φE,n = 0 (along lc,PEC) (2.87)

and n× ~Et,n = 0 which is automatically fulfilled if (2.87) holds:

n× ~Et,n = −k−2c,nn×∇∇· ~Et,n = −k−1c,nn×∇φE,n = −k−1c,nz∂φE,n∂τ

= 0 (2.88)

Consider now the lc,PMC contour, both n·~Et,n = 0 and n×∇×~Et,n = 0 are satisfied by thetransverse electric field. The second condition is automatically fulfilled by the assumptionthat ~Et,n belongs to the third group. On the other hand, the first condition implies:

n· ~Et,n = −k−2c,nn·∇∇· ~Et,n = −k−1c,nn·∇φE,n ⇒∂φE,n∂n

= 0 (along lc,PMC) (2.89)

Finally, let us consider the lp,n ∪ lp,s section of the contour and rewrite the four pe-riodic conditions of table 2.3 in terms of φE,n. In the first place, we notice that the∇× ~Et,n

∣∣∣lp,n

= χy ∇× ~Et,n∣∣∣lp,s

condition is automatically satisfied by the assumption

of ~Et,n belonging to group III . The other three boundary conditions can be rewritten as:


= χy n× ~Et,n∣∣∣lp,s⇒ n×∇φE,n|lp,n = χy n×∇φE,n|lp,s


= χy n· ~Et,n∣∣∣lp,s⇒ n·∇φE,n|lp,n = χy n·∇φE,n|lp,s

∇· ~Et,n∣∣∣lp,n

= χy ∇· ~Et,n∣∣∣lp,s⇒ φE,n|lp,n = χy φE,n|lp,s

where the first condition is automatically satisfied if the third identity holds, as analogousto what happened with (2.88). Therefore, along periodic contours the electric potentialφE,n fulfills:

φE,n|lp,n = χy φE,n|lp,s (2.90)

n·∇φE,n|lp,n = χy n·∇φE,n|lp,s (2.91)

The differential equation and the boundary conditions that the φE,n’s satisfy are summa-rized in table 2.6.

Suppose that φE,n and φE,m are two electric potentials corresponding to ~Et,n and ~Et,m

41


Table 2.6: Equivalent Eigenfunction problem for TM z modes.

∆φE,n + k2c,nφE,n = 0 (on S)

φE,n = 0 (along lc,PEC)

n·∇φE,n = 0 (along lc,PMC)

φE,n|lp,n = χy φE,n|lp,s (along lp,n ∪ lp,s)

n·∇φE,n|lp,n = χy n·∇φE,n|lp,s (along lp,n ∪ lp,s)

respectively. Then we have∫S

φE,nφ∗E,mdS = (kc,nkc,m)−1

∫S

∇· ~Et,n∇· ~Et,mdS

= (kc,nkc,m)−1[−∫S

~Et,n · ∇∇· ~Et,mdS +

∮l

(I•n,m

)∗dl

](2.92)

where the contour integral vanishes under the |χx| = |χy| = 1 assumption as stated by(2.10). Thus, ∫

S

φE,nφ∗E,mdS =

kc,mkc,n

∫S

~Et,n · ~E∗t,mdS (2.93)

it follows that if the ~Et,n’s are orthonormal to each other, so are the φE,n’s.

Therefore, from the set of eigenfunctions belonging to group III, a set of or-thonormal eigenfunctions of the problem defined in table 2.6 can be obtained. Con-versely, we can derive the ~Et,n’s belonging to group III from the φE,n’s using the relation:

−∇φE,n = kc,n ~Et,n (2.94)

Obviously, the ~Et,n’s thus defined are irrotational. By substituting this equation in∇(2.86)we obtain∇∇· ~Et,n + k2c,n ~Et,n = 0, which is equivalent to the wave equation, (2.2), since∇× ~Et,n = 0 for TM z modes. Consider the boundary conditions that the ~Et,n’s have tosatisfy along the lc,PEC contour:

n× ~Et,n = −k−1c,nn×∇φE,n = −k−1c,n∂φE,n∂τ

= 0

∇· ~Et,n = −k−1c,n∇·∇φE,n = kc,nφE,n = 0

using the PEC boundary condition on table 2.6. Furthermore, the boundary conditions

42


that the ~Et,n’s have to fulfill along lc,PMC contour are very simply checked taking intoaccount the PMC boundary condition on table 2.6 and the irrotational nature of the ~Et,n’sdefined through (2.94). Finally, using the periodic boundary conditions for the electricpotential, the periodic boundary conditions for the ~Et,n’s are easily checked. Therefore,the ~Et,n’s derived from the eigenfunctions of table 2.6 through (2.94) belong to groupIII . Moreover,∫

S

~Et,n · ~E∗t,mdS = (kc,nkc,m)−1∫S

∇φE,n · ∇φ∗E,mdS

= (kc,nkc,m)−1[−∫S

φE,n∇·∇φ∗E,mdS +

∮l

φE,n(n·∇φ∗E,m

)dl

]Consider the contour integral, along the lc,PEC and lc,PMC contours the integral vanishessince φE,n = 0 and n·∇φE,n = 0 respectively. By using the wave equation on the surfaceintegral and the periodic boundary conditions for the electric potential on the line integralalong the lp,n ∪ lp,s contour, the former equation can be rewritten as:∫

S

~Et,n · ~E∗t,mdS =kc,mkc,n

∫S

φE,nφ∗E,mdS +

1− |χy|2

kc,nkc,m

∫lp,s

φE,n(n·∇φ∗E,m

)dl (2.95)

Therefore, if the |χx| = |χy| = 1 conditions holds, the above equation shows that the~Et,n’s obtained though (2.94) are orthonormal if the φE,n’s are selected to be orthonormal.

As analogous what happened with the solutions belonging to group II , there is aone-to-one correspondence between the ~Et,n’s in group III and the solutions of theproblem defined in table 2.6. These functions can form a complete set of orthonormaleigenfunctions. The variational expression necessary for the proof is given by:

k2c (φE) =

[∫S

|∇φE|2 dS − 2Re

∫lc,PEC

φ∗E (n×∇φE) dl

−Re∫lp

φ∗E (n×∇φE) dl −Re∫lp,s

χ∗y φ∗E|lp,s n×∇φE|lp,n dl

+Re

∫lp,s

χy φ∗E|lp,n n×∇φE|lp,s dl

]/∫S

|φE|2 dS (2.96)

43


Its first order variation is:

δk2c

∫S

|φE|2 dS = −2Re

∫S

δφ∗E[∆φE + k2cφE

]dS

+ 2Re

∫lc,PEC

φ∗E (n·∇δφE) dl − 2Re

∫lc,PMC

δφ∗E (n·∇φE) dl

+Re

∫lp,s

δφ∗E|lp,s[n·∇φE|lp,s − χ

∗y n·∇φE|lp,n

]dl

+Re

∫lp,s

δφ∗E|lp,n[χy n·∇φE|lp,s − n·∇φE|lp,n

]dl

−Re∫lp,s

n·∇δφE|lp,s[φ∗E|lp,s − χy φ

∗E|lp,n

]dl

−Re∫lp,s

n·∇δφE|lp,n[χ∗y φ

∗E|lp,s − φ∗E|lp,n

]dl (2.97)

which, once more, shows that if φE is an eigenfunction of the problem defined in ta-ble 2.6, then the first order variation δk2c corresponding to any small variation δφE fromφE vanishes. Conversely, if the first order variation δk2c vanishes for every possible smallvariation, δφE , from φE , then φE is an eigenfunction.

As analogous to the previous section we have not considered the eigenfunction corre-sponding to k2c,n = 0. If k2c,n vanishes, ∇φE,n = 0 in S from (2.94) and φE,n is constantin S. However, consider the periodic boundary conditions,

φE,n|lp,n = χy φE,n|lp,s ; φE,n|lp,e = χx φE,n|lp,w

in general (χx, χy ∈ C), the only constant function that fulfills this boundary conditionsis φE,n = 0. Only if χx = χy = 1, this constant function could exist. But, even inthat case, if the waveguide contour has some PEC section, then the constant functionsvanishes once more, since along the lc,PEC contour φE = 0. Thus, we conclude that

an arbitrary piecewise-continuous square-integrable scalar function defined in S can be

expanded in terms of the(k−1c,n∇· ~Et,n

)’s except for a constant term needed when the

periodicity constants are simultaneously χx = χy = 1 and when the waveguide contour

has no PEC section.

Finally, let us, once more, consider the relation between the transverse electric and

44

2.7. GENERAL THEORY OF WAVEGUIDES

magnetic fields. From the general relation, (2.66),

~Ht,n =−1

ωµkz,nz ×

∇∇· ~Et,n − k2z,n ~Et,n

and noting that eigenfunctions belonging to group II are irrotational with non-vanishingeigenvalues, we write, making use of the wave equation (2.2):

~Ht,n =ωε

kz,nz × ~Et,n (2.98)

where we can define for TM z modes:

Zn =kz,nωε

(2.99)

which has impedance dimensions. Taking into account (2.67), (2.83) and (2.98) we con-clude that, in general, the relation between the transverse electric and magnetic fields canbe written as:

~Ht,n =z × ~Et,nZn

(2.100)

where the impedance, Zn, is:

Zn =

√µ

εfor TEM z modes.

ωµ

kz,nfor TEz modes.

kz,nωε

for TM z modes.

(2.101)

2.7 General Theory of Waveguides

In this section, we shall derive the most general form of an electromagnetic field in astraight lossless waveguide with a uniform cross section, such as the one depicted in fig-ure 2.1(c). It must be highlighted that, although the preceding sections have consideredthat the contour lc could be any combination of PEC and PMC boundaries, the analysisin this section is restricted to waveguides where the complete lc section of the contouris PEC. To do so, we first observe that ~E, ~H , ∇× ~E and ∇× ~H in Maxwell’s equationsare all well behaved (i.e., piecewise-continuous and square-integrable) functions, each ofwhich can be expanded in terms of an appropriate set of functions (section 2.6). Then,

45


by substituting their expanded forms into Maxwell’s equations, all the expansion coeffi-cients and, hence, the electric and magnetic fields will be determined. Since no a priori

assumption is required on the functional forms of the fields such as the exponential vari-

ation with z, this method should give all the possible solutions of Maxwell’s equations in

the waveguide.

In the first place, we expand the transverse components of ~E in terms of the completeset of ~Et,n’s as shown by (2.46) (note that the transverse components of ~E could also havebeen expanded in terms of the z× ~Et,n’s, (2.48)) and the longitudinal component in termsof the

(k−1c,n∇· ~Et,n

)’s as demonstrated in section 2.6.4 (note that the longitudinal electric

field could also have been expanded in terms of the(k−1c,n∇× ~Et,n

)’s plus a constant term):

~E = ~Et + zEz =∑

~Et,n

∫S

~E · ~E∗t,ndS + z∑ ∇· ~Et,n

kc,n

∫S

~E · z∇· ~E∗t,nkc,n

dS (2.102)

where the summation is from n = 1 to n =∞. The election of expanding the transverse

electric field in terms of the ~Et,n’s rather than in terms of the z× ~Et,n’s, so as the election of

the set of(k−1c,n∇· ~Et,n

)’s rather than the set of

(k−1c,n∇× ~Et,n

)’s, is based on the similarity

between boundary conditions of the function to be expanded and the set of eigenfunctions

in terms of which it is expanded. Let us now use the sets of z× ~Et,n’s and(k−1c,n∇× ~Et,n

)’s

to expand the magnetic field, ~H ,

~H =∑

z × ~Et,n

∫S

~H ·(z × ~E∗t,n

)dS

+∑ ∇× ~Et,n

kc,n

∫S

~H ·∇× ~E∗t,nkc,n

dS +z√A

∫S

~H · z√A∗

dS (2.103)

where it shall be noted that ∇× ~Et,n is a z directed vector and the last term correspondsto the constant term which is necessary to expand an arbitrary scalar function in terms ofthe(k−1c,n∇× ~Et,n

)’s, when the periodicity conditions are simultaneously χx = χy = 1

and the contour of the waveguide has no PMC section, as stated in section 2.6.3. Thenormalizing factor is

√A.

Next, noting that ∇× ~E is an ~H-like function, (A.1), we expand it in terms of the

46


z × ~Et,n’s and the(k−1c,n∇× ~Et,n

)’s:

∇× ~E =∑

z × ~Et,n

∫S

∇× ~E · z × ~E∗t,ndS

+∑ ∇× ~Et,n

kc,n

∫S

∇× ~E ·∇× ~E∗t,nkc,n

dS +z√A

∫S

∇× ~E · z√A∗

dS (2.104)

Let us rewrite each expansion coefficient in the last equation so that the resulting expres-sion can be easily compared to (2.103). To this end, we write ∇× ~E in the form:

∇× ~E =

[∇t + z

∂

∂z

]×[~Et(z) + zEz(z)

]= ∇t × ~Et(z) +∇tEz(z)× z +

∂

∂zz × ~Et(z) (2.105)

where ~Et(z) and zEz(z) indicate the transverse and longitudinal components of ~E em-phasizing that they are functions of z in contrast to the ~Et,n’s which are independent.The first term on the right-hand side of (2.105) is longitudinal, i.e. it is a z-directed vec-tor, while the second and third terms are transverse vectors. By using (2.105), the firstexpansion coefficients in (2.104) can be rewritten, taking into account that z × ~E∗t,n is atransverse vector, in the following form:

∫S

∇× ~E · z × ~E∗t,ndS =

∫S

[∇tEz(z)× z +

∂

∂zz × ~Et(z)

]· z × ~E∗t,ndS

=∂

∂z

∫S

~Et(z) · ~E∗t,ndS +

∫S

Ez(z)∇· ~E∗t,ndS −∮l

Ez(z) n· ~E∗t,ndl (2.106)

where the first term has been integrated by parts using the following expression:∫S

∇tEz(z)× z · z × ~E∗t,ndS = −∫S

∇tEz(z) · ~E∗t,ndS

=

∫S

Ez(z)∇· ~E∗t,ndS −∮l

Ez(z) n· ~E∗t,ndl

The second expansion coefficients in (2.104) can be rewritten, taking into account that∇× ~E∗t,n is a z-directed vector,

∫S

∇× ~E ·∇× ~E∗t,nkc,n

dS =

∫S

∇t × ~Et(z) ·∇× ~E∗t,nkc,n

dS

47


By using the following identity:

∇·

[~Et(z) ·

∇× ~E∗t,nkc,n

]= ∇t × ~Et(z) ·

∇× ~E∗t,nkc,n

− ~Et(z) ·∇×∇× ~E∗t,n

kc,n

and the wave equation, (2.2) (noting that the(k−1c,n∇× ~Et,n

)’s are derived from eigenfunc-

tions belonging to group II), the final expression for the second expansion coefficientsis: ∫

S

∇× ~E ·∇× ~E∗t,nkc,n

dS = kc,n

∫S

~Et(z) · ~E∗t,ndS +

∮l

n× ~Et(z) ·∇× ~E∗t,nkc,n

dl (2.107)

Finally, the third expansion coefficients in (2.104) are easily rewritten as:

z√A

∫S

∇× ~E · z√A∗

dS =z√A

∫S

∇t × ~Et(z) · z√A∗

dS =z√A

∮l

n× ~Et(z) · z√A∗

dl

(2.108)Consider now the contour integrals of (2.106)-(2.108). The electric field, ~E, must fulfillthe boundary conditions of the waveguide, i.e.,

Ez(z) = 0 (along lc) Ez(z)|lp,n = χy Ez(z)|lp,s (along lp)

n× ~Et(z) = 0 (along lc) n× ~Et(z)∣∣∣lp,n

= χy n× ~Et(z)∣∣∣lp,s

(along lp)(2.109)

where the x-direction periodic boundary conditions are obtained from the y-direction onesby appropriately interchanging quantities. Then, using the boundary conditions of ~Et,nwe can write:∮

l

Ez(z) n· ~E∗t,ndl =(1− |χx|2

)∫lp,w

Ez(z) n· ~E∗t,ndl +(1− |χy|2

)∫lp,s

Ez(z) n· ~E∗t,ndl

∮l


dl =(1− |χx|2

)∫lp,w


dl

+(1− |χy|2

)∫lp,s


dl

As a result, if the |χx| = |χy| = 1 condition is assumed these two contour integrals vanish.The last contour integral also vanishes since it is only needed in the expansion if χx =

48


χy = 1 (as commented in section 2.6.3):∮l

n× ~Et(z) · zdl = (1− χx)∫lp,w

n× ~Et(z) · zdl + (1− χy)∫lp,s

n× ~Et(z) · zdl = 0

To sum up, by substituting in (2.104) the rewritten and simplified expressions for theexpansion coefficients (2.106)-(2.108), the expansion of ∇× ~E is finally:

∇× ~E =∑

z × ~Et,n

∂

∂z

∫~Et(z) · ~E∗t,ndS +

∫Ez(z)∇· ~E∗t,ndS

+∑

kc,n∇× ~Et,nkc,n

∫~Et(z) · ~E∗t,ndS (2.110)

For the expansion of ∇× ~H , we use the ~Et,n’s and the(k−1c,nz∇· ~Et,n

)’s,

∇× ~H =∑

~Et,n

∫∇× ~H · ~E∗t,ndS + z

∑ ∇· ~Et,nkc,n

∫∇× ~H · z∇·

~Et,nkc,n

dS (2.111)

By using,

∇× ~H = ∇t × ~Ht(z) +∇tHz(z)× z +∂

∂zz × ~Ht(z) (2.112)

similar to (2.105), the expansion coefficients of (2.111) can be rewritten as follows. Forthe first one, we have∫

∇× ~H · ~E∗t,ndS =

∫ [∇tHz(z)× z +

∂

∂zz × ~Ht(z)

]· ~E∗t,ndS

=∂

∂z

∫~Ht(z) ·

[~E∗t,n × z

]dS +

∫~E∗t,n · [∇t ×Hz(z) z] dS

by integrating by parts the second term:∫∇× ~H · ~E∗t,ndS =

∂

∂z

∫~Ht(z) ·

[~E∗t,n × z

]dS

+

∫Hz(z) z ·

[∇× ~E∗t,n

]dS −

∮l

Hz(z) z ·(n× ~E∗t,n

)dl (2.113)

where, as analogous to (2.106), the contour integral vanishes since n× ~Et,n = 0 alonglc and both Hz(z) and n× ~Et,n are periodic along the lp contour. The second expansion

49


coefficient of (2.111) is rewritten as,

∫∇× ~H · z∇·

~Et,nkc,n

dS = kc,n

∫~Ht(z) ·

(z × ~E∗t,n

)dS

+

∮l

(n× ~Ht(z)

)· z∇· ~E∗t,nkc,n

dl (2.114)

where the following identity and the wave equation (2.2) (noting that the(k−1c,nz∇· ~Et,n

)’s

are derived from eigenfunctions belonging to group III) have been used,

∇t ·

[~Ht(z)× z

∇· ~E∗t,nkc,n

]= ∇t × ~Ht(z) · z

∇· ~E∗t,nkc,n

− ~Ht(z) ·

(∇∇· ~E∗t,nkc,n

× z

)

Once more, the contour integral of (2.114) vanishes, since along the lc contour∇·~Et,n = 0

and both n× ~Ht(z) and∇· ~Et,n are periodic along the lp contour. To sum up, the rewrittenand simplified expansion of ∇× ~H is:

∇× ~H =∑

~Et,n

∂

∂z

∫~Ht(z) ·

[~E∗t,n × z

]dS +

∫Hz(z) z ·

[∇× ~E∗t,n

]dS

+ z∑

kc,n∇· ~Et,nkc,n

∫~Ht(z) ·

(z × ~E∗t,n

)dS (2.115)

By substituting (2.102) and (2.115) into ∇× ~H = jωε ~E and writing the transverseand longitudinal components separately, we have:

∑~Et,n

[∂

∂z

∫~Ht(z) · ~E∗t,n × zdS +

∫Hz(z) z · ∇× ~E∗t,ndS

]=

jωε∑

~Et,n

∫~E · ~E∗t,ndS (2.116)

∑kc,n∇· ~Et,nkc,n

∫~Ht(z) ·

(z × ~E∗t,n

)dS = jωε

∑ ∇· ~Et,nkc,n

∫ (~E · z

) ∇· ~E∗t,nkc,n

dS

(2.117)

50


Similarly, using (2.103) and (2.110), we obtain two equations from ∇× ~E = −jωµ ~H ,

∑z × ~Et,n

∂

∂z


∫Ez(z)∇· ~E∗t,ndS

= −jωµ

∑z × ~Et,n

∫~H ·(z × ~E∗t,n

)dS (2.118)

∑kc,n∇× ~Et,nkc,n

∫~Et(z) · ~E∗t,ndS

= −jωµ∑ ∇× ~Et,n

kc,n

∫~H ·∇× ~E∗t,nkc,n

dS − jωµ z√A

∫~H · z√

A∗dS (2.119)

Equating the coefficients of the corresponding ~Et,n on both sides of (2.116), we have:

∂

∂z

∫~Ht(z) · ~E∗t,n × zdS +

∫Hz(z) z · ∇× ~E∗t,ndS = jωε

∫~E · ~E∗t,ndS (2.120)

Analogously, (2.117) gives:

kc,n

∫~Ht(z) ·

(z × ~E∗t,n

)dS = jωε

∫ (~E · z


dS (2.121)

where the set of(k−1c,n∇· ~Et,n

)’s is derived from eigenfunctions belonging to group III ,

thus∇· ~Et,n 6= 0. By doing exactly the same with (2.118) and (2.119), we have

∂

∂z


∫Ez(z)∇· ~E∗t,ndS = −jωµ

∫~H ·(z × ~E∗t,n

)dS (2.122)

from the former, and

kc,n

∫~Et(z) · ~E∗t,ndS = −jωµ

∫~H ·∇× ~E∗t,nkc,n

dS (2.123)

and, ∫~H · z√

A∗dS = 0 (2.124)

from the later. It shall be remarked that (2.123)-(2.124) are obtained assuming that∇×~Et,ndoes not vanish. Indeed, the set of the

(k−1c,n∇× ~Et,n

)’s is derived from eigenfunctions

belonging to group II , thus ∇× ~Et,n 6= 0. Let us now investigate what happens with

51


(2.120)-(2.124) when the ~Et,n’s belong to each one of the three groups.

Group I: Suppose that ~Et,n belongs to group I , (2.6). In the first place we note that aneigenfunction from group I has no longitudinal fields, thus only the transverse expansioncoefficients are of interest here. Particularizing (2.120) and (2.122) for this case (making∇× ~Et,n = ∇· ~Et,n = 0) we have:

∂

∂z

∫~Ht(z) · ~E∗t,n × zdS = jωε

∫~E · ~E∗t,ndS (2.125)

∂

∂z


∫~H ·(z × ~E∗t,n

)dS (2.126)

By substituting the second equation in the first, a differential equation is obtained, notingthat ~Ht(z) · ~Et,n = ~H · ~Et,n:

∂2

∂z2

∫~E · ~E∗t,ndS + ω2µε

∫~E · ~E∗t,ndS = 0

The most general solution is then,∫~E · ~E∗t,ndS = Cne

−jkz,nz +Dnejkz,nz

where Cn, Dn ∈ C are constants and kz,n is given by (2.3). The eigenvalue, kc,n, associ-ated with eigenfunctions belonging to group I vanishes, thus:

kz,n = ω√εµ

By substituting this solution back into (2.126), we have:∫~H ·(z × ~E∗t,n

)dS = Z−1n

[Cne

−jkz,nz −Dnejkz,nz

](2.127)

where used has been made of (2.101).

Group II: Let us now suppose that ~Et,n belongs to group II . Since the eigenfunc-

52


tions belonging to group II are solenoidal, we rewrite (2.120)-(2.123),

∂

∂z

∫~Ht(z) · ~E∗t,n × zdS +

∫Hz(z) z · ∇× ~E∗t,ndS = jωε

∫~E · ~E∗t,ndS

∂

∂z


∫~H ·(z × ~E∗t,n

)dS

kc,n


∫~H ·∇× ~E∗t,nkc,n

dS

Please note that, since eigenfunctions belonging to group II have no longitudinal electricfield, (2.121) is not of relevance here. By substituting the second and third equations intothe first one,

∂2

∂z2


(ω2µε− k2c,n

) ∫~Et(z) · ~E∗t,ndS = 0 (2.128)

Once more, the solution becomes,∫~Et(z) · ~E∗t,ndS = Cne

−jkzz +Dnejkzz

provided that (2.3), k2z = ω2µε− k2c,n, is not equal to zero. By substituting this solutioninto the original equations, we have∫

~H ·(z × ~E∗t,n

)dS = Z−1n

(Cne


)∫

~H ·∇× ~E∗t,nkc,n

dS =jkc,nωµ

(Cne


) (2.129)

where use has been made of (2.101). When kz = 0, the solution to the differential equa-tion (2.128) is a lineal function of z. Substituting this lineal solution in the original equa-tions we obtain: ∫

~Et(z) · ~E∗t,ndS = Fn +Gnz∫~H ·(z × ~E∗t,n

)dS =

jGn

ωµ∫~H ·∇× ~E∗t,nkc,n

dS =jkc,nωµ

(Fn +Gnz)

(2.130)

53


Group III: Finally, if ~Et,n belongs to group III , then the eigenfunction is irrota-tional and (2.120) simplifies to,

∂

∂z

∫~Ht(z) · ~E∗t,n × zdS = jωε

∫~E · ~E∗t,ndS

By substituting (2.121) into (2.122),

∂

∂z


k2c,njωε

∫~Ht(z) ·

(z × ~E∗t,n

)dS = −jωµ

∫~H ·(z × ~E∗t,n

)dS

Taking the derivative with respect to z and substituting the simplified (2.120) we obtainthe exact same differential equation (2.128):

∂2

∂z2


(ω2µε− k2c,n

) ∫~E · ~E∗t,ndS = 0

Again, assuming that (2.3), k2z = k2 − k2c,n, does not vanish, the most general solution is:∫~Et(z) · ~E∗t,ndS = Cne


∫~Ht(z) · z × ~E∗t,ndS = Z−1n

(Cne


)∫ (

~E · z) ∇· ~E∗t,n

kc,ndS =

kc,njkz,n

(Cne


)(2.131)

where use has been made of (2.101). However, if kz = 0, then we can advantageouslywrite the following differential equation:

∂2

∂z2

∫~Ht(z) · z × ~E∗t,ndS = 0

and replace (2.131) by, ∫~Et(z) · ~E∗t,ndS =

jGn

ωε∫~Ht(z) · z × ~E∗t,ndS = Fn +Gnz∫ (~E · z


dS =kc,njωε

(Fn +Gnz)

(2.132)

54

2.8. COMPLEX POWER FLOW

Therefore, the expansion coefficients for (2.102) and (2.103) have been determinedfor all possible cases if ω 6= 0. We conclude that, if none of the kz,ns vanish, the mostgeneral expression for the electromagnetic field in the waveguide when ω 6= 0 is givenby:

~E=∑

~Et,n(Cne

−jkz,nz+Dnejkz,nz

)+z∇· ~Et,njkz,n

(Cne

−jkz,nz−Dnejkz,nz

)(2.133)

~H=∑

z× ~Et,nZn

(Cne


)−∇×

~Et,njωµ

(Cne

−jkz,nz+Dnejkz,nz

)(2.134)

where (2.3), k2z,n = ω2µε − k2c,n, and (2.101) are used. If some of the kz,ns are equal tozero, the coefficients of the corresponding terms must be replaced by either (2.130) or(2.132) depending on the particular group concerned. This result is identical to the one

obtained when closed PEC waveguides are considered [Kurokawa, 1969, p. 136].

In section 2.6, it was shown that electromagnetic fields in a waveguide could be di-vided into three groups, TEM z, TEz and TM z modes. However, since at the beginningof the chapter, (2.1), it was assumed the exponential field variation with z, it was not clearwhether or not all possible electromagnetic fields in a waveguide could be expressed aslinear combination of those modes. In this section, the exponential field variation wasnot assumed (indeed, non-exponential field variation appeared in the particular case ofkz = 0), and we have demonstrated that every possible electromagnetic field in a waveg-uide is given by (2.133)-(2.134) and no other functional forms need to be considered.

2.8 Complex Power Flow

To this point we have found that, under the |χx| = |χy| = 1 condition, transversely pe-riodic cylindrical structures (figure 2.1) can be regarded as waveguides. The preced-ing sections have dealt with several important properties of transversely periodic waveg-uides highlighting the similarities and differences with traditional PEC or PMC closedwaveguides. This section studies the complex power flow in the considered type of struc-tures, since some peculiarities arise that must be noted. Hereafter we shall assume that

none of the kz,n’s vanish, thus we will use (2.133) and (2.134) to expand the electric andmagnetic fields.

In order to study the complex power flow, we need to write an appropriate expressionfor the complex Poynting vector, [Stratton, 1941, pp. 131-137]. Let us first split the

55


complex Poynting vector into its transverse and longitudinal components,

~S(z) = ~St(z) + zS(z) = ~E × ~H∗ =(~Et + zEz

)×(~H∗t + zH∗z

)= ~Et × zH∗z + zEz × ~H∗t + ~Et × ~H∗t (2.135)

where, self-evidently, the first two terms are perpendicular to the z direction and belongto ~St(z) and the third one is a longitudinal term and belongs to zS(z).

Consider the well known identity [Kurokawa, 1969, p. 72], also known as PoyntingTheorem,

∇·[~E × ~H∗

]= −σ ~E · ~E∗ − jω

[µ ~H · ~H∗ − ε ~E · ~E∗

]and integrate it over the waveguide cross section,∮

l

[~E × ~H∗

]·(−n) dl =

∫S

σ ~E · ~E∗dS + jω

∫S

[µ ~H · ~H∗ − ε ~E · ~E∗

]dS (2.136)

By equating the real and imaginary parts of this equation we have that one half of thereal part of the left-hand side of (2.136) represents the average power flowing into thewaveguide through the contour per unit length and that one half of the imaginary part rep-resents 2ω times the difference between the average magnetic and electric stored energiesper unit length in the waveguide. Let us now evaluate this integral. Using the transversecomponents of the Poynting vector, (2.135),∮

l

[~E × ~H∗

]·(−n) dl =

∮l

[~Et × zH∗z + zEz × ~H∗t

]·(−n) dl (2.137)

The first term ~Et × zH∗z can be expanded, using (2.133) and (2.134), as follows:∮l

~Et × zH∗z · (−n) dl =

∮l

∑n

~Et,n(Cne

−jkz,nz+Dnejkz,nz

)×∑m

∇×~E∗t,m−jωµ

(C∗me

jk∗z,mz+D∗me−jk∗z,mz

)·ndl

56

2.8. COMPLEX POWER FLOW

by rearranging the above expression,∮l

~Et × zH∗z · (−n) dl =∑n,m

(Cne

−jkz,nz+Dnejkz,nz

)C∗me

jk∗z,mz+D∗me−jk∗z,mz

−jωµ

∮l

n× ~Et,n · ∇× ~E∗t,mdl (2.138)

Since both ~Et,n and ~Et,m fulfill the same boundary conditions, by using (2.10) it becomesapparent that, if the |χx| = |χy| = 1 condition is satisfied, (2.138) vanishes. On the otherhand, the second term of (2.137) can be expanded as follows:

∮l

zEz × ~H∗t ·(−n) dl =

∮l

∑n

z∇· ~Et,njkz,n

(Cne


)×∑m

z× ~E∗t,mZ∗m

(C∗me

−jk∗z,mz−D∗me−jk∗z,mz)·(−n) dl

which can be rewritten as:∮l

zEz × ~H∗t ·(−n) dl =∑n,m

Cne−jkz,nz−Dne

jkz,nz

jkz,n

C∗me−jk∗z,mz−D∗me−jk

∗z,mz

Z∗m

∮l

n· ~E∗t,m∇· ~Et,ndl (2.139)

Once more, assuming the |χx| = |χy| = 1 condition and using (2.10), the contour integralvanishes. Therefore, since both terms of the right-hand side of (2.137) are zero, thereis no average net power leakage in the transverse directions. However, this does notmean that the power flow across one of the periodic boundaries, for instance lp,s, vanishes.It means that the power that flows outwards from S through lp,s, is the same as the powerthat enters S through lp,n. Obviously, this opposes the behavior of closed PEC or PMC

waveguides, where the power flow is zero all along the contour. Furthermore, (2.136)being zero also implies that the average magnetic and electric stored energies per unitlength are equal, i.e. ∫

ε′ ~E · ~E∗dS =

∫µ′ ~H · ~H∗dS (2.140)

which is the resonance condition that is also fulfilled by any propagating closed waveg-uide mode. This is one of the key theoretical results that constitute the theoretical base onwhich the generalized Transverse Resonance Technique, described in chapter 3, relies.

57


Let us now consider the time-average complex power flow across the surface S,[Collin, 1960, p. 10], i.e. the transmitted complex power,

P =1

2

∫~E × ~H∗ · zdS =

1

2

∫~Et × ~H∗t · zdS (2.141)

where (2.135) has been used. By substituting the general field expansions in the aboveexpression we have:

P =1

2

∫ ∑n

~Et,n(Dne

−jkz,nz+Cnejkz,nz

)×∑m

z× ~E∗t,mZ∗m

(C∗me

−jk∗z,mz−D∗mejk∗z,mz)·zdS

=1

2

∑n,m

(Cne

−jkz,nz+Dnejkz,nz

) C∗me−jk∗z,mz−D∗mejk∗z,mzZ∗m

∫~Et,n · ~E∗t,mdS

where, thanks to the orthogonality property (2.14), and assuming that the ~Et,n’s have beennormalized, we obtain,

P =1

2

∑n

1

Z∗n

|Cn|2e2Im(kz,n)z−|Dn|2e−2Im(kz,n)z+2jIm

[C∗nDne

j2Re(kz,n)z]

(2.142)

Therefore, as analogous to closed PEC or PMC waveguides, the total time-averagecomplex power flow across the transverse section, S, of the waveguide is equal to thesum of the time-average complex power flow of each waveguide mode.

2.9 Conclusions

In this chapter, it has been demonstrated for the first time that any field in a transverselyperiodic cylindrical structure, such as the one depicted in figure 2.1(a), can be describedby means of an infinite sum of eigenfunctions or modes, as long as the periodicity con-stants linking the periodic contours satisfy the |χx| = |χy| = 1 condition. Furthermore,the eigenvalues associated to this eigenfunctions are real and these modes can be classi-fied into three different groups, TEM z, TEz and TM z modes. These results show thatperiodic structures, under the aforementioned condition, can be analyzed and understoodby means of any closed waveguide theory or technique. This result is the fundamen-tal scientific contribution in this dissertation. The following chapters are devoted to thedevelopment of a periodic structure analysis tool based on the Mode Matching (MM) tech-nique, widely used in the analysis of closed-waveguide structures and devices. Finally, itshall be highlighted that the theoretical results obtained in this Chapter, in combination

58

2.9. CONCLUSIONS

with the practical implementation detailed in Chapter 4, have given rise to [Varela andEsteban, 2012].

59

3Characterization of Transversely Periodic

Waveguides

3.1 Introduction

The objective of this chapter is to introduce a semi-analytical technique for the characteri-zation of transversely periodic homogeneous waveguides, i.e. the obtention of the cut-offfrequencies and potential distributions of the TEM z, TEz and TM z modes.

The Transverse Resonance Technique (TRT ) was originally proposed in [Ramo andWhinnery, 1944], and was successfully applied to the computation of cut-off frequenciesin ridge waveguides, [Hopfer, 1955], and to the analysis of printed circuits, [Yee, 1985].The original technique used a single-mode circuital model to compute the cut-off frequen-cies. With the appearance of modern computers, it was possible to generalize the TRT tothe usage of multi-mode circuital models, [Montgomery, 1971, Sorrentino and Leuzzi,1982, Sorrentino and Itoh, 1984]. More recently, the TRT concept, in combination withother full-wave methods such as the method of moments [Deslandes and Wu, 2006] orthe multimodal variational method [Zeid and Baudrand, 2002], together with commercialsolvers [Mencarelli and Rozzi, 2006], has been used to characterize periodic structures.It shall be commented that, since the multi-mode version of the TRT that will be usedin this dissertation makes implicit use of the Mode Matching technique, the knowledgeregarding the relative convergence phenomenon, [Mittra et al., 1972, Leroy, 1983, Shihand Gray, 1983], and the pros and cons of the different possible formulations, [Vassallo,1976, Safavi-Naini and MacPhie, 1981, Mansour and Macphie, 1985, Mansour and Mac-phie, 1986, Shih et al., 1985, Overfelt and White, 1989], are of direct use here.

This chapter extends the usage of the generalized TRT to the characterization of trans-versely periodic homogeneous waveguides and its contents have led to [Varela et al.,

61

CHAPTER 3. CHARACTERIZATION OF TRANSVERSELY PERIODICWAVEGUIDES

2010] and [Varela and Esteban, 2011b]. Among all the closed structure characterizationtechniques, the generalized TRT has been chosen in this dissertation because of its sim-plicity, efficiency, accuracy and because it provides great versatility. On the other hand,the generalized TRT can only be used when the waveguide cross section can be segmentedin the cartesian coordinate system. Of course, a great number of structures that can be seg-mented in the cartesian coordinate system are of practical interest. However, in order tointroduce the method in a more general way, the next paragraph will explain the staircaseapproximation procedure.

Consider a transversely periodic waveguide such as the one shown in figure 2.1 ofthe preceding chapter, the TRT is based on the fact that the solutions of the structure (themodes) are resonant, (2.140), at any transverse section. Since the aforementioned waveg-uide can not be segmented in the cartesian coordinate system, we will use a staircaseapproximation to obtain the modes of the waveguide. Figure 3.1(a) shows the trans-versely periodic waveguide under consideration, where the vertical dashed lines show theplanes at which the waveguide will be segmented. Figure 3.1(b) shows the result of thestaircase approximation. The resulting waveguide is split into sixteen different regions inorder to establish the resonant condition. Of course, the approximation can be as good asdesired by adding more regions. An example of the usage of this staircase approximation

(a) (b)Regions I-VII Regions XI-XVIRegion VIII

xy

S(VIII-XV)S(I-VII)

(c)

B(i)2B(i)

1

Γ (i) Γ (or)

l (i)

B(or)1

A(or)2

B(or)2

l (or)

Γ (ol)

B(ol)1

A(ol)1

B(ol)2

l (ol)

S (L) S(R)

lc,PEClp,e

lp,n

lp,s

lp,w

Figure 3.1: (a) Cross section of a generic bi-periodic waveguide, analogous to figure 2.1(c). (b)Staircase approximation of the bi-periodic waveguide. The cross section has been segmented insixteen regions and the discontinuities between regions are characterized by means of fifteen GSMs, S(I−XV ). (c) Generalized transverse equivalent circuit of (b), where S(L) and S(R) representS(I−V II) and S(V III−XV ) respectively.

62

3.2. THE GENERALIZED TRANSVERSE RESONANCE TECHNIQUE FORTEZ AND TMZ MODES

can be found in chapter 5. Inside each region, the electric and magnetic potentials areexpanded as a sum of terms that fulfill both the wave equation and the boundary condi-tions of the region, as described in section 3.2.1. Once appropriate potential expansionshave been written in each region, the discontinuities between regions are characterized,section 3.2.2, by a Generalized Scattering Matrix (GSM ), S(I−XV ) in figure 3.1(b). Then,by successively cascading GSMs , as described in section 3.2.3, the generalized equiv-alent circuit shown in figure 3.1(c) is obtained, where S(L) and S(R) represent S(I−V II)

and S(V III−XV ) respectively. The solutions or resonances of the equivalent circuit are themodes of the transversely periodic waveguide, section 3.2.4.

It is important that the difference between the modes of the transversely periodicwaveguide and the terms of the potential expansions in each region is clear. This clarifi-cation is relevant since each term of the potential expansions can be regarded as a modeof the region when it is considered as a waveguide (this will be further commented insection 3.2.1). Furthermore, it shall be remarked that, although the generalized TRT canbe very easily adapted to the characterization of inhomogeneous waveguides, this disser-tation will only use homogeneous waveguides and, thus, the generalized TRT formulationthat will be presented in this chapter is particularized for that case.

Finally, this chapter is divided into four sections. The first one, section 3.2, describesthe characterization procedure for TEz and TM z modes. Because of the special natureof the transversely periodic TEM z modes, their characterization is described separatelyin section 3.3. After that, the potential distributions and cut-off frequencies of someexamples will be shown in section 3.4. Finally, there will be a conclusion section.

3.2 The Generalized Transverse Resonance Tech-

nique for TEz and TM z modes

It was demonstrated in the last chapter, sections 2.6.3 and 2.6.4, that the complete setof eigenfunctions belonging to groups II and III can be obtained by the resolution ofthe simpler scalar eigenfunction problems defined in table 2.5 and table 2.6. The readershould recall at this point that the scalar potentials are function only of the transversecomponents and that the electric and magnetic potentials were named that way becauseof the similarity of their definitions, (2.69) and (2.85), with the longitudinal electric andmagnetic fields, (2.49). The scalar eigenvalue problems to be solved have been repeated

63


Table 3.1: Equivalent Eigenfunction problem for TEz and TM z modes.

TEz modes TM z modes

∆φH,n + k2c,nφH,n = 0 ∆φE,n + k2c,nφE,n = 0 on S

n·∇φH,n = 0 φE,n = 0 along lc,PEC

φH,n = 0 n·∇φE,n = 0 along lc,PMC

φH,n|lp,n=χy φH,n|lp,s φE,n|lp,n=χy φE,n|lp,s along lp,n ∪ lp,sn·∇φH,n|lp,n=χy n·∇φH,n|lp,s n·∇φH,n|lp,n=χy n·∇φH,n|lp,s along lp,n ∪ lp,s

in table 3.1 for convenience, where

k2c,n = ω2µε− k2z,n (3.1)

and the x-direction periodic boundary conditions are obtained from the y-direction onesby interchanging χy with χx and lp,n and lp,s with lp,e and lp,w, respectively.

3.2.1 Potential Expansions

As previously commented, in order to solve the eigenvalue problems in the transversesection of a transversely periodic waveguide, for instance the one shown in figure 3.1(b),the cross section S is split into different regions. Inside each region, r, the electric andmagnetic potentials of the n-th waveguide mode are expanded as the following sum ofterms,

φ(r)H,n

φ(r)E,n

=∑p

φ(r)n,p =

∑p

Ψ(r)p (y)

Λ(r)p

[∓B(r)

1,n,pe−jk(r)x,px +B

(r)2,n,pe

−jk(r)x,p(l(r)−x)]

(3.2)

where Ψ(r)p (y) is a set of orthogonal functions that satisfy the y-direction boundary condi-

tions of region r, Λ(r)p is a normalization constant and the ∓ holds for the magnetic and

electric potentials respectively. The factor in brackets of each term represents (or can beunderstood as) two waves, traveling in the ± x-direction, where k(r)x,p is the x-directionpropagation constant, l(r) is the length of the region as shown in figure 3.2(a)-(b) andB

(r)1,n,p and B

(r)2,n,p are the unknown amplitudes of the progressive and regressive waves

respectively. In the waveguide cross section, shown in figure 3.1(b), there are two dif-ferent types of regions. The parallel-plate regions (regions I − V II and XI − XV I),

64


figure 3.2(a), and the periodic-boundary region (region V III), figure 3.2(b). The partic-ularizations of the potential expansion, (3.2), for each type of region are described below.Before that, there are some common characteristics that will be commented.

On the one hand, the normalization constant, Λ(r)p , is defined as,

Λ(r)p =

∫C(r)

Ψ(r)p (y) Θ(r)

p (y) dy (3.3)

where the Θ(r)q ’s are orthogonal, over the line C(r), to the Ψ

(r)p ’s, except for the case of

q = p. The integration contour, C(r), is defined as the vertical line that goes from thelower to the upper boundaries in each region. On the other hand, for a fixed value ofthe z-direction propagation constant, kz, the x-direction propagation constant, k(r)x,p, isobtained from:

k(r)x,p =

√ω2µε−

(k(r)y,p

)2− k2z (3.4)

where k(r)y,p is a separation constant specific of each region and the sign of the squareroot must be chosen so that the imaginary part of k(r)x,p is negative. This way numericalrobustness is preserved when cascading discontinuities (see section 3.2.3).

Parallel-Plate Region: The first type of region, or ”parallel-plate region”, is shownin figure 3.2(a). The boundary conditions are two PEC planes at y = d(r) and y =

d(r) + h(r). Hence, each term of the expansion in this region can be understood as a modeof a parallel-plate waveguide of height h(r), and Ψ

(r)p is chosen as:

Ψ(r)p (y) =

cos(k(r)y,p

[y − d(r)

]), for φH,n terms

sin(k(r)y,p

[y − d(r)

]), for φE,n terms

(3.5)

(d)x

y

B(r)2

A(r)2

B(r+1)1

A(r+1)1

(c)x

yB(r)

2

A(r)2

B(r+1)1

A(r+1)1

h(r)

l (r)

(b)

B(r)1 B(r)

2

A(r)1 A(r)

2

xy

B(r)1 B(r)

2

A(r)1 A(r)

2

l (r)

h(r)

d(r)

(a)x

y

Figure 3.2: (a) Parallel-plate region. (b) Periodic-boundary region. (c) Discontinuity betweentwo parallel-plate regions. (d) Discontinuity between a periodic-boundary and a parallel-plateregions.

65


where the y-direction separation constant is,

k(r)y,p =pπ

h(r),

p = 0, 1, 2, . . .∞ for φH,n terms

p = 1, 2, 3, . . .∞ for φE,n terms(3.6)

For the parallel-plate regions, Θ(r)p = Ψ

(r)p . Thus, the normalization constant is defined

in this type of region as Λ(r)p = h(r)/2 for both magnetic and electric potential expansion

terms, except the p = 0 magnetic term for which the normalization constant isQ(r)0 = h(r).

Furthermore, for this type of region, the values of k(r)y,p are determined by the geometry. Fora given frequency and kz, the k(r)x,p values can be determined from (3.4) and the potentialis fully defined in the region through (3.2). As aforementioned, the terms of the potentialexpansions are identical to the longitudinal electric and magnetic fields of the modes ofa parallel-plate waveguide (particularized for kz = 0). This is why it was important tohighlight the difference between terms and modes.

Periodic-Boundary Region: The second type of region or ”periodic-boundary re-

gion”, is shown in figure 3.2(b). If the potential expansions in this type of region areunderstood as the longitudinal magnetic and electric fields (to which they are very simi-lar), then the terms of the expansions can be regarded as Floquet harmonics [Clarke andBrown, 1980] and, thus, the complete expansion would be a discrete plane wave spec-trum. By imposing the periodicity condition to the potential itself and its derivative withrespect to the y direction, table 3.1, between the y = 0 and the y = h(r) planes, the set ofΨ

(r)p ’s is found to be,

Ψ(r)p = e−jk

(r)y,py , where

k(r)y,p = −ξy −

2pπ

h(r)

ξy =arg (χy)− jln |χy|

h(r)

(3.7)

with p = −∞, . . . ,−1, 0, 1, . . . ,∞ for both magnetic and electric potentials. It is worthhighlighting that (3.7) is a general solution, ∀χy ∈ C. The y-direction separation con-stants, k(r)y,p, are real under the |χx| = |χy| = 1 condition. In this type of region, the or-thogonal set of Θ

(r)p ’s is chosen to be,

Θ(r)p (y) = ejk

(r)y,py (3.8)

Therefore, the normalization constant for this type of region is Λ(r)p = h(r). Finally, it

shall be noted that in a periodic-boundary region, in addition to the kz value, a value forthe periodicity constant, χy, must be specified. Once these two parameters are specified,

66


the k(r)y,p can be obtained from (3.7) and the k(r)x,p’s from (3.4) for a given frequency. Onlythen the potentials are completely specified through (3.2).

3.2.2 Discontinuity Characterization

Now that we have appropriate potential expansions for every type of region let us charac-terize the discontinuities between regions by means of a GSM . The boundary conditionsto be imposed to both potentials along the discontinuities are the continuity of the poten-tials themselves and their derivatives with respect to the normal direction, i.e.

φH,n|x− = φH,n|x+ φE,n|x− = φE,n|x+∂φH,n∂x

∣∣∣∣x−

=∂φH,n∂x

∣∣∣∣x+

∂φE,n∂x

∣∣∣∣x−

=∂φE,n∂x

∣∣∣∣x+

(3.9)

where x± refers to the right and left side of the discontinuity. Throughout this disserta-tion we will need to characterize two different types of discontinuity. The first one is thediscontinuity between two parallel-plate regions, figure 3.2(c). The second type is a dis-continuity between a periodic-boundary region and a parallel-plate region, figure 3.2(d).

By using the notation of figure 3.2(c)-(d), the continuity of the potential at the discon-tinuity between any two regions is written as,

∑p

Ψ(r)p (y)

Λ(r)p

[∓A(r)

2,n,p +B(r)2,n,p

]=∑q

Ψ(r+1)q (y)

Λ(r+1)q

[∓B(r+1)

1,n,q + A(r+1)1,n,q

]

where h(r) > h(r+1). Multiplying the equation by Θ(r)k (y) and integrating along the dis-

continuity we have:

∑p

∓A(r)2,n,p +B

(r)2,n,p

Λ(r)p

∫C(r)

Ψ(r)p Θ

(r)k dy =

∑q

∓B(r+1)1,n,q + A

(r+1)1,n,q

Λ(r+1)q

∫C(r)

Ψ(r+1)q Θ

(r)k dy

because of the orthogonality property, the right-hand side reduces to the k-th term,

[∓A(r)

2,n,p +B(r)2,n,p

]=∑q

∓B(r+1)1,n,q + A

(r+1)1,n,q

Λ(r+1)q

∫C(r)

Ψ(r+1)q Θ(r)

p dy (3.10)

where k has been renamed as p. As a result, we have obtained the equation for thecontinuity of the potential.

Let us now consider the continuity of the potential’s derivative with respect to the

67


normal,

∑p

k(r)x,pΨ

(r)p (y)

Λ(r)p

[±A(r)

2,n,p +B(r)2,n,p

]=∑q

k(r+1)x,q

Ψ(r+1)q (y)

Λ(r+1)q

[±B(r+1)

1,n,q + A(r+1)1,n,q

]

By multiplying the equation by Θ(r+1)k and integrating the resulting expression over the

C(r+1) line,

∑p

k(r)x,p±A(r)

2,n,p +B(r)2,n,p

Λ(r)p

∫C(r+1)

Ψ(r)p Θ(r+1)

p dy = k(r+1)x,q

[±B(r+1)

1,n,q + A(r+1)1,n,q

](3.11)

where the orthogonality property has been used and, once more, k has been renamed as q.

In order to characterize the discontinuity, it is customary to rewrite the linear systemof equations formed by (3.10) and (3.11) in a matrix form. By truncating the infinite sumsof the r-th and (r + 1)-th regions to P and Q terms respectively:[

∓A(r)2 +B

(r)2

]= Y

[∓B(r+1)

1 + A(r+1)1

](3.12)

X tK(r)x

[±A(r)

2 +B(r)2

]= K(r+1)

x

[±B(r+1)

1 + A(r+1)1

](3.13)

where A(r)2 , B(r)

2 , A(r)1 and B(r)

1 are vectors containing the unknown term amplitudes im-pinging on (As) and reflected from (Bs) the discontinuity (figure 3.2(c)-(d)), K(r)

x andK

(r+1)x are diagonal matrices with the x-direction separation constants of the r-th and

(r + 1)-th regions respectively, the superscript t refers to the matrix transpose and Y andX are full matrices defined as,

Yp,q =(

Λ(r+1)q

)−1∫C(r)

Ψ(r+1)q Θ(r)

p dy ; Xp,q =(

Λ(r)p

)−1∫C(r+1)

Ψ(r)p Θ(r+1)

p dy (3.14)

The objective here is to rewrite (3.12) to obtain a GSM representation of the discontinuityof the form,

B(r)2 = S11A

(r)2 + S12A

(r+1)1 (3.15)

B(r+1)1 = S21A

(r)2 + S22A

(r+1)1 (3.16)

by defining

J =(K

(r+1)x

)−1X tK

(r)x ; H = 2 (JY + U)−1 (3.17)

68


the objective is accomplished as follows,

S21 = HJ S22 = ±H − U

S11 = ∓ [Y S21 + U ] S12 = Y [U ∓ S22](3.18)

where the upper and lower signs hold for the magnetic and electric potentials respectivelyand U is the unity matrix of appropriate dimensions.

Discontinuity between two parallel-plate regions: On the one hand, let us nowparticularize (3.14) to the case of a discontinuity between two parallel-plate regions, fig-ure 3.2(c). In order to avoid the relative convergence phenomenon, [Leroy, 1983, Sor-rentino et al., 1991], the value of Q should be kept as close as possible to P · h(r+1)/h(r).The integrals of the full matrices X and Y are identical to each other,

Λ(r+1)q Yp,q=Λ(r)

p Xp,q=

∫ h(r+1)+d(r+1)

y=d(r+1)

cos(k(r)y,p[y − d(r)

])cos(k(r+1)y,q

[y − d(r+1)

])dy (3.19)

for the magnetic potential case and,

Λ(r+1)q Yp,q=Λ(r)

p Xp,q=

∫ h(r+1)+d(r+1)

y=d(r+1)

sin(k(r)y,p[y − d(r)

])sin(k(r+1)y,q

[y − d(r+1)

])dy (3.20)

for the electric potential.

Discontinuity between a periodic-boundary and a parallel-plate region: On theother hand, for the particular case of the discontinuity of figure 3.2(d), between a periodic-boundary and a parallel-plate region, 2P +1 andQ terms are used in the r-th and (r + 1)-

th respectively. The value of Q has to be kept as close as possible to 2P · h(r+1)/h(r). Inthis case, the integrals of the Y and X matrices are not identical to each other. The formeris,

Λ(r+1)q Yp,q =

∫ h(r+1)+d(r+1)

y=d(r+1)

cos(k(r+1)y,q

[y − d(r+1)

])sin(k(r+1)y,q

[y − d(r+1)

]) ejk

(r)y,pydy (3.21)

and the later,

Λ(r)p Xp,q =

∫ h(r+1)+d(r+1)

y=d(r+1)

cos(k(r+1)y,q

[y − d(r+1)

])sin(k(r+1)y,q

[y − d(r+1)

]) e−jk

(r)y,pydy (3.22)

where the upper and lower expressions hold for the magnetic and electric potentials re-spectively. Self-evidently, when the |χx| = |χy| = 1 condition is imposed (3.22) is the

69


complex conjugate of (3.21),

Λ(r+1)q Yp,q = Λ(r)

p X∗p,q

Therefore, only one of them has to be computed.

3.2.3 Discontinuity Cascading

The well-known GSM cascading procedure [Patzelt and Arndt, 1982] is easily adaptedfor the cascading of discontinuities in the transverse direction y. Consider the general-ized equivalent circuit of figure 3.1(c). Let us obtain the GSM that characterizes bothdiscontinuities and region i.

Let S(L)ij and S(R)

ij be the sub-matrices of the GSMs that characterize the left and rightdiscontinuity respectively, A(r)

1,2 and B(r)1,2 the vectors containing the complex amplitudes

of each term in the r-th region (on reference planes 1 and 2), and

Γ(i) = diag[exp

(−jk(i)x,pl(i)

)](3.23)

is the so-called propagation matrix. Then, the GSM that relates vectors A(ol)2 and B(ol)

2

with A(or)1 and B(or)

1 is given by the following expressions:

SC11 = S(L)11 + S

(L)12 Γ(i)S

(R)11 Γ(i)HS

(L)21

SC12 = S(L)12 Γ(i)

[S(R)11 Γ(i)HS

(L)22 Γ(i) + U

]S(R)21

SC21 = S(R)21 Γ(i)HS

(L)21

SC22 = S(R)21 Γ(i)HS

(L)22 Γ(i)S

(R)12 + S

(R)22

(3.24)

whereH =

(U − S(L)

22 Γ(i)S(R)11 Γ(i)

)−1(3.25)

and U is the unity matrix of appropriate dimensions.

3.2.4 Characteristic Equation

At this point, all the elements of the generalized equivalent circuit of figure 3.1(c) areknown. It is time to obtain the resonance condition of the circuit. Let us first summarizethe known relations. In the first place, we have the scattering relations at the discontinu-

70


ities B

(ol)2 = S

(L)11 A

(ol)2 + S

(L)12 A

(i)1

B(i)1 = S

(L)21 A

(ol)2 + S

(L)21 A

(i)1

;

B

(i)2 = S

(R)11 A

(i)2 + S

(R)12 A

(or)1

B(or)1 = S

(R)21 A

(i)2 + S

(R)21 A

(or)1

(3.26)

Next, we have the relations between the amplitude vectors at the different referenceplanes,

A(ol)2 = Γ(ol)B

(ol)1

A(ol)1 = Γ(ol)B

(ol)2

;

A

(i)2 = Γ(i)B

(i)1

A(i)1 = Γ(i)B

(i)2

;

A

(or)2 = Γ(or)B

(or)1

A(or)1 = Γ(or)B

(or)2

(3.27)

where Γ(ol), Γ(i) and Γ(or) are particularizations of the Γx matrix in (4.33). Finally, thex-direction periodicity is imposed by,

A(or)2 = χxB

(ol)1 ; B

(or)2 = χxA

(ol)1 (3.28)

It shall be remarked here that, self-evidently, the proposed characterization method con-siders the x and y-direction periodicities in two separate steps. On the one hand, they-direction periodicity is introduced as part of the potentials expansion in the periodic-

boundary regions. On the other hand, the x-direction periodicity is forced as part of theresonance condition of the generalized equivalent circuit.

After some algebraic manipulations, the resonant condition of the equivalent circuitcan be written in two different ways: as a generalized eigenvalue problem and as a char-acteristic equation. The former looks as follows,[

S(L)21 Γ(ol)Γ(or)Ψ21 0

S(R)11 Γ(i) + S

(R)12 Γ(or)Γ(ol)Ψ11 −U

][B

(i)1

B(i)2

]

= χx

[U −S(L)

21 Γ(ol)Γ(or)Ψ22 − S(L)22 Γ(i)

0 −S(R)12 Γ(ol)Γ(or)Ψ12

][B

(i)1

B(i)2

](3.29)

and the later turns out to be,[χ−1x S

(L)21 Γ(ol)Γ(or)Ψ21 − U S

(L)21 Γ(ol)Γ(or)Ψ22 + S

(L)22 Γ(i)

S(R)11 Γ(i) + S

(R)12 Γ(ol)Γ(or)Ψ11 χxS

(R)12 Γ(ol)Γ(or)Ψ12 − U

][B

(i)1

B(i)2

]= 0 (3.30)

71


where,

Ψ11 = HS(L)11 Γ(ol)Γ(or)S

(R)21 Γ(i) Ψ12 = HS

(L)12 Γ(i)

Ψ21 = S(R)21 Γ(i) + S

(R)12 Γ(ol)Γ(or)Ψ11 Ψ22 = S

(R)22 Γ(ol)Γ(or)Ψ12

(3.31)

andH =

(U − S(L)

11 Γ(ol)Γ(or)S(R)22 Γ(ol)Γ(or)

)−1(3.32)

The usage of (3.29) or (3.30) depends on the type of results that are needed. When usingthe eigenvalue problem, the solutions are all the possible x-direction periodicity constants,χx, that fulfill (3.29) at a particular frequency. When using the characteristic equation,the x-direction periodicity constant value χx is fixed and the solutions are the frequenciesat which (3.30) is satisfied. For the particular case of waveguide characterization, i.e.the obtention of the cut-off frequencies of the modes, we will use (3.30) all along thisdissertation. In order to find these cut-off frequencies, the frequency axis must be scanned.Once this scan has provided approximate solutions, Muller’s method [Atkinson, 1989] isused to efficiently provide more accurate values.

Most of the transversely periodic waveguides that are used as example in this dis-sertation are symmetric. Please note, that the structure’s symmetry does not imply thepotential symmetry, since periodicity conditions are not symmetric in general. However,since transversely periodic symmetric waveguides are very common in practice let us par-ticularize (3.30) for that case. Let us highlight that the scattering matrices of symmetricstructures are related as follows, S(L)

11 = S(R)22 , S(L)

12 = S(R)21 , S(L)

21 = S(R)12 and S(L)

22 = S(R)11 .

Moreover, Γ(ol) = Γ(or) thus the outer propagation matrix will be renamed as Γ(o). Tak-ing all that into account, the generalized characteristic equation of the equivalent circuitshown in figure 3.1(c) is,[

χ−1x S(R)12

(Γ(o))2

Ψ12 − U Π

Π χxS(R)12

(Γ(o))2

Ψ12 − U

][B

(i)1

B(i)2

]= 0 (3.33)

where,

Π = S(R)11 Γ(i) + S

(R)12

(Γ(o))2

Ψ11 H =

[U −

S(R)22

(Γ(o))22

]−1Ψ11 = HS

(R)22

(Γ(o))2S(R)21 Γ(o) Ψ12 = HS

(R)21 Γ(o)

(3.34)

72

3.3. THE GENERALIZED TRANSVERSE RESONANCE TECHNIQUE FORTEMZ MODES

3.3 The Generalized Transverse Resonance Tech-

nique for TEM z modes

The TEM z mode characterization by means of the generalized transverse resonance tech-nique is described in this section. Although, the characterization procedure described inthe preceding section is basically the same for TEM modes, there are some important pe-culiarities that require special consideration. In section 2.6.2 of chapter 2, we found thatTEM z modes can be obtained from a scalar potential that satisfies the Poisson equation,∆φ = 0, under the ∂φ/∂τ = 0 boundary condition along the lc,PEC contour and

φ|lp,n + const = χy φ|lp,s ;∂φ

∂n

∣∣∣∣lp,n

= χy∂φ

∂n

∣∣∣∣lp,s

(3.35)

along lp,s ∪ lp,n contour. The reader should recall that the constant in (3.35) vanishesexcept for the very particular case of χx = χy = 1. As aforementioned, the ∂φ/∂τ = 0

boundary condition along the lc,PEC contour implies that the electrostatic potential is aconstant along each PEC contour. A periodic solution of a periodic structure requires

a periodic excitation. Therefore the conductors must be set to periodic potential values

using the periodicity constants χx and χy, as shown in figure 3.3(a).

(a)

xy

V0

V0χyχx

V0χxV0χx-1

V0χyχx2

V0χy-1χx

-1 V0χy-1

V0χyχx-1 V0χy

V0χy-1χx V0χy

-1χx2

V0χx2

(b)

V0χy V0χyχx

V0χxV0

lc,PEClp,e

lp,n

lp,s

lp,w

CI

Figure 3.3: (a) Transverse section (z = constant plane) of a generic transversely periodic waveg-uide. The red dashed line shows the chosen unit cell and the blue dash-dotted line shows thecurrent integration contour that will be used in section 3.3.6. (b) Segmented unit cell of the trans-versely periodic waveguide in (a).

73


3.3.1 Electrostatic Potential Expansion

Let us consider figure 3.3(b). In each region, r, the electrostatic potential of the n-th

waveguide TEM z mode can be expanded as follows:

φ(r)n = Ψ

(r)l (x, y) +

∑p

Ψ(r)p

Λ(r)p

[B

(r)1,n,pe

−jk(r)x,px +B(r)2,n,pe

jk(r)x,p(l(r)−x)

](3.36)

where Ψ(r)l (x, y) is a lineal function of x and y, Ψ

(r)p is a set of orthogonal functions that

fulfill the y-direction boundary conditions of the r-th region, Λ(r)p is a normalization con-

stant defined in the same way as (3.3), B(r)1,n,p and B(r)

2,n,p are the unknown term amplitudesand the x and y-direction separation constants are related by,

(k(r)x,p)2

= −(k(r)y,p)2

(3.37)

It shall be remarked here that, although the mathematical formalism of (3.36) is almostidentical to (3.2), the physical interpretation of both expressions differs. For instance, thefactor in brackets in (3.36) can not be understood as a couple of propagating waves.

Parallel-Plate Region: This type of region is bounded by two PEC planes at y = d(r)

and y = d(r) +h(r). As opposed to the previous section, the boundary conditions are now,

φ(r)n

(y = d(r)

)= V

(r)lo ; φ(r)

n

(y = d(r) + h(r)

)= V (r)

up (3.38)

where V (r)lo and V

(r)up stand for the potentials at which the lower and upper conductors

are set. For instance, V (I−V II)lo = V0 and V (I−V II)

up = χyV0, using the numeration of fig-ure 3.1(b) and the potentials of figure 3.3(b). The linear function that fulfills this boundarycondition is:

Ψ(r)l (y) = α(r)

y

(y − d(r)

)+ β(r)

y (3.39)

with α(r)y =

(V

(r)up − V (r)

lo

)/h(r) and β(r)

y = V(r)lo . Besides, the set Ψ

(r)p is chosen as:

Ψ(r)p = sin

(k(r)y,p

[y − d(r)

])(3.40)

where k(r)y,p = pπh(r)

, p = 1 . . .∞ and Θ(r)p = Ψ

(r)p . Finally, the normalization constant is

Λ(r)p = h(r)/2.

Periodic-Boundary Region: In a periodic-boundary region the periodicity betweenthe y = 0 and y = h(r) planes is enforced by (3.35). The electrostatic expansion for thistype of region in then determined by the periodicity constant χy, just in the same way as in

74


section 3.2.1. Furthermore, the set of orthogonal functions for the electrostatic expansionis the same as the one of the electric and magnetic potential expansions, (3.7),

Ψ(r)p = e−jk

(r)y,py , where

k(r)y,p = −ξy −

2pπ

h(r)

ξy =arg (χy)− jln |χy|

h(r)

(3.41)

with p = −∞ . . . ,−1, 0, 1, . . .∞ and Ψ(r)l = 0 as long as χy 6= 1. It is in the process

of obtaining this orthogonal set of functions where the constant in (3.36) is found to

vanish. This is the only way to obtain a set of functions that simultaneously fulfill both

the differential Poisson equation and the boundary conditions. The set of orthogonalweighting functions for this region is Θ

(r)p =

(Ψ

(r)p

)∗, as long as |χy| = 1. If χy = 1,

then the p = 0 term of the sum has to be replaced by a lineal function of x with two newunknowns, α(r)

x and β(r)x ,

Ψ(r)l (x) =

(Λ(r)

)−1 [α(r)x

(x− x(r)ini

)+ β(r)

x

](3.42)

Both α(r)x and β

(r)x play the role of B(r)

1,n,0 and B(r)2,n,0 as the unknown amplitudes to be

determined, and p reduces to p = −∞, . . . ,−1, 1, . . . ,∞ in (3.41). Even in this casethe constant in (3.36) vanishes. Finally, the normalization constant is chosen as to beΛ

(r)p = h(r) in (3.36) and (3.42).

3.3.2 Discontinuity Characterization

In order to characterize the discontinuity between regions, the electrostatic boundary con-ditions have to be imposed, taking into account that we will be only considering homoge-neous waveguides,

φ(r)n

∣∣x−

= φ(r+1)n

∣∣x+

;∂φ

(r)n

∂x

∣∣∣∣∣x−

=∂φ

(r+1)n

∂x

∣∣∣∣∣x+

(3.43)

As analogous to the preceding section, this equations imply the continuity of both thepotential and its derivative with respect to the normal direction.

Consider a generic discontinuity between any two regions. In the first place, let us im-pose the continuity of the potential by using the generic electrostatic potential expansions,

75


(3.36):

Ψ(r)l

(x−, y

)+∑p

Ψ(r)p

Λ(r)p

[A

(r)2,n,p +B

(r)2,n,p

]

= Ψ(r+1)l

(x+, y

)+∑q

Ψ(r+1)q

Λ(r+1)q

[B

(r+1)1,n,q + A

(r+1)1,n,q

]where the notation of figure 3.2(c)-(d) has been used. In order to obtain the unknownterm amplitudes, i.e. the A’s and B’s, let us multiply the equation by Θ

(r)k and integrate

the resulting expression over discontinuity contour C(r+1),

∫C(r+1)

(Ψ

(r)l

(x−, y

)+∑p

Ψ(r)p

Λ(r)p

[A

(r)2,n,p +B

(r)2,n,p

])Θ

(r)k dy

=

∫C(r+1)

Ψ(r+1)l

(x+, y

)Θ

(r)k dy +

∑q

B(r+1)1,n,q + A

(r+1)1,n,q

Λ(r+1)q

∫C(r+1)

Ψ(r+1)q Θ

(r)k dy

In order to extend the integral on the right-hand side of the equation to the complete C(r)

contour, so that the orthogonality property can be used, we have to subtract the contribu-tions of the non-vanishing potentials at the upper and lower conductors. By defining,

Ic,k = V (r+1)up

∫Cup

Θ(r)k dy + V

(r+1)lo

∫Clo

Θ(r)k dy (3.44)

where Clo and Cup are the vertical lines that go from the lower boundary of the r-thregion to the lower boundary of the (r + 1)-th region and from the upper boundary of the(r + 1)-th region to the upper boundary of the r-th region, respectively. Moreover, bydefining

Il,k =

∫C(r+1)

Ψ(r+1)l Θ

(r)k dy −

∫C(r)

Ψ(r)l,y (y) Θ

(r)k (y) dy (3.45)

where Ψ(r)l (x, y) = Ψ

(r)l,x (x) · Ψ

(r)l,y (y). Then, the potential continuity equation can be

rewritten as follows,

Ψ(r)l,x

(x−) ∫

C(r)

Θ(r)k (y) dy +

∑p

A(r)2,n,p +B

(r)2,n,p

Λ(r)p

∫C(r)

Ψ(r)p Θ

(r)k dy

= Il,k + Ic,k +∑q

B(r+1)1,n,q + A

(r+1)1,n,q

Λ(r+1)q

∫C(r+1)

Ψ(r+1)q Θ

(r)k dy

76


On the one hand, if χy 6= 1, by using the orthogonality property and renaming k as p, theabove equation can be rewritten as:

A(r)2,n,p +B

(r)2,n,p = Il,p + Ic,p +

∑q

B(r+1)1,n,q + A

(r+1)1,n,q

Λ(r+1)q

∫C(r+1)

Ψ(r+1)q Θ(r)

p dy (3.46)

On the other hand, if region r is a periodic-boundary region and χy = 1, then the p = 0

term of (3.46) has to be replaced with:

α(r)x

(x− − x(r)ini

)+ β(r)

x = Il,p + Ic,p +∑q

B(r+1)1,n,q + A

(r+1)1,n,q

Λ(r+1)q

∫C(r+1)

Ψ(r+1)q dy (3.47)

Let us now consider the continuity of the derivative of the potential with respect to x.By taking a look at the definitions of Ψ

(r)l for each type of region, we realize that the only

way that ∂Ψ(r)l /∂x does not vanish is in a periodic-boundary region when χy = 1. Since a

discontinuity between two periodic-boundary regions does not make sense, ∂Ψ(r)l /∂x =

α(r)x /Λ(r) will only appear in the right-hand side of the equality when region r is a periodic

boundary region and χy = 1. Therefore, in general, the continuity of the derivative of thepotential is,

−jα(r)x

Λ(r)+∑p

k(r)x,pΨ

(r)p

Λ(r)p

[−A(r)

2,n,p +B(r)2,n,p

]=∑q

k(r+1)x,q

Ψ(r+1)q

Λ(r+1)q

[A

(r+1)1,n,q −B

(r+1)1,n,q

]

Once more, multiplying the above equation by Θ(r+1)k and integrating over the C(r+1) line,

∑p

k(r)x,pB

(r)2,n,p − A

(r)2,n,p

Λ(r)p

∫C(r+1)

Ψ(r)p Θ(r+1)

p dy − jα(r)x

Λ(r)

∫C(r+1)

Θ(r+1)p dy

= k(r+1)x,q

[A

(r+1)1,n,q −B

(r+1)1,n,q

](3.48)

where k has been renamed as q.

Considering for now, the χy 6= 1 case, the linear equation system formed by (3.46)and (3.48) can be written in matrix form as:

A(r)2 +B

(r)2 = I + Y

[A

(r+1)1 +B

(r+1)1

]X tK(r)

x

[−A(r)

2 +B(r)2

]= K(r+1)

x

[A

(r+1)1 −B(r+1)

1

] (3.49)

77


where the vector I is defined as Ip = Il,p + Ic,p, K(r)x = diag

(k(r)x,p

)and X and Y are the

following complete matrices,

Yp,q =(Λ(r+1)q

)−1 ∫C(r+1)

Ψ(r+1)q Θ(r)

p dy

Xp,q =(Λ(r)p

)−1 ∫C(r+1)

Ψ(r)p Θ(r+1)

p dy

(3.50)

For convenience, (3.49) is rewritten as,[B

(r)2

B(r+1)1

]=

[S11 S12

S21 S22

][A

(r)2

A(r+1)1

]+

[S01

S02

](3.51)

with

S11 = Y S21 − U S12 = Y (S22 + U) S01 = I + Y S02

S21 = 2HX tK(r)x S22 = 2HK

(r+1)x − U S01 = −HX tK

(r)X I

(3.52)

and H =(X tK

(r)x Y +K

(r+1)x

)−1. I should be noted that the matrix formed by S11, S12,

S21 and S22 is completely analogous to the electrodynamical scattering matrix. However,the physical interpretation differs, as far as there is not actual transverse propagation ofthe electrostatic expansion terms.

Discontinuity between two parallel-plate regions: Let r be the bigger parallel-plateregion and r+1 the smaller one, just as figure 3.2(c) shows. As analogous to section 3.2.2,the electrostatic potential expansions in the r-th and (r + 1)-th regions have been trun-cated to P and Q terms, respectively. Furthermore, the value of Q has to be kept as closeas possible to P · h(r+1)/h(r). When considering this type of discontinuity the followingintegrals have to be solved. In the first place,

Ic,p = V(r+1)lo

∫ d(r+1)

d(r)sin(k(r)y,p

[y − d(r)

])dy + V (r+1)

up

∫ d(r)+h(r)

d(r+1)+h(r+1)

sin(k(r)y,p

[y − d(r)

])dy (3.53)

78


next,

Il,p = α(r+1)y

∫ d(r+1)+h(r+1)

d(r+1)

y · sin(k(r)y,p

[y − d(r)

])dy − α(r)

y

∫ d(r)+h(r)

d(r)y · sin

(k(r)y,p

[y − d(r)

])dy

+[β(r+1)y − α(r+1)

y d(r+1)]∫ d(r+1)+h(r+1)

d(r+1)

sin(k(r)y,p

[y − d(r)

])dy

−[β(r)y − α(r)

y d(r)] ∫ d(r)+h(r)

d(r)sin(k(r)y,p

[y − d(r)

])dy (3.54)

and, finally,


p Xp,q =

∫ d(r+1)+h(r+1)

d(r+1)

sin(k(r)y,p[y − d(r)

])sin(k(r+1)y,q

[y − d(r+1)

])dy (3.55)

Discontinuity between a periodic-boundary region and a parallel-plate region:Let r be the periodic-boundary region, where the potential expansion has 2P + 1 terms,and let r+1 be the parallel-plate region, whereQ terms are used in the potential expansion,as shown in figure 3.2(d). In order to avoid the relative convergence phenomenon, thevalue of Q has to be kept as close as possible to 2P · h(r+1)/h(r). The elements of the Ivector are obtained from:

Ic,p = V(r+1)lo

∫ d(r+1)

0

ejk(r)y,pydy + V (r+1)

up

∫ h(r)

d(r+1)+h(r+1)

ejk(r)y,pydy (3.56)

Il,p = α(r+1)y

∫ h(r+1)+d(r+1)

d(r+1)

yejk(r)y,pydy +

[β(r+1)y − d(r+1)α(r+1)

y

] ∫ h(r+1)+d(r+1)

d(r+1)

ejk(r)y,pydy (3.57)

Additionally, the complete matrices X and Y are,


p X∗p,q =

∫ d(r+1)+h(r+1)

d(r+1)

sin(k(r+1)y,q

[y − d(r+1)

])ejk

(r)y,pydy (3.58)

as long as |χy| = 1.


Due to the presence of the vector S0 in (3.51), an ad-hoc procedure to cascade disconti-nuities in the characterization of TEM modes has been developed. Let us consider, oncemore, the two discontinuities shown in figure 4.3. By using the same notation, (4.33) and

79


(4.33), we have:B(r−1) = SC11A

(r−1) + SC12A(r+1) + SC01

B(r+1) = SC21A(r−1) + SC22A

(r+1) + SC02

(3.59)

where,

SC11 = SL11 + SL12ΓlΥ11 SC12 = SL12Γ

lΥ12 SC01 = SL12ΓlΥ01 + SL01

SC21 = χ−1y SR21ΓuΥ21 SC22 = χ−1y SR21Γ

uΥ22 + SR22 SC02 = χ−1y SR21ΓuΥ02 + SR02

with

Υ11 = HSR11ΓuSL21 Υ12 = χyHS

R12 Υ01 = H

(χyS

R01 + SR11Γ

uSL02)

Υ21 = SL21 + SL22ΓlΥ11 Υ22 = SL22Γ

lΥ21 Υ02 = SL22ΓlΥ01 + SL02

3.3.4 Inhomogeneous Linear System Resolution

At this point, we have all the necessary elements to build up the generalized equivalentcircuit of figure 3.1(c). The GSMs in figure 3.1(c), SL and SR, are replaced here by therelations in (3.51), although they will keep their names. These matrices are obtained bysuccessively applying the cascading procedure in section 3.3.3. When comparing withsection 3.2.4, the relations in (3.26) are now replaced by,

B(ol)2 = S

(L)11 A

(ol)2 + S

(L)12 A

(i)1 + S

(L)01

B(i)1 = S

(L)21 A

(ol)2 + S

(L)21 A

(i)1 + S

(L)02

;

B

(i)2 = S

(R)11 A

(i)2 + S

(R)12 A

(or)1 + S

(R)01

B(or)1 = S

(R)21 A

(i)2 + S

(R)21 A

(or)1 + S

(R)02

(3.60)

where, of course, the S sub-matrices are the ones defined in section 3.3.2, rather thansection 3.2.2. However, (3.27) and (3.28) are identical and will not be repeated here.

The biggest difference between the TE and TM mode characterization and the TEMone is that the former case needs, in the first place, to find the eigenvalues and then theunknown term amplitudes and the later, directly solves an inhomogeneous linear systemof equations, since its eigenvalue is zero by definition. From the above relations, the

80


electrostatic expansion term amplitudes are found by solving:[U − χ−1x S

(L)21 Γ(ol)Γ(or)Υ21 −S(L)

21 Γ(ol)Γ(or)Υ22 − S(L)22 Γ(i)

−S(R)11 Γ(i) − S(R)

12 Γ(or)Γ(ol)Υ11 U − χxS(R)12 Γ(or)Γ(ol)Υ12

][B

(i)1

B(i)2

]=

[S(L)02 + χ−1x S

(L)21 Γ(ol)Γ(or)Υ02

χxS(R)12 Γ(or)Γ(ol)Υ01 + S

(R)01

](3.61)

where,

Υ21 = HS(R)21 Γ(i) Υ22 = HS

(R)22 Γ(or)Γ(ol)S

(L)12 Γ(i)

Υ11 = S(L)11 Γ(ol)Γ(or)Υ21 Υ12 = S

(L)11 Γ(ol)Γ(or)Υ22 + S

(L)12 Γ(i)

Υ02 = H(χxS

(R)22 Γ(or)Γ(ol)S

(L)01 + S

(R)02

)Υ01 = χ−1x S

(R)11 Γ(ol)Γ(or)Υ02 + S

(L)01

and H =(U − S(R)

22 Γ(or)Γ(ol)S(R)11 Γ(ol)Γ(or)

)−1.

As analogous to section 3.2.4, it is of great interest the particularization of (3.61) forsymmetric structures. It shall be noted that the relations between the sub-matrices of theleft and right discontinuities are identical to the ones in section 3.2.4, i.e. S(L)

11 = S(R)22 ,

S(L)12 = S

(R)21 , S(L)

21 = S(R)12 and S(L)

22 = S(R)11 . However, the vectors S(L)

01 , S(L)02 , S(R)

01 andS(R)02 are unrelated and, thus, they must be computed for each side. As a result, (3.61) is

rewritten as follows:[U − χ−1x M11 M21

M21 U − χxM11

][B

(i)1

B(i)2

]=

[S(L)02 + χ−1x S

(R)12

(Γ(o))2

Υ02

S(R)01 + χxS

(R)12

(Γ(o))2

Υ01

](3.62)

where,

M11 = S(R)12

(Γ(o))2

Υ21 M21 = −S(R)11 Γ(i) − S(R)

12

(Γ(o))2

Υ11

Υ21 = HS(R)21 Γ(i) Υ11 = S

(R)22

(Γ(o))2

Υ21

Υ02 = H(χxS

(R)22

(Γ(o))2S(L)01 + S

(R)02

)Υ01 = χ−1x S

(R)11

(Γ(o))2

Υ02 + S(L)01

and H =

(U −

[S(R)22

(Γ(o))2]2)−1.

81


3.3.5 The Special χy = 1 Case

When χy = 1 the procedure described in section 3.3.2, 3.3.3 and 3.3.4 can not be applied.For the χy = 1 case, the system of equations that characterizes a discontinuity is,

α(r)x

(x− − x(r)ini

)+ β(r)

x = I0 + Y0

[A

(r+1)1 +B

(r+1)1

][A

(r)2 +B

(r)2

]= I + Y

[A

(r+1)1 +B

(r+1)1

]X tK(r)

x

[−A(r)

2 +B(r)2

]− jX t

0α(r)x = K(r+1)

x

[A

(r+1)1 −B(r+1)

1

] (3.63)

where I0, Y0 and X0 are the particularizations of (3.44), (3.45) and (3.50), substitutingΨ

(r)p (y) by the unity. Thus, I0 is a scalar and Y0 and X0 are vectors. The system of equa-

tions (3.63) does not seem to allow a scattering-matrix-like representation. Therefore, thescattering matrix formalism can not be used. Each discontinuity is characterized by itsown equation system, (3.63), and the inhomogeneous linear system of equations is ob-tained by imposing the x-direction boundary condition to a generalized equivalent circuitsimilar to the one depicted in figure 3.1(c). The obtention of the inhomogeneous linearsystem of equations is an easy but tedious task and will not be reproduced here. How-ever, all this trouble is rarely necessary since the characterization approach described forχy 6= 1 works with periodicity constant values very close to the unit (χy = ej10

−6).

On the other hand, when both χx and χy are simultaneously unity, the described modi-fication of the generalized transverse resonance technique can not be applied at all. In thiscase, the constant in (3.36) does not vanish and plays an important role. If that constantis not taken into account, the four conductors in figure 3.3(b) would be set to the samepotential, V0, and the unique solution for the potential would be a constant. Consequently,the trivial solution would be obtained, with a vanishing electric field. On the other hand,by using the correct boundary conditions, the four conductors are set to four differentpotentials. The lower-left conductor would be set to V0, the lower-right conductor wouldbe set to V0 + Vx, the upper-left conductor would be set to V0 + Vy and the upper-rightconductor would be set to V0 + Vx + Vy (where Vx, Vy ∈ C). This way two independentsolutions can be obtained and, thus the waveguide has two TEM modes. However, thisparticular case has not been implemented, as far as the described method has proved togive useful results with χx = 1; χy = ej10

−6 .

82

3.4. SOME EXAMPLES

3.3.6 Characteristic Impedance

The transmission line representation of a TEM mode requires the knowledge of both thepropagation constant (determined by the dielectric in which the wires are embedded) andthe characteristic impedance. Although, this Ph. D. dissertation will make no use of thecharacteristic impedance definition detailed in this section, the reader may feel curiousabout how to define such impedance in a transversely periodic waveguide.

On the one hand, a voltage between conductors seems, in principle, difficult to be uni-vocally defined in this case. On the other hand, the current I0 on a wire can be computedas the circulation of the magnetic field over any contour which contains the wire section,such as the contour CI shown in figure 3.3. Using the periodicity properties, the integralmay be written in the following way:

I0 =(1− χ−1x

)√ ε

µ

(∫ p2

y=0

z × ~Et

∣∣∣x= p

2

· ydy +χ−1y

∫ p

y= p2

z × ~Et

∣∣∣x= p

2

· ydy

)

−(1− χ−1y

)√ ε

µ

(∫ p2

x=0

z × ~Et

∣∣∣y= p

2

· xdx +χ−1x

∫ p

x= p2

z × ~Et

∣∣∣y= p

2

· xdx

)(3.64)

On the other hand, P is defined as the TEM-mode transmitted power in the transversesection of a periodic cell. Therefore, the characteristic impedance of the TEM mode canbe defined as:

Z =2P

|I0|2(3.65)

It has been numerically checked that this definition is identical to the voltage-currentdefinition,

Zn =V0I0

(3.66)

where V0 is not the voltage between any pair of wires, but the potential at which the wireis set. Furthermore, when V0 is chosen as real the current flowing through the wire that isobtained is also real, and so is the characteristic impedance.

3.4 Some examples

This section is devoted to show the results obtained by the generalized Transverse Res-onance Technique when applied to the characterization of transversely periodic waveg-uides. In particular, the cut-off frequencies and the magnetic, electric and electrostaticpotential distributions will be presented for the three waveguides depicted in figure 3.4.

83


In addition to the waveguide dimensions, transversely periodic waveguides require thatthe periodicity constants are specified too. This is important because, self-evidently, thecut-off frequencies and potential distributions of the waveguide modes vary as a functionof χx and χy.

Periodic Wire Waveguide: A transversely periodic wire waveguide is depicted in fig-ure 3.4(a). For simplicity, this example uses squared wires. If, for instance, circular wireswere to be analyzed, a staircase approximation of the contour would have been requiredfor the analysis of the structure. An example of such case can be found in chapter 5. Thefirst five TEz and TM z mode cut-off frequencies, fc, of a periodic wire waveguide withpx = py = 21mm and lx = ly = 6mm at χx = ejπ/3 and χy = ejπ/4 are,

TEz: fc(GHz) 2.83 11.25 12.55 15.48 15.97

TM z: fc(GHz) 7.38 13.54 15.26 17.42 19.07

(3.67)

In addition, figure 3.6 shows the real and imaginary parts of the potential distribution ofthe third TM z mode and figure 3.7 shows the real and imaginary parts of the electrostaticpotential, from which the electric and magnetic fields of the TEM z mode are obtained,[Varela and Esteban, 2011b].

Self-evidently, the adaptation of the generalized TRT derived in this chapter is notonly useful for the computation of the cut-off frequencies and potential distributions oftransversely periodic waveguide modes. For instance, the TRT can be used to analyzethe transverse wave propagation in cylindrical structures. However, this type of analysisis not one of the objectives of this dissertation and not much space will be devoted to

(a) (b) (c)ε, µ ε, µ ε, µ

lx

lyly

lxw

lx

lywy

wx

px

py

Figure 3.4: Transversely periodic waveguides used in this chapter. The upper row shows the morecommonly chosen unit cell, as depicted in figure 3.1(b). The lower row the unit cell as chosenin this dissertation. (a) Periodic Wire Waveguide. (b) Periodic Cross Waveguide. (c) PeriodicH-shaped Waveguide.

84

3.4. SOME EXAMPLES

it. Nevertheless, two transverse dispersion diagrams along the contour of the irreducibleBrillouin zone, defined in figure 3.6(a), are shown in figure 3.7. On the one hand, fig-ure 3.8(a) shows the transverse dispersion diagram of a thin, l/p = 0.01, squared-wirewaveguide. On the other hand, figure 3.8(b) shows the analogous diagram for a thick,l/p = 0.333 squared-wire waveguide. Both diagrams have been obtained by computingthe cut-off frequencies of the first four modes of the periodic wire waveguides under dif-ferent combinations of the periodicity constants. For instance, along the Γ−X segment,the y-direction periodicity constant is fixed to a constant value, χy = 1 and the x-directionone, χx = ejθ, varies as θ ranges from zero to π.

Finally, figure 3.5 shows the computed values for the characteristic impedance of theTEM mode over the boundaries of the irreducible Brillouin zone (see figure 3.6(a)) as afunction of the size of the wires (lx = ly = l) compared to the period (px = py = p).The characteristic impedance ranges from very low values (6 Ω) for thick wires to highvalues (270 Ω) for thin wires. The impedance on the Γ point is theoretically infinite (sinceI0 = 0), and thus the growing values of the impedance in its vicinity.

Periodic Cross Waveguide: Figure 3.4(b) shows the cross section of a transverselyperiodic cross waveguide. The following results have been obtained for χx = ejπ/6 and

69

14

23

38

62

100

269

440 440

1179

Γ X M Γ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Z ( Ω )TEM

pl

164

Figure 3.5: Characteristic impedance of the TEM mode over the boundaries of the irreducibleBrillouin zone (figure 3.6(a)) as a function of the size of the wires (lx = ly = l) compared to theperiod (px = py = p).

85


(a) (b)

X

M

Γ

Figure 3.6: Distribution of the third electric potential eigenfunction of a Periodic Wire waveguidewith dimensions dx = dy = 21mm and lx = ly = 6mm under the χx = ejπ/3 and χx = ejπ/4

periodic boundary conditions. The potential distribution is normalized to its maximum modulus,the dark-red and dark-blue zones stand for +1 and −1 values respectively. (a) Real part of theelectric potential and boundaries of the irreducible Brillouin Zone. (b) Imaginary part of theelectric potential.

(a) (b)

Figure 3.7: Distribution of the electrostatic potential of a Periodic Wire waveguide with dimen-sions dx = dy = 21mm and lx = ly = 6mm under the χx = ejπ/3 and χx = ejπ/4 periodicboundary conditions. The potential distribution is normalized to its maximum modulus, the dark-red and dark-blue zones stand for +1 and−1 values respectively. (a) Real part of the electrostaticpotential. (b) Imaginary part of the electrostatic potential.

86

3.5. CONCLUSIONS

χy = ejπ/9 in a px = py = 21mm, lx = ly = 10.5mm and w = 2.1mm cross waveguide.The cut-off frequencies of the first five TE and TM modes are,

TEz: fc(GHz) 1.28 11.46 11.82 14.30 14.80

TM z: fc(GHz) 8.25 15.54 16.24 17.13 19.65

(3.68)

Additionally, the real and imaginary parts of the potential distribution of the second TEz

mode are shown in figure 3.9. The reader should recall that, as stated in the previouschapter, and as opposed to closed PEC or PMC waveguides, the electric and magneticfields of a transversely periodic waveguide are, in general, complex. They can not benormalized to be real.

Periodic H-shaped Waveguide: The cross section of an H-shaped periodic waveg-uide is shown in figure 3.4(c). A lx = 12mm, ly = 8mm and wx = wy = 2mm metal-His in a square unit cell, px = py = 18mm. The first five TE and TM mode cut-offfrequencies are,

TEz: fc(GHz) 2.41 10.46 11.61 16.21 17.94

TM z: fc(GHz) 12.54 18.98 22.32 23.53 25.77

(3.69)

Furthermore, figure 3.10 and figure 3.11 show the real and imaginary parts of the potentialdistributions of the second TEz mode and the TEM mode respectively.

3.5 Conclusions

An extension to the well-known generalized Transverse Resonance Technique to the char-acterization of transversely periodic waveguides has been detailed in this chapter. Theproposed approach has been introduced in two steps, the obtention of the electric andmagnetic eigenfunctions, section 3.2, and the obtention of the electrostatic potential, sec-tion 3.3. Finally, the computed cut-off frequencies and potential distributions of threeparticular transversely periodic waveguides have been shown in section 3.4.

Now that a way to obtain the complete set of modes that describe transversely periodicwaveguides has been described we are prepared to face the analysis of non-cylindricaltransversely periodic structures in the next chapter.

87


Γ X M0

5

10

15

(

)

Γ Γ X M Γ0

2

4

6

8

10

12

freq

uen

cy (G

Hz)

(a) (b)

Figure 3.8: Transverse dispersion diagram of a thin, (a), and thick, (b) periodic wire waveguidesalong the contour of the irreducible Brillouin zone. The TEz modes are represented with blue linesand the TM z modes with red lines. The ratio l/p of the thin and thick squared-wire waveguidesis 0.01 and 0.33 respectively.

(a) (b)

Figure 3.9: Distribution of the second magnetic potential eigenfunction of a Periodic Crosswaveguide with dimensions dx = dy = 21mm, lx = ly = 10.5mm and wx = wy = 2mmunder the χx = ejπ/6 and χx = ejπ/9 periodic boundary conditions. The potential distributionis normalized to its maximum modulus, the dark-red and dark-blue zones stand for +1 and −1values respectively. (a) Real part of the magnetic potential. (b) Imaginary part of the magneticpotential.

88

3.5. CONCLUSIONS

(a) (b)

Figure 3.10: Distribution of the second magnetic potential eigenfunction of a periodic H-shapedwaveguide with dimensions px = py = 18mm, lx = 12mm, ly = 8mm and wx = wy = 2mmunder the χx = ejπ/5 and χx = ejπ/3 periodic boundary conditions. The potential distributionis normalized to its maximum modulus, the dark-red and dark-blue zones stand for +1 and −1values respectively. (a) Real part of the magnetic potential. (b) Imaginary part of the magneticpotential.

(a) (b)

Figure 3.11: Distribution of the electrostatic potential of a periodic H-shaped waveguide withdimensions px = py = 18mm, lx = 12mm, ly = 8mm and wx = wy = 2mm under the χx = ejπ/5

and χx = ejπ/3 periodic boundary conditions. The potential distribution is normalized to itsmaximum modulus, the dark-red and dark-blue zones stand for +1 and −1 values respectively.(a) Real part of the electrostatic potential. (b) Imaginary part of the electrostatic potential.

89

4Analysis of Transversely Periodic

Structures

4.1 Introduction

The previous discussion, chapter 2, has provided an alternative interpretation of peri-odic cylindrical structures. Now, it is clear that if |χx| = |χy| = 1 these structures can beconsidered as closed waveguides. Assuming that condition, chapter 3 introduced a gen-eralization of the generalized Transverse Resonance Technique capable of obtaining theTEM z, TEz and TM z modes of transversely periodic waveguides. The next logical stepis to consider a discontinuity between two different periodic waveguides, just in the sameway as if they were discontinuities between closed waveguides. This chapter proposesthe Mode Matching (MM) technique as an efficient, accurate and versatile method for theanalysis of these discontinuities. The MM technique is a mature and well-known methodof analysis, with a wide span of different formulations [Wexler, 1967, Safavi-Naini andMacPhie, 1981, Chu et al., 1985, Alessandri et al., 1988, Arndt et al., 1997]. It is im-portant to note, as implied by (2.14), that the modes of the periodic waveguides are notorthogonal in self-reaction, as defined in [Eleftheriades et al., 1994]. Therefore, a powerorthogonal MM formulation, such as the conservation of the complex power techniquepresented in [Safavi-Naini and MacPhie, 1981], has to be used.

This chapter has two well differentiated parts. In the first part, section 4.2, we willinvestigate if the MM technique can be used to obtain the Generalized Scattering Ma-trix (GSM) representation of a discontinuity between two different generic transverselyperiodic waveguides. The reader will see that both the procedure and the resulting ex-pressions are completely analogous to those obtained when the MM technique is appliedto a closed waveguide discontinuity. The second part of this chapter, sections 4.3 and 4.4,

91

CHAPTER 4. ANALYSIS OF TRANSVERSELY PERIODIC STRUCTURES

is devoted to the application of the aforementioned technique for the analysis of morecomplex transversely periodic structures such as wire-medium slabs, fishnets or texturedsurfaces. It is very important to highlight that, this chapter is restricted to the morepractical case of transversely periodic waveguides with a combination of PEC andperiodic boundary conditions and that the |χx| = |χy| = 1 condition will be hence-forth assumed.

4.2 The Mode Matching Technique

Let us expand the transverse electric and magnetic fields of a generic waveguide using(2.133) and (2.134) as a sum of TEM z, TEz and TM z modes, defined in sections 2.6.2,2.6.3 and 2.6.4,

~E(g)t =

∑p

~E(g)t,p

[C

(g)0,pe

−jk(g)z,pz+D(g)0,pe

jk(g)z,pz]

+∞∑m=1

∇φ(g)H,m×zk(g)c,m

[C

(g)H,me

−jk(g)z,mz +D(g)H,me

jk(g)z,mz]

−∞∑n=1

∇φ(g)E,n

k(g)c,n

[C

(g)E,ne

−jk(g)z,nz +D(g)E,ne

jk(g)z,nz]

(4.1)

~H(g)t =

∑p

z× ~E(g)t,p

Z(g)0,p

[C

(g)0,pe

−jk(g)z,pz−D(g)0,pe

jk(g)z,pz]

+∞∑m=1

∇φ(g)H,m

k(g)c,mZ

(g)H,m

[C

(g)H,me

−jk(g)z,mz −D(g)H,me

jk(g)z,mz]

+∞∑n=1

∇φ(g)E,n×z

k(g)c,nZ

(g)E,n

[C

(g)E,ne

−jk(g)z,nz −D(g)E,ne

jk(g)z,nz]

(4.2)

where we will assume that the transverse electric field of the TEM z modes, ~E(g)t,p , has

been normalized, the impedances Z(g) are those defined in (2.101) and φ(g)H,m and φ(g)

E,n arethe normalized solutions of the potential problems defined in tables 2.5 and 2.6.

Consider the discontinuity between two transversely periodic waveguides as depictedin figure 4.1(a). Figure 4.1(b) shows the transverse view of the discontinuity, where theSA surface is bounded by the darker grey and periodic contours and SB is bounded by thelighter grey and periodic contours (SB ⊂ SA). Finally, figure 4.1(c) shows the longitu-

92

4.2. THE MODE MATCHING TECHNIQUE

dinal view of the discontinuity, where the mode amplitudes C(g) and D(g) stand for thewaves impinging at and reflected from the discontinuity. In order to obtain a GSM char-acterization of the discontinuity, the transverse electric and magnetic fields of waveguidesA and B will be matched at the discontinuity (plane z = 0).

4.2.1 Transverse Electric Field Continuity

By using the electric field expansion (4.1) for both waveguides, the continuity of thetransverse electric field is written, in the aperture surface SB, as follows,

∑p

~E(A)t,p

[C

(A)0,p +D

(A)0,p

]+∞∑m=1

∇φ(A)H,m×zk(A)c,m

[C

(A)H,m+D

(A)H,m

]−∞∑n=1

∇φ(A)E,n

k(A)c,n

[C

(A)E,n+D

(A)E,n

]=∑v

~E(B)t,v

[D

(B)0,v +C

(B)0,v

]+∞∑s=1

∇φ(B)H,s×zk(B)c,s

[D

(B)H,s+C

(B)H,s

]−∞∑u=1

∇φ(B)E,u

k(B)c,u

[D

(B)E,u+C

(B)E,u

](4.3)

If (4.3) is multiplied by the complex conjugate transverse electric field of the k-th

TEM z mode of waveguide A and integrated over the cross section of waveguide B, SB,expanding the integrals of the left-hand side of the resulting equation to the complete SAcross section (taking into account that the transverse electric field must vanish in SA−SB)

x

y

z

Waveguide A

Waveguide B

SB

SA

lp,wlp,e

lp,n

lp,sD(A)

2

C(A)2

D(B)1

C(B)1

(a) (b) (c)

Figure 4.1: (a) Discontinuity between two generic transversely periodic waveguides. (b) Trans-verse view of the discontinuity in (a). (c) Longitudinal view of the discontinuity in (a).

93


and using the orthonormality property we have,

[C

(A)0,k +D

(A)0,k

]=∑v

[D

(B)0,v +C

(B)0,v

] ∫SB

~E(B)t,v ·

(~E(A)t,k

)∗dS

+∞∑s=1

[D

(B)H,s+C

(B)H,s

]k(B)c,s

∫SB

∇φ(B)H,s×z ·

(~E(A)t,k

)∗dS

−∞∑u=1

[D

(B)E,u+C

(B)E,u

]k(B)c,u

∫SB

∇φ(B)E,u ·

(~E(A)t,k

)∗dS (4.4)

The first integral on the right-hand side of (4.4) can be rewritten as follows,∫SB

~E(B)t,v ·

(~E(A)t,k

)∗dS =

∫SB

∇φ(B)0,v ·

(∇φ(A)

0,k

)∗dS =

∮lB

φ(B)0,v

(n·∇φ(A)

0,k

)∗dl

where lB is the contour of waveguide B. In addition, the second integral simplifies to,∫SB

∇φ(B)H,s×z ·

(~E(A)t,k

)∗dS =

∫SB

∇×φ(B)H,sz ·

(∇φ(A)

0,k

)∗dS =

∮lB

(φ(A)0,k

)∗∇φ(B)

H,s · τdl

where the fact that the divergence of a rotational vanishes has been used. Finally, the thirdintegral vanishes,∫

SB

∇φ(B)E,u ·

(~E(A)t,k

)∗dS =

∫SB

∇φ(B)E,u ·

(∇φ(A)

0,k

)∗dS =

∮lB

φ(B)E,u ·

(∇φ(A)

0,k · n)∗

dl = 0

because φ(B)E,u = 0 along the PEC section of the lB contour and both φ(B)

E,u and ∇φ(A)0,k · n

are periodic with the same periodicity, and where (2.58) has been used. To sum up, (4.4)can be rewritten as,

[C

(A)0,k +D

(A)0,k

]=∑v

[D

(B)0,v +C

(B)0,v

] ∮lB

φ(B)0,v

(n·∇φ(A)

0,k

)∗dl

+∞∑s=1

[D

(B)H,s+C

(B)H,s

]k(B)c,s

∮lB

(φ(A)0,k

)∗∇φ(B)

H,s · τdl (4.5)


TEz mode of waveguide A and integrated over the cross section of waveguide B, SB,expanding the integrals of the left-hand side of the resulting equation to the complete SA

94


cross section and using the orthonormality property we have,

[C

(A)H,k+D

(A)H,k

]=∑v

[D

(B)0,v +C

(B)0,v

]k(A)c,k

∫SB

~E(B)t,v ·

(∇φ(A)

H,k

)∗×zdS

+∞∑s=1

[D

(B)H,s+C

(B)H,s

]k(A)c,k k

(B)c,s

∫SB

∇φ(B)H,s×z ·

(∇φ(A)

H,k

)∗×zdS

−∞∑u=1

[D

(B)E,u+C

(B)E,u

]k(A)c,k k

(B)c,u

∫SB

∇φ(B)E,u ·

(∇φ(A)

H,k

)∗×zdS (4.6)

The first surface integral on the right-hand side of (4.6) can be rewritten as,∫SB

∇φ(B)0,v ·

(∇φ(A)

H,k

)∗×zdS =

∫SB

(∇φ(A)

H,k

)∗· z×∇φ(B)

0,v dS =

∮lB

(φ(A)H,k

)∗∇φ(B)

0,v · τdl = 0

since along the PEC contour∇φ(B)0,v · τ vanishes and both φ(A)

H,k and∇φ(B)0,v · τ are periodic

with the same periodicity along the periodic contour lp. In addition, the second integralon the right-hand side of (4.6) can be rewritten as,∫

SB

∇φ(B)H,s ·

(∇φ(A)

H,k

)∗dS =

∮lB

(φ(A)H,k

)∗∇φ(B)

H,s · ndl +(k(B)c,s

)2∫SB

(φ(A)H,k

)∗· φ(B)

H,sdS

where the contour integral vanishes as it is readily seen by using the boundary conditions.Finally, the third surface integral in the right-hand side of (4.6) is zero as follows,∫SB

∇φ(B)E,u ·

(∇φ(A)

H,k

)∗×zdS =

∮lB

φ(B)E,u

[(∇φ(A)

H,k

)∗· τ]dl−

∫SB

φ(B)E,u∇·

[∇×

(φ(A)H,k

)∗z]dS = 0

where the surface integral vanishes since the divergence of a rotational is always zero andthe line integral also vanishes since φ(B)

E,u = 0 along the PEC contour and both ∇φ(A)H,k · τ

and φ(B)E,u are periodic with the same periodicity. To sum up, substituting the non-vanishing

rewritten integral in (4.6) we have,

[C

(A)H,k+D

(A)H,k

]=∞∑s=1

[D

(B)H,s+C

(B)H,s

] k(B)c,s

k(A)c,k

∫SB

(φ(A)H,k

)∗· φ(B)

H,sdS (4.7)


TM z mode of waveguide A and integrated over the cross section of waveguide B, SB,expanding the integrals of the left-hand side of the resulting equation to the complete SA

95


cross section and using the orthonormality property we have,

[C

(A)E,k+D

(A)E,k

]= −

∑v

[D

(B)0,v +C

(B)0,v

]k(A)c,k

∫SB

~E(B)t,v ·

(∇φ(A)

E,k

)∗dS

−∞∑s=1

[D

(B)H,s+C

(B)H,s

]k(A)c,k k

(B)c,s

∫SB

∇φ(B)H,s×z ·

(∇φ(A)

E,k

)∗dS

+∞∑u=1

[D

(B)E,u+C

(B)E,u

]k(A)c,k k

(B)c,u

∫SB

∇φ(B)E,u ·

(∇φ(A)

E,k

)∗dS (4.8)

The first surface integral on the right-hand side of (4.8) can be rewritten as,∫SB

∇φ(B)0,v ·(∇φ(A)

E,k

)∗dS =

∮lB

(φ(A)E,k

)∗∇φ(B)

0,v · ndl −∫SB

(φ(A)E,k

)∗∇·∇φ(B)

0,v dS

where, on the one hand, the surface integral vanishes but, on the other hand, the contourintegral does not. In addition, the second integral in (4.8) can be rewritten as,∫

SB

∇φ(B)H,s×z ·

(∇φ(A)

E,k

)∗dS =

∮lB

(φ(A)E,k

)∗∇φ(B)

H,s · τdl −∫SB

(φ(A)E,k

)∗∇·∇×φ(B)

H,szdS

where, self-evidently, the surface integral vanishes. Finally, the third integral can berewritten as,∫

SB

∇φ(B)E,u ·

(∇φ(A)

E,k

)∗dS =

∮lB

φ(B)E,u

(∇φ(A)

E,k

)∗· ndl +

(k(A)c,k

)2∫SB

φ(B)E,u

(φ(A)E,k

)∗dS

where the contour integral vanishes as it is readily seen by applying the boundary condi-tions. To sum up, by using the simplified integrals (4.8) can be rewritten as,

[C

(A)E,k+D

(A)E,k

]= −

∑v

[D

(B)0,v +C

(B)0,v

]k(A)c,k

∮lB

(φ(A)E,k

)∗∇φ(B)

0,v · ndl

−∞∑s=1

[D

(B)H,s+C

(B)H,s

]k(A)c,k k

(B)c,s

∮lB

(φ(A)E,k

)∗∇φ(B)

H,s · τdl

+∞∑u=1

[D

(B)E,u+C

(B)E,u

] k(A)c,k

k(B)c,u

∫SB

φ(B)E,u

(φ(A)E,k

)∗dS (4.9)

96


4.2.2 Transverse Magnetic Field Continuity

By using the magnetic field expansion (4.1) for both waveguides, the continuity of thetransverse magnetic field is written, in the aperture surface SB, as follows,

∑p

z× ~E(A)t,p

Z(A)0,p

[C

(A)0,p −D

(A)0,p

]+∞∑m=1

∇φ(A)H,m

k(A)c,mZ

(A)H,m

[C

(A)H,m−D

(A)H,m

]+∞∑n=1

∇φ(A)E,n×z

k(A)c,n Z

(A)E,n

[C

(A)E,n−D

(A)E,n

]=∑v

z× ~E(B)t,v

Z(B)0,v

[D

(B)0,v −C

(B)0,v

]+∞∑s=1

∇φ(B)H,s

k(B)c,s Z

(B)H,s

[D

(B)H,s−C

(B)H,s

]+∞∑u=1

∇φ(B)E,u×z

k(B)c,u Z

(B)E,u

[D

(B)E,u−C

(B)E,u

](4.10)

If (4.10) is multiplied by the complex conjugate rotated transverse electric fieldof the k-th TEM z mode of waveguide B, i.e. z×

(~E(B)t,k

)∗, and integrated over the cross

section of waveguide B, SB, and using the orthonormality property we have,

(Z

(B)0,k

)−1 [D

(B)0,k −C

(B)0,k

]=∑p

[C

(A)0,p −D

(A)0,p

]Z

(A)0,p

∫SB

z× ~E(A)t,p · z×

(~E(B)t,k

)∗dS

+∞∑m=1

[C

(A)H,m−D

(A)H,m

]k(A)c,mZ

(A)H,m

∫SB

∇φ(A)H,m · z×

(~E(B)t,k

)∗dS

+∞∑n=1

[C

(A)E,n−D

(A)E,n

]k(A)c,n Z

(A)E,n

∫SB

∇φ(A)E,n×z · z×

(~E(B)t,k

)∗dS (4.11)

By rewriting the first and third integrals of the above equation as,∫SB

z× ~E(A)t,p · z×

(~E(B)t,k

)∗dS =

∫SB

∇φ(A)0,p ·

(∇φ(B)

t,k

)∗dS∫

SB

∇φ(A)E,n×z · z×

(~E(B)t,k

)∗dS = −

∫SB

∇φ(A)E,n ·

(∇φ(B)

t,k

)∗dS

it is clear that the three integrals on the right-hand side of (4.11) were already simpli-fied when dealing with the transverse electric field continuity. Therefore, by using the

97


simplified integrals, (4.11) can be rewritten as,

(Z

(B)0,k

)−1 [D

(B)0,k −C

(B)0,k

]=∑p

[C

(A)0,p −D

(A)0,p

]Z

(A)0,p

∮lB

(φ(B)0,k

)∗∇φ(A)

0,p · ndl

−∞∑n=1

[C

(A)E,n−D

(A)E,n

]k(A)c,n Z

(A)E,n

∮lB

φ(A)E,n ·

(∇φ(B)

0,k · n)∗

dl (4.12)

If (4.10) is multiplied by the complex conjugate rotated transverse electric fieldof the k-th TEz mode of waveguide B, i.e. z×

(~E(B)t,k

)∗, and integrated over the cross

section of waveguide B, SB, and using the orthonormality property we have,

(Z

(B)H,k

)−1 [D

(B)H,k−C

(B)H,k

]=∑p

[C

(A)0,p −D

(A)0,p

]Z

(A)0,p k

(B)c,k

∫SB

z× ~E(A)t,p ·

(∇φ(B)

H,k

)∗dS

+∞∑m=1

[C

(A)H,m−D

(A)H,m

]k(A)c,mZ

(A)H,mk

(B)c,k

∫SB

∇φ(A)H,m ·

(∇φ(B)

H,k

)∗dS

+∞∑n=1

[C

(A)E,n−D

(A)E,n

]k(A)c,n Z

(A)E,nk

(B)c,k

∫SB

∇φ(A)E,n×z ·

(∇φ(B)

H,k

)∗dS (4.13)

Once more, the three integrals of the right-hand side of (4.13) were already simplified insection 4.2.1. Thus (4.13) can be rewritten as,

(Z

(B)H,k

)−1[D

(B)H,k−C

(B)H,k

]=∑p

[C

(A)0,p −D

(A)0,p

]Z

(A)0,p k

(B)c,k

∮lB

φ(A)0,p ·

(∇φ(B)

H,k · τ)∗

dl

+∞∑m=1

[C

(A)H,m−D

(A)H,m

] k(B)c,k

k(A)c,mZ

(A)H,m

∫SB

φ(A)H,m ·

(φ(B)H,k

)∗dS

−∞∑n=1

[C

(A)E,n−D

(A)E,n

]k(A)c,n Z

(A)E,nk

(B)c,k

∮lB

φ(A)E,n

(∇φ(B)

H,k · τ)∗

dl (4.14)

If (4.10) is multiplied by the complex conjugate rotated transverse electric field ofthe k-th TM z mode of waveguide B and integrated over the cross section of waveguide

98


B, SB, and using the orthonormality property we have,

(Z

(B)E,k

)−1[D

(B)E,k−C

(B)E,k

]=∑p

[C

(A)0,p −D

(A)0,p

]Z

(A)0,p k

(B)c,k

∫SB

z× ~E(A)t,p ·

(∇φ(B)

E,k×z)∗

dS

+∞∑m=1

[C

(A)H,m−D

(A)H,m

]k(A)c,mZ

(A)H,mk

(B)c,k

∫SB

∇φ(A)H,m ·

(∇φ(B)

E,k

)∗×zdS

+∞∑n=1

[C

(A)E,n−D

(A)E,n

]k(A)c,n Z

(A)E,nk

(B)c,k

∫SB

∇φ(A)E,n×z ·

(∇φ(B)

E,k

)∗×zdS (4.15)

Again, the integrals of the right-hand side of the equation were already dealt with. There-fore, the above equation can be rewritten as,

(Z

(B)E,k

)−1[D

(B)E,k−C

(B)E,k

]=∞∑n=1

[C

(A)E,n−D

(A)E,n

] k(B)c,k

k(A)c,n Z

(A)E,n

∫SB

φ(A)E,n ·

(φ(B)E,k

)∗dS (4.16)

4.2.3 Linear System of Equations - Matrix Formulation

The infinite sums of the three equations obtained from the continuity of the transverseelectric field, (4.5), (4.7) and (4.9), and the ones obtained from the continuity of thetransverse magnetic field, (4.12), (4.14) and (4.16), have to be truncated in order to solvethe linear system of equations that they form. Thus, the infinite sums of TEM z, TEz

and TM z modes are truncated to Mp, Mm and Mn modes in waveguide A and to Mv, Ms

and Mu modes in waveguide B, respectively. It is customary to rewrite these equationsin matrix notation in order to facilitate the resolution of the system. By defining theamplitude vectors,

C(A) =[C

(A)0,1 , . . . , C

(A)0,Mp

, C(A)H,1, . . . , C

(A)H,Mm

, C(A)E,1 , . . . , C

(A)E,Mn

]t(4.17)

D(A) =[D

(A)0,1 , . . . , D

(A)0,Mp

, D(A)H,1, . . . , D

(A)H,Mm

, D(A)E,1, . . . , D

(A)E,Mn

]t(4.18)

C(B) =[C

(B)0,1 , . . . , C

(B)0,Mv

, C(B)H,1 , . . . , C

(B)H,Ms

, C(B)E,1 , . . . , C

(B)E,Mu

]t(4.19)

D(B) =[D

(B)0,1 , . . . , D

(B)0,Mv

, D(B)H,1, . . . , D

(B)H,Ms

, D(B)E,1, . . . , D

(B)E,Mu

]t(4.20)

99


where the superscript t indicates the vector transpose, the diagonal matrices,

Z(A) = diag[Z

(A)0,1 , . . . , Z

(A)0,Mp

, Z(A)H,1, . . . , Z

(A)H,Mm

, Z(A)E,1 , . . . , Z

(A)E,Mn

](4.21)

Z(B) = diag[Z

(B)0,1 , . . . , Z

(B)0,Mv

, Z(B)H,1 , . . . , Z

(B)H,Ms

, Z(B)E,1 , . . . , Z

(B)E,Mu

](4.22)

(4.23)

the linear system of equations obtained from the continuity of the transverse electric andmagnetic field can be written as, [

C(A) +D(A)]

= Y[C(B) +D(B)

]Y †Z−1(A)

[C(A) −D(A)

]= Z−1(B)

[D(B) − C(B)

] (4.24)

where the † superscript means the matrix conjugate transpose, Y is the full matrix

Y =

Y0,0 Y 0,H 0

0 Y H,H 0

Y E,0 Y E,H Y E,E

(4.25)

and its submatrices are defined as,

Y 0,0p,v =

∮lBc

φ(B)0,v

(n·∇φ(A)

0,p

)∗dl Y 0,H

p,s =[k(B)c,s

]−1∮lBc

(φ(A)0,p

)∗(∇φ(B)

H,s · τ)

dl

Y H,Hm,s =

k(B)c,s

k(A)c,m

∫SB

(φ(A)H,m

)∗φ(B)H,sdS Y E,0

n,v = −[k(A)c,n

]−1∮lBc

(φ(A)H,m

)∗(n·∇φ(B)

0,v

)dl

Y E,En,u =

k(A)c,n

k(B)c,u

∫SB

(φ(A)E,n

)∗φ(B)E,udS Y E,H

n,s = −[k(A)c,n k

(B)c,s

]−1∮lBc

(φ(A)E,n

)∗(∇φ(B)

H,s · τ)

dl

It shall be remarked here that, by applying the periodic boundary conditions along the

periodic sections of the contour (which are shared by both waveguides), the contour inte-

grals need only to be computed along the conductor section, lBc , of the contour. In addi-tion, the surface integrals of Y H,H

m,s and Y E,En,u can be rewritten as contour integrals by using

(13) of [Collin, 1960, p. 333]. However, the implementation of the proposed approachthat has been used to compute the results shown in the examples of section 4.4, makesuse of surface integrals in all six submatrices. Since the transversely periodic waveguidemodes have been obtained by means of the generalized Transverse Resonance Technique,the surface integrals are all analytic and no increase of performance is achieved by usingcontour integrals.

The linear equation system (4.24) can be rewritten in order to obtain a GSM represen-

100

4.3. ANALYSIS OF TRANSVERSELY PERIODIC STRUCTURES

tation of the discontinuity as follows,[D(A)

D(B)

]=

[S11 S12

S21 S22

][C(A)

C(B)

](4.26)

By defining X = Z(B)Y†Z−1(A), the submatrices of the GSM are obtained from,

S21 = HX ; S22 = H − I ; S11 = Y S21 − I ; S12 = Y (S22 + I) (4.27)

where I is the identity matrix of adequate dimensions and H = 2 (XY + I)−1.

To sum up, in this section we have confirmed that the Mode Matching techniquecan be used to obtain a GSM representation of a discontinuity between two differenttransversely periodic waveguides. Moreover, the reader can observe that the integrals tobe solved, (4.25), are identical to the ones of a discontinuity between two generic closedwaveguides, see for instance [Conciauro et al., 1999, p. 79].

4.3 Analysis of Transversely Periodic Structures

Now that the GSM representation of a discontinuity has been obtained, let us considerhow to analyze more complex transversely periodic structures. In order to introduce theproposed approach, let us consider the wire-medium slab shown in Fig. 4.2(a) that, forsimplicity, is made up of square-section wires (some results will be given for this structurein Section 4.4.1). To use the proposed analysis method, the structure has to be appropri-ately split into different periodic waveguides. Figure 4.2(b) shows the longitudinal sectionof a unit cell of the slab. It is evident that the structure can be split into three waveguides

Waveguides A, C

(c)

Waveguide B

(d)

ε , µ0 0 ε , µB B

ΓB

z

ε , µB B

ε , µ0 0

dz

SII

SIε , µ0 0

(b)(a)

x

y

Figure 4.2: Segmentation of a square-wire medium slab. (a) Three dimensional view of a bi-periodic wire-medium slab. (b) Longitudinal section of the unit cell and generalized equivalentcircuit. (c) Cross section of a periodic rectangular waveguide. (d) Cross section of the periodicWire waveguide (also shown in Fig. 4.4(b)).

101


of two different types. On the one hand, Fig. 4.2(c) shows the cross section of a peri-odic rectangular waveguide, which constitutes waveguides A and C on the analysis ofthe wire-medium slab. On the other hand, Fig. 4.2(d) shows a periodic Wire waveguide(waveguide B). Once the structure has been split into periodic waveguides, the disconti-nuities between the different waveguides are characterized, using MM, by means of theGSM, [Patzelt and Arndt, 1982, Chu and Itoh, 1986]. A wire-medium slab has two dis-continuities (characterized by the GSMs SI and SII) and, since they are identical to eachother, only one of them has to be computed. The reflection and transmission coefficientsof the wire-medium slab are then obtained from the cascade connection of SI and SII

through the matrix ΓB of the propagation factors of the waveguide B modes, as repre-sented by the generalized equivalent circuit shown on the right-hand side of Fig. 4.2(b).

When dealing with periodic layered surfaces, the condition |χx| = |χy| = 1 is ful-filled when a homogeneous plane wave impinges on the surface at an arbitrary anglesince, as stated in Section 4.3.1, the modes of the rectangular periodic waveguide canbe regarded as homogeneous plane waves in such a case. Therefore, the proposed ap-proach is not usable for non-homogeneous plane-wave excitation since, in such a case,at least one periodicity constant would not have unitary modulus. In addition, it shallbe highlighted that the presence of losses in the structure under analysis does not violatethe |χx| = |χy| = 1 condition anyhow. Dielectric losses can be easily dealt with by usingcomplex permittivity values, and there are MM formulations that account for the losseson the transverse metallic walls [Wade and MacPhie, 1990].

4.3.1 The Periodic Rectangular Waveguide

Consider the particular case of a transversely periodic waveguide in which the lc contour isremoved. A rectangular contour under entirely periodic boundary conditions is obtained,figure 4.2(c). This particular case of periodic waveguide is the simplest possible crosssection, and will be herein referred to as periodic rectangular waveguide. The modes ofthis periodic waveguide are analytical, thus making such waveguide a perfect example toillustrate some of the properties described in Chapter 2. These properties are much moredifficult to clarify in more general periodic waveguides, for which it is necessary to resortto an explicit formulation of the modes, that would obscure the discussion.

Solving the wave equation (2.2) by means of the separation of variables method leads

102


to the longitudinal fields of the TE and TM modes:

hz|mx,my

ez|mx,my

= e−jkx,mxxe−jky,myy (4.28)

where mx, my = −∞ . . .∞ and the transverse propagation constants are:

kx,mx = −ξx −2mxπ

dx; ky,my = −ξy −

2myπ

dy

ξx =Arg (χx)− j ln |χx|

dx; ξy =

Arg (χy)− j ln |χy|dy

, (4.29)

and being dx and dy the sizes of the unit cell in the x and y directions respectively. Self-evidently, the TE and TM modes are degenerate. As aforementioned in section 2.6.2, atransversely periodic waveguide without a conductor does not have any TEM modes.However, if χx = χy = 1 the mx = my = 0 modes of each TE and TM sets have to bereplaced with two TEM modes whose fields are:

~Et =x

~Ht =y

η

;

~Et =y

~Ht =− x

η

(4.30)

The reader should note that the TEM field distribution in (4.30) can not be derived from

a periodic potential but from a linear one. Thus, the importance of the constant in theboundary conditions, (2.62), for the scalar potential is once more observed. The fieldsdescribed by (4.28)-(4.29) are direct TE and TM solutions of (2.2) under the periodicboundary conditions and hold for any complex value of the periodicity constants, χx andχy. The reader may have already identified these modes with the well-known Floquet

harmonics. Each mode of this waveguide can be regarded as a plane wave propagating

in the complex direction given by:

~kmx,my = kx,mxx+ ky,my y + kz,mx,my z (4.31)

where kz,mx,my is obtained from (2.3) taking into account that k2c,mx,my= k2x,mx

+ k2y,my.

Let us consider the eigenvalues k2c,mx,myof these plane waves or periodical waveguide

modes. Self-evidently, they are, in general, complex and hence producing a set of non-

homogeneous plane waves. It is easily seen that these non-homogeneous plane waves arenot orthogonal to each other in the sense of (2.14). Furthermore, (2.15) is not a valid

103


variational expression and the completeness proof, as given in chapter 2, does not hold.

However, if |χx| = |χy| = 1, then the transverse wave-numbers (4.29) are real and theeigenfunctions become homogeneous plane waves. These homogeneous plane waves areorthogonal to their complex conjugates and form a complete set of eigenfunctions. There-fore, the cross section can be considered a closed periodic waveguide. As aforementioned,section 2.8, the complex power flow of each mode on the left- and right-hand-side and onthe lower- and upper-side contours are equal to each other. Thus, the complex power flowson the cross section of the waveguide in the z direction. However, the Poynting vector isnot in the z direction, but in the direction determined by ~kmx,my . Therefore, although thepower is transmitted in the ~kmx,my direction, the net complex power flow of any mode inthe transverse waveguide cross-section is z directed.


In general, the well-known GSM cascading procedure [Patzelt and Arndt, 1982] has beenused in this Ph. D. dissertation. In addition, a modification to that procedure is herebyproposed for the analysis of a particular type of structure. Consider the particular case oftwo displaced waveguides, linked by a Periodic Rectangular one, as detailed in figure 4.3.Figure 4.3(a) and (b) show the top and lateral views of the aforementioned situation. Letus chose the red dash-dotted line as the unit-cell of the structure. Then the cross-sectionof the (g − 1) waveguide is shown in figure 4.3(c) and the cross-section of the (g + 1)

waveguide is shown in figure 4.3(d). As a result the (g + 1) waveguide does not fit thegeneric transversely periodic waveguide as shown in figure 2.1(c). In order to analyzesuch situation some modifications should be considered when obtaining the GSM of thediscontinuity between waveguides (g) and (g + 1). To avoid this difficulty, and by virtueof the periodicity, the cascading procedure can include an x- and a y-direction displace-ment very easily, thus switching from the red dash-dotted unit-cell to the green dashedunit-cell, where the (g + 1) waveguide (figure 4.3(e)) fits the generic cross-section infigure 2.1(c). In the standard cascading procedure a propagation matrix Γ is used to con-sider the longitudinal propagation of the terms between discontinuities in the z-direction.The proposed modification also takes into account the x- and y-direction displacement bymodifying the propagation matrix, so that it considers the wave propagation in the x- andy-directions as well. To sum up, the discontinuity between the (g − 1) and (g) waveg-uides is characterized in the red dash-dotted unit-cell and the discontinuity between the(g) and (g + 1) waveguides is characterized in the green dashed line. Then, the displace-ment between both unit-cells is accounted for in the propagation matrix of the Periodic

104


Rectangular waveguide, (g). This procedure is particularly useful when both waveguides,(g − 1) and (g + 1) are identical.

Let S(L)ij and S(R)

ij be the sub-matrices of the GSMs that characterize the left and rightdiscontinuities in figure 4.3(b) respectively, C(g)

1,2 and D(g)1,2 the vectors containing the com-

plex amplitudes of each term in the g-th waveguide (on reference planes 1 and 2), and Γu

and Γl the propagation matrices in the g-th waveguide, defined as

Γu = ΓzΓuxy , Γl = ΓzΓ

lxy , (4.32)

where Γz, Γuxy and Γlxy are the following diagonal matrices:

Γz = diag[e−jk

(r)z,kl

(r)]

Γuxy = diag[e−jk

(g)x,k(px−Dx) · e−jk

(g)y,k(py−Dy)

]Γlxy = diag

[e−jk

(g)x,kDx · e−jk

(g)y,kDy

] (4.33)

The relations between the complex amplitude vectors at the two reference planes of the

(c)

l (g)

xy

Dx

DyΓxy

D(g)1

C(g)1C(g-1)

D(g-1)

C(g+1)

D(g+1)D(g)2

C(g)2

Γxy

(a)

(d) (e)

(b)

ε , µ(g)(g)

ε , µ(g+1)(g+1)

ε , µ(g-1)(g-1)

Figure 4.3: Schematic of the GSM cascading procedure with discontinuities displaced in the xand y directions. (a) Top view of the two discontinuities to be cascaded. (b) Lateral view of thetwo discontinuities and the intermediate region. (c) Cross-section of the (g − 1) waveguide. (d)Cross-section of the (g + 1) waveguide as seen when using the red dash-dotted unit cell. (e)Cross-section of the (g + 1) waveguide as seen when using the green dashed unit cell.

105


g-th waveguide are:

C(g)1 = ΓlD

(g)2 , C

(g)2 = χ−1x χ−1y ΓuD

(g)1 (4.34)

Then, the GSM that relates vectors C(g−1)2 and D(g−1)

2 with C(g+1)1 and D(g+1)

1 is given bythe following expressions:

SC11 = S(L)11 + S

(L)12 ΓlS

(R)11 ΓuHS

(L)21

SC12 = S(L)12 Γl

[S(R)11 ΓuHS

(L)22 Γl + U

]S(R)21

SC21 = S(R)21 ΓuHS

(L)21

SC22 = S(R)21 ΓuHS

(L)22 ΓlS

(R)12 + S

(R)22

(4.35)

whereH =

(χxχyU − S(L)

22 ΓlS(R)11 Γu

)−1(4.36)

and U is the unity matrix of appropriate dimensions. As is self-evident, the Γxy matricesare the ones that introduce the displacement along the x and y axes. Taking advantageof both the plane-wave expansion in the g-th region and the periodicity, the Γxy matriceshave been written in terms of negative exponentials, thus ensuring numerical robustness[Patzelt and Arndt, 1982] while requiring the usage of two different propagation matricesΓu and Γl.

It is important to note that (4.35) and (4.36) simplify into the usual expressions whenDx = Dy = 0. Obviously, in a periodic environment, (4.35) and (4.36) also hold forun-displaced discontinuities, using Dx = mpx and Dy = npy with m,n ∈ Z.

4.4 Numerical and Experimental Results

This section contains six examples in which the proposed analysis method shows its ver-satility, accuracy and efficiency. The first three structures are periodic surfaces that fitthe generalized equivalent circuit shown in figure 4.2(b). The fourth example is a dis-placed pair of metallic cut wires that will show the usefulness of the displaced cascadingprocedure described in section 4.3.2. The fifth example is a waveguide simulator of a pe-riodic surface that has been built and measured. Finally, a parallel-plate waveguide witha bi-periodically textured surface is analyzed as a periodic waveguide resonator.

106

4.4. NUMERICAL AND EXPERIMENTAL RESULTS

4.4.1 Wire-Medium Slab

The analysis of wire media has attracted a lot of attention because of the important rolethat it plays in the Sievenpiper’s high impedance surface [Sievenpiper et al., 1999]. Ho-mogenization techniques have been developed to study the behavior of wire media [Sil-veirinha et al., 2008] and multilayered mushroom-type structures [Kaipa et al., 2011].The MoM is also an efficient alternative if the wire radius is small compared to the unitcell dimensions. The analysis method proposed in this chapter can analyze thin and thickwires and deals with arbitrarily long wires (arbitrarily thick wire-medium slabs) with noincreased effort.

The wire-medium slab has already been introduced in section 5.2 and figures 4.2 and4.4(b). To compare the proposed approach with the results of the MoM in [Silveirinha,2006] a square unit cell has been considered with dx = dy = 21 mm, lx = ly =

√2dx/100,

dz = 42 mm and εBr = 2.2 (see Figs. 4.4 and 4.2). Note that square wires of√

2dx/100

side have been used instead of circular wires of dx/100 radius. Fig. 4.5(a) shows themodulus of the reflection coefficient for the first TM mode, i.e., the amplitude of the TMpolarized wave reflected when a similar TM wave of unit amplitude impinges on the wireslab, for three different incidence angles, θinc = 15, 45 and 75, at φinc = 0. Theseincidence angles are related to the transverse periodicity constants as follows:

ξx =√ω2µ0ε0 sin (θinc) cos (φinc) (4.37)

ξy =√ω2µ0ε0 sin (θinc) sin (φinc) . (4.38)

(a) (b) (c) (d)ε, µ ε, µ ε, µ

ε, µ

lx

lyly

lxw

lx

lywy

wx

lx

ly

Figure 4.4: Periodic waveguides used in this chapter. The upper row shows the common way inwhich this structures are analyzed. The lower row shows its unit cells as chosen in this dissertation.(a) Closed rectangular waveguide. (b) Periodic Wire waveguide. (c) Periodic Cross waveguide.(d) Periodic H-shaped waveguide.

107


Excellent agreement between both analysis methods has been obtained for the three dif-ferent incidence angles. Thus, the equivalence between a circular and square wires isobserved for thin wires.

The considered example presents excellent convergence properties. The results shownin Fig. 4.5(a) were computed using 20 modes in every periodic waveguide. The compu-tation of the complete scattering matrix of the wire slab required 120 ms of CPU perfrequency point in a 2.8 GHz processor. This period includes the time spent by the gener-alized TRT to characterize the periodic Wire waveguide at each frequency point.

Considering the very fast convergence shown by this example, Fig. 4.5(b) comparesthe MM-TRT results, computed using only the first two and three modes, with the ho-mogenization theory approximation of [Silveirinha, 2006, Fig. 2]. The homogenizationresults, presented by the red dashed line, are derived in [Silveirinha, 2006] by matchingat the discontinuities the impinging wave with two homogenized modes, the TEM andthe first TM mode, of the wire medium. In order to obtain a solvable equation system, an

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

Normalized wavenumber, k0dx

Am

plit

ude

of ρ

θ =15ºinc

θ =45ºinc

θ =75ºinc

MM-TRTMoM

0 2 4 6 8 10Frequency (GHz)

MM: 20 modes

MM: 1 TEM and 1 TM modeMM: 1 TEM and 2 TM modes

Approx. of [Silveirinha, 2006]

(a) (b)

Figure 4.5: (a) Modulus of the reflection coefficient of the first TM mode under three differentincidence angles (θinc = 15, 45 and 75). The continuous lines show the MM-TRT results and thecircles, crosses and triangles show the results taken from [Silveirinha, 2006]. (b) Modulus of thereflection coefficient of the first TM mode under θinc = 45 incidence. The relative permittivityis unity and the dimensions are identical to those of the previous example. The continuous lineshows the MM-TRT results using 20 modes.

108


additional boundary condition has to be introduced. This homogenization theory showsgood agreement with the full-wave solution (continuous line) up to 8 GHz. The reasonwhy the homogenization theory behaves so well is easily understood when its results arecompared to the MM-TRT using only the TEM and the first TM mode of the periodic Wirewaveguide (shown by the green dash-dotted line). The considered structure is very welldescribed by means of only these two modes, especially in the long wavelength regime.Moreover, the blue dashed line shows that by adding just the next TM mode, i.e., usingthe TEM and the first two TM modes in the periodic Wire waveguide, the wire slab canbe accurately described up to almost 10 GHz.

Finally, figure 4.6 shows the transmission and reflection coefficients of a TM planewave impinging on a lossy wire medium slab, ε = 2.2 (1− j0.02), compared to thenumeric HFSS results. The reader maybe has already anticipated that HFSS has greattrouble dealing with the very thin wires used in the previous examples. Therefore, thislossy wire medium example has been computed for thicker wires, i.e. dx = dy = 21 mm,lx = ly = 2 mm and dz = 42 mm. This example has also been computed using 20

modes of the rectangular periodic waveguide and its analysis time is very similar to theone presented for the lossless case.

-20

-15

-10

-5

0

20 lo

g,

|Si

j|(

dB)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5-7-6-5-4-3-2

Frequency (GHz)

Si,

j(r

ad)

∀

Figure 4.6: Transmission (red line) and reflection (blue line) coefficients for a TM wave impingingin a lossy Wire medium (θinc = 15 and φinc = 0). The continuous lines show the results of theproposed approach and the circles and triangles show the numerical HFSS results.

109


4.4.2 Fishnet

As a second example, the so-called fishnet structure [Jelinek et al., 2010, Ding et al.,2008] has been considered. This structure has attracted a lot of interest as a way ofachieving negative refraction at high frequencies because of its ease of manufacture. In[Jelinek et al., 2010] a MM approach was already adopted to analyze the wave propagationthrough periodically stacked fishnets along arbitrary directions and [Marques et al., 2009]proposed an approximated analytical method for the normal wave propagation of the samestructure, even for metallic lossy structures [Delgado et al., 2010].

The fishnet surface fits the generalized equivalent circuit shown in Fig. 4.2(b), byreplacing waveguide B with the closed rectangular waveguide shown in Fig. 4.4(a). TheMM-TRT reflection coefficient for the fundamental TE mode, i.e., a TE polarized waveimpinging at θinc, is compared to the FEM results (using HFSS) in Fig. 4.7. The analyzedfishnet has a dx = dy = 21 mm square unit cell, and a closed rectangular waveguide withlx = 10 mm, ly = 5 mm and dz = 100 µm. Once more, excellent agreement between bothanalysis methods is found for the three incidence angles. The results shown in Fig. 4.7were computed using 200 periodic rectangular waveguide modes. This large numberof modes come from the fact that the MM technique is well suited for finite thickness

10 12 14 16

-20

-15

-10

-5

0

2

3

4

Frequency (GHz)

|S

|

(dB

)11

2S

(ra

d)

11

∀

θ = 0ºinc

θ = 15ºinc

θ = 30ºinc

MM-TRTHFSS

Figure 4.7: Reflection coefficients for the first TE mode under three different incidence angles(θinc = 0, 15 and 30). The continuous lines show the results of the proposed approach and thecircles, triangles and crosses show the numerical HFSS results.

110


problems (i.e., finite waveguide lengths dz), and the use of reduced dz values slows itsconvergence significantly. Nevertheless, only 3.4 ms of CPU is needed per frequencypoint. When comparing this analysis time with that of the previous example, it should benoted that the closed rectangular waveguide modes are analytical. The analysis time isthen greatly reduced, since there is no need for the generalized TRT eigenvalue search.

4.4.3 Cross-Shaped Metal Patches

The third example is also a periodic surface, this time filled with the cross-shaped metalpatches shown in Fig. 4.4(c). The results of a hybrid MoM-BIRME (Boundary Inte-gral Resonant Mode Expansion) method, presented in [Bozzi et al., 2010, Fig. 5a], havebeen reproduced Fig. 4.8, along with the results of the MM-TRT, for a symmetric cross(lx = ly = 4.5 mm, wx = wy = 1 mm) in a square unit cell (dx = dy = 10 mm), for twodifferent metal thicknesses (dz = 50 and 100 µm), and under normal wave incidence.

A slight shift of the resonant frequency, 0.145 GHz, representing a 0.5% relative er-ror, can be observed in Fig. 4.8 when comparing the MM-TRT with the MoM-BIRMEmethods. However, very good agreement is found between the MM-TRT and the HFSS

27 27.2 27.4 27.6 27.8 28-30

-25

-20

-15

-10

-5

0

Frequency (GHz)

tran

smis

sion

coe

ffic

ien

t (d

B)

MM-TRTMoM-BIRME [Bozzi et al., 2010]HFSS

Figure 4.8: Transmission coefficient of a periodic surface made up of cross-shaped metal patchesfor two different metal thicknesses. The continuous lines show the MM-TRT results, the dashedlines show the MoM-BIRME results and the circles show the numerical HFSS results. Red lines:dz = 100 µm. Blue lines: dz = 50 µm.

111


numerical solution. The zero metallization thickness curve shown in [Bozzi et al., 2010]is not reproduced here since the proposed approach is not well suited for that particularunrealistic situation. The MM-TRT results were computed using 200 modes in the peri-odic rectangular waveguide. It took 9 s to compute the periodic Wire waveguide modesby means of the generalized TRT and 72 ms, per frequency point, to compute the resultsshown in Fig. 4.8. These analysis times are comparable to those reported in [Bozzi et al.,2010] for the same example.

4.4.4 Rectangular Displaced Patches

The purpose of this fourth example is to show the usefulness and accuracy of the displacedcascading procedure described in section 4.3.2. To that end, the structure proposed in[Burokur et al., 2009] for the obtention a negative refraction index has been used. Thetop and lateral views of the considered structure are shown in figure 4.9 as an inset. Thestructure is formed by a pair of rectangular metallic patches with lateral displacement,Dx, separated by a dielectric spacer. The dimensions used in this example are thosestated in [Burokur et al., 2009], i.e. lx = 9.5mm, ly = 0.3mm, tm = 35µm, the lossydielectric spacer has a relative permittivity of εr = 3.9 (1− j0.02) and its thickness is

Frequency (GHz)8 9 10 11 12 13

-25

-20

-15

-10

-5

0

20 lo

g(d

B)

|S

|i,j

MM-TRT: D =2MM-TRT: D =0

HFSSxx

ε, µ

lylx

Dx

ε, µ td

tm

Figure 4.9: Transmission and Reflection coefficients of a displaced pair of rectangular metallicpatches under normal wave incidence with its electric field parallel to the wires. The top andlateral views of the considered structure are shown in the figure as an inset. The continuous linesshow the reflection coefficient and the dashed lines show the transmission coefficients.

112


td = 1.2mm, finally the unit-cell has the following dimensions px = 19mm and px =

9.5mm. The transmission and reflection parameters of a normal incidence plane wavewith its electric field parallel to the wires are shown in figure 4.9 for two different cases,Dx = 0 and Dx = 2mm. The scattering parameters obtained by means of the MM-TRThybrid are compared to the home-made HFSS simulation results. Very good agreementcan be observed between both simulations. It shall be commented here that the displacedpatch results significantly differ from those reported in [Burokur et al., 2009]. The HFSShome-made simulation used a 18 adaptive mesh iterations, thus a erratum in the structuredimensions reported in [Burokur et al., 2009] is the most likely cause of the differences.

Regarding the computational effort, the hybrid MM-TRT approach has used 200 waveg-uide modes in the rectangular periodic waveguides for the analysis. The obtention of theperiodic wire waveguide has taken 9s and the obtention of the scattering parameters used860ms per frequency point in both cases. When comparing this value to the one reportedfor the previous example the reader should note that this example requires the computa-tion of two different scattering matrices to characterize two discontinuities and the GSM

cascading procedure has to be used twice. It must be highlighted here that the usage ofthe displaced discontinuity GSM cascading procedure described is section 4.3.2 saves usfrom half the computational burden. Please note that by using the standard cascadingprocedure four different discontinuities would need to be characterized and the GSM cas-cading procedure would have to be used four times. Therefore, the displaced examplewould take twice the reported time.

4.4.5 Periodic H-shaped Waveguide Simulator

The last of this set of five periodic-surface examples is a waveguide simulator [Hannanand Balfour, 1965] that has been built and measured to test the proposed approach. AnH-shaped thick metal patch on top of a ROHACELL dielectric (εr = 1.045) was cho-sen for this example (see figure 4.4(d)) because of its resonant response, which providesa good test for the convergence of the method. The longitudinal section and the gener-alized equivalent circuit of the structure is shown in the right-hand side of figure 4.10.The periodic H-shaped waveguide is waveguide B and the dielectric slab is waveguideC. As opposed to previous examples, the SI and SII matrices are not identical. There-fore, the three GSMs have to be computed in order to analyze the structure. However,the last one, SIII , is a dielectric discontinuity between periodic rectangular waveguidesand its solution is analytical and uncomplicated. Once the three discontinuities are char-acterized, the GSM of the complete structure is computed by cascading SI , SII and SIII .

113


Obviously, more intricate structures with multiple dielectric layers and metallizations canalso be analyzed by just simply cascading either dielectric regions (as already done in thespectral domain approach [Cwik and Mittra, 1987]), or metallization regions, understoodas periodic waveguides (and then from the point of view of [Patzelt and Arndt, 1982]).

A photograph of the metal H-shaped patch is shown in figure 4.10 as an inset. AWR-90 waveguide was used for the measurements. Note that the structure is rotated 90

to fit figure 4.4(d) (dx = 10.16 mm and dy = 22.86 mm). Since the ultimate purposeof this measurement is to check the precision of the proposed approach, the H-shapedmetal patch was built and the dimensions simulated by MM-TRT were those measured onthe manufactured patch (lx = 7.79 mm, ly = 19.81 mm, wx = 2.37 mm, wy = 7.47 mm,dBz = 3.21 mm and dCz = 3.1 mm). The measurements (continuous lines) are comparedwith simulation results (dashed lines) in figure 4.10. The computed results have beenobtained by combining the scattering parameters of the first and second TE modes, andusing arg (χx) = π and arg (χy) = 0 as periodic boundary constants (invariant with fre-quency). Excellent agreement is found between theory and measurements.

A convergence study has been carried out. This study was not carried out in the previ-

-30

-25

-20

-15

-10

-5

0

7 8 9 10 11 12 13

-2

0

2

4

S11S21S22meas.

Frequency (GHz)

|S

|

(dB

) ij

2S

(ra

d)

ij

∀

ΓB

z

ε µB

B, dz

B

SII

SIε , µ0 0

ε , µC C

ε , µ0 0

ΓC

SIII

dzC

Figure 4.10: Scattering parameters of the waveguide simulator of a periodic surface made upof H-shaped metal patches on top of a ROHACELL substrate. Continuous lines: measurements.Dashed lines: computed results. A photograph of the built breadboard is included as an inset.The right-hand side of the figure shows the longitudinal section of a unit cell of the measuredwaveguide simulator and its generalized equivalent circuit.

114


ous examples because it required an abnormally low number of modes in the first exampleand an abnormally high number of modes in the second and third examples. The relativeerror on the transmission and reflection resonant frequencies is shown in figure 4.11, asa function of the number of modes used in the periodic rectangular waveguide. This rel-ative error considers the values obtained with 150 modes as reference. An error of lessthan 0.5% is obtained by using more than 61 modes. For this reason the results shown inFig. 4.10 were computed using 61 modes. It took 1.71 s to compute the modes of the fourwaveguides and 13.3 ms per frequency point to compute the scattering parameters of thestructure.

4.4.6 Parallel-plate with a bi-periodic textured surface

To illustrate the variety of situations in which the MM-TRT with periodic waveguides canbe used, a last example has been borrowed from [Kildal et al., 2011], and consists of aparallel-plate waveguide with one of the plates made up of square pins. The unit cell ofsuch a bi-periodic structure is shown as an inset in Fig. 4.12, in addition to its generalizedequivalent circuit. The method of analysis proposed in this chapter considers the struc-ture as a periodic waveguide resonator. Therefore, the |χx| = |χy| = 1 condition impliesun-attenuated transverse propagation in this case. By analogy with that previously com-mented, this condition is not incompatible with the presence of losses in the structure. If,for instance, dielectric losses were included in the analysis, the resonant frequencies ofthe structure would no longer be real but complex. The physical interpretation of suchsolutions is beyond the scope of this dissertation. Nevertheless, the proposed approach is

21 41 61 81 101 1210

0.5

1

1.5

2

2.5

3

Number of Periodic Rectangular Waveguide Modes

Rel

ativ

e E

rror

(%)

S resonant freq.11S resonant freq.21

Figure 4.11: Relative error of the transmission and reflection resonant frequencies of the waveg-uide simulator of Fig. 4.10, with respect to the values computed with 150 modes, as a function ofthe number of modes used in the periodic rectangular waveguide.

115


perfectly capable of providing those complex solutions. For its analysis the structure issplit into two different waveguides, a periodic rectangular waveguide (waveguide A) anda periodic Wire waveguide (waveguide B) cascaded to each other and shortcircuited atboth ends. Given the GSM that characterizes the discontinuity between both waveguides,S, (similar to the GSMs used in the example of Section 4.4.1) the resonant frequenciesof the structure are obtained from the resonance condition of the generalized equivalentcircuit, i.e.,

det

(U + S11

(ΓA)2

S12

(ΓB)2

S21

(ΓA)2

U + S22

(ΓB)2)

= 0 (4.39)

where U is the unity matrix and ΓA and ΓB are the propagation matrices of waveguidesA and B, respectively.

Figure 4.12 expands [Kildal et al., 2011, Fig. 4] to the complete contour of the irre-ducible Brillouin zone. Each point in the figure is a resonant frequency of the structureunder particular periodic boundary conditions, or the phase constant of a propagatingparallel-plate waveguide mode at that frequency. The complete Brillouin diagram showsthat the structure presents a stopband from 10.9 to 22.3 GHz, where the lower limit is

Γ X M Γ0

5

10

15

20

25

30

35

Freq

uen

cy (G

Hz)

ΓB

S

ΓA

Γ

X

M

Figure 4.12: Transverse dispersion diagram in the contour of the irreducible Brillouin zone ofa periodic waveguide resonator. Continuous lines show the proposed approach results and blackdots are the results taken from [Kildal et al., 2011]. A three-dimensional representation of theresonator and its generalized equivalent circuit are shown as an inset.

116

4.5. CONCLUSION

higher than the one predicted by only looking at the Γ−X segment (9.4 GHz). Very goodagreement is observed between the proposed approach and the numerical solution of [Kil-dal et al., 2011] (using CST Microwave Studio) in the fundamental parallel-plate mode.However, some discrepancies are found in the first two higher-order modes, namely, asmall frequency displacement and the presence/absence of some coupling between thesetwo modes.

The computation of the four resonant frequencies at each point of the irreducible Bril-louin contour in Fig. 4.12 used an average of 1.6 s, considering 81 modes in the periodicrectangular waveguide.

4.5 Conclusion

This chapter has, on the one hand, studied the possibility of using a closed waveguidetheory (the Mode Matching technique) in the characterization of discontinuities betweendifferent transversely periodic waveguides and concluded that this approach works. Onthe other hand, a hybrid MM-TRT method has been proposed as an efficient, accurate andversatile method for the analysis of transversely periodic structures. A variety of exampleshave been presented. Those focused on periodic layered surfaces have shown the abilityof the proposed method to consider arbitrary dielectric and metallization thicknesses, andto obtain the reflected fields when plane waves impinge at arbitrary incidence angles. Afinal example has been provided to show that the proposed method is not constrained tothe analysis of layered surfaces, but can deal with other problems that include bi-periodicstructures such as a parallel-plate waveguide characterization.

In addition, a modification of the well-known procedure to cascade the GSMs thatmodel the longitudinal discontinuities has also been proposed. This new cascading proce-dure has proved to be useful when the GSMs to be cascaded are displaced in the transversedirections, opening up the opportunity to analyze some different geometries, with no in-creased effort.

All the examples show up the efficiency of the method and the accuracy of its results.Consequently, the fact that periodic structures can be successfully analyzed by means ofwaveguide concepts can be stated.

Finally, it shall be highlighted that the the practical implementation of the proposedapproach detailed in this Chapter, in combination with the theoretical results obtained inChapter 2, have given rise to [Varela and Esteban, 2012].

117

5Analysis of Laterally Open Periodic

Waveguides

5.1 Introduction

The extension of the generalized Transverse Resonance Technique described in chapter 3is used in this chapter for the characterization of laterally open periodic waveguides. Thereason as to why this constitutes a separate chapter, is that there are some important dif-ferences between the objectives and procedures of the characterization of transverselyperiodic waveguides and the analysis of laterally open periodic waveguides. First andforemost, this chapter will release the χy periodicity constant from the |χy| = 1 condi-tion. The implications of this releasement are detailed here so as the new generalizedequivalent circuit and its associated characteristic equation. The contents of this chapterhave given rise to [Varela and Esteban, 2011a].

In this chapter four particular structures have been selected to demonstrate the capa-bilities of the proposed approach. These structures are similar to the post wall waveg-uide [Hirokawa and Ando, 1998], to the waveguiding defect on electromagnetic bandgapstructures [Simpson et al., 2004], and to the Substrate Integrated Waveguide (SIW) [Des-landes and Wu, 2001]. For convenience, they will herein be referred to as post-wallwaveguides. The analyzed structures are made up of a homogeneous parallel-plate waveg-uide of height s, with its plates interconnected by one or more rows of vertical posts,periodically placed along the y axis (Fig. 5.1 shows a schematic of the first structure an-alyzed in section 5.3). Therefore, post-wall structures are periodic along the y axis, andlaterally open (infinite in ±x directions). The material filling the structure can be lossy(with complex ε or µ) although only examples with lossless dielectric are presented (realµ and ε). The post-wall waveguide solutions, as homogeneous open periodic structures

119

CHAPTER 5. ANALYSIS OF LATERALLY OPEN PERIODIC WAVEGUIDES

(i.e., its Bloch modes), will be, in principle, forward waves, and therefore improper leakymodes, since they will always have at least one fast space harmonic.

The analysis of this class of structure has attracted a lot of attention in the recent pastand has been carried out by means of CPU time-consuming numerical methods (such asfinite differences in time or frequency domains [Xu et al., 2003, Kokkinos et al., 2006] orthe method of lines [Yan et al., 2005]) or commercial packages (such as Ansoft HFSS orCST Microwave Studio). When efficient semi-analytical methods are used, it is custom-ary to resort to fictitious walls that close the structure [Deslandes and Wu, 2006, Cassiviet al., 2002]. On the contrary, the method of analysis proposed in this chapter is highlyefficient and takes into account the open and periodic character of the analyzed structureswithout any approximation. Although the proposed approach is restricted to the anal-ysis of structures that can be segmented in the Cartesian coordinate system, a staircaseapproximation has been successfully applied to the characterization of circular contours,thus circumventing the basic limitation of the proposed approach.

An overview of the proposed analysis method for laterally open periodic waveguidesis detailed in Section 5.2. This section also contains a description of the differences withrespect to chapter 3 regarding the field expansion and the derivation of the new charac-teristic equation. Section 5.3 shows the results for four post-wall waveguides, for whicha breadboard has been built and measured. The dispersion diagram for the first modesand the propagation constant for the fundamental mode are compared in this Section bothwith the measurements and the already published numerical approaches. The conclusions

s

z y

x

Figure 5.1: Three-dimensional schematic of a single-row post-wall waveguide.

120

5.2. METHOD OF ANALYSIS

are summarized in Section 5.4.

5.2 Method of Analysis

Fig. 5.2 shows the top-view of a generic multi-row structure and its generalized equivalentcircuit. The horizontal dashed lines delimit the chosen period in which the analysis isgoing to be carried out. The vertical dash-dotted line shows the structure symmetry planethat will be used to simplify the generalized equivalent circuit discussed in Section 5.2.2.

Back in chapter 3, the obtention of transversely periodic waveguide TEz and TM z

modes was carried out by the resolution of simpler scalar eigenfunction problems (sec-tion 3.2). The solutions of these simpler eigenfunction problems were named magneticand electric potentials because of their similitude with the longitudinal magnetic and elec-tric fields of the waveguide modes that were obtained form them. The reader should recallthat the difference between a potential and the actual longitudinal field is a normalizationconstant. In other words, the solutions of table 3.1 can be understood as potentials andthen (2.69) and (2.85) can be used to obtain the transverse electric field or, the solutionsof table 3.1 can be understood as the actual longitudinal magnetic and electric field andthen, the expressions in table 2.1 can be used to derive an expression for the transverseelectric field. The former, is the approach used in chapter 3. The latter, is the one that ismore appropriate to the analysis of laterally open periodic waveguides.

The first, and one of the most important differences between this chapter and chapter 3is that, while in chapter 3 we were looking for solutions with fixed transverse and generic

GSM

B(I)2B(I)

1 B(N)1

Γ (I)

l (I)

PECPMC

Γ (I)

xy

GSM

A(N)1

ε, µ

Figure 5.2: Top view of a laterally open periodic multiple-row waveguiding structure and itsgeneralized transverse equivalent circuit.

121


longitudinal (z-direction) behavior, here we will fix the longitudinal behavior and look forthe values of the periodicity constant χy that are solutions of the eigenfunction problem.Therefore, we are no longer looking for transversely periodic modes, i.e. modes witha fixed transverse periodic behavior fulfilling the |χx| = |χy| = 1 condition. We arenow looking to Bloch modes, i.e. modes that are periodic (as defined in table 2.3) ontheir direction of propagation. In this chapter, only the TM z modes will be considered,since there are no relevant results for TEz Bloch modes. Please note that the coordinatesystem presented in Fig. 5.1 does not coincide with the usual coordinate system used forthe analysis of the rectangular waveguide or the SIW. The solutions that will be soughtare the TM z

m,n set, where, as analogous to the rectangular waveguide, m and n denote thenumber of variations in the x and z directions, respectively. Therefore, the Bloch modesimilar to the fundamental TE10 mode of the rectangular waveguide will henceforth bedenoted as TM z

10 mode.

In order to introduce the proposed approach, consider the structure shown in Fig. 5.3(a),which is the top view Fig. 5.1. In this figure only the right-hand half of a periodic cell isdisplayed. In order to analyze this structure by means of the proposed approach severalsteps have to be accomplished. In the first place, the structure has to be split into regionsin which the longitudinal electric field expansion is known. This particular structure hasbeen split into three regions of two different types as shown in Fig. 5.3(a). The modifi-cations to be made to the electric potential expansions of section 3.2.1 will be detailed insection 5.2.1. Once the structure has been appropriately split, the discontinuities betweenregions may be characterized. For this particular example, two discontinuities have to beconsidered. Since both discontinuities are identical to each other, only one of them hasto be characterized. In order to do so, a field matching procedure is used. As analogousto the Mode Matching technique, the discontinuities will be characterized by a GSM .This process is identical to the one described in section 3.2.2 and will not be repeatedhere. Finally, using the cascading procedure described in section 3.2.3, a single GSM

(a) (b)

Region I Region II Region III

yx x

y

Region I Region XI

Regions II-X

Figure 5.3: Structure segmentation. (a) Top view of the right-hand half of a single-row square-post periodic cell. (b) Top view of the right-hand half of a five-step staircase approximation of asingle-row circular-post periodic cell.

122

5.2. METHOD OF ANALYSIS

that characterizes both discontinuities and region II is obtained. At this point the gener-alized equivalent circuit of the structure under analysis, shown in Fig. 5.2, may be built.From the resonances of the generalized equivalent circuit in the z = 0 plane, a general-ized characteristic equation is obtained for a given kz (section 5.2.2). The solutions to thisequation will be the y-direction propagation constants of the structure.

Most practical applications use circular instead of square vertical posts. Althoughthe proposed analysis method is restricted to Cartesian coordinate system segmentablestructures, circular geometries can be analyzed by means of a staircase approximation.Consider the structure shown in 5.3(b), which is the last example of this chapter. The fig-ure shows a five-step staircase approximation of a circular post waveguide. In this case,the structure has been split into eleven regions and, taking into account the symmetry ofthe post, five discontinuities of two different types have to be characterized. By repeat-edly using the cascading procedure, a single GSM that represents all discontinuities andintermediate regions may be obtained. This way, the same generalized equivalent circuitof Fig. 5.2 can be used to obtain the propagation constants of the structure.

Consequently, to analyze the proposed examples, some modifications have to be madeto the potential expansions defined back in section 3.2.1 and the generalized characteristicequation for the equivalent circuit of Fig. 5.2 has to be obtained, since both the disconti-nuity characterization and the cascading procedure are identical to the ones described inchapter 3.

5.2.1 Modifications to the Field Expansions

As aforementioned, this section details the slight changes and physical reinterpretationsto the field expansions described in chapter 3. The first important difference between thischapter and chapter 3 is that we will now be looking for the actual z-direction electric fieldand not for a electric potential. This means that the solution must have some variationin the z-direction. Considering the type of structures that will be analyzed, with PECboundary conditions at two z = constant planes, the z-direction electric field can bewritten as (3.2) plus a cosine z variation,

E(r)z =

∑p

Ψ(r)p (y)√Q

(r)p

[B

(r)1,pe

−jk(r)x,px +B(r)2,pe

−jk(r)x,p(l(r)−x)

]cos (kzz) (5.1)

where kz = nπ/s (with n = 0, 1 . . .). Please note that the addition of the z variationdoes not alter the discontinuity characterization procedure described in section 3.2.2 of

123


chapter 3 since it is identical for every region.

The second important difference is related to the obtention the the k(r)x,p by means of(3.4),

k(r)x,m =

√ω2µε− (k

(r)y,m)2 − k2z (5.2)

in a periodic-boundary region. Unlike in the parallel-plate region, the expansion terms inthe periodic-boundary regions are space harmonics. The sign of the square root in (5.2)must now be chosen so that the imaginary part of k(r)x,m is positive for forward, fast spaceharmonics, and negative otherwise [Collin and Zucker, 1969, Ch. 19], [Oliner, 1963].This detail was irrelevant when the |χy| = 1 condition was imposed. However, it hereplays an important role in the physical interpretation of the solutions.

Finally, it shall be highlighted that the rest of the contents in sections 3.2.1 to 3.2.3remain unaltered and can be directly used here.

5.2.2 Characteristic Equation

By means of the discontinuity characterization and cascading procedures described inchapter 3, any structure may be reduced to a convenient generalized equivalent circuit.For the particular examples considered in this chapter the most convenient generalizedequivalent circuit is that shown in Fig. 5.2. Once the appropriate boundary conditionsare imposed, the characteristic equation is obtained from the resonance condition of thistransverse circuit. In the generalized equivalent circuit of Fig. 5.2 the transverse GSMrepresents the result of cascading all the discontinuities of the structure.

Both PEC and perfect magnetic conductor (PMC) have been used as x-directionboundary conditions at the symmetry plane to obtain odd and even Bloch modes, re-spectively. On the right-hand side boundary (Fig. 5.2), A(N)

1 = 0 is enforced. Thus,only outgoing waves in the x-direction are used in the outer region and a laterally openstructure is simulated. This is equivalent to a matched load in the generalized equivalentcircuit.

The generalized characteristic equation becomes

det[U ± S11

(Γ(I))2]

= 0 (5.3)

where Γ(I) is a diagonal matrix whose elements are e−jk(I)x,ml

(I) and the plus and minus signshold for odd and even modes, respectively. For a given frequency and kz value, Muller’s

124


method [Atkinson, 1989] is applied to the search for the solutions of the characteristicequation (5.3) in the ξy complex plane.

The constant kz is zero for the fundamental mode of the proposed structures sinceit presents no variation in the z-direction. For lossless structures, once the propagationconstants of the z-direction invariant modes are known, the remaining modes may beobtained analytically by means of a simple frequency mapping. Equation (5.2) shows thatfor each term and region the values of k(r)x,m (and therefore ξy) coincide, for a TMz

m,n modeat a given frequency fm,n, with the same values for the TMz

m,0 mode at a frequency fm,0,when

fm,n =

√f 2m,0 +

1

µε

( n2s

)2(5.4)

To sum up, for the proposed structures, the knowledge of the z-invariant modes is enoughto obtain the complete dispersion diagram. Obviously, (5.4) only holds in the losslesscase (real µ and ε) and practical SIW structures are filled with lossy dielectrics. In thelossy case it is unavoidable to look for the solutions of (5.3) for the different values ofn. However, (5.4) serves to show that, contrary to that mentioned elsewhere [Xu et al.,2003, Deslandes and Wu, 2006], z-direction variant modes (n 6= 0) can be as stronglyguided as the modes without vertical variation (n = 0).

5.3 Numerical and Experimental Results

In Fig. 5.4 the top view of the periodic cells of the four analyzed structures is presented.The horizontal dashed lines delimit the unit cell and the vertical dash-dotted line showsthe symmetry plane. In all four breadboarded structures l(I) = 18 mm, h(I) = 21 mm andh(II) = 15 mm. For the first three l(II) = 6 mm and d(II) = 3 mm.

These dimensions have been chosen for two reasons. Firstly, breadboarding was lim-ited by the commercially-available square and circular post dimensions. Secondly, ourultimate purpose was to check the accuracy of the analysis method. Therefore, we wantedto reduce the uncertainty of the measurements as much as possible. In order to do that,we have used air instead of a dielectric to fill the waveguide, chosen a sufficiently lowfrequency range, and a separation between posts large enough so that the leakage losseswould be high enough to be measured.

The dispersion diagrams shown throughout this Section present, for each Bloch mode,the propagation constant of the space harmonics whose phase constants are in the rangebetween 0 and π/h(I) rad/m. Accordingly, both progressive and regressive Bloch modes

125


are displayed in each diagram (traveling in the +y and −y directions with attenuationconstants, Im(ξy), positive and negative, respectively, as follows from (3.7)). This repre-sentation has been chosen so that the mode coupling between the fundamental and higher-order modes can be more clearly seen (as in Figs. 5.6 and 5.8).

All the Bloch modes of the analyzed structures have at least one simultaneously for-ward and fast space harmonic. Thus, they are improper leaky modes. In the frequencyranges displayed, there is only one of these improper harmonics, which is always them = 0 one. Regarding the dispersion diagrams of this chapter, in the progressive fre-quency range of each mode the m = 1 proper backward harmonic is shown. In theregressive frequency range the m = 0 improper harmonic is displayed.

The proposed approach shows good convergence properties. The theoretical resultspresented in the following subsections have been computed with 21 terms in the periodic-boundary regions unless otherwise stated.

The propagation constant of the fundamental mode of the proposed structures has beenmeasured using a statistical method similar to which can be found in [Marquez-Segura andCamacho-Penalosa, 1995]. This method is not capable of measuring phase constants nearzero or π/h(I), and therefore the measurements at the corresponding frequency rangesare not shown. In the dispersion diagrams the standard deviation bars are shown onlywhere they are high enough to be appreciable. For each structure a photograph of thebuilt breadboard, when the upper plate has been removed, is also shown as an inset. Thesliding short-circuit used for the measurements can be seen in the photographs.

h(I)h(II)

d

l (II) l (I)

(II)

(a) (b)l (III)

(c)

D(IV)

(d)

φ

Figure 5.4: Unit cell schematic of the four proposed examples. a) Single-row waveguide. b)Multiple-row waveguide. c) Displaced-row waveguide. d) Circular-post single-row waveguide

126


5.3.1 Single-Row Waveguide

The first structure (Fig. 5.4(a)) corresponds to a waveguide made up of two parallel rowsof square metallic posts. The theoretical propagation constants ξy of the first three Blochmodes are presented in Fig. 5.5, where they have been numbered accordingly with itsanalogous modes on the rectangular metallic waveguide. Figure 5.5 also includes themeasured propagation constant for the fundamental mode. The numerical results shownin this figure include the computation of the propagation constants of three modes in twohundred frequency points, and have required 3.2 CPU seconds on a single core of an Intelprocessor at 2.66 GHz, which illustrates the efficiency of the method.

Excellent agreement is obtained between the TMz10 theoretical and measured phase

and attenuation constants. At the 3.8 − 7.5 GHz band the theoretical and measured at-tenuation constant is approximately 0.4 Np/m. The method of analysis does not considerconductor losses. Hence, this attenuation constant is mainly due to the energy leakage ofthe mode, which radiates through the arrays of square posts.

5.3.2 Multiple-Row Waveguide

The second structure considered in this chapter is a four-row waveguide, Fig. 5.4(b),where the distance between post rows is l(III) = 15 mm. The four post rows can beregarded as a rodded or wired medium [Rotman, 1962, Pendry et al., 1996] with a cut-offfrequency of around 7 GHz. Below this frequency, transverse (XY -plane) propagation isforbidden in the region of the posts, thus preventing radiation. The computed propagationconstants of the first six even Bloch modes, compared with the measurements for the fun-damental mode, are presented in Fig. 5.6 (the odd modes and the standard deviation barsof the measurements are not shown for clarity). Once more, the theoretical and measuredpropagation constants of the fundamental mode are in very good agreement.

Additional even and odd modes propagate between 7 and 10 GHz. The phase con-stants of the first three even of these additional modes are shown in dashed lines inFig. 5.6. These modes have not been given a TMz

m,n label, since they are guided inthe inside of the multiple-row structure as shown in Fig. 5.7. In this figure the modes(a)-(c) are ordered by ascending cut-off frequency. It is worth noting that the fluctuationsof the fundamental mode propagation constant in Fig. 5.6, because of the coupling withthe higher-order modes, have been accurately predicted.

The fundamental mode computed attenuation constant is approximately 10−7 Np/m atthe 3.8− 7.5 GHz band, much lower, as expected, than for the single-row guide. The fun-

127


-5

0

5

4

20

40

60

80

100

120

140

Frequency (GHz)

Im(

) (N

p/m

)ξ y

TM theoryz10

TM theoryz30

TM theoryz20

TM meas.z10

6 8 10

2mπ

Re(

) +

(ra

d/m

)ξ y

h(I)

light li

ne

Figure 5.5: Theoretical and measured single-row waveguide propagation constants. Attenuationand phase constants are presented in the upper and lower figures, respectively. The standarddeviation of the measurements is presented with vertical bars.

128


0

4

0

20

40

60

80

100

120

140

Im(

) (N

p/m

)ξ y

Re(

) +

(ra

d/m

)ξ y

2mπ

h(I)

4 6 8 10Frequency (GHz)

TM theoryz10

TM theoryz11

TM theoryz30

TM meas.z10

12

2

light

line

Figure 5.6: Theoretical and measured four-row waveguide propagation constants. Attenuationand phase constants are presented in the upper and lower figures, respectively. In dashed lines thephase constants of additional higher-order modes. Only even modes are presented.

129


damental mode measured attenuation constant is around 0.05 Np/m. The difference be-tween these values is justified by the fact that the conductor losses, not accounted for in thetheory, are much higher than the leakage losses because of the strongly-guiding waveg-uide that the four-post structure constitutes. It is not possible to extend the measurementsbeyond 13.15 GHz because of the appearance of the TMz

11 mode, which is generated bythe asymmetric coaxial-to-post-wall transition. The perturbation of the measurements isexplained by the propagation of this mode. This becomes an indirect experimental proofof the existence, and non-weakly guiding, of a non-invariant mode in the z-direction in apost-wall waveguide.

5.3.3 Displaced-Row Waveguide

A two displaced-row structure has been analyzed and measured. Once more, this examplewill show the usefulness and accuracy of the displaced waveguide cascading proceduredescribed in section 4.3.2. The aforementioned cascading procedure is very easily rewrit-ten for its use in a generalized TRT scheme interchanging waveguides by regions, modesby terms and taking into account that here the direction of propagation is x and not z. Thedistance between rows is l(III) = 4.5 mm, and the second row is displaced a half-period,D(IV ) = 10.5 mm (Fig. 5.4(c)).

Theoretical and measured propagation constants are shown in Fig. 5.8. Excellentagreement between theoretical and measured phase constants is obtained below 10.5 GHz.Over 10.5 GHz the first mode couples with the third, thus producing a complex pair of

(c)

(a)

(b)

Re (E )z

Figure 5.7: Contour lines of the real part of the z-component electric field for the first threehigher-order modes of Fig. 5.6. The darker contours hold for negative Re(Ez) values and thelighter ones hold for positive values ((a) f = 8 GHz, (b) and (c) f = 8.5 GHz).

130


non-propagating modes. Therefore, no reliable measurements are obtained in this band.The TMz

10 theoretical attenuation constant is very similar to the measured one between4 and 7 GHz. This attenuation constant is, approximately, 0.06 Np/m. This value is anorder of magnitude smaller than for the single-row waveguide. Over 7 GHz, no reliablemeasurements have been obtained for the attenuation constant (see the error bars).

5.3.4 Substrate Integrated Waveguides

This Section deals with the analysis of the SIW schematized in Fig. 5.4(d). As alreadymentioned, the proposed analysis method is restricted to the analysis of structures whenthey are segmentable in a Cartesian coordinate system. In order to analyze SIWs, a stair-case approximation has to be used to avoid this limitation. The contour of the approxima-tion to the circle is defined by considering one quarter of the circumference, and imposingthe 45o symmetry. The number of steps in the quarter circle will be used henceforth todenote the order of the approximation. For instance, a five-step approximation was shownin Fig. 5.3(b).

The proposed method is successfully compared with two different approaches alreadyreported in the literature and with the measurements of a breadboard, thus demonstratingits capability to analyze this type of structure by means of the staircase approximation.A convergence analysis with both the number of steps needed to represent accurately thecircular boundary and with the number of terms in (5.1) are presented.

In the first place, the proposed approach is compared with the Boundary Integral -Resonant Mode Expansion (BI-RME) method [Cassivi et al., 2002]. The dimensions ofthe structure are l(I) = 2.225 mm, h(I) = 1.5 mm, = 0.8 mm, h(II) = 0.7 mm and thepermittivity is εr = 2.2.

In order to determine the minimum number of steps needed to analyze a circular-postwaveguide accurately, a convergence analysis has been carried out for this structure at30 GHz, with 41 terms in the periodic-boundary regions. Fig. 5.9(a) shows the real andimaginary part of the fundamental mode propagation constant as a function of the numberof steps. The phase and attenuation constants, extrapolated from Fig. 5.9(a) for an infinitenumber of steps (perfectly circular post), are 658 rad/m and 11 · 10−3 Np/m respectively.Using a twenty-five-step approximation, relative errors below 0.5% and 11% in the phaseand attenuation constants are warranted. The rather high relative error of the attenuationconstant is due to the very small separation between the posts, which is actually smallerthan the diameter of the posts (h(II) < ). This small separation greatly reduces the

131

CHAPTER 5. ANALYSIS OF LATERALLY OPEN PERIODIC WAVEGUIDESIm

( )

(Np/

m)

ξ yRe

( )

+

(

rad/

m)

ξ y2m

πh(I)

4 6 8 10Frequency (GHz)

TM theoryz10

TM theoryz30

TM theoryz20

TM meas.z10

0

20

40

60

80

100

120

140

-1

0

1lig

ht line

Figure 5.8: Theoretical and measured two-displaced-row waveguide propagation constants. At-tenuation and phase constants are presented in the upper and lower figures, respectively. Thestandard deviation of the measurements is presented with vertical bars.

132


energy leakage of the waveguide. With an attenuation constant five orders of magnitudebelow the phase constant, an 11% error on the former cannot be considered as remarkable.It is worth noting that Fig. 5.9(a) can be understood as the variation of the propagationconstant from a square-post waveguide (i.e., the one-step approximation) to a perfectlycircular-post waveguide (the extrapolation to an infinite-step approximation). While theenergy leakage is higher in the circular-post structure, the Bloch-mode cut-off frequencyis lower. The area of the square-post waveguide is bigger than the area of its inscribedcircle, therefore better preventing the energy leakage. On the other hand, a smaller postarea increases the equivalent width of the waveguide, thus lowering the Bloch-mode cut-off frequency and increasing the phase constant at a given frequency.

Once the number of steps has been fixed, a convergence analysis with the numberof expansion terms in (5.1) is carried out. Since the number of terms in the parallel-plate regions is determined by means of fixed relations to avoid the relative convergencephenomenon, the number of terms in the periodic-boundary regions is the only variableof this analysis. Fig. 5.9(b) shows the value of both the attenuation and phase constantsof the structure at 30 GHz as a function of the number of terms used in the periodic-boundary regions. The phase and attenuation constants extrapolated for a infinite numberof terms are 653 rad/m and 8.4·10−3 Np/m. The relative error is kept below 0.3% and 13%using 41 terms in the periodic-boundary regions for the phase and attenuation constantsrespectively.

The dispersion diagram of the SIW introduced in [Cassivi et al., 2002, Fig. 3] ispresented in Fig. 5.10, where the results of [Cassivi et al., 2002] by means of the BI-RME

640

660

-0.01

0

Re(

)|

(r

ad/m

)ξ y

f=30

GH

z

0 100Number of steps

20 40 60 80 0

654

658

662

Number of terms in the periodic-boundary regions101

-0.016

-0.012

-0.008

Im( )| (N

p/m)

ξy

f=30GH

z

21 41 61 81

(a) (b)

Figure 5.9: Real and imaginary parts of the fundamental Bloch mode propagation constant at30 GHz for the SIW introduced in [Cassivi et al., 2002, Fig. 3] as a function of (a) the numberof steps used in the staircase approximation and (b) the number of expansion terms used in theperiodic-boundary regions.

133


method are compared to the results obtained with the method proposed in this dissertation.Good agreement in the phase constant values is found between methods for both thefundamental and the first higher-order mode (no data is provided in [Cassivi et al., 2002]regarding the attenuation constant).

The second approach to which the proposed method is compared is the Method ofLines (MoL), reported in [Yan et al., 2005]. The dispersion diagram of [Yan et al., 2005,Fig. 8] has been computed for the = 1 mm and = 1.5 mm cases (the rest of thedimensions can be found at Fig. 5.11 caption). Both examples have been computed with atwenty-five-step approximation and with 41 terms in the periodic-boundary regions. Theresults are compared in Fig. 5.11, where very good agreement can be observed, not onlyin the phase constant of the propagating modes, but also in the stop-band frequency range,and in the attenuation constant value in this band. Please note that the results from [Yanet al., 2005] have been adapted to match the dispersion diagram convention used in thisdissertation.

For the circular-post waveguide, or SIW, a breadboard has also been built and mea-sured. Its dimensions are the same as the square-post single-row waveguide presented inSection 5.3.1, but with circular posts of diameter = 6 mm instead of square posts of6 mm per side. The effect of the square posts in the first example (Figs. 5.3(a), 5.4(a)and 5.5) will be compared with the effect of the inscribed-circle posts (Figs. 5.3(b) and5.4(d)).

A convergence analysis for this structure has been carried out at 4 GHz. The phaseand attenuation constant values, extrapolated for an infinite number of steps (perfectly

10 20 30 40 50 60

200

600

1000

1400

1800

Frequency (GHz)

Light line

TM - BI-RME20

TM - BI-RME10

TM - This theory10TM - This theory20

Re(ξ

) (r

ad/m

)y

Figure 5.10: Dispersion diagram for a structure of dimensions l(I) = 2.225 mm, h(I) = 1.5 mm, = 0.8 mm, h(II) = 0.7 mm and εr = 2.2. Comparison between this theory and the BI-RMEmethod [Cassivi et al., 2002].

134


circular post) are 38.15 rad/m and 0.72 Np/m, respectively. Compared to the square-postwaveguide 22% and 39% differences are obtained. Using a twenty-five-step approxima-tion the relative error is kept below 1.5% and 2.2% in the phase and attenuation constants,respectively.

On the other hand, the phase and attenuation constants extrapolated to an infinitenumber of terms in the periodic-boundary regions are 36.6 rad/m and 0.67 Np/m. Using 41

terms the relative errors are kept below 2.5% and 5.2%, respectively. With respect to theconvergence analysis of the first structure (Fig. 5.9(b)), it is noted that although the relativeerror of the phase constant has increased, the relative error of the attenuation constant hassignificantly decreased. Energy leakage is much higher in this case, with an attenuationconstant only two orders of magnitude lower than the phase constant. Therefore, theerrors in both constants tend to be more similar to each other.

The measured and theoretical propagation constants of the fundamental mode areshown in Fig. 5.12. To compute the results shown in Fig. 5.12 twenty-five different dis-continuities and GSM concatenations must be carried out per iteration of Muller’s method.A single core of an Intel processor at 2.66 GHz spends 7.8 ms to find a propagation con-stant of the square-post waveguide per frequency (including the complex plane search).The staircase approximation to the circular-post waveguide requires 84 ms to completethe same task. This is 10 times slower, but still efficient enough.

For comparison purposes, the computed propagation constant for the square-postwaveguide is also shown in Fig. 5.12. Only the monomode band of the structure is shownhere for clarity. Once more, excellent agreement between theoretical and measured prop-agation constants has been obtained. The cut-off frequency of the fundamental mode,compared to square-post waveguide, has dropped from 3.73 GHz to 3.58 GHz. Besides,

-25

-20

-15

-10

-5

0

5

Im(ξ

) (N

p/m

)y

16 16.5 17 17.5 18Frequency (GHz)

φ = 1.5mm - MoLφ = 1.5mm - This theoryφ = 1mm - MoLφ = 1mm - This theory

Re(

) +

(ra

d/m

)ξ y

2mπ

h(I)

940

960

980

1000

1020

1040

16 16.5 17 17.5 18Frequency (GHz)

Figure 5.11: Dispersion diagram for a SIW. h(I) = 3 mm, εr = 10.2. Dimensions of the =1 mm case: l(I) = 3.056 mm, h(II) = 2 mm. Dimensions of the = 1.5 mm case: l(I) =2.806 mm, h(II) = 1.5 mm. Comparison between this theory and the MoL [Yan et al., 2005]

135


the attenuation constant has increased from 0.4 Np/m to 0.65 Np/m. Thus, more energyleakage occurs in the circular-post waveguide than in the square-post one.

Finally, it shall be remarked that both the propagation constant and the field distribu-tion of the fundamental mode of this breadboard have been used in [Navarro-Tapia et al.,2012] for the characterization of obstacles in periodic waveguides.

5.4 Conclusion

This chapter has introduced some modifications to the generalized TRT described inChapter 3 in order to extend its usage to the characterization, i.e. the computation of thedispersion diagrams, of a class of open periodic waveguides. The proposed modificationshave been validated by means of measurements and by comparison with different numer-ical approaches found in the technical literature. Four breadboards have been built andthe fundamental-mode propagation constants have been measured. Excellent agreementbetween the theoretical results and measurements has been obtained.

In addition, the usefulness and accuracy of the displaced waveguide cascading proce-dure described in Chapter 4 has been, once more, demonstrated by means of the Displaced-Row Waveguide breadboard (section 5.3.3).

It shall also be highlighted that, while technical literature has not paid too much at-tention to higher-order modes with vertical variation (along z axis, with n 6= 0), both thetheoretical results and indirect experimental confirmation of its presence (as modes withlow leakage losses) have been obtained.

Finally, the Substrate Integrated Waveguide section can be used as an example of howto use the staircase contour approximation introduced in Chapter 3. Moreover, this sectionshows that the proposed approach can obtain accurate results without a determinant lossof efficiency.

136

5.4. CONCLUSION

0

-1

0

20

40

60

80

100

120

140

Im(

) (N

p/m

)ξ y

Re(

) (r

ad/m

)ξ y

-2

4Frequency (GHz)

5 6 7 8

TM staircase approx.z10

TM square-post theoryz10

TM circular-post meas.z10

Figure 5.12: Theoretical and measured circular-post single-row waveguide propagation con-stants. Attenuation and phase constants are presented in the upper and lower figures, respectively.The standard deviation of the measurements is presented with vertical bars.

137

6Conclusions

6.1 Original Contributions

For the first time, the wave equation under a combination of conductor and periodic

boundary conditions has been studied. This study has revealed that, under the |χx| =

|χy| = 1 condition, transversely periodic cylindrical structures can be described by meansof a complete set of TEM , TE, and TM modes. Furthermore, it has been demonstratedthat these modes share many interesting properties with the modes of any closed waveg-uide. As a result, a novel method for the analysis of periodic structures has been proposed.

On the one hand, a generalization of the well known Transverse Resonance Techniquehas been developed with the aim of obtaining the complete set of modes that character-ize any transversely periodic waveguide, such as the periodic wire, cross or H-shapedwaveguides, rigorously taking into account the periodic boundary conditions.

On the other hand, the Mode Matching technique has been applied, in conjunctionwith the aforementioned TRT, to the characterization of abrupt discontinuities betweenwaveguides. Several structures, for instance a wire-medium slab, a fishnet surface and aparallel-plate waveguide with a bi-periodic textured surface, have been analyzed by meansof this hybrid approach. The accuracy of the obtained results show that, as predicted bythe initial wave equation study, periodic structures can be, not only analyzed, but also

understood by means of closed waveguide concepts and techniques.

In addition, a modification to the conventional GSM cascading procedure has beenproposed for the analysis of structures with displaced waveguides. It has been shown thatthis procedure can be applied to both the generalized Transverse Resonance and ModeMatching techniques by means of the displaced-row post waveguide and the rectangulardisplaced patch surface examples, respectively. In both cases the accuracy of the proposedmodification has been demonstrated.

139

CHAPTER 6. CONCLUSIONS

Finally, a novel technique for the characterization of laterally open periodic waveg-uides has been proposed. It shall be highlighted that, this technique accounts for the openand periodic nature of the structure with no approximation. Alternative approached haveto resort to some fictitious walls to close the structure.

6.2 Future Research Lines

• Further research is needed to overcome the |χx| = |χy| = 1 limitation both in thetheoretic and practical fields. This extension would allow us to, for instance, con-sider the non-homogeneous plane wave incidence on periodic structures or charac-terize the attenuated transverse wave behavior in structures such as the one of theexample in section 4.4.6.

• A comprehensive study of transversely periodic in-homogeneously filled waveg-uides is also of great interest because of the wide range of potential applications. Forinstance, some dielectric post structures similar to the wire-medium or the multiple-row post-wall waveguide, can be found in the literature.

• Alternative closed waveguide theories should also be tested for the analysis of pe-riodic structures. For instance, the Boundary Integral - Resonant Mode Expansioncould be used instead of the generalized Transverse Resonance Technique. Thiscould allow an easier analysis of any structure that is not segmentable in a cartesiancoordinate system. In particular, circular or ring-like patch periodic surfaces couldbe analyzed in a more efficient way.

• The inclusion of conductor losses to the analysis method should be a priority. Thecharacterization of the conductor losses has always been an important topic in theelectromagnetic analysis. Nowadays, because the increasing number of Terahertzsystems, where this losses are particularly important, this point has gathered evenmore importance.

• As stated in the introduction this Ph. D. dissertation was, in part, motivated by theneed of a highly efficient mothod of analysis for periodic structures. The proposedMode Matching - Transverse Resonance Technique analysis method can be nowused for the synthesis of High Impedance or Frequency Selective Surfaces, anykind of textured surface or, in general, any kind of transversely periodic structure.

140

6.3. FRAMEWORK

6.3 Framework

The work reported in this Ph. D. dissertation has been carried out at the Departamento deElectromagnetismo y Teorıa de Circuitos in the Escuela Tecnica Superior de Ingenierosde Telecomunicacion of the Universidad Politecnica de Madrid with explicit consent ofthe Universidad de Malaga.

One of the Future Research Lines previously mentioned, namely the usage of alter-native closed waveguide analysis techniques, has already been initiated within a researchperiod at the Dipartimento di Elettronica, Facolta di Ingegneria, Universita degli Studi di

Pavia, Italy.

The research projects for which the work reported in this Ph. D. dissertation hasgenerated results are:

• Aplicaciones de Nuevos Conceptos de Metamateriales en el Diseno de CircuitosActivos y Pasivos de Microondas y Milimetricas (TEC2006-04771). Subvencionadopor el Ministerio de Educacion y Ciencia y fondos FEDER de la Union Europea.Octubre 2006 a Septiembre 2009.

• Engineering Metamaterials (CSD2008-00066) del programa Consolider-Ingenio 2010.Subvencionado por el Ministerio de Ciencia e Innovacion. Enero 2008 a Diciembre2012.

• Nuevos Circuitos de Comunicaciones Basados en Metamateriales (P10-TIC-6883).Subvencionado por la Junta de Andalucıa. Febrero 2011 a Enero 2015.

The main financial support for the development of this Ph. D. dissertation has comeform the project ”Engineering Metametarials (EMET)”.

6.4 Publications

This Ph. D. dissertation has given rise to the following publications:

6.4.1 Journal Articles

• J. E. Varela and J. Esteban, ”Analysis of Laterally Open Periodic Waveguides byMeans of a Generalized Transverse Resonance Approach”. IEEE Trans. Microw.

Theory Tech., 59(4) : 816− 826.

141

CHAPTER 6. CONCLUSIONS

• J. E. Varela and J. Esteban, ”Characterization of Waveguides with a Combinationof Conductor and Periodic Boundary Contours: Application to the Analysis of Bi-Periodic Structures”. IEEE Trans. Microw. Theory Tech., 60(3) : 419− 430.

• M. Navarro-Tapia, J. Esteban, J. E. Varela and C. Camacho-Penalosa, ”Simulationand Measurement of the S parameters of Obstacles in Periodic Waveguides”. IEEE

Trans. Microw. Theory Tech., 60(4) : 1146− 1155.

6.4.2 Conference Proceedings

• J. E. Varela, J. Esteban and C. Camacho-Penalosa, ”New Approach to the Anal-ysis of Bi-periodic Cylindrical Structures and its Application to a Wire Medium”.2010 Proceedings of the Fourth European Conference on Antennas and Propaga-

tion (EuCAP), pp. 1-4

• J. E. Varela and J. Esteban, ”Analysis of Periodic Structures by Means of a Gen-eralized Transverse Resonance Approach”. 2010, IEEE MTT-S Int. Microwave

Symposium Digest (MTT), pp. 21-24.

• J. E. Varela and J. Esteban, ”Computation of the Wire Medium TEM Mode byMeans of a Transverse Resonance Technique”. 2011, IEEE International Sympo-

sium on Antennas and Propagation (APSURSI), pp. 1506-1509.

142

AAppendix A

A.1 The Wave Equation for the Transverse Electric

Field

In this section the wave equation, for the transverse electric field, will be derived startingfrom Maxwell’s equations. Let us begin substituting (2.1) into Maxwell’s equations

∇× ~H = jωε ~E; ∇× ~E = −jωµ ~H (A.1)

where σ is assumed to be zero. In the first place, by substituting in the first equation, wehave

∇×(~Ht + zHz

)e−jkzz = jωε

(~Et + zEz

)e−jkzz (A.2)

The left hand side consists of two terms:

∇× ~Hte−jkzz = e−jkzz∇× ~Ht − jkze−jkzz z × ~Ht (A.3)

∇× zHze−jkzz = e−jkzz∇Hz × z (A.4)

The second termon the right-hand side of (A.3) and the term on (A.4) are perpendicularto z. In order to determine the direction of the first term of (A.3) consider:

z ×(∇× ~Ht

)= ∇

(~Ht · z

)− (z · ∇) ~Ht

Since ~Ht has no z component and no z variation, both terms on (A.5) are zero. Therefore,∇× ~Ht is parallel to z. Equation (A.2) can be now decomposed in two equations, one for

143

APPENDIX A. APPENDIX A

the z-component and the other for the transverse component:

∇× ~Ht = jωεzEz (A.5)

z ×∇Hz + jkz z × ~Ht = −jωε ~Et (A.6)

Similarly, from the second Maxwell equation of (A.1) two equations arise:

∇× ~Et = −jωµzHz (A.7)

z ×∇Ez + jkz z × ~Et = jωµ ~Ht (A.8)

Next, taking the divergence of the first equation of (A.1):

∇ · ~E = 0

substituting (2.1) into this equation gives

e−jkzz(∇ · ~Et +∇Ez · z

)− jkz ze−jkzz ·

(~Et + zEz

)= 0

since Ez is independent of z, this equation simplifies to:

∇ · ~Et = jkzEz (A.9)

Similarly, considering the divergence of the second equation of (A.1) we have:

∇ · ~Ht = jkzHz (A.10)

At this point, an equation of ~Et alone will be obtained, eliminating ~Ht, Ez and Hz

from the above equations. On the one hand, calculate ∇×(A.7):

∇×∇× ~Et = jωµz ×∇Hz

Using (A.6), we have

∇×∇× ~Et − ω2µε~Et = ωµkz z × ~Ht

144

A.2. DERIVATION OF THE PERIODIC BOUNDARY CONDITIONS FOR THETRANSVERSE ELECTRIC FIELD

Subtracting∇(A.9)

∇×∇× ~Et −∇(∇ · ~Et

)− ω2µε~Et = ωµkz z × ~Ht − jkz∇Ez (A.11)

On the other hand, consider −jkz z×(A.8)

jkz∇Ez − k2z ~Et = kzωµz × ~Ht

Substituting now this equation in (A.11), the desired equation for ~Et is obtained

∇×∇× ~Et −∇(∇ · ~Et

)− k2c ~Et = 0 (A.12)

wherek2c = ω2µε− k2z (A.13)

A.2 Derivation of the Periodic Boundary Conditions

for the Transverse Electric Field

This section deals with the derivation of the periodic boundary conditions for the trans-verse electric field. Let A and B be two different sections of a waveguide contour, theperiodic boundary conditions that the electric and magnetic fields have to fulfill are:

n× ~E∣∣∣B

= χ n× ~E∣∣∣A

; n× ~H∣∣∣B

= χ n× ~H∣∣∣A

(A.14)

where χ is the periodicity constant which is in general complex, i.e. χ ∈ C, and n is theunitary-outward-normal vector to the contour at each point. By defining a local coordinatesystem in each point of the considered contour formed by the set [n, τ , z] of orthogonalvectors, the n× ~E term may be rewritten as follows:

n× ~E = n× (nEn + τEτ + zEz) = zEτ − τEz (A.15)

In this way, (A.14) may be slit into four different equations:

Eτ |B = χ Eτ |A Hτ |B = χ Hτ |AEz|B = χ Ez|A Hz|B = χ Hz|A

(A.16)

On the one hand, the boundary conditions for the electric field components may be

145


easily written in terms of the transverse electric field using (A.9):

n× ~Et

∣∣∣B

= χ n× ~Et

∣∣∣A

; ∇ · ~Et∣∣∣B

= χ ∇ · ~Et∣∣∣A

(A.17)

On the other hand, the ~H field equations are analogous:

n× ~Ht

∣∣∣B

= χ n× ~Ht

∣∣∣A

(A.18)

Hz|B = χ Hz|A (A.19)

Substituting Hz from (A.7) into (A.19):

∇× ~Et∣∣∣B

= χ ∇× ~Et∣∣∣A

(A.20)

which is the first periodic boundary condition for the magnetic field. Consider nown× (A.8)

n×z ×∇Ez + jkzn×z × ~Et = jωµn× ~Ht ⇒

z[n·∇Ez + jkzn· ~Et

]= jωµn× ~Ht (A.21)

Substituting the gradient of (A.9) into this equation:

z[n·∇

(∇· ~Et

)− k2z n· ~Et

]= −ωµkzn× ~Ht (A.22)

using the wave equation (2.2),

z[n·(∇×∇× ~Et

)− k2n· ~Et

]= −ωµkzn× ~Ht (A.23)

Taking into account that ∇× ~Et is a z directed vector, (A.7), the first term of (A.23) maybe rewritten as:

z

[z · ∂

∂τ

(∇× ~Et

)− k2n· ~Et

]= −ωµkzn× ~Ht (A.24)

By substituting this expression into (A.18) and noting that since∇×~Et is periodic, (A.20),its derivative with respect to τ must also be periodic with the same periodicity, the finalperiodicity boundary condition in terms of the transverse electric field is:

n· ~Et∣∣∣B

= χ n· ~Et∣∣∣A

(A.25)

146

A.3. SOME MATHEMATICAL LEMMAS

A.3 Some Mathematical Lemmas

Lemma A.3.1 Let ~f and ~g be singled valued complex vector functions of the transverse

components, then ∣∣∣∣∫ ~f ~g ∗ dS

∣∣∣∣2 ≤ 2

(∫ ∣∣∣~f ∣∣∣2 dS

∫|~g |2 dS

)(A.26)

where the integrals are over a certain area S.

Proof Noting that the modulus of a function is always positive,

0 ≤∫ ∣∣∣∣~f ∫ |~g|2 dS − ~g

∫~f · ~g ∗dS

∣∣∣∣2 dS

=

∫ ∣∣∣∣~f ∫ |~g|2∣∣∣∣2 dS +

∫ ∣∣∣∣~g ∫ ~f · ~g ∗dS∣∣∣∣2 dS

− 2Re

∫ [~f ∗ · ~g

∫|~g|2 dS

∫~f · ~g ∗ dS

]dS =

∫ ∣∣∣~f ∣∣∣2 dS

∫|~g |4 dS

+

∫|~g |2 dS

∣∣∣∣∫ ~f · ~g ∗dS∣∣∣∣2 − 2Re

∫|~g |2 dS

∣∣∣∣∫ ~f · ~g ∗dS∣∣∣∣2

=

∫ ∣∣∣~f ∣∣∣2 dS

∫|~g |2 dS −

∣∣∣∣∫ ~f · ~g ∗dS∣∣∣∣2∫

|~g |2 dS

since |~g |2 is positive, this last identity proves (A.26).

Lemma A.3.2 Let ~f and ~g be singled valued complex vector functions of the transverse

components, then ∫ ∣∣∣~f ± ~g ∣∣∣2 dS ≤ 2

(∫ ∣∣∣~f ∣∣∣2 dS +

∫|~g |2 dS

)(A.27)

where the integrals are over a certain area S.

Proof From,

0 ≤∫ ∣∣∣~f ± ~g ∣∣∣2 dS =

∫ ∣∣∣~f ∣∣∣2 dS +

∫|~g |2 dS ± 2Re

∫~f · ~g ∗dS

147


we have2

∣∣∣∣Re ∫ ~f · ~g ∗dS∣∣∣∣ ≤ ∫ ∣∣∣~f ∣∣∣2 dS +

∫|~g |2 dS

by substituting this last equation into∫ ∣∣∣~f ± ~g ∣∣∣2 dS ≤∫ ∣∣∣~f ∣∣∣2 dS +

∫|~g |2 dS + 2

∣∣∣∣Re ∫ ~f · ~g ∗dS∣∣∣∣

equation (A.27) is obtained.

148

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