PhDThesis_WeiSong

Modelling of Electromagnetic Bandgap Structuresusing an Alternating Direction Implicit

(ADI)/Conformal Finite-Difference Time-DomainMethod

Wei Song

A Thesis Submitted in Partial Fulfilment of the Requirements

for the Degree of

Doctor of Philosophy

Department of Electronic Engineering

c© Queen Mary University of London

February, 2008

To my family

Abstract

In recent years, there have been growing interests in utilizing Electromagnetic bandgap

(EBG) structures in the antenna and microwave engineering. Due to their performance in

guiding and efficiently controlling electromagnetic (EM) waves, EBGs allow us to manip-

ulate the propagation of electromagnetic waves to an extent that was not possible with

materials readily exist in nature.

Extensive studies in ways of both numerical modellings and experiments have driven

rapid advances in the development of EBG technology. Among the various numerical

methods which have been applied to study the EBGs, the Finite-Difference Time-Domain

(FDTD) method is one of the most popular numerical methods due to its simplicity in

algorithm and its accuracy in the solution for complex structures in a wide range of

applications. However, most of the FDTD approaches in modelling EBGs employ the

conventional Yee’s algorithm. Consequently, a high spatial resolution is required to min-

imize the numerical dispersion caused by the staircase approximation when curved in-

clusions/oblique surfaces are involved. As a result of the high spatial resolution, more

computer resources are needed in the FDTD simulation.

In this study, the nonorthogonal FDTD (NFDTD) method is used to numerically in-

vestigate the EBG structures. With all the inclusions and surfaces being modelled con-

formally, the requirement on the spatial resolution is much lower and the EBGs can be

modelled more accurately with less computer memory and a shorter computation time.

The efficiency improvement of the NFDTD method over the Yee’s FDTD method is stud-

ied. Moreover, an Alternating Direction Implicit (ADI) scheme is introduced into the

NFDTD method and the proposed ADI-NFDTD method has greatly improved the late

time stability of the NFDTD method.

i

Acknowledgements

The author would like to express her deepest gratitude to her supervisor, Professor Yang

Hao, for his patient and helpful supervision. His wide knowledge and deep academical

insights has guided and supported the author in every step of her research. Deep thanks

are also due to Professor Clive G. Parini for his invaluable advice and the precious en-

couragement towards this study.

The author wish to express her warm and sincere thanks to the following people and

institutions that have kindly offered their help:

Dr. Vassileios Douvalis for his valuable detailed guidance in the basis of FDTD and

conformal FDTD.

Dr. Yan Zhao, for his wide knowledge he always generously shared and his patient

help in every step in programming.

Dr. Sunil Sudhakaran, for patiently sharing his wide knowledge in antennas and

EBGs.

Mr Nikolis Giannakis, for his detailed discussion in PWE method.

Dr Akram Alomainy, Miss Maria-anna Setta, Mr. John Wang, Dr. Pavel Belov, Dr.

Yue Chen, Dr. Chantaraskul, Ms Xu Yang, Dr. Crystal Tang, for their helpful discussions,

suggestions and valuable friendship and encouragements.

Mr Kok Huen, for his friendly help in IT supporting, Mrs Lynda Rolfe, Ms Melissa

Yeo and other office people for their kindness and efficient management work in every

stage of her study.

The author owes the loving thanks to Lin Xiao, for his every support in both life and

study.

The financial support of the Electronic Engineering, Queen Mary, University of Lon-

don is gratefully acknowledged.

ii

List of Publications

W. Song, Y. Hao, C.G. Parini, “An ADI-FDTD Algorithm in Curvilinear Coordinates”,

2005 Asia-Pacific Microwave Conference, December 4-7, 2005

W. Song, Y. Hao, C.G. Parini, “ADI-FDTD Algorithm in curvilinear co-ordinates”,

Electronics letters, Vol. 41, No. 23, Nov. 2005, pp. 1259-1261.

W. Song, Y. Hao, C.G. Parini, “Calculating the Dispersion Diagram Using the Nonorthog-

onal FDTD Method”, The Institution of Engineering and Technology Seminar on Meta-

materials for Microwave and (Sub) Millimetrewave Applications: Electromagnetic Bandgap

and Double Negative Designs, Structures, Devices and Experimental Validation. Sep.

2006.

W. Song, Y. Hao, C.G. Parini, “Comparison of Nonorthogonal FDTD and Yee’s Algo-

rithm in Modelling Photonic Bandgap Structures”, International Sympsium on Antennas

and Propagation (ISAP2006), Singapore, Nov. 2006.

P.S. Hall, Yang Hao, V.I. Nechayev, A. Alomainy, C.C. Constantinou, C. Parini, M.R.

Kamarudin, T.Z. Salim, D.T.M. Heel, R. Dubrovka, A.S. Owadall, Wei Song, A. Serra, P.

Nepa, M. Gailo, M. Bozzetti, “Antennas and propagation for On-Body Communication

Systems”, Antennas and Propagation Magazine, IEEE Vol. 49, Issue 3, 41 - 58, 2007.

iii

Contents

Abstract i

Acknowledgements ii

List of Publications iii

Contents iv

List of Abbreviations viii

List of Figures x

List of Tables xvii

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Research Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Basics and an Overview of Electromagnetic Bandgap Structures 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Bloch’s Theorem and the Dispersion Diagram . . . . . . . . . . . . . . . . . 6

2.2.1 Translational Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.2 Bloch’s Theorem and Periodic Boundary Condition . . . . . . . . . 10

2.2.3 Brillouin Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.4 Dispersion Diagram and Photonic Band Gap . . . . . . . . . . . . . 13

2.3 An Overview of Numerical Methods for the Modelling of EBGs . . . . . . 17

2.3.1 The Generalized Rayleigh’s Identity Method and the Korringa-Kohn-

Rostoker (KKR) Method . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Plane Wave Expansion method . . . . . . . . . . . . . . . . . . . . . 19

iv

2.3.3 Modelling EBGs using the Transfer-Matrix Method . . . . . . . . . 22

2.3.4 Modelling EBGs using the Finite-Different Time-Domain Method . 25

2.4 An Overview of EBG Applications . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.1 In-Phase Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.2 Suppression of Surface Waves . . . . . . . . . . . . . . . . . . . . . . 34

2.4.3 EBGs working in Defect Modes . . . . . . . . . . . . . . . . . . . . . 38

2.4.4 Subwavelength Imaging from the passband of the EBGs . . . . . . 55

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3 A Brief Introduction to the Finite-Difference Time-Domain Method for Mod-

elling the EBG Structures 59

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Formulations of the Yee’s FDTD algorithm . . . . . . . . . . . . . . . . . . 60

3.2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.2 Yee’s Orthogonal Mesh . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2.3 Time Domain Discretization: the Leapfrog scheme and the Courant

stability condition (CFL condition) . . . . . . . . . . . . . . . . . . . 63

3.3 Other Spatial Domain Discretization Schemes . . . . . . . . . . . . . . . . . 66

3.3.1 Subgridding Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3.2 Nonorthogonal Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3.3 Hybrid FDTD Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.4.1 Mur’s Absorbing Boundary Conditions (ABCs) . . . . . . . . . . . 76

3.4.2 Perfect Matched Layers (PML) . . . . . . . . . . . . . . . . . . . . . 78

3.4.3 Periodic Boundary Condition (PBC) . . . . . . . . . . . . . . . . . . 79

3.5 Band Gap Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.5.1 Source Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.5.2 Dispersion Diagram Calculation . . . . . . . . . . . . . . . . . . . . 85

3.5.3 Transmission and Reflection Coefficient Calculation . . . . . . . . . 86

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

v

4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD

Method 90

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2 A Brief Introduction to the Nonorthogonal FDTD Method . . . . . . . . . 91

4.2.1 The Curvilinear Coordinate Systems . . . . . . . . . . . . . . . . . . 92

4.2.2 The Conventional Nonorthogonal FDTD . . . . . . . . . . . . . . . 96

4.2.3 Lee’s NFDTD Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2.4 Douvalis’ NFDTD Algorithm . . . . . . . . . . . . . . . . . . . . . . 101

4.2.5 Numerical Stability of the Nonorthogonal FDTD Method . . . . . . 103

4.3 Alternating Direction Implicit Finite-Difference Time-Domain Method . . 108

4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.3.2 The Idea of Alternating Direction Implicit (ADI) Method . . . . . . 108

4.3.3 ADI Applied to the Finite-Difference Time-Domain Method . . . . 109

4.3.4 Numerical Stability of the ADI-FDTD Method . . . . . . . . . . . . 114

4.4 Alternating Direction Implicit Nonorthogonal FDTD Method . . . . . . . 117

4.4.1 Derivation of the ADI-NFDTD Formulation . . . . . . . . . . . . . 117

4.4.2 Reduction of the ADI-NFDTD to the Conventional ADI-FDTD . . 124

4.4.3 Periodic Boundary Condition Incorporated in the ADI-NFDTD . . 126

4.5 Validation of the ADI-NFDTD Method . . . . . . . . . . . . . . . . . . . . . 131

4.5.1 Removal of the CFL Stability Criteria . . . . . . . . . . . . . . . . . 131

4.5.2 Numerical Efficiency and the Late Time Instability Improvement

over the Conventional NFDTD . . . . . . . . . . . . . . . . . . . . . 135

4.5.3 Numerical Efficiency Improvement over the Orthogonal ADI-FDTD 145

4.5.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5 NFDTD and ADI-NFDTD modelling of EBG Structures 160

5.1 NFDTD Modelling of the Infinite Electromagnetic Bandgap Structures . . 161

5.1.1 The Model of the EBG Structures . . . . . . . . . . . . . . . . . . . . 161

5.1.2 Mesh of the Unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.1.3 Settings in the FDTD Simulations . . . . . . . . . . . . . . . . . . . . 166

vi

5.1.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.1.5 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . 185

5.2 NFDTD Modelling of the (Semi-)Finite EBG Structure . . . . . . . . . . . . 187

5.2.1 The EBG Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

5.2.2 FDTD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

5.2.3 Mesh of the Unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5.2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191


5.3 NFDTD Modelling of an EBG-like Waveguide . . . . . . . . . . . . . . . . 199

5.4 EBG-like Negative Refractor Modelling . . . . . . . . . . . . . . . . . . . . 200

5.5 ADI-NFDTD Modelling of the EBG Unit Cell . . . . . . . . . . . . . . . . . 203

5.5.1 NFDTD Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

5.5.2 Comparison of the ADI-NFDTD and NFDTD Simulation Results . 205


6 Conclusions and Future work 207

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

6.2.1 Further Numerical Validations . . . . . . . . . . . . . . . . . . . . . 209

6.2.2 Enhanced NFDTD/ADI-NFDTD Algorithms . . . . . . . . . . . . . 209

Bibliography 212

A The Formulation of Plane-Wave-Expansion (PWE) Method For Two-Dimensional

EBG Modelling 229

B The Detailed Comparison of Douvalis’ and Lee’s NFDTD Equations 235

B.1 Douvalis’ Final NFDTD Equations . . . . . . . . . . . . . . . . . . . . . . . 235

B.2 Lee’s NFDTD Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

B.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

C Towards the Derivation of the Three-Dimensional ADI-NFDTD Equations 242

vii

List of Abbreviation

2-D: Two-dimensional

3-D: Three-dimensional

ABC: Absorbing boundary condition

ADI: Alternating direction implicit

ADI-DNFDTD: ADI-NFDTD based on the Douvalis’ NFDTD

ARER: Averaged relative error rate

CAF: Computer applied fluid

CCW: Coupled cavity waveguide

CFL: Courant-Friedrich-Levy

CLWM: Capacitively loaded wire medium

CPU: Central processing unit

EBG: Electromagnetic bandgap structure

EM: Electromagnetic

FDTD: Finite-difference time-domain

FFT: Fast Fourier transformation

FSS: Frequency-selective surface

HSR: High spatial resolution

ITO: Indium tin oxide

KKR: Korringa-Kohn-Rostoker

LD-NFDTD: Local distorted NFDTD

LSR: Low spatial resolution

MD-EBG: Metallo-dielectric electromagnetic band-gap

NFDTD: Nonorthogonal FDTD

PBC: Periodic boundary condition

PBG: Photonic bandgap structure

viii

PC: Photonic crystal

PEC: Perfect electric conductor

PMC: Perfect magnetic conductor

PML: Perfectly matched layer

PWE: Plane wave expansion

RER: Relative error rate

RHS: Right-hand-side

SPP: Surface plasmon polariton

TE: Transverse electric

TM: Transverse magnetic

TMM: Transfer-Matrix Method

UC-EBG: Uniplanar compact electromagnetic bandgap

—————

ix

List of Figures

2.1 A part of a one-dimensional EBG structure. . . . . . . . . . . . . . . . . . . 7

2.2 (a) A part of the infinitely long two-dimensional EBG structure. (b) Unit

cell marked on the x-y cut plane of the EBG structure. . . . . . . . . . . . . 8

2.3 (a) Two angles of view of the woodpile EBG structure. (b) Unit cell for the

woodpile EBG material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 A dielectric configuration with discrete translational symmetry. . . . . . . 11

2.5 (a) The physical lattice of an EBG made using a square lattice. (b) The

Brillouin zone of the reciprocal lattice. . . . . . . . . . . . . . . . . . . . . . 12

2.6 The one-dimensional Dispersion Diagram for a one-dimensional EBGs. . . 14

2.7 A two-dimensional dispersion diagram for a two-dimensional EBGs. . . . 15

2.8 A three-dimensional dispersion diagram for a two-dispersion EBG with

hexagonal lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.9 (a) Geometry of a mushroom-like EBG structure. (b) The Antenna with the

PEC or PMC ground plane. (c) The Antenna with the EBG ground plane

[63]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.10 FDTD simulated return loss results of the dipole antenna over the EBG,

PEC, and PMC ground planes of the same dimension. . . . . . . . . . . . . 32

2.11 The FDTD result of (a)return loss of the dipole; (b)The reflection phases of

the mushroom-like EBG surface versus frequency. . . . . . . . . . . . . . . 32

2.12 Schematic, cross section of the proposed slot antenna loaded with UC-PBG

reflector, and the top view of the UC-PBG. [81]. . . . . . . . . . . . . . . . 33

2.13 Configuration of a square curl antenna over an EBG surface. . . . . . . . . 33

2.14 Radiation pattern comparison of dipoles near the thin grounded high di-

electric constant slab and the EBG surface. . . . . . . . . . . . . . . . . . . . 35

2.15 A scanning electron micrograph of the hexagonal array of dots. . . . . . . 37

2.16 A sample set of reflectivity data recorded as a function of the photon energy. 37

2.17 Geometry of a patch antenna with a strip dipole FSS superstrate [111]. . . 39

x

2.18 Field distribution of the EBG antenna [111]. . . . . . . . . . . . . . . . . . . 40

2.19 Comparison of the radiation patterns of the FSS antenna composite and

the patch antenna only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.20 Three dimensional dispersion diagram of the two dimensional EBG with

circular rods lying in vacuum in a triangular lattice [35]. . . . . . . . . . . . 42

2.21 The dispersion diagram of the expanded EBG [35]. . . . . . . . . . . . . . . 42

2.22 Constant-frequency dispersion diagram of the expanded EBG for λ = 7.93

[35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.23 Total field modulus radiated by the structure excited by the wire source at

λ = 7.93 [35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.24 Polar emission diagram for the structure excited by the wire source at λ =

7.93 [35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.25 (a) Measured and calculated transmission spectra and (b) Calculated field

distribution of a zig-zag CCW waveguide. . . . . . . . . . . . . . . . . . . . 45

2.26 (a) Measured and calculated transmission spectra of a Y-shaped coupled-

cavity based splitter. (b)Calculated power distribution inside the input and

output waveguide channels of the splitter. . . . . . . . . . . . . . . . . . . . 46

2.27 (a) Measured and calculated transmission spectra of a coupled-cavity switch-

ing structure. (b)Calculated field pattern of the switching structure. . . . . 47

2.28 the coupled cavity waveguides (CCW) in 3D photonic crystals. . . . . . . 48

2.29 Dielectric EBG structure having air-voids filled with liquid crystal as defect

elements [100]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.30 Transmission coefficient for various numbers of defect cylinder rows when

γ = 45 [100]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.31 Transmission coefficient for various values of the tilt angle γ in the case of

two defect cylinder rows [100]. . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.32 Dielectric EBG structure with a liquid crystal defect layer [100]. . . . . . . 52

2.33 Director orientation profile across the liquid crystal defect layer. . . . . . . 52

2.34 Transmission coefficient versus normalized frequency for various values

of the applied voltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

xi

2.35 (a) Metallic EBG structure with a liquid crystal defect layer. (b) Transmis-

sion coefficient versus normalized frequency for various values of the ap-

plied voltage when Lc = 4a, Ls = a [100]. . . . . . . . . . . . . . . . . . . . 54

2.36 The EBG structure composed of capacitively loaded wire medium (CLWM). 56

2.37 Isofrequency contours for the CLWM. . . . . . . . . . . . . . . . . . . . . . 57

2.38 Simulated distribution of electric field (a) amplitude and (b) intensity for

the sub-wavelength lens formed by the CLWM operating in the canaliza-

tion regime [103]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1 Yee’s spatial grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Leapfrog scheme - the temporal scheme of the FDTD method. . . . . . . . 64

3.3 A cross section of a computational domain meshed according to the sub-

gridding algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4 Enlarged view of the top-left corner of figure 3.3. . . . . . . . . . . . . . . . 70

3.5 A part of a three dimensional nonorthogonal mesh showing the covariant

vectors and the contravariant vectors. . . . . . . . . . . . . . . . . . . . . . 71

3.6 Meshes of the cross section of a cylindrical cavity. . . . . . . . . . . . . . . 74

3.7 Periodic boundary conditions when calculating the infinite EBGs. . . . . . 81

3.8 The FDTD procedure in modelling EBG structures. . . . . . . . . . . . . . 83

3.9 The modulated Gaussian pulse. . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.10 The super cell of the EBG waveguide. . . . . . . . . . . . . . . . . . . . . . 86

3.11 Numerical Model for an EBG structure of semi-finite size. . . . . . . . . . . 87

3.12 Numerical Model for an EBG structure of finite size. . . . . . . . . . . . . . 88

4.1 The definition of basic vectors. . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.2 The covariant and the contravariant basic vectors in a nonorthogonal FDTD

cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.3 The NFDTD iteration scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.4 The facet (a1 × a2) (or a3) on a NFDTD cell. . . . . . . . . . . . . . . . . . . 100

4.5 A two-dimensional nonorthogonal FDTD cell. . . . . . . . . . . . . . . . . 103

4.6 Co- and contravariant basis vectors in primary grid G and dual grid G[127]. 106

xii

4.7 Electric field at a degenerated cell simulated by the use of the primary and

the dual basis vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.8 The working scheme of the proposed ADI-NFDTD algorithm. . . . . . . . 123

4.9 A mesh of the (0.3m × 0.3m) free space domain. . . . . . . . . . . . . . . . 132

4.10 Hz field temporal results for the modelling of a single frequency wave

propagation in free space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.11 The mesh of the cut plane of the cylindrical cavity. . . . . . . . . . . . . . . 136

4.12 The NFDTD modelling of the cavity. . . . . . . . . . . . . . . . . . . . . . . 136

4.13 An example of the excitation signal in time and frequency domain. Am=1,

f = 7GHz and dt = 1ps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.14 The comparison of the H field temporal results at early stage from the con-

ventional NFDTD and the ADI-NFDTD schemes. . . . . . . . . . . . . . . 138

4.15 Hz field temporal results with dt = 1ps. . . . . . . . . . . . . . . . . . . . . 140

4.16 The calculated resonant frequency spectra of the cavity resonator. . . . . . 141

4.17 Resonant frequency spectrum of the cavity resonator calculated from the

ADI-NFDTD with dt = 17ps, referenced by that from the conventional

NFDTD with dt = 1ps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.18 The averaged relative error rate of the NFDTD and ADI-NFDTD simula-

tion results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.19 The modulated Gaussian pulse as excitation in (a) time domain and (b)

frequency domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.20 In ADI-NFDTD, the computation domain is meshed by a (46 × 46) grid. . 147

4.21 In ADI-FDTD, the computation domain is meshed by a (60 × 60) grid. . . 148



4.24 The resonant frequency spectra calculated by the ADI-NFDTD (with mesh

by 46 by 46 grids) and the ADI-FDTD (with mesh by 60 by 60 grids). . . . 156



xiii



5.1 The cylindrical metallic rods in free space in square lattice. . . . . . . . . . 162

5.2 The cylindrical metallic rods in free space in triangular/rhombic lattice. . 163

5.3 The mesh schemes for an unit cell from the square lattice in the NFDTD

and the Yee’s FDTD modelling. . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.4 The mesh schemes for an unit cell from the triangular lattice in the NFDTD

and the Yee’s FDTD modelling. . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.5 The modelling schemes of the unit cell approach for EBG structures. . . . 166

5.6 The dispersion diagram of the first few TE modes, for the copper rods in

free space in square lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.7 The dispersion diagram of the first few TM modes, from the copper rods

in free space in square lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.8 The dispersion diagram of the first few TE modes, from the copper rods in

free space in triangular/rhombic lattice. . . . . . . . . . . . . . . . . . . . . 170

5.9 The dispersion diagram of the first few TM modes, from the copper rods

in free space in triangular/rhombic lattice. . . . . . . . . . . . . . . . . . . 171

5.10 The TE mode frequency spectra of the wave vector k at Γ point, for the

modelling of the cylindrical copper rods in triangular lattice using the

NFDTD method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

5.11 The TE mode frequency spectra of the wave vector k at Γ point of the cylin-

drical copper rods in triangular lattice using Yee’s FDTD and the NFDTD

method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

5.12 The TE mode frequency spectra of the wave vector k at Γ point, for the

modelling of the cylindrical copper rods in square lattice using the NFDTD

method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5.13 The TE mode frequency spectra of the wave vector k at Γ point the cylin-

drical rods in square lattice simulated by the Yee’s FDTD with adequate

spatial resolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

xiv


drical rods in square lattice, simulated by the Yee’s FDTD method using

reduced spatial resolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 179


drical rods in square lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

5.16 The TM mode frequency spectra of the wave vector k at Γ point in recipro-

cal space of the cylindrical rods in square lattice. . . . . . . . . . . . . . . . 181

5.17 The minimum spatial resolution (a) and mesh size (b) required for the Yee’s

FDTD and NFDTD algorithms for the triangular lattice TE mode unit cell

simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

5.18 The dielectric cylinders surrounded by air, forming a square lattice. . . . . 187

5.19 The cut plane of the EBG structure consists of 8 by infinite dielectric cylin-

drical rods array in square lattice. . . . . . . . . . . . . . . . . . . . . . . . . 188

5.20 The conformal mesh of the unit cell with spatial resolution of 48X48 per

unit cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5.21 The conformal mesh of the unit cell with different spatial resolutions. . . . 190

5.22 The NFDTD temporal results after a Gaussian pulse excitation. . . . . . . 191

5.23 The NFDTD temporal results with different spatial resolutions. . . . . . . 192

5.24 The NFDTD temporal results with different spatial resolutions after a Mod-

ulated Gaussian pulse plane wave excitation at one side of the EBG slab.

(a) Mesh 5.21(c) is used; (b) Mesh 5.21(d) is used. . . . . . . . . . . . . . . . 193

5.25 The transmission coefficients (TE mode) of the dielectric rods (figure 5.18)

calculated using the NFDTD method. r is the radius of the cylinder. . . . . 195

5.26 The theoretical transmission coefficients (TE mode) of the dielectric rods

(figure 5.18) predicted by the transfer-matrix technique [47]. . . . . . . . . 195

5.27 The transmission coefficients (TE mode) of the EBGs . . . . . . . . . . . . . 196

5.28 The transmission coefficients (TM mode) of the dielectric rods (figure 5.18)

calculated using the NFDTD method with different spatial resolutions. r

is the radius of the cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

xv

5.29 The theoretical transmission coefficients (TM mode) of the dielectric rods

(figure 5.18) predicted by the transfer-matrix technique [47]. . . . . . . . . 197

5.30 Comparison of the FDTD and NFDTD simulation parameters. . . . . . . . 198

5.31 The NFDTD field plot near the bend of the EBG-like waveguide. . . . . . . 200

5.32 The electric field plot from the EBG like structure simulated by the NFDTD

method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

5.33 The electric field plot from the EBG like structure simulated by the Ansoft

HFSS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

5.34 The unit cell of the EBGs (figure 5.1) and the Brillouin Zone in the recipro-

cal lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

5.35 The dispersion diagram of the square lattice EBG, TE mode results, calcu-

lated using the NFDTD methods. . . . . . . . . . . . . . . . . . . . . . . . 204

5.36 The frequency spectra of different probes when calculating k = k1. . . . . 204

5.37 The temporal response at one probe position when calculating k = k1. . . 205

5.38 The comparison of the frequency spectra in calculating k = k1 using the

conventional NFDTD and the ADI-NFDTD . . . . . . . . . . . . . . . . . . 206

xvi

List of Tables

4.1 Comparison of the computational time and accuracy of the simulation re-

sults on a Pentium IV 2.40GHz PC with RAM of 1.5GB. With the same

level of accuracy, the ADI-NFDTD shows a saving rate of 1.6 in total com-

putational time compared with the conventional NFDTD. . . . . . . . . . . 133

4.2 Comparison of the accuracy of the resonant modes of the aforementioned

PEC resonator calculated by the conventional NFDTD and the ADI-NFDTD

with the theoretical results. In both simulations, dt = 1ps. The modes with∗ are spurious modes introduced in the meshing of the cylindrical cavity.

They are compared with the nearest genuine mode when RER is evaluated

and hence they cause big numerical errors. In the first deviation of RER

calculation, these errors are not included. In the second deviation of RER

calculation (with ∗), these errors are taken into consideration. . . . . . . . 142


copper resonator calculated by the ADI-NFDTD (Mesh 46 × 46) with the

theoretical results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152


copper resonator calculated by the conventional ADI-FDTD (Mesh 60×60)

with the theoretical results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 153







xvii

4.7 Comparison of the accuracy and computer resources ( computer memory

and computing time) of the ADI-NFDTD and ADI-FDTD in calculating

the resonant modes of the copper cavity. The simulations are run under

programming environment of Matlab 2007b. . . . . . . . . . . . . . . . . . 155

5.1 Comparison of the relative error rate on the dispersion diagrams at M

point for (a) TE mode, (b) TM mode, calculated by the Yee’s FDTD and

the NFDTD with low spatial resolutions. This error rate is approximated

by the difference of the low-spatial-resolution result compared with the

high-spatial-resolution result. . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.2 Comparison of the minimum spatial resolution required by NFDTD and

the Yee’s FDTD when simulating the copper rods infinite EBGs in square

lattice or triangular lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

5.3 Comparison of the memory, the computational time used (on average) in

calculating one k vector in dispersion diagram, and the accuracy of the

results using the Yee’s FDTD and the NFDTD method. . . . . . . . . . . . 184

xviii

Chapter 1

Introduction

1.1 Introduction

Electromagnetic bandgap structures (EBGs) are periodically structured artificial electro-

magnetic media. They generally possess band gaps, a range of frequency in which the

electromagnetic (EM) waves cannot propagate. As the principle of ’bandgap’ applies to

photonics engineering, it is also termed as photonic bandgap (PBG) materials or photonic

crystals (PC). Throughout this thesis, these materials are referenced as EBGs.

EBGs structures have attracted a lot of attention for their versatility in controlling

the propagation of electromagnetic waves [1, 2]. Numerical methods have been used

to predict their performance and assist their design. Among them, the Finite-Difference

Time-Domain (FDTD) method [3] is one of the most popular numerical techniques for

modelling EBGs. As a simple way to discretize the Maxwell’s equations, FDTD does

not require model symmetry or complex mathematical formulation, and hence it can be

applied to model inhomogeneous structures such as EBGs which include defects. As a

straitforward solution to the Maxwell’s equations, FDTD provides accurate temporal re-

sults, which enable the study of EBGs over a wide frequency band. After it was first intro-

duced by Yee[4] in 1966, the FDTD algorithm has been going through continuous modi-

fication, refinements and extensions[5–8], which further enhance the method’s capability

and broaden its appeal. As computer costs keep decline, this versatile method gains more

1

Chapter 1 Introduction 2

popularity and wider range applications in electromagnetic community [9, 10], including

EBGs [11, 12].

In a tremendous amount of FDTD approaches in modelling EBGs, an overwhelming

majority is based on the Yee’s scheme [13–16], using uniform orthogonal meshes. There

is also an alternative FDTD approach developed in the nonorthogonal coordinate system

[17], in which a uniform rhombic grid is employed when modelling the rhombic unit

cell. In that approach, the formulae are derived from the conventional Yee’s scheme with

adjustments for the fixed skewed angle in the grid. However, when the curved unit cell

element is considered, staircasing approximation is employed, either with an orthogonal

grid or with a rhombic grid. It is anticipated that the staircasing approximation will

cause numerical errors when the wavelength of interest is small with regard to the grid

size. Consequently, a dense grid with high spatial resolution is required and this leads to

extensive computation with a large computer memory requirement.

On the other hand, the nonorthogonal FDTD (NFDTD) scheme originated by Holland

in 1983 [5] uses structured conformal meshes when modelling curved structures. Com-

pared to the staircase FDTD scheme, fewer meshes are needed to represent the curved or

oblique boundary of electromagnetic structures. As a result, the computational efficiency

can be greatly improved by using the NFDTD method[18]. This thesis will demonstrate

the computational efficiency of the NFDTD method in modelling EBGs with curved in-

clusions.

However, the NFDTD scheme inherently suffers the late time numerical instability[19–

21]. Since it has been successfully applied to the orthogonal FDTD and removes the

Courant-Friedrich-Levy (CFL) stability condition of FDTD method[7, 22], the alternating

direction implicit (ADI) scheme will be extended to the curvilinear coordinates in this

study and a novel Alternating-Direction Implicit Nonorthogonal Finite-Difference Time-

Domain (ADI-NFDTD) method is proposed. The ADI-NFDTD method has demonstrated

that the time increment (dt) used in the simulation is no longer constrained by the CFL

stability condition. Numerical simulations show that with an increased time increment

(dt), the numerical efficiency of the NFDTD can be improved. Additionally, the numerical

simulations show that the inherent late time instability of the NFDTD method is greatly


reduced in the ADI-NFDTD algorithm. The occurrence of the unstable results is delayed

to a much later time. This is quite beneficial in modelling EBGs when wave propagating

with high wave numbers is more prone to suffer the late time instability.

1.2 Research Objective

The objective of this study is to compare the efficiency and accuracy of the nonorthog-

onal FDTD (NFDTD) method and the Yee’s FDTD method when modelling EBGs with

curved inclusions. In light of the fact that the NFDTD suffers the late time instability, an

additional objective is to modify the NFDTD into a more stable, efficient and accurate

method.

1.3 Outline of the Thesis

In Chapter Two, the basics of the EBGs will be briefly introduced. Bloch’s theory and the

dispersion diagrams are presented providing a theoretical insight of EBG structures. This

is followed by a brief review of the numerical methods used for the study of EBGs, in-

cluding the generalized Rayleigh’s method, the Korringa-Kohn-Rostoker (KKR) method,

the Plane Wave Expansion (PWE) method, the Transfer-Matrix Method (TMM) and the

Finite-Difference Time-Domain (FDTD) method. Then EBG configurations and applica-

tions are reviewed.

In Chapter Three, the foundation of FDTD, i.e., the Yee’s algorithm is briefly reviewed.

As far as the mesh generation is concerned, different FDTD grid schemes are presented

in this chapter. Techniques for modelling the EBGs are also presented.

Chapter Four firstly presents the extensions of the Yee’s FDTD method, including the


nonorthogonal FDTD (NFDTD) approach, in which a generalized curvilinear coordinate

system is applied. The ADI-FDTD algorithm, in which an alternating-direction implicit

(ADI) method is applied to the orthogonal FDTD method. As the NFDTD suffers late

time instability, ADI scheme is introduced to the general nonorthogonal FDTD method

and the ADI-NFDTD method is proposed. In particular, the formulae of the periodic

boundary condition (PBC) incorporated in the ADI-NFDTD scheme is presented. In the

final part of this chapter, the novel ADI-NFDTD scheme is validated by numerical simu-

lations. The efficiency and the late-time instability improvements are demonstrated.

Then numerical simulation results for the NFDTD and ADI-NFDTD modelling of

electromagnetic bandgap structures are presented and discussed in Chapter Five.

Conclusions are drawn in Chapter Six with the suggestions on future work.

Chapter 2

Basics and an Overview of

Electromagnetic Bandgap Structures

2.1 Introduction

Periodic structures have been studied for many years. In the late 1980s, a fully three-

dimensional periodic structure, operating at microwave frequencies, was realized by

Yablonovitch et al. [1] by mechanically drilling holes into a block of dielectric material.

This material prevented the propagation of EM radiation in any three dimensional spa-

tial direction whereas the material is transparent in its solid form at these wavelengths.

Such artificially engineered periodic materials are generically known as electromagnetic

bandgap (EBG) structures.

The main feature of these materials is the existence of a gap (stopband) in the fre-

quency spectrum of propagating EM waves [2, 23–26]. This bandgap frequency depends

on the permittivity of the dielectric inclusions or background used, the dimensions of the

inclusions/defects, their periodicity and the incidence angle of electromagnetic waves

[27]. This feature leads to a variety of phenomena of both fundamental [28, 29] and practi-

cal [1, 30] interests in the way which fascinated artists and scientists alike. In this chapter,

the basic theory, numerical methods and applications of the EBGs will be reviewed.

5

Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 6

2.2 Bloch’s Theorem and the Dispersion Diagram

Symmetry in an electromagnetic structure or system is important to the analysis of wave

behaviour. Most theoretical studies on EBGs are based on an interesting symmetry prop-

erty as the EBGs show periodicity in the dielectric of the material. As a consequence,

Bloch’s theory is developed accordingly to describe the modes in such structures. Based

on Bloch’s theory, a dispersion diagram can be derived to describe the frequency be-

haviour of EBGs. In this section, basic concepts related to the study of EBGs will be

introduced, including: Translational Symmetry, Bloch’s Theorem and Periodic Boundary

Condition (PBC), Brillouin Zones and Dispersion Diagram and Photonic Band Gap.

2.2.1 Translational Symmetry

Definition:

A system with translational symmetry is unchanged by a translation through a displace-

ment d[2].

Suppose we have a function ε(r) in a translationally invariant system. Then, when

we do a translation with an operator Td:

Td[ε(r)] = ε(r + d) = ε(r) (2.1)

A system that has continuous translational symmetry is invariant under the T d’s of

any displacement. An example system which has continuous translational symmetry in

all three directions is free spaceε(r) = 1.

All of the EBGs do not have continuous translational symmetry; instead, they all have

discrete translational symmetry. In other words, they are not translational invariant

under any distance, but only under certain distances - which are a multiple of some fixed


step length (period). The basic step length is termed as the lattice constant a and the basic

step vector is called the primitive lattice vector a or the fundamental translation vector

a. So, we have ε(r) = ε(r + a) and ε(r) = ε(r + R) where R = sa and s is an integer.

Consequently, these structures can be considered as one unit being repeated over and

over. This unit is known as the unit cell.

Figures 2.1, 2.2 and 2.3 show examples of EBG structures that are periodic in one

dimension, two dimensions and all three dimensions respectively. In figure 2.1, the di-

electric property of the material is repeated in one direction at a period of a. The unit

cell is marked using the blue lines. In figure 2.2, the infinitely long cylindrical rods are

periodically loaded in both x and y direction at a distance of a respectively. The unit cell

is marked using the red dashed lines. The material in figure 2.3 is termed as the woodpile

EBG structure and is also referred to as a layer-by-layer photonic crystal in the physics

literature [31]. A unit cell of this material is shown in figure 2.3(b).

(a)

(b)

Figure 2.1: A part of a one-dimensional EBG structure. This EBG consists of alternating layersof materials of different dielectric properties. These materials are periodically stacked on topof each other. a is the lattice constance. (a) the three-dimensional view of the EBG; (b) theone-dimensional view of the EBG.


Figure 2.2: (a) A part of the infinitely long two-dimensional EBG structure. (b) Unit cellmarked on the x-y cut plane of the EBG structure.


(a)

(b)

Figure 2.3: (a) Two angles of view of the woodpile EBG structure. (b) Unit cell for the wood-pile EBG material.


2.2.2 Bloch’s Theorem and Periodic Boundary Condition

Bloch studied wave propagation in three-dimensionally periodic media in 1928, followed

by an extending theorem in one dimension by Floquet in 1883 [24]. In Bloch’s study, he

proved that waves in such a medium can propagate without scattering, their behaviour

is governed by a periodic envelop function multiplied by a planewave. This means that,

in a photonic crystal with periodic dielectric function:

ε(r) = ε(r + s1a1 + s2a2 + s3a3), si = 0,±1,±2, ... (2.2)

where ai(i = 1, 2, 3) are three primitive lattice vectors. For a crystal periodic in all three

dimensions, the wave can be expressed in forms of combinations of planewaves and a

periodic function [2]:

H(r) = eik·ru(r) (2.3)

where u is a periodic envelope function.

Equation 2.3 is commonly known as Bloch’s theorem. It is known in solid-state

physics as a Bloch State [32] and in mechanics as a Floquet mode [33].

A simple example is shown in figure 2.4. The structure is repetitive in the y-direction,

and invariant and infinite in the x-direction. So this structure has continuous translational

symmetry in the x-direction and discrete translational symmetry in the y-direction. The

unit cell is highlighted in the figure with a blue box. The primitive lattice vector in this

case is a = ay.

According to Bloch’s theorem, fields in these structures are in Bloch state

H(r) = eikxx·eikyy·uky(y, z), (2.4)

where u(y, z) is a periodic function in y satisfying

u(y + s · a, z) = u(y, z) si = 0,±1,±2, ... . (2.5)


Figure 2.4: A dielectric configuration with discrete translational symmetry. (a is the latticeconstant.)

According to equation (2.4), this state can be considered as a plane wave, modulated

by a periodic function because of the periodic lattice.

As a result, the field in Bloch’s state can be studied by a so called unit cell approach,

in which only elements in one unit cell are investigated and outside-cell elements are

related and expressed using the following relationship:

H(r + say) = eikxx·eiky(y+sa)·uky(y + sa, z)

= eikysa·(eikxx·eikyy·uky(y, z)

)= eikysa H(r) (2.6)

Equation (2.6) is used as the periodic boundary condition (PBC) in the study of the

infinite EBG structures.


2.2.3 Brillouin Zone

A key fact about Bloch’s states is that a Bloch state with wave vector ky and a Bloch

state with wave vector ky + mb are identical, where b = 2π/a and m is an integer. That

means the mode frequencies are also periodic in ky : ω(ky) = ω(ky + mb). So we only

need to consider ky to exist in the range −π/a < ky ≤ π/a. This region of important,

nonredundant values of ky is called the Brillouin zone.

Another example shown in figure 2.5 is made up of cylindrical elements periodically

laid in a square lattice. Figure 2.5(a) shows the physical lattice, and the Brillouine Zone

of the reciprocal lattice is plotted in figure 2.5(b). This structure shows not only a discrete

translational symmetries along the x and y directions, but also rotation, mirror-reflection

and inversion symmetry. It is shown [2] that when these symmetries are shown in phys-

ical space, the reciprocal space (k space) shows the same kinds of symmetry. As a result,

in the Brillouin zone shown in figure 2.5(b), every k-point included in it need not to be

considered. Only a smallest region within the Brillouin zone for which the k vector are

not related by symmetry need to be considered. That smallest region is termed as the irre-

ducible Brillouin zone, shown in light blue color; the rest of the Brillouin zone contains

redundant copies of the irreducible zone.

(a) (b)

Figure 2.5: (a) The physical lattice of a EBG made using a square lattice. An arbitrary vectorr is shown. (b) The Brillouin zone of the reciprocal lattice, centered at the origin (Γ). Anarbitrary wave vector k is shown. The irreducible Brillouin zone is the light blue triangularwedge. The special points at the center, corner and face are conventionally known as Γ, Mand X [2].


2.2.4 Dispersion Diagram and Photonic Band Gap

With the knowledge of the irreducible Brillouin zone, the possible modes against the

wave vector k can be plotted. This curve can be plotted in one dimension, two dimen-

sions or three dimensions. These plots provide the dispersion relation and information on

the flow of energy in an intuitive way. Figure 2.6 shows two examples of one-dimensional

diagrams of one-dimensional EBG materials.

Figure 2.6(a) is the dispersion diagram for a uniform dielectric medium, to which a

periodicity of a has been artificially assigned. It is known that in a uniform medium, the

speed of light is reduced by the index of refraction. So the plot is just the light-line given

by

ω(k) =ck√

ε(2.7)

Because the wavevector k repeats itself outside the Brillouin zone, the lines fold back

into the zone when they reach the edges. The dashed lines show the folding effect of

applying Bloch’s theorem with an artificial periodicity a.

In figure 2.6(b), there is a gap in frequency between the upper and lower branches of

the lines - a frequency gap in which no mode, regardless of k, can propagate through the

structure. This gap is called an electromagnetic band gap, which can be further classified

as:

1. Complete Electromagnetic Band Gap: A complete photonic bandgap is a range of ω

in which there are no propagating (real k) solutions of Maxwell’s equations for any k,

surrounded by propagating states above and below the gap.

2. Incomplete Electromagnetic Band Gap: This bandgap only exist over a subset of all

possible wavevectors, polarizations, and/or symmetries.

Figures 2.7 and 2.8 show two-dimensional and three-dimensional dispersion dia-

grams of two two-dimensional EBGs respectively. In this thesis, only the two-dimensional

dispersion diagram is used as an example for numerical validation.


(a) (b)

Figure 2.6: (a) Dispersion relation (band diagram), frequency ω versus wave number k of auniform one-dimensional medium, (b) Schematic effect on the bands of a physical periodicdielectric variation (inset), where a gap has been opened by splitting the degeneracy at thek = ±π/a. The right-up corner shows the physical lattice and the wave vector. a is the latticeconstant [34].


(a)

(b)

Figure 2.7: (a) A two-dimensional dispersion diagram for a two-dimensional EBGs (illus-trated in figure 2.5), in which a complete bandgap is shown by the light blue color; (b) Atwo-dimensional dispersion diagram where an incomplete bandgap can be found. For ex-ample, the bandgap from normalized frequency 0.404 - 0.552 is only valid with k vector fromΓ to M , and a subset from X to Γ, indicated by the light blue color.


Figure 2.8: A three-dimensional dispersion diagram for a two-dimensional EBG. The EBGis made of circular rods of radius ρ = 0.6, with optical index of 2.9, lying in vacuum on ahexagonal lattice with period of 4. The horizontal plane gives the Bloch wave vector k. Thevertical axis gives 1/λ. The triangle corresponding to the irreducible Brillouin Zone has beendrawn in the (kx, ky) plane. The parameters about the EBGs can be found in [35].


2.3 An Overview of Numerical Methods for the Modelling of

EBGs

Numerical methods are used to predict the performance of EBGs, either for the pur-

pose of theoretical understanding of EBGs, or for the assistance of EBGs related designs.

These numerical methods include the generalized Rayleigh Identity Method [36–39],

the Korringa-Kohn-Rostoker (KKR) approach [40–42], the Plane Wave Expansion (PWE)

method [43–45], the Transfer Matrix Method (TMM) [46, 47], and the Finite-Difference

Time-Domain (FDTD) method [3], etc. Among them, the Plane Wave Expansion (PWE)

method and the FDTD method are two of the most popular methods. In this section, these

numerical methods will be briefly discussed and the details about the FDTD method will

be presented in the following chapters.

2.3.1 The Generalized Rayleigh’s Identity Method and the Korringa-Kohn-

Rostoker (KKR) Method

Nicorovici and McPhedran et al. extended Rayleigh’s technique from electrostatic to full

electromagnetic problems, for singly[37], doubly[38] and triply[39] periodic systems.

Consider EBGs consisting of an array of cylinders or spheres in an isotropic homoge-

neous dielectric host medium. Denote the fundamental translation vectors of the lattice

as ei(i = 1, 2) for cylinder array or (i = 1, 2, 3) for sphere array. The electromagnetic wave

has the wave number k, the equations for the components of the electric and magnetic

fields decouple and each field component satisfies the Helmholtz equation

(∇2 + k2)f(r) = 0 (2.8)


The solution f(r) has to fulfill the quasiperiodicity condition

f(r + RP ) = ei ki· RP f(r) ∀p (2.9)

where ki = (ki, θi, ϕi), is the wave vector of the incident radiation and RP stands for the

vectors from the origin of coordinates to the center of the pth cylinder (equation (2.10))

or sphere (equation (2.11)):

RP = p1e1 + p2e2 ≡ (p1, p2) , pi ∈ Z (2.10)

RP = p1e1 + p2e2 + p3e3 ≡ (p1, p2, p3) , pi ∈ Z (2.11)

Green’s function G, which obeys the inhomogeneous Helmholtz equation in the pe-

riodic systems, can be expressed as:

(∇2 + k2)G = −c∑

p

δ(r − RP − ρ)eik0· RP . (2.12)

This satisfies the following quasiperiodicity conditions:

G(r + RP , ρ) = eik0· RP G(r, ρ) (2.13)

and

G(r, ρ + RP ) = e−ik0· RP G(r, ρ) (2.14)

where c = 2π for the cylinder array and c = 1 for the sphere array.

By introducing the lattice sums, Nicorovici et al. obtained a representation of the

Green’s function in terms of a rapidly convergent Neumann series [36–39].

If the solution for the periodic lattice is approximated through a variation-iteration

method instead of being calculated directly, the generalized Rayleigh’s Identity method

becomes the Korringa-Kohn-Rostoker (KKR) method to photonics[39, 48], which is achieved

by Korringa, Kohn and Rostoker [40] using different approaches.

In light of this, the Generalized Rayleigh’s Identity method and the KKR method


share some common features. The most attractive feature of these two methods is that

the greater part of the work of calculating energy bands consists of the calculation of

certain geometrical ’structure constants’, which need only to be calculated once for each

type of lattice. This leads to a more compact and faster convergent scheme than the PWE

method in the case where the potential V (r) is spherically or cylindrically symmetrical

within the inscribed elements and constant in the space between them. However, the

application of the Generalized Rayleigh’s Identity method and the KKR method is limited

to modelling structures with the spherical or cylindrical symmetry, and for ε(r) being

piecewise constant as well. When the actual potential violates these conditions, these

procedures are not suitable[42].

Compared to the Generalized Rayleigh’s Identity method, the KKR method has even

a limited range of applications. For complex ε(r), the variational KKR method cannot be

used. On the other hand, the Generalized Rayleigh’s Identity method is capable of solv-

ing problems in which the dielectric constant taking on finite or infinite values, or imag-

inary values. The Generalized Rayleigh’s Identity method can also be applied when the

cylinders are composed of an arbitrary number of coaxial circular shells filled with mate-

rials having different complex dielectric constants. However, being variational, the KKR

method can be expected to converge more rapidly within its range of application[48].

2.3.2 Plane Wave Expansion method

Compared with the Generalized Rayleigh’s Identity method or the KKR method, the

Plane Wave Expansion (PWE) is much simpler and the computer code is much easier to

write and runs significantly faster. In terms of convergence, the PWE technique gener-

ally demonstrates slower convergence compared with the two aforementioned methods.

However, there are also numerical examples in which the PWE demonstrates fairly rapid

convergence [42]. Moreover, PWE method is not limited to spherically or cylindrically


symmetric modulation of the potential. Instead, the PWE method can readily handle

almost all sorts of modulations [42, 49].

In the plane wave expansion method, the work is done in the Fourier space. Denote

k = x1k1 + x2k2 as the two-dimensional wave vector of the wave and

G(h) = h1b1 + h2b2 (2.15)

as a reciprocal-lattice vector. In other word, G is the vector in the Fourier space. The

component of the electric field (or magnetic field) is expanded in plane waves as equation

(2.16).

E(r) =∑

G

B Gei(k+ G)·r (2.16)

where B Gis the Fourier coefficients.

Once the electric field (or magnetic field) is known, the other EM field vectors are

determined.

The inverse dielectric constant is also expanded as:

1ε(r)

=∑

G

κ Gei G·r (2.17)

in which ε is a periodic function in physical space satisfying: ε(r) = ε(r + R); κ G is the

Fourier coefficient which can be written in certain form for certain EM properties and

configuration.

Substituting (2.16) and (2.17) into Maxwell’s equations, the following eigenvalue ex-

pression is obtained: ∑G′

κ G− G′ |k + G|2B G′ =ω2

c2B G . (2.18)

Equation (2.18) can be solved using a standard eigenvalue solver [50, 51].

Unlike some other method (e.g. the KKR method), the PWE method does not rely

on the assumption of spherical or cylindrical symmetry of the structure being modeled

hence it a more general method.

The PWE method is also going through continuous developments which:


• release the method from restrictions of the complex and frequency-dependent di-

electric function [44, 52];

• and enhance the PWE towards a faster convergence rate [50, 53].

In the original PWE method, the calculation of components characterized by frequency-

dependent, complex dielectric functions is more challenging than calculating purely di-

electric materials, since it requires the solution of a generalized nonlinear eigenvalue

problem. This eigenvalue problem can be solved by a linearization scheme, which re-

quires the diagonalization of an equivalent, enlarged matrix, which means more compu-

tational loads. Kuzmiak et al. reported an alternative PWE approach (as presented in the

previous section) to incorporate the frequency-dependent dielectric function, in which

the generalized eigenvalue problem is reduced to the problem of diagonalizing of a set

of matrices whose size equals the number of plane waves kept in the expansions for the

components of the EM field in the system [44]. Then the PWE method is generalized by

them again in order to handle complex, frequency-dependent dielectric function [52].

The PWE method solves Maxwell equations in the whole region including the inclu-

sion element (region II) and the embedding background (region I). However, a step in

the dielectric constant occurs at the boundary. This makes the expansion of the dielec-

tric constant in (2.17) badly convergent. As a result, huge basis sets may be needed to

accurately describe a photonic band structure [44]. To avoid this problem, an embedding

method is introduced into the PWE method [50, 53] in which the wave equation is only

solved in region I (between the elements), with region II (inside the elements) replaced

by an embedding potential defined on the boundary. Much more rapid convergency is

demonstrated by the use of the embedding method [50, 53].

Although the PWE method is a general method and can treat arbitrary shapes of

the inclusion elements in EBGs, it is more efficient when the shape of element shows

spherical or cylindrical symmetry. Otherwise, large number of basis sets will be required

in the expansion and the PWE method will then become computationally expensive as

the computational time is proportional to the cube of the number of the plane waves [76].

An interest topic of the study of EBGs is the behaviour of impurity modes associated with

the introduction of defects into the EBG structure. While this problem can be tackled


within a PWE approach using the supercell method in which a single defect is placed

within each supercell of an artificially periodic system, the calculations require a lot of

computer time and memory. Besides, when the EBG is of finite size, it is more difficult to

expand the parameters in the PWE method. On the other hand, the following numerical

method presented is readily to calculate the transmission/reflection coefficients for a EBG

slab with finite thickness.

2.3.3 Modelling EBGs using the Transfer-Matrix Method

Pendry and MacKinnon introduced a complementary technique, i.e. the transfer-matrix

method (TMM), of studying EBG structures in 1992[46]. In the TMM, the total volume of

the system is divided into small cells and the fields in each cell are coupled to those in the

neighboring cells. Then the transfer matrix can be defined by relating the incident fields

on one side of the EBG structure with the outgoing fields on the other side.

Expressing Maxwell’s equations:

∇× E = −∂ B

∂t, ∇× H =

∂ D

∂t(2.19)

in (k, ω) space yields:

k × E = ω B , k × H = −ω D . (2.20)

Assuming that

D = ε E = ε0εrE and B = µ H = µ0µr

H, (2.21)


equations (2.20) then become equations (2.22) and (2.23):

⎡⎢⎢⎢⎣

x y z

kx ky kz

Ex Ey Ez

⎤⎥⎥⎥⎦ = ωµ

⎡⎢⎢⎢⎣

Hxx

Hyy

Hzz

⎤⎥⎥⎥⎦ (2.22)

⎡⎢⎢⎢⎣

x y z

kx ky kz

Hx Hy Hz

⎤⎥⎥⎥⎦ = −ωε

⎡⎢⎢⎢⎣

Exx

Eyy

Ezz

⎤⎥⎥⎥⎦ (2.23)

Equation (2.22) yields:1

ωµ(kxEy − kyEx) = Hz , (2.24)

and equation (2.23) yields:

kyHz − kzHy = −ωεEx . (2.25)

Substituting Hz in equation (2.25) with the expression in (2.24) yields:

ky[1

ωµ(kxEy − kyEx)] − kzHy = −ωεEx (2.26)

By making the substitution:

H ′ =i

aωε0H (2.27)

then equation (2.28) can be obtained:

(iakz)H ′y =

iakyc2µ−1

r

a2ω2[(iakx)Ey − (iaky)Ex] − εrEx . (2.28)

Applying the simple cubic mesh, the fields are defined by vectors a, b, c of length a in the

x, y, z directions, respectively. Transforming equation (2.28) back into real space yields


equation (2.29) from which the z components of the vectors are eliminated:

H ′y(r + c) = −εr(r + c)Ex(r + c) + H ′

y(r)

− c2µ−1r (r − b + c)

a2ω2[Ey(r + a − b + c) − Ey(r − b + c) − Ex(r + c) + Ex(r − b + c)]

+c2µ−1

r (r + c)a2ω2

[Ey(r + a + c) − Ey(r + c) − Ex(r + b + c) + Ex(r + c)]

(2.29)

Another three equations (2.30)- (2.32) can be obtained in a similar way:

H ′x(r + c) = −εr(r + c)Ey(r + c) + H ′

x(r)

− c2µ−1r (r − a + c)

a2ω2[Ey(r + c) − Ey(r − a + c) − Ex(r − a + b + c) + Ex(r − a + c)]

+c2µ−1

r (r + c)a2ω2

[Ey(r + a + c) − Ey(r + c) − Ex(r + b + c) + Ex(r + c)] (2.30)

Ex(r + c) = +a2ω2

c2µr(r)H ′

y(r) + Ex(r)

+ ε−1r (r)[H ′

y(r − a) − H ′y(r) − H ′

x(r − b) + H ′x(r)]

− ε−1r (r)[H ′

y(r) − H ′y(r + a) − H ′

x(r + a − b) + H ′x(r + a)] (2.31)

Ey(r + c) = −a2ω2

c2µr(r)H ′

x(r) + Ey(r)

+ ε−1r (r)[H ′

y(r − a) − H ′y(r) − H ′

x(r − b) + H ′x(r)]

− ε−1r (r + b)[H ′

y(r − a + b) − H ′y(r + b) − H ′

x(r) + H ′x(r + b)] (2.32)

Equations (2.29) and (2.30) express the H fields on the next plane of cells in terms of

the E fields on the same plane, and the H fields on the previous plane. Equations (2.31)

and (2.32) express the E fields on the next plane in terms of the E and H fields on the

previous plane. Thus, given the x, y components of the E and H fields on one side of

a dielectric structure, the x, y components of the E and H fields on the other side can

be found by integrating through out the structure. For a dielectric structure containing


(L × L × L) cells, the dimensions of the transfer matrix are 4L2.

The TMM has the advantage that the transmission coefficients and attenuation coeffi-

cients for incident electromagnetic waves of various frequencies can be obtained directly

from the calculations. This method can also be efficiently used in cases when the PWE

method fails or becomes too time consuming. In particular, when the dielectric func-

tion ε is frequency dependent, or when ε has large imaginary values, Fourier expansion

methods are not efficient [47]. However, disordered systems and periodic systems with

imperfections can be easily studied by the TMM method.

Using the TMM, the band property of an infinite periodic system can be calculated,

but the main advantage of this method is the calculation of the transmission and reflec-

tion coefficients for EM waves of various frequencies incident on a finite-thickness slab

of the EBG material. In that case, the material is assumed to be periodic in the directions

parallel to the interfaces.

On the other hand, the Finite-Different Time-Domain (FDTD) method is more flexible

to model arbitrary shaped configurations and complicated dielectric properties in finite

or infinite structures. The computational effort in the FDTD method is proportional to

the number of the representative points on the spatial mesh. In the following topic, the

applications of the FDTD method in modelling the EBGs is briefly reviewed. Details

about the FDTD method and its extensions will be presented in the next two Chapters.

2.3.4 Modelling EBGs using the Finite-Different Time-Domain Method

The FDTD method is one of the most popular numerical methods for the solution of prob-

lems in electromagnetics. As a simple way to discretize Maxwell’s equations, FDTD does

not require model symmetry or complex mathematical formulation. As a straitforward

solution to the Maxwell’s equations, it provides accurate temporal results, which enable

the study of wide frequency range problems. The first FDTD algorithm was introduced

by Yee[4] in 1966. Since then, modification and extensions of this method [5–7, 54–56]


have been going through continuously.

The FDTD approach has several advantages for modeling EBG structures. It can read-

ily incorporate the very complex geometries associated with the EBG structures them-

selves and their integration with other devices. It can deal with a variety of complex

materials. It also allows the investigations of the broad and narrow bandwidth responses

of an EBG structure from a single simulation. There are over 3600 papers published by

September 2007, reporting EBGs related research by means of FDTD simulations, and

approximately 79% of them are published after the year 2000. A small fraction of these

researches is reviewed in the following section.

• Kesler et al. reported an antenna design with the use of EBG structures as planar

reflectors in 1996 [57]. Field patterns calculated using the FDTD method are in good

agreement with the measured patterns.

• Qian, Radisic and Itoh reported the investigation of four types of EBG structures as

synthesized dielectric materials which possess distinctive stopbands for microstrip lines

experimentally and by simulations using FDTD in 1997 and found an agreement[58].

• Boroditsky, Coccioli and Yablonovitch analyzed the dispersion diagram of an EBG

consisting of a perforated dielectric slab and the properties of a micro-cavity formed by

introducing a defect into such a crystal using the FDTD method in 1998 [59].

• Chutinan and Noda studied the waveguide created by either removing one stripe

from a three-dimensional woodpile EBG or filling up or decreasing the sizes of air holes

from a two-dimensional EBG slab using the FDTD method in 1999 [60] and 2000 [61].

• Yang and Rahmat-Samii analyzed the mushroom-like EBGs using the FDTD simula-

tions in regards to their reflection phase characterizations with applications as a ground

plane for low profile antenna design [62](2001) [63](2003) and lowering mutual coupling

for antenna array [64](2003).

• Ozbay et al. presented a study of the localized coupled-cavity modes in two-dimensional

dielectric EBG with the field patterns and the transmission spectra obtained by the FDTD

simulations in 2002 [65].

• Weily and Bird et al. reported the study of a planar resonator antenna based on a

woodpile EBG material in 2005 [66] and a linear array of EBG Horn sectoral antennas in


2006 [67].

• Kantartzis et al. presented an analysis of double negative metamaterial-based waveg-

uide and antenna devices based on a three-dimensional ADI-FDTD method in 2007 [68].

• Pinto and Obayya reported the study of an EBG cavities using an improved complex-

enveloped ADI-FDTD method in 2007 [69].

• There have been a number of attempts to verify the perfect lens concept realized by

EBG materials [101] using the FDTD method [70–73]. Zhao et al. studied EBGs with ma-

terial frequency dispersion by means of an auxiliary differential equation (ADE) based

dispersive FDTD methods with averaged permittivity along the material boundaries im-

plemented [74, 75] and with spatial dispersion effects considered [12].

It is worth noting that among the vast application of FDTD in modelling the EBG

related structures, there are three approaches based on nonorthogonal coordinate system:

• The finite difference method developed by Chan et al. [76](1995) and Pendry et al.

[77](1998) [78](1999) is often termed as the Order-N method by its authors. This method

has been applied to EBGs with either pure dielectric inclusions [77] or pure metallic inclu-

sions [78]. However, in their present form, they cannot be applied to some complicated

cases such as an EBG whose inclusions contain both dielectric and metallic components

[17].

• Qiu and He developed an FDTD algorithm that is based on a nonorthogonal coordi-

nate system to study EBGs consisting of a skew lattice in 2000 [17, 79]. This method does

not rely on any assumption on the dielectric property of the material to be modelled and

is ready to tackle complex structures such as combinations of dielectric and metallic com-

ponents in the EBG cell. However, the use of a globally uniformed skew grid imposes

the staircase approximation when a curved surface is modelled.

• Roden and Gedney et al. studied the transmission coefficient of a two-dimensional

EBG at oblique incidence using FDTD algorithm under orthogonal and nonorthogonal

grids [154]. The EBG consists of four dielectric rods in y-direction and infinitely laid rods

in x-direction. A set of auxiliary variables is introduced which implicitly accounts for

the phase shift between corresponding points in different unit cells when a plane-wave

source is obliquely incident.


2.4 An Overview of EBG Applications

The first attempts of EBG applications were based on EBG substrates that were fabricated

by drilling a periodic pattern of holes in the substrate, or by etching a periodic pattern

of circles in the ground plane [1]. However, nowadays, the regularly generated novel

ideas expand this rapidly developing scientific area with extremely high rate. Enormous

novel designs and wide range applications can be found in microwave and radio engi-

neering, optical circuit designs and optics spectroscopy. The potential of EBGs attracts

researchers from communities which stand aside from electromagnetics like acoustics,

hydrodynamics, mechanics, etc [80] .

When EBGs are applied to antennas as substrate, high impedance ground planes[64,

81, 82], or reflector [57, 83–85], their band-gap features are revealed mainly in two ways:

the in-phase reflection, and the suppression of surface-wave propagation. The in-phase

reflection feature leads to low profile antenna designs [62, 81, 82]. Meanwhile, the feature

of surface-wave suppression helps to improve antenna’s performance such as increasing

the antenna gain and reducing back radiation [86–89].

There is another well-known characteristic of EBGs, which is the ability to support

localized electromagnetic modes inside the frequency gap by introducing defects in the

periodic lattice. This leads to the development of two important group of applications:

the highly-directive antennas [27] and the EBG waveguide. The former group has high

directivity due to the limited angular propagation allowed within the EBG material, in-

cluding EBG resonator/superstrate antennas [35, 66, 90, 91] and EBG cavities [92]. The

devices in the latter group can efficiently transmit electromagnetic waves, even for 90

bands with zero radius of curvature[60, 79], including EBG waveguide [93–96], power

splitters, directional couplers [97, 98], switches [65](and the references therein), and the

EBG filters [99, 100].

There are also applications utilizing the passband of the EBGs, e.g., the subwavelength

imaging canalization which is studied extensively by Belov et al. [80, 101–103] (and the

references therein).


2.4.1 In-Phase Reflection

Ground planes are used in many antennas. They redirect one-half of the radiation into

the opposite direction, improving the antenna gain, and partially shield objects on the

other side. The reflection phase is of special interest when designing the ground plane

of the antennas. The reflection phase is defined as the phase of the reflected electric field

at the reflecting surface. It is normalized to the phase of the incident electric field at the

reflecting surface.

A perfect electric conductor (PEC) has a 180 reflection phase for a normally incident

plane wave. That means, in a conventional antenna having a flat metal sheet as a ground

plane, if the radiating element is too close to the conductive surface, the 180 out of phase

reflected waves will cancel the radiation waves, resulting in poor radiation efficiency.

This radiation efficiency reduction can also be explained as the image currents generated

by a smooth conducting surface cancel the currents in the antenna if the radiating ele-

ment is too close to the ground plane. This problem is often addressed by including a

quarter-wavelength space between the radiating element and the ground plane, so that

the reflected wave is in-phase with the radiation wave at the radiating element. However,

such a structure then requires a minimum thickness of λ/4 [23].

Ideal perfect magnetic conductor (PMC) ground plane will have a 0 reflection phase

for a normally incident plane wave. However, no natural material has ever been found

to realize the magnetic conducting surface[82].

On the other hand, an EBG structure can be designed to realize a PMC-like surface in a

certain frequency band [15, 23]. Ma et al. demonstrated a magnetic surface realized using

a two-dimension Uniplanar Compact Electromagnetic Bandgap (UC-EBG) structure ex-

perimentally and numerically [104]. The UC-EBG has the merit of easy fabrication, and

most UC-EBGs have been designed by etching a periodic pattern on the ground plane

[105, 106]. Except for a stopband over a wide range of frequency observed, the slow-

wave effect is verified when investigating the propagation characteristics of a UC-EBG

structure in the passband.

However, an EBG surface is more than a PMC surface. The reflection phase of a EBG


surface varies continuously from 180 to −180 versus frequency, not only 180 for a PEC

surface or 0 for a PMC surface. Yang et al. found through their reflection phase study of

a mushroom type EBG structure, that the EBG ground plane requires a reflection phase

in the range of 90±45 for a low profile wire antenna to obtain a good return loss. Figure

2.11(b) demonstrates an example of the reflection phase of a mushroom-like EBG surface

working as ground plane for a dipole antenna. FDTD method is used in this study.

A finite EBG ground plane with 1λ12GHz × 1λ12GHz size is used in their analysis. This

configuration as shown in figure 2.9(a) is termed as metallo-dielectric electromagnetic

band-gap (MD-EBG) structure and is often referred to as the mushroom-like EBG struc-

ture. The height of the dipole over the top surface of the EBG ground plane is 0.02λ 12GHz

so the overall height of the dipole antenna from the bottom ground plane of the EBG

structure is 0.06λ12GHz . The input impedance is matched to a 50Ω transmission line.

The return loss of the dipole antenna over the EBG ground plane is compared with

those of a dipole antenna over a PEC and PMC ground plane of the same dimension (see

figure 2.9(b)(c)). It is seen that the best return loss of −27dB is achieved by the dipole

antenna over the EBG ground plane. By varying the length of the dipole from 0.26λ12GHz

to 0.60λ12GHz , the return loss changes, which is plotted in figure 2.11(a). The frequency

band of the dipole model is 11.5−16.6 GHz according to−10dB return criteria. It is nearly

the same frequency region (11.3−16GHz) as the reflection phase of the EBG surface varies

from 90 + 45 to 90 − 45 (shown in figure 2.11(b)).

A dipole antenna over a thin grounded high dielectric constant slab can also provide

a similar reflection phase curve against frequency and a similar return loss performance.

However, with the same dimensions with the EBG ground plane, the dielectric constant

of the thin slab need to be increased to εr = 20. More to the point, the EBG antenna

also demonstrates a higher gain and lower back lobe in terms of return loss due to the

suppression of surface waves, which will be discussed in the next section.

More detailed discussion about this EBG antenna can be found in reference [63].

As demonstrated in the above example, with the in-phase reflection, the radiation

element of the antennas can be placed very close to the EBG structure. In this way, many

low profile antenna designs can be realized. A low-profile cavity backed slot antenna on


(a)

(b)

(c)

Figure 2.9: (a) Geometry of a mushroom-like EBG structure, which consists of a latticeof metal plates, connected to a solid metal sheet by vertical conducting vias. The EBGsis with the following parameters: W = 0.12λ12GHz , g = 0.02λ12GHz , h = 0.04λ12GHz ,r = 0.005λ12GHz, εr = 2.20, where W is the patch width, g is the gap width, h is the substratethickness, r is the radius of the vias, and εr is the substrate permittivity. (b) The Antenna withthe PEC or PMC ground plane. (c) The Antenna with the EBG ground plane [63].


Figure 2.10: FDTD simulated return loss results of the dipole antenna over the EBG, PEC,and PMC ground planes of the same dimension. The dipole hight is 0.04λ12GHz with thePEC and PMC ground plane and the overall antenna height is 0.06λ12GHz [63].

(a) (b)

Figure 2.11: The FDTD result of (a)return loss of the dipole with its length varying from0.26λ12GHz to 0.60λ12GHz ; (b)The reflection phases of the mushroom-like EBG surface versusfrequency. The frequency band of the dipole model is 11.5 − 16.6 GHz according to −10dBreturn criteria. The frequency band of the plane wave model is 11.3 − 16GHz for 90 ± 45

reflection phase region. [63].


a UC-PBG structure was proposed in [81](figure 2.12). The cavity depth of the proposed

antenna (35 mil) is 16 times thinner than that of a conventional λ/4 wavelength cavity

slot antenna (559 mil). Similarly, a low-profile circularly-polarized patch antenna was

proposed in [62] (figure 2.13). The EBG ground plane size can be as small as (0.82λ ×0.82λ). The overall height of the EBG structure in conjunction with the antenna proposed

can be 0.1λ (λ corresponds to the working frequency in free space).

Figure 2.12: Schematic, cross section of the proposed slot antenna loaded with UC-PBG re-flector, and the top view of the UC-PBG. [81].

Figure 2.13: Configuration of a square curl antenna over an EBG surface. The size of theground plane is 1λ×1λ where λ is the free-space wavelength at working frequency 1.57GHz.The low profile curl antenna is with height of 0.06λ. [62].


2.4.2 Suppression of Surface Waves

Another property of metals is that they support surface waves. These are propagating

electromagnetic waves that are bound to the interface between two dissimilar materials,

e.g. metal and free space. At microwave frequencies, they are nothing more than the

normal currents that occur on any electric conductor [23]. The surface waves decay ex-

ponentially into the surrounding materials and will not couple to external plane waves

if the metal surface is smooth and flat [107]. However, the presence of bends, discontinu-

ities or surface texture will result in the surface waves radiating vertically [23].

On a finite ground plane, surface waves propagate until they reach an edge or corner,

where they can radiate into free space. More to the point, if multiple antennas share the

same ground plane, surface currents can cause unwanted mutual coupling.

By incorporating a special texture on a conducting surface, it is possible to alter its EM

properties. In the circumstance where the period of the surface texture is much smaller

than the wavelength, the structure can be described using an effective medium model,

and its qualities can be summarized and expressed by the surface impedance [107]. A

smooth conducting sheet has low surface impedance, but with a specially designed ge-

ometry, a textured surface can have high surface impedance.

The surface impedance of the textured metal surface can be characterized by an equiv-

alent parallel resonant LC circuit. At low frequencies, it is inductive, and supports trans-

verse magnetic (TM) waves. At high frequencies it is capacitive, and supports transverse

electric (TE) waves. Near the LC resonance frequency, the surface impedance is very

high. In this region, waves are not bound onto the surface; instead, they radiate into the

surrounding space [23].

With the suppression of surface waves, the radiation pattern of the antenna can be

improved by the EBG ground plane. Figure 2.14 shows a higher gain and lower back

lobe than the dipole antenna over thin grounded high dielectric constant (εr = 20) slab

with the same dimension.

The suppression of surface wave has been done with several geometries, such as a

metal sheet covered with small bumps [108], or a corrugated metal slab [107, 109] (and


(a)

(b)

Figure 2.14: Radiation pattern comparison of dipoles near the thin grounded high dielectricconstant slab and the EBG surface. (a) E-plane pattern. (b) H-plane pattern. The patterns arecalculated at the resonant frequency of 13.6 GHz. Since the high dielectric constant substrateis used in the grounded slab and strong surface waves are excited, the dipole on the slabshows a lower gain and higher back lobe [63].


the references therein), or an array of lumped-circuit elements to produce a thin two-

dimensional structure that should generally be described by band structure concepts,

even though the thickness and periodicity are both much smaller than the operating

wavelength [107]. The following shows an example of bumpy surfaces.

Kitson et al. used triangular lattice of bumps, patterned on a glass substrate coated

with a thin film of silver and found that propagation is prohibited in all directions for

modes with designed energy band [108].

The following are the details about their experiment. An excited atom can only relax

via the spontaneous emission of a photon if its energy matches that of an available op-

tical mode. The interaction between light and materials that are periodically modulated

on the scale of the wavelength of light leads to photonic band gaps. In light of these,

if such a gap coincides in energy with the excited state of an atom within the material

then spontaneous emission will be inhibited. In Kitson’s experiment, the surface consists

of hexagonal array of photoresist dots on a glass substrate (figure 2.15) coated with a

thin film of silver. The thin silver layer is used to support the propagation of the surface

plasmon polariton (SPP), which is a non-radiative transverse magnetic (TM) mode that

propagates at the interface between a metal and a dielectric. The repeat distance is 300 nm

and the dots have a radius of around 100 nm. A white light source and a computer con-

trolled spectrometer were used to produce a collimated TM-polarized monochromatic

beam in the wavelength range 400 to 800 nm. Monochromatic light incident on the prism

at a suitable internal angle θ resonantly excites SPPs that propagate on the silver/air in-

terface. This absorbs energy from the beam, reducing the reflectivity. The intensity of the

light reflected from the prism was monitored.

Figure 2.16 plots the reflectivity data recorded as a function of the photon energy and

kx, the component of the photon momentum in the plane of the silver/air interface. This

data is for a propagation direction Ψ of 100 (inset). The light regions represent high re-

flectivity and the dark regions correspond to low reflectivity. The regions of low reflectiv-

ity (dark) are a result of photons that have been absorbed through the resonant excitation

of SPPs. The dispersion curve of the SPP is directly mapped out by the reflectivity graph.

There is a clear gap in the dispersion curve, centered at 1.98 eV.


Figure 2.15: A scanning electron micrograph of the hexagonal array of dots. The dots arecomposed of photo-resist on a glass substrate. The surface has been coated with a thin filmof silver to support the propagation of the SPP [108].

Figure 2.16: A sample set of reflectivity data recorded as a function of the photon energy andkx, the component of the photon momentum in the plane of the silver/air interface. This datais for a propagation direction Ψ of 100 (inset) [108].


2.4.3 EBGs working in Defect Modes

Although EBG structures are based on periodicity, many interesting applications have

been generated because the defect EBG has the ability of supporting localized electro-

magnetic modes inside the frequency gap. Defects can be realized in many ways, includ-

ing removing segments of elements from a EBG, changing the shape or EM properties of

segments of elements, or replacing a segment of EBG with other materials. In this sec-

tion, examples of EBG superstrate, EBG waveguides, EBG-splitter and coupler and EBG

tunable filter will be presented.

• EBG Superstrates

In contrast to an EBGs being placed below an antenna as a substrate to miniaturize

the antenna and reduce backward radiation, EBGs can also be placed above the antenna

to enhance antenna performance in terms of directivity [35, 90, 110–112].

Figure 2.17 shows an example of a high directive EBG resonator antenna utilizing a

frequency-selective surface (FSS) superstrate designed by Lee et al.[111]. FSSs are chosen

in Lee’s design because of the fact that they are easy to fabricate using the etching pro-

cesses and they can help achieving a more compact EBG antenna design, especially in

terms of the antenna thickness. Figure 2.18 shows the field distribution of the EBG an-

tenna. A substantial enhancement on the directivity of the antenna by utilizing the EBG

superstrate is demonstrated in figure 2.19.

Enoch et al. reported in [35] in 2003 a device that radiates energy into a very narrow

angular range, based on a two dimensional EBG which is made of circular rods of radius

r = 0.6, with optical index nr = 2.9, lying in vacuum. The rods are arranged on a

triangular/hexagonal lattice with period a = 4 (distance between the centers of the rods).

The dispersion diagram of this EBG is presented in figure 2.20.

When the cell of this original EBG is expanded in the y direction, i.e., the vertical spac-

ing between two grids is enlarged from√

3a/2 ≈ 3.46 to 3.9, a new EBG is formed with

dispersion diagram shown in figure 2.21. At the wavelength λ = 7.93 corresponding to


Figure 2.17: Geometry of a patch antenna with a strip dipole FSS superstrate [111].


Figure 2.18: Field distribution of the EBG antenna [111].


Figure 2.19: Comparison of the radiation patterns of the FSS antenna composite and the patchantenna only. The directivity of the antenna is substantially enhanced [111].

the horizontal plane at the bottom of figure 2.21, the constant-frequency dispersion dia-

gram of the expanded EBG reduces to a small curve (see figure 2.22). As a consequence,

any source embedded in a slice of this expanded EBG will radiate only with a small range

of wavevector k.

By placing the original (unexpanded) EBG which exhibits bandgap at the same wave-

length, backwards radiation is eliminated. In this way, a device with field radiated

in a narrow angular range using almost any excitation is achieved. The half-power

beamwidth is equal to ±4.0(see figure 2.24), in comparison to the ±4.5 achievable by

an aperture having the same width illuminated by a field with constant amplitude and

phase. By increasing the lateral size of the structure and sacrificing the range of possible

working wavelengths, even narrower radiation pattern can be obtained [35].


Figure 2.20: Three dimensional dispersion diagram of the two dimensional EBG with circularrods lying in vacuum in a triangular lattice [35].

Figure 2.21: The dispersion diagram of the expanded EBG [35].


Figure 2.22: Constant-frequency dispersion diagram of the expanded EBG for λ = 7.93 [35].

Figure 2.23: Total field modulus radiated by the structure excited by the wire source at λ =7.93 [35].


Figure 2.24: Polar emission diagram for the structure excited by the wire source at λ = 7.93[35].

• EBG Waveguide, Splitters and Couplers

The efficient guiding and bending of light by integrated photonic devices is important

to the design of optical circuits for technological and optical computing applications.

Conventional dielectric or metallic waveguides have large scattering losses when sharp

bends are introduced. However, EBG studies enable an efficient way of guiding wave

even for sharp bends[60, 94–96].

Ozbay et al. demonstrated a zig-zag coupled cavity waveguide (CCW) formed by

removing consecutive rods from a two-dimensional EBG with rods loaded in free space

in triangular lattice (shown in figure 2.25(a)). A defect band is observed between 0.857ω0

to 0.949ω0. Complete transmission is seen for certain frequencies within the defect band.

Since the defect band shows sharp band edges compared to the EBG edges, it is suggested

in [65] that this property can be used to construct photonic switches by changing the

position of the defect band.

A Y-shaped splitter (shown in figure 2.26) is also presented in paper [65] in order

to demonstrate the splitting of EM power. The splitter consists of one input CCW and

two output CCWs. The input and output waveguide ports contain five and six coupled

cavities, respectively. As seen in figure 2.26(a), the propagating mode inside the input

CCW splits equally into two output CCW ports for all frequencies within the defect band.

The electric field distribution inside the splitter is computed for frequency ω = 0.916ω0


and is shown in figure 2.26(b).

If a single rod is placed to the left side of the junction of the Y-splitter, as shown in

figure 2.27(b), the splitter structure becomes a photonic switch. Because the symmetry of

the Y-shaped structure is broken, the power at each output waveguide port is drastically

changed. In this way, the amount of power flow into the output ports can be regulated.

(a) (b)

Figure 2.25: (a) Measured (solid line) and calculated (dotted line) transmission spectra of azig-zag CCW waveguide which contains 16 cavities. (b) Calculated field distribution of thezig-zag CCW waveguide. (Reference: figure 5 in [65].)


(a) (b)

Figure 2.26: (a) Measured (solid line) and calculated (dotted line) transmission spectra of a Y-shaped coupled-cavity based splitter. (b)Calculated power distribution inside the input andoutput waveguide channels of the splitter for frequency ω = 0.916ω0 . (Reference: figure 7 in[65].)


(a) (b)

Figure 2.27: (a) Measured (solid line) and calculated (dotted line) transmission spectra of acoupled-cavity switching structure. (b)Calculated field pattern clearly indicates that most ofthe power is coupled to the right port. (Reference: figure 8 in [65].)

EBG waveguide can also be realized based on the three-dimensional layer-by-layer

dielectric EBG structures [60, 96]. Figure 2.28(a) shows the experimental setup. The

woodpile EBG constructing the CCW’s consists of square shaped alumina rods hav-

ing a refractive index 3.1 at the microwave frequencies. The dimension of each rod is

0.32cm×0.32cm×15.25cm. The offset between the rods is 1.12cm. The bandgap of the

EBG extends from 10.6 to 12.8GHz. The defect is formed by removing a single rod from

a unit cell of the EBG crystal. The electric-field vector of the incident EM field was par-

allel to the rods of the defect lines. When the defects exists in adjacent unit cells, very

high transmission of the EM wave was observed within a frequency range inside the

bandgap of the otherwise perfect EBG, which is hereinafter referred to as the waveguide

band. Nearly a complete transmission was observed within the waveguide band for a

straight EBG waveguide (figure 2.28(b)) and greater than 90% transmission for an EBG

waveguide with 40 bend (figure 2.28(c)).


(a)

(b)

(c)

Figure 2.28: (a) Experimental setup for measuring the transmission-amplitude andtransmission-phase spectra of the coupled cavity waveguides (CCW) in 3D photonic crys-tals. (b) A mechanism to guide light through localized defect modes in a woodpile EBGs. (c)Bending of the EM waves around sharp corners [96].


• EBG Tunable Filters

A variety of filtering devices, based on the two-dimensional dielectric and metallic

EBGs containing nematic liquid crystal materials as defect elements or defect layers have

been explored by Kosmidou, et al. [100]. The FDTD simulation revealed that the defect

states originating from the liquid crystal impurities are tunable by the application of a

local static electric field. Narrow mode linewidths, almost 0.2 nm, and tuning ranges in

the order of tens of nanometers, covering in some cases both the C and L bands, can be

achieved with low operating voltages (0 − 4 V). So the proposed devices are rendered

suitable to operate as a spectral filter in modern optical communication systems.

Depicted in figure 2.29 is the defect EBG filter with air voids filled with liquid crystal

as defect elements. It is based on a photonic crystal consisting of a triangular lattice of

infinitely long air cylinders embedded in silicon. The radius of the circular cross section

of the air rods is set to 0.3a, where a is the lattice constant. The relative permittivity of Si

is considered to be εr = 11.4. The defect air voids at the centre of the device are filled with

E7, a typical nematic liquid crystal material characterized by ordinary and extraordinary

refractive indexes equal to 1.49 and 1.66, respectively. The defect area is surrounded by

five periods of the EBG cells, whereas the lateral width (in y− direction) of the device is

presumed to be infinite. The optical axis direction inside the liquid crystal, lying in the

xy plane, is defined by the tilt angle γ which can be altered by means of applying a local

static electric field.

Figure 2.30 and 2.31 show how defect modes can be tuned by changing the number

of defect void rows and by tuning the tilt angle γ in the defect cylinder rows.


Figure 2.29: Dielectric EBG structure having air-voids filled with liquid crystal as defect ele-ments [100].

Figure 2.30: Transmission coefficient for various numbers of defect cylinder rows when γ =45 [100].


Figure 2.31: Transmission coefficient for various values of the tilt angle γ in the case of twodefect cylinder rows [100].

In figure 2.32, it can be seen that the discrete defect cylinders are replaced by an E7

layer interposed between two blocks of the EBGs. Between the EBG and the E7 layer,

there exist two thin films of indium tin oxide (ITO) with relative refractive index equal to

1.9 and thickness 0.2a, on which electrods are attached in order to provide a local static

voltage across the liquid crystal slab. Figure 2.33 shows how the tilt angle γ in the E7 slab

varies according to the spatial position and to the different values of the applied voltage.

By altering the optical axis orientation inside of the defect slab, the thickness of E7 layer

Lc, or the distance Ls, the effective permittivity will changes, which in turn causes the

change of the position of the defect modes. Figure 2.34 demonstrates how the position

of the defect modes can be tuned by tuning the applied voltage, assuming Lc = 3a and

Ls = 0.567a.


Figure 2.32: Dielectric EBG structure with a liquid crystal defect layer [100].

Figure 2.33: Director orientation profile across the liquid crystal defect layer for differentvalues of the applied voltage [100].


Figure 2.34: Transmission coefficient versus normalized frequency for various values of theapplied voltage when Lc = 3a, Ls = 0.567a [100].

Another two-dimension EBG tunable filter is based on EBGs with infinite long metal-

lic rods loaded in a background material with low refractive index (Nr = 1.32) in a square

lattice. The radius of each cylinder cross section is set to 0.2a. As illustrated in figure

2.35(a), the E7 layer is placed between two blocks of EBGs, each with four periods of

the EBG cells in the x− direction, and with infinite long assumed in y− direction. Fig-

ure 2.35(b) shows the tuning effect of the defect modes by changing the applied voltage,

assuming Lc = 4a and Ls = a.

More tuning results can be found in paper [100].


(a)

(b)

Figure 2.35: (a) Metallic EBG structure with a liquid crystal defect layer. (b) Transmissioncoefficient versus normalized frequency for various values of the applied voltage when Lc =4a, Ls = a [100].


2.4.4 Subwavelength Imaging from the passband of the EBGs

Resolution of common imaging systems is restricted by the so-called diffraction limit [80].

This effect limits the minimum diameter d of spot of light formed at the focus of a lens,

given as:

d = 1.22λf

a(2.33)

where λ is the wavelength of the light, f is the focal length of the lens, and a is the diam-

eter of the beam of light, or (if the beam is filling the lens) the diameter of the lens. As a

result, even if one could fabricate an imperfection-free optical system, there would still

be a limit to the resolution of an image created by the conventional optical lens. In order

to overcome the diffraction limit, an artificial material (EBGs) with electromagnetic prop-

erties dramatically different from the materials from nature was proposed as a candidate

for a perfect lens and the theoretical possibility of sub-wavelength imaging was demon-

strated by Pendry in his seminal work [101]. Belov et al. experimentally demonstrated

a possibility to channel the near field distribution of a line source with sub-wavelength

details through a EBG crystal. Channelled intensity maximum having radius of λ/10

has been achieved by the use of an electrically dense lattice of capacitively loaded wires

[80, 103].

Figure 2.36 shows the experimental implements of the EBGs composed of capaci-

tively loaded wire medium (CLWM) and the EBG lens. Figure 2.37 shows isofrequency

contours for the frequency region ka = 0.43 ∼ 0.47. The isofrequency contour of the

host material for ka = 0.46 is shown as the small circle around Γ point. The part of the

isofrequency contour for the EBG corresponding to ka = 0.46, and located within the first

Brillouin Zone, is practically flat. This part is perpendicular to the diagonal of the first

Brillouin Zone. Thus, in order to achieve channeling regime, the interfaces of the slab is

oriented orthogonally to (11)-direction as shown in figure 2.36(b).

Figure 2.38 depicts the simulated amplitude and intensity distribution of a line source

working on ka = 0.46 is excited near the interface of the CLWM slab. A clear channel

through the slab can be observed together with a bright spot having a radius of λ/6

(determined from the intensity distribution at level max(intensity)/2) behind the slab.


An experimental verification of sub-wavelength imaging using CLWM slabs (shown in

figure 2.36(c) ) demonstrated an impressive resolution of λ/10. This lens can be designed

thick, since the required tunnel thickness is not related with the distance to the source.

Application of this CLWM lens being used in the near-field microscopy in the optical

range is suggested in [80], when the needle of a microscope used as a probe can be located

physically far from the tested source.

(a) (b)

(c) (d)

Figure 2.36: (a) A schematic illustration of the EBG structure composed of capacitively loadedwire medium (CLWM). (b) A schematic illustration of the lens formed by the EBG (CLWM).(c) The implemented CLWM EBG sample and the probe used in the measurements. (d) Aschematic illustration of the loaded wires (A piece of it) [103].


Figure 2.37: Isofrequency contours for the CLWM. The numbers correspond to values ofnormalized frequency ka [103].

(a) (b) .

Figure 2.38: Simulated distribution of electric field (a) amplitude and (b) intensity for thesub-wavelength lens formed by the CLWM operating in the canalization regime [103].


2.5 Summary

The periodic structures are presently one of the most rapidly advancing sectors in the

electromagnetic arena. In this chapter, the basic theory about the EBG structures, the

numerical methods that are popular in modelling EBGs, and examples from the vast

applications of the EBGs are reviewed. These applications include EBG antennas with

EBGs working as substrate or superstrate of the antennas, EBG waveguide, splitters and

couplers, EBG filters and EBG subwavelength imaging channels.

Chapter 3

A Brief Introduction to the

Finite-Difference Time-Domain

Method for Modelling the EBG

Structures

3.1 Introduction

The Finite-Difference Time-Domain (FDTD) Method [3] has been proven to be one of

the most effective numerical methods in the study of EBGs. As a direct solution to the

Maxwell’s equations, FDTD is simple and straightforward to solve complex EBG struc-

tures. As a time domain solution, it is convenient in dealing with the wide frequency

band characteristics of EBGs.

The foundation of FDTD is laid down by Yee in 1966 [4]. Yee chose a geometric rela-

tion for his spatial sampling of the vector components of the electric and magnetic fields

that robustly represent both the differential and integral forms of Maxwell’s equations.

The original Yee’s FDTD algorithm is second-order accurate in both time and space. Nu-

merical dispersion can be kept small by having a sufficient number of grid spaces per

59

Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 60

wavelength. This chapter gives out a brief review of the FDTD fundamentals. The tech-

niques of FDTD in EBG structure modelling are also presented.

3.2 Formulations of the Yee’s FDTD algorithm

Yee’s algorithm deals with both electric and magnetic fields in time and space using the

coupled Maxwell’s curl equations rather than solving for the electric field (or the mag-

netic field) alone with a wave equation.

3.2.1 Maxwell’s Equations

Consider a region of space that has no electric or magnetic current sources, but may have

materials that absorb electric or magnetic field energy. The time-dependent Maxwell’s

equations are given in differential and integral form by:

∂ B

∂t= −∇× E − σ∗ H (3.1)

∂ D

∂t= ∇× H − σ E (3.2)

∇ · D = 0 (3.3)

∇ · B = 0 (3.4)

and∂

∂t

∫∫A

B · dA = −∮

l

E dl −∫∫

Aσ∗ H · dA (3.5)

∂

∂t

∫∫A

D · dA =∮

l

H dl −∫∫

Aσ E · dA (3.6)

©∫∫

A

D dA = 0 (3.7)

©∫∫

A

B dA = 0 (3.8)


where σ∗ : equivalent magnetic loss (ohms/meter)

σ : electric conductivity (siemens/meter)

E : electric field, also called the electric flux density (volt/meter)

H : magnetic field strength (ampere/meter)

D : electric displacement field (coulomb/meter2)

B : magnetic field, also called the magnetic flux density (tesla, or volt-seconds/meter2)

A : surface area (meter2)

In linear, isotropic, nondispersive materials, D and B are related to E and H by:

D = ε E = ε0εrE and B = µ H = µ0µr

H (3.9)

where ε and µ are the medium permittivity and permeability, ε0 and µ0 are the per-

mittivity and permeability in free space, and εr and µr are the relative permittivity and

permeability.

Substituting (3.9) into (3.1) and (3.2) yields Maxwell’s curl equations in linear, isotropic,

nondispersive materials:∂ H

∂t= − 1

µ∇× E − 1

µσ∗ H (3.10)

∂ E

∂t=

1ε∇× H − 1

εσ E (3.11)

Writing out the vector components of the curl operators of (3.10), (3.11) yields the

following system of six coupled scalar equations under Cartesian coordinate:

∂Hx

∂t= − 1

µ

[∂Ez

∂y− ∂Ey

∂z

]− σ∗

µHx (3.12)

∂Hy

∂t= − 1

µ

[∂Ex

∂z− ∂Ez

∂x

]− σ∗

µHy (3.13)

∂Hz

∂t= − 1

µ

[∂Ey

∂x− ∂Ex

∂y

]− σ∗

µHz (3.14)


∂Ex

∂t=

1ε

[∂Hz

∂y− ∂Hy

∂z

]− σ

εEx (3.15)

∂Ey

∂t=

1ε

[∂Hx

∂z− ∂Hz

∂x

]− σ

εEy (3.16)

∂Ez

∂t=

1ε

[∂Hy

∂x− ∂Hx

∂y

]− σ

εEz (3.17)

The system of six coupled partial differential equations (3.12) - (3.17) forms the basis of

the FDTD numerical algorithm for electromagnetic wave interactions with general three-

dimensional objects.

Yee’s FDTD scheme discretizes Maxwell’s curl equations by approximating the time

and space first-order partial derivatives with centered differences using mesh and the

leapfrog scheme.

3.2.2 Yee’s Orthogonal Mesh

Yee’s algorithm centers its E and H components in a three-dimensional space so that

every E component is surrounded by four H components, and every H component is

surrounded by four E components. This provides a beautifully simple picture of three-

dimensional space being filled by an interlinked array of Faraday’s Law and Ampere’s

Law contours. It is possible to identify E components associated with displacement cur-

rent flux linking with H loops, as well as H components associated with magnetic flux

linking with E loops. This is shown in figure 3.1.

Utilizing Yee’s spatial gridding scheme, the spatial partial derivatives in equations

(3.12)-(3.17) can be approximated by central differential operators:

∂Ey

∂z|(i+ 1

2,j,k) ≈

Ey(i + 12 , j, k + 1

2 ) − Ey(i + 12 , j, k − 1

2)∆z

, etc. (3.18)

Consequently equation (3.12) becomes:


Figure 3.1: Yee’s spatial grid

∂Hx(i + 12 , j, k)

∂t+

1µ(i + 1

2 , j, k)· σ∗(i +

12, j, k)Hx(i +

12, j, k) = − 1

µ(i + 12 , j, k)

·[

Ez(i + 12 , j + 1

2 , k) − Ez(i + 12 , j − 1

2 , k)∆y

− Ey(i + 12 , j, k + 1

2 ) − Ey(i + 12 , j, k − 1

2 )∆z

]

(3.19)

3.2.3 Time Domain Discretization: the Leapfrog scheme and the Courant sta-

bility condition (CFL condition)

Yee’s algorithm also centers its E and H components in time. It is termed a leapfrog

arrangement (shown in figure 3.2). All of the E components in the modelled space are

computed and stored in memory using the previous E and the newly updated H data.

Then H is recomputed based on the previous H and the newly obtained E. This process

continues until time-stepping is concluded.

A central differential approximation is applied to equation (3.19):

∂Hx

∂t|(n∆t) ≈ H

(n∆t+ 12∆t)

x − H(n∆t− 1

2∆t)

x

∆t(3.20)


Figure 3.2: Leapfrog scheme - the temporal scheme of the FDTD method.

With approximation:

H(n∆t)x ≈ H

(n∆t+ 12∆t)

x + H(n∆t− 1

2∆t)

x

2(3.21)

equation (3.19) becomes a discretization equation (3.22) which can be solved easily

using computer program.

Hx(i +12, j, k)(n∆t+ 1

2∆t) =

1∆t − σ∗

2µ1

∆t + σ∗2µ

Hx(i +12, j, k)(n∆t− 1

2∆t)

− 1( 1∆t + σ∗

2µ) · µ∆y

[Ez(i +

12, j +

12, k)(n∆t) − Ez(i +

12, j − 1

2, k)(n∆t)

]

+1

( 1∆t + σ∗

2µ) · µ∆z

[Ey(i +

12, j, k +

12)(n∆t) − Ey(i +

12, j, k − 1

2)(n∆t)

]

(3.22)

Numerical stability of the Yee algorithm requires a bounding of the time-step (∆t) ac-

cording to the space increments (∆x, ∆y and ∆z). This is Courant-Friedrich-Levy (CFL)

stability condition, given in three dimensions by

∆t ≤ ∆tmax =1

c√

1∆x2 + 1

∆y2 + 1∆z2

(3.23)


In a cubic grid ( where ∆x = ∆y = ∆z = ∆), equation (3.23) can be expressed as

∆t ≤ ∆tmax =∆

c√

3(∆x = ∆y = ∆z = ∆) (3.24)

This upper bounding on ∆t enabled the successful application of FDTD methods to a

wide variety of electromagnetic wave modeling problems with moderate electrical size.

However, there are important potential applications of FDTD modelling which find the

Courant (CFL) stability bound too restrictive. For example, when simulating the fine-

scale geometric structures problems, the cell size ∆ need to be much less than the shortest

wavelength λmin. So for a fixed total time of simulation, the time step ∆t limited by CFL

can cause the total number of time-steps Nsim required to be very large:

Nsim =Tsim

∆t(3.25)

Hence, it makes the simulation computationally intensive and sometimes it is impos-

sible to achieve results in a reasonable simulation time.

Many attempts have been made to relax or even remove the stability constraint. Early

work involves applying alternating-direction-implicit (ADI) technique to the Yee’s grid

in order to formulate an implicit FDTD scheme [5]. In the first attempt of ADI-FDTD

in 1984 [5], the finite-difference operator was factored into three operators with each of

them being performed implicitly in respect to the three coordinate directions (namely x,

y, or z). However, this scheme was proved difficult to demonstrate numerical stability

at that time at that time [3]. It was from 1999, when a 2-D FDTD algorithm free of the

Courant stability conditions was proposed for a 2D-TE wave [22], the ADI method was

again applied. The resulting FDTD formulation was found to be unconditionally stable

[7, 22]. Consequently, the CFL constraint of the FDTD modeling is completely removed

and hence the selection of the time step is only dependent on the model accuracy required

[7].


3.3 Other Spatial Domain Discretization Schemes

Since the FDTD method is a grid based algorithm, mesh generation is very important.

A properly defined mesh will reduce numerical errors and increase computational effi-

ciency.

Orthogonal uniform meshes in the Cartesian coordinates are the most simple and

straightforward meshing scheme and they are most commonly used in FDTD modelling.

An orthogonal FDTD grid generally introduces the minimum numerical errors [113]. As

a result, even in a conformal mesh generation scheme, a boundary-orthogonal mesh is

preferred. However, the staircase approximation for curve structures often causes nu-

merical inaccuracy (or numerical dispersion) [114] and inefficiency.

A variety of mesh generation schemes have been developed, leading to many modi-

fied FDTD schemes. Beside the orthogonal staircase mesh, subgridding, nonorthogonal

meshes and hybrid meshes are all interesting candidates and have a wide range of appli-

cations, including problems where large structures with fine details and objects compris-

ing of curved or oblique surfaces.

3.3.1 Subgridding Mesh

In order for the numerical computation to yield accurate results, the spatial increment

∆x used in FDTD need to be much smaller (less than λ/10) than the structure of interest.

Consequently, simulating an electrically large domain with locally fine structures using

overall an fine mesh (small ∆x) and hence a small ∆t is computationally costly.

A subgridding mesh scheme is introduced to alleviate this problem. The basic idea

is to divide the computational volume into sub-regions and simulate them with variable

step sizes. A coarse grid is used in a large volume and fine meshes are applied only

around fine structures or discontinuities. To minimize the numerical reflections from an

abrupt transition from coarse meshes to fine ones, small scaling factors and a sequence


of subgrids can be applied when necessary [115].

The field updates can be explained by referring to figure 3.3. The fields inside the

coarse mesh sub-region and the fine mesh sub-region ( denoted in figure 3.3 by × and

black respectively), are calculated using the conventional FDTD equations (3.26) and

(3.27). The time increments in each sub-regions can be chosen according to the smallest

spatial increment or can be related to the spatial increments in each sub-region. On the

coarse-fine grid boundary, an interpolation is utilized to calculate the tangential electric

field (⊗) and the boundary layer magnetic fields in fine mesh (blue ). For the subgrid-

ding FDTD, various mesh and interpolation schemes can be found in [115–118].

The subgridding mesh scheme decreases the required computer memory and there-

fore expands the capability of the FDTD method. Meanwhile, it shows good numerical

stability [8].

Hn+ 1

2x (i, j, k) = H

n− 12

x (i, j, k)

− ∆t

µ

[En

z (i, j, k) − Enz (i, j − 1, k)

∆y− En

y (i, j, k) − Eny (i, j, k − 1)

∆z

]

Hn+ 1

2y (i, j, k) = H

n− 12

y (i, j, k)

− ∆t

µ

[En

x (i, j, k) − Enx (i, j, k − 1)

∆z− En

z (i, j, k) − Enz (i − 1, j, k)

∆x

]

Hn+ 1

2z (i, j, k) = H

n− 12

z (i, j, k)

− ∆t

µ

[En

y (i, j, k) − Eny (i − 1, j, k)

∆x− En

x (i, j, k) − Enx (i, j − 1, k)

∆y

](3.26)


En+1x (i, j, k) = En

x (i, j, k)

+∆t

ε

⎡⎣H

n+ 12

z (i, j + 1, k) − Hn+ 1

2z (i, j, k)

∆y− H

n+ 12

y (i, j, k + 1) − Hn+ 1

2y (i, j, k)

∆z

⎤⎦

En+1y (i, j, k) = En

y (i, j, k)

+∆t

ε

⎡⎣H

n+ 12

x (i, j, k + 1) − Hn+ 1

2x (i, j, k)

∆z− H

n+ 12

z (i + 1, j, k) − Hn+ 1

2z (i, j, k)

∆x

⎤⎦

En+1z (i, j, k) = En

z (i, j, k)

+∆t

ε

⎡⎣H

n+ 12

y (i + 1, j, k) − Hn+ 1

2y (i, j, k)

∆x− H

n+ 12

x (i, j + 1, k) − Hn+ 1

2x (i, j, k)

∆y

⎤⎦

(3.27)

Equations (3.28) and (3.29) show examples of how the field at the coarse-fine grid

boundary are updated using the neighbouring averaging scheme, subjected to the same

time interval calculated according to the fine mesh being used in all the sub-regions (see

figure 3.4).

Ef (2, 1, 1) =14Ec(1, 1) +

34Ec(2, 1) (3.28)

Hc(2, 2) =14Hf (2, 1, 2) +

14Hf (2, 2, 2) +

14Hf (2, 3, 2) +

14Hf (2, 4, 2) (3.29)

where Ef and Hf denote the fields in the fine mesh sub-region and Ec and Hc denote the

fields in the coarse mesh sub-region.


Figure 3.3: A cross section of a computational domain meshed according to the subgriddingalgorithm. Positions where the field quantities are calculated are shown. Since the spatialincrement in the fine mesh is only half that of the coarse grid, the time increment for the finemesh domain is equal to half of that in the coarse domain [8].


Figure 3.4: Enlarged view of the top-left corner of figure 3.3. In the coarse-fine grid boundary,the electric field in fine mesh (Ef (2, 1, 1)) is updated by electric field in coarse mesh region(Ec) using the neighbouring averaging equation (3.28); and the magnetic field in coarse mesh(Hc(2, 2)) is updated by magnetic field in the fine mesh region (Hf ) using the neighbouringaveraging equation (3.29).


3.3.2 Nonorthogonal Mesh

Many real world electromagnetic problems are characterized by geometries with oblique

angles or curved boundaries. Representing an oblique (or curved) surface using stair-

cased mesh usually requires very fine meshes which results in very small time steps in

the FDTD algorithm and consequently the computation becomes extensive.

As Maxwell’s equations are vector equations and can be implemented in any coordi-

nate system, they can be expressed in the nonorthogonal coordinate system as is shown in

[119]. In 1983, Holland introduced Maxwell’s equations in the nonorthogonal coordinate

system into the FDTD method and put forward a nonorthogonal FDTD (NFDTD) algo-

rithm, meshed with general nonorthogonal grids in a nonorthogonal coordinate system

[5]. In this mesh scheme, oblique surfaces or curved structures are meshed conformally

and more accurately with a coarser mesh. Since then, the NFDTD method has been re-

fined by many researchers including Yee [120], Lee [121], Mittra [122], Jurgens [123, 124],

Railton [55], Hao [18] and Douvalis [125] etc..

In nonorthogonal coordinate systems, an arbitrary vector can be expressed as a lin-

ear combination of two types of components according to two bases: the covariant and

contravariant components of this vector. In the FDTD modelling of an electromagnetic

(EM) problem, the covariant component relates physically to the flow of the vector along

the contour of an arbitrary surface while the contraviant component represents the flux

density of this vector passing through this surface.

Figure 3.5: A part of a three dimensional nonorthogonal mesh showing the covariant vectorswith the blue arrows and the contravariant vectors with the orange arrows.


However, compared with the conventional Cartesian FDTD method, the global curvi-

linear FDTD must store many additional metric tensors which are essential parameters in

the NFDTD scheme and are calculated from the spatial increment of each grid. While the

contravariant components are being updated in a Maxwellian way as in Yee’s scheme,

the covariant components have to be computed from the contravariant ones using two

additional projection equations applying the metric tensors.

As a result, the global curvilinear FDTD method is more computationally intensive

than the conventional Cartesian FDTD method. To alleviate this problem, a hybrid mesh

scheme, i.e. the Local Distorted NFDTD (LD-NFDTD) [18, 55] can be used. More details

of NFDTD method can be found in Chapter 4.

A mesh generating software GENGRID V2.2 is employed in this study. It is a free

software distributed by Computer Applied Fluid (CAF) Lab for educational purpose. It

is convenient in generating 2-D meshes, with four optional mesh stretching algorithms

provided.

3.3.3 Hybrid FDTD Meshes

Since each mesh generating scheme has its own advantages and drawbacks, it is quite in-

teresting to combine different mesh generation schemes to achieve efficient and accurate

FDTD grids for numerical simulations. An overview of example hybrid mesh generation

schemes is presented below.

• A Hybrid Conformal and Orthogonal Grid

This kind of mesh can be implemented by employing conformal cells at the curved

boundary within an underlying Cartesian coordinate system [126]. In other words, the

curvilinear meshes are used in the immediate vicinity of the curved boundary, while the

vast majority of the mesh away from the curved boundary can be rectangular/square.


Consequently, the additional computations from the NFDTD method are reduced. The

so-called ’Local-distorted nonorthogonal FDTD (LNFDTD)’ scheme also benefits from

greater accuracy and versatility than the conventional Yee’s Cartesian FDTD method but

with preserved efficiency [18].

• A Conformal Grid with the subgridding meshes

The subgridding in space domain can be applied combined with the conformal grid

leading to a subgridding NFDTD method. The computational efficiency is expected to

be improved in comparison with a subgridding scheme based on orthogonal mesh, or a

pure NFDTD scheme. Besides the subgridding in space, a time subgridding scheme can

also be employed in the NFDTD method and is reported to be helpful in reducing the

late time instability in the NFDTD method [126].

• A Conformal Grids with a triangular meshes

In [127], Schuhmann et al. observed that degenerated cells on the nonorthogonal

FDTD meshes are responsible for local field errors, leading to an irregular convergence

behaviour and also making the late time instability worse. To overcome this problem,

it is suggested that the NFDTD be combined with the triangular mesh fillings. Figure

3.6 demonstrates the staircase approximation, the nonorthogonal mesh, the degenerated

cell in the nonorthogonal mesh, and the proposed triangular fillings NFDTD respectively,

when meshing the cylindrical cavity. This scheme is regarded as a more flexible scheme

and may considerably improve the efficiency and accuracy of the NFDTD [127].


Figure 3.6: Meshes of the cross section of a cylindrical cavity. (a) the staircase approximation(b) nonorthogonal mesh (c) details of the degenerated cell in the nonorthogonal mesh whichis most responsible for the numerical error and late time instability; (d) the triangular fillingsNFDTD [127].


3.4 Boundary Conditions

In numerical modelling, many geometries of interest are defined in ’open’ regions where

the spatial domain of the computed field is unbounded in one or more coordinate di-

rections. Computers can only store and calculate a limited amount of data, therefore

an absorbing boundary condition (ABC) is often used to truncate the computational do-

main and to suppress spurious reflections of the outgoing numerical wave analogs to an

acceptable level.

Generally, there are two groups of ABCs: some derived from differential equations

and others employing absorbing materials. The most popular ABCs in the first group is

the one derived by Engquist and Majda [128] with the discretisation given by Mur[129].

These are based on approximating the outgoing wave equation by linear expressions

using a Taylor approximation. Material-based ABCs are realized by surrounding the

computational domain with a lossy material that dampens the outgoing fields. In this

group, the perfectly matched layer (PML) technique [130–135] which was put forward

by Berenger in 1994, demonstrated significantly better accuracy than most other ABCs so

it is widely used in the FDTD simulations.

An ideal EBG structure which has infinite periodicity does not exist in the real world.

Any EBG structure used in the real world is with boundaries. However, it is always

interesting and beneficial to study an infinite EBGs before the application of its finite-

sized dual. Periodic boundary condition (PBC) is the tool which enable the computation

of an infinite EBG through a limited computational domain. So EBGs which are infinite

in one-dimension, two-dimension and three-dimension can be modelled efficiently by

studying a few unit cells (or one unit cell) and applying proper PBCs.


3.4.1 Mur’s Absorbing Boundary Conditions (ABCs)

Engquist and Majda derived a theory of one-way wave equations which permits wave

propagation only in certain directions. For example, consider the two-dimensional wave

equations in Cartesian coordinates:

∂2U

∂x2+

∂2U

∂y2− 1

c2

∂2U

∂t2= 0 (3.30)

where U is a scalar field component and c is the wave phase velocity. We can define the

partial differential operator as:

L =∂2

∂x2+

∂2

∂y2− 1

c2

∂2

∂t2= D2

x + D2y −

1c2

D2t (3.31)

The wave equation is then written as

LU = L−L+U = 0 (3.32)

with the wave operator L been factored by L− and L+:

L− ≡ Dx − Dt

c

√1 − S2 and L+ ≡ Dx +

Dt

c

√1 − S2 (3.33)

with

S =Dy

Dt/c(3.34)

Solution of equation (3.35)

L−U = 0 (3.35)

is a solution of the wave function (3.30) however the wave only propagates towards the

−x direction. So, if tangential field components at the boundary of x = 1 satisfy equation

(3.35), this wave is not bouncing back into the computational domain.

As the space increment ∆ being much smaller than the smallest operating wave-

length, the variation of the the field components along grids should be small. So when


S is small, Taylor series can be used to approximate the square-root function in (3.33) by

two-term expansion: √1 − S2 ∼= 1 − 1

2S2 (3.36)

Substituting equation (3.36) into equation (3.33) yields:

L− = Dx − Dt

c

[1 − 1

2

(cDy

Dt

)2]

= Dx − Dt

c+

cD2y

2Dt(3.37)

Then substituting (3.37) into equation into (3.35), multiplying by Dt we get a second-

order accurate ABC at the x = 0 boundary.

DtL−U =

∂2U

∂x∂t− 1

c

∂2U

∂t2+

c

2∂2U

∂y2= 0 (3.38)

Mur used a simple central-difference scheme to interpret (3.38) in Yee’s space (with

spatial increment ∆x,∆y) and time (with time step ∆t) domain. For example, the mixed

x and t derivative in the second order ABC is written as:

∂2U |n1/2,j

∂x∂t=

12∆t

[(U |n+1

1,j − U |n+10,j

∆x

)−(

U |n−11,j − U |n−1

0,j

∆x

)](3.39)

In this way, the tangential field under discretization at the boundary (e.g. U |n+10,j ) is calcu-

lated as follows ( suppose ∆x = ∆y = ∆):

U |n+10,j = −U |n+1

1,j +c∆t − ∆c∆t + ∆

(U |n+1

1,j + U |n−10,j

)+

2∆c∆t + ∆

(U |n1,j + U |n0,j

)+

(c∆t)2

2∆(c∆t + ∆)(U |n0,j+1 − 2U |n0,j + U |n0,j−1

)+

(c∆t)2

2∆(c∆t + ∆)(U |n1,j+1 − 2U |n1,j + U |n1,j−1

)(3.40)

Remove the y-directive term, the first-order Mur’s ABC at x = 0 boundary is ob-

tained:

U |n+10,j = −U |n+1

1,j +c∆t − ∆c∆t + ∆

(U |n+1

1,j + U |n−10,j

)+

2∆c∆t + ∆

(U |n1,j + U |n0,j

)(3.41)


3.4.2 Perfect Matched Layers (PML)

In the perfect matched layer (PML)[130, 131] medium, each component of the electro-

magnetic field is split into two parts. In Cartesian coordinates, the six components yield

12 subcomponents denoted as Exy , Exz , Eyx, Eyz , Ezx, Ezy, Hxy, Hxz , Hyx, Hyz, Hzx, Hzy,

and Maxwell’s equations are replaced by 12 equations,

ε∂Exy

∂t+ σyExy =

∂Hzx + Hzy

∂y(3.42)

ε∂Exz

∂t+ σzExz = −∂Hyz + Hyx

∂z(3.43)

ε∂Eyz

∂t+ σzEyz =

∂Hxy + Hxz

∂z(3.44)

ε∂Eyx

∂t+ σzEyx = −∂Hzx + Hzy

∂x(3.45)

ε∂Ezx

∂t+ σzEzx =

∂Hyz + Hyx

∂x(3.46)

ε∂Ezy

∂t+ σzEzy = −∂Hxy + Hxz

∂y(3.47)

µ∂Hxy

∂t+ σ∗

xHxy = −∂Ezx + Ezy

∂y(3.48)

µ∂Hxz

∂t+ σ∗

yHxz =∂Eyz + Eyx

∂z(3.49)

µ∂Hyz

∂t+ σ∗

xHyz = −∂Exy + Exz

∂z(3.50)

µ∂Hyx

∂t+ σ∗

yHyx =∂Ezx + Ezy

∂x(3.51)

µ∂Hzx

∂t+ σ∗

xHzx = −∂Eyz + Eyx

∂x(3.52)

µ∂Hzy

∂t+ σ∗

yHzy =∂Exy + Exz

∂y(3.53)

where the parameters (σx, σy, σz , σ∗x, σ∗

y , σ∗z ) are homogeneous electric and magnetic con-

ductivities. Applying the central discretization to the temporal and spatial partial dif-

ferential operator yields FDTD equations with PML absorbing boundary. For example,


equation (3.42) becomes

En+ 1

2xy (i, j, k) =

1∆t − σy

2ε1

∆t + σy

2ε

En− 1

2xy (i, j, k) +

1( 1∆t + σy

2ε ) · ε∆y

· [Hnzx(i, j + 1, k) − Hn

zx(i, j, k) + Hnzy(i, j + 1, k) − Hn

zy(i, j, k)](3.54)

It is proven that for any propagating plane wave at an interface normal to a (a, b, c =

x, y, z) lying between PML media of same ε and µ, if the transverse conductivities σb,σ∗b ,σc,σ∗

c

are equal and all couples of conductivities (σx, σ∗x),(σy, σ

∗y),(σz, σ

∗z ) satisfy the matching

impedance condition (σ/ε = σ∗/µ), the wave will be transmitted into and in between the

PML layers with no reflection.

This approach is based on the splitting of the field components into two sub-components.

Sacks et al. [132], Gedney [133] etc. were able to formulate the PML based on a Maxwellian

formulation which removed the need to split the field. Veihl and Mittra presented a sim-

ilar work in a different way [134]. The application of the unsplit field PML to the FDTD

method is more computationally efficient, and it can be extended to nonorthogonal and

unstructured grid techniques.

3.4.3 Periodic Boundary Condition (PBC)

As is mentioned previously, in an infinite EBG structure, the wave or field is in Bloch’s

state and can be studied by the unit cell approach, in which only elements in one unit cell

are modelled and the fields in the adjacent unit cells are expressed explicitly using the

periodic boundary condition[76] (PBC):

F (r + la) = eik·la F (r) (3.55)


where a is the lattice constant, l is any vector with the same dimensions with a and all

entries integers, and F can be the covariant EM field components or the contravariant

EM fluxes.

Equation (3.55) is an important boundary condition in EBG modelling because it en-

ables an efficient numerical approach to analyze the infinite EBG structures. When the

EM field components of the cells on the boundary of the FDTD computational domain

are updated, the field components outside the computational domain will be referred to,

which, by utilizing PBCs, can be expressed using the field components inside the com-

putational domain and introducing a phase shift calculated from the dimension of the

unit cell. For example, equation (3.56) shows how Hx at boundary k = 1 is updated us-

ing PBCs when variable Ey(i, j, 0) is not available in the FDTD domain. Equation (3.57)

shows the updating equation for Ex at boundary k = maxz (where maxz denotes the

maximum number in z direction in the FDTD domain).

Hn+ 1

2x (i, j, 1) = H

n− 12

x (i, j, 1)

−∆t

µ

[En

z (i, j, 1) − Enz (i, j − 1, 1)

∆y− En

y (i, j, 1) − Eny (i, j,maxz) · e−ik·maxz∆z

∆z

]

(3.56)

En+ 1

2x (i, j,maxz) = E

n− 12

x (i, j,maxz)

+∆t

ε

[Hn

z (i, j + 1,maxz) − Hnz (i, j,maxz)

∆y− Hn

y (i, j, 1) · eik·maxz∆z − Hny (i, j,maxz)

∆z

]

(3.57)

The other equations for the field components at all the boundaries can be updated in

this way. Figure 3.7 (a) and (b) illustrate the interpolations of PBCs in EBGs with square

and triangular lattices respectively.


(a)

(b)

Figure 3.7: Periodic boundary conditions when calculating the infinite EBGs. The inclusionof the EBGs (marked by the solid black line) can be of any shape. The FDTD computationaldomain is limited to one unit cell/ super cell, marked by the green color. To calculate the fieldof the boundary layer of the computational domain (Layer 1 or Layer maxv in the graph),fields at the adjacent unit cell/ super cells are needed but they are outside the computationaldomain (marked with the yellow layer). However, they can be expressed using the fieldvalue within the domain (Layer maxv or Layer 1) applying equation (3.55). (a) PBCs in EBGswith rectangular lattice. The interpolations between cells are marked with the orange andthe pink lines. (b) PBCs in EBGs with triangular lattice. The interpolations between cells aremarked with the orange lines.


3.5 Band Gap Calculation

Through the FDTD modelling of the bandgap structures, some EBG characteristics can

be obtained in an intuitive way. For example, the pass-band/stop-band behaviour and

the transmission/reflection coefficients can be examined readily by the time-domain field

response. However, to study the dispersion relation (bandgap characteristics) of an EBG

structure, certain procedures involving post processing are necessary (see figure 3.8) and

will be presented in this section.

3.5.1 Source Excitation

A Modulated Gaussian Pulse (also termed as the Gabor pulse) with the following form

is used in the simulation of EBG structures:

S(t) = e−t2

2σ2 cos(2πξt + φ) (3.58)

where σ controls the (effective) time width of the pulse and consequently controls

the bandwidth of the source [136]. ξ and φ are the frequency and phase of the single

frequency wave that modulates the Gaussian pulse. Figure 3.9 shows a typical modulated

Gaussian waveform with σ = 8 × 10−6 s (= 40∆t; ∆t: sampling time), ξ = 100kHz, and

φ = 0 rad.

A modulated Gaussian pulse is chosen because it is efficient in time-frequency reso-

lution. Compared with a truncated pure sine pulse, the pulse energy of the modulated

Gaussian pulse is more concentrated near the center time of the pulse and the center fre-

quency. By controlling the time width σ, a desired form of a modulated Gaussian pulse

for a certain application can be defined.


Figure 3.8: The FDTD procedure in modelling EBG structures.


(a)

(b)

Figure 3.9: The modulated Gaussian pulse. (a)The shape of the modulated Gaussian pulsein the time domain and (b) the magnitude of its Fourier transform [136].


3.5.2 Dispersion Diagram Calculation

The dispersion diagram is very useful in the study of the bandgap characterization of an

infinite EBGs. As is introduced in last chapter, the dispersion diagram plots the possible

modes against the wave vector in the irreducible Brillouin zone. With proper periodic

boundary conditions, the infinite EBGs can be modelled by only one cell, or only a group

of cells if there are defects in the EBGs. The cell or the group of cells will be referred to in

the following paragraphs as the unit cell or the super cell respectively [61].

• The Unit Cell Approach

After a mesh is set up for the unit cell, one can randomly choose a few points in the

mesh as source points [76] and several points as probe (or monitored) points. The probe

points should be evenly located and dense enough to capture all the possible modes that

will be generated. The k vector from the irreducible Brillouin zone is used to set up the

periodic boundary conditions.

A modulated Gaussian pulse is applied at the source points to excite all the possible

electromagnetic (EM) modes of the EBG over a wide range of frequencies. As the FDTD

time evolution proceeds, only the true transmission modes remain in the computational

domain, and the pseudo transmission modes will eventually vanish [79]. After the tem-

poral responses of the probe points are recorded at every time step for a proper period

of time, the temporal signatures are Fourier transformed to obtain frequency spectra on

which peaks at certain frequency values can be found. These peaks indicate the exis-

tence of the supported transmission modes (eigenmodes of the EBGs) corresponding to

the wave vector k. Plotting these frequency values against the wave vector k gives the

dispersion diagram of the EBGs.

• The Super Cell Approach

The super cell approach works almost the same way as the unit cell approach does,


only the modelling domain consists of the defected cell(s) normally surrounded by reg-

ular unit cells. Periodic boundary conditions (PBCs) are also used to terminate the mod-

elling domain if the structure is assumed to be infinite. In the direction in which the

defects appears periodically, the PBCs are placed one period (of the defects) away from

each other. In the direction where the defects do not show periodicity, a number of unit

cells will be necessary in between the PBCs. Since the use of PBCs will represent the

defect periodically where it does not actually appear, the number of EBG unit cell layers

should be large enough to isolate the EM modes from the spurious defects in the neigh-

bouring super cell. On the other hand this unit cell layer should be minimized as much

as possible in order to maximize the efficiency of the computation. This number is often

determined experientially and is normally more than 10 [79]. Figure 3.10 shows the su-

per cell used by Chutinan et al. when they model the waveguide created by filling up one

column of the air holes in the EBGs [61].

Figure 3.10: The super cell of the waveguide created by filling up one column of the air holesin the EBGs [61]. In the y direction, the defects are periodic with period of one unit cell, soone unit cell is used in between the PBCs. In x direction, five unit cells are used between thePBCs to isolate the modes from the neighbouring spurious defects.

3.5.3 Transmission and Reflection Coefficient Calculation

If the bandgap structure is not infinite in its periodic direction, then transmission coeffi-

cient is helpful in order to find the band gap of the structure of interest.


Take the two-dimensional EBG with 4 arrays of cylindrical rods in x-direction and

infinitely loaded rods in y-direction in free space for example. Since the rods are infinite

in y-direction, the periodic boundary condition (PBC) can still be used to terminate the

computational domain in this direction. In x-direction, in which there exist 4 arrays of

rods, absorbing boundary condition (ABCs) can terminate the computational domain at

a proper distance away from the scattering structure. In this way of combining PBCs

with ABCs, the computational efficiency of the finite EBGs modelling can be increased.

The boundary condition setup is shown in figure 3.11.

Figure 3.11: Numerical Model for a two-dimensional EBG structure of semi-finite size. Thereare 4 arrays of cylindrical rods in x-direction and infinitely loaded rods in y-direction. So thecomputation domain is terminated by the PBCs in y-direction and by ABCs in x-direction. Aplane wave source in form of Modulated Gaussian Pulse is defined at one side of the EBGs.The responses at the other side of the EBGs are collected as Probe set 1 for calculating thetransmission coefficient (S21). The responses at the same side of the EBGs are collected asProbe set 2 for calculating the reflection coefficient (S11).

A plane wave travelling in the x-direction in form of a Modulated Gaussian pulse

is excited from a line source at one end of the structure. Temporal signatures of two

lines of probes at both sides of the structures (as shown in figure 3.11) are recorded and

analyzed to calculate the transmission and reflection coefficients respectively. For probe

sets 1, the Fourier Transformation can be applied directly to the time domain signal.

The averaged frequency spectra along the probe line shows the transmission coefficient

against frequency and bandgap can be found. To calculate the reflection coefficient, the

process is similar only the signal is recorded after the first pulse passed.

If the whole structure is finite in size, then the whole computational domain should

be terminated with ABCs in all directions. A modulated Gaussian pulse is excited in one

side of the EBGs (point source or line source) and the probes are defined at the other side.


Another simulation is required with the EBG structure replaced by free space. Then the

band gap can be found with the transmission coefficient from the EBGs calibrated by that

from the free space model (figure 3.12).

Figure 3.12: Numerical Model for an EBG structure of finite size. The computational domainis terminated by the ABCs in all the directions. A Modulated Gaussian Pulse is excited at oneside of the EBGs. The responses at Probe set 1 and Probe set 2 with the existence of the EBGare calibrated by those without EBG in the calculation of the transmission coefficient (S21)and the reflection coefficient (S11).


3.6 Summary

The Finite-Difference Time-Domain (FDTD) method is widely used because of its sim-

plicity for numerical implementation. It is a flexible means of directly solving Maxwell’s

time-dependent curl equations or their equivalent integral equations using the finite dif-

ference approximations as well. It can be used to solve various types of electromagnetic

problems, including the anisotropic and nonlinear problems. This chapter briefly re-

viewed the fundamentals about the FDTD method, including Yee’s spatial and temporal

grid, the updating formulation and two important boundary conditions - the absorbing

boundary conditions (ABCs) and the periodic boundary conditions (PBCs). The comput-

ing techniques specifically in calculating the EBG related parameters are also presented,

including the dispersion diagram calculation, and the transmission and reflection coeffi-

cient calculation.

Developments on the FDTD method towards a more accurate and more computa-

tionally efficient method never stop evolving. The next chapter will review two major

enhanced schemes over the Yee’s FDTD approach. A novel FDTD approach based on

these enhanced schemes is proposed.

Chapter 4

The Development of the

Alternating-Direction Implicit

Nonorthogonal FDTD Method

4.1 Introduction

Despite the simplicity and flexibility of the original Yee’s FDTD algorithm, the FDTD

applications require large amount of memory and central processing unit (CPU) time to

obtain accurate solutions when solving electrically large structure problems. Theoretical

studies on the FDTD show that the intensive memory and CPU time requirement mainly

come from the following two modelling constrains[3, 137]. Firstly, the spatial increment

step must be small enough in comparison with the wavelength (usually 10 ∼ 20 steps

per smallest wavelength) in order to obtain accurate field component values. Secondly,

the time step must be small enough to meet the Courant-Friedrich-Levy (CFL) stability

condition. If the time step is beyond the CFL bound, the FDTD scheme will become

numerically unstable leading to a spurious increase of the field values without limit as a

FDTD solution marches [138].

90

Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 91

There are two major extensions of the FDTD method addressing these two aforemen-

tioned constrains - the Nonorthogonal FDTD (NFDTD) method [5, 121] which improves

modelling accuracy while decreasing the requirement of the spatial increment, and the

FDTD algorithm based on the Alternating Direction Implicit method (ADI-FDTD) which

releases the CFL restriction on the FDTD time step[138].

Based on these two methods, a novel Alternating-Direction Implicit Nonorthogonal

Finite-Difference Time-Domain method (ADI-NFDTD) is developed. In the proposed

method, the curved structures are modelled based on the NFDTD algorithm. However,

by the introduction of the implicit method, the ADI-NFDTD method has demonstrated

an improvement in terms of the late time instability which is inherent in the NFDTD

method. This chapter will firstly review the NFDTD and the ADI-FDTD methods. Then

the ADI-NFDTD method is introduced and validated using numerical simulations.

4.2 A Brief Introduction to the Nonorthogonal Finite-Difference

Time-Domain Method

In 1983, Holland developed a nonorthogonal FDTD (NFDTD) algorithm and opened up

the possibilities of a more general, efficient and accurate numerical method [5]. In this

method, the FDTD technique is no longer restricted to an orthogonal cartesian grid. In-

stead, a generalized curvilinear coordinate system is used. As a direct consequence, an

arbitrary structure with curved boundary or oblique surface can be meshed conformally

and be modelled accurately, without employing the staircase approximation as is the

case in the Yee’s algorithm. Since the NFDTD method was proposed, it has been success-

fully applied to analyze optical dielectric waveguide, dielectric-loaded resonant cavity,

microstrip discontinuities and periodic structures at oblique incidence, etc.


In this section, the basic formulations of the NFDTD [5, 121] and its recent develop-

ment [139] will be reviewed. The numerical instability problem is also discussed.

4.2.1 The Curvilinear Coordinate Systems

Since Maxwell’s equations can be implemented in any coordinate system, the nonorthog-

onal coordinate system is introduced into the conventional FDTD method and a global

distorted grid conformal to the curved boundary is generated [119]. Therefore geometri-

cal characteristics involved in the spatial domain are clearly defined without the staircase

approximation. This section reviews the basics of the nonorthogonal coordinate system.

• Base vectors

(a) (b)

Figure 4.1: The definition of basic vectors (a) covariant vector; (b) contravariant vector.

There are two types of base vectors in the nonorthogonal coordinate system: covari-

ant and contravariant bases [113, 119]. Consider a coordinate line along which only the

coordinate ξ varies (see figure 4.1(a)), a tangential vector to the coordinate line is given

by:

limdξ→0

r(ξ + dξ) − r(ξ)dξ

= rξ (4.1)

The covariant bases in the nonorthogonal coordinates are tangential to the three co-

ordinate lines along which only ξi(i = 1, 2, 3) varies, designated as:

ai = rξi (4.2)


A normal vector to a coordinate surface on which the coordinate ξ is constant is given

by (see figure 4.1(b)). These normal vectors to the three coordinate surfaces are the three

contravariant base vectors of the curvilinear coordinate system, designated as:

ai = ∇ξi (4.3)

Figure 4.2 illustrates two types of base vectors for a nonorthogonal FDTD cell with

six sides.

Figure 4.2: The covariant basic vectors (a1, a2 and a3), and the contravariant basic vectors(a1, a2 and a3) in a nonorthogonal FDTD cell.

Then, an arbitrary vector can be expressed by:

E =3∑

i=1

eiai =3∑

i=1

eiai (4.4)

And the relationship holds:

ai · aj = δij (4.5)

where δ is the Kronecker delta:

δij =

⎧⎨⎩ 1 for i = j

0 for i = j, (4.6)


• Metric tensor

The metric tensor plays an important role and is also sufficient to characterize com-

pletely the geometrical properties of the object. Using the notation of Stratton [119], in the

curvilinear space a vector can be represented by its covariant and contravariant tensors,

as defined in equation (4.7)-(4.8).

¯gij =

⎡⎢⎢⎢⎣

g11 g12 g13

g21 g22 g23

g31 g32 g33

⎤⎥⎥⎥⎦

¯gij =

⎡⎢⎢⎢⎣

g11 g12 g13

g21 g22 g23

g31 g32 g33

⎤⎥⎥⎥⎦

where

gij = ai · aj (4.7)

andgij = ai · aj (4.8)

From equations (4.4) - (4.6) and equations (4.7), (4.8), the covariant and contravariant field

components are related to each other by the use of the matric tensors:

ei =3∑

j=1

gijej (4.9)

ei =3∑

j=1

gijej (4.10)

Denote g as :

√g = a1 · a2 × a3 . (4.11)

The reciprocal basis related to the unitary basis as

ai =aj × ak√

g( i,j,k in ascending order). (4.12)


• Curl in the Nonorthogonal Coordinates

The operation of curl in the nonorthogonal coordinates can be written as [113]:

∇× A =1√g

3∑i=1

[(∂Ak

∂uj

)ai −

(∂Ak

∂ui

)aj

]

( i,j,k in ascending order. ) (4.13)

If a contravariant base vector am(m = 1, 2, 3) is used to perform a scalar dot product

of equation (4.12), then combining with equation (4.5), equation (4.12) becomes:

∇× A · am =1√g

3∑i=1

[(∂Ak

∂uj

)(ai · am) −

(∂Ak

∂ui

)(aj · am)

]

=1√g

3∑i=1

[(∂Ak

∂uj

)δim −

(∂Ak

∂ui

)δjm

]

( i,j,k in ascending order and m = 1, 2, 3.) (4.14)

Taking m = 1 for example:

∇× A · a1 =1√g

3∑i=1

[(∂Ak

∂uj

)(ai · a1

)− (∂Ak

∂ui

)(aj · a1

)]

=1√g

3∑i=1

[(∂Ak

∂uj

)δi1 −

(∂Ak

∂ui

)δj1

]

=1√g

[∂A3

∂u2− ∂A2

∂u3

](4.15)

Equation (4.14) is the basis of Holland’s differential equations. Lee reformulated Hol-

land’s scheme and presented a more efficient time marching scheme in the nonorthogonal

FDTD algorithm in 1992 [121]. Lee’s scheme dominates ever since, until recently when

an alternative scheme was developed by Douvalis [139], which is also presented in the

following.


4.2.2 The Conventional Nonorthogonal FDTD

Based on the aforementioned basic concepts of nonorthogonal coordinates, Maxwell’s

equations can go through discretization in a nonorthogonal coordinate system. In a

source free and loss free medium, Maxwell’s curl equations are expressed as:

−µ∂ H

∂t= ∇× E (4.16)

ε∂ E

∂t= ∇× H (4.17)

Following equation (4.14), Maxwell’s curl equation can be written as:

−µ∂h1

∂t=

1√g

(∂e3

∂u2− ∂e2

∂u3

)

−µ∂h2

∂t=

1√g

(∂e1

∂u3− ∂e3

∂u1

)

−µ∂h3

∂t=

1√g

(∂e2

∂u1− ∂e1

∂u2

)

ε∂e1

∂t=

1√g

(∂h3

∂u2− ∂h2

∂u3

)

ε∂e2

∂t=

1√g

(∂h1

∂u3− ∂h3

∂u1

)

ε∂e3

∂t=

1√g

(∂h2

∂u1− ∂h1

∂u2

)(4.18)


In a similar procedure as in the Cartesian FDTD scheme, equations (4.18) are dis-

cretized using the central-difference approximations in both of the time and space do-

main. So equations (4.18) become:

h1(n+ 12)(i +

12, j, k) = h1(n− 1

2)(i +

12, j, k)

− ∆t

µ(i + 12 , j, k)

√g(i + 1

2 , j, k)·(

e(n)3 (i +

12, j +

12, k) − e

(n)3 (i +

12, j − 1

2, k))

+∆t

µ(i + 12 , j, k)

√g(i + 1

2 , j, k)·(

e(n)2 (i +

12, j, k +

12) − e

(n)2 (i +

12, j, k − 1

2))

h2(n+ 12)(i, j +

12, k) = h2(n− 1

2)(i, j +

12, k)

− ∆t

µ(i, j + 12 , k)

√g(i, j + 1

2 , k)·(

e(n)1 (i, j +

12, k +

12) − e

(n)1 (i, j +

12, k − 1

2))

+∆t

µ(i, j + 12 , k)

√g(i, j + 1

2 , k)·(

e(n)3 (i +

12, j +

12, k) − e

(n)3 (i − 1

2, j +

12, k))

h3(n+ 12)(i, j, k +

12) = h3(n− 1

2)(i, j, k +

12)

− ∆t

µ(i, j, k + 12)√

g(i, j, k + 12 )

·(

e(n)2 (i +

12, j, k +

12) − e

(n)2 (i − 1

2, j, k +

12))

+∆t

µ(i, j, k + 12)√

g(i, j, k + 12 )

·(

e(n)1 (i, j +

12, k +

12) − e

(n)1 (i, j − 1

2, k +

12))

e1(n+1)(i, j +12, k +

12) = e1(n)(i, j +

12, k +

12)

+∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2 )·(

h(n+ 1

2)

3 (i, j + 1, k +12) − h

(n+ 12)

3 (i, j, k +12))

− ∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2 )·(

h(n+ 1

2)

2 (i, j +12, k + 1) − h

(n+ 12)

2 (i, j +12, k))


e2(n+1)(i +12, j, k +

12) = e2(n)(i +

12, j, k +

12)

+∆t

ε(i + 12 , j, k + 1

2 )√

g(i + 12 , j, k + 1

2 )·(

h(n+ 1

2)

1 (i +12, j, k + 1) − h

(n+ 12)

1 (i +12, j, k)

)

− ∆t

ε(i + 12 , j, k + 1

2 )√

g(i + 12 , j, k + 1

2 )·(

h(n+ 1

2)

3 (i + 1, j, k +12) − h

(n+ 12)

3 (i, j, k +12))

e3(n+1)(i +12, j +

12, k) = e3(n)(i +

12, j +

12, k)

+∆t

ε(i + 12 , j + 1

2 , k)√

g(i + 12 , j + 1

2 , k)·(

h(n+ 1

2)

2 (i + 1, j +12, k) − h

(n+ 12)

2 (i, j +12, k))

− ∆t

ε(i + 12 , j + 1

2 , k)√

g(i + 12 , j + 1

2 , k)·(

h(n+ 1

2)

1 (i +12, j + 1, k) − h

(n+ 12)

1 (i +12, j, k)

)

(4.19)

Furthermore, in order to fulfill an explicit time-marching nonorthogonal FDTD scheme,

the covariant field components (ei/hi) must be calculated using its dual field (ei/hi). So,

additional projection equations (4.10) are needed. As the covariant field values should

be in the same space position with the contravariant ones, an interpolation scheme (i.e. a

neighbouring averaging projection scheme) is introduced as follows:

v1(i, j, k) = g11(i, j, k)v1(i, j, k)

+g12(i, j, k)

4(v2(i − 1, j, k) + v2(i − 1, j + 1, k) + v2(i, j, k) + v2(i, j + 1, k)

)+

g13(i, j, k)4

(v3(i − 1, j, k) + v3(i − 1, j, k + 1) + v3(i, j, k) + v3(i, j, k + 1)

)(4.20)

where v stands for either e or h. Thus the contravariant components v1, v2, and v3 can be

obtained from equations (4.19). The working scheme of a nonorthogonal FDTD can be

summarized and illustrated by the following chart:


Figure 4.3: The NFDTD iteration scheme.

4.2.3 Lee’s NFDTD Algorithm

Lee reformulated Holland’s nonorthogonal FDTD algorithm based on the integral form

of Maxwell’s equations [121]. This modification results in a more efficient time marching

procedure with respect to the one presented by Holland.

Consider a source free and loss free isotropic medium. Taking Ampere’s Law integral

form in one facet of a nonorthogonal cell, for example:

∂

∂t

∫∫S

ε E · dS =∮

l

H dl (4.21)

When equation (4.21) is applied to the the facet of (a1 ×a2) (or a3, figure 4.4), after the

discretization, we obtain:

ε∂ E

∂t· (a1 × a2) =

2∑i=1

H · ai (4.22)

From equation (4.12):

a3 =a1 × a2√

g(4.23)

So equation (4.22) becomes:

ε∂ E

∂t· (a3√g) =

2∑i=1

H · ai (4.24)


Figure 4.4: The facet (a1 × a2) (or a3) on a NFDTD cell is highlighted using the light orangecolour.

Combining equation (4.24) with (4.4) finally yields an equation which is equivalent to

equations (4.18) from the Holland’s scheme:

∂e3

∂t=

1ε√

g

2∑i=1

hi (4.25)

The difference of Holland’s and Lee’s scheme in terms of the central-difference equa-

tions (4.19) stands in that Holland transformed e and h in equations (4.19) into E and H

which stand for the exact value of the field components using equations (4.26); while Lee

kept e and h in equations (4.19) for the neat expression.

Ei =√

giiei

Ei =√

giiei

H i =√

giihi

Hi =√

giihi (4.26)


4.2.4 Douvalis’ NFDTD Algorithm

Douvalis proposed a new nonorthogonal FDTD algorithm [139] based on the curl func-

tion in the nonorthogonal coordinates:

∇× A =1√g

3∑i=1

[(∂Ak

∂uj

)ai −

(∂Ak

∂ui

)aj

]

( i,j,k in ascending order.) (4.27)

In Holland’s algorithm, a contravariant base vector am ( m = 1, 2, 3.) is used to per-

form a scalar dot product of equation (4.13), leading to equation (4.14). This results in

contravariant field vector at the left hand side of the equation and covariant field vector

at the right hand side. To unify the type of base vectors, Douvalis used a covariant base

vector am to perform the scalar dot product and successfully eliminate the contravariant

base vector [139]. Then equation (4.27) becomes:

∇× A · am =1√g

3∑i=1

[(∂Ak

∂uj

)(ai · am) −

(∂Ak

∂ui

)(aj · am)

]

=1√g

3∑i=1

[(∂Ak

∂uj

)gim −

(∂Ak

∂ui

)gjm

]

(i,j,k in ascending order and m=1,2,3.) (4.28)

An example of equation (4.28) in calculating e1 is given in equation (4.29):

ε∂e1

∂t=

1√g

3∑i=1

[(∂hk

∂uj

)gi1 −

(∂hk

∂ui

)gj1

](4.29)

Then discretization and central difference approximation on the partial differential

operator are performed as shown in equation (4.30). Thus all field components involved

are covariant variables. In this way, the inaccuracy introduced by neighbour averaging

of contravariant components when calculating covariant ones are eliminated.


e1(n+1)(i, j +

12, k +

12) = e1

(n)(i, j +12, k +

12)

+g11(i, j + 1

2 , k + 12)∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

3 (i,j+1,k+ 12) − h

(n+ 12)

3 (i,j,k+ 12)

)

− g21(i, j + 12 , k + 1

2)∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

3 (i+ 12,j+ 1

2,k+ 1

2) − h

(n+ 12)

3 (i− 12,j+ 1

2,k+ 1

2)

)

+g21(i, j + 1

2 , k + 12)∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

1 (i,j+ 12,k+1) − h

(n+ 12)

1 (i,j+ 12,k)

)

− g31(i, j + 12 , k + 1

2)∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

1 (i,j+1,k+ 12) − h

(n+ 12)

1 (i,j,k+ 12)

)

+g31(i, j + 1

2 , k + 12)∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

2 (i+ 12,j+ 1

2,k+ 1

2) − h

(n+ 12)

2 (i− 12,j+ 1

2,k+ 1

2)

)

− g11(i, j + 12 , k + 1

2)∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

2 (i,j+ 12,k+1) − h

(n+ 12)

2 (i,j+ 12,k)

)(4.30)

However, equation (4.30) is not the final NFDTD equation because field components

such as h1 (i,j,k+ 12) are not defined at the referred spatial position. So another neighbour

averaging scheme has to be introduced in order to interpret these variables. For example,

h1 (i,j,k+ 12)

=14

[h1(i +

12, j, k) + h1(i +

12, j, k + 1) + h1(i − 1

2, j, k) + h1(i − 1

2, j, k + 1)

](4.31)

It can be proven (see Appendix B) that compared with the conventional Holland’s

or Lee’s scheme, the neighbour averaging is restricted to only the curl operator of the

Maxwell’s equation. That explains the fact that in NFDTD simulations, Douvalis’ NFDTD

scheme demonstrated longer stable results in comparison with the conventional Hol-

land’s or Lee’s scheme [139].


4.2.5 Numerical Stability of the Nonorthogonal FDTD Method

• Generalized Courant-Friedrich-Levy (CFL) stability condition

As in the conventional Yee’s FDTD algorithm, the time interval ∆t used in the nonorthog-

onal FDTD method is also constrained by the Courant-Friedrich-Levy (CFL) stability con-

dition. Consider a two-dimensional NFDTD cell shown in figure 4.5, ξi and ηj denote the

skewed coordinates, and θij denote the rotating angle from ξi to ηj . As is analyzed in

[140], the CFL stability condition for a nonorthogonal mesh should be extended as equa-

tion (4.32), the minimum value is taken so that the simulation time interval dt should

agree with (4.32) for all the cells in the NFDTD grid. The orthogonal CFL condition is

the special case when θ = 90. (Throughout this thesis dt denotes the time interval used

in the simulation while ∆t denotes the time interval corresponding to the CFL stability

condition or relating to formulae derivation.)

Figure 4.5: A two-dimensional nonorthogonal FDTD cell.

dt ≤ ∆t = min [∆ti,j] = min

[sin θi,j

c ·√

(∆ξi)2 + (∆ηi)2

](4.32)

For an orthogonal mesh, it is straightforward to calculate ∆t. However, with a nonorthog-

onal mesh, it is not trivial to find the minimum value as diverse cells exist. Moreover, for

the grid containing universally large and locally very small or skewed cells, the choice of

∆t for the smallest ∆ti,j may result in low efficiency in the NFDTD simulation.


• Late time instability in the NFDTD algorithm

It has been reported [19, 125, 133, 141] that the nonorthogonal FDTD algorithm often

suffers from the late time instabilities. Here, the unstable solution means that an unlim-

ited increase of non-physical electric or magnetic charges. The so-called late time insta-

bility arises from the projecting scheme for the covariant and contravariant components

in locally distorted cells, and it has been proved to be inherent in the NFDTD scheme

[19]. More specifically, it can be explained in two points of view:

1. In terms of the divergence free condition, the locally conformal mesh renders the base vec-

tor spatial variation [125], which causes the violation of the divergent free condition of

Maxwell’s equations in the source free domain [19]. As a consequence, the divergence works

as a numerical source/error which amplifies itself by each iteration and hence causes the late

time instability;

2. Mesh containing highly elongated or very narrow cells with interior angles breaching 0

or 180 leads to the projection matrices losing their positive definite properties [133], [141]

etc.

Many efforts have been put forward to reduce the late time instability [142–145]. Ged-

ney et al. suggested an approximation to force a symmetry in the projection operator and

rendered it positive definite[133].

For simplicity in deriving a sufficient stability condition in [133], a lossless and non-

magnetic domain is assumed. Expressing the NFDTD explicit formula in matrix form:

bn = bn−1 − ∆tCeDεAddn− 1

2 (4.33)

dn+ 12 = dn− 1

2 + ∆tChAbbn (4.34)

where d and b are vectors of the discrete vector flux densities, the superscripts refer to

discrete time, Ce and Ch represent the discrete contour integrals of the electric and mag-

netic fields about the primary and the secondary cell faces, respectively. Dε is a diagonal


matrix with entries representing the inverse of the relative permittivity and Ab and Ad

are the projection matrices.

Introducing the vector

wn =

⎡⎣ bn

dn+ 12

⎤⎦

Then equation (4.33) and (4.34) can be reposed as

wn = Gwn−1 (4.35)

where

G =

⎡⎣ I − ∆tCeDεAd

−∆tChAb I − ∆t2ChAbCeDεAd

⎤⎦

To ensure stability for the passive recursive system w, it is necessary that G must have

a complete set of distinct eigenvalues and eigenvectors and all the eigenvalues of G must

satisfy |λG| ≤ 1. To that end, it is proved in [133] that the matrix M where

M = ChAbCeDεAd (4.36)

which represents the discrete curl-curl operation should be positive definite. However,

in the NFDTD method, if εr is inhomogeneous, then M = ChAbCeDεAd may no longer

be positive definite since DεAd will be nonsymmetric.

It is suggested in [133] that equation (4.33) can be approximated as:

bn = bn−1 − ∆tCeD1/2ε AdD

1/2ε dn− 1

2 (4.37)

Then M is rewritten as

M ≈ ChAbCeD1/2ε AdD

1/2ε (4.38)

in which positive definite is forced for inhomogeneous medium because D1/2ε AdD

1/2ε is

symmetric positive definite.


Using this technique, the projection operation in NFDTD (equation (4.20)) is stated

as:

Ei = giiDi

εri

+gij

4

4∑p=1

D2p√

εriεrp

+gik

4

4∑q=1

D3q√

εriεrq

It is demonstrated that the time for instability occurring is dramatically postponed by

the use of this approximation while the accuracy of the NFDTD method is not signifi-

cantly degraded [133].

Weiland suggested that the dual mesh can be used to generate the matrix tensor g to

give a more stable result. In his proposed method [127],

gij = Ai · Aj (4.39)

is used instead of the conventionally used

gij = Ai · Aj (4.40)

where Ai and Ai are covariant basis vectors in primary grid G and dual grid G (see figure

4.6).

Figure 4.6: Co- and contravariant basis vectors in primary grid G and dual grid G[127].

As created by calculating the centers of the primary grid cells, the dual grid is a more


’averaged’ grid which is not as distorted as the primary one (see figure 4.6). Figure 4.7

shows that simulation based on the dual grid demonstrated a much longer stable tempo-

ral iteration than that based on the primary grid.

Figure 4.7: Electric field at a degenerated cell: Stable ’D’-case (dual basis vectors for theevaluation of

√g) and unstable ’P’-case, where the primary basis vectors are used [127].

A hybrid FDTD method with Matrix Pencil Technique was developed by Adve et al.

to overcome the late time stability [144]. In this method, the time-domain response in

a short period of time after the excitation has died down is obtained using the FDTD

method. Usually during this short period the instabilities have not set in. Then this re-

sponse is modelled as a sum of complex exponentials and used as an input to the matrix

pencil algorithm which results in a stable time-domain response for all the time [146]. In

this way, the late time instability is eliminated by the use of the Matrix Pencil Technique.

Douvalis’ NFDTD scheme which was introduced in the previous section has demon-

strated an improvement in the late time instability in the numerical simulations [139].

Theoretical analysis indicates that an unconditional stable LNFDTD algorithm is possi-

ble if the inherent non-divergence of the fields could be eliminated [19].


4.3 Alternating Direction Implicit Finite-Difference Time-Domain

Method

4.3.1 Introduction

As introduced in last Chapter, Yee’s leapfrogging scheme in the time domain leads to an

accurate and elegent scheme in the FDTD. However, to assure a numerically stable op-

eration during an FDTD simulation, the CFL stability condition must be satisfied, which

sometimes restricts the computational efficiency. In light of this, a computationally ef-

ficient candidate approach for the release of the CFL condition is to use the alternating

direction implicit (ADI) algorithm rather than the explicit Yee’s leapfrogging. In fact,

work with ADI-FDTD in the early 1980’s has already achieved promising results for

two-dimensional models [150, 152]. However, using the ADI-FDTD of [150, 152], it was

proved difficult to demonstrate the required numerical stability for the three-dimensional

case, and research in this area was largely discontinued [3].

However, there was a revival of interest in the use of the ADI-FDTD to obtain uncon-

ditional numerical stability around year 2000 [7, 22, 147]. In this section, Zheng et al.’s

ADI-FDTD formula, as well as the derivation of the unconditional numerical stability for

the full three-dimensional modelling are briefly introduced.

4.3.2 The Idea of Alternating Direction Implicit (ADI) Method

Consider a two-dimensional diffusion equation:

∂u

∂t= D

(∂2u

∂x2+

∂2u

∂y2

)(4.41)

The ADI scheme requires each time step ∆t to be divided into two sub-steps ∆t/2. In


each sub-step, a different dimension is treated implicitly:

un+ 1

2j,l = un

j,l +12α

(δ2xu

n+ 12

j,l + δ2yu

nj,l

)(4.42)

un+1j,l = u

n+ 12

j,l +12α

(δ2xu

n+ 12

j,l + δ2yu

n+1j,l

)(4.43)

Here,

α≡D∆t

∆2and ∆ ≡ ∆x = ∆y (4.44)

The advantage of this approach is that each substep requires only the solution of a

simple tridiagonal system [138].

4.3.3 ADI Applied to the Finite-Difference Time-Domain Method

In this section, the ADI-FDTD formulae for the two-dimensional (2-D) transverse electric

(TE) mode and the three-dimensional (3-D) wave propagation are presented. The ADI-

FDTD formulation for the 2-D Transverse Magnetic (TM) modes can be obtained easily

by the dual procedure of the TE formulation.

• The Two-Dimensional ADI-FDTD Formulae for TE Modes Modelling

In order to simplify the problem, it is assumed that the medium in which the wave

propagates to be vacuum, and all cells in the computational domain to have the same

size. The electromagnetic-field components are arranged on the cells in the same way as

that using the conventional FDTD method.

The calculation for one discrete time step is performed using two procedures as pre-

sented by equation set (4.45 -4.50) in two procedures.


First Procedure:

En+ 1

2x (i +

12, j) = En

x (i +12, j)

+∆t

2ε∆y

[Hn

z (i +12, j +

12) − Hn

z (i +12, j − 1

2)]

(4.45)

En+ 1

2y (i, j +

12) = En

y (i, j +12)

− ∆t

2ε∆x

[H

n+ 12

z (i +12, j +

12) − H

n+ 12

z (i − 12, j +

12)]

(4.46)

Hn+ 1

2z (i +

12, j +

12) = Hn

z (i +12, j +

12)

+∆t

2µ∆y

[En

x (i +12, j + 1) − En

x (i +12, j)]

− ∆t

2µ∆x

[E

n+ 12

y (i + 1, j +12) − E

n+ 12

y (i, j +12)]

(4.47)

Second Procedure:

En+1x (i +

12, j) = E

n+ 12

x (i +12, j)

+∆t

2ε∆y

[Hn+1

z (i +12, j +

12) − Hn+1

z (i +12, j − 1

2)]

(4.48)

En+1y (i, j +

12) = E

n+ 12

y (i, j +12)

− ∆t

2ε∆x

[H

n+ 12

z (i +12, j +

12) − H

n+ 12

z (i − 12, j +

12)]

(4.49)

Hn+1z (i +

12, j +

12) = H

n+ 12

z (i +12, j +

12)

+∆t

2µ∆y

[En+1

x (i +12, j + 1) − En+1

x (i +12, j)]

− ∆t

2µ∆x

[E

n+ 12

y (i + 1, j +12) − E

n+ 12

y (i, j +12)]

(4.50)


In the first procedure, the Ey components and the Hz components as shown in equation(4.46)

and (4.47), are defined as synchronous variables. Equation (4.46) cannot be used for direct

numerical calculation, so that equation(4.51) derived from equations (4.46) and (4.47) by

eliminating the Hn+ 1

2z components need to be solved instead. There after, equation(4.47)

can be calculated directly using the En+ 1

2y components calculated by (4.51) as follows:

En+ 1

2y (i − 1, j +

12) −

[(2√

εµ∆x

∆t

)2

+ 2

]E

n+ 12

y (i, j +12) + E

n+ 12

y (i + 1, j +12)

= −(

2√

εµ∆x

∆t

)2

Eny (i, j +

12)

−(

2µ∆x

∆t

)·[H

n+ 12

z (i +12, j +

12) − H

n+ 12

z (i − 12, j +

12)]

+∆x

∆y·[En

x (i +12, j + 1) − En

x (i +12, j) − En

x (i − 12, j) + En

x (i − 12, j + 1)

](4.51)

In the second procedure, the Ex component and the Hz components are defined as

synchronous variables. Similarly equation (4.52) derived from equations (4.48) and (4.50)

by eliminating the Hn+1z components is shown as follows:

En+1x (i +

12, j − 1) −

[(2√

εµ∆x

∆t

)2

+ 2

]En+1

x (i +12, j) + En+1

x (i +12, j + 1)

= −(

2√

εµ∆y

∆t

)2

Enx (i +

12, j)

−(

2µ∆y

∆t

)·[H

n+ 12

z (i +12, j +

12) − H

n+ 12

z (i +12, j − 1

2)]

+∆y

∆x·[En

y (i + 1, j +12) − En

y (i, j +12) − En

y (i + 1, j − 12) + En

y (i, j − 12)](4.52)

Once the En+ 1

2y (En+1

x ) component is calculated, Hn+ 1

2z (Hn+1

z ) can be updated using

equation (4.47) ( equation (4.50)) explicitly. Since the simultaneous linear equations (4.51)

and (4.52) can be written in tri-diagonal matrix form, the computational costs are not very


significant.

• The Three-Dimensional ADI-FDTD Formula

Extension of ADI to a general three-dimensional case once faced difficulty and de-

layed the deployment of ADI to FDTD. However, a different three-dimensional ADI-

FDTD method was reported by Zheng et al. [7] which caused the revival of interest in the

use of ADI-FDTD.

In the method proposed by Zheng et al. [7], the ADI scheme is applied in terms of

the sequence of the terms on the right-hand-side (RHS) of the equations (the first and

the second terms), rather than in terms of the coordinate directions. It then leads to

only two alternations in the computation directions in three dimensions. As a result, at

each substep, the computations are performed with respect to all the three coordinate

directions but with different terms or components. For instance, in (4.53), one can see

that x-component (Ex) is implicitly computed by the z directional component, while in

(4.56) the x-component (Hx) is explicitly calculated from the z directional components.

First Procedure:

En+ 1

2x (i +

12, j, k) = En

x (i +12, j, k)

+∆t

2ε∆y

[H

n+ 12

z (i +12, j +

12, k) − H

n+ 12

z (i +12, j − 1

2, k)]

− ∆t

2ε∆z

[Hn

y (i +12, j, k +

12) − Hn

y (i +12, j, k − 1

2)]

(4.53)

En+ 1

2y (i, j +

12, k) = En

y (i, j +12, k)

+∆t

2ε∆z

[H

n+ 12

x (i, j +12, k +

12) − H

n+ 12

x (i, j +12, k − 1

2)]

− ∆t

2ε∆x

[Hn

z (i +12, j +

12, k) − Hn

z (i − 12, j +

12, k)]

(4.54)


En+ 1

2z (i, j, k +

12) = En

z (i, j, k +12)

+∆t

2ε∆x

[H

n+ 12

y (i +12, j, k +

12) − H

n+ 12

y (i +12, j, k − 1

2)]

− ∆t

2ε∆y

[Hn

x (i, j +12, k +

12) − Hn

x (i, j − 12, k +

12)]

(4.55)

Hn+ 1

2x (i, j +

12, k +

12) = Hn

x (i, j +12, k +

12)

+∆t

2µ∆z

[E

n+ 12

y (i, j +12, k + 1) − E

n+ 12

y (i, j +12, k)]

− ∆t

2µ∆y

[En

z (i, j + 1, k +12) − En

z (i, j, k +12)]

(4.56)

Hn+ 1

2y (i +

12, j, k +

12) = Hn

y (i +12, j, k +

12)

+∆t

2µ∆x

[E

n+ 12

z (i + 1, j, k +12) − E

n+ 12

z (i, j, k +12)]

− ∆t

2µ∆z

[En

x (i +12, j, k + 1) − En

x (i +12, j, k)

](4.57)

Hn+ 1

2z (i +

12, j +

12, k) = Hn

z (i +12, j +

12, k)

+∆t

2µ∆y

[E

n+ 12

x (i +12, j + 1, k) − E

n+ 12

x (i +12, j, k)

]

− ∆t

2µ∆x

[En

y (i + 1, j +12, k) − En

y (i, j +12, k)]

(4.58)

The Second Procedure is similarly obtained. Then using a process similar to the 2-D

case, the E and H components can be solved by means of tri-diagonal matrix computa-

tion.


4.3.4 Numerical Stability of the ADI-FDTD Method

The numerical stability of the two cases is also presented in [7, 147] and [149]. Uncondi-

tionally stable characteristics are shown regardless of the time step ∆t by calculation of

eigenvalue of the time-marching matrix of the recursive system. It is demonstrated in the

following way.

The approach of the ADI-FDTD can be summarized as a matrix form:

M1X

(n +

12

)= P1X(n)

(for advancement from nth to (n + 12)th time step.)

M2X (n + 1) = P2X(n +12) (4.59)

(for advancement from (n + 12)th to (n + 1)th time step.)

Here, X(n) contains all the field components at the nth time step. M1, M2, P1 and P2

are the coefficient matrices with their elements related to values of spatial and temporal

steps. They are sparse matrices. M1 and M2 are associated with the left hand side of

equations(4.45),(4.51),(4.47), (4.52),(4.49),(4.50) while P1 and P2 are associated with the

right hand side of those equations.

X

(n +

12

)= M−1

1 P1X(n)

X (n + 1) = M−12 P2X(n +

12) (4.60)

Combining them yields:

X (n + 1) = M−12 P2M

−11 P1X(n) (4.61)

OrX (n + 1) = ΛX(n) (4.62)

with ΛX(n) = M−12 P2M

−11 P1


For the recursive system of (4.62), if the magnitudes of all the eigenvalues are less

than unity, then the system can be considered to be stable [7]. Zheng et al. claimed that

by using the software of Maple (a symbolic computation software) [148], the eigenvalues

of the matrix Λ are calculated and they gained all magnitudes of unity regardless of dt

value used [7, 147]. Thus the Courant stability condition is removed. The followings

briefly presents their analysis.

Assuming the spatial frequencies to be kx, ky , and kz along the x, y, and z directions

respectively, the field components in the spatial spectral domain can be written as:

Ex|ni+ 12,j,k

= Enxe−j(kx(i+ 1

2)∆x+kyj∆y+kzk∆z)

Ey|ni,j+ 12,k

= Eny e−j(kxi∆x+ky(j+ 1

2)∆y+kzk∆z)

Ez|ni,j,k+ 12

= Enz e−j(kxi∆x+kyj∆y+kz(k+ 1

2)∆z)

Hx|ni,j+ 12,k+ 1

2

= Hnx e−j(kxi∆x+ky(j+ 1

2)∆y+kz(k+ 1

2)∆z)

Hy|ni+ 12,j,k+ 1

2= Hn

y e−j(kx(i+ 12)∆x+kyj∆y+kz(k+ 1

2)∆z)

Hz|ni+ 12,j+ 1

2,k

= Hnz e−j(kx(i+ 1

2)∆x+ky(j+ 1

2)∆y+kzk∆z) (4.63)

Denote the field vector in the spatial spectral domain as:

Xn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Enx

Eny

Enz

Hnx

Hny

Hnz

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Substitution of equations 4.63 into the first procedure of the three-dimensional ADI-

FDTD equations 4.53-4.58 leads to

X

(n +

12

)= Λ1 · Xn (4.64)


with

Λ1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1Qy

Wx·Wy

µεQy0 0 jWz

εQy

−jWy

εQy

0 1Qz

Wy·Wz

µεQz

−jWz

εQz0 jWx

εQz

Wx·WzµεQx

0 1Qx

jWy

εQx

−jWx

εQx0

0 −jWz

µQz

jWz

µQz

1Qz

0 Wx·WzµεQz

jWz

εQx0 −jWx

εQx

Wx·Wy

µεQx

1Qx

0−jWy

εQy

jWx

εQy0 0 Wz ·Wy

µεQy

1Qy

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Wα =∆t

∆α· sin(

kα∆α

2) , Qα = 1 +

W 2α

µε, α = x, y, z

Λ2, which denotes the recursive coefficients for the second procedure of ADI-FDTD,

can be obtained in dual process. Then Λ in equation 4.62 is calculated as:

Λ = Λ1 · Λ2 (4.65)

By use of software Maple V5.2, Zheng et al. were able to obtain the expression of Λ

and its eigenvalues, which are expressed as:

λ1 = λ2 = 1 , λ3 = λ5 =√

R2 − S2 + jS

R

λ4 = λ6 = λ∗3 =

√R2 − S2 − jS

R(4.66)

with

R = (µε + W 2x )(µε + W 2

y )(µε + W 2z )

S =√

4µε(µεW 2x + µεW 2

y + µεW 2z + W 2

xW 2y + W 2

y W 2z + W 2

z W 2x )(µ3ε3W 2

xW 2y W 2

z ).

Since R ≥ S, the square roots in the numerator of the expressions for λ3, λ4, λ5 and λ6

are real numbers. Consequently, all the eigenvalues have a magnitude of unity regardless

of the time step ∆t. Therefore, the conclusion that the ADI-FDTD is unconditionally

stable is reached [7].


4.4 Alternating Direction Implicit Nonorthogonal Finite-Difference

Time-Domain Method

The alternating direction implicit (ADI) scheme has been successfully applied to the or-

thogonal FDTD instead of Yee’s leapfrogging one, which removes the CFL stability con-

dition and improved the efficiency of FDTD method. However, to the best of the author’s

knowledge, no report is available on ADI implemented on the nonorthogonal FDTD. In

this section, the ADI-FDTD method is extended into the curvilinear coordinate system

and the Alternating-Direction Implicit Nonorthogonal Finite-Difference Time-Domain

(ADI-NFDTD) method that is free of the CFL stability condition is proposed. Conse-

quently, with an increased dt, the numerical efficiency of the NFDTD can be improved.

The ADI-NFDTD also demonstrates improved late time stability compared with the con-

ventional NFDTD scheme. It is demonstrated that the ADI-NFDTD can be reduced to

the conventional ADI-FDTD scheme under the Cartesian coordinate.

4.4.1 Derivation of the ADI-NFDTD Formulation

Denoting the covariant electric and magnetic field components which represent the flow

of field along the grid as Ei and Hi (i = 1, 2, 3), and the contravariant electric displace-

ment fluxes and magnetic field which represent the flow go through the facets of the grid

as Di and H i (i = 1, 2, 3), the NFDTD differential equations in an isotropic medium with

the medium permeability µ are written in the generalized curvilinear coordinate system

as:

∂D1

∂t=

1√g

[∂H3

∂u2− ∂H2

∂u3

](4.67)

∂D2

∂t=

1√g

[∂H1

∂u3− ∂H3

∂u1

](4.68)

∂D3

∂t=

1√g

[∂H2

∂u1− ∂H1

∂u2

](4.69)


∂H1

∂t=

1µ√

g

[∂E2

∂u3− ∂E3

∂u2

](4.70)

∂H2

∂t=

1µ√

g

[∂E3

∂u1− ∂E1

∂u3

](4.71)

∂H3

∂t=

1µ√

g

[∂E1

∂u2− ∂E2

∂u1

](4.72)

where, g is the computed metric tensor calculated from the local curvilinear coordinates.

The two-dimensional ADI-NFDTD scheme can be demonstrated by a TE mode for-

mulation derivation. The TM mode formula can be derived in exactly the same proce-

dure.

By applying the ADI principle to equations (4.67) -(4.72), the computation of equa-

tions (4.67)-(4.72) for the FDTD marching from the n-th time step to the (n + 1)-th time

step is broken up into two computational sub-advancements: the advancement from the

n-th time step to the (n + 12)-th time step and the advancement from the (n + 1

2)-th time

step to the (n + 1)-th time step.

In the first advancement, the first spatial partial derivatives on the right-hand side of

equations (4.67)-(4.72) are replaced with an explicit approximation of its known values at

the n-th time step; while the second spatial partial derivatives on the right-hand side of

equations (4.67)-(4.72) are replaced with an implicit approximation of its unknown values

at the (n + 12)-th time step. This is the first procedure.

In the second advancement, the first spatial partial derivatives on the right-hand side

of equations (4.67)-(4.72) are replaced with an implicit approximation of its unknown

pivotal values at the (n + 1)-th time step, while the second spatial partial derivatives on

the right-hand side of equations (4.67)-(4.72) are replaced with an explicit approximation

of its known values at the (n + 12 )-th time step. This is the second procedure.

For TE mode modelling, D3, H1 and H2 equal to zero. Taking this into consideration,

equations for the ADI-NFDTD in advancing from the n-th to the (n + 1)-th time step are

written as:


First Procedure (from the n-th to the (n + 12)-th time step):

D1n+ 12 (i +

12, j) − D1n

(i +12, j)

=dt

2√

g(i + 12 , j)

·[Hn

3

(i +

12, j +

12

)− Hn

3

(i +

12, j − 1

2

)](4.73)

D2n+ 12 (i, j +

12) − D2n

(i, j +12)

= − dt

2√

g(i, j + 12)

·[H

n+ 12

3

(i +

12, j +

12

)− H

n+ 12

3

(i − 1

2, j +

12

)](4.74)

H3n+ 12 (i +

12, j +

12) − H3n

(i +12, j +

12) =

dt

2µ(i + 12 , j + 1

2)√

g(i + 12 , j + 1

2)

·⎡⎣ En

1

(i + 1

2 , j + 1) − En

1

(i + 1

2 , j)

−En+ 1

22

(i + 1, j + 1

2

)+ E

n+ 12

2

(i, j + 1

2

)⎤⎦ (4.75)

Second Procedure (from the (n + 12)-th to the (n + 1)-th time step):

D1n+1(i +

12, j) − D1n+ 1

2 (i +12, j)

=dt

2√

g(i + 12 , j)

·[Hn+1

3

(i +

12, j +

12

)− Hn+1

3

(i +

12, j − 1

2

)](4.76)

D2n+1(i, j +

12) − D2n+1

2 (i, j +12)

= − dt

2√

g(i, j + 12)

·[H

n+ 12

3

(i +

12, j +

12

)− H

n+ 12

3

(i − 1

2, j +

12

)](4.77)

H3n+1(i +

12, j +

12) − H3n+ 1

2 (i +12, j +

12) =

dt

2µ(i + 12 , j + 1

2)√

g(i + 12 , j + 1

2 )

·⎡⎣ En+1

1

(i + 1

2 , j + 1)− En+1

1

(i + 1

2 , j)

−En+ 1

22

(i + 1, j + 1

2

)+ E

n+ 12

2

(i, j + 1

2

)⎤⎦ (4.78)


In the first procedure, the covariant E2 and the contravariant H3 as shown in equation

(4.74), are defined as synchronous variables. The covariant H3 and the contravariant D2

as shown in equation (4.75) are defined as synchronous variables.

In the second procedure, the covariant E1 and the contravariant H3 as shown in equa-

tion (4.76) are defined as synchronous variables. The covariant H3 and the contravariant

D1 as shown in equation (4.78) are defined as synchronous variables.

In the ADI-FDTD scheme, the two synchronized equations can be combined into one

equation by eliminating one variable (H3), so that E2 (or E1) can be solved by a simple

tri-diagonal matrix calculation. However, in the ADI-NFDTD, there are four variables

defined synchronous in one procedure (E2, H3, H3, and D2 in the first procedure, and

E1, H3, H3, and D1 in the second procedure), so additional equations are needed.

In order to calculated the covariant variables (E1, E2 and H3) from their contravariant

pairs (D1, D2 and H3) in the ADI-NFDTD, a same interpolating scheme as in a conven-

tional NFDTD is used. Equations (4.79)-(4.81) are examples of the interpolating equations

in a two-dimension TE case.

Et1(i +

12, j) =

g11(i + 12 , j)

ε1(i + 12 , j)

D1t(i +

12, j) +

g12(i + 12 , j)

4ε1(i + 12 , j)

·⎡⎣ D2t (

i, j − 12

)+ D2t (

i + 1, j − 12

)+D2t (

i, j + 12

)+ D2t (

i + 1, j + 12

)⎤⎦ (4.79)

Et2(i, j +

12) =

g22(i, j + 12)

ε2(i, j + 12)

D2t(i, j +

12) +

g12(i, j + 12)

4ε2(i, j + 12)

·⎡⎣ D1t (

i − 12 , j)

+ D1t (i − 1

2 , j + 1)

+D1t (i + 1

2 , j)

+ D1t (i + 1

2 , j + 1)⎤⎦ (4.80)

Ht3(i +

12, j +

12) = H3t

(i +12, j +

12) (4.81)

where t = n ( or t = n + 12 ).

Substituting the expressions for En+ 1

22 from equation (4.80) and H3n+ 1

2 from equation


(4.81) into (4.75), and the resulting expression for Hn+ 1

23 into equation (4.74) yields equa-

tion (4.82). This is a tridiagonal system of equation that can be easily solved. Thus D2n+ 12

in the first procedure is updated. Then the Hn+ 1

23 in the first procedure can be calculated

by equations (4.75) and (4.81).

− g22(i − 1, j) · dt2

4µ(i − 1, j) · ε2(i − 1, j) ·√g(i, j)√

g(i − 1, j)· D2n+ 1

2 (i − 1, j)

+

[1 +

g22(i, j) · dt2

4µ(i − 1, j) · ε2(i, j) ·√

g(i, j)√

g(i − 1, j)+

g22(i, j) · dt2

4µ(i, j) · ε2(i, j) · g(i, j)

]

·D2n+ 12 (i, j) − g22(i + 1, j) · dt2

4µ(i, j) · ε2(i + 1, j) · g(i, j)· D2n+ 1

2 (i + 1, j)

= D2n(i,j) − dt

2√

g(i, j)[Hn

3 (i, j) − Hn3 (i − 1, j)]

− dt2

4√

g(i, j)·[

En1 (i, j + 1) − En

1 (i, j)µ(i, j)

√g(i, j)

− En1 (i − 1, j + 1) − En

1 (i − 1, j)µ(i − 1, j)

√g(i − 1, j)

]

+dt2

16µ(i, j)g(i, j)

·⎡⎣g12(i + 1, j)

ε2(i + 1, j)

a=i+1;b=j+1∑a=i;b=j

D1n+ 12 (a, b) − g12(i, j)

ε2(i, j)

a=i;b=j+1∑a=i−1;b=j

D1n+ 12 (a, b)

⎤⎦

− dt2

16µ(i − 1, j)√

g(i, j)√

g(i − 1, j)

·⎡⎣g12(i, j)

ε2(i, j)


D1n+ 12 (a, b) − g12(i − 1, j)

ε2(i − 1, j)

a=i−1;b=j+1∑a=i−2;b=j

D1n+ 12 (a, b)

⎤⎦ (4.82)

Once the value of D2n+ 12 is obtained, E

n+ 12

2 can be updated by equation (4.80). Then,

H3n+ 12 can be calculated explicitly by equation (4.75). As a direct consequence, H

n+ 12

3 can

be calculated through equation (4.81).


In a similar way, substituting the expressions for En+11 from equation (4.79) and H3n+1

from equation (4.81) into equation (4.78), and the resulting expression for Hn+13 into (4.76)

yields equation (4.83).

− g11(i, j − 1) · dt2

4µ(i, j − 1) · ε1(i, j − 1) ·√g(i, j)√

g(i, j − 1)· D1n+1

(i, j − 1)

+

[1 +

g11(i, j) · dt2

4µ(i, j − 1) · εx(i, j) ·√g(i, j)√

g(i, j − 1)+

g11(i, j) · dt2

4µ(i, j) · εx(i, j) · g(i, j)

]

·D1n+1(i, j) − g11(i, j + 1) · dt2

4µ(i, j) · ε1(i, j + 1) · g(i, j)· D1n+1

(i, j + 1)

= D1n+ 12 (i,j) − dt

2√

g(i, j)

[H

n+ 12

3 (i, j) − Hn+ 1

23 (i, j − 1)

]

− dt2

4√

g(i, j)·⎡⎣E

n+ 12

2 (i + 1, j) − En+ 1

22 (i, j)

µ(i, j)√

g(i, j)− E

n+ 12

2 (i + 1, j − 1) − En+ 1

22 (i, j − 1)

µ(i, j − 1)√

g(i, j − 1)

⎤⎦

+dt2

16µ(i, j)g(i, j)

·⎡⎣g12(i, j + 1)

εx(i, j + 1)


D2n+1(a, b) − g12(i, j)

ε1(i, j)

a=i;b=j−1∑a=i+1;b=j

D2n+1(a, b)

⎤⎦

− dt2

16µ(i, j − 1)√

g(i, j)√

g(i, j − 1)

·⎡⎣g12(i, j)

ε1(i, j)

a=i;b=j−1∑a=i+1;b=j

D2n+1(a, b) − g12(i, j − 1)

ε1(i, j − 1)

a=i+1;b=j−1∑a=i;b=j−2

D2n+1(a, b)

⎤⎦ (4.83)

This is a tridiagonal system of equation that can be easily solved. Thus D1n+1in the

second procedure is updated. Then En+11 , H3n+1

and Hn+13 in the second procedure can

be updated by equations (4.79), (4.78) and (4.81) respectively.

Figure 4.8 shows a flow chart of the working scheme of ADI-NFDTD method.


Figure 4.8: The working scheme of the proposed ADI-NFDTD algorithm.


4.4.2 Reduction of the ADI-NFDTD to the Conventional ADI-FDTD

The proposed ADI-NFDTD algorithm is an extension of the ADI-FDTD algorithm to a

nonorthogonal grid. As a result, if the grid is reduced to a uniform orthogonal grid, this

algorithm will reduce to the orthogonal ADI-FDTD. Equations (4.84) - (4.89) illustrate

how this happens. Firstly, the field components defined by Lee’s algorithm will reduce

into the orthogonal FDTD components as follows (two-dimension TE mode):

Dm(i, j)√

gmm(i, j)εm(i, j)

= Em⊥(i, j) (m = 1 or 2)

Em(i, j)√gmm(i, j)

= Em⊥(i, j) (m = 1 or 2)

H3(i, j) = H3⊥(i, j) (4.84)

where the Em⊥ and Hm⊥ (m = 1, 2 or 3) denotes the field components in the orthogonal

(1, 2, 3) coordinate system. The g tensors have the relationship of (4.85) in two-dimension

case.

g(i, j) = g11(i, j) · g22(i, j) − g12(i, j)2 (4.85)

Since in the orthogonal FDTD method the grids are uniform and orthogonal, with

spatial increment ∆x and ∆y in x and y direction, equations (4.86) and (4.87) are obtained.

g12(i, j) = 0 (4.86)

g11(i, j) = ∆x2

g22(i, j) = ∆y2

g(i, j) = ∆x2 · ∆y2 (4.87)

Substituting equations (4.85) and (4.86) into (4.82), yields equation (4.88).


−√

g22(i − 1, j) · dt2

4µ(i − 1, j) ·√g(i, j)√

g(i − 1, j)·√

g22(i − 1, j) · D2n+ 12 (i − 1, j)

ε2(i − 1, j)

+

[ε2(i, j)√g22(i, j)

+

√g22(i, j) · dt2

4µ(i − 1, j) ·√g(i, j)√

g(i − 1, j)+

√g22(i, j) · dt2

4µ(i, j) ·√g(i, j)√

g(i, j)

]

·√

g22(i, j) · D2n+12 (i, j)

ε2(i, j)

−√

g22(i, j) · dt2

4µ(i, j) · g(i, j)·√

g22(i + 1, j) · D2n+ 12 (i + 1, j)

ε2(i + 1, j)

=ε2(i, j)√g22(i, j)

·√

g22(i, j) · D2n(i − 1, j)

ε2(i, j)

− dt

2√

g11(i, j) ·√

g22(i, j)· [Hn

3 (i, j) − Hn3 (i − 1, j)]

− dt2

4√

g11(i, j) ·√

g22(i, j)· [ 1

µ(i, j) ·√g22(i, j)· En

1 (i, j + 1) − En1 (i, j)√

g11(i, j)

− 1µ(i − 1, j) ·√g22(i − 1, j)

· En1 (i − 1, j + 1) − En

1 (i − 1, j)√g11(i − 1, j)

] (4.88)

Then by substituting (4.84) and (4.87) into equation (4.88) to change the variants to or-

thogonal FDTD variants and by multiplying −4∆x2·∆y·µ(i,j)

dt2on both sides of the resulting

equation, equation (4.89) is derived, which is ADI-FDTD formula under uniform orthog-

onal grid.

− µ(i, j)µ(i − 1, j)

· En+ 12

y⊥ (i − 1, j) − [4∆x2εy(i, j)µ(i, j)

dt2+

µ(i, j)µ(i − 1, j)

+ 1] · En+ 12

y⊥ (i, j)

+En+ 1

2y⊥ (i + 1, j)

=4∆x2εy(i, j)µ(i, j)

dt2· En

y⊥(i, j) − 2∆x · µ(i, j)dt

· [Hnz⊥(i, j) − Hn

z⊥(i − 1, j)]

+∆x

∆y· [En

x⊥(i, j + 1) − Enx⊥(i, j)] +

∆x

∆y· µ(i, j)µ(i − 1, j)

· [Enx⊥(i − 1, j + 1) − En

x⊥(i − 1, j)]

(4.89)

Equation (4.89) is the general equation for the orthogonal ADI-FDTD scheme. It can

be applied in homogeneous or inhomogeneous, isotropic or anisotropic media.


4.4.3 Periodic Boundary Condition Incorporated in the ADI-NFDTD Method

The periodic boundary condition (PBC) for numerically modelling the infinite EBGs can

be easily introduced into the proposed ADI-NFDTD.

The PBCs in the explicit formulae in the ADI-NFDTD (equations (4.73), (4.75), (4.77)

and (4.78)) can be realized in the same way as in a conventional FDTD (or NFDTD).

The formulation of PBCs in the matrix calculation (equations (4.82) and (4.83) which are

derived from the implicit equations (4.74) and (4.76)) is presented in the following. x, y, z

are used to denote the three directions (dimensions) in stead of 1, 2, 3 in the following

part of the thesis. They do not indicate that the variables are under Cartesian coordinate.

Assume an infinite EBG modelling with elements aligned in square lattice. Assume

the unit cell consists of xmax cells in x-direction and ymax cells in y-direction respectively,

so that the computational domain is (1 ≤ i ≤ xmax, and 1 ≤ j ≤ ymax). Recall the

ADI-NFDTD implicit equation for the calculation of Dy for example:


− gyy(i − 1, j) · dt2

4µ(i − 1, j) · εy(i − 1, j) ·√g(i, j)√

g(i − 1, j)· Dyn+ 1

2 (i − 1, j)

+

[1 +

gyy(i, j) · dt2

4µ(i − 1, j) · εy(i, j) ·√

g(i, j)√

g(i − 1, j)+

gyy(i, j) · dt2

4µ(i, j) · εy(i, j) · g(i, j)

]

·Dyn+ 12 (i, j) − gyy(i + 1, j) · dt2

4µ(i, j) · εy(i + 1, j) · g(i, j)· Dyn+ 1

2 (i + 1, j)

= Dyn(i,j) − dt

2√

g(i, j)[Hn

z (i, j) − Hnz (i − 1, j)]

− dt2

4√

g(i, j)·[

Enx (i, j + 1) − En

x (i, j)µ(i, j)

√g(i, j)

− Enx (i − 1, j + 1) − En

x (i − 1, j)µ(i − 1, j)

√g(i − 1, j)

]

+dt2

16µ(i, j)g(i, j)

·⎡⎣gxy(i + 1, j)

εy(i + 1, j)


Dxn+12 (a, b) − gxy(i, j)

εy(i, j)


Dxn+12 (a, b)

⎤⎦

− dt2

16µ(i − 1, j)√

g(i, j)√

g(i − 1, j)

·⎡⎣gxy(i, j)

εy(i, j)


Dxn+ 12 (a, b) − gxy(i − 1, j)

εy(i − 1, j)

a=i−1;b=j+1∑a=i−2;b=j

Dxn+12 (a, b)

⎤⎦ (4.90)

When updating fields in any cell (i, j) (1 < i < xmax and 1 < j < ymax) in the domain

of the unit cell except for the boundary (referred to as the ’main area’ in the following text

for short), the entries in the coefficient matrix W1(i, j), W2(i, j), W3(i, j) can be calculated

using (4.91)-(4.93) directly from equation (4.90):

W1(i, j) = − gyy(i − 1, j) · dt2

4µ(i − 1, j) · εy(i − 1, j) ·√g(i, j)√

g(i − 1, j)(4.91)


W2(i, j) = 1 +gyy(i, j) · dt2

4µ(i − 1, j) · εy(i, j) ·√

g(i, j)√

g(i − 1, j)

+gyy(i, j) · dt2

4µ(i, j) · εy(i, j) · g(i, j)(4.92)

W3(i, j) = − gyy(i + 1, j) · dt2

4µ(i, j) · εy(i + 1, j) · g(i, j)(4.93)

However, as far as the boundary cell (i = 1, i = xmax, j = 1 or j = ymax) need

to be calculated, equation (4.90) cannot be applied directly. So the periodic boundary

relationships (equations (4.94) and (4.95)) need to be introduced into equation (4.90).

F (1 − 1, j) = eik·(xmaxdx)x·F (xmax, j)

F (1 − 2, j) = eik·(xmaxdx)x·F (xmax − 1, j)

F (xmax + 1, j) = e−ik·(xmaxdx)x·F (1, j)

F (i, 1 − 1) = eik·(ymaxdy)y·F (i, ymax)

F (i, ymax + 1) = e−ik·(ymaxdy)y ·F (i, 1) (4.94)

G(1 − 1, j) = G(xmax, j)

G(xmax + 1, j) = G(1, j)

G(i, 1 − 1) = G(i, ymax)

G(i, ymax + 1) = G(i, 1) (4.95)

where F denotes the field value and G stands for the spatial and material parameters.

The left hand side of equation (4.94) is the field expression for the cells outside the

computational domain. So, when these values are required, their equivalent right hand

side pair which consists of value of cells inside the computational domain can be referred

to. As for the spatial and material parameters, they can be simply displaced due to the

periodicity of the EBGs. Thus the coefficient matrix W1(1, j), W2(1, j), W3(1, j) can be


calculated using equation (4.96), (4.97) and (4.98). [W1, W2, W3] (i = xmax, j = 1, j =

ymax) can be calculated in a similar way.

W1(1, j) = − gyy(xmax, j) · dt2

4µ(xmax, j) · εy(xmax, j) ·√g(1, j)√

g(xmax, j)(4.96)

W2(1, j) = 1 +gyy(1, j) · dt2

4µ(xmax, j) · εy(1, j) ·√

g(1, j)√

g(xmax, j)

+gyy(1, j) · dt2

4µ(1, j) · εy(1, j) · g(1, j)(4.97)

W3(1, j) = − gyy(1 + 1, j) · dt2

4µ(1, j) · εy(1 + 1, j) · g(1, j)(4.98)

The calculation of the the resulting matrix S can be performed in the same way. When

calculating the main area of the unit cell (2 < i < xmax and 1 < j < ymax), the entries

in the resulting matrix S(i, j) can be calculated using equation (4.99) which is readily

provided in the right hand side of equation (4.90).

S(i, j) = Dyn(i,j) − dt

2√

g(i, j)[Hn

z (i, j) − Hnz (i − 1, j)]

− dt2

4√

g(i, j)·[

Enx (i, j + 1) − En

x (i, j)µ(i, j)

√g(i, j)

− Enx (i − 1, j + 1) − En

x (i − 1, j)µ(i − 1, j)

√g(i − 1, j)

]

+dt2

16µ(i, j)g(i, j)

·⎡⎣gxy(i + 1, j)

εy(i + 1, j)


Dxn+12 (a, b) − gxy(i, j)

εy(i, j)


Dxn+12 (a, b)

⎤⎦

− dt2

16µ(i − 1, j)√

g(i, j)√

g(i − 1, j)

·⎡⎣gxy(i, j)

εy(i, j)


Dxn+12 (a, b) − gxy(i − 1, j)

εy(i − 1, j)

a=i−1;b=j+1∑a=i−2;b=j

Dxn+12 (a, b)

⎤⎦ (4.99)

However, when calculating S(i, j) for (i = 1, 2, xmax and j = 1, ymax), equation (4.94)


and equation (4.95) need to be introduced into equation (4.90) when a value from outside

the computational domain is required. After all the entries are expressed using values

inside the computational domain, equation (4.90) can be presented in matrix form as:

W(j) · Dy(j)n+ 1

2 = S(j)n+ 12 (4.100)

where W is the coefficient matrix, Dy is the fields to be updated, and S is the resulting

matrix:

W(j) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

W(i=1,j)2 W

(i=2,j)1 0 ... 0 W i=xmax,j

3

W(i=1,j)3 W

(i=2,j)2 W

(i=3,j)1 ... 0 0

0 W(i=2,j)3 W

(i=3,j)2 ... 0 0

0 0 W(i=3,j)3 ... 0 0

0 0 0 ... 0 0

... ...

0 0 0 ... W(i=xmax−1,j)1 0

0 0 0 ... W(i=xmax−1,j)2 W i=xmax,j

1

W(i=1,j)1 0 0 ... W

(i=xmax−1,j)3 W i=xmax,j

2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Dy(j) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Dy(i=1,j)

Dy(i=2,j)

Dy(i=3,j)

...

Dy(i=xmax−1,j)

Dy(i=xmax,j)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4.101)


S(j) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

S(i=1,j)

S(i=2,j)

S(i=3,j)

...

S(i=xmax−1,j)

S(i=xmax,j)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4.102)

4.5 Validation of the ADI-NFDTD Method

In this section, the ADI-NFDTD algorithm is verified by the modelling of wave propaga-

tion in free space, in which, a much larger dt which violates the CFL condition is used. In

the second simulation, the ADI-NFDTD and the NFDTD algorithm are used in the mod-

elling of a two dimensional cylindrical perfect conductor (PEC) cavity resonator. Simula-

tion results are compared in terms of time and frequency domain, including the numer-

ical errors and the late time instability. In the third simulation, a two dimensional cylin-

drical copper cavity resonator is modelled using the ADI-NFDTD and the ADI-FDTD

methods. The numerical accuracy and the computing efficiency, including the computer

memory and computation time, are compared.

4.5.1 Removal of the CFL Stability Criteria

In this section, radio propagation in free space is modeled using both the conventional

NFDTD and the proposed ADI-NFDTD scheme. This is done as a pilot simulation to


verify that the ADI-NFDTD algorithm is free from the CFL stability criteria. The compu-

tational efficiency of the proposed ADI-NFDTD is also demonstrated.

• Parameters Of The Free Space Wave Propagation Model

The computational domain is (0.3m × 0.3m) in free space meshed by (30 × 30) cells

(figure 4.9). A point source with a continuous sine wave at 1GHz is excited at the center

of the domain (15, 15). The magnetic field (Hz) is probed at position (6, 9). Mur’s first

order absorbing boundary condition is used.

0 0.1 0.2 0.30

0.1

0.2

0.3

Figure 4.9: A mesh of the (0.3m× 0.3m) free space domain.

• Simulation Results

With a time interval dt = 1e−10s and a simulation period of 5000 time steps, the ADI-

NFDTD algorithm produces a stable output. The first 40 time steps output is plotted in

figure 4.10(a). The resolution of the Fast Fourier Transform (FFT) to the time domain

signal is 0.002GHz. The relative error of the computed operating frequency is 0.1%.

Hence the proposed ADI-NFDTD demonstrates very low numerical dispersion. How-

ever, in the conventional NFDTD algorithm, the same modelling parameters will result in

an unstable temporal output at a very early stage, shown in figure 4.10(b). This indicates

that the dt of 1e − 10s has already violated the CFL condition for the NFDTD.

By decreasing dt step by step in the simulation, dt = 5e − 12s is found to be the


maximum value which satisfies the CFL condition and gives a stable result for the same

physical period (up to 100000 time steps). The same FFT resolution with the previous

ADI-NFDTD one can be achieved (0.002GHz). The relative error rate of the frequency

spectrum is also 0.1%.

The computational efficiency of the two NFDTD schemes is also compared. With the

same FFT resolution (0.002GHz) and the numerical error rate of 0.1%, the simulation

time is compared. On a Pentium IV 2.40GHz PC with RAM of 1.5GB, under computing

environment of Matlab, the ADI-NFDTD program runs 5000 time steps at a rate of 20

time-steps/second; and the conventional NFDTD one runs 100000 time steps at a rate of

250 time-steps/second. So 250 seconds are used in the ADI-NFDTD simulation and 400

seconds are used in the conventional NFDTD one for achieving the same stability in the

same period of physical time and the same level of accuracy. The ADI-NFDTD bears a

saving factor of 1.6 in simulation time with reference of the conventional NFDTD in this

set of simulations. These results are summarized in Table 4.1.

NFDTD ADI-NFDTD

dt (second) 5e − 12 1e − 10

Number of Iteration 100000 5000

Computing time for one iteration(second)

1/250 1/20

Total Computer Run Time (second) 400 250

Saving rate of the Total ComputerRun Time by the ADI-NFDTD

- 1.6

Stability Status for the results Stable Stable

FFT resolution 0.002GHz 0.002GHz

Relative Error Rate in Frequency 0.1% 0.1%

Table 4.1: Comparison of the computational time and accuracy of the simulation results ona Pentium IV 2.40GHz PC with RAM of 1.5GB. With the same level of accuracy, the ADI-NFDTD shows a saving rate of 1.6 in total computational time compared with the conven-tional NFDTD.


0 10 20 30 40-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Time steps (time interval dt = 100ps)

Ma

gn

etic F

ield

Hz (

A/m

) ADI-NFDTD

(a)

0 5 10 15 20

-2

0

2

4

6

8

10x 10

29

Time steps (time interval dt = 100ps)

Ma

gn

etic fie

ld H

z (

A/m

) conventional NFDTD

(b)

Figure 4.10: Hz field temporal results for the modelling of a single frequency wave propa-gation in free space, with time interval dt = 100ps in both the ADI-NFDTD and the NFDTDsimulations. (a) the ADI-NFDTD results; (b) the conventional NFDTD results.


4.5.2 Numerical Efficiency and the Late Time Instability Improvement over

the Conventional NFDTD

In this section, the conventional NFDTD and the proposed ADI-NFDTD schemes are ap-

plied to calculate the resonant frequency of a two-dimensional cavity resonator. Numeri-

cal efficiency of the ADI-NFDTD is demonstrated and the late time instability of NFDTD

is greatly improved.

• Parameters Of The Cylindrical PEC Cavity Resonator

A cylindrical perfect electric conductor (PEC) cavity resonator is assumed to be infi-

nite long. The radius of the cavity is 0.15m. The cavity is filled with vacuum of permit-

tivity εr = 1. The outer material is PEC, which means the skin depth of the material is

small enough so that penetration of the EM field through the material can be neglected.

• The FDTD Model

Since the metallic layer enclosing the cavity is thick enough to be looked as ’infinitely’

thick from inside the cavity, the whole computational domain can be simplified into a two

layer model, with an outside metallic layer terminated at an adequate thickness (much

greater than the skin depth). The computational region is set to be (0.52m × 0.5333m)

meshed by (25 × 27) cells (shown in figure 4.11).

A sine modulated cosine pulse (equation( 4.103)) is excited at selected source posi-

tions inside the cavity (see figure 4.12) to provide a wide band excitation to excite all the

possible TE modes:

Source(n) = Am · sin(2πf · n · dt) · (1 − cos(2πf · n · dt)) (4.103)

where Am relates to the amplitude of the signal, f is the frequency parameter, dt is the

time increment and 1 ≤ n ≤ 1/(dt · 2πf) is the iteration index. Figure 4.13 shows an

example of the pulse when Am=1, f = 7GHz and dt = 1ps.


After a period of time, the modes that are not supported by the cavity will vanish

and only those supported by the cavity will resonate. Then temporal signatures at sam-

pled probing points will go through a Fast Fourier transformation (FFT) and the resonant

modes will be found as peaks in frequency spectra. The position of the probing points is

also illustrated in figure 4.12.

0.1 0.2 0.3 0.4 0.5

0.1

0.2

0.3

0.4

0.5

Figure 4.11: The mesh of the cut plane of the cylindrical cavity with radius r = 0.15m, withthe air-metal boundary indicated by the red line.

Figure 4.12: The NFDTD modelling of the cavity.


0 1 2 3 4 5

x 10−9

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

Am

plitu

de

(a)

0 2 4 6 8 10

x 109

0

20

40

60

80

100

Frequency (Hz)

Mag

nitu

de

(b)

Figure 4.13: An example of the excitation signal in time and frequency domain. Am=1, f =7GHz and dt = 1ps.



The magnetic fields inside the resonator at the same probe location modelled by the

ADI-NFDTD and the NFDTD schemes are compared. Both simulations use the same

mesh profile (figure 4.11) and timestep dt = 1ps. It is expected that the ADI-NFDTD

results are less accurate than the conventional NFDTD ones. The first 8000 time steps

results calculated from both algorithms are normalized by their own peaks and are com-

pared in figure 4.14. The resemblance of the two curves can be taken as one of the verifi-

cations of the proposed ADI-NFDTD algorithm.

Figure 4.14: The comparison of the normalized H field time domain results of the first8000 time steps (dt = 1ps) from the conventional NFDTD (solid line) and the ADI-NFDTDschemes (dotted line).

The comparison of the ADI-NFDTD and the NFDTD schemes shows that the late

time instability occurs much later in the ADI-NFDTD scheme than in the conventional

NFDTD one. For instance, in the conventional NFDTD simulation using the dt of 1ps,

the result began to become unstable after 12000 time steps, which corresponds to 12ns

in physical time (details can be found in figure 4.15(a)). After that, the energy in the

simulation domain will increase exponentially due to the late time instability ( figure


4.15(b)). The ADI-NFDTD result is stable until 100000 time steps, which corresponds to

100ns in physical time ( figure 4.15(c)).

More simulations were carried out using different dt in order to investigate the sta-

bility of the proposed ADI-NFDTD scheme. It is shown that unlike the conventional

ADI-FDTD, the proposed ADI-NFDTD scheme is not an unconditionally stable scheme.

Instability in the ADI-NFDTD temporal results is observed. The CFL stability condition

on dt in the conventional NFDTD is also observed in this sets of simulations. When

dt > 16ps, the result becomes unstable at a very early stage. In other words, as soon

as the energy propagates to the cell which requires the smallest ∆t ij , dt does not satisfy

the local CFL condition anymore and causes instability immediately. However, with the

ADI-NFDTD, this instability does not occur with the same or greater dt value.

In order to study the relationship of the accuracy of the simulated results with the

time interval dt used in the simulation, the resonant frequencies calculated from FFT are

compared with the theoretical resonant frequencies. For those ADI-NFDTD results which

become unstable in the later time steps, the data in their stable period are used. There is

a tradeoff when considering the amount of data to be used. If only the data in the stable

period are used, the frequency resolution in FFT will be low limiting the accuracy of the

frequency result. In this case, some data at the beginning of the unstable period will have

to be involved. The more unstable data are used, the higher frequency resolution can be

achieved after FFT. However, too many data in the unstable period will result in a high

noise level in the frequency spectra. As is mentioned previously, for the dt value of 1ps,

the result of the conventional NFDTD becomes unstable from around 12000 time step.

After that, the energy of the field increases gradually and the number of the NFDTD

temporal samples used for FFT are chosen to be 40000 to balance the aforementioned

tradeoff. The number of the ADI-NFDTD temporal samples is chosen to be 100000. So

the frequency resolution of the spectra for the conventional NFDTD and the ADI-NFDTD

are 0.025GHz and 0.01GHz respectively.

The calculated resonant frequency spectra with dt = 1ps is plotted in figure 4.16. The

resonant frequencies are compared with the theoretical results in Table 4.2. The absolute

value of the difference of the calculated results with the theoretical results[151] can be


0 0.5 1 1.5 2 2.5 3

x 104

-15

-10

-5

0

5

10

15

Magnetic f

ield

Hz(A

/m)

Time step (dt = 1ps)

(a)

0 1 2 3 4 5

x 104

-5000

0

5000

Time steps ( time interval: 1e-12s)

Magnetic fie

ld H

z

conventional NFDTD

(b)

0 2 4 6 8 10

x 104

-15

-10

-5

0

5

10

15

Time steps ( dt = 1ps)

Magnetic f

ield

Hz (

A/m

)

(c)

Figure 4.15: Hz field temporal results with dt = 1ps. (a) the first 30000 time steps by theNFDTD; (b) the first 50000 time steps by the NFDTD. (c) the first 100000 time steps by theADI-NFDTD.


firstly obtained. Then it is divided by the theoretical value and expressed in percentage

form giving the relative error rate (RER) of each resonant mode. Finally, the standard

deviation of the calculated resonant frequencies for all the resonant modes (from 0GHz

to 2.732GHz) are taken as a parameter which will be referred in the following as the

’averaged relative error rate (ARER)’. (Note that the theoretical frequency is used as the

expected value in the standard deviation calculation, and this applies to all the error rate

calculations in this chapter.)

0 0.5 1 1.5 2 2.5 3

x 109

0

1000

2000

3000

4000

5000

Frequency (Hz)

Mag

netic

fiel

d H

z(A

mpe

r/m

eter

)

NFDTDADI−NFDTD

Figure 4.16: Resonant frequency spectra of the cavity resonator calculated from the twoschemes, with dt = 1ps. Solid line: the conventional NFDTD scheme; dotted line: the ADI-NFDTD scheme.

As can be seen in figure 4.16, the accuracy of the conventional NFDTD is limited by

this frequency resolution while by using the ADI-NFDTD scheme, the resonant frequency

can be located in the frequency spectra with a higher resolution. Some resonant modes

have very close frequencies. For example, it is hard to distinct different resonant modes

at around 2.5GHz by the conventional NFDTD simulation while it is easy to do so in

the ADI-NFDTD one with a higher frequency resolution. It can also be seen that the

noise level of the ADI-NFDTD result is lower than the conventional NFDTD one. That


is because the time domain data from the former include less unstable numerical errors

than those from the latter.

NFDTD NFDTD ADI-NFDTD ADI-NFDTDTheoretical results (GHz) Relative results (GHz) Relativeresults[151] FFT resolution Error Rate FFT resolution Error Rate

(GHz) = 0.025GHz (RER)(%) = 0.01GHz (RER)(%)0.5863 0.575 1.93 0.59 0.630.9721 0.925, 1.075∗ 4.85,10.59∗ 0.92,1.08∗ 5.36, 11.10∗

1.2198 1.2 1.62 1.2 1.621.3372 1.4 4.70 1.39 3.951.693 1.675 1.06 1.68 0.771.7 1.675, 1.825∗ 1.47, 7.35∗ 1.68,1.83∗ 1.18,7.65∗

2.043 2.07 1.32 2.07 1.322.136 2.15 0.66 2.14 0.192.234 2.15 3.76 2.17 2.862.389 2.5 4.65 2.49 4.232.553 2.6 1.84 2.56 0.272.718 2.825 3.94 2.62 3.612.732 2.825 3.40 2.84 3.95

Averaged Relative Error Rate (ARER)from all the Genuine Modes 2.85 - 2.85ARER from all the Modes 4.70 - 4.75

Table 4.2: Comparison of the accuracy of the resonant modes of the aforementioned PECresonator calculated by the conventional NFDTD and the ADI-NFDTD with the theoreticalresults. In both simulations, dt = 1ps. The modes with ∗ are spurious modes introducedin the meshing of the cylindrical cavity. They are compared with the nearest genuine modewhen RER is evaluated and hence they cause big numerical errors. In the first deviation ofRER calculation, these errors are not included. In the second deviation of RER calculation(with ∗), these errors are taken into consideration.

In ADI-NFDTD, it is observed that the high frequency components will attenuate with

the increment of the time step dt used. So with big dt, the high frequency components be-

come hard to distinguish among the noise. This is illustrated in figure 4.17, in which the

frequency spectra calculated from the ADI-NFDTD result with dt = 17ps are compared

with the spectra from the conventional NFDTD with dt = 1ps. Because of the attenuation

in high frequency in ADI-NFDTD, the amplitude of the spectra of ADI-NFDTD result is


multiplied uniformly by a factor of 3.5 to give a better view of comparison with the con-

ventional NFDTD spectrum result. It can be seen that the lower frequency components

can still be precisely evaluated in ADI-NFDTD results; and the higher frequency compo-

nents also can be identified by the trends but not as precisely as in a smaller dt case (e.g.

figure 4.16 in which dt = 1ps).

So in the ADI-NFDTD scheme, the choice of the dt value depends largely on the re-

quirement on accuracy. It is worth noticing that this dt value of 17ps used in this ADI-

NFDTD simulation has violated the CFL condition of the conventional NFDTD. That

means the same dt, in conventional NFDTD scheme, will result in a temporal result un-

stable at a very early stage and no reasonable frequency spectrum is possible to obtain.

The averaged relative error rate of the resonant frequency by the proposed ADI-

NFDTD scheme and the conventional NFDTD scheme are plotted against the relative

time interval in figure 4.18. Theoretically, the ADI-NFDTD may not necessarily provide

better accuracy than the conventional NFDTD.

However, as far as this set of simulations are concerned, the ADI-NFDTD scheme

can always provide longer stable temporal results to perform the FFT with a frequency

resolution which is small enough. So with some dt, the ADI-NFDTD result shows smaller

relative error rate. It can be seen that from the 16th unit, the relative error rate of the

conventional NFDTD results increases dramatically. That is the dt value required by the

CFL stability condition. From that value, the resonant frequencies become undetectable

with the conventional NFDTD, while in the ADI-NFDTD, the error rate only increases

slowly with the increase of the dt.


0 0.5 1 1.5 2 2.5 3

x 109

0

1000

2000

3000

4000

5000

Frequency (Hz)

Mag

netic

fiel

d H

z(A

mpe

r/m

eter

)

NFDTDADI−NFDTD

Figure 4.17: Resonant frequency spectrum of the cavity resonator calculated from temporalresults by the ADI-NFDTD with dt = 17ps (dotted line), compared with the spectrum cal-culated from temporal results by the conventional NFDTD with dt = 1ps (solid line). Theamplitude of the ADI-NFDTD result is uniformly amplified by 3.5 times.

0 5 10 15 16 202

3

4

5

6

7

8

Relative Time Interval dt ( in the unit of 1ps)

Rel

ativ

e E

rror

Rat

e (%

)

NFDTDADI−NFDTD

Figure 4.18: The averaged relative error rate of the NFDTD and the ADI-NFDTD simulatedresonant frequency spectra as the function of relative time interval dt/∆t(∆t = 1ps). Whendt > 16∆t, the CFL condition is violated. The direct consequence is that the relative errorrate of the conventional NFDTD increases to 100%, while ADI-NFDTD still provides resultswith a relative error rate of less than 5%.


4.5.3 Numerical Efficiency Improvement over the Orthogonal ADI-FDTD

In this section, a two-dimensional cavity resonator is modelled using both the conven-

tional Yee’s FDTD and the proposed ADI-NFDTD scheme to demonstrate the numerical

efficiency improvement of the novel ADI-NFDTD over the conventional orthogonal ADI-

FDTD.

• Parameters Of The Cylindrical Copper Cavity

A cylindrical copper cavity resonator is assumed to be infinitely long. The radius

of the cavity is 0.15m. The cavity is filled with vacuum of permittivity εr = 1. The

outer material is chosen to be copper, with relative permittivity εr = 1 and conductivity

σ = 5.8 × 107 S/m. The computational domain to be modelled is (0.6m × 0.6m).

• The FDTD Model

The modelling procedure is quite similar to that in 4.5.2 where the ADI-NFDTD and

the NFDTD are compared by the modelling of a PEC resonator. Only in this modelling,

the mesh files (figures 4.20 - 4.23) are different and the cavity is excited by a wide band

modulated Gaussian pulse (equation (4.104)). Once the temporal responses of the probe

points go through the FFT, the resonant frequencies are identified as peaks in the spectra

and they are compared with the theoretical results. Figures 4.20, 4.21, 4.22 and 4.23 show

the meshes of the ADI-NFDTD and the ADI-FDTD model under different spatial resolu-

tions. The mesh size are (46×46) for the ADI-NFDTD modelling, (60×60), (70×70), and

(80×80) for the ADI-FDTD ones respectively. The time step is chosen to be 2ps and 25000

iterations are run in each simulations. This ensures the same FFT resolution (0.02GHz) in

every modelling.

Source(n) = Amej2πf ·n·dt · e−((n−delay)·dt

τ2)2 (4.104)

where Am is the maximum amplitude, f is the operating frequency, delay = 1dt·f is the


delay in time of the pulse, τ2 = 18f is the pulse half-duration at the 1/e points, dt is the

time increment and n is the iteration index. In this simulation, Am = 1, f = 0.5GHz,

dt = 2ps and the modulated Gaussian pulse is plotted in time domain and frequency

domain in figure 4.19.

0 1 2 3 4 5 6 7 8 9 10

x 10−9

0

0.05

0.1

time (s)

Am

plitu

de

real part of the excitation functionimaginary part of excitation function

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 109

0

5

10

15

20

frequency (Hz)

Mag

nitu

de

(b)

Figure 4.19: The modulated Gaussian pulse as excitation in (a) time domain and (b) fre-quency domain.


0 0.1 0.2 0.3 0.4 0.5 0.60

0.1

0.2

0.3

0.4

0.5

Figure 4.20: In ADI-NFDTD, the computation domain is meshed by a (46 × 46) grid.


10 20 30 40 50 60

10

20

30

40

50

60

Figure 4.21: In ADI-FDTD, the computation domain is meshed by a (60 × 60) grid.


10 20 30 40 50 60 70

10

20

30

40

50

60

70



10 20 30 40 50 60 70 80

10

20

30

40

50

60

70

80




Table 4.3 tabulates the resonant frequencies obtained from the FFT of the temporal

results calculated by the ADI-NFDTD with mesh (46 × 46). Table 4.4, 4.5, and 4.6 show

the resonant frequencies calculated from the ADI-FDTD with mesh (60 × 60), (70 × 70),

and (80 × 80) respectively.

The relative error rate of each mode is calculated by the absolute value of the dif-

ference of the calculated frequency with the theoretical one[151] divided by the latter in

expression of percentage. Then the standard deviation of the relative error rates from all

the frequencies (from 0GHz to 3GHz ) are taken as a measure of an overall relative error

rate.

The frequency spectra calculated from the ADI-FDTD with different spatial resolu-

tions and the ADI-NFDTD are compared in figures 4.24, 4.25 and 4.26. The theoretical

results are denoted by dashed lines vertical to x-axis at their designated frequencies.

Then the numerical error including the overall relative error rates and the number of

spurious modes from the ADI-NFDTD and the ADI-FDTD modellings are summarized

in table 4.7. The computational effort including the computer memory employed in the

simulation and the computation time for each modelling are also compared in this table.

It can be clearly seen from this table that to model the same structure with the same dt and

iteration number (and the same FFT resolution) the ADI-NFDTD method can provide

results with less numerical errors using less computer memory and computation time

compared with the ADI-FDTD method. This comparison demonstrates the accuracy and

computing efficiency of the ADI-NFDTD over the ADI-FDTD method.

However, compared to the ADI-FDTD method, the ADI-NFDTD method is not an

unconditionally stable method. When the time step is small, the temporal result still

suffers the late time instability which is inherited from the NFDTD scheme. This is the

drawback of the ADI-NFDTD method.


Theoretical The Genuine Relative Error The Spurious Relative ErrorResults [151] Modes Rate (RER) Modes Rate (RER)

(GHz) (GHz) (%) (GHz) (%)0.5863 0.6 2.340.9721 0.95 2.27 1.08 11.101.2198 1.22 0.021.3372 1.4 4.701.69 1.69 01.7 1.73 1.76

2.043 2.09 2.302.136 2.14 0.192.234 2.18 2.422.389 2.5 4.652.5533 2.5 2.072.718 2.86 4.852.732 2.86 4.312.956 2.86 1.56

Averaged Relative Error RateIncluding Only The Genuine 3.04 - -Modes (ARER-G)Averaged Relative Error RateIncluding All the Modes - - 4.25(ARER)

Table 4.3: Comparison of the accuracy of the resonant modes of the aforementioned copperresonator calculated by the ADI-NFDTD (Mesh 46 × 46) with the theoretical results.



(GHz) (GHz) (%) (GHz) (%)0.5863 0.56 4.490.9721 0.86 11.531.2198 1.16 4.90 1.1 9.821.3372 1.3 2.78 1.38 3.201.69 1.64 2.96 1.48 , 1.6 12.43 , 5.331.7 1.7 0

2.043 2.06 0.83 1.94 5.042.136 2.1 1.692.234 2.26 1.162.389 2.38 0.38 2.34 2.052.5533 2.58 1.062.718 2.66 2.132.732 2.66 2.632.956 2.98 0.81 2.86 3.25


Table 4.4: Comparison of the accuracy of the resonant modes of the aforementioned cop-per resonator calculated by the conventional ADI-FDTD (Mesh 60 × 60) with the theoreticalresults.



(GHz) (GHz) (%) (GHz) (%)0.5863 0.56 4.490.9721 0.88 9.471.2198 1.18 3.26 1.12 8.181.3372 1.34 0.211.69 1.62 4.141.7 1.78 4.71

2.043 2.1 2.79 1.98 3.082.136 2.16 1.122.234 2.28 2.062.389 2.34 2.052.5533 2.5 2.07 2.62 2.622.718 2.72 0.072.732 2.72 0.442.956 2.82 4.60





(GHz) (GHz) (%) (GHz) (%)0.5863 0.56 4.490.9721 0.88 9.471.2198 1.16 4.901.3372 1.38 3.20 1.42 6.191.69 1.64 2.96 1.5 , 1.56 11.24 , 7.691.7 1.72 1.18 1.8 5.88

2.043 2.0 2.10 1.88 7.982.136 2.1 1.692.234 2.2 1.522.389 2.36 1.212.5533 2.54 0.512.718 2.74 0.812.732 2.74 0.292.956 2.92 1.22



ADI-NFDTD ADI-FDTD

Mesh size 46 × 46 60 × 60 70 × 70 80 × 80dt (s) 2 × 10−12 2 × 10−12 2 × 10−12 2 × 10−12

nmax 25000 25000 25000 25000Frequency Resolution (GHz) 0.02 0.02 0.02 0.02Number of Spurious Mode 1 7 3 5

ARER-G (%) 3.04 3.91 3.84 3.49ARER (%) 4.24 6.31 4.55 6.022

Computer Memory (MB) 84.7 105.5 132.1 179.2Computer Run Time (minutes) 50 123 204 294

Table 4.7: Comparison of the accuracy and computer resources ( computer memory andcomputing time) of the ADI-NFDTD and ADI-FDTD in calculating the resonant modes of thecopper cavity. The simulations are run under programming environment of Matlab 2007b.


0.5863 0.9721 1.2198 1.69 2.043 2.3892.5532.718 2.9560

500

1000

1500

2000

2500

3000

ADI−FDTD 60X60 P4 (42,21)ADI−NFDTD 46X46 P4 (30,18)

Figure 4.24: The resonant frequency spectra calculated by the ADI-NFDTD (with mesh by 46by 46 grids) and the ADI-FDTD (with mesh by 60 by 60 grids).

0.5863 0.9721 1.2198 1.69 2.043 2.3892.5532.718 2.9560

500

1000

1500

2000

2500

3000

3500

4000




0.5863 0.9721 1.2198 1.69 2.043 2.3892.5532.718 2.9560

500

1000

1500

2000

2500

3000



4.5.4 Discussions

In this section, the proposed ADI-NFDTD is compared with the conventional NFDTD

scheme by modelling the free space wave propagation and a resonanting PEC cylindri-

cal cavity. The ADI-NFDTD scheme is also compared with the ADI-FDTD algorithm

by modelling a copper cylindrical cavity. The observations from these simulations are

discussed below.

• Stability Characteristics

The simulation results firstly show that the CFL condition of the conventional NFDTD

algorithm is removed by using the ADI-NFDTD scheme. The chosen value of the time

step dt is only limited by the accuracy requirement.

Unlike the conventional ADI-FDTD algorithm, the ADI-NFDTD scheme is not an un-

conditionally stable algorithm. This algorithm suffers the late time instability. However,


compared with the conventional NFDTD simulation with the same dt, the unstable re-

sult occurs at a later stage. That means the late time instability of the NFDTD is largely

improved by the use of the ADI technique.

• Accuracy

Compared with the NFDTD results, the ADI-NFDTD time domain results suffer at-

tenuation in amplitude with time going and a delay in phase. Increasing the time interval

dt results in a higher attenuation speed and a larger phase delay. The choice of dt will af-

fect the accuracy of the frequency spectrum. When a smaller dt is used, the ADI-NFDTD

is able to provide a more precise frequency spectrum with a high FFT resolution and a

low noise level. However, with an increase of dt, the amplitude attenuation in high fre-

quency components will increase. Consequently, higher frequency components tend to

be buried by the noise in the frequency spectra. So the accuracy in the higher frequency

band will decrease with the increase of dt, although the frequency resolution remains

high.

Compared with the ADI-FDTD method, the ADI-NFDTD method does not employ

the staircase approximation. The curved or oblique structures can be modelled confor-

mally, which means less grids can be applied and higher accuracy can be obtained in the

ADI-NFDTD modelling than in the ADI-FDTD one.

• Computational Efficiency

As a larger dt can be used in the ADI-NFDTD simulation than in a conventional

NFDTD one, the former can be more computationally efficient than the latter. However,

this increase in efficiency may be limited with the consideration of the accuracy issue

with a relatively large dt. The mesh file is another factor that affects the improvement on

efficiency in the ADI-NFDTD scheme. The improvement may be more significant with

those kinds of meshes that have locally very small and/or distorted cells.

Compared with the ADI-FDTD method, the ADI-NFDTD also demonstrated an im-

proved computational efficiency. As a direct result of coarser grids, computational effort

in terms of computer memory and computation time can be saved.


4.6 Summary

In this chapter, two extensions of FDTD, namely the nonorthogonal FDTD (NFDTD) and

the ADI-FDTD are briefly reviewed. A novel two-dimensional ADI-NFDTD in curvilin-

ear coordinate system that is free of the CFL stability condition is then presented. This is

followed by presenting three sets of numerical simulations that served as validations of

the ADI-NFDTD method.

1. The CFL stability condition is proven to be removed in the ADI-NFDTD method.

This is done by the free space wave propagation modelling using both the ADI-NFDTD

and the NFDTD. Therefore the time step dt in the ADI-NFDTD simulation is no longer

restricted by the numerical stability.

2. Comparison of ADI-NFDTD with the NFDTD: These two methods are compared

when modelling a PEC cavity resonator. It is demonstrated that the ADI-NFDTD tem-

poral result resembles the NFDTD result when dt is small. When a larger dt is used, the

error on the amplitude and phase of the temporal results will increase but the late time

stability of the NFDTD scheme is significantly improved. Since the dt is no longer re-

stricted by the CFL condition, the computational efficiency can be improved by the use

of a larger dt. This improvement is more significant if the mesh contains locally very

small and/or distorted cells.

3. Comparison of ADI-NFDTD with the ADI-FDTD: Since an oblique or curved sur-

face is modelled conformally instead of employing the staircase approximation, the re-

quirement of the spatial resolution in the ADI-NFDTD modelling is considered to be

lower than that in the ADI-FDTD modelling. This is verified by the modelling of a cop-

per cavity resonator using these two methods. The numerical error ( including relative

error rate and the number of spurious modes) and the computational resource (includ-

ing the computer memory and the CPU time) are compared. The ADI-NFDTD scheme

demonstrated a better accuracy and a saving on the computer resource. However, the

ADI-NFDTD scheme is not unconditionally stable like the ADI-FDTD method. It does

suffer the late time instability.

Chapter 5

NFDTD and ADI-NFDTD modelling

of EBG Structures

In this chapter, numerical experiments for modelling two-dimensional structures with

curved surfaces using NFDTD method will be reported. The comparison of the NFDTD

and Yee’s FDTD algorithm in modelling EBG structures is done in the first two simu-

lations: the calculation of the dispersion diagram of an infinite EBG structure and the

transmission coefficient calculation of a (semi-)finite bandgap structure. From these com-

parisons, the efficiency and accuracy of the NFDTD method is demonstrated. NFDTD

modelling of waveguide mode observed from a defect EBG structure and a prism shaped

EBG-like refractor are also presented.

In the last section of this chapter, numerical performance of the proposed ADI-NFDTD

method is demonstrated by modelling an EBG unit cell with curved inclusions. With a

reduction in the late time instability demonstrated by the ADI-NFDTD method, the ADI-

NFDTD results are shown with a high frequency resolution and a low noise level.

160

Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 161

5.1 NFDTD Modelling of the Infinite Electromagnetic Bandgap

Structures

In this section, two-dimensional (2-D) infinite EBG structures with curved inclusions are

studied numerically by both the NFDTD and the Yee’s FDTD schemes. By comparing

the numerical accuracy of the two schemes, the requirement on spatial resolutions, com-

puter memory, processing time, the efficiency and accuracy of the NFDTD method is

demonstrated. In this study, the unit cell approach that was previously discussed was

employed.

5.1.1 The Model of the EBG Structures

Metallic rods periodically loaded in square and triangular/rhombic lattices in free space

are modelled using both the NFDTD and the Yee’s schemes. These EBG structures are

infinite in x and y direction with a lattice constant (period) a. In the z direction, the rods

are infinitely long. Each rod is made from copper with relative permittivity εr = 1 and

conductivity σ = 5.8 × 107 S/m. The ratio of the radius r to the lattice constant a is cho-

sen to be r/a = 0.2. Numerical simulations are performed to determine the dispersion

diagrams for both the transverse electric (TE) and transverse magnetic (TM) polariza-

tion in square (figure 5.1) and triangular/rhombic lattices (figure 5.2). Results from both

the NFDTD and Yee’s FDTD methods are compared in terms of accuracy, efficiency and

robustness.


(a)

Γ X

M

a

r

unit cell

(b)

Figure 5.1: (a) The cylindrical metallic rods in free space in square lattice with radius r,spacing a and r/a = 0.2. (b) The x-y cut plane of the EBGs. The unit cell is the area within thedashed red line. The Brillouin Zone in the reciprocal lattice is shown in the light blue area.


(a)

Γ

X J

a r

unit cell

unit cell

unit cell

(b)

Figure 5.2: (a) The cylindrical metallic rods in free space in triangular/rhombic lattice withradius r, spacing a and r/a = 0.2. (b)The x-y cut plane of the EBGs. The unit cell is the areawithin the dashed red line. The Brillouin Zone in the reciprocal lattice is shown in the lightblue area.


5.1.2 Mesh of the Unit cell

The unit cells in square and triangular/rhombic lattice (see figure 5.1(b), figure 5.2(b)) are

meshed with different spatial resolutions using Yee’s scheme and the NFDTD scheme,

with examples shown in figure 5.3 and figure 5.4.

5 10 15 20 25 30

5

10

15

20

25

30

(a) (b)

10 20 30 40 50

10

20

30

40

50

20 40 60 80

10

20

30

40

50

60

70

80

(c) (d)

Figure 5.3: Examples of the mesh schemes for an unit cell from the square lattice in theNFDTD and the Yee’s FDTD modelling. (a) (18 × 18) NFDTD cells. (b) (30 × 30) FDTD cells.(c) (50 × 50) FDTD cells. (d) (80 × 80) FDTD cells.


(a) (b)

(c) (d)

Figure 5.4: Examples of the mesh schemes for an unit cell from the triangular lattice in theNFDTD and the Yee’s FDTD modelling. (a) (18 × 15) NFDTD cells. (b) (30 × 26) FDTD cells.(c) (50 × 42) FDTD cells. (d) (80 × 69) FDTD cells.


5.1.3 Settings in the FDTD Simulations

Because the rods in the EBGs are periodically located and infinite in the x and y directions,

the whole structure can be studied by investigating only one unit cell and by applying pe-

riodic boundary conditions (PBC). The excitation, the probe settings, the computational

domain and the boundary conditions are illustrated in figure 5.5.

(a)

(b)

Figure 5.5: The modelling schemes of the unit cell approach for EBG structures. (a) for squarelattice of figure 5.1; (b) for triangular/rhombic lattice of figure 5.2.

A modulated Gaussian pulse is used to provide a wide band excitation at different

positions inside the unit cell domain. These source points, as well as the probe points

introduced in the following, are randomly chosen but need to excite and cover all the

possible modes inside the structure. After all the possible modes are excited by the wide


band excitation, most modes will die out soon as evanescent waves. Only the modes

that are supported by the EBGs as propagating modes will propagate and hit the unit

cell again and again, resulting in accumulated energy, which can be detected as peaks

in the frequency spectra in the post processing. Temporal signals at the aforementioned

selected probe positions are used to detect all the possible transmission modes. With

the Fast Fourier Transformation (FFT) applied to these temporal signature, the frequency

spectra can be obtained and dispersion diagrams can be plotted.

As a direct solution of Maxwell’s equations, the FDTD method is an accurate time do-

main method in EM modelling. However, this accuracy depends on the spatial resolution

used in the simulation. To achieve an accurate, convergent result, the spatial resolution

should has an appropriate value. This means that the mesh should be fine enough and

a small spatial increment (∆x,∆y) will consequently require a small time increment (∆t)

by the CFL stability condition.

On the other hand, the spatial resolution and the corresponding time increment (∆t)

directly related to the computer memory and the computational time required in the sim-

ulation. These two factors are the main concerns and restrictions of the FDTD method. In

this way, a low spatial resolution is preferred in order to save computer resources (mem-

ory and time) and to enable electrically large models to be simulated. As a consequence,

it is interesting to investigate the relationship of the accuracy/convergency of the results

with the spatial resolution used in the simulation.

5.1.4 Simulation Results

The simulation is performed with various mesh densities and operating frequencies. The

dispersion diagrams of the aforementioned EBGs calculated by the NFDTD and Yee’s

FDTD method are compared in figures 5.6-5.9.

The simulation results demonstrate that the dispersion diagrams are converged when

the spatial resolution is adequate. The results obtained with high spatial resolutions (e.g.

a FDTD grid with spatial increment equals to one-eightieth of the operating wavelength


in free space dxY EE = λcfs

80 ) can be considered to be accurate and are taken as a reference

to validate the other FDTD schemes.

It is further noticed that with adequate spatial resolutions, results from the NFDTD

and Yee’s schemes exhibit similar error rates. Take the simulation result of EBGs with a

rectangular lattice as an example. It can be seen that the maximum error of the results

obtained from Yee’s FDTD with low spatial resolution occur near M and Γ point in the

dispersion diagram. In Table 5.1, a comparison of numerical results from various FDTD

schemes is demonstrated for the first few modes in figure 5.6 and figure 5.7.

0

0.5

1

1.5

2

Fre

quen

cy (ω

a/(2

πc))

Copper rods (r/a=0.2) in Square Lattice

Γ X M Γ

TE Mode Yee

80 80X80 NFF=1 (λ

cfs/80)

TE Mode Yee30

30X30 NFF=1 (λcfs

/30)

TE Mode NFDTD18

18X18 NFF=1(λcfs

/18)

Figure 5.6: The dispersion diagram of the first few TE modes, for the copper rods in freespace in square lattice. The Yee’s FDTD simulation result with high spatial resolution isplotted in purple line as a reference result. The Yee’s FDTD and the NFDTD result withthe low spatial resolution is plotted to compare with the high spatial resolution result. Itcan be seen that the maximum disagreement appears at M point. So the error rate at Mpoint is listed in table5.1(a). (NFF: Normalized central Feeding Frequency parameter of themodulated Gaussian pulse. It is a frequency parameter in the modulated Gaussian pulse.)


0

0.5

1

1.5

2

Fre

quen

cy (ω

a/(2

πc))


Γ X M Γ

TM Mode YEE

80 NFF=1 (λ

cfs/80)

TM Mode YEE30

NFF=1 (λcfs

/30)

TM Mode NFDTD18

NFF=1 (λcfs

/18)

Figure 5.7: The dispersion diagram of the first few TM modes, from the copper rods in freespace in square lattice. The Yee’s FDTD simulation result with high spatial resolution isplotted as a reference result. The Yee’s FDTD and the NFDTD result with the low spatialresolution is plotted to compare with the high spatial resolution result. It can be seen thatthe maximum disagreement appears at M point. So the error rate at M point is listed intable5.1(b). NFF: Normalized central Feeding Frequency parameter of the modulated Gaus-sian pulse. It is a frequency parameter in the modulated Gaussian pulse.


0

0.5

1

1.5

2

Dispersion Diagram of the Copper Rods in Triangular Lattice

Γ X J Γ

Fre

quen

cy (ω

a/(2

πc))

TE mode Yee80

NFF=1 (λcfs

/80)

TE mode Yee28

NFF=1 (λcfs

/28)

TE mode NFDTD18

NFF=3 (λcfs

/6)

Figure 5.8: The dispersion diagram of the first few TE modes, from the copper rods in freespace in triangular/rhombic lattice. The Yee’s FDTD simulation result with high spatial res-olution is plotted in blue line as a reference result. The Yee’s FDTD result with the minimumspatial resolution is plotted in red. The NFDTD result with the minimum spatial resolutionis plotted in green.


0

0.5

1

1.5

2

Γ X J Γ

Copper rods (r/a=0.2) in Triangular Lattice

Fre

quen

cy (ω

a/(2

πc))

TM Mode YEE80

80X69 NFF=1 (λcfs

/80)

TM Mode YEE30

30X26 NFF=1 (λcfs

/30)

TM Mode NFDTD18

18X15 NFF=1 (λcfs

/18)

Figure 5.9: The dispersion diagram of the first few TM modes, from the copper rods in freespace in triangular/rhombic lattice. The Yee’s FDTD simulation result with high spatial res-olution is plotted in blue line as a reference result. The Yee’s FDTD and the NFDTD resultwith the low spatial resolution is plotted to compare with the high spatial resolution result.


Results by the Results by the Relative Error Rate (RER)TE high spatial low spatial approximated by the

resolution (HSR) resolution (LSR) difference of HSR and LSR.Mode

Y ee80 Y ee30 NFDTD18|Y ee30−Y ee80|

Y ee80

|NFDTD18−Y ee80|Y ee80

1st Mode 0.6131 0.5977 0.5977 2.58% 2.58%2nd Mode 0.6131 0.5977 0.5977 2.58% 2.58%3rd Mode 0.6794 0.66 0.695 2.86% 2.23%4th Mode 0.8757 0.921 0.8757 5.17% 0%5th Mode 1.4 1.435 1.418 2.50% 1.29%6th Mode 1.497 1.48 1.487 1.14% 0.67%7th Mode 1.539 1.508 1.529 2.01% 0.65%

MaximumRER

5.17% 2.23%

Averaged RER2.92% 1.72%

(ARER)

(a)

Results by the Results by the Relative Error Rate (RER)TM high spatial low spatial approximated by the

resolution (HSR) resolution (LSR) difference of HSR and LSR.Mode

Y ee80 Y ee30 NFDTD18|Y ee30−Y ee80|

Y ee80

|NFDTD18−Y ee80|Y ee80

1st Mode 0.7297 0.7367 0.7367 0.41% 0.41%2nd Mode 0.8666 0.8514 0.834 0.81% 3.76%3rd Mode 0.8666 0.8514 0.834 0.81% 3.76%4th Mode 1.386 1.358 1.307 2.02% 5.7%5th Mode 1.588 1.581 1.571 0.44% 1.07%6th Mode 1.588 1.581 1.571 0.44% 1.07%7th Mode 1.601 1.595 1.64 0.37% 2.43%

MaximumRER

2.02% 5.7%

Averaged RER0.93% 3.14%

(ARER)

(b)

Table 5.1: Comparison of the relative error rate on the dispersion diagrams at M point for(a) TE mode, (b) TM mode, calculated by the Yee’s FDTD and the NFDTD with low spatialresolutions. This error rate is approximated by the difference of the low-spatial-resolutionresult compared with the high-spatial-resolution result.


However, the NFDTD scheme exhibits an overwhelming advantage when the mini-

mum spatial resolutions required by these two algorithms are compared. Take the mod-

elling of the copper rods in a triangular lattice (TE mode calculation) as an example. Fre-

quency spectra at k = OΓ (Γ point on the dispersion diagram) is calculated by NFDTD

method using different density of meshes (size: 18× 15, 18× 15 and 10× 8) and different

normalized central feeding frequencies (NFF = 1, 3, 2). These parameters corresponds

to spatial resolutions of ( λcfs

18 , λcfs

6 and λcfs

5 ) respectively (see figure 5.10).

Results with dxNFDTD = λcfs

18 are used as a reference as it has been proved to be

adequate to provide converged and accurate results. It can be seen that dxNFDTD = λcfs

6

is the minimum spatial resolution required by NFDTD for this model to provide a correct

dispersion relation with all the genuine modes identified. Any reduction on the spatial

resolution upon this value will result in wrong dispersion relationship due to the fact that

strong spurious modes are mixed with the genuine modes in the calculated frequency

spectra.


0 0.5 1 1.5 20

10

20

30

40

50

60

70

80

Normalized Frequency (ωa/2πc)

Mag

netic

Fie

ld H

z (A

/m)

Frequency Spectra of wave vector k at Γ pointCopper rods (r/a=0.2) in Triangular Lattice

NFDTD 18X15 NFF=1 (λ/18)NFDTD 18X15 NFF=3 (λ/6)NFDTD 10X8 NFF=2 (λ/5)

Figure 5.10: The TE mode frequency spectra of the wave vector k at Γ point, for the modellingof the cylindrical copper rods in triangular lattice. The green line and the pink line showthe frequency spectra calculated by NFDTD using a high spatial resolution (dxNFDTD =λcfs

18 ) and a low spatial resolution (dxNFDTD = λcfs

6 ) respectively. The positions of peaks inthese two lines agree well, indicating all the modes founded by the low spatial resolution(dxNFDTD = λcfs

6 ) are all genuine. The blue line shows the frequency spectra using spatialresolution of dxNFDTD = λcfs

5 . On the blue line, spurious modes (marked by the dotted redcircles) are strong enough to compete with a genuine mode (marked by the dashed orangecircle) , which will cause a wrong dispersion diagram. The comparison of these three linesindicates that dxNFDTD = λcfs

6 is the minimum spatial resolution required by NFDTD forthis modelling.


Figure 5.11 shows that Yee’s FDTD simulation with dx = λcfs

26 provides a result with

spurious spectrum in low frequency band. It is found from more simulations that ( λcfs

28 )

is the mininum spatial resolution required by the Yee’s scheme for this model.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

10

20

30

40

50

60

70

80

90

100


Mag

netic

Fie

ld H

z (A

/m)

Frequency Spectra of wave vector k at Γ point Copper rods (r/a=0.2) in Triangular Lattice TE mode

NFDTD 18X15 NFF=1 (λ/18)

Yee’s FDTD 26X23 NFF=1 (λ/26)

Figure 5.11: The TE mode frequency spectra of the wave vector k at Γ point of the cylindricalcopper rods in triangular lattice. Yee’s FDTD results with a spatial resolution of dx = λcfs

26provides a result with spurious energy in low frequency band (marked by the dotted redcircles) which are strong enough to compete with a genuine mode (marked by the dashedorange circle). The NFDTD results with a spatial resolution of dx = λcfs

18 which provide allthe genuine modes is used as a reference.


More examples are shown in order to compare the minimum spatial resolutions re-

quired by the NFDTD and the Yee’s algorithms. Figure 5.12 illustrates the frequency

spectra at Γ for TE mode modelling of the copper rods in a square lattice, calculated by

using the NFDTD method with different spatial resolutions. Figures 5.13 - 5.15 show the

spectra calculated using Yee’s FDTD scheme with different spatial resolutions.

It can be seen from figures 5.12, 5.13 and 5.14, although some spurious resonances

occur at some probe positions (marked by the dotted red circles), the amplitude of them

averaged by all the probes are much smaller than that of a genuine mode. Since the

dispersion diagram is calculated by an average of frequency spectra of all the probe po-

sitions, these spurious modes can be numerically filtered out as noise and will not affect

the dispersion diagram calculation. This is referred to in this thesis as the ”system tol-

erance”. However, as shown in figure 5.15, if the spatial resolutions are reduced further,

the spurious modes become as strong as a genuine one. This means the spatial reso-

lution is beyond the system tolerance and consequently results in inaccurate dispersion

calculations for EBGs.

Figure 5.16 compares the TM mode frequency spectra for Γ point in a square lattice of

EBGs, calculated by the NFDTD and the Yee’s FDTD algorithms.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

1

2

3

4

5

6

7

8

Normalized Frequency (ωa/(2πc))

Mag

nitu

de o

f Mag

netic

Fie

ld H

z (A

/m)

Frequency Spectra for wave vector k from the center of the Brillioun Zone to Γ point (TE) (NFF: Normalized Central feeding frequency; λ

cfs: wavelength at central frequency in free space)

NFDTD 18X18 NFF=1 (λcfs

/18)

NFDTD 18X18 NFF=2 (λcfs

/9)

NFDTD 18X18 NFF=2.25 (λcfs

/8)

Figure 5.12: The TE mode frequency spectra of the wave vector k at Γ point, for the modellingof the cylindrical copper rods in square lattice. NFDTD simulation result with a low spatialresolution (dxNFDTD = λcfs

9 ) agrees with the high spatial resolution (dxNFDTD = λcfs

18 ).With a spatial resolution (dxNFDTD = λcfs

8 ) close to the minimum one (dxNFDTD = λcfs

6 ),spurious mode (marked by the dotted red circle) can be observed with a weaker energy levelthan a genuine mode (marked by the dashed orange circle). At other probe positions the spu-rious energy at this frequency is much weaker than a genuine mode, so the average energyat this frequency is much weaker than a genuine mode, which results in this spurious modecan be filtered out numerically as noise and will not affect the calculation of the dispersiondiagram.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5


Mag

nitu

de o

f Mag

netic

Fie

ld H

z (A

/m)

Frequency Spectra for wave vector k from the center of the Brillioun Zone to Γ point (TE) (NFF: Normalized Central feeding frequency)

Yee’s FDTD 80X80 NFF=2.2 (λcfs

/36)

Yee’s FDTD 36X36 NFF=1 (λcfs

/36)

Figure 5.13: The TE mode frequency spectra of the wave vector k at Γ point in reciprocalspace of the cylindrical rods in square lattice simulated by the Yee’s FDTD with adequatespatial resolutions (dxFDTD = λcfs

36 ). Spurious mode (marked by the dotted red circle) canbe observed with a much weaker energy level than a genuine mode (marked by the dashedorange circle). As a result, this spurious mode can be filtered out numerically as noise andwill not affect the calculation of the dispersion diagram.


0 0.5 1 1.5 2 2.50

2

4

6

8

10

12


Mag

nitu

de o

f Mag

netic

Fie

ld H

z (A

/m)

Frequency Spectra for k vector on Γ point (TE) (NFF: Normalized Central feeding frequency)


/36)


/32)

Figure 5.14: The TE mode frequency spectra of the wave vector k at Γ point of the cylindricalrods in square lattice, simulated by the Yee’s FDTD method using reduced spatial resolutions.By reducing the spatial resolution from (dxFDTD = λcfs

36 ) to (dxFDTD = λcfs

32 ), which is closeto the required minimum value dxFDTD = λcfs

30 , spurious mode (marked by the dotted redcircle) can be observed with a weaker energy level than a genuine mode (marked by thedashed orange circle). Since averaging this energy at this frequency upon all the probespositions gives a much lower energy level than a genuine mode, this spurious mode can befiltered out numerically as noise and will not affect the calculation of the dispersion diagram.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

1

2

3

4

5

6

7

8

9

10

11


Mag

nitu

de o

f Mag

netic

Fie

ld H

z (A

/m)

Frequency Spectra for k vector on Γ point (TE) (NFF: Normalized Central feeding frequency)

Yee’s FDTD 50X50 NFF=2 (λ

cfs/25)

Yee’s FDTD 36X36 NFF=1 (λcfs

/36)

Figure 5.15: The TE mode frequency spectra of the wave vector k at Γ point in reciprocalspace of the cylindrical rods in square lattice. The blue line plots the Yee’s FDTD resultswith a spatial resolution lower than the minimum spatial resolution. Consequently, spuriousmode (marked by the dotted red circle) can be observed with the blue line with a comparableor higher energy level than that of a genuine mode (marked by the dashed orange circle). Asa result, this spurious mode can not be filtered out numerically as noise and will result in awrong dispersion diagram. The purple line plots the Yee’s FDTD results with an adequatespatial resolution.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

5

10

15

20

25

30


Mag

nitu

de o

f Ele

ctric

Fie

ld E

z (V

/m)

Frequency Spectra for wave vector k from the center of the Brillioun Zone to Γ point (TM) for the copper rods in square lattice

( NFF: Normalized Central feeding frequency; λ

cfs: wavelength at central frequency in free space)


/80)


/30)

NFDTD 18X18 NFF=2.0 (λcfs

/9)

Figure 5.16: The TM mode frequency spectra of the wave vector k at Γ point for the cylindri-cal rods in square lattice, modelled using the NFDTD and the Yee’s FDTD with different spa-tial resolutions. Yee’s FDTD result with spatial resolution of (dxFDTD = λcfs

80 ) is used as a ref-erence. In Yee’s simulation result with spatial resolution of dxFDTD = λcfs

30 , spurious energy(marked by the red dotted circle) is observed with an amplitude comparable with a genuinemode (marked by the orange dash circle). At the same probe position, NFDTD simulationprovides all the genuine modes with a much lower spatial resolution (dxNFDTD = λcfs

9 ).


The minimum spatial resolutions above the system tolerance for modelling the afore-

mentioned structures (figures 5.1 and 5.2) required by the NFDTD and the Yee’s FDTD

methods is summarized in Table 5.2. The minimum values vary from different models

and operating frequencies, etc. (see figure 5.17). FDTD simulations with finite or infinite

EBG structures must meet their corresponding requirements and a high spatial resolu-

tion is always preferred for the consideration of accuracy within the capability of the

computer resources.

Table 5.3 shows how the computer resources can be saved by using NFDTD with a

lower spatial resolution. (The data are from the NFDTD and FDTD simulations for one

k-vector on the dispersion diagram of the copper rods in triangular lattice, TE mode.)

Spatial Resolution(dx) Mesh Size for 1 cell ( adx )

Model NFDTD YEE’s FDTD NFDTD YEE’s FDTD

Square Lattice λcfs

6λcfs

306·f · 2πa

c 30 · f · 2πacTE Mode

Square Lattice λcfs

9λcfs

309·f · 2πa

c 30 · f · 2πacTM Mode

Triangular Lattice λcfs

6λcfs

286·f · 2πa

c 28 · f · 2πacTE Mode

Triangular Lattice λcfs

9λcfs

309·f · 2πa

c 30 · f · 2πacTM Mode

λcfs: Wavelength at central working frequency in free space.

a:lattice constant; f : working frequency; c: speed of light in vacuum.

Table 5.2: Comparison of the minimum spatial resolution required by NFDTD and the Yee’sFDTD when simulating the copper rods infinite EBGs in square lattice or triangular lattice.


0 0.5 1 1.5 2 2.5 30

10

20

30

40

50


Min

imum

Spa

tial R

esol

utio

n

(cel

ls/fr

ee s

pace

wav

elen

gth)

Copper rods array (r/a=0.2) Triangular Lattice. TE polarization

Yee’s FDTDNFDTD

(a)

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100

120

Dom

ain

size

(in

x−

dire

ctio

n)

fo

r th

e un

it ce

ll

Normalized frequency (Normalized Frequency (ωa/2πc))

Copper rods array (r/a=0.2) Triangular Lattice. TE polarization

Yee’s FDTDNFDTD

(b)

Figure 5.17: The minimum spatial resolution (a) and mesh size (b) required for the Yee’sFDTD and NFDTD algorithms for the triangular lattice TE mode unit cell simulation.


Unit NFDTD Yee’s FDTD

Operating FrequencyNormalized frequency 3 3

(NFF)Mesh size

cell2 18 × 18 72 × 72(Minimum required)

Spatial resolutioncells per free-space

6 24wavelength

Frequency resolution Normalized frequency 0.0139 0.0139

dt second 1.7187e− 11 1.7187e− 11

Total number of iteration10360 10360

(nmax)Computer Memory Used

MB 6.87 11.1(maximum value)

Computation timesecond 15.1 42.3

(on average)Relative Error Rate

% 7.99 4.25(Maximum)

Relative Error Rate% 1.53 0.38

(Mean)Averaged Relative Error

% 2.09 0.74Rate (ARER)

Table 5.3: Comparison of the memory, the computational time used (on average) in calculat-ing one k vector in dispersion diagram, and the accuracy of the results using the Yee’s FDTDand the NFDTD method.


5.1.5 Discussion and Conclusion

All comparisons demonstrate the efficiency of the NFDTD algorithm over the Yee’s FDTD

algorithm when modelling EBGs with curved inclusions.

When the spatial resolution in the simulation is adequate, the dispersion diagram is

convergent from both algorithms. With the spatial resolution being decreased gradually,

the error increases and spurious resonances start to occur. The spurious energy is seen

to increases gradually as the spatial resolution being decreased further, until its energy

is comparable to that of a genuine eigenmode and hence result in a spurious eigenmode

of the EBG. As a result, for each model a minimum spatial resolution is required and the

spatial resolution used in the simulation need to be high enough to avoid the spurious

modes violating the dispersion diagram.

In the unit cell approach, the size of the computational domain is small as only a single

EBG element is included in the simulation. Consequently the computational burden is

small for both FDTD schemes and a dense mesh is affordable by both schemes. The

approach of finding the minimum spatial resolution using both the Yee’s FDTD and the

nonorthogonal FDTD helps to achieve the following conclusions:

• Knowledge of the spurious eigenmodes can be obtained. When predicting the

spurious modes due to an inadequate spatial resolution before any simulation is run, it is

quite likely to relate the small physical dimension (dx) to a frequency component in the

higher frequency band. However, simulation results show that spurious modes always

appear in lower frequency bands (among the first few genuine modes) for the staircase

approximation. Higher frequency bands are ’noise-free’, with all the modes genuine or

all the genuine modes much stronger than the spurious ones.

• To simulate any finite bandgap structure in the real world, the spatial resolution

should be high enough to avoid spurious modes in the simulation results. So a pre-

knowledge about the minimum spatial resolution required is necessary. This knowledge

can be gained by the study of the corresponding infinite EBGs via the unit cell approach.

• The minimum spatial resolutions depends on the configurations of EBGs, the

mode calculation and the FDTD scheme. For the same model, the NFDTD algorithm


always requires a much lower minimum spatial resolution than the Yee’s algorithm.

• As can be seen in figure 5.17, the mesh size required in modelling one unit

cell increases (nearly) proportionally with the increase of the operating frequency in the

Yee’s FDTD simulation. Consequently, when simulating a finite sized bandgap structure

operating at a high frequency in the real world, substantial computer resources (memory

and CPU time) are required. These facts make it impossible to simulate electrically large

EBG structures using the Yee algorithm.

• Despite the requirement of the additional variables and calculations in the NFDTD

method, the computer resource (memory and CPU time) used in the simulation can still

be reduced compared to Yee’s scheme by the use of a much coarser grid.

• When the mesh size is restricted by the computer memory, NFDTD method can

always deal with high order modes with less numerical errors. In this way, NFDTD is

more robust in terms of high frequency response than the Yee’s FDTD scheme.

• Due to the late time instability of the NFDTD algorithm, the energy of the tem-

poral signal will increase dramatically after a certain physical time in the simulation.

Consequently, the temporal signatures used for frequency analysis in post processing

should be truncated. However, a limited physical length of the time period will reduce

the frequency resolution after the Fourier transformation is performed. In order to pro-

vide the required frequency resolution, the truncated signal must be long enough. As

a result, the high power numerical errors will be involved in the post process which

will increase noise in the frequency spectra. At this point, the proposed ADI-NFDTD is

expected to reduce the late time instability of NFDTD algorithm. The ADI-NFDTD sim-

ulation results shown in Chapter 4 demonstrate the expected improvements in lowering

the noise level and delaying the instability into a later time.


5.2 NFDTD Modelling of the (Semi-)Finite EBG Structure

In this section, a semi-finite two-dimension EBG structure (finite in one direction and infi-

nite in another direction) is studied by the NFDTD and the conventional Yee algorithms.

The bandgaps of the EBG structure are determined by a transmission coefficient calcu-

lation and they are compared with analytical results calculated from the transfer-matrix

method (TMM)[47].

5.2.1 The EBG Model

The EBG, as shown in figure 5.18, consists of arrays of dielectric cylindrical scatterers.

These cylinders have a radius r = 0.48cm and a dielectric constant εr = 9. They form a

square lattice with a lattice constant (period) a = 1.27cm. These rods are infinitely loaded

in the x direction. In the y direction, there are 8 layers of elements. In the z direction,

the rods are infinitely long. The ratio of the radius r to the lattice constant a (r-a ratio) is

r/a = 0.378. Numerical simulations are performed to determine the bandgaps for both

the TE and TM polarization.

Figure 5.18: The dielectric cylinders with radius r = 0.48cm, and dielectric constant εr = 9,surrounded by air, forming a square lattice with a lattice constant a=1.27cm; Infinite in z-direction; Infinite in x-direction; Finite (8 elements) in y-direction;


5.2.2 FDTD Model

Since the rods in the EBGs are periodic and infinite in the x direction, the entire struc-

ture can be studied by investigating only one row of EBG cells and by applying peri-

odic boundary conditions (PBCs) at the ends of x direction. In the y direction, a Perfect

Matched Layer (PML) is used to terminate the computational domain. The model is il-

lustrated in figure 5.19.

Figure 5.19: The cut plane of the EBG structure consists of 8 by infinite dielectric cylindricalrods array in square lattice, modelled by the NFDTD method. The rods are assumed to beinfinitely long, so a two-dimension model is applied. In the direction which the 8 elementsaligned in, PML is used to terminate the computational domain. In the direction which thearray is infinitely loaded in, the unit cell approach is applied with PBCs used to terminatethe computational domain.

A wide band plane wave propagating along the y direction is excited from a line

source at one side of the EBG rods. The wave form is Gaussian pulse which covers fre-

quency range 0 ∼ 20GHz.

At the other side, temporal signal is collected and Fourier Transformation is per-

formed. Then, after calibration, the transmission coefficients which indicating the ex-

istence of bandgaps can be calculated.


5.2.3 Mesh of the Unit cell

The meshing of the unit cell under various spatial resolutions are shown in figure 5.20

and figure 5.21.

Figure 5.20: The conformal mesh of the unit cell of the structure described in figure 5.18 inthe x-y cut plane with spatial resolution of 48X48 per unit cell.


(a) (b)

5

2

5

3

5

4

5

5

(c) (d)

Figure 5.21: The conformal mesh of the unit cell of structure (figure 5.18) in the x-y cut planewith different spatial resolutions ((a) 26X26; (b) 16X16; (c) 12X12; (d) 10X10 per unit cell).


5.2.4 Simulation Results

The temporal responses at the probes are plotted in figure 5.22, figure 5.23 and figure

5.24.

0 1 2 3 4 5 6

x 104

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Time steps

Magnetic f

ield

Hz (

A/m

)

Temporal result (source signal after time step 2e4 is all-zeros)

source

probe 1

Figure 5.22: The NFDTD temporal results (using mesh 5.20) after a Gaussian pulse planewave excitation at one side of the EBG slab.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-7

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Time (s)

Magnetic F

ield

Hz (

A/m

)

Mesh: mycircle542

smaller

source

allprobes1,1(1,1:19500)

(r = 8 cells; a=26 cells)

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time (s)

Magnetic F

ield

Hz (

A/m

)

Mesh: mycircle546

smaller

source

allprobes1,1(1,:)


(b)

Figure 5.23: The NFDTD temporal results with different spatial resolutions after a ModulatedGaussian pulse plane wave excitation at one side of the EBG slab. (a) Mesh 5.21(a) is used;(b) Mesh 5.21(b) is used. )


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-7

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Time (s)

Magnetic F

ield

Hz (

A/m

)

Mesh: mycircle547

smaller

source

allprobes1,1(1,:)


(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-7

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Time (s)

Magnetic F

ield

Hz (

A/m

)

Mesh: mycircle548

smaller

source

allprobes1,1(1,:)

(r = 3 cells ; a=10 cells)

(b)

Figure 5.24: The NFDTD temporal results with different spatial resolutions after a ModulatedGaussian pulse plane wave excitation at one side of the EBG slab. (a) Mesh 5.21(c) is used;(b) Mesh 5.21(d) is used.


The transmission coefficients versus frequency for TE polarization calculated by the

NFDTD algorithm are plotted in figure 5.25. It can be seen that when the mesh size for

an unit cell is greater than agrid = 12 cells, the results are convergent. The transmission

coefficient shows considerate reduction around 11GHz and 15.5GHz. The first stopband

is not seen because the EBG layer in the y direction is relatively too thin (in terms of

electrical length) for prohibiting the lower frequency (longer wavelength) waves. The

NFDTD results show good agreement with the the TMM results (figure 5.26) and the

Yee’s FDTD results (figure 5.27). Good agreement is also seen in the TM mode results (

see figure 5.28 and figure 5.29).


1 2 3 4 5 6 7 8 9.52 12 12.8 13.614.3 15.7216.5 20-100

-80

-60

-40

-20

0

20

40

60nfdtd v63 S21

Frequency (GHz)

S21 (

Magnetic f

ield

Hz)

(dB

)

r = 3 cells

r = 4 cells

r = 5 cells

r = 8 cells

r = 16 cells

Figure 5.25: The transmission coefficients (TE mode) of the dielectric rods (figure 5.18) calcu-lated using the NFDTD method. r is the radius of the cylinder.

Figure 5.26: The theoretical transmission coefficients (TE mode) of the dielectric rods (figure5.18) predicted by the transfer-matrix technique [47].


2 4 6 8 10 12 14 16 18 20-60

-50

-40

-30

-20

-10

0

10

Frequency (GHz)

Tra

nsm

issio

n C

oeff

icie

nt,

T (

dB

)

3

6

12

24

(a)

1 2 3 4 5 6 7 8 9.52 12 12.8 13.614.3 15.7216.5 20-100

-80

-60

-40

-20

0

20

40

60Compare of the Yee's FDTD and NFDTD results

Frequency (GHz)

S21

(Magnetic f

ield

Hz)

(dB

)

r = 8 cells

FDTD r =12 cells

(b)

Figure 5.27: The transmission coefficients (TE mode) of the EBGs consisting of dielectric rods(figure 5.18). (a) the Yee’s FDTD simulation results (results from Dr Yan Zhao) and (b) thecomparison of NFDTD and Yee’s FDTD results with similar spatial resolutions.


Figure 5.28: The transmission coefficients (TM mode) of the dielectric rods (figure 5.18) cal-culated using the NFDTD method with different spatial resolutions. r is the radius of thecylinder.

Figure 5.29: The theoretical transmission coefficients (TM mode) of the dielectric rods (figure5.18) predicted by the transfer-matrix technique [47].



Both the NFDTD and Yee’s FDTD results show good agreement with the analytical results

predicted by the transfer-matrix technique when the spatial resolution is large enough.

However, there exist small differences in the parameters used by these two algorithms.

As it can be seen from figure 5.30, the NFDTD model uses an exact parameter set

by the physical model. Since the discretization in Yee’s scheme uses uniform cells, to

accurately model a r-a ratio of 0.48/1.27 = 48/127, a resolution of at least agrid = 127

cells and rgrid = 48 cells are needed. This is a too heavy computational burden for FDTD

simulation. So, a similar r-a ratio is used to reduce the necessity of a high resolution.

a = 1.28 cm is used in Yee’s FDTD model to make a r-a ratio of 0.48/1.28 = 3/8. As a

result, agrid can be any integral multiple of 8, with rgrid being the corresponding integral

multiple of 3.

r_grid = 8

a_grid = 26

r_grid = 12

a’_grid = 32

r_grid =16

a_grid = 46

r_grid = 24

a’_grid = 64

r_grid = 4

a_grid = 12

r_grid = 6

a’_grid = 16

r_grid = 5

a_grid = 16

r_grid = 9

a’_grid = 24

r_grid = 3

a_grid = 10

r_grid = 3

a’_grid = 8

NFDTD

a = 1.27cm

Yee’s FDTD

a’=1.28cmYee’s FDTD:

r = 0.48cm;

a’ = 1.28cm;

r/a’ = 3/8;

NFDTD:

r = 0.48cm;

a = 1.27cm;

r/a = 48/127;

Figure 5.30: Comparison of the FDTD and NFDTD simulation parameters.


This comparison shows another advantage of the NFDTD algorithm over the Yee

algorithm:

• The spatial increments (dx or dy) of the Yee algorithm in one simulation are

fixed. Even with the subgridding scheme, dx (or dy) in each subdomain is fixed. On the

other hand, the spatial increments of the NFDTD algorithm are flexile. As a result, even

when the inclusions of the modelled problems are orthogonal, the NFDTD algorithm can

model structures with any dimensions with any ratios more accurately, more easily and

by using a simpler and coarser mesh.

5.3 NFDTD Modelling of an EBG-like Waveguide

A two-dimension EBG-like structure with defects working as waveguide is modeled us-

ing both the NFDTD algorithm and the Yee’s FDTD algorithm. (15 × 16) dielectric rods

with relative permittivity of εr = 11.56 and r-a ratio r/a = 0.2 are aligned in square lat-

tice, with a bended pass on which the rods are missing. In the simulations, PML is used

to terminate the computational domain. The waveguide structure is working in bandgap

frequency of a corresponding non-defect EBGs with a sine wave excited as a point source

on the defected pass. The wave is seen propagating along the waveguide with almost

all energies constrained within the defected pass. A snapshot of the field distribution in

the TM mode from NFDTD algorithm is shown in figure 5.31. It is similar to the FDTD

result. However, the NFDTD algorithm saves a lot of computation time by using a mesh

size of (550 × 590) compared to the mesh size of (890 × 940) in the Yee’s FDTD method.


Figure 5.31: The NFDTD field plot near the bend of the EBG-like waveguide.

5.4 EBG-like Negative Refractor Modelling

Another EBG-like structure consisting of infinite long metallic wires which are arranged

in a prism shape was studied (ref. Figure (6) in [153]) to investigate the negative refraction

effect of the PBG-like structure. The wires are of radius r = 0.63 cm and r/a = 0.36.

The wires are arranged in a prism shape with 10 elements in the base and vertical sides,

respectively. A sine wave is excited from a point source in the middle from the left side

of the prism. PML is used to terminate the simulation domain. The structure is modelled

using the NFDTD method with a feeding frequency of 6.8 GHz and 7.6 GHz. The positive

and negative refraction reported in [153] from Ansoft’s HFSS simulation is verified by the

NFDTD simulation (see figure 5.32).


(a)

(b)

Figure 5.32: The electric field plot from the prism-shaped EBG like structure consisting ofmetallic wires, simulated by NFDTD method. (Unit is in V/m.) (a) the positive refractionseen with feeding frequency of 6.8 GHz. (b) the negative refraction seen with feeding fre-quency of 7.6 GHz.


(a)

(b)

Figure 5.33: Direction of the electric-field-propagation refraction from the prism-shaped PBGlike structure with positive refraction (a) at 6.8 GHz, and the negative refraction (b) at 7.6 GHz(Unit is in V/m), simulated by Ansoft HFSS. [153]


5.5 ADI-NFDTD Modelling of the EBG Unit Cell

In this section, the performance of the proposed ADI-NFDTD method is demonstrated by

the modelling of an EBG unit cell ( figure 5.1) in comparison with the NFDTD method. A

reduction in the late time instability is demonstrated by the ADI-NFDTD method. Con-

sequently the ADI-NFDTD results are with a higher frequency resolution and a lower

noise level. As a result, some modes that are very hard to detect by NFDTD modelling

can be found easily in the ADI-NFDTD simulation results.

5.5.1 NFDTD Limitations

In the first numerical model of chapter 4, a dispersion diagram of an infinite 2-D EBG

formed by metallic rods periodically loaded in free space is examined by the NFDTD

simulations. However, with some k vector, the unstable NFDTD results occurs earlier

than others. This means that in order to keep the minimum frequency resolution, more

errors are involved in the truncated temporal signals which leads to a higher noise level

in the frequency spectrum. As a result, some genuine modes are ’buried’ in the noise and

become very hard to detect.

Take the previous copper rods in square lattice TE mode simulation for example. Both

the physical and reciprocal lattices are plotted in figure 5.34. The dispersion diagram cal-

culated by NFDTD is also recalled and re-plotted in figure 5.35. In those simulations,

while calculating k points near k = ΓM , the time domain signals exhibit instability

quickly. Figure 5.37(a) shows the temporal signature at one probe position for calcualt-

ing k = k1. Figure 5.36 shows the frequency spectra from different probe positions. As

a consequence of the exponential power numerical error, it is hard to find the mode at

frequency 1.35(normalized) from the spectra. As is shown in figure 5.36, this frequency

peak is missing with most probe positions (blue ones), and only with few probe positions

(red one as an example), this genuine mode is strong enough to compete with the noise

and form a very small peak (emphasized by a green circle).


Figure 5.34: The unit cell of the EBGs (figure 5.1) and the Brillouin Zone in the reciprocallattice with k = k1 shown by a red vector.

0

0.5

1

1.5

2

X: 23Y: 1.356

Fre

quen

cy (ω

a/(2

πc))


Γ X M k1 Γ

Figure 5.35: The dispersion diagram of the square lattice EBG, TE mode results, calculatedusing the NFDTD methods.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

50

100

150

Frequency Spectra at k1 point by NFDTD


Mag

netic

Fie

ld H

z

Figure 5.36: The frequency spectra of different probes when calculating k = k1. It is hard todetect the eigenfrequency as a peak at frequency 1.356.


5.5.2 Comparison of the ADI-NFDTD and NFDTD Simulation Results

A simulation is carried out by the ADI-NFDTD method for the same problem using the

same mesh file and the same time step dt. Temporal and frequency results are compared

with the NFDTD results (figure 5.37 and figure 5.38). As it can be seen, the resemblance of

the first 0.5 picosecond signals verifies the ADI-NFDTD algorithm. After that, the energy

of the ADI-NFDTD signal reduces gradually while that of the NFDTD signal increases

dramatically. After the FFT, the ADI-NFDTD frequency spectra show a lower noise level

and mode of 1.36 (emphasized by the purple circle in figure 5.38) is clearly shown.

0 2 4 6 8 10 12 14 16

x 10−8

−50

0

50

100

Time (s)

Mag

netic

Fie

ld H

z (A

/m)

Temporal signal at one Probe Point at k=k1 Point

Probe 1 by NFDTDProbe 1 by ADI−NFDTD

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−8

−2

−1

0

1

2

Time (s)

Mag

netic

Fie

ld H

z (A

/m)

Temporal signal at one Probe Point at k=k1 Point

Probe 1 by NFDTDProbe 1 by ADI−NFDTD

(b)

Figure 5.37: The temporal response at one probe position when calculating k = k1. (a)Thecomparison showing the reduction of the late time instability by ADI-NFDTD. (b)The en-larged view of the first 0.5 picosecond signals, which shows the resemblance of the ADI-NFDTD results and the NFDTD results.


0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

50

100

150

200

Frequency Spectra at one k point by NFDTD and ADI−NFDTD


Mag

netic

Fie

ld H

z

Probe of NFDTDProbe of ADI−NFDTD

Figure 5.38: The comparison of the frequency spectra in calculating k = k1 using the con-ventional NFDTD and the ADI-NFDTD. The NFDTD spectra which best indicates the modeat 1.36 is plotted in red line. The ADI-NFDTD spectra with an overall low noise level andclearly showing the mode at 1.36 is plotted in green line.


In this section, the advantage of the ADI-NFDTD scheme over the conventional NFDTD

scheme in calculating the dispersion diagram of an EBG by means of the unit cell ap-

proach is demonstrated. When modelling the unit cell using the periodic boundary con-

dition based on different wave number k, the temporal responses become unstable much

earlier based on some k than others. This will cause an inadequate length of the time

domain result, a high noise level in the result, or both. As a consequence, some peaks

in the frequency spectra which stand for the corresponding eigenmodes of the EBGs are

very hard or impossible to detect.

To addresse this issue, the proposed ADI-NFDTD algorithm is used to model the

same problem. Much longer stable temporal results are achieved. Due to the longer

temporal result, the FFT is performed with a higher resolution. Because the high energy

unstable numerical error is avoided, the noise level in the frequency spectra is very low

and every eigenmode is clearly indicated by a distinct peak.

Chapter 6

Conclusions and Future work

6.1 Summary

Electromagnetic bandgap (EBG) structures have attracted world wide attention in the

electromagnetics and antenna communities due to their ability to guide and efficiently

control electromagnetic waves. Among a variety of numerical methods being applied to

study wave propagation and the bandgap properties of the EBGs, the Finite-Difference

Time-Domain (FDTD) method is popular because of its ability to deal with wideband

simulations from a time domain method and its ability to model complex structures. In

this thesis, the basics of EBG structures and the numerical methods for modelling EBGs

were reviewed in Chapter Two. Then the basics of the conventional Yee’s FDTD and the

techniques in relate to the modelling of the EBGs were presented in Chapter Three.

Two main aims were targeted in this study:

• To compare the numerical efficiency and accuracy of the nonorthogonal FDTD

method (NFDTD) and the Yee algorithm in modelling EBG structures.

• To modify the NFDTD algorithm into a more stable, efficient and accurate algo-

rithm.

The author’s main contributions with regard to these two aims start from Chapter

Four.

207

Chapter 6 Conclusions and Future work 208

In Chapter Four, the ADI-FDTD method is extended into the curvilinear system and

a novel ADI-NFDTD is proposed.

• The formulae of the ADI-NFDTD method (including the ADI-NFDTD incorpo-

rating the PBCs) are derived.

• Numerical simulation validates that in the ADI-NFDTD method, the CFL con-

dition of the FDTD method is removed. The computational efficiency may be increased

by using a larger dt in the ADI-NFDTD scheme than the conventional NFDTD scheme.

• Numerical simulation shows that the inherent late time instability of the NFDTD

method is largely reduced by the use of the ADI method. The ADI-NFDTD is still not an

unconditionally stable scheme. However, the ADI-NFDTD scheme is stable over a much

longer period of time.

In Chapter Five, the efficiency of NFDTD over FDTD when modelling EBGs with

curved inclusions is demonstrated in the numerical simulations:

• Both Yee’s FDTD and the NFDTD algorithm require a certain minimum spatial

resolution when modelling EBGs. To simulate a finite bandgap structure, a pilot simu-

lation can be run to find out the minimum spatial resolution required by modelling the

corresponding infinite EBGs with the unit cell approach. The minimum spatial resolution

required by the NFDTD algorithm is always much lower than that required by the Yee

algorithm.

• To reduce the late time instability of the NFDTD algorithm, the proposed ADI-

NFDTD scheme is applied in the modelling of the EBG unit cell. The ADI-NFDTD simu-

lation results demonstrate the expected reduction in the late time stability of the NFDTD

algorithm, including the delay of the numerical instability into a later time, an increase

of the frequency resolution and reduction of the noise floor in the frequency spectra.

• However, there are a few factors restricting the improvement in the computa-

tional efficiency of the ADI-NFDTD algorithm. Firstly, matrix calculation is introduced

in the ADI-NFDTD scheme and results in a more complex calculation scheme than the

conventional NFDTD. Secondly, the increase of the time step will result in a decrease in


numerical accuracy. Therefore, in ADI-NFDTD, the time step should be carefully cho-

sen for the combined consideration of efficiency and accuracy. The choice of the NFDTD

mesh is also a factor which affects the numerical efficiency. With meshes locally fine and

distorted, the numerical efficiency improvement using the ADI-NFDTD is expected to be

more significant.

6.2 Future Work

6.2.1 Further Numerical Validations

So far in this work, the NFDTD method and the ADI-NFDTD method have demonstrated

their feasibility and advantage in modelling the EBG structures. Simulation criteria have

been found and the in-house NFDTD/ADI-NFDTD codes have been verified by some

pilot EBGs simulations. Novel EBG structures can be designed with the help of these

simulation tools. Three dimensional EBGs can also be modelled as further validations of

the NFDTD and the ADI-NFDTD method.

6.2.2 Enhanced NFDTD/ADI-NFDTD Algorithms

Improvements can also be made in terms of the NFDTD/ADI-NFDTD algorithms and

the associated programming.

• Enhancement in the NFDTD Method

The frequency and spatial dispersive FDTD is successfully developed for modelling

EBGs with frequency and spatial dispersions [12, 75]. At present, the method is based on

Cartesian coordinates. It will be interesting to extend it into the generalized curvilinear


coordinate system and hence aim to achieve a frequency and spacial dispersive NFDTD

method.

• Extending the Two-Dimensional ADI-NFDTD Method into Three Dimensions

The ADI-NFDTD presented in this thesis is two-dimensional only. The following

derivations illustrate the difficulties in the attempts of extending the two-dimensional

ADI-NFDTD into three dimensions.

In Chapter 4, the partial differential maxwell’s equations (4.67)-(4.72) are written in

two procedures based on the ADI approximation. For simplicity, the first procedure is

presented here as an example. Assuming an isotropic medium with permittivity ε, the

three-dimensional ADI-NFDTD equation based on the updating equation for Ex is (de-

tailed derivation can be found in Appendix C):

−Coefxx1En+ 1

2x (i +

12, j, k − 1) + (1 + Coefxx2)Exn+ 1

2 (i +12, j, k)

− Coefxx3En+ 1

2x (i +

12, j, k + 1)

− Coefxy1En+ 1

2y (i, j, k − 1

2) + Coefxy1E

n+ 12

y (i + 1, j, k − 12)

+ Coefxy2En+ 1

2y (i, j, k +

12) − Coefxy2E

n+ 12

y (i + 1, j, k +12)

− Coefxz1En+ 1

2z (i +

12, j − 1

2, k − 1

2) + Coefxz1E

n+ 12

z (i +12, j +

12, k − 1

2)

+ Coefxz2En+ 1

2z (i +

12, j − 1

2, k +

12) − Coefxz2E

n+ 12

z (i +12, j +

12, k +

12)

=6∑

i=1

Sni (i, j, k) (6.1)

where the expressions for Si(i = 1..6) and the coefficients Coefxxp (p = 1, 2, 3) and

Coefxmp (m = y, z; p = 1, 2) can be found in Appendix C. There are two other up-

dating equations presented in a similar form relating to En+ 1

2x , E

n+ 12

y and En+ 1

2z . It is non-

trivial to solve these equations because a lot more variables are coupled to each other

compared with the two-dimensional ADI-NFDTD or the three-dimensional ADI-FDTD.


Consequently, the feasibility of solving these equations as well as the necessary approxi-

mations need to be further studied.

• Develop the ADI-NFDTD Method Base on Douvalis NFDTD Scheme

The ADI-NFDTD method presented in this thesis is based on the widely used Lee’s

NFDTD algorithm. However, it has been demonstrated by numerical simulation and

in theory ( Appendix B), the Douvalis’ NFDTD algorithm presents better late time nu-

merical stability. So the ADI-NFDTD based on the Douvalis’ NFDTD algorithm (ADI-

DNFDTD) is expected to show better numerical performance.

• Development of the Envelop ADI-NFDTD Method

Since unconditional numerical stability of the ADI-FDTD for the full three-dimensional

case was derived [7, 22, 147], there was a revival of interest in the use and the develop-

ment of the ADI-FDTD. The Envelop ADI-FDTD is one of its extensions [156]. In the

Envelop ADI-FDTD, the envelope technique was coupled with the conventional ADI-

FDTD. It has demonstrated in simulation results, the superior performance over the ADI-

FDTD in terms of numerical accuracy. More to the point, this method retains the same

level of complexity in terms of the matrix calculations as the conventional ADI-FDTD.

As a result, this method could be a candidate to improve the numerical accuracy of the

ADI-NFDTD scheme.

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Appendix A

The Formulation of

Plane-Wave-Expansion (PWE)

Method For Two-Dimensional EBG

Modelling

In this section, the PWE method used for solving a periodic array of parallel dielectric

(or metallic) rods of circular cross section and dielectric function ε(ω) is presented. The

formulation are from paper [45] and [44].

As in [45] the rods are assumed parallel to the x3 axis. Further assumption is the rods

do not overlap. The intersections of the axes of these rods with the x1x2 plane form an

arbitrary two-dimensional Bravais lattice whose sites are given by the vectors

x‖(l) = l1a1 + l2a2 (A.1)

Here a1 and a2 are the two noncollinear primitive translation vector of the lattice,

while l1 and l2 are any two integers, positive, negative or zero, which we denote collec-

tively by l. The area of the primitive unit cell of this lattice is ac = a1 × a2.

229

Appendix A The Formulation of Plane-Wave-Expansion (PWE) Method For Two-Dimensional EBG Modelling 230

The lattice reciprocal to the direct lattice is defined by the translation vectors

G‖(h) = h1b1 + h2

b2 (A.2)

The primitive translation vectorsb1 andb2 of the reciprocal lattice are the solutions of

the equations

ai ·bj = 2πδij , i, j = 1, 2 (A.3)

where h1 and h2 are arbitrary two integers that we denote collectively by h.

The dielectric function of this composite system is position dependent, and will be

denoted by ε(x‖), where x‖ = (x1, x2) is a position vector in the x1x2 plane. ε(x‖) is a

periodic function of x‖ and satisfy the relation

ε(x‖ + x‖(l)) = ε(x‖) (A.4)

ε−1(x‖) can be expanded in a two-dimensional Fourier series according to

1ε(x‖)

=∑G‖

κ(G‖)ei G‖·x‖ (A.5)

where the Fourier coefficient κ( G‖) can be calculated according to the material dielectric

function.

κ(G‖) =1ac

∫ac

d2x‖1

ε(x‖)e−i G‖·x‖ (A.6)

Here the electromagnetic waves of H polarization are considered. E polarization can

be analyzed in a very similar way. In the case of H polarization the solutions of Maxwell’s

equations have the forms:

H(x; t) = (0, 0,H3(x1, x2|ω))e−jωt (A.7)

E(x; t) = (E1(x1, x2|ω), E2(x1, x2|ω), 0)e−jωt (A.8)


The Maxwell’s curl equations for the three nonzero field components are

∂E2

∂x1− ∂E1

∂x2=

iω

cH3 (A.9)

∂H3

∂x1=

iω

cD2 =

iω

cε(x‖)E2 (A.10)

∂H3

∂x2= − iω

cD1 = − iω

cε(x‖)E1 (A.11)

When we eliminate E1 and E2 from these equations we obtain the equation satisfied

by H3, which can be written as

∂

∂x1

[1

ε(x‖)∂H3

∂x1

]+

∂

∂x2

[1

ε(x‖)∂H3

∂x2

]+

ω2

c2H3 = 0 (A.12)

Then H3(x‖) is expanded as:

H3(x‖|ω) =∑G‖

A(k‖| G‖)ei(k‖+ G‖)·x‖ (A.13)

where k‖ = x1k1+x2k2 is the two-dimensional wave vector of the wave and G‖ is defined

in equation (A.2).

Substituting equation (A.13) and equation (A.5) into equation (A.12), we obtain the

equation for the coefficients A(k‖| G‖):

∑G′‖

(k‖ + G‖) · (k‖ + G′‖)κ(G‖ − G′

‖) A(k‖| G‖) =ω2

c2A(k‖| G‖) (A.14)

which has the form of a standard eigenvalue problem for a symmetric matrix.

To solve this eigenvalue problem, the variational theorem/variational method is ap-

plied. In this theorem, it is stated that the true eigenvectors of a Hermitian operator are

the ones that minimize the variational energy, subject to the restriction that it is orthog-

onal to the eigenvectors below it. The variational energy for this eigenvalue problem


is

Evar =

∑G′‖

∑G‖

A(k‖| G‖)∗(k‖ + G‖) · (k‖ + G′‖)κ(G‖ − G′

‖) A(k‖| G′‖)∑

G‖A(k‖| G‖)∗ A(k‖| G′

‖)(A.15)

Then beginning with some guess for A(k‖| G‖), the computer calculates the variational

energy and updates its guess so as to lower the variational energy. The guesses are en-

forced to be orthogonal to any eigenvectors that were found previously. Eventually the

algorithm converges on the true A(k‖| G‖) and moves on to the next one.

It can be seen that the Fourier coefficients κ( G‖) of ε−1(x‖) play a central role in the

determination of the photonic band structures. The following shows two examples of the

expansion coefficients κ( G‖) for the aforementioned rods array with different dielectric

properties. The dielectric rods array is firstly studied followed by the investigation of

conductive rods with simple form of dielectric function.

• Dielectric rods array

In this case the dielectric rods with dielectric constant εb is hosted in material with

dielectric constant εa. So ε−1(x‖) is expanded:

1ε(x‖)

=1εb

+(

1εa

− 1εb

)∑l

S(x‖ − x‖(l)) (A.16)

with

S(x‖) =

⎧⎨⎩ 1 for x‖ ∈ R

0 for x‖ ∈/R, (A.17)

where R is the region of the x1x2 plane defined by the cross section of the rod whose axis

intersects that plane at x‖ = 0.

The Fourier coefficient κ(G‖) is then given by

κ(G‖) =1ac

∫ac

d2x‖e−iG‖x‖ 1ε(x‖)

=1εb

δG‖,0 +(

1εa

− 1εb

)1ac

∫d2x‖e−iG‖x‖S(x‖) (A.18)


where the integration in the second line of this equation is over the entire x1x2 plane.

Taking the definition of S(x‖) (equation (A.17)) into the consideration of equation (A.18),

we obtain:

κ(G‖) =1εa

f +1εb

(1 − f) , G‖ = 0

κ(G‖) =(

1εa

− 1εb

)1ac

∫R

d2x‖e−iG‖x‖ , G‖ = 0 (A.19)

where f is the filling faction, i.e., the fraction of the total volume occupied by the rods. It

is given by f = aRac

, where aR is the area of the domain R, i.e., the cross sectional area of

the rod. In the case when the rods have a circular cross section of radius R:

κ(G‖) =1εa

f +1εb

(1 − f) , G‖ = 0

κ(G‖) =(

1εa

− 1εb

)f

2J1(G‖R)G‖R

, G‖ = 0 (A.20)

where J1(x) is a Bessel function.

• Conductive rods array

In the second sensorial the rods is conductive and the dielectric function has the sim-

ple, free-electron form

ε(ω) = 1 − ω2p

ω2(A.21)

where ωp is the plasma frequency of the conduction electrons. The host is the free space.

In the general case of metal cylinders of arbitrary cross section, the Fourier coefficients

κ(G‖) are given by

κ(G‖) = 1 + fω2

p

ω2 − ω2p

, G‖ = 0

κ(G‖) =ω2

p

ω2 − ω2p

1ac

∫d2x‖S(x‖)e−iG‖x‖ , G‖ = 0 (A.22)

In the case of metallic cylinders whose cross section is a circle of radius R, we obtain

for κ(G‖)


κ(G‖) = 1 + fω2

p

ω2 − ω2p

, G‖ = 0

κ(G‖) =ω2

p

ω2 − ω2p

f2J1(G‖R)

G‖R, G‖ = 0 (A.23)

Appendix B

The Detailed Comparison of

Douvalis’ and Lee’s NFDTD

Equations

In this section, the Douvalis’ and the Lee’s NFDTD Equations will be fully interpreted

in order to demonstrate the reduction on the neighbouring averaging projection that the

former scheme involved.

B.1 Douvalis’ Final NFDTD Equations

Recall Douvalis’ equation (B.1) which appeared once in Chapter 3.

235

Appendix B The Detailed Comparison of Douvalis’ and Lee’s NFDTD Equations 236

e1(n+1)(i, j +

12, k +

12) = e1

(n)(i, j +12, k +

12)

+g11(i, j + 1

2 , k + 12)∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

3 (i,j+1,k+ 12) − h

(n+ 12)

3 (i,j,k+ 12)

)

− g21(i, j + 12 , k + 1

2)∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

3 (i+ 12,j+ 1

2,k+ 1

2) − h

(n+ 12)

3 (i− 12,j+ 1

2,k+ 1

2)

)

+g21(i, j + 1

2 , k + 12)∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

1 (i,j+ 12,k+1) − h

(n+ 12)

1 (i,j+ 12,k)

)

− g31(i, j + 12 , k + 1

2)∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

1 (i,j+1,k+ 12) − h

(n+ 12)

1 (i,j,k+ 12)

)

+g31(i, j + 1

2 , k + 12)∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

2 (i+ 12,j+ 1

2,k+ 1

2) − h

(n+ 12)

2 (i− 12,j+ 1

2,k+ 1

2)

)

− g11(i, j + 12 , k + 1

2)∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

2 (i,j+ 12,k+1) − h

(n+ 12)

2 (i,j+ 12,k)

)

(B.1)

It can be rewritten in form of:

e1(n+1)(i, j +

12, k +

12) = e1

(n)(i, j +12, k +

12)

+ G11(i, j +12, k +

12)

+ G21(i, j +12, k +

12)

+ G31(i, j +12, k +

12) (B.2)

where G11, G21 and G31 can be represented as:

G11(i, j +12, k +

12) =

g11(i, j + 12 , k + 1

2 )∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)

·(

h(n+ 1

2)

3 (i,j+1,k+ 12) − h

(n+ 12)

3 (i,j,k+ 12) + h

(n+ 12)

2 (i,j+ 12,k) − h

(n+ 12)

2 (i,j+ 12,k+1)

)(B.3)


G21(i, j +12, k +

12) =

g21(i, j + 12 , k + 1

2)∆t

ε(i, j + 12 , k + 1

2 )√

g(i, j + 12 , k + 1

2)

·(

h(n+ 1

2)

1 (i,j+ 12,k+1) − h

(n+ 12)

1 (i,j+ 12,k) − h

(n+ 12)

3 (i+ 12,j+ 1

2,k+ 1

2) + h

(n+ 12)

3 (i− 12,j+ 1

2,k+ 1

2)

)(B.4)

G31(i, j +12, k +

12) =

g31(i, j + 12 , k + 1

2)∆t

ε(i, j + 12 , k + 1

2 )√

g(i, j + 12 , k + 1

2)

·(

h(n+ 1

2)

2 (i+ 12,j+ 1

2,k+ 1

2) − h

(n+ 12)

2 (i− 12,j+ 1

2,k+ 1

2) − h

(n+ 12)

1 (i,j+1,k+ 12) + h

(n+ 12)

1 (i,j,k+ 12)

)(B.5)

The h components required in G21 and G31 are not defined in their referred spatial po-

sition. As a result, the neighbouring averaging scheme as shown in Chapter 3 is applied

and the expression for G11, G21 and G31 can be obtained using the NFDTD field compo-

nents at their defined spatial positions. For example, h1 (i,j,k+ 12) and h2 (i+ 1

2,j+ 1

2,k+ 1

2) are

approximated as:

h1 (i,j,k+ 12)

∼= 14

[h1(i +

12, j, k) + h1(i +

12, j, k + 1) + h1(i − 1

2, j, k) + h1(i − 1

2, j, k + 1)

](B.6)

h2 (i+ 12,j+ 1

2,k+ 1

2)

∼= 14

[h2(i, j +

12, k) + h2(i, j +

12, k + 1) + h2(i + 1, j +

12, k) + h2(i + 1, j +

12, k + 1)

](B.7)

So equation (B.2) can be written as (B.8) which is the final NFDTD equations under


Douvalis’ scheme.

e1(n+1)(i, j +

12, k +

12) = e1

(n)(i, j +12, k +

12)

+g11(i, j + 1

2 , k + 12)∆t

ε(i, j + 12 , k + 1

2 )√

g(i, j + 12 , k + 1

2)· H(n+ 1

2)

11 (i, j +12, k +

12)

+g21(i, j + 1

2 , k + 12)∆t

4ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)· H(n+ 1

2)

21 (i, j +12, k +

12)

+g31(i, j + 1

2 , k + 12)∆t

4ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)· H(n+ 1

2)

31 (i, j +12, k +

12)

(B.8)

where

H11(i, j +12, k +

12) =

[h

(n+ 12)

3 (i, j + 1, k +12) + h

(n+ 12)

3 (i, j, k +12)]

−[

h(n+ 1

2)

2 (i, j +12, k + 1) − h

(n+ 12)

2 (i, j +12, k)]

H21(i, j +12, k +

12) =

m=i+ 12;n=j+1∑

m=i− 12;n=j

[h1(m,n, k + 1) − h1(m,n, k)]

−m=i+1;n=j+1∑

m=i;n=j

[h3(m,n, k +

12) − h3(m − 1, n, k +

12)]

H31(i, j +12, k +

12) =

m=i+1;n=k+1∑m=i;n=k

[h2(m, j +

12, n) − h2(m − 1, j +

12, n)]

−m=i+ 1

2;n=k+1∑

m=i− 12;n=k

[h1(m, j + 1, n) − h1(m, j, n)]

(B.9)


B.2 Lee’s NFDTD Equations

The following part interprets Lee’s scheme through an example of e1 calculation. In

order to obtain a final NFDTD equation in which ei and hi(i = 1, 2, 3) are eliminated,

the Maxwell equations (B.10) are combined with the neighbouring averaging projection

equations (B.11).

e1(n+1)(i, j +12, k +

12) = e1(n)(i, j +

12, k +

12)

+∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

3 (i, j + 1, k +12) − h

(n+ 12)

3 (i, j, k +12))

− ∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)·(

h(n+ 1

2)

2 (i, j +12, k + 1) − h

(n+ 12)

2 (i, j +12, k))

e2(n+1)(i +12, j, k +

12) = e2(n)(i +

12, j, k +

12)

+∆t

ε(i + 12 , j, k + 1

2)√

g(i + 12 , j, k + 1

2)·(

h(n+ 1

2)

1 (i +12, j, k + 1) − h

(n+ 12)

1 (i +12, j, k)

)

− ∆t

ε(i + 12 , j, k + 1

2)√

g(i + 12 , j, k + 1

2)·(

h(n+ 1

2)

3 (i + 1, j, k +12) − h

(n+ 12)

3 (i, j, k +12))

e3(n+1)(i +12, j +

12, k) = e3(n)(i +

12, j +

12, k)

+∆t

ε(i + 12 , j + 1

2 , k)√

g(i + 12 , j + 1

2 , k)·(

h(n+ 1

2)

2 (i + 1, j +12, k) − h

(n+ 12)

2 (i, j +12, k))

− ∆t

ε(i + 12 , j + 1

2 , k)√

g(i + 12 , j + 1

2 , k)·(

h(n+ 1

2)

1 (i +12, j + 1, k) − h

(n+ 12)

1 (i +12, j, k)

)

(B.10)


e(n+1)1 (i, j +

12, k +

12) = g11(i, j +

12, k +

12)e1(n+1)(i, j +

12, k +

12)

+g21(i, j + 1

2 , k + 12)(i, j, k)

4

m=i+ 12;n=j+1∑

m=i− 12;n=j

e2(n+1)(m,n, k +12)

+g31(i, j + 1

2 , k + 12)(i, j, k)

4

m=i+ 12;n=k+1∑

m=i− 12;n=k

e3(n+1)(m, j +12, n) (B.11)

Substitute equations (B.10) into (B.11) yields:

e(n+1)1 (i, j +

12, k +

12) = E

(n)1 (i, j +

12, k +

12)

+g11(i, j + 1

2 , k + 12 )∆t

ε(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)· H(n+ 1

2)

11 (i, j +12, k +

12)

+g21(i, j + 1

2 , k + 12)∆t

4ε(i + 12 , j, k + 1

2 )√

g(i + 12 , j, k + 1

2)· H(n+ 1

2)

21 (i, j +12, k +

12)

+g31(i, j + 1

2 , k + 12)∆t

4ε(i + 12 , j, k + 1

2 )√

g(i + 12 , j, k + 1

2)· H(n+ 1

2)

31 (i, j +12, k +

12)

(B.12)

where

E1(i, j +12, k +

12) = g11(i, j +

12, k +

12)e1(n)(i, j +

12, k +

12)

+g21(i, j + 1

2 , k + 12 )

4

m=i+ 12;n=j+1∑

m=i− 12;n=j

e2(n)(m,n, k +12)

+g31(i, j + 1

2 , k + 12 )

4

m=i+ 12;n=k+1∑

m=i− 12;n=k

e3(n)(m, j +12, n)

H11(i, j +12, k +

12) =

[h3(i, j + 1, k +

12) − h3(i, j, k +

12)]

−[h2(i, j +

12, k + 1) + h2(i, j +

12, k)]


H21(i, j +12, k +

12) =

m=i+ 12;n=j+1∑

m=i− 12;n=j

[h1(m,n, k + 1) − h1(m,n, k)]

−m=i+1;n=j+1∑

m=i;n=j

[h3(m,n, k +

12) − h3(m − 1, n, k +

12)]

H31(i, j +12, k +

12) =

m=i;n=k+1∑m=i−1;n=k

[h2(m + 1, j +

12, n) − h2(m, j +

12, n)]

−m=i+ 1

2;n=k+1∑

m=i− 12;n=k

[h1(m, j + 1, n) − h1(m, j, n)]

(B.13)

B.3 Conclusion

When Lee’s NFDTD formulation ( equations (B.12) and (B.13)) is compared with Douvalis

one ( equations (B.8) and (B.9)), one can see that the last three entries at the right-hand-

side (RHS) of the updating equations ( relating to H11,H21 and H31) are calculated using

exactly the same formula. The only difference stands in the first entry at the RHS of the

updating equations.

This entry can be understood as origining from the temporal partial operator in Maxwell’s

equations. In Lee’s scheme, a neighbouring averaging scheme is applied to this entry (see

E1 in equations (B.13)), because the contravariant field components have to be projected

to the covariant components which are defined in different spatial positions. On the other

hand, in Douvalis’ scheme, this averaging is avoided because only covariant components

are introduced into the algorithm. It is worth noting that the neighbouring averaging

scheme is not completely eliminated from Douvalis’ NFDTD scheme, as it appears in the

calculation of H11,H21 and H31, which relates to the curl operator in Maxwell’s equa-

tions. However, the use of the neighbouring averaging scheme is reduced in Douvalis’

NFDTD algorithm compared with Lee’s one.

Appendix C

Towards the Derivation of the

Three-Dimensional ADI-NFDTD

Equations

In Chapter 4, the partial differential maxwell’s equations (4.67)-(4.72) are written after

discretization based on the ADI approximation in two procedures. For simplicity, the first

procedure is presented here as an example. Assuming the permittivity of the material is

ε, the ADI-NFDTD method is written in the first procedure as:

(from the n-th to the (n + 12)-th time step)

242

Appendix C Towards the Derivation of the Three-Dimensional ADI-NFDTD Equations 243

Exn+ 12 (i+

12, j, k)−Exn

(i+12, j, k) =

dt

2ε(i + 12 , j, k)

√g(i + 1

2 , j, k)

·⎡⎣ Hn

z

(i + 1

2 , j + 12 , k)− Hn

z

(i + 1

2 , j − 12 , k)

−Hn+ 1

2y

(i + 1

2 , j, k + 12

)+ H

n+ 12

y

(i + 1

2 , j, k − 12

)⎤⎦ (C.1)

Eyn+ 12 (i, j+

12, k)−Eyn

(i, j+12, k) =

dt

2ε(i, j + 12 , k)

√g(i, j + 1

2 , k)

·⎡⎣ Hn

x

(i, j + 1

2 , k + 12

)− Hnx

(i, j + 1

2 , k − 12

)−H

n+ 12

z

(i + 1

2 , j + 12 , k)

+ Hn+ 1

2z

(i − 1

2 , j + 12 , k)⎤⎦ (C.2)

Ezn+ 12 (i, j, k+

12)−Ezn

(i, j, k+12) =

dt

2ε(i, j, k + 12)√

g(i, j, k + 12 )

·⎡⎣ Hn

y

(i + 1

2 , j, k + 12

)− Hny

(i − 1

2 , j, k + 12

)−H

n+ 12

x

(i, j + 1

2 , k + 12

)+ H

n+ 12

x

(i, j − 1

2 , k + 12

)⎤⎦ (C.3)

Hxn+ 12 (i, j +

12, k +

12) − Hxn

(i, j +12, k +

12) =

dt

2µ(i, j + 12 , k + 1

2)√

g(i, j + 12 , k + 1

2)

·⎡⎣ En

y

(i, j + 1

2 , k + 1) − En

y

(i, j + 1

2 , k)

−En+ 1

2z

(i, j + 1, k + 1

2

)+ E

n+ 12

z

(i, j, k + 1

2

)⎤⎦ (C.4)

Hyn+ 12 (i +

12, j, k +

12) − Hyn

(i +12, j, k +

12) =

dt

2µ(i + 12 , j, k + 1

2)√

g(i + 12 , j, k + 1

2)

·⎡⎣ En

z

(i + 1, j, k + 1

2

)− Enz

(i, j, k + 1

2

)−E

n+ 12

x

(i + 1

2 , j, k + 1)

+ En+ 1

2x

(i + 1

2 , j, k)⎤⎦ (C.5)

Hzn+ 12 (i +

12, j +

12, k) − Hzn

(i +12, j +

12, k) =

dt

2µ(i + 12 , j + 1

2 , k)√

g(i + 12 , j + 1

2 , k)

·⎡⎣ En

x

(i + 1

2 , j + 1, k) − En

x

(i + 1

2 , j, k)

−En+ 1

2y

(i + 1, j + 1

2 , k)

+ En+ 1

2y

(i, j + 1

2 , k)⎤⎦ (C.6)


Applying the NFDTD projection scheme:

vm(i, j, k) = gxx(i, j, k)vm(i, j, k)+gmy(i, j, k)vy(i, j, k)+gmz(i, j, k)vz(i, j, k) (m = x, yorz)

(C.7)

and substitute Hn+ 1

2y into equation (C.1), then substituting Hxn+ 1

2 , Hyn+ 12 , Hzn+1

2 from

equations (C.4), (C.5) and (C.6) into the resulting equation, one can finally obtain:

Exn+ 12 (i +

12, j, k) =

6∑i=1

Sni (i, j, k) + Coefxx1(i, j, k)E

n+ 12

x (i +12, j, k − 1)

− Coefxx2(i, j, k)En+ 1

2x (i +

12, j, k) + Coefxx3(i, j, k)E

n+ 12

x (i +12, j, k + 1)

+ Coefxy1(i, j, k)En+ 1

2y (i, j, k − 1

2) − Coefxy1(i, j, k)E

n+ 12

y (i + 1, j, k − 12)

− Coefxy2(i, j, k)En+ 1

2y (i, j, k +

12) + Coefxy2(i, j, k)E

n+ 12

y (i + 1, j, k +12)

+ Coefxz1(i, j, k)En+ 1

2z (i +

12, j − 1

2, k − 1

2) − Coefxz1(i, j, k)E

n+ 12

z (i +12, j +

12, k − 1

2)

− Coefxz2(i, j, k)En+ 1

2z (i +

12, j − 1

2, k +

12) + Coefxz2(i, j, k)E

n+ 12

z (i +12, j +

12, k +

12)

(C.8)

where the expressions for Si(i = 1..6) and the coefficients Coefxxp (p = 1, 2, 3), Coefxmp (m =

y, z, p = 1, 2) can be expressed by denoting:

Co1(i, j, k) =dt

2ε(i + 12 , j, k)

√g(i + 1

2 , j, k)(C.9)

Co2(i, j, k) =dt

2µ(i + 12 , j, k + 1

2)√

g(i + 12 , j, k + 1

2)(C.10)

s4(i, j, k) = −Co2(i, j, k)gyy(i, j, k)[Ez

n(i + 1, j, k +12) − Ez

n(i, j, k +12)]

(C.11)

s5(i, j, k) = −Co2(i, j, k)gxy(i, j, k)[Ey

n(i +12, j, k + 1) − Ey

n(i +12, j, k)

](C.12)

s6(i, j, k) = −Co2(i, j, k)gyz(i, j, k)[Ex

n(i +12, j +

12, k +

12) − Ex

n(i +12, j − 1

2, k +

12)]

(C.13)


Then:

S1(i, j, k) = Exn(i +12, j, k) (C.14)

S2(i, j, k) = Co1(i, j, k)[Hn

z (i +12, j +

12, k) − Hn

z (i +12, j − 1

2, k)]

(C.15)

S3(i, j, k) = −Co1(i, j, k)gyy(i, j, k)Hyn(i +12, j, k +

12)

− Co1(i, j, k)gxy(i, j, k)Hxn(i +12, j, k +

12)

− Co1(i, j, k)gyz(i, j, k)Hzn(i +12, j, k +

12)

+ Co1(i, j, k)gyy(i, j, k − 1)Hyn(i +12, j, k − 1

2)

+ Co1(i, j, k)gxy(i, j, k − 1)Hxn(i +12, j, k − 1

2)

+ Co1(i, j, k)gyz(i, j, k − 1)Hzn(i +12, j, k − 1

2) (C.16)

S4(i, j, k) = Co1(i, j, k) [s4(i, j, k) − s4(i, j, k − 1)] (C.17)



Coefxx1(i, j, k) = Co1(i, j, k − 1)gyy(i, j, k − 1)Co2(i, j, k − 1) (C.20)

Coefxx2(i, j, k) = Co1(i, j, k) [gyy(i, j, k)Co2(i, j, k) − gyy(i, j, k − 1)Co2(i, j, k − 1)]

(C.21)

Coefxx3(i, j, k) = Co1(i, j, k)gyy(i, j, k)Co2(i, j, k) (C.22)

Coefxy1(i, j, k) = Co1(i, j, k − 1)gxy(i, j, k − 1)Co2(i, j, k − 1) (C.23)

Coefxy2(i, j, k) = Co1(i, j, k)gxy(i, j, k)Co2(i, j, k) (C.24)

Coefxz1(i, j, k) = Co1(i, j, k − 1)gyz(i, j, k − 1)Co2(i, j, k − 1) (C.25)

Coefxz2(i, j, k) = Co1(i, j, k)gyz(i, j, k)Co2(i, j, k) (C.26)

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